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Response function analysis of time-dependent diffusional defect processes
OOOI-6160186$3.00 + 0.00 Pergamon Journals Ltd
Acfa merall. Vol. 34, No. 7, pp. 1303-1305, 1986 in Great Britain
Printed
RESPONSE
FUNCTION ANALYSIS DIFFUSIONAL DEFECT
OF TIME-DEPENDENT PROCESSESI-
A. D. BRAILSFORD Research Staff, Ford Motor Company, Dearborn, MI 45121-2053,
U.S.A.
and L. K. MANSUR Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge. TN 37831, U.S.A. (Received
5 July 1985; in revised,form
II October
1985)
Abstract-The loss rate of defects to cavities in a material subject to spatially uniform but arbitrarily time-dependent irradiation is re-derived by means of Response Function theory. The result coincides with that of our earlier derivation. R6s11m&Nous avons obtenu la vitesse de perte de defauts sur les cavites dans un mattriau soumis a une irradiation spatialement uniforme, mais dependant du temps d’une manitre quelconque, ri I’aide de Ia theorie de la fonction de rtponse. Le resultat est en accord avec celui de notre demonstration anterieure. Z~~~f~ung-MitteIs der Theorie der Antwortfunktion wird die Einfangsrate von Fehlstellen an Hohlr~umen in einem Werkstoff, der einer r~umiich gleichm~~igen, aber zeitlich beliebig variierenden Bestrahlung unterliegt, nochmals abgeleitet. Das Ergebnis stimmt nit unserer friiheren Ableitung iiberein.
In a recent publication
[l], we have derived the form of the defect loss rate to cavities and dislocations in a material wherein self point defects are produced in a spatially homogeneous but arbitrarily timedependent manner. We found, in particular, that in addition to there being a component of the loss rate having the steady state form 121,there was an additional contribution dependent upon the entire prior history of the irradiation. However, we were subsequently able to establish, by means of detailed examinations of specific examples, that the consequences of the corrections to the steady state loss rate form were of very minor practical significance. Thereby we were able to validate the significant body of earlier analysis wherein such corrections bad not even been contemplated. This we view as a highly significant conclusion, the importance of which gains even further prominence when the inevitable future demands for analysis engendered by fusion research are borne in mind. Consequently we have stayed alert to the possibility that a somewhat simpler derivation of our earlier result might be available and that this new approach, --tResearch sponsored in part by the Ford Motor Company, Dearborn, Michigan and in part by the Division of Materials Sciences, U.S. Department of Energy, under contract DE-ACO5-84OR~I400 with Martin Marietta Energy Systems, Inc.
whatever its nature, might be more readily generalizable to sink types other than voids or dislocations alone [I]. Such a new approach, namely Response Function Analysis, has now been identified. Its application to the void loss rate problem is the subject of this note. Inspection of Ref. [l] will soon reveal that the source of much of the complexity contained therein resides in the use of the Laplace Transform technique. This method inherently contains details of the initial conditions at time 1 = 0. In the context of Effective Medium theory [2]-the basis for computing defect loss rates-this means that details of the microscopic initial state around each sink-type must be specified, and their consequences explored. Such a program was carried through in our earlier work. And, as we have said, the consequences of such detail proved of no practical importance in the loss rate context. It is natural then to look for methods where the initial microscopic state never enters in the first place. The Response Function analysis given here is one such. It has only one conceptual difficulty associated with it. Namely that a void or cavity is conceived, in the first piace, as having a radius which is fixed in time from the remote past to the infinite future. Granted, voids do grow slowly on some time scales, this is nevertheless a significant mental hurdle to overcome. The point, however, is the following. The loss rate of defects to a void in the Adiabatic Growth Regime [3] is a relatively insensitive function of the
1303
1304
and MANSUR:
BRAILSFORD
TIME-DEPENDENT
void growth velocity. Thus it is quite legitimate to calculate the loss rate as a power series in the growth rate. The lowest order term (i.e. that which arises when the void size is regarded as fixed) provides a good first approximation. Moreover, the dependence upon the past history is found to depend upon events of only the recent past [l]. Thus, in the end, one can justify the approach by its eventual predictions. It should be emphasized that the present derivation is significantly shorter than the earlier one largely because one is now fore-armed with an extensive body of additional insight. Thus, although the ensuing analysis is largely self-contained, it is perhaps most appropriate to view it as an addendum to our earlier discussion of the time-dependent rate theory problem.
