摘要:

GENERAL DISCUSSIONS OFTHE FARADAY SOCIETYDate190719071910191 119121913191319131914191419151916191619171917191719181918191819181919191919201920192019201921192119211921192219221923192319231923I923I92319241924I9241924192519251926192619271927SubjectOsmotic PressureHydrates in SolutionThe Constitution of WaterHigh Temperature WorkMagnetic Properties of AlloysColloids and their ViscosityThe Corrosion of Iron and SteelThe Passivity of MetalsOptical Rotary PowerThe Hardening of MetalsThe Transformation of Purs IronMethods and Appliances for the Attainment of High Temperatures in aRefractory MaterialsTraining and Work of the Chemical EngineerOsmotic PressurePyrometers and PyrometryThe Setting of Cements and PlastersElectrical FurnacesCo-ordination of Scientific PublicationThe Occlusion of Gases by MetalsThe Present Position of the Theory of IonizationThe Examination of Materials by X-RaysThe Microscope : Its Design, Construction and ApplicationsBasic Slags : Their Production and Utilization in AgriculturePhysics and Chemistry of ColloidsElectrodeposition and ElectroplatingCapillarityThe Failure of Metals under Internal and Prolonged StressPhysico-Chemicai Problems Relating to the SoilCatalysis with special reference to Newer Theories of Chemical ActionSome Properties of Powders with special referaice to Grading byThe Generation and Utilization of ColdAlloys Resistant to CorrosionThe Physical Chemistry of the Photographic ProcessThe Electronic Theory of ValencyElectrode Reactions and EquilibriaAtmospheric Corrosion.First ReportInvestigation on Oppau Ammonium Sulphate-NitrateFluxes and Slags in Metal Melting and WorkingPhysical and Physico-Chemical Problems relating to Textile FibresThe Physical Chemistry of Igneous Rock FormationBase Exchange in SoilsThe Physical Chemistry of Steel-Making ProcessesPhotochemical Reactions in Liquids and GasesExplosive Reactions in Gaseous MediaPhysical Phenomena at Interfaces, with special reference to MolecularAtmospheric Corrosion. Second ReportLaboratoryElutriationOrientationThe Theory of Strong Elcctrolytes1927 Cobcsion and Related ProblemsVolumeTrans. 33678999101011121213131314141414151516161616171717171818191919191920202020202121222223232GENERAL DISCUSSIONS OF THE FARADAY SOCIETYDri:eI928I929192919299309309319329329339339311934I935193519361936193119371398193819391939194019411941194219431944194519451946194619471947194719471948193819391949I9491950195019501950195119511952195219521953195319541954SubjectHomogeneous CatalysisCrystal Structure and Chemical ConstitutionAtmospheric Corrosion of Metals.Third ReportMolecular Spectra and Molecular StructureOptical Rotatory PowerColloid Science Applied to BiologyPhotochemical ProcessesThe Adsorption of Gases by SolidsThe Colloid Aspects of Textile MaterialsLiquid Crystals and Anisotropic MeltsFree RadicalsDipole MomentsColloidal Elecctroly tesThe Structure of Metallic Coatings, Films and SurfacesThe Phenomena of Polymerization and CondensationDisperse Systems in Gases : Dust, Smoke and FogStructure and Molecular Forces in (a) Pure Liquids, and (b) SolutionsThe Properties and Functions of Membranes, Na turd and ArtificialReaction KineticsChemical Reactions Involving SolidsLuminescenceHydrocarbon Chemistry35The Hydrogen Bond 36The Mechanism and Chemical Kinetics of Organic Reactions in LiquidThe Structure and Reactionsof RubberMolecular Weight and Molecular Weight Distribution in High Polymers.(Joint Meeting with the Plastics Group, Society of Chemical Industry) 40The Application of Infra-red Spectra to Chemical Problems 41Oxidation 42Dielectrics 42 ASwelling and Shrinking 42 BElectrode Processes Disc.1The Labile Molecule 2Surface Chemistry.Colloidal Electrolytes and SolutionsThe Interaction of Water andporous MaterialsThe Electrical Double Layer (owing to the outbreak of war the meetingwas abandoned, but the papers were printed in the Transactions)The Oil-Water Interface 37Systems 37Modes of Drug Action 3938(Jointly with the SociCt6 de Chide Physique atBordeaux.) Published by Butterworths Scientific Publications, Ltd.Trans. 43Disc. 34Lipo-Proteins 6Heterogeneous Catalysis 8Physico-chemical Properties and Behaviour of Nuclear AcidsSpectroscopy and Molecular Structure and Optical Methods of In-vestigating a11 Structure Disc.Trans.47 Electrical Double LayerHydr(icarbons Disc. 1011The Physical Chemistry of Proteins 13The Reactivity of Free Radicals 14The Equilibrium Properties of Solutions of Non-Electrolytes 15I6The Study of Fast Reactions 17Coagulation and Flmlation 18The Physical Chemistry of Process MetallurgyCrystal Growth 5Chromatographic Analysis 7Trans. 46The Size and Shape Factor in Colloidal SystemsRadiation Chemistry 12The Physical Chemistry of Dyeing and TanningVolume2425252526262728292930303131323233333434353GENERAL DISCUSSIONS OF THE FARADAY SOCIETYDate Subject VoizimcI9551955195619561957195719581958195919591960I960196119611962I962196319631964I964Microwave and Radio-Frequency SpectroscopyPhysical Chemistry of EnzymesMembrane PhenomenaPbysical Chemistry of Processes at High PressuresMolecular Mechanism of Rate Processes in SolidsInteractions in Ionic SolutionsConfigurations and Interactions of Macromolecules and Liquid CrystalsIons of the Transition ElementsEnergy Transfer with special reference to Biological SystemsCrystal Imperfcctions and the Chemical Reactivity of SolidsOxidation-Reduction Reactions in Ionizing SolventsThe Physical Chemistry of AerosolsRadiation Effects in Inorganic SolidsThe Structure and Properties of Ionic MeltsTneiastic Collisions of Atoms and Simple MoleculesHigh Resolution Nuclear Magnetic ResonanceThe Structure of Electronically-Excited Species in the Gas-PhaseFundamental Processes in Radiation ChemistryChemical Reactions in the AtmosphereDislocations in Solids1920212223242526272829303132333435353738For ciirreni availability of Discussionvolumes, see back cover

ISSN:0366-9033    

DOI:10.1039/DF964380X001

出版商:RSC    

年代:1964

数据来源: RSC

摘要:

