mechanism of the koehler dislocation multiplication process

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Mechanism of the Koehler dislocation multiplication process J. J. Gilmanaa Department of Materials Science and Engineering, University of California at Los Angeles, LosAngeles, California, USA

To cite this Article Gilman, J. J.(1997) 'Mechanism of the Koehler dislocation multiplication process', PhilosophicalMagazine A, 76: 2, 329 336To link to this Article DOI 10.1080/01418619708209978URL http://dx.doi.org/10.1080/01418619708209978

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Mechanism of the Koehler dislocation multiplication

process

By J O H N . G I L M A N

Department of Materials Science and Engineering, U niversity of California atLos Angeles. Los Angeles, California 90024, USA

[Rrccirad Scptemher. 1996 and ncceprad in revised fo rm 18 . member 1994

ABSTRACTIn the Koehler dislocation multiplication process, segments of screw

dislocations glide off the primary glide plane onto secondary planes, and thenback onto a primary plane parallel to the first plane. This generates trailingdipoles for small excursions between the two primary planes, or the pinningpoints of Frank read mills for large excursions. The concern of this paper isthe mechanism that causes the lines to cross-glide, and th e distribution of dipolesheights that results. It is proposed that self-excited oscillations of movingdislocation lines, induced by thermally induced shear-strain fluctuations, causethe cross-gliding. The properties of this process are described. It is similar toatwo-dimensional randomwalk, an d this determines the dipole height distribution.

The Koehler process is dominant for dislocation multiplication in structuralmaterials, especially at high strain rates. The resulting dipole height distributionisan important factor in determining various propertiesof the cold-worked state.Some of these are specific heat, the thermal conductivity, fatigue degradation.

strain hardening. Bordoni internal friction and corrosion resistance.

8 1. INTRODUCTION

When a turbulent gas passes over a stretched stringit is likely to cause self-excited oscillations. This is fam iliar as the singing or galloping of electrical trans-mission wires in high winds, and th e flutter of decorative streamers of athletic events.Randomly excited vibrations also appear as the flutter of aircraft wings and thevibrations of a clarinet reed. In engineering it is known as flow-induced vibration.Various examples have been given in texts on vibrations such as that of Den Hartog(1956, 1985) or of Pippard 1989). For the case of dislocation lines, the present

autho r mentioned this phenomenon some years ago (Gilman 1968) but did notdevelop the subject. In the meantime, the importance of the Koehler multiple-cross-glide mechan ism for dislocation m ultiplication has come to be generally recog-nized. A consequence of the K oehler process is that it produces a high co ncentrationof dislocation dipoles in cold-worked crystals. Since these dipoles strongly affectsome of the physical properties, the process takes on special importance.

The Koeh ler (1 952) process is outlined in fig.1 Its experimen tal verification hasbeen given elsewhere (Johnston and Gilman1960).Its importance has two aspects.First, starting from only a single dislocation half-loop, a nearly arbitrary length ofdislocation line can be continuously generated.Thus, from a loop that is a fewmicrons in lengths in a cube of 1 cm3 volume, a continuous length (with a linedensity of 1013 cm-) can be generated that equals the distance from the Earth tothe Sun. Second, the process generates a plethora of dislocation dipoleswhich trail

0141 8610 97 512 00 1997 Taylor Francis Ltd.


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330 J. J. Gilman

Fig. 1

Schematic diagrams of the Koehler multiple-cross-glide process. a) A segment Py of a screwdislocation with Burgers vector b and velocity vector v cross-glides from the lowerprimary glide plane up the (filled) secondary glide plane and then cross-glides againonto the upper primary glide plane. Note that the jogs ap and y6 lie perpendicular tob; so they are edge dislocations and are constrained to move only on their currentsecondary glide plane. (b )As the dislocation lines of a) continue to move, the segmenton the upper plane becomes a semicircle and then a heart-shaped configuration untilthe lobes o f the heart meet, and parts of the lines annihilate one another. At the sametime, on the bottom plane the lines swing forwards as well as laterally until a portionof them becomes annihilated. Note that the upper and lower lines must be able to passover one another in order for the indicated motions to occur. This requires that theseparation H of the planes be greater than a value that depends on the applied stress.c) Further motion of the dislocation lines of (b). The configuration at the centre has

been restored to its form in a). The original line on the lower glide plane is moving offto the right. There is a new loop on the upper glide plane with one segment moving offto the left, and the other to the right. d) Continuation of the motion of the dislocationlines in ( a ) when the separation h of the primary glide planes is too small for the lines

to pass over one another under the given applied stress. Two edge dislocation dipolesare left behind as the original line and the segment that underwent double cross-glidemove forwards. Note that in a homogeneous stress field there is no net force on eithero f the dipoles; so they do not move laterally.


