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On the mechanism of the inhibition of diffusional creep by second phase particles
337
Materials Science and Engineering, 11 (1973) 337-343 © American Society for Metals, Metals Park, Ohio, and Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
On the Mechanism of the Inhibition of Diffusionai Creep by Second Phase Particles B. B U R T O N C.E.G.B., Berkeley Nuclear Laboratories, Berkeley, Glos. GL13 9PB (Gt. Britain) (Received September 11, 1972)
Summary* Diffusional creep can be'inhibited by the presence of second phase particles on grain boundaries. This inhibition is discussed in terms oJ the limitation of grain boundaries as vacancy sources and sinks by particles. I f vacancies condense on particle-free regions of a boundary then stress is concentrated at particles. This stress concentration ean be relieved if defect loops nucleate and grow in the particle/ matrix interface. Since the ease of nucleation will depend upon the line energy of the defect and this in turn will depend on the shear modulus of the particle, it is predicted that elastically hard particles will be most effective in inhibiting diffusional creep. This model is developed and its predictions are compared with reported experimental observations.
1. INTRODUCTION
It is well established 1'2 that the deformation of pure metals at low stresses and elevated temperatures can occur by the stress directed diffusion of vacancies. Lattice or grain boundary diffusion can predominate depending on grain size and test temperature and the corresponding creep rates and ~gb have been calculated 3- 5 to be: -
Baf2D d2k~
(1)
and ~gb --
B' a f2 wDgb d a kT '
(2)
where a is the applied stress, t2 the atomic volume, d the grain size, w the grain boundary width, D the lattice and Dg b the grain boundary diffusion coefficient and kT has the usual meaning. B and B' are * R6sum6 en franqais/l la fin de l'article. Deutsche Zusammenfassung am SchluB des Artikels.
numerical constants which take the values ~ 10 and ~ 150/n respectively. Equations (1) and (2) predict creep rates of material of similar grain size to vary only as the diffusion coefficient. However, recent experiments have shown 6- l o that diffusion creep rates can be significantly inhibited by the presence of certain second phase particles. Since the presence of particles should not affect diffusion rates their influence is probably best interpreted in terms of some limitation of grain boundaries as vacancy sources and sinks 8'11A2, thus invalidating one of the basic assumptions in the derivation of eqns. (1) and (2). In reported studies of the influence of particles two outstanding features are recurrent. Firstly, the linear relationship between ~ and a is lost and appears to be replaced by a relationship of the form ~ oc a - a o where ao is a threshold stress below which creep does not occur, ao appears to increase linearly with the volume fraction of second phase 9'1°. Secondly, in many cases creep rate progressively decreases with strain. Since particles act as inert markers during diffusion creep they tend to collect on boundaries parallel to the tensile stress axis and this has been observed metaUographically 8'13. The progressive decrease in creep rate appears to be associated with this particle collection emphasising that grain boundary particles are most important in inhibiting creep. The present paper considers the role of particles in grain boundaries and suggests ways in which they may influence creep by affecting vacancy condensation and evaporation at boundaries. Only condensation is considered in detail but results obtained are equally applicable to evaporation. 2. VACANCY CONDENSATION BOUNDARIES
ON
PARTICLE-FREE
A grain boundary is a region of high atomic disorder and can act as an efficient vacancy sink.
338
B. B U R T O N
Ashby ~ envisages the condensation process occurring by the motion of a line defect through the boundary analogous to a dislocation having a component of its Burger's vector perpendicular to the boundary. The description of this defect must be somewhat vague since the precise atomic configuration at the boundary is unknown. However, a reasonable approximation is a process such as that depicted schematically in Fig. 1 where the line dividing "collapsed" from "non-collapsed" regions represents the defect. It is reasonable to assign a line energy to this defect proportional to/d) 2 where ~t is the shear modulus and b the atomic size.
f
gb
I I I I I I I I I"
-gb
!
I
collapsed
I!
Fig. 1. Condensation of vacancies on a grain boundary under a compressive stress.
For continuing creep to occur requires continual vacancy condensation and thus for these line defects or "grain boundary dislocations" to multiply. This process was considered in more detail in a previous paper ~4. It was concluded that while in principle the multiplication process could occur by the classical nucleation of vacancy loops the stresses required for this process are too high. The problem of twodimensional nucleation is overcome however if some continuous source such as a spiral defect exists. Defects of this type can absorb vacancies even at low supersaturations and thus a grain boundary free from particles can continuously absorb vacancies even at low stresses.
by bowing out between them. The stress required for such a process is given by Ashby 11 as
where E is the line energy of the defect and 2 the interparticle spacing. If the particle/matrix interface does not act as a vacancy sink a loop must be left around particles in the wake of the migrating defect. This is depicted in Fig. 2. The effect of this process continuing will be to relax the applied stress on particle-free regions of the boundary and concentrate it at particles. It is evident that if the particle/ matrix interface remains inactive as a sink then a limiting creep strain will rapidly be approached. (Since vacancy condensation is only being accommodated elastically this limiting strain will be of the order of the elastic strain of the specimen for that particular applied stress.) For creep to continue requires the stress concentration to be relaxed. This can be achieved by the nucleation and growth of defect loops in the particle/ matrix interface if the stress concentration is sufficiently high. These loops will annihilate those initially left in the grain boundary and so permit steady
IIIIIIIli~l l l l ]
Fig. 2. The nucleation of a defect loop on a grain boundary particle.
