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A simulation study of ostwald ripening of gas bubbles in metals accounting for real gas behaviour
Acta metall, mater. Vol. 39, No. 8, pp. 1845-1852,1991 Printed in Great Britain.All rights reserved
0956-7151/91 $3.00+ 0.00 Copyright © 1991 PergamonPress pie
A SIMULATION STUDY OF OSTWALD RIPENING OF GAS BUBBLES IN METALS ACCOUNTING FOR REAL GAS BEHAVIOUR P. F. P. FICHTNER~, H. SCHROEDER and H. TRINKAUS Institut fur Festkoerperforschung, Forschungszentrum Juelich, P.O. Box 1913, D-5170 Juelich, Germany (Received 20 August 1990; in revised form 8 January 1991)
Abstract--The Ostwald ripening kinetics of bubbles containing non-ideally behaving gas is studied by a computer simulation approach. The temporal evolution of mean radii and bubble size distributions under transient and steady state conditions is examined. In comparison with ideal gas, the results for real gas show large differences for mean radii rm~<50 nm, which is the usual range of Transmission Electron Microscopy investigations. Rrsmnr--49n etudie par simulation sur ordinateur la cinrtique du mfirissement d'Ostwald de buUes contenant un gaz au comportement non idral. On examine l'rvolution dans le temps du rayon moyen et des distributions de taille des bulles, en rrgime transitoire et en rrgime stationnaire. Par rapport au gaz idral, les rrsultats obtenus dans le cas d'un gaz rrel prrsentent de grandes diffrrences pour les rayons moyens rm~<50 nm, ce qui est le domaine courant des recherches en microscopic 61ectronique en transmission.
Zusammenfassung--Die Kinetik der Ostwald-Reifung von Blasen, die nicht-ideales Gas enthalten, wird mit Hilfe von Computer-Simulation studiert. Die zeitliche Entwicklung der mittleren Blasen-Radien und der Blasen-GrrBenverteilung werden f/Jr den anf'~inglichenObergangszustand und den sp/iteren quasistation/iren Zustand untersucht. GroBe Abweichungenvom Verhalten f/Jr ideales Gas werden fiir mittlere Blasen-Radien kleiner als 50nm, also fiir den iiblichen Bereich der Transmissions-ElektronenMikroskopie gefunden.
1. INTRODUCTION A two phase system consisting of a distribution of dispersed second-phase precipitates within a primary parent phase tends to reduce its total interfacial free energy and/or to relax the pressure within the precipitates by virtue of a coarsening process in which the average size of precipitates increases while its number density decreases. There are two main types of mechanisms resulting in precipitate coarsening; precipitate migration and coalescence on one side and resolution and reabsorption of the second-phase component on the other side. In solids, the latter mechanism, called "Ostwald ripening" (OR), is commonly considered to be dominant. After an earlier discussion by Greenwood [1], the theory of OR was systematically developed by Lifshitz and Slyozov [2] and by Wagner [3], with particular emphasis on the coarsening of solid precipitates. The coarsening of gas bubbles in solids was first discussed by Greenwood and Boitax [4] and analysed in terms of the Lifshitz-Slyozov-Wagner (LSW) theory by Markworth [5]. In both studies, ideal gas behaviour was assumed. tOn leave from Escola de Engenharia, Universidade Federal do Rio Grande do Sul, 90000 Porto Alegre, Brazil.
For small bubbles with diameters 2r < 10nm the condition for equilibrium of the gas with the surface of the bubble, p = 27/r (where), is the specific surface free energy of the matrix), yields values of the pressure p in the GPa-range where the ideal gas law is a poor approximation and more sophisticated equations of state are needed. An analytical theory of OR for such conditions has been outlined and preliminary results for He bubbles in metals have been given previously [6]. A systematic theoretical analysis is, however, still lacking. On the other hand, experimental results on the coarsening of inert gases in metals have been interpreted in terms of both mechanisms, migration and coalescence [7, 8] as well as OR kinetics [4, 9]. A procedure to discriminate the two main coarsening mechanisms by testing for specific invariances has been proposed [6] and applied successfully recently [10]. The partially controversial discussion about coarsening of inert gas bubbles has stimulated interest in details of the two main mechanisms. In the present paper we make an attempt to provide guide lines for further experimental and theoretical work on the OR kinetics of inert gases in metals, with special emphasis on the effects of deviations from the ideal gas behaviour resulting from extremely high pressure. Using computer simulation,
1845
1846
FICHTNER et al.: OSTWALD RIPENING OF GAS BUBBLES
we study the evolution of a population of helium bubbles in an otherwise homogeneous medium. 2. BASIC ASSUMPTIONS AND EQUATIONS
The mean gas concentration in the matrix, Cg, is determined by gas conservation. Assuming that during coarsening no gain or loss of gas atoms occurs and that dCg/dt is small we obtain from equation (1) (d/dt)Y-,iN ~ = Zl4nrDg [•g -- C s (rl)] ~ 0
In simulating gas bubble coarsening in metals we follow the basic assumptions made in the LSW approach: (1) The parent phase forms an ideal solution of gas; (2) the fraction of gas in solution is negligible compared with the fraction in bubbles; (3) the volume fraction occupied by bubbles is small such that a diffusional mean field approach may be used; (4) the bubbles are spherical; (5) the pressure within the bubbles is in equilibrium with the thermal equilibrium vacancy concentration. On the basis of these assumptions, the diffusion controlled rate of change of the number of gas atoms or vacancies in a given bubble of radius r may be written as d(Ng.v)/dt = 4nr O,.v [Cg.v - C,,v(r)]/~.
(1)
The suffixes g, v label gas atoms or vacancies, respectively; Ds, v are the corresponding diffusivities in the matrix; Cg, v are the mean field concentrations; Cg.v (r) are the concentrations close to the bubble surface (in units of particles per matrix atoms) and is the matrix atomic volume. The local concentrations are given by Cs, v = e x p ( - Gg, v / k T + pg, v/kT)
(2)
where G~.,v are the Gibbs free enthalpies of solution, #g,v are the chemical potentials of gas atoms and vacancies and k T has the usual meaning. The chemical potentials #g,v=(OFs/c~Ng.v)~v.~ may be derived from the bubble formation free energy FB = FG + ?S, where FG is the free energy of the gas phase, S is the total bubble surface and 7 is the specific surface free energy. Disregarding elastic effects, we may write for the vacancy chemical potential close to the surface of a bubble I.tv = (~FB/~N~)~¢, =
tl(~FB/dr') = t l ( - p
+ 27 /r).
(3)
V represents the bubble volume and p the gas pressure inside the bubble. Using equation (3) in equation (2) we find Cv = C p e x p [ - (p - 27/r)f~/KT].
(4)
At low gas concentrations at which the sink strength of the bubble structure is small compared to the sink strength of the vacancy sources and at high temperatures at which C~ ~ C~ is readily established everywhere the pressure inside a bubble is always close to its equilibrium value, p = 2?/r. In this case, which we shall assume in the following, the vacancy fluxes are completely linked with the gas fluxes. Otherwise, the bubbles become overpressurized and a different treatment is required [11].
