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Materials Science and Engineering, 6, pp. 260-264, 1972-05-01
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A rate theory analysis of the temperature dependence of dislocation
velocity
Krausz, A. S.
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https://nrc-publications.canada.ca/eng/view/object/?id=8962d843-2992-4fbd-9950-3c5da862707a https://publications-cnrc.canada.ca/fra/voir/objet/?id=8962d843-2992-4fbd-9950-3c5da862707aA Rate Theory Analysis
of the Temperature Dependence of DislocA. S . KKAUSZ"
Nario~zrrl Rrscnrclr Courzril (!/'Canndn, Dirision o/'B~~ildirry Resenrch, Ortcirva, Orfi. (Cnnafla)
(Received January 5. 1970; in revised form April 20. 1970)
S U M M A R Y
A previous studj, .rhowed that uAen the appar.ent activation energy is n 1inem.futzction o f t h e stress the relation between t l ~ e dislocntion velocity, v , and thc stress, t , can be described \t,ell for nzanj* ct;~~stal.s on the basic. o f N I I N S J ' I ~ I I I ~ C ~ ~ I . ~ C ( I / /7ot~ntinl et1erg.v ha/.rici., rlc.c.o~tizt heltzjj f n A ~ n of' t / l ~ for.\t'rlrd rind reverse processes. The pi'esent mcestigation was carried out to determine whether the tenzperatur.e dependence of dislocation velocity could be described on the same basis, assuming that the activation para- meters are independent ofthe temperature. It is shou9i1 that with thzse simple conditions the rate theoi:v i.r in
good agreement wit11 the e.~perimental results obtain- ed on the stress and temper.ature dependence of the di.rlocation velocirj~for Ge, S i and CaF,. A relation- ship was also clericed fiom the absolute rate theory for the description o f t k e stress sensitivity, alnvlaln~.
The expre.ssion explains fully the experimental re.c.ults obtained in Al, C u and Ag in terms ofphysical quantities over the whole trmnperature range. It is concluded thnt the asymmett.ica1, triangular energy bar.i.ier slznpe is a usefirl approximation for the analy- sis of the tl~ermally activated movement of disloca- tions.
In einer fiiiheren Arbeit wurde gezeigr. dajl rler Zitsammenhang zrvischen Versetzut~g.sgesch~t~i~~cIiy- keit v und Schubspnnnung Tfiir vie/. Kristtillr~ gut mit der Vorstellung eines asymmetrischen Versetzungs- hindernisses beschrieben ~serden katzn, wenn die scheinbare Aktivierungsener.gie eine lineare Fiolk- tion der Sprinnung ist; dnbei Ir1erdc.n Vot.wiirt.r- und Riickwiii.fspro~esse beriicksichtigt. Die rorliege~zde Unter.c.uchung wurde du1.chgefii1a.t urn heraus-u- finden, ob die Tenzperatur~abhiingigkeit der Verset-
zungsgeschwindigkeit auf derselben Grundlage be- schrieben werden kann wenn mnn anninirnt, daJ die Aktivierun3spnrameter nicht von der Temperafur abhlingen. Es wird gezeigt, daJ dir Ratentheorie
unter diesen einfcichen Antlahmen in i u t e r
ber rein-
stirnmung mit esperinlentellen Restimrnungen der Spannungs- und Tempei~atui~ablziingigkeit der Ver- setzungsyescli\vinciigkc~it in G e , S i und CaF, ist. Fiir die S p a n i ~ u i i g s e t ~ ~ ~ ~ f b ~ d l i c l ~ k e i t alnv/alnt wurde aits der ahsoluten Ratentlieorie eine Beziehung abgeleitet. Diese Beziehung erklart die experimen- tellen Ergehnisse an Al, C u und Ag im gesamten Tenipc.ratu~.bereicll allein mit pl~ysikalischen Groben. Es t v i ~ ~ l die S c h l u ~ f o l g e ~ ~ u n g gezogen, dab die a.rymtnetrische dreieckige Form der Energiebarriere eine niitzliche Niiherunq Jir die Analyse der ther- misch aktiuierten Versetzungsbewegimg ist.Dans une etude anterieure le nzouvement des disloca- tions a Pte' dkcrit dans le cas de barriPres d'hnergie potentielle dissymktriques, la possibitite' de sauts inverses des dislocations ktant prise en con.c.idkrafion. Lorsque I'knergie d'act ivtrrion apparente est unefonc- tion lineaire de la cont~zinte, ce mod2le esplique bien, d a m le cas de beaucoup de c~.israux, In variation de la vitesse v des dislocatiol~s acec lu conti.ainte T .