state deformation. This process is also depicted in Fig. 2. (An alternative way in which the stress can be relaxed is to punch out dislocation loops into the adjacent grain 15. However, lattice defects have a higher line energy than interfacial defects and this process may be more difficult.) If the stress concentrated at particles is ap, the energy to nucleate a loop in the particle/matrix interface can be written : AG = 2rcrU-
3. VACANCY C O N D E N S A T I O N ON B O U N D A R I E S CONT A I N I N G PARTICLES
When vacancies ~/re condensed on to a boundary containing particles movement of the line defect in the boundary will soon become impeded by the particles. If the vacancy supersaturation is sufficiently high the line defect may by-pass the particles
(3)
a]} = 2 E / b 2 ,
rcr 2 try b .
(4)
The first term represents the elastic energy of the loop where U is the line energy and the second term is the work done by the applied stress on creation of the loop. Putting ~ A G / t 3 r = 0 the critical radius of curvature above which nucleated loops will grow spontaneously can be calculated to be" r e = U/trvb
(5)
339
I N H I B I T I O N OF D I F F U S I O N A L C R E E P
and the activation energy for nucleation of loops of size r~ is: AGc =
xU2/apb.
(6)
The rate of nucleation of these loops per unit area of particle is thus : (]Q)rc = V exp - rcU2/apbkT
(7)
where v is some frequency factor, and the time taken to nucleate one loop per particle is therefore : tN = (rcr2 v exp -- rcU2/apbkT) -1 ,
(8)
where rp is the particle radius. For steady state conditions one loop must be nucleated on each particle in the time taken to condense one layer of vacancies on the adjacent particle-free region of boundary. Now ~d=jvf2 where j~ is the vacancy flux to the grain boundary and taking ~ from eqn. (1) gives the time taken to condense one layer of vacancies on a boundary under a stress o b to be •
(13)
= ( a - Vap)JvfJ/d,
where the factor (1 - V)- 1 is taken to be unity and is omitted. At low stresses Op --~ ff/V and ab "" 0 demonstrating that nearly all the applied stress is concentrated at particles, the rate of nucleation of loops is extremely small and ~ is effectively zero. At higher stresses more rapid nucleation is possible, ab becomes nonzero, (a/V) > ap > a and ~ becomes significant. On further increasing stress creep is not limited by nucleation and the applied stress is more evenly distributed. The dependence of creep rate upon applied stress is shown in Fig. 3 for the value of ........
I
........
I
........
I
.......
1.0
(9)
tc = d k T / B D b 2 ab,
0.1
thus : Z
nrpZv exp - T t U 2 / a p b k T = B D b 2 a b / d k T .
(10)
Now ap and ab are related to the applied stress by the equation : O p / r r p2 H A -J- O'b(1 - - Xrp2 HA) :
19,
0.01
(11)
where n A= V/Izr 2 is the number of particles per unit area of boundary and V is the volume fraction. It is possible to solve these two equations for ap and ab for various values of U and V if the appropriate constants are known. For any particular system the only other constant which is not known precisely is the frequency factor v. However, this may be assessed approximately as follows. Since the rate of nucleation is equal to the product of the number of critical loops per unit area and the rate of arrival of new vacancies per loop: ,,, 2xr c bjv (Cv/b 2) exp - AGc/k T ,
,~
(12)
where 2rtrcb is the area available per loop for accepting vacancies,jv ,-, 101 s m - 2 s- 1 under typical creep conditions and Cv "~ 10-* is the vacancy concentration. This gives v ~ 2r~rcbjv(Cv/b 2) ~ 1018 m - 2 s-1. Using this value for v and values of the other constants appropriate to a previous study on Cu/A1203 alloys 1° (namely d = 13 #m, T = 1173°K, rp=0.3 ktm and D = 3 x 10 -1'* m2/s) eqns. (10) and (11) have been solved numerically for ap and a b and the creep rate calculated from the equation:
0.001
~=,6= ........
Iv=,d ~
I ........ I ........ I 0.01 0.1 '.0 STRESS. M N I m 2
Fig. 3. Creep rate-stress relationship calculated from eqn. (13) for various volume fractions of second phase. The creep rate is normalised by dividing by the proportionality constant in the N a b a r r o - H e r r i n g equation.
U = 1 0 pN and for V = 1 0 - I , 10 -2 and 10 -3. In each case the creep rate drops sharply at low stresses to very low values becoming effectively zero below a certain stress, and the form of the curves is a very good approximation to the simplified expression : BI2D - dZkT (a-go),
(14)
where ao depends linearly upon V. The creep rate-stress relationship has also been computed for various values of n U 2 / b k T and the
340
B. BURTON
form of the relationship is similar to the curves in Fig. 3. The dependence between a o and rcU2/bkT is very close to linear with the proportionality constant being ,-~ V/55. It should be noted that the value of cro is very insensitive to the value of the frequency factor v (and also to rp, D) since, as in all nucleation processes, the rate of nucleation is a very strong function of the driving force, in this case av and the value of the exponential in eqn. (10) completely dominate the pre-exponential factors. Indeed, changing the pre-exponential by ten orders of magnitude shifts the position of the curves.in Fig. 3 by less than a factor of 2 thus justifying the use of the very approximate value of v. Thus in order to condense vacancies on a boundary containing particles a critical stress has to be exceeded given by: (ao),o = zcU2 V/55bkT.
