Long-range internal stresses and low temperature phonon scattering in plastically deformed niobium and tantalum single crystals

Materials Science and Engineering, 96 (1987) 167-183

167

Long-range Internal Stresses and Low Temperature Phonon Scattering Plastically Deformed Niobium and Tantalum Single C r y s t a l s

in

W. WASSERBACH

Institut fiir Physik am Max-Planck-Institut fi~r Metallforschung, D-7000 Stuttgart 80, Heisenbergstrasse 1 (F.R.G.) (Received June 1, 1987; in revised form June 23, 1987)

ABSTRACT

The thermal conductivity of niobium and tantalum single crystals plastically deformed in steps at intermediate temperatures has been measured in the temperature range 0.3-20 K. From these measurements the thermal resistivity Wa induced by the plastic deformation was evaluated in the temperature range 0.3-5 K. After each deformation step a thermal resistivity Wd proportional to T -2 was found, indicating that phonons are scattered strongly by static strain fields rather than by dynamic phonon-dislocation processes. X.ray topography and transmission electron microscopy studies revealed that the dislocation arrangement is heterogeneous, consisting o f kink walls and dislocation sheets. Since classical dislocation pile-ups have not been observed by the lattice thermal conductivity measurements nor by the dislocation arrangement studies, the long-range in ternal stresses causing phonon scattering have to be attributed to the heterogeneous dislocation arrangement as proposed according to Mughrabi's composite model.

1. INTRODUCTION

The work hardening of metals is predominantly determined by the arrangement of dislocations and the internal stresses associated with them. According to Seeger et al. [1], long-range internal stresses due to pile-ups of primary dislocations play a central role and are held responsible for the flow stress in the linear stage II of the work-hardening curve. In Hirsch's [2] theory, stage II work hardening is attributed to the blocking action of obstacles formed by short-range secondary slip, which is generated in turn to relax the stresses around piled-up groups of primary dislocations. The 0025-5416/87/$3.50

additive effect of stress concentration around the piled-up groups is sometimes sufficient to exceed the yield stress on secondary systems under circumstances in which the applied stress alone is insufficient to activate secondary sources [3]. By contrast, in a more recent theoretical model, Mughrabi [4] ascribes the development of significant long-range internal stresses to the heterogeneity of the dislocation distribution even when classical dislocation pile-ups are absent. The existence of substantial long-range internal stresses in plastically deformed metal single crystals has been proved experimentally by magnetic measurements on nickel [5] and iron [6] single crystals, by observations on magnetic domain patterns on nickel single crystals [7], by magnetic small-angle scattering of neutrons in iron single crystals [8] and by transmission electron microscopy (TEM) studies of copper single crystals in which the dislocations had been pinned by neutron irradiation both in the unloaded and in the stress-applied state [ 9-12]. The results obtained by the magnetic investigations have been explained as due to long-range internal stresses caused by groups of piled-up dislocations of the same sign, with approximately 20 dislocations per group. In fact, TEM studies in deformed copper crystals proved the existence of primary dislocation groups and pile-ups, although not in sufficient number [11, 12]. In contrast, however, TEM studies in the unloaded state failed to show dislocation pile-ups (see for example refs. 9, 10 and 13-16). There might be the possibility that because of the preparation of thin foils for TEM experiments the internal stress equilibrium of such specimens is much more disturbed than it is in the bulk material. Another possible explanation of the apparent absence of the dislocation pile-ups in TEM studies might be that they are © Elsevier Sequoia/Printed in The Netherlands

168

disguised b y localization in the dislocation sheets formed during deformation into stage II of the work-hardening curve [ 1 7 - 2 0 ] . In order to investigate long-range stress fields in plastically deformed crystals, a nondestructive technique is necessary as in the case of the magnetic methods [5, 6, 8] which, in addition, are sensitive to large~cale dislocation arrangements. Such a powerful tool is the measurement of the lattice thermal conductivity at low temperatures, since phonon scattering depends strongly on the type of crystal lattice defects and their arrangement (see for example ref. 21), as pointed o u t in more detail in Section 2. In the present paper a review is given of an experimental and theoretical programme in which the control of dislocation character and arrangement is emphasized. The use of single crystals of refractory b.c.c, metals (niobium and tantalum) allows us to vary the dominant t y p e of dislocation and the dislocation arrangement [22]; after deformation at low temperatures, the dislocation arrangement consists mainly of long screw dislocations, whereas deformation at intermediate temperatures leads to braids or walls of edge dislocations. Another advantage of the use of the superconducting metals niobium and tantalum is that, at temperatures below a b o u t 0.2T c (To = 9.25 K is the critical temperature for niobium and T¢ = 4.48 K for tantalum), nearly all heat is carried b y phonons because the number of normal conducting electrons decreases drastically with decreasing temperature [ 23 ]. Therefore the lattice thermal conductivity can be easily determined from the total conductivity in the superconducting state. The thermal conduction of niobium and tantalum single crystals deformed at intermediate temperatures ( 2 9 5 , 3 5 5 , 3 7 0 , 4 2 0 or 470 K) was measured between 0.3 and 20 K. From these measurements the thermal resistivity due to the plastic deformation was determined. The dislocation arrangement was studied by X-ray topography and TEM. The scattering of the phonons by the dislocations was calculated with the aid of non-linear elasticity theory, taking fully into account the elastic anisotropy of the crystals. With this model of phonon-dislocation scattering, the dislocation densities after each deformation step were determined and compared with data in the literature.

