The kinetics of void annealing in neutron irradiated aluminum

AC,<, ,M~rollurgr~s. Vol. 28. pp 15! to 161 Pergamon Press Ltd. 1980 Prmted in Great Britan

THE KINETICS OF VOID ANNEALING IN NEUTRON IRRADIATED ALUMINUM* R. A. VANDERMEER Metals and Ceramics

Division, (Receiwd

and J. C. OGLE

Oak Ridge National

Laboratory,

Oak Ridge, TN 37830, U.S.A.

4 June 1979; in revised form 6 August 1979)

Abstract-The kinetics of void annealing in neutron-irradiated, high purity aluminum containing 0.006 void volume fraction was studied by means of dilatometry. Over the temperature range 3588573 K, the isothermal recovery of void volume fraction exhibited a logarithmic kinetic behavior. The data were analyzed in terms of a recovery rate equation of the form

A theory for void annealing in irradiated aluminum is described hased on the premise that voids shrinking under the action of interfacial forces emit single vacancies which migrate by diffusion to internal sinks where they are annihilated. In the theory, Q,, is equal to the self diffusion activation agreement between theory and experiment was energy. For the parameters Q0 and K,, satisfactory achieved. The theory as presented cannot account for the parameter /?. Its influence on recovery was speculated to be related to the damage state of the material. The ratio-of-slopes method was used to analyze recovery rate data from isochronal experiments. In common with the isothermal results, the measured activation energy for void annealing was found to be a function of how much void volume fraction remained to be annealed. An approximate void size distribution function was derived which agreed satisfactorily with experiment. R6sum&Nous avons ttudit par dilatometrie la cinttique du revenu des cavites dans I’aluminium de haute purete contenant une fraction volumique &levee de cavites Cgale a 0,006. Entre 358 et 573 K, la restauration isotherme de la fraction volumique de cavites prtsentait une cinitique logarithmique. Nous avons analysi: les resultats par une equation de la vitesse de restauration sous la forme:

df

z

Nous prtsentons une theorie de la restauration des cavitts dans I’aluminium irradie qui repose sur I’hypothtse selon laquelle les cavites qui rttrecissent sous l’action des forces d’interface emettent des monolacunes qui migrent par diffusion vers des pitges internes oit elles s’annihilent. Dans cette theorie, Q. est tgal a I’tnergie d’activation d’autodiffusion. Nous avons obtenu un accord acceptable entre la theorie et I’experience, en ce qui concerne les parametres Q. et K,. La thtorie, telle que nous la presentons, ne peut toutefois pas rendre compte du parametre p. Nous pensons que son influence sur la iestauration est lice a I’etat des dommages dans le materiau. Nous avons utilise la methode du rapport des pentes pour analyser la vitesse de restauration a partir d’experiences isochrones. Comme dans le cas des mesures isothermes, I’energie d’activation du revenu des cavitts dependait de la fraction volumique de cavites restantes. Nous avons calcule une fonction de repartition de la taille des cavites, qui est en accord satisfaisant avec les valeurs experimentales. Zusammenfassnng-Mittels Dilatometrie wurde die Kinetik des Ausheilens von Hohlrlumen in neutronenbestrahltem, hochreinem Aluminium (mit 0,006 Volumanteil der Hohlrlume) untersucht. Uber den Temperaturbereich 358 bis 573 K zeigte die isotherme Erholung des Volumanteiles ein logarithmisches kinetisches Verhalten. Die Daten wurden mit folgender Gleichung der Erholungsrate beschrieben:

df

z=

-fK,exp-

[

~QoRT- Bf-

1

Es wird eine Theorie fiir das Ausheilen von Hohlrlumen in bestrahltem Aluminium auf der Voraussetzung aufgebaut, daf3 Hohlraume schrumpfen, indem sie unter dem EinfluB von Grenzflachenkraften einzelne Leerstellen emittieren, welche dann zu internen Senken diffundieren und dort ausheilen. In der Theorie entspricht Q. der Aktivierungsenergie ftir Selbstdiffusion. Bei den Parametern Q. und K, wurde befriedigende tjbereinstimmung zwischen Theorie und Experiment erzielt. Die vorgelegte Theorie kann den Parameter j nicht erklaren. Es wird vermutet, daR dessen EinfluB auf die Erholung mit dem Schadigungszustand zusammenhangt. Die Methode der Verhaltnisse zwischen Steigungen wurde zur Auswertung der Erholungsraten aus isochronen Experimenten verwendet. Die Methode des Steigungsverhaltnisses wurde benutzt, urn die Daten zur Erholungsgeschwindigkeit aus den isochronen Experimenten zu analysieren. Ubereinstimmend mit den isothermen Messungen waren die ermittelten Aktivierungsenergien fiir das Ausheilen von Hohlraumen abhlngig davon, wieviel Hohlraum-Volumanteil noch auszuheilen war. Die GrGl3enverteilung der Hohlrlume wurde naherungsweise abgeleitet und stimmte hinreichend gut mit dem Experiment iiberein. * Research sponsored by the Division with Union Carbide Nuclear Corp.