(5)
In the above expression, the sum runs over all bubbles. For a homogeneous concentration field of dissolved gas atoms equations (2) and (4) lead to Cg = e x p [ - G ~ k T ] ( r exp[#8/kT])/
(6)
where the brackets denote ensemble averaging. The number of gas atoms inside a given bubble may be correlated with the size dependent pressure, p(r), by use of an appropriate equation of state N s = n(p = 27/r, T ) V
(7)
where n is the number density of gas atoms in the bubble. Following the procedure outlined in Ref. [6] we condense the quantities characterizing the parent phase in a reduced time t* = (3/f~) (2?/kT)-2Ogt e x p [ - G ~ / k T ]
(8)
and account for the gas equation of state in the reduced variables N* = n-2 z-3
=
3/4n(27/kT)-3N
r* = (nz) -1 = (2? /kT) -1 r.
(9) (10)
z(n) = p / ( n k T ) is the gas compressibility factor. Introducing these reduced variables (t* and r* have the dimensions of a volume, N* has the dimension of a square of a volume) into equations (1) and (6) we obtain a "master equation" for Ostwald ripening of equilibrium bubbles dN*/dt* = r*Acg
(11)
Ac8 = [(r exp(#~(r)/kT)>/(r>] -
exp(/~g(r)/k T).
(12)
The ensemble averages introduced by equation (l 2) renders equation (11) an integro-differential equation. The solution of the latter is complicated when a real gas equation of state is used instead of the ideal gas law.
3. GAS EQUATION OF STATE The behaviour of a real gas is best described by the compressibility factor, z = p / ( n k T ) , which has a value of 1 at low densities (ideal gas behaviour) but increases continuously with increasing n. For hard sphere equations of state such as the simplified Van der Waals or Carnaham-Starling equations, z becomes formally infinite at closest packing. It has, however, turned out that at high densities the repulsive atomic potentials are not well described by hard spheres but are relatively soft, in particular for gases of low atomic number such as helium.
FICHTNER et al.: OSTWALD RIPENING OF GAS BUBBLES To our knowledge, bubble coarsening in metals is experimentally best investigated for the inert gas helium [9, 10]. Taking this as a respresentative example we use the semi-empirical high density equation of state given in Ref. [6] for 4He
1847
Pressure p (5Po) 107
101 I
100 '
'
106
a - 600 k
i0s'
b-lOOOk
10-1
I
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... j
lO"1oc-l+ook .f
z = (1 - p) (l + p - 2p 2) + (1 - p)2pB/v
I
+ (3 - 2p)p2zl + (1 - p ) p 2 z ; v where
"~ 10z
p =v,V
101
v = 56T-1/+exp[-O.145T -1/4) in units of/~3
100
Zl = 0.1225 v T °-~55
z~v ~ - 5 0 . The chemical potential follows from /~ = f + p v , w h e r e f i s the free energy per atom ( f = S - P dv), as
#g. k T =/a,a - (27 - 16p)p2/6 + (6 - 9p + 4p2)B/3v + (27 - 16p)p2zl/6 + (9 - 8p)p2z2v/6 where #id is the chemical potential for ideal gas. The gas equation of state affects the O R kinetics in two ways: via the dissolved gas concentration Cs(r ) = exp[l~s(r)/kT] and via the relation between bubble radius and corresponding content of gas atoms N = (4n/3)r3n. Figures 1 and 2 illustrate the behaviour of cg (r) and N as a function of r for both, ideal gas and helium as a real gas in a metallic matrix with 7 = 2 N/m. These large differences, in particular in Fig. 1, are expected to manifest themselves in the coarsening kinetics.
107 105
Pressure p (GPo) 10o 10"1 = i
OK'~K6OOK 100
=t. 101 x
--
rear gas ideot gns
\
10-1
~
~
10-3 10-5 10-1
,
t
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i
101
10-1
-
-
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,
l
100
rear gas
,dmt gas
,
,
I
,
101
,
102
Radius r (nm) Fig. 2. Number density of He atoms in equilibrium bubbles at different temperatures (600 ~< T ~< 1400 K) as a function of the bubble radius r or internal pressure p = 27/r for a metallic matrix with V = 2 N/m.
B = 170 T -v3 - 1750/T in units of A 3
101 \
'i :"~"
:..:.. . ] "
o..
,
10z
Radius r (nm) Fig. 1. Dissolved He concentration cg = exp(#ne/kT ) around an equilibrium bubble as a function of the bubble radius r or internal pressure p for a typical metallic matrix (7 = 2 N/m) at temperatures T in the range 600 ~< T ~< 1400 K.
4. NUMERICAL ANALYSIS An analytical analysis of the integro-differential equation (11) describing O R kinetics of real gas bubbles is difficult [6]. In the following we simulate the evolution of a bubble size distribution (BSD) starting from a given initial distribution. The results are considered to provide guide-lines for more detailed analytical approximations. The numerical analysis starts from an appropriate BSD defined by 6000 bubbles. F o r this, the mean field concentration is calculated by use of equations (6) or (12). Then, each bubble is allowed to change size according to equation (11) during a small increment of time At*. To avoid substantial changes in the mean field during At*, the length of the latter is chosen such that only the smallest member of the bubble population is lost. Then, the procedure is repeated. When the number of bubbles has decreased to 4000 the population is increased to the initial number of 6000. In order to repopulate, we first construct the BSD using about 100 points and then rebuild the bubble population in such a way that bubbles with the same size are not allowed, but the shape of the distribution is maintained. With this scheme it is possible to account for large variations in mean radius while maintaining a good definition of the BSD and a reasonable numerical efficiency. Furthermore, this procedure also stepwise removes artificial structures in the size distribution associated with the numerical discretization. On the other hand, the system of equations which we use to describe the evolution of bubble populations is, strictly spoken, deterministic. In a real physical situation stochastic fluctuations will always occur. In the LSW approach the selected analytical solution is that which is stable against fluctuations. In a numerical stimulation study some fluctuations are actually required to obtain reasonable convergence to
FICHTNER et al.: OSTWALD RIPENING OF GAS BUBBLES
1848
such a solution. Our repopulation scheme, along with some numerical inaccuracy, provide the necessary fluctuations. Within the mean field approach spatial correlations are not accounted for. These would manifest themselves in a broadening of the BSD and an increase of the amplitude of the mean radius growth law, but would not affect its logarithmic slope [12]. Spatial correlation between particles in a homogeneous system becomes increasingly important for increasing particle number density and, in general, becomes noticeable for volume fractions above 5%. We have tested our simulation code by applying it to the ideal gas case for which the stable asymptotic quasi-steady-state behaviour has been analysed analytically previously [5].
