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The structure of kinks at dislocation interphase boundaries and their role in boundary migration—II. Kinetic analyses including kink motion
Vol. 38, No. 6, pp. 889-896, 1990 Printed in Great Britain.All rights reserved
0956-7151/90$3.00+ 0.00 Copyright © 1990PergamonPress plc
Acta metall, mater.
THE STRUCTURE OF KINKS AT DISLOCATION INTERPHASE BOUNDARIES A N D THEIR ROLE IN B O U N D A R Y MIGRATION--II. KINETIC ANALYSES I N C L U D I N G KINK MOTION N. P R A B H U and J. M. H O W E Department of Metallurgical Engineering, and Materials Science, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A. (Received
15 D e c e m b e r 1989)
Akstraet--A kinetic model was formulated to account for the influence of the density of orientation, thermal and intersection kinks on the velocity of dislocation ledges during precipitate growth. The relative contributions of each mechanism to the ledge velocity are characterized and the kinetic model is compared with existing experimental data on dislocation ledge motion in the A1-Ag system. R6snm6--Un modulecin~tiqueest pr6sent6pour tenir compte de l'influence de la densit6 de d~crochements dus fi l'orientation, fi la temp6rature et fi l'intersection sur la vitesse des marches de dislocations au cours de la croissance des pr6cipit6s. Les contributions relatives de chaque m6canisme fi la vitesse des matches sont caract6ris6es et le mod61e cin6tique est compar6 aux donn6es exp6rimentales disponibles sur le mouvement des marches de dislocations dans le syst6me AI-Ag. Zasammenfassmng--EinkinetischesModell wird forrnuliert, um den Einflul3der Dichte von Orientierungsthermischen und Schneidprozel3kinkenauf die Geschwindigkeit von Versetzungsstufen w/ihrend des Wachstums einer Ausscheidung zu beschreiben. Der relative Beitrag eines jeden Mechanismus zur Geschwindigkeit der Stufe wird dargestellt; das kinetische Modell wird mit vorhandenen experimentellen Beobachtungen zur Bewegung von Versetzungsstufen im AI-Ag-System verglichen.
1. INTRODUCTION The results described in Part I of this investigation provided an additional level of detail about the growth mechanisms of precipitate plates in f.c.c.-,h.c.p, transformations. They showed that the ledgewise migration of dislocation interphase boundaries can be limited by kink nucleation and migration, and that kinks can be produced by orientation, thermal and intersection effects. Incorporation of these mechanisms into kinetic models should allow more accurate expressions to be developed for the growth kinetics of ledges at interphase boundaries. In this study, a kinetic model which directly connects kink and ledge motion with macroscopic thermodynamics is formulated, and the predictions from this model are compared with the experimental observations discussed in Part I and other data available in the literature. 2. KINETIC MODEL FOR DISLOCATION MOTION BY KINKS The velocity v~ of a single ledge (or step) migrating across and parallel to a planar interphase boundary by random atomic attachment to the edge has been calculated by several investigators [1-3], and is given by expressions such as vj = D ( C o -
C~)/[hlct(p)(Cp
-
Ce)]
(1)
where D is the solute interdiffusivity, Co is the average matrix composition far from the interface, C, is the matrix composition in equilibrium with the precipitate, Cp is the precipitate composition, ht is the ledge height and ~(p) is a constant ( ~ 2 ) which represents an effective diffusion distance [l, 2]. Equation (1) is suitable for describing a disordered ledge migrating according with diffusion-controlled kinetics. However, if a fairly straight ledge lies along or near to a low-energy boundary orientation (within ~10°), where a significant portion of the interphase boundary is immobile due to the absence of kinks, equation (l) may not accurately describe the growth velocity. Furthermore, equation (1) does not have any parameters which can account fo'r the dependence of the ledge velocity on the kink density due to thermal nucleation and interaction with other ledges. These mechanisms were all seen to be operative and affecting the ledge velocities in Part I. A simple formulation which alleviates most of these problems can be obtained by letting the velocity of kinks in a dislocation ledge vk be given by v~ in equation (1). This is a reasonable assumption since if the kinks are large, such as the 35 nm kinks in Fig. 5 of Part I, the solute flux across the kinks should be essentially the same as that across a disordered ledge except at the corners, which comprise only a small portion of the kink. Hence, migration of the
889
890
PRABHU and HOWE: KINKS AT INTERPHASE BOUNDARIES--II
(a) z
/[~__~_
solute
h I edcje / /
~//
/leage
I,/
I~
hkink
h
"I
. ~ " Direction of ~" motionof kink
= Direction of motion of ledge
<110> It < 1120> a
T
(b)
zI x
y
Fig. 1. (a) Illustration of the solute flux across a large kink in a ledge. The height of the kink is large compared to the height of the ledge so that the flux is assumed to be constant along the kink. (b) A profile of the kink along the y axis illustrating that equation (I), which describes the velocity of a step in an infinitely long ledge is suitable for describing the diffusion flux within the xz plane for the large kink. kink will be limited by the flux across the edge as for a disordered ledge, as illustrated in Fig. l(a) and (b). The velocity of the dislocation ledge can then be calculated using the expression [4] VI : h k Ck/)k (2) where h k is the height of the kink [Fig. l(a) and Fig. 2 in Part I], and Ck = Ck~ + Ck is the kink concentration or average interkink spacing, as in Part I. Substituting equations (1), (2) an (5) of Part I for ck due to orientation, thermal and intersection kinks, respectively, the velocity of a dislocation ledge migrating by the diffusive motion of kinks is given by vl = [tan 0 + (2hk/a)e (-~P/kr) + h k)L - l I D (Co - C~)/hj ~ (p)(Cp - Ce)
(3)
where 0 is the angle between the dislocation line and close-packed direction, a is the width of a single kink, AF* is the free-energy for double-kink formation, k is Boltzmann's constant, T is temperature and 2 is the interledge spacing. While it appears that the kink density described by equation (3) can exceed one kink per atom because the three source terms are summed, this is unlikely since one mechanism usually dominates the kink concentration for possible combina-
tions of orientation, temperature, etc. so that the sum of Ck < 1, as will be demonstrated in Sections 4 and 5. Also note that when ck -- 1, equation (3) reduces to equation (1). This provides further justification for letting Vk in equation (2) be given by equation (1). Although equation (3) accounts for a ledge velocity which is limited due to an absence of kinks along a Peierls valley and also for a reduced ledge velocity in a Peierls valley due to the thermal nucleation of double kinks and intersection kinks, it should be recognized that this equation is an approximation to the three-dimensional diffusion problem which needs to be solved in order to describe fully the lateral growth of ledges by kinks at solid-solid interfaces. Such a treatment has been made for solid-vapor interfaces [5]. This more detailed analysis may be particularly important for very small kinks, where the assumption that the solute flux across a kink is similar to that across a disordered ledge is probably not accurate. 3. N U M E R I C A L BASES FOR
CALCULATIONS
Ledge velocities in this study were calculated using composition and diffusivity data available for 7
PRABHU and HOWE: KINKS AT INTERPHASE BOUNDARIES--II
891
Table I. Parameters used to calculate ledge velocities from diminishing supersaturation (driving force) for equation(3) growth as the solvus temperature ( ~ 7 3 0 K ) is Temperature Diffusivity Cc CO C~, ~(p)2 approached. Thus, the ledge velocity depends (K) (cm2's) (atomic fraction) 573 2.86 × 10 ~2 0.008 0.042 0.667 2.0 strongly on both orientation and temperature and 623 2.03× I0 ]1 0.014 0.042 0.667 2.4 increases rapidly as a ledge curves away from a 673 1.08 × 10 ,0 0.024 0.042 0.667 2.8 low-energy orientation. Note that the ledge velocity is independent of the kink height for orientation kinks because of a cancellation in the equation for v~. (or 7') precipitate plates in an AI-Ag alloy. Diffusivities were calculated from the expression [6] 4.2. Thermal kinks D =0.11~,e I 27 '~3 ×1, 3/ .987 T) ( c m 2 / s ) and the equiIn order to understand the dependence of ledge librium matrix composition adjacent to the interface velocity on temperature due to the thermal nucleation Ce was estimated from the solvus in a recent version of double kinks, it is first necessary to understand the of the AI-Ag phase diagram [7]. The average matrix dependence of the activation energy for double-kink composition C0 was taken as 4 . 2 a t . % Ag and values nucleation on kink size. This is because the ledge for e(p) were obtained from the literature [2]. The velocity depends on the kink size through the activavalue of C~ of the ?, phase was chosen as 66.7 at.% Ag tion energy for double-kink formation. Figure 3, (Ag2AI) independent of temperature [8]. Table 1 lists which is similar to Fig. 4 in Part I, illustrates the all of the ~alues used to calculate ledge velocities from increases in dislocation length, interfacial area and equation 13) at T = 573, 623 and 673 K. precipitate volume which accompany nucleation of a The ledge height h was taken as the interplanar double kink on a ledge. An equation that describes spacing of two { 111 = matrix planes [9] or 0.4676 nm. the total free-energy change A F which occurs during The spacing between close-packed rows of atoms the formation of a double kink is given as (Peierls valleys) is 0.247 nm, which was assumed to be A F = -- VAFv + Ave + Lvs (4) the m i n i m u m kink height hk; higher kinks were specified as nh k where n = integer. The m i n i m u m kink where AFv is the free-energy change per unit volume separation a was taken as the interatomic spacing of precipitate phase formed, 7c is the chemical compobetween atoms along a close-packed row of AI atoms nent of the interracial energy of the ledge, ~,,~is the or 0.286 nm. 4. RESULTS
~
The next three sections analyze the effects of each of the three kink-density terms in the square brackets in equation (3) on the velocity of a dislocation ledge, Comparison among the limiting velocities due to orientation, thermal and intersection kinks as well as between the calculated growth kinetics and experimental measurements of these rates will be reported in the Discussion section.