2. THEORETICAL BACKGROUND OF PHONONDISLOCATION INTERACTION

Measurements of the thermal resistivity of crystals at low temperatures can be used as a tool for studying lattice defects in a quantitative way (e.g. for determining dislocation densities). The important feature of the technique is that the p h o n o n spectrum in a crystal depends strongly on temperature and that the dominant wavelength of the phonons varies with temperature. Theoretical studies [24-34] and experimental investigations [35-48] have shown that, at temperatures below a b o u t 10 K, dislocations have a pronounced effect on the lattice thermal resistivity. The interaction between phonons and dislocations may be of both a dynamic and a static nature; phonons are scattered b y mobile dislocations and by the elastic strain field around a fixed dislocation. The static scattering of phonons from dislocations requires a non-linear interaction between lattice distortions and phonons. Since experimental data concerning high order elastic moduli were lacking, in early theoretical calculations [24, 25] the anharmonic interaction was described in terms of a single parameter, e.g. the average Griineisen parameter. As a consequence, these calculations of dislocation scattering clearly could not describe the rather complicated situation adequately. Bross and coworkers [26-29] introduced the tools of non-linear elasticity, since within its framework the coupling between the phonons and the dislocations is described b y welldefined and independently measurable thirdorder elastic constants. Since experimental data concerning high order elastic moduli of single crystals were lacking (only polycrystalline data were available), the calculations were confined to elastically isotropic crystals which have three independent third-order elastic constants, whereas more recent investigations [30-33] t o o k advantage of the fact that, in recent years, complete sets of six third-order elastic constants for a number of anisotropic crystals have become available. From theoretical calculations [24-26], it is expected that the static scattering of phonons by randomly distributed dislocations should produce a thermal resistivity Wd proportional t o T -2, i.e. the thermal conductivity K d = 1 / W d varies with T 2 (Fig. 1, curve R).

169

T2

than with T 2 with decreasing temperature [50, 51] (Fig. 1, curve G). On the assumption that the dislocation arrangement consists of groups of N edge dislocations with the same sign (with N = 20), it is found according to Bross [51] t h a t this scattering process leads to an increase AWd in the thermal resistivity

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Fig. 1. Influence of dislocations on the lattice thermal conductivity: curve R, the influence of randomly distributed dislocations; curve D, the influence of dislocation dipoles according to ref. 49; curve G, the influence of piled-up groups of primary edge dislocations according to refs. 50 and 51. It should be noted that, for dipole scattering, the lattice thermal conductivity of a sample of finite dimensions will be limited by surface scattering.

In contrast, the scattering of phonons by edge dislocation dipoles leads to deviations from the quadratic temperature dependence of the lattice thermal conductivity (Fig. 1, curve D). In the dipole configuration, two parallel edge dislocations of opposite signs are separated by distances small compared with other dislocation distances. Since the strain fields of the dislocations in such an arrangement partially cancel, it is expected, at least at very low temperatures where the d o m i n a n t wavelength of the phonons is large compared with the distance of the two dislocations of a dipole, that the thermal resistance is smaller than that of two isolated dislocations [49]. By contrast, if phonons are scattered by groups of dislocations of the same sign, the thermal conductivity decreases more strongly

(i)

A similar result is obtained if the scattering of phonons by a uniform distribution of parallel dislocations of the same Burgers vector is investigated, i.e. this scattering mechanism gives rise to a thermal resistivity proportional to T -a [321. Several models have been used to describe the dynamic phonon-dislocation interaction [ 34, 52-59]. If the dynamic scattering process consists of the interaction of phonons with kinks in screw dislocations [52-54, 58], a thermal resistivity proportional to T -3 is expected [48]. In contrast, the interaction of phonons with vibrating dislocations leads to a resonant scattering process [34, 55-57, 59]. In particular, phonons with frequencies in the vicinity of the resonance frequency ¢00 of the vibrating dislocation are scattered strongly. In Fig. 2, a typical example of the influence of a resonant-type phonon-dislocation scattering process on the lattice thermal conductivity is presented for extended dislocations [48]. According to Kronmiiller [57], extended dislocations may be excited into internal periodic oscillations and may therefore interact with lattice waves. Kroupa [60] has investigated the internal vibrations of the core of a screw dislocation in a b.c.c, lattice and f o u n d that the eigenfrequencies are of the order of 10 m Hz. Because the d o m i n a n t p h o n o n energy in the thermal current is f o u n d to be (h~/kT)dom ~ 4, extended dislocations with a resonance frequency of about 10 ~2Hz strongly scatter phonons at about 1 K (h = h/2~r where h is Planck's constant and k is Boltzmann's constant). 3. SPECIMENPREPARATION The specimens used were single crystals grown by electron beam melting of polycrystalline rods of niobium and tantalum. The specimens had orientations in the central part