of Materials

Sciences, U.S. Department

151

of Energy

under Contract

W-7405-eng-26

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KINETICS OF VOID ANNEALING

I. INTRODUCI’ION Many non-fissionable metals and alloys when subjected to intense neutron irradiation at certain temperatures will “swell”. It is well known that this volume increase is due to the nucleation and growth of small, three-dimensional vacancy clusters called voids. The supply of vacancies arises from the continual atom displacements produced by the neutron bombardment. Because of the obvious technological importance of this phenomenon in structural members of nuclear reactors, extensive theoretical and experimental research on the subject has been carried out. A number of studies [14] have demonstrated that post-irradiation annealing can cause the irradationproduced voids to shrink and disappear @inter) and correspondingly, the material densifies. The most extensive investigations of void annealing have been concerned with aluminum and its alloys. The body of literature available on this subject has recently been reviewed by McHargue [S]. Most of the work up to now has been aimed at identifying by TEM the general microstructural aspects of void annealing behavior and ascertaining the temperature range where the recovery of swelling occurs. Newer techniques such as small angle X-ray scattering [6] and positron annihilation studies [7] have also been useful in these respects. Voids may also be developed under certain conditions in rapidly-quenched metals. Volin and Balluffi [S] and Westmacott, Smallman and Dobson [9] have investigated by TEM the in situ annealing behavior of individual well-isolated voids in quenched aluminum. The observed void shrinkage rates were in excellent agreement with a diffusion-limited, quasisteady-state vacancy migration mechanism. Knowledge regarding the kinetics of void annealing in irradiated aluminum is, by contrast, considerably more primitive. Packan and Braski [l], utilizing TEM in the same manner as cited above, observed that the void shrinkage rates were quite variable. They made no attempt at quantitatively interpreting the results from a mechanistic point of view. Furthermore, reversible transients were noted when the temperature was changed upward or downward and the usual vacancy diffusion mechanism could not account for the magnitude of this effect. Jostsons et al. [3] determined the activation energy of the swelling recovery process in a neutron irradiated commercial grade aluminum. From measurements of density recovery in bulk specimens, they found a value considerably higher than the expected aluminum self-diffusion activation energy. the annealing of voids in irradiated Clearly, aluminum is much more complex than it is in quenched aluminum. Hence, mechanistic details of the phenomenon remain vague. The purpose of the present investigation was to measure in more detail the swelling recovery kinetics of neutron-irradiated, high purity aluminum and to attempt an interpretation of the results in terms of

basic point defect mechanisms. In this work, the densification of irradiated aluminum caused by the annealing of a large void population was followed by means of a “fast-response” dilatometer. Both isochronal and isothermal annealing experiments were performed. 2. EXPERIMENTAL The irradiated aluminum was supplied to us by K. Farrell of the Oak Ridge National Laboratory. The aluminum had been purchased from Cominco American, Inc. (Grade 69, zone-refined), swaged to rods 3.17 mm in diameter and recrystallized. Lengths of this material were irradiated at 328 f 5 K in the High Flux Isotope Reactor at ORNL to a fast neutron fluence of 1.2 x 10” neutrons/cm’. After irradiation approximately 780 at. ppm Si and a few ppm of He and H were present as a result of transmutation reactions. Experiments on similarly irradiated aluminum of comparable purity have been reported previously [I, 4,6]. The irradiated rods were sawn into 10-mm long cylindrical specimens. Parallel-sided flats were then lapped on the ends of each specimen with a faceting machine using a 17pm diamond paste. A small 0.8 mm hole was drilled normal to the cylindrical axis midway along the length of each specimen to accommodate the bead of a thermocouple. The dilatometer was a commercial instrument manufactured by Theta Industries, Inc. (the Dilatronit III). The cylindrical specimens were springmounted horizontally between two silica rods one of which (the push rod) was frictionlessly seated to a linear-variable-differential-transformer (LVDT). The expansion or contraction of the specimen which was under a slight compressive stress (0.1 MNmm2) was transferred to the push rod where its movement through the core of the LVDT was converted to an electrical signal (voltage) that was amplified, demodulated and then continuously displayed on a strip chart recorder. A micrometer mounted to the LVDT-push rod module permitted calibration of the length change signal. The specimen was surrounded by a water-cooled copper coil and heated by induction. A Pt-Pt 10% Rh thermocouple (wire diam. 0.13 mm) was press fitted to the hole in the specimen. It was used to measure the temperature and to act in conjunction with a stepless SCR power supply and high frequency induction generator to control the temperature. The entire specimen assembly measuring module and induction coil were situated inside an evacuated enclosure operating at a pressure of less than 5 N me2 (5 x 10m5 torr). The isochronal experiments consisted of increasing the temperature of the specimen stepwise in increments of 18-25 K from room temperature up to 673 K. The specimen was kept at each holding temperature for 0.9 k s. Achieving a new temperature

