Low temperature deformation properties of crystalline mercury

LOW TEMPERATURE DEFORMATION PROPERTIES OF CRYSTALLINE MERCURY J. S. ABELL* Physics Department. University of Surrey. Guildford. England (Rrceiwd

30 April

1975)

Abstract-Single crystals of mercury of square and circular cross-section have been subjected to tensile tests at 4.2K and iive distinct types of deformation mode have been identified. Three of these. [1Ii; ( li0)slip, '[i%:'(i21) twinning and ( li0; ‘( 332)’ kinking are modes which have been observed at 77 K. The two new shear modes which have been detected at 4.2 K are (il3)(iiO), g = 0.47 which has been associated with the x-7 phase transformation occurring in mercury at this temperature, and (i13}( 110). g = @2-t for which the mechanism involved has not been determined. Metallographic observations on these shear ‘modes are presented and the crystallography of the two new modes is discussed in detail. The orientation dependence of the occurrence of these deformation modes is interpreted using Schmid factor contour plots of the most highly stressed variants. and it is found that such an analysis can be satisfactorily extended to include the transformation shear process. It is deduced that the resolved shear stress on the transformation shear mode need only be half that on the conventional slip mode for the transformation to occur at 4.2K. The transformation habits usually occur in pairs and this feature of the transformation can be understood when the special crystallographic relationship existing for this shear mode is examined in the light of the Schmid factor analysis. Rbum&--On a soumis des monocristaux de mercure de section carrte et circulaire B des essais de traction B 4.2 K et on a identifie cinq modes de deformation distincts. Trois d’entr’eux. glissement ; 1Ii: (110). maclage :i%l (721) et piiage : IiOl (332). avaient eti observes B 77K. Les deux nouveaux modes de cisaillement observes B 4,2 K sont (f13j (ii0). g = 0,47 que l’on associe a la transformation de phase 1 - 7 qui se produit dans le mercure a cette temperature, et (i13j (110). g = 0.24 pour lequel on n’a pu determiner le mecanisme mis en jeu. On presente des observations metallographiques de ces modes de cisaillement et I’on discute en detail la cristallographie des deux nouveaux modes. Grace auxcastes du facteur de Schmid des variantes Ies plus fortement sollicitees. on interpmte l’apparition de ces modes de deformation en fonction de l’orientation. et on trouve que Ton peut etendre l’analyse pour y inclure le cisaillement de transformation. On en d$duit qu’il sufbt que la c&ion reduite dans le mode de cisailtement de transformation atteigne la moitie de la valeur dans le mode de glissement classique pour que la transfor~tion se produise B 4.2 K. Les plans d’eccolement de la transformation se prisentent e~n~ralement par paires et tbn peut comprendre cette particularit& de la tmnsformation en examinant dc;apr&st’analyse du facteur de Schmid. la relation cris~llographique particulibe qui existe dans ce mode de cisaillement. Zusammenfassung-Quecksilbereinkristaiie rechtwinkligen und runden Querschnitts wurden bei 4.2 K gedehnt. Fiinf verschieden Verformungsmoden konnten aufgedeckt werden. Drei davon. ( 1li)( liO)-Gleitung, ‘{i_ji’(i21)-Zwillingsbildung und { li0; ‘( 332)’ Kinkbildung wurden schon bei 77K beobachtet. Zwei neue Schermoden wurden bei 4.2K gefunden. nlmlich { 113}(iiO). g = 0.47. welche der bei dieser Temperatur in Quecksilber ablaufenden 2-y Phasentransformation zugeschrieben wurde, und fil3}( 1 IO), g = 0.24. fur die der verantwortliche Mechanismus nicht bestirmnt wurde. Metallografische Reobachtungen iiber diese Schermoden werden vorgelegt und die Kristahografie wird in Einzeiheit diskutiert. Die Orientierungsabhdngigkeit im Auftreten dieser Verformungsmoden wird mit Hilfe von Konturdiagrammen der Schmidfaktoren der am stlrksten belasteten Varianten interpretiert. Eine solche Analyse kann zufriedenstellend auf den Transformations-Scherprozess ausgedehntwerden. Es wird abgeleitet. daB die kritische FlieBspannung t& den Transfor~tions-~he~roz~ nur die HtilRe derjenigen ftir die konventioneile Gieitmode Transformations bei 4.2 K ist. Die Transformationen entstehen gewiihnbch paarweise. Das kann mit einer Analyse der kristal~ografis~hen Verhghnisse bei dieser Schermode auf der Basis der Schmidfaktoren verstanden werden.

stress-induced r-7 transition in mercury. no systematic investigation of the deformation characteristics at low temperatures has been reported. The purpose of this paper is to present and discuss the results of uniaxial deformation tests performed on single crystals of mercury at 42 K. The results on slip. twinning and kinking are compared with those observed at higher temperatures. Detailed observations on the ‘*_y transfo~ation are presented and the orientation dependence of the occurrence of the various possible deformation modes is discussed.