b
15 to)
g ~o ~ 5 "6 -
~
/
~
0.0
~
,
O. I
0.2
,
,
0.3
0.4
0.5
0.6
Orientation Angle (rod)
4.1. Orientation kinks
Figure 21a) shows the dependence of ledge velocity on orientation angle for the case of geometric kinks of arbitrary height at three different temperatures. The ledge velocity is zero when the angle is zero due the absence of kinks, and it increases as tan 0 with increasing angle away from a low-energy ( 110 )= Ir ( 1120 )~, Peierls valley, as expected from the form of the first term in equation (3). The ledge velocity at 673 K increases about 2.2 x 1 0 - 6 cm/s per 0.1 rad ( ~ 6 ) awa3, from a (110)=11 ( l l ] 0 ) . . , o r i e n t a tion and reaches a value of 1.0 × 10-Scm/s for an orientation that is slightly greater than 0.4 rad ( ~ 23 °) away from the close-packed direction. Furthermore, as illustrated in Fig. 2(b), the ledge velocity increases rapidly with temperature in the range of 573-695 K (300-42T'C) due to a greater increase in diffusivity than decrease in driving force with temperature. The rapid decrease in velocity above 695 K is due to
~ -/ " ~
/
~, Io -(b) E b ~
5
"6 ~o -~ >
500
55o
soo
65o
7oo
Temperature (K)
Fig. 2. (a) Dislocation ledge velocity vs orientation angle for T = 573, 623 and 673 K (0.1 rad = 5.7°). (b) Dislocation ledge velocity vs temperature for several different ledge orientations.
892
PRABHU and HOWE: KINKS AT INTERPHASE BOUNDARIES--II to solve for the critical kink height for nucleation h* as
,~,l
h~== (5.40 × 10 -~° ;)c + 1.155 ~Tsl~hk)/ "
p_L///
'
-- 1.62 x 10 9AFv.
/,, ,- /. j/ / L~/ / " ini 2 { I II } =0.4676 nm ,=.~ ~._~
~
Fig. 3. Illustration of double-kink formation showing the increase in dislocation line and ledge area. structural component of the interfacial energy of the ledge and L, A and V are the length, area and volume increases, respectively, associated with formation of a kink of height hk. The dependence of the increase in length, area and volume as a function of the kink height can be derived by geometric analysis of Fig. 3, which gives the following relationships L =hk/0.866 A = 4.676 x 10-1°L = 5.40 x 10-1° hk
(5a) (5b)
V = (4.676 x 10-~°)(0.866 L)(1.5 L) = 8.099 x 10-1°h~.
(5c)
Substituting equations (5a)-(5c) into equation (4) and taking the derivative of A F with respect to h k at constant AFv and 7~ gives ~ A F / ~ h k = - 1.62 x 10 -9 AFv
+5.40 x 10-]°~c+ 1.155d?Jdh k (6) where the aTs/tThk has been included since ~ depends on hk as described by equation (3) in Part I. Equation (6) can be set equal to zero and rearranged in order
This procedure is analogous to derivation of the critical radius of a spherical precipitate used in classical nucleation theory [10], the main difference here being in the geometry and structural energy of the double kink. Normally, h~' in equation (7)would be substituted back into a further developed version of equation (4) to obtain AF* as a function of h*. However, because hk appears in a logarithm in the expression for d),s/dh k, the resulting equation cannot be solved analytically. Hence, the following alternate approach was employed. An expression for the free-energy change AF* associated with the nucleation of a double kink of critical height h ~ can be found by rearranging equation (7) to find AFv, and this expression for AFv is substituted into equation (4). In order to use either equation (7) or the resulting equation for AF*, values for the chemical and structural components of the interfacial free-energy ~c and dTs/~hk need to be included. A reasonable value for dL/Ohk is found by inserting the appropriate values for the elastic constants, etc. into equation (3) in Part I and differentiating equation (3) with respect to hk to yield t3'ys/63h k
=
4.232 x 10 -1l ln(6.985 x 109 hk)
- 3.142 x 10-11. (8) A value of 7c for a coherency dislocation ledge is difficult to estimate but 25 mJ/m 2 is considered to be a reasonable value [11]. This is two and one-half times that of 10 mJ/m 2, which was estimated for a coherent { I l l } f.c.c./f.c.c. A1-Ag interface at about 280°C [12], and is an order of magnitude less than the value of 350 mJ/m 2, which was estimated for an incoherent ~t/7 interface [13]. By substituting 25 mJ/m 2 for 7~ and equation (8) for 6~'~s/t~h k in equation (7), the following relationship between h~' and AF~ is obtained h ' A F t - 0 . 0 3 ln(6.985 x 109 h~')= 1.610 × 10 -2.
E o
25
Equation (9) shows that the dependence of the critical kink height h~' on AFv is nonlinear due to the expression for c3'ys/dhk . Since equation (9) is difficult to solve explicitly for h~ as a function ofAFv, AF~ can be obtained as a function of h ~' instead. Then, using h* for the other quantities in equation (4) and substituting AF~ from equation (9), the free-energy change AF* for a critical-size double kink h~' is given by
2o I.~
Io "~ •~ -
(9)
O.13eV
o
=
(7)
//
L,
500
'
'
550
600 Temperature
650
700
(K)
Fig. 4. Dislocation ledge velocity vs temperature for a ledge that is migrating by 1.24 nm (n = 5) double kinks with an activation energy of formation of 0.13 eV. The velocity of a ledge with an activation energy of 0.34 eV is also indicated for comparison with experiment.
