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Kinetic Theory of Dislocation Climb. II. Steady State Edge Dislocation ClimbR. W. Balluffi and R. M. Thomson Citation: Journal of Applied Physics 33, 817 (1962); doi: 10.1063/1.1777172 View online: http://dx.doi.org/10.1063/1.1777172 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/33/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Stress and Displacement Fields of an Edge Dislocation that Climbs with a Uniform Velocity J. Appl. Phys. 38, 2612 (1967); 10.1063/1.1709956 Kinetic Theory of Dislocation Climb. I. General Models for Edge and Screw Dislocations J. Appl. Phys. 33, 803 (1962); 10.1063/1.1777171 Theory of Dislocation Climb in Metals J. Appl. Phys. 31, 1077 (1960); 10.1063/1.1735749 SteadyState Creep through Dislocation Climb J. Appl. Phys. 28, 362 (1957); 10.1063/1.1722747 Theory of SteadyState Creep Based on Dislocation Climb J. Appl. Phys. 26, 1213 (1955); 10.1063/1.1721875
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JOURNAL OF APPLIED PHYSICS VOLUME 33, :-;rUMBER 3 MARCH, 1962
Kinetic Theory of Dislocation Climb. II. Steady State Edge Dislocation Climb*
R. W. BALLUFFI AND R. M. THOMSON
University of Illinois, Urbana, Illinois
(Received July 14, 1961)
Quantitative expressions for the steady state climb rate of a straight unconstrained edge dislocation are obtained. The results represent specific solutions to a general kinetic model of climb developed in preceding work (Part I). Solutions are obtained for the case where interstitials in the lattice can be neglected compared to vacancies, where the dominant point defects responsible for fast diffusion along dislocation cores are vacancies, and where unlike jogs attract one another. No a priori assumptions are made about the ability of the dislocation or its jogs to maintain local point defect equilibrium. A wide range of conditions is treated including positive and negative climb where the vacancy concentrations may be either near or far from equilibrium. Positive and negative climb are shown to be inherently different processes, and it is found that dislocations tend to get joggy in crystals far from equilibrium. Several brief applications to problems of current interest are given. A need for critical experiments is emphasized.
1. INTRODUCTION
KINETIC models for the climb of edge and screw dislocations were developed in the preceding
paper (Part I). General equations describing the climb of a straight unconstrained edge dislocation were established there, and a general method for solving these equations was indicated. In order to obtain complete solutions of the climb problem, however, it is necessary to simplify the general formalism of Part I to some extent by making some specific physical assumptions about the climb model. In the present paper we carry out complete solutions for steady state edge dislocation climb for the case where interstitial defect concentrations in the lattice are unimportant and can be neglected, and where vacancies are taken to be the dominant point defects responsible for fast diffusion along the dislocation core. We also continue the assumption used in Part I that there is an attraction between unlike jogs and a repulsion between like jogs. Solutions are obtained for both positive and negative climb. Results are also obtained for a wide range of lattice vacancy supersaturations and subsaturations and include cases where the jog population on the line becomes strongly perturbed. Fortunately, the final expressions for climb are relatively simple, and a complete summary and discussion of results is given in Sec. V. Brief applications of the results to several problems of current interest are also made in several places. A discussion of certain relevant aspects of diffusion along dislocation cores is given in an Appendix.
II. POSITIVE AND NEGATIVE CLIMB WITH DOMINANT DEFECT ASSUMPTION
A number of physical assumptions must be made before the steady state jog equations [Eqs. (5), (21), (22), and (24) of Part IJ can be integrated. We have already discussed the build-up of jog energy with size I in Part I on the plausible assumption that the two unlike jogs of a jog pair attract one another at all
* Supported by the United States Air Force Office of Scientific Research.
distances up to the well-formed limits A or A' (see Fig. 5 of Part I). Equations (2) and (3) of Part I only define the build-up energy, but in order to obtain solutions to the jog equations the detailed nature of the dependence of jog energy on I must be known. In view of our general ignorance about the analytic behavior of this function, a reasonable approximation is to use the simple linear relations [see Eqs. (2) and (3) of I],
8j -Ej=K (j=2,3,.· ·A) (1)
8/- E/=K' (j=2,3,.· ·A')
as shown graphically in Fig. 1. The double jog energy 28J is seen to be greater than the formation energy of either a single vacancy or extra atom on the line. If this assumption is not satisfied, some of the physical consequences of the solutions to be discussed later will be inverted. When 8f<28.r> 8/, then for reasonable values of the jog energy, the condition on the vacancy formation energy is not very restrictive. However, it does mean that the interstitial formation energy on the dislocation must be much less than in the lattice for cases like the close-packed metals where the interstitial formation energy in the lattice is thought to be quite high. In these cases, one is led to a high binding energy of the extra atoms to the dislocations.
t 2& ------;,;l;,""--,---"'v-----=--r-.....:: , ::-
£(2) Ef
I i , I , -------r---i , , I
123... A 1-
A: •.. 32 I -£
. FIG. 1. Diagram of the energy vs length of a jog pair formed by ~lther vacancy aggregation or interstitial aggregation. The solid lme represents the expected curve, and the dotted lines represent the linear build-up approximations in Eqs. (1).
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818 R. W. BALLUFFI AND R. M. THOMSON
A second assumption is that, in any particular crystal, only one type of defect is dominant. We shall take the apparently common case where the vacancy is the dominant defect. The concentration of interstitials in the lattice is assumed to be negligible and may be completely neglected compared to the vacancy concentration. In the core we allow a concentration of extra atoms which is small compared to the vacancy concentration but which still plays an important part in establishing the boundary conditions at the extra atom end of the 7J(l) spectrum. We therefore have extra atoms in the core which play a role in the kinetic processes of aggregation in the core, but there is no exchange of these extra atoms with the lattice in the form of interstitials. The climb rate is then given solely in terms of the rate of vacancy emission or absorption at the dislocation.
