An electron microscope study of plastic deformation in single crystals of synthetic quartz

Tecfono~hysics, 33 (1976) 43-79 @ Elsevier Scientific Publishing Company, Amsterdam - Printed in The Nethertands

AN ELECTRON MICROSCOPE STUDY OF PLASTIC DEFORMATION SINGLE CRYSTALS OF SYNTHETIC QUARTZ

D.J. MORRISON-SMlTH

l, M.S. PATERSON

Research School &Earth (Australia)

Sciences, Australian National Uniuersity, Canberra, A.C.T.

(Submitted

43

IN

and B.E. HOBBS *

May 27, 1975; revised version accepted March 23, 1976)

ABSTRACT Morrison-Smith, D.J., Paterson, M.S. and Hobbs, B.E., 1976. An electron microscope study of plastic deformation in single crystals of synthetic quartz. Tectonophysics, 33: 43-79. Specimens cut from a single synthetic quartz crystal and oriented normal to r (IlOl) were deformed in a gas apparatus at 3 kbar confining pressure and temperatures from 475” to 900°C. There was a large variation in stress-strain behaviour between specimens, which could be correlated with inhomogeneity in OH concentration as revealed by infrared absorption. This inhomogeneity was also revealed by the optical microscope as a banding in the distribution of strain parallel to the (0001) growth suriaces. Other optical features in the specimens were deformation bands parallel to the (1010) prism planes, undulatory extinction and, on a finer scale, optical lamellae of several orientations, especially prismatic and rhombohedral. Detailed transmission electron microscope observations were made at each temperature after various amounts of strain. The dislocation structures revealed can be broadly grouped into those produced up to 550°C which are suggestive of relatively low temperature behaviour, and those above 550” C which indicate that climb and other high-temperature recovery processes are active. A notable feature in the low temperature range is the generation of clusters of dislocation loops around submicroscopic inclusions already present in the initial material: this effect provides an alternative to the multiplication of grown-in dislocations and gives the equivalent of a much higher initial dislocation density than would the grown-in dislocations alone. Other low-temperature features include intense banding of tangles of dislocations, some of which can be associated with optical lamellae. At higher temperatures, the dislocations become more curved and intertwined and form many small, isolated loops and dipoles and occasional networks: also, bubbles appear above 800°C. At the lower temperatures, dislocation densities of the order of lo9 cm+ are soon reached in the yield region, rising to a maximum of over 1O’O crnv2 at larger strains; the maximum densities reached at high temperatures are one to two orders of magnitude less. The most active slip planes appear to be m {lOTO} and z {lOil}, with c 10001 1 and a (2fiO) slip directions, respectively. In discussion, constraints that the electron microscope observations can place on ’ Now at: Wurstpierpoint College, Hurstpierpoint, Sussex, England. 2 Now at: Department of Earth Sciences, Monash University, Clayton, Victoria, Australia.

44

microdynamical theories of quartz behaviour are considered, especiatly in regard to muftiplication laws, effective initial dislocation densities and strain hardening. The need for homogeneous crystals in future experiments is emphasized.

INTRODUCTION

There have been a number of studies of the plastic deformation of quartz since Griggs and co-workers demonstrated that natural quartz could be deformed appreciably in the laboratory (Carter et al., 1961, 1964). At first, stress-strain curves were not obtained and attention was directed mainly to optical features resulting from the deformation, especially deformation lamellae, in a quest for information about active slip systems. Later, through use of an improved solid pressure medium apparatus (Griggs and Blacic, 1964), stress-strain curves became available and the phenomenon of hydrolytic weakening was discovered (Griggs and Blacic, 1965; Griggs, 1967). In the initial experiments on hydrolytic weakening, OH (or, at least, hydrogen) was introduced into the dry, natural quartz from the dehydrating talc pressure medium but most later experiments have been performed on synthetic quartz crystals containing known amounts of grown-in OH, as detested by infra-red absorption spectroscopy. The synthetic crystals have generally shown marked heterogeneity in deformation, which has been attributed to heterogeneity in the distribution of the grown-in OH in the crystals (Blacic, 1971; 1975), a heterogeneity that is also indicated by optical observations (Dodd and Fraser, 1967). Nevertheless, synthetic crystals have still been used in the more recent studies on quartz deformation. These studies have involved two important developments, namely, the use of gas-medium deformation apparatus to give more precise stress-train measurements and the use of the electron microscope to observe the dislocation substructures. The first two of these recent single crystal studies were those of Baeta and Ashbee (1970a,b), performed at atmospheric pressure, and those of Hobbs et al. (1972), done at 3 kbar confining pressure. From the observation of pronounced yield point drops similar to those observed in germanium and some ionic crystals, the latter authors suggested that a microdynamical theory of deformation, such as that of Haasen (1964), might be appropriately used to describe the behaviour of quartz, and such a theory has since been developed by Griggs (1974) using particular assumptions about the role of OH. Both the recent experiments of Balder-man (1974) and the present work stem, in part, from the need for further precise stress-strain data for the microdynamical theory. However, since additional constraints can be placed on the theory by detailed observations on the dislocation substructures in the electron microscope, and since these observations are in any case essential for a fuller understanding of the mechanisms of deformation, a primary aim of the present work has been to explore these structural aspects in detail.

45

The application of the transmission electron microscope to the study of minerals became feasible with the development of suitable procedures for specimen preparation. The first studies on quartz depended on crushed fragments (McLaren and Phakey, 1965a, 1966a, b) but in later work ion thinning techniques allowed much closer control of specimen orientation and observable area. The first major study of experimentally deformed quartz was made by McLaren et al. (1967) on the natural quartz specimens deformed by Carter et al. (1964); but later observations by McLaren and Retchford (1969) were made on synthetic quartz specimens deformed by Hobbs (1968). The latter specimens showed very high densities of tangled dislocations, even at very small strains, suggesting that there is a large multiplication rate and that the individual dislocations do not move very far; at higher temperatures the dislocation densities were lower and there was less tangling. In subsequent studies, similar observations have been made by others, and effects due to recovery and the submicroscopic nature of deformation lamellae have been explored in some detail (Ardell et al., 1973; Baeta and Ashbee, 1973; White, 1973; Christie and Ardell, 1974; Twiss, 1974). The present study set out to explore in greater detail than hitherto the progressive change in stress-strain behaviour and dislocation substructure with increase in tempemture, restricting attention to one orientation of one synthetic crystal in order to ensure comparability of other variables. The aim has been to gain further understanding of the mechanisms of deformation and to provide further constraints on the theory of the defo~ation. However, the stress-strain results are, as in past studies, affected by the usual heterogeneity of a synthetic crystal and special attention, therefore, has also been given to correlating the heterogeneity in deformation with the banding in OH content. EXPERIMENTAL

DETAILS

All specimens were obtained from a single crystal of synthetic quartz, designated W2, supplied by Dr. D.W. Rudd of Western Electric. The main impurity contents, in ppm atomic ratio, were 800 H/Si and 15’7 Na/Si (Hobbs et al., 1972; note the reduction by a factor of 2 in H/Si following revision of the value of the standard crystal Xo by S.H. Kirby, personal communication, 1975). The specimens were cored normal to an exposed r {%lOl} face at one end of the crystal and then from successive layers cut parallel to this surface. Cores were taken in two rows on either side of the seed in each layer in such a way as to exclude the seed and the X-growth regions. The seed Iies approximately 5” off the (0001) plane. Experiments were performed using the gas apparatus designed by Paterson (1970), with conditions similar to those described by Hobbs et al. (19721, that is, 3 kbar confining pressure and a strain rate of approximately lo-’ set-’ over a temperature range from 475 to 900°C. Following Hobbs (1968), all specimens were preheated for 1% hours at 600°C and 3 kbar con-