Figure 3(a) shows a double logarithmic plot of the mean radius rm as a function of time obtained by starting from the analytical quasi-steady-state BSD [5]. Because of the finite value of the initial r m at t = 0, the steady state behaviour becomes evident only after an apparent transient. This effect can be removed by plotting r mz_ rm2 (t = 0) [Fig. 3(b)]. Figure 3(b) shows clearly that the simulated values follow the analytically predicted behaviour characterized by a slope dlog(rm)/ dlog(t) = 0.5. G o o d agreement between simulated and analytical results is also found for the evolution of the BSD as illustrated in Fig. 4, where the histogram points obtained from the simulations and the analytical BSD (continuous line) are presented. 5. RESULTS 5.1. Ostwald ripening under real gas behaviour
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103
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To illustrate the effect of the real gas (RG) behaviour at high gas densities we start from an
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ew
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10L,
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. . . . . . 0.8 1.2
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100 10-I
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i
103
10~
i
I
I
I0s
106
107
I0e
t" Fig. 3. (a) Temporal evolution of the mean radius r,~ expressed in reduced variables r* [equation (10)] and t* [equation (8)] assuming ideal gas behaviour and a typical metallic matrix 0' = 2N/m) at T = 1123 K. The crosses corresponds to the calculated values which follows the trend suggested by the continuous line. An arbitrary choice of parameters D m = 3.56 I0 -t°m2s, f l = 10.95,~3 and G~ = 2.5 eV renders the time scale on the top in units of seconds. (b) r .2 (t*) - r*2(0) vs t* corresponding to the data in (a). The simulated values denoted by + lie in a straight line showing that the curvature in Fig. 3(a) is due to the finite value rm(0) = I nm. In addition, the slope of the line corresponds to the analytical value of 1 for such a plot.
J
\
o/,° "fl
,
,
,
,
,
,
,
i
,
,
,o~
i
i
0 gO 80 1201602002/,0280320
r(nm) Fig. 4. Comparison between the analytical bubble size distribution (continuous line) and the simulated results (O) for the ideal gas case. The good agreement between the curves observed over a large interval of mean radius r= show that the simulation code is working properly.
FICHTNER et al.:
OSTWALD RIPENING OF GAS BUBBLES
arbitrary distribution of an approximately gaussian shape with rm = 0 . 2 5 n m and standard deviation tr = 0.03 n m [see Fig. 6(a)]. The temporal evolution of r m for both, real and ideal gas, are presented in Fig. 5 for two temperatures (973 and 1273 K) but the same gas dissociation energy (3.5 eV), and for a typical metallic matrix 0' = 2 N/m). Obviously the R G behaviour results in a much faster initial growth. With increasing time, however, the growth rate slows down and the curves converge above 60 nm. More precisely, the logarithmic derivative dlog(rm)/dlog(t) for the R G case continuously increases from ~ 0.1 at short times, and converges to 0.5 at long times. Because of the finite value of r m (t = 0), precise values of the slope cannot be obtained without the knowledge of the growth law, as was done for the I G case [see Fig. 3(a) and (b)]. On the other hand, the curvature in r m (t) obviously means that the R G behaviour cannot be represented by a unique power law. Similar as the growth rate, the BSD is also changing continuously. This is illustrated in Figs 6 and 7 for the low and high temperatures cases, respectively. Starting from the arbitrary BSD [Fig. 6(a)], the R G - B S D quickly changes into a shape characterized by a short tail towards smaller bubbles and a relatively sharp cut towards larger bubbles [Fig. 6(b)]. With increasing time [Fig. 6(c-d)], the RG-BSD becomes continuously broader but still much sharper than an IG-BSD with the same r m and n u m b e r of bubbles. Figure 7 illustrates how the R G - B S D approaches the IG-BSD shape. However, even for rm-~ 60 n m [Fig. 7(d)] the R G behaviour still influences the distribution shape. To check whether the arbitrary choice of the initial BSD in the above simulations has influenced the evolution of the RG-BSD, we also started the simulations from the analytical quasi-steady-state BSD for ideal gas as in Fig. 4 (rm = 1 nm), which is much
- -
102
I
I
I
x
reaL gas 1223K
* o
reaL gas 923K ,dealgos 1223K
+
IdeaL gas 923K
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i
,
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i
,
,
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i
,
,
'=0
,
a)
rm= 0.25nm
1849
!r ~ i
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,
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'
'
r~ =2Anm t =1oo /,~,,
'
'
c)
.=~ 01 0.2 0.3 rm=lnm ~ t=l /',
"~
1 2 015 b) rm= 6.1nm f = 50OO
0./,
3
/* d)
/I 0./*
0.8
1.2
r (nm)
1.6
20 0
2
~
6
8
10
r (nm)
12
Fig. 6. Time evolution of the bubble size distribution (BSD)
assuming real gas behaviour and a arbitrary initial BSD showed in (a) for the T = 9 2 3 K case in Fig. 5. The continuous line corresponds to a quasi-steady-state ideal gas BSD with the same rm and number of bubbles. The real gas BSD are much sharper than the corresponding ideal gas BSD and are characterized by a short tail towards small bubbles and a relatively sharp cut at the large bubble side. broader than the characteristic R G - B S D for the same rm [see Fig. 6(b)]. This is illustrated in Fig. 8. After a short transient (not shown), the R G - B S D quickly recovers its characteristic shape [Fig. 8(b)] and evolves towards the IG-BSD shape in a very similar way as in the previous cases (see Figs 6 and 7). Hence, we may conclude that the characteristic shape of the RG-BSD, as discussed above, reflects a property of the bubble system which is determined by the real gas behaviour. We emphasize that the bubble size range commonly considered in Transmission Electron Microscopy (TEM) is between 1 and 50nm. The above results show significant quantitative discrepancies between I G and R G behaviour in this size interval.
I
=
j
/
"
',
f b 0 ~u
J,, ........... •
10-2
10-1
___l
100
2
~
rm=13nm t = 50
6
8
8
10
b) /'~',
16
24
32
rm=6Onm t = 2000
~0 d)
J l__
I
101
102
_
I
_
103
I
10~
105
f ( a r b i t r a r y unifs)
Fig. 5. Time evolution of the mean radius rm for ideal and real gas behaviour assuming the same matrix properties. The real gas system shows a much faster initial growth but converges to the ideal gas behaviour at large rm.
4
8
12
r (nm)
16
20
2t.O
20
z.O 60
80
100 120
r (nm)
Fig. 7. Time evolution of the bubble size distribution (BSD) assuming real gas behaviour for the T = 1223 K case in Fig. 5. This figure illustrates the convergence of the real gas BSD to the ideal gas shape (continuous curves).
1850
FICHTNER et aL:
OSTWALD RIPENING OF GAS BUBBLES
rm=lnm f=O
r m= 16.8 nm f = 100
o)
d)
,~
/'\ ./', .... \, 0 03, 0.8 1.2 1.6
21o 0 L, 8 12 16 20 N 28 32
g
rm=3.Snm f:l
m 0
1
2
3
r m : 7./+ nm
f = 10
0
2
,d~ "
~
b)
!
L, 5
6
7 c)
///~\
6 8 10 12 r(nm)
I r m= 32.6nm f =500
0
10 20
e) ~-
30
I.~. rn~= 60"Snrn f =2000
0- 20 ~
~0
i\ 50
60' f)
~-
100 120
60 80 r(nm)
Fig. 8. Time evolution of the real gas bubble size distributions (BSD) starting with a BSD corresponding to the ideal gas case (represented by continuous line). The real gas BSD quickly assumes its characteristic shape and evolves in the same way as in the cases illustrated in Figs 6 and 7.