Under the above conditions the system of jog equations is appreciably simplified. Equations (24) and (5) of Part I become respectively
nl-l Pm 7J(l-1)-7J(l) /lj=O (l=2,3," ·L), (2a)
L-A'
(L-A-A') L '1/(1)= 1. (2b) I=A+J
The vacancy boundary condition, Eq. (21) of Part I, neglecting the extra atom terms, becomes
¢-nw= (L+l)o=x. (2c)
The extra atom boundary condition, Eq. (22) of Part I, becomes
Pm(nono'-nn')+ (r-L)o/ cr-l) = 0, (2d)
where r is defined by Eq. (23) of Part I. As pointed out in Sec. III of Part I, the jog equations
can only be integrated and solved when additional information concerning the defect density maintained at the permanent jogs relative to the over-all average defect density on the line is obtained. We do this by solving the diffusion problem for the defect distribution along the dislocation. It turns out, as will be seen in Sec. IV, that the required information consists of knowing the ratio of the average relative vacancy supersaturation on the line to the relative vacancy supersaturation at a permanent jog. We may write this required ratio in the form
(ii-no)/ (nt-no) =ii/cr(O), (3)
where ii=(n/no)-l, cr(0)=(nl/no)-1 (for all l of permanent size).
In the following section we set up the diffusion problem and solve for the required vacancy supersaturation ratio.
III. DIFFUSION SOLUTION
When the jog energy builds up linearly as in Eq. (1), and when vacancies are the dominant defects, the
equation defining jog permanency, Eq. (25) of Part I, becomes
vmno[nl/no-exp(K/kT)J~O (4)
for jog pairs smaller than A. Later results will show that the vacancy supersaturation on the line never becomes very large and that exp(K/kT»nl/nO under usual conditions. Consequently, the above relation is not satisfied for jog pairs of size less than A, and they are, therefore, incipient. In the case of negative climb we conclude that jogs of size l' <A' are also incipient in the present linear model.
The diffusion problem, therefore, consists of finding the vacancy concentration along a dislocation possessing incipient jogs of size 1 <A (or l' <A') which do not perturb the vacancy concentration locally, and permanent jogs of size l>A (or 11>A') which act as steady vacancy sources or sinks. We take the particular case of a dislocation lying along the z axis of a cylindrical region of a crystal containing supersaturated vacancies. The dislocation possesses permanent jogs spaced a distance L'::::::.L-A-A' apart, where L is given by Eq. (2b). The fraction of vacancies along the cylindrical surface of r=R is maintained at a fixed value, N(R). This model should have rather wide applicability and should be a good approximation for actual cases where the dislocation is fairly isolated and is not sharply curved. The solution will be found to have a logarithmic dependence on R, and is, therefore, not very sensitive to the geometry of the boundary.
It is first necessary to consider the diffusion of vacancies to the dislocation and the rate at which they jump onto the line. The dislocation acts as a sink of unknown effectiveness, and the steady state vacancy concentration, N(ro,z), established in a small cylindrical region around the dislocation enclosed by the radius ro may be found by equating the lattice diffusion flux into this region to the vacancy flux lost to the dislocation line. The result is
2nDv
I/>(z)-w n(z)= [N(R)-cV(ro,z)]. (5) a2 In(R/ro)
Here I/>(z) and n(z) are local functions of position along the line. The right-hand side of Eq. (5) is the inward radial diffusion flow from the lattice, where a is the interatomic spacing and Dv is the vacancy lattice diffusivity. In deriving Eq. (5) we have neglected diffusion in the z direction in the crystal. This approximation is justified, since diffusion is assumed to be very rapid along the dislocation core, and the gradients in the z direction in the crystal will generally be smaller than the radial gradients.l/>(z) may be further expressed as [see Eq. (9) Part I]
(6)
Having these relations it is now possible to set up and solve the diffusion equation for vacancies on the disloca-
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KIKETIC THEORY OF DISLOCATION CLIMB. II 819
tion line. The steady state equation for this is
(v",/2)[d2n(z)/ dz2J+«i>(z) -w n(z) -f)=O. (7)
The first term is the usual diffusion accumulation term. The second and third terms are the gain and loss terms to the lattice, and the last term () is a constant loss term for vacancies already on the line which represents the continuous flux of vacancies which is needed for the creation of new jogs on the line and to maintain other aspects of the steady state jog population. Since () is not a function of z it will play no role in the determination of cr /0'(0) and need not be specified further. Our assumption that vacancies are the dominant diffusing defects allows us to neglect interstitial terms in Eq. (7). By combining Eqs. (5) and (6) and putting the results in Eq. (7) we have
where (v m/2)[d2n(z)/ dz2J-Qn(z)+ 8= 0,
8=,B«i>0(S+1)-f)
Q=(3w
,B= (1+[Zd In (R/ro)/21l'J}-1
S=[N(R)/No-1J.
(8)
(9)
No is here the vacancy concentration in the lattice which would be in local equilibrium with the dislocation line, and «i>o is the corresponding value of «i>. Therefore, S is the relative vacancy supersaturation maintained at r=R.