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of the stereographic triangle, except for specimen T a l , oriented in the vicinity of [001]. In order to ensure slip exclusively on the plane which carries the m a x i m u m resolved shear stress (MRSS) (see for example ref. 61), the specimens were doped with about 150 at.ppm C (Nb) or with about 250 at.ppm N (Ta). The crystals were deformed in steps at strain rates 4 = 3 × 10 -5 s-z at temperatures of 2 9 5 , 3 5 5 , 3 7 0 , 420 or 470 K. After each deformation step the thermal conductance was measured, the slip-line pattern investigated, and the crystallographic orientation determined by Laue X-ray back reflection. The thermal conductance was measured with a steady state heat method using a SHe refrigerator (for details, see ref. 62). The data characterizing specimens are collected in Table 1. In order to minimize dislocation rearrangement during the preparation of thin foils, after the final deformation step the niobium specimens were irradiated with about 2 × 1022 fast neutrons m -2 at the Forschungsreaktor Mfinchen [9]. Slices parallel to selected crystallographic planes were cut from each crystal and mechanically and chemically treated as described elsewhere [63]. X-ray topographs and TEM micrographs were taken from each slice. Because of the activity of the isotope ~S2Ta the tantalum specimens were not irradiated.

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4. E X P E R I M E N T A L R E S U L T S

4.1. Work hardening and dislocation arrangem en t

Figure 3 shows the work-hardening curves of the different specimens. Except for speci-

TABLE 1

Characterization of the specimens

Specimen

Diameter (ram)

Residual resistance ratio

Nbl Nb2 Nb3

3.2 4.0 4.0

1200 185 185

Tal Ta2 Ta3

3.9 3.3 3.8

60 111 185

L o w temperature heat conductivity constant A

Normal state heat conductivity constants (eqn. (3))

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b

(W m K -a)

(Xl0-5mWK-I) (X10-2mK2W-I)

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0.435 2.81 2.81

295 370 470

16 20 25

2.26 2.2 1.9

9 4.86 2.9

295 355 420

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men T a l th e curves were calculated for single slip on th e plane t hat carries the MRSS. The specimens show t he typical behaviour of b.c.c. metals. The work-hardening curves resemble those o f f.c.c, single crystals; however, the experimental findings suggest t hat a gradual transition occurs f r o m stage I into stage III and th at the characteristic features of stage II are missing. Specimen T a l , oriented for multiple slip [ 6 4 ] , exhibits no stage I at all but an almost parabolic behaviour o f the true tensile stress-strain curve (Fig. 3(b)). The slip-line patterns after d e f o r m a t i o n into stage II-III are similar for all specimens oriented for single slip. On the side surface, t he primary slip lines are straight and long whereas, on the t o p surface, however, t h e y are curved and wavy (Fig.

Fig. 3. Work-hardening curves of the specimens: (a) niobium specimens Nbl, Nb2 and Nb3 and the inset shows the orientations of the undeformed specimens (the shear stress ~ and the shear strain a have been resolved on the plane that carries the MRSS); (b) tantalum specimen Tal which deforms by multiple slip (therefore the axial stress has been plotted vs. the axial strain); (c) tantalum specimens Ta2 and Ta3 which deform by single slip on MRSS. It should be noted that, at the strain values marked with open circles, the deformation was interrupted and the thermal conductivity measured.

4), indicating f r e q u e n t cross-slip o f screw dislocations with an (a/2) <111> Burgers vector [65]. For the [001]-oriented specimen T a l , at least two di fferent slip systems could be observed (not shown). TEM observations and X-ray t o p o g r a p h y led t o similar pictures of dislocation arrangements in t he different specimens; the majority of the dislocations are accumulated in multipole bundles lying in the glide plane or in dislocation kink walls perpendicular to the slip plane and the glide direction and in dislocation sheets roughly parallel to t he macroscopic glide plane. The kink walls are parallel to the [121] line direction o f primary edge dislocations (Fig. 5). The walls have a thickness of a b o u t 0.5/~m and are separated by 2.5/~m.

