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The influence of dissociation of small helium-vacancy clusters on the growth of helium bubbles, the accumulation of helium at the grain boundaries and the kinetics of helium desorption from irradiated materials
263
Journal of Nuclear Materials 154 (1988) 263-267 North-Holland, Amsterdam
THE INFLUENCE OF DISSOCIATION OF SMALL HELIUM-VACANCY CLUSTERS OF ALA ON THE GROWTH OF ~~ BUBBLES, THlZ AC AEON AT THE GRAIN BOUNDARIES AND THE KINETICS OF HELIUM DESORFTION FROM IRRADIATED MATERIALS V.A. BORODIN, V.M. ~NICHEV and A.I. RYAZANOV I. V.Kurchatov Atomic Energy Institute, 123182 Moscow, USSR Received 20 October 1987; accepted 9 February 1988
This paper deals with the effect of thermal dissociation of small helium-vacancy clusters on the currents of helium atoms to gas fiied bubbles and to gram boundaries. The theory proposed demonstrates the possibility of accelerated accumulation of helium in the bubbles and at the gram boundaries. Also the desorption of helium from metat foils being irradiated with alpha-particles is considered and the effective coefficient of helium diffusion in the material is determined.
1. Introduction
One of the important problems of radiation physics of materials is the investigation of the interaction of helium with radiation point defects and their clusters which are produced in irradiated materials. The development of helium bubble structures in these materials at elevated temperatures requires the analysis of helium diffusion through the damaged lattice. The diffusion of helium to the gram boundaries results in its accumulation at them and leads ot the high-temperature irradiation embrittlement of materials [l]. The rate of diffusion of helium atoms depends on the positions occupied by these atoms. The coefficient of diffusion of interstitial helium, Dp, is significantly larger than that of substitutional helium atoms, D, (i.e. Dp z+ D,). However, the interstitial helium atoms are easily captured by vacancies, small helium-vacancy clusters being formed as a result. The experiments on
helium desorption from irradiated materials indicate that the majority of such clusters contains one vacancy and one helium atom [2] thus being simply substitutional helium atoms. The energy of binding E,” of these simplest helium-vacancy clusters is rather high (e.g. for Ni E,” =r3.0 eV [2,3]) and so the concentration C, of substitutions helium atoms exeeds considerably that of interstitial helium, Ct,. Consequently, in the theoretical investigation of the kinetics of microstructural evolution in irradiated materials only helium diffusion by the substitutional mechanism is usually taken into account.
0022-3115/88/$03.50 0 Elsevier Science Publishers (North-Holland
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However, experiments on the desorption of h&m from irradiated materials indicate that the diffusion of helium atoms in the lattice containing irradiation induced point defects can occur considerably faster [3,4]. A possible mechanism explaining this effect invokes the helium atoms passing from the substitutional position to the interstitial one according to the relation HeVeHe+V,
(1)
and for some time the helium atoms diffuse as interstitials until they are recaptured by vacancies. Although the concentration of interstitial helium atoms is considerably lower than that of substitutional helium, the accumulation of helium at sinks (grain boundaries, helium bubbles, etc.) is determined by the product of the concentration by the corresponding diffusion coefficient. Therefore, the higher diffusional mobility of interstitial helium (e.g. for Ni the co~~pon~ng activation energy of migration is EF = 0.08 eV [2]) can compensate for its lower concentration and the overall rate of helium diffusion will be determined by the combined mechanism. In section 2 of the present paper this mechanism is used to describe quantitatively the experiments such as that presented in ref. [4] and its applicability is also discussed. If the combined mechanism of helium diffusion is actually responsible for the results of [4] then under certain conditions (especially when the helium content in a material is high) it should manifest itself in the processes which require the diffusional inflow of helium. B.V.
V.A. Borodin et al. / Influence of dissociation of small He-vacancies
264
As an example we consider the acceleration of helium accumulation at the grain boundaries and in the helium bubbles (sections 3 and 4, respectively).