A simple and useful expression for the climb rate may be obtained from the transformed diffusion equation, Eq. (8). The term Mo(S+1) in Eq. (8) represents vacancies jumping on the line, and the term ,Bw n(z) represents vacancies jumping off. We therefore have
x =¢-iiw=,B[«i>o(S+ 1) -wiiJ=,B«i>o[S-cr]. (10)
We next require boundary conditions for the solution of Eq. (8). The jogs represent effective sinks for vacancies, and the concentration will be symmetric about a point midway between the jogs. Therefore,
[dn(Z)] -- =0
dz z~L/2 (11)
The solution satisfying Eqs. (8) and (11) is
n(z) =c{exp(az)+exp(a{L-z} )J+ 8/Q, (12)
where (13)
The boundary value of the concentration nCO) at a jog may be determined by taking a small dislocation length oz directly adjacent to the jog and finding the steady state concentration there by equating the vacancy flux in by pipe and lattice diffusion to the vacancy loss into
:JQ5 .! -
o 4 aL
FIG. 2. leaL) plotted as a function of aL. When aL<3, leaL) is approximated by aL/6. "Vhen aL>3, leaL) is approximated by (1-2/aL).
the jog. The boundary condition at a jog at z=O is then
-!n(O) vm+!v.J+!vm[dnJ dz Z~O
-oz{Q n(0)-8}=0, (14)
where V.J is the frequency with which the jogs emit vacancies on the line. Fitting Eg. (12) to Eq. (14) and realizing that v.J=nOVm and a2«1 (see the Appendix), we obtain
[no-8/QJ[exp(az)+exp(a{L-z})J 8 n(z) = +-. (15)
[exp(aL)+1J+a[exp(aL)-1J Q
Defining exp(az)+exp(a{L-z} )
g(z) (16) [exp (aL) + 1 J+a[ exp (aL) -1 J'
we may express the desired ratio in the form
[cr/O' (O)J= (ii-no)/[n(O)-noJ = (1-g)/[1-g(0)J, (17)
where L
g=L-l i g(z)dz.
Using Egs. (16), (17), and (18) we find
~=1+~1[exp(aL)+1J 2 }=1+f (aL).
0'(0) a [exp(aL)-1J aL a
(18)
(19)
The function f(aL) is plotted vs aL in Fig. 2, and the results indicate that the supersaturation ratio can be adequately represented by either of the two following simple forms depending upon the value of aL :
cr/0'(0)=L/6, (aL<3);
cr/O"{O)=a-1(1-2/aL), (aL>3). (20)
We note that this diffusion problem has been discussed in terms of vacancy supersaturation. It may be seen that the results also hold for subsaturation if suitable changes of sign are made.
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R20 R. W. BALLUFFI AND R. M. THOMSON
TABLE r. Summary of final results for edge dislocation climb in crystal containing vacancies. -----------
Positive climb (S> 1) Negative climb (S < 1) aL X L X L
------------------"L<3 MoS Lo[1 +Sa2L2/12]-Al2 (34>oS Lo
aL>3 2MoS/aL Lo[1 +S(I-2/aL)]-A12 2i1c/>oS/aL Lo[I+S(I-2/aL)]<A' 1)12
The assumption has been made throughout that the dislocation line is stationary, whereas actually, of course, it is climbing. It is important, therefore, to consider how good this approximation is under expected conditions. Let us consider the extreme case of a rapidly quenched crystal with an extremely large vacancy supersaturation containing a dislocation climbing at the maximum possible rate, x={3cpoS. Let T be the time for an average vacancy to travel the distance R. Our approximation of neglecting the dislocation motion during climb is permissable if the distance Xd climbed by the dislocation during the time T is less than R. The ratio of these distances is
Xa/ R={3Zd:VUS(R/ a)""-'(R/ a)· N(R),
where N(R) is the quenched-in vacancy concentration. Since usually N (R) <: 10-4, we conclude that the stationary model is a good approximation. Lothel has analyzed further aspects of the assumption of a stationary model with similar conclusions.
IV. SOLUTION OF JOG EQUATIONS IN STEADY STATE
We now have all the information required for the solution of the jog equations. There are various types of solutions which can be obtained: (1) slow climb, where the equilibrium is assumed to be perturbed by only a small amount; (2) positive fast climb in which vacancy supersaturations are high; and (3) negative fast climb where the lattice possesses far fewer than the equilibrium density of vacancies. The detailed analysis for each case is carried out below, and the results are tabulated in Table I. It is found there that, although for purposes of analysis it is necessary to make the breakdown into the three cases, the formulas developed for fast and slow climb can be put into single unified equations for practical use.
1. General Method
The jog equations for steady state climb are now Eqs. (2a,b,c,d) in the form
nl_1 Vm 71(l-1)-vtr}(I)=o (1=2,3," ·L)
ci)-iiw=(1cpo(S-ii) = (L+1)o=x
iim(nono' -iiii')+ (r- L)o/(r-1)= 0 (21)
L-A'
(L-A-A') L 71(1)= 1. I=A+I
1 J. Lothe, J. Appl. Phys. 31, 1077 (1960).
The first equation is the difference equation for the clusters, subject to the boundary conditions of the second and third equations. The fourth equation is the collision condition. The second equation is written in the form found in Eq. (10). The difference equation can be rewritten, valid for aliI up to L-A'+l, as
g(l) 71(l)-71(l-1) = -'}'(l-1)
71(1) =ii
g(l)= vl/nl_1 Vm
'}'(l)=o/nl V m ; (1=2,3,·· ·L-A'+1).