172 the primary edge dislocations accumulated in the kink walls is about 10 ~ m -2 inside and about 10 ~ m -2 outside. The dislocation sheets roughly parallel to the macroscopic (MRSS) slip plane are seen in Fig. 6. Their separation perpendicular to the slip plane is about 1-2 #m. Dislocation densities up to 2 × 1014 m -2 have been f o u n d in niobium [66, 67]. The dislocation sheets consist of a network of primary and secondary screw dislocations [66-70] (Fig. 7). Short links of a(100) dislocations appear to stabilize the sheets [67]. The X-ray topography micrograph in Fig. 8 gives a good survey of the macroscopic dislocation arrangement. 4.2. Thermal conductivity measurements In Fig. 9(a) and Fig. 9(b) the thermal conductivities K of one niobium and one tantalum specimen respectively are plotted logarithmicaUy against temperature. For the undeformed specimens, all curves are similar in shape and exhibit a m a x i m u m at about 0.2Tc and a minim u m at about 0.3Tc. Below the maximum the data are consistent with a T 3 relationship for all specimens, as would be expected from boundary scattering of lattice waves [71]. The values of the constant A in the relationship Kg,b = A T a

(2)

obtained at the lowest temperatures investigated are given in Table 1. Kg.b denotes the lattice thermal conductivity of the undeformed specimens limited by surface (boundary) scattering. At temperatures above the critical temperature To, the electronic thermal resistivity 1/Ke,n in the normal state is described by 1 - -

Ke, n Fig. 4. Slip-line patterns of specimen Nb2 after 14% deformation at 370 K (TA, tensile axis): (a) long and straight slip lines on the side surface which are parallel to the primary slip direction [111], showing that the slip-line pattern is homogeneous which indicates homogeneous single slip on the MRSS plane; (b) slipline pattern on the top surface, in which the slip bands are curved and wavy, indicating frequent crossslip of primary screw dislocations. Between the kink walls, screw dislocations, dislocation loops and dislocations of a nonscrew character are visible. The density of

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according to Klemens [21 ]. The values of the constants a and b are given in Table 1. The constant b is calculated from the residual resistance ratio [ 72 ], and a is determined from the conductivity measurements in the normal state. Below the critical temperature T¢, the electronic contribution K~. ~ decreases rapidly with decreasing temperature, whereas the lattice contribution Kg, s increases because of the reduction in the phonon-electron interaction. Therefore, in the superconducting state

173

Fig. 5. Dislocation arrangement of specimen Nb3, 22% deformed at 470 K (foil parallel to the primary slip plane (101);g = (020)). Most of the primary edge dislocations are accumulated in kink walls (at A) of high density (N w > 1015 m-2). The walls have a thickness of 0.3-0.5 ]~m and are separated by about 3 pm. The dislocation density in the cell interior between the kink walls is rather small (N c ~. 1013 m-2). Part of a dislocation network (grid) can be seen at B.

t h e t h e r m a l c o n d u c t i v i t y Ku o f t h e u n d e f o r m e d specimens is d e s c r i b e d b y Ku = Kg + Ke,s

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T h e c o n t r i b u t i o n s Kg. s and Ke, s m a y be calculated f r o m the t h e o r y o f Bardeen e t al. [23]. Figure 10 shows t h e d i f f e r e n t c o n t r i b u t i o n s Ke,n, Ke,s, K~,s and Kg, b t o t h e t h e r m a l conductivity of the undeformed specimen Tal. A f t e r plastic d e f o r m a t i o n t h e t h e r m a l c o n d u c t i v i t y in t h e t e m p e r a t u r e region b e l o w a b o u t 0 . 3 T c decreases w i t h increasing shear strain owing to t h e increase in t h e dislocation density. As well as t h e r e d u c t i o n in t h e absolute values o f t h e t h e r m a l conductivities, a change in t h e t e m p e r a t u r e d e p e n d e n c e was

observed w i t h increasing d e f o r m a t i o n o f t h e specimens (Fig. 9). A f t e r a small d e f o r m a t i o n a cubic t e m p e r a t u r e d e p e n d e n c e is observed. A f t e r a strong d e f o r m a t i o n t h e t e m p e r a t u r e d e p e n d e n c e b e c o m e s w e a k e r and f o l l o w s Kd oc T 2. In o r d e r to investigate t h e i n f l u e n c e o f the plastic d e f o r m a t i o n , t h e lattice t h e r m a l resistivity Wd d u e to t h e plastic d e f o r m a t i o n was calculated f r o m t h e t h e r m a l c o n d u c t i v i t y m e a s u r e m e n t s . T h e conductivities o f t h e und e f o r m e d and d e f o r m e d specimens are d e n o t e d b y Ku and Kd respectively. T h e t h e r m a l resistivity 1 / K d o f a d e f o r m e d s p e c i m e n is the sum o f t h e resistivity 1 / K u o f t h e u n d e f o r m e d specimen and the a d d i t i o n a l resistivity Wd d u e to the deformation: 1

1 -

Kd

Ku

4- Wd

(6)

174

Fig. 6. TEM micrograph of a foil parallel to the conjugate slip plane (101) of specimen Nb2, 14% deformed at 370 K. The dislocation arrangement consists mainly of dislocation sheet pairs roughly parallel to the trace of the primary slip plane (101). The average distance between the two sheets forming a pair is about i b~m; the distance between two adjacent pairs is about 1 ~tm. The crystal lattices of two neighbouring layers to one sheet of a pair are misoriented. This is manifest from the alternate changes in the black-white background contrast. The original orientation of the crystal lattice is restored when one couple of a particular sheet is crossed.