1. IS~RODLCI’ION A recent series of experimental and theoretical investigations of the mechanical behaviour of crystalline mercury have firmly established the operative deformation modes at 77K and above. Apart from some preliminary observations on the occurrence of the

* Present address: Department of Physical Metallurgy and Science of Materials. University of Bi~ingham, Bimxingham 15. England. If

12

ABELL:

DEFOR,MATION

OF CRYSTALLINE

MERCURY

Mercury crystallises in the r-phase whose structure is most conveniently referred to a face-centred rhombohedral unit cell of axial angle 98’ 22”“. On this basis. the predominant slip mode at all temperatures has been established as { lli,\( liO)“-j) except near the melting point where non-crystallographic slip may occur (6’together with wavy slip in the closest-packed (011; direction!3*6’ Below * 170 K deformati& hvinning may occur. the habit plane being irrational and close to {i’%} with shear direction (i21) and shear strain 0.63!7-g’ Kinking defining the { li0) mirror plane, which is perpendicular to the predominant (ITO) slip direction, has been observed at 77 K15)with an effective shear direction parallel to the operative { llf} slip plane normal, which is irrational but approx. (332). The r-phase of mercury is stable under normal conditions down to 4.2K(‘“*11) but two other forms of crystalline mercury are known to exist; j-Hg having a tetragonal structure, produced under high pressure and stable below 79K,“‘*“’ and PHg of unknown structure obtained by tensile deformation at 4.2KJ4) The existence of the martensitic y-phase has been demonstrated by means of superconductivity,‘i4’ electrical resistivity(‘*i4) and metallographic observations!4q’5) These measurements served to distinguish the y-phase from j-Hg, and this has been conhrmed

Table 1. Summary of the letter notation used for the shear elements of the operative modes. Approximations to rational elements are indicated by apostrophes

by bold lines and the regions are labelled with the code letter of the shear direction.

the letter notation is surmnarised in Table 1. Also included here are the variants L. M and N of the

in a more direct manner by low temperature X-ray diffraction.“ ‘) A modification (16’ to a conventional X-ray cryostat allowed a specimen to be deformed in situ both in tension and by hammering at 4.2K. Tensile deformation produced the ;-phase and hammering the j-phase, and a tentative explanation for this result was proposed on the basis of the relative molar volumes of the three phases of mercury.“‘) Preliminary measurements of the shear elements associated with the r-y transformation”*” indicate that the habit plane and shear direction are close to (i13) and (110) with a macroscopic shear strain of 05; detailed results reported in the present paper essentially cot&m these shear elements except that the shear direction should be written (no), i.e. the opposite sense to that previously reported. In a previous ,paper!j’ the deformation geometry for single crystals of mercury was discussed in terms of the established slip, twin and kink modes. A con111 venient letter-notation for the shear elements of the 1 operative deformation modes was introduced and 2 used in presenting contour plots for the Schrnid fac3 tors of the modes for crystals oriented in the standard L III stereographic unit triangle. This scheme was found particularly useful in explaining the orientation dependence of the occurrence of the various deformation modes at 77 K and it is appropriate in this paper to summarise the essential features of the scheme since it will be used in the interpretation of the pres2ii 2ii ent results and will be further developed to include Fig. 1. Stereographic plots for crystalline ix-mercury showthe transformation shear process. ing (a) the standard 111 stereogram illustrating the letter Thus, Fig. l(a) shows the standard 111 stereogram notation for the elements of the slip, twinning kinking for crystalline mercury which is divided into six unit and transformation modes defined in Table 1. The stantriangles by the three :liOl mirror planes of the face dard unit triangle is shown shaded and is used in (b) and (cl to plot Schmid factor contours (in units >f l_O-l) for centered rhombohedral unit cell. The positions of the the most highly stressed variants of_(b) (1 ll)( 110) slip crystallographic variants of the various shear elein tension or compression and (cl ‘( 1351’twinning in tenments of the operative slip, twinning and kinking sion. The boundaries between regions in which different variants of the deformation modes operate are indicated modes are shown in letter form on the stereogram and

ABELL:

DEFORM,ATION

OF CRk’ST.ALLIYE MERCURI

x-;’ transformation habit plane uhich wiil be incorporated into the schcmc in this paper. By computing curves of constant Schmid factor for the possible shear modes it has proved possible to determine ahich variants of these modes arc likely to occur in single crystals of particular orientations. THO such examples are reproduced in Figs. lib) and (c) which show the most hikay stressed variants of ib) the j I Ii:< Ii@ slip mode and ICI the -ii%; twinning mode in tension represented in the standard unit triangle. The trianpls is sub-divided in regions xhsre ditTercnt variants of the particular mode are favoursd. The variants are denoted by the code letter of their shear directions. Note that all variants of ths slip mode have zero Schmid factor for crystals oriented along [l 1l] and [l IO], and one variant of the tivin mode is favoured in tension for most crystal orientations. This scheme has been successfully applied to the analysis of the orientation dependence of the operative deformation modes at 77K’” and the present uork will show that it can be used with squal success to account for the observed deformation behnviour at -1.2K. 2. ESPERI>lEXl-AL

