from single defects to a structure-property relationship for polycrystals

UltramIcroscopy 40 (1992) 321-329 North-Holland

From single defects to a structure-property relationship for polycrystals

D a v i d A Smi th TJ Watson Research Center, IBM Research Dwtston, Yorktown Heights, N Y 10598, USA

Received 4 January 1991, at Editorial Office 7 August 1991

The challenge confronting the microscopist seeking to elucidate the connection between the properties and mlcrostructure of a polycrystalhne material or the engineer seeking to design a material is immense This is because of the diversity of interface structure and the complexity of the interactions possible between the various defects in interfaces and in the grain interiors It is argued that so far the defects observed in interfaces seldom provide a mechanism for interfaclal processes and that existing experimental approaches are fundamentally hmited to a subset of the totality of interfaces which may be formed However, to establish a structure-property correlation for a polycrystalhne material as distinct from a blcrystal it may not be necessary to have a detailed knowledge of structure for each individual interface Instead, a differentiation of interfaces into a limited number of classes each with distinctive properties which can be analysed utlhsing the insights of percolation may provide the means to connect the properties of individual interfaces to the behavior of the polycrystallme ensemble

1. Introduction

It is known that interfaces dominate many aspects of the behavior of a polycrystalhne ensemble and that particular interfaces have a char- acterlstlc and unique structure [1-3] The link between the structure and propert ies has proved very elusive for grain boundaries and interphase interfaces alike It has long been recognlsed that the existence of five macroscopic degrees of free- dom renders it impracticable to charactertse the whole spectrum of boundary structures even for a particular material Instead, the direction of research efforts has been to a t tempt the elucidation of general rules through the study of particular boundaries This approach has enjoyed only hmited success For low-angle grain boundaries the principles are established, even though it is not always obvious quite what dislocations will occur It IS incontrovertible that a dislocation network in which the overall Burgers vector is normal to the rotation axis will be the essence of the grain boundary structure [4] Fur thermore, It is rela-

twely straightforward to deduce the structure of a mixed boundary from those of the corresponding tilt and twist boundaries This IS done by project- lng the dislocation structures of the simple boundaries onto the desired plane and then re- constructing unstable nodes [4-6] No such pro- cedure can be developed for high-angle grain boundaries This is because, in contrast to the low-angle case where crystal coordination is pre- served between dislocations, the variation of the boundary structure between the dislocations in a high-angle boundary as the orientation of the boundary plane changes is not known High-resolution transmission electron microscopy and computer simulation are both hmlted to relatwely simple structures so that the nature of the arbi- trary boundary remains inaccessible to structural characterisatlon However, lrrespectwe of the particular context the notion of some kind of ordered structure in which defects may be embedded is prevalent for interfaces [1,2,5]

Some progress has been made in defining the generic features of interfaces in the various classes

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322 D A S m t t h / A structure-property relattonshlp for polycrystals

of materials, for individual interfaces this involves defining the nature of allowed defects and the core structure in which they are embedded [7] Even the relatively simple task of understanding the behavior of a single interface in a btcrystal has met with hmlted success The amsotropy of grain boundary diffusion [8,9] and the mlsorlentatlon dependence of critical current in high-temperature superconductors [10] are examples of the successful elucldatlon and explanation of a structure property relationship, a counter-example is provided by grain boundary migration

2. Interfacial defects

There are numerous observations of mterfaclal defects together with a taxonomy of the admissi- ble kinds of defects [7] However, the elucidation of the allowable reference structures for rater- faces has yet to be completed and may prove to be an unattainable or even meaningless objective

The sahent features of the character and be-

havlor of mterfaclal defects from the experlmen- tahst 's point of view are (a) mterfaclal defects adopt a regular configuration in local equilibrium [11-13], (b) interracial defects can move and multiply [14-16], (c) lnterfaclal defects have the pos- slblhty of dual step and dislocation character [17], (d) frequently, lnterfaclal defects are observed In a density which is insufficient to account for the lattice relationship [18,19], (e) Burgers vectors of interracial defects need not be crystal lattice translation vectors [20-22]

