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Sink strengths for thin film surfaces and grain boundaries
Journal of Nuclear Materials 90 (1980) 44-59 0 North-Holland Publishing Company
SINKSTRENGTHSFORTHINFILMSURFACESAND
GRAINBOUNDARIES
R BULLOUGH, M.R HAYNS and M.H. WOOD Theoretical Physics Division,AERE, Hat-well,Oxfordshire, OX11 ORA, UK Received 1 November 1979
The formulation of a “lossy” continuum rate theory model for the representation of the evolvement of radiation damage in either a thin film or in a polycrystalline body requires a knowledge of the sink strengths for these surface sinks. The relative merits of the embedding and cellular models for the calculation of such sink strengths will be discussed and suggested modifications when non-linear bulk recombination is significant are examined by appropriate numerical proce dures. It is concluded that such modificationsare unnecessarywhen the “correct” sink strengths are used. The importance of obtaining such reliable rate theory representation of the microstructure to the efficiency of our exploration of the factors influencing growth and swelling in irradiated materials is obvious.
1. Introduction
space, although as we shall see below it can be judiciously modified to encompass even a regular distribution of sinks such as the voids in a perfect void lattice [5]. ‘I% distinction between ordered sinks and random sinks raises the question of the relation between sink strengths derived using cellular models (in which spatial ordering is implied) and those using the embedding model which precludes ordering. We believe the cellular model is often inappropriate for the derivation of useful sink strengths and even when appropriate has sometimes been used incorrectly. To illustrate this point we derive in section 2 the grain boundary sink strength * using both the embedding and cellular model and indicate the rather special relation between them for this particular sink; the sink strength for arbitrary rate limitation at the boundary is then presented for use in the general rate theory continuum. Attention is then focussed in section 3 on the foil surface sink strength and we obtain this sink strength using the cellular model; the inappropriateness of an embedding procedure for
Surface defects, such as the external surfaces of a thin foil or the internal grain boundaries, can be significant sinks (or sources) of intrinsic point defects during irradiation and the net point defect losses at such sinks must be quantitatively understood if the observed internal microstructure is to be interpreted correctly. Further, for purely practical computational purposes the sink strengths of such surface defects are required for explicit inclusion in the general rate theory model of the total microstructure. Brailsford and Bullough [ 1 ] and Bra&ford, Bullough and Hayns [2] have discussed an embedding procedure * within a uniform lossy medium for the derivation of such sink strengths and have obtained sink strengths for a wide range of sink types such as voids, dislocations and precipitates. Brailsford [4] has discussed the general validity of this procedure and has obtained various “interactive” corrections for the sink strengths; however for the present discussion the essential point is that the embedding procedure is only strictly appropriate when the sink type is randomly distributed in
* Various expressions for the sink strength of a grain boundary have been published using either the embedding model [6] or the cellular model [ 71 but there has been no discussion of their relative merits - particularly in the limit of dominant defect loss at the grain boundaries compared with the loss at internal sinks.
* Based on the Maxwell [ 31 embedding procedure for the continuum representation of an inhomogeneous dielectric. 44
R. BulIough et al. /Sink strengths for film surfaces and grain boundaries
this “completely periodic” situation will be emphasized. Again the general sink strength with arbitrary surface rate limitation will be presented. The significance of ordering in determining the procedure for sink strength evaluation is then further elaborated in section 4 where we discuss the void sink strength when only voids are present. In particular by a judicious choice of sink free region in the embedding procedure we show that this can yield the correct sink strength for a void in an ordered distribution of voids up to swelling values of 100% or higher. In section 5 we revert back to the foil surface situation and examine the sensitivity of the sink strength to bulk recombination. We conclude, in agreement with perturbation [8] and iterative [9] analyses that the sink strength is insensitive and may be used with strong bulk recombination without modification. Finally we demonstrate the accuracy and usefulness of the complete rate theory, in which such foil surface sink strengths are included, by comparing the growth predictions in electron irradiated foils of zirconium using either the full rate theory model or the spatially varying model in which the surface losses are explicitly included. The agreement for the foil surface sink is excellent and furthermore we would also expect the grain boundary sink strength to be independent of bulk recombination.
2. The grain boundary sink strength We will begin by deriving a sink strength for the grain boundary by using the obvious cellular model with the simplest boundary condition at the boundary; this boundary condition will be generalised later to include arbitrary rate limitation there. The embedding procedure will then be described and used to obtain an alternative sink strength. The two sink strengths are then carefully compared with each other and with alternative published sink strengths based on the incorrect use of the cellular model. 2. I. The cellular model This model for the polycrystalline body considers the single spherical isolated grain, indicated in fig. 1a, of radius R. There is no identified surrounding medium, and the conditions in all the grains are assumed to be
45
a
Internal
sink
b Fig. 1. (a) Isolated grain of radius R - the cellular model The single crystal sink strength within the spherical grain is $c. The boundary condition at the grain boundary on the fractional defect concentration c is indicated; the only defect flow to the boundary comes from within this spherical grain, (b) The grain of radius R is embedded in the rate theory “lossy” medium - the embedding model. The single crystal sink strength within the spherical ain is k& and the lossy medium P has a total sink strength of kBb + k&. There is now defect flow to the boundary both from within the grain and from the surrounding lossy medium.
similar. It is this assumed similarity of the grains and the isolated nature of this typical grain that is consistent with the term “cellular”. However, the assump tion of a spherical form with its inability to “fill” space and the implied random impingement of surrounding grain boundaries upon it also introduces an implied randomness into the actual represented grain distribution - we shall stress this point when we relate the results of this cellular model to the corresponding embedding model. To obtain an effective sink strength for the grain boundary using this model we proceed as follows: The steady state atomic concentration of point defects (either interstitials or vacancies *) c = c(r) within the * The distinction between the two will only be ma& where necessary by an appropriate i or v subscript.
R. Bullough et al. /Sink strengths for film surfaces and grain boundaries
46
grain is given by the continuity equation
D[$+y
+K-Dk&c=O,
rate to eq. (4) in the continuum is
(1)
0
F gt, = Dk&
(7)
>
which, from the continuity eq. (6), yields where spherical polar co-ordinates with their origin at the centre of the spherical grain have been adopted, D is either point defect diffusion coefficient, K is the rate of production in dpa/sec of either defect and k& is the total sink strength from all the “single crystal” microstructure within the grain. For simplicity of derivation point defect thermal emission processes will not be explicitly included but the appropriate final vacancy emission rates will simply be presented. If the grain boundary is an ideal sink c=O
r=R,
at
(2)
ADkh
L l--
R sinh kscr
and the point defect loss rate/atom of the grain out of the grain, to the grain boundary is Rout = _ -3 cell
RD;i
dC I ,.zR
= kEz sC
[k,$
coth k,,R - l] . (4)
We wish to replace the polycrystalline real body by a complete lossy continuum with a total sink strength k*=k$,+k;,,
k$, = k& [k,,R coth k,,R - l]
X [l + k&R*/3 - k,,R coth k,,R] -’ ,
and we note the extreme values: (1) when the losses to the internal sinks dominate over the losses to the grain boundaries, i.e. when
(5)
k& = 15/R*
k,,R + 00,
(10)
* The impliciteffect of bulk recombination will be discussed in section 5. ** The significance of the - superscript for the point defect concentration in the lossy or rate theory continuum will become apparent when we discuss the embedding model below.
k,,R + 0 .
(11)
3K c(r) dr = Dk4SC R* + 1 - k,,R coth k,$
(6)
where coo is the point defect concentration ** in the lossy continuum and the equivalent point defect loss
when
It is important to emphasize that the sink strength (9) can only be justified by this procedure when k& is non-zero, since otherwise the flow (8) loses its dependence on k$, and equating the flows cannot yield an expression for k& It is to examine further the validity of eq. (9), particularly in this low internal sink limit, that we wish to discuss the equivalent embedding model, but before doing so however we point out that the sink strength (9) could also be obtained by replacing cm in eq. (7) by the mean point defect concentration in the grain
where k& is the required grain boundary sink strength. The continuity equation for this continuum is (neglecting thermal emission and bulk recombination * loss) K-Dk*c”=O,
(9)
and (2) when the internal sink losses are very small compared to the grain boundary losses (3)
r Gnh k,,R 1 ’
(8)
Equating the flows (4) and (8) gives the required expression for the grain boundary sink strength (subject to the ideal sink assumption (2)):
k& = 3k,,/R
then the concentration within the grain is given by the solution of eq. (1) subject to eq. (2) and that it be bounded at r = 0; that is c(r) =
Fgi, = K&,/(k;t, + k:c) .
I
.
(12)
It follows, of course, that the sink strength (9) is consistent with a precise equality between the mean concentration in the isolated grain (12) and the concentration co in the lossy continuum given by eq. (6). Furthermore this consistency does not apply if cm in eq. (7) is replaced by the concentration at the centre of the grain c(O)=
D+ 1 -s;;RR SC
[
1 .
SC
(13)
41
R. Bullough et al. /Sink strengths for film surfaces and grain boundaries
In this case equating the flows yields a spurious grain boundary sink strength that has been used by some authors [7] : glJ = [3/~21kcR cash kscR - sinh k,,R] X [sinh k,,R - k,,R]-’
.
(14)
when k,,R + m this sink strength has the same limiting values as (9) given by eq. (10) but on the other hand, when the internal sink losses are very small compared to the grain boundary losses its limiting value is k;,, = 6/R’
when
k,,R + 0 ,
(15)
which does not agree with eq. (1 l), the corresponding limiting value of eq. (9). 2.2. i?re embedding model [IO,11 ]
(16)
exp[-k(r
- R)]
,
(18)
is the point defect concentration in the lossy continuum well away (r + a) from the spatial perturbations caused by the actual embedded grain. The point defect loss rate into the grain boundary T = R from the lossy continuum is thus
1r=R=$$P+k%
3D&
Ein=xdr
(17)
(1%
from eqs. (17) and (18). The mean point defect flow rate into the grain boundary is thus [l 1) F= i[Pn 3K
tP] k,,R coth k,,R - 1 + 1 + kR
2R2 C
and remove from it a spherical volume of radius R. This sphere is then replaced by a sphere of equal volume of single crystal material with a sink strength of k$, as indicated in fig. 1b. A grain boundary sink is identified at the surface r = R and since, in the real polycrystalline body, adjacent grain boundaries will always impinge on this surface, it is not sensible to identify a “sink free zone” [2] outside this surface as is necessary for the derivation of sink strengths of other sink types [2] (see section 4 for example). The grain is thus directly embedded in the lossy continuum. To obtain the required gram boundary sink strength we must obtain the mean point defect loss rate into the surface r = R from both outflow from within the grain and from the inflow from the surrounding “lossy” continuum region. The point defect concentration within the gram r
c m = K/ok2
=-
In this model we begin with the “lossy” continuum with a total sink strength of k2 = k& + k;,, ,
where k2 is now the total sink strength (16) and
kf c
k2
1’ (20)
The equivalent point defect loss rate to eq. (20) in the lossy medium away from the embedded grain is given by eq. (7) and equating these two flow rates yields a quadratic equation for k$,R2 with the explicit solution
x
(2_y-1)a2 1-2,
(21)
where fl=a!cotha!,
(22)
(lr2= k&R2 .