TECHXQUES

Rod single crystals of square and circular crosssection bvere grown from ths high purity mercury ( < 3 ppm non-gaseous impwit> J used in the previous work. The crystal moulds were made of precision bore glass tubing. but in the case of the square crystals the mould bvas specially adapted so as to provide scratch-free crystal surfaces. Into the outer 1Omm square bore tube tit four accurateI>. machined and polished glass slides which reduce the bore to 6 mm square. The slides are lubricated with ethyl alcohol and a bung holds the plates and mercury in position during gro\vth. After growth. the set of slides containing the single crystal is extracted as a unit from the outer tube, thus eliminating any scratching of the crystal surfaces on the walls of the tube. The slides are then simply lifted away from the crystal one by one leabing the crystal undamped, with faces particularly suitable for metallographic observation. Both types of crystal were grown by a modified Bridman technique. The mould was lowered at a speed of 1 mm~min into a dewx of ethyl alcohol maintained at a temperature of _ LYOK with liquid nitrogen’. A nucleation point for solidification uas provided by inserting a copper pin through the centre of the bung into the liquid mercury. The diameter of the cylindrical crystals and the facial dimension of the square crystals were both 6 mm and the crystals were grown at least ?Omm long. Thsy uere oriented by the Laue back reflection method and annealed at XOK for at least a week. Ths crystal preparation. mounting procedure and metallographic techniques used in these experiments have been descritxd in detail elseM-herc.‘3.‘.“’The uniasial tensile tests were peiformed in a tensile jig designed to fit insids a con-

13

centric glass double dewr s>-stem. The inner de&at was precooled to 77K and then liquid helium was slowly transferred. the crystal at first being cooled by cold gds and finally by liquid. Observation of the crystal was possible during deformation through a window in the silvered glass walls of the dewars. Xo attempt was made to measure the stress imparted to the crystals during the tests, but the overall strain u’as dcduccd on completion of the test from thz displacement of ta-o fiducial marks on the crystal surface. 3. RESULTS The orientations of the crystal axes of the thirty specimens subjected to uniaxial tests are shown in the standard stereographic triangle of Fig. 2. The testtemperature for all but &vo of the crystals was +2K, the exceptions being crystals 29 and 30 kvhich were tested at 55 K. Se\en of the crystals uere of the cylindrical type and arz represented by circular symbols; the remainder had square cross-sections. One crystal was tested in compression undsr impact and is denoted b> a bar thus. i?; the other crystals were all deformed in tension. Crystallographic slip on the i LITI ( ITO) mode. the predominant operative mode at 77 K and above, was observed on all but four crystals, the exceptions being represented by open symbols and numbered 1. 2. 4 and 5. For crystals S, 9. 10. 13. 19 and 30. slip was the only observed deformation mechanism. Thz operative variant of the slip mode was in all crses that predicted by the maximum Schmid factor according to the scheme outlined in section I. and discussed in detail in a preiious paper.“’ A typical micrograph illustrating this crystallographic slip is shoun in Fig. 3. The slip lines were evenly spaced indicating a more

2ii

Fig. 2. Orientation of the axes of the crystals subjected to uniaxial tests represented on the standard unit trian$e. The key explains the s\mbols used to illustrate the detormation behaviour in different areas of the triangle. Ke!,: 0, circular cross-section crystal: W. quar_e cross-section crysgl: J. no detectable she: -.
ABELL:

Fig. 3. Micrograph Fig. 4. Micrograph

DEFORM.%TIOS

of crystallographic

ofa twin intersection

OF CRYSTALLIYE

: Eli:
on crystal S deformed in teniion at 4.2 I(

ix 30). showing ~~ccommo~~tion slip OR crystal f dcform~~ in tension a1 4.2 R [ x 301.

180,

I60 1 ILO!

Fig. 5. Resistance as a function of increasing temperature for crystal 16 after tensile deformation at -1.1K. showing the 7-z transformation at about 55 K.

homogeneous deformation than occurs at 77 K where bands of dip were more common.(5’ Six crystals, numbered 1, 2, 3. 4, 5 and 6 and denoted by symbols with horizontal bars, three grouped near [l Il] and three close to [I IO]. deformed by twinning on variant (a’) of the ‘:i%j’ mode observed at 77 K.‘s) Reference to Fig. l(c) shows that this is the most highly stressed variant in tension. Four of these specimens 1, 7. 4 and 5 which showed no evidence of prior slip, also twinned on the complementary variant (a) which was shown to be the second-most highly stressed variant in tension. (j’ Specimen 11 which was tested in compression, twinned on variant (h’). the most highly stressed system in compression.‘” A typical twin formation is shown in the micrograph in Fig. -t taken from a crystal on which two complementary variants of the ‘:i%I’ mode occurred. Note the accommodation slip at the twin intersection. a feature entremely common at 77 K which has been discussed in detail elsewhere!” Accommodation kinking was Table 2. Results of resistance measurements on four crystals undergoing the r-; transformation