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Koehler dislocationmulliplicationprocess 33 1

behind moving screw dislocations. These dipoles cause strain hardening, thermalresistance, fatigue, strength reduction, acoustic attenuation and other propertychanges (Gilman 1964). With increasing time and plastic strain, the dipoles tendto aggregate into cell walls (Kuhlman-Wilsdorf 1985).

The concern of this paper is the initiating mechan ism for multiple cross-glide. I tis proposed that it results from self-excited oscillations (flutter) that are induced bycollisions of moving screw dislocations with thermal shear-strain fluctuations.Previously proposed mechanisms have involved the Lorentz force on a movingdislocation (Nabarro 1987) (since shown not to exist), and internal stress sourcessuch as precipitates, other dislocations (Li 1961) and surfaces (Gilman 1961).However, the process appears to be intrinsic which eliminates the second of thesepossibilities. The surface effect plays a secondary role.

From time to time it has been argued that cross-gliding results from thermalactivation. This is inconsistent, however, with the fact that the rate of multiplication

(loops per centimetre) increases with increasing dislocation velocityfig. 2) (Johnstonand Gilman 1960). If it resulted from thermal activation, it would be expected tooccur less frequently at higher velocities. Note that the average velocity oftenincreases many orders of magnitude when the applied stress increases by oneorder of magnitude. T h u s velocity, and not stress, appears to be the independentvariable.

Two other factors do not seem to be consistent with thermal activation. Oneisthat the dislocation density within a widening glide band increases with decreasingtemperature. The other is that dislocation multiplication occursin LiF (at highdislocation velocities) at78 K which is well below the Debye temperature.

A condition for the generation of self-excited oscillations is that the dam ping inthe system be small. This condition has been verified in crystals of pure simple metalsand pure salts, by means of direct measurements of dislocation velocities and by

Fig.

.

.. i.

I I I I I I I I I..0.2 0.4 0.6 0.8 1 o

Effect of stress (velocity)on the rate of dislocation multiplicationin a lithium fluoride crystal.The logarithmof the multiplicationrate is given as a function of the reciprocal appliedshear stress. The full circles are experimental measurements [data from Johnston andGilman I 960)).


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332 J. J. Gilman

ultrasonic attenuation measurements. In less pure crystals it is true for local regionsbetween impurities and defects. The 'Peierls force' on the primary glide planes inthese crystals is vanishingly sm all (Nadg orny i 1987). Therefo re, the da mp ing coeffi-cients for dislocation motion are small, of the order of 3x lop4 dyn cm-* whichgives dra g forces on dislocations tha t a re small comp ared with driving forces.

0 2. DISLOCATIONSS SELF-EXCITED OSCILLATORSWe start with the equ ation of motion ofa dislocation line and co nsider the effect

of impulsive forces on it. T o minimize complications, the m aterial tha t contains thedislocation line is taken to be cubic. Th e line must be screw type ifit is to multiplycross-glide. Therefore, only three parameters are needed to characterize it: theBurgers displacementb, the shear modulus G and the mass density p. Th e effectivemass per unit length will be taken to bern = pb2, an d the viscosity coefficient of themedium 7 he line will sup po rt vibrational m odes of lengthL up to L = A where A

represents a segment lying between points that are pinned by impurities, dipoles, orother crystal defects. The tensionof the line is T which is approximately equal toGh2/2rc.

The line tension T is not isotropic because the shear stiffness is a second-ordertensor. Thus the tension (change in energy with curvature) will be different for acurved line lying on the primary as against the secondary glide plane. We assumesimple one-axis curvatures and, to keep the discussion simple, consider only twotensions: one, T, on the primary plane, and the other,T,, on the secondary plane.These correspond to two shear moduliG and G, ( b being the same for both planes).

