The Kink Picture of Dislocation Mobility and Dislocation-point-defect Interactions

The paper gives a coherent account of the theory of d i s loca t ion mobil i ty i n terms of kink-pair formation and kink migration. Three leve l s of descr ip t ion a r i s e i n a n a t u r a l way, namely those of the motion of s t r a i g h t d i s loca t ions , of kink-pair formation on d i s loca t ions , and of kink migration. The i n t e r r e l a t i o n s h i p s between these leve l s ("hierarchies") a r e discussed. The i n t e r a c t i o n with phonons and foreign atoms ("impurities") i s t rea ted on the t h i r d l e v e l , t h a t of the kink mobility. Experimental information on the proper t i es of kinks may be obtained from measurements of d i s loca t ion v e l o c i t i e s , of flow s t r e s s , and of i n t e r n a l f r i c t i o n and modulus e f f e c t . The comparison between theory and experiment i s i l l u s t r a t e d by examples from valence c r y s t a l s (d i s loca t ion ve loc i ty i n Ge) and from body-centred cubic t r a n s i t i o n metals (y re laxa t ion = kink-pair formation on screw d is loca t ions ; dislocation-enhanced Snoek e f f e c t ; SnoekK6ster re laxa t ion i n Nb and Ta). 1 . Introduct ion and General Background.The "discovery" of d i s loca t ions during the period 1929-34 provided the explanat ion f o r the observation t h a t i n many c r y s t a l l i n e mate r ia l s the c r i t i c a l shear s t r e s s a. f o r p l a s t i c deformation by g l i d e i s several orders of magnitude lower than the t h e o r e t i c a l shear s t reng th oth as calculated from the model of per fec t c r y s t a l s . Already i n some of t h e e a r l y work i t was recognized t h a t the resis tance-to-gl ide due t o the d i s c r e t e nature of c r y s t a l s was not completel y removed by the movement of s t r a i g h t d i s loca t ions running p a r a l l e l t o one of the major c rys ta l lographic d i rec t ions ( t h i s was the model used throughout i n the e a r l y days of d i s loca t ion theory) . Based on what i s now ca l led the PRANDTL-DEHLINGERFRENKEL-KONTOROVA model [1,6,7] an attempt t o est imate the s t r e s s required to move a s t r a i g h t "dis locat ion" i n a c r y s t a l i s already contained i n u.DEHLINGER'~ "Verhakungen" paper of 1929 [ l ] . (A b r i e f h i s t o r i c a l account of e a r l y d i s loca t ion models and t h e i r re la t ionsh ips t o the problem of d i s s i p a t i o n of mechanical energy ( = i n t e r n a l f r i c t i o n ) i n c r y s t a l s has appeared elsewhere [81.) Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981531 C5-202 JOURNAL DE PHYSIQUE A t the suggestion of E.OROWAN, i n 1939 R.PEIERLS [g] performed the f i r s t det a i l e d ca lcu la t ion of the resolved shear s t r e s s required t o move a s t r a i g h t d i s loca t ion through a c r y s t a l i n the absence of thermal f luc tua t ions . This s t r e s s is now ca l led the P e i e r l s s t r e s s op. The l i n e energy Ed of a s t r a i g h t d i s loca t ion on a given g l i d e plane i s a minimum i n the so-called "Peier ls va l leys" and a maximum on the "Peier ls h i l l s " (comp. Fig.1). The o r i g i n a l est imate of P e i e r l s ( f o r a h i s t o r i c a l descr ip t ion see [10]) was corrected and improved by var ious authors; f o r the methods used and the r e s u l t s obtained the reader i s re fe r red t o the review l i t e r a t u r e []l-141. The bes t ca lcu la t ions ava i lab le ind ica te t h a t i n many instances the observed c r i t i c a l shear s t r e s s e s , a t l e a s t a t not too low temperatures, a r e subs t a n t i a l l y lower than the calculated P e i e r l s s t r e s s e s . I n the post-war development of d i s loca t ion theory i t became soon c l e a r t h a t d i s loca t ion l i n e s must be considered f l e x i b l e . W.SHOCKLEY [l51 pointed out t h a t together with the exis tence of the P e i e r l s p o t e n t i a l t h i s may lead t o the formation of *, i . e . , shor t segments of d i s loca t ion l i n e s i n which these cross over from one P e i e r l s va l ley t o a neighbouring one (comp. Fig. l ) , and estimated t h a t kinks may form i n thermal equilibrium. A.SEEGER [12,16] recognized t h a t the thermally ac t iva ted formation of p a i r s of kinks of opposite s ign ( o r i g i n a l name "double kinks", i n the following b r i e f l y ca l led kink p a i r s ) under the ac t ion of an o s c i l l a t i n g applied s t r e s s might lead t o a re laxa t ion e f f e c t i n i n t e r n a l f r i c t i o n . He proposed t h a t the re laxa t ion process observed by P.G.BORDON1 [17,18] i n various deformed f c c metals below room temperature might be due t o t h i s process. This suggestion was developed f u r t h e r i n numerous theore t ica l and experimental papers (see, e .g . , [19,20]) and appears now to be general ly accepted as the cor rec t explanat ion of the Bordoni re laxa t ion . From a h i s t o r i c a l point of view i t i s i n t e r e s t i n g t o note t h a t q u i t e a l a rge number of a l t e r n a t i v e explanations involving d i s loca t ions have been proposed during the l a s t twenty-five years (see, e . g . , [20,211) but t h a t the f i r s t explanation i n terms of disLocation theory proved t o be c o r r e c t (BORDONI himself [l81 suggested an explanat ion not involving d i s loca t ions) . According t o the kink-pair formation model the Bordoni re laxa t ion i s an i n t r i n s i c d i s loca t ion property d i r e c t l y r e l a t e d to the exis tence of the P e i e r l s p o t e n t i a l . I n most c r y s t a l s i t s t i l l provides the bes t information f o r est imating the numerical value of the P e i e r l s s t r e s s 0 or the height of the P e i e r l s b a r r i e r s of d i s loca t ions P lying p a r a l l e l t o c rys ta l lographic d i r e c t i o n s . Recently, the modification of the thermally ac t iva ted kink-pair formation process by atomic defects ( i n p a r t i c u l a r by foreign i n t e r s t i t i a l atoms) has become the subject of much i n t e r e s t , a s i s w i t nessed by t h e present conference. In add i t ion t o kinks t h a t a r e formed by thermal a c t i v a t i o n we have to consider geometrical kinks 1221. These a re kinks t h a t a r e present even a t the lowest temperatu res i n d i s loca t ion l i n e s t h a t a r e approximately p a r a l l e l t o low-index c rys ta l lo graphic d i rec t ions and f ixed by anchoring points not lying i n the same P e i e r l s Peierls hill valley hill