MERfL-R\

frequently associated with these intersections and was also observed at the tips of twins terminating in the matrix These were again a common feature for certain orientations at 77 K. Macroscopic kinking occurred in three specimens numbered 6. 7 and 11 and are represented by symbols with vet-t&l bars in Fig. 2. The operative variant in all cases was A iA) corresponding to the associated (.A) A slip system. On fifteen specimens. numbered I-t-28 and denoted by symbols with oblique bars in the lo~r region of the standard triangle. the predominant surfaoz markings were those associated with the X-;. rransformation.‘r5’ The deformation of ail these specimens was accompanied by an audible click or cry usually associated with diffusionless deformation mechanisms. The electrical resistance of four of these deformed crystals, whose orientations are numbered 16, B-26 in Fig. 2, was measured as a function of temperature in the range -&%I!-IOOK. .-t typical resultant curve is shown in Fig. 5. All the curves have one feature in common: an abrupt increase in &stance over a welldefined temperature range in an othenvise smooth curve. This discontinuity corresponds to the transition from the ;-phase. produced by the strain at 4.2 K. back to the parent x-phase. The measured tempemture range of the discontinuity, the associated proportional resistance increase and the measured macroscopic strain on these crystals arc given in Table 2. The general appearance of a transformed crystal is shown in the micrograph reproduced as Fig. 6 which was taken at + XOK, and it is iborth noting that, to the naked eye at least. the surface morphology did not change at all during the rise in temperature from 4.2 K to this observation temperature. The transformation traces were straight and usually extended through the cross-section of the sp+zimen. They usually occurred in pairs as wn in Fig. 6 but in some cases only one set of parallel markings was observed. The density and the number of sets of traces appears to vary with orientation in the lower half of the standard triangle. Thus. crystal 23 exhibited just two or three traces on one system corresponding to one habit plane. and deformed predominantly by : 1lfi ( 110) slip as already described. whereas the surface of crystal ‘8 on which slip was soar&y detectable. was cot-ered with transformation traces on four different systems. which defined two pairs of habit planes.

.ABELL: DEFORM\TIOY

OF CRYST.-\LLINE MERCCR\-

Fig. 6. Micropraph of the overall appsarance of one tke of crystal 26 which transformed when tested in tension at 4.2 K l x 1). Fig. 7. Micrograph of obtuse-angled intersections between plates of y-mercury on crystal 26 illurrratin~ the displacement of slip lines in the transformed regions. Yote that acute-angJrd intersections appear to bc avoided I x 30). Fig. 8. Micrograph of unustlal surface features on crystal 3 with axis orientation close to illi indiiarmg interactions at obtuse angles (x 301. Fig. 9. Xlxrograph of cr>stA I7 shoain,0 the interaction between two rqions of y-mercur! detining tu’o :TI3], habit planes. The plant of interjection is a : IiO;, ti%rror plane of r-mercurr_ ( x 701.

A higher magnification microgaph of the crystal shown in Fig. 6 appears as Fig. 7 where the intersections between transformed regions are clear. The displacement of the : lli;( li0; slip lines across the regions of ;,-Hg demonstrates the existence of a shear process associated with the transformation. As previously noted,“” the pairs of traces alwavs intersect at obtuse angles typified by the morphology in Fig. 7: acute-angled intersections have never been observed and indeed appear to be avoided as seen in this figure. The transference of the transformation shear process from one plane to another and back again clearly indicated on Figs. 6 and 7 is reminiscent of cross-slip formation in cubic metals and to a lesser extent the phenomenon of cross-twinning seen in certain metals including mercury.‘9’ Reduction in thickness of the ;-Hg region by a series of intersections in the well-defined manner is also a consistently observed feature of the transformed crystals, a phenomenon which again has a parallel in tu-inning in mercury.‘” The habit plane of the transformation traces was identified using two-surface and circumferential analyses”,” and in all cases was found to be within a few degrees of the crystallographic [?I31 plane. A detailed circumferential analysis of seien sets of traces on four cylindrical crystals revealed that the poles of the associated shear directions corresponded closely to the close-packed direction (110) of the r-structure. Holvever, the sense of the surface tilts indicates that the shear on the Ii13: habit plane is in fact in the opposite direction. <:iiO). This result was confirmed by accurate surface tilt measurements of six sets of traces on the three square crystals numbered 17. 21 and 2. which were particularly suitable

for such measurements. These metallographic measurements n-ere then processed by two methods,‘“’ one stereographic. the other algebraic to determine the magnitude of the shear strain associated with the x-;’ transformation The results of the two procedures were found to be consistent to within YO and only the algebraic values are shown in Table 3 together with the result of the shear direction measurement on these crystals. The sense of the surface tilts on the remaining square cross-section crystals was consistent with this (no) shear direction. Crystals 39 and 30 had orientations lying in the same part of the triangle as those just considered. However these crystals were tested at 55 K and no evidence of the transformation was observed. The only detected deformation mode was the most highly stressed variant of the f 117)( li0) slip mode. Closer examination of crystals 1. 2 and 3 souped near Ill revealed. in addition to slip and twin traces, some unusual surface traces: an example of these appears as Fig. 8. taken from crystal 3. The morphology of these traces involving intersections of pairs at obtuse angles is similar to that observed on crystals undergoing the r-y transformation. Moreover, the habit plane of these traces \vas again determined to be within a few degrees of li13:. the observed r-; transformation habit. Stereographic analysis of surface tilt measurements on two sets of these traces yielded the most remarkable result that the associated shear direction \\as close to (I IO) as indicated in Table 3: that is the opposite sense of the macroscopic shear direction involved in the I-;: transformation. The correspondins values of the shear strain magnitude of this deformation process deduced by the stereographic and algebraic methods already men-