(N)rp oc exp - (2rCrp U - rcr2 trp b)/kT
(16)
and the corresponding threshold stress will be:
-~r2p-ff / V.
(17)
Thus for small rp, ao should be a function of rp and the overall dependence is shown in Fig. 4. It should
~r c
. .ix\. \
55kT/21rU
\
\
-" ~ "" (~.)rp PARTICLE
4. C O M P A R I S O N TIONS
WITH
EXPERIMENTAL
OBSERVA-
(15)
The model predicts rro to be essentially independent of the particle size since it only appears in the pre-exponential factor and this will be valid for relatively large particles. However, for very small particles the situation is different since rp may be less than the critical radius re. In this case the nucleation event can be achieved by nucleating sub-critical loops of r = rp at the rate given by :
((r°)'P ~- r.b
rp < 55 kT/zcU, ao decreases with rv in a manner described by eqn. (17). Eventually ao becomes zero at rp= 55 kT/2rrU when 2U/rvb= 55 kT/zcr2pb, thus indicating that very small particles should not inhibit vacancy condensation. This is typically when rp ~< 10 nm however and since for most systems rp is greater than this a o is expected to be essentially independent of rp. For still larger particles it is likely that the stress concentration can be relaxed more easily by the punching of dislocation loops x5 or other plastic flow processes and the present model will break down.
RADIUS
Fig. 4. Variation of the threshold stress for vacancy condensation with particle radius.
be noted that eqn. (17) reaches a maximum value at rv= 55 kT/lrU and eqns. (15) and (17) intersect at this maximum. For rp > 55 kT/nU, ao is independent of rp and is given by eqn. (15); and for
A successful model which describes the influence of particles on diffusion creep must explain three main features. Firstly, why some particles have an effect and others apparently do not. Secondly, it must explain the origin of threshold stress and the parameters upon which this depends and finally how the creep rate diminishes as particles accumulate on longitudinal boundaries. 4.1. The effectiveness of different particles Equation (7) predicts the ease of nucleation of defect loops on particles to depend strongly upon U the line energy of the defect in the particle/matrix interface. Since the energy of a line defect in a grain boundary is proportional t o #m b2 where #m is the shear modulus of the matrix then the energy in a particle/matrix interface can probably be written Uoc½(/~m+#p ) where ½(#m+#v) is some average shear modulus and #p is the shear modulus of the particle. Thus in cases where #p > #m, U will be large and inhibition is expected. When #p < #m however nucleation should be easy and the material behave as though no particles were present. Reported data on two-phase systems are represented on a volume fraction-relative melting point diagram in Fig. 5 where the subscripts p and m refer to particle and matrix respectively. (~ values are not available for some of the systems and so the ratio of the melting temperatures is plotted instead. However, since # oc TM this ratio is equivalent to /~p/#~.) The vertical bars on the points represent the range of compositions studied and the data are taken from the literature as follows: Mg/ZrH2 6's, M g / M g O 7'16, M g / ~ M n X 7 J s, A u / A 1 2 O s 9, M g / M g 2 -
341
INHIBITION OF DIFFUSIONAL CREEP Si 19, Mg/~Zr 19, Cu/A120310, Cu/Cu2O20, Cu/ GeO221, A1/Fe 22, CuZn/Pb and CuZn/voids 2a. It is clearly indicated in this Figure that the conditions for inhibition are large volume fractions of elastically hard (high TM) particles*. 4.2. The threshold stress for creep The theory in section 3 predicts that a critical stress has to be exceeded before a boundary can absorb (or emit) vacancies at a significant rate and above this stress creep rate should be given by the equation :
Bf2D (~ i -- d2kT
rcU2V ~ 55bkT/
~ sP .ot,~,ol~ o.I
,6s
cuz .b
~
(18)
DIFFUSION
~'Ai o CREEP if 2 a INHIBITED
~ \ x± eM~/.=Z, I "1. 1. o :uz,/vo,d, T ' ~ / . , M . , ,r.~ cutc~a°l o~ 1
.~
Mgl~CMn
~,/~,
o ,a"
.k I I
t 2
~,~go]; I 3
I
4
Fig. 5. Volume fraction-relative melting point diagram showing
the r6gimewhere diffusioncreep is inhibited. when lattice diffusion predominates and a corresponding equation can also be written for the grain boundary diffusion case. This type of creep equation has indeed been observed for many of the reported systems (Au/AI 203, Cu/Al2Os, Mg/Mg2Si, Mg/~Mn, Mg/aZr and Mg/ZrH2) although the variation of ao with the parameters, U, V and T has not been determined in most cases. The linear variation between a o and V has, however, been demonstrated for Au/A12 O3 and Cu/AI203 and the associated values for U calculated from eqn. (15) are 27 pN and 80 pN respectively. The smaller value for the gold alloy is to be expected since U oc (/am+/ap) and #Au
higher temperature and the temperature dependence of #m and #p will tend to increase the difference in the values for U. It should be noted that the values of U are in the range of 0.1-0.3 of the value of a line defect in the lattice (0.5 #b 2) and thus are of the correct order. This effect of temperature has also been noted in Mg/aMn by Jones is where a 0 again decreased when temperature was increased. Since ao oc U2/T the variation of a o with temperature should vary approximately in the same way as /a2/T. Shear moduli usually decrease by a factor of ~ 2 between 0.5 TMand the melting point and thus a0 is expected to decrease by a factor of ~ 8 over this same temperature range. 4.3. The decrease in creep rate with strain As particles accumulate on longitudinal boundaries the critical supersaturation for condensation on these boundaries must increase. Initially the number of particles per unit area of boundary is n A = V/~r 2 and the threshold stress for creep is that associated with this value of V. On transverse boundaries nA remains constant but on longitudinal boundaries nA increases with strain. Thus the threshold stress for creep will always be that one associated with the concentration of particles on longitudinal boundaries. The number of particles collected on a grain edge of area d 2 at a strain e is ed3nv where nv=3V/4nr 3 is the number of particles per unit volume. Thus the total number of particles per unit area of boundary at a strain e is: V ( 3d t (nA)~ = rcr---~ 1 + 4~rpe .