(23)
The limiting values of this sink strength are: k$, = 3k,,/R
when
ksJ2 + 00,
(24)
in exact agreement with eq. (lo), and k& = (3/2R2)(5
+ 421) = 14.4/R2 ,
(25)
which is very close to the corresponding cellular limit (11).
48
R. Bullough et al. / Sink strengths for film surfaces and grain boundaries
I
I
10
100
I
I
1000
too00
k:,R2 Fig. 2. A comparison of the dependence of the grain boundary sink strengths on the sink strengths within the grains k& and the average grain radius R. Curve A is the sink strength derived using the cellular model and the mean point defect concentration, curve B is the sink strength using the cellular model and the concentration at the centre of the grain and curve C is the sink stiength using the embedding modet
The variations of the embedding grain boundary sink strength (21), the cellular sink strength using the mean concentration (9) and the cellular sink strength using the concentration at the centre of the grain (14) with internal sink strength k& and grain radius R are shown in fig. 2. We see at once that the (mean concentration) cellular sink strength is really quite indistinguishable from the embedding result for this very wide range of parameters. On the other hand the (central concentration) cellular result (14) shows considerable deviation from the other two and must be regarded as unacceptable. The almost exact agreement between the cellular sink strength (9) and the embedding sink strength (21) although gratifying requires some explanation, which will become much clearer after our discussion of the thin film surface sink strengths and the void strengths in the next two sections. At this stage it should suffice to comment that since all grains are the same and are essentially independent the mean point defect concentration F (eq. (12)) in any one grain in the cellular model must be identified with
the physically meaningful point defect concentration in the lossy medium cm where all the gram boundaries have been smeared out. However, the grains are randomly disposed and since the embedding procedure is only appropriate for random distributions of sinks it should be appropriate for this situation. As we shah see below this coincidence of the cellular and embedding models only occurs for the grain boundary sink. Having argued that the result (9) is an acceptable grain boundary sink strength we now present the corresponding general result, using the mean cellular point defect concentrations, when the ideal sink boundary condition (2) is replaced by the general rate limited form: -Ddc/&=K[c-ce]
at
r=R,
(26)
where K is the point defect transfer velocity into the boundary sink and ce is the point defect equilibrium concentration; note for interstitials ce = ce = 0. The required, general grain boundary sink strengths are
R. BuIlough et al. / S’ink strengths for film surfaces and grain boundaries
then:
tion since its construction in a similar fashion to the previous grain boundary analysis would imply the random impingement on any one embedded foil of adjacent foil surfaces and not a set of parallel identical “foils”. To obtain the required foil surface sink strength k,” we fust obtain the steady point defect concentration c = c(x) within the foil given by the continuity equation
vacancy sink strength k&b = %
X
[(
[k,R
k,,,R
coth k,,,R - I]
- F
WC
coth kVsCR - 1
3sv -l +&R +k&R I ’
(27)
interstitial sink strength kfgb = T
X
[ki,J? coth kiscR - l]
kiscR - p)
coth kiscR - 1
isc
3Si -’ + SiR + k&R I ’
(28)
and, in the usual notation [6], the rate of vacancy emission from the grain boundaries is Kevgb = ~vk:,b~~ .
D d*c/dx* + K - Dk,2,c = 0 ,
(31)
where the Cartesian coordinate x-axis is perpendicular to the foil surfaces with the coordinate origin x = 0 on one foil surface and the other surface located at x = 21. Again k& is the sink strength from all the microstructure within the foil and thermal vacancy emission processes will not be explicitly included but will be presented at the end of the section. The solution of eq. (31) in the half foil region 0 Q x < I, subject to the ideal sink boundary condition * c=o
at
x=0,
(32)
and the symmetry condition (29)
d&lx= 0
In eqs. (27) and (28)
Sv = Ky/Dv p Si = Ki/Di
49
(30)
at
x =I
is cash k&x - I) cash k,,l I ’
and the zero rate limitation result (9) follows from eq. (27) or (28) by letting Sv, St + O0s
3. The foil surface sink strength ln this case of a single foil of thickness 21, it is
clear that the point defect concentration in the lossy continuum, where the surface sink strength is required, must be equal to the actual mean concentration in the foil. The continuum may be thought of as replacing a one dimensional periodic array of foil surfaces but where each “foil” is completely independent of its “neighbours”; there is no useful realization * of the embedding model for this situa* However, as we shall see in the next section, it may be poasible to discover a suitable “sink free region” that would
(33)
(34)
and the point defect loss rate/atom of the foil across the two foil surfaces is F OUdD& l
1 =5
;i;; x=o
k,,l
tanh k,,Z .
(35)
The total sink strength in the equivalent lossy continuum is k*=k;+&,
(36)
where kt is the required foil surface sinkstrength. The steady state continuity equation for the continuum is K - Dk2coo = 0,
(37)
facilitate such a realization but since this would, at best, only reproduce the cellular model result it does not seem worthwhile pursuing further. * The results for general rate limitation at the surfaces will be presented at the end of the section.
R. Bullough et al. /Sink strengths for firm surfacesand grain boundaries
50
where cm is the point defect concentration in the continuum and losses due to bulk recombination are not included; the significance of bulk recombination will be discussed in detail for this sink in section 5. The equivalent point defect loss rate to eq. (35) in the continuum is F, = Dk,2c” ,
(38)
which, from the continuity equation (37) yields F, = Kk;/(k:
+ kzC) .
(39)
Equating the flows (35) and (39) gives the required expression for the foil surface sink strength (subject to the ideal sink assumption (32)):
1. SC
k:=+
w
coth k,,l - &IL
kz=s
when
when
k,,l + 00 ,
(41)
(46)
The discrepancy between eq. (46) and the result (42) should be noted. With the conviction that eq. (40) is a consistent foil surface sink strength we can now generalize the result to include the possibility of rate limitation at the foil surfaces; that is we replace the ideal sink boundary condition (32) by Ddc/dx=R[c-ce]
at x=0,
(47)
and proceed as before. We thus obtain the general results: vacancy sink strength
1’
(48)
3
(49)
sv
coth k,,,l t 1 -
We note the two extreme values of this sink strength: kz = k,,/l
k,,l+O.
kzvsc
interstitial sink strength
and coth kiscl + 1 - & isc 1
kz = 3/12
when
k,,l+
0 .
(42)
The mean point defect concentration in the foil is C=-
1 1 s lo
c(x)dx=D+ SC
[
l--
tanh k,,l k,J
1’
K”VS =Dk2ce v vs (43)
which for the sink strength (40) is identical to cm, given by eq. (37). Hence the sink strength (40) is completely consistent with the previously stated notion that the point defect concentration in the lossy continuum must be identical to the mean point defect concentration in the foil. Again, as in the case of the grain boundary sink strength derivation, some authors [ 121 have used the concentration c(l) at the foil centre in eq. (38) instead of the mean concentration cm = F. Such an assump tion yields the foil sink strength [ 121 k,2 = (k,,/l)
coth k,,1/2 .
WI
The error involved in adopting this sink strength for
the ,foil surfaces will be discussed in section 5. It will suffice here to give the two extreme values taken by eq. (44): kz = k,,/l
when
k,,l + - ,
and, in the usual notation [6], the rate of vacancy emission from the foil surfaces is
(45)
wa
v,
where ct is the vacancy equilibrium concentration. In eqs. (48) and (49) Sv.= K&
,
Si = KJDi ,
(51)
where K, and Ki are respectively the transfer velocities of vacancies and interstitials across the foil surfaces. Finally the zero rate limitation result (40) follows from eq. (48) or eq. (49) by letting Sv, St + * e
4. The void sink strength To further clarify the importance of sink ordering on the relation between the embedding and cellular model estimates of sink strengths we now discuss the void sink strength and particularly emphasize the various interactive corrections that relate to this sink. The results of this section should help to distinguish
R. Buliough et al. / Sink strengths for pm surfdces and grain boundaries
51
between various published and sometimes erroneous sink strengths based on the cellular model and the more correct results of the embedding model. Before discussing the relative merits of the two models it is convenient to obtain the various sink strengths and for simplicity we omit considerations of vacancy thermal emission and treat the void as an ideal sink. We begin with the cellular model.
void
4.1. The cellularmodel
c(r) = -E(G
In this model the voids (all of radius a) are assumed to be regularly distributed in the body such that each can be considered to lie at the centre of a spherical cell, as shown in fig. 3, of radius R,. When such voids are the only sinks present in the body the steady state point defect concentration within the cell is given by the solution of the continuity equation:
Quantities required are the mean concentration in the cell:
Dd -_ rl&
r2dc +K=() dr 1 (
c=O
at
r=a,
(53)
and the zero flow condition at the cell boundary: dc/dr=O
r=Rc,
at
(54)
where spherical polar coordinates have their origin at the centre of thk spherical cell. The solution is
C=--R:Ta’[
[2R,3 - (r+a)ar] .
Rc
r2 c(r) dr =
K 1SDa(Rf - a3)
X [5R6C - 9aR’C + 5a3R3 - a61 9 C
(56)
the concentration at the cell boundary:
(52)
9
subject to the ideal sink boundary condition at the
c(R3 = &(R,
-a)2(W+a),
and the number of point defects per second that enter the void =E (R: _ as), 352
(58)
where R is the atomic volume. In the lossy continuum where this regular distribution of voids is smeared out the number of point defects/atom lost at voids is Dk$cm, where kt is the required total void sink strength and co is the point defect concentration in the continuum. The number of atoms associated with each void is [GX$]-‘, where
a
Continuity
Lonsy continuum
c”, = 3/41rR: ,
(Ri=Rz
Fig. 3. (a) The cellular model, appropriate for a regular distribution of voids. The radius of the cell is related to the current void volume concentration by eq. (78). (b) The embedding model, appropriate for a random distribution of voids. RSF is the radius of the sink free region surrounding the void and the composite sphere is embedded in the 10s~ continuum.