DIFFUSIONAL
equation (2) implies that the loss rate at time t also depends upon the time rate of change of c(r, t) at times in the future (i.e. t’ > t). That is, causality is not yet built into equation (2). It must, of course, turn out that r$(t) = 0 for t c 0. Furthermore, we must show that 9(0/(2n) ‘I*, for t > 0, is identical with the function $(t) we defined in equation (I) of Ref. [l]. That is the basic object of the present exercise. In Effective Medium Theory one finds the loss rate to one sink and imposes the condition that it be consistent with the initial Ansatz. Applied here to a cavity, one must solve
ac
at-“V2~+(LD+LCf-G=0
m #)(t J
-
t$$Lt~
-m
1
-S(t).
(2)
In this relation the term proportional to kz is the steady state loss rate to cavities; kf being the steady state sink strength for cavities [2]. This term alone survives in steady state (&/at = 0) if S(l) denotes any possible source term arising from time-dependent corrections to the steady state thermal emission rate embodied in the term -Dkfe in equation (2). The tatter arises from a boundary condition of the type * ;
= vr[c, E]; r = rC
(3)
being imposed at the void surface. Here ur is a transfer velocity across the void surface. This boundary condition is retained in the time-dependent problem considered below. Note that at this stage, the void (cavity) is presumed not to grow-nor to have ever grown-so that E is time-independent. We have remarked upon this conceptual point earlier. The Ansatz contained in equation (2) differs from that derived in (1) most signifi~ntly through the range of integration in the integral term. As it stands,
(5)
and solve equation (3) using Fourier Transforms. Specifically, if f, is the Fourier Transform off(t)
-- 1 m S(t)e’@‘dr. fur - (2n)“Z s _-ou
(1)
Here D is the defect diffusivity, cfr, t) the defect concentration per unit volume, c, the concentration in thermal equilibrium and ki is the steady state sink strength for dislocations: (ki N Zp, where 2 is a constant, of different magnitude for each point defect type and pe is the dislocation density.) We here hypothesize that the loss rate to voids (cavities) is of the form
X
rate per
c = c, + c,(r, t)
As in [I], we shall neglect the effects of intrinsic recombination. Moreover, we shall avail ourselves of the result there established that the toss rate of defects to dislocations, L,, is adequately given by
1 L,(r, t) = D kf(c - c) + (2n)“~ [
(4)
where G is the time-dependent production unit volume. To this end, we put
RESPONSE FUNCTION ANALYSIS
L,(r, t) = Dkb[c(r, 1) - c,].
DEFECT PROCESSES
(6)
Thus equation (4) leads to V2Clo- q*cto = - q%;a,
(7)
where q2 = k2 - iw(& + l/D)
(8)
c;*, = {G,,, + S, + Dkf(E - c,)8(o)~jDq2 k2=kf+kf,
(9) W)
and d(w) is the Dirac delta function, multiplied by (2n)“Z. The solution of equation (7) that satisfies the Fourier Transform of equation (3), viz. D2
= u~[c,~ - (65- c,)lj(W)],
at r = r,,
(IV
is cl0 = cz - A(rC/r)e-d’-rc)
(12)
where (13) and s = v,r,/D. This solution can be used to find the net loss rate to the cavity. For C such cavities per unit volume, distributed at random, the total loss rate is C times this quantity. The latter must be the same as the asymptotic form in the medium if the approach is to be self-consistent. Hence, with kz 3 4nr,C, one finds the further relation Dki(l + qr,)A = D[-kf(E
- c,)~(o)
+(k:--io&)cp”,]-S,
the right-hand member equation (2) for large r.
being
the transfo~
(14)
of
BRAILSFORD
and MANSUR:
TIME-DEPENDENT
Using the derived form of A, equation (13) we equate terms linearly dependent upon the transform of the production rate, G,,, and those independent of the latter, separately. Thereby we obtain k f - iw.),,, =
k&(1
+ qr,.)