16

XBELL: DEFORM4TION OF CRYSTALLINE MERCURY

Table 3. Measurrd salues of shear strnin and shear direction fcs three crystals undergoing the z-;’ transformation. and one or~stal close to Ill which deformed on

iil3;(lloj Shear Directkm Deviation from

Crystal

Snear Strain

17

0.46

24O

0.56

3O

0.51

2O

0.42

2O

0.42

4O

22

0.46

4O

3

0.23

z”

0.24

P

21

I

tioned are also given in the table. The average value of _ 0.24 serves to distinguish this mechanism from the r-7 transformation esen though the other shear elements are surprisingly similar. 4. DISCUSSIOX 4. I Cr_nfal~oyrcrp/zy of’ rh de$.mrratifx~ rmtles The results presented in Section 3 have revealed five types of deformation behaviour occurring in crystalline mercury at +2K. Three of the deformation modes involved. i1Ii) (iTO}slip, =(i?S)'(i3 I j twinning and : ifO)=( 332)’ kinking. are those which occur at 77 K and have been studied in detail.“’ In common with results at 77K, twinning is confined to orientations close to ( 1IO} and the occurrence of kinking defines an intermediate orientation region where neither twinning nor slip can occur exclusively; this confirms a previous observation’“’ that kinking plays an important role in the deformation of mercury at 4.9 and 77K and can be regarded as a macroscopic deformation mode and not just as an accommodation mechanism. A detailed study of these kinks has led to a gene& analysis of the crystallography of kinks which applies to all crystalline materials.’ Iat However. the present experimental results show that at 41K there exist two more competitive defo~ation modes. One having shear crystallographic elements :Ti3;-( 1IO}, g = G2.4 occurs for a limited range of orientations near I II, and the other with elements IT 13j(~O), g = 047 associated with the z-;’ transformation. is observed for the majority of orientations remote from 11I. The predominant mechanism of deformation for mercury crystals at 42K is by means of undergoing the r-;’ transformation on the ;il3~-(i-iO) shear mode. If we assume for the moment that these rational indices apply to this transformation. then we

can consider the crystallography of this shea.r mode. There are six crystallographically equivalent variants of the iil3i plane. one of which lies in each of the siu unit triangles of the standard 111 projection for r-mercury. However, there are only three variants of the close-packed direction (no>. Thus. each closepacked direction lies in two variants of this plane. If the assumption of rational indices is correct, then the occurrence of pairs of habit planes is readily expfained; the two operative variants are those containing a common (RO) shear direction. This is confirmed in practice when the specific variants of a pair are considered, The lack of any accommodation etTects at the intersections and the well-defined nature of the plane of intersection shown in Fig. 9 indicate the ease of self-accommodation of the pairs of traces, supporting the view that assignment of rational indices to the process is justified. The plane of intersection is then a [ITO:, mirror plane of the parent structure as depicted in Fig. 9. It has been noted that the two variants were never observed to intersect at acute angles: indeed, it appeared that such intersections were actually avoided as is clear in Fig. 7. The reason for this now becomes apparent. Such an intersection involving the a~~ommo~tion of opposite shears would be extremely difficult without the occurrence of the usual accommodation effects as are observed at complementary twin intersections in mercury.“’ The occurrence of the transformation habits in pairs provides a convenient notation scheme for the crystallographic variants; this constitutes an extension of that invoked for the slip. kink and twin modes summarised in Section 1. Thus, the habit planes (113) and (ii31 are denoted (L) and (& respectively, and their common shear direction [ii01 is Iabelled 5 in accordance with the scheme already outlined. Simifariy the pair of planes (13i) and fi31), and the pair (3if) and (31-i) with corresponding shear directions [iOil and [Oii] are represented by the letters (M), (M’), (IV), (N’) and fl, 7. respectively. Modes such as (LB and (CR, which occur as a pair. we term complementary modes. T’he positions of these planes and directions in the standard 111 projection are shown in Fig. t(a) and a summary of the notation is contained in Table 1. The foregoing crystaliograph~ applies equally well to the other deformation mode :i13:( 110) observed at +2 K, provided the assumption of rational indices is again made. It should be noted that these two modes {i13f(‘i_fO) and :fl31( 110) involving essentially the same habit plane but confined to quite distinct parts of the standard triangle are not crystallographically equivalent. This arises from the fact that the (110) direction is not a two-fold axis of the z-mercury structure and thus the two senses of this direction are not crystaliographicdl~y equivalent. The observation of the $13: habit plane in two distinct shear mechanisms involving the same macroscopic shear direction but opposite sense shears and

ABELL:

DEFOR,MATION

OF CRYSTALLINE

different shear strain magnitudes is somewhat surprising. It is now well-established that the mode having shear elements $13:(nO), g = 047 is associated with the r-7 transformation but the origin of the $131( 110). g = O-24 shear mechanism is unknown. A theoretical study of the crystallography of possible twinning modes in the r-mercury structure gave rise to five modes involving no atomic shuffles with shears less than unitv.“’ One of these. involving the $i%,ji’ habit plane is the observed mode at 77K(8.9’ and, in the present study. has been shown to be the operative mode at 4.2 K. Another predicted mode possesses a fi13) composition plane, but the corresponding irrational shear direction ‘( 141)’ and the shear strain magnitude of 0%9 serve to distinguish the observed deformation mode from this possible twin mode. Further. it is unlikely that such a twin mode would operate when the -:Tfj}’ mode, possessing a much smaller shear strain. has been shown to occur on these crystals at 4.2 K. An alternative explanation in terms of a possible association with the r-j transformation is difficult to substantiate. The p-phase of mercury was originally produced only under conditions of high pressure”” although a later discussion of the nature of the transition suggested that certain characteristics could be qualitatively explained in terms of a martensitic behaviour.“” More recently, X-ray diffraction evidence has been presented”” which shows that a shear process can be a significant factor in the production of j-Hg, and that this dependence of shear increases as the temperature is lowered below 79K. the temperature below which fi-Hg is stable. However, the shear process which contributes to the z-j transformation has always been found to be of a compressive nature. In particular, in the X-ray diffraction experiments, tensile stresses on a polycrystalline sample of r-Hg always produced ;:-Hg(“*” and the compressive stresses imparted by surface hammering yielded the #I-phase.(*‘I Operation of the $13) (110) shear mode has been limited in the present tensile tests to orientations near 111, and reference to Fig. 10(c) shows that this shear mode could only occur in compression for a small range of orientations close to 112, where it would be competing with the highly stressed (1 lij( 170) slip mode (see Fig. lb). Thus it seems unlikely that this mode is associated with the r-8 transformation, since the j-phase can be so readily produced at 5 K by the application of a compressive stress: the mechanism involved in this shear process therefore remains unexplained. The possible theoretical shear elements of the x-7 transformation in mercury cannot at present be reported and compared with the experimentally observed elements because the structure of the y-phase is not yet determined. A possible structure for y-mercury has been proposed using the pseudopotential theory of metals(zO) but an attempt to relate the theoretical habit planes predicted by the application of the current theories of martensite crystallo-

Ill

0 1 2 3

MERCURY

111

1

3 112

112

2 1

L

2ii

2ii

Fig. 10. Stereographic plots of Schmid factor contours in units of 10-r for (a) the (il3i(‘iiO) transformation shear mode on the standard 111 stereogram. Negative factors are indicated by bars and correspond to values for the (il3j( 110) mode operative for orientations close to 111. The standard unit triangle has been used to represent the most hJgl-11~ stressed variants in tension of (b) the (i131( 110) transformation mode and (c) the :i13:( 110) deformation mode. Bold lines indicate regions where different variants of these modes operate. For orientations above the b_oundary through [llO] in (b) and below that through (101) in (c) the factors become negative indicating that the mode cannot operate in tension.

graphy to the x-7 transformation, assuming this y-structure, has proved unsuccessful.(J~ *w I) This failure may be due to inadequacies in the theories, but the results of X-ray diffraction experiments” t.li) aimed at determining the structure of the y-phase suggest that a more likely explanation is that the predicted structure is incorrect. Thus. the assi_enment of the rational indices (i13f to the habit plane remains essentially an experimental result. The actual habit plane may be irrational although the pair morphology involving a common shear direction and no accommodation effects at the intersection suggests a crystallographic process. The shear strain magnitude associated with a particular deformation process is a measure of the strains that have to be accommodated in the crystal. The measured value of 047 for the shear strain magnitude associated with the z-7 transformation is remarkably large compared with those for other martensitic transformations. Thus, in order that the transformation be energetically favourable, the atomic mechanism involved is likely to be particularly simple. The fact that the transformation is stress-induced suggests that it might be instructive to compare it with deformation twinning in r-mercury for which the shear strain of @63 is also large. No atomic shuffling following the

18

ABELL:

DEFORMATION

OF CRYSTALLINE

MERCURY

4.2 Orientation dependence of deformation modes

Table 1. Schmid factors (x 100) for the six variants of ;i131(iiO) for crystals lG-38 undergoing the z-y transformation, and of :1131( 110) for crystals l-3 close to 111. Negative indices are indicated by a bar over the number and operative modes are underlined

The present results show that the orientation of a single crystal of mercury has a significant effect on its deformation behaviour at 42K; a similar behaviour has been found at higher temperatures. The orientation dependence of the occurrence of slip, kinking and twinning has been discussed extensively for deformation at 77K.“’ and similar arguments appear appropriate for these modes when operative at 4.2 K and will not be repeated in detail here. Suffice it to say that when illi; slip, (li0) kinking and ‘(i35)’ twinning occur as macroscopic deformation modes, the most highly stressed system is always operative in accordance with the scheme outlined in Section 1. However, when slip and kinks are observed as accommodation modes at twin intersections this criterion does not always apply, the operative variant here depending on which twin mode occurs. Thus, on crystal 4, twins (b) and (b’) occur with accommodation slip on mode (B) although mode (A) is just as highly stressed. The special crystallographic relationship underlying the occurrence of these twin intersections and the associated accommodation effects has been reported in detail elsewhere.“’ A quantitative appraisal of the orientation dependence of the deformation behaviour at 42K can be found by regarding the $13)(iiO) transformation shear process as a conventional deformation mode for r-mercury at this temperature. The behaviour can then be analysed in terms of the relative Schmid factors of the various possible modes for any particular orientation. This can best be done by comparing the S&mid factor contour plots for the various competing deformation modes. Such an analysis has been successfully applied to the orientation dependence of the operative modes at 77K as described in Section 1. The most highly stressed slip and twinning systems in tension have been illustrated in Fig. 1, and similar plots can be constructed for the two extra modes (il3}(iiO) and {i13)(110) encountered at 4.2K. Since the crystallographic elements of these two modes are the same apart from the sense of the shear direction involved, the contour plots can be derived from the same stereographic plot shown in Fig. 10(a), care being taken to determine the sign of the Schmid factor in each case. The most highly stressed variants of the two modes for orientations within the standard triangle are shown in Figs. 10(b) and (c), only variants with positive values, corresponding to the present tensile tests, being considered in each case. For the transformation shear mode, variant (L)Z is the most highly stressed for the majority of orientations, with variant (N’fi predominant in the lower left-hand comer. The boundary through [llO] shows where the factors reduce to zero and above this level they become negative indicating that this mode cannot operate in tension. In the case of the (i13)( 110) mode, variant