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required only a few seconds. After reaching 673 K, whereupon all the voids had been eliminated, the furnace was turned off and the specimen allowed to cool to room temperature. Without disturbing the specimen, a second run was carried out. Its purpose was to provide base line information for comparison and to make corrections for the slight thermal lag of the measuring system. In the isothermal phase of this study, the irradiated specimens were heated directly to the annealing temperature and held for periods of up to 600 k s. Generally, less than 0.02 k s were required to attain a predetermined annealing temperature. Thereafter, thermal fluctuations were never more than + 3 K ; usually they were less. Upon completion of a run the specimen was cooled to room temperature. Sometimes a repeat test on the same but now annealed specimen was performed. This was necessary at the higher annealing temperatures where void annealing was rapid, to correct the length change data for thermal lag in the measuring module. 3. RESULTS 3.1 fsuchronul studies Figure 1 is a plot of A//I, versus holding temperature for an isochronally step-heated specimen initially containing voids (open circles) and then with the voids

removed (closed circles). The length lo is the starting length at room temperature for each run. The plotted length change data were measured at the end of each 0.9 k s holding period. The shrinkage of the specimen with the voids relative to the one containing no voids began at about 425K and continued to about 575K. Thereafter, the two curves were parallel. The difference between these two curves at the higher temperatures can be related to the total void volume fraction originally present after irradiation according to the formula: AV

-.__ E 3 :I_

Void volume fractions calculhted from this and similar data of both an isothermal and isochronal nature, varied between 0.0045 and 0.0065. A value of 0.006 had been anticipated based on the research of Packan [lo] who studied void swelling in identical material using TEM and an interpolation to a fluence of 1.2 x 1021 neutrons~~2 using his Fig. 6. The recovery of swelling on a fractional basis is shown in Fig. 2 plotted versus holding temperature. Here the data obtained from five separate specimens are summarized and compared with the findings of other investigators using different tools. Each study was on compatable purity aluminum irradiated to about the same fluence, viz. 0.006 to 0.01 void volume fraction. The slight differences between the various studies are probably caused by variations in void volume fraction. The irradiated aluminum used in the present study contains a significant concentration of transmuted silicon which could precipitate and/or coarsen during the anneals that eliminate the voids. This raises the possibility that A& might be affected by the presence of the silicon. A calculation, however, revealed that the maximum effect on Al/lo to be expected from pre-

m

A

TEM

14)

0.6 &

t

“400

.

450 ABSOLUTE

400 ABSOLUTE

500 TEMPERATURE

6cO

700

t ,C )

Fig. 1. Fractional change in length versus holding temperature for high purity. neutron-irradiated aluminum with (open circles) and without (closed circles) voids.

r I

l

s

300

(1)

0

VO

-

500 TEMPERATURE

.

550

600

CK1

Fig. 2. Fractional recovery of swelling versus annealing temperature for high purity aluminum, neutron irradiated to nominally 12, void volume fraction. Note comparison with the independent studies by small angle scattering (SAX), transmission electran microscopy (TEM) and positron annihilation.

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cipitation shouid be 50 times smaller than the effect from the voids. Indeed in separate annealing experiments on a quenched aluminum-silicon alloy containing seven times more silicon than that obtained from transmutation during irradiation, no effects on Al/l0 from silicon precipitation were detected. Thus, it was concluded that the effect of silicon precipitation on the At/l,, data during annealing could be neglected. During most of the isochronal annealing runs, Al/l0 was continuously recorded as a function of time. Because the temperature changes were accomplished abruptly and the system settled down at the new temperature in just a few seconds, it was possible to determine the swelling recovery rate, i.e. the slope of Al/l, versus time curve immediately preceding and very soon after a temperature change. Thus, in the temperature range where voids were disappearing, an apparent activation energy, QA could be estimated from the following equation :

I

o400 MEAN

I

I

I

I

450

500

550

ABSOLUTE

TEMPERATURE

t = 0, i.e. A~~/~~,was determined experimentally from the thermal expansion characteristics of the aluminum as the temperature was raised from room temperature to the annealing temperature. The thermal expansion coefficient was found to be independent of whether or not voids were present provided they do not shrink. The quantity Alfile represents the fractional length change remaining when all the voids were removed. The difference between Ali/&, and A/f/l, can be related to the total void volume fraction originally present in the aluminum through equation 1. The fraction of void volume unrecovered after annealing for time, r, in an isothermal experiment may be defined as

3.2 Isothermal studies

where A/t/f, is the instantaneous fractional length change. Equation 3 was used to convert the isothermal length change data to void volume fraction unrecovered which was then plotted versus time in various functional forms, viz. exponential, hyperbolic, power

A typical void annealing curve generated under isothermal conditions is shown in Fig. 4. Here A~/l* is plotted versus time on a linear scale. The annealing temperature was 498K. The length change value at

T=498:2K

I 0.36

600 (IO

Fig. 3. Apparent activation energy for void annealing plotted versus temperaturedetermined by the ratio of slopes method during isochronal experiments.