AF* = -h*[1.304
x l0
II
+ 2.444 x 10-]= ln(6.985
x 10 9
h~')]
+ 4.889 x 10 -H h~' ln(6.985 x 109 h~') - 2.279 x 10 -tl h~
(10)
PRABHU and HOWE: KINKS AT INTERPHASE BOUNDARIES--It which shows that the free-energy change associated with the formation of a critical kink increases with the kink height h*. Due to the relationship in equation (9), it is also apparent that nucleation of small kinks is favored under conditions of large supersaturation and vice versa. Table 2 shows the values of AF* for various h* obtained from equation (10) for multiple kinks with heights of nh*, where n = 5, 10 and 20. The value of AF~ that would be required to nucleate the kink with height h* and activation energy AF* was obtained from equation (9) and is shown in the last column in Table 2. It is not possible to calculate these quantities for n < 5 because the equation for ~ in Part I becomes negative. This represents the limit of validity of the equation, which is most accurate when L >>p4. Figure 4 shows the dependence of ledge velocity on temperature for a ledge which is advancing by the nucleation of 1.24nm (n = 5) double kinks with an activation energy of formation of 0.13 eV. The ledge velocity increases ~ith temperature up to about 695 K and then rapidly decreases above this temperature, The increase in ledge velocity with temperature up to 695 K is due to an increase in thermal activation for nucleation of double kinks while the rapid decrease above this temperature is due to overbalancing by the diminishing supersaturation (driving force) for growth. The ledge ~elocities for 2.47 (n = 10) and 4.94nm (n =20) kinks are not shown in Fig. 4 because their maximum velocities at 695 K were only about 7.5 × 10 8 and 6.2 × 10-~5 cm/s, respectively. Thus, the ledge velocity decreases rapidly as the kink size increases due to the increasing activation energy for double-kink formation. Even though the ledge velocity is multiplied by the kink height in equation (3) the activation energy in the exponent dominates the velocity~ Therefore, it is unlikely that ledge migration will be controlled by the nucleation of thermal double-kinks unless the supersaturation is high and the thermal kinks are small, i.e. n < 10 and AF* < 0.5 eV. This is discussed further in Sections 5.1 and 5.2.2.
4.3. Intersection kinks Figdre 5(a)shows the dependence of ledge velocity on interledge spacing (intersection kinks) for the same three temperatures as in Fig. 2(a), assuming that the kink height is 1.24 nm (n = 5) for comparison with the thermal kink in Fig. 4. From Fig. 5(a) it is apparent that the ledge velocity increases linearly with decreasing interledge spacing as expected from equation (3). The ledge velocity also increases with Table 2. Calculatedvaluesof AF* and AF~ for critical kinks with h~' = 1.24, 2.47 and 4.94 nm h~' AF* AF~ n (nm) (eV)~ (J/cm3) 5 1.24 0.13o -65.22 l0 2.47 0.521 -38.96 20 4.94 1.563 -24.76 ~1 e V =
1.602
×
10-
~ J.
,~ ~ 'o ~. g ~
893
30 (a) 673 K z5 / 2o 15
"
~ l0 ,, 5 573K ~, -I I I I 1 I I L ~ o.oz 0.03 o.04 o.o5 0.o6 0,07 0,08 o.09 o.I I/(rnterledge Spacing) (nm) "1
-~ ~ b ~ g *'
la
3o -tb) 25 2o J5
"~ Io _-, 5 _~ > 500
550
6o0 650 Temperature (K)
7o0
Fig. 5. (a) Dislocation ledge velocity vs interledge spacing for T=573, 623 and 673K (0.02nm-~=50nm)and (b) dislocation ledge velocity vs temperature for interledge (interdislocation) spacings of 1, 10 and 50 nm. temperature up to 695 K through the diffusivity term, as illustrated in Fig. 5(b) and discussed with reference to orientation and thermal kinks for Figs 2 and 4. For an interledge spacing of 50 nm (0.02 nm -~) the ledge velocity reaches a maximum of about 6.0 x 10 -7 c m / s at 695 K and as the interledge separation decreases to 10 nm (0.1 n m - ~) the velocity increases by 5 times to 3.0 × 10 -6 c m / s . As the interledge spacing further decreases to 1 nm the velocity increases to about 2.9 x 10-Scm/s [Fig. 5(b)], approaching that of a disordered ledge. It is important to note that the ledge velocity is multiplied by the kink height for the case of intersection kinks [equation (3)] and that the values illustrated in Fig. 5 for a 1.24 nm kink with n = 5 will change proportionally with the value ofn. The height of an intersection kink is probably determined by factors such as the Burgers vectors of the crossing dislocation ledges and the height of the ledges, i.e. the elastic interaction between the crossing ledges, but this has not been investigated quantitatively. 5. DISCUSSION
ft. 1. Comparison of ledge velocities among orientation, thermal and intersection kinks Table 3 compares the ledge velocities due to orientation, thermal nucleation and intersection events for
894
PRABHU and HOWE: KINKS AT INTERPHASE BOUNDARIES--II Table 3. Ledgevelocitiesdue to orientation,thermal and intersectionkinks vs kink size and temperature Ledge velocity ( × 10 6cm/s) n 5
Height (nm) 1.235
10
2.470
20
4.940
Temperature Orientation (K) (0.05, 0.4 rad) 573 0.1, 0.8 695 1.2, 10.0 573 0.|, 0.8 695 1.2, 10.0 573 0.1, 0.8 695 1.2, 10.0
the same kink heights and temperatures. From these data, it is apparent that under conditions of high supersaturation, ledge migration by the nucleation of double kinks only a few atoms high is likely to set the velocity except when a ledge is nearly 0.6 rad away from a low-energy orientation or intersection kinks are only 1.0 nm apart. This conclusion is apparent from the data for n = 5 in the first two rows of Table 3. Since a ledge with an orientation of 0.6 rad away from a low-energy direction approximates a disordered ledge, a ledge moving by the nucleation of small double kinks should move continuously at a velocity similar to that of a disordered ledge migrating by long-range volume diffusion. Under conditions of low supersaturation where the activation energy for nucleation of thermal double kinks is large, such as in rows three and four of Table 3, the ledge velocity will be set by the formation of orientation kinks unless the ledge lies within a few degrees ( < 0.10 rad) of a low-energy orientation and there is a corresponding abundance of intersection kinks. The same trend continues for the largest kink in rows five and six in Table 3. When the intersection kinks are scarce or widely separated, the orientation kinks will control the ledge velocity. As the interledge spacing decreases the influence of the intersection kinks on the ledge velocity increases and becomes nearly twice that of the orientation kinks for spacings less than about 50.0 nm. Therefore, high solute supersaturations and intersecting dislocations can limit the ledge velocity when they are present; however, in their absence the ledge velocity will be controlled by its orientation with respect to the low-energy Peierls barrier in the lattice. An important problem which remains to be investigated is whether there is a characteristic kink size for each of the three different kink mechanisms, 5.2. Comparison between theoretical and experimental ledge velocities 5.2.1. Orientation kinks. To date, the orientation dependence of the ledge velocity shown in Fig. 2 has not been systematically examined experimentally, However, behvavior in the limiting cases of no kinks (0°) and many kinks ( > 20 °) has been documented, Several investigators have observed that dislocation ledges appear to become largely immoble when they align along low-energy (110),[[(11~0)¢ orientations on the faces of 7" and 7 (Ag2AI) plates during
Thermal (eV) I.I (0.13) 23.9(0.13) -0.08 (0.52) -6.2 × 10 ~5(1.56)
Intersection (50.0, 1.0 nm) 0.2, 2.1 0.6, 28.6 0.4, 4.2 1.2, 57.2 0.8, 8.4 2.4, 114.4