36

fining pressure, although it is now clear that this does not homogenize the OH content within specimen. The program involved deforming several specimens by different amounts at each temperature in an attempt to gain as much information as possible on deformation processes through the various stages of the stress-strain curve. After the experiments, each specimen was sectioned ~ong~tudin~~y parallel to a fll!?O) plane and a standard 30~ optical thin section was made parallel to this plane, as well as an additional thin section for electron microscope study. A projection of important planes and directions on this plane is given in Fig. 1 for convenient reference. The electron microscope specimens were first studied under the optical microscope and brass rings (3 mm o.d., 2.6 mm i.d,, 0.5 mm thick) were glued around interesting areas. These ~ng-supposed areas were then sepa-

~(TIoI,

0

POLES

X

DIRECTIONS

e

POLES TO PLANES = DIRECTIONS

TO PLANES

Fig. 1. Stereographic projection illustrating the relationship between important planes and directions and the direction of applied stress (shown by the vertical arrows), The indexing scheme shown here is used throughout this paper.

47

rated and thinned using an ion-bombardment thinner of the type described by Gillespie et al. (1971). Most of the electron microscopy was performed with an Hitachi HUBOOF microscope in the Research School of Biological Sciences, Australian National University, while later observations were performed with a J.E.O.L. JEM 200 microscope in the Physics Department of Monash University. Both microscopes were generally operated at 200 kV. In neither case was a highangle tilting stage available at the time of this work, so tilts were limited to *lo”. STRESS-STRAIN

RESULTS

Stress-strain curves were calculated from load and displacement measurements as described by Hobbs et al. (1972). Likely errors in the plotted stress-strain curves are similar to those quoted by these authors, and are marked on one of the curves plotted in Fig. 2. The basic features of the curves for most specimens (Figs. 2 and 3) resemble those described by Hobbs et al., and their generalized picture of a stressstrain curve appears again applicable. The yield stress (or upper yield stress where there is a yield drop) decreases approximately exponentially with increasing temperature. Yield point drops are restricted to specimens deformed

c 5

IO

15

20

STRAIN %

Fig. 2. Stress-strain curves for a series of specimens containing narrow deformation bands (see text and Fig. 5b). These specimens consistently appear softer and more frequently exhibit yield drops than specimens containing broad deformation bands. Error bars marked on the curve for 575°C are expected to be typical for all the stress-train curves plotted in this and subsequent figures.

48

between 550 and 700°C. The extent of the region of low work hardening after yielding varies from being effectively absent at 475°C to covering much of the curve at high temperatures. The rate of work hardening in this stage decreases with increasing temperature, the rate of decrease being much more rapid between 475 and 700°C than above 700°C. Following this region of low work hardening, there is commonly a region of increased work hardening, the gradient of which decreases with increasing temperature. Occasionally a later stage with an apparently decreasing rate of work hardening is observed which appears to be associated with failure of the specimen. A notable feature is the marked variation in behaviour of specimens deformed under apparently identical conditions, as shown by the two groups of results plotted separately in Figs. 2 and 3. Figure 4 shows a series of experiments performed at 550°C where there is also a distinct arrangement of the curves into two groups, one significantly harder than the other. This variation in strength is found to be approximately constant throughout the temperature range - the hardest specimens supporting approximately twice the stress supported by the softest for a given strain within the plastic region. The explanation of this variation in terms of heterogeneity of OH concentration is discussed in the next section.

STRAIN

%

STRAIN

%

Fig. 3. Stress-train curves for a series of specimens containing broad deformation bands (see text and Fig. 5a). These specimens are generally harder, shows higher rates of work hardening and fails at lower strains than those of Figure 2. Fig. 4. Stress-strain curves for a complete range of experiments at 550°C. Note how the curves tend to fall into two groups. The harder specimens (2319 and 2322) show broad main deformation bands, while the softer specimens (2320, 2321, 2323) contain narrow main deformation bands.

49

OPTICAL

MICROSCOPE

AND INFRARED

OBSERVATIONS

The main optical features are deformation bands, undulatory extinction and deformation lamellae, similar in appearance to those in naturally and experimentally deformed natural quartz. Their characteristics are as follows: (1) Basal deformation bands (Figs. 5a and b). These are sharply defined bands of concentrated deformation approximately parallel to the basal plane. Some are nearly 1 mm wide, others only about 0.1 mm, the width not varying significantly with strain or temperature. They are normally straight or, at most, only slightly curved, although very occasionally the boundary may be stepped; the boundaries of the narrower bands are generally slightly wavy. The bands appear to account for high proportions of the strain in the specimens since they show large rotations of the lattice (up to about 40”) and of the specimen edge (up to 20”). They cannot be regarded as simple kink bands for, despite their sharp boundaries, there is a continuous curvature of the lattice across the band. (2) Prismatic deformation bands (Fig. 5~). These lie approximately parallel to the (liO0) prism planes and are subsidiary to the basal deformation bands and usually more irregularly shaped. They appear more akin to areas of undulatory extinction except for a marked and complex fine structure. The lattice rotations are in the opposite sense to those in the basal bands and are generally relatively small (
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By noting that the undeformed bands characteristically occurred either at a corner or across the central area of a specimen and recalling that the specimens were cored from two different levels in the crystal, it is possible to r-e-establish the relative position of all cores in the crystal in a way that is consistent with the undeformed bands being a pair of adjacent layers of abnormally low OH content parallel to the growing surface. This reconstruction in turn suggested that the basai defo~ation bands represent aboveaverage fluctuations in OH content parallel to the growing surface, an interpretation that is consistent both with the present microscope observations and with the evidence reported in the literature that OH banding parallel to growth surfaces is common in synthetic quartz crystals (Dodd and Fraser, 1967), and which would explain the orientation of the deformation banding

Fig. 5. Optical micrographs under crossed polarizers of specimens compressed parallel to their long axes: (a) 2319 {OS% strain at 550°C) showing broad basal deformation band A-A, prismatic deformation band B-B and some undulatory extinction. (b) 2318 (1.2% strain at 525’C) showing narrow basal deformation band C-C and wavy undeformed bands D-D.

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(c) 2302 (0.6% strain at 500°C) showing offset in broad main band and two irregular prismatic bands E-E. (d) 2389 (7.0% strain at 800°C) with pair of wavy undeformed bands F-F still evident at this temperature and showing development of “chequerboard” pattern of undulatory extinction. Scale mark represents 1 mm.

usually observed (Hobbs et al., 1966; Hobbs, 1968; BaBta and Ashbee, 1969a; Blacic, 1971,1975; Balder-man, 1974). There is, therefore, a strong suggestion that the basal deformation features represent local variations in purity of the crystal and are not related to the experimental conditions. In order to confirm the presumption of OH banding, an infrared absorption study was made, following Brunner et al. (1959,196l) and Dodd and Fraser (1967). The measurements were made at room temperature with a Perkin Elmer 180 infra-red spectrometer by Dr. J.T. Gourley of the Department of Solid State Physics, Australian National University. The area of the broad absorption peak lying within the 2000-4000 cm-’ region was taken as being proportional to OH content, but since the width of the peak was nearly the same for all observations, the maximum height of the peak could also be taken as the measure of OH content. Measurements were made on a

Fig. 6. Area of specimen 2320 (550°C) bands. (a) Crossed polarizers. (b) Plane polarized light; note lamellae, Scale mark represents 0.1 mm.

showing

nearly

pair of wavy, apparently

horizontal

undeformed

in this photograph.