2. The kinetics of helium desorption during the irradiation of material with alpha-particles For describing the initial stage of helium desorption from the metal foil being irradiated with a-particles in a an accelerator [4], it is necessary to seek the solution of the corresponding one-dimensional diffusion equations for a plate whose thickness R, coincides with the projected range of implanted helium. Since usually such experiments are carried out at sufficiently high temperatures then the vacancy concentration C, in the material can be considered practically constant and identical with the thermal equilibrium value, C,,, whereas the diffusion of vacancies can be neglected. Since the substitutional helium has an energy of migration, which is much higher than that observed in ref. [4], it can also be regarded as immobile. Correspondingly, the problem is described by the set of equations:
Finally, as initial conditions we take C,(t = 0) = 0 (where a: denotes the type of point defects: a = s for substitutional and a! = p for interstitial helium atoms). Differentiating the second equation in eq. (2) with respect to time, substituting into it the aC,/ar from the first equation and introducing new variables 0 = u&t, x = x/R, and y = C, exp( pt), we obtain the following equation for y:
~+(l-+-cy=q+$-, X where n = D,/(uC,,Ri) = (L/R,)‘, z =/~/UC,, and L, = (Q’eC,,) ‘/* is the mean path length of interstitial helium before recombination with a vacancy. It is seen that E = exp((E,‘- E:)/kT), where E,f is the vacancy formation energy. Usually E,” > E$, e.g. for Ni E,” = 3.0 eV and Et = 1.6 eV, and therefore c +z 1. Fq. (4) can be easily solved [5] and for the most interesting case of efficient capture of interstitial helium by free vacancies (q +Z 1) we obtain, taking into account that c< 1:
c, =
jd, 7
P x
(2)
clusters
(x
+
xj
_
i
4
e k-1
cosIclk(x- 111exP(-a@) 142clk + sinQk)(l+ v-4) i ’
(5) where fi = tr exp( - Esb/kT) and a = va exp( - EF/kT) are, respectively, the rate of dissociation of substitutional helium and the rate of recombination of vacancies with interstitial helium atoms; v, is the oscillation frequency of a helium atom in the lattice and kT has its usual meaning. We shall use the Cartesian system of coordinates with the origin at the surface of the sample and the x-axis along the normal to the surface. Then the boundary conditions for eq. (2) will be written as
dCP dx
=AC,(x=O),
x=0
(3)
where X = (AR,)-‘, qk = en&(1 +nr:>-’ and pk is the k th root of the ordered in increase system of positive roots of the equation pk tan pk = l/X. If there is no energy barriers to hinder the desorption of helium for the free surface of material then A r+:a-‘, where a is the interatomic distance, and so X << 1. Consequently, at not too large k, which provide the main contribution to Cp, pk - a(k - l/2). In the experiments on desorption one is usually interested in determination of the current of implanted helium desorbing from the free surface of the material In our model the expression for the current jr, of helium from the free surface (x = 0) is of the following form
=jo, r=R,
where A is the rate of surface desorption and j. is a prescribed helium current out of the helium stopping region. Here j. is considered wnstant, negIecting the initial stage of attainment of steady-state diffusion current at x = R, after the beginning of irradiation. Such imposition of the boundary conditions at x = R, corresponds to the case where the range R, is much greater than the straggling of implanted helium [5].
Xexp[ -(2k
-t l)‘D&/R$]
,
where D,.f = s2j3L~/4. It is seen that the characteristic time +rcof attainment of steady-state current at the surface of the material is T== Rc/Der (for k = 0) so that De, plays the role of effective coefficient of helium
KA. Borodin et al. / Influence ofdissociationof small He-vacancies clusters
diffusion.
Substitution
D,, a exp( - Ez/kT), tion energy E,7 is
of /3 and L, into D,, gives where the effective helium migra-
This relation, obtained for the special case of the small recombination length (L, QCRP) and of weak dissociation of substitutional helium, is identical with the result obtained in ref. [4] from simple qualitative considerations and can be used to explain the value E,7; = 0.8 eV obtained there for nickel. It should be noted, however, that, under the conditions of ref. [4], the combined mechanism is only one of the possible mechanisms of accelerated helium diffusion and this rather low value of E,7 can be attributed as well to some other mechanism (e.g. to the helium pipe diffusion along dislocations [S]).
3. The effect of dissociation of helium-vacancy
clusters on the accumulation of helium at the grain boundaries
265
Here the energy of binding of helium atom with the grain boundary is supposed to be high (see e.g. ref. [6]), so that we may neglect the evaporation of helium from the boundary. The variation of the concentration of helium at the grain boundary, Cs, is determined by the evident condition:
(9) The solution of eq. (2) with the boundary condition eq. (8) and the initial conditions C,( t = 0) = C,, C,( t = 0) = 0 can be found in the form of a series just like it is done in the previous section. However, here we will restrict ourselves to the qualitative demonstration of the possible effect of the combined helium diffusion mechanism and so the problem will be simplified. Only shorttime kinetics (t 5 r,,, where r8 = 8-l is the characteristic time of dissociation of substitutional helium) will be regarded, when C, is approximatly equal to C,. Then it is easy to obtain the following relation for the current of helium atoms to the boundary, jr: jp=Dp%
The results of the previous section can be easily extended to the case of helium accumulation at the grain boundaries, which promotes the nucleation of grain-boundary helium bubbles and leads to a considerable loss of plasticity of the strained material [l]. In the present model the grain boundary is regarded as a thin plane (of the thickness 1, = 3-4~) in an infinite medium. It is assumed that the diffusion of helium to the boundary occurs from the region of the material where apart from vacancies other traps for helium atoms are practically absent. Due to the symmetry of the problem we can seek its solution only at x > 0. As in the previous section we shall neglect substitutional mechanism of helium diffusion in comparison with the combined one and regard substitutional helium as immobile. Also, the concentration of vacancies is assumed to be the thermal equilibrium one, C,,. This assumption is valid not only under the conditions of annealing, but also e.g. during the irradiation with alpha-particles if the temperature is high enough to neglect the irradiation generated vacancies. Now it is necessary to find the finite at x + co solution of eq. (2) with the following condition at the grain boundary
dCP dx
x-o
=+,(x=0).