In this equation PI is the recovery rate at temperature 7’t just before the temperature was raised to T2 and Pz is the recovery rate at T, immediately after the change. R is the gas constant. This procedure, sometimes called the ratio of slopes method, was applied to the data from four of the isochronal runs. The results are displayed in Fig. 3. The temperature plotted here is the mean of Tl and T2. While there was considerable spread from specimen to specimen, it was evident that no single activation energy could describe the void annealing process. The general trend was for QA to increase as the temperature was raised and more and more recovery took place and to approach a saturation value at the higher temperatures.

0

I

I 0.72 ANNEALING

I

I

1.08 TIME

1.44

I 1.80

(ksect

Fig. 4. Typical change in length versus time during isothermal void annealing.

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155

t.0

B E5 & 0.6 E

5

3 0.6

6 t,

2 0.4 u-

4 i d

0.2

a 0 ro-2

2

5

ro-’

2

5

roe

2

ANNEALING

5

IO’

2

5

102

2

5

(03

TIME ( kc)

Fig. 5. Swelling fraction unrecovered plotted versus time on a logarithmic scale for selected isothermal annealing temperatures spanning the temperature range studied. Curves drawn according to equation 4 in text.

law. logarithmic. etc. The various functional forms of time were chosen on the basis of the predictions of different recovery theories. In our view, the best description of the data was a logarithmic equation of the kind

with I, and t, being constants that depend only on the annealing temperature. A few examples, selected because they span the temperature range studied, are given in Fig. 5 where ,f is plotted versus time on a logarithmic scale. The lines were drawn according to equation 4 with the constants k and to determined by a trial and error curve fitting procedure. Table 1 lists the experimental k and IntO values for all the temperatures studied. Only the data from the very highest annealing temperature could not be readily interpreted using equation 4. In that case, two sets of constants were calculated based on fitting either the later or the earlier portion of the recovery curve. Logarithmic kinetics like those given by equation 4 suggested a recovery equation of the type

temperature. If we let K(T) = K, exp - Q,JRT and r(f, t) = exp /3f/R T, then a short hand way of writing equation 5 is

df ~ =

-,fK(t)rU‘.

r)

dt

Still, at this point in our discussion, equation 5 (6) may be regarded as empirical. On the other hand, it may also be viewed as a first order, diffusion-controlled kinetic rate equation modified by the a(J t) term. The r(f; t) modification may be substantial, if not overriding, when /If+ R?: Being exponential in nature, u(X T) could vary much more rapidly with time than f itself does. With this extreme in mind, it can be shown in much the same way as Astrom [ll] did, that certain approximations are valid which allow equation 5 to be integrated to yield a result having the form of equation 4. In that case. the constants k and t, become

Vb)

(5) where

K,,

Q. and /j are constants Table Recovery temperature (K) 358 359 402 418 420 456 461 498 507 526 539 573

independent

of

1. Experimentally

The exponential character of g(f; t) might arise if the defect population is intense enough to modify the activation energy barrier that needs to be surmounted measured

lnt, In(ksec) +3.61 +5.19 + 4.28 + 2.25 + 1.39 -2.04 +0.26 -4.21 - 3.86 - 3.65 - 5.84 -5.95 (- 3.69)

k and Int,

k 0.08 0.1 0.06 0.20 0.12 0.105 0.155 0.14 0.12 0.22 0.14 0.217 (0.5)

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KINETICS OF VOID-ANNEALING

be a straight line whose slope and intercept are related to Q. and K, respectively. Figure 7 depicts the experimental to’s plotted in this manner. Using the value of fl calculated above, Ki and Q. were estimated from the intercept and slope determined by a least-squares curve fitting analysis. Their values are given in Table 2. In calculating the line shown plotted in Fig. 7, the two lowest temperature points were excluded. At these temperatures only the earliest recovery events had occurred. Based on the isochronal results (Fig. 3) where the apparent activation energy was extremely small for low temperatures and early recovery stages, negative deviations from equation 8 might be expected at low temperatures in the isothermal experiments as the case appeared to be.

I

0.5

:

0.4 k

cl

i ioo

0

200 ABSOLUTE

,300

400

TEMPERATURE

500

600

4. THEORY

(K)

Fig. 6. Plot of the experimental recovery parameter, versus absolute annealing temperature.

k,

in a diffusion process. Similar equations have been proposed for the recovery behavior of cold worked aluminum as measured by the recovery of stored energy El l] or flow stress [12]. Deviations from equation 4 are to be expected when f approaches zero, i.e. long annealing times, because the /If+ RT condition is violated and the approximations utilized above are no longer valid. According to equation 7a, an experimental value for /I can be obtained from a plot of li versus absolute temperature. Figure 6 is such a plot for the data at hand. Table 2 lists the value of ~LS calculated from the slope of the least squares straight line forced to pass through the origin in this figure. Of the two values of k determined from the annealing kinetics at the highest temperature only one (the one relating to the later recovery events) showed reasonable agreement with the remainder of the k’s. The other deviated substantially and, hence, was not included in the least-squares calculation of the slope. An evaluation of the constants Ki and Q. was accomplished by considering the temperature dependence of r, which has Arrhenius equation characteristics. Rewriting equation 7b in the form 1nt

suggests

0

-InT=LIn2R+!_ Q”-P BK,

0 T

R

(8)