5.2. M a s t e r curves
In considering gas diffusivity and solubility on the matrix in a reduced time t*, we have scaled out the quantitites which are most sensitive to temperature. The effect of temperature on bubble coarsening via the gas equation of state is much weaker. This is illustrated in Fig. 9, which shows the same data presented in Fig. 5 in terms of reduced time t* [equation (8)] and radius r* [equation (10)]. The shape of the BSD also follow a general behaviour depending mainly on the mean radius but only weakly on the temperature. Figure I0 illustrates this point showing the BSD standard deviation (or) as a function of r= from the cases presented in Fig. 5 (IG and RG at two temperatures), for the simulations starting with the IG-BSD (Fig. 8) and for an empirical distribution to be discussed in next section.
103 x reaL gas 1223K * reaL gas 923K 10 2
/
/
102
o ideat ga, l m K
lo1 =G 10 100 _ . . . . . . . . . . . ;l
10-1
I
I
101
I
I
103 t" (~3)
I
I
10s
I
I
107
Fig. 9. Time evolution of the mean radius rmin terms of the reduced variables r~*[equation (10)] and t* [equation (8)] for the same data set as in Fig. 5. This figure illustrates the scaling provided by such reduced variables.
FICHTNER et al.: OSTWALD RIPENING OF GAS BUBBLES Because the quasi-steady-state IG-BSD is self-similar, tr/rm is independent of r~. For RG, the characteristic shape of the distribution quickly achieved for the cases where the simulations started with a small rm (e.g. as in Fig. 5) seems to best represent a quasisteady-state R G behaviour, where the broadening of the RG-BSD, parametrized in terms of tr/rm, increases quasi linearly with Iog(rm) in the range rm ~< 100 nm. Deviations from the quasi-steady-state behaviour and their dependence on the initial shape of the BSD, for both ideal and real gas, are also illustrated in Fig. 10.
The initial BSD is determined by the details of the nucleation kinetics and/or by the migration and coalescence of small bubbles. Therefore, depending on the situation, different initial BSD can be expected. In this section we try to evaluate the influence of the initial BSD shape on the temporal evolution of the mean radius, rm, and its standard deviation, a, by performing the simulations starting from different initial BSD. Figure 11 shows a comparison of rm (t) curves for three cases: LSW type [see Fig. 8(a)], gaussian distribution (r~ = 1 nm and tr = 0.2 nm) and an empirical distribution which has a tail towards large bubbles [see Fig. 12(a)]. There exists no significant differences between these curves, showing that the master curve presented in Fig. 9 has a rather general validity. The observed behaviour in a/rm also shows a convergence to a general behaviour (see Fig. 10), but obviously strongly depends on the initial shape of the distribution. The temporal evolution of the empirical distribution mentioned above is shown in Fig. 12. The tail towards large radii quickly disappears giving rise to
0.38
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l
0.3/+ 0.30
~
0,26 0.22 0.18 real gas
0.1/+ 0.10 10-I
,
I
i
100 Mean
i
I
101
,
,
10 2
Radius rm (nm)
Fig. 10. Evolution of the standard deviation of the bubble size distributions, ¢r, normalized to the corresponding mean radius rm as a function of r m. The symbols + and * corresponds to the cases pre~nted in Fig. 5, * refers to the case of Fig. 12, x and • refers to the real gas simulations starting with the ideal gas BSD as in Fig. 8, at two different temperatures and O and • corresponds to pure ideal gas cases.
t (arbitrary units} 10o 101 102
10-1 [
I
I
103
I
r
,o2 T:l123K
1
/[
/"
+ ,sw
~
* GAUSS
1
.-'/
I i --
101
t 10 2
5.3. Transient behaviour
1851
i
J
J
103
10~
I0s
t"
,J10 °
106
(~3)
Fig. 1I. Evolution of r* vs t* showing transient behaviours introduced by starting the simulations from different bubble size distributions (see text).
a "standard" RG-BSD showing a characteristic sharp cut at large bubble side and a tail to the small bubble side. For comparison, the quasi-steady-state IG-BSD normalized to the same r m and the same number of bubbles is also shown. We conclude that transients associated with initial shape of the BSD are not of fundamental importance for the Ostwald ripening of equilibrium gas bubbles, although the distribution may present different shapes during a transient period. 6. CONCLUSIONS The simulation approach presented in this paper is suited to describe Ostwald ripening kinetics of bubbles containing non-ideally behaving gas at low gas concentration and at high temperature at which the pressure inside the bubbles may be assumed to be close to its thermal equilibrium value. In comparison with ideal gas, the results for real gas show large differences for mean radii rm ~< 50 nm, which is the range of TEM investigations. In this range, the mean radii reached within a given annealing time are substantially larger than for ideal gas. The growth law cannot be described by a single power law. In a double-logarithmic plot of the mean radius vs time the slope increases continuously from about 0.1 to the asymptotic ideal gas value of 0.5. The bubble size distributions for the real gas case are found to be significantly narrower than for the ideal gas case. The temporal evolution of the mean radius is only initially and then only weakly dependent upon the assumed shape of the initial size distribution. The time-dependence of the mean radius can, therefore, be condensed into a kind of master curve which is independent of the gas dissociation energy and only weakly dependent upon temperature. An approximate fitting procedure on the basis of the presented results may be used to analyse bubble
1852
FICHTNER et al.: OSTWALD RIPENING OF GAS BUBBLES
rm : 0 . 7 8 r i m
t:0
rm= &,.56nm f =10
o)
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/°.\
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rm=25.62nm f = 2000
t=5 l
e)
fOx
/
-~-~_--C . . . .
rm=3.85nm
8
,o \ i
13
1
, K,
3 z~ 5 6 7
rm= 8.47nm t:lO0
b)
i\
,oo
i
f) ~.-o-,o
o
\\ ,
0
1
2
3 4 r(nm)
5
6
L ^ . ~ - ~- ° "
0
. . . . . .
'~ \ ,
10
40
20 30 r(nm)
Fig. 12. Time evolution showing the transients on the real gas bubble size distribution introduced by assuming the empirical distribution in (a) as a starting distribution. annealing data and, in particular, to extract from them the gas dissociation energy. The simulation approach is also suited to test future analytical approaches to the Ostwald ripening kinetics of real gas bubbles. Acknowledgements---One of us, P.F.P.F., would like to acknowledge the support from the Alexander yon Humbold Foundation and from Conselho Nacional de Pesquisa (CNPq-Brazil).
REFERENCES 1. G. W. Greenwood, Acta metall. 4, 243 (1956).