x-o
= s{
EerfJ$
-1 +e’lTO erfcK},
(10) where erf(x) is the error function; erfc(x) = 1 - erf(x); ri = a’/D ; r2 = (UC,,)-’ and ~0~ = 7r-l - rF1. Since a2 = IO-13 cm2 and Dp = 10-l cm2/s then TV2: lo-l4 s, and therefore times t X=71 are of interest for practical purposes. Taking into account that r1/r2 = C,, g: 1, we can see that up to r8 = r2/e 3* r2 the kinetics of helium accumulation at the grain boundary is determined by the relation jr = aBCSoCVi1/2 erf fi.
(11)
Using eqs. (9) and (ll), we easily obtain
The eqs. (lo)-(12) are valid in the case when the concentration of substitutional helium is not too high and the vacancies created by the dissociation of substitutional helium do not disturb the initial distribution of vacancies, i.e. when Cr g C,,. Since Cr, 5 EC+ we obtain the following restriction: C, +z CT=exp( E,b/kT).
266
KA. Barodin et al. / Influence
4. The influenceof dissociation of hehuu-vacancy
ofdissociationofsmallHe-vacancies
clus-
ters ou the kinetics of growth of helium bubbh In this section we investigate the effect of the combined helium diffusion mechanism on the accumulation of helium in bubbles. The growth of helium bubbles occurs in a supersaturated solution of helium atoms and self point defects, such as vacancies and interstitials. However, here we shall neglect the possible presence of interstitials, because they do not interact with helium atoms and are therefore of no interest for the present investigation. For determining the steady-state currents of helium atoms and vacancies to helium bubbles we write, taking into account the thermal dissociation reaction (I), the following equations of diffusion of point defects near the bubble D,
AC,=&-oC&
(o=v,p),
I>,(C,Ac,-c,AC,)=PC,-aC,C,,
(13)
where DV is the diffusion coefficient of vacancies. Note that here, in contrast to the previous sections, the spatial variations of vacancy and substitutional helium concentration are taken into account, as well as the fact that the diffusion of a substitutional helium atom requires the presence of a free vacancy in one of the newt-n~~bour lattice sites. Let us separate out around each bubble the so called “sphere of influence” of the bubble with the radius L. In general this radius is determined by the concentration and sizes of the remaining bubbles, as well as by the density of the other sinks for point defects. Such model enables one to consider each bubble separately and is well applicable for a sufficiently rarified system of bubbles, i.e. when L >> R, where R is the average radius of bubbles. Using the spherical coordinate system with the origin at the centre of the bubble, we write the boundary conditions as
clusters
ached at quasi-steady equilibrium between the processes of dissociation and formation of HeV clusters, i.e.
c
pL<_=zP cSL oc,,
(15)
L*
It is evident from this relation that since C,, is always higher than the thermally equilibrium vacancy concentration, then cL d e G 1. Since even far from bubbles the ~ncentration of interstitial helium is much less than that of substitutional helium then the analogous relation between them should take place everywhere in the influence region. It means that to zero order in tL the profiles of concentration of vacancies and substitutional helium near bubbles are practically insensitive to interstitial helium. If we shall seek the concentrations of point defects as an expansion in the series expansion parameter eL, then the expansion of the concentrations C, and C, will start from the terms of zero order in eL and the expansion of Ct, from the term of first order in eL: C,=CJ”)+O(tL)
(a=v,s),
(16)
c, = fLCp(l)+ o( c;>.
Substituting the expansion eq. (16) into the system eq. (13) and retaining the therms of the same order in fL, we obtain the following equations for CT,!‘),Ci’“) and C(l). P . AC:“=0
(Q.=v,
s),
)
Dp “Cp’l) - CT( C,(‘)Cp(‘) - Cs(*‘CVL = 0.