(15)
I + qrc + s
DIFFUSIONAL
Evidently,
DEFECT PROCESSES
1305
in the large s limit now used
Consequently, the function 4(t), evaluating the inverse transform
which is given by
(22) S,,,+ Dk;(c
- c,)~(o)
=
Dk&(l + qr,)(? - c,)S(w)
(16)
1 +qr,,+s Now when w = 0, q = k by equation in this limit, equation (15) reproduces cavity sink strength [2] k2
=
’
(8). {Similarly, the steady state
k&7(1+ kr, 1 l+kr,+s
it will be noted.} But more to the point, we may eliminate 4,,, between equations (8) and (15) to find
is trivially evaluated for t < 0. We may obtain it by contour integration in the upper half of the complex w plane. In this region q is analytic and 4,” has no singularities on the real axis. Hence 4(t) = 0 for t < 0. This establishes causality. The further correspondence with our prior analysis is most readily shown by computing the Laplace Transform of 4(t) directly from equation (22). Since 4(t) = 0 for t < 0. this transform, Qp say, is
s x
@‘r=
0
q?=k2;+k~;;-;). (
s
(18)
This is a cubic equation for q as a function of w. Of its solutions, we need the root with positive real part [see equation (12)], after which 4,,, is determined via equation (15). Hence the loss rate function will be completely determined. Actually, the subsequent analysis can be made significantly simpler in the void problem provided the ultimate growth of interest is diffusion limited. This hinges upon the observation that in this instance the parameter s is of order (r, lb), where b is a diffusion jump distance. Consequently, s is much greater than unity. Moreover, one observes from equation (18) that as w -+ cc, q -+ (- h/D)“*. Consequently, the combination (1 + qr< + s) appearing in the last equation differs from s only for frequencies so large that w > D/b’, i.e. for frequencies greater than the elementary atomic jump frequency. This range is of absolutely no interest in cavity growth problems of practical concern. The approximation qrc
iw/D}“*
+ikgr,
(19)
where fl = {ki + k2, + (ikir,)2}“2.
ti(t)e-P’dt
I =(2n)1.2
s
X 4,,d,,, ~~ p +io’
(23)
But since 4,,, is analytic in the upper half of the contour integration yields complex 0 plane, Qp = (2n)“‘b,,, or &=$[{p’+gj”*-/I]
(24)
from equations (22) and (21). This completes the re-derivation of one of the salient results of Ref. [l]. For 4(t) is now seen to be simply (27r)“*$(f), where $(t) is the loss rate function derived in the earlier work.? Inserting the preceding in the starting equation used here shows that the two approaches are thus exactly equivalent. For the sake of completeness, we turn briefly to the content of equation (16). In the limit of large s, it reduces simply to S,,, = 0. This is a further result that is contained in Ref. [I] when details of the initial state are neglected. Lastly, there is the question of reaction control at the void interface (i.e. or+ D/b). The limiting frequency at which the simplification leading to equation (19) now applies is correspondingly somewhat lower. However, it is evident that the transfer rate into the void would have to be very slow indeed for this to significantly alter the dynamic characteristics of the loss rate for cycle frequencies envisaged in fusion reactor components,
(20) REFERENCES
We now couple the above equation (17) to derive 4,
with the content of from equation (IS).
TCompare equations (20) and (24) above with equations (43) and (48) of Ref. [I].
AM
W--K
1. A. D. Brailsford and L. K. Mansur, Acta metaN. 33, 1425 (1985). 2. A. D. Bra&ford and R. Bullough. Phil. Trans. R. SOC. Lond. A302, 87 (1981).