(Ad)/3is the most highly stressed for all orientations where its operation in tension is allowed. For the crystals undergoing the z-7 transformation, the predominant associated (il3;(flO) mode is, in all cases, the one with the largest positive factor according to Fig. IO(b). This observation provides striking confirmation of the validity of the (]13i(iiO) shear process. Since many of these crystals exhibited more than one set of $13) traces, it is instructive to examine the Schmid factors of all the possible variants for the fifteen crystals tested. It can be seen from Table 4 that for each crystal. four or five and in some cases six variants have favourable factors for operation in tension. As indicated in the table, many of these possible variants were in fact observed. As we have already seen the transformation habits frequently occur in pairs and the second of the habits operative in this way is determined by the common shear direction even where the resolved shear stress on this second system is unfavourable. For example, in crystal 20 the second system (N) occurs in preference to (L) which has a high S value because (N’)? having the highest S factor, is the predominant mode and (N) and (N’) possess a common shear direction 7. Closer examination of the Schmid factors for crystal 28 proves particularly interesting in connection with the consequences of this special crystallographic relationship of two paired variants. Modes (N) and (N’) occur as might be expected. However, the pair of habits (L) and (C) are also operative even though another mode (&f’) is over five times more highly stressed than (C). A possible reason for this can be found by noting that the Schmid factor for mode (M), the complementary mode to (W), is in fact negative and thus cannot be operative in tension. This fact, therefore, appears to prohibit not only the operation of mode (M) but also of the complementary mode (M’) although this is favourably stressed. This implies that the pair morphology facilitates the accommodation of the plates of 7-Hg more readily than the occurrence of a single variant, and consequently if a pair is prevented from forming. as in this case, another pair may be operative even though one of the pair is relatively lowly stressed

twinning shear is necessary, and the same seems likely to apply to the z-7 transformation.

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Fig. 11. Comparison of Schmid factor contours for competing deformation modes in tension at 4.2K. The bold boundaries separating regions where different modes operate have been constructed as_suming that the critical resolved shear stress on the (11_31(110) transformation mode is half that on the (113( 110) slip mode, which is itself a quarter that on the [ 113:( 110) mode. The most highly stressed variants are labelled according to the notation of Table 1. The orientations of the crystal axes and the observed modes have been represented by the symbois described in Fig. 2.

As might be expected crystals 29 and 30 both deformed by (1 li) slip, although favourably orientated for the z-‘J transformation. These two crystals were tested at 55K in contrast to 492K for the rest of the crystals. and their behaviour indicates that the transformation only occurs at temperatures below 55 K. The observed reversion temperature for the transformation derived from the resistance measurements is consistent with this result (see Table 2). It is clear from this discussion that the assumption that the transfo~ation shear process can be regarded as a conventional deformation mode for mercury at low temperatures is justified. A similar argument can be applied to crystals 1, Z and 3 exhibiting the {il3}(110) deformation mode. The Schmid factors for all six variants of this mode for these crystals are shown in Table 4. The values are all positive and high and, in each case, the predominant operative mode is the most highly stressed variant illustrated in Fig. lo(c); this resuit provides confirmation of the observed sense of the ( 110) operative shear direction in this region. Reference to Fig. lqb) shows that the (ii0) shear direction is not allowed for crystaIs of this orientation tested in tension. Although the transformation shear mode is highly favoured in the lower regions of the standard triangle, [11-i!{ 170) slip is the predominant mode for orientations in the middle of the triangle. The Schmid factors for [i131(ii’O) decrease rapidly in this region (see Fig. lob) and by comparing these contours with those for slip in Fig. l(b), we can determine quantitatively the boundary separating the occurrence of these modes. If we consider that the resolved shear stress on $13: (no) has only to be half that on [ 112 ( 110) for the former to operate, then by comparing the corresponding Schmid factors in the same ratio we obtain Fig. 11. The boundary between modes A and