The verification of the Read-Shockley dislocation model for low-angle grain boundaries [4] depended on establishing a correlation between dislocation density and mlsonentat lon Explicitly this meant showing that the observed Burgers vector density satisfied one of the equivalent equations given below

S=(uXV)O, (1)

where S is the Burgers vector traversed by a probe vector V lying in a grain boundary with a

Fig 1 Electron mmrograph of a gram boundary in hghtly deformed zirconium m which some but not all of the geometrically necessary &slocatlons are visible The visible dislocations are the reszdue of a small plashc strata

D A Srmth / A structure-property relattonshtp for polycrystals 323

rotation angle 0 about a rotation ax is / /which for small angles reduces to

d = O/b , (2)

or Bollmann's 0-lattice equation [5]

(I -- A - 1 ) x (°) = b (L) ( 3 )

where b (L) is the Burgers vector traversed by a probe vector x <°) in an interface characterised by a transformation A It is well estabhshed that interfaces between phases where the principal strains or rotations are small are comprised of dislocations plus a homogeneous elastic strain which together account for the measured relationship between lattices (1) and (2) Comphca- tlons become apparent when it is no longer physical to use one of the crystal lattices as reference lattice For example, fig 1 is an electron micro- graph of a gram boundary m lightly deformed zirconium in which some but not all of the geometrically necessary dislocations are visible This is a very common occurrence, but the explanation is more eluswe in a material with a hexagonal structure since the coincidence site lattice approach has only limited apphcabihty yet dislocations are quite clear

Another very familiar example where all the geometrically necessary dislocations are not observed is the first-order twin in cubic crystals It is found that the dislocations observed by electron microscopy are those which accommodate any small deviations from the preose twin disorientation, a 60 ° rotation about (111) or are a residue of a deformation process [23], there is no physical reality to the extremely dense array of dislocations which would m principle generate the twinning relationship by a 60 ° rotation A variation of the same phenomenon is found In the type-B or hetero-twm interfaces, between NIS1 z or CoSI 2 and sdlcon [24,25] but in this situation the dislocations accommodate misfit rather than mlsorien- ration from the Ideal twinned configuration A further illuminating example is provided by com- parison of observations of grain boundaries where the disorientation has a value such that on a Read-Shockley basis wlth lattice (1) as reference the dislocation spacing xs a few Burgers vectors

Here different techniques gwe complementary insights to the nature of the boundary Two-beam (strong or weak) images reveal networks of dislocations similar in geometry to low-angle grain boundaries [11-16,18,19] but with a spacing which can be related to the disorientation from some reference lattice (usually a coinodence slte lattice) other than the perfect crystal lattice High- resolution transmission electron microscopy observations [26-31] and X-ray scattering experiments [32], however, reveal features which can be Interpreted as an almost, but not perfectly, periodic array of lattice dislocations gwing a Burgers vector density which does, however, accommodate the disorientation expressed with lattice (1) as reference The two sets of observations are reconciled when it is reahsed that (a) coincidence site lattices may be regarded as periodic arrays of lattice dislocations, and (b) the two-beam observations and the high-resolution observations probe the long- and short-range elastic fields of the grain boundary respectwely

There are many other situations which prove much more difficult to resolve, for example, the nature of the misfit accommodation in the class of interfaces which join phases where the unit cells are related by a close approxamanon to a rational fraction, e g , epltaCtlC A1-SI where the

4 lattice parameter ratio is 3 [33] The nature of the reference structures in general remains to be elucidated with the aid of experiments or computer modelling It will be apprecmted that al- though the Burgers vector denstty can be calcu- lated, provided the reference structure is known, the Burgers vector dtstrtbutton must be deter- mined experimentally