(17)
The solution of eqs. (17) for vacancies and substitutional helium is str~~tfo~~d and we can easily obtain the following well-known expressions for their currents j,( (Y= v, s) to the bubble:
C=(i-= R) = CaR, Ca(r
= L)
= Car,
(14)
where C,, is some prescribed concentration at the outer boundary of the influence cell and C,, is the thermally equilibrium concentration at the bubble surface for the point defects of type a. We shall further take C,, = CPR = 0, for the helium is usually practically unsolvable in metal lattices. It is assumed in the present model that the formation of interstitial helium in material is only associated with the dissociation of substitutional helium. Therefore its concentration far from bubbles cannot exceed that re-
Substituting the solutions for Ci”’ into the equation for Cjl) and supposing R a L, C,, * C,, after some transformations we find
W’(2 r/L,, 2R/L, ) W(2-%5,,2R/.&) ’
(19)
V.A. Borodin et al. / Influence of dissociation of small He-vacancies clusters
where
-M
U and M are the degenerate hypergeometric functions and L, = (D,/uC,,)‘/~. As the maximum concentration of vacancies in the material usually does not exceed 10-3-10-4, the minimum path length of interstitial helium atom before its capture by a vacancy is Lr,min = 10*a. It is evident that the interstitial helium formed at a distance less then - L, from the bubble surface will be captured predominantly by the bubble and not by free vacancies. Therefore in the case of L < L, practically no helium atoms formed through reaction (1) will be recaptured by vacancies and the resulting form of C$) in this case is trivial c(1) =
P
CPL. E .
( 1 1 -
CL
r
In the case where L 3* L, (more accurately, when L = RL213 [7]) thermodynamic equilibrium is maintained betkeen dissociation and formation of HeV clusters, so that CpL.= cLCSLand the second term in the right-hand side of eq. (19) may be omitted. The current of interstitial helium to bubble in this case is c~YpDp
.ip=
-C
SLY
R
where y = P
1
+
& _ 2
4
4
W’@/Lr ,2R/Lr) W(2L/L,
3
2R/L,)
L, < R Yp = R/2L,.
As it can be seen from eqs. (18) and (21), the effective diffusion coefficient of helium in the latter case in De, = D,C,, + /3Lf. If we assume C,, = C,,, then the temperature T * above which the current of interstitial helium arising from the dissociation of helium-vacancy clusters exceeds that of substitutional helium to the bubble is -
E,b+E;-Ez k
I
'
The accelerated helium accumulation in bubbles due to the combined mechanism of helium diffusion under certain conditions can result in creation of high gas pressures in bubbles. For example, let us consider the behaviour of rather small (R - lOa) bubbles in the case of high-temperature annealing of material, so that C,, = C,,. Let us also restrict ourselves to t 5 7s. Then the initial concentration of substitutional helium C, varies little and the change of bubble volume can also be neglected [7]. In the most interesting case of rarified enough system of small bubbles (i.e. R << L, -ZKL) the density of helium in the bubble with the volume V= 4rrR3/3 by the moment t = rs will be n > 4=R2jpTb/V High pressure P of helium in the = 3a*C,/R*C,,. bubble (Pw 2 kT, where w is the atomic volume) corresponds to the density n >, 0.3 [7], which can be achieved if C,, > 10( R,‘a)*C,, . With T - lo3 K and R/a - 10 the development of overpressurized bubbles is possible during the annealing of material containing the concentration of preimplanted helium C, 2 100 ppm. Recently experimental indication has been obtained [8] of such overpressurizing of small (R 5 1 nm) helium bubbles in aluminium with - 1000 ppm of pre-implanted helium. Naturally, this overpressure can exist only limited time and is eventually relaxed as the annealing proceeds [8]. Relaxation of the high pressure in a bubble may occur either by punching out the interstitial dislocation loops [9] or by changing the bias of absorption of point defects by the bubble as the result of its elastic interaction with them [7].
(22)
and W’(x, y) = dW(x, y)/dy. It can be easily shown that as long as R/L, -=K1 we have Yp= 1, whereas for
T*z
267
(23)
where Ez is the activation energy for self-diffusion. Rutting for nickel E,” = 3.0 eV, EF = 0.08 eV, Ez = 2.8 eV and C,, = 10m4 we obtain T * = 700 K. The lower concentration C,, results in the lower value of T *.
References [l] H. Trinkaus, J. Nucl. Mater. 118 (1983) 39. [2] A. Goland, in: Point Defects in Solids (in Russian) (Mir, Moscow, 1979) p. 243. [3] W. SchiUing, in: Point Defects and Defect Interactions in Metals, Proc. of Yamada Conf. V (Univ. of Tokyo, Tokyo, 1982) p. 303. [4] V. PhiIIips, K. Sonnenberg and J.M. Williams, J. Nucl. Mater. 107 (1982) 271. [S] V.A. Borodin, V.M. Manichev and A.I. Ryazanov, Poverkhnost N7 (1987) 20. [6] M.I. Baskes and V. Vitek, Met. Trans. Al6 (1985) 1625. [7] V.A. Borodin, V.M. Manichev and A.I. Ryazanov, Fiz. Met. MetalIoved. 63 (1987) 435. [S] H.E. Hansen, H. Rajainmaki, R. TaIja, M.D. Bentson, R.M. Niemenen and K. Petersen, J. Phys. F15 (1985) 1. [9] J.H. Evans, A. van Veen and L.M. Caspers, Radiat. Eff. 78 (1983) 105.