(22)
This equation is most easily solved by an iteration procedure. That is, the solution of the equation for 71(2) is 71(2) = [ii-'}'(1)]/g(2). The next term 71(3) is found in terms of 71(2), and so forth. The solution can thus be written in the form of the continued fraction
{ ii-,},(l) '}'(2)}
g(2) 71(l) = '}' (3)
g(3)
g(l) (23)
This expression takes a more manageable form if the factor {g(2)'" g(l)} is taken out of the fraction. The result is
71(l) = {g(2)g(3)· .. g(l)} -I {ii - [ '}'(1) +,},(2) g(2) +'}'(3)g(2)g(3)+·· . +'}'(l-1)g(2)· .. g(l-1)]}, (24)
or 1
71(1)={exp[-L Ing(q)J} q=2
I-I p
X {ii-['}'(l)+ L '}'(p) eXPL lng(q)]}. (25) p=2 q=2
The function g(q) is related to the various energies of formation of vacancies by Eq. (9) of Part I,
g(l)= exp[(0f- 01)/kT]
(0"1_1+1) (26)
The relative vacancy supersaturations which we employ are defined by Eqs. (3) and
(27)
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KINETIC THEORY OF DISLOCATION CLIMB. II 821
The sum over Ing(q) yields
I I I-I
L: lng(q)= L: (Of- Oq)/kT- L: In(l +O"q) q=2
I-I
=[28(l)- 8f ]/kT-ln II (1+O"q) q=1
1-1
=2&(l)/kT+lnno-ln II (1+O"q). (28)
Substitution into Eq. (25) yields
1-1
'Y)(l) = (ii/no) II (O"p+l) exp[ -28(l)/kT] p=1
1-1
-(O/lImno){no-l (0"1+1)-1+L: (1+O"p)-1 p=2
p-I
Xexp(2&(p)/kT) II (1+0",,)-1} ,[=i
I-I
X{exp(-2&(l)jkT)· II (O"q+1)}; (29)
1=2,3, ... L+1-A'.
This equation is the general solution for I<:L-A'+l but it can be further reduced when th~ functions 0"1 and S(l) are known. It has been pointed out that the basic assumption introduced to solve the jog equations IS
O"I=a (/=1,2,···A-l)
0"1=0"(0) (I=A, A+l, .. ·L-A'). (30)
The ratio a/O"(O) is given by Eq. (20). Also, it is assumed that the jog energy function is linear up to I=A and then becomes constant, Eq. (1) (see Fig. 1). With these assumptions, the summations in Eq. (29) break into sums over incipient jogs and sums over well-formed jogs, with the result that
'Y)(l)= {A -o/vm(B+C+D)}[O"(O) + 1Jl-A+Do/llm
A = (6+ 1).1./ L02
B= (1 +a)A-2/no2Lo~
exp(-K/kT) C
X{ (a+1)A-2 exp[ -(A-2)K/kTJ-l}
(1 +a) exp( -K/kT)-l
D=[no 0"(0)]-1; I=A,A+1," ·L+1-A'.
(31)
This equation is the general solution of the jog difference equation for l <: L+ 1-A', and is subject to the boundary conditions in Eqs. (21).
Equation (31) does not give the interstitial density
which is needed for the interstitial boundary condition in Eqs. (21). To obtain the interstitial density, the difference equation must be summed up to L through the interstitial incipient jogs, but we have found it more convenient to sum the difference equation backwards from the single interstitials through the interstitial clusters. After the reverse summation is performed, complementarity is invoked to make vacancy clusters equivalent to interstitial clusters. In order to make the reverse summation simpler, we make the change of variable from vacancy clusters to interstitial clusters,
r/(I)=1](L+1-/)
1'=L+1-1. (32)
The jog difference equation in Eqs. (21) then becomes
nL-I' 11m 1]'(I'+1)-IIL+l-l' 'Y)'([') =0. (33)
As before, the difference equation takes the normal form
g' (I' + 1) 'Y)' (l' + 1) -'Y)' (I') = - '"y' (l')
g'(l'+ 1) = nL-I" Vm/VL-I'+1
'"y'(l') = -O/V£+I_I'
1'=2,3, .. ·L-A+1.
(34)
The solution is a continued fraction, written in the form
I'
'Y)'(!') = {exp[ - L: lng'(p)]} p=2
p=1'-1 p
X{n'-['"y'(l)+ L '"y'(p)exp L lng'(q)]}. (35) p=2 q=2
The sum over the g'(q) is easily performed, and the general result is
I'
'Y)'(l') = (n'/no') II [0""(P)+1J-l exp( -2&'(l')/kT) p=2
1'-1 p
X L exp[2&'(p+1)/kT} II [1 +0"" (q)]} p=2 q=2
I'
X{exp[(&/-2&'(l'»/kT] II[0""(p)+1J-l}. (36) p=2
We have used the definition 0""(P)=u(L+1-p) in Eq. (36). The jog energy build-up in the incipient clusters is again linear as in Eq. (1), and
2&'(l')= 8/+CI'-1)K'; l'<:A'
28'(l')=28J = 8/+(A'-1)K'; I~A'. (37)
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R22 R. W. BALLUFFT AND R. M. THOMSON
L
FIG. 3. Typical jog population density 'YJ (I) as a function of I.
The solution for the interstitial clusters is then given by
'I/'(l') = {A'n'+o/Jlm(B'+C'+D')}
X[O'(O)+ 1J-(I'+H!) +D'o/Jlm[O' (0) + 1J
A'= (a+1)2-A'/Lo2no'
B' = [exp(K'/kT)J(O'+ 1)"-A'/ Lo2no'no
(0'+1)3-A' exp(2K'/kT) (38) C'=--------·----------
no'noLr?