19

(Lb corresponds almost exactly to that found by experiment. Crystals close to the boundary exhibit both deformation modes and the boundary indicates the relative predominance of the two modes. If the operation of the iil3r(iiO) mode in single crystals is controlled by the conventional Schmid and Boas criterion. then this result indicates that the critical resolved shear stress on this mode is approx~ately half that for { 1li: ( li0) slip at 4*2K. For orientations near [I 10-Jthis boundary becomes meaningless since the S&mid factors for both modes are low and crystals deform predominantly by ‘;i%; ‘ twinning. Treating (i131(110) in a similar manner, we have to assume that the critical resolved shear stress on this mode is four times that on the slip mode for a reasonable approximation to the experimental boundary to be achieved; the result is shown in Fig. 11. Stress measurements were not made in the present experiments but it uas observed that these three crystals were particularly hard on testing which supports the above analysis. Crystals with orientations near [ 1111 are very rare. and when they do occur, their deformation characteristics are often found to be unusual. In particular. non-crystallographic slip has been observed at 77K during bend tests on such specimens :(9bin the present tests, slip only occurred on crystal 3 and then it was on the most highly stressed Illi){ li0) mode together with a few twins on the most favoured *$?ii’ system but the unusual behaviour of Uystals with such orientations is amply demonstrated by rhe occurrence of the (i13:( 110) mode. The Schmid factor analysis thus offers a possible explanation for the occurrence of the observed deformation modes at &2K for particular orientations within the unit triangle. An alternative approach involves the application of anisotropic elasticity theory to determine the shear moduli associated with various deformation modes. The results are only strictly applicable to small strains and are probably best interpreted in terms of ease of elastic accommodation of the sheared regions in the matrix. Such an analysis has shown that mercury is an exceptionally anisotropit metal. the masimum and minimum shear moduii differing by a factor of 7 using compliance constants and 11 using stiffnesses.‘“’ Calculation of the shear moduli associated with various possible slip and twinning modes for the z-mercury structure showed that the values for the experimentally observed modes were consistent with their operation. The values for the (i13) (iiio) transformation shear mode were also given and were comparable with those for the slip and twinning modes considered. In particular, the values for the (il3)( 1lo} mode, occurring for orientations ilose to I1 1. are considerably lower than those for a possible competitive slip mode involving the ctose-packed (011) shear direction. which has been observed for these orientations at higher temperatures.‘3’ Thus, while these results do not show convincingly why this mode shoutd operate. they do

20

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indicate that shearing on this mode can be readily accommodated elastically in the adjacent material. Although a full realisation of why a particular habit plane and macroscopic shear direction occur in connection with a martensitic transformation cannot be found without considering the nature of the product crystal structure and the crystallographic relationship it has with the parent, it has proved extremely vaiuable in this case to regard these shear elements as a conventionai deformation mode and to compare it with other possible modes. This is particularly valid in this case, of course, since the transformation is stress-induced and, in single crystals, the geometry of the crystallographic planes with respect to the applied stress is a controlling factor in the production of the pphase. Acknowledgements-The author is greatly indebted to Dr. A. G. Cracker for his guidance and encouragement throughout this work. Financial assistance from the University of Surrey is gratefully acknowled~d.

REFERENCES I. Bacon D. J., Heckscher F. and Cracker A. G.. Acra

Cryst. 17, 760 (1964). 2. Cracker A. G., Heckscher F. and Bevis M., Phil. Mag. 8, 1863 (1963). 3. Rider J. G. and Heckscher F., Phil. Msg. 13, 687 (1966).

MERCURY

4. Abel1 J. S. and Cracker A. G.. Scripta .Cfer. 2. 419 (1968). 5. Abel1 J. S.. Cracker A. G. and Guyoncourt D. IM. M.. J. &Zar. Sci. 6. 361 (1971). 6. Cracker A. G. and Aytas’I.. Reports of Second International Conference on the Strength of Metals and Alloys (American Society of IMetals, Cleveland 1970). 7. Cracker A. G., Heckscher F.. Bevis M. and Guyoncourt D. M. M.. Phil. &fug. 13. 1191 (1966). 8. Guyoncourt D. IM. M. and Cracker A. G.. Acta Jb!et. 16, 523 (1968). 9. Guyoncourt D. M. M. and Cracker A. G.. Acru iVet. 18, 805 (1970). 10. Barrett C, S.. Acfa Crysfal/ogr 10, 5s (19571. I I. Abeli J. S.. Cracker A. G. and King H. W.. Phil. 1Vfag. 21, 207 (1970). 12. Swenson C. A., Phys. Rev. 111, 82 (1958). 13. Atoji M., Shirber J. E. and Swenson C. A.. J. Chem. Phys. 31, 1628 (1959). 14. Doidge P. R. and Eastham A. R., Phil. .Vag. 18, 655 (1968). 15. Abell J. S. and Cracker A. G., 7% &fechanism of Phase Transformations in Crystalline Solids. p. 192. Institute of Metals, London (1969). 16. Abell J. S. and King H. W., Cryogenics 10. 119 (1970). 17. Abe11J. S. and Barrett C. S.. iMer. Trans. 1. 2647 (1970). 18. Cracker A. G.. Abeli J. S. and Flewitt P. E. J., to be published. 19. Schirber J. E. and Swenson C. A., Acta Xer. 10, 511 (1962). 20. Weaire D.. Phil. Mug. 18, 213 (1968). 21. Ross N. D. H. and Abel1 J. S., Ph_u. Starus Solidi (A) I. K33 (1970). 22. Cracker A. G. and Singleton G. A. A. M.. Phys. Status Solidi (A) 6. 635 (1971).