10 mm thick slab of the crystal, cut normal to the seed, with the infra-red beam parallel to the seed and therefore to the growing face in the region from which the specimens had been taken. The percentage absorption was obtained from areas approximately 3 mm X 0.5 mm at 0.5 mm intervals across the slab. The results are plotted against distance from the seed for one half of the crystal in Fig. 7, which can therefore be taken as a plot of OH content versus position. Absolute values of OH content cannot be given because the absorption was not calibrated and, although the figure of 800 H atoms per lo6 Si atoms quoted above for this crystal should represent more or less an average value because of the large beam used in the previous measurement, the exact location of that measurement is not known. Also represented in Fig. 7 are the positions of the optical basal bands on the same distance scale. The results given in Fig. 7 confirm the deductions given above. Although the slit width of 0.5 mm limits the resolution, the main optical basal features clearly mirror the variation in OH concentration, high and low OH content correlating with high and low amounts of deformation, respectively. The measurement in the widest band is not affected by lack of resolution from

53 RELATIVE 0.

20

(OH) ABSORPTION(%) POSITIONS OF CORES AND OPTICAL FEATURES q0

q3

Bp I NARROW MAIN BAND

Fig. 7. Diagram showing the relationship between OH distribution and optical deformation bands. ‘Ike left-hand side of the figure shows a plot of relative infrared absorption against distance from the seed for one half of the crystal. Relative absorption is approximately proportional to the relative OH content. Plotted next to this is a diagram of the average positions of cores with respect to the seed, measured from the cored slabs, and the positions of the main deformation features, measured from optical micrographs. It can be seen that there is good correlation between the positions of these bands and the local variations in OH content.

the slit width and indicates an OH content approximately 50% higher than in most of the specimens. A rough attempt to correct for broadening due to breadth of slit, using “unfolding” procedure, suggests a similar OH concentration in the main narrow basal deformation band and at least a 25% relative depletion in OH content in the undeformed bands. It may therefore be concluded that the major basal banding in degree of deformation in the deformed specimens is clearly correlated with fluctuations in OH concentration (cf. Blacic, 1971,1975). A one-toone correspondence is now obvious between the pattern of OH banding and the stress-strain behaviour. The specimens with the prominent narrow basal deformation bands (inner row of specimens, Fig. 7) are those giving the softer stress-strain curves shown in Fig. 2 in which the yield point drops are more prominent, while the harder specimens (Fig. 3) correspond to outer layer specimens cont~ning broad basal bands. The relative weakness of the specimens from the inner row is also correlated with a greater tendency for areas of undulatory extinction to develop outside the deformation bands at the higher temperatures. The prismatic deformation bands, which are larger and more common at higher temperatures, tend to be more common in specimens that show yield point drops and a more nearly uniform distribution of undulatory extinction but their occurrence is not strongly correlated with the two classes of specimens just distinguished. The undulatory extinction also tends to be

54

distributed more nearly uniformly at higher temperatures, and above about 800°C the effects of OH banding tend to break down, although they are still discernible at 900” C. ELECTRON

Starting

MICROSCOPE

OBSERVATIONS

material

The average grown-in dislocation content of the specimens has previously been shown by X-ray topography to be approximately lo3 cm-* (Hobbs et al., 1972). Additional observations have now been made by electron microscopy of the defect structures in specimens at three preparatory stages prior to the actual deformation: (1) as cored; (2) after subjecting to 3 kbar hydrostatic pressure at room temperature; (3) after the normal preheat treatment at 6OO”C, 3 kbar confining pressure for 90 minutes. In all three cases no dislocations were observed by electron microscopy. Since the lower observable density is approximately lo6 cm-*, any effect of specimen preparation must have left the dislocation density below this level. While in many areas no features were observed at all, most specimens contained some areas of contrast features such as those shown in Fig. 8. Such features would probably not be resolved by X-ray topography. They normally occur as small complicated images, approximately 200 to 2000 R in diameter, which are similar to those produced by dislocation loops or inclusions. Most show symmetrical images, although a small proportion are markedly asymmetric. An important feature of these images is their wide variation in structure, not only for different diffracting conditions but also within the same area as can be seen in Fig. 8. Some of this complexity of image structure could be ascribed to the fact that they were not usually imaged under strict two-beam conditions, the best that can normally be achieved being a row of systematics. Another major feature is the occurrence and variation of a line of no contrast. Some images appear symmetric with no line of no contrast, others show it perpendicular to g and others occasionally parallel to [liOO] but the majority of images show a line of no contrast that is parallel to [OOOl] regardless of operating reflection. It has not been possible to put these features completely out of contrast. The small features just described are observed to be present at all stages prior to and during the deformation process, up to the highest temperatures. There is little variation in their nature throughout the range of experimental Fig. 8. Electron tions. (a)-(c) almost (d) g = portion

micrographs

of inclusions

in undeformed

specimens

under various

condi-

g = (ilO1) systematic row, with large deviation from Bragg condition in (a), at exact Bragg condition in (b) and small deviation from Bragg condition in (c). (1100) systematic row, large deviation from exact Bragg condition. Note high proof images show lines of no contrast parallel to [OOOl].

d

56

conditions. Their density, however, varies sharply with position; they are either absent, or occur in regions containing between 1Or2 and 1Ol4 cme3, with little apparent gradation in density between these two extremes. (The measured densities were not corrected for overlap of images as suggested by Hirsch et al. (1965), but, with the densities observed here, miscounting due to overlap would not appear to be a serious problem; also only those features are included which are larger than the electron beam damage centres, that is, greater than about 200.4, and which can be readily distinguished from them.) The characteristics of these images are best explained in terms of the stress fields round inclusions, where both the particle shape and the elastic properties of the matrix exhibit less than spherical symmetry (see, for example, Ashby and Brown, 1963a and b; Hirsch et al., 1965; Yoffe, 1970; Degischer, 1972). The general complexity of the images precludes a detailed analysis, but their observed properties do not appear to be compatible with dislocation loops, bubbles, or ion or electron beam damage centres. It has been suggested that the sodium impurity in synthetic quartz crystals is present in the form of acmite (NaFeSi~O~) which is trapped during the growth process (B. Sawyer, Quality in Cultured Quartz, Sawyer Research Products Inc., Eastlake, Ohio). Normally crystals containing observable inclusions are rejected, but inclusions on the scale proposed here would not be detectable by normal inspection procedures. Using the measured Na content (no analyses were made for Fe) to calculate the approximate volume of acmite that could be produced, the observed average density of inclusions would be produced if their linear dimensions were a few hundred ~gstroms. Considering the approximations in the calculations and the uncertainty as to the actual particle sizes, there would appear to be sufficient sodium available to produce the observed density of inclusions. It is, therefore, assumed that these contrast features arise from impurity inclusions, possibly of acmite, and that their distribution is a function of the growth conditions. The assumption may also explain the variety of forms of these images, in that the inclusions could precipitate in a variety of orientations and habits although, of course, there must be some preferred orientation to produce the observed concentrations of lines of no contrast approximately parallel to [OOOl] . Specimens

deformed

at 4 75-6OO”C

The chief features of these specimens are: (1) marked heterogeneity of deformation, with very rapid changes in dislocation densities over short distances; (2) dislocation clusters; (3) elongate dislocation loops; and (4) largescale features, especially bands of high dislocation density and dislocation tangling. The principal featuxs observed in areas containing low average dislocation densities are clusters of dislocation loops up to 0.51.1in diameter, as shown in

Fig. 9. Areas of specimen 2288 (deformed 0.3% at 475” C). Type-A dislocation clusters are the simplest type observed, type-B appear to consist of relatively simple arrays of dislocation loops but are rare and type-C are more complicated but much more common than either of the other types.