XJ1-(0'+1)A'-2 eXP[(A'-2)K'/kTJ}
l 1- (0'+1) exp(K'/kT)
D'= (0'(0)+ 1)/noO' (0) ; [=A', A'+l, .. ·L-1\.
Equations (38) and (31) are equivalent expressions for the cluster densities. When these two solutions are made to satisfy the boundary conditions, the climb problem is solved. Thus the climb problem for both positive supersaturation and negative supersaturation is now reduced to the following set of equations.
{1(S-O')nQW=Lo (39a)
(r-L)o/(r-l)- vmnono'(1 +0'+0"+0'0") = 0 (39b)
(39c)
L-A (L-A-A') L 'I/'(l') = 1. (39d)
A'+1
The equations for 'I/(l) and 'Y}'(l') are given in Eqs. (31) and (38). The expected form of a typical 'I/(l) vs l curve is shown in Fig. 3. Due to the transcendental nature of Eqs. (31) and (38), Eqs. (39) can be solved in closed form only in a linear approximation, or for very strong deviations from equilibrium.
Our results will be equally valid whether the supersaturation of vacancies is positive or negative. However, the jog nucleation is physically different in these two cases. In positive supersaturation, the vacancy clusters are formed by the collision of single vacancies to form divacancies, etc. As a vacancy cluster grows in size, according to the complementarity relation, the final result is the formation of an interstitial at the cutoff
length L. Thus both vacancies and interstitial populations are perturbed by climb.
In negative climb, the process is reversed in a curious way. It is first of all necessary to start with an interstitial. The interstitial then lives long enough to emit a vacancy because of the low density of the vacancies. By this process the interstitial becomes a double interstitial. Then another vacancy is emitted, and the interstitial grows to three interstitials, etc. The reverse process thus depends upon a net production of interstitials by some process on the dislocation. According to Eq. (39b), there are two sources of interstitials on the line. The first is emission of interstitials from already formed jogs, and the second is homogeneous pair production. The first interstitial source therefore causes jog nucleation by a sort of boot-strap process which depends upon jogs already present which act as interstitial sources.
2. Cases where I L41(O) 1< 1
It is sufficient in slow climb to work only with the first three equations of (39) because the change in L is a second-order effect in the climb rate. We expand (39) to first order in a, 0'(0), a', and S. The final results are
0'[1+2 (A+A' -1)/ Lo2a2J+20' (0)/ Lrfl2=S
x=cf>o{3(S-ii). (40)
The climb rate is found when the ratio O'(O)/ii is taken from Eq. (20).
(a) Case I: aL<3: 0'(0)=6a/L.
S/a= 1+2(A+A'+5)/(aLo)2
x=SnQW{1{ 1 + (aLo)2/2(A+A'+5)}-I.
(b) Case II: aL> 3: 0"(0) =aii'/ (1-2/aL).
(41)
S/ii= 1 +2(A+A' -1)/(aLo)2+2/(aL-2)
nQW{1S{2(A+A' -1)/ (aLO)2+2/ (aL- 2)} x= (42)
{2(A +A' -1)/ (aLo)2+aL/ (aL- 2)}
These results are valid for both positive and negative climb, corresponding to positive or negative a. In the expansion of the transcendental Eqs. (39), the range of validity for slow climb is given by the conditions
1 LO'(O) 1 <1
IMI<1 IA'al <1.
(43)
We note that these conditions are satisfied for supersaturation values of S which might be quite high in case 1. That is, even though the supersaturation of vacancies on the dislocation must be small, the supersaturation in the lattice S can be large compared to unity. In case II, slow climb is valid only when the lattice supersaturation S is small compared to unity.
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KINETIC THEORY OF DISLOCATION CLIMB. II 823
The results of Eqs. (40) do not contain the ratio r and are therefore not sensitive to the method by which the jogs are nucleated. Physically, in slow climb, the perturbation of the jog population is negligible and sufficient jogs are always present for climb to proceed, so long as the conditions (43) are satisfied. The results of slow climb can be written very simply when some of the finer features of Eqs. (41) and (42) are neglected:
Case I: x~l/>o{3; ~Lo
Case II: x""251/>0f3 / aL; L~Lo. (44)
3. Cases where \ L(1(O) \ > 1
In fast climb we again start from Eqs. (39), with the basic assumption now that I LO' (0) I < 1. In fast positive climb (0'>0), the dominant term in 7/'(l') of Eq. (38) is the term in D'. The sum over 7/'(1') in Eq. (39d) becomes
L-I= (L-A -A' -1)0/novmO'(0). (45)
In similar fashion, in fast negative climb (0' <0) the sum in Eq. (39d) can be taken over 7/ instead of 7/',
and again D is the dominant term. The result of the sum is again Eq. (45). With the use of Eq. (39a) and the ratios for 0'/0'(0), the climb rate (valid for both positive and negative 0') becomes
(a) Case I: aL <3
0'/5= (1+12/a2L2}-1
x= 5{31/>0(1-a2D/12),
(b) Case II: aL>3
0'/5= 1-2/aL
X = 25{31/> 0/ aL.
(46)
(47)
In these equations, we have neglected A and A' with respect to L. In fast positive climb case I will normally be valid, while in fast negative climb, case II will usually hold.