(19)
Thus the material behaves as though it contained a volume fraction V~=(nA)~nrp 2 of particles and so according to the present model ao will increase with strain in a manner described by the equation • rrU2 ( 3d ) ( a o ) - 55bkT 1 + ~ e Vo, (20) where V0 is the original volume fraction. The dependence of creep rate on creep strain, again taking the lattice diffuaion case, is then given by :
=~
cr
55bkr
1+ ~
Vo
(21)
and the rate of decrease of ~ with e is :
Oi ~3e --
(~_~) 2 DVo drp '
A \'~,/
where A = 3~BO/220b .~ 7 × 10-19 m 2.
(22)
342
B. BURTON
Thus in order for O~/~3e to be large Vo must be large or d and rp small (for a fixed temperature). However, the parameter which is more important in choosing alloys for technological applications is the limiting creep strain eL. This is the strain at which ~ 0 as a0 approaches a and can be calculated by putting rrU2
55bkT
(1
3d) + 4 r p e V°=a
case of the magnesium alloy where large particlefree zones were formed during creep on boundaries perpendicular to the stress axis, the faster creep of these "pure" zones may have contributed to overall extension. Since this contribution will increase with zone size and thus with strain this may well explain the non-linear behaviour in Fig. 6.
5. CONCLUSIONS
to be:
~55bkTa
4rp =
,- 6 V00
) 1
.
(23)
Now eL oc rp/d and thus the limiting creep strain is smaller for larger grain size material containing a constant volume fraction of small particles. U
~2 x uJ a:
~/
Cu/AI203
z
~
Z
r
2
H
2
4 6 8 CREEP STRAIN x I02
I0
Fig. 6. Variation of strain rate with creep strain for two alloys.
In order t o compare these predictions with experimental results the creep rate of Cu/l~/o A 1 2 0 3 1 ° and Mg/0.5~ ZrH2 s is plotted versus strain in Fig. 6. The values of eL are 0.03 and 0.11 respectively and the line energy U can be calculated from these values using eqn. (23). For the copper alloy putting rv=0.3 pm, d = 1 3 #m, a = l . 1 6 MN/m 2 and T = 1173°K gives U = 6 5 pN, a value close to that calculated earlier in subsection 4.2 from absolute measurements of go. For the magnesium alloy taking V=0.005, rp~-1 #m (by inspection from ref. 8, Fig. 1), T=673°K, d=100 /an and a = 1.1 MN/m 2 gives U - 3 0 pN again a value of the correct order. Thus in this respect behaviour is consistent with the accumulation of particles on longitudinal boundaries. However, Fig. 6 shows that 0~/ae is not constant as predicted but varies by a factor of ~ 2 for the magnesium alloy and ~ 3 for the copper alloy. Such curvature is not easily quantifiable but several possibilities exist. For instance, any primary creep contribution or "settling in" at the beginning of test tends to over-exaggerate the creep rate in the initial stages. Alternatively, in the
Vacancy condensation on grain boundaries requires the movement of line defects through the boundary. If no particles are present this movement is relatively easy and boundaries can act as vacancy sinks (and sources) at small supersaturations. When particles are present, movement of the line defect can be impeded and the defect must bow out in order to by-pass these particles. If the particle/ matrix interface is not a good vacancy sink then any continuing condensation on particle-free regions of the boundary rapidly leads to stress concentration at the particles. This stress concentration can be relaxed if defect loops nucleate and grow on the partiele/matrix interface. For elastically soft particles this process should be easy and these particles have a negligible effect on diffusional creep. For elastically hard particles the nucleation process may be difficult and a critical stress must then be exceeded before boundaries can absorb vacancies at a significant rate. Thus a corresponding threshold stress for diffusional creep exists. When the particle size is relatively large this threshold stress is independent of particle size but below a certain particles size it drops rapidly to zero, indicating that nucleation on very small particles is easy. The collection of particles on longitudinal boundaries during creep leads to a progressive increase in the threshold stress and causes creep rate to diminish with strain. Creep eventually stops when the stress required for condensation approaches the applied stress.
ACKNOWLEDGEMENTS
The author is grateful to Mr. I. G. Crossland, Dr. J. E. Harris, Mr. G. F. Hines, Dr. R. B. Jones of Berkeley Nuclear Laboratories and Mr. R. E. Lewis
INHIBITION OF DIFFUSIONAL CREEP
and Prof. G. W. Greenwood of Sheffield University for permission to quote their unpublished work and to Mr. M. V. Speight for useful discussion. This paper is published by permission of the Central Electricity Generating Board.