The kinetic recovery equations (equations 4 and 5) have, up to now, been viewed largely as empirical although certain broad mechanistic implications have been alluded to. The purpose of this section is to provide a firmer mechanistic foundation for these equations and to gain some appreciation for the physical significance of Q,,, Ki and p. 4.1 Void size distrib~tion~nctio;~ In a large population of spherically shaped voids, the number of voids per unit volume with radii between r and r f dr remaining after annealing for time, t, may be defined as dN(r, t) = n(r, t) dr,

(9)

where n(r, t) is the void size distribution function. The total number of voids per unit volume N(t), and the total void volume per unit volume, V(t), that remain

0

I

I

t.9

2.t

I

-2

-4

+ 5

-6

I P 5 -8

that a plot of In to - In T versus l/T should -so

Table 2. Experimental recovery rate equation parameters for void annealing in aluminum

Parameter

QO (kJ/mole)

P (kJ/mole)

KI (set- ‘)

144.5

27.3

1.41 x loto

1.7

‘Ooo,,

2.3

2.5

2.7

iK1

Fig. 7. Plot of Int, - InT versus the reciprocal of the absolute annealing temperature.

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after annealing for time, t are then given by the following integrals: N(t) = /dNjr,t)

(lOa)

= 10% n(r,r)dr

157

KINETICS OF VOID ANNEALlNG

On combining equations 12, 13 and 14 and performing the indicated differentiation, the following differential equation results:

Wr, t f ---=

-n(r.i)D{$expp+

;[cxp$

- lil

and V(t) =

s

u, dN(r, t) =

sm4

17r3n(r, t)

’

dr. (lob)

0 3

The fractional void volume left unrecovered

at time, t, is

where dr/dt of r only,

has been considered

A=---

RT

where V(0) is the initial void volume fraction. The crux of the problem in evaluatingf(t) is to know the distribution function, n(r, t). To derive n(r, 1) we will further assume the following: 1. The voids are basically unstable and, given sufhcient thermal activation, will shrink so as to reduce the total interfacial energy of the system. 2. The effects of trapped gases (He or H) in the voids may be ignored. 3. Voids shrink by emitting single vacancies which migrate to sinks in the matrix and are eliminated in a quasi-steady-state fashion [8]. 4. Vacancy migration is a diffusion-controlled process. With these assumptions, it can be shown following Volin and Balluffi [S] and Westmacott et al. [93, that the shrinkage rate for a single void of radius, r, is given by. (12) where D,. is the vacancy diffusion coefficient, R and V, are the atomic and molar volumes respectively and y is the specific interfacial (surface) energy of the material. If C(x) = C,,, where C,, is the thermodynamic equilibrium vacancy concentration of the material, then D&C (2) = DrRC,, z Dso where Dso is the self-diffusion coefficient (ignoring the relatively minor correlation factor). Equation 12 with this substitution has been very successfulIy employed [S, 93 to describe the shrinkage kinetics of individual voids in quenched aluminum. Thus far, we have considered the shrinkage rate of an individual void. In a large population of voids of many different radii, we need to be concerned with the flux of voids through size space. This can be treated as a diffusion problem such that the net number of voids leaving the size class between r and r + dr during the time period between t and t + df is J = n(r,f)

$.

J, however, must satisfy the continuity dtr(r. tl \(,

dt

equation

d.1 dr ’

(14)

to be a function

and D = D,RC(r_

j.

The solution to this differential equation is n(r, t), the size distribution function that is required to calculate f(f). By applying the separation of variable methods to solve this differential equation, it can be derived that n(r, t) = K. exp - 12Dt exp -

where K0 and i.’ are integration

$09 dr

s0

constants

(16)

and

We have been unable to find an exact analytical solution to the integral in equation 16. It was noted in the experimental work by Packan [ 13) and Houston and Farrell [4], however, that even the smallest void sizes were such that most voids had radii, r > A. This suggested the possibility of approximating the exponentials in J/(r) by the appropriate series expansions. Carrying this out, and neglecting all but the first two terms in each series, yields

It is recognized that this is a rather drastic procedure, but as will be seen later when compared with experiment, it seems to be adequate for our purposes. Thus, equation 16 gives, after integration n(r. t) 2 K,r* exp -

c

.1

R2 D + ki

1

(19)

The determination of K,, and i2 requires that equation 19 be evaluated under some known conditions. The obvious one would be at t = 0 where H (r.0) would be the initial size distribution function. Then, if the initial void density N(O) and initial void volume V(0) are known, equation 19 with a t = 0 may be substituted into equations 1Oa and b and the constants determined. It is not evident, however. that the as-irradiated void size distribution should have exactly the same radius functionality. i.e. ll/(r), as a partially annealed size distribution. On the other hand, the growth of voids during radiation is prob-