An important subset of interfaclal dlslocanons is that with dislocation shift complete (DSC) or transformation character DSC is the nomencla- ture due to Bollmann [5] and conveys the idea that DSC dislocations are perfect lnterfaclal dislocations If we designate the lattices of the two phases which meet at the interface (1) and (2), the DSC dislocations have as Burgers vectors translation vectors which map a site in lattice (1) onto a site in lattice (2) Consequently the motion of such lnterfaclal dislocations, which conserves the interracial structure, causes one phase to grow

324 D A Smtth / A structure-property relattonshtp for polycrystals

Jill] 1

ITLITLITLITJ ITLIT ITLITJ

ITH II,LTI T ITII .[T1,LTILII

[0001]

Fig 2 Sketch of a transformation dlslocaUon in a fcc-hexagonal mterphase boundary Plane of the projection is (110)/(1120), (3 and × symbols define atom poslUons m A and B layers of stacking sequences of planes parallel to plane of projection Note step at core and requirement for shuffles

in alternate {111}/(0001) planes when &slocatlon ghdes

at the expense of the other This crystallography is well known for the twinning dislocation in the fcc structure and the transformation dislocation in the hcp-fcc interface Fig 2 is a sketch of a transformation dislocation in an fcc-hexagonal lnterphase boundary The two kinds of symbols represent atom sites in different layers normal to the plane of the projection The interface is indicated by a bold line Note the step at the core of the &slocatlon and the requirement for shuffles in alternate {111}/(0001) planes when the dislocation ghdes Conventionally, the term transformation dislocation IS reserved for lnterphase boundaries and DSC dislocation for grain boundaries Motion in general is by a mLxed climb-glide process King and Smith [34] have analysed the relationship between the Burgers vector and the step height for coincidence boundaries in cubic materials

An irregular arrangement of dislocations is characteristic of grain boundaries which have been involved in some dynamic process The at- tainment of equlhbrlum may involve annihilation and rearrangement, possibly resulting in a change

in the crystallographic parameters describing the interface Inteffaclal dislocations are expected to respond to external forces similarly to crystal lattice dislocations but their movement is subject to two additional constraints These are that the lnterfaclal dislocation may be confined to the interface plane (because the Burgers vector would introduce an energetically forbidden fault in a grain Interior) and any movement involves over- coming the Pelerls barrier for the &slocatlon concerned together with an additional barrier which is related to the energy required to perturb the network of which it is a part [35] In many cases the interface plane does not contain the Burgers vector and thus pure glide motion will not be possible The dislocations discussed above are embedded in a core structure which may be the perfect crystal for epltaCtlC and low-angle boundaries or some other reference structure when the transformation A does not relate near neighbors at the origin [5]

The conclusion of both modelling and experimental studies [26-31,36-39] is that the grain boundary core structure is as crystal-hke as possible, thus in metals, low-energy relaxed structures are associated with maximal coordination which is equivalent to mmlmisatlon of the grain boundary excess volume, m semiconducting materials reconstruction at grain boundaries mlnlmlses the density of dangling bonds and antl-Slte defects [40,41], in ionic materials the guiding principle is the preservation of charge neutrality concomitant with a coordination shell of unhke species [42]

3. Properties of single interfaces

The llst of interracial properties which have been shown to be structure-sensitive is extenswe even though in general the data are limited and only refer to a subset of the total interface popu- lation However, the task of developing a synthesis relating structure to properties for lndlvldual interfaces IS simplified by recognislng that invari- ably the behavior in any particular case involves a core process possibly in series with an interaction with a long-range field of the boundary There seems to be a general trend in grain boundary