343
1 R. B. Jones, Nature, 207 (1965) 70. 2 H. Jones, Mater. Sei. Eng., 4 (1969) 106. 3 F. R. N. Nabarro, Rept. of Conf. on the Strength of Solids, 1948, The Physical Society, London, p. 75. 4 C. Herring, J. Appl. Phys., 21 (1950) 437. 5 R. L. Coble, J. Appl. Phys., 34 (1963) 1679. 6 P. Greenfield and W. Vickers, J. Nucl. Mater., 22 (1967) 77. 7 W. Vickers and P. Greenfield, J. Nucl. Mater., 27 (1968) 73. 8 J. E. Harris, R. B. Jones, G. W. Greenwood and M. J. Ward, J. Nuel. Mater., 29 (1969) 154.
9 F. K. Sautter and E. S. Chen, in C. S. Ansell (ed.), Oxide Dispersion Strengthening, Gordon and Breach, New York, 1968, p. 495. 10 B. Burton, Metal Sci. J., 5 (1971) 11. 11 M. F. Ashby, Scripta Met., 3 (1969) 837. 12 G. W. Greenwood, Scripta Met., 4 (1970) 171. 13 R. L. Squires, R. T. Weiner and M. Phillips, J. Nucl. Mater., 8 (1963) 77. 14 B. Burton, Mater. Sci. Eng., 10 (1972) 9. 15 J. E. Harris, to be published. 16 I. G. Crossland and R. B. Jones, personal communication. 17 R. B. Jones, in D. G. Brandon and A. Rosen (eds.), Quantitative Relation between Properties and Micro-Structure, Israel Universities Press, 1969, p. 343. 18 R. B. Jones, personal communication. 19 G. F. Hines, personal communication. 20 B. Burton, Ph. D. Thesis, University of Sheffield, 1969. 21 M. F. Ashby, Proc. 2nd Intern. Conf. on Strength of Metals and Alloys, Vol. II, Am. Soc. Metals, 1970. 22 B. Ya Pines and Ye Ye Badyan, Fiz. Metal. i Metalloved., 29 (1970) 847. 23 R. E. Lewis and G. W. Greenwood, personal communication.
MOcanisme d'inhibition du fluage-diffusion par des particules arune seconde phase
Zum Mechanismus der Behinderuny des Diffusionskriechens durch Teilchen einer zweiten Phase
Le fluage-diffusion peut ~tre inhib~ par la pr6sence de particules d'une seconde phase darts les joints de grains. L'auteur 6tudie ce ph6nom6ne en consid6rant les limitations apport6es par les particules au r61e de sources et de puits de lacunes jou6 par les joints de grains. Si les lacunes se condensent dans des zones du joint d6pourvues de particules, il en r6sulte une concentration de contrainte sur les particules. Cette concentration de contrainte peut &re relax6e par la cr6ation et la croissance de boucles de d6fauts ~t l'interface particule-matrice. Comme la facilit6 de formation de ces boucles d6pend de l'6nergie de ligne des d6fauts et que celle-ci d6pend du module de cisaillement de la particule, on peut pr6voir que les particules dures d'un point de vue 61astique seront les plus efficaces pour inhiber le fluage-diffusion. Cette id6e est d6velopp6 et les pr6dictions du module sont compar6es avec des observations exp6rimentales publi6es.
Diffusionskriechen kann durch die Gegenwart von Teilchen einer zweiten Phase an Korngrenzen behindert werden. Diese Behinderung wird unter dem Gesichtspunkt einer Einschr~nkung der Funktion der Korngrenzen als Leerstellenquelle und -senke durch die Teilchen diskutiert. Kondensieren Leerstellen in teilchenfreien Bereichen einer Korngrenze, so entsteht an den Teilchen eine Spannungskonzentration, die durch die Bildung und das Wachstum von Versetzungsringen in der TeilchenMatrix-Grenzfl~iche gemildert werden kann. Da die Keimbildungswahrscheinlichkeit von der Linienenergie des Defektes und diese wiederum vom Schermodul des Teilchens abh~ingt kann man vorhersagen, dab Diffusionskriechen durch elastisch harte Teilchen besonders wirkungsvoll behindert wird. Dieses Modell wird entwickelt und seine Vorhersagen werden mit ver6ffentlichten experimentellen Beobachtungen verglichen.
REFERENCES
Materials Science and Engineering, 11 (1973) 337-343 © American Society for Metals, Metals Park, Ohio, and Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
On the Mechanism of the Inhibition of Diffusionai Creep by Second Phase Particles B. B U R T O N C.E.G.B., Berkeley Nuclear Laboratories, Berkeley, Glos. GL13 9PB (Gt. Britain) (Received September 11, 1972)
Summary* Diffusional creep can be'inhibited by the presence of second phase particles on grain boundaries. This inhibition is discussed in terms oJ the limitation of grain boundaries as vacancy sources and sinks by particles. I f vacancies condense on particle-free regions of a boundary then stress is concentrated at particles. This stress concentration ean be relieved if defect loops nucleate and grow in the particle/ matrix interface. Since the ease of nucleation will depend upon the line energy of the defect and this in turn will depend on the shear modulus of the particle, it is predicted that elastically hard particles will be most effective in inhibiting diffusional creep. This model is developed and its predictions are compared with reported experimental observations.