Fig. 9. These clusters are most frequently observed in regions containing high densities of inclusions and commonly contain a dark central core. A good example of such a feature is shown in the cluster marked A in Fig. 9a, where it is apparent that the outer loop is connected to this central feature by a closely spaced pair of dislocations. These type-A clusters are the simplest form observed. The type marked B (Fig. 9b) are similar but the connection between the loops and any core is not so obvious. However, these simpler types are not as common as the type marked C in Fig. 9a, which appear to consist of closely spaced pairs of central loops, with smaller elongated subsidiary loops apparently growing from them. The area between clusters generally contains negligibly few dislocations. However, because of the high density within clusters, the average dislocation density is frequently as high as lo9 cm -2. Major increases in dislocation density generally appear to be achieved by the growth and generation of loops within clusters. As the loops grow they tend to develop crystallographically straight sides which permit easy identification of the loop plane. The majority of loops encountered lie in (01x1) planes (Fig. 10a) while smaller numbers are in (0110) (Fig. lob) or (i2iO) planes (Fig. 10~). Generally these loops appear to grow anisotropically, so that extensive loop growth tends to lead to the development of bands of loops elongated

loops appear to be lying in the (0111) plane with straight Fig. 10. (a) Large cluster of dislocation loops The regular straight-sided _ edges parallel to [1123][2110] and [2113] directions. Specimen 2287 (0.1% strain at 475°C). (b) Boundary between deformed (bottom right) and virtually undeformed regions (top left). Deformation apparently spreading by growth of regular straight-sided loops into the undeformed region. Loops appear to be lying in (0170) plane with long edges parallel to [OOOl] direction. The fringe pattern is associated with a crack. Specimen 2275 (0.5% strain at 575°C). (c) Areas of straight dislocation segments forming large loops. The loops appear to lie in the (2jiO) pl anr with long edges parallel to ~ [OOOl], [Olll] and (Olll] directions. Specimen 2335 (0.5% strain at 575°C).

(0)

b)

Fig. 11. (a) High density of small clusters which have grown until they begin to coalesce. Associated with the growth of these clusters is the development of narrow bands of dislocations lying in (1100) planes. Specimen 2304 (0.25% strain at 500°C). (b) (liO0) section showing dislocations lying in that plane. Most segments are straight and parallel to [1123], [1120] and [1123] directions. They appear to be parts of loops lying in (l’iO0) planes. Specimen 2356 (2.5% strain at 600°C).

parallel to (Olil) due to the predominance of loops in clusters on these planes. In addition, there are other bands of dislocation loops, not obviously developed from loops in clusters, which frequently form parallel to a (1300) plane (Fig. 1 la) and contain straight-sided loops also lying in (IiOO) planes (Fig. llb). With increasing dislocation density or increasing temperature, interaction between dislocations from these different sources and loop systems becomes important, and there is extensive tangling of dislocations (Fig. 12). In these regions of tangled dislocations, the density tends to greater uniformity, the most obvious systematic variations in density being associated with optical lamellae. It appears that the optical lamellae correspond to bands of tangled dislocations of rather higher density than in the regions through which they pass (c.f. McLaren et al., 1970). Typical dislocation densities within the lamellae vary between 5 - lo9 and 10” cmS2, while the density of surrounding dislocations is restricted to the range 2-5 - lo9 cmS2. Lamellae were ob-

Fig. 12. Various distrjbutions of dislocation loops in bands. (a) Loops lying in (IlOO) planes, such that there is an almost uniform density of disiocations across the field of view. Specimen 2305 (0.5% strain at SOO°C). (b) Narrow, very high density band2 of (1300) toops, separated by regions containing negtigibly few (1x00) loops, and intersecting rather poorly defined bands of (0111) loops. Spe$nen 2320 (2% strain at 55O_“C). (c) Boundary between regions containing only (1100) dislocation loops or (0111) loops. The boundary is quite sharp and runs approx imately parallel to the (IiOO) plane, There is marked banding of the loops in the two regions. Specimen 2305 (0.5”+ strain UC(100” (‘I

61

served only in areas containing tangled dislocations range, and only in areas of multiple slip.

in the above density

Specimens deformed at 600--9OO*C Above 550”, features begin to appear that are suggestive of thermally activated processes such as climb of dislocations. Thus in the temperature range from 550 to 650°C there is a gradual transition to structures typical of high temperatures, although the transition is not wholly restricted to this temperature range since some features typical of low temperatures (e.g., straight-sided loops) can be found in proximity to networks at higher temperatures, and inhomogeneity persists in some degree up to 900°C. However, the small dislocation clusters described previously are observed only up to 600°C. In the range of temperatures between 550 and 650”) most specimens still show structures similar to those typical of low temperatures. However, the bands and clusters of dislocations are generally of rather lower density and are not so well defined. In addition, tangling commences at a significantly lower dislocation density than in specimens deformed below 550” C. Typical high-temperature structures consist of generally curved and tangled dislocations which show only occasional straight segments or other similarities to low-temperature loop structures (see Figs. 13 and 14). Occasionally areas of dislocation networks are observed, although they are not very extensive at any temperature, being observed first in small patches at 550°C and only slightly more frequently at 900°C. The structures of the networks are basically either square or hexagonal but, in practically all cases, the shapes of the constituent dislocations are not regular or straight, indicating that the component dislocations have not reached equilibrium positions. One of the most common features of high-temperature structures is the presence of large numbers of small loops (see Fig. 13) ranging in size from 200 to 5000 A. The size range is fairly wide in any given specimen, the distribution moving to larger sizes with increasing temperature or decreasing local dislocation density. These loops appear to be present in practically every area of tangled dislocations in these specimens but their density appears to be widely variable, with no simple relation to temperature or strain. Generally the loops appear eliptical, suggesting that they are inclined at a large angle to the foil, with a marked alignment of the major axes parallel to [OOOl] , [ITOO] and occasionally [Ii011 .These small loops appear to be debris produced by pinching off dipoles, as the latter are frequently observed with a similar alignment (see Figs. 13a and 13d). In specimens deformed above about 8OO*C, spherical bubbles approximately 0.1-0.21~ in diameter are occasionally observed (Fig. 14). Generally these bubbles are of similar form and appearance to those described by McLaren and Phakey (1966a) and deduced by them to contain HzO. The main difference is that none of the present bubbles had achieved an equilibri-

62

63

Fig. 14. Areas of specimen 2394 deformed 4% at 9OO”C, showing low dislocation densities with typical high-temperature structure. The main new feature is the presence of several small bubbles (marked A). These bubbles are thought to contain Hz0 and have been observed to form only at temperatures above 800” C.

Fig. 13. Dislocation structures in specimens deformed at high temperatures. (a) Typical structure for high temperature, showing mostly curved and tangled dislocations with a high density of debris loops and dipoles. There is a marked linear trend to the dislocation distribution caused by the alignment of dipoles and loops parallel to [OOOl]. Specimen 2353 (0.16% strain at 600°C). (b) Similar area to (a) but with lower dislocation density and negligible directional trend. There is a high density of debris loops parallel to [ 00011, [ 1100 1, [ liOl] directions but few dipoles. Several large inclusions appear to act as pinning points for dislocations. Several dislocation- segments are showing weak contrast for this (1101) reflection, suggesting [2fi3] and [ 12131 as Burgers vectors for these dislocations. Specimen 2391, (1.25% strain at 800°C). (c) A region containing high density of dislocations. In such areas debris loops appear to be rather smaller and dipoles more common than in other areas. In this area there is a relatively high proportion of straight dislocation segments parallel to the traces of (1700) and (0001) planes. Specimen 2393 (7 .O% strain at 900°C). (d) This area contains many features common to high temperature specimens such as dipoles and debris loops, but with a very marked lineation to the structure, due to the alignment of these dipoles and debris loops parallel to the [OOOl] direction. Specimen 2354 (0.33% strain at 600°C).