(59)
is the initial volume concentration of voids (when a = 0). Thus the effective number of point defects per second that enter each void in the lossy continuum F cant= Dk$c”liX$
b
-a3),
.
@)
By equating the two flows (58) and (60) we obtain the general expression for the required void sink strength k$ =
4nCcK (Rf - a3)/3Dc” .
(61)
For the lossy continuum representation of this regular
R. BuIlough et al. /Sink strengths for film surfaces and grain boundaries
52
distribution of voids to be at all physically meaningful the point defect concentration in it must be equal to the mean point defect concentration in the cell, as discussed above, i.e. c =- -r, (62) with this equality, eq. (61) yields from eq. (56) k$ = haC$fF-, where + 5a3Rz - a61
+K-Dk$c=O.
(64)
r=a,
at
(69)
the concentration and its radial derivative are continuous across r = R, and the concentration in the lossy continuum a long way from the embedded region is bounded and, from eq. (68), is given by
is the interactive correction term on the sink strength when the equality with the mean concentration i? in eq. (62) has been used. Several authors [ 131 have derived an alternative correction term fccRd by using the equality
c(r)=A/r+B-Kr2/6D,
c O”= c(R3.
and
(65)
This has the explicit form, from eqs. (57) and (61) fc(Rc)
= 2@,3
-
a3M&
-
a)*@&
+ 4
*
(68)
The boundary conditions are that the void is an ideal point defect sink c=O
(63)
fT= 5(Rz - a3)‘/[5R,6 - 9&
and in the lossy continuum r > R~F
C OD=
K/Dk$ .
(70)
Subject to these boundary conditions the complete solution is
c(r)-K/Dk$
a
+(~/r)exP[kv(&
-r)]
(71)
,
~>RSF,
(66)
(72)
where the constants A, B and Pare 4.2. The embedding model
A=
This model has been discussed in detail by Brailsford, Bullough and Hayns [2] and by Brailsford [4]. It specially applies when the distribution of voids is random and we will, for comparative purposes with the above cellular model results, concentrate on the relatively simple situation when only voids are present, all are of equal radius a, present ideal sinks for the point defects and there is no vacancy thermal emission from them. The model is displayed in fig. 3 and is constructed by removing a sphere of radius RsF from the lossy continuum and replacing it by a sphere of the perfect material with a single finite void at its centre; the sphere is thus “sink free” apart from the central void. The net point defect flow rate into this void is then calculated and again equated to the equivalent flow rate in the lossy continuum. To obtain this flow rate into the central void we solve the following steady state continuity equations for the point defect concentration c(r) in the two regions. In the sink free region a < r < R~F , 2
Ka 6Dk$[l + kv&F
x [-3(k;R&
- a)]
+ 2kvRsF + 2)
+ k$(a*+a*RsFkv - kvRbF)] ,
B=
K 6Dk$ [l + kv(Rs~ - a)] X [3(k$R& + 2kvRsF + 2) + k:(R;F
P=
- a3)] , (73)
K 6Dk$[l+ kv(RsF - a)] X [-6.a + k$(RsF - a)*(2RsF + a)] .
The number of point defects per second that enter the void in the embedded sphere is F emb
=-
ha20 dc 52
dr
1r=a
= slk$
f X (1 + kvRsF) + k ‘“7
(67) + kv&F
- a&F
+ a)]
4naK [l + kv(RsF - a)] - ‘) [3(Rs~ + fZ)
I
.
R. Bullough et al. /Sink strengths for film surfaces and grain boundaries
We notice that in this model, in contrast to the corresponding cellular expression (58), the required void sink strength appears explicitly in eq. (74). As for the cellular model we obtain k$ by equating the flow rate (74) to the corresponding flow rate (60) in the continuum a long way from the embedded sphere, where cm in eq. (60) is now given by eq. (70). This yields the cubic equation for kv : kt = 4~&k,b
(kv) ,
(75)
where 6
&F
- a) 6
X [~(RsF + a) + ~kv(RsF-
x [l
+ kv(RsF - a)]-’
a)&~
+
%)I
(76)
is the interactive correction term which can only be evaluated when the cubic eq. (75) has been solved for kv. The choice of the radius of the sink free region RSF in eq. (76) is, of course, important and has been discussed previously by Brailford, Bullough and Hayns [ 21. These authors argued that when a second sink type is present, such as dislocations, the value of RSF can be reduced down to the void radius a without much loss of accuracy. In the present situation where only voids are present it is clear from the construction of the embedding model when the sphere of radius RSF was removed from the lossy continuum and replaced by the sphere with the void of radius (I at its centre, that this radius must reflect the volume of material/void when the voids are of radius a. We may therefore logically choose RSF
=R, ,
(77)
where R, is the average cell radius associated with each void and is related to the actual void concentration Cv by C, = 3/41rR,3,
(78)
which is to be distinguished from the initial II = 0 void concentration CG given by eq. (59). The choice of RSF in eq. (77) then enables us to compare the three void sink strength corrections fT,f,.cRd and em,, with the understanding that the femb reSUlt f with the choice of eq. (77) for RSF is the best esti-
53
mate for a random distribution of voids. This comparison is given in fig. 4 where the variation of the three correction terms is shown against the swelling S=a3/R%
.
(79)
The definition of swelling is somewhat arbitrary; we have, in eq. (79), chosen to identify the swelling with the total number of vacancies in voids/atom of the body or in other words we refer the volume change due to the voids to the initial volume (when u = 0) of the body and not to the current volume of the body (when the voids are of radius a). It is, of course, important that when experimental swelling values are presented the reference volume used should also be indicated. In terms of the swelling S the three correction terms have the explicit form. From eqs. (64) and (79) 5(1 +x+x2)2 fT=(, -X)(5+6X+3x*+xs)’
(80)
from eqs. (66) and (79) 2(1 +x+x2)
few
= (1 -X)(2 tx) ’
031)
and from eqs. (75) (76) and (79) f emb =
x [xt y(l
-x)J -l ,
(82)
where y is the positive root of the cubic femb(Y) =Y2/3S
(83)
and x = [S/(1 + AS)]V3.
(84)
We also include in fig. 4 the variation of fimb obtained from eq. (76) with no sink free region: RsF=a,
(85)
which has the explicit form j&,,,={2+3S+[9S2+l~]“*}/2.
(86)
It is apparent from fig. 4, and may be seen analytically by solving the cubic (83), that when the sink free radius is taken equal to the cell radius, as in eqs. (82) and (83), that
femb
F fc(R3 *
(87)
54
R. BuUough et al, /Sink strengths for film surfaces and grain boundaries
Fig. 4. The variation of the various void sink strength correction andfimb (eq. (86)) with the swelling S.
This rather remarkable, but fortuitous, result tells us that when a sphere of lossy continuum is replaced by the sink free sphere (with a single void at its centre) of radius equal to the average cell radius then the constant concentration cm in the lossy continuum is not perturbed near the sphere; mathematically P=O
038)
in eq. (73). Thus in this case when only voids are present the void sink strength in a random distribution of voids can be obtained from the cell model by taking the cell boundary concentration as the lossy continuum concentration. We believe however that such a cellular approach is dangerous since it strictly only applies to the regular spatial distribution of sinks and then it is the mean concentration in the cell that must be identified with the constant concentration in the lossy medium. We shall demonstrate that a sink free radius can be found to permit the embedding model to model just this situation. Thus the embedding model is the only sensible approach when the distribution is random and is particularly so when other sinks are present since in that case a
termsf&eq. (80% fc(R,) (e¶. (81))pfemb (eqs. (82) and (83))
consistent coDcould not be defined. Furthermore, the embedding model always ensures consistency since all the sinks are simultaneously present in the surrounding lossy continuum. The sink free region can either be neglected for each sink type [2] or a value for the radius of the region consistent with the highest density of remaining sink types in the microstructure can be chosen. The fZ)& has the lowest value, in fig. 4, and has been included there for iUustrative purposes only. An interesting aspect of fig. 4 is that
fF >femb
(89)
over the whole range of swelling. One could interpret this to mean that the sink strength of a void as a member of a regular distribution of voids (for which fr would be appropriate) is always greater than the sink strength of a similar void in a random distribution of voids of the same density (for which fmb would be appropriate). Perhaps this is a contributary reason for the formation of perfect void lattices in many materials. To conclude this discussion of the void sink
R. Bullough et al. /Sink strengths for film surfaces and grain boundaries
55
I
I
1.30-
1.25 -
1.20 -
.l
1
100
sl%)
Fig. 5. The variation of Q with the swelling S required to ensure that the void sink strength using the embedding model agrees with the cellular result appropriate for a regular distribution of voids. The radius of the sink free zone is defined by RSF = aRg and the swelling S = a ‘/R 3.
strength we ask whether a sink free radius can be found that enables the embedding model to encompass the regular distribution of voids. Thus we take RSF
=d0,
(90)
and obtain the variation of femb with swelling S, from eqs. (75) and (76) for a range of 01.We can then deduce the dependence of (IIon S to ensure that the f embvalue always coincides with fr, which we know to be correct for the regular distribution. The required variation of cr with S is shown in fig. 5. It is thus clear that although the embedding procedure is only strictly appropriate for random distributions of sinks it can be “fixed up” for the regular distribution by a judicious choice of variable sink free region. We now revert back to the foil surface sink situation and examine the sensitivity of this sink strength to bulk recombination. 5. The foil surface sink strength and bulk recombination In any practical use of the foil surface sink strength (40) or the more general results (48) and (49) it is
essential to ensure that the presence of significant bulk recombination does not explicitly change the form or magnitudes of these sink strengths. This question has been discussed previously by White, Fisher and Miller [ 121, but they based their argument upon the use of the sink strength (44) and found it was necessary to modify (empirically) the sink strength (44) when the bulk recombination was significant. On the other hand Bra&ford, Matthews and Bullough [8] and Hayns [9] have presented both perturbation and iterative analyses respectively that indicate that the direct modifications on the (correct) sink strengths should probably always be negligible. It thus seemed worthwhile to check directly the validity of eq. (40) when point defect losses by bulk recombination were not negligible by comparing results obtained from the lossy continuum rate theory with exactly equivalent results using the full spatial solution for a thin foil. The generalized forms of the continuity equation (13) for the vacancy (cv) and interstitial (ci) concentrations within the irradiated foil are d2cv
D
VG
+K+Ke,,, - Dvk$,Cv - WiCv = 0 3
(91)
56
R. Bullough et al. /Sink strengths for firm surfaces and grain boundrrries
d2c.