1. INTRODUCTION
It is well established 1'2 that the deformation of pure metals at low stresses and elevated temperatures can occur by the stress directed diffusion of vacancies. Lattice or grain boundary diffusion can predominate depending on grain size and test temperature and the corresponding creep rates and ~gb have been calculated 3- 5 to be: -
Baf2D d2k~
(1)
and ~gb --
B' a f2 wDgb d a kT '
(2)
where a is the applied stress, t2 the atomic volume, d the grain size, w the grain boundary width, D the lattice and Dg b the grain boundary diffusion coefficient and kT has the usual meaning. B and B' are * R6sum6 en franqais/l la fin de l'article. Deutsche Zusammenfassung am SchluB des Artikels.
numerical constants which take the values ~ 10 and ~ 150/n respectively. Equations (1) and (2) predict creep rates of material of similar grain size to vary only as the diffusion coefficient. However, recent experiments have shown 6- l o that diffusion creep rates can be significantly inhibited by the presence of certain second phase particles. Since the presence of particles should not affect diffusion rates their influence is probably best interpreted in terms of some limitation of grain boundaries as vacancy sources and sinks 8'11A2, thus invalidating one of the basic assumptions in the derivation of eqns. (1) and (2). In reported studies of the influence of particles two outstanding features are recurrent. Firstly, the linear relationship between ~ and a is lost and appears to be replaced by a relationship of the form ~ oc a - a o where ao is a threshold stress below which creep does not occur, ao appears to increase linearly with the volume fraction of second phase 9'1°. Secondly, in many cases creep rate progressively decreases with strain. Since particles act as inert markers during diffusion creep they tend to collect on boundaries parallel to the tensile stress axis and this has been observed metaUographically 8'13. The progressive decrease in creep rate appears to be associated with this particle collection emphasising that grain boundary particles are most important in inhibiting creep. The present paper considers the role of particles in grain boundaries and suggests ways in which they may influence creep by affecting vacancy condensation and evaporation at boundaries. Only condensation is considered in detail but results obtained are equally applicable to evaporation. 2. VACANCY CONDENSATION BOUNDARIES
ON
PARTICLE-FREE
A grain boundary is a region of high atomic disorder and can act as an efficient vacancy sink.
338
B. B U R T O N
Ashby ~ envisages the condensation process occurring by the motion of a line defect through the boundary analogous to a dislocation having a component of its Burger's vector perpendicular to the boundary. The description of this defect must be somewhat vague since the precise atomic configuration at the boundary is unknown. However, a reasonable approximation is a process such as that depicted schematically in Fig. 1 where the line dividing "collapsed" from "non-collapsed" regions represents the defect. It is reasonable to assign a line energy to this defect proportional to/d) 2 where ~t is the shear modulus and b the atomic size.
f
gb
I I I I I I I I I"
-gb
!
I
collapsed
I!
Fig. 1. Condensation of vacancies on a grain boundary under a compressive stress.
For continuing creep to occur requires continual vacancy condensation and thus for these line defects or "grain boundary dislocations" to multiply. This process was considered in more detail in a previous paper ~4. It was concluded that while in principle the multiplication process could occur by the classical nucleation of vacancy loops the stresses required for this process are too high. The problem of twodimensional nucleation is overcome however if some continuous source such as a spiral defect exists. Defects of this type can absorb vacancies even at low supersaturations and thus a grain boundary free from particles can continuously absorb vacancies even at low stresses.
by bowing out between them. The stress required for such a process is given by Ashby 11 as
where E is the line energy of the defect and 2 the interparticle spacing. If the particle/matrix interface does not act as a vacancy sink a loop must be left around particles in the wake of the migrating defect. This is depicted in Fig. 2. The effect of this process continuing will be to relax the applied stress on particle-free regions of the boundary and concentrate it at particles. It is evident that if the particle/ matrix interface remains inactive as a sink then a limiting creep strain will rapidly be approached. (Since vacancy condensation is only being accommodated elastically this limiting strain will be of the order of the elastic strain of the specimen for that particular applied stress.) For creep to continue requires the stress concentration to be relaxed. This can be achieved by the nucleation and growth of defect loops in the particle/ matrix interface if the stress concentration is sufficiently high. These loops will annihilate those initially left in the grain boundary and so permit steady
IIIIIIIli~l l l l ]
Fig. 2. The nucleation of a defect loop on a grain boundary particle.
state deformation. This process is also depicted in Fig. 2. (An alternative way in which the stress can be relaxed is to punch out dislocation loops into the adjacent grain 15. However, lattice defects have a higher line energy than interfacial defects and this process may be more difficult.) If the stress concentrated at particles is ap, the energy to nucleate a loop in the particle/matrix interface can be written : AG = 2rcrU-
3. VACANCY C O N D E N S A T I O N ON B O U N D A R I E S CONT A I N I N G PARTICLES
When vacancies ~/re condensed on to a boundary containing particles movement of the line defect in the boundary will soon become impeded by the particles. If the vacancy supersaturation is sufficiently high the line defect may by-pass the particles
(3)
a]} = 2 E / b 2 ,
rcr 2 try b .