64

urn configuration of negative crystals with planar faces parallel to low-index crystallographic planes of the matrix material. Nevertheless, the presence of the bubbles suggests that Hz0 may be precipitated in specimens annealed under pressure if the temperature is sufficiently high (c.f. McLaren and Retchford, 1969). Identification of slip systems The identification of slip systems in quartz has so far been derived from several approaches: (1) Energy arguments that the Burgers vectors should be the shortest repeat vectors in the lattice, viz. a. (ii20>, c lo001 J and ia + c> (e.g., McLaren and Phakey, 1965b; McLaren et al., 1967; Baeta and Ashbee, 1969a). (2) Observation of lattice rotations or of the orientation of deformation lamellae, assuming the latter to represent slip planes (Christie and Green, 1964; Carter et al., 1964; Hobbs, 1968; Heard and Carter, 1968; Blacic, 1971; Hobbs and McLaren, 1972). (3) Observation of slip lines on the surface (Christie et al., 1964; Baeta and Ashbee, 1969a, b, 1970a, b; Twiss, 1974). (4) Electron microscope observations. In the case of the electron microscope studies, there have so far been relatively few complete dete~inations of siip systems because of difficulties in determining the Burgers vectors and where Burgers vectors have been assigned it has generally been on circumstantial evidence (for example, McLaren and Phakey, 196533; McLaren et al., X967,1970). In some cases, the positive identification of the Burgers vector has not been possible because of rapid damage in the electron beam (McLaren et al., 1970). In other cases, there has been difficulty in achieving invisibility under the g. b = 0 criterion. However, Burgers vector determinations under this criterion, especially with g. b X u = 0 also, have been made by Baeta and Ashbee (1968) and Ardell et al. (1974). In making such determinations, possible effects of anisotropic elasticity should, of course, be taken into account, if necessary (Head et al., 1973; specific application of contrast image simulation to quartz was first made by J.A. Retchford (unpublished M.Sc thesis, Month university, 1973); however, in practice it appears that the criteria for isotropic elasticity are adequate (3-W. McCormick, personal communication, 1975). In the present work, attempts were made to identify Burgers vectors using the simple isotropic criteria, but these attempts were not wholly successful, at least in part because of the limitations of specimen orientation without a high-angle tilting stage; often it was possible to observe dislocations with very weak contrast but only rarely was complete invisibility achieved. Slip planes have, on the other hand, been deduced in many instances from the orientation of dislocation loops, Most of the loops formed at low temperature appear to have grown by glide alone, in which case the loop plane

65

defines the slip plane. The most common such loops appear to lie on the (1100) or (Olil)/(lOii) planes. The (Olil) and (1Oii) loops are symmetrically oriented with respect to the (1120) section plane and their distinction relies on identification of dislocation intersections with the top and bottom surfaces of the specimen; frequently this distinction is difficult to make but in some areas both planes seem to- be active. Other loops have been observed -on (OOOl), (OliO)/(lOiO) and (2110)/(1210) planes in small numbers throughout the temperature range, although there is some tendency for (0001) loops to be more common at the higher temperatures. The (1100) loops tend to show weak diffraction contrast with all {lOjO} and {11~0} reflections in (1120) and (0001) sections, although no complete invisibility has been achieved, suggesting that b = [OOOl] . Further support for b = [OOOl] is provided by the observation that dislocation segments parallel to [OOOl] appear to have moved out of their slip plane, probably due to cross-slip at the low temperatures. For the other important low temperatures loops, on {Oli 1)) there is marked reduction in contrast for g = (OOOl), which suggests that the Burgers vectors could be [2iiO] or [i2iO]. This suggestion is supported by observations of dipoles being dragged out from the expanding loop segments. ‘The orientation of the dipoles is such as to indicate the same Burgers vectors. For the other types of loops the assignment of Burgers vectors is generally less certain. At the higher temperatures, identification of slip planes becomes more difficult as loops are less well defined but, in many cases, approximation to invisibility conditions and alignments of dipoles and debris loops were ob-

TABLE I Schmidt factors (>0.25) plane

of possible slip systems for applied stressed normal to (ilO1)

Slip system

Schmidt factor

Slip system

Schmidt factor

(ioii?)

0.491

(oiio) [2iG] (IToo) [i1%3] --

0.360

(1Oii) (0111)

0.317

[iaio]

(OlT2) [ 2iiO]

(IToo) [OOOI1

0.486

- (lo=) [ 12101 (OlT3) [ 2iiO]

0.483

(iofi) [izio] (0111) 121101

0.470

(0001) [i2io]

0.421

(0001)

[ 2iiO]

(i2io) [oooi 1 (2iiO) [OOOl]

(l_dO) [1213] (1100) [1123] [1123] [ llF3]

- (lOJO) [lZO] (0110) [2110]

0.267

66

served which are compatible with the occurrence of all three of the previously proposed Burgers vectors. The indications for the activity of the (c + aX1123) Burgers vector were restricted to specimens deformed above 550°C. It is concluded that large amounts of slip occur on the m prism (l%OO)[OOOl] and z rhomb (Olil)[ZiiO] and/or (lOii)[i2iO] systems, and possibly minor amounts of slip- on- the following basal and prism systems: - (00~1)[2iiO] and (0001)[1210] ; (oii0)]2=3] and (ioio)]i213]; - (OllO)[ZiiO] and (lOiO)[i_2iO] ; and (1210)[0001] and (2110)[0001]. These relative degrees of activity correspond in general to the relative values of the Schmidt factors (Table I) except for the absence of the ilOi21 and {1013) slip planes, which were observed by Baeta and Ashbee (1969b) to be active at atmospheric pressure and high temperature. VARJATION

OF DISLOCATION

DENSITY WITH STRAIN AND TEMPERATURE

Attempts to relate the measured dislocation densities to strain or temperature are hampered by the marked inhomogeneity of deformation because the measured average values of strain are difficult to relate to local strains on the scale of the dislocation density variations, which are in turn also affected by the local OH content. Only rather general comments can therefore be made. At low temperatures (475~-550”), the average dislocation density in the areas showing development of clusters and bands rises very rapidly to 10’ cm-* in the yield region and continues to increase more slowly beyond this to reach a maximum of 10” cm-* or so. With increasing temperature the dislocation density at each stage is lower. For example, the density at which tangling commences falls from approximately 5 * log cm-’ at 475°C to approximately 2 * lo9 cm-* at 55O”C, while the maximum density observed at the largest strains falls from greater than 10” crnw2 to approximately 8 . 10’ cm-* in spite of the specimens being taken to markedly larger strains at the higher temperatures. For temperatures higher than 55O”G, there is a noticeable drop in measured densities along the full length of the stress-strain curve, decreasing from those mentioned above to a maximum density of approximately 5 - lo9 cm-’ (at high average strains), to tangling at below 5 * 10’ cm-” and to a density at yield which is difficult to define but probably approximates 2 - lo8 cm-‘. Thus dislocation densities at a given plastic strain are, in general, reduced substantia.lly with increasing temperature. Any exceptions to this general trend can usually be attributed to anomalously low-temperature-like behaviour, probably brought about by local reductions in the OH content. These densities are of the same order as in most previous observations (Baeta and Ashbee, 1968; McLaren and Retchford, 1969; McLaren et al., 1970; Hobbs et al., 1972) although densities as high as lO’*--lo’*, not ob-

67

served in the present experiments, have been reported (McLaren and Retchford, 1969; Hobbs et al., 1972). These higher densities probably result from the specimens investiga~d having been taken to signific~tly higher strains at relatively lower temperatures than those in this study. RELATIONSHIP

BETWEEN OPTICAL AND ELECTRON MICROSCOPE OBSERVA.