L'objet de la pre'sente etude etait de voir si la varia- tion de la vitesse des dislocations en fonction de la tempkr.ature pouvair 6t1.e de'crite a I'aide du m6me modgle, les paran12tres de I'actication thermique etant suppo.c.6~ indhpendants de la temperature. O n trouve qu'uvec ce.r 11)pothbes simples, la thdorie du
- -.- -- - - .. .-.p---.---.-.-.p-p
* Present address: Department of Chemical Engineering, University of Ottawa, Ottawa, Canada.
Materials Science and E~zgineering
TEMPI:KATUKE DEPENDENCE OF I>ISLC)C'ATIC)N V I : L ~ C ' I ' I Y 26 1
glissement activk est en hon accortl arec les rksu1tat.v 111e17t et ~1~1n.s to~rte I'krhelle (les ter~7/1k1.rrtur.e.v, ci I'aide
expkrimentaux relat ijs ri la z;ur.irrtion de 10 uite.vs.cJ (1' qrcrnr1c~ur.s 1) h~~.riclues. 1e.s rl;.slr 1tcrt.v e.~pc;r.ir?~c~ntau.~ des dislocations en,fonction de Ici contraintr et ck lu obter7~r.s clcrns Al, Cu et Ao. 0 1 7 en c o n c l ~ ~ r q~re ICI temp6r.atur.e duns Ge, S i et CaF,. La t11e'ul.i~' ~10s har.ri2r.e (fincr.qie di.s.s~~r7zc;tricl~ie, do ,forr?le tricrngu- vitesses ahsolues de re'action a e'galen7ent conduit ci Illire, con.rtitue une ri/)pr.oxit~7atiurrpern~ettant danri- 1'6tablisse1.lzent d'lrne r.elation your Iu sensibilit~; ri 10 Ijaer utilemerzt 1. rlzourernent ther.moactici cles contrainte alnv/2lnr. Cette r.elation expliq~rc entiPrc- dislocatior~s.
The establishment of the temperature and stress dependence of the activation rate is of central interest in the study of dislocation mobility. In a previous study', an investigation was carried out to determine whether the effect of the stress on the dislocation velocity, c, could be described by a linearly stress-dependent apparent activation energy
AE. The linear stress dependence is mathematically
advantageous and physically reasonable as there are several mechanisms for which
where AE' is the activation energy, V is an activa- .tion volume and r e f , is the effective stress acting on the dislocation. It was shown in this study that at the above condition the stress dependence of the dislocation velocity in Ge, Si, InSb. GaSb, NaC1, LiF, Fe-3% Si, W, Mo, Ni can be described well with the asymmetrical form of the rate equation
(
AE6tTvbrcffbr.ll)- v, exp -
Ineqn. (2)subscripts fand b indicate that the quantity is associated with the forward and the backward movements of the dislocation respectively. The other symbols have their usual meaning.
In terms of the absolute rate theory2 eqn. (2) was expressed in explicit form for a quantum statistical system as
where QZ and Qr are the partition functions for the activated state and the reactant state respectively,
i
is the average distance travelled by the dislocation after each activation, K is the transmission coefficient;the other symbols have their usual meaning.
The analysis showed that eqn. (3) described well the stress dependcnce of the dislocation velocity for all of the investigated crystals. It is the purpose of the present paper to report the results ofa study that was carried out to develop an understanding of the temperature dependence of the dislocation mobility.
The mathe~natically simplest temperature depen- dence can be expressed as :
These conditions are well satisfied by several mechanisms for which the apparent activation energy is a linear function of the stress. In these cases the stress dependence represented by eqn. (2) is preserved and the investigation of the temperature dcpendencc predicted by eclns. (2) and (3) can follow iinmediatcly when the restrictions of expres- sioil (4) are superimposed.