As a dislocation line moves, it continually collides with randomly distributedshear-strain fluctuations which exert forces on it tha t are ra ndo m in time a nd space.These forces are distributed isotropically for a stationary dislocation but have resul-tant directions opposed to the velocity vector of a moving line. Their magnitudesincrease as the velocity increases. No te th at this process is different from the therm alexcitation. Th e effects of the latte r w ould decrease with increasing velocity becausethere would be less time for an excitation as the velocity increased. In contrast, thebuffeting forces increase with the velocity because the changes in quasimomentumbecome larger, and the collision frequency increases.

3. T H E LUCTUATIONSA standa rd resultof fluctuation theory is tha t mean square shear-strain fluctua-

tions y is the shear strain) are given by

where k is the Boltzmann constant, T is the temperature and V s the volume ofinterest (Parrinello an d Ra hm an 1982, Weiner 1983). Th us the fluctuations are smallunless V0is small (atomic dimensions), and they increase in proportionto the squareroot of the temperature. The buffeting forces can be substantial. TakingV, = b L,and L = lob , G = 5 x 10 dyn cmP 2 and b = 2.5 A he stress fluctuations (strainmultiplied by modulus) given by eqn. (1) have magnitudes of about1 G P a a troom temperature. For smaller values ofL they are larger, up to the intrinsicshear strength of the material (i.e. aboutG/10). The magnitudes of the fluctuatingforces can also be estimated from the rate of change in momentum when a disloca-tion line collides with a fluctuation. The collision cross-section is approximately


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Koehlrr dislocation multiplicationprocexy 3 3 3

equal to b (Hirth and Lothe 1968). Equation (1) gives the stress fluctuationsT = G Ay; so the force fluctuation per unit length is

f z b AT. ( 2 )

Moving at a velocityvd, a dislocation line of lengthL and collision cross-sectionh sweeps out a volume Lbvdper second. The phonon occupation n umbersnp for theatoms in this volume are

I

np = [exp (&) - 11 ( 3 )where cp = (h /2n)w, is the phonon energy withh the Planck constant andwq hefrequency of a phonon with wavenumberq. At temperatures above the Debye tem-perature 0 , the phonon energy becomeskO, and the bracketed term in eqn. (3)becomes approximately equal to its argument (Walton1983); so the number of

phonon s per unit volume becomesn=

(9/2h3)(T / O ) .Then, the number of collisionswith the segment of lengthL is

L v ~n, = 4.5---,

b 8 4)

confirming that the collision rate increases with increasing temperature and disloca-tion velocity. ForL = cm, the co llision rate is4 x l o 1 * - when the dislocationvelocity is 3x lo4 cm s-, Thus the collision frequency is approximately equalto thelocal atomic vibration frequency.

3 4. RESPONSEF A DISLOCATION L I N E TO RANDOM EXCITATIONSUsing the given definitions, the equation of motion for the dislocation segment is

a2y av a2yax a t a t2-

Ti -5 v2 m- T(a, t)S x - a ) ,

where j s the displacementof the line (0 < .c < L ) , is time, br(a . t ) is the drivingstress function and 6(.u a ) is the delta function which places the driving stressfunction at .i = u and i = p or s. The terms on the left-hand side are the curvatureforce, the viscous drag force and the intertial force (all per unit length) (Newland

1984). This is the stand ard starting point for the theoryof the random vibrations ofastring.There are two resonant frequencies = n/L) T,/m)2 with i = p, s. In some

cases. they will be equal; so the dislocation line will tend to walk randomly fromone plane to the other. This will tend to maximize the rate of dislocation multi-plication, leading to high ductility (Gilman 1995). Since the dipoles left behind by theKoehler process will strain harden the initial regions of the glide planes, dislocationmotion will be inhibited there. However, the motionwill remain easy on both sidesof these central regions;so glide bands will spread sidewise into virgin regions u ntilthe bands begin to impinge on one another (Johnston and Gilman 1960).

The frequency spectrum of the fluctuationsis broad, ranging from zeroto theatom ic cut-off frequency which canbe estimated from the Debye frequency which isequal to k O / h (if 0 = 200 K, this is 4.2 x 10 s-). Thu s line segmen t oscillators ofvarious lengths can be excited by the motion.


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334 J . J. Gilman

Note that an interesting but speculative possibility is that the fluctuationsbecome correlated by the strain field of the dislocation enough to form vortices.Then a fast-moving dislocation might leave Karman vortex trails (Tietjens 1990)in its wake (for more details see Blevins (1990)). This speculative possibility wouldresult in semiregular buffeting in addition to the more localized random buffeting.Taking the size of the dislocation line to be 2b, according to hydrodynamic obser-vations (Tietjens 1990), the spacing of vortices of the same sign would be about 8b;so impulses in opposite directions would arrive about every 4b, or at a frequency ofabout vd/4b.