TIONS

Most of the deductions from optical microscopy relating to the inhomogeneity of defo~ation are confirmed by the electron microscope. At low temperaturesY particularly in those specimens containing broad defamation bands, much of the specimen remains as if undeformed, with no observable dislocations and only occasional areas of inclusions. Most of the dislocations in such specimens are ~on~entra~d in the main defo~atio~ band, indicating that it has accommodated most of the strain within the specimen. The band generally contains high densities of dislocations on at least two slip systems, so that there is commonly a good deal of tangling. This might be expected to result in a high density of deformation lamellae, as was found to be the case by optical microscopy. With increasing ~em~erat~e the dislocation density associated with undulatory extinction outside deformation bands increases significantly but the densities’inside and outside main deformation bands do not become comparable except at very high temperatures when the optical contrast between the band and surrounding areas is also significantly reduced and there are no longer any marked differences in the dislocation structures. At temperatures greater than ~p~x~rna~ly 600” C, recovery evidently becomes important and the mobility of dislocations significantly higher, leading to more uniform densities and more nearly homogeneous structures. Inhomogeneity in OH content, however, still appears to have an influence even up to 9OO”C, Thus, in specimens deformed between 600” and 700” I regions of lower OH concent~tion, as detrained by infrared absorption, show dislocation structures typical of low temperatures (e.g., straigbtsided loops) whereas neighbou~ng regions of higher OH concentration show structures, typical of high temperatures, that imply climb (e.g., curved and tangled dislocations with networks, dipoles and “debris” loops); evidently, on the time scale of the experiment, recovery is proceeding at a significant rate in the OH-rich areas but at a negligible rate in the drier areas. No significant differences were observed in operative slip systems between the major optical features, the latter being correlated with wide v~iations in dislocation density and apparent mobility of dislocations. Apparently these variations are related to the local OH content which does not affect different slip systems differently.

6X DISCUSSION

General

Although the aim of studying the variation in deformational behaviour with temperature alone has been complicated by the heterogeneity of deformation, interesting conclusions can be drawn. Thus, by referring to particular bands, such as the broad deformation band in the harder crystals, approximate comparability of the other conditions should be ensured since, as noted earlier, any diffusion of OH is evidently too restricted in range to bring about significant redistribution of OH content on the scale of the defo~ation bands except at the highest temperatures. However, in relating the electron microscope observations to strain, it has to be borne in mind that the actual local strain may be many times greater than the quoted average strain for the whole specimen. The nature of the variation in dislocation substructures suggests that distinct low-temperature and high-temperature regimes of behaviour can be distinguished, the transition from one to the other occurring in the region of 550-65O”C. The regimes can be characterized as follows: (1) Low-temperature regime. Where the dislocation density is still low, the dislocations commonly are straight and lie along simple crystallographic directions and dislocation loops tend to lie in low index planes that are known slip planes, although sometimes “bowing-out” is observed on otherwise straight dislocations, suggesting cross-slip; also a striking feature is the development of clusters of dislocations at the inclusions. However, after relatively small strains, high dislocation densities develop, commonly but not always in bands, with a high degree of tangling. (2) High tem~erut~re regime. The dislocations are generally curved and often do not he in a single plane, and lower densities of dislocations are built up with increasing strain, especially at the higher temperatures. There are many small loops, similar in appearance to those usually described as “debris” loops in metals. Also dipoles are common and networks sometimes appear. Thus the low-temperature regime appears to involve mainly the conservative movement of dislocations, with the expansion of loops within slip planes playing a predominant part in the rapid increase in dislocation density with strain; the straightness of the dislocations suggest a controlling role of the Peierls stress but elastic interactions may contribute ~po~~tly to workhardening at the high dislocation densities. On the other hand, in the hightemperature regime there appears to be a considerable amount of climb and possibly annotation or other recovery processes; that is, strongly thermally activated processes now play an important part. The observations of Baeta and Ashbee (1973) fall largely in the latter regime. The transition between the two regimes occurs in a temperature region that is approximately equal to or a little above the “hydrolytic weakening

69

temperature” for this crystal, as defied by Griggs and Blacic (Griggs, 1967; see also Hobbs et al., 1972). The considerable breadth of the transitional region may reflect, in part, an effect of the heterogeneity in the OH content. This association of the transition with the hydrolytic weakening temperature suggests that the transition depends in a vital way on change in the OH diffusion rate with temperature. However, there are possibly two different explanations or, at least, views of the transitions in terms of mechanism. On the one hand, while accepting that OH participates in the dislocation motion at all stages, the transition may be viewed primarily as a change, through thermal activation, in the deformation processes as such somewhat analogous to the change from cold-work to hot-work regimes in other crystalline materials. On the other hand, the transition may be viewed as an expression of change in thermally activated processes involving primarily the OH itself, such as a change in mechansim of diffusion (e.g. from a predominance of pipe diffusion along dislocations to a general lattice diffusion; c.f. Griggs, 1974). In either case, it would be desirable to clarify what factors control the microstructural changes that distinguish the low- and high-temperature regimes and to relate them to the corresponding changes in mechanical behaviour, especially the marked decrease in rate of work hardening at small strains that defines the hydrolytic weakening temperature. Dislocation

multiplication

As has been pointed out previously (Baeta and Ashbee, 1968,1973; Hobbs et al., 1972) and can be seen again in this work, the rate at which the dislocation density increases with strain in quartz appears exceptionally high when compared with that of other materials, for example, germanium (Berner and Alexander, 1967). This high rate of dislocation multiplication is especially obvious in the low-temperature regime where high dislocation densities are soon attained during straining; as noted earlier, the initial dislocation density is of the order of lo3 cm-’ but a density of the order of lo9 cm-* is already reached in the yield region, followed by slower increase to the order of 10” cme2 with further straining. However, the observation of the dislocation clusters around inclusions suggests that there is an alternative mode of dislocation generation to breeding from the grown-in dislocations, which may help to explain, at least in these specimens, the apparent extraordinarily high multiplication rate through the yield region. There are, in general, two roles which small inclusions can play in dislocation multiplication (Hirsch, 1957,1972): (1) An inclusion can interact with a moving grown-in dislocation; for example, a Frank-Read source can be formed by double cross-slip (Brown and Stobbs, 1971). Such a source will tend to produce dislocation loops in a slip plane parallel to the one in which the initial intersecting dislocations moves. The type of dislocation cluster typified by A in Fig. 9a may have formed in this way since these clusters often appear to consist of loops lying