Journal of Nuclear Materials 154 (1988) 263-267 North-Holland, Amsterdam
THE INFLUENCE OF DISSOCIATION OF SMALL HELIUM-VACANCY CLUSTERS OF ALA ON THE GROWTH OF ~~ BUBBLES, THlZ AC AEON AT THE GRAIN BOUNDARIES AND THE KINETICS OF HELIUM DESORFTION FROM IRRADIATED MATERIALS V.A. BORODIN, V.M. ~NICHEV and A.I. RYAZANOV I. V.Kurchatov Atomic Energy Institute, 123182 Moscow, USSR Received 20 October 1987; accepted 9 February 1988
This paper deals with the effect of thermal dissociation of small helium-vacancy clusters on the currents of helium atoms to gas fiied bubbles and to gram boundaries. The theory proposed demonstrates the possibility of accelerated accumulation of helium in the bubbles and at the gram boundaries. Also the desorption of helium from metat foils being irradiated with alpha-particles is considered and the effective coefficient of helium diffusion in the material is determined.
1. Introduction
One of the important problems of radiation physics of materials is the investigation of the interaction of helium with radiation point defects and their clusters which are produced in irradiated materials. The development of helium bubble structures in these materials at elevated temperatures requires the analysis of helium diffusion through the damaged lattice. The diffusion of helium to the gram boundaries results in its accumulation at them and leads ot the high-temperature irradiation embrittlement of materials [l]. The rate of diffusion of helium atoms depends on the positions occupied by these atoms. The coefficient of diffusion of interstitial helium, Dp, is significantly larger than that of substitutional helium atoms, D, (i.e. Dp z+ D,). However, the interstitial helium atoms are easily captured by vacancies, small helium-vacancy clusters being formed as a result. The experiments on
helium desorption from irradiated materials indicate that the majority of such clusters contains one vacancy and one helium atom [2] thus being simply substitutional helium atoms. The energy of binding E,” of these simplest helium-vacancy clusters is rather high (e.g. for Ni E,” =r3.0 eV [2,3]) and so the concentration C, of substitutions helium atoms exeeds considerably that of interstitial helium, Ct,. Consequently, in the theoretical investigation of the kinetics of microstructural evolution in irradiated materials only helium diffusion by the substitutional mechanism is usually taken into account.
0022-3115/88/$03.50 0 Elsevier Science Publishers (North-Holland
Physics
Publishing
Division)
However, experiments on the desorption of h&m from irradiated materials indicate that the diffusion of helium atoms in the lattice containing irradiation induced point defects can occur considerably faster [3,4]. A possible mechanism explaining this effect invokes the helium atoms passing from the substitutional position to the interstitial one according to the relation HeVeHe+V,
(1)
and for some time the helium atoms diffuse as interstitials until they are recaptured by vacancies. Although the concentration of interstitial helium atoms is considerably lower than that of substitutional helium, the accumulation of helium at sinks (grain boundaries, helium bubbles, etc.) is determined by the product of the concentration by the corresponding diffusion coefficient. Therefore, the higher diffusional mobility of interstitial helium (e.g. for Ni the co~~pon~ng activation energy of migration is EF = 0.08 eV [2]) can compensate for its lower concentration and the overall rate of helium diffusion will be determined by the combined mechanism. In section 2 of the present paper this mechanism is used to describe quantitatively the experiments such as that presented in ref. [4] and its applicability is also discussed. If the combined mechanism of helium diffusion is actually responsible for the results of [4] then under certain conditions (especially when the helium content in a material is high) it should manifest itself in the processes which require the diffusional inflow of helium. B.V.
V.A. Borodin et al. / Influence of dissociation of small He-vacancies
264
As an example we consider the acceleration of helium accumulation at the grain boundaries and in the helium bubbles (sections 3 and 4, respectively).