+ K - bikfscci - MiCv = 0 , Di 1 dir2
(92)
FiN = 6s
where CY is the bulk recombination coefficient and K&, is the rate of vacancy emission from the internal microstructure [ 1,2] which has a total sink strength of k:,, and k& for vacancies and interstitials respectively. This pair of non-linear equations have to be solved numerically subject to appropriate boundary conditions at the foil surfaces or equivalently at one foil surface (x = 0) and a symmetry condition halfway through the foil (X = Z). The ideal sink boundary condition * is C, =I$
,Ci
=O
at
(93
at
X=1.
(94)
For simplicity we now suppose that only one type of sink exists inside the foil and that these sinks are network dislocations and are uniformly distributed through the foil. In this case we may write [ 1,2] &
= PN,
&
= ZiPN 9
(96)
x=0 ’
which must be equal to the net loss rate per atom of interstitials at the dislocations within the foil I
W$PN&)
- &PNC&)
FiN 3
+K:NN)~
,(97)
0
if dislocations are the only sink type present, otherwise the equality would be with the net interstitial loss rate per atom at the total internal microstructUre. Here, pvN is the rate Of vacancy emiSSiOnfrom * Computations have been successfully made with general rate limitation boundary conditions imposed on the foil surfaces but for the present purposes the ideal sink conditions will suffice.
(99)
where A = Kf and we will use this computed growth rate, with physical parameters appropriate to zirconium, to investigate the validity of the foil sink strengths and bulk recombination effects. The general rate theory equations corresponding to eqs. (9 1) and (92) for the lossy continuum point defect concentrations are * K + K&c + Kvs - Dv(k&c + k&) cv - (YciCv= 0 ,
(95)
where PN is the dislocation density and 21 el) is the bias parameter defining the preferential interaction between interstitials and dislocations comparedwith that between vacancies and dislocations. In terms of these point defect concentrations, the net loss rate per atom of vacancies across the two surfaces
FiN=fJ
(98)
can easily be seen by substracting eqs. (9 1) and (92) and integrating the difference over half the foil thickness in conjunction with the symmetry condition (94). In a hexagonal material such as zirconium the edge dislocations all lie on the prismatic planes and thus the growth/dpa in the a-axis direction caused by the climb of such dislocations due to absorption of the excess interstitials is [ 141 de/dA= (l/X)
X = 0,
and the symmetry conditions is &v/dX=&i/dX=O
the network dislocations [ 1,2]. The equality
(100) K - Di(kfsc + kiL,)Ci - OLCiCv =0 7
(101)
where now the foil surface sink strengths have been incorporated into the total microstructure sink strengths. If we again assume that only network dislocations are present then the equivalent net flow rate of vacancies to the foil surface (96) is F vs = Dvk$v
- DikTsCi- K& 3
(102)
and of interstitials to the dislocations (97) is F iN
=
WiPNG
- DVPNCV t GN 9
(103)
where again we see the equality (98) by substracting eqs. (100) and (101) and the growth rate is again given by eq. (99). The foil sink strengths k&, kfs were derived in section 3 and from eq. (40) are, for the present situa-
* In practice for computational convenience we always avoid assuming the quasi-steady state approximation and include the appropriate time derivatives on the right hand sides of the contimity equations. However,for presentational purposes we will omit them here.
R. Bullough et al. /Sink strengths for film surfaces and gmin boundaries
L 10'2
I
57
I
10"
PN
(me’)
10"
Fig. 6. The variation of‘predicted growth strain rate in a foil of thickness 10B6 m, with dislocation density pi for the zirconium parameters in table 1 when bulk recombination is neglected. Curve A is the correct variation obtained by using either the rate theory with the foil surface sink strengths (104) or by integrating the spatial equations (91) and (92) directly. Curve B is the predicted growth strain rate using the erroneous foil surface sink strength (44) and the other curve indicates the error resulting from such a choice.
tion k:,=$[coth/$/2L---]-1, _$104)
k;s = (Z,PN)1’2 I
coth (ZipN)“2Z [
1
1
(ZipN)r’2Z
*
These sink strengths were derived by identifying the
mean point defect concentration in the foil with the point defect concentration in the lossy continuum and it is clear from eqs. (97) and (103) that in the absence of bulk recombination they will yield growth rates in exact agreement with the result of integrating the full spatial equations (91) and (92). The alternative foil sink strength (44), uses the point defect concentration at the foil mid-point instead
Table 1 Physical and irradiation parameters used to model the electron irradiation response of zirconium [ 141 D, = 0,” exp[ -EF/kT] Di = Dp exp[ -E$kT] ct = exp[-EQkT]
,
, ,
0,” = 10e6 m2/s, Dp a 7.5 X10-a m2/s,
with EF = 0.65 eV with Ep = 0.3 eV with bf = 1.4 eV
Bulk recombination coefficient u given by a/oi = 2 X 10zo mm2. Interstitial dislocation preference Zt = 1.10 relative to vacancy 2, = 1.0. Displacement damage rate K = 3 X 10q3 dpa/s.
R. Bullough et al. / Sink strengths for film surfaces and grain boundaries
58
I
10"
10'3
I
PN
Imw21
10'L
1
10'5
Fig. 7. The variation of predicted growth strain rate in a foil of thickness 10m6 m with dislocation density PN for the zirconium parameters in table 1 when bulk recombination is included. The dotted curve is the accurate variation obtained by integrating the non-linear spatial equations (91) and (92) with eq. (97). Curve A is the rate theory result with the foil surface sink strengths (104) and curve B is with the sink strengths (44).
and the error on the growth rate that ensues is shown in fig. 6 for a foil of thickness 2Z= lo-” m, when bulk recombination is completely neglected. The physical and radiation parameters used in this section, and purporting to relate to zirconium, are given in table 1. The justification for claiming that these relate to zirconium is discussed in detail in an analysis [ 141 of the observed growth in that material and will not be pursued further in the present paper. We conclude from fig. 6 that the sink strengths for the foil surfaces (104) are certainly correct for zero bulk recombination and it only remains to check their accuracy in the presence of such recombination. This we have done by numerically integrating the non-linear spatial equations (91) and (92) across the foil by a discretization procedure; in practice we find the integration can be achieved with sufficient accuracy by adopting a minimum finite difference mesh of l/40. Increasing the mesh size from 30 to 40 points, results in an improvement in the calculated growth strain rate of less than 1%. Again the predicted growth strain rate for a foil of thickness 2Z= 10V6 m and using the parameters in table 1 but with bulk recombination included is given in fig. 7 (the dotted curve). To compare with this “accurate” variation we have used the rate theory, but with the same bulk recombination included (eqs. (100) and (lol)), with both
the foil surface sink strengths (104) (curve A) and the alternative strengths (44) (curve B); the influence of bulk recombination is thus simply included in the modifications to the lossy continuum point defect concentrations obtained from the quadratic solution of eqs. (100) and (101) and no explicit modification to the sink strengths themselves are included. We see from this figure that the error using the rate theory with the unmodified sink strengths (104) is always small (55%) compared with very large errors when the unmodified alternative sink strength is used. It is thus not surprising that authors [ 121 who chose to use the sink strength (44) are forced to modify it empirically in the presence of bulk recombination.
6. Conclusions (1) The foil surface sink strength (104) should be used in a rate theory representation of damage in foils. (2) This sink strength does not require modification (either empirically or by perturbation analysis) to allow for the presence of bulk recombination. It is thus only necessary to include the effects on the point defect concentrations of bulk recombination losses and not explicitly on the sink strengths (3) We expect the same conclusions to be true for
R. Bullough et al. /Sink strengths for firm surfaces and grain boundaries
the grain boundary sink strength (9) in preference to the alternative (14). (4) For random distributions of sinks, the embedding model is preferable and, indeed, if more than one sink type is present the embedding model is the only sensible model, since it permits a logical choice for RSF and ensures a consistent point defect concentration in the lossy continuum. (5) If the sink distribution is regular then the cellular model is appropriate and the mean point defect concentration can be identified with the concentration in the equivalent lossy continuum. However, we have shown that even in this case the embedding model can be constructed to yield equivalent results by varying RSF appropriately. (6) The best foil sink strength is obtained by regarding the sink distribution as perfectly periodic (regular) and thus the mean concentration in the cell model is obviously appropriate. An embedding model would be inappropriate for this situation. On the other hand the grain boundary sink combines the features of regularity and randomness in such a way that both models are appropriate for the derivation of its strength. (7) The void sink strength in a regular distribution is always greater than it would be if the same density of voids were randomly distributed. A feature that could have relevance to the formation of void lattices so frequently observed in many materials. (8) Finally, although in the present paper we have explicitly only demonstrated the reliability of the foil surface sink strength for the simple situation of a constant dislocation network density in a foil, we have, in fact, made many additional comparative computations including other internal sinks such as growing
59
interstitial loops, voids and traps together with complex rate limited boundary conditions at such sinks and at the foil surfaces. In all cases the general foil surface sink strengths (48) and (49) provide an accurate representation of the surface losses and require no modification due to bulk recombination; we thus would argue that any such modifications are unnecessary when these sink strengths are used.