Details o f t h e analysis have b e e n p r e s e n t e d elsewhere [48]. I n Figure 11 t h e t h e r m a l resistivity Wd o f s p e c i m e n N b 3 is p l o t t e d l o g a r i t h m i c a l l y against t e m p e r a t u r e . F o r e a c h d e f o r m a t i o n step t h e lattice t h e r m a l resistivity Wd s h o w s a t e m p e r a t u r e d e p e n d e n c e p r o p o r t i o n a l to T -2. T h e analysis f o r t h e o t h e r n i o b i u m a n d t a n t a l u m s p e c i m e n s yields a similar p i c t u r e ( n o t s h o w n ) . F r o m t h e t h e r m a l resistivity values at low temperatures, the dislocation density of the s p e c i m e n s a f t e r e a c h d e f o r m a t i o n step can b e d e t e r m i n e d , using t h e e q u a t i o n

Wd = B N d T - 2

(7)

w i t h B = 1.4 × 10 -14 m s K s W -1. In Fig. 12 t h e t o t a l d i s l o c a t i o n d e n s i t y N a has b e e n p l o t t e d l o g a r i t h m i c a l l y against T - - r0. T h e b r o k e n lines w e r e f i t t e d b y e y e to t h e e x p e r i m e n t a l d a t a . F r o m t h e s e fits, we o b t a i n r - - TO = O . 1 3 G b N d 112

(8)

w h e r e G ( = 4.73 × 10 l° N m -2 a n d 6 . 5 9 × 10 l° N m -2 f o r n i o b i u m and t a n t a l u m r e s p e c t i v e l y ) is t h e shear m o d u l u s a n d b ( = 2.86 × 10 -t° m ) is t h e Burgers v e c t o r .

175

Fig. 7. TEM micrograph of a foil parallel to the primary slip plane (i01) of specimen Ta2, 24% deformed at 355 K. The dislocation network parallel to the primary slip plane consists of a crossed grid of primary and secondary dislocations. The dislocation density is about (2-4) × 1014 m-2.

5. DISCUSSION The lattice thermal resistivity after deformatio n shows the t e m p e r a t u r e d e p e n d e n c e e x p e c t e d f r o m the scattering of p h o n o n s by th e static strain field surrounding the dislocations. In addition, the magnitude o f the ther-

mal resistivity is increased with increasing d e f o r m a t i o n because o f the i n t r o d u c t i o n o f dislocations. These results are in agreement with a series o f thermal conductivity experiments on di fferent copper alloys [35, 38, 39, 4 5 - 4 7 , 73], pure nickel crystals [74] and superconducting samples of niobium [36],

176

tantalum [42], Ti-Nb and Nb-V alloys [75]. By contrast, several researchers have reported deviations of the above-mentioned behaviour of the thermal resistivity (or conductivity), especially in copper alloys [40, 41, 43, 44, 46, 47, 74, 76] and in superconducting samples [77-80]. For copper alloys the experimental results reveal that a strong scattering of phonons is present in the deformed specimens and, in addition, a change in the slope of the thermal conductivity is observed in the temperature

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conjugate slip plane (101) of specimen Nbl, 8 % deformed at 295 K. Extinction contrast (vertical) is parallel to the trace of the primary slip plane (TPSP) (101) from dislocation sheets, and orientation contrast (horizontal) is parallel to the trace of the (111) plane from kink walls perpendicular to the primary slip plane, c~1 and (~2 denote the components of the characteristic radiation of the copper X-ray tube. At A, the component c~1 lies above the component ~2; at B, ~1 lies below (~2.This indicates a characteristic bending of the primary slip plane caused by kink walls.

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\ \1 region between 2 and 3 K. This "kink" in the thermal conductivity data is thought to be an effect of the lattice thermal conductivity because there was no variation in the electrical resistivity in the temperature range 1 - 4 K which could p r o d u c e such a large variation in the electronic conductivity. The appearance of the kink has been attributed to static phonon scattering b y edge dipoles [41, 44], to the influence of a dense array of edge dislocations [43, 81, 82] or to dynamic scattering b y vibrating dislocations [40, 43, 46, 47, 83]. As pointed o u t b y Klemens and coworkers [ 8 3 - 8 5 ] , solute atoms in an alloy tend to rearrange themselves a b o u t a dislocation, forming Cottrell atmospheres. There is a strong interaction between the dilatational strain of an edge dislocation and the solute which increases the static p h o n o n scattering thermal resistivity [84, 85] and gives rise to resonance scattering b y edge dislocations with a resonance frequency ~00 of a b o u t 10 z2 Hz [83]. The combined scattering of phonons both by the static strain field of a dislocation and by