2. The kinetics of helium desorption during the irradiation of material with alpha-particles For describing the initial stage of helium desorption from the metal foil being irradiated with a-particles in a an accelerator [4], it is necessary to seek the solution of the corresponding one-dimensional diffusion equations for a plate whose thickness R, coincides with the projected range of implanted helium. Since usually such experiments are carried out at sufficiently high temperatures then the vacancy concentration C, in the material can be considered practically constant and identical with the thermal equilibrium value, C,,, whereas the diffusion of vacancies can be neglected. Since the substitutional helium has an energy of migration, which is much higher than that observed in ref. [4], it can also be regarded as immobile. Correspondingly, the problem is described by the set of equations:
Finally, as initial conditions we take C,(t = 0) = 0 (where a: denotes the type of point defects: a = s for substitutional and a! = p for interstitial helium atoms). Differentiating the second equation in eq. (2) with respect to time, substituting into it the aC,/ar from the first equation and introducing new variables 0 = u&t, x = x/R, and y = C, exp( pt), we obtain the following equation for y:
~+(l-+-cy=q+$-, X where n = D,/(uC,,Ri) = (L/R,)‘, z =/~/UC,, and L, = (Q’eC,,) ‘/* is the mean path length of interstitial helium before recombination with a vacancy. It is seen that E = exp((E,‘- E:)/kT), where E,f is the vacancy formation energy. Usually E,” > E$, e.g. for Ni E,” = 3.0 eV and Et = 1.6 eV, and therefore c +z 1. Fq. (4) can be easily solved [5] and for the most interesting case of efficient capture of interstitial helium by free vacancies (q +Z 1) we obtain, taking into account that c< 1:
c, =
jd, 7
P x
(2)
clusters
(x
+
xj
_
i
4
e k-1
cosIclk(x- 111exP(-a@) 142clk + sinQk)(l+ v-4) i ’
(5) where fi = tr exp( - Esb/kT) and a = va exp( - EF/kT) are, respectively, the rate of dissociation of substitutional helium and the rate of recombination of vacancies with interstitial helium atoms; v, is the oscillation frequency of a helium atom in the lattice and kT has its usual meaning. We shall use the Cartesian system of coordinates with the origin at the surface of the sample and the x-axis along the normal to the surface. Then the boundary conditions for eq. (2) will be written as
dCP dx
=AC,(x=O),
x=0
(3)
where X = (AR,)-‘, qk = en&(1 +nr:>-’ and pk is the k th root of the ordered in increase system of positive roots of the equation pk tan pk = l/X. If there is no energy barriers to hinder the desorption of helium for the free surface of material then A r+:a-‘, where a is the interatomic distance, and so X << 1. Consequently, at not too large k, which provide the main contribution to Cp, pk - a(k - l/2). In the experiments on desorption one is usually interested in determination of the current of implanted helium desorbing from the free surface of the material In our model the expression for the current jr, of helium from the free surface (x = 0) is of the following form
=jo, r=R,
where A is the rate of surface desorption and j. is a prescribed helium current out of the helium stopping region. Here j. is considered wnstant, negIecting the initial stage of attainment of steady-state diffusion current at x = R, after the beginning of irradiation. Such imposition of the boundary conditions at x = R, corresponds to the case where the range R, is much greater than the straggling of implanted helium [5].
Xexp[ -(2k
-t l)‘D&/R$]
,
where D,.f = s2j3L~/4. It is seen that the characteristic time +rcof attainment of steady-state current at the surface of the material is T== Rc/Der (for k = 0) so that De, plays the role of effective coefficient of helium
KA. Borodin et al. / Influence ofdissociationof small He-vacancies clusters
diffusion.
Substitution
D,, a exp( - Ez/kT), tion energy E,7 is
of /3 and L, into D,, gives where the effective helium migra-
This relation, obtained for the special case of the small recombination length (L, QCRP) and of weak dissociation of substitutional helium, is identical with the result obtained in ref. [4] from simple qualitative considerations and can be used to explain the value E,7; = 0.8 eV obtained there for nickel. It should be noted, however, that, under the conditions of ref. [4], the combined mechanism is only one of the possible mechanisms of accelerated helium diffusion and this rather low value of E,7 can be attributed as well to some other mechanism (e.g. to the helium pipe diffusion along dislocations [S]).
3. The effect of dissociation of helium-vacancy
clusters on the accumulation of helium at the grain boundaries
265
Here the energy of binding of helium atom with the grain boundary is supposed to be high (see e.g. ref. [6]), so that we may neglect the evaporation of helium from the boundary. The variation of the concentration of helium at the grain boundary, Cs, is determined by the evident condition:
(9) The solution of eq. (2) with the boundary condition eq. (8) and the initial conditions C,( t = 0) = C,, C,( t = 0) = 0 can be found in the form of a series just like it is done in the previous section. However, here we will restrict ourselves to the qualitative demonstration of the possible effect of the combined helium diffusion mechanism and so the problem will be simplified. Only shorttime kinetics (t 5 r,,, where r8 = 8-l is the characteristic time of dissociation of substitutional helium) will be regarded, when C, is approximatly equal to C,. Then it is easy to obtain the following relation for the current of helium atoms to the boundary, jr: jp=Dp%
The results of the previous section can be easily extended to the case of helium accumulation at the grain boundaries, which promotes the nucleation of grain-boundary helium bubbles and leads to a considerable loss of plasticity of the strained material [l]. In the present model the grain boundary is regarded as a thin plane (of the thickness 1, = 3-4~) in an infinite medium. It is assumed that the diffusion of helium to the boundary occurs from the region of the material where apart from vacancies other traps for helium atoms are practically absent. Due to the symmetry of the problem we can seek its solution only at x > 0. As in the previous section we shall neglect substitutional mechanism of helium diffusion in comparison with the combined one and regard substitutional helium as immobile. Also, the concentration of vacancies is assumed to be the thermal equilibrium one, C,,. This assumption is valid not only under the conditions of annealing, but also e.g. during the irradiation with alpha-particles if the temperature is high enough to neglect the irradiation generated vacancies. Now it is necessary to find the finite at x + co solution of eq. (2) with the following condition at the grain boundary
dCP dx
x-o
=+,(x=0).