References [l] A.D. Brailsford and R. Bullough, J. Nucl. Mater. 44 (1972) 121. [2] A.D. Brailsford, R. Bullough and M.R. Hayns, J. NucL Mater. 60 (1976) 246. [ 31 J.C. Maxwell, A Treatise on Electricity and Magnetism, Vol. 1, 3rd ed. (Clarendon, Oxford, 1892) p. 440. [4] A.D. Bra&ford, J. Nucl. Mater. 60 (1976) 257. [5] J.H. Evans, Nature 229 (1971) 403. [6] R. Bullough and M.R. Hayns, in: Proc. 2nd Intern. Symp. on Continuum Models of Discrete Systems, Mont Gabriel, Canada, 1977. [ 71 P.T. Heald and J.E. Harbottle, J. Nucl. Mater. 67 (1977) 229. [ 81 A.D. Bra&ford, J.R. Matthews and R. BuBough, J. Nucl. Mater. 79 (1979) 1. [9] MR. Hayns, J. NucL Mater. 79 (1979) 323. [lo] A.D. Bra&ford and R. Bullough, in: Proc. Intern. Conf. on Physical Metalhugy of Reactor Fuel Elements, Berkeley Castle, 1973 (The Metals Society, London) p. 148. [ 111 A.D. Bra&ford and R. Bullough, J. NucL Mater. 69/70 (1978) 434. [ 121 R.J. White, S.B. Fisher and K.M. Miller, CEGB Report RD/B/N4162, Nov. 1977. [13] A.J.E. Foreman and M.J. Makin, J. Nucl. Mater. 79 (1979) 43. (141 S. Buckley, R. BuBough and M.R. Hayns, J. Nucl. Mater. 89 (1980) 283.
SINKSTRENGTHSFORTHINFILMSURFACESAND
GRAINBOUNDARIES
R BULLOUGH, M.R HAYNS and M.H. WOOD Theoretical Physics Division,AERE, Hat-well,Oxfordshire, OX11 ORA, UK Received 1 November 1979
The formulation of a “lossy” continuum rate theory model for the representation of the evolvement of radiation damage in either a thin film or in a polycrystalline body requires a knowledge of the sink strengths for these surface sinks. The relative merits of the embedding and cellular models for the calculation of such sink strengths will be discussed and suggested modifications when non-linear bulk recombination is significant are examined by appropriate numerical proce dures. It is concluded that such modificationsare unnecessarywhen the “correct” sink strengths are used. The importance of obtaining such reliable rate theory representation of the microstructure to the efficiency of our exploration of the factors influencing growth and swelling in irradiated materials is obvious.
1. Introduction
space, although as we shall see below it can be judiciously modified to encompass even a regular distribution of sinks such as the voids in a perfect void lattice [5]. ‘I% distinction between ordered sinks and random sinks raises the question of the relation between sink strengths derived using cellular models (in which spatial ordering is implied) and those using the embedding model which precludes ordering. We believe the cellular model is often inappropriate for the derivation of useful sink strengths and even when appropriate has sometimes been used incorrectly. To illustrate this point we derive in section 2 the grain boundary sink strength * using both the embedding and cellular model and indicate the rather special relation between them for this particular sink; the sink strength for arbitrary rate limitation at the boundary is then presented for use in the general rate theory continuum. Attention is then focussed in section 3 on the foil surface sink strength and we obtain this sink strength using the cellular model; the inappropriateness of an embedding procedure for
Surface defects, such as the external surfaces of a thin foil or the internal grain boundaries, can be significant sinks (or sources) of intrinsic point defects during irradiation and the net point defect losses at such sinks must be quantitatively understood if the observed internal microstructure is to be interpreted correctly. Further, for purely practical computational purposes the sink strengths of such surface defects are required for explicit inclusion in the general rate theory model of the total microstructure. Brailsford and Bullough [ 1 ] and Bra&ford, Bullough and Hayns [2] have discussed an embedding procedure * within a uniform lossy medium for the derivation of such sink strengths and have obtained sink strengths for a wide range of sink types such as voids, dislocations and precipitates. Brailsford [4] has discussed the general validity of this procedure and has obtained various “interactive” corrections for the sink strengths; however for the present discussion the essential point is that the embedding procedure is only strictly appropriate when the sink type is randomly distributed in
* Various expressions for the sink strength of a grain boundary have been published using either the embedding model [6] or the cellular model [ 71 but there has been no discussion of their relative merits - particularly in the limit of dominant defect loss at the grain boundaries compared with the loss at internal sinks.
* Based on the Maxwell [ 31 embedding procedure for the continuum representation of an inhomogeneous dielectric. 44
R. BulIough et al. /Sink strengths for film surfaces and grain boundaries
this “completely periodic” situation will be emphasized. Again the general sink strength with arbitrary surface rate limitation will be presented. The significance of ordering in determining the procedure for sink strength evaluation is then further elaborated in section 4 where we discuss the void sink strength when only voids are present. In particular by a judicious choice of sink free region in the embedding procedure we show that this can yield the correct sink strength for a void in an ordered distribution of voids up to swelling values of 100% or higher. In section 5 we revert back to the foil surface situation and examine the sensitivity of the sink strength to bulk recombination. We conclude, in agreement with perturbation [8] and iterative [9] analyses that the sink strength is insensitive and may be used with strong bulk recombination without modification. Finally we demonstrate the accuracy and usefulness of the complete rate theory, in which such foil surface sink strengths are included, by comparing the growth predictions in electron irradiated foils of zirconium using either the full rate theory model or the spatially varying model in which the surface losses are explicitly included. The agreement for the foil surface sink is excellent and furthermore we would also expect the grain boundary sink strength to be independent of bulk recombination.
2. The grain boundary sink strength We will begin by deriving a sink strength for the grain boundary by using the obvious cellular model with the simplest boundary condition at the boundary; this boundary condition will be generalised later to include arbitrary rate limitation there. The embedding procedure will then be described and used to obtain an alternative sink strength. The two sink strengths are then carefully compared with each other and with alternative published sink strengths based on the incorrect use of the cellular model. 2. I. The cellular model This model for the polycrystalline body considers the single spherical isolated grain, indicated in fig. 1a, of radius R. There is no identified surrounding medium, and the conditions in all the grains are assumed to be
45
a
Internal
sink
b Fig. 1. (a) Isolated grain of radius R - the cellular model The single crystal sink strength within the spherical grain is $c. The boundary condition at the grain boundary on the fractional defect concentration c is indicated; the only defect flow to the boundary comes from within this spherical grain, (b) The grain of radius R is embedded in the rate theory “lossy” medium - the embedding model. The single crystal sink strength within the spherical ain is k& and the lossy medium P has a total sink strength of kBb + k&. There is now defect flow to the boundary both from within the grain and from the surrounding lossy medium.
similar. It is this assumed similarity of the grains and the isolated nature of this typical grain that is consistent with the term “cellular”. However, the assump tion of a spherical form with its inability to “fill” space and the implied random impingement of surrounding grain boundaries upon it also introduces an implied randomness into the actual represented grain distribution - we shall stress this point when we relate the results of this cellular model to the corresponding embedding model. To obtain an effective sink strength for the grain boundary using this model we proceed as follows: The steady state atomic concentration of point defects (either interstitials or vacancies *) c = c(r) within the * The distinction between the two will only be ma& where necessary by an appropriate i or v subscript.
R. Bullough et al. /Sink strengths for film surfaces and grain boundaries
46
grain is given by the continuity equation
D[$+y
+K-Dk&c=O,
rate to eq. (4) in the continuum is
(1)
0
F gt, = Dk&
(7)
>
which, from the continuity eq. (6), yields where spherical polar co-ordinates with their origin at the centre of the spherical grain have been adopted, D is either point defect diffusion coefficient, K is the rate of production in dpa/sec of either defect and k& is the total sink strength from all the “single crystal” microstructure within the grain. For simplicity of derivation point defect thermal emission processes will not be explicitly included but the appropriate final vacancy emission rates will simply be presented. If the grain boundary is an ideal sink c=O
r=R,
at
(2)
ADkh
L l--
R sinh kscr
and the point defect loss rate/atom of the grain out of the grain, to the grain boundary is Rout = _ -3 cell
RD;i
dC I ,.zR
= kEz sC
[k,$
coth k,,R - l] . (4)
We wish to replace the polycrystalline real body by a complete lossy continuum with a total sink strength k*=k$,+k;,,
k$, = k& [k,,R coth k,,R - l]
X [l + k&R*/3 - k,,R coth k,,R] -’ ,
and we note the extreme values: (1) when the losses to the internal sinks dominate over the losses to the grain boundaries, i.e. when
(5)
k& = 15/R*
k,,R + 00,
(10)
* The impliciteffect of bulk recombination will be discussed in section 5. ** The significance of the - superscript for the point defect concentration in the lossy or rate theory continuum will become apparent when we discuss the embedding model below.
k,,R + 0 .
(11)
3K c(r) dr = Dk4SC R* + 1 - k,,R coth k,$
(6)
where coo is the point defect concentration ** in the lossy continuum and the equivalent point defect loss
when
It is important to emphasize that the sink strength (9) can only be justified by this procedure when k& is non-zero, since otherwise the flow (8) loses its dependence on k$, and equating the flows cannot yield an expression for k& It is to examine further the validity of eq. (9), particularly in this low internal sink limit, that we wish to discuss the equivalent embedding model, but before doing so however we point out that the sink strength (9) could also be obtained by replacing cm in eq. (7) by the mean point defect concentration in the grain
where k& is the required grain boundary sink strength. The continuity equation for this continuum is (neglecting thermal emission and bulk recombination * loss) K-Dk*c”=O,
(9)
and (2) when the internal sink losses are very small compared to the grain boundary losses (3)
r Gnh k,,R 1 ’
(8)
Equating the flows (4) and (8) gives the required expression for the grain boundary sink strength (subject to the ideal sink assumption (2)):
k& = 3k,,/R
then the concentration within the grain is given by the solution of eq. (1) subject to eq. (2) and that it be bounded at r = 0; that is c(r) =
Fgi, = K&,/(k;t, + k:c) .
I
.
(12)
It follows, of course, that the sink strength (9) is consistent with a precise equality between the mean concentration in the isolated grain (12) and the concentration co in the lossy continuum given by eq. (6). Furthermore this consistency does not apply if cm in eq. (7) is replaced by the concentration at the centre of the grain c(O)=
D+ 1 -s;;RR SC
[
1 .
SC
(13)
41
R. Bullough et al. /Sink strengths for film surfaces and grain boundaries
In this case equating the flows yields a spurious grain boundary sink strength that has been used by some authors [7] : glJ = [3/~21kcR cash kscR - sinh k,,R] X [sinh k,,R - k,,R]-’
.