growth in in situ hot-stage TEM experiments [14, 15], in agreement with the zero velocity expected from equation (3) and illustrated in Fig. 2. At the other extreme, ledge velocities of 1-3 × 10-5 cm/s at 673 K (400°C) and 2-8 × 10 -5 cm/s at 698 K (425°C) have been reported for y plates during in situ hot-stage TEM experiments [14]. These are nearly equal to velocities predicted for angular deviations greater than about 0.5 rad away from a low-energy orientation in Fig. 2 and yield values similar to pure diffusion control. The orientation dependence in between these two extremes is an area which is deserving of further, careful in situ hot-stage TEM experiments, 5.2,2. Thermal kinks. For a 4.0 nm kink with n = 16, equation (10) predicts an activation energy for double-kink formation of 1.14 eV. Although this is a high value and it is 3.4 times greater than the experimentally determined value of 0.34 eV for the 4.0 nm kink in Fig. 3 of Part I, it demonstrates that the value of 0.34eV is not unreasonable. An explanation for the difference between the calculated and experimental value for a 4,0 nm kink may be that limitations in the detectability of double-kink nucleation by TEM, i.e. it is not possible to observe a kink much smaller than about 2.5 nm due to the width of the dislocation image using weak-beam dark-field imaging at temperature, combined with errors in estimating the kink height in and the magnification of the TEM image, have resulted in an overestimate of the nucleation kink size experimentally from Fig. 3 in Part I. For example, if the critical kink height h* were 2.5 nm instead of 4.0 nm, the value for AF* would decrease to 0.52 eV (Table 2), which is in fairly close agreement with the experimentally determined value of 0.34 eV. Also, the value of AF* is very sensitive to values used for the chemical and structural components of the interfacial free-energy and cannot be overly reliable despite a best effort. Given the limitations associated with experimental measurement of the kink height and the assumptions in equation (10), this can be considered as good agreement. It should be remembered that the experimental activation energy was determined from the kink density using equation (2) in Part I, which does not depend on a measurement of the double-kink size and that this is probably a reasonably accurate value. The validity of 0.34 eV as an activation energy for double-kink nucleation can also be examined by comparing the values of AFv in Table 2 for h* = 2.47
PRABHU and HOWE: KINKS AT INTERPHASE BOUNDARIES--II and 4.94nm with the values of AF~ determined for the y' phase from the A1-Ag phase diagram assuming an ideal solution model. From Table 2, AFv = -38.96 J/cm 3 for a 2.47 nm kink and - 2 4 . 7 6 J / c m 3 for a 4.94nm kink. Using an ideal solution model [16] and the compositions for the and V' phases in Table 1 yields AFv = -38.23 and --21.27 J/cm 3 at 573 and 673 K, respectively. These values indicate that nucleation of a double kink in the size range of 2.5-5.1) nm is reasonable below a ternperature of about 673 K, and in fact, this is in agreement with the experimental observation of double-kink nucleation in Fig. 3 of Part I, which was recorded at 648 + 15 K (375 + 15°C). The experimentally determined activation energy of 0.34eV and the immobility of ledges along ( l l 0 ) ~ ] l (11~0)~, orientations provide strong evidence that thermal nucleation of double kinks has a relatively minor influence on the growth kinetics of ledges under conditions of relatively low supersaturation in the AI-Ag s3stem. This is further illustrated in Fig. 4, where the ledge velocity for a double-kink activation energy of 0.34 eV reaches a maximum of about 0.8 ~ 10 6cm s at 700 K. However, ledge migration limited b2¢ the nucleation of double kinks a few atoms high under conditions of high supersaturation, where AF~ is greater than about - 4 0 . 0 J/cm 3 and the activation energy for double-kink nucleation is less than 0.13 eV, is certainly possible. Although ledges have been observed to appear to move continuously under conditions of high supersaturation during in situ hot-stage TEM studies [17], thereby providing experimental evidence that such behavior ls occurring, it should be remembered that it is not possible to see atomic kinks using amplitude contrast techniques. This problem needs to be addressed further using in situ high-resolution TEM techniques, 5.2.3. Intersection kinks. The only experimental data on the velocity of ledges possibly due to intersection kinks is for 7 plates in an AI-Ag alloy [14]. The velocity of dislocation ledges on the plate faces were measured by in situ hot-stage TEM techniques and the interkink spacing was calculated from a kinetic analysis which assumed that ledge motion across the plate faces occurred by a kinked dislocation mechanism. Thesc results produced an interkink spacing of 39 nm for a ledge velocity of 8.2 × 10-7 cm/s at 425°C (698 K). Since the measured interledge spacing on the same precipitates was 20-40 nm, it was suggested that intersection kinks could be responsible for the observed velocity. The experimental observations in Part I have shown that kinks can be produced when dislocation ledges cross. For a kink spacing of 30 nm, equation (31 predicts a dislocation ledge velocity of about 1.3 × 10-6cm/s for a kink height of 1.24nm (n = 5). This is in agreement with the experimental data to within a factor of two, which is reasonable considering the limitations of the experimental technique and the assumptions regarding the kink height
895
and velocity calculation. A velocity of about 1.3 × 10 -6 c m / s is also an order of magnitude less than a ledge velocity of about 5 × 10 5cm/s, whichis the velocity expected for a disordered ledge at this temperature. 6. CONCLUSIONS The results from this investigation demonstrate that: (i) A kinetic model has been formulated which accounts for the kinetics of atomic attachment at orientation, thermal and intersection kinks and their effects on the velocity of dislocation ledges. (ii) Several features in in situ TEM experiments that could not be accounted for by previous kinetic models can be described using the present expression for the ledge velocity; predicted velocities agree well with experimental data. (iii) The ledge velocity depends strongly on orientation and achieves values approaching a disordered interface for angles greater than about 20 ° away from a low-energy Peierls orientation due to the high density of orientation kinks. (iv) When ledges lie along Peierls valleys they can move by the formation of double kinks and intersection kinks, whose velocities depend on the activation energy for double-kink nucleation and the interledge (or interdislocation) spacing. (v) Under conditions of low supersaturation, the ledge velocity is most likely to be limited by orientation effects unless the ledge lies nearly parallel to a close-packed direction and/or there is a closelyspaced network of intersecting dislocations. (vi) Under conditions of high supersaturation, the ledge velocity is likely to be dominated by the nucleation of small thermal double-kinks. Intersection kinks play a major role when the ledges are closely spaced and the other two mechanisms are largely inoperative. research was supported by grants from the National Science Foundation (Grant No. DMR-8610439) and Alcoa. The authors are grateful to Professor H. I. Aaronson for a critical review of the manuscript. Acknowledgements--This
1. 2. 3. 4. 5. 6. 7.
REFERENCES G. J. Jones and R. Trivedi, J. appl. Phys. 42, 4299 (1971). C. Atkinson, Proc. R. Soc. A384, 107 (1982). M. Enomoto, Acta metall. 35, 935 (1987). J. P. Hirth and J. Lothe, Theory of Dislocations, 2nd edn, p. 537. Wiley, New York (1982). W. K. Burton, N. Cabrera and F. C. Frank, Phil. Trans. g. Soc. A243, 299 (1950-51). R. C. Weast (editor) Handbook of Chemistry and Physics, 63rd edn, p. F-52. CRC Press, Boca Raton, Fla (1982). R. P. Elliott and F. A. Shunk, Bull. Alloy Phase Diag. 1, 36 (1983).
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PRABHU and HOWE:
KINKS AT INTERPHASE BOUNDARIES--II
8. J. M. Howe and R. Gronsky, Scripta metall. 20, 1165 (1986). 9. J. M. Howe, U. Dahmen and R. Gronsky, Phil. Mag. A 56, 31 (1987). 10. J. W. Christian, The Theory of Transformations in Metals and Alloys, p. 420. Pergamon Press, Oxford (1965). 11. R. V. Ramanujan, unpublished research, Carnegie Mellon Univ., Pittsburgh, Pa (1988). 12. F. K. LeGoues, R. N. Wright, Y. W. Lee and H . I . Aaronson, Acta metall. 32, 1865 (1984).