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ably governed by the same basic surface energy and diffusion-control principles as in void shrinkage. Therefore, a similar rate equation (perhaps containing added terms) but opposite in sign would play an important role in void growth. Thus, it could be argued qualitatively that the as-irradiated void size distribution might have enough similarity to an annealed size distribution to be useful in evaluating K,, and 1’. This would have to be carefully scrutinized experimentally, however. Allowing the leeway then to calculate K,, and 1’ as outlined above, the results are

and

(21)

KINETICS

OF VOID ANNEALING

results point to the possibility of such an effect during void annealing. However, for the present case, we can only speculate as to its origin. Neutron irradiated aluminum contains other damage besides voids. This damage takes the form of substantial dislocation arrangements-loops, presumed to have arisen from interstitialcy aggregation and tangles as in cold work. In irradiated nickel [2] and aluminum [S], such dislocation microstructures tended to recover in about the same temperature range as did voids. Interstitial loops should be excellent sinks, in fact, for vacancies. In addition, dislocations could provide short circuit diffusion paths thus enhancing vacancy diffusion rates. Lacking additional mechanistic insights we will merely assume that the lowering of the activation energy barrier may be included as a multiplying factor, c~i, such that D = cr,D,QC(m),

4.2 Theoretical

where

void annealing rate

void volume unrecovered after time, by entering equations 20 and 21 into equation 19 which, in turn, is substituted into equation 11. After performing the necessary integration we have

Bf

The fractional

tl’ = exp E’

t, may be obtained

f=-=

V(t) V(0)

N(O)

exp-4nAADt.

- = -f4IIA dt

(22)

V(0)

The recovery rate then may be determined

df

as

$,

which predicts simple, first order reaction kinetics for void annealing. This result is in obvious disagreement with the experimental observations (see Section 3.2 especially equations 4, 5 and 6) which require a not inconsequential modification to a first order kinetic equation in order to be explained. Of the quantities entering into equation 23, only D, the diffusivity term, seems to be capable of sufficient lattitude to alter the theory adequately. In the annealing of voids in quenched aluminum, Volin and Balluffi [8] found that D = Dso. In the present work we have chosen to represent D as D = D,RC(co).

(25)

A second way in which D may be influenced is through the C(cc) term. Thus, C(co) could be larger than C,, if the void density is high. In that situation, so many vacancies would be emitted that they cannot be considered to be annihilated instantaneously, as was approximately true for the case of void annealing in quenched aluminum where the void density was significantly lower. Hence, in the irradiated aluminum each emitted vacancy would spend a finite period of time roaming the matrix until it was accepted by a sink. Consequently, an excess vacancy concentration, C,, could exist such that

C(m) =

c, + c,, = c,,

(26)

This contribution could be estimated, following an approach by Damask and Dienes [IS], by assuming that vacancy emission from shrinking voids and the corresponding vacancy annihilation at sinks can be treated as a steady state process. Thus dC 2 dt

= k, - k,C, = 0

(24)

There appears to be at least two ways in which D may be affected by a large defect population and damage state in the case of void annealing after irradiation. First, D may be influenced in the vacancy diffusivity term, D,. It is known, in the case of recovery in coldworked aluminum, that the presence of a dislocation microstructure lowers the activation energy barrier that must be surmounted by the diffusing specie. The barrier is assumed to be lowered by an amount proportional to the defect concentration [ll, 121. Kelly [14] has alluded to the importance of an analogous effect in heavily damaged irradiated solids. Our

where k

= ”

‘(‘) df Q

dt

is the vacancy emission rate and k,, the annihilation rate constant is assumed to be the product of a sink density ps and the vacancy diffusion coefficient. Equation 27 can be solved and rearranged to yield 1 1 - f. 4IlAN(O)/p,

(28)

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OF VOID ANNEALING

159

Equation 28 only has physical meaning if the denominator remains positive, a condition necessitating ps > f4nAN(O). Substituting appropriate values for A and N(0) (see next section) into this inequality and taking.f = 1. i.e., the as-irradiated state, a sink density of >2 x lo9 cm-’ is required. If the dislocation loops of the damaged state may be considered as the sinks, then from the data of Packan [I 33, e.g. loop density 2 x 10’4cm-3 and average loop diameter - IO-’ cm, an actual sink density of three times that amount is calculated for irradiated aluminum. Under these conditions, the contribution of equation 28 to the overall recovery equation would correspond to a relatively minor effect. Finally, substituting equations 26 and 28 into 25, recognizing that D$C,, = DsD and then inserting the answer into equation 23 we obtain for the void annealing rate

df = - f’4nA N$&, dr

1 ! 1 - f417AN(0)/ps >

(29)

Comparing this result with equations 5 and 6 and realizing that D,, = D, exp - QsD/RT, the following relationships become evident: K(T) = 4llA NgDsD,

x(f; I) =

1 Bf 1 - f4L’AN(O)/p, > exp fi’

(Job)

(3Oc)

Qo = Qso.