2. I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids 19, 35 (1961). 3. C. Wagner, Z. El. Chem. 65, 581 (1961). 4. G. W. Greenwood and A. Boltax, J. nucl. Mater. 5, 234 (1972). 5. A. J. Markworth, Metall. Trans. 4, 2651 (1973). 6. H. Trinkaus, Rad. Effects 78, 189 (1983). 7. P. J. Goodhew and S. K. Tyler, Proc. R. Soc. Lond. A 377, 151 (1981). 8. Z. H. Luklinska, G. yon Bradsky and P. J. Goodhew, J. nucl. Mater. 135, 206 (1985). 9. J. Rothaut, H. Schroeder and H. Ullmaier, Phil. Mag. A 47, 781 (1983). 10. H. Schroeder and P. F. P. Fichtner, J. NucL Mater., Conf. Proceedings Kyoto (1989). In press. 11. H. Trinkaus, Scripta metall. 23, 1773 (1989). 12. M. Marder, Phys. Rev. A 36, 858 (1987).
0956-7151/91 $3.00+ 0.00 Copyright © 1991 PergamonPress pie
A SIMULATION STUDY OF OSTWALD RIPENING OF GAS BUBBLES IN METALS ACCOUNTING FOR REAL GAS BEHAVIOUR P. F. P. FICHTNER~, H. SCHROEDER and H. TRINKAUS Institut fur Festkoerperforschung, Forschungszentrum Juelich, P.O. Box 1913, D-5170 Juelich, Germany (Received 20 August 1990; in revised form 8 January 1991)
Abstract--The Ostwald ripening kinetics of bubbles containing non-ideally behaving gas is studied by a computer simulation approach. The temporal evolution of mean radii and bubble size distributions under transient and steady state conditions is examined. In comparison with ideal gas, the results for real gas show large differences for mean radii rm~<50 nm, which is the usual range of Transmission Electron Microscopy investigations. Rrsmnr--49n etudie par simulation sur ordinateur la cinrtique du mfirissement d'Ostwald de buUes contenant un gaz au comportement non idral. On examine l'rvolution dans le temps du rayon moyen et des distributions de taille des bulles, en rrgime transitoire et en rrgime stationnaire. Par rapport au gaz idral, les rrsultats obtenus dans le cas d'un gaz rrel prrsentent de grandes diffrrences pour les rayons moyens rm~<50 nm, ce qui est le domaine courant des recherches en microscopic 61ectronique en transmission.
Zusammenfassung--Die Kinetik der Ostwald-Reifung von Blasen, die nicht-ideales Gas enthalten, wird mit Hilfe von Computer-Simulation studiert. Die zeitliche Entwicklung der mittleren Blasen-Radien und der Blasen-GrrBenverteilung werden f/Jr den anf'~inglichenObergangszustand und den sp/iteren quasistation/iren Zustand untersucht. GroBe Abweichungenvom Verhalten f/Jr ideales Gas werden fiir mittlere Blasen-Radien kleiner als 50nm, also fiir den iiblichen Bereich der Transmissions-ElektronenMikroskopie gefunden.
1. INTRODUCTION A two phase system consisting of a distribution of dispersed second-phase precipitates within a primary parent phase tends to reduce its total interfacial free energy and/or to relax the pressure within the precipitates by virtue of a coarsening process in which the average size of precipitates increases while its number density decreases. There are two main types of mechanisms resulting in precipitate coarsening; precipitate migration and coalescence on one side and resolution and reabsorption of the second-phase component on the other side. In solids, the latter mechanism, called "Ostwald ripening" (OR), is commonly considered to be dominant. After an earlier discussion by Greenwood [1], the theory of OR was systematically developed by Lifshitz and Slyozov [2] and by Wagner [3], with particular emphasis on the coarsening of solid precipitates. The coarsening of gas bubbles in solids was first discussed by Greenwood and Boitax [4] and analysed in terms of the Lifshitz-Slyozov-Wagner (LSW) theory by Markworth [5]. In both studies, ideal gas behaviour was assumed. tOn leave from Escola de Engenharia, Universidade Federal do Rio Grande do Sul, 90000 Porto Alegre, Brazil.
For small bubbles with diameters 2r < 10nm the condition for equilibrium of the gas with the surface of the bubble, p = 27/r (where), is the specific surface free energy of the matrix), yields values of the pressure p in the GPa-range where the ideal gas law is a poor approximation and more sophisticated equations of state are needed. An analytical theory of OR for such conditions has been outlined and preliminary results for He bubbles in metals have been given previously [6]. A systematic theoretical analysis is, however, still lacking. On the other hand, experimental results on the coarsening of inert gases in metals have been interpreted in terms of both mechanisms, migration and coalescence [7, 8] as well as OR kinetics [4, 9]. A procedure to discriminate the two main coarsening mechanisms by testing for specific invariances has been proposed [6] and applied successfully recently [10]. The partially controversial discussion about coarsening of inert gas bubbles has stimulated interest in details of the two main mechanisms. In the present paper we make an attempt to provide guide lines for further experimental and theoretical work on the OR kinetics of inert gases in metals, with special emphasis on the effects of deviations from the ideal gas behaviour resulting from extremely high pressure. Using computer simulation,
1845
1846
FICHTNER et al.: OSTWALD RIPENING OF GAS BUBBLES
we study the evolution of a population of helium bubbles in an otherwise homogeneous medium. 2. BASIC ASSUMPTIONS AND EQUATIONS
The mean gas concentration in the matrix, Cg, is determined by gas conservation. Assuming that during coarsening no gain or loss of gas atoms occurs and that dCg/dt is small we obtain from equation (1) (d/dt)Y-,iN ~ = Zl4nrDg [•g -- C s (rl)] ~ 0
In simulating gas bubble coarsening in metals we follow the basic assumptions made in the LSW approach: (1) The parent phase forms an ideal solution of gas; (2) the fraction of gas in solution is negligible compared with the fraction in bubbles; (3) the volume fraction occupied by bubbles is small such that a diffusional mean field approach may be used; (4) the bubbles are spherical; (5) the pressure within the bubbles is in equilibrium with the thermal equilibrium vacancy concentration. On the basis of these assumptions, the diffusion controlled rate of change of the number of gas atoms or vacancies in a given bubble of radius r may be written as d(Ng.v)/dt = 4nr O,.v [Cg.v - C,,v(r)]/~.
(1)
The suffixes g, v label gas atoms or vacancies, respectively; Ds, v are the corresponding diffusivities in the matrix; Cg, v are the mean field concentrations; Cg.v (r) are the concentrations close to the bubble surface (in units of particles per matrix atoms) and is the matrix atomic volume. The local concentrations are given by Cs, v = e x p ( - Gg, v / k T + pg, v/kT)
(2)
where G~.,v are the Gibbs free enthalpies of solution, #g,v are the chemical potentials of gas atoms and vacancies and k T has the usual meaning. The chemical potentials #g,v=(OFs/c~Ng.v)~v.~ may be derived from the bubble formation free energy FB = FG + ?S, where FG is the free energy of the gas phase, S is the total bubble surface and 7 is the specific surface free energy. Disregarding elastic effects, we may write for the vacancy chemical potential close to the surface of a bubble I.tv = (~FB/~N~)~¢, =
tl(~FB/dr') = t l ( - p
+ 27 /r).