The values for the cutoff length L are also easily obtained. In positive climb, the sum over the jogs in Eq. (39d) expressed in terms of 7/(l) becomes
L-I= [O'(O)]-I{A -0/ I'm(B+C+D) }[1+0'(0)]L-A-A'+1
+ (L-A-A'+1)DO/v m • (48)
Since Eq. (45) is approximately valid, the term in the brace must be zero. Again, the term in D is the dominant term, so in positive climb L satisfies the relations
(a) Case I: aL<3: 0'>0
(L/Lo)2= (1+0')-'\= [1+5a2D/12]-A, (49)
(b) Case II: aL>3: 0'>0
(L/Lo)2=[1+5(1-2/aL)]-A. (50)
In a similar manner for negative climb, the sum over 7/' (l) yields
(a) Case I: aL<3: 0'<0. This fast climb case apparently cannot be attained. (b) Case II: aL>3: 0'<0
(L/ LO)2= (0'+ 1).\'-1= [1+5(1- 2/aL)JA'-I. (51)
Equations (46) to (51) are the final results for fast climb, and show that in these cases the dislocation is usually a very effective source or sink for the vacancies. Also, the dislocation becomes very joggy far from equilibrium.
V. RESULTS
1. Final Results and Discussion
A summary of final results is given in Table 1. Here, we have combined the various results attained in Sec. IV into a number of quite simple expressions which should apply over a wide range of conditions. The climb rate is generally seen to depend upon the incident equilibrium vacancy flux 1/>0, the lattice supersaturation 5, the geometrical factor {3, and the parameter aL. A discussion concerning the magnitude of a is given in the Appendix. L is, of course, the average spacing between jogs, and the diffusion solution shows that the expressions for X in Table I hold regardless of whether L is determined by the thermally activated processes of the present kinetic model or whether it is primarily determined by other mechanisms such as, for example, dislocation cutting. If L is determined by other mechanisms, its value must be derived by methods beyond the scope of the present paper.
The results are seen to depend strongly upon the magnitude of the parameter aL. When aL<3, all excess vacancies above the equilibrium flux impinging upon the line are permanently captured causing climb. In this case the activation energy for climb is simply that of self-diffusion. When aL>3, many of these vacancies jump back off the line before they are annihilated, and the climb rate is slowed down by the factor 2/aL. The activation energy for climb then contains the self-diffusion energy, the jog energy, which is contributed by Lo=exp(0J/kT), and further terms contributed by Land a. In general, we see that the apparent activation energy for climb is quite complicated in this case and contains a number of energy parameters. For crystals with high jog energies the self-diffusion and jog energy terms will probably dominate, however.
The reason for the strong dependence of the climb rate on aL is easily seen. The mean distance which a vacancy would travel on a smooth dislocation before jumping off is x= (Vm/W)!~a-I. The condition that aL<3, or alternatively, x>L/3, therefore, corresponds approximately to the case where the average impinging vacancy is able to travel far enough along the line so that it is destroyed at a jog before jumping back off.
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~24 R. lvV. BALLUFFI AKD R. M. THOMSON
This is the so-called condition of "saturation" which has been discussed by Lothe, I and our results in this respect are quire similar to his [see Eq. (27) of reference 1].2,:;
The values of L depend upon a, S, and the size of the jog nucleus (A or A'). When the crystal is near equilibrium and S is small, L~Lo and the jog population is barely perturbed. However, when S is large, as for example in a quenched crystal, the dislocations may become highly jogged compared to equilibrium:l
According to Table I, L becomes progressively smaller than Lo as the system gets further from equilibrium for both positive and negative climb. We note that the effects of nonequilibrium vacancies in perturbing the jog densities are not symmetrical for positive and negative climb. For example, when aLo<3 it is apparently impossible to perturb the jog density significantly from equilibrium in negative climb. This comes about because O>S>-l,5
In many problems it is desirable to know the probability of capture and destruction of an impinging vacancy at a dislocation. This quantity, denoted by
2 J, Friedel, Les Dislocations (Gauthier-Villars, Paris, France, 1956), p. 74.
3 In reference 2, Friedel has given a different condition for saturation which states that it will occur when L is smaller than the distance between the climbing dislocation and the sources or sinks in the crystal supplying vacancies to the dislocation. We do not believe that this condition is realistic, since Friedel's model does not take into account the fast diffusion of defects along the dislocation to the jogs. We note that several attempts have been made to interpret the kinetics of polygonization in bent crystals in terms of Friedel's saturation condition [see p. 194 of reference 2; also, S. Amelinckx and R. Strumane, Acta Met. 8, 312 (1960)]. There it is suggested that polygonization occurs under saturation conditions at high temperatures with a lattice self-diffusion activation energy QL, and at lower temperature with an activation energy given by (QI,+ 8J). A close examination of available polygonization rate data indicates, however, that the polygonization process actually proceeds with a continuously increasing activation energy, and it does not seem possible to describe it in terms of a single rate process. At the moment it appears that the polygonization process is more complicated than indicated above, and that changes in apparent activation energies must be due to other causes.
4 Let us take, as an example, a crystal quenched from SOO° to 25°C, where E,=0.9 ev, 8J=0.S ev, (Eb-E",- 8m)=0.9 ev, and A=S. In this case S~SXlO'°, a~2.SXlO-8, and Lo=2.5XlO'3 . This means that in equilibrium there would be essentially no jogs on a dislocation segment of reasonable length. In the quenched crystal, however, solution of the expression for L in positive climb when aL<3 gives L~7X1Q3. The dislocation segments are then jogged, and the condition aL<3 is satisfied. The climb rate is given by x={3</>oS, and climb is rapid.