x-o
= s{
EerfJ$
-1 +e’lTO erfcK},
(10) where erf(x) is the error function; erfc(x) = 1 - erf(x); ri = a’/D ; r2 = (UC,,)-’ and ~0~ = 7r-l - rF1. Since a2 = IO-13 cm2 and Dp = 10-l cm2/s then TV2: lo-l4 s, and therefore times t X=71 are of interest for practical purposes. Taking into account that r1/r2 = C,, g: 1, we can see that up to r8 = r2/e 3* r2 the kinetics of helium accumulation at the grain boundary is determined by the relation jr = aBCSoCVi1/2 erf fi.
(11)
Using eqs. (9) and (ll), we easily obtain
The eqs. (lo)-(12) are valid in the case when the concentration of substitutional helium is not too high and the vacancies created by the dissociation of substitutional helium do not disturb the initial distribution of vacancies, i.e. when Cr g C,,. Since Cr, 5 EC+ we obtain the following restriction: C, +z CT=exp( E,b/kT).
266
KA. Barodin et al. / Influence
4. The influenceof dissociation of hehuu-vacancy
ofdissociationofsmallHe-vacancies
clus-
ters ou the kinetics of growth of helium bubbh In this section we investigate the effect of the combined helium diffusion mechanism on the accumulation of helium in bubbles. The growth of helium bubbles occurs in a supersaturated solution of helium atoms and self point defects, such as vacancies and interstitials. However, here we shall neglect the possible presence of interstitials, because they do not interact with helium atoms and are therefore of no interest for the present investigation. For determining the steady-state currents of helium atoms and vacancies to helium bubbles we write, taking into account the thermal dissociation reaction (I), the following equations of diffusion of point defects near the bubble D,
AC,=&-oC&
(o=v,p),
I>,(C,Ac,-c,AC,)=PC,-aC,C,,
(13)
where DV is the diffusion coefficient of vacancies. Note that here, in contrast to the previous sections, the spatial variations of vacancy and substitutional helium concentration are taken into account, as well as the fact that the diffusion of a substitutional helium atom requires the presence of a free vacancy in one of the newt-n~~bour lattice sites. Let us separate out around each bubble the so called “sphere of influence” of the bubble with the radius L. In general this radius is determined by the concentration and sizes of the remaining bubbles, as well as by the density of the other sinks for point defects. Such model enables one to consider each bubble separately and is well applicable for a sufficiently rarified system of bubbles, i.e. when L >> R, where R is the average radius of bubbles. Using the spherical coordinate system with the origin at the centre of the bubble, we write the boundary conditions as
clusters
ached at quasi-steady equilibrium between the processes of dissociation and formation of HeV clusters, i.e.
c
pL<_=zP cSL oc,,
(15)
L*
It is evident from this relation that since C,, is always higher than the thermally equilibrium vacancy concentration, then cL d e G 1. Since even far from bubbles the ~ncentration of interstitial helium is much less than that of substitutional helium then the analogous relation between them should take place everywhere in the influence region. It means that to zero order in tL the profiles of concentration of vacancies and substitutional helium near bubbles are practically insensitive to interstitial helium. If we shall seek the concentrations of point defects as an expansion in the series expansion parameter eL, then the expansion of the concentrations C, and C, will start from the terms of zero order in eL and the expansion of Ct, from the term of first order in eL: C,=CJ”)+O(tL)
(a=v,s),
(16)
c, = fLCp(l)+ o( c;>.
Substituting the expansion eq. (16) into the system eq. (13) and retaining the therms of the same order in fL, we obtain the following equations for CT,!‘),Ci’“) and C(l). P . AC:“=0
(Q.=v,
s),
)
Dp “Cp’l) - CT( C,(‘)Cp(‘) - Cs(*‘CVL = 0.