I. GENERAL DISLOCATION THEORY, STRUCTUREAND PROPERTIESRelation Between the Fine Structure of Dislocations and theCrystal StructureBY S. AMELINCKXS.C.K.-C.E.N., Mol (Belgium)Received 13th July, 1964The detailed structure of dislocations reflects the crystal structure especially if the Burgers vectorbecomes large, which is likely to be the case in relatively complicated crystals. The dissociationof dislocations into ribbons of stacking faults is most easily observed in layer structures because theglide planes are then parallel to the plane of observation in electron microscopy. For a number oflayer structures striking combinations of stacking fault ribbons and dislocations, which can readilybe interpreted on the basis of the crystal structure, have been observed.Ribbons consisting of two,four and six partials are discussed in relation with the structures of tin disulphide, talc and thechromium halides respectively. The observation of ribbons allows to deduce values for the stackingfault energy.Originally a dislocation line was thought of as a linear discontinuity withoutmuch structure. This simple picture is a reasonably good description as long asone ignores the details of the crystal structure. However, dislocations in realcrystals present usually a fine structure which has a close connection with the crystalstructure.The first successful attempt to relate the dislocation structure to the crystalstructure is due to Heidenreich and Shockley.1 These authors pointed out thatglide dislocations with a Burgers vector a/2 [ l i O ] in (111) planes of face-centredcubic metals should under certain circumstances be extended, i.e., should consistof two partial dislocations with Burgers vectors a/6 [2ii] and a/6 flTI].Betweenthe two partials the normal stacking sequence is perturbed: these is a stackingfault. In face-centred cubic metals this fault is equivalent to one lamella in thehexagonal arrangement. The simplest glide dislocation in one of the simpleststructures thus turns out to have already a fine structure : two partial dislocationsand the associated ribbon of stacking fault. The whole assembly is called anextended dislocation or a ribbon. The concept has been very fruitful since it allowedone to explain qualitatively and even quantitatively the differences in behaviourwith respect to plastic deformation of different face-centred cubic crystals.Theelectron microscope has brought striking direct confirmation of the existence ofextended dislocations. Moreover, as a result of the electron microscopic observ-ations in a large number of substances, it became clear that more complicatedarrangements of partials and stacking faults occur. In most cases, an obviousand direct relation between the dislocation fine structure and the crystal structureis found. It is the purpose of this paper to discuss this relationship on the basis ofthe electron microscopic evidence.8 FINE STRUCTURE OF DISLOCATIONSGEOMETRY OF GLIDE PLANESThe movement of dislocations usually takes place on those planes for which thebinding between successive layers is weakest.These planes are usually the closest-packed planes. In a number of crystals the geometry of the glide planes can beapproximated by a close-packed planar arrangement of rigid spheres (fig. 1,a). Ifthis is the case the relative glide movement of two such planes from one position tothe next crystallographically equivalent one takes place in two steps as shown infig. 1. This can immediately be " felt " if one tries to slide the two rigid layers ofspheres one with respect to the other. The two steps result because the spheres inone layer tend to follow, during their movement, the valleys in the other layer ofspheres. This picture is the basis of the Shockley partial scheme, each glide stepcorresponding to one partial.In a number of crystal structures the glide planeshave this structure. A number of such structures are tabulated in table 1. Themost probable position of the glide plane is indicated by means of an arrow.TABLE 1structure stacking symbolcadmium iodidecadmium chloridemolybdenum sulphide aBatbubta/3atbab. . .chromium chloridediamondwurtzite aatbbtaatbp . . .aylbtcPlatbalc . . .aa t bB t CY t aa t bB t CYA complication immediately arises if these layers are "defective ". Let usconsider for example the arrangement of fig. 1,b which occurs in silicate layer struc-tures. The presence of regularly arranged " vacancies " increases the repeat distanceO r O 0A A 80 0 0 0(4" O O O O (6)0 0 0 00 0 00 0 .0 .0 0(c) . "AX2. 0xt0 . 0 0 A0 0 .FW. 1 . 4 ~ ) Close-packed layer of spheres-a glide path is indicated; (b) ring pattern of spheresof the type occurring in silicate layer structures ; (c) closepacked layer with two different kinds ofatoms ; (d) graphite ring pattern.in the plane and as a result the number of elementary glide steps may become largerthan two. In the particular case shown, the nearest crystallographically equivalentposition is reached after four steps. Longer glide paths may be considered; oneconsisting of eight steps is indicated in fig. 1,bS. AMELINCKX 9The same type of complication may arise because all atoms in the close-packedlayer are not of the same chemical nature (fig. 1,c) or because layers adjacent to theclose-packed layers, between which glide takes place, are defective.As a resultof this, the atoms which make up the close-packed layers are no longer equivalent ;the repeat distance increases and hence the number of glide steps increases as well.An example of this is met, e.g., in the chromium chloride structure (fig. 2). Thechromium ions form a hexagonal ring pattern in the octahedral interstices betweentwo close-packed chlorine layers. As a result two kinds of chlorine ions are tobe considered. The shortest repeat distances are now of the type X1X2 and follow-ing the valleys this distance can be covered in four unit steps as shown in fig. 2.0, 9, chlorine; 0, chromiumFIG. 2.-Sandwich layer of the chromium chloride and chromium bromide structure.The unitcell is outlined by means of a double line.One therefore expects the dislocations to be split into four partials, each partialperforming one slip step. The details of this process are discussed below.The idealized titanium oxide structure offers another example. The structureconsists of a hexagonally-packed arrangement of oxygen atoms containing titaniumions in octahedral interstices. The filling of the interstices by titanium ions is asfollows. In one plane filled and empty rows of octahedral interstices alternate.In successive layers the filled rows alternate as well. This idealized structure isshown in fig. 3 4 b. It is now evident that the repeat distance depends on the direc-tion of the Burgers vector. For the [Ool] Rurgers vector it remains the same as forthe oxygen lattice, but in the directions [ l O i ] and [ O i l ] it has doubled.In the lattercase, dissociation into partials takes place.Filling some of the interstices between the layers which glide one with respectto the other may also result in an increased repeat distance. Let us consider, forexample, the structure of aluminium oxide whereby 3 of the octahedral intersticesin one layer are filled according to the ring pattern of fig. 4. It is clear that theshortest identity vectors have become of the type X1X2. They can be decomposedin four unit steps [2].We shall now apply these principles to a number of structures and comparethe predictions with the observations10 FINE STRUCTURE OF DISLOCATIONSCLOSE-PACKED STRUCTURESNO DISSOCIATIONWe shall first consider structures where glide takes place between close-packedplanes and where as a result the geometrical conditions for dissociation in Shockleypartials are satisfied.Even in such cases it is possible that undissociated dislocationsare observed ; this is due to the fact that the stacking fault energy may be too large.ab CFIG. 3.-Idealized titanium oxide structure projected on the (100) plane of rutile.FIG. 4.-Aluminium oxide structure projected on the basal plane.Examples of crystals where this occurs are aluminium,3 antimonium telluride andbismuth telluride.4In other close-packed structures the glide plane may be different from the close-packed plane. This is, e.g., the case in crystals with the sodium chloride structure,such as magnesium oxide and nickel oxide.In these crystals the (1 10) plane func-tions as a glide plane. If a dislocation with a a/2 [110] Burgers vector and a (1iO)glide plane should dissociate the atom configuration in the faulted region wouldbe such as to bring ions with the same sign in contact. This leads to an excessivelylarge stacking fault energy of Coufomb origin and hence to undissociated dislocationsS. AMELINCKX 11A similar reasoning applies to the calcium fluoride structure ; as a result dislocationsare found to be undissociated in substances like uranium dioxide and calciumfluoride.5~ 6DISSOCIATION INTO TWO PARTIALSLOW STACKING FAULT ENERGY ALLoys.-The Simplest example Of dissociationinto two partials is found in the low stacking fault energy alloys.7 Such alloys areformed if to a face-centred cubic metal an impurity is added which, when presentin sufficient concentration, would give rise to a hexagonal phase.Adding aluminiumto copper or tin to silver decreases, for instance, the stacking fault energy. In suchalloys the dislocations are visibly extended (fig. 5). From contrast experiments itwas possible to establish that these faults are of the intrinsic type.8CADMIUM IODIDE smucmm.-The A X 2 cadmium iodide structure offers a par-ticularly simple example of glide between close-packed planes. This structurecan be described by the stacking symbol aybaybayb . . . The italic lettersrepresent the anions or X-ions, while the greek letters represent the cations or A-ions.The close-packed planes of cations are sandwiched between two close-packed layers of anions. They occupy the octahedral interstices. The bindingbetween two successive X-layers is probably weaker than between an A- and an X-layer. This results in easy cleavage and glide between the two close-packed X-layers. As a consequence partials with Burgers vectors of the type Ao, Bo or aA,aB are to be expected.* They have indeed been found in cadmium iodide and tindisulphide.9 The ribbons are particularly wide in tin disulphide, of which an exampleis shown in fig. 5. Glide on the X-X glide plane produces a stacking fault of thetype ayb ayb cpa cpa . . . which can be considered as one lamella in the cadmiumchloride structure for which the stacking symbol is ayb cpa bac .. . The sand-wiches XAX remain unsheared in this process. Similar ribbons are observed inthe closely related cadmium chloride structure as, for instance, in nickel bromideby Price.10 In this case the stacking fault is one lamella of the cadmium iodidestructure.MOLYBDENUM smPmE.-Dislocations in this substance have been studied bya number of investigators.11-13 At least two different kinds of dislocations havebeen found : undissociated dislocations and widely dissociated dislocation ribbons(fig. 7). The structure of molybdenum sulphide can be represented by the stackingsymbol apa bab afla bab . . . The italic letters refer to sulphur ions and the greekletters to molybdenum ions.The binding between two sandwiches limited bysulphur layers is mainly of the van der Waals type and hence considerably weakerthan the covalent bonds between molybdenum and sulphur. It is therefore reason-able to expect that s-S glide, i.e., glide between two sulphur layers will be easierthan MoS-glide. A priori it might even seem unlikely that glide on MoS-glide planeswould take place at all. However, as we shall see, the observations are consistentwith the two types of glide planes. The structure of molybdenum sulphide is shownschematically in fig. 8 ; the complete Burgers vectors are of the type AB as referredto fig. 8. For both assumptions concerning the location of the slip plane in thestructure, glide takes place between close-packed layers and on a purely geometricalbasis dissociation into partials is possible in either case.Whether a visible dis-sociation occurs or not will depend on the value of the stacking fault energy. Inthe same specimens one finds an almost equal fraction of very wide ribbons on the* Use is made of a modified Thompson notation for denoting Burgers vectors, see, e.g., ref. (25)12 FINE STRUCTURE OF DISLOCATIONSone hand and of undissociated dislocations on the other hand. The Burgers vectorsof these different types of dislocations were determined using the extinction criteriumg.b. = 0. In this way one finds that the undissociated dislocations or the singledislocations have Burgers vectors of the type AB of the partials in the ribbons haveBurgers vectors of the type Aa+oB.The simplest way to account for these twoFIG. 8.4tructure of molybdenum sulphide as projected on the c-plane. 2 I ’’ \ I ‘ /’_--_--i a ) tbl (C)FIQ. 9.--(a) View in space of the molybdenum sulphide structure emphasizing the bond directions ;(b) projection on the c-plane of the normal environment of the molybdenum ions ; (c) projectionon the c-plane of the distorted environment of the molybdenum ion resulting from Mo-S glide.FIG. 10.-Analysis in terms of Burgers vectors of fig. 7. Use is made of an adapted Thompsonnotation.types of dislocations or stacking faults is to assume that MoS-glide as well as SS-glide takes place. On SS-glide the environment of the molybdenum ions will notchange, i.e., the normal orientation of the covalent bonds along the diagonals ofthe triangular prism will be conserved (fig.9a). On the other hand, MoS-glideover a vector like Aa would give rise to a stacking sequence a/3ababaybcpc . . .whereby the environment of the molybdenum is wrong in one layer ; this environ-ment is shown in fig. 9 c ; and it is clear that the MoS bonds are now orienteS. AMELINCKX 13differently. It is therefore reasonable to assume that the high-energy stacking faultis associated with glide on an MoS type glide plane whereas the low-energy stackingfault is associated with SS-glide. The interaction between a perfect dislocation anddislocation ribbons is visible in fig. 7. The analysis in terms of Burgers vectorsof this photograph is shown in fig. 10. It is now clear that for one family of partialsthe Burgers vectors at the crossing perfect dislocations are mutually perpendicularexplaining the small degree of interaction whilst for the other family they are atan angle of 30" giving rise to strong interaction.This analysis confirms indirectlythe model given above.SUPER-LATTICES IN CLOSE-PACKED STRUCTURES : ORDERED ALLOYSConsider for simplicity the dislocations in the ordered AuCu3 superlattice.The passage of a dislocation with Burgers vector a/2-[110] restores the geometricalorder but leaves a so-called anti-phase boundary as represented schematically on atwo-dimensional model in fig. 11. A certain energy per unit area is associated witho e o e o e o e o e oe o e o o o e o e o e0 . 0 e o e o e 0 .00 e o o 0 0 e 0 e 0 .o e o e o e o e o e o e 0e o e o e o e o e o o o eFIG. 11 .-Two-dimensional model for a strip of anti-phase boundary limited by two dislocationsin an ordered alloy.such an anti-phase boundary. A second perfect dislocation in the same glide planeand following the first one restores both geometrical and chemical order. Fromthese considerations it will be evident that the two perfect dislocations are coupledby means of a strip of anti-phase boundary. Depending on the specific energy ofthis boundary closely or more widely coupled perfect dislocations will result. Sincemoreover each perfect dislocation may split into Shockley partials, ribbons of fourcomponents tend to form. A model for such a ribbon is shown in fig. 12.If thestacking fault energy is large and the anti-phase boundary energy small one willonly observe coupled perfect dislocations. The distance between these perfectdislocations depends on the anti-phase boundary energy in the same way as thedistance between partials depends on the stacking fault energy. The anti-phaseboundary energy itself will depend on the ordering parameter ; in completely dis-ordered alloys there will be no coupling since the ordering parameter is zero; thedistance would be minimum in the completely ordered alloy. An example of so-called super-dislocations in the ordered alloy Ni3Mn is shown in fig. 13.14 Couplingof dislocations due to ordering also takes place in the suboxide of niobium whichresults from the ordering of interstitial oxygen in niobium metal 146 (fig.14). Thesuperlattice unit cell of this suboxide as referred to the body-centred niobiumstructure is outlined in fig. 15. For two of the [ l l l ] directions of the niobiumlattice the repeat distance is not doubled in the suboxide lattice. In the two other[ill] directions, this is the case. This remark explains the co-existence of singledislocations and super-dislocations visible in fig. 1414 FINE STRUCTURE OF DISLOCATIONSDISSOCIATION I N MORE THAN TWO PARTIALS: CHROMIUM CHLORIDEAND CHROMIUM BROMIDEThe structure of chromium chloride can be described as a cubic close-packingof chlorine ions with the chromium ions in octahedral interstices. The chromiumions are arranged in layers and build a ring pattern similar to that in graphite leaving6 of the sites unoccupied.The chromium layers alternate with two chlorine layersfollowing the scheme, aylb cpla balc, where the greek letters represent chromiumand the italic letters chlorine. The index indicates the 3 of the sites which are leftempty. The structure can be referred to a hexagonal unit cell which is shown inprojection on the c-plane in fig. 2. The unit cell is indicated by the double line;Stacking faultAnti -Phase boundaryFIG. 12.-Model for a ribbon of four dislocations in an ordered alloy of the type AuCu3. Stackingfaults and anti-phase boundary are indicated by a different cross-hatching.it is three such sandwiches high. Since all observed dislocation patterns are situatedin the foil plane, which is the c-plane, it is justified to conclude that the c-plane isthe main glide plane.It is almost obvious then to locate the glide plane betweentwo chlorine layers since the binding is weakest there. Ball model considerationsshow that the possible glide movements have perfect Burgers vectors of the typeXlX2 and its five equivalent vectors.From the projection of fig. 2 these vectors can dissociate into four partial vectorsof the Shockley type according to the scheme,- -X& = a+&+;+>.A crystallographically equivalent position can also be reached after 6 steps in thedirection XI X3 and in the crystallographically equivalent ones. This large vectorcan be decomposed into 6 partial vectors according to the schemS. AMELINCKX 15Whereas in the first case the glide path has to “ go around ” one atom, the glide pathhas zig-zag shape in the second case.The observations show a majority of six-foldribbons which seems to suggest that most glide is according to the second scheme.This can be understood on the basis that the zig-zag path is a better approximationto a straight line than the “ around the corner” path. All simple glide paths areindicated in fig. 2. The conclusion from the previous paragraph can be formulatedin dislocation language by saying that perfect dislocations in chromium chlorideform either multi-ribbons of four partials separated by three stacking faults or ofsix partials linked together by five stacking faults. Both types of multi-ribbonsFIG. 15.-Unit cell of niobium suboxide referred to the parent body-centred structure of niobium.have in fact been observed ; six-fold ribbons are shown in fig.16. The modelsfor these multi-ribbons are represented in fig. 17. Whereas in the six-fold ribbonsonly partials with two different Burgers vectors occur, three different Burgers vectorsoccur in the four-fold ribbons. The models for these different ribbons have beenconfirmed by contrast experiments. It has further been shown 15 that the relativespacing of the partials is directly related to the relative energies of the differentstacking faults present in the ribbons. It turns out that a wrong stacking of thechlorine ions as well as a wrong stacking of the chromium ions contributes to thestacking fault energy.The stacking fault with the highest energy results whenthe stacking of the chlorine ions is wrong and when moreover the two layers ofchromium ions come one vertically above the other. For a detailed analysis werefer to ref. 15.The structure of chromium bromide is closely related to that of chromiumchloride. The bromine ions are hexagonally close-packed with the chromium ionsin octahedral interstices. The structure consists again of the same sandwiches asin the chromium chloride structure, only differently stacked. In this case againthe high-energy stacking fault results when in successive layers the chromium ionscome all one on top of the other. Again six- and four-fold ribbons are observed,however, the succession of stacking faults of different energies is different, givingrise to ribbons of a different aspect (fig.16,b)16 FINE STRUCTURE OF DISLOCATIONSCOVALENT STRUCTURESDIAMOND STRUCTUREThe diamond structure can be represented by the symbol aab j3cy aabpcy . . .Glide is presumably easier between the planes a and a (21 and j3, or c and y) becausethis implies breaking the smallest number of bonds. However, dissociation intopartials has to take place between two close-packed layers, i.e., between a and b, pand c, or y and a. The stacking fault resulting from glide on the (1 11) plane wouldthen have the structure ctabpcy bbc yaa bpc yaa . . ., i.e., it would consist of onelamella of the wurtzite structure. An undissociated dislocation is shown in fig.18,a. A dissociated dislocation containing an intrinsic fault is represented in fig.18,b.Dissociation with formation of a ribbon containing an extrinsic fault leadsto the structure shown in fig. 18,c.FIG. 17.-Models for multi-ribbons in the FIG. 18.Dislocations in the diamond structure :chromium halides. (a) undissociated dislocations ; (b) dislocationribbons containing an intrinsic fault; (c) dis-location ribbons containing an extrinsic fault.In silicon crystals twisted about a [1 1 11 axis at 1200°C and then slowly cooled,hexagonal networks lying in the (111) plane perpendicular to the twisting axis wereobserved.16 In these networks all nodes are dissociated. One family of nodescontains an intrinsic fault the other an extrinsic fault. A schematic view of thegeometry as well as the lettering pattern are shown in fig.19, while fig. 20 representsan observed region. A cut along the lines XY and UV of fig. 19 is given in fig. 18.This cross-section shows the structure of the intrinsic and extrinsic faults respectively.The conclusion from these observations is that in silicon and also in germanium,intrinsic as extrinsic stacking faults have small energy.16, 17 Observation on stackingfault triangles in silicon are in accord with this conclusion.1FIG. 5.-Extended nodes in a copper 4 % aluminium alloy (by courtesy of A. Art).[See page 11.FIG. 6.-Wide dislocation ribbons in tin disulphide. Note the presence of a triple ribbon.[See page 11.[To face page 1 6 FIG. 7.-Dislocation ribbons and undissociated dislocations in molybdenum sulphide.The analysisin terms of Burgers vectors is given in fig. 10.[See page 11FIG. 13.-Super-dislocations in the ordered alloy Ni3Mn (by courtesy of Marcinkowski).[See page 13.FIG. 14.--Single and super-dislocations in ordered niobium suboxide (by courtesy of J. Van Landuyt).[See page 13FIG. 16.--Multi-ribbons in chromium chloride (a) and chromium bromide (b).[Seepage 15FIG. 20.-Network of extended nodes in silicon crystal deformed by twisting about the [ l l l ] axis[See page 16.perpendicular to the foil plane. The inset shows extended nodes in germanium.PIG. 23.-Extended node in aluminium nitride seen under three different contrast conditions.[See page 18FIG. 25.-Ribbons in indium selenide seen under different contrast conditions (by courtesy ofMarinkovic).[Seepage 19.FIG.28. Dislocation configurations in graphite seen under two different contrast conditions.Note the presence of a triple ribbon, which is completely out of contrast in (b).[Seepage 20FIG. 33.-Multi-ribbon in talc seen under different contrast conditions showing that alternatingpartials of the quartet have the same Burgers vectors as in the model of fig. 32.[See page 23FIG. 37..-Change in width of ribbons in tealite on crossing an anti-phase boundary AB of the[See page 24.type shown in fig. 36 (by courtesy of V. Marinkovic)S. AMELINCKX 17THE WURTZITE STRUCTUREAluminium nitride, for instance, crystallizes in the wurtzite structure whichcan be described in terms of the close-packing of spheres by means of the symbolsequence aab paa bp.Here a or p denotes nitrogen, and Q or b aluminium. Fig.214 is a schematic view of the structure as seen along the c-direction while the cross-section is shown in fig. 21,b. Since all observed dislocation patterns are situatedFIG. 19.-Geometry of network of extended nodes in silicon. Cross-sections across the two typesof ribbons are shown in fig. 18,b and c.a-P-a-d-b-FIG. 21.-The wurtzite structure : (a) projection on the c-plane ; (b) side view along the directionAU showing the two types of glide planes I and LI.in the c-plane this plane must be the main glide plane.19 Like in the diamond struc-ture there are two possible locations for the glide planes ; either between a and p,or a and b, or between b and p (or a and a) ; these will be called type I and type I1glide planes respectively.They are indicated in fig. 21,b. The perfect 60" dis-location in type I1 glide planes can be represented as in fig. 2 2 4 In this dislocationmodel the number of AlN bonds to be broken on motion is smaller than that for thetype I glide plane. However, it is not clear how such a dislocation could dis-sociate into partials. However, this becomes possible if the glide plane is of type I18 FINE STRUCTURE OF DISLOCATIONSThe extended 60" dislocation in type I glide planes would then appear as representedin fig. 22,b or c. The glide vectors of the partials are indicated as Aa and aB withrespect to fig. 21. The fault within the dislocation ribbon would have the stackingb/3aabpcyaacyaa which is equivalent to one lamella of the sphalerite structure,which can be symbolized as aorb/Icyaab/3cy, as shown in fig.22. The observationsprove that dissociation takes place; fig. 23 shows an extended node in the sub-limation-grown platelet of aluminium nitride.8FIG. 22.-Dislocations in the wurtzite structure : (a) undissociated dislocations ; (b) and (c) twopossible ways of dissociating into patials for the dislocations (a). The arrows indicate the atommovements.Sublimed crystals of zinc sulphide were studied by Blank et aZ.20 The crystalswere in the wurtzite (high-temperature form) when grown ; on cooling the stackingfault energy becomes effectively negative and all dislocations which are present inthe basal plane split into infinitely wide stacking fault ribbons transforming in thisway one lamella of wurtzite into the sphalerite structure. This could be deducedfrom the diffraction pattern which shows streaking in the c-direction due to faultson the c-planes.Moreover, it shows the spots due to the sphalerite modification.21In view of the polar nature of crystals of the sphalerite and of the wurtzite type,dislocations of opposite sign can also be distinguished from a chemical point ofview. In AlN, for instance, one can distinguish " Al" dislocations and " N "dislocations; by this we mean that for an edge dislocation the supplementary halfplane would end in the one case on a row of Al-ions-in the other case, on a rowof N-ions.It would be of interest to compare the width of " aluminium ribbons "with the width of " nitrogen ribbons ". Contrast experiments distinguish the twokinds of ribbons ; it is sufficient to determine the sign of the dislocations.STRUCTURES RELATED TO THE WURTZITE STRUCTUREThe structure of the sulphides and selenides of gallium and indium is closelyrelated to that of wurtzite ; 22 it consists of four-layered lamellae which can be de-scribed by a symbol like aPPa. A schematic view of such a four-layer lamella isshown in fig. 21. In gallium sulphide and in indium selenide, e.g., the stackingsymbol is a p fi a b a a b a p /3 a. The binding within the four layer lamellae is oS . AMELINCKX 19the covalent type and hence very strong. However, the binding between four layerlamellae is of the van der Waals' type.Glide will therefore preferentially take placebetween two close-packed sulphur or selenium layers and again dissociation intoShockley partials becomes possible. An example of dislocation ribbons observedin indium selenide is visible in fig. 25.23 The photograph shows two different con-trasts of the same area allowing a Burgers vector determination. The resultingvectors are indicated on the photograph.FIG. 24.-Four-layered lamella occurring in the structures of InSe, Gas and GaSe.GLIDE BETWEEN CLOSE-PACKED LAYERS WITH OMISSIONSTWO PARTIALS; GRAPHITEThe structure of graphite is shown schematically in fig. 1,d and in fig. 26. Itconsists of a stacking of hexagonally linked layers of carbon atoms as shown inprojection in fig.26,a. The normal stacking can be described by the symbol a b a b a b . . ., i.e., half of the atoms come vertically above each other. X-ray evidence forthe occurrence of rhombohedra1 graphite with a stacking symbol a b c a b c (fig.26,b) is found occasionally in deformed samples. The c-plane is the glide andcleavage plane. In cleaved specimens all dislocation arrangements are thereforeparallel to the plane of observation. It is found that all basal dislocations aresplit into 0.1 p wide ribbons.24325 This can be interpreted in the following way.The vectors in the c-plane which connect one atom to the nearest crystallographicallyequivalent one are ABAC and BC (fig. 26) as well as their negatives; these are thepotential glide vectors. They can be decomposed into two partial vectors accordingto the reaction AB = Aa+aB; in other words, the perfect dislocations can splitinto two partial dislocations having Burgers vectors enclosing an angle of 120".The stacking fault between the two partials is then one lamella in the rhombohedralstacking as represented in fig.27. Geometrically, a second type of dissociationwould be possible, for instance, AC = Afl+flC as shown in fig. 26,c. The corres-ponding first partial would bring one layer on top of the second one. The energyassociated with such an A-over-A stacking is, however, considerably higher thanthat of the rhombohedral stacking fault. The second type of dissociation probablydoes not take place for that reason.Burgers vector determinations have confirmedthe picture outlined here. A model of an extended dislocation having the proposedBurgers vectors is represented in fig. 27. No CC-bonds are broken only deforma-tion of the hexagons is required to take up the strain.As can be judged fromfig. 28 the three partials go out of contrast simultaneously which shows that theyThree-fold ribbons are frequently observed in graphite20 FINE STRUCTURE OF DISLOCATIONShave the same Burgers vectors. The Burgers vectors are indicated in fig. 28. Thethree-fold ribbons result from the fusion of two single ribbons according to thereactions :(Aa+oB) +(oC+ An) = AoS Ao+ Aa.Hexagonat RhombOnedral.( d )( @ IFIG. 26.-Ctructure of graphite : (a) cross-sections for the hexagonal form ; (6) cross-section forthe rhombohedra1 form ; (c) projection on the c-plane showing the a, b and c positions ; (d) schemefor denoting Burgers vectors in the c-plane ; (e) scheme for denoting Burgers vectors perpendicularto the c-plane.a aIa ab bb bFIG.27.-Model for 60" dislocation in graphite. No C-C bonds are broken, only distorted.It is clear that the three Burgers vectors are the same. A detailed discussion ofthe geometry of three-fold ribbons is given in ref. (25). In boron nitride, which hasthe same structure as graphite except that two kinds of atoms alternate along aring, the dislocations appear to be undissociatedS. AMELINCKX 21MORE THAN TWO PARTIALS; TALCTalc is a complicated silicate layer structure; it can be considered as a stackingof multilayers in which the succession of single layers is 0-Si-(0-O€€)--Mg-(0-OH)-Si-0. A cross-section through such a rnultilayer is shown in fig.29.The (0-OH) layers are close-packed and magnesium is in the octahedral inter-stices. The silicon atoms are tetrahedrally surrounded by oxygen atoms. It isassumed that on glide these multilayers remain unsheared. The structure of the0SiOand OH:O;; A1 OH0FIG. 29.-Cross-section through a talc multi-layer.FIG. 30.-Structure of the oxygen layers limiting a talc multilayer. The monoclinic unit cell oftalc is outlined.limiting oxygen layers is represented in fig. 30. The bonding between successivemultilayers is due to van der Waals bonding and hence is weak; as a result, glideand cleavage take place between two such oxygen sheets.The stacking of the oxygensheets is not known unambiguously26 but a model which seems to explain theobserved features is shown in fig. 31.27 The shortest glide vector that will repro-duce the same configuration is along the a-direction in fig. 30. Using a ball modelit will be realized that glide proceeds in at least four steps which are indicated in fig.31. Taking into account the monoclinic symmetry glide in the b-direction takese i a t steps (fig. 30). In dislocation language it means that the dislocation wit22 FINE STRUCTURE OF DISLOCATIONSBurgers vector splits into four partial dislocations forming a ribbon of which amodel is shown in fig. 32. On the other hand, glide in the direction A A 2 wouldWI C )FIG.31.-Model for the stacking of the oxygen layers in the talc structure.( a ) < b )FIG. 32.-Model for dislocation ribbons in talc. The Burgers vectors forma zig-zag sequence : (a) edge ribbon ; (b) mew ribbon.produce eight-fold ribbons. The dissociation will be observable provided the energyof the stacking faults concerned is small enough. That this is the case can be judgeS. AMELINCKX 23from fig. 33. The zig-zag configuration of partial Burgers vectors has been con-firmed by contrast experiments (fig. 33).27GLIDE BETWEEN NON-CLOSE-PACKED PLANESWe shall only discuss tin sulphide and the closely related tealite (tin-lead sulphide).Tin sulphide crystallizes in the orthorhombic structure shown in fig.34; 28 thiscan be considered a deformed sodium chloride structure. It has a pronouncedlamellar character giving rise to perfect cleavage along the c-plane. The glideplane is also parallel to the c-plane according to Hoffmann.28 The probable loca-tion of the cleavage plane and also of the glide plane is indicated by arrows in fig. 34which represents the structure as viewed along the a-direction. The projectionon the c-plane of two layers of the tin sulphide structure, one on each side of theFIG. 34.-Structure of tin sulphide (SnS) and of tealite (SnPbSz). The arrows indicate the probablelocation of the glide plane.glide plane, is shown in fig. 35. From ball model considerations, possible glidevectors are ii and g. The glide vector ii could dissociate into the two partial vectorsiil and ii2 with gain in energy. Other dissociations are not obvious.The interestof these compounds from a dislocation point of view lies in the fact that the glideplane is not a close-packed plane and hence that the potential Burgers vectors arenot at angles of 120" as it is usually the case for close-packed planes, but form anangle of 90".The structure of tealite is the same as that of tin sulphide except that the tin sitescan be occupied as well by tin as by lead. Hoffmann 28 did not determine whetheror not tin and lead form an ordered arrangement. However, from the observationThe probable complete Burgers vectors are indicated in fig. 3524 FINE STRUCTURE OF DISLOCATIONSof anti-phase domains in tealite one can conclude that an ordered arrangementmust be present.In ref. (29) a detailed discussion is presented on the possible anti-phase boundary structures in this compound. From the diffraction pattern andfrom the characteristics of the fringe pattern associated with these anti-phaseboundaries one can conclude that the ordered structure must be as shown schem-atically in fig. 36 in which are considered tin and lead planes: in one type of c-plane the cations consist of tin ions whereas in the other type of c-plane they consistFIG. 35.-Projection on the c-plane of two planes of the tin sulphide structure one each side ofthe glide plane.Pb+Sn-I } 2y=0.236- S n v P b - ,1 i - 2 ~ ~ 0 . 2 6 4I - Sn Fb-- c IIPb -------I-------!% - i 2yIFIG. 36.-Cchematic view of anti-phase boundary in tealite.of lead ions.In a given type of c-plane only one kind of cation is present29 Fromfig. 36 it is now clear that the glide plane in one crystal part is situated between twotin layers whereas the corresponding glide plane in the other crystal part is situatedbetween two lead layers. As we shall see, the behaviour of ribbons on crossingan anti-phase boundary reflects the change in composition.The dislocations in tealite appear to be dissociated presumably according to thescheme shown in fig. 32, Z = 51 + 52. If such a ribbon crosses an anti-phase boundaryof the type discussed above (fig. 36) the chemical composition of the planes on bothsides of the glide plane, changes for the same dislocation ribbon from tin into leador vice versa ; it is therefore expected that also the stacking fault energy will change.This is what is observed in fig.37 where the ribbon changes its width on crossingthe anti-phase boundary AB. Since the foil has constant thickness the change inwidth can only be due to a change in stacking fault energy.All illustrations used in this paper are taken from papers by Dr. Delavignetteand the author, unless stated otherwise. I wish to thank Dr. Delavignette for hispermission to use them. I am also grateful to Dr. A. Art, V. Marinkovic, M.Marcinkowski and J. Van Landuyt for the use of photographsS . AMELINCKX 251 Heidenreich and Shockley, Rept. ConJ Strength of Solids (Bristol, 1947), p. 57.2 Kronberg, Acta Met., 1961,9, 970.3 Hirsch, Horne and Whelan, Phil.Mag., 1956, 1, 677. Hirsch, Kelly and Menter, Proc. 3rdInt. Conf. Electron Micruscopy (London, 1954), p. 231.4 Delavignette and Amelinckx, Phil. Mag., 1960, 5, 729.5 Schuller and Amelinckx, Naturwiss., 1960, 21, 591.6 Blank and Amelinckx, J. Appl. Physics, 1963, 34,2200.7 Howie and Swann, Phil. Mag., 1961, 6, 1215.8 Art, Gevers and Amelinckx, Phys. Stat. Solidi, 1963, 3, 697.8b Howie and ValdrC, Phil. Mag., 1963, 8, 1981.9 Siems, Delavignette and Amelinckx, Phil. Mag., 1964, 9, 121.10 Price and Nadeau, J. Appl. Physics, 1962, 33, 1543.11 Amelinckx and Delavignette in Direct Observation of Imperfections in Crystals (Newkirk12 Pashley and Presland, Proc. European Reg. Con5 Electron Microscopy (Delft, 1960), 1, 417.13 Boswell, Proc. European Reg. Conf. Electron Microscopy (Delft, 1960), 1, 409. Kamiya,14Marcinkowski and Fisher, J. Appl. Physics, 1960, 31, 1687. Marcinkowski and Brown,146 Van Landuyt and Amelinckx, Appl. Physics Letters, 1964, 4, 15.15Delavignette and Amelinckx, Trans. Brit. Ceramic SOC., 1963, 62, 687. Amelinckx andDelavignette, J. Appl. Physics, 1962, 33, 1458.16 Aerts, Delavignette, Siems and Amelinckx, J. Appl. Physics, 1962, 33, 3078.17 Art, Aerts, Delavignette and Amelinckx, Appl. Physics Letters, 1963, 2,40.18Booker and Stickler, Acta Met., 1962, 10, 993. Booker and Stickler, Proc. 5th Int. Con$Electr. Microscopy (Philadelphia, 1962), 1, B-8. Mendelson, J. Appl. Physics, 1964, 35, 1570.19 Delavignette, Kirkpatrick and Amelinckx, J. Appl. Physics, 1961, 32, 1098.20 Blank, Delavignette and Amelinch, Phys. Stat. Solidi, 1962,2, 1660. Chadderton, Fitzgerald21 Blank, Delavignette and Amelinckx, Phys. Stat. Solidi, in press.22 Basinski, Dove and Mooser, Helv. physic. Acta, 1961, 34, 373.23 Marinkovic, V., private communication.24 Amelinckx and Delavignette, J. AppZ. Physics, 1960,31, 2126. Williamson, Proc. Roy. SOC. A ,25 Delavignette and Amelinckx, J. Nucl. Materials, 1962, 5, 17.26 Hendrickx, 2. Krist. Min. Petr. Abstr. A ; 2. Krist., 1938, 99, 264.27 Amelinckx and Delavignette, J. Appl. Physics, 1961, 32, 341.28 Hoffmann, Z. Krist., 1935, 92, 161.29 Marinkovic and Amelinckx, Phys. Stat. Solidi, in press.and Wernick, ed. Wiley (Interscience), New York, 1962), p. 295.Gillet, Bull. Microscop. Appl., 1960, 10, 83.Ando, Nonoyama and Uyeda, J. Physic. Soc. Japan, 1960,15,2025.Acta Met., 1961, 9, 764.and Yoffb, Phil. Mag., 1963, 8, 167. Chikawa, Japan J. Appl. Physics, 1964, 3, 229.1960,257,457

ISSN:0366-9033    

DOI:10.1039/DF9643800007

出版商:RSC    

年代:1964

数据来源: RSC

摘要:

Some Basic Problems Concerning Subgrain BoundariesBY W. BOLLMANNBattelle Memorial InstituteGeneva-Carouge, SwitzerlandReceived 4th June, 1964To render the paper self-consistent a summary is given of the dualistic representation of disloca-tion reactions and its application to subgrain boundaries. Some basic problems involvingthe definition of the Burgers vector, Frank‘s Formula, irregular tilt boundaries and “ dislocated ”dislocation networks are discussed. An expansion of the dualistic method to three dimensions isgiven.1. INTRODUCTIONRegular networks of dislocations are of great importance as subgrain boundaries.The theory of dislocation networks has been developed by Frank1 and applied andreviewed by Amelinckx and Dekeyser.2 The stress field has been calculated by Li.3These authors, as far as the geometry of dislocation arrangements was involved,applied Thomson’s notation.A notation based on a dualism between the con-figuration of dislocation lines and the configuration of the corresponding Burgersvectors was introduced by the author.4 Here we shall first give a brief summaryof that method and will then discuss some special problems. Finally, the dualisticmethod will be extended to the three-dimensional case.2. THE DEFINITION OF THE BURGERS-VECTORTwo definitions of the Burgers vector (BV) based on Burgers loops are in use,Burgers’ and Frank’s definition. The physical meaning of the BV is most clearlyshown by the Volterra cut (the cut through the crystal limited by the dislocation lineand the displacement of one face of the cut with respect to the other by b followedby rejoining the crystal).We shall show that Burgers’ and Frank‘s definition areequivalent, i.e., they use exactly the same assumptions, partly explicitly stated,partly implicitly applied. The only difference is the difference in sign.An imaging process (as introduced by Frank) is assumed between a dislocatedcrystal and a perfect crystal. This imaging has to be done coordinate wise, wherebythe coordinate system is determined by the position of the atoms. The one-coordinate system is curved (elastically distorted), the other one is straight.First, the sense of the dislocation line is determined arbitrarily, then a loop(right hand screw) is imaged from one crystal on to the other (fig.1).(a) In BURGERS’ DEFINITION a closed loop in the perfect crystal is imaged on tothe dislocated crystal where it is open as described by Read.5 The definition statesthat the Burgers’ loop has to be drawn such that it would be closed in a perfectcrystal, which effectively means an imaging.(b) In FRANK’S DEFINITION a closed loop in the dislocated crystal is imaged intothe perfect crystal, where it is open. In the imaging process the distorted atom-to-atom steps of the dislocated crystal are implicitly considered equivalent to the un-distorted steps of the perfect crystal.26W. BOLLMANN 27We can write the BV in the following form :-+ + - + + + - + +b* = b,a,(r)+ b,a,(r)+ b3a3(r) - ai(r) (i = 1, 2, 3) are the basic vectors of the elementary cell of the crystal.If thecrystal is elastically distorted, the ag depend on the position r. On the other hand,b = (bl, bz, b3) is the coordinate triple independent of the position r and thus in-variable for a given dislocation, i.e., b is the BV attributed to the dislocation, while + 3 3 + +b* is its image as the closure failure of aspecific Burgers’ loop (bB = - bF, but bg =!= - bz ;3 +-bg = Burger’s definition, b~ = Frank‘s definition).+ --f3 +-3- -(a)+ -bFIG. 1 .-Definition of the Burgers’ vector. (a) Burgers’ delinition b~ ; (b) Frank’s definition b ~ .3. SUMMARY OF THE DUALISTIC METHODWe use here Burgers’ definition of the BV. By using Frank‘s node condition@bin = X:bo,t) the following dualities can be derived between the configurations ofthe dislocation lines L and the corresponding Burgers’ vectors B (fig.2).3 +D- 1D-2D-3D-4LAn L-nodeAn L-field (a fieldlimited by dislocation lines)An L-Line separatestwo fieldsIf an LLine points in acounterclockwise sense,Bcorresponds to B-polygoncorresponds to a B-node (node in the B-net)The corresponding BV connectsthe two corresponding nodesthe corresponding BV flowsinto the corresponding nodeand vice vers28 SUBGRAIN BOUNDARIESWith these relations the B-net can easily be constructed starting with an L-field andfixing the corresponding B-node, and from there on using D-3 and D-4 for construct-ing the net mesh by mesh. The dislocation network divides the space into two parts,the one below, the other above the net.The 23-net represents a map of all displace-ments introduced by the dislocations between the different L-fields referred to thespace above the network, the part below the network being considered fixed. Detailsfor the study of dislocation reactions are given in ref. (4).L -~ /FIG. 2.Dualistic correlations between the L- and the B-net. The L-net consists of screw dislocationswith Burgers vectors pointing in the direction of the line sense.4. DISLOCATION NETWORKS AS SUBGRAIN BOUNDARIESFirst, we consider a fully general subgrain boundary where the two joiningcrystals may be slightly different in orientation as well as in lattice constant andstructure.Such a boundary can be thought to be formed in 3 steps (fig.3).1. A perfect crystal is cut into two parts, a lower- and an upper one. The lowerpart is Ieft unchanged, while the upper one is rotated and/or its lattice distorted.The face of the upper crystal is then cut so that it fits the surface of the lowercrystal again.2. The crystals are brought together purely geometrically without interaction of theatoms. The superposition of the two periodic lattices will produce a two-dimen-sional moire pattern at the boundary (moire case).3. The atoms interact such as to form a continuous crystal as far as it is possible.The moire pattern changes to a dislocation network.We discuss here 6irst the moire case. Two vectors r(1) and r(2) are defined thef i l l is supposed to be lying in the crystal below the boundary (in the unchangedcrystal) and r(2) in the one above the boundary.If there were no boundary (Le.,if both parts form one continuous crystal), r(l) and r(2) would be identical, i.e., they+ +following way (fig. 3).++-+ W. BOLLMANN 29could be transformed into each other by " translation ". In an elastically deformed(curved) crystal, this " translation " would have to follow the curvature of the co-ordinate system.If the second crystal is rotated and/or distorted such that a crystal boundary isformed r(2) is also rotated and/or distorted. Now the two vectors differ by a vectorB such that3- b + +(4.1) g = # 2 ) - p ) .+- +- -+The vector B for$xed r(l) and r(2) is characteristic for the boundary, i.e., it is in-dependent of the choice of side (I) and side (2) of the boundary (except that the sign of Bchanges when the numbers of side (1) and (2) are interchanged.+3+ +FIG.3.--Formation of a subgrain boundary. (a) Cutting of a complete crystal r(2) = r(1); (6)changing of part 2; (c) rejoining the two parts geometrically (moird case); (d) rejoining the twoparts physically.5. FRANK'S FORMULAIf the two crystals which are joined at the boundary are of the same nature butdiffer in orientation, a second condition for B is given by Frank's formula, whichcan be approximated for small angles (sin I3 = 8, cos 13 = 1) as33 3 3B = [ex r ( 9 . (5.1) -+6 is a vector in the direction of the axis of rotation (right-hand screw) with the lengthcorresponding to the angle of rotation (radians).The sequence of the factors, i.e.30 SUBGRAIN BOUNDARIESthe sign of the product has to be determined according to the chosen definition ofthe Burger's vector (in our case for Burger's definition). B as a vector product hasto be perpendicular to both factors, to 8 as well as to r(1). The first holds generally,the second is only approximately true.For general rotations Frank's formula can be written :--f+ 4Bi = (a, - 8ik)r$1),where aik is a general orthogonal tensor and Sik is the unit tensor. (The general formof Frank's formula is of interest for the study of coincidence lattices.)Up to now we have considered the mor6 case, which, for a subgrain boundaryis described by the relations (4.1) and (5.1). Fora fixed 6 a continuous B is related to a continuous r(1).When the two crystals interact,most of the atoms are forced into equilibrium positions and the moirC bands con-tract to a regular network of discrete dislocation lines (fig. 34. In that case Bbecomes the sum of the BVs of the dislocations crossed by ~(11, i.e., B acquires dis-Crete values Bn (nodes of the B-net) and (4.1) and (5.1) become quantized :Both relations are continuous.-+ +- ++3 --f+ -++r(1) also takes discrete values rJ1) which can be attributed to the centre of the L-fields,i.e., to the points of unstressed matching of the two rotated crystals or, as Frank 6pointed out, rn(l) marks the points of the coincidence lattice of the two crystals.The coincidence lattice is defined for two lattices interpenetrating each other by allthe atom positions common to both lattices.In this case it consists of rows of atomsparallel to the axis of rotation passing through the centre of the L-fields.--+6. TWIST BOUNDARY+-To apply Frank's formula (5.4) rJ1) can be taken from the L-net between twoarbitrary numbers and Bn from the B-net between the corresponding numbers (fig. 2).The vectors have to be compared at the same scale. (For better visibility the scaleof the L- and the B-net usually are drawn markedly different.) A consequence of(5.1) or (5.4) is that the axis of rotation 8 is always perpendicular to the B-net. Asthe relative position of two crystals grains is determined by one 6 alone, the B-netof a sub-grain-boundary has to be a plane or to approximate to a plane as nearly aspossible.A B-net may be a plane when the dislocations of the corresponding L-net allhave coplanar BVs, which is only possible if 8 is parallel to a simple crystallographicorientation. In this case we speak of a pure twist boundary.This twist boundaryin addition may be fuZ2 if the L-net is parallel to the B-net or partial if the L-net is+3+-W. BOLLMANN 31inclined with respect to it (fig. 4). The partial twist boundary is essentially obtainedby projecting the full twist boundary in the direction of the axis (along the rows ofatoms of the coincidence lattice) on to the inclined plane or surface. In contrast tothe B-net, the L-net does not have to form a plane as long as it is a projection of thefull twist boundary.~ _ _FullTwtstBoundory_L-R r t r o fTwistBoundsry_I___-TiltboundaryFure bundaryL BGeneml BoundaryL 0FIG.4.-Relative orientation of the L- and the B-net in different types of subgrain boundaries.A foreign dislocation (one with a non-coplanar BV) introduced into a pure twistboundary produces a step in the corresponding B-net.For an arbitrary orientation of 0 the plane of the B-net has to be approximatedby a stepped surface, in general with two independent sets of steps (because theBVs have fixed orientation and cannot be inclined). The appropriate foreign dis-locations have to be introduced into the L-net. We call the corresponding boundarya general twist boundary (as compared to the pure twist boundary).This, in addi-tion, can be full or partial depending on whether the L-net is parallel or inclined tothe B-net (fig. 4). When the L-net is perpEndicular to the B-net, i.e., parallel to 0,we speak of a tilt boundary.--f--f7. TILT BOUNDARYA tilt boundary has to consist of an array of parallel edge dislocations. It canbe considered as a limiting case of a partial twist boundary. A tilt boundary canbe regular or irregular depending on whether its plane lies in a special orientationpassing through the centre of the fields so as to cut through a regular array of dis-locations in the corresponding full twist boundary, or in an arbitrary orientation?cutting through different kinds of dislocations. The corresponding cut can be de-termined in the B-net.In the first case, the cut passes through an array of nodesin the B-net and the corresponding dislocations lie between the corresponding field-centres (atom rows of the coincidence lattice) (fig. 4). In the second case the cuthas to be approximated by a stepped line through the nearest B-nodes and the cor-responding path through the L-net is such that an irregular tilt boundary also appearsas " stepped " (fig. 5). Amelinckx and Dekeyser 2 give examples of stepped tiltboundaries32 SUBGRAIN BOUNDARIES8. " DISLOCATED " DISLOCATION NETWORKA new problem arises when a dislocation line breaks out of a regular network.In the periodic structure of the L- as well as of the B-net an irregularity appears as atype of " dislocation " (fig.6). At first sight this " dislocation " in the B-net seemsLB_1-- -FIG. 5.-Irregular tilt boundary.FIG. 6.-" Dislocated " dislocation network, due to the break away of one dislocation line from thenetwork.paradoxical because the B-net was constructed by assuming (a) the invariance of theBV and (b) Frank's node condition (continuity), and one of these conditions at leastappears to be violated. Interaction of dislocations cannot be the reason, becauseby choosing 6 arbitrarily small, the meshes of the L-net become large (i.e., the inter-action small) without change of the B-net. By referring to the description of the+W. BOLLMANN 33BV in 8 2 as of the type,- b - b + + + - + +b* = blcl(r)+ b2c,(r)-t b3c3(r),3we see that b = (bl, b2, b3) (i.e., the BVs of all the dislocation lines) may remainunchanged while the distortion in cl(r) is the image by means of (5.4) of the distortionof the coincidence lattice due to the breakaway of a dislocation line.The BV ofthe " lost " dislocation may be determined by a Burger's loop in the B-net.++9. EXTENSION TO THREE-DIMENSIONAL CONFIGURATIONSThe question arises whether the concept of the B-net as a fully connected con-figuration could be extended to the three-dimensional case, i.e., for example, thejunction of three subgrain boundaries in a line. Every subgrain boundary and itsB-net is a periodic structure. The junction of two periodic structures with differentmesh sizes or different orientations has to occur by means of" dislocations ".These" dislocations " of the L- and the B-net represent the dislocation lines broken awayfrom the network to contribute to the formation of the third boundary. Thus, itis no longer possible to construct a connected B-configuration for all three boundariestogether.c - SpoctTm'sl baundoryright hand screw(TW. r ) =-L 2 =-FIG. 7.-Description of different kinds of boundaries by the C-8-dualism.An extension of the dualistic method to three dimensions is possible by represent-ing a whole boundary by its vector 8. The dualistic correspondence (see page 34) be-tween the crystal-space (C-space) and the &space can be only given for small angles8, since only for small angles can two consecutive rotations be approximated bythe addition of two 0 vectors.The sense of rotation may be determined from the relations of the numberingin the C- and in the 0-space (fig.7). The domain attributed to the first number,NI (whichever is chosen as the first) is considered as fixed and the second, N2 is ro-tated in the sense of a right-hand screw by passing from Nl-+N2 in the &pattern.-++4--f34 SUBGRAIN BOUNDARIESC-Space1. A complete subgrain (3dimensions-3d)2. A subgrain boundary (2 d )3. An intersection of bound-aries in a line (1 d)4. An intersection of severalboundaries in a point (0 d )8-Spacea point junction of &vectors+corresponds tocorresponds to 8-vector (1 d)corresponds to polygon formed by &vectorscorresponds to polyhedron formed by 8-(0 4 -+++ (2 4vectors (3 d)An example of how to study the possible character of the subgrain boundariesat a linear intersection is given in fig. 8. By these means it is possible to study therelations between subgrains in a polygonized material.FIG. 8.-Some special possibilities of a junction of three subgrains, with 161 = const. and full twist-or tilt boundaries. An infinity of other arrangements in space is possible.The author thanks Prof. F. C. Frank, Dr. D. G. Brandon for fruitful discussionsand Mrs. F. Bruy6re for the execution of the drawings as well as the BattelleMemorial Institute for financially supporting the work.1 Frank, Conference on Plastic Deformations of Crystalline Solids (MelIon Institute, 1950), p. 150.L Amelinckx and Dekeyser, Solid State Physics, 1959, 8, 325.3 Li, in Electron Microscopy and Strength of Crystals, ed. G. Thomas and J. Washburn5 Read, Dislocations in Crystals (London, 1953), pp. 32-33.6 Frank, F. C., private communication.(John Wiley & Sons, New York, 1963), p. 713. 4 Bollmann, Phil. Mag., 1962, 7, 1513

ISSN:0366-9033    

DOI:10.1039/DF9643800026

出版商:RSC    

年代:1964

数据来源: RSC

摘要:

Shape of Three-Fold Extended NodesBY L. M. BROWN AND A. R. THOLGNCavendish Laboratory, CambridgeReceived 15th June, 1964T he self-stress of a dislocation is defined in such a way that it can be calculated at the dislocationcore in terms of line integrals over the dislocation. The relationship between this definition andother definitions is studied. The shape of extended nodes is calculated by an iterative method,whereby points on the dislocation core are displaced successively until the total stress on them (self-stress plus stress due to stacking fault and other partials) vanishes. The solutions are presentedg raphically.In an isotropic medium, it is found that the self-stress of curved screw dislocations can be muchgreater than that of similarly curved edge dislocations.Thus there is a difference in appearancebetween nodes formed from screw and edge dislocations. The relationship between the stackingfault energy and the radius of curvature of the partials in the node depends on whether the partialsare edge-type or screw-type. It is possible, however, to determine the stacking fault energy approxi-mately by measuring the width of the node, for the width of the node is much less dependent on thecharacter of the dislocations which make it up than it is on the curvature of the partials.The calculations have so far been restricted to isotropic media, and this constitutes their greatestdrawback.This paper contains a risumk of a calculation of the shape of extended, three-foldnodes, as well as a careful comparison of theory and experiment in a system of alloys.The formation of such nodes, and the use which can be made of them to measurethe stacking fault energy of crystals in which it is not otherwise possible to do so,were first explained by Whelan.1 Since that time, there have been several papersdealing with the subject, notably by Howie and Swann,z who examined variousf.c.c. metal alloys, and Siems, Delavignette and Amelinckx,3 who examined layerstructures.Apart from the interest in stacking fault energies, verification of the theory pro-posed here represents a check on the essential correctness of Wulff's theorem as itapplies to dislocations-indeed, there can be few physical systems in which it is pos-sible to test Wulff's theorem so directly, knowing at the outset the orientation de-pendence of the interface energy (in our case, the dislocation energy).THEORYUsing the approximation of isotropic elasticity, detailed calculations have beenmade of the shape of extended nodes.The calculation is based on a definition ofthe stress on an element of dislocation due to the rest of the dislocation-the so-called " self-stress ".4 If o(r), calculated from the ordinary elastic theory of dis-locations, is the component of stress which does work when the dislocation elementglides, the self-stress us is defined as2oS(ro) = a h + e) + o(r0 - E). (1)Here, ro is a point on the dislocation line, and e is a vector normal to the line; thelength of E is an arbitrary parameter in the theory, but it is related to the core radius336 SHAPE OF THREE-FOLD EXTENDED NODESof the dislocation. Using this definition, one finds that the self-stress around acircular shear loop is given byos=Here, G is the shear modulus ; v is Poisson's ratio ; r is the loop radius and a measuresthe distance around the loop ; a gives the character of the dislocation at the elementwhose self-stress is as; a measures the angle between the Burgers vector b and thetangent to the element. When the loop becomes large, the log term predominates,and the dislocation behaves as if it had an energy # per unit lengthGb2 1 - v cos2 a rE(a) =- In-471 1 - v E'and thus a " line tension " given byGb 2-v 3~T = E + - d2E =a:br ( 0% =- [ - + - cos 2 x 1 In k).da2 8zr 1 - v 1 - v(3)(4)Eqn. (4) might be called " de Wit and Koehler's approximation ".5 The relationshipbetween tension and energy given by eqn.(4) is a general result, and can be provedsimply by considering the forces on an element of line, provided the forces tendingto rotate the line are also included.6The approximation of eqn. (4) is not very good for a loop radius of only a fewthousand times the cut-off radius; and the approximation is particularly bad forlarge v. In spite of this, it is useful for qualitative understanding of problems likethe node problem, and it is therefore of some interest to generalize the results justquoted to include an arbitrary configuration of dislocation.FIG. 1.-A shear loop of arbitrary shape in the plane of the paper.For an arbitrary configuration of dislocation line, the self-stress on an element,defined by eqn. (l), will approach the value given by eqn.(4), provided r is interpretedas the radius of curvature at the element ; and will approach it more and more closelyas the scale of the configuration is increased, without a change in its shape. Thiscan be seen from fig. 1. The contribution to the self-stress at X has been brokenup into a part due to the dislocation outside AB, and the part within. The partwithin can be given as yev+ 8nr 1 - v 1 - v cos Za] In :L . M. BROWN AND A . R . THOLBN 37One way of showing this 4 is to sum the contributions of the kinks causing the cur-vature at X over the length AB = $. The contribution to the self-stress from thepart outside AB can be put into the form, const.Gb/r, which simply expresses thefact that stresses from dislocations fall off inversely as the distance from them.Increasing the scale of the configuration increases r in the same proportion; so, forany p, a scale can be found which makes the log term arbitrarily large compared tothe other terms. Hence, a first approximation to problems of dislocation shapecan be found by using a line tension T = akbr ; this approximation is essentiallyWulff’s theorem for dislocations. A complete theory of the self-stress would alsoinclude terms due to the core of the dislocation. For the present, the arbitraryassumption of taking E = b has been followed, and no additional terms have beenadded. This procedure gives the same value for the average self-stress of the shearloop as the value deduced from the elastic energy of the loop, including an energyGb2(2-v)/87t(l -v) for the core.But further work is required to clarify the role ofthe core; in a proper treatment, the atomic structure of the core would be reflectedin the equations.We are now in a position to use the general theory outlined above to tacklethe node problem. A symmetrical, three-fold node is shown in fig. 2. Evidently,FIG. 2.-An extended, threefold node, and the parameters used to describe it.a first approximation to the problem is to neglect all stresses except the self-stress,and to write for the curvature at points L, M and N( y = stacking fault energy)Ylb = os,,Although this equation is not very accurate, it does show that screw nodes (thosenodes whose partial dislocations are in the screw orientation at L, M and N ; orequivalently, those nodes whose arms are screw) will have a larger radius than edgenodes.However, within the framework of the theory, it is possible to do an exactcalculation.4 A trial shape is assumed for the partials, and the total stress cal-culated at points such as those shown in iig. 2. The total stress includes the self-stress, the stress due to the other partial dislocations, and the stress y/b due to thefault. Each point is then moved a distance proportional to the stress at the point ;the stresses are recalculated, and the points moved again ; and so on until all stressesare zero.The node is then in its equilibrium shape38 SHAPE OF THREE-FOLD EXTENDED NODESThe results of the calculations can be summarized in two equations :P -- - -27 - -08( &) cos 2a + { *104rS) + *%I( &) cos 2a} log,, i,Gb2P -- z2 - .055~~;)-.06((&2) cos 2a+{.018~~)+.036(+--) cos 2ct} log,, -. E(7)Here, b refers to the Burgers vector of the partial dislocation ; p and w are definedin fig. 2; a refers to the character of the partials at points L, M and N.Eqn. (6) and (7) have a form based on eqn. (5), with terms added to take accountof the complicating effects due to the finite size of the node. The numerical co-efficients have been chosen to fit all the computed values to better than 10 % overthe range of a and v of experimental interest.For v = 0.5, and edge nodes (a = n/2),eqn. (6) and (7) may be in error by as much as 20 %. The equations can be regardedas interpolation formulae for the computed data.EXPERIMENTALThe radius, width and character of symmetrical nodes were measured in severalNi-Co-Cr alloys. The alloys N, T and U (table 1) are in the f.c.c. region, while the otheralloys are in the two-phase region, f.c.c. and b.c.c.7 The alloys were supplied by the MondNickel Company in strip form, about 250 microns thick. They were annealed at 1200°Cfor 4 h, cold-rolled to a thickness of about 150 microns for ease of polishing, and thenannealed again under the same conditions. The specimen was deformed by bending arounda thin rod before electro-thinning in Jacquet’s solution (85 acetic acid, 10 perchloric acid,5 water).The nodes were not generally in a plane pefpendicular to the electron beam.Thus itwas necessary to correct the node dimensions for inclination. The true radius and truewidth are given bywhere $ is the angle between the electron beam and the normal to the plane containing thenode ; Cp,b, is the sum of the three projected radii ; CWLbs is the sum of the three projectedlengths w’ (fig. 2).These formulae are an adequate approximation for angles I/I less than 45”. Nodeslying in planes more steeply inclined were not used. It is more convenient to use this methodthan to correct each radius separately, and if no correction is applied, a systematic errorresults.The character of the node, i.e., the angle between one arm of the node and its Burgersvector (for the whole dislocation) was determined by Burgers vector analysis.Usually,in spite, of the multiplicity of available Burgers vectors, it is possible to determine the char-acter of a node from only one micrograph (and diffraction pattern). This is because, withthe reflections normally used, one arm of the node vanishes and an approximate knowledgeof the slip plane k e s the Burgers vector of the arm.For each alloy, the bulk Poisson’s ratio, and Young’s modulus were measured withstrain gaugesL. M. BROWN AND A. R . THOLI~N 39RESULTSCOMPARISON OF THEORY WITH EXPERIMENTIn the theory, there are three parameters which completely determine the shapeof the node : y/Gb, v and E. For the purposes of this section, we have determinedy/Gb from the observed radius of screw nodes ; but we have taken v = 0.24, whichis the mean observed bulk value of Poisson's ratio, and we have taken E = b = 1-44 A.It is impossible to justify the latter two assumptions.If the material were iso-tropic, the assumption concerning v would be exact; but it is unlikely that thematerial is even nearly isotropic. However, if one compares the bulk value of vwith a value determined to give correctly the energies of screw and mixed dislocationsin anisotropic cubic crystals, one finds that there is a rough correspondence betweenthe two. Therefore the assumed value will probably approximately reflect the varia-tion in energy between dislocations in screw and edge orientation for our alloys.The assumption regarding E is equally artificial, but the node parameters probablydepend only weakly on E in any case.TABLE 1approximate compositionAlloy % Ni % co % Cr Pscrew(') (;)expt.6)theoryN 40 40 20 350 3.50 4-37K 45 15 40 550 4.15 4.43L 45 25 30 625 4.00 4.46T 30 50 20 875 4.40 4.48U 30 60 10 1350 4.00 4.45P 30 30 40 1400 4.10 4.56S 30 40 30 1 700 4-80 4-57Table 1 lists the alloys, their composition, the observed radius of screw nodes, andthe observed ratio (plw) for the screw nodes, together with the value predicted byeqn. (6) and (7). The agreement between the experimental and theoretical quantitiesis quite good, except for alloy N. In alloy N, w-100 A ; which is of the same orderas the dislocation image width.Therefore, the measurements are particularly diffi-cult for this alloy, and the disagreement probably does not reflect a failure of thetheory. From Table 1, the mean (p/w) for all the alloys, omitting alloy N, is 4.24 ;the theoretical mean is 4.51 ; and for comparison, the theory of Siems et aZ.3 gives4-04. The ratio (p/w) is not a particularly sensitive test of any node theory, providedthe relation between p and y/Gb is approximately correct.In fig. 3 and 4 the variation of p and (p/w) with a is plotted, and compared withtheory for alloy T. Once more, the agreement is adequate, although the scatter in pis large. The theory of Siems et aZ. fails to predict the decrease inp with increasing a.Another effect not accounted for in the theory is the effect of the free surfacesof the foil.These can affect the nodes in two distinct ways. The first consists inaltering the angles between the arms of the node, and thus changing its shape, thesecond, in directly affecting the self-stress of the partials. If a node has arms whichall intersect a free surface, it is unstable, and will glide out, leaving a single dis-location. If it is prevented from doing so by other dislocations, it may still becomedistorted.If the radius of a node is appreciably larger than the distance of the node from asurface, the self-stress is directly affected. For the r in the log term of eqn. (3) shouldthen be replaced by a length roughly equal to the distance of the node from thesurface ; this affects a& in the same way, and hence the relationship between p an40 SHAPE OF THREE-FOLD EXTENDED NODESy/Gb from eqn. (5).For the nodes considered here, calculation shows that thiseffect is very small. An indication that the effect is small is that the (plw) values forlarge nodes do not deviate more from the theory than they do for small ones. Onlyfor nodes whose radius is equal to the foil thickness or more will this effect beimportant.5 0 4 0 70 8 0 9b 01 o 10 20 30 4baFIG. 3.-A plot of (p/w) as a function of a, for comparison with theory ; alloy T.aFIG. 4.-A plot of p as a function of a, for comparison with theory ; alloy T.The scatter in p which is observed is undoubtedly due to internal stresses, of whichthe first type of surface effect may be taken as an example. A stress of much lessthan y/b can cause appreciable distortion of the node by changing the position ofits arms ; the effect that the stress can have depends on the length of the arms. Scatterdue to this effect can be minimized by discarding markedly asymmetrical nodesL. M. BROWN AND A . R. THOLBN 41Generally, then, the theory is confirmed. The test of the theory is not verysevere, in that the parameter (plw) is probably not very sensitively dependent uponthe detailed shape of the node; hut one can use the theory to determine stackingfault energies with some confidence.The authors are grateful to Dr. P. B. Hirsch for his encouragement.1 Whelm, Proc. Roy. SOC. A., 1958, 249, 114.2 Howie and Swam, Phil. Mag, 1961, 6, 1215.3 Siems, Delavignette and Amelinckx, 2. Physik, 1961,165, 502.4 Brown, L. M. Phil. Mag., in press.5 de Wit and Koehler, Physic. Rev., 1959, 116, 1113.6 Chou and Eshelby, J. Mech. Physics Solids, 1962, l0,27.7 Koster and Nowikov, Freiberger Forschungshefte, 1960, 38, 68

ISSN:0366-9033    

DOI:10.1039/DF9643800035

出版商:RSC    

年代:1964

数据来源: RSC

摘要:

Shape of Extended NodesBY DR. R. SIEMSInstitut fur Theoret. Physik C der Tech. Hochschule, AachenReceived 12th October, 1964An analytical treatment of the extended node problem is presented which takes the orientationdependence of the line energy of the partial dislocations into account. As in former calculationsan approximate expression is used for the interaction energy between different partial dislocations.The results are compared with electron microscope observations.The geometry of extended nodes can be used to determine stacking-fault energiesas was first shown by Whelan.1 The method was applied to various f.c.c. alloysby Howie and Swann.2 In a previous paper 3 the theory of these nodes was workedout in more detail, especially by taking the interaction between the partial disloca-tions approximately into account, and applied to nodes in layer structures (graphite,AlN, etc.).Thornton et aZ.4 reviewed Howie and Swann’s data by making use ofthe improved theory. Brown 5 calculated the shape of extended nodes numericallyby an iterative method and Brown and ThWn 6 compared the results of this calculationwith observations on nodes in Ni-Co-Cr alloys. In Brown’s treatment a trialshape was assumed for the node ; for this trial shape the total stress (self-stress plusstress due to the stacking-fault plus stress due to the other partials) was calculated fora number of points on the dislocation core. Each of these points was then displaceda distance proportional to the stress at the point. The stresses were then recalculatedfor the new dislocation configuration and the points moved again.The procedurewas repeated until all stresses were zero. This numerical treatment of the problem,which avoids some of the approximations employed in ref. (3), explains the dependenceof the radius of curvature on the character of the node, whereas ref. (3) gives a wrongdependence. It seemed worthwhile to see if one could not improve the accuracyof the theory 3 and still obtain analytical solutions.The main shortcoming of the calculations presented in ref. (3) was that thevariation of the line energy with the character of the partial dislocations along theirlines was neglected, e.g., for a screw node * the line energy at any point was assumedto be that of a pure screw dislocation.Under this assumption only the length ofthe dislocation line and the line energy at the node count. A dislocation having alarge line energy at the node (i.e., an edge dislocation) tends to have a short lineand the respective node has a large radius of curvature, whereas a screw node (lowline energy) has a small radius of curvature. In the present paper the old theory 3is improved by taking the variation of the line energy along the line into account.The problem can be treated in a way analogous to that presented in ref. (3) andanalytic solutions for the width of the node and for the radius of curvature can beobtained.* A node is called a screw node if the partial dislocations at the node and the total dislocationsat a large distance from the node are in screw position.Edge nodes are defined in an analogousway.4R. SIBMS 43THEORYIn order to find the equilibrium form of the node presented in fig. 1, expressionsfor the total energy are derived as functionals of the curve y(x) representing thepartial dislocation line. The stable (minimum energy) shape is then obtained asa solution of the Euler-Lagrange equation of the corresponding variational problem.FIG. 1.-Schematic drawing of an extended node.The notations used in the text are indicated.The energies connected with dislocation a from P to positive infinite x will becalculated. They are counted from the energy of the configuration represented bydashed lines in fig. 1. The system of co-ordinates which was used is indicated infig. 1.It is non-rectangular. The elements of arc length and of area areds = J1 +y‘+y’2 dx and dF = 3 JFydx,respectively.The energy due to the stacking-fault isE, = $ y I m ( y -y,)dx (y = stacking fault energy)0That part of the interaction energy which is dependent on the function y(x) isapproximated by 3withK=---- ltb2 [2-v(1+2p)],Sn(1- v)(p = shear modulus, b = Burgers vector of the partials, v = Poisson’s number,D = 1, - 1 for screw and edge nodes, respectively)44 SHAPE OF EXTENDED NODESThe line energy at a point where the Burgers vector forms an angle 4 with thedislocation line is approximated by the expression,~ ( 4 ) = 2Q(l -V C O S ~ ~ ) = Q(2-V-v cos 24),withpb2 In (Rlr)8n 1 - vQ=--,The contributions to the line energy will be important only for elements of thedislocation line close to P, i.e., for elements from a region where the dislocation linehas a well-defined radius of curvature.R is identified with this radius of curvature(cf. e.g., ref. (6)), and r with the length of the Burgers vector. Eqn. (1) is supposedto represent the line energy in good approximation for large values of R/r. Inthe variational procedure the change of energy due to a change in In (R/r) will beneglected. For any point at a partial dislocation of an edge node 4 is equal to theangle Q formed by the direction of the dislocation line at this point and the y-axis,for a screw node 4 is equal to o = 4 2 - SZ. One hasThustan cc) = cotan i’2 = (1/43)(1+ 2y’).With the abbreviation ,/1 +y‘+y’2 = w, one obtains for the integrated line energy :E, = Srn Q[(2 - v[w + vp(w - 3/(2w) - 2 + v + vp/2]dx.0The total energyE = El 2 + E, + E, =withK Y .J3 V = -- In - + - p ( y - y , ) + Q[(2 - v)w +vp(w - 3/(2w)- 2+ v + vp/2]2 Y lhas to be minimized with the boundary conditions,y = y1 for x = m, and y‘ = -3 for x = 0.The Euler-Lagrange equationaV d dV ----=()dy dxdy’leads tor - 3 v - 9 ~ ~ -+-vp--J.1 27 I ] y” = 0.) f f + J % - Q w 3YSince Vdoes not depend explicitly on x, a first-order equation for y(x) can be given ;8V8Y’V - y’- = const. (3)The constant is determined from the boundary condition at x = MI ( y = y1, y‘ = 0).t is zero. The equilibrium condition for the partials at x = 00 yields y1 = K/J?;rR.SIEMS 45Introducing $ = y/yi, t = X/YI, d$/dt = $' = y', one obtains from eqn. (3) : 1 1 3vp 1((2 - v)(2 + $ I ) - vp(1- $')]- - -($'+ 2f2)--, -4+ 2v + vp = 0. w 2 WFor x = 0, one obtains with $'= -3 and with the definition @O = @(x = 0) :$O-l-ln$o = T,with111098II/ V=O.2<,/''screw nodesp= 1edoe nodesp=-1(4)7 ------TI---&----< w4 - R F5 6 7 8 9 10 1°F- O i-- __102FIG. 2.-The parameter $0 = y o / y ~ as a function of the radius of curvature R, for different valuesof Poisson's ratio v. The inner cut-off radius Y is about equal to the length of the Burgers vector.Fig. 2 shows the solution $0 of eqn. (4) as a function of v and R/r for screw nodes( p = 1) and for edge nodes (p = -1).The values of $0 for 45" nodes ( p = 0)and any value of v are given by the curve marked v = 0." (For v = 0 the value of$0 is independent of p , for p = 0 it is independent of v.)* The above derivation, by requiring y' = -9 at x = 0, presupposes a symmetry of the dis-location lines about the y and x axes (fig. I). Of all node shapes having this symmetry, that whichhas the lowest energy is determined. In general, there will be other, unsymmetric, nodes havinga lower energy. Only for pure screw and pure edge nodes can one be sure that the actual nodeis indeed symmetric. Since, however, also the observed nodes of mixed character are usuallyrather symmetric, the calculated minimum energy symmetric shape will be a good approximationto the actud (unsymmetric) one.For such mixed nodes of character cc the quantity p in the aboveequations is given by cos 2%46 SHAPE OF EXTENDED NODESThe width yo is given byYO = $ 0 ~ 1 = $ o K / J ~ Y .-5n: 2-1 -0 ,From eqn. (2) one obtains (with y' = -4 and y = yo) for the radius of curvaturea t x = O :R = - - 2 J l + y ' + y ' 2 3J3 Y" 5I I I IDISCUSSIONThree characteristic lengths are conveniently used to describe an extended node.The radius yo of the inner circle (the width of the node), the radius R of curvature,and the width y 1 J3 of the straight part of the dislocations (see fig. 1). Since thestacking-fault energy y is not known a priori, the absolute values of these quantitiescannot be calculated. Their ratios, however, are independent of y * and can beobtained from theory and compared with observed values.In fig. 3-7, the theor-etical results of Brown and TholCn (broken lines) and of the present calculation(full lines) are compared with values observed at nodes in Ni-Co-Cr alloys byBrown and Tholkn and at nodes in graphite (ref. (3)).UFIG. 3.-A plot of R/yo as a function of the character u of graphite nodes.Broken line: Brown and TholCn's theory; full line: present theory. The points represent theresults of measurements 3 on two nodes.In graphite the width y1 J3 of extended dislocations can be determined in theelectron microscope. In this case, therefore, all three lengths are measurable.For two of the graphite nodes (a and h) shown in fig. 16 of ref. (3) the directions ofthe Burgers vectors had been determined (a was an almost pure screw node, h hadan ax60" orientation).In fig. 3 and 4 the values of R/yo and yo/yl are plottedagainst a for graphitte (v = 0-24). An accurate determination of y1 is dificultbecause of the width of the dislocation image ; bearing this in mind, the agreementis satisfactory.In metals and alloys the stacking-fault energy is usually so high that the widthy1 J3 of extended dislocations cannot be resolved. We shall use, as Brown and* With exception of the slight y-dependence of the In (R/r) termR. SIEMS 47TholCn, Rlyo and the variation of R as a check on the theory. In fig. 5 values ofR,/yo for screw nodes in alloys with different y-values are plotted against R.* The76-’I 731 410O L 1 I 0 W 20 30 40 SO 60 h kaFIG.4.-A plot of yo/yl against tc for graphite.U 0Broken line : Brown and Tholh’s theory ; full line : present theory. The points represent theresults of measurements 3 on two nodes.I51- - r- -I -6 8 10 12 14 j6 l @ fRFIG. 5.-A plot of Rlyo against R for screw nodes.Broken line : Brown and Tholtn’s theory ; full line : present theory. The points represent Brownand Tholtn’s measurements 6 on screw nodes in various Ni-Co-Cr alloys with different stackingfault energies (v = 0.24).* (cf. table 1 of ref. (6).48 SHAPE OF EXTENDED NODESabsolute values and the general trend are predicted by both theories. In fig. 6 and 7,R/yo and R at nodes in one of the alloys (T) are plotted against a and comparedwith Brown and Tholen’s eqn. (6) and (7) of ref. (6) and with the present theoreticalresults.5421IaFIG. 6.-A plot of Rlyo against a.Comparison of the present theory (full line) and of Brown and Tholh’s theory (eqn. (6) and (7)of ref. (0, broken line) with their measurements on nodes in a Ni-Co-Cr alloy (T).00 10 20 30 40 50 60 w 80 90.CCFIG. 7.-Variation of R with the character a of the node.Broken line : Brown and Tholhn’s theory ; full line : present theory. For both curves y has beenchosen to give the observed value of R = 875 8, for screw nodes. The points represent measure-ments 6 on the same Ni-Co-Cr alloy as in fig. 6.1 Whelan, Proc. Roy. SOC. A, 1958, 249, 114.2 Howie and Swann, PhiZ. Mag., 1961, 6, 1215.3 Siems, Delavignette and Amelinckx, 2. Physik, 1961, 165, 502.4Thornton, Mitchell and Hirsch, Phil. Mag., 1962, 7, 1349.5 Brown, Phil. Mag., 1964, 10, 441.6 Brown and ThoMn, this Discussion

ISSN:0366-9033    

DOI:10.1039/DF9643800042

出版商:RSC    

年代:1964

数据来源: RSC

摘要:

Elastic Interaction Between Prismatic Dislocation Loops andStraight DislocationsBY F. KROUPA * AND P. B. HIR~CH -fReceived 15th June, 1964The contribution of the elastic interaction between dislocations moving in slip planes and ofrandomly distributed prismatic dislocation loops and of cavities to the critical shear stress in f.c.c.metals is estimated. The results are applied to quench hardening in aluminium.An increase of the critical shear stress, i.e., quench hardening, takes place in puremetals after quenching and successive ageing13 2, 3 The critical shear stress inaluminium increases by quenching by AT - 0.6 kglmm2 ; Kino found an approx. 50 %decrease of AT after annealing at a temperature of about 200°C. Tanner 4 investigatedthe temperature dependence at lower temperatures and found that it does not substan-tially differ from the temperature dependence of the critical shear stress of slowlycooled aluminium.Immediately after quenching the hardening is very small and increases only in thecourse of ageing.It follows that the hardening is not caused by individual vacanciesbut more complicateddefects which are formedduring thecondensation of the excess ofvacancies during ageing and which can act as more effective obstacles to the motionof the dislocations.Observations by transmission microscopy have revealed the presence of a highdensity of prismatic dislocation loops in quenched aluminium.5~ 63 7 For superpureA1 these loops are of the unfaulted type, with Burgers vector *[I 101 ; for zone-refined Al, quenched under certain conditions, they are faulted with Burgers vector*[111].8 Some of the vacancies may be in the form of small clusters too small tobe revealed by microscopy.6 In other metals or alloys other types of defects areformed; for example, in quenched Au the predominant defect is a tetrahedron ofstacking fault 9 while in some alloys the vacancies condense on dislocations to formhelices.10Several mechanisms of quench hardening have been proposed, which may beimportant under different conditions. Thus, in alloys in which the vacancies annealout at dislocations to produce a high density of jogs (making screw dislocationshelical) the sources may be hardened due to the frictional force from the jogs. Whenthe predominant defect is a cluster, or a prismatic loop, or a tetrahedron, the inter-action between the moving dislocation and these defects must be considered.Thisinteraction is mainly elastic, but jogs may also be produced.9~ 11, 123 13 Estimatesof the strengths of the elastic interaction with loops and cavities have been made bySaada and Washburn,l3 Friedel 14 and Coulomb and Friedel.15 The purpose of thepresent paper is to make new estimates of these interactions and of the hardeningusing the calculations available on the stress fields and interaction energies appropriateto these problems.16S179 18* Institute of Physics, Czechosl. Acad. Sci., Praguej. Cavendish Laboratory, Cambridge450 ELASTIC INTERACTION BETWEEN DISLOCATIONELASTIC INTERACTION BETWEEN A DISLOCATION LOOP AND ASTRAIGHT DISLOCATIONThe stress fields from circular glide loops and prismatic loops have been studiedby Kroner 16 and Kroupa 19 respectively. When the dislocation is at distances fromthe loop comparable to the loop radius R, the nature of the stress field is rathercomplicated.At distances r>R the stress around a loop decreases as l/r3 ; the totalforce exerted by the loop on the dislocation decreases as 1/r2 and the interaction energyas l / r . The variations with distance of the total force Fl and the interaction energyEOl are illustrated for two cases in fig. 1 and 2.FIG. 1.-Interaction of edge dislocation, Burgers vector bl, with dislocation loop, Burgers vector bo.In fig. 1 the edge dislocation (Burgers vector bl) moves in the slip plane parallelto the plane of the prismatic loop (Burgers vector bo) normal to the plane of the loopat a distance ~ 3 0 = R.In this case the short-range interaction is repulsive, the re-pulsive force reaching a maximum at a distance x20 - R. The interaction energy is amaximum at x20 = 0. For a dislocation with bl of opposite sign the force is attractiveand the interaction energy negative. In fig. 2, a screw dislocation moves in the x2x3plane and intersects the loop with Burgers vector bo. In this case the force is repulsiveon one side of the loop and attractive on the other, and the interaction energy reachesthe maximum value (of opposite sign) on opposite sides of the loop.The case of f.c.c. metals was calculated in detail.Prismatic loops on (111)planes with 3[110] Burgers vectors and pure edge and pure screw straight dislocationswith 3[110] Burgers vectors moving on { 11 l} slip planes were considered ; there are14 different relative orientations. On averaging over all relative positions in whicF. KROUPA AND P. B. HIRSCH 51the maximum possible interaction occurs (in most cases this position corresponds tothe intersection of the loop and the dislocation) the average maximum interactionenergy EZix = GboblR/4 and the average maximum component of the force in the slipplane of the dislocation Pmax= Gbobl/4, where G is the shear modulus.The calculations show that the long-range elastic interaction between the loop andthe dislocation is negligible, while when the dislocation is very near to the dislocationloop, a strong short-range elastic interaction occurs, which results in the loop being aserious obstacle to the motion of the dislocation.B It"FIG. 2.Interaction of screw dislocation, Burgers vector bl, with dislocation loop, Burgers vector b*.The forces acting on elements of the dislocation line decrease rapidly with thedistance of the elements from the loop ; the total force acting on the dislocation is,therefore, concentrated in the neighbourhood of the loop and can be replaced approxi-mately by a point force.The calculation assumes that the dislocation remainsstraight, and the loop immobile. The changes of the shape of the dislocation and ofthe loop can be estimated from the forces acting on elements of the loop and of thedislocation.17 When the dislocation does not intersect the loop the changes arelikely to be small and the above assumption seems reasonable.When the dislocationintersects the loop jogs are formed in most cases, and contact interactions can takeplace.14 Thus, for a loop with the same Burgers vector as that of the dislocation,part of the loop will combine with the dislocation, causing it to become heavily jogged,leaving behind a smaller prismatic loop. For other combinations of Burgers vectors(excepting those at right-angles to each other) attractive junctions are formed over asmall segment of the loop. For these Friedel14 estimates the interaction energyA Uw2TR/3 (where Tis the line tension) by analogy with Saada'szo result for a rando52 ELASTIC INTERACTION BETWEEN DISLOCATIONforest, with trees of length 2R.This estimate neglects the fact that the loop is curvedand the appropriate value of the line tension Tis also somewhat uncertain ; however,putting T- Gb2/2, AU- Gb2R/3, which is of the same order as the average maximumelastic interaction E::,, quoted above. Saada’s calculations for trees indicate thatfor half the cases the maximum repulsive force on intersecting a tree occurs when thejunction is completely constricted. In these cases the maximum force Pmax quotedabove should be equal to the maximum force on the junction model. In other cases,the node becomes unstable before it is completely constricted, but even then themaximum force, although smaller is often of the same order as that correspondingto complete constriction.It appears therefore that our estimate of Pmax should bereasonable, if somewhat too low, even if the loop changes its configuration to theextent of forming a junction.ESTIMATE OF CRITICAL SHEAR STRESS DUE TO ELASTICINTERACTION WITH LOOPSSince the forces exerted by a loop on a straight dislocation are concentrated veryclosely near the loop, they can be regarded effectively as point forces. However,although the maximum total force varies as 1/r2 for distances r> R, it has large valuesover a distance r - R from the centre of the loop. Calculations indicate that it issufficient to consider the interaction of loops within a plate like region of thickness2R about the slip plane.The total force averaged over this thickness is PkBX--S;Pmax, i.e., PA,, = Gbobl/8.If I’ is the average separation of interacting loops along the dislocation the shearstress for plastic flow is then Gbol8I’. I’ will be larger than Z, the average distancebetween loops in the plate ( I = 1/ $ ~ n , where n is the number of loops per unitvolume) and depends on the interaction energy. Now most of the loops attract thedislocation somewhere, and we shall treat them all as attractive centres. By minimiz-ing the energy Friedel 14 has shown that I f - l(ZT/EOl)*, where T is the line tension ofthe dislocation. Putting T- Gb2/2 and EOl - Gb2RI8, we obtain1’- 1(41/R)* = R- ‘n-+Azg = +GboRn g.givingUsing the dislocation density, p = 2nRn, this result can be put into the formwhere lo is the average distance between loops, i.e., lo = n-3.Comparing this resultwith the relation z = aGb,/p usual for hardening theories 20, 21322, 23 we see thatfor loops the effective a is rather smaller than for forest, long-range stress, or sessile joghardening.Using the values n- 1015 cm-3, R - 150 A for A1 quenched from 600°C into icedbrine, and aged at room temperature,24 corresponding to p - lOlo/cm2 we obtainAzg - 0.1 5 kg/mm2 compared with typical experimental values of 0.6 kg/mmZ. 192It is likely that our estimate is somewhat too low because the elastic interaction ofjunction reactions is underestimated, and because there is also some contribution tothe flow stress from loops at distances up to -2R. These factors might increase ourestimate by a factor of -2, so that this type of interaction makes a substantialcon-tribution to quench hardness.This may, however, not be the only contributionF. KROUPA AND P . B . HIRSCH 53and it is likely that source hardening by jogs and possibly by impurities transportedto the dislocations during ageing may be equally important.Since the interaction energy is very high (for loops of radius 150 A, EOl-25 eV)the temperature dependence of Arg is expected to be very small, and, apart from thetemperature dependence of G, to be controlled by formation and movement of jogs.25The small temperature dependence of Arg is in agreement with the results of Tanner,4although it is not at all obvious why the temperature dependence should be exactly thesame as for annealed Al.It should be mentioned that for small loops, R- a fewatomic diameters, the interaction energy is a few eV, and the temperature dependencebecomes much greater.A drop in Arg is expected around 200°C when the loops anneal out, in agreementwith experiment.2 However, Kino’s results 2 show that only about half the quench-hardness anneals out at this temperature; the remainder anneals out at a muchhigher temperature, and may be controlled by impurity-dislocation interaction.ESTIMATE OF ELASTIC INTERACTION WITH SMALL VACANCY CLUSTERSWe shall now consider the contribution to hardening from small vacancy clusterswhich may be formed after quenching 6 and which may be invisible in the electronmicroscope.These will be considered as spherical cavities, radius R,, density nc/cm3.Coulomb and Friedel 15 considered large cavities with radius much greater than thecore radius, assuming that the interaction energy E, is equal to the dislocation energystored in the volume of the cavity, E,-+Gb2 x 2R, and AT- Gb/Z,, where Zc is thedistance between pinning cavities along the dislocation. For small cavities (R, - a few Burgers vectors), however, the relaxation of the stress field should be rathersmaller.Bullough and Newman 18 have calculated the elastic interaction between adislocation and a cavity. For a screw dislocation the interaction energy Es is given by- 5Gb2Rz(1 - V ) E, =2n(7 - 5v)r2(the result given by Bullough and Newman 18 has been corrected by a factor +). Thisformula holds on the assumption that the stress near the cavity changes little, i.e., atlarge distances from the dislocation.Nevertheless, it is reasonable to use this expres-sion to give an approximate estimate of the interaction energy at r-RC, when thedislocation is tangent to the cavity. For this value of Rc, and v = 4, E8- Gb2RC/lO.The actual maximum value of the interaction energy is likely to be somewhat larger,when the dislocation passes through the cavity. For an edge dislocation the inter-action energy varies with the position of the cavity relative to the slip plane. Theinteraction is a maximum in the slip plane, and at I--&, this is E-Gb2Rc/5. If thecavity is so small that only the core energy is relaxed the maximum interaction energyis -Gb22Rc/10 = Gb2RC/5, which is of the same order as the estimate obtained byconsidering the elastic interaction.At present, it does not seem possible to calculate a reliable force distance curve toenable the maximum force to be derived.However, for the core energy contributionthe force will be a maximum when the dislocation leaves the cavity; the same isprobably also nearly true for the elastic interaction. Using the expression of E8given above or by considering the core energy it is easy to see that the maximumforce on a dislocation in a slip plane through the centre of the cavity is Pmax- Gb2/5.The force will be appreciable only for dislocations on planes intersecting the cavity ;since the force varies as r-3, dislocations bypassing the cavity will experience only 54 ELASTIC INTERACTION BETWEEN DISLOCATIONnegligible force.We therefore need consider only cavities within a plate like regionof thickness 2Rc, just as for loops. The average force over this region is -Gb2/10.Proceeding as before we find the contribution to the flow stress to bewhich is of the same form and order of magnitude (but somewhat smaller) than thecorresponding estimate for loops. For a given vacancy concentration and the sameconcentration of cavities and loops the hardening for loops is greater by a factorThe expression AT# can be rewritten in terms of the vacancy concentration cArg- Gb/lOZf - GbR,nt/lO,(Rclb)&.to giveGbcS 110(4x/3): R,'AT# - -From this it might appear that the smaller R, the larger AT#. But for small R,the flow stress is thermally activated.Thus for Rc-4b (i.e. -1OA radius),E-Gb2RC/lO- 1-5 eV. The precise temperature dependence of AT# depends on theparticular force-distance curve, but the critical temperature Tc above which thiscontribution to the flow stress becomes negligible should be nearly independent ofthe form of this curve, and is given byT,-Elk In { ~ ~ b ~ p ( 2 R , n , l ' ~ 8 ) - ' )where vo is the atomic frequency, p the moving dislocation density and i. the strain rate(see, e.g., Friedel ref. (14)). With VO-1013 sec-1, p-108 cm-2, 6-10-5 sec-1,T,- E/2OA - 900'K. However, even at room temperature AT# will be substantiallyreduced by thermal activation for this size cavity, and the reduction will be greaterfor smaller cavities.Thus, sub-electron microscopic cavities (say 5 20 A in diameter)should give rise to a temperature sensitive contribution to AT. For larger cavitiesour estimates no longer apply ; the elastic interaction is larger, and for cavities withR,? lOOb the estimate made by Coulomb and Friedel 15 becomes applicable. Thereis as yet no direct electron microscope evidence for the existence of cavities in quenchedaluminium* ; the small cavities which might exist could contribute to the temperaturesensitive part of AT, which does not appear to be important.In this discussion we have neglected the possibility of interaction between thedislocation and the cavity causing the former to climb. For edge dislocation this islikely to result in the formation of a double superjog and a decrease in the hardeningeffect; for screws the situation is more complicated; a small prismatic segment ofradius Rc is likely to be formed, which should be able to glide along the screw.26If the prismatic segment becomes trapped, the dislocation may tear away leaving asmall prismatic loop, but the hardening effect is expected to be of the same order asfor the original cavity.This mechanism is therefore unlikely to cause more hardening,although it suggests that the cavities will eventually be absorbed by the dislocations,some being transformed into loops. (For a detailed discussion on some possibleinteraction between cavities and dislocations see ref. (12), (27))CONCLUSIONS(i) Elastic interaction between dislocations and loops in quenched A1 can accountfor a substantial part of the observed yield stress. (ii) The contribution from smallunobservable cavities is unlikely to be important, since it would be rather temperaturesensitive (which is not observed). It would, however, be desirable to have some* Note added inproofi See, however, ref.(28)€7. KROUPA AND P. B . HIRSCH 55experimental evidence on the temperature dependence at very low temperatures.(iii) It is likely that part of the quench-hardening effect is due to hardening of thesources. This may be due to a high density of jogs, but impurity locking may alsobe important since during ageing the impurities are likely to be transported to thedislocations.1 Maddin and Cottrell, Phil.Mag., 1955,46, 735.2 %no, J. Sci. Hiroshima Univ. A, 1958, 22,259.3 Kimura, Maddin and Kuhlmann-Wilsdorf, Acta Met., 1959, 7, 145.4 Tanner, Acta Met., 1960, 8, 730.5 Hirsch, Silcox, Smallman and Westmacott, Phil. Mag., 1958, 3, 897.6 Kuhlmann-Wilsdorf and Wilsdorf, J. Appl. Physics, 1960, 31, 616.7 Smallman, Westmacott and Coiley, J. Inst. Met., 1959-60,88, 127.8 Cotterill and Segall, Phil. Mug., 1963, 8, 1105.9 Silcox and Hirsch, Phil. Mag., 1959, 4, 72.10 Thomas and Whelan, Phil. Mug., 1959, 4, 511.11 Hirsch, J. Inst. Met., 1960-61, 89, 303.12 Kuhlmann-Wilsdorf, Maddin and Wilsdorf, A.S.M. Philadelphia Seminar, Strengthening13 Saada and Washburn, Int. Conf. Crystal Lattice Defects, Kyoto, 1962 (J. Physic. SOC. Japan,14 Friedel, Proc. 1961 Berkeley ConJ Electron Microscopy and Strength of Crystals, 1963 (New15 Coulomb and Friedel, Dislocations and Mechanical Properties of Crystals (Wiley, New York,16 Kroner, Kontinuumstheorie der Versetzungen und Eigenspannungen, Berlin, 1958.17Kroupa, Phil. Mag., 1962, 7, 783.18 Bullough and Newman, Phil. Mag., 1962,7,529.19Kroupa, Czech. J. Phys. B, 1960,10,284.20 Saada, Acta Met., 1960, 8,841.21 Seeger, Handbook of Physics, 1958, VII-2.22 Bailey and Hirsch, Phil. Mag., 1960, 5,485.23 Kuhlmann-Wilsdorf, Trans. A.I.M.E., 1962,224, 1047.24 Silcox and Whelan, Phil. Mag., 1960, 5, 1.25 Silcox and Hirsch, Phil. Mug., 1959, 4, 1356.26 Strudel and Washburn, Phil. Mag., 1964,9, 491.27 Kuhlmann-Wilsdorf and Wilsdorf, Proc. Berkeley ConJ Electron Microscopy and StrengthZsKiritani, J. Phys. SOC. Japan 1964, 19, 618.Mechanisms in Solids, 1960.1963, 18, suppl. I, 43).York, Interscience), p. 605.1957, Lake Placid Conference, 1956), p. 555.of Crystals, 1963 (New York, Interscience), p. 575