S When dislocations are pressed up against barriers by applied stresses, there may be strong climb forces, and it is of interest to see whether the departure from equilibrium in such cases could he sufficient to jog the dislocations and promote easy climb. Present theories of high-temperature creep are based upon the climb of dislocations over barriers, and this jogging process could conceivably account for observations that the activation energy for high-temperature creep in even high jog energy metals like copper is close to that of self-diffusion, For diffusion-controlled dislocation climb we require that aL<3. According to the results in Table I, it is then not possible to perturb the jog population in the case of negative climb. For the case of positive climb a value of S of unity or greater is required to noticeably jog the dislocations. Local stresses of ~1O'0 dynes/em? or larger are therefore required. These stresses appear high, and it is therefore doubtful that many jogs could result from this mechanism.
P d, can be readily calculated from the present results using Eq. (10) to yield
Pd=X/(34JoS.
Pd is therefore unity when aL <3 and is equal to 2/aL when aL>3.
An important result of the present work is that the equations for X in Table I are essentially the same as those which can be obtained by a considerably simpler and more intuitive method than the exceedingly detailed present method. A brief description of this simpler approach follows. 6 If the average drift velocity of jogs spaced an average distance L apart on the line is V.r, then as an approximation
X=V.r/ L= lImnou(O)/ L. (52)
The right-hand side of Eq. (52) was developed with the aid of Eq. (25) of Part I. The simple diffusion model in Sec. III gives the result that
X={34Jo(S-U). (53)
Simultaneous solution of Eqs. (52) and (53), and use of the relations between u and u(O) in Eqs. (20), yields the results for X in Table I to a good approximation. The approximations involved in writing Eq. (52) can be brought out by deriving a corresponding relation by means of the more exact jog equations. If Eq. (2a) is summed from A to A', then some manipulation yields
L-A'
X"'-'(L-A-A')o= L [n(0)lIm1](i-1)-1](1)1I.r] 1~.Hl
L-A'
X"'-'v.r L 1](l)+lImn(O)1](A)-lI.J1](L-A') (54) A+l
One of the terms in the last equation is the same drift velocity obtained in the simple diffusion result, Eq. (52), but in addition there are two additional terms due to the diffusion of jogs in "cluster space." The jog diffusion terms come about because of population gradients among the boxes of Fig. 8 of Part I, and all such effects are neglected in the simplified climb treatment. The present results indicate that it is a good approximation to neglect these additional terms. It was not fully apparent at the beginning that this simpler approach would yield essentially correct results, and it was necessary, unfortunately, to demonstrate this by carrying out the full analysis. Of course, no information about L can be obtained from the simpler approach, so it cannot be considered the complete solution of the problem.
6 This simpler approach has been used in references 1 and 2 (pp. 67 et. seq,).
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KINETIC THEORY OF DISLOCATION CLIMB. II 825
2. Growth of Platelets
Some initial discussion of platelet growth has occurred in Sec. IV of Part I. Our conclusion there that the physical situation for a growing loop is very different from that of a decaying one is in agreement with a similar statement by Silcox and Whelan.7 Their experiments on bulk annealing also bear out this conclusion, since loops in the former case are very polygonal in shape while loops in the latter are more circular. Silcox and Whelan suggest that the polygonal shape might be due to the equilibrium effects discussed by Saada,8 but our opinion is that they are due rather to the requirements of jog nucleation during kinetic growth. In fact, we feel that the case of platelet growth is the most clear-cut situation where the homogeneous climb theory proposed in this paper can be applied, and it is unfortunate that more detailed data on platelet growth are not available. Because of its importance as a test of the present theory and as a method for the determination of some of the parameters involved in it, we write the radial growth rate G for a polygon:
{(3cf>~; aL<3
G= 2{3cf>~(aLo)-! [1+S(1-2/aL)J\/2; aL>3.
By measuring the ditTerence between rates where aL <3 is valid and where aL> 3 is valid, the diffusion activation energy can be separated from the jog energy. In addition, in the second case, the value of A is of interest. These formulas hold, of course, only when the length of a polygon side is greater than the cutoff L+ 1 given in Table I. When this condition is not satisfied, the climb rate is oscillatory in time. That is, when a jog runs out to a comer, the supersaturation on the line builds up until another jog is nucleated. Then the jog reduces the dislocation line supersaturation to a lower value. Obviously our present formalism does not fit this more complex time-dependent case. We also note that {3 contains the size of the polygon in a logarithmic expression, but the variation is probably never an important contribution to the problem.
APPENDIX
On the Magnitude of a
The parameter a is one of the most important parameters in the present model, and we shall consider its magnitude in this Appendix. From Eqs. (13) and (9), and Eg. (9) of Part I
a= (2{3w/jJm)!~exp[ - (Eb+E m - 8m )/2kT], (A1)
7 J. Silcox and M. Whelan in Structure and Properties of Thin Films, edited by C. A. Neugebauer et at. (John Wiley & Sons, Inc., New York, 1959), p. 164.
8 G. Saada, Acta Met. 7,367 (1959).
since (Zdm!'"'-J1. Now Eb is positive, and, in general, we expect Em> 8m. Therefore, we conclude that a«l at all temperatures. At low and intermediate temperatures, of course, a will be exceedingly small.
The above conclusions are based on our expectation that a point defect in the core of a dislocation should experience considerable relaxation in the direction of the line itself, in much the same way as that postulated for the crowdion. Thus, depending upon the amount of relaxation, the motion energy of the point defect may be made arbitrarily small. This basic assumption is untested by actual computations, but there are good reasons for suggesting that it actually exists to some degree. The most important reason is simply that the core of a dislocation is a region of considerable deviation from the relatively rigid crystal lattice so that relaxations will be more extensive in a dislocation core than in a perfect crystal lattice. To the extent that the concept of line tension retains any meaning in the core region (and line tension does not lose its meaning entirely on the atomic scale) the relaxation should exist mainly in the direction of the dislocation line.