13. H. I. Aaronson, K. C. Russell and G. W. Lorimer, Metall. Trans. 8A, 1644 (1977). 14. C. Laird and H. I. Aaronson, Acta metall. 17, 505 (1969). 15. J. M. Howe, Phase Transformations '87, p. 637. Inst. of Metals, London (1988). 16. D. A. Porter and K. E. Easterling, Phase Transformations in Metals and Alloys, p. 15. Van Nostrand Reinhold, New York (1981). 17. J. M. Howe, unpublished research, Carnegie Mellon Univ. Pittsburgh, Pa (1988).
0956-7151/90$3.00+ 0.00 Copyright © 1990PergamonPress plc
Acta metall, mater.
THE STRUCTURE OF KINKS AT DISLOCATION INTERPHASE BOUNDARIES A N D THEIR ROLE IN B O U N D A R Y MIGRATION--II. KINETIC ANALYSES I N C L U D I N G KINK MOTION N. P R A B H U and J. M. H O W E Department of Metallurgical Engineering, and Materials Science, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A. (Received
15 D e c e m b e r 1989)
Akstraet--A kinetic model was formulated to account for the influence of the density of orientation, thermal and intersection kinks on the velocity of dislocation ledges during precipitate growth. The relative contributions of each mechanism to the ledge velocity are characterized and the kinetic model is compared with existing experimental data on dislocation ledge motion in the A1-Ag system. R6snm6--Un modulecin~tiqueest pr6sent6pour tenir compte de l'influence de la densit6 de d~crochements dus fi l'orientation, fi la temp6rature et fi l'intersection sur la vitesse des marches de dislocations au cours de la croissance des pr6cipit6s. Les contributions relatives de chaque m6canisme fi la vitesse des matches sont caract6ris6es et le mod61e cin6tique est compar6 aux donn6es exp6rimentales disponibles sur le mouvement des marches de dislocations dans le syst6me AI-Ag. Zasammenfassmng--EinkinetischesModell wird forrnuliert, um den Einflul3der Dichte von Orientierungsthermischen und Schneidprozel3kinkenauf die Geschwindigkeit von Versetzungsstufen w/ihrend des Wachstums einer Ausscheidung zu beschreiben. Der relative Beitrag eines jeden Mechanismus zur Geschwindigkeit der Stufe wird dargestellt; das kinetische Modell wird mit vorhandenen experimentellen Beobachtungen zur Bewegung von Versetzungsstufen im AI-Ag-System verglichen.
1. INTRODUCTION The results described in Part I of this investigation provided an additional level of detail about the growth mechanisms of precipitate plates in f.c.c.-,h.c.p, transformations. They showed that the ledgewise migration of dislocation interphase boundaries can be limited by kink nucleation and migration, and that kinks can be produced by orientation, thermal and intersection effects. Incorporation of these mechanisms into kinetic models should allow more accurate expressions to be developed for the growth kinetics of ledges at interphase boundaries. In this study, a kinetic model which directly connects kink and ledge motion with macroscopic thermodynamics is formulated, and the predictions from this model are compared with the experimental observations discussed in Part I and other data available in the literature. 2. KINETIC MODEL FOR DISLOCATION MOTION BY KINKS The velocity v~ of a single ledge (or step) migrating across and parallel to a planar interphase boundary by random atomic attachment to the edge has been calculated by several investigators [1-3], and is given by expressions such as vj = D ( C o -
C~)/[hlct(p)(Cp
-
Ce)]
(1)
where D is the solute interdiffusivity, Co is the average matrix composition far from the interface, C, is the matrix composition in equilibrium with the precipitate, Cp is the precipitate composition, ht is the ledge height and ~(p) is a constant ( ~ 2 ) which represents an effective diffusion distance [l, 2]. Equation (1) is suitable for describing a disordered ledge migrating according with diffusion-controlled kinetics. However, if a fairly straight ledge lies along or near to a low-energy boundary orientation (within ~10°), where a significant portion of the interphase boundary is immobile due to the absence of kinks, equation (l) may not accurately describe the growth velocity. Furthermore, equation (1) does not have any parameters which can account fo'r the dependence of the ledge velocity on the kink density due to thermal nucleation and interaction with other ledges. These mechanisms were all seen to be operative and affecting the ledge velocities in Part I. A simple formulation which alleviates most of these problems can be obtained by letting the velocity of kinks in a dislocation ledge vk be given by v~ in equation (1). This is a reasonable assumption since if the kinks are large, such as the 35 nm kinks in Fig. 5 of Part I, the solute flux across the kinks should be essentially the same as that across a disordered ledge except at the corners, which comprise only a small portion of the kink. Hence, migration of the
889
890
PRABHU and HOWE: KINKS AT INTERPHASE BOUNDARIES--II
(a) z
/[~__~_
solute
h I edcje / /
~//
/leage
I,/
I~
hkink
h
"I
. ~ " Direction of ~" motionof kink
= Direction of motion of ledge
<110> It < 1120> a
T
(b)
zI x
y
Fig. 1. (a) Illustration of the solute flux across a large kink in a ledge. The height of the kink is large compared to the height of the ledge so that the flux is assumed to be constant along the kink. (b) A profile of the kink along the y axis illustrating that equation (I), which describes the velocity of a step in an infinitely long ledge is suitable for describing the diffusion flux within the xz plane for the large kink. kink will be limited by the flux across the edge as for a disordered ledge, as illustrated in Fig. l(a) and (b). The velocity of the dislocation ledge can then be calculated using the expression [4] VI : h k Ck/)k (2) where h k is the height of the kink [Fig. l(a) and Fig. 2 in Part I], and Ck = Ck~ + Ck is the kink concentration or average interkink spacing, as in Part I. Substituting equations (1), (2) an (5) of Part I for ck due to orientation, thermal and intersection kinks, respectively, the velocity of a dislocation ledge migrating by the diffusive motion of kinks is given by vl = [tan 0 + (2hk/a)e (-~P/kr) + h k)L - l I D (Co - C~)/hj ~ (p)(Cp - Ce)
(3)
where 0 is the angle between the dislocation line and close-packed direction, a is the width of a single kink, AF* is the free-energy for double-kink formation, k is Boltzmann's constant, T is temperature and 2 is the interledge spacing. While it appears that the kink density described by equation (3) can exceed one kink per atom because the three source terms are summed, this is unlikely since one mechanism usually dominates the kink concentration for possible combina-
tions of orientation, temperature, etc. so that the sum of Ck < 1, as will be demonstrated in Sections 4 and 5. Also note that when ck -- 1, equation (3) reduces to equation (1). This provides further justification for letting Vk in equation (2) be given by equation (1). Although equation (3) accounts for a ledge velocity which is limited due to an absence of kinks along a Peierls valley and also for a reduced ledge velocity in a Peierls valley due to the thermal nucleation of double kinks and intersection kinks, it should be recognized that this equation is an approximation to the three-dimensional diffusion problem which needs to be solved in order to describe fully the lateral growth of ledges by kinks at solid-solid interfaces. Such a treatment has been made for solid-vapor interfaces [5]. This more detailed analysis may be particularly important for very small kinks, where the assumption that the solute flux across a kink is similar to that across a disordered ledge is probably not accurate. 3. N U M E R I C A L BASES FOR
CALCULATIONS
Ledge velocities in this study were calculated using composition and diffusivity data available for 7
PRABHU and HOWE: KINKS AT INTERPHASE BOUNDARIES--II
891