(3)
V represents the bubble volume and p the gas pressure inside the bubble. Using equation (3) in equation (2) we find Cv = C p e x p [ - (p - 27/r)f~/KT].
(4)
At low gas concentrations at which the sink strength of the bubble structure is small compared to the sink strength of the vacancy sources and at high temperatures at which C~ ~ C~ is readily established everywhere the pressure inside a bubble is always close to its equilibrium value, p = 2?/r. In this case, which we shall assume in the following, the vacancy fluxes are completely linked with the gas fluxes. Otherwise, the bubbles become overpressurized and a different treatment is required [11].
(5)
In the above expression, the sum runs over all bubbles. For a homogeneous concentration field of dissolved gas atoms equations (2) and (4) lead to Cg = e x p [ - G ~ k T ] ( r exp[#8/kT])/
(6)
where the brackets denote ensemble averaging. The number of gas atoms inside a given bubble may be correlated with the size dependent pressure, p(r), by use of an appropriate equation of state N s = n(p = 27/r, T ) V
(7)
where n is the number density of gas atoms in the bubble. Following the procedure outlined in Ref. [6] we condense the quantities characterizing the parent phase in a reduced time t* = (3/f~) (2?/kT)-2Ogt e x p [ - G ~ / k T ]
(8)
and account for the gas equation of state in the reduced variables N* = n-2 z-3
=
3/4n(27/kT)-3N
r* = (nz) -1 = (2? /kT) -1 r.
(9) (10)
z(n) = p / ( n k T ) is the gas compressibility factor. Introducing these reduced variables (t* and r* have the dimensions of a volume, N* has the dimension of a square of a volume) into equations (1) and (6) we obtain a "master equation" for Ostwald ripening of equilibrium bubbles dN*/dt* = r*Acg
(11)
Ac8 = [(r exp(#~(r)/kT)>/(r>] -
exp(/~g(r)/k T).
(12)
The ensemble averages introduced by equation (l 2) renders equation (11) an integro-differential equation. The solution of the latter is complicated when a real gas equation of state is used instead of the ideal gas law.
3. GAS EQUATION OF STATE The behaviour of a real gas is best described by the compressibility factor, z = p / ( n k T ) , which has a value of 1 at low densities (ideal gas behaviour) but increases continuously with increasing n. For hard sphere equations of state such as the simplified Van der Waals or Carnaham-Starling equations, z becomes formally infinite at closest packing. It has, however, turned out that at high densities the repulsive atomic potentials are not well described by hard spheres but are relatively soft, in particular for gases of low atomic number such as helium.
FICHTNER et al.: OSTWALD RIPENING OF GAS BUBBLES To our knowledge, bubble coarsening in metals is experimentally best investigated for the inert gas helium [9, 10]. Taking this as a respresentative example we use the semi-empirical high density equation of state given in Ref. [6] for 4He
1847
Pressure p (5Po) 107
101 I
100 '
'
106
a - 600 k
i0s'
b-lOOOk
10-1
I
'
.'~
... j
lO"1oc-l+ook .f
z = (1 - p) (l + p - 2p 2) + (1 - p)2pB/v
I
+ (3 - 2p)p2zl + (1 - p ) p 2 z ; v where
"~ 10z
p =v,V
101
v = 56T-1/+exp[-O.145T -1/4) in units of/~3
100
Zl = 0.1225 v T °-~55
z~v ~ - 5 0 . The chemical potential follows from /~ = f + p v , w h e r e f i s the free energy per atom ( f = S - P dv), as
#g. k T =/a,a - (27 - 16p)p2/6 + (6 - 9p + 4p2)B/3v + (27 - 16p)p2zl/6 + (9 - 8p)p2z2v/6 where #id is the chemical potential for ideal gas. The gas equation of state affects the O R kinetics in two ways: via the dissolved gas concentration Cs(r ) = exp[l~s(r)/kT] and via the relation between bubble radius and corresponding content of gas atoms N = (4n/3)r3n. Figures 1 and 2 illustrate the behaviour of cg (r) and N as a function of r for both, ideal gas and helium as a real gas in a metallic matrix with 7 = 2 N/m. These large differences, in particular in Fig. 1, are expected to manifest themselves in the coarsening kinetics.
107 105
Pressure p (GPo) 10o 10"1 = i
OK'~K6OOK 100
=t. 101 x
--
rear gas ideot gns
\
10-1
~
~
10-3 10-5 10-1
,
t
100
,
,
i
101
10-1
-
-
,
,
l
100
rear gas
,dmt gas
,
,
I
,
101
,
102
Radius r (nm) Fig. 2. Number density of He atoms in equilibrium bubbles at different temperatures (600 ~< T ~< 1400 K) as a function of the bubble radius r or internal pressure p = 27/r for a metallic matrix with V = 2 N/m.
B = 170 T -v3 - 1750/T in units of A 3
101 \
'i :"~"
:..:.. . ] "
o..
,
10z
Radius r (nm) Fig. 1. Dissolved He concentration cg = exp(#ne/kT ) around an equilibrium bubble as a function of the bubble radius r or internal pressure p for a typical metallic matrix (7 = 2 N/m) at temperatures T in the range 600 ~< T ~< 1400 K.
4. NUMERICAL ANALYSIS An analytical analysis of the integro-differential equation (11) describing O R kinetics of real gas bubbles is difficult [6]. In the following we simulate the evolution of a bubble size distribution (BSD) starting from a given initial distribution. The results are considered to provide guide-lines for more detailed analytical approximations. The numerical analysis starts from an appropriate BSD defined by 6000 bubbles. F o r this, the mean field concentration is calculated by use of equations (6) or (12). Then, each bubble is allowed to change size according to equation (11) during a small increment of time At*. To avoid substantial changes in the mean field during At*, the length of the latter is chosen such that only the smallest member of the bubble population is lost. Then, the procedure is repeated. When the number of bubbles has decreased to 4000 the population is increased to the initial number of 6000. In order to repopulate, we first construct the BSD using about 100 points and then rebuild the bubble population in such a way that bubbles with the same size are not allowed, but the shape of the distribution is maintained. With this scheme it is possible to account for large variations in mean radius while maintaining a good definition of the BSD and a reasonable numerical efficiency. Furthermore, this procedure also stepwise removes artificial structures in the size distribution associated with the numerical discretization. On the other hand, the system of equations which we use to describe the evolution of bubble populations is, strictly spoken, deterministic. In a real physical situation stochastic fluctuations will always occur. In the LSW approach the selected analytical solution is that which is stable against fluctuations. In a numerical stimulation study some fluctuations are actually required to obtain reasonable convergence to
FICHTNER et al.: OSTWALD RIPENING OF GAS BUBBLES
1848
such a solution. Our repopulation scheme, along with some numerical inaccuracy, provide the necessary fluctuations. Within the mean field approach spatial correlations are not accounted for. These would manifest themselves in a broadening of the BSD and an increase of the amplitude of the mean radius growth law, but would not affect its logarithmic slope [12]. Spatial correlation between particles in a homogeneous system becomes increasingly important for increasing particle number density and, in general, becomes noticeable for volume fractions above 5%. We have tested our simulation code by applying it to the ideal gas case for which the stable asymptotic quasi-steady-state behaviour has been analysed analytically previously [5].