3. A. D. Brailsford, J. Nucl. Maw. To be published
Acfa merall. Vol. 34, No. 7, pp. 1303-1305, 1986 in Great Britain
Printed
RESPONSE
FUNCTION ANALYSIS DIFFUSIONAL DEFECT
OF TIME-DEPENDENT PROCESSESI-
A. D. BRAILSFORD Research Staff, Ford Motor Company, Dearborn, MI 45121-2053,
U.S.A.
and L. K. MANSUR Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge. TN 37831, U.S.A. (Received
5 July 1985; in revised,form
II October
1985)
Abstract-The loss rate of defects to cavities in a material subject to spatially uniform but arbitrarily time-dependent irradiation is re-derived by means of Response Function theory. The result coincides with that of our earlier derivation. R6s11m&Nous avons obtenu la vitesse de perte de defauts sur les cavites dans un mattriau soumis a une irradiation spatialement uniforme, mais dependant du temps d’une manitre quelconque, ri I’aide de Ia theorie de la fonction de rtponse. Le resultat est en accord avec celui de notre demonstration anterieure. Z~~~f~ung-MitteIs der Theorie der Antwortfunktion wird die Einfangsrate von Fehlstellen an Hohlr~umen in einem Werkstoff, der einer r~umiich gleichm~~igen, aber zeitlich beliebig variierenden Bestrahlung unterliegt, nochmals abgeleitet. Das Ergebnis stimmt nit unserer friiheren Ableitung iiberein.
In a recent publication
[l], we have derived the form of the defect loss rate to cavities and dislocations in a material wherein self point defects are produced in a spatially homogeneous but arbitrarily timedependent manner. We found, in particular, that in addition to there being a component of the loss rate having the steady state form 121,there was an additional contribution dependent upon the entire prior history of the irradiation. However, we were subsequently able to establish, by means of detailed examinations of specific examples, that the consequences of the corrections to the steady state loss rate form were of very minor practical significance. Thereby we were able to validate the significant body of earlier analysis wherein such corrections bad not even been contemplated. This we view as a highly significant conclusion, the importance of which gains even further prominence when the inevitable future demands for analysis engendered by fusion research are borne in mind. Consequently we have stayed alert to the possibility that a somewhat simpler derivation of our earlier result might be available and that this new approach, --tResearch sponsored in part by the Ford Motor Company, Dearborn, Michigan and in part by the Division of Materials Sciences, U.S. Department of Energy, under contract DE-ACO5-84OR~I400 with Martin Marietta Energy Systems, Inc.
whatever its nature, might be more readily generalizable to sink types other than voids or dislocations alone [I]. Such a new approach, namely Response Function Analysis, has now been identified. Its application to the void loss rate problem is the subject of this note. Inspection of Ref. [l] will soon reveal that the source of much of the complexity contained therein resides in the use of the Laplace Transform technique. This method inherently contains details of the initial conditions at time 1 = 0. In the context of Effective Medium theory [2]-the basis for computing defect loss rates-this means that details of the microscopic initial state around each sink-type must be specified, and their consequences explored. Such a program was carried through in our earlier work. And, as we have said, the consequences of such detail proved of no practical importance in the loss rate context. It is natural then to look for methods where the initial microscopic state never enters in the first place. The Response Function analysis given here is one such. It has only one conceptual difficulty associated with it. Namely that a void or cavity is conceived, in the first piace, as having a radius which is fixed in time from the remote past to the infinite future. Granted, voids do grow slowly on some time scales, this is nevertheless a significant mental hurdle to overcome. The point, however, is the following. The loss rate of defects to a void in the Adiabatic Growth Regime [3] is a relatively insensitive function of the
1303
1304
and MANSUR:
BRAILSFORD
TIME-DEPENDENT
void growth velocity. Thus it is quite legitimate to calculate the loss rate as a power series in the growth rate. The lowest order term (i.e. that which arises when the void size is regarded as fixed) provides a good first approximation. Moreover, the dependence upon the past history is found to depend upon events of only the recent past [l]. Thus, in the end, one can justify the approach by its eventual predictions. It should be emphasized that the present derivation is significantly shorter than the earlier one largely because one is now fore-armed with an extensive body of additional insight. Thus, although the ensuing analysis is largely self-contained, it is perhaps most appropriate to view it as an addendum to our earlier discussion of the time-dependent rate theory problem.