(304

Thus, two of the three experimental constants, KI and Qo, have well-defined theoretical meanings. Enough is known about the parameters in equations 3Oc and d from other experiments to make a comparison of theory and experiment in these cases.

5. COMPARISON OF THEORY AND EXPERIMENT 5.1 Void size distribution function Void size distribution histograms for neutron irradiated aluminum of purity comparable to that used in this study have been determined by Packan [13]. Figure 8 reproduces the histogram for a specimen irradiated to a fluence of 1 x 10” neutrons/cm’ at 328 K and containing N(0) = 3.9 x lOI voids per cm3 at V(0) = 0.0086 but with no post radiation annealing. This histogram was chosen for comparison with the void size distribution function derived in the present work because: 1. the irradiation conditions were nominally the same, 2. the procedure for evaluating the integration constants K, and 1’ with the as-irradiated void size distribution could be tested

90

160

300

230

VOID RADIUS

( xrd81

km)

Fig. 8. Example of a void adapted etically

size distribution histogram from Packan [14] and superimposed by the theorderived void size distribution function (equation 24).

and validated, and 3. the severity of the approximations used to obtain an analytical solution to n(r, t) could be scrutinized. The distribution function calculated for these conditions corresponds to the solid line superimposed on the histogram of Fig. 8. This curve was calculated from equations 19-21 with a dr size increment of 15 x 10-s cm and the N(0) and I/(O) values stated by Packan [13]. No other adjustable parameter was necessary because no post-irradiation annealing had occurred and t in equation 19 could be taken to be zero. In view of the experimental uncertainties in N(O), i.e. *35%, we consider the agreement between the calculated function and the measured histogram to be adequate if not excellent. We further conclude that the integration constants K, and 1,* may. in the present case at least, be determined from the characteristics of the as-irradiated but pre-annealed void size distribution. The assumptions employed to derive an analytical form for n(r, t) also seem justified. Because of the nature of the approximation, the fit of the calculated curve to the data is expected to be worst at small values of r, i.e., r < A = (2Ry/kT). This is also the region where the experimental values were in greatest error. The value of A depends on T and 7, the surface energy. Taking T as the irradiation temperature, n = 16.6 x 10-24cm3 and :’ = 1140 ergs/cm* [9] a value of 84 x lo-’ cm may be estimated for A. It is well known, however, that surface energy can be significantly lowered by the segregation of impurities to the surfaces. The y value used in the above determination of A was that estimated from void shrinkage studies in quenched, high purity aluminum where impurity effects were probably small. On the other hand, because 700ppm Si was generated by transmutation during neutron-irradiation and because Si has an affinity for the void interfaces in aluminum [ 161, it is expected that ;’ could be substantially lower than 1140 ergs/cm ’ in the case of voids in irradiated aluminum. Hendricks PC al. [ 171 have determined surface void properties in neutron-irradiated aluminum by means of a small angle neutron diffraction study. They found that 7 reductions of >26”/;, were necessary to account for their results. Thus, we conclude that r ,< A = 84 x IO-@ cm is an overesti-

160

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KINETICS OF VOID ANNEALING

mate and should be at least 26% lower implying that the approximations used to derive n(r, t) may be better than first thought. Regarding possible changes in the void size distribution during annealing, equation 19 makes certain predictions. Since time and size were separable as variables in this treatment, these equations imply that the mean void radius should not change as the population of voids anneals out. The findings of Houston and Farrell [4] on aluminum are contrary to this prediction. They found that the small voids annealed first while the larger voids disappeared later. On the other hand, the results of Kulcinski et al. [2] on nickel and Epperson et al. [6] support the prediction except perhaps during the very final stage of annealing. 5.2. Recovery equation parameters

It was shown in Section 4.2 that two of the three parameters entering the recovery equation (equation 5) have predicted relationships with known material parameters. Thus, the experimentally measured values of K, and Q. may be compared to values calculated with equations 3Oc and d. The measured Qe value of 144 f 19 kJ/mole (Table 2) was slightly greater than, but within experimental error of, the self diffusion activation energy of 126 kJ/mole determined by Volin and Balluffi [8] from void shrinkage studies. When compared to the self diffusion measurements of Lundy and Murdock [18] who found Qsr, = 143 kJ/mole, the agreement of Q. was almost perfect. Taking A = 62 x lo-* cm (the previous estimate reduced by 26%) N(0) = 2.5 x 1014 voids/cm3 from Packan [6, 131, DO= 0.176 cm’/? and V(0) = 0.006, a value of 5.7 x 10” s-i was calculated for Ki using equation 30~. This was approximately four times larger than the experimental value (Table 2). Generally, when experimental pre-exponential constants such as this are compared with theoretical values, not much better than order of magnitude agreement can be expected. As yet, no theory has been developed for the constant p in the recovery equation. By analogy with cold-worked metals, we believe it to be a function of the state of damage in the material. Values of 102 [ 121 and 63 kJ/mole [ 1l] were found experimentally for recovery in cold worked aluminum. Our value of 27 kJ/ mole is substantially lower implying that the damage state in irradiated aluminum may not be as intense as for cold working. In spite of being that low, /I nevertheless has an important effect on the kinetics of void annealing in irradiated aluminum. Future experiments like these on material irradiated to different void populations and damage states may provide more substantive clues as to the nature of p. 5.3 Isothermal versus isochronal activation energies The equation used to describe the recovery of voids in irradiated aluminum, i.e. equation 5, intimates that the apparent activation energy, QA, at any point in the annealing process will depend on how much void