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vibrating dislocations gives rise to a T 2 temperature dependence of the thermal conductivity with a drastic influence of the resonant scattering in the temperature region between 2 and 3 K [83] (similar to the curve in Fig. 2). Another possible explanation for the low temperature lattice thermal conductivity behaviour in concentrated alloys has been suggested [74, 75]. In these papers, deviations of the p h o n o n conductivity from the normal T 2 dependence are associated with a short elec-

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Nb---=.l,D~ I o

I013

/ Ioi i I •/ o

/ /

/

/

I0 72 -A / I

I

I

I I illl

5

I

tO

20

] ] r - r o (b4Po)

I

Fig. 12. T o t a l ' d i s l o c a t i o n d e n s i t y Net as a f u n c t i o n o f t h e r e s o l v e d s h e a r s t r e s s T -- T0: - - - , f i t t e d b y e y e to the experimental data.

tronic mean free path according to the theory of Lindenfeld and Pennebaker [86]. As pointed o u t by Anderson and coworkers [47, 87, 88], one major problem of the interpretation of experimental thermal conductivity measurements of alloys is that the electronic thermal conductivity Ke must be subtracted from the total conductivity K to obtain the lattice conductivity Kg. Since Kg is approximately 10 -2 K, it is essential to determine Ke very precisely. However, in particular at very low temperatures, the use of the Wiedemann-Franz law may be in question [47, 87] and, in addition, the uncertainties in the T5s temperature scale might have caused additional errors [87, 88] (the Tss temperature scale is based on the vapour pressure of liquid helium). The above-mentioned errors in the determination of the lattice thermal conductivity of alloys can be avoided by using superconducting specimens. Anderson and coworkers [77-80] have measured the thermal conduc-

tivities of different superconductors. For impure polycrystalline specimens of niobium and tantalum with a low residual resistance ratio which have been doped with hydrogen, resonances have been observed in the phonon mean free paths at a b o u t 0.8 K and 0.1 K for niobium and tantalum respectively. Anderson and coworkers [80] conclude that these resonances are due to the resonant scattering of phonons by dislocations produced by precipitation of the 13-phase hydride. In contrast, no resonances have been observed after hydrogen doping of pure niobium single crystals [89], whereas resonances occur if the specimens are doped with nitrogen before the hydrogen doping. These results have been explained by tunnelling states of trapped hydrogen. Besides the fact that the magnitudes of the dynamic phonon scattering models are uncertain, all dynamic theories assume a random distribution of the dislocations, i.e. that all dislocations interact dynamically with the phonons. However, this cannot be the case since the dislocation arrangement is heterogeneous. A heterogeneous dislocation arrangem e a t consisting either of kink walls and dislocation sheets or of a cellular microstructure is a c o m m o n feature of uniaxial or cyclicaUy deformed crystals [90] and of cold-rolled or swaged specimens [91]. Since most of the dislocations are located in the walls and the sheets, their motion is hindered because of their high density. Possibly only a small fraction (about 1%) of free dislocations located inside the cell interiors can vibrate and thus dynamically scatter the phonons. So far we assumed in our consideration of the dynamic phonon-dislocation interaction that all p h o n o n modes are scattered equally strongly. This assumption results in a thermal conductivity curve as shown in Fig. 2. If, however, only one p h o n o n mode is strongly scattered resonantly, the thermal conductivity decreases by only a factor of roughly 2 independent of the dislocation density [87, 88]. Because in this case the thermal current is carried b y the two other phonon modes which are scattered b y the sample surface, a thermal conductivity proportional to T ~ is expected. Our experiments clearly indicate that the thermal resistivity Wd due to the plastic deformation increases with increasing strain and is throughout proportional to T -2. Therefore, it must be concluded that, if dynamic scatter-

179 ing is present, it has only a weak influence on the lattice thermal conductivity of the strained niobium and tantalum samples and that the static scattering of the phonons by the strain field of the dislocations has to be taken into account. From a theoretical point of view, it might be expected that both dislocation walls and dislocation sheets which accumulate roughly 99% of the dislocations scatter the phonons only weakly. The dislocation walls consist of a multipolar arrangement of edge dislocations of a high density (Nw ~ 1015 m-2). As mentioned in Section 2, the strain fields of a dipole arrangement partially cancel, and phonons with a sufficiently long wavelength will not be scattered [49]. According to Carruthers [92], the effective wavelength X characterizing these phonons most important in conduction at low temperatures is defined by 0.6a0 = --T

state have shown that in stage II of the workhardening curve a high density of groups of dislocations with the same sign of the primary glide system are formed and are mainly located in the space between the dislocation sheets [ 11 ]. These piled-up groups of dislocations of the same sign give rise to a strong scattering of phonons. The possible influence of pile-up groups on the lattice thermal conductivity of specimen Nb3 is shown in Fig. 13. From the present thermal conductivity experiments, it must be concluded that in b.c.c, metals deformed into stage II-III the excess density of edge dislocations of one sign is rather

!