Figure 3(a) shows a double logarithmic plot of the mean radius rm as a function of time obtained by starting from the analytical quasi-steady-state BSD [5]. Because of the finite value of the initial r m at t = 0, the steady state behaviour becomes evident only after an apparent transient. This effect can be removed by plotting r mz_ rm2 (t = 0) [Fig. 3(b)]. Figure 3(b) shows clearly that the simulated values follow the analytically predicted behaviour characterized by a slope dlog(rm)/ dlog(t) = 0.5. G o o d agreement between simulated and analytical results is also found for the evolution of the BSD as illustrated in Fig. 4, where the histogram points obtained from the simulations and the analytical BSD (continuous line) are presented. 5. RESULTS 5.1. Ostwald ripening under real gas behaviour
f(sl 10-1
100
101
10z
103
I
I
I
I
I
I
,
/
a)
10z
/"
/ .~
To illustrate the effect of the real gas (RG) behaviour at high gas densities we start from an
10~'
fo-*.
ro=lnm / \
I0 z
//
/
I01 =_E
II
100 I
I
I
I
10 3
10a
10 5
106
107
I0s
I0 s
101
10z
103
/
f
10 ~
Ii
\
b)
10~ 103
\
10~
i0 3
ew
/
101
"f i0 z
,
I
101
/ 10 0
rm:17.6nm /
10L,
b)
o
%
210
I
~o
•~
2, 1.6
0./,
t-
t(s) 100
. . . . . . 0.8 1.2
0
t" (A3)
I
\
o
I
10-1
/
/
101 ~ + ~
o,
t:0 /°/ \
E
J
S j
~0
,
,
i
,
,
J
,
I
J
J
,
io~._
I
i
/, 8 12 16 20 2/, 28 32
100 10-I
I
i
103
10~
i
I
I
I0s
106
107
I0e
t" Fig. 3. (a) Temporal evolution of the mean radius r,~ expressed in reduced variables r* [equation (10)] and t* [equation (8)] assuming ideal gas behaviour and a typical metallic matrix 0' = 2N/m) at T = 1123 K. The crosses corresponds to the calculated values which follows the trend suggested by the continuous line. An arbitrary choice of parameters D m = 3.56 I0 -t°m2s, f l = 10.95,~3 and G~ = 2.5 eV renders the time scale on the top in units of seconds. (b) r .2 (t*) - r*2(0) vs t* corresponding to the data in (a). The simulated values denoted by + lie in a straight line showing that the curvature in Fig. 3(a) is due to the finite value rm(0) = I nm. In addition, the slope of the line corresponds to the analytical value of 1 for such a plot.
J
\
o/,° "fl
,
,
,
,
,
,
,
i
,
,
,o~
i
i
0 gO 80 1201602002/,0280320
r(nm) Fig. 4. Comparison between the analytical bubble size distribution (continuous line) and the simulated results (O) for the ideal gas case. The good agreement between the curves observed over a large interval of mean radius r= show that the simulation code is working properly.
FICHTNER et al.:
OSTWALD RIPENING OF GAS BUBBLES
arbitrary distribution of an approximately gaussian shape with rm = 0 . 2 5 n m and standard deviation tr = 0.03 n m [see Fig. 6(a)]. The temporal evolution of r m for both, real and ideal gas, are presented in Fig. 5 for two temperatures (973 and 1273 K) but the same gas dissociation energy (3.5 eV), and for a typical metallic matrix 0' = 2 N/m). Obviously the R G behaviour results in a much faster initial growth. With increasing time, however, the growth rate slows down and the curves converge above 60 nm. More precisely, the logarithmic derivative dlog(rm)/dlog(t) for the R G case continuously increases from ~ 0.1 at short times, and converges to 0.5 at long times. Because of the finite value of r m (t = 0), precise values of the slope cannot be obtained without the knowledge of the growth law, as was done for the I G case [see Fig. 3(a) and (b)]. On the other hand, the curvature in r m (t) obviously means that the R G behaviour cannot be represented by a unique power law. Similar as the growth rate, the BSD is also changing continuously. This is illustrated in Figs 6 and 7 for the low and high temperatures cases, respectively. Starting from the arbitrary BSD [Fig. 6(a)], the R G - B S D quickly changes into a shape characterized by a short tail towards smaller bubbles and a relatively sharp cut towards larger bubbles [Fig. 6(b)]. With increasing time [Fig. 6(c-d)], the RG-BSD becomes continuously broader but still much sharper than an IG-BSD with the same r m and n u m b e r of bubbles. Figure 7 illustrates how the R G - B S D approaches the IG-BSD shape. However, even for rm-~ 60 n m [Fig. 7(d)] the R G behaviour still influences the distribution shape. To check whether the arbitrary choice of the initial BSD in the above simulations has influenced the evolution of the RG-BSD, we also started the simulations from the analytical quasi-steady-state BSD for ideal gas as in Fig. 4 (rm = 1 nm), which is much
- -
102
I
I
I
x
reaL gas 1223K
* o
reaL gas 923K ,dealgos 1223K
+
IdeaL gas 923K
I
I
i
,
,
i
,
,
,
i
,
,
'=0
,
a)
rm= 0.25nm
1849
!r ~ i
,
,
,
,
,
'
'
r~ =2Anm t =1oo /,~,,
'
'
c)
.=~ 01 0.2 0.3 rm=lnm ~ t=l /',
"~
1 2 015 b) rm= 6.1nm f = 50OO
0./,
3
/* d)
/I 0./*
0.8
1.2
r (nm)
1.6
20 0
2
~
6
8
10
r (nm)
12
Fig. 6. Time evolution of the bubble size distribution (BSD)
assuming real gas behaviour and a arbitrary initial BSD showed in (a) for the T = 9 2 3 K case in Fig. 5. The continuous line corresponds to a quasi-steady-state ideal gas BSD with the same rm and number of bubbles. The real gas BSD are much sharper than the corresponding ideal gas BSD and are characterized by a short tail towards small bubbles and a relatively sharp cut at the large bubble side. broader than the characteristic R G - B S D for the same rm [see Fig. 6(b)]. This is illustrated in Fig. 8. After a short transient (not shown), the R G - B S D quickly recovers its characteristic shape [Fig. 8(b)] and evolves towards the IG-BSD shape in a very similar way as in the previous cases (see Figs 6 and 7). Hence, we may conclude that the characteristic shape of the RG-BSD, as discussed above, reflects a property of the bubble system which is determined by the real gas behaviour. We emphasize that the bubble size range commonly considered in Transmission Electron Microscopy (TEM) is between 1 and 50nm. The above results show significant quantitative discrepancies between I G and R G behaviour in this size interval.