D A Smtth / A structure-property relattonshtp for polycrystals 325

q ~ . m

r I a ~" 8s LU a- 0

(3_

3 ' r

I I I

I ,.E 0 s ~

MISORIENTATION Fig 3 Schematic illustration of behavior of properties such as energy, electrical resistance, suscept~bdlty to corrosion, pre- ctpltat~on kinetics, moblhty, dlffusw~ty, segregation and vacancy sink efficiency which are rnmtmlsed at both special and low-angle boundaries, and critical current, fracture stress and dielectric breakdown field which are greatest for specml boundaries The behawor of a general boundary is indicated

by a dashed hne

phenomena m which essentially two classes of behawor are exhibited, the two classes are associated with low-angle and special boundaries on the one hand and general boundaries on the other These generahsatlons are illustrated schematically in Figs 3a and 3b for a low-angle boundary and the vicinity of a special boundary There are two kinds of dependency of property on structure, there ~s always an extremum m behawor for special and low-angle boundaries (for the low-angle case the singularity is at a mlsorlentatlon of 0 °, l e , the perfect crystal) The extremum may, m prmclple, involve a local mini- mum, a local maxamum or a saddle point Energy, electrical resistance, barrier height, susceptlbdlty

to corrosion, precipitation kinetics, moblhty, dlf- fUSlVlty, segregation and vacancy sink efficiency are mmlmlsed at both special and low-angle boundaries [10,43-51] Critical current, fracture stress and dielectric breakdown field are greatest for special boundaries [10,52,53]

Key dependencies which provide a hnk between many of these properties are the grain boundary excess energy and the association between energy and excess volume Gram boundary energy has a direct connection with gram boundary precipitation, corrosion and fracture, the excess volume is directly related to diffusion

It ~s approprmte at this point to d~scuss the nature of the insights to mechanisms which come from studies of structure At the quahtatlve level there are the simple correlations of the kind referred by Hoffmann and Turnbull [8] which were further emphaslsed by Krakow and Smith [9] to account for the amsotropy of grain boundary diffusion m tilt blcrystals The experimental observations are that the diffusion coefficient parallel to the tilt axis DII is greater than that perpendicular to the tilt axis D ± and the ratto D I I / D . decreases as the mlsortentatlon 0 m- creases The correlation with structure ~s that DII revolves transport along the cores of the edge dislocations which accommodate the gram mlsonentat~on whereas D± revolves transport mainly in "good" crystal when the dislocation spacing is large, 1 e , the rotatton angle is small An exphclt insight to the origin of the large value of Oil comes from experimental data whtch show that the lnteratomlc dlstances in the dislocation core are expanded relative to the lattice and the core ~s comprised of atoms with decreased coordination numbers More quant~tatwe correlations have been attempted, to relate the behawor of defects in interfaces to the properties of the interface as a whole Such developments depend on a postu- lated relationship at the very simplest level of the form

n = ,do, (4)

w h e r e / / i s the property of the interface contain- ing a density of defects P each with properties ~'d An alternate expression ts a law of mixtures

326 D A S m t t h / A structure-property relattonshlp for polycrystals

where the Interface property H is written as

H = ~ d X + ( 1 - X ) ~ ,, (5)

where X is the effective area fraction of defects and ~-, represents the behavior of the interface between the defects

Certain propert ies such as the amsotropy of grain boundary dlffuslon [8], the mlsonentat lon dependence of Jc in high-temperature superconductors [10] and the normal reslstwlty of metals [44] are relatively well described by equations like eqs (4) and (5) at least for certain simple boundary configurations Other propert ies are known to be structure dependent but the connection between structure and behavior in many instances remams obscure An example of this latter kind is grain boundary m~gratlon for which no correlation has been found in general, between the flux of grain boundary dislocations or steps and the distance migrated by the boundary [54,55] Sur- prisingly, the diffusion-controlled movement of mterphase interfaces during precipitation is well described in terms of addmon of atoms to ledges [47] However, this may be mainly a consequence of the importance of lathce diffusion to transport material to the interface and the concomitant decrease in the importance of the role of the interface other than as a sink An apter compari- son would appear to be with an interface-controlled process [56] such as the polymorphlc crys- talhsatlon of CoSI 2, but in this case the growth process is ~sotroplc)