ISSN:0366-9033    

DOI:10.1039/DF9643800049

出版商:RSC    

年代:1964

数据来源: RSC

摘要:

Interaction of Cavities and Dislocations in CrystalsBY A. J. FORTYH. H. Wills Physics Laboratory, University of BristolReceived 16th June 1964Dislocations may be expected to interact strongly with voids or cavities in crystals. The inter-action may be physical, through the stress fields of both the dislocation and the cavity, or chemical,by a transfer of point defects. The mechanisms for such interactions are discussed in detail, andillustrated by direct observations made by transmission electron microscopy on crystals of leadiodide. These show elastic interactions between mobile dislocations and cavities and the " climb "of sessile dislocations in the immediate vicinity of cavities. The paper discusses the possible effectson the mechanical and chemical processes associated with dislocations when cavities are present incrystals.The formation of cavities (or voids) in crystals and the interaction of these withdislocations can have a significant effect on both mechanical and chemical properties.For example, small voids, formed preferentially along dislocations or dispersedrandomly throughout a crystal, can impede the motion of dislocations and therebylead to hardening.Indeed, it is possible to account for some forms of radiation-hardening in this way ; the aggregation or condensation of lattice vacancies createdby irradiation produces voids and these interact with dislocations. A quantitativediscussion of radiation-hardening can be formulated along these lines.1Clearly it is important to understand the nature of interactions between voidsand dislocations.Whilst the cavities observed in electron microscope studies oflead iodide crystals are generally larger than those thought to be responsible for theradiation hardening of metals, and a precise quantitative treatment is therefore notrelevant in this case, a qualitative study of dislocationfcavity interactions in thismaterial does seem to be useful. Indeed, the special arrangements of dislocations andcavities in lead iodide offer important advantages to be gained from such a study.Specimens suitable for electron microscopy are readily prepared by growth fromaqueous solution in the form of platelets with well-developed (0001) habit planes.The crystallography ensures that both dislocations and the plate-like cavities formedby electron bombardment in the microscope lie in the plane of the specimen, an idealgeometrical situation for the direct observation of interactions.Interactions of cavities and dislocations can be broadly classified as physical orchemical ; the mutual interaction of stress fields gives rise to a mechanical or physicalforce, whilst the transfer of point defects between cavities and dislocations leads to achemical interaction. These two classes of interaction will be discussed in turn in thefollowing sections. The discussion will be in general terms but, wherever possible,it will be illustrated by examples of interactions observed in lead iodide.The specialcrystallographic orientation and the marked anisotropy of dislocation motion and themigration of point defects in lead iodide make it possible to separate the two forms ofinteraction with little ambiguity.GENERAL FEATURES OF DISLOCATIONS AND CAVITIES I N LEAD IODIDELead iodide crystallizes with a hexagonal layer structure, composed of close-packed layers of I- ions arranged in hexagonal close-packing, with the Pbz+ ions5A.J . FORTY 57placed in the octahedral interstices between alternate pairs of I- layers. Deformationoccurs most readily by slip in the basal plane (OOOl), presumably through the planesseparating the molecular sandwiches of PbI2. Dislocations introduced by deformationduring preparation of specimens for electron microscopy therefore lie in the basalplane and have their Burgers vectors in this plane.Crystals prepared by growthfrom aqueous solutions have the form of platelets in basal plane orientation. " Slipdislocations " in these appear under the electron microscope as sharply imaged linesextending across the specimen plane.The exposure of a crystal of lead iodide to the electron beam during examinationin the microscope leads to structural damage and, in some cases, to decomposition.2Under low intensities of illumination only structural damage occurs. This producessmall loops of dislocation which appear to lie in inclined planes, with Eloil) orienta-tions. The Burgers vectors of these have not been unambiguously determined but itis thought that there should be a large component in the [OOOl J direction ; for this isexpected if the loops are sessile dislocations formed by the collapse of {lOil) discs ofvacancies.Large, irregularly shaped loops of sessile dislocation appear in the crystals underhigh-intensity irradiation.These lie in the basal plane (0001) and have diametersvarying between one and a hundred microns. Again, the Burgers vector of theseloops has not been determined unambiguously but it is thought that they are formedby the aggregation and subsequent collapse of vacancies in plate-like clusters in thebasal plane. These loops appear under conditions of irradiation which lead simul-taneously to the formation of cavities. They grow into irregular shapes (see fig. 5,for example) by chemical interaction with cavities (see later). The collapse of asingle cavity into a series of concentric basal plane loops has been observedoccasionally.The most striking result of irradiation at high intensity is the appearance of electron-transparent regions, or bright patches.These patches have been interpreted 3 asthe images of plate-like cavities, lying in the basal plane, and formed by the aggrega-tion of vacancies created by irradiation. The cavities have diameters ranging from afew hundred A to several microns, and thicknesses between ten and a hundred A.The bright contrast due to enhanced transmission is predominant, but in some patchesthe brightness is considerably reduced and a narrow fringe of enhanced diffractionappears around them. Examples of the two kinds of contrast may be seen in fig.5.It is thought that the bright contrast arises from reduced absorption within a cavity,whilst the diffraction contrast reveals the existence of elastic strain of the crystalstructure surrounding it.There is some indirect evidence that cavities contain iodine gas and, during laterstages of the decomposition of a crystal, precipitates of metallic lead. Cavities clearlyplay an important part in the decomposition reaction, by providing sites within thecrystal for the precipitation of the products of decomposition. The relative amountsof absorption contrast and diffraction contrast probably depend on the pressure ofiodine gas in the cavities. For the mechanical stability of a thin plate-like cavitymust depend on the internal support provided by the gas.Strong diffraction contrastmight therefore be associated with a tendency towards collapse.Cavities are formed in the centre of decomposition (i.e., the centre of the irradiatedarea) and subsequently drift radially outwards with velocities of several microns persecond into the cooler parts of the crystal. It is impossible to account for this kindof movement simply but an explanation has been sought in terms of an electrolytictransfer of ions under the electric field set up by secondary emission from the crystalunder irradiation.4 An iodine atmosphere assists such a transfer58 CAVITIES AND DISLOCATIONSPHYSICAL INTERACTION BETWEEN CAVITIES AND DISLOCATIONSThe migration of a cavity through a crystal of lead iodide provides an ideal experi-mental situation for the study of interactions with dislocations. It is possible toobserve directly the movements of dislocations in close proximity to cavities, and,moreover, to investigate the variation of the strength of interaction with separation.By suitable adjustment of the electron beam it is possible to produce cavities anddirectthem towards dislocations in other parts of the crystal.A sequence of observationsmade as a cavity moves past a dislocation in this way provides a striking illustrationof mutual interaction.When a cavity in the basal plane approaches a ‘‘ slip dislocation ” (a dislocationlying in the basal plane and having its Burgers vector in that plane) the latter appearsto be drawn towards it and is eventually “ sucked ” into the cavity (see fig.1, forexample). The dislocation moves in its slip plane and the interaction must thereforebe physical rather than chemical; that is, the movement is caused by a mechanicalforce on the dislocation rather than by a transfer of point defects and climb of thedislocation.A dislocation in the neighbourhood of a cavity will experience an attractive force,either through the mutual interaction of the two stress fields or through the tendencyfor relaxation of the stress field around the dislocation to occur by deformationof the free surface at the wall of the cavity. The force due to mutual interaction ofstress fields is likely to be small in the present case because the long-range stress fieldof a plate-like cavity will have a negligible shear component in the basal plane (i.e., theplane in which the dislocation lies and moves).The situation is very similar to thatof two parallel dislocations with Burgers vectors perpendicular to one another. It ispossible to derive an attractive force due to the mutual relaxation of stress fields alongthe lines discussed by Newman and Bullough 5 for small voids or vacancies. How-ever, in the present situation, where the cavity provides a free surface close to thedislocation, the force can probably be accounted for almost entirely by a surfaceattraction. The analysis of surface image forces made by Head 6 might be appliedto this particular problem but it should be modified to take into account the limitedfree surface offered by a cavity.It is perhaps surprising that interaction occurs over such a long range (the dis-location starts to move when the separation is a few microns), especially since theobservations are made on a thin crystal.The stress field around an elastic singularitymight be expected to decay within a distance of the order of specimen thicknessbecause of relaxation of stresses at the crystal surfaces. However, a longer range ofinteraction might exist in this particular case because the slip dislocations lie in thebasal plane and have Burgers vector in that plane. The stress field is therefore notrelaxed so readily by the free surfaces, particularly if the dislocation is predominantlyin the screw orientation.The subsequent association of the cavity and dislocation after interaction is interest-ing.If the cavity is not forced to move further through the crystal it remains firmlyanchored to the dislocation. Dislocations can be “ decorated ” with cavities, asshown in fig. 2, and are thereby pinned inside the crystal. This provides an illustra-tion of the form of radiation-hardening discussed by Coulomb and Friedel.1 How-ever, if the cavity is forced to move beyond the dislocation (by increasing the beamintensity, for example) this remains firmly attached and extends or trails behind thecavity. Several examples of dislocations trailing behind migratory cavities are shownin fig. 3FIG. 1 .-Electron micrograph showing the physical interaction betweencavities and “ slip dislocations ” in a crystal of lead iodide.x 20,000.FIG. 2.-The “ decoration ” ofFIG. 3.-The trailing of dislocations behind migratory cavities. FIG. 4.-The ‘‘ decoration ” ofx 10,000. migratorFIG. 5a, b, c.-Sequence of electronment of a large sessile dislocationwith cavitiesA. J . FORTY 59The elongation of pre-existing dislocations in this manner is interesting. Theperturbation of the dislocation line is not smoothed out by line tension effects and thismight be taken to indicate a hardening of the structure in the wake of a cavity. Thereare no visible defects here, but it would be surprising if the " re-crystallized " materialin the wake of a cavity migrating at a velocity of several microns per second werehighly imperfect. The aggregation or condensation of point defects in the initialstages of annealing of the imperfect structure might well preserve the elongatedconfiguration of the trailing dislocation.This pinning effect is shown on a moremacroscopic scale in fig. 4 where small cavities have appeared along the trailingdislocations in the wake of large cavities.CHEMICAL INTERACTIONS OF CAVITIES AND DISLOCATIONSThe shapes and relative positions of cavities and dislocations change markedlyif they are sufficiently close for a transfer of point defects and, subsequently, climb ofthe dislocations to take place. The rate of transfer must depend primarily on thetemperature of the crystal (about 200°C in those crystals which undergo rapid de-composition), but the direction of flow of material will be determined by the strainfields of the dislocations and cavities, andwill depend also on the nature of the diffusingspecies of point defect.The exchange of vacancies or interstitials between neigh-bouring sessile dislocation loops, and the importance of this in the annealing behaviourof loops in irradiated or quenched metals, has been discussed already by Kroupa,Silcox and Whelan.7 The corresponding general problem of the transfer of defectsbetween sessile dislocation loops and cavities could be discussed in a similar manner.However, the particular case of chemical interaction in crystals of lead iodide underirradiation with an electron beam is more difficult to analyze; both the anisotropyof the strain fields and of the diffusion of defects in this material and also the large,undefined temperature gradients that are established by the non-uniform exposureto the electron beam must be taken into account.Therefore the observationspresented below, although of considerable general interest, cannot be analyzedsatisfactorily at present.Fig. 5 shows a sequence of electron micrographs of a part of a crystal containinga large sessile dislocation loop (i.e., a dislocation loop lying in the basal plane andhaving its Burgers vector perpendicular to that plane). At a number of places theirregularities in the shape of the loop can be associated with the presence of cavities,either in contact with the loop or close to it. The shape and size of the loop changesmarkedly in this sequence. A sessile loop can move in this fashion only by climb inthe basal plane.This requires a transfer of point defects, most probably molecularvacancies or associated Pb2f and I- vacancies, either along the dislocation line itselfor through the crystal structure between the dislocation and cavities. The apparentcorrelation between irregularities in the dislocation and the proximity of cavitiessuggests that the latter process, involving a chemical interaction, is most likely. Thecavities will exert a physical force on the dislocation because of the mutual interactionof the stress ficlds (this situation is similar to that of parallel edge dislocations inparallel slip planes with similar Burgers vectors) but the loop cannot glide under thisforce. However, the existence of the force will affect the climb of the dislocation.The direction of the force will depend on whether a cavity lies inside or outsidethe sessile loop and, by a transfer of vacancies, the loop can climb away fromsome cavities and towards others under the influence of this force. This accountsfor the irregular behaviour that is commonly observed when chemical interactiontakes place60 CAVITIES AND DISLOCATIONS1 Coulomb and Friedel, Dislocations and Mechanical Properties of C r y & f s (ed. Fisher, Johnson,2 Forty, Disc. Faraday SOC., 1961, 31,247.3 Forty, Phil Mag., 1960,5,787.4 Forty, Phil. Mag., 1961, 6, 895.5 Bulloiigh and Newman, Phil. Mug., 1962,7, 529.6 Head, Phil. Mag., 1953,43,433.7 Kroupa, Silcox and Whelan, Phil. Mag., 1961, 6, 971.Thomson and Vreeland), Wiley (New York), 1956, p. 555

ISSN:0366-9033    

DOI:10.1039/DF9643800056

出版商:RSC    

年代:1964

数据来源: RSC

摘要:

General Theory of Surface DislocationsBY B. A. BILBY," R. BULLOUGH~ AND DORA K. DE GRINBERGSReceived 7th July, 1964The general formulae for the description of the dislocation content of a boundary betweentwo crystals rotated relatively to one another are reviewed. It is pointed out that the formula forthe dislocation tensor of the boundary does not take into account the symmetry of the crystalstructure, and a procedure is outlined by which the rotation generating the boundary may be re-duced to allow for these symmetry operations. Some preliminary results of calculations on theStretch computer showing the situation for selected boundaries between crystals of the holo-symmetric cubic group are reported.1. INTRODUCTIONThe concept of the surface dislocation was introduced in 1954 1 but except foran application to the theory of martensite crystallography 2 and some unpublishedwork 3 discussing the analysis of dislocation networks by Amelinckx 4 little use hasbeen made of the explicit description of dislocation arrays as entities by the approp-riate tensor. The general formula for the dislocation content of a boundary betweentwo different structures with a general relative orientation is required in the applica-tion to martensite crystallography.2 It has been shown that,5 when the two struc-tures are the same so that the material on one side of the boundary differs fromthat on the other by a rotation only, the description in terms of a dislocation tensoris equivalent to the original analysis of the dislocation content given by Frank.6In the analysis of Frank, as in the general formula given for the dislocation tensorof a boundary,l the relative orientation of the two pieces of crystal separated bythe dislocation boundary is specified by giving the rotations necessary to generatethe material on each side from a reference lattice.It is, however, implicitly assumedthat account has been taken of the crystal symmetry in specifying these rotations;i.e., no explicit account is taken, e.g., of the fact that if the rotation happens to beabout a tetrad axis of the structure and to be of magnitude 4 2 then physically thestructure has not been rotated at all. The rotations used to describe the relativeorientations of the material on the two sides of the boundary must in some sensebe reduced to take account of the symmetry operations of the crystal structurepossessed by the material.It is the problem of modifying the general theory ofthe boundary to take account of this effect which is the central topic discussed inthe present paper.The nature of the problem can be clearly seen when the axis of relative rotationof the material on the two sides of the boundary coincides with a rotation axis ofthe structure and no other symmetry need be considered. The generating rotationand the symmetry rotation are then commutative operations and the process ofreduction of the resultant Burgers vector of the dislocation lines cutting a givenline drawn in the boundary can be represented diagrammatically (see $2).In orderto discuss a more general situation it is first necessary to derive some further resultsfrom the formula giving the dislocation tensor of the boundary (5 3). As in theanalysis of Frank we consider the resultant Burgers vector of the dislocation lines* Department of Metallurgy, University of Sheffield. t Theoretical Division, A.E.R.E., Harwell. t Theoretical Division, A.E.R.E., Harwell.$ Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Ares, Argentina.662 THEORY OF SURFACE DISLOCATIONSin the interface that are cut by an arbitrary vector in the interface. The arbitraryvectors in the interface that lead to stationary values of the resultant Burgers vectorare then deduced and we show that the maximum resultant Burgers vector is a functiononly of the angle of mis-orientation across the interface.This last result is thenused in 5 4 to show how the true dislocation content of an arbitrary grain boundarymay be deduced, when allowance is made for the particular point group symmetryof the structure. The general analysis has been applied to obtain some preliminaryresults for the structure with the highest symmetry, viz., the liolo-symmetric cubicgroup. This is of considerable importance in considering boundaries in metalssince many of them have either the face-centred cubic or the body-centred cubicstructure. The application so far considered has been to select various axes forthe rotation which determines the mis-orientation of the structures and to use theStretch computer to derive the correct reduced dislocation content for rotationsabout these axes at intervals of two degrees.Finally in 6 5 the implications of thisanalysis are discussed briefly, and future applications to materials of lower symmetryand to boundaries separating materials of different structure are outlined.2. GENERAL FORMULAELet PI, kz, 43 be a set of orthogonal unit vectors defining a system of orthogonalCartesian co-ordinates. All components of vectors and matrix representations ofdeformations are referred to this Cartesian system. Let DQ(+) and Dji(-) bethe representations in the 3, system of two deformations that respectively generatetwo lattices with bases a&+) and a&-) from a references system ai. Then Bilby,Bullough and Smith 1 have shown that the dislocation content of a planar interfacewith normal v =v &g separating the two lattices is specified bywhere the matrices Eat(+) and Et&-) are the reciprocals of D$,(+) and Di&-) respectively.Bi& is the resultant Burgers vector of dislocation lines cutting unit length of aline in the boundary perpendicular to the XI direction.There is no loss of generalityif the (-) lattice is taken as the reference lattice and this will be done throughout ;thus Dil(-) = &j or in matrix notation * D(-) = I. If b = b&i is the resultant Burgersvector of dislocation lines which are cut by an arbitrary vector p in the boundary,then, with the ( -) lattice as the reference lattice, (2.1) may be expressed in the form :where E is the reciprocal of the matrix representation D of the deformation thatgenerates the (+) lattice from the (-) reference lattice.In particular, if the twolattices on either side of the boundary differ only by a rigid rotation then D = R,whereis the yatrix representation of a rotation through an angle 8 about the unit vectorn = nix(. For a general planar grain boundary, expression (2.2) becomes :Bi j = &jklvk[Ei: - E$F)] (2.1)b = [E-I]p, (2.2)R,, = cos B+ninj(l -cos 9)-&ijknk sin 8 (2-3)b = [RT-I]p,or in component form:b, = [(l-cos B)(ninj-6,j)+Eii,nk sin O]pjThis is equivalent to the result given by Frank.6* In this work matrices will be represented by capital letters, either in italics with explicit orderedsuffixes indicating rows and columns respectively, or in heavy type.Similarly, vectors will be writtenas small letters. Transposition will be denoted by a T as superscriptB . A . BILBY, R. B. BULLOUGH AND D. K. DE GRINBBRG 63We now discuss the simple situation when the rotation axis of the rotation Rcoincides with a rotation axis of symmetry of the structure. It is apparent firstthat there is no change in the physical nature of the boundary if we replace thematrix RT by one obtained by pre- and post-multiplication by rotation matricesG1 and G2 which are any operations of the symmetry group. Physically this isequivalent to applying to the structure both before and after the actual rotationgenerating the boundary, a symmetry operation, which, of course, leaves it in-variant.Thus the same physical boundary would be described for any choice ofthe vector p by a series of Burgers vectors given by the formulab = [G,RTG2-I]p. (2.6)If provisionally we assume that the vector p has been chosen, the appropriate descrip-tion is that which minimizes the modulus of the vector b in this equation. If therotations GI, RT and Gz are all about the same axis then the matrices commutewith one another so thatGlRTG2 = G1G2RT = G3RT (2.7)where GJ is some other operation of the symmetry group. This procedure willnow be illustrated in two simple cases with the aid of a diagram. Suppose firstthat the common rotation axis is a diad axis of symmetry. Then we have to dowith the cyclic point group with two elements only, viz., I and G such that G2 = I.The only possibilities are thenandThe situation is illustrated in fig.1 where the vector p is shown as OA and the vectorRTp as OB or OB’. So long as the angle of rotation 8 determined by the rotationRT is less than 4 2 , the Burgers vector AB obtained by the application of formulaFIG. 1.-Illustrating the reduction of the resultant Burgers vector when the rotation axis is a diadaxis of symmetry.(2.8) has the smallest modulus possible. If, however, the angle 8 exceeds 4 2 sothat the situation is as indicated by a vector b2 = AB’ then the modulus of theBurgers vector can be reduced by using the operation GRT to describe the rotation.This converts the vector AB’ to the vector AD, and so the resultant Burgers vecto64 THEORY OF SURFACE DISLOCATIONSof the lines cut by the vector p now becomes AD = b:.Here we are thus usingthe formula (2.9). As a further example, if the rotation axis is a tetrad, the cyclicgroup has now the elements I, G, G2, 6 3 and 6 4 = I. By an elaboration of the pre-vious discussion the reduction process becomes necessary now when the angle ofrotation 8 determined by the matrix RT exceeds n/4. The situation is summarizedin fig. 2 which shows diagrammatically the variation of the modulus of the vectorb for this last case. The curve A is obtained from the original formula (2.4) withoutany reduction of the rotation matrix by operations of the symmetry group, and the2Ib' 1/ \ - I 20"460"DeFIG. 2.-Curve B shows the variation of the modulus of the resultant Burgers vector I b I and thereduced angle of rotation + as a function of the angle of rotation 8 about a [loo] axis.Curve Ais the unmodified result with no allowance for symmetry.curve B shows the effect of this reduction. In this simple situation the resultantBurgers vector is zero if the rotation operation is a symmetry operation, i.e., if 6= n/2. This is a general result. If the rotation is a symmetry operation itselfit is only necessary to combine with it the inverse symmetry operation to give theidentity as the resultant and so to obtain a zero value for the resultant Burgersvector.In this discussion the variation of the analysis with the choice of the vector phas not been considered, but only the choice of that operation of the symmetrygroup needed to reduce the Burgers vector b for fixed p.In the general situationthe effect of a possible variation of p must be considered and this point is nowdiscussed.3. GENERAL ROTATION BOUNDARYIn order to examine how the resultant Burgers vector b depends on the choiceof p, we look for the particular unit p vectors, denoted by p(1) and p(d, that leadto resultant Burgers vectors b(1) and b(9 respectively, whose moduli are stationary.The general expression for I b 12 from eqn. (2.5) isI b I = 4(1- ninjpipi] sin2 (8/2) (3.1B. A . BILBY, R . BULLOUGH AND D. K . DE GRINBERG 65We require that I b I should be stationary, subject to the auxiliary conditionsthat p is confined to the boundary, and that it is a unit vector.This means thatand thatApplication of Lagrange multipliers leads to the following results :PTV = 0 (3.3)pTp = 1 (3.2)(i) I b I has a maximum value I b(1)l given byI b(l) I = f 2 sin (8/2),p = p(" = (n A v)/sin a.I b(2) I = k12 cos a sin (8/2),when(ii) I b I has a minimum value I b(2) 1 given bywhenHere a is the angle between the boundary normal v andThe resultant Burgers vector b(1) and b(2) are given byp = g(2) == [n - cos av]/sin acos a = nTv.(3.4)(3.5)(3.6)(3.7)the rotation axis n ; thus(3.8)b(1) = -2 sin (8/2)[(n A V) sin (8/2)-(~-cos an) cos (O/Z)]/sin a, (3.9)andb(2) = 2 sin (812) cos a[(n A Y) cos (O/~)+(Y-COS an) sin (8/2)]/sin u. (3.10)From (3.5) we see that the maximum resultant Burgers vector, given by (3.9), arisesfrom the dislocation lines cut by the vector p(1) in the boundary which is orthogonalto the axis of rotation n.The minimum resultant Burgers vector given by (3.10),arises from the vector p(2) given by eqn. (3.7), and this vector is orthogonal to ~(1).Also, the maximum and minimum resultant Burgers vectors bcl) and b(2) are ortho-gonal to each other and to the axis of rotation. There are two special cases whicharise if the angle a, between the rotation axis and the boundary normal, is zero orninety degrees.(i) a = 0. SIMPLE TWIST BOUNDARYIn this case I b 1 does not have discrete stationary values and expressions (3.4),(3.9, (3.6) and (3.7) mean thatI b I = &2sin(8/2) (3.1 1)for any unit vector p in the boundary.Since the resultant Burgers vector is alwaysorthogonal to the axis of rotation n (expressions (3.9) and (3.10) indicate thatnTb(1) = nTb(2) = 0 and thus nTb is aiways zero), it must lie in the boundary. Thedislocation structure of the boundary is thus isotropic.(ii) a = 90". SIMPLE TILT BOUNDARYIn this case the minimum resultant Burgers vector b(2) is identically zero andthus the vector p(2) does not cut any dislocations. It follows that the dislocationlines in the boundary are all parallel and orthogonal to the vector ~ ( 1 ) ; i.e., thedislocation lines are all parallel to the axis of rotation n. Furthermore, since b(1)is, from (3.59, orthogonal to n, it follows that the dislocations must be pure edge.Thus if the rotation axis lies in the boundary then the boundary must contain anarray of edge dislocations all of which are parallel to the rotation axis.66 THEORY OF SURFACE DISLOCATIONSFor the simple tilt boundary we have thereforeIPA V , pfl, =b(') = 2 sin (8/2)(v cos (8/2)-(n~ v ) sin (012))I b(') I = fi2 sin (6j2),and(3.12)(3.13)4.EFFECT OF CRYSTAL SYMMETRYThe most important result of the previous paragraph is that expressed by eqn.(3.4). This states that the maximum value of the modulus of the Burgers vectorfor any rotation boundary depends only on the relative angular rotation of thematerial on the two sides of the boundary. To search then for a description ofthe boundary which gives the lowest physical dislocation density, it is only necessaryto examine the relative rotation across the boundary, and not to consider explicitlythe variation of the vector p.The general formula for the dislocation content ofthe boundary is given as before by eqn. (2.6). The matrices G1 and GZ are properrotation matrices so that the resultant matrix GlRTG2 is thus also a proper rotationmatrix. It is therefore only necessary to determine which of the possible symmetryoperations GI and Gz lead to a resultant rotation matrix GlRTG2 which has theminimum 8 value. Then, by the formula (3.4) we can immediately calculate fromthat value the least value I b(1) 1 and this will be the lowest possible value for themaximum of the modulus of the Burgers vector for any position of the unit vectorp in the boundary.The calculation is simplified by the fact that the trace of any rotation matrixwith rotation angle 8 is equal to 1 +2 cos 8.Thus we have only to compare the valuesof the traces of the various matrix products. This means that the actual axes aboutwhich the rotations in the symmetry operations and in the generating rotation occuronly enter the problem in so far as they determine the forms of the various matrices,but do not have to be considered explicitly in arriving at the least maximum valueof the Burgers vector. This is a great simplification in the analysis ; indeed, withoutthis result the analysis would become prohibitively lengthy even with the computingfacilities now available. Let us now consider the concrete steps necessary for theanalysis of a given dislocation boundary.The relative rotation of the material onthe two sides of the boundary is first specified by a rotation matrix RT correspondingto a rotation 8 about an axis n. The transpose of this rotation matrix is then pre-multiplied by one operation of the symmetry group and post-multiplied by another,not necessarily a different one. The traces of 211 the matrices resulting from thismultiplicatioii are then calculated. The combination of symmetry elements GIand G2 which lead to the maximum value of the trace is then determined. TheBurgers vector calculated by using the transpose of the original rotation matrixmodified by these two selected symmetry operations is then the Burgers vectorwhich has the least value of its maximum modulus for any orientation of the vectorp in the boundary, and is thus the one which, at least formally, correctly describesthe boundary as a dislocation structure.* If the order of the symmetry group isN then there are N2 matrix products to be considered in this way.When theappropriate pair of operations for the rotation considered has been obtained andthe resulting rotation matrix calculated, the axis of rotation corresponding to this* Assuming that the least maximum modulus criterion adopted is appropriate (cp. 5 5)B . A . BILBY, R. BULLOUGH AND D . K. D E GRINBERG 67matrix may be found, and this is the one which together with the angle 8 most cor-rectly represents the boundary in physical terms. The variation of the boundarynormal may now be considered.The variation of the boundary normal affectsthe vectors p(1) and p(2) and the actual Burgers vectors b(1) and b(2) calculated fromthem. However, because of formula (3.4) it does not affect the maximum value ofthe modulus of b. Formula (3.6) shows in fact that the choice a = 4 2 always leadsto a value of 0 for the minimum value of the Burgers vector. This means that theboundary is a pure tilt boundary.As a preliminary example of the application of the above analysis we considera series of boundaries specified by rotations at increments of 2 deg. about selectedaxes in a cubic crystal. These results are obtained from a general programmewhich has been written for the Stretch computer. The programme has as its initialdata the generating rotation axis (n) for the matrix R and the matrices representingthe symmetry operations of the gencrators of the group to be considered.It examinesall rotations about n at increments of 2 deg. For each one it performs the oper-ations of forming all the matrix products, calculating the traces and selecting theappropriate pair of symmetry elements for the selected generating rotation. Itthen calculates the least value of the maximum modulus of the Burgers vector,i.e., the quantity I b(1) 1. It also calculates the resultant rotation axis and the angleof rotation, which give the correct physical description of the boundary if theunreduced formula (2.4) is to be used.5U"30° QeFIG. 3.The variation of I b 1 and 4 when the rotation axis is [ 11 11.In the examples which have been so far examined by the machine the holo-symmetric proper cubic group 0 with 24 symmetry elements has been used.Thereare thus 576 matrix products to be examined for any given generating deformation.Fig. 2, 3 and 4 show the graphs plotted out by the machine for the variation of theleast value of the maximum modulus of the Burgers vector for the selected boun-daries. The rotation axes chosen were the [loo], [ l l l ] and [321]. It is apparentfrom the figures that the maximum Burgers vector varies rapidly in the neighbour-hood of certain generating rotations and falls to zero when this rotation coincide68 THEORY OF SURFACE DlSLOCATIONSwith the symmetry operation. At each discontinuity of slope in the curve for theresultant Burgers vector there is a change in the appropriate pair of symmetryoperations employed and so in the axis and angle of the effective generating rotation.1.01 10FIG.4.-The variation of I b 1 and + when the rotation axis is [321].5. DISCUSSION AND FUTURE WORKThe results so far reported in t b s paper represent a preliminary survey of theproblem of taking note of the symmetry of the material on the two sides of a rotationboundary, or indeed of a boundary between two different crystaI structures. Itis planned to carry out a systematic survey of boundaries in the different crystalsystems and also of those between different crystals, and to make some assessmentof what proportion of such boundaries in a randomly oriented collection of crysta1-lites can properly be regarded as boundaries of large angle. The effect of varyingthe orientation of the boundary normal has also to be further examined. Therelation of the theory to other descriptions of the disorder in large angle boundaries,and to the idea of the coincidence lattice between two differently oriented crystalsare other problems for future study,Finally the least maximum modulus criterion here adopted to select the mostsatisfactory physical description of a given boundary almost certainly coincideswith the criterion of lowest free energy for the boundaries described by simple dis-location arrays and for which 8 is small. Its validity requires further justificationunder other conditions. However, this involves the further complication of con-sidering the discrete dislocation structure rather than the mean dislocation tensor,and at the largest 0, of the significance of the dislocation description itself.1 Bilby, Report Conf. Defects in Crystalline Solids (Bristol, 1954, London ; The Physical Society,3 de Grinberg, Dora K., M. Met. Thesis (University of Sheffield, 1960).4 Amelinckx, Physica, 1957,23, 663.6 Frank, Symp. Plastic Deformation of Crystalhe Solids, NA VEXOS-P-834 (Pittsburgh), 1950,1955, p . 123) 2 Bullough and Bilby, Proc. Physic. SOC. B, 1956,69, 1276.5 Smith, Ph.D. Thesis (University of Sheffield, 1957).p. 150

ISSN:0366-9033    

DOI:10.1039/DF9643800061

出版商:RSC    

年代:1964

数据来源: RSC

摘要:

Low Temperature Thermal Resistivity due to Dislocations inInsulatorsBY A. SEEGER, H. BROSS AND P. GRUNERMax-Planck-Institut fur Metallforschung, Stuttgart, and Institut fur Theoretischeund angewandte Physik der Technischen Hochschule Stuttgart,Stuttgart, GermanyReceived 5th October, 1964The advantages of measurements of the phonon heat conductivity at low temperatures forstudying defects in crystals are briefly outlined. A continuum tbeory treating the simultaneousscattering of pho nons from dislocations, point defects, and by three-phonon processes (N-pro-cesses) is described. The influence of the simultaneous action of more than one scattering mechanismand of the dislocation arrangement is discussed. Comparison with experiments shows satisfactoryagreement for alloys, but striking disagreement for ionic crystals in which the dishcation densitywas estimated by etch-pit counts.Possible implications of this discrepancy are discussed.Measurements of the thermal resistivity of crystals at low temperatures can beused 192 as a tool for studying lattice defects in a quantitative way. The importantfeature is that the phonon spectrum of a crystal depends strongly on temperature,and that the dominant wavelength of the phonons varies with temperature, thewave-number being approximately proportional to the absolute temperature T.1 log TFIG. 1.-Temperature dependence of the phonon heat conductivity K on a double logarithmic scale ;for details see text.The phonon scattering cross-sections of various lattice defects, such as dislocations,point defects, grain-boundaries, etc., are characteristic functions of the phononwavelength.This wavelength dependence of the scattering cross-section is re-flected by a characteristic temperature dependence of the phonon heat conductivityof a crystal containing lattice defects. Fig. 1, which is based on the results to bediscussed later, shows in a schematic way the variation of the low-temperature670 LOW TEMPERATURE THERMAL RESISTIVITYthermal conductivity due to phonons as a function of temperature for a crystal inwhich three types of scattering processes have been taken into account : scatteringof phonons from dislocations, scattering of phonons from point defects, andscattering of phonons by phonons (so-called three-phonon processes, normal processes,or N-processes).In agreement with experiment, the thermal conductivity exhibits amaximum, usually around or somewhat above liquid-hydrogen temperatures. Ina single crystal of large enough dimensions, the thermal resistivity below themaximum is dominated by scattering of phonons from dislocations, which onaccount of their far-reaching strain-field scatter long-wavelength phonons (i.e.,those dominating at low temperatures) rather effectively. The short-range strain-fields of point defects, however, become of importance only at much highertemperatures.It is interesting to compare the heat conductivity measurements with electricalresistivity measurements in metals. In the latter technique, the wave-numbers ofthe electrons participating in the conduction process are fixed by the Fermi energyand do not depend significantly on temperature.This means that the wavelengthdependence of the scattering cross-section, which for electrons also exists, cannotbe exploited in such experiments. Electrical resistivity measurements in metalsdo not enable us to differentiate in a straightforward way between different types ofdefects and are therefore much less powerful for the diagnosis of lattice defectsthan heat conductivity measurements .In order to utilize the phonon heat conductivity for the study of lattice defects,we must avoid any overshadowing by electronic heat conductivity, which wouldbring in similar disadvantages as those just discussed for the metallic electricalconductivity. In insulators, e.g., the alkali halides, this problem does not arise.In semiconductors, very pure materials have to be used in order to make theminsulating at the temperatures involved.Since the number of techniques availablefor the study of defects in pure semi-conductors at low temperatures is limited,the study of the phonon heat conductivity is likely to become an important toolfor investigating the defects produced by low temperature irradiation of pure semi-conductors,3~ 4 particularly under conditions where electron spin resonances arenot applicable.In metals, special measures must be taken in order to suppress the contributionof the conduction electrons to the thermal conductivity. In metals with exceptionallyhigh magnetoresistance, such as Bi, this can be done by applying very high magneticfields.In superconducting metals, the contribution of the electrons to the thermalconductivity can be suppressed by going to temperatures sufficiently below thetransition temperature (see, e.g., ref. (5)). In practice, however, these temperaturesmust be so low that this technique is not yet a convenient tool. The most commonlyapplied technique, particularly by Klemens and his collaborators, is the suppressionof the electronic contribution to the heat conductivity of metals by alloying. Thealloying elements have to be chosen so that they are effective scatterers of electrons(which means a valency different from the matrix) but have little effect on thephonons (small mass difference between alloying elements and matrix atoms).Atypical example of such an alloy is a-brass. The estimates to be given later forcopper are intended to be applied to a-brass. Since in the alloying method, pointdefects (viz., impurity atoms) are introduced, it is suitable for the study of dislocations(and other extended defects) but not of point defects such as vacancies or interstitials.Huebner 6 has shown how the phonon scattering from vacancies (and other pointdefects) in pure metals can be studied by utilizing measurements of the temperaturedependence of the thermoelectric power. The presence of phonon scatterers supA . SEEGER, H . BROSS AND P . GRUNER 71presses the phonon-drag contributions to the thermoelectric power of the pure metal,and this alters the temperature dependence of the thermoelectric power of metalssuch as gold considerably over the temperature range of about 20-100°K.The applications of the thermal conductivity measurements to the study of dis-locations in crystals have been limited, probably because comparison betweenexperiment and theory were not encouraging.Particularly for ionic crystals largediscrepancies exist.A number of years ago, we started a programme of theoretical research on thclow temperature thermal conductivity by phonons with the aim to investigate inmore detail the cause of these discrepancies. The present communication discussesthe present status of the field and the main directions in which further research mightbe undertaken.BASIC FEATURES OF THE THEORYWe confine our theory to low temperatures.Here the wavelengths of the latticevibrations are large compared with the lattice parameter. The strain-fields of thedislocations are slowly varying over the interatomic distances. We thereforeemploy a continuum treatment throughout, similar to Debje’s theory of specificheats. The application of the continuum theory, which disregards the dispersionof the lattice vibrations, to three-phonon processes has been justified.7 A com-parison with other theoretical approaches to the theory of thermal conductivitycan be found in the literature.* We confine ourselves to a listing of the principaladvantages of our treatment over previous theories, particularly those of Klemens 1 ~ 2and Carruthers.9 Our treatment does not require any knowledge of atomic forceconstants, which are difficult to measure. The doubtful approximation of replacingthe third-order force constants by Gruneisen’s constant is avoided.The treatmentallows fully for the polarization of the elastic waves. As a measure of the mechanicalnon-linearity of the crystal, which is essential for obtaining scattering of the phonons,the third-order elastic constants are employed. These can be obtained experi-mentally by a number of techniques ; numerical values are known for some metals,semi-conductors and ionic crystals.The theory allows fully for three-phonon processes, which are always presentbut which, by themselves, do not give rise to a finite thermal resistivity.In treatingthe effect of the various scattering mechanisms (three-phonon processes, dislocationscattering, and point defect scattering) no use is made of the doubtful assumptionthat relaxation times exist. The transport equations are solved by the variationalprocedure, which ensures that the computed heat conductivity is a good approximationto the exact value.The theory does not require an explicit knowledge of the strain-field of the dis-locations or the point defects. It has been formulated in such a way that the know-ledge of the spatial distribution of the incompatibilities suffices. This is of con-siderable advantage when more complicated dislocation arrangements than that ofa single straight dislocation are to be investigated.All the calculations done sofar have been carried out for elastically isotropic media. This is a restriction whichhas been introduced mainly for practical reasons, since otherwise the computationsbecome involved.The starting point of the quantitative treatment is the energy densityHere A1 and A2 are the two second-order elastic constants of an isotropic medium(e.g. shear modulus G and modulus of compression K ) ; A3, A4 and A5 are the thre72 LOW TEMPERATURE THERMAL RESISTIVITYthird-order constants (e.g., Murnaghan's constants Z, m, n). e0 is the strain tensorof the (static) lattice defects, 7 the strain tensor of the elastic waves, and s the dis-placement vector associated with the elastic waves. The subscripts I, 11 and IIIdenote the first, second and third invariants of the tensors involved.Denotingthe mass density in the (statically) deformed state by p', the Hamiltonian is given bywhere df is the volume element of the crystal in the statically deformed state. ThisHamiltonian is quantized and written in the formH = Ho+Vl+V2. (3)The first term in eqn. (3) is the harmonic approximation (which does not give riseto phonon scattering)w, = C h o k ( a t , a , + $ ) . (4)kHers A is Planck's constant h divided by 2n, m k the circular frequency of the vibra-tion of wave-vector k and polarization index j , with k standing for simplicity forboth wave-vector k and index of polarization j . Nk is the occupation number ofihe mode (L); a: is the creation operator defined bya: $(.. . , Nk, . . .) = JNk+I $(. . . , N k 3 - 1 , . . .),ak($(. . . , Nk, . . .) = J N , $(. . . , Nk-l,. . .)( 5 4(5b)and ak is the destruction operator defined by-($ = wave-function).The scattering of the plzonons by the static defects is described by the termv, = c V,,.(a,a~. . . .),k,k'where the coefficients of the individual terms of the sum, describing the scatteringof a phonon of wave-vector k into one of wave-vector k', are of the form,Vlkkt = Ai(ek . ekt)(k . k')E:(k + k')+ . . . etc. (6b)(ek = polarization vector, €"(k) = Fourier transform of the static strain tensor.)The termwithdescribes the interaction of three phonons with each other, e.g., the destructionof one phonon of wave-vector k" and the creation of two phonons with wave-vectors k and k .In eqn.(1) and also in the following treatment we have confined ourselves to thelowest order of the non-linear effects which permit a consistent treatment. Adetailed investigation shows that this should be quite a good approximation in thepresent case, whereas for the scattering of electrons from edge dislocations an ana-logous procedure cannot be justified.10V2k+e,ku = Ai(ek. k)(e,, . k')(ekw , k") . . . etC.9 (76A. SEBGER, H . BROSS AND P. GRUNER 73We shall omit a detailed account of the solution of the transport problem, i.e.,the calculation of the thermal conductivity by the variational procedure, and referto the literature.7~ 8 9 1 1 9 12 The final result has the formwhere IC = thermal conductivity, k = Boltzmann's constant, Q = atomic volume,VT = sound velocity of transversal moves andIn deriving eqn.(8) it was assumed that the crystal contained a random arrange-ment of dislocations of density Nd and dislocation strength b, and a concentrationcp of point defects, described as centres of dilatation with strength A F (volumeexpansion per centre in an infinite crystal). g { x ; y ) is a function characteristicfor the material under consideration; it depends on the ratios of the elastic con-stants and also on the character of the dislocation lines. g ( x ; y } has to be deter-mined numerically for a given material. It is nevertheless remarkable that theknowledge of this one function suflices to describe the thermal conductivity as afunction of the temperature and of the strength and number of the defects.RESULTS AND DISCUSSIONFig.2 shows the result of computations for copper (as a model for a-brass)plotted on reduced scales for the thermal conductivity and the temperature. Thereduced point defect concentrationserves as a parameter for the curves, which exhibit the characteristic shape dis-cussed in connection with fig. 1. For copper,C2 = 11.86 10-24 cm3 ; b = 2.56 10-8 cm;For the particular example of cp = 10-4 A P / Q = 0.3, Nd = 1011 cm-2, (Ndb2) =(9 x lO-2)4,and IC = 0.74 k [watt/deg. cm].The maximum occurs then at the temperature Tmax = 39°K and has the magnitudeKmm = 0.704 [wattldeg. cm]. Consider the limiting cases of fig. 2. Fig. 3 showsthe result for the scattering by edge dislocations and three-phonon processes.Atlow temperatures one finds the same result as in the earlier treatment,g in which thethree-phonon processes were neglected. At high temperatures a P-law is alsofound, but with a different numerical factorVT = 2.293 105 cmlsec.= 10-4, the reduced temperature is8 = 2.3 x 10-2 T/"K,Z = 2.1295e2. (10)The transition between the two TZ-laws had to be computed numerically.13 Sinceit constitutes a violation of the temperature dependence usually attributed to dis-locations, it should be taken into account in evaluating experimental data.The other limiting case is that of simultaneous scattering by three-phononprocesses and point defects. This case is particularly interesting, since, takenseparately, both types of processes do not give rise to a finite heat resistivity.Oper-ating simultaneously, the three-phonon processes prevent the divergence of thepoint defect scattering conductivity integral at long wavelengths, and the poin74FIG. 2.-sistivityLOW TEMPERATURE THERMAL RESISTIVITYr---k-Reducedaveragedafter Gruner.12 /-- T- - - ----1/5 10' 10'FIG. 3.-Temperature dependence of the phonon heat conductivity of copper on reduced scales,taking into account scattering from isolated edge dislocations of random orientation and threephonon processes, after Bross, Gruner and Kirschenmann.1A . SEEGER, H. BROSS AND P. GRUNER 15defects destroy the conservation of quasi-momentum, which is responsible for thevanishing of the resistivity due to the three-phonon processes.A detailed analysis(fig. 4) shows that at low temperatures the temperature variation of K follows aT-1-law, characteristic for the point defects, whereas at higher temperatures thethree-phonon processes become more and more important, leading finally to a T-2-law. In order to observe such a high-temperature law, the reduced temperaturewould have to exceed the value 500. Since our continuum theory fails as soonas Umklapp processes become important, i.e., well below the Debye temperature,10io-10-10-FIG. 4.-'I'emperature dependence of the phonon heat conductivity of copper on reduced scales,taking into account point defect scattering and three phonon processes (PO = mass density of thematerial), after Gruner.12the quantity cp(AV*)2, which determines the point defect scattering, must be verysmall in order to make the T-Llaw experimentally detectable.As indicated infig. 1, for low point-defect concentrations and high enough dislocation density,the T-1 region will be followed on its low temperature side by a temperature rangein which the thermal conductivity varies more rapidly with temperature than T-1before the maximum of thermal conductivity is reached.The heat resistivity due to dislocations depends to some extent on the disloca-tion arrangement, particularly at low temperatures, at which the dominant wave-length becomes comparable with the distance between the dislocations. This problemhas been investigated by studying various arrangements of groups of screw disloca-tions on the same glide plane.14 The results depend on the temperature throughthe parameterkRTf = -,hvTwhere R is a measure of the extension of the group, e.g., the pile-up length.A76 LOW TEMPERATURE THERMAL RESISTIVITYone would expect, for a given dislocation density a group of dislocations of the samesign has always a higher resistivity than a random arrangement, the increase be-coming more significant the smaller the parameter t. The dependence on t doesnot obey a simple power law; in rough approximation a TZ- or Ts12-law for theconductivity holds in the region where the group-effects are significant.Considering together the effects of the three-phonon processes discussed aboveand of the dislocation arrangement, we see that the classical P l a w for the dislocationscattering is unlikely to hold over a wide temperature range.In accurateevaluations these deviations should be taken into account.Unfortunately, the number of experiments with which the theory can be com-pared (i.e., in which the dislocation density has been determined independently) isvery limited.8 In experiments on a-brass, Kemp, Klemens and Tainshl5 haveestimated the dislocation density by stored energy measurements. If the scatteringby isolated edge dislocation only is considered, the theoretical resistivity per unitlength of the dislocations is by about a factor of two smaller than the experimentalvalue. This is presumably well within the uncertainties of the experiment and thetheory (the inclusion of the three-phonon processes and of the dislocation arrange-ment would both increase the theoretical value).The situation is similar for theexperiments of Lomer and Rosenberg,l6 in which the dislocation density was estim-ated from transmission electron micrographs. We may therefore conclude thatthe experiments on metals (or alloys) are essentially in agreement with the theory.A more detailed comparison between theory and experiment will have to awaitfurther experiments under better defined conditions.The situation is far less satisfactory for ionic crystals. In two papers on LiFand NaCl, the dislocation density has been determined by etch-pit counts.179 18The experimental heat resistivity per unit length comes out at least a factor of 103bigger than the theoretical value.This discrepancy cannot be explained in termsof impurities or other point defects, kinks, jogs, etc., since these could not giverise to the observed temperature dependence of the thermal conductivity. Anyexplanation of this discrepancy must be such as not to violate the satisfactory agree-ment for the alloys as discussed above. It appears therefore somewhat unlikelythat the suggestion of Ishioka and Suzuki 18 is able to remove the discrepancy.These authors wish to attribute the main part of the thermal resistivity of dislocationsto the scattering of phonons by the vibrations of the dislocations. Nevertheless,it is recognized that treating the dislocations as static imperfections constitutes themain defect of our treatment.Our treatment of dislocation groups shows that the experimental results on thealkali halides could not be accounted for even if it were assumed that each etch-pitcontained a group of, say, 6 dislocations of the same sign with distances of about50A.We must therefore conclude that the main cause of the thermal resistivity inalkali halides is an extended defect which is not detected by etch-pit techniques.Such defects may be isolated dislocations or dislocation dipoles. The latter willgive rise to the observed temperature dependence only if the two dislocations of adipole are farther apart than the dominant wavelength at the lowest temperaturesinvestigated.An investigation of the scattering of phonons from edge dislocation dipoles isplanned.Without detailed calculations it can already be said that in order toaccount for the observations, the density of dislocations undetected by etch-pitswould have to be about 103 times higher than that of those detected by etch-pits,irrespectively of whether the extra dislocations are present as single dislocations oras dipoles. Such a situation would cast some doubt on the validity of the attemptA . SBEGER, H. BROSS AND P. GRUNER 77to correlate the observed etch-pit density with physical properties of deformed alkalihalide crystals. Further experiments to study this situation (e.g., thermal con-ductivity measurements and transmission electron microscopy on the same specimens)appear desirable.The authors acknowledge gratefully the support of the Deutsche Forschungs-gemeinschaft for part of the work reported here.1 Klemens, Proc. Physic. SOC. A, 1955,68,1113.2 Klemens, Solid State Physics, 1958,7.3 Vook, Radiation Damage in Semiconductors (Royaumont, 1964, Dunod, Paris, 1965).4 Albany and Vandevyver, Rahtion Damage in Semiconductors (Royaumont, 1964, Dunod,5 Mendelssohn, J. Physic. SOC. Japan, 1963, 18, suppl. 11, 17.6 Buebner, Physic. Rev. A, 1964,135, 1281.7 Bross, Phys. stat. sol., 1962, 2,481.8 Bross, Seeger and Haberkorn, Phys. stat. sol., 1963, 3, 1126.9 Carruthers, Rev. Mod. Physics, 1961, 33, 92.10 Seeger and Bross, Z. Narurforschg., 1960,15a, 663.11 Bross, Seeger and Gruner, Ann. Physik, 1963, 11,230.12 Gruner, to be published.13 Bross, Gruner and Kirschenmann, to be published.14 Gruner, to be published.15 Kemp, Klemens and Tainsh, Phil. Mag., 1959,4,845.16 Lomer and Rosenberg, Phil. Mag., 1959,4,467.17 Sproull, Moss and Weinstock, J. Appl. Physics, 1959, 30, 334.18 Ishioka and Suzuki, J. Physic. SOC. Japan, 1963,18, suppl. 11, 93.Paris, 1965)