coo r---l 25t50 & E

z .g

A

0

too -o

OA <“g-Bf” A

0

5oTb

b

A

0

b

0

0

9

q

_ 0

F F ::

b

-

A

0.2

0

0.4

FRACTION

VOID

0.6 VOLUME

0.6

1.0

RECOVERED

Fig. 9. Isochronally measured activation energies of Fig. 3 plotted versus fraction of the void volume recovered. Straight line corresponds to equation 34 with values of Q. and b determined from the isothermal experiments. annealing has yet to transpire. relationship should hold

Thus,

the following

QA = Qo - Bf:

(31)

In Fig. 9 the isochronally determined QA values of Fig. 3 are replotted, this time versus f rather than temperature. Superimposed on this plot is a graph of the straight line representing equation 31 with the Q. and /I values determined experimentally from the isothermal runs. With the exception of the earliest (f1) and perhaps latest (f-0) stages of recovery, the correspondence of the activation energies is reasonable. The isochronal results displayed deviations from the proposed recovery equation at the temperature extremes as was already noted in the isothermal experiments. 6. CONCLUSIONS 1. The kinetics of void annealing in neutron irradiated, high purity aluminum have been successfully measured by means of a fast response dilatometer. 2. Over the temperature range 358-573 K, the isothermal recovery of void volume fraction was best represented by logarithmic kinetics. 3. Analysis of the data suggested a recovery rate equation of the type

df

Tit=

-fK,exp-

-

c Qo

Bf ___ RT

1 ’

for void annealing. 4. A diffusion-controlled, vacancy migration theory of void annealing was described which related Q. to the self diffusion activation energy and K, to other known material constants. 5. A satisfactory agreement between theory and experiment was realized for Q,, and K,. 6. The parameter B remains undefined theoretically, but is thought to be related to the nature of the damage state of the material.

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KINETICS OF VOID ANNEALING

7. Activation energies estimated by the ratio-ofslopes technique from isochronal experiments demonstrated that the activation energy for void annealing was a function of how much recovery had taken place in agreement with the isothermal results. 8. An approximate void distribution function was derived and found to adequately portray experimental observation. Ackno~~edge~e~ts-be authors are indebted to K. Farrell for supplying the irradiated aluminum. Helpful discussions with R. W. Hendricks. M. H. Yoo and C. J. McHargue are gratefully acknowledged.

REFERENCES 1. N. H. Packan and D. N. Braski, J. nucl. Mater. 34, 307 (1970). 2. G. L. Kufcinski, B. Mastel and H. E. Kissinger, Acra Metall. 19, 27 (1971). 3. A. Jostsons, E. L. Long, Jr, J. 0. Stiegler, K, Farrell and D. N. Braski, in Radiation-Induced Voids in Metals, P. 363, J. W. Corbett and L. C. Ianniello, eds., U.S. Atomic Energy Commission, Washington, D.C. (1972).

161

4. J. T. Houston and K. Farrell, J. nucl. Muter. 40, 225

(1971). 5. C. J. McHargue, to be published. 6. J. E. Epperson, R. W. Hendricks and K. Farrell, Phil. May. 30, 803 (I 974). 7. K. Peterson, N. Thrane, G. Trumpy and R. W. Hendricks, Appl. Phys. 10, 85 (1976). 8. T. E. Volin and R. W. Balluffi. Phvs. stat. sol. 25. 163 (1968). 9. K. H. Westmacott, R. E. Smallman and P. S. Dobson, Metal/. Sci. J. 2, 177 (1968).

IO. N. H. Packan, J. nucl. Mater. 40, 1 (197 I). Ii. H. U. Astriim, Ark. Fys. 10, I97 (195.5). 12. D. Kuhlmann, G. Masing and J. Raffelsieper, 2. ~~~ru~~~.40, 241 (1949). 13. N. H. Packan, Voids in Neutron Irradiated Aluminum. Ph.D. Thesis, University of Missouri, Rolla, MO (1970). ORNL/TM-3109. 14. B. T. Kelly, Irradiation Damage to Solids, p. 192, Pergamon Press, Oxford (1966). 15. A. C. Damask and G. J. Dienes, Point Defects in Metals. D. 285. Gordon and Breach. New York (1963). 16. K. Far;&, J. Bentley and D. N. Braski, Scriptu. retail. 11,243 (1977). 17. R. W. Hendricks and J. Schelten, Phil. Mug. 36, 907 (1977). 18. T. S. Lundy and J. F. Murdock, J. Appl. Phys. 33, 1671 (1962).