(14)
when k,,R + m this sink strength has the same limiting values as (9) given by eq. (10) but on the other hand, when the internal sink losses are very small compared to the grain boundary losses its limiting value is k;,, = 6/R’
when
k,,R + 0 ,
(15)
which does not agree with eq. (1 l), the corresponding limiting value of eq. (9). 2.2. i?re embedding model [IO,11 ]
(16)
exp[-k(r
- R)]
,
(18)
is the point defect concentration in the lossy continuum well away (r + a) from the spatial perturbations caused by the actual embedded grain. The point defect loss rate into the grain boundary T = R from the lossy continuum is thus
1r=R=$$P+k%
3D&
Ein=xdr
(17)
(1%
from eqs. (17) and (18). The mean point defect flow rate into the grain boundary is thus [l 1) F= i[Pn 3K
tP] k,,R coth k,,R - 1 + 1 + kR
2R2 C
and remove from it a spherical volume of radius R. This sphere is then replaced by a sphere of equal volume of single crystal material with a sink strength of k$, as indicated in fig. 1b. A grain boundary sink is identified at the surface r = R and since, in the real polycrystalline body, adjacent grain boundaries will always impinge on this surface, it is not sensible to identify a “sink free zone” [2] outside this surface as is necessary for the derivation of sink strengths of other sink types [2] (see section 4 for example). The grain is thus directly embedded in the lossy continuum. To obtain the required gram boundary sink strength we must obtain the mean point defect loss rate into the surface r = R from both outflow from within the grain and from the inflow from the surrounding “lossy” continuum region. The point defect concentration within the gram r
c m = K/ok2
=-
In this model we begin with the “lossy” continuum with a total sink strength of k2 = k& + k;,, ,
where k2 is now the total sink strength (16) and
kf c
k2
1’ (20)
The equivalent point defect loss rate to eq. (20) in the lossy medium away from the embedded grain is given by eq. (7) and equating these two flow rates yields a quadratic equation for k$,R2 with the explicit solution
x
(2_y-1)a2 1-2,
(21)
where fl=a!cotha!,
(22)
(lr2= k&R2 .
(23)
The limiting values of this sink strength are: k$, = 3k,,/R
when
ksJ2 + 00,
(24)
in exact agreement with eq. (lo), and k& = (3/2R2)(5
+ 421) = 14.4/R2 ,
(25)
which is very close to the corresponding cellular limit (11).
48
R. Bullough et al. / Sink strengths for film surfaces and grain boundaries
I
I
10
100
I
I
1000
too00
k:,R2 Fig. 2. A comparison of the dependence of the grain boundary sink strengths on the sink strengths within the grains k& and the average grain radius R. Curve A is the sink strength derived using the cellular model and the mean point defect concentration, curve B is the sink strength using the cellular model and the concentration at the centre of the grain and curve C is the sink stiength using the embedding modet
The variations of the embedding grain boundary sink strength (21), the cellular sink strength using the mean concentration (9) and the cellular sink strength using the concentration at the centre of the grain (14) with internal sink strength k& and grain radius R are shown in fig. 2. We see at once that the (mean concentration) cellular sink strength is really quite indistinguishable from the embedding result for this very wide range of parameters. On the other hand the (central concentration) cellular result (14) shows considerable deviation from the other two and must be regarded as unacceptable. The almost exact agreement between the cellular sink strength (9) and the embedding sink strength (21) although gratifying requires some explanation, which will become much clearer after our discussion of the thin film surface sink strengths and the void strengths in the next two sections. At this stage it should suffice to comment that since all grains are the same and are essentially independent the mean point defect concentration F (eq. (12)) in any one grain in the cellular model must be identified with
the physically meaningful point defect concentration in the lossy medium cm where all the gram boundaries have been smeared out. However, the grains are randomly disposed and since the embedding procedure is only appropriate for random distributions of sinks it should be appropriate for this situation. As we shah see below this coincidence of the cellular and embedding models only occurs for the grain boundary sink. Having argued that the result (9) is an acceptable grain boundary sink strength we now present the corresponding general result, using the mean cellular point defect concentrations, when the ideal sink boundary condition (2) is replaced by the general rate limited form: -Ddc/&=K[c-ce]
at
r=R,
(26)
where K is the point defect transfer velocity into the boundary sink and ce is the point defect equilibrium concentration; note for interstitials ce = ce = 0. The required, general grain boundary sink strengths are
R. BuIlough et al. / S’ink strengths for film surfaces and grain boundaries
then:
tion since its construction in a similar fashion to the previous grain boundary analysis would imply the random impingement on any one embedded foil of adjacent foil surfaces and not a set of parallel identical “foils”. To obtain the required foil surface sink strength k,” we fust obtain the steady point defect concentration c = c(x) within the foil given by the continuity equation
vacancy sink strength k&b = %
X
[(
[k,R
k,,,R
coth k,,,R - I]
- F
WC
coth kVsCR - 1
3sv -l +&R +k&R I ’
(27)
interstitial sink strength kfgb = T
X
[ki,J? coth kiscR - l]
kiscR - p)
coth kiscR - 1
isc
3Si -’ + SiR + k&R I ’
(28)
and, in the usual notation [6], the rate of vacancy emission from the grain boundaries is Kevgb = ~vk:,b~~ .
D d*c/dx* + K - Dk,2,c = 0 ,
(31)
where the Cartesian coordinate x-axis is perpendicular to the foil surfaces with the coordinate origin x = 0 on one foil surface and the other surface located at x = 21. Again k& is the sink strength from all the microstructure within the foil and thermal vacancy emission processes will not be explicitly included but will be presented at the end of the section. The solution of eq. (31) in the half foil region 0 Q x < I, subject to the ideal sink boundary condition * c=o
at
x=0,
(32)
and the symmetry condition (29)
d&lx= 0
In eqs. (27) and (28)
Sv = Ky/Dv p Si = Ki/Di
49
(30)
at
x =I
is cash k&x - I) cash k,,l I ’
and the zero rate limitation result (9) follows from eq. (27) or (28) by letting Sv, St + O0s
3. The foil surface sink strength ln this case of a single foil of thickness 21, it is
clear that the point defect concentration in the lossy continuum, where the surface sink strength is required, must be equal to the actual mean concentration in the foil. The continuum may be thought of as replacing a one dimensional periodic array of foil surfaces but where each “foil” is completely independent of its “neighbours”; there is no useful realization * of the embedding model for this situa* However, as we shall see in the next section, it may be poasible to discover a suitable “sink free region” that would
(33)
(34)
and the point defect loss rate/atom of the foil across the two foil surfaces is F OUdD& l
1 =5
;i;; x=o
k,,l
tanh k,,Z .
(35)
The total sink strength in the equivalent lossy continuum is k*=k;+&,
(36)
where kt is the required foil surface sinkstrength. The steady state continuity equation for the continuum is K - Dk2coo = 0,
(37)
facilitate such a realization but since this would, at best, only reproduce the cellular model result it does not seem worthwhile pursuing further. * The results for general rate limitation at the surfaces will be presented at the end of the section.
R. Bullough et al. /Sink strengths for firm surfacesand grain boundaries
50
where cm is the point defect concentration in the continuum and losses due to bulk recombination are not included; the significance of bulk recombination will be discussed in detail for this sink in section 5. The equivalent point defect loss rate to eq. (35) in the continuum is F, = Dk,2c” ,
(38)
which, from the continuity equation (37) yields F, = Kk;/(k:
+ kzC) .
(39)
Equating the flows (35) and (39) gives the required expression for the foil surface sink strength (subject to the ideal sink assumption (32)):
1. SC
k:=+
w
coth k,,l - &IL
kz=s
when
when
k,,l + 00 ,
(41)
(46)
The discrepancy between eq. (46) and the result (42) should be noted. With the conviction that eq. (40) is a consistent foil surface sink strength we can now generalize the result to include the possibility of rate limitation at the foil surfaces; that is we replace the ideal sink boundary condition (32) by Ddc/dx=R[c-ce]
at x=0,
(47)
and proceed as before. We thus obtain the general results: vacancy sink strength
1’
(48)
3
(49)
sv
coth k,,,l t 1 -
We note the two extreme values of this sink strength: kz = k,,/l
k,,l+O.
kzvsc
interstitial sink strength
and coth kiscl + 1 - & isc 1
kz = 3/12
when
k,,l+
0 .
(42)
The mean point defect concentration in the foil is C=-
1 1 s lo
c(x)dx=D+ SC
[
l--
tanh k,,l k,J
1’
K”VS =Dk2ce v vs (43)
which for the sink strength (40) is identical to cm, given by eq. (37). Hence the sink strength (40) is completely consistent with the previously stated notion that the point defect concentration in the lossy continuum must be identical to the mean point defect concentration in the foil. Again, as in the case of the grain boundary sink strength derivation, some authors [ 121 have used the concentration c(l) at the foil centre in eq. (38) instead of the mean concentration cm = F. Such an assump tion yields the foil sink strength [ 121 k,2 = (k,,/l)
coth k,,1/2 .
WI
The error involved in adopting this sink strength for
the ,foil surfaces will be discussed in section 5. It will suffice here to give the two extreme values taken by eq. (44): kz = k,,/l
when
k,,l + - ,
and, in the usual notation [6], the rate of vacancy emission from the foil surfaces is
(45)
wa
v,
where ct is the vacancy equilibrium concentration. In eqs. (48) and (49) Sv.= K&
,
Si = KJDi ,
(51)
where K, and Ki are respectively the transfer velocities of vacancies and interstitials across the foil surfaces. Finally the zero rate limitation result (40) follows from eq. (48) or eq. (49) by letting Sv, St + * e
4. The void sink strength To further clarify the importance of sink ordering on the relation between the embedding and cellular model estimates of sink strengths we now discuss the void sink strength and particularly emphasize the various interactive corrections that relate to this sink. The results of this section should help to distinguish
R. Buliough et al. / Sink strengths for pm surfdces and grain boundaries
51
between various published and sometimes erroneous sink strengths based on the cellular model and the more correct results of the embedding model. Before discussing the relative merits of the two models it is convenient to obtain the various sink strengths and for simplicity we omit considerations of vacancy thermal emission and treat the void as an ideal sink. We begin with the cellular model.
void
4.1. The cellularmodel
c(r) = -E(G
In this model the voids (all of radius a) are assumed to be regularly distributed in the body such that each can be considered to lie at the centre of a spherical cell, as shown in fig. 3, of radius R,. When such voids are the only sinks present in the body the steady state point defect concentration within the cell is given by the solution of the continuity equation:
Quantities required are the mean concentration in the cell:
Dd -_ rl&
r2dc +K=() dr 1 (
c=O
at
r=a,
(53)
and the zero flow condition at the cell boundary: dc/dr=O
r=Rc,
at
(54)
where spherical polar coordinates have their origin at the centre of thk spherical cell. The solution is
C=--R:Ta’[
[2R,3 - (r+a)ar] .