ISSN:0366-9033    

DOI:10.1039/DF9643800069

出版商:RSC    

年代:1964

数据来源: RSC

摘要:

GENERAL DISCUSSIONProf. P. Haasen (Universitat Gottingen) said: In describing slip in the wurtzitestructure in Amelinckx's paper, it is recognized that a perfect 60"-dislocation inthe widely spaced type 11 glide planes cannot dissociate into partials. It is difficultto believe that the slip plane will be of type I, the closely-spaced plane of the partialshear. The same problem arises in the diamond cubic and the sphalerite structureand how it can be solved. In the latter we have some evidence (from the differencein mobility of In- and Sb-dislocations in InSb 1) that slip occurs on type 11 planes.Dislocations appear, however, to be " dissociated " in extended nodes of equilibrateddislocation networks in both structures.2 I doubt that moving dislocations carrystacking faults in these structures.Maybe they just '' associate " themselves, bydiffusive rearrangements, with one of two partial dislocations of opposite signwhen they come to rest.3 The faults formed in this way could be inti-insic orextrinsic ones. This could lead to the usual gain in energy by '' dissociation "if the distance between the partials is large compared to that of the perfect disloca-tion and its associated partial dislocation. Such a behaviour would have importantconsequences for the plasticity of these structures.4Dr. H. Alexander (Institut f u r Metallphysik, Universitat Gottingen) said : To-gether with Mr. Siethoff I observed dislocation networks with extended nodes, similarto those discussed in Amelinckx's paper, in single crystals of Ge, InSb and (recently)of InAs (fig.l*). Most nodes consisted of screw dislocations. From the width ofthe nodes we derived the stacking fault energies. They theory of Siems, Delavignetteand Amelinckx 5 was used, but instead of the line energy the line tension was put intothe calculation. We obtained the following stacking fault energies 6 : Ge, 66 : InSb,31 ; InAs, 25 erg cm-2. These values are somewhat surprising, since the ioniccontribution to the binding becomes larger in the order (Ge, InSb, InAs). InAs isthe only one of our three substances which shows clearly a difference of the width ofneighbouring nodes, i.e., the energy of intrinsic and extrinsic faults must be ratherdifferent.Dr. K. H. G. Ashbee (BerkeZey Nuclear Laboratories, Glos.) (contributed) : Thebonding in U02 is mainly covalent and not ionic.Consequently, unlike CaF2which has the same crystal structure but ionic bonding, U02 can be expected tocontain dissociated dislocations. Direct evidence for such extended dislocationsin U02 is presented later in this Discussion.Dr. G. R. Booker (Cavendish Laboratory, Cambridge) said: During the last fewyears, we have examined a large number of dislocation networks in a variety of" pure "Si specimens by the transmission electron microscope method. The dislocationsforming the nodes were mostly close to screw orientation. In some instances thenodes appeared to be extended, while in other instances little if any extension couldbe detected. The overall variation in node size observed was at least 5 : 1.One possible explanation is that the networks with unextended nodes occurredbecause the dislocations in these instances did not all lie in precisely the same plane.* Between pp.80-81.1 Peissker, Haasen and Alexander, PJzil. Mw., 1961, 7, 1279.2 Siethoff and Alexander, Phys. stat. sol., 1964, 6, K 165.3 Haasen and Seeger, Halbleiterprobleme, vol. IV (Vieweg, Braunschweig, 1954), p. 68.4 Haasen, Festkorperprobleme, vol. I11 (Vieweg, Braunschweig, 1964), p. 167.5 Siems, Delavignette and Amelinckx, 2. Physik., 1961, 165, 502.6 Siethoff and Alexander, Phys. stat. sol., 1964, 6, K 165.7FIG. 1 .-Extended dislocation nodes in indium arsenide ( x 9,000).(The unusually light contrast present in the extended nodes is a consequence of the reproduction,and does not appear in the original prints.)[See Alexander, page 78FIG.1.-Surface structures along the trace of twist or mixed boundaries. (a) Frank spirals ; (b) cleavagestructure. The mean distance between the monatomic cleavage steps, i.e., between the emergence pointsof dislocations amounts to 370 A.FIG. 2.-Monoatomic cleavage steps having a mean distance of50 A startzfrom emergence points of subboundary dislocations.FIG. 3.-(u) On a cleavage face eachemergence point of edge dislocationsforming a tilt boundary is marked bya gold nucleus. The cleavage steps run-ning to the upper right are caused byforeign dislocations. (b) Ion arrange-ment at emergence points of edge dis-locations.b[See Bethge, page 79FIG.1 .-Transmission electron micrograph of section parallel to principal glide plane ofNi+40 % Co single crystal. Stage I ; shear strain E = 0.20 ; resolved shear stress T = 1.7 kg/mm2after Thieringer3). The arrows point to examples of contrast from extended dipoles. U =direction of trace of unpredicted plane. C = direction of trace of conjugate plane.[See Seeger, page 82FIG. l.-Ni+60 at. % Co, foil normal = (Oll), operating g-vector = (200). A = stacking-faultnook, B = stacking-fault node.[See Diflert, page 85GENERAL DISCUSSION I9However, this is unlikely because many such networks were in specimens which hadbeen annealed at 1200°C for several hours in hydrogen and slowly cooled. Anotherpossibility is that varying amounts of impurity were present and the observationscan be accounted for by the Suzuki effect.This again is unlikely because it does notseem possible that the small amounts of impurity involved could produce such largevariations in node size. A rigorous investigation of these nodes is currently beingmade.Prof. H. Bethge, Dr. R. Scholz and Dr. V. Schmidt (Deutsche Akad. Wissen-schaften, Berlin) said : Concerning the structure of dislocations in sub-boundarieswe have excellent theoretical conceptions from the papers of Frank,l Amelinckx 2and Bollmann.3 After Vogel and co-workers' first works on germanium,4 experi-mental material is available especially by the works of Amelinckx,5 on colouredalkali halide crystals? and recently in numerous papers the observation of dislocationnetworks in crystals imaged by transmission electron microscopy is reported.Inthe following some of our recent investigations on the NaCl crystal will be dealtwith as a contribution to the discussion of the paper presented by Bollmann. Thearrangement of dislocations in sub-boundaries between subgrains of greater mis-orientation was electron microscopically investigated. For this, the dualisticmethod and the step rule given by Bollmann proved to be a useful means.There are two possibilities for detecting dislocations which emerge to the surfaceof NaCl crystals.6 Fig. 1" shows both the possibilities. On the left-hand side,we see structures obtained by evaporation in high vacuum. A spiral forms aroundeach emergence point of a dislocation with a component of the Burgers vectorperpendicular to the surface.Imaging of the monatomic steps is done by thegold decoration technique, first described by Bassett.7 If there were dislocationswith Burgers vectors parallel to the surface, we should have found annular lamellaearound each emergence point. Limited by the lamellar width, we can only detectdistances between dislocations which are not smaller than 0.2 or 0.3 p. If thedislocations lie closer together, we consider the cleavage structure created by thesub-boundary. In the right-hand picture, the sub-boundary lies perpendicular.The arrow marks the direction of cleavage? and starting from each intersected dis-location, whose Burgers vector in this case again has a component perpendicularto the surface, a cleavage step originates. The step height is equal to the mag-nitude of the Burgers vector component. The cleavage steps finally join to formhigher steps, thus giving rise to the river pattern.The mean dislocation distanceis 370 A. This corresponds to a twist angle of misorientation of 0.44".We have grown special bicrystals from the mclt by a method published recently.These crystals contain boundaries with a predetermined misorientation. Fig. 2"and 3 describe two examples. In fig. 2 the distance between the steps starting fromthe dislocation is 5OA. At this extremely high magnification the path of the stepshas been traced for better recognition. In this case palladium was used for decor-ation? because thereby narrower step distances can be resolved.A more exactdiscussion of the dislocation arrangement reveals that the distance between the dis-locations which emerge oblique to the surface-as we here only see its projection* Between pp. 80-81.1 Frank, Cunf. Plastic Deformation of Crystalline Solia3 (Mellon Institute, 1950), p. 150.2 Amelinckx and Dekeyser, Solid State Physics, 1959, 8, 325.3 Bollman, Phil. Mag., 1962, 7, 1513.4 Vogel, Pfann, Corey and Thomas, Physic. Reu., 1953, 90,489.5 Amelinckx, Dislocations and Mechanical Properties of Crystals (ed. Fisher, Johnston, Thomson6 Bethge, Phys. stat. sol., 1962, 2, 3, 775. 7 Bassett, Fhil. Mag., 1958, 3, 1042.and Vreeland), Wiley (New York), 1956, p. 380 GENERAL DISCUSSION-is 27 A only. The corresponding angle of misorientation between the sub-grains is 8.5".In fig. 3a*, the sub-boundary runs perpendicular. The orientationof the sub-boundary was light-microscopically determined, and the exact mis-orientation as well as the position of the axis were determined by X-ray measure-ments. From Frank's formula the directions and the distances between thedislocations, building up the bomdary, could now be calculated. From this wecan conclude that dislocations with Burgers vectors parallel to the surface shouldemerge to the surface with distances of 200 A. Along the trace of the sub-boundarywe see an equidistant arrangement of the decorating metal nuclei. This equi-distant arrangement essentially differs from the decoration at the cleavage steps.The distance amounts to 200 A, i.e., it is in best agreement with the expected distancebetween edge dislocations.We conclude from this that each emergence point wasmarked by a gold nucleus. This is comprehensible. The illustration in fig. 3b*shows the ion arrangement disturbed at the emergence point of an edge dislocation.The excess charge should give rise to the preferable nucleation. The monatomiccleavage steps starting from the trace of the sub-boundary must be attributed toforeign dislocations which additionally to the edge dislocations lie in the sub-boundary. The Burgers vector of the foreign dislocations must have a componentperpendicular to the cleavage face, and together with the Burgers vector of the edgedislocations as vector of the type a/2(110) it can form an angle of 60 or 120".Atthe points where the foreign dislocations emerge to the surface, k i n k s can beobserved in the course of the sub-boundary (see arrow in fig. 3a). Following thedualistic method given by Bollmann we could explain the dislocation reactionbetween the foreign dislocation and the edge dislocations. The kinks in the traceof the sub-boundary are then in good agreement with Bollmann's step rule. Thedislocation arrangement, found in this way, is represented in fig. 4. The dottednFIG. 4.-Dklocation arrangement of the tilt boundary shown in fig. 3a.line ABC corresponds to the trace of the sub-boundary of fig. 3a in the region of akink. The dislocation reaction between the foreign dislocations lying betweenDB, and the edge dislocations is described by a/2[10l]+a/2[Oli] = a/2[110], wherea/2[110] is the Burgers vector of the edge dislocations.As a result we have thefine structure represented, and following from this, the presence of kinks in thetrace of the sub-boundary in the region of the emergence points of the foreign* Between pp. 80-81GENERAL DISCUSSION 81dislocations. It may further be mentioned that for the determination of the disloca-tion arrangement also the trace of the sub-boundary on the (010) face was electronmicroscopically investigated. A publication in detail is planned for Phys. stat. sol.Dr. R. Bullough (A.E.R.E., Harwell) said: I would like to comment on thedefinition of the Burgers vector. The true vector (Frank‘s definition) is a vectorof a perfect reference system and is obtained by means of a directional correspondencefrom a closed circuit around the dislocation in the final crystal to the perfect referencelattice.The closure failure in the reference lattice is the true Burgers vector. Thesign of this vector is purely conventiona1,l for example, FSIRH. If a closed circuitin the reference lattice is mapped on to the final crystal then the closure failure inthe dislocated crystal will not be a perfect lattice vector and its magnitude willdepend on the starting position of the correspondence. This is the local Burgersvector (or Burgers definition). Clearly if the circuit is very large and a long distancefrom the dislocation the local Burgers vector will be almost a lattice vector. Againits sign is purely a matter of convention and if we wish the local Burgers vector tohave the same sign as the true Burgers vector then for the former we should adoptSFIRHinstead of FS/RH as used for the latter.I therefore do not see the significanceof putting bg = - b p when the sign is purely a matter of convention.Dr. W. Bolimam ( 1 s t . Battelle, Gen2ve) said: In answer to Dr. Bullough, threedifferent co-ordinate systems have to be considered : (a) the Euclidean co-ordinatesystem of the surroundings; (b) the curved (strained) co-ordinate system of thecrystal, which is determined by the actual positions of the atoms ; (c) the “ perfectcrystal ” co-ordinate system of the reference lattice.The imaging process implicitly states the equivalence of (b) and (c).Thus theBurgers vector can be looked at as a co-ordinate difference in either one of thesetwo systems. On the other hand, what is called the “local Burgers vector” isactually a local atomic spacing with respect to system (a), but is not a Burgers vectorin its proper sense with its invariance and conservative properties.Concerning the sign, I fully agree that it is purely a question of convention, butthis sign convention has to be stated clearly before dealing with geometrical problems.Dr. W. Bournam (Inst. Battelle, GenBve) said: in answer to Dr. Sleeswick Ibelieve that there is a difference in the concept of a dislocation. Dr. Sleeswickseems to use a more general concept in a continuous medium, while I am strictlylimiting myself to a dislocation with discrete Burgers vector in a crystal.In addition,his objections against “ geometrical ” and “ physical ” joining of the two parts of acrystal do not seem to be justified. Relaxation only occurs within the period of thedislocation network forming the boundary, as the moirC bands contract to dislocationlines. There is no overall displacement of the two crystals.Dr. J. Hornstra (Philips Research Laboratories, Eindhoven) said: In his paper,Bollmann prefers Burgers’ definition of the BV, but especially for grain boundaries,this definition has certain disadvantages. This disadvantage is clear from his fig. 3where the resultant Burgers vector B is defined as the difference of two vectorsr(1) and r(2).Arbitrarily, r(1) is drawn parallel to the boundary; one might alsodraw r(2) parallel to the boundary or choose the vectors in such a way that theboundary bisects the angle between them. All these cases lead to different vectorsB and hence to different BVs of the dislocations forming the boundary. In Frank‘sdefinition both vectors are parallel to the boundary, so there is no ambiguity. Thesevectors are transferred to a perfect reference lattice, where their difference definesthe resultant Burgers vectors in a unique way. It only depends on the length ofr and on its angle with the rotation axis.1 see Bilby, Bullough and Smith, Proc. Roy. SOC. A , 1955, 331, 26382 GENERAL DISCUSSIONDr. W . Bollmanu ( I s t . Battelle, Gen2ve) said: In answer to Dr.Hornstra, Ido not think that there is a basic contradiction between the two definitions for smallangle boundaries, i.e., as long as Frank‘s formula can be applied in the form+ 4+=B = p x r ]4 -+ -+ -+and r can be identified with r(l). The difference between r(l) and r(2) is very smalland if r(l) is chosen parallel to the boundary, r(2) is nearly parallel too.Prof. A. Seeger (Max-Planck-Institut fur Metallforschung, Stuttgart) said : Czjzek 1carried out calculations of the shape of stacking fault nodes which might be com-pared with the work of Brown. Czjzek avoided the use of a variable line energyby setting up the total energy of a node of arbitrary shape, using Kroner’s theory.For the computation, he employed a trial function for the shape of the node which,in the asymmetric case, contained as much as thirteen parameters.For a givenstacking-fault energy and for a given orientation of the node, he minimized the totalenergy with respect to these parameters. The main differences from Brown’s calcula-tion, apart from the difference of an energy against a stress calculation, appear to bethat Czjzek conhed himself to a finite number of parameters describing the dis-location shapes and that he did not have to make what Brown and Tholkn callthe “ de Wit and Koehler ” approximation. As the authors point out, this ap-proximation is of doubtful validity for small nodes. On the other hand, the re-striction to a fixed number of degrees of freedom enabled Czjzek to treat alsoasymmetric nodes, whereas the present calculations are confined to symmetric nodeshapes.With regard to the main features, the calculations of Czjzek and Brownare equivalent, and they give the same results for practical purposes. It is gratifyingto see that several methods exist which give reliable results for the node shapes,provided there are no external influences (other dislocations, image forces, etc.)which interfere.In their transmission electron microscopy studies of thin foils parallel to theprincipal glide plane prepared from plastically deformed Ni-Co single crystals,Maderz and Thieringer 3 observed and described in some detail faint lines whichfollowed accurately the trace of either the conjugate or the unpredicted glide plane,and which not rarely zig-zagged between these two directions (fig.1 *). This phenom-enon has also been observed in copper.4-6 Hirsch and Steeds 5 showed that thecontrast effects of these faint lines can be accounted for by a dipole of two Frankdislocations. On theoretical grounds, i L is more likely that these “extended di-poles ” consist of a pair of Shockley partial dislocations and a pair of stair-roddislocations. As shown in fig. 2, two different configurations are possible, whichwe term “ S-shaped ” and “ Z-shaped ”. Both of them can be visualized as havingbeen generated by a dislocation reaction between two extended 60”-dislocationson parallel (1 1 i)-planes (separation yo). If the reacting complete dislocations arenot exactly 60°-dislocations, the above-mentioned zig-zag pattern may result.Comparing the S- and 2-configurations for the same separation yo, it is seen thatthe former is the more stable one.Compared with a dipole formed of two com-plete (though extended) dislocations, the extended dipoles are stable only up to acertain maximum value of yo. Wobser has calculated this maximum separation* Between pp. 80-81.1 Czjzek, unpublished results.2 Mader, Electron Microscopy and Strength of Crystals (Interscience Publishers, New York,4 Essmann, Phys. stat. sol., 1963,3, 932.5 Hirsch, The Relation between the Structure and Mecfianicaf Properties of Metals (H.M.S.O.),--f 31963), p. 183. 3 Thieringer, D@lumarbeit T.H. (Stuttgart, 1961).p. 40. 6 Basinski. this DiscussionGENERAL DISCUSSION 83as a function of stacking-fault energy.Fig. 3 gives the stability limit and the cor-responding width d'= as seen from an [l 1 f]-direction for an S-dipole as a functionof y/Gb for the special case of an isotropic medium with Poisson's ratio v = 1/3.Calculations for individual f.c.c. metals taking into account the elastic anisotropywill be published elsewhere. Such curves can be used to deduce upper limits forthe stacking-fault energies of f.c.c. metals from the observed widths of extendeddipoles. The finite extension of extended dipoles (separation of the contrast intotwo parallel lines) has been seen in silver (J. Steeds, private communication),FIG. 2.-S- and 2-shaped configurations of extended stacking-faults. The hatched areas denoteintrinsic stacking-faults.1000 II I I .5 15 0 1 I0 20 25I I103FIG.3.--dm,,/b and yo/b as a function of the stacking-fault energy y. (b = dislocation strengthof complete dislocations, G = shear modulus.)The phenomena described above are of particular interest in the above-mentionedexamples of Cu or Ni-40 % Co alloys, where no structure is seen in the electron micro-scope contrast of the extended dipoles. Under these conditions the intensity of thecontrast varies strongly with the extension of the configuration, and therefore alsowith the stacking-fault energy. The reason for this is that the coniiguration hasthe resultant Burgers vector zero. Therefore the contrast must vanish for suf-ficiently large stacking-fault energies. In order to exploit this situation for deter-mining stacking-fault energies quantitatively, we must have a quantitative theor84 GENERAL DISCUSSIONfor the intensity of the contrast and also measurements of the intensity of the dipolecontrast relative to that of ordinary dislocation contrast.Both kinds of investigationsare under way in Stuttgart.While the results of these investigations are still pending, we can obtain pre-liminary estimates of the stacking-fault energies in the following way. From theinvestigations of Wilkens and Hornbogen I (in particular fig. 8) on screw dislocationdipoles we may conclude that the intensity of the contrast decreases very rapidlywith decreasing separation d of the dislocations, if d becomes smaller than aboutone-quarter of the extinction distance. In the cases of interest here, the extinctiondistances are about lOOA. The dipole contrast in Cu is fainter than that in Ni-40 % Co alloys.A reasonable estimate appears to be that in Ni-40 x Co alloysthe maximum width dmaX of the extended dipoles is about 20A and somewhatless in Cu. These estimates are in good agreement with the determination of thestacking-fault energies of these materials by the zm-method.21 3Dr. P. R. Swam (Institut fur Metallphysik, Gottingen") said: Recently someresults have been obtained concerning the effect of temperature on the apparentstacking fault energy of a copper-7 % aluminium alloy. It is found that extendednodes formed in this alloy by room temprature deformation will always shrinkin the range 200-300°C.There is, therefore, an apparent increase in stacking faultenergy y on heating. The results, shown in fig. 1, indicate that there is no furtherchange in the radii of curvature of nodes on heating above about 300°C. Oncooling the change in radius of curvature is not reversible, for cooling rates varyingfrom 103"C/sec to 10-5"Clsec and it is therefore believed that neither Suzukisegregation nor ordering are responsible for the apparent change in y. Instead,there is a solute impedance force acting on the partials which is preventing theequilibrium configuration from being adopted.It is considered that the true stacking fault energy of these alloys is better cal-culated from heated nodes than from nodes formed at room temperature.Further-more, the difference between the high temperature (yt) and low temperature (yapp)measurements of stacking fault energy is directly related to the solute impedancestress zf of the alloy at the lower temperature and at very low dislocation velocities,by the equation,where 61 is the Burgers vector of a Shockley partial dislocation. Using thisequation the frictional stress of the Cu-7.33 % A1 alloy was calculated to be 1-4 kg/mm2. This stress is equal to the difference between the critical resolved shearstresses of the alloy and pure copper measured at room temperature 5 and suggeststhat the flow stress of solute rich copper-aluminium alloys is determined mainlyby a solute impedance having a weak dependence on dislocation velocity than bythe grown-in-dislocation substructure.The presence of a solute impedance of this magnitude can also explain theappearance of highly-unsymmetrical, isolated nodes which are sometimes ob-served in solute-rich alloys.The extended nodes observed after heating are nearlyalways symmetrical and would therefore be more suited to the measurements of?f = ( b P P - ?t)/bl,* Until August, 1965.1 Wilkens and Hornbogen, Phys. stat. sol., 1964, 4, 557.2 Berner and Krontniiller, Moderne Probleme der MetalEphysik (Springer, Berlin-Giittingen-3 Seeger, Berner and Wolf, 2. Physik., 1959, 155, 247.4 Christian and Swann, A.I.M. E. Cot$ series, Alloying Effects in Conceiitrated Solid Solutions,5 Koppenaal and Fine, Trans. A.I.M.E., 1962, 224, 347.Heidelberg, 1964), Kap.2.to be publishedGENERAL DISCUSSION 85the type discussed by Brown and Tholen. In the determination of absolute valuesof stacking fault energy '' the solute impedance error " can probably be neglectedin alloys with electron concentration less than about 1.15 but the error becomesquite serious in more concentrated alloys.Discussions with J. W. Christian are gratefully acknowledged. The results shownin the figure were obtained at the U.S. Steel Fundamental Research Laboratory,Monroeville, Pa., U.S.A.6. I I I 1 ICU - 7.33 '/o Ai5 -TRUE STACKING FAULT ENERGY044-9x ? CONTRIBUTION2,KING FAULT ENERGY- -I -0 I00 200 300 400 500Differt, Prof. A.Metallforschung, Stuttgart)energies in Ni-Co-alloys wetemp., "CFIG.1.Seeger and Dr. M. Wilkens (Max-Planck-lnstitut fursaid : During investigations on the stacking-faultfound a stacking-fault configuration which seems tohave been not yet described in the literature. The micrograph fig. 1" shows anexample. Two straight dislocations join at an angle of about 60°, suspending astacking-fault between them. Owing to the rarity of these events we were not yetable to determine the Burgers vectors of the dislocations involved. Neverthelesswe may discuss possible explanations. One of them allows us to deduce reasonablevalues for the stacking-fault energy. Another possible explanation will be brieflymentioned below.Diffraction patterns show that the straight dislocations are parallel to <llO)-directions. This might indicate that they are of Lomer-Cottrell or Hirth-type.With this assumption, fig.2 gives schematically the geometry of the stacking-faultsand the partial dislocations. In addition to the large stacking-fault area there aretwo further small ribbons on the inclined (1 11 )-planes.The Burgers vectors of the partial dislocations (A), (B), (C), (E), (F) are con-nected by the following relations :bA -I- bB 4- bc = bB 4- bE + bFbA 4- bc = bE 4- bF.In addition, the corner of the 60" angle may be separated into two points 0 and 0'connected by the dislocation G. Since the dislocation B has moved from the corner* Between pp. 80-8186 GENERAL DISCUSSION0, it may be assumed to be a Shockley partial. Taking into account the criteriaof stability for the different types of Lomer-Cottrell and Hirth dislocations (Hirth I),there remains only one possibility for indexing the Burgers vectors, shown in table 1.All other possibilities can be excluded.FIG.2.