It is of interest to see what information about a can be obtained from the measurements of self-diffusion along dislocation cores. Lothe! has shown that the activation energy for self-diffusion along the core, in terms of the formation and motion energies of defects, is strongly dependent upon the degree of correlation between defect jumps. Bardeen and Herring9 were the first to notice that in the single string diffusion model a single vacancy will make many jumps back and forth along the line during its lifetime on the line but will only displace a given atom by at most a net random displacement of a. Applying Lothe's analysis! of this problem to our single string model we find an activation energy for dislocation core self-diffusion of
ExperimentslO indicate the Q~Qd2, where QL is the activation energy for self-diffusion in the lattice. Therefore,
or
This result is unreasonable since E j > Eb, and we conclude, with Lothe, that the correlated single-string model is invalid as a self-diffusion model. We emphasize, however, that the single-string model for diffusion of vacancies is not invalidated by this result. According to Lothe a more reasonable pipe self-diffusion model
9 J. Bardeen and C. Herring, Imperfections in Nearly Perfect Crystals (John Wiley & Sons, Inc., New York, 1952), p. 261.
10 D. Turnbull and R. E. Hoffman, Acta Met. 2, 419 (1954).
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826 R. W. BALLUFFI A"\'D R. M. THOMSON
would have two or more strings of easy diffusion between which vacancies could easily jump.
In the latter case in the absence of strong correlation effects we may simply write
Q~Sf+Sm=Ef-Eb+S", or
E",+Eb - S",=QL-Qd.
In this case if Qd~Qd2, a~exp( -Qd4kT), and we conclude again that a« 1. This result is not unreasonable but must be regarded with caution, since so little is
known about the geometry of dislocation core selfdiffusion.
Several of the comments in this Appendix have, of course, been speculative, and point up the great need for more knowledge of the atomic configuration of dislocation cores. In the case of edge dislocations, we need simply postulate a motion energy for the vacancy, and it does not matter by what mechanism the motion is achieved. Unfortunately, however, the case for the screw dislocation climb is much more involved, as has been seen in Part 1.
JOURNAL OF APPLIED PHYSICS VOLUME 33, NUMBER 3 MARCH, 1962
Alumina Whisker Growth on a Single-Crystal Alumina Substrate
P. L. EDWARDS* AND R. J. HAPPEL, JR.
U. S. Naval Ordnance Laboratory, White Oak, Silver Spring, Maryland
(Received August 25, 1961)
Alumina whiskers have been grown on single-crystal alumina substrates by heating aluminum filings, located near the crystals, to 1400°C in a stream of wet hydrogen. The whiskers grew crystallographically coherent with the substrate, and had their axes either parallel to the c axis or in one of 12 equally spaced directions in the basal plane. These 12 directions divide into two distinct sets, a (1120> set and a (1100) set, each having sixfold symmetry, and with the directions of one set midway between the directions of the other. These growth directions are the screw-dislocation directions in alumina, hence it seems plausible that the whiskers grew coherently with the substrate at the site of emergent screw dislocations.
INTRODUCTION
WHILE studying sapphire whiskers grown on an alumina substrate by heating aluminum filings
to 1400°C in the presence of moist hydrogen, we observed occasional groups of parallel whiskers. Since the whiskers grew parallel to each other, and their only relation to each other was through the substrate, presumably an alumina crystallite, it seemed likely that the whiskers grew coherently with the substrate and in directions determined by it. To test this idea we grew alumina whiskers on single-crystal alumina substrates. They were found to grow only in certain directions on the base crystal.
Webb and Forgeng1 grew alumina whiskers as described above, and reported that their axes were parallel to c-crystallographic axis and that they had an axial pore. They were able to show that the atomic layers were in a ramp-like structure characteristic of the atomic layers around a screw dislocation. Frank2 has shown that in certain cases a pore at the site of a screw dislocation might be an energetically favorable crystallographic configuration. Webb and Forgeng assumed that this was the case for the alumina whiskers with axial pores.
* Now at Texas Christian University, Fort Worth, Texas. 1 W. W. Webb and W. D. Forgeng, J. App!. Phys. 28, 1449
(1957). 2 F. C. Frank, Acta C!),st, 4, 497 (19$1).
Sears and DeVries3 have grown alumina whiskers by heating an alumina rod to 1800-2000°C in the presence of dry hydrogen. They attribute the growth to the reduction of the alumina to a more volatile oxide in the hottest region and a subsequent reversal of the reduction reaction in a colder region. Whiskers grew having a (0001) and (1100) orientations, and were called c whiskers and a whiskers, respectively. The a whiskers broadened into platelets at their free end.
Gibbs4 reports the following as the Burgers vector directions in alumina:
(a) [1120J, [1:210J, [2110J, (b) [1100J, [0110J, [i010J, (c) [0001].
Burgers vectors (a) and (b) were determined from a study of plastic flow in alumina and (c) was inferred from spiral growth steps on the basal plane and from the growth of c-type whiskers. The Burgers vector directions (b) correspond to the directions of the a whiskers of Sears and DeVries.
EXPERIMENTAL
The alumina whiskers were grown by placing a few aluminum filings in the bottom of an alumina boat. A
3 G. W. Sears and R. C. DeVries, J. Chern. Phys. 32, 93 (1960). 4 P. Gibbs, Kinetics of High-Temperature Processes, edited by
W. D. Kingery (John Wiley & Sons, Inc., New York, 1959), p. 21.
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