DIFFUSIONAL
equation (2) implies that the loss rate at time t also depends upon the time rate of change of c(r, t) at times in the future (i.e. t’ > t). That is, causality is not yet built into equation (2). It must, of course, turn out that r$(t) = 0 for t c 0. Furthermore, we must show that 9(0/(2n) ‘I*, for t > 0, is identical with the function $(t) we defined in equation (I) of Ref. [l]. That is the basic object of the present exercise. In Effective Medium Theory one finds the loss rate to one sink and imposes the condition that it be consistent with the initial Ansatz. Applied here to a cavity, one must solve
ac
at-“V2~+(LD+LCf-G=0
m #)(t J
-
t$$Lt~
-m
1
-S(t).
(2)
In this relation the term proportional to kz is the steady state loss rate to cavities; kf being the steady state sink strength for cavities [2]. This term alone survives in steady state (&/at = 0) if S(l) denotes any possible source term arising from time-dependent corrections to the steady state thermal emission rate embodied in the term -Dkfe in equation (2). The tatter arises from a boundary condition of the type * ;
= vr[c, E]; r = rC
(3)
being imposed at the void surface. Here ur is a transfer velocity across the void surface. This boundary condition is retained in the time-dependent problem considered below. Note that at this stage, the void (cavity) is presumed not to grow-nor to have ever grown-so that E is time-independent. We have remarked upon this conceptual point earlier. The Ansatz contained in equation (2) differs from that derived in (1) most signifi~ntly through the range of integration in the integral term. As it stands,
(5)
and solve equation (3) using Fourier Transforms. Specifically, if f, is the Fourier Transform off(t)
-- 1 m S(t)e’@‘dr. fur - (2n)“Z s _-ou
(1)
Here D is the defect diffusivity, cfr, t) the defect concentration per unit volume, c, the concentration in thermal equilibrium and ki is the steady state sink strength for dislocations: (ki N Zp, where 2 is a constant, of different magnitude for each point defect type and pe is the dislocation density.) We here hypothesize that the loss rate to voids (cavities) is of the form
X
rate per
c = c, + c,(r, t)
As in [I], we shall neglect the effects of intrinsic recombination. Moreover, we shall avail ourselves of the result there established that the toss rate of defects to dislocations, L,, is adequately given by
1 L,(r, t) = D kf(c - c) + (2n)“~ [
(4)
where G is the time-dependent production unit volume. To this end, we put
RESPONSE FUNCTION ANALYSIS
L,(r, t) = Dkb[c(r, 1) - c,].
DEFECT PROCESSES
(6)
Thus equation (4) leads to V2Clo- q*cto = - q%;a,
(7)
where q2 = k2 - iw(& + l/D)
(8)
c;*, = {G,,, + S, + Dkf(E - c,)8(o)~jDq2 k2=kf+kf,
(9) W)
and d(w) is the Dirac delta function, multiplied by (2n)“Z. The solution of equation (7) that satisfies the Fourier Transform of equation (3), viz. D2
= u~[c,~ - (65- c,)lj(W)],
at r = r,,
(IV
is cl0 = cz - A(rC/r)e-d’-rc)
(12)
where (13) and s = v,r,/D. This solution can be used to find the net loss rate to the cavity. For C such cavities per unit volume, distributed at random, the total loss rate is C times this quantity. The latter must be the same as the asymptotic form in the medium if the approach is to be self-consistent. Hence, with kz 3 4nr,C, one finds the further relation Dki(l + qr,)A = D[-kf(E
- c,)~(o)
+(k:--io&)cp”,]-S,
the right-hand member equation (2) for large r.
being
the transfo~
(14)
of
BRAILSFORD
and MANSUR:
TIME-DEPENDENT
Using the derived form of A, equation (13) we equate terms linearly dependent upon the transform of the production rate, G,,, and those independent of the latter, separately. Thereby we obtain k f - iw.),,, =
k&(1
+ qr,.)