4. Collective behavior

The propert ies of single interfaces are a key to such fields as band-gap engineering [57,58] and the supermodulus effect [59] However, for many materials the mare issue is the understanding of the behavior of a polycrystal, not a blcrystal This involves, m principle, consideration of the couphng of grain boundaries to other grain boundaries, gram-to-gram couphng, and the couphng of grain boundary propert ies to those of grain interiors It is well known that materials in polycrystalhne form have propert ies that are quahtatwely

and quantltatwely different from those of the same material in single-crystal form Examples include stress-strain curves, dfffuslonal creep, co- ercwlty, and reslstwlty

Previous work on polycrystalhne materials has sought to establish a connection between a single measure of the grain size, d, and physical properties of the polycrystal For example, the Ha l l - Petch relation [60] in which the yield stress is proportional to the inverse square root of the grain size, Coble's [61] relation for the creep strain rate being proportional to the reverse of the cube of the grain size, the Her r lng -Nabar ro [62] relation for the creep strata rate being pro- porhonal to the inverse of the square of the gram size, and the Fuchs [63] expression in which the res~stwlty is proportional to the Inverse of the gram size In all cases, the boundary-to-boundary variations of behavior were neglected in developing the models The description of gram boundary propert ies in terms of a single length scale gwes a remarkable insight to trends m the relationship between mlcrostructure and properties [60-63] Nevertheless, a full synthesis requires the incorporation of boundary-to-boundary variations in propert ies and the couphng of processes in the grain boundaries and the grain interiors

The rest of this section is devoted to an introduction to a novel approach using the theory of percolation to account for the behavior of a polycrystal in terms of the constituent grain boundaries An initial structure consisting of (001) fiber-textured randomly oriented equmxed hexagonal grains with cubic symmetry is generated by the computer Each grain is assigned a random azimuthal orientation The mlsorlentat~on of ad- joining grains across each grain boundary ~s calcu- lated The gram boundaries are then placed into two distinct categories depending upon their mls- orientation angle which for present purposes are low-angle and high-angle, the cut-off angle, 0c, between these two classes is a computational variable Having dwlded the grain boundaries Into two classes, it is then possible to search for clusters of grams which share like boundaries We focus on clusters of grams connected by low-angle grain boundaries The properties of clusters that are of interest are their content or mass and

D A Smtth / A structure-property relattonshtp for polycrystals 327

a l inear dimension The mass in the present work is gwen by the number of grams per cluster, and the l inear dimension is given by the radius of gyration The microstructure as a whole, then, may be character ized in terms of the average propert ies of the clusters present From experiment, the division between low-angle and high- angle boundar ies is generally taken to be ~ 10- 15 ° Since the density distr ibution of misorlenta- tion angles is constant in two dimensions and noting that for [001] tilt boundaries the maximum mlsorientat lon angle is 45 °, a value of 0 c = 15 ° lmphes that the microstructure is very close to the percolat ion threshold At the percolat ion threshold (Pc), the ratio at which the number of low-angle boundaries to the total number equals 0 347 [64], there exasts a cluster which spans the width of the sample and has a fractal dimension of 1 896 [65] In turn, the proximity to Pc means that the mlcrostructure tends to be ramtfied, that is, character lsed by clusters" of very large average weighted mass and radius of gyration An example of a mlcrostructure and the clusters which comprise It is shown in fig 4, where the centers of grains with a common low-angle boundary are connected This part icular structure was gener-

( ( ( (

) ) )

Fig 4 Sample mxcrostructure showing hexagonal grams and clusters of low-angle boundaries Grams in same cluster are connected by hne through the center of gram A sample that ~s smaller than is generally used in s~mulanons ~s shown so

that the bonds may be wslble

ated for 0 = 15 ° and it is notable for the ramified nature of the clusters and their in terpenetra t ion For this part icular value of 0 c, the mlcrostructure is close to (but below) the percolat ion threshold The new insight which comes from the work described in this section is that there is a structural unit larger than the grain size, and this is the cluster size The cluster characteristics offer an addit ional descriptor of the mIcrostructure which embodies a length scale, connectivity and gram boundary character