Equation (5) i s difficult to solve in general because impacts on the line arerandom in both x and t . Newland (1984) gave a particularly good discussion ofgeneral solutions. In the dislocation case, a particular problem is that the natureof Ti is not clear for dynamic conditions where time retardation effects are impor-tant. The dependence of Ti on curvature is also not clear for dynamic conditions.

Therefore, to simplify the problem, the line tension term is linearized, becoming anelastic restoring force proportional to y , and given by By with ,d = Gb2/4xL .This isequivalent to associating the effective line tension with only the lines core energy.For further simplification, let the random impacts along the length be replaced byeffective values F ( t ) at the midpoint of the line at random times and which last forshort times dT. Then eqn. 5 ) becomes

my rli Py = F ( t ) , ( 6 )

where the dots mean differentiation with respect to time, and F ( t ) is a drivin force.Dividing through by m, recognizing tha t the resonant frequency is w = (p/m)/, andsetting cy = 7/2mw, he solution of this can be written as the Dyhamel integral (Smith

1988)

y ( t )= 1 F ( t ) sin [ w ( t T ) ] exp [ - a w ( t r ) ] r.Since (and therefore a ) s small, the exponential damping term can be neglected.Furthermore, since the moving line will be subjected to constant average buffetingforces, F ( t ) = Fo = constant. Then, eqn. (7) becomes

(7)1

mw 0

y ( t ) = s in [ w ( t T ) ] dr.m w o

Integrating this and substituting for the resonant frequency give

FOWy ( t )=-[I - cos (wt)] = yo[l os (wt)]P (9)

where y o is the average displacement, and y ( t )oscillates around the average with anamplitude equal to 2y0.

8 5. THE ISTRIBUTIONOF DIPOLE HEIGHTSRandom excitation of the vibrational modes of a moving screw dislocation will

excite all the modes of X = 2L to X = 2b where X is the mode wavelength. We areinterested in the transverse modes whose corresponding energies range from hv,/2Lto hv,/2b where h is the Planck constant and v = ( G / P ) ~s the transverse soundspeed. If the maximum mode length to be considered is l o3 A he mode energyvaries from about 6 x lop6 to about 6 x lo p2 eV, a reasonable range.


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Koehler dislocation multiplication process 335

The overall Koehler process occurs in two stages: firstly, glideoff the primaryplane; seco ndly glideoff the secondary plane back onto the primary plane. Fo r eachposition held by the screw dislocation, its next move is either forward byb on theprimary plane, or forward obliquely onto the secondary glide plane. The inverseprobabilities are small. Let the probability of the former process bep , and theprobability of the latter process bes In structural materials,p will be somewhatgreater than s except for isotropic materials wheres = p = i. For strongly layeredmaterials, p >> s which completely suppresses the Koehler mechanism.

For the simplest (isotropic) case wherep = s = i, and the shear stresses on theprimary and secondary p lane (as well as any drag stresses) are equal, the probabilityP , of a multiple-cross-glide event in which the excursion on th e secondary plane isone atom ic height is the productof the probability of thep s step, and the prob-ability for the p step. That is, for a one-step excursion,P I = ( ) 2 = i ; or a two-step excursion, P1 = ( )3 = Q and so on to an n-step excursion, withP,, = (4Thu s the most probable dipole height is one atomic distance, and dipoleswith largeheights are improbable. For example,P l o 0-001, or 250 times less probable th anP I . This is consistent with the fact that only a few large dipoles are seen directly,whereas many dipoles (mostly not large) are inferred from changes in thermal con-ductivity (ordinary electron microscopy does not see dipoles with small heights).

It should be remembered that, in specimens th at a re undergoing plastic flow, thedipole population is in a state of considerable flux. Not only are dipoles beingcreated by the Koehler process, but also they are being dissociated by the localshear stresses. being re-formed through trapping interactions, and being annihilatedby collisions.

Let a be the glide plane spacing. The maximum dipole spacingh* is determinedby the value of the heighth at which the applied stress decomposes dipoles. Thenthe range of heights is (withC the shear modulus, v the Poisson ratio and b theBurgers displacement)

Gba < h

mechanism of the koehler dislocation multiplication process

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