(4)
The first term represents the elastic energy of the loop where U is the line energy and the second term is the work done by the applied stress on creation of the loop. Putting ~ A G / t 3 r = 0 the critical radius of curvature above which nucleated loops will grow spontaneously can be calculated to be" r e = U/trvb
(5)
339
I N H I B I T I O N OF D I F F U S I O N A L C R E E P
and the activation energy for nucleation of loops of size r~ is: AGc =
xU2/apb.
(6)
The rate of nucleation of these loops per unit area of particle is thus : (]Q)rc = V exp - rcU2/apbkT
(7)
where v is some frequency factor, and the time taken to nucleate one loop per particle is therefore : tN = (rcr2 v exp -- rcU2/apbkT) -1 ,
(8)
where rp is the particle radius. For steady state conditions one loop must be nucleated on each particle in the time taken to condense one layer of vacancies on the adjacent particle-free region of boundary. Now ~d=jvf2 where j~ is the vacancy flux to the grain boundary and taking ~ from eqn. (1) gives the time taken to condense one layer of vacancies on a boundary under a stress o b to be •
(13)
= ( a - Vap)JvfJ/d,
where the factor (1 - V)- 1 is taken to be unity and is omitted. At low stresses Op --~ ff/V and ab "" 0 demonstrating that nearly all the applied stress is concentrated at particles, the rate of nucleation of loops is extremely small and ~ is effectively zero. At higher stresses more rapid nucleation is possible, ab becomes nonzero, (a/V) > ap > a and ~ becomes significant. On further increasing stress creep is not limited by nucleation and the applied stress is more evenly distributed. The dependence of creep rate upon applied stress is shown in Fig. 3 for the value of ........
I
........
I
........
I
.......
1.0
(9)
tc = d k T / B D b 2 ab,
0.1
thus : Z
nrpZv exp - T t U 2 / a p b k T = B D b 2 a b / d k T .
(10)
Now ap and ab are related to the applied stress by the equation : O p / r r p2 H A -J- O'b(1 - - Xrp2 HA) :
19,
0.01
(11)
where n A= V/Izr 2 is the number of particles per unit area of boundary and V is the volume fraction. It is possible to solve these two equations for ap and ab for various values of U and V if the appropriate constants are known. For any particular system the only other constant which is not known precisely is the frequency factor v. However, this may be assessed approximately as follows. Since the rate of nucleation is equal to the product of the number of critical loops per unit area and the rate of arrival of new vacancies per loop: ,,, 2xr c bjv (Cv/b 2) exp - AGc/k T ,
,~
(12)
where 2rtrcb is the area available per loop for accepting vacancies,jv ,-, 101 s m - 2 s- 1 under typical creep conditions and Cv "~ 10-* is the vacancy concentration. This gives v ~ 2r~rcbjv(Cv/b 2) ~ 1018 m - 2 s-1. Using this value for v and values of the other constants appropriate to a previous study on Cu/A1203 alloys 1° (namely d = 13 #m, T = 1173°K, rp=0.3 ktm and D = 3 x 10 -1'* m2/s) eqns. (10) and (11) have been solved numerically for ap and a b and the creep rate calculated from the equation:
0.001
~=,6= ........
Iv=,d ~
I ........ I ........ I 0.01 0.1 '.0 STRESS. M N I m 2
Fig. 3. Creep rate-stress relationship calculated from eqn. (13) for various volume fractions of second phase. The creep rate is normalised by dividing by the proportionality constant in the N a b a r r o - H e r r i n g equation.
U = 1 0 pN and for V = 1 0 - I , 10 -2 and 10 -3. In each case the creep rate drops sharply at low stresses to very low values becoming effectively zero below a certain stress, and the form of the curves is a very good approximation to the simplified expression : BI2D - dZkT (a-go),
(14)
where ao depends linearly upon V. The creep rate-stress relationship has also been computed for various values of n U 2 / b k T and the
340
B. BURTON
form of the relationship is similar to the curves in Fig. 3. The dependence between a o and rcU2/bkT is very close to linear with the proportionality constant being ,-~ V/55. It should be noted that the value of cro is very insensitive to the value of the frequency factor v (and also to rp, D) since, as in all nucleation processes, the rate of nucleation is a very strong function of the driving force, in this case av and the value of the exponential in eqn. (10) completely dominate the pre-exponential factors. Indeed, changing the pre-exponential by ten orders of magnitude shifts the position of the curves.in Fig. 3 by less than a factor of 2 thus justifying the use of the very approximate value of v. Thus in order to condense vacancies on a boundary containing particles a critical stress has to be exceeded given by: (ao),o = zcU2 V/55bkT.
(N)rp oc exp - (2rCrp U - rcr2 trp b)/kT
(16)
and the corresponding threshold stress will be:
-~r2p-ff / V.
(17)
Thus for small rp, ao should be a function of rp and the overall dependence is shown in Fig. 4. It should
~r c
. .ix\. \
55kT/21rU
\
\
-" ~ "" (~.)rp PARTICLE
4. C O M P A R I S O N TIONS
WITH
EXPERIMENTAL
OBSERVA-
(15)
The model predicts rro to be essentially independent of the particle size since it only appears in the pre-exponential factor and this will be valid for relatively large particles. However, for very small particles the situation is different since rp may be less than the critical radius re. In this case the nucleation event can be achieved by nucleating sub-critical loops of r = rp at the rate given by :
((r°)'P ~- r.b
rp < 55 kT/zcU, ao decreases with rv in a manner described by eqn. (17). Eventually ao becomes zero at rp= 55 kT/2rrU when 2U/rvb= 55 kT/zcr2pb, thus indicating that very small particles should not inhibit vacancy condensation. This is typically when rp ~< 10 nm however and since for most systems rp is greater than this a o is expected to be essentially independent of rp. For still larger particles it is likely that the stress concentration can be relaxed more easily by the punching of dislocation loops x5 or other plastic flow processes and the present model will break down.