70

in a {Ol~illi pyramidal plane with a (2iiO) Burgers vector in the plane of the or, more rarely, in a (Ol’%.OJprismatic plane with a c [OOOl] Burgers vector. However, such interactions cannot be expected to be common on account both of the very low density of grown-in dislocations and, at low temperatures, of the relatively low mobility of dislocations, as indicated by the small size of the clusters. (2) If the inclusion has different elastic properties from the matrix, then under applied stress there will be a stress concentration at the inclusion. In particular, an applied shear stress will give a stress concentration favouring the formation of prismatic dislocation loops in the neighbourhood of the inclusion. Parts of such prismatic loops can then act as Frank-Read sources to give expanding dislocation loops in various slip planes containing the Burgers vector of the prismatic loop. If the inclusions were acmite they would be significantly stiffer elastically than the quartz (Clark, 1966) and so dislocation generation in this matter would be feasible. Such an explanation may apply to dislocation clusters of type C (Fig. 9) which form the majority of the clusters. Although the details are not altogether clear, the central loops could be prismatic loops with a (Z!i?iO) Burgers vectors, from which the outer loops are generated (Fig. 15). This structure is similar to features described by Hirsch (1972) and interpreted by him in terms of prismatic loops, segments of which act as generators for expanding loops in highly stressed slip planes. The evidence that there are such mechanisms of dislocation multiplication involving the inclusions is obvious only in the low-temperature regime. I-Iowever, the absence of the small clusters at higher temperat~es may be loop

(olio

2T

[2iio]

(olio)

oiit

a

b

C

Fig. 15. Schematic diagram illustrating possible modes of development of type-C disloca tion clusters. (a) Illustrates a frequently observed structure where --loops occur as closely spaced pairs. (b) A prismatic loop is formed with Burgers vector El2101 whose edges lying parallel to the (lOii_) plane move to glide because of the high resolved shear stress on the system

(1011) fi2ioj. (c)A shear loop is formed lying on (0110)

with Burgers vector [2-O 1. The screw components can cross-slip into (013 1) as the system (011-i) [ 2=0] also has a high resolved shear stress on it.

71

merely the result of high mobility of the dislocations whereby the loops generated expand beyond the thickness of the foil, and so a similar role of inclusions may exist at higher temperatures as well. On the other hand, it does not appear that all dislocation multiplication is dependent on inclusions since there are many areas in the specimens in which inclusions are not visible, but in which the dislocation density nevertheless increases with strain, even in the low-temperature regime. There must, therefore, be other mechanisms of dislocation multiplication as well, the nature of which is at present unknown; generation from grown-in dislocations or from surface sources are two possibilities, although at low temperatures dislocations from surface sources may not penetrate far into the specimen. Returning to the question of rate of dislocation multiplication, the existence of several possible multiplication mechanisms implies that there may be no single simple multiplication law. In fact, it is not even obvious what magnitude should be assigned to the multiplication rate at low strains where the multiplication is thought to occur from sources generated at inclusions. If it is assumed that at each inclusion a length of dislocation of the order of 0.1~ is available for growth by loop expansion, then the total length of dislocation available for expansion in each cm3 is lo7 cm if the inclusion density is 101* cme3; in this case, the initial dislocation density from which to reckon the multiplication is, in effect, 10’ cmm2, or even higher where the inclusion density is higher, that is, much greater than the lo3 cm-’ density of grown-in dislocations. Thus, where generation from inclusions is occurring, the same extraordinarily high rate of multiplication need not be invoked as when the lo3 cmm2 dislocations are required to be multiplied up to 10’ or lOlo cm-*. Yet another multiplication rate may apply to the dislocations in the prismatic deformation bands. The latter dislocations do not obviously originate at inclusions and they often appear to have a c [OOOl] Burgers vector, which has been noted as being uncommon amongst the grown-in dislocations in quartz (Lang and Miuscov, 1967; McLaren and Retchford, 1969; McLaren et al., 1971); generation from surface sources has again the problem of penetration but it is perhaps supplemented by a Frank-Read multiplication from sources formed by a double cross-slip process like that suggested by Mendelson (1972) for silicon since the prismatic deformation bands are morphologically similar to those in silicon. At high temperatures, the slower rate of increase in dislocation density with strain may be due either to a lower effectiveness of the multiplication mechanisms or to a counterbalancing annihilation, in each case because of the greater climb mobility of the dislocations. However, it is difficult to draw conclusions directly from the electron microscope observations because the dislocations now presumably move much greater distances than are contained in the normal field of view and accordingly it becomes very difficult to i-e-construct the history of their motion and so discover the multiplication mechanisms.

72

Constraints

on microdynamical

theories

It was suggested by Hobbs et al. (1972) that the yield behaviour of quartz, particularly the yield point drop, could be explained by a microdynamical theory such as that of Haasen (1964; see also Alexander and Haasen, 1968), and that such a theory could also be applied to the later stages of the stressstrain curve if suitably modified to include recovery processes. The revelance of a theory of this type is thought to rest on the Peierls stress being a primary factor in the resistance to dislocation motion and hence on the strain rate being determined primarily by the intrinsic dislocation velocity rather than by the rate at which extrinsic obstacles are surmounted. Appropriate examples are germanium and silicon which, because of their strong directional or covalent bonding, have high dislocation core energies and consequently tend to have a low initial dislocation density. By analogy, similar considerations might therefore be expected to apply to quartz. This microdynamical approach was taken up by Griggs (1974) who developed and extended the Haasen theory by introducing expressions for dislocation velocities and rates of recovery that incorporate explicit dependences on OH content. The Griggs model is described by the following equations :

i, = gv,b(u,

+ u2)

fi = N, [6CUI+ u2) - PC+ + u4)l

(1) (2)

where : E.P = rate of increase in plastic resolved shear strain, N = rate of increase in dislocation density N, N, = density of mobile dislocations, = magnitude of Burgers vector, ; -- a constant governing the rate of depletion in dislocation density due to recovery, 6 = a parameter governing the rate of dislocation multiplication, taken to be equal to KT, where K is a constant and 7, = T - AN1l2 is the effective resolved shear stress acting on a dislocation (7 being the applied resolved shear stress and A equal to pb/2n (1 - V) where I-(is the shear modulus and v Poisson’s ratio). The quantities u1 and ul, represent the dislocation glide velocities for the cases where the velocity is limited by the rate of diffusion of OH to the dislocation in a radial direction and along the dislocation core, respectively, as expressed by : u1 =

COC T

1

cmr,m

u2= --

T/J

-QI

Bol exp XT_

Be2expm

-Q2

(3) (4)

73

where m,Bol,Bo2,Q1 and Qz are constants, T is the absolute temperature, and C is the average OH concentration. The quantity co is related to C by the expression:

where (Yis a constant; co is intended to represent the OH concentration outside the core region of the dislocation, that is, in the reservoir from which the radial OH diffusion occurs. The quantities u3 and u4 represent the effect of two recovery processes, each of which also depends on OH concentration, as follows:

C AN1j2 -Q, =- Bo4 exp -RT /.i

u4 T

where E&, Bo4, Q3 and Q4 are constants and o is the applied stress. By fitting the equations systematically to particular experimental data of Blacic (1971) and Balderman (1974) and assuming that Q2 = Q3 = Q4 = i Q,, iV, = N and No = lo* cm-* and that the defo~ation is homogeneous, Griggs showed that the above model gives a general fit to all the observations on the 0’ orientation with a set of parameters that includes m = 1.75, K = 3 * lo-’ cm.dyn;‘, Q1 = 32,000 calmol-’ and l/a = 1.3 - 10 I3 cm’. This good fit to observed stress-strain curves extends also to the observations on the same orientation by Hobbs et al. (1972), at least up to 65O”C, although not to the other orientations studied by them. Further study of the model is therefore clearly called for, However, here we shall only discuss particular aspects of the model, especially where comparison can be made with the electron microscope obser%ations relating to constituent processes embraced by the model. The first obviously questionable feature in the model is the assumption of homogeneous deformation, in view of the almost invariable observation of grossly heterogeneous defo~ation arising from the heterogeneity in composition of synthetic crystals. The effect of this assumption is that the strain rate and OH concentration are underestimated in the regions in which the defo~ation is concentrated. Qu~itatively, the behaviour should still be the same but, since the observed stressstrain curve now represents an average over different strain rates, finer details may be expected to be smeared out. In support of this expectation, some preliminary computer simulations of a modified Haasen model applied to a specimen containing regions to two different OH concentrations showed that the intensity of the yield drop is diminished in comparison with that for a homogeneous specimen of the same mean OH concentration (Paterson, unpublished work). It has also been

possible, by taking into account the actual strains in the regions of different to arrive at an approximate description of a homogeneous specimen (Morrison-Smith, in press). One of the crucial elements in the Griggs model is the manner of simulating the outst~dingly high rate of work hardening in the low-temperature regime. The Haasen model incorporates only the work-hardening &sing from a given dislocation having to move through the long-range stress field of all other dislocations, an effect that is taken into account through the use of the effective stress, 7 - AN “’ . However, inserting the observed dislocation densities for N, Griggs showed that this Taylor type of work-hardening effect fell far short of the measured work-hardening. He therefore introduced another type of hardening, due to OH exhaustion arid expressed through the quantity co in eqs. 3 and 5. He argued that OH will segregate to the core and immediate vicinity of dislocations, rendering them mobile, and that, as the dislocation density increases, the matrix of the crystal will become correspondingly depleted in OH. This depletion will lead to slower growth of dislocations since new dislocations must in turn be saturated with OH drawn from the matrix. Therefore, since the growth of dislocations during their motion is regarded as the rate-limiting process, the dislocation velocity will be reduced as the dislocation density increases and the matrix is depleted in OH, leading to the hardening effect. Explanation of the observed rate of work-h~en~g in this way depends critically on the value of the parameter cy in eq. 5 and requires that a cloud of about 50-100 OH groups are accommodated around the dislocation per lattice spacing along it. However, it is difficult to see how this accommodation can be achieved interstitially (cf., discussion by Buerger, 1954, on “stuffing” the quartz structure). An alternative to the OH exhaustion hypothesis for the work hardening is to return to some sort of dislocation interaction hardening that is more effective than that in the Taylor model. Such an alternative is possibly suggested by the observation that multiple slip accompanies the high rates of hardening, which may, therefore, result from the tangling and mutual obstruction of dislocations on different slip systems {cf., Friedel and Saada, 1968). The occurrence of multiple slip is not recognized in the Haasen and Griggs models. However, while the electron microscope observations do reveal complex tangling in the high work hardening stage, further study is required to elucidate the particular interactions that are most effective between the dislocations on the different slip systems and to give quantitative expression to the effect. The microstructural observations also bear on several other aspects of a microdynamical theory for quartz: (1) Dislocation velocities. Using Griggs’ expression (eq. 3) for VI, the calculated dislocation velocities in the low-temperature regime at small to moderate strains are of the order of lO+j cm.sec-’ ; for example, using N = lo9 to 1O1* crne2 ,7 = 5 kbar, T = 550°C and rd = 1200 - lo-’ H/Si for an OHrich band, the value of u1 is 1.1 to 2.6 - 10e6 cm.sec-‘. Thus the calculated OH concentration,

75

velocities are of the same order of magnitude as those deduced from the electron microscope observations on the size of the loops in the vicinity of inclusions, assuming that the loops expanded steadily throughout the plastic straining. The agreement gives some confidence in the validity of the model at low strains or, at least, in the underlying assumption here that the motion of the dislocation in the vicinity of the inclusions can be regarded as typical of the remainder. (2) Initial dislocation density. Under “Dislocation multiplication” it was pointed out that the value to be chosen for No is far from clear. Griggs’ choice of N,, = 10’ cm-‘, influenced by the present observations on the role of inclusions, must obviously be applied with caution to other samples of quartz until a similar microstruct~~ basis is demonstrated in them. On the other hand, if a mobile dislocation population is to be built up from lo3 initial dislocations, Griggs points out that extraordinarily high values of R are implied and yield point drops much more pronounced than those observed would be predicted (although in practice the latter effect may be mitigated to some extent by smearing due to heterogeneity of composition, as noted above). (3) Mobile dislocation density. In the application of microdynamical theories it has often been assumed, usually for lack of information, that all the dislocations are mobile, that is, that N, = N. However, it is not clear whether this assumption is always valid. In the present experiments in the low-temperature regime, an average value for the velocity was deduced above to be around 1gm6 cm.sec-’ at 550°C at small strains. Then calculation of N, from & = N, bu (Hirth and Lothe, 1968), taking 6 = 5 - lob5 see-’ in an OH-rich band, gives N, = lo9 cm-*. This figure agrees in order of magnitude with the observed density in the yield region but is possibly an order of magnitude too low in the region of strong work-h~ening, The latter discrepancy may arise because the velocity is now being overestimated, having been deduced from observations at low strain, or because a proportion of the dislocations has been immobilized by the tangling interactions that develop at higher densities. The latter immobilization would tend to contribute a workhardening effect. However, it would be unwise to accept these indications of a reduction in proportion of mobile dislocations without further verification. A similar effect has been observed in germanium by Springer (1971), but the opposite effect has also been observed by Sumino and Kojima (1971); the latter authors deduced that only a few percent of the dislocations are mobile in stage I of the work-h~den~g of geranium while almost all dislocations move in stage II. (4) Multiplication law. As noted above, there may be more than one multiplication mechanism and so the use of a single all-embracing law needs testing. The law fi = N, US is reported to describe satisfactorily the dislocation multiplication in germanium (Alexander and Haasen, 1968), but the rate in quartz appears to be rather higher than in germanium and so other components may also contribute. A pointer to this difference may lie in the

76

observations suggesting a substantial role of multiplication of the FrankRead type, especially at inclusions, as opposed to the kinetic type of multiplication source thought to be the basis of the law used in the Haasen model. From this discussion, it can be concluded that further attempts to apply microdynamic theory to quartz would be much helped by more detailed experimental work, paying special attention to the following aspects: (1) The use of homogeneous specimen material, well characterized in composition and initial defect content. (2) Independent me~urements on dislocation velocities and muItiplication rates. (3) Further study of work-hardening and recovery processes. ACKNOWLEDGEMENTS

We wish to thank Dr. J.N. Boland for generous assistance in electron microscopy and for extensive discussions during the work, Mr. A.W. Geatley for maintenance of the deformation apparatus and Mr. G.T. Milburn both for specimen and thy-action preparation and for adviee on photographic techniques and materials. Discussions with Drs. A.C. McLaren, J.M. Christie, S.H. Kirby and J.W. McCormick, and the late Professor D.T. Griggs were also very useful. We are indebted to Professor G.A. Horridge of the Research School of Biological Sciences, Australian National University for permission to use the Hitachi electron microscope and to Mr. R.G. Whitty and his assistants for maintaining this machine. Similarly, we thank Professor R. Street and Dr. A.C. McLaren of the Physics Department, Monash University, for permission to use that Department’s electron microscope during the last stages of this project; we also thank Mr. R.W. Bryant for photo~aphic assistance during this period. Finally, we thank Dr. J.T. Gourley of the Department of Solid State Physics, Australian National University for his contribution to the infrared studies. Financial support for one of us (D.J.M.S.) was provided through an Australian National University Research Scholarship.

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