Arltilj~.si.s tile direct dislocation velocity t7leasur.e- r?letits
I t has becn s h o w i ~ ~ that eqn. (3) represents well the stress dependence of the dislocation velocity in Ge3.4 and in Si5. The same experimental results and the measurements of Keig and CobleQn CaF, were now analyzed with respect to the temperature dependence of the dislocation velocity using cqns. (3) and (4). Good matching was obtained within the experimental scatter over the whole temperature range for all of the investigated measurements. A typical example is shown in Fig. I . It is concluded from these azalyses that when the restrictions of expression (4) are cuperimposcd eqn. (3) is a good approximation for thi. description of the thermally activated dislocation movement in Ge, Si and CaF,. Although i t is more reliable to use the direct dislocation velocity n~easurements. it is difficult to obtain sufficient experimental data. This necessitates
A. S. KRAUSZ
Fig. 1. The stress and temperature dependence of the dislocation velocity in Si. The shading indicates the experimental scatter5. The curve was calculated from eqn. (3). The activation para- meters are
depends on the temperature according to the following equation which has been derived from eqns. (2) and ( 4 ) :
V b
vb
AE: -AE; - (&+ v b ) ~ , f f 1 + - - exp - - - K c VI K k T v AE: - AE: -(K
+
v b ) ~ r r r kT I - e x p (5)In explicit form, according to eqns. ( 3 ) and ( 4 ) , the temperature dependence of the stress sensitivity is
v b l b ~ b Qbf Q; AE: -AE: - ( K + V ~ ) T , I ~ 1
+---
KT,,, T/, I f K f Q:aexp
= - kT kT Q: Q; AE: -AE: - ( K + +b)~,ff (6) I - - - exp If 4 Q: Qb k TEquation (5) describes the temperature depen- dence of the stress sensitivity when, as a first approximation, v b / v f is considered to be independent of the temperature. For a detailed expression of the temperature dependence, the explicit form given by eqn. ( 6 ) should be considered. In this equation the statistical mechanical terms can be transformed readily into the usual thermodynamic quantities.
In eqn. (5):
k Q: k Q,t
l f ~ f - 7
=
l b ~ b - -=
2 xlo5
cm sec-' (AE: - AEZ) <(&+
V b ) r e f f.
h Qf h Q;,From this condition it follows immediately that
AE: = 2.235 eV when T approaches zero :
AEZ = 2.215 eV
N o particular effort was made to obtain the best fit. If the temperature-independent part ( v , Vb/vf
&
using the observations obtained in indirect dis- and v b / v f ) of the pre-exponential factor in the location velocity measurements and analyzing the numerator of eqn. (5) is denoted by cc and in the temperature dependence of the stress sensitivity. denominator byP,
two cases have to be considered : Analysis of the indirect dislocation velociry measure- ( I ) B = lAs the temperature increases, nT increases ments
indefinitely and the slope of the nT versus T relation may approach
It has often been observed that the slope of the 1 2
j
In v versus In z relation is a function of the tempera- - -
(&+
vb) T e f fture7. For a triangular, asymmetrical energy barrier AEZ
-AE?
+(Vb+ & ) z e f f (7)the rate theory of dislocation mobility [eqns. ( 2 ) ,
(3), ( 4 ) ] predicts that the stress sensitivity A schematic representation of the nT versus T
relation is shown in Fig. 2 for
P =
1. When the(
a
ref,)
T , structure = n eqn. (7) has to be modified according to eqn. ( 6 ) . temperature dependence of v b / v f is significant,TEMPERATURE DEPENDENCE OF DISL,OC'ATION VIiLOC17'1' 263
0
T O K
Fig. 2. Schematic representat~on of the rrT us. T relation for P = I.
(2) b < I
As the temperature increases, nT increases asyinptotically
T / l ~ , f f l + x n T + - -
k 1 - p '
An inflection occurs at the temperature
IT;
at which tanh Inb
exp - AE:+
AE: -(vb+
T / ~ ) T , ~ ~k T
A schematic representation of the n T rersus T relation for
b<
1 is shown in Fig. 3. The possible temperature dependence of the pre-exponential factor vb/vf in eqn. (8) can also be considered as before.These conclusions have now to be compared with the information available in the literature on the temperature and stress dependence of 11.