Rc
r2 c(r) dr =
K 1SDa(Rf - a3)
X [5R6C - 9aR’C + 5a3R3 - a61 9 C
(56)
the concentration at the cell boundary:
(52)
9
subject to the ideal sink boundary condition at the
c(R3 = &(R,
-a)2(W+a),
and the number of point defects per second that enter the void =E (R: _ as), 352
(58)
where R is the atomic volume. In the lossy continuum where this regular distribution of voids is smeared out the number of point defects/atom lost at voids is Dk$cm, where kt is the required total void sink strength and co is the point defect concentration in the continuum. The number of atoms associated with each void is [GX$]-‘, where
a
Continuity
Lonsy continuum
c”, = 3/41rR: ,
(Ri=Rz
Fig. 3. (a) The cellular model, appropriate for a regular distribution of voids. The radius of the cell is related to the current void volume concentration by eq. (78). (b) The embedding model, appropriate for a random distribution of voids. RSF is the radius of the sink free region surrounding the void and the composite sphere is embedded in the 10s~ continuum.
(59)
is the initial volume concentration of voids (when a = 0). Thus the effective number of point defects per second that enter each void in the lossy continuum F cant= Dk$c”liX$
b
-a3),
.
@)
By equating the two flows (58) and (60) we obtain the general expression for the required void sink strength k$ =
4nCcK (Rf - a3)/3Dc” .
(61)
For the lossy continuum representation of this regular
R. BuIlough et al. /Sink strengths for film surfaces and grain boundaries
52
distribution of voids to be at all physically meaningful the point defect concentration in it must be equal to the mean point defect concentration in the cell, as discussed above, i.e. c =- -r, (62) with this equality, eq. (61) yields from eq. (56) k$ = haC$fF-, where + 5a3Rz - a61
+K-Dk$c=O.
(64)
r=a,
at
(69)
the concentration and its radial derivative are continuous across r = R, and the concentration in the lossy continuum a long way from the embedded region is bounded and, from eq. (68), is given by
is the interactive correction term on the sink strength when the equality with the mean concentration i? in eq. (62) has been used. Several authors [ 131 have derived an alternative correction term fccRd by using the equality
c(r)=A/r+B-Kr2/6D,
c O”= c(R3.
and
(65)
This has the explicit form, from eqs. (57) and (61) fc(Rc)
= 2@,3
-
a3M&
-
a)*@&
+ 4
*
(68)
The boundary conditions are that the void is an ideal point defect sink c=O
(63)
fT= 5(Rz - a3)‘/[5R,6 - 9&
and in the lossy continuum r > R~F
C OD=
K/Dk$ .
(70)
Subject to these boundary conditions the complete solution is
c(r)-K/Dk$
a
+(~/r)exP[kv(&
-r)]
(71)
,
~>RSF,
(66)
(72)
where the constants A, B and Pare 4.2. The embedding model
A=
This model has been discussed in detail by Brailsford, Bullough and Hayns [2] and by Brailsford [4]. It specially applies when the distribution of voids is random and we will, for comparative purposes with the above cellular model results, concentrate on the relatively simple situation when only voids are present, all are of equal radius a, present ideal sinks for the point defects and there is no vacancy thermal emission from them. The model is displayed in fig. 3 and is constructed by removing a sphere of radius RsF from the lossy continuum and replacing it by a sphere of the perfect material with a single finite void at its centre; the sphere is thus “sink free” apart from the central void. The net point defect flow rate into this void is then calculated and again equated to the equivalent flow rate in the lossy continuum. To obtain this flow rate into the central void we solve the following steady state continuity equations for the point defect concentration c(r) in the two regions. In the sink free region a < r < R~F , 2
Ka 6Dk$[l + kv&F
x [-3(k;R&
- a)]
+ 2kvRsF + 2)
+ k$(a*+a*RsFkv - kvRbF)] ,
B=
K 6Dk$ [l + kv(Rs~ - a)] X [3(k$R& + 2kvRsF + 2) + k:(R;F
P=
- a3)] , (73)
K 6Dk$[l+ kv(RsF - a)] X [-6.a + k$(RsF - a)*(2RsF + a)] .
The number of point defects per second that enter the void in the embedded sphere is F emb
=-
ha20 dc 52
dr
1r=a
= slk$
f X (1 + kvRsF) + k ‘“7
(67) + kv&F
- a&F
+ a)]
4naK [l + kv(RsF - a)] - ‘) [3(Rs~ + fZ)
I
.
R. Bullough et al. /Sink strengths for film surfaces and grain boundaries
We notice that in this model, in contrast to the corresponding cellular expression (58), the required void sink strength appears explicitly in eq. (74). As for the cellular model we obtain k$ by equating the flow rate (74) to the corresponding flow rate (60) in the continuum a long way from the embedded sphere, where cm in eq. (60) is now given by eq. (70). This yields the cubic equation for kv : kt = 4~&k,b
(kv) ,
(75)
where 6
&F
- a) 6
X [~(RsF + a) + ~kv(RsF-
x [l
+ kv(RsF - a)]-’
a)&~
+
%)I
(76)
is the interactive correction term which can only be evaluated when the cubic eq. (75) has been solved for kv. The choice of the radius of the sink free region RSF in eq. (76) is, of course, important and has been discussed previously by Brailford, Bullough and Hayns [ 21. These authors argued that when a second sink type is present, such as dislocations, the value of RSF can be reduced down to the void radius a without much loss of accuracy. In the present situation where only voids are present it is clear from the construction of the embedding model when the sphere of radius RSF was removed from the lossy continuum and replaced by the sphere with the void of radius (I at its centre, that this radius must reflect the volume of material/void when the voids are of radius a. We may therefore logically choose RSF
=R, ,
(77)
where R, is the average cell radius associated with each void and is related to the actual void concentration Cv by C, = 3/41rR,3,
(78)
which is to be distinguished from the initial II = 0 void concentration CG given by eq. (59). The choice of RSF in eq. (77) then enables us to compare the three void sink strength corrections fT,f,.cRd and em,, with the understanding that the femb reSUlt f with the choice of eq. (77) for RSF is the best esti-
53
mate for a random distribution of voids. This comparison is given in fig. 4 where the variation of the three correction terms is shown against the swelling S=a3/R%
.
(79)
The definition of swelling is somewhat arbitrary; we have, in eq. (79), chosen to identify the swelling with the total number of vacancies in voids/atom of the body or in other words we refer the volume change due to the voids to the initial volume (when u = 0) of the body and not to the current volume of the body (when the voids are of radius a). It is, of course, important that when experimental swelling values are presented the reference volume used should also be indicated. In terms of the swelling S the three correction terms have the explicit form. From eqs. (64) and (79) 5(1 +x+x2)2 fT=(, -X)(5+6X+3x*+xs)’
(80)
from eqs. (66) and (79) 2(1 +x+x2)
few
= (1 -X)(2 tx) ’
031)
and from eqs. (75) (76) and (79) f emb =
x [xt y(l
-x)J -l ,
(82)
where y is the positive root of the cubic femb(Y) =Y2/3S
(83)
and x = [S/(1 + AS)]V3.
(84)
We also include in fig. 4 the variation of fimb obtained from eq. (76) with no sink free region: RsF=a,
(85)
which has the explicit form j&,,,={2+3S+[9S2+l~]“*}/2.
(86)
It is apparent from fig. 4, and may be seen analytically by solving the cubic (83), that when the sink free radius is taken equal to the cell radius, as in eqs. (82) and (83), that
femb
F fc(R3 *
(87)
54
R. BuUough et al, /Sink strengths for film surfaces and grain boundaries
Fig. 4. The variation of the various void sink strength correction andfimb (eq. (86)) with the swelling S.
This rather remarkable, but fortuitous, result tells us that when a sphere of lossy continuum is replaced by the sink free sphere (with a single void at its centre) of radius equal to the average cell radius then the constant concentration cm in the lossy continuum is not perturbed near the sphere; mathematically P=O
038)
in eq. (73). Thus in this case when only voids are present the void sink strength in a random distribution of voids can be obtained from the cell model by taking the cell boundary concentration as the lossy continuum concentration. We believe however that such a cellular approach is dangerous since it strictly only applies to the regular spatial distribution of sinks and then it is the mean concentration in the cell that must be identified with the constant concentration in the lossy medium. We shall demonstrate that a sink free radius can be found to permit the embedding model to model just this situation. Thus the embedding model is the only sensible approach when the distribution is random and is particularly so when other sinks are present since in that case a
termsf&eq. (80% fc(R,) (e¶. (81))pfemb (eqs. (82) and (83))
consistent coDcould not be defined. Furthermore, the embedding model always ensures consistency since all the sinks are simultaneously present in the surrounding lossy continuum. The sink free region can either be neglected for each sink type [2] or a value for the radius of the region consistent with the highest density of remaining sink types in the microstructure can be chosen. The fZ)& has the lowest value, in fig. 4, and has been included there for iUustrative purposes only. An interesting aspect of fig. 4 is that
fF >femb
(89)
over the whole range of swelling. One could interpret this to mean that the sink strength of a void as a member of a regular distribution of voids (for which fr would be appropriate) is always greater than the sink strength of a similar void in a random distribution of voids of the same density (for which fmb would be appropriate). Perhaps this is a contributary reason for the formation of perfect void lattices in many materials. To conclude this discussion of the void sink
R. Bullough et al. /Sink strengths for film surfaces and grain boundaries
55
I
I
1.30-
1.25 -
1.20 -
.l
1
100
sl%)
Fig. 5. The variation of Q with the swelling S required to ensure that the void sink strength using the embedding model agrees with the cellular result appropriate for a regular distribution of voids. The radius of the sink free zone is defined by RSF = aRg and the swelling S = a ‘/R 3.
strength we ask whether a sink free radius can be found that enables the embedding model to encompass the regular distribution of voids. Thus we take RSF
=d0,
(90)
and obtain the variation of femb with swelling S, from eqs. (75) and (76) for a range of 01.We can then deduce the dependence of (IIon S to ensure that the f embvalue always coincides with fr, which we know to be correct for the regular distribution. The required variation of cr with S is shown in fig. 5. It is thus clear that although the embedding procedure is only strictly appropriate for random distributions of sinks it can be “fixed up” for the regular distribution by a judicious choice of variable sink free region. We now revert back to the foil surface sink situation and examine the sensitivity of this sink strength to bulk recombination. 5. The foil surface sink strength and bulk recombination In any practical use of the foil surface sink strength (40) or the more general results (48) and (49) it is
essential to ensure that the presence of significant bulk recombination does not explicitly change the form or magnitudes of these sink strengths. This question has been discussed previously by White, Fisher and Miller [ 121, but they based their argument upon the use of the sink strength (44) and found it was necessary to modify (empirically) the sink strength (44) when the bulk recombination was significant. On the other hand Bra&ford, Matthews and Bullough [8] and Hayns [9] have presented both perturbation and iterative analyses respectively that indicate that the direct modifications on the (correct) sink strengths should probably always be negligible. It thus seemed worthwhile to check directly the validity of eq. (40) when point defect losses by bulk recombination were not negligible by comparing results obtained from the lossy continuum rate theory with exactly equivalent results using the full spatial solution for a thin foil. The generalized forms of the continuity equation (13) for the vacancy (cv) and interstitial (ci) concentrations within the irradiated foil are d2cv
D
VG
+K+Ke,,, - Dvk$,Cv - WiCv = 0 3
(91)
56
R. Bullough et al. /Sink strengths for firm surfaces and grain boundrrries
d2c.