-DisIocation configuration for a stacking-fault nook, schematically.The plane containing the axes s and t, is a symmetry plane of the configuration,which is in accordance with the observations.TABLE BU BURGERS VECTORS OF THE PARTIAL DISLOCATIONS IN FIG. 2crystallographie notation Thompson cotationbA = a/3DIO] yD+ CdbB = a/6[211] 6Bbc = a/6[21I] 4bF = a/6[2Il] cubE = a/3[lOI] a D f A6bo = a/3[011] ya+ CA = Ca+ yAThe Burgers vector of the complete dislocation is b = a[I00]; a = cubic lattice constant.In order to calculate the shape of the configuration as a function of the stacking-fault energy, we have used a method similar to the one of Siems et aZ.2 Elasticisotropy was assumed throughout.The shape of the dislocation B was assumedto be an arbitrary function v = v(x), where v and x are the co-ordinates as shownin fig. 2. The interaction energy between B on the one side and A and C or E and Fon the other side was expressed as integrals over the dislocation lines accordingto Kroner’s theory.3 The integrations over the straight dislocations A and E areelementary. For the integrations over C and F this is also true, since we assumedfor simplicity that C and F are straight dislocations. Finally, the energy of thestacking-fault area and the essential parts of the interaction-energy and the self-energy (Chou and Eshelby4) can be expressed as an integral over the unknown1 Hirth, J.AppL Physics, 1961,32, 700.2 Siems, Delavignette and Amelinckx, 2. Physik, 1961, 165, 502.3 Kroner, Kontinuumstheorie der Versetzungen und Eigenspannungen (Berlin, Springer, 1958).4 Chou and Eshelby, J. Mech. Physics Solidr, 1962,10,27GENERAL DISCUSSION 87curve u(x) of the dislocation B. Variational methods yield a differential equationF(V”, v’, v, x, v, y ) = 0,where v is Poisson’s ratio, y the stacking-fault energy per unit area, and v’ and v”the first and second derivative of tl with respect to x. (The dependence of the self-energy per unit length of the dislocation B on v’ has been allowed for in the vari-ational procedure.) For practical evaluations we may confine ourselves to thesolution around the point P (x = 0, v = V), see fig.2. Here, the first derivativeu‘(0) is determined by symmetry. For the coordinate Y of the point P an equationresults. The second derivative u”(0) appears on account of the line tension ofthe dislocation B. Since the interaction-energy terms are found to be about threetimes larger than the self-energy term, y depends only weakly on the value of u”(0).We were therefore allowed to employ semi-empirical values for ~ “ ( 0 ) .From three of the observed “ stacking-fault nooks ”, which could be analyzedwith respect to the length V, we get the following results (v = 0.36) :Ni+60 at. % CoNi+67 at. % ColO3y/Gb0 = 1-65lO3yIGbo = 0.75(1-60 (M), 1-0 (H+S))(0.72 (M), 0.42 (H+S)).G is the shear modulus and bo the nearest atomic distance in an f.c.c.lattice. Forcomparison we have included the corresponding values of Mader 1 (=(M)) andof Howie and Swann 2 (= (H + S)), which were derived from node measurements.Another configuration might possibly explain the micrograph of fig. 1. If wedo not assume the limiting straight dislocation lines to be Hirth- or Lomer-Cottrell-dislocations, a second possible configuration can be found as follows. We startfrom a dislocation with a Burgers vector b = +fOll], lying on the slip planes (111)and (ill). If both parts of the dislocation are aligned with the trace of the plane(ill), the dislocation can split off a Shockley partial, moving in the (111)-planeand leaving behind a Frank sessile dislocation.This model can be visualized fromfig. 2 by removing the dislocation C, F and G and replacing bA and bE in table 1by bA = bE = a/3[111]. h the approximation introduced above (elastic isotropy),there is no elastic interaction between the Frank and the Shockley dislocation.This implies that the shape of the stacking-fault area is determined only by theline tension of B and the energy of the stacking-fault area itself. Using the samemethod as described, the second model yields values of y which are smaller by abouta factor of four than the results mentioned above. This seems to be unreasonably low.Furthermore, if the model under discussion were correct, the contrast of the twocomplete dislocations in fig.1 reaching out from the stacking fault should be of theresidual contract type (g . b = 0), i.e., much weaker and considerably broader thanthat of other dislocations. Comparison with neighbouring dislocations showsthat this is not the case. An additional argument against the second model comesfrom a comparison of stacking-fault nooks and stacking-fault nodes in the samesamples. This point is discussed below. Summarizing these arguments we con-clude that the first model is much more likely to be correct than the second. Furtherinvestigations to decide in a final manner between the two models are in progress.DIsCuSSIoN.-The application of the theory of elasticity to stacking-fault nooksis much more straightforward than to stacking-fault nodes.Whereas in a node1 Mader, in Electron Microscopy and Strength of Crystds, ed. Washburn and Thomas (Inter-2 Howie and Swann, Phil. Maq., 1961,61215.science, New York, 1962), Chap. 488 GENERAL DISCUSSIONthe shape of all three partial dislocations comes out only as the result of detailedcomputations, in the stacking-fault nook problem the symmetry of the configurationis fixed from the beginning and the shape of only one of the partial dislocationshas to be calculated. In order to facilitate the analytic treatment of the interactionenergy between two of the (curved) partial dislocations of a node, Sierns et aZ.1(in an improved form, Siems2) assumed one of them to be straight. Whereas fora node this requires additional justifications, it is certainly permitted for the stacking-fault nook. We expect therefore our calculations on the stacking-fault nook tobe reliable.From the experimental point of view, it is of considerable advantage that for agiven value of the stacking-fault energy the length Y is larger by a factor of 3 or 4than the bisectrix w, which is a useful measure of the size of a stacking-fault node.This means that under given conditions the experimental error is smaller for astacking-fault nook than for a node.By combining our theoretical results withthose of Brown and ThOlh,3 we get for the ratio Vlw the theoretical values 3.7and 3.8 for Ni+60 at. % Co and Ni3.67 at. % Co. As a consequence of theparticular choice of the logarithmic term in ref. (S), this ratio is almost the same foredge and screw nodes.Using our own node measurements on the alloys men-tioned above, we found experimentally that this ratio deviates less than about 10 %from the expected values. These deviations are within the experimental uncertainty.We consider this as a confirmation of our dislocation model for the stacking-faultnook.Our results on the stacking-fault energy agree well with those of Mader,4 whohas adopted the method of Siems et aZ.1 to asymmetric nodes and who also usedthe calculations of Czjzek5 mentioned in the contribution of Seeger to this Dis-cussion. On the other hand, our results are higher by a factor of 1-75 than thoseof Howie and Swann.6 This is not surprising since the latter authors used a formulawhich neglects the repulsive forces between the partial dislocations of a node.Consequently their formula yields values of the stacking fault energy which aretoo low.Dr.M. F. Ashby (Institut fur Metallphysik, Universitat Gottingen) said : I wouldlike to comment on the contrast of the lens-shaped cavities shown in Dr. Forty’spaper. Contrast can arise in several ways. Consider a lens-shaped cavity in athin foil, as shown in fig. 1. First, contrast arises because the optical path ofelectrons is shorter along paths passing through the cavity than along paths whichdo not. This type of contrast has been discussed by Ashby and Br0~n,7 who showedthat the image of a small cavity (of thickness of the order of 0.1 to, where to is theoperating extinction distance 8) may be either lighter or darker than the background,depending on the depth of the cavity in the foil, and, more important, on the devi-ation of the foil from the exact Bragg angle.Larger cavities, of the order of toor more in thickness, will show circular fringes (thickness fringes2); one suchfringe is shown schematically at (1) in the figure.The effect of depth in the foil probably accounts for the range of contrast fromcavity to cavity seen in fig. 5 of Dr. Forty’s paper, and fringe effects could account1 Seims, Delavignette and Amelinckz, 2. Physik, 1961, 165, 502.2 Siems, this Discussion.4 Mader, in Electron Microscopy and Strength of Crystals, ed. Washburn and Thomas (Inter-5 Czjzek, unpublished7 Ashby and Brown, Phil. Mag., 1963,8,1649.8 Hirsch, Howie and Whelan, Phil.Trans, A , 1960, 252,499,3 Brown md Tholkn, this Discussion.science, New York, 1962), Chap. 4.6 Howie and Swam, Phil. Mag., 1661, 6, 1215GENERAL DISCUSSION 89for most of the contrast change within a single cavity. It is important to realizethat this is not absorption, but diffraction contrast.Secondly, contrast arises if atoms near the cavity are displaced from the positionsthey would occupy in a perfect lattice. (Displacements may be caused by an excesspressure of gas inside the cavity, or by surface tension.) This sort of contrast canbe very complicated. Some examples are shown in fig. 1. If the displacementsare small, contrast of the type shown at (2) will arise. This contrast is similar tothat shown by a dislocation loop lying in the plane of the foil with Burgers vectornormal to the foil-a situation discussed in detail by Howie and Whelan 1-r tothat caused by the strain field of certain precipitate particles.2 These authors haveshown this contrast to have the following properties.It is strongest when the foilis orientated so that the Bragg condition is satisfied for one set of diffracting planes,when a line of no contrast, shown as a dotted line on the figure at (Z), always liesnormal to the reciprocal lattice vector g defining the diffracting plane.2 (This vectorcan be determined quickly from a diffraction pattern.) Experimental determinationof this relation between the line of no contrast and g shows the existence of latticedisplacements round precipitates or cavities.As the foil is tilted steadily awayfrom the exact Bragg angle, the contrast becomes less pronounced, and finally dis-appears. (A cavity which lies particularly close to one surface of the foil showsspecially strong contrast, due partly to the increased displacements resulting fromrelaxation of stress at the surface.) Cavities which show contrast which can beexplained in this way are visible in fig. 5 of Dr. Forty's paper.The interference of electrons diffracted from above and below a cavity withlarge displacements ( > b where b is the lattice constant) can cause other contrasteffects. When the cavity is roughly parallel to the foil surface, this interferencegives rise to a set of nearly parallel fringes, also normal to the reciprocal latticevector g, as shown in (3), superimposed on the contrast shown in (2).When, onthe other hand, the cavity is indined to the electron beam by tilting the specimenthrough a large angle, say 20°, then the displacements normal to the plane of thecavity (which up to now have not influenced the contrast) have a component parallelto the reciprocal lattice vector, and give rise to contrast in the form of concentricellipses as shown at (4), whose spacing should decrease as the inclination of thecavity to the beam increases. (It is probable that superimposed " depth fringes ",1 Howie and Whelan, Proc. Roy. SOC. A , 1962,267,206.2 Ashby and Brown, Phil. Mag., 1963,8, 108390 GENERAL DISCUSSIONparallel to the intersection of the plane on which the cavity lies with the foil surface,will also be visible.These are formed in the same way that stacking-fault fringesarc formed.) Such contrast is not visible on Dr. Forty’s micrographs but mightbecome visible if the foils were tilted through a sufficiently large angle and wouldconstitute strong evidence for the existence of displacements round the cavities.Most, but not all, of the contrast shown by Dr. Forty’s cavities can be explainedby causes illustrated at (1) and (2) of fig. 1. The remaining contrast effects mustbe due either to the superposition of contrast such as that illustrated at (3) and(4), or-more probably-to deviation from a perfect lens shape.Dr. K. €3. G. Ashbee (Berkeley Nuclear Labs., Glos.) said: Observations ofcavities similar to those observed in lead iodide by Dr.Forty have been made 1in quenched U02. In U02, however, the cavities are thought to be hollow cylinderspassing right through the foil since (a) the contrast within them is identical withthat outside the foil edge as evidenced when they join the foil edge, and (b) the degreeof brightness within a cavity is independent of its diameter d (d varies from 0.2 t to> t , where t, the foil thickness, is estimated from the number of fringes at extendeddislocations).The cavities are formed at high-beam intensities (large condenser aperture) andtheir appearance coincides with the onset of evaporation at the foil edge. Theymigrate, without change of size, at rates of up to 0.2 p per sec (attempts to movethem faster results in catastrophic evaporation) presumably towards hotter areasof the foil, although rarely towards the centre of the field of view.If a migratingcavity passes to within 0.5 p of a dislocation line it experiences an elastic inter-action. At this distance, which is about 5 times the foil thickness and up to 6 timesthe cavity diameter, the cavity is deviated from its original path, moves towards andjoins the dislocation. Further heating causes the cavity to heal without furthermovement of either itself or the dislocation (see fig. l(a), (b)). This presumablymeans that the temperature reached is below that required to permit dislocationglide and/or climb, i.e., below N 1000°C. The contrast left joining the two segmentsof dislocation line originally separated by a cavity which has healed, is reminiscentof a dislocation loop (fig. l(b)).Both the circumferences of the cavities and the evaporating foil edge are oftencharacterized by a fringe of dark contrast.This must be an absorption effect dueto ledges at an evaporation surface.The migration of cavities in quenched U02 is by mass transport through thevapour phase. The rate of migration is too fast for self-diffusion to be the control-ling mechanism. Also, surface diffusion can only be of minor importance otherwisethe jagged foil edges would be smoothed out.Since attempts to produce identical cavities in slowly cooled U02 have beenunsuccessful, it seems likely that the nuclei for the present cavities arise from quench-ing and not from localized heating or irradiation damage by the electron beam.An ideal nucleus would be formed by migration to the same point on the surfaceof excess vacancies retained by the quench.Dr.D. A. Young (Imperial College) said: It is not clear why Dr. Ashbee prefersvapour transport to surface diffusion. Using the best available vapour pressuredata for U02,2 which extend only down to 1600°K but for the purposes of this noteare extrapolated to the temperatures mentioned by Dr. Ashbee, one obtains -3 x10-20 mm at 1000°K and -6 x 10-10 mm at 1500°K. Rates of evaporation, even1 Ashbee, Phil. Mag., 1964, in press.2 Rand and Kubaschewski, The Therrnochemical Properties of Uranium Compounds (Oliverand Boyd, Edinburgh, 1963)FIG.1.-(a) Cavity A trapped at a dislocation. (b) After heating, the cavity heals leaving a loopof dislocation contrast.[To face Ashbee, page 90GENERAL DISCUSSION 91calculated using the Polanyi-Wigner equation, are then orders of magnitude too lowfor cavity migration by this mechanism.If one does the calculation backwards, calculating the expected activationenergy on this basis, one obtains 25-30 kcal mole-1, i.e., about 1/5 the enthalpy ofsublimation. Besides being roughly equal to the activation energy for inter-stitialcy migration in U02+s, this is a reasonable value for surface migration.Applying Tanunann’s old-fashioned idea that surface diffusion becomes importantfor 0.25 Tm < T<0.33Tm with self-diffusion dominating at T> 0.6 Tm, one shouldlook for evidence that surface migration is inadequate before invoking evaporation.Finally, it is possible that the electron microscope environment is “ stronglyoxidizing ” for U02, thus cavity motion could be an extrinsic effect found only insuper-stoichiometric material.Dr. J.Hornstra (Philips Research Laboratories, Eindhoven) said: In order tofind the rotation matrix with the minimum value of 8, or the maximum value of thetrace, Bilby et al. examined 576 products of the form G1RTG2. When 1 met the sameproblem some years ago 1 I examined only 24 products of the form GIRT and 4:still believe that tbis is correct. The equivalence of the two methods can easily beproved by noting that a similarity transformation Q-lRTQ leaves the trace invariant.Hencetrace GlRTG2 = trace G~G~RTGzG;~ = trace GZGlRT = trace G3RT,in which G3 is another symmetry operation.In 5 5, the authors doubt whether the criterion of lowest 6 is the decisive pointin selecting the most favourable rotation matrix.If the boundary plane is notgiven, I can see no better criterion. As soon as the boundary plane is known, thedislocation density of the boundary may be calculated. In the paper mentionedabove, a case is discussed where not 8 but the dislocation density is the decisivefactor.Dr. R. Bullough (A.E.R.E., Harwell) said: We realize that only the 24 productsneed be evaluated at a given rotation angle to determine the minimum equivalentrotation, but we are grateful to Dr. Hornstra for presenting his simple proof.Inthe paper we first show that the maximum resultant Burgers vector of dislocationlines in the boundary can be directly related to the equivalent angle of rotation anddoes not depend on the axis of rotation. Thus by minimizing the rotation angle 8we are not saying that 8 is necessarily the decisive factor but that when 0 is a minimumthe maximum resultant Burgers vector will have minimum value. The total dis-location density in the interface will also depend on the magnitude of the orthogonalminimum resultant Burgers vector and this value depends on the boundaryorientation.Prof. P. Haasen (Universitiit Gottingen) said: Our flow stress measurementsand etch pit counts on NaCl2 support Seeger’s idea that there must be unetcheddefects present in these crystals to account for their thermal resistivity.As is alsodescribed later in this Discussion the observed etch pit density is too small by afactor of 103 to 104 to explain the low temperature critical shear stress of NaClin a forest intersection model. We thought of attributing this discrepancy to joghardening which, however, does not help in the thermal resistivity problem. Dis-location dipoles may be helpful3 in explaining thermal resistivity as well as thestage I1 flow stress in NaCl which is athermal. Dipoles are then unlikely, however,1 Hornstra, Physicu, 1960,26,198.2 Haasen and Hesse, Symp. no. 15 (N.P.L., Teddington, 1963), p. 137.3 Gilman, this Discussion92 GENERAL DISCUSSIONto be responsible for the low temperature critical shear stress. The problemremains unsolved.Dr. R. W. Davidge (Metallurgy Dept., Imperial College, London) said: Prof.Seeger finds that the thermal resistivity of NaCl as calculated from various terms,including effects of point defects and dislocations, is some three or four ordersof magnitude below the experimentally observed value. In deformed NaCl thetotal length of dislocation dipole, as determined from bulk density data, is abouttwo orders of magnitude greater than the screw dislocation density. Could theeffect of dipoles explain some of the resistivity anomaly?Prof. P. L. Pratt (Metallurgy Bept., Imperial College, London) said: While itis possible that the marked difference between experimental and calculated thermalresistivities of deformed ionic crystals is due to dipole debris there is evidence whichsuggests that this is not so. At the meeting of the American Physical Society inMarch, 1963, Pohl and Taylor claimed that the recovery of the increased thermalresistivity “happens at temperatures that are roughly 100°K higher than thoserequired for the recovery of the mass density”. We know that the recovery ofdensity is associated with the annealing out of dipole debris, and thus it appearsthat debris is not an important source of phonon scattering.Dr. L. B. Griffiths (Tyco Lab. Inc., Mass.), said: One possible reason for thelarge discrepancy (102-1 03 crn-z), reported by Seeger between theoretical predicationand experimentally determined (by etch pit count) dislocation densities was indicatedby Prof. Gilman in discussion. In essence this refers to the linking together ofparallel edge dislocations beneath the crystal surface thus rendering the defectsundetectable by etching methods.I merely wish to add that some recent experimental results on a-Sic single crystalsdemonstrate Gilrnan’s point quite clearly. We found, using a copper decorationtechnique 1 and also X-ray metallography, that the dislocation densities determinedfrom etch pit counts, gave values of as much as 104 times lower than was the casein reality. This result is a direct manifestation of the linking of parallel edgedislocations.1 Trickett and Griffiths, J . Appl. Physics, to be published

ISSN:0366-9033    

DOI:10.1039/DF9643800078

出版商:RSC    

年代:1964

数据来源: RSC