(17)
The solution of eqs. (17) for vacancies and substitutional helium is str~~tfo~~d and we can easily obtain the following well-known expressions for their currents j,( (Y= v, s) to the bubble:
C=(i-= R) = CaR, Ca(r
= L)
= Car,
(14)
where C,, is some prescribed concentration at the outer boundary of the influence cell and C,, is the thermally equilibrium concentration at the bubble surface for the point defects of type a. We shall further take C,, = CPR = 0, for the helium is usually practically unsolvable in metal lattices. It is assumed in the present model that the formation of interstitial helium in material is only associated with the dissociation of substitutional helium. Therefore its concentration far from bubbles cannot exceed that re-
Substituting the solutions for Ci”’ into the equation for Cjl) and supposing R a L, C,, * C,, after some transformations we find
W’(2 r/L,, 2R/L, ) W(2-%5,,2R/.&) ’
(19)
V.A. Borodin et al. / Influence of dissociation of small He-vacancies clusters
where
-M
U and M are the degenerate hypergeometric functions and L, = (D,/uC,,)‘/~. As the maximum concentration of vacancies in the material usually does not exceed 10-3-10-4, the minimum path length of interstitial helium atom before its capture by a vacancy is Lr,min = 10*a. It is evident that the interstitial helium formed at a distance less then - L, from the bubble surface will be captured predominantly by the bubble and not by free vacancies. Therefore in the case of L < L, practically no helium atoms formed through reaction (1) will be recaptured by vacancies and the resulting form of C$) in this case is trivial c(1) =
P
CPL. E .
( 1 1 -
CL
r
In the case where L 3* L, (more accurately, when L = RL213 [7]) thermodynamic equilibrium is maintained betkeen dissociation and formation of HeV clusters, so that CpL.= cLCSLand the second term in the right-hand side of eq. (19) may be omitted. The current of interstitial helium to bubble in this case is c~YpDp
.ip=
-C
SLY
R
where y = P
1
+
& _ 2
4
4
W’@/Lr ,2R/Lr) W(2L/L,
3
2R/L,)
L, < R Yp = R/2L,.
As it can be seen from eqs. (18) and (21), the effective diffusion coefficient of helium in the latter case in De, = D,C,, + /3Lf. If we assume C,, = C,,, then the temperature T * above which the current of interstitial helium arising from the dissociation of helium-vacancy clusters exceeds that of substitutional helium to the bubble is -
E,b+E;-Ez k
I
'
The accelerated helium accumulation in bubbles due to the combined mechanism of helium diffusion under certain conditions can result in creation of high gas pressures in bubbles. For example, let us consider the behaviour of rather small (R - lOa) bubbles in the case of high-temperature annealing of material, so that C,, = C,,. Let us also restrict ourselves to t 5 7s. Then the initial concentration of substitutional helium C, varies little and the change of bubble volume can also be neglected [7]. In the most interesting case of rarified enough system of small bubbles (i.e. R << L, -ZKL) the density of helium in the bubble with the volume V= 4rrR3/3 by the moment t = rs will be n > 4=R2jpTb/V High pressure P of helium in the = 3a*C,/R*C,,. bubble (Pw 2 kT, where w is the atomic volume) corresponds to the density n >, 0.3 [7], which can be achieved if C,, > 10( R,‘a)*C,, . With T - lo3 K and R/a - 10 the development of overpressurized bubbles is possible during the annealing of material containing the concentration of preimplanted helium C, 2 100 ppm. Recently experimental indication has been obtained [8] of such overpressurizing of small (R 5 1 nm) helium bubbles in aluminium with - 1000 ppm of pre-implanted helium. Naturally, this overpressure can exist only limited time and is eventually relaxed as the annealing proceeds [8]. Relaxation of the high pressure in a bubble may occur either by punching out the interstitial dislocation loops [9] or by changing the bias of absorption of point defects by the bubble as the result of its elastic interaction with them [7].
(22)
and W’(x, y) = dW(x, y)/dy. It can be easily shown that as long as R/L, -=K1 we have Yp= 1, whereas for
T*z
267
(23)
where Ez is the activation energy for self-diffusion. Rutting for nickel E,” = 3.0 eV, EF = 0.08 eV, Ez = 2.8 eV and C,, = 10m4 we obtain T * = 700 K. The lower concentration C,, results in the lower value of T *.
References [l] H. Trinkaus, J. Nucl. Mater. 118 (1983) 39. [2] A. Goland, in: Point Defects in Solids (in Russian) (Mir, Moscow, 1979) p. 243. [3] W. SchiUing, in: Point Defects and Defect Interactions in Metals, Proc. of Yamada Conf. V (Univ. of Tokyo, Tokyo, 1982) p. 303. [4] V. PhiIIips, K. Sonnenberg and J.M. Williams, J. Nucl. Mater. 107 (1982) 271. [S] V.A. Borodin, V.M. Manichev and A.I. Ryazanov, Poverkhnost N7 (1987) 20. [6] M.I. Baskes and V. Vitek, Met. Trans. Al6 (1985) 1625. [7] V.A. Borodin, V.M. Manichev and A.I. Ryazanov, Fiz. Met. MetalIoved. 63 (1987) 435. [S] H.E. Hansen, H. Rajainmaki, R. TaIja, M.D. Bentson, R.M. Niemenen and K. Petersen, J. Phys. F15 (1985) 1. [9] J.H. Evans, A. van Veen and L.M. Caspers, Radiat. Eff. 78 (1983) 105.