I
=
j
/
"
',
f b 0 ~u
J,, ........... •
10-2
10-1
___l
100
2
~
rm=13nm t = 50
6
8
8
10
b) /'~',
16
24
32
rm=6Onm t = 2000
~0 d)
J l__
I
101
102
_
I
_
103
I
10~
105
f ( a r b i t r a r y unifs)
Fig. 5. Time evolution of the mean radius rm for ideal and real gas behaviour assuming the same matrix properties. The real gas system shows a much faster initial growth but converges to the ideal gas behaviour at large rm.
4
8
12
r (nm)
16
20
2t.O
20
z.O 60
80
100 120
r (nm)
Fig. 7. Time evolution of the bubble size distribution (BSD) assuming real gas behaviour for the T = 1223 K case in Fig. 5. This figure illustrates the convergence of the real gas BSD to the ideal gas shape (continuous curves).
1850
FICHTNER et aL:
OSTWALD RIPENING OF GAS BUBBLES
rm=lnm f=O
r m= 16.8 nm f = 100
o)
d)
,~
/'\ ./', .... \, 0 03, 0.8 1.2 1.6
21o 0 L, 8 12 16 20 N 28 32
g
rm=3.Snm f:l
m 0
1
2
3
r m : 7./+ nm
f = 10
0
2
,d~ "
~
b)
!
L, 5
6
7 c)
///~\
6 8 10 12 r(nm)
I r m= 32.6nm f =500
0
10 20
e) ~-
30
I.~. rn~= 60"Snrn f =2000
0- 20 ~
~0
i\ 50
60' f)
~-
100 120
60 80 r(nm)
Fig. 8. Time evolution of the real gas bubble size distributions (BSD) starting with a BSD corresponding to the ideal gas case (represented by continuous line). The real gas BSD quickly assumes its characteristic shape and evolves in the same way as in the cases illustrated in Figs 6 and 7.
5.2. M a s t e r curves
In considering gas diffusivity and solubility on the matrix in a reduced time t*, we have scaled out the quantitites which are most sensitive to temperature. The effect of temperature on bubble coarsening via the gas equation of state is much weaker. This is illustrated in Fig. 9, which shows the same data presented in Fig. 5 in terms of reduced time t* [equation (8)] and radius r* [equation (10)]. The shape of the BSD also follow a general behaviour depending mainly on the mean radius but only weakly on the temperature. Figure I0 illustrates this point showing the BSD standard deviation (or) as a function of r= from the cases presented in Fig. 5 (IG and RG at two temperatures), for the simulations starting with the IG-BSD (Fig. 8) and for an empirical distribution to be discussed in next section.
103 x reaL gas 1223K * reaL gas 923K 10 2
/
/
102
o ideat ga, l m K
lo1 =G 10 100 _ . . . . . . . . . . . ;l
10-1
I
I
101
I
I
103 t" (~3)
I
I
10s
I
I
107
Fig. 9. Time evolution of the mean radius rmin terms of the reduced variables r~*[equation (10)] and t* [equation (8)] for the same data set as in Fig. 5. This figure illustrates the scaling provided by such reduced variables.
FICHTNER et al.: OSTWALD RIPENING OF GAS BUBBLES Because the quasi-steady-state IG-BSD is self-similar, tr/rm is independent of r~. For RG, the characteristic shape of the distribution quickly achieved for the cases where the simulations started with a small rm (e.g. as in Fig. 5) seems to best represent a quasisteady-state R G behaviour, where the broadening of the RG-BSD, parametrized in terms of tr/rm, increases quasi linearly with Iog(rm) in the range rm ~< 100 nm. Deviations from the quasi-steady-state behaviour and their dependence on the initial shape of the BSD, for both ideal and real gas, are also illustrated in Fig. 10.
The initial BSD is determined by the details of the nucleation kinetics and/or by the migration and coalescence of small bubbles. Therefore, depending on the situation, different initial BSD can be expected. In this section we try to evaluate the influence of the initial BSD shape on the temporal evolution of the mean radius, rm, and its standard deviation, a, by performing the simulations starting from different initial BSD. Figure 11 shows a comparison of rm (t) curves for three cases: LSW type [see Fig. 8(a)], gaussian distribution (r~ = 1 nm and tr = 0.2 nm) and an empirical distribution which has a tail towards large bubbles [see Fig. 12(a)]. There exists no significant differences between these curves, showing that the master curve presented in Fig. 9 has a rather general validity. The observed behaviour in a/rm also shows a convergence to a general behaviour (see Fig. 10), but obviously strongly depends on the initial shape of the distribution. The temporal evolution of the empirical distribution mentioned above is shown in Fig. 12. The tail towards large radii quickly disappears giving rise to
0.38
I
l
0.3/+ 0.30
~
0,26 0.22 0.18 real gas
0.1/+ 0.10 10-I
,
I
i
100 Mean
i
I
101
,
,
10 2
Radius rm (nm)
Fig. 10. Evolution of the standard deviation of the bubble size distributions, ¢r, normalized to the corresponding mean radius rm as a function of r m. The symbols + and * corresponds to the cases pre~nted in Fig. 5, * refers to the case of Fig. 12, x and • refers to the real gas simulations starting with the ideal gas BSD as in Fig. 8, at two different temperatures and O and • corresponds to pure ideal gas cases.
t (arbitrary units} 10o 101 102
10-1 [
I
I
103
I
r
,o2 T:l123K
1
/[
/"
+ ,sw
~
* GAUSS
1
.-'/
I i --
101
t 10 2
5.3. Transient behaviour
1851
i
J
J
103
10~
I0s
t"
,J10 °
106
(~3)
Fig. 1I. Evolution of r* vs t* showing transient behaviours introduced by starting the simulations from different bubble size distributions (see text).
a "standard" RG-BSD showing a characteristic sharp cut at large bubble side and a tail to the small bubble side. For comparison, the quasi-steady-state IG-BSD normalized to the same r m and the same number of bubbles is also shown. We conclude that transients associated with initial shape of the BSD are not of fundamental importance for the Ostwald ripening of equilibrium gas bubbles, although the distribution may present different shapes during a transient period. 6. CONCLUSIONS The simulation approach presented in this paper is suited to describe Ostwald ripening kinetics of bubbles containing non-ideally behaving gas at low gas concentration and at high temperature at which the pressure inside the bubbles may be assumed to be close to its thermal equilibrium value. In comparison with ideal gas, the results for real gas show large differences for mean radii rm ~< 50 nm, which is the range of TEM investigations. In this range, the mean radii reached within a given annealing time are substantially larger than for ideal gas. The growth law cannot be described by a single power law. In a double-logarithmic plot of the mean radius vs time the slope increases continuously from about 0.1 to the asymptotic ideal gas value of 0.5. The bubble size distributions for the real gas case are found to be significantly narrower than for the ideal gas case. The temporal evolution of the mean radius is only initially and then only weakly dependent upon the assumed shape of the initial size distribution. The time-dependence of the mean radius can, therefore, be condensed into a kind of master curve which is independent of the gas dissociation energy and only weakly dependent upon temperature. An approximate fitting procedure on the basis of the presented results may be used to analyse bubble
1852
FICHTNER et al.: OSTWALD RIPENING OF GAS BUBBLES
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