S /

(9)

where a is the lattice constant and 0 the Debye temperature. With a = 0.33 nm and 0 = 275 K for niobium,we obtain an effectivewavelength of 55 nm at T = 1 K, which is larger than the mean spacing d = 1 / N ~ 12 of about 18 nm of two dislocations in a wall. The dislocation sheets consist of a cross-grid of primary and secondary dislocationswith a mean distance between two dislocationsof each type of less than 70 nm (see for example refs. 69 and 70). As pointed out by Ackermann and Klemens [81], dislocationstend to arrange themselves so as to minimizethe free energy and, in a dense array, this results in cancellationof the long-range strain field. Phononsof wavelengths comparable with the dislocation spacings are thus only weakly scattered as observed in the samples of the commercialalloy Evanohm [43, 82]. In addition, multiple slip in a [001]oriented sample leads to dislocation cell structures which are generallybelieved to be virtually free of long-range internal stresses and are therefore considered to be low energy dislocation arrays [93, 94], which scatter phonons only weakly. The thermal resistivities in uniaxially[38, 73] and cyclically[73] deformed copper alloys have been attributed to the static scattering of phonons by dislocation pile-ups having the same sign. In fact, TEM studies of copper single crystals in the stress-applied

10'

ii I 10c

II II

/

/

I 10~

i

I

I

I

I

10 L 0.1

0.5 1

5 T(KI

Fig. 13. Lattice thermal conductivity of specimen Nb3, 22.2% deformed at 470 K. The experimental data (I) are compared with a theoretically calculated curve (curve G) assuming that the phonons are scattered by groups of piled-up edge dislocations of the same sign (with 20 dislocations per group). S denotes the lattice thermal conductivity as limited by the crystal surface.

180 small compared with the total dislocation density. This is consistent with our TEM and X-ray topography observations because the angle of tilt of the kink walls and the dislocation sheets is found to be only a few minutes. Therefore, it is concluded that priedup dislocation groups play a much smaller role in b.c.c, metals than in f.c.c, metals. According to gest~k and Seeger [22], in b.c.c. metals the formation of dislocation sheets composed of primary and secondary dislocations and the accompanying long-range stresses are mainly responsible for the increase in the work-hardening rate b e y o n d stage I. The dislocations of the primary slip system move under the action of the applied stress approximately along the glide plane of MRSS; thus the slip is rather homogeneous. The dislocation sheets originate from primary glide dislocations that were immobilized b y reactions with secondary dislocations where local secondary slip could occur. These barriers are overcome b y extensive cross-slip of screw dislocations. The resulting cylindrical slip surfaces are reflected in the typical wavy slip-line patterns shown in Fig. 4(b). Our experimental findings of phonon scattering by internal stresses, even in the unloaded state, can only and easily be explained with the composite model according to Mughrabi [4, 95-97], originally developed for uniaxial multiple slip and for cyclic single slip. In crystals in which a heterogeneous dislocation distribution develops during deformation, substantial long-range internal stresses arise unavoidably as a natural consequence of the compatibility requirements in the stressapplied state. The dislocation arrangement, consisting of kink walls and dislocation sheets in single-slip-oriented crystals or of a cell structure in the multiple-slip-oriented specimen T a l , can be viewed as a composite of relatively hard walls of high local dislocation density separated by relatively soft cell interior regions of much lower local dislocation density. The dense dislocation walls act as obstacles to glide dislocations in the cell interiors. As a consequence, a certain number of glide dislocations are held up at the interfaces between the walls and the cell interiors. The effect of the interface dislocations is that they provide tensile internal stresses in the cell walls and compressive internal stresses in the cell interiors. These deformation-induced internal

stresses sum up locally with the applied stress to equal the flow stresses required locally for the simultaneous plastic straining of the hard walls and the soft cell interiors. Thus the interface dislocations, even though of a low density compared with the total density, maintain compatibility during deformation at the boundaries between the soft and the hard regions. According to Mughrabi [97 ], the mean dislocation density Nd is given by gd = fwgw +fcNe

(10)

neglecting the density of the interface dislocations. The area fractions fw (wall) and fc (cell interior) are related through fw + fc -- 1

(11)

The local flow stresses rw and rc are given by f w = The t "~- A T w

(12)

and re = r ~

+ ~Xrc

(13)

where rh~t is the macroscopic mean flow stress. The quantities Arw and Are denote the longrange internal stresses in the walls (or sheets) and in the cell interiors respectively. The mean macroscopic flow stress T ~ = f -- f0 is assumed to be given by [97] r~

= fwrw + f~rc

(14)

or

rh~ = 2 ~ ( f c f w ) l l 2 G b N d 112

(15)

where ~ is a constant approximately equal to 0.3-0.4. For the long-range internal stresses, Mughrabi [97] reveals that h r w = oeVbfc(Nw 112 - - Nc1/2)

(16)

and A r c = o~Gbfw(Nc 112 - - Nw1/2)

(17)

The local stress profile from wall to wall has been analysed in detail by Mughrabi [98] in the case of cyclic deformation of copper crystals. The tensile internal stress field of the walls (with a maximum value Arw as given in eqn. (16)) extends into the cell interior with a length of approximately 0.1d where d is the separation of t w o adjacent wafts. The compressive internal stress field of the cell interior extends over a distance of approximately 0.7d.