Table I. Parameters used to calculate ledge velocities from diminishing supersaturation (driving force) for equation(3) growth as the solvus temperature ( ~ 7 3 0 K ) is Temperature Diffusivity Cc CO C~, ~(p)2 approached. Thus, the ledge velocity depends (K) (cm2's) (atomic fraction) 573 2.86 × 10 ~2 0.008 0.042 0.667 2.0 strongly on both orientation and temperature and 623 2.03× I0 ]1 0.014 0.042 0.667 2.4 increases rapidly as a ledge curves away from a 673 1.08 × 10 ,0 0.024 0.042 0.667 2.8 low-energy orientation. Note that the ledge velocity is independent of the kink height for orientation kinks because of a cancellation in the equation for v~. (or 7') precipitate plates in an AI-Ag alloy. Diffusivities were calculated from the expression [6] 4.2. Thermal kinks D =0.11~,e I 27 '~3 ×1, 3/ .987 T) ( c m 2 / s ) and the equiIn order to understand the dependence of ledge librium matrix composition adjacent to the interface velocity on temperature due to the thermal nucleation Ce was estimated from the solvus in a recent version of double kinks, it is first necessary to understand the of the AI-Ag phase diagram [7]. The average matrix dependence of the activation energy for double-kink composition C0 was taken as 4 . 2 a t . % Ag and values nucleation on kink size. This is because the ledge for e(p) were obtained from the literature [2]. The velocity depends on the kink size through the activavalue of C~ of the ?, phase was chosen as 66.7 at.% Ag tion energy for double-kink formation. Figure 3, (Ag2AI) independent of temperature [8]. Table 1 lists which is similar to Fig. 4 in Part I, illustrates the all of the ~alues used to calculate ledge velocities from increases in dislocation length, interfacial area and equation 13) at T = 573, 623 and 673 K. precipitate volume which accompany nucleation of a The ledge height h was taken as the interplanar double kink on a ledge. An equation that describes spacing of two { 111 = matrix planes [9] or 0.4676 nm. the total free-energy change A F which occurs during The spacing between close-packed rows of atoms the formation of a double kink is given as (Peierls valleys) is 0.247 nm, which was assumed to be A F = -- VAFv + Ave + Lvs (4) the m i n i m u m kink height hk; higher kinks were specified as nh k where n = integer. The m i n i m u m kink where AFv is the free-energy change per unit volume separation a was taken as the interatomic spacing of precipitate phase formed, 7c is the chemical compobetween atoms along a close-packed row of AI atoms nent of the interracial energy of the ledge, ~,,~is the or 0.286 nm. 4. RESULTS
~
The next three sections analyze the effects of each of the three kink-density terms in the square brackets in equation (3) on the velocity of a dislocation ledge, Comparison among the limiting velocities due to orientation, thermal and intersection kinks as well as between the calculated growth kinetics and experimental measurements of these rates will be reported in the Discussion section.
b
15 to)
g ~o ~ 5 "6 -
~
/
~
0.0
~
,
O. I
0.2
,
,
0.3
0.4
0.5
0.6
Orientation Angle (rod)
4.1. Orientation kinks
Figure 21a) shows the dependence of ledge velocity on orientation angle for the case of geometric kinks of arbitrary height at three different temperatures. The ledge velocity is zero when the angle is zero due the absence of kinks, and it increases as tan 0 with increasing angle away from a low-energy ( 110 )= Ir ( 1120 )~, Peierls valley, as expected from the form of the first term in equation (3). The ledge velocity at 673 K increases about 2.2 x 1 0 - 6 cm/s per 0.1 rad ( ~ 6 ) awa3, from a (110)=11 ( l l ] 0 ) . . , o r i e n t a tion and reaches a value of 1.0 × 10-Scm/s for an orientation that is slightly greater than 0.4 rad ( ~ 23 °) away from the close-packed direction. Furthermore, as illustrated in Fig. 2(b), the ledge velocity increases rapidly with temperature in the range of 573-695 K (300-42T'C) due to a greater increase in diffusivity than decrease in driving force with temperature. The rapid decrease in velocity above 695 K is due to
~ -/ " ~
/
~, Io -(b) E b ~
5
"6 ~o -~ >
500
55o
soo
65o
7oo
Temperature (K)
Fig. 2. (a) Dislocation ledge velocity vs orientation angle for T = 573, 623 and 673 K (0.1 rad = 5.7°). (b) Dislocation ledge velocity vs temperature for several different ledge orientations.
892
PRABHU and HOWE: KINKS AT INTERPHASE BOUNDARIES--II to solve for the critical kink height for nucleation h* as
,~,l
h~== (5.40 × 10 -~° ;)c + 1.155 ~Tsl~hk)/ "
p_L///
'
-- 1.62 x 10 9AFv.
/,, ,- /. j/ / L~/ / " ini 2 { I II } =0.4676 nm ,=.~ ~._~
~
Fig. 3. Illustration of double-kink formation showing the increase in dislocation line and ledge area. structural component of the interfacial energy of the ledge and L, A and V are the length, area and volume increases, respectively, associated with formation of a kink of height hk. The dependence of the increase in length, area and volume as a function of the kink height can be derived by geometric analysis of Fig. 3, which gives the following relationships L =hk/0.866 A = 4.676 x 10-1°L = 5.40 x 10-1° hk
(5a) (5b)
V = (4.676 x 10-~°)(0.866 L)(1.5 L) = 8.099 x 10-1°h~.
(5c)
Substituting equations (5a)-(5c) into equation (4) and taking the derivative of A F with respect to h k at constant AFv and 7~ gives ~ A F / ~ h k = - 1.62 x 10 -9 AFv
+5.40 x 10-]°~c+ 1.155d?Jdh k (6) where the aTs/tThk has been included since ~ depends on hk as described by equation (3) in Part I. Equation (6) can be set equal to zero and rearranged in order
This procedure is analogous to derivation of the critical radius of a spherical precipitate used in classical nucleation theory [10], the main difference here being in the geometry and structural energy of the double kink. Normally, h~' in equation (7)would be substituted back into a further developed version of equation (4) to obtain AF* as a function of h*. However, because hk appears in a logarithm in the expression for d),s/dh k, the resulting equation cannot be solved analytically. Hence, the following alternate approach was employed. An expression for the free-energy change AF* associated with the nucleation of a double kink of critical height h ~ can be found by rearranging equation (7) to find AFv, and this expression for AFv is substituted into equation (4). In order to use either equation (7) or the resulting equation for AF*, values for the chemical and structural components of the interfacial free-energy ~c and dTs/~hk need to be included. A reasonable value for dL/Ohk is found by inserting the appropriate values for the elastic constants, etc. into equation (3) in Part I and differentiating equation (3) with respect to hk to yield t3'ys/63h k
=
4.232 x 10 -1l ln(6.985 x 109 hk)
- 3.142 x 10-11. (8) A value of 7c for a coherency dislocation ledge is difficult to estimate but 25 mJ/m 2 is considered to be a reasonable value [11]. This is two and one-half times that of 10 mJ/m 2, which was estimated for a coherent { I l l } f.c.c./f.c.c. A1-Ag interface at about 280°C [12], and is an order of magnitude less than the value of 350 mJ/m 2, which was estimated for an incoherent ~t/7 interface [13]. By substituting 25 mJ/m 2 for 7~ and equation (8) for 6~'~s/t~h k in equation (7), the following relationship between h~' and AF~ is obtained h ' A F t - 0 . 0 3 ln(6.985 x 109 h~')= 1.610 × 10 -2.
E o
25
Equation (9) shows that the dependence of the critical kink height h~' on AFv is nonlinear due to the expression for c3'ys/dhk . Since equation (9) is difficult to solve explicitly for h~ as a function ofAFv, AF~ can be obtained as a function of h ~' instead. Then, using h* for the other quantities in equation (4) and substituting AF~ from equation (9), the free-energy change AF* for a critical-size double kink h~' is given by
2o I.~
Io "~ •~ -
(9)
O.13eV
o
=
(7)
//
L,
500
'
'
550
600 Temperature
650
700
(K)
Fig. 4. Dislocation ledge velocity vs temperature for a ledge that is migrating by 1.24 nm (n = 5) double kinks with an activation energy of formation of 0.13 eV. The velocity of a ledge with an activation energy of 0.34 eV is also indicated for comparison with experiment.