(15)
I + qrc + s
DIFFUSIONAL
Evidently,
DEFECT PROCESSES
1305
in the large s limit now used
Consequently, the function 4(t), evaluating the inverse transform
which is given by
(22) S,,,+ Dk;(c
- c,)~(o)
=
Dk&(l + qr,)(? - c,)S(w)
(16)
1 +qr,,+s Now when w = 0, q = k by equation in this limit, equation (15) reproduces cavity sink strength [2] k2
=
’
(8). {Similarly, the steady state
k&7(1+ kr, 1 l+kr,+s
it will be noted.} But more to the point, we may eliminate 4,,, between equations (8) and (15) to find
is trivially evaluated for t < 0. We may obtain it by contour integration in the upper half of the complex w plane. In this region q is analytic and 4,” has no singularities on the real axis. Hence 4(t) = 0 for t < 0. This establishes causality. The further correspondence with our prior analysis is most readily shown by computing the Laplace Transform of 4(t) directly from equation (22). Since 4(t) = 0 for t < 0. this transform, Qp say, is
s x
@‘r=
0
q?=k2;+k~;;-;). (
s
(18)
This is a cubic equation for q as a function of w. Of its solutions, we need the root with positive real part [see equation (12)], after which 4,,, is determined via equation (15). Hence the loss rate function will be completely determined. Actually, the subsequent analysis can be made significantly simpler in the void problem provided the ultimate growth of interest is diffusion limited. This hinges upon the observation that in this instance the parameter s is of order (r, lb), where b is a diffusion jump distance. Consequently, s is much greater than unity. Moreover, one observes from equation (18) that as w -+ cc, q -+ (- h/D)“*. Consequently, the combination (1 + qr< + s) appearing in the last equation differs from s only for frequencies so large that w > D/b’, i.e. for frequencies greater than the elementary atomic jump frequency. This range is of absolutely no interest in cavity growth problems of practical concern. The approximation qrc
iw/D}“*
+ikgr,
(19)
where fl = {ki + k2, + (ikir,)2}“2.
ti(t)e-P’dt
I =(2n)1.2
s
X 4,,d,,, ~~ p +io’
(23)
But since 4,,, is analytic in the upper half of the contour integration yields complex 0 plane, Qp = (2n)“‘b,,, or &=$[{p’+gj”*-/I]
(24)
from equations (22) and (21). This completes the re-derivation of one of the salient results of Ref. [l]. For 4(t) is now seen to be simply (27r)“*$(f), where $(t) is the loss rate function derived in the earlier work.? Inserting the preceding in the starting equation used here shows that the two approaches are thus exactly equivalent. For the sake of completeness, we turn briefly to the content of equation (16). In the limit of large s, it reduces simply to S,,, = 0. This is a further result that is contained in Ref. [I] when details of the initial state are neglected. Lastly, there is the question of reaction control at the void interface (i.e. or+ D/b). The limiting frequency at which the simplification leading to equation (19) now applies is correspondingly somewhat lower. However, it is evident that the transfer rate into the void would have to be very slow indeed for this to significantly alter the dynamic characteristics of the loss rate for cycle frequencies envisaged in fusion reactor components,
(20) REFERENCES
We now couple the above equation (17) to derive 4,
with the content of from equation (IS).
TCompare equations (20) and (24) above with equations (43) and (48) of Ref. [I].
AM
W--K
1. A. D. Brailsford and L. K. Mansur, Acta metaN. 33, 1425 (1985). 2. A. D. Bra&ford and R. Bullough. Phil. Trans. R. SOC. Lond. A302, 87 (1981).
3. A. D. Brailsford, J. Nucl. Maw. To be published