+ K - bikfscci - MiCv = 0 , Di 1 dir2
(92)
FiN = 6s
where CY is the bulk recombination coefficient and K&, is the rate of vacancy emission from the internal microstructure [ 1,2] which has a total sink strength of k:,, and k& for vacancies and interstitials respectively. This pair of non-linear equations have to be solved numerically subject to appropriate boundary conditions at the foil surfaces or equivalently at one foil surface (x = 0) and a symmetry condition halfway through the foil (X = Z). The ideal sink boundary condition * is C, =I$
,Ci
=O
at
(93
at
X=1.
(94)
For simplicity we now suppose that only one type of sink exists inside the foil and that these sinks are network dislocations and are uniformly distributed through the foil. In this case we may write [ 1,2] &
= PN,
&
= ZiPN 9
(96)
x=0 ’
which must be equal to the net loss rate per atom of interstitials at the dislocations within the foil I
W$PN&)
- &PNC&)
FiN 3
+K:NN)~
,(97)
0
if dislocations are the only sink type present, otherwise the equality would be with the net interstitial loss rate per atom at the total internal microstructUre. Here, pvN is the rate Of vacancy emiSSiOnfrom * Computations have been successfully made with general rate limitation boundary conditions imposed on the foil surfaces but for the present purposes the ideal sink conditions will suffice.
(99)
where A = Kf and we will use this computed growth rate, with physical parameters appropriate to zirconium, to investigate the validity of the foil sink strengths and bulk recombination effects. The general rate theory equations corresponding to eqs. (9 1) and (92) for the lossy continuum point defect concentrations are * K + K&c + Kvs - Dv(k&c + k&) cv - (YciCv= 0 ,
(95)
where PN is the dislocation density and 21 el) is the bias parameter defining the preferential interaction between interstitials and dislocations comparedwith that between vacancies and dislocations. In terms of these point defect concentrations, the net loss rate per atom of vacancies across the two surfaces
FiN=fJ
(98)
can easily be seen by substracting eqs. (9 1) and (92) and integrating the difference over half the foil thickness in conjunction with the symmetry condition (94). In a hexagonal material such as zirconium the edge dislocations all lie on the prismatic planes and thus the growth/dpa in the a-axis direction caused by the climb of such dislocations due to absorption of the excess interstitials is [ 141 de/dA= (l/X)
X = 0,
and the symmetry conditions is &v/dX=&i/dX=O
the network dislocations [ 1,2]. The equality
(100) K - Di(kfsc + kiL,)Ci - OLCiCv =0 7
(101)
where now the foil surface sink strengths have been incorporated into the total microstructure sink strengths. If we again assume that only network dislocations are present then the equivalent net flow rate of vacancies to the foil surface (96) is F vs = Dvk$v
- DikTsCi- K& 3
(102)
and of interstitials to the dislocations (97) is F iN
=
WiPNG
- DVPNCV t GN 9
(103)
where again we see the equality (98) by substracting eqs. (100) and (101) and the growth rate is again given by eq. (99). The foil sink strengths k&, kfs were derived in section 3 and from eq. (40) are, for the present situa-
* In practice for computational convenience we always avoid assuming the quasi-steady state approximation and include the appropriate time derivatives on the right hand sides of the contimity equations. However,for presentational purposes we will omit them here.
R. Bullough et al. /Sink strengths for film surfaces and gmin boundaries
L 10'2
I
57
I
10"
PN
(me’)
10"
Fig. 6. The variation of‘predicted growth strain rate in a foil of thickness 10B6 m, with dislocation density pi for the zirconium parameters in table 1 when bulk recombination is neglected. Curve A is the correct variation obtained by using either the rate theory with the foil surface sink strengths (104) or by integrating the spatial equations (91) and (92) directly. Curve B is the predicted growth strain rate using the erroneous foil surface sink strength (44) and the other curve indicates the error resulting from such a choice.
tion k:,=$[coth/$/2L---]-1, _$104)
k;s = (Z,PN)1’2 I
coth (ZipN)“2Z [
1
1
(ZipN)r’2Z
*
These sink strengths were derived by identifying the
mean point defect concentration in the foil with the point defect concentration in the lossy continuum and it is clear from eqs. (97) and (103) that in the absence of bulk recombination they will yield growth rates in exact agreement with the result of integrating the full spatial equations (91) and (92). The alternative foil sink strength (44), uses the point defect concentration at the foil mid-point instead
Table 1 Physical and irradiation parameters used to model the electron irradiation response of zirconium [ 141 D, = 0,” exp[ -EF/kT] Di = Dp exp[ -E$kT] ct = exp[-EQkT]
,
, ,
0,” = 10e6 m2/s, Dp a 7.5 X10-a m2/s,
with EF = 0.65 eV with Ep = 0.3 eV with bf = 1.4 eV
Bulk recombination coefficient u given by a/oi = 2 X 10zo mm2. Interstitial dislocation preference Zt = 1.10 relative to vacancy 2, = 1.0. Displacement damage rate K = 3 X 10q3 dpa/s.
R. Bullough et al. / Sink strengths for film surfaces and grain boundaries
58
I
10"
10'3
I
PN
Imw21
10'L
1
10'5
Fig. 7. The variation of predicted growth strain rate in a foil of thickness 10m6 m with dislocation density PN for the zirconium parameters in table 1 when bulk recombination is included. The dotted curve is the accurate variation obtained by integrating the non-linear spatial equations (91) and (92) with eq. (97). Curve A is the rate theory result with the foil surface sink strengths (104) and curve B is with the sink strengths (44).
and the error on the growth rate that ensues is shown in fig. 6 for a foil of thickness 2Z= lo-” m, when bulk recombination is completely neglected. The physical and radiation parameters used in this section, and purporting to relate to zirconium, are given in table 1. The justification for claiming that these relate to zirconium is discussed in detail in an analysis [ 141 of the observed growth in that material and will not be pursued further in the present paper. We conclude from fig. 6 that the sink strengths for the foil surfaces (104) are certainly correct for zero bulk recombination and it only remains to check their accuracy in the presence of such recombination. This we have done by numerically integrating the non-linear spatial equations (91) and (92) across the foil by a discretization procedure; in practice we find the integration can be achieved with sufficient accuracy by adopting a minimum finite difference mesh of l/40. Increasing the mesh size from 30 to 40 points, results in an improvement in the calculated growth strain rate of less than 1%. Again the predicted growth strain rate for a foil of thickness 2Z= 10V6 m and using the parameters in table 1 but with bulk recombination included is given in fig. 7 (the dotted curve). To compare with this “accurate” variation we have used the rate theory, but with the same bulk recombination included (eqs. (100) and (lol)), with both
the foil surface sink strengths (104) (curve A) and the alternative strengths (44) (curve B); the influence of bulk recombination is thus simply included in the modifications to the lossy continuum point defect concentrations obtained from the quadratic solution of eqs. (100) and (101) and no explicit modification to the sink strengths themselves are included. We see from this figure that the error using the rate theory with the unmodified sink strengths (104) is always small (55%) compared with very large errors when the unmodified alternative sink strength is used. It is thus not surprising that authors [ 121 who chose to use the sink strength (44) are forced to modify it empirically in the presence of bulk recombination.
6. Conclusions (1) The foil surface sink strength (104) should be used in a rate theory representation of damage in foils. (2) This sink strength does not require modification (either empirically or by perturbation analysis) to allow for the presence of bulk recombination. It is thus only necessary to include the effects on the point defect concentrations of bulk recombination losses and not explicitly on the sink strengths (3) We expect the same conclusions to be true for
R. Bullough et al. /Sink strengths for firm surfaces and grain boundaries
the grain boundary sink strength (9) in preference to the alternative (14). (4) For random distributions of sinks, the embedding model is preferable and, indeed, if more than one sink type is present the embedding model is the only sensible model, since it permits a logical choice for RSF and ensures a consistent point defect concentration in the lossy continuum. (5) If the sink distribution is regular then the cellular model is appropriate and the mean point defect concentration can be identified with the concentration in the equivalent lossy continuum. However, we have shown that even in this case the embedding model can be constructed to yield equivalent results by varying RSF appropriately. (6) The best foil sink strength is obtained by regarding the sink distribution as perfectly periodic (regular) and thus the mean concentration in the cell model is obviously appropriate. An embedding model would be inappropriate for this situation. On the other hand the grain boundary sink combines the features of regularity and randomness in such a way that both models are appropriate for the derivation of its strength. (7) The void sink strength in a regular distribution is always greater than it would be if the same density of voids were randomly distributed. A feature that could have relevance to the formation of void lattices so frequently observed in many materials. (8) Finally, although in the present paper we have explicitly only demonstrated the reliability of the foil surface sink strength for the simple situation of a constant dislocation network density in a foil, we have, in fact, made many additional comparative computations including other internal sinks such as growing
59
interstitial loops, voids and traps together with complex rate limited boundary conditions at such sinks and at the foil surfaces. In all cases the general foil surface sink strengths (48) and (49) provide an accurate representation of the surface losses and require no modification due to bulk recombination; we thus would argue that any such modifications are unnecessary when these sink strengths are used.
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