AF* = -h*[1.304
x l0
II
+ 2.444 x 10-]= ln(6.985
x 10 9
h~')]
+ 4.889 x 10 -H h~' ln(6.985 x 109 h~') - 2.279 x 10 -tl h~
(10)
PRABHU and HOWE: KINKS AT INTERPHASE BOUNDARIES--It which shows that the free-energy change associated with the formation of a critical kink increases with the kink height h*. Due to the relationship in equation (9), it is also apparent that nucleation of small kinks is favored under conditions of large supersaturation and vice versa. Table 2 shows the values of AF* for various h* obtained from equation (10) for multiple kinks with heights of nh*, where n = 5, 10 and 20. The value of AF~ that would be required to nucleate the kink with height h* and activation energy AF* was obtained from equation (9) and is shown in the last column in Table 2. It is not possible to calculate these quantities for n < 5 because the equation for ~ in Part I becomes negative. This represents the limit of validity of the equation, which is most accurate when L >>p4. Figure 4 shows the dependence of ledge velocity on temperature for a ledge which is advancing by the nucleation of 1.24nm (n = 5) double kinks with an activation energy of formation of 0.13 eV. The ledge velocity increases ~ith temperature up to about 695 K and then rapidly decreases above this temperature, The increase in ledge velocity with temperature up to 695 K is due to an increase in thermal activation for nucleation of double kinks while the rapid decrease above this temperature is due to overbalancing by the diminishing supersaturation (driving force) for growth. The ledge ~elocities for 2.47 (n = 10) and 4.94nm (n =20) kinks are not shown in Fig. 4 because their maximum velocities at 695 K were only about 7.5 × 10 8 and 6.2 × 10-~5 cm/s, respectively. Thus, the ledge velocity decreases rapidly as the kink size increases due to the increasing activation energy for double-kink formation. Even though the ledge velocity is multiplied by the kink height in equation (3) the activation energy in the exponent dominates the velocity~ Therefore, it is unlikely that ledge migration will be controlled by the nucleation of thermal double-kinks unless the supersaturation is high and the thermal kinks are small, i.e. n < 10 and AF* < 0.5 eV. This is discussed further in Sections 5.1 and 5.2.2.
4.3. Intersection kinks Figdre 5(a)shows the dependence of ledge velocity on interledge spacing (intersection kinks) for the same three temperatures as in Fig. 2(a), assuming that the kink height is 1.24 nm (n = 5) for comparison with the thermal kink in Fig. 4. From Fig. 5(a) it is apparent that the ledge velocity increases linearly with decreasing interledge spacing as expected from equation (3). The ledge velocity also increases with Table 2. Calculatedvaluesof AF* and AF~ for critical kinks with h~' = 1.24, 2.47 and 4.94 nm h~' AF* AF~ n (nm) (eV)~ (J/cm3) 5 1.24 0.13o -65.22 l0 2.47 0.521 -38.96 20 4.94 1.563 -24.76 ~1 e V =
1.602
×
10-
~ J.
,~ ~ 'o ~. g ~
893
30 (a) 673 K z5 / 2o 15
"
~ l0 ,, 5 573K ~, -I I I I 1 I I L ~ o.oz 0.03 o.04 o.o5 0.o6 0,07 0,08 o.09 o.I I/(rnterledge Spacing) (nm) "1
-~ ~ b ~ g *'
la
3o -tb) 25 2o J5
"~ Io _-, 5 _~ > 500
550
6o0 650 Temperature (K)
7o0
Fig. 5. (a) Dislocation ledge velocity vs interledge spacing for T=573, 623 and 673K (0.02nm-~=50nm)and (b) dislocation ledge velocity vs temperature for interledge (interdislocation) spacings of 1, 10 and 50 nm. temperature up to 695 K through the diffusivity term, as illustrated in Fig. 5(b) and discussed with reference to orientation and thermal kinks for Figs 2 and 4. For an interledge spacing of 50 nm (0.02 nm -~) the ledge velocity reaches a maximum of about 6.0 x 10 -7 c m / s at 695 K and as the interledge separation decreases to 10 nm (0.1 n m - ~) the velocity increases by 5 times to 3.0 × 10 -6 c m / s . As the interledge spacing further decreases to 1 nm the velocity increases to about 2.9 x 10-Scm/s [Fig. 5(b)], approaching that of a disordered ledge. It is important to note that the ledge velocity is multiplied by the kink height for the case of intersection kinks [equation (3)] and that the values illustrated in Fig. 5 for a 1.24 nm kink with n = 5 will change proportionally with the value ofn. The height of an intersection kink is probably determined by factors such as the Burgers vectors of the crossing dislocation ledges and the height of the ledges, i.e. the elastic interaction between the crossing ledges, but this has not been investigated quantitatively. 5. DISCUSSION
ft. 1. Comparison of ledge velocities among orientation, thermal and intersection kinks Table 3 compares the ledge velocities due to orientation, thermal nucleation and intersection events for
894
PRABHU and HOWE: KINKS AT INTERPHASE BOUNDARIES--II Table 3. Ledgevelocitiesdue to orientation,thermal and intersectionkinks vs kink size and temperature Ledge velocity ( × 10 6cm/s) n 5
Height (nm) 1.235
10
2.470
20
4.940
Temperature Orientation (K) (0.05, 0.4 rad) 573 0.1, 0.8 695 1.2, 10.0 573 0.|, 0.8 695 1.2, 10.0 573 0.1, 0.8 695 1.2, 10.0
the same kink heights and temperatures. From these data, it is apparent that under conditions of high supersaturation, ledge migration by the nucleation of double kinks only a few atoms high is likely to set the velocity except when a ledge is nearly 0.6 rad away from a low-energy orientation or intersection kinks are only 1.0 nm apart. This conclusion is apparent from the data for n = 5 in the first two rows of Table 3. Since a ledge with an orientation of 0.6 rad away from a low-energy direction approximates a disordered ledge, a ledge moving by the nucleation of small double kinks should move continuously at a velocity similar to that of a disordered ledge migrating by long-range volume diffusion. Under conditions of low supersaturation where the activation energy for nucleation of thermal double kinks is large, such as in rows three and four of Table 3, the ledge velocity will be set by the formation of orientation kinks unless the ledge lies within a few degrees ( < 0.10 rad) of a low-energy orientation and there is a corresponding abundance of intersection kinks. The same trend continues for the largest kink in rows five and six in Table 3. When the intersection kinks are scarce or widely separated, the orientation kinks will control the ledge velocity. As the interledge spacing decreases the influence of the intersection kinks on the ledge velocity increases and becomes nearly twice that of the orientation kinks for spacings less than about 50.0 nm. Therefore, high solute supersaturations and intersecting dislocations can limit the ledge velocity when they are present; however, in their absence the ledge velocity will be controlled by its orientation with respect to the low-energy Peierls barrier in the lattice. An important problem which remains to be investigated is whether there is a characteristic kink size for each of the three different kink mechanisms, 5.2. Comparison between theoretical and experimental ledge velocities 5.2.1. Orientation kinks. To date, the orientation dependence of the ledge velocity shown in Fig. 2 has not been systematically examined experimentally, However, behvavior in the limiting cases of no kinks (0°) and many kinks ( > 20 °) has been documented, Several investigators have observed that dislocation ledges appear to become largely immoble when they align along low-energy (110),[[(11~0)¢ orientations on the faces of 7" and 7 (Ag2AI) plates during
Thermal (eV) I.I (0.13) 23.9(0.13) -0.08 (0.52) -6.2 × 10 ~5(1.56)
Intersection (50.0, 1.0 nm) 0.2, 2.1 0.6, 28.6 0.4, 4.2 1.2, 57.2 0.8, 8.4 2.4, 114.4