The mathematics of percolat ion provides a means to include binary or higher differentiation in grain boundary propert ies and leads to the Introduction of new mlcrostructural features and measures clusters joined by like boundaries, their mass and their radii of gyration [66-68] A first success of this new synthesis is in accounting for the Jc of a polycrystalhne superconductor in terms of an orientat ion distribution function and the misorientat ion dependence of critical current for individual boundaries [69]

5. Discussion

The most exphclt experimentally based general s tatement that can be made concerning grain boundary structure is that the grain boundary coordination is as crystal-like as possible The boundary then is distinguished from the grain interior by a local difference in density, usually a decrease This decrease in density has been pro- posed as a major contributor to the grain boundary excess energy [70] This raises the possibility of measuring the associated change in mean in- ner potential in the electron microscope and gaining an insight to the local expansion and thus excess energy of grain boundaries through Fres- nel imaging [71] It is significant that this approach would not be limited to simple periodic boundaries Fur thermore , the widespread observations of grain boundary dislocations [11-21] and the generali ty of the crystal rotat ion phenomenon [72-75] suggest that grain boundary structures resist perturbat ion, a corollary of this result IS that the energy surface, in disorientat ion space, for all boundaries is curved with the spe-

328 D.A Smith / A structure-property relattonshtp for polycrystals

clal boundar i e s at cusped min ima The s tructural implicat ions of this deduc t ion are not yet clear If the unifying factor is grain bounda ry dislocations, i e , all grain bounda r i e s may be descr ibed physi- cally in terms of arrays of dislocations which can move and multiply, much of the phenomeno logy of interracial processes can be ra t lonahsed Since the dislocations requi red for the mot ton of particular boundar i e s are characterist ic of that boundary, as is the activation energy for the cl imb c o m p o n e n t of the dislocation mot ion, so it ts mevl table that lnterfaclal proper t ies will be a func t ion of the crystal lography of the b o u n d a r y Presen t u n d e r s t a n d i n g does not extend to making this connec t ion explicit The problems are legion and Include the uncer ta in ty of the genera l validity of the grain bounda ry dislocation model , the van - a t lon in dis locat ion con ten t of an interface as It migrates and sweeps up dislocations or partici- pates in some o ther process such as point defect absorpt ion or deformat ion , the anlsotropy and bounda ry - to -bounda ry var ia t ions in act ivation energy for diffusion and the effects of solutes

Paradoxically, a l though much less ts known about the s t ructure of in te rphase interfaces than about grain boundar ies , at least in the context of their movemen t in phase t ransformat ions , there is a much be t te r and quant i ta t ive u n d e r s t a n d i n g of their behavior This may reflect an mslghtful focus on the essential fea tures of this set of phe- n o m e n a steps, t ranspor t and crystallography and the avoidance of fascinat ing but u l t imately in- significant detail [47]

6. Conclusions

The e lucidat ion of the mechantsms by which interfaces affect the proper t ies of mater ia ls requires an u n d e r s t a n d i n g of the behavior of individual interfaces and that of ensembles of grains In a very few relatively simple cases such as the anlsotropy of grain bounda ry diffusion, the mls- o r ien ta t ion d e p e n d e n c e of Jc In ceramic superconductors or the no rma l resistivity in metals and the diffusion-control led growth of second phases there is a quant i ta t ive s t r u c t u r e - p r o p e r t y rela- t ionship However, the role of interracial struc-

ture in proper t ies such as grain bounda ry migra- t ion or shdmg remains elusive There is a need to develop techniques to study the full range of interface structures inc luding their chemical com- posi t ion at the atomic level

Acknowledgements

The author is grateful to the following for permiss ion to reproduce figures Figure 1 - Dr M E Wel land , F igure 4 - Dr C S Nichols and TMS and to Dr C S Nichols for many invaluable discussions

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