Fig. 3. Schematic representation of the rrT cs. T relation for P < 1.
Li8.9 showed from a thermodynamic argument that as the temperature approaches O°K the stress sensitivity
and
This is in full agreement with the conclusions derived from the rate theory given by eqn. (6). Li's theory does not explain the temperature dependence of the stress sensitivity above absolute zero ; eqns. ( 5 ) and (6). however, describe the 11 versus T relation over
the whole temperature range.
The ,IT versus T measurements of Basinskilo carried out on Al, Cu and Ag single crystals were analyzed using eqn. (5). Figure 4 shows that the rate theory describes well the observed behaviour*.
3
50 ICI: i s 23C 250 300
TELI'Ej:.-JRE 'k
Fig. 4. 'l hc ( l 117) u s . 7 rclarion measured at constant strain rate level in Al single cyrstals". The curve was calculated from eqn.
(6) using the form
and
where z is the usual geometrical factor
* In an isolated study. using a n expression similar in form, Schmid" i~nalyzcd the strcss sensitivity of Cu. His analysis, however. was incorrect ;~ntl led to erroneous results.
364 A. S. KRAUSZ
These results are considered as good evidence in support of the investigated form of the rate theory and it is concluded that the linear approximation represented by eqns. ( 5 ) and (6) describes iully the previcusly unexplained -behaviour of the stress sensitivity in terms of physical quantities over the whole temperature range.
CONCLUSIONS AND SUMMARY
It has already been shown that for a linearly stress- dependent apparent activation energy the stress dependence of the dislocation velocity can be described well with the absolute rate theory if the energy barrier is asymmetrical. As a continuation of this study an investigation was carried out to determine whether the same rate equation could describe the temperature dependence of the dis- location velocity when the activation parameters are independent of the temperature. The investiga- tion led to the following conclusions:
(1) When the energy barrier is approximated with an asymmetrical triangle the absolute rate theory predicts well the experimentai results obtained for both the stress and temperature dependence of the dislocation velocity in Ge, Si and CaF,.
(2) The analysis indicates that the predicted tem- perature dependence of the stress sensitivity is also in good agreement with the experimental results obtained with Al, Cu and Ag. The thermodynamic argument of Li on the value and slope of the nT
versus T relation at absolute zero is supported by the rate theory. The previously unexplained relation between the stress sensitivity and temperature is fully described by the rate theory in terms of physical quantities over the whole temperature range.
ACKNOWLEDGEMENTS
The author is much in debt to K. J. Laidler for the many discussions which contributed materially to this paper. For Lhe kind comments the author expresses his gratitude to H. Eyring. This paper is a contribution from the Division ofBuilding Research, National Research Council of Canada, and is published with the approval of the Director of the Division.
REFERENCES I A. S. KRAUSZ, Acta Met., 16 (1968) 897.
2 S. GLASSTONE, K. J. LAIDLER A N D H. EYRING, The Theory of Rate Processes, McGraw-Hill, 1941.
3 A. R. C;i,~u~xunr, J . R. PATEL AND.L. G. R U E I N , .I. Appl.
Pkys., 33 (1962) 2736.
4 V. CELLI, M. KABLER, T. NINOMIYP. A N D R . TIIOMSO~T, PIlyj. Rev., 131 (1963) 58..
5 T. SUZUKI A N D H. KUJIMA, A C I U Met., 14 (1966) 913.. 6 G. A. KEIG A N D R. i. COBLE. J. Appl. Phys., 39 (1966) 6090. 7 J . W . CHRISTIAN, Acta Met., 15 (1967) 1257.
8 J . C. M. LI, Trans. AIME. 233 (1965) 219.
9 J. C. M. LI, in A. R. ROSENFIELD E I al. (eds.), Dislocation Dyr~amics, McGraw-Hill, 1968, p. 87.
10 Z. S. BASINSKI, Pllil. Mag., 4 (1959) 393. l l G . SCHMID, Phys. Status Solidi, 18 (1966) 829.