growth in in situ hot-stage TEM experiments [14, 15], in agreement with the zero velocity expected from equation (3) and illustrated in Fig. 2. At the other extreme, ledge velocities of 1-3 × 10-5 cm/s at 673 K (400°C) and 2-8 × 10 -5 cm/s at 698 K (425°C) have been reported for y plates during in situ hot-stage TEM experiments [14]. These are nearly equal to velocities predicted for angular deviations greater than about 0.5 rad away from a low-energy orientation in Fig. 2 and yield values similar to pure diffusion control. The orientation dependence in between these two extremes is an area which is deserving of further, careful in situ hot-stage TEM experiments, 5.2,2. Thermal kinks. For a 4.0 nm kink with n = 16, equation (10) predicts an activation energy for double-kink formation of 1.14 eV. Although this is a high value and it is 3.4 times greater than the experimentally determined value of 0.34 eV for the 4.0 nm kink in Fig. 3 of Part I, it demonstrates that the value of 0.34eV is not unreasonable. An explanation for the difference between the calculated and experimental value for a 4,0 nm kink may be that limitations in the detectability of double-kink nucleation by TEM, i.e. it is not possible to observe a kink much smaller than about 2.5 nm due to the width of the dislocation image using weak-beam dark-field imaging at temperature, combined with errors in estimating the kink height in and the magnification of the TEM image, have resulted in an overestimate of the nucleation kink size experimentally from Fig. 3 in Part I. For example, if the critical kink height h* were 2.5 nm instead of 4.0 nm, the value for AF* would decrease to 0.52 eV (Table 2), which is in fairly close agreement with the experimentally determined value of 0.34 eV. Also, the value of AF* is very sensitive to values used for the chemical and structural components of the interfacial free-energy and cannot be overly reliable despite a best effort. Given the limitations associated with experimental measurement of the kink height and the assumptions in equation (10), this can be considered as good agreement. It should be remembered that the experimental activation energy was determined from the kink density using equation (2) in Part I, which does not depend on a measurement of the double-kink size and that this is probably a reasonably accurate value. The validity of 0.34 eV as an activation energy for double-kink nucleation can also be examined by comparing the values of AFv in Table 2 for h* = 2.47
PRABHU and HOWE: KINKS AT INTERPHASE BOUNDARIES--II and 4.94nm with the values of AF~ determined for the y' phase from the A1-Ag phase diagram assuming an ideal solution model. From Table 2, AFv = -38.96 J/cm 3 for a 2.47 nm kink and - 2 4 . 7 6 J / c m 3 for a 4.94nm kink. Using an ideal solution model [16] and the compositions for the and V' phases in Table 1 yields AFv = -38.23 and --21.27 J/cm 3 at 573 and 673 K, respectively. These values indicate that nucleation of a double kink in the size range of 2.5-5.1) nm is reasonable below a ternperature of about 673 K, and in fact, this is in agreement with the experimental observation of double-kink nucleation in Fig. 3 of Part I, which was recorded at 648 + 15 K (375 + 15°C). The experimentally determined activation energy of 0.34eV and the immobility of ledges along ( l l 0 ) ~ ] l (11~0)~, orientations provide strong evidence that thermal nucleation of double kinks has a relatively minor influence on the growth kinetics of ledges under conditions of relatively low supersaturation in the AI-Ag s3stem. This is further illustrated in Fig. 4, where the ledge velocity for a double-kink activation energy of 0.34 eV reaches a maximum of about 0.8 ~ 10 6cm s at 700 K. However, ledge migration limited b2¢ the nucleation of double kinks a few atoms high under conditions of high supersaturation, where AF~ is greater than about - 4 0 . 0 J/cm 3 and the activation energy for double-kink nucleation is less than 0.13 eV, is certainly possible. Although ledges have been observed to appear to move continuously under conditions of high supersaturation during in situ hot-stage TEM studies [17], thereby providing experimental evidence that such behavior ls occurring, it should be remembered that it is not possible to see atomic kinks using amplitude contrast techniques. This problem needs to be addressed further using in situ high-resolution TEM techniques, 5.2.3. Intersection kinks. The only experimental data on the velocity of ledges possibly due to intersection kinks is for 7 plates in an AI-Ag alloy [14]. The velocity of dislocation ledges on the plate faces were measured by in situ hot-stage TEM techniques and the interkink spacing was calculated from a kinetic analysis which assumed that ledge motion across the plate faces occurred by a kinked dislocation mechanism. Thesc results produced an interkink spacing of 39 nm for a ledge velocity of 8.2 × 10-7 cm/s at 425°C (698 K). Since the measured interledge spacing on the same precipitates was 20-40 nm, it was suggested that intersection kinks could be responsible for the observed velocity. The experimental observations in Part I have shown that kinks can be produced when dislocation ledges cross. For a kink spacing of 30 nm, equation (31 predicts a dislocation ledge velocity of about 1.3 × 10-6cm/s for a kink height of 1.24nm (n = 5). This is in agreement with the experimental data to within a factor of two, which is reasonable considering the limitations of the experimental technique and the assumptions regarding the kink height
895
and velocity calculation. A velocity of about 1.3 × 10 -6 c m / s is also an order of magnitude less than a ledge velocity of about 5 × 10 5cm/s, whichis the velocity expected for a disordered ledge at this temperature. 6. CONCLUSIONS The results from this investigation demonstrate that: (i) A kinetic model has been formulated which accounts for the kinetics of atomic attachment at orientation, thermal and intersection kinks and their effects on the velocity of dislocation ledges. (ii) Several features in in situ TEM experiments that could not be accounted for by previous kinetic models can be described using the present expression for the ledge velocity; predicted velocities agree well with experimental data. (iii) The ledge velocity depends strongly on orientation and achieves values approaching a disordered interface for angles greater than about 20 ° away from a low-energy Peierls orientation due to the high density of orientation kinks. (iv) When ledges lie along Peierls valleys they can move by the formation of double kinks and intersection kinks, whose velocities depend on the activation energy for double-kink nucleation and the interledge (or interdislocation) spacing. (v) Under conditions of low supersaturation, the ledge velocity is most likely to be limited by orientation effects unless the ledge lies nearly parallel to a close-packed direction and/or there is a closelyspaced network of intersecting dislocations. (vi) Under conditions of high supersaturation, the ledge velocity is likely to be dominated by the nucleation of small thermal double-kinks. Intersection kinks play a major role when the ledges are closely spaced and the other two mechanisms are largely inoperative. research was supported by grants from the National Science Foundation (Grant No. DMR-8610439) and Alcoa. The authors are grateful to Professor H. I. Aaronson for a critical review of the manuscript. Acknowledgements--This
1. 2. 3. 4. 5. 6. 7.
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8. J. M. Howe and R. Gronsky, Scripta metall. 20, 1165 (1986). 9. J. M. Howe, U. Dahmen and R. Gronsky, Phil. Mag. A 56, 31 (1987). 10. J. W. Christian, The Theory of Transformations in Metals and Alloys, p. 420. Pergamon Press, Oxford (1965). 11. R. V. Ramanujan, unpublished research, Carnegie Mellon Univ., Pittsburgh, Pa (1988). 12. F. K. LeGoues, R. N. Wright, Y. W. Lee and H . I . Aaronson, Acta metall. 32, 1865 (1984).
13. H. I. Aaronson, K. C. Russell and G. W. Lorimer, Metall. Trans. 8A, 1644 (1977). 14. C. Laird and H. I. Aaronson, Acta metall. 17, 505 (1969). 15. J. M. Howe, Phase Transformations '87, p. 637. Inst. of Metals, London (1988). 16. D. A. Porter and K. E. Easterling, Phase Transformations in Metals and Alloys, p. 15. Van Nostrand Reinhold, New York (1981). 17. J. M. Howe, unpublished research, Carnegie Mellon Univ. Pittsburgh, Pa (1988).