181 The magnitude of Arw is roughly six times the magnitude of Z~rc. When the sample is unloaded until the applied shear stress is zero, the residual stresses Arw and Arc which are opposite in sign remain frozen in the specimen. Because of these residual microstresses the phonons are scattered in a static sense which explains the temperature dependence of the observed thermal resistivities (W0 cc T -2) after each deformation step in stage II-III. With increasing strain the residual microstresses increase because of an increase in the dislocation density N0. In order to estimate the phonon scattering, the crystal can be envisaged to consist of t w o regions with internal stresses of different constant values. Further, it is assumed that Nc "~ Nw. The compressive microstress extends over a distance of ~ d with a constant value of Arc as given in eqn. (17), whereas the tensile microstress extends over a distance of -~d with a constant value of roughly 1 A~ w as given in eqn. (16). Since the spatial extensions of the residual microstresses are large compared with the mean p h o n o n wavelength (from our TEM micrographs, we obtain d equal to 3 pm), we assume that the thermal resistivities due to the scattering of phonons b y both residual microstresses are additive. Thus the phonons are scattered by an effective microstress Ar ---[ ½Z~rw [ + [ ~z~ [. With fw = 0.2 and f~ = 0.8 as observed from the TEM micrographs, we obtain, with eqn. (10), N 0 = 0.23Vw and with eqns. (16) and (17) the effective residual microstress

From the experimental results obtained after deformation into stage I of the workhardening curve, it might be expected that the dislocation arrangement consists of rand o m l y distributed dislocations since W0 cc T -2 is found. However, the dislocation arrangement after deformation into stage I of the work-hardening curve consists of multipolar bundles of primary edge dislocations (see for example ref. 90). Again, as for dislocation kink walls, it is expected that such a dislocation arrangement scatters the phonons only weakly (i.e. because of the weak strain field of the dipoles). By contrast, our experiments (tantalum specimens Ta2 and Ta3) clearly demonstrate that the thermal resistivities W0 are proportional to T -2 and their magnitudes depend on the a m o u n t of deformation, even in stage I. Therefore, sufficiently large strain fields must be present in order to scatter the phonons. These results are in agreement with magnetic measurements on iron single crystals deformed into stage I [6]. In a recent paper, Comins and Jackson [99] have pointed out that realistic calculations of the stresses of planar edge dislocation multipoles show that, although inside the multipoles the stresses are much larger than the yield stress, they remain unrelaxed because there is insufficient r o o m for secondary dislocations to expand. From our experimental results, we conclude that in crystals deformed into stage I the phonons are scattered by these stresses inside the multipole bundles and therefore give rise to the observed thermal resistivity W0 cc T -2.

A r ~. 0.275o~GbNw 112 0 . 6 2 a G b N d 112

(18)

Our estimation reveals that the effective residual microstress for p h o n o n scattering is comparable with the mean macroscopic flow stress rh~t in eqn. (15): rh~ "~ 0.6o~GbNd 112

(19)

With a ~ 0.3 the values of both the effective residual microstress for p h o n o n scattering ~T in eqn. (18) and the mean macroscopic flow stress rr~¢ in eqn. (19) as determined from Mughrabi's composite model are in rather good agreement with the flow stress values r -- r0 in eqn. (8) as determined from the thermal conductivity measurements. Equation (8) corresponds well to the data in the literature obtained from TEM investigations [67].

6. SUMMARY Thermal conductivity measurements have been performed on niobium and tantalum single crystals plastically deformed at intermediate temperatures. The lattice thermal resistivity W0 due to the plastic deformation is proportional to T -2, indicating that the phonons are scattered b y stress fields rather than b y vibrating dislocations. Our experiments reveal that long-range internal stresses prevail also in unloaded specimens deformed into stage II-III of the work-hardening curve. In addition, even in a crystal which deforms b y multiple slip and contains dislocation cell structures which have so far been considered as energetically favourable dislocation patterns

182

of negligible internal stress, these stresses are present in the unloaded state (specimen Tal). Another result of our experiments is that the sources of the stresses cannot be due to the classical arrangement of pile-up groups of primary dislocations. By contrast, our experiments can be explained in a quantitative way using Mughrabi's model in which the dislocation arrangement is viewed as a composite consisting of relatively hard walls of high local dislocation density separated by relatively soft cell interior regions of much lower local dislocation density. A necessary consequence of the heterogeneity of the dislocation distribution which has been confirmed by our TEM and X-ray topography investigations is that, during deformation, long-range internal stresses are built up. On unloading, these longrange internal stresses are frozen in as residual microstresses which scatter the phonons strongly and thus give rise to a lattice thermal resistivity Wd cc T -2. ACKNOWLEDGMENTS

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