RADIUS
Fig. 4. Variation of the threshold stress for vacancy condensation with particle radius.
be noted that eqn. (17) reaches a maximum value at rv= 55 kT/lrU and eqns. (15) and (17) intersect at this maximum. For rp > 55 kT/nU, ao is independent of rp and is given by eqn. (15); and for
A successful model which describes the influence of particles on diffusion creep must explain three main features. Firstly, why some particles have an effect and others apparently do not. Secondly, it must explain the origin of threshold stress and the parameters upon which this depends and finally how the creep rate diminishes as particles accumulate on longitudinal boundaries. 4.1. The effectiveness of different particles Equation (7) predicts the ease of nucleation of defect loops on particles to depend strongly upon U the line energy of the defect in the particle/matrix interface. Since the energy of a line defect in a grain boundary is proportional t o #m b2 where #m is the shear modulus of the matrix then the energy in a particle/matrix interface can probably be written Uoc½(/~m+#p ) where ½(#m+#v) is some average shear modulus and #p is the shear modulus of the particle. Thus in cases where #p > #m, U will be large and inhibition is expected. When #p < #m however nucleation should be easy and the material behave as though no particles were present. Reported data on two-phase systems are represented on a volume fraction-relative melting point diagram in Fig. 5 where the subscripts p and m refer to particle and matrix respectively. (~ values are not available for some of the systems and so the ratio of the melting temperatures is plotted instead. However, since # oc TM this ratio is equivalent to /~p/#~.) The vertical bars on the points represent the range of compositions studied and the data are taken from the literature as follows: Mg/ZrH2 6's, M g / M g O 7'16, M g / ~ M n X 7 J s, A u / A 1 2 O s 9, M g / M g 2 -
341
INHIBITION OF DIFFUSIONAL CREEP Si 19, Mg/~Zr 19, Cu/A120310, Cu/Cu2O20, Cu/ GeO221, A1/Fe 22, CuZn/Pb and CuZn/voids 2a. It is clearly indicated in this Figure that the conditions for inhibition are large volume fractions of elastically hard (high TM) particles*. 4.2. The threshold stress for creep The theory in section 3 predicts that a critical stress has to be exceeded before a boundary can absorb (or emit) vacancies at a significant rate and above this stress creep rate should be given by the equation :
Bf2D (~ i -- d2kT
rcU2V ~ 55bkT/
~ sP .ot,~,ol~ o.I
,6s
cuz .b
~
(18)
DIFFUSION
~'Ai o CREEP if 2 a INHIBITED
~ \ x± eM~/.=Z, I "1. 1. o :uz,/vo,d, T ' ~ / . , M . , ,r.~ cutc~a°l o~ 1
.~
Mgl~CMn
~,/~,
o ,a"
.k I I
t 2
~,~go]; I 3
I
4
Fig. 5. Volume fraction-relative melting point diagram showing
the r6gimewhere diffusioncreep is inhibited. when lattice diffusion predominates and a corresponding equation can also be written for the grain boundary diffusion case. This type of creep equation has indeed been observed for many of the reported systems (Au/AI 203, Cu/Al2Os, Mg/Mg2Si, Mg/~Mn, Mg/aZr and Mg/ZrH2) although the variation of ao with the parameters, U, V and T has not been determined in most cases. The linear variation between a o and V has, however, been demonstrated for Au/A12 O3 and Cu/AI203 and the associated values for U calculated from eqn. (15) are 27 pN and 80 pN respectively. The smaller value for the gold alloy is to be expected since U oc (/am+/ap) and #Au
higher temperature and the temperature dependence of #m and #p will tend to increase the difference in the values for U. It should be noted that the values of U are in the range of 0.1-0.3 of the value of a line defect in the lattice (0.5 #b 2) and thus are of the correct order. This effect of temperature has also been noted in Mg/aMn by Jones is where a 0 again decreased when temperature was increased. Since ao oc U2/T the variation of a o with temperature should vary approximately in the same way as /a2/T. Shear moduli usually decrease by a factor of ~ 2 between 0.5 TMand the melting point and thus a0 is expected to decrease by a factor of ~ 8 over this same temperature range. 4.3. The decrease in creep rate with strain As particles accumulate on longitudinal boundaries the critical supersaturation for condensation on these boundaries must increase. Initially the number of particles per unit area of boundary is n A = V/~r 2 and the threshold stress for creep is that associated with this value of V. On transverse boundaries nA remains constant but on longitudinal boundaries nA increases with strain. Thus the threshold stress for creep will always be that one associated with the concentration of particles on longitudinal boundaries. The number of particles collected on a grain edge of area d 2 at a strain e is ed3nv where nv=3V/4nr 3 is the number of particles per unit volume. Thus the total number of particles per unit area of boundary at a strain e is: V ( 3d t (nA)~ = rcr---~ 1 + 4~rpe .
(19)
Thus the material behaves as though it contained a volume fraction V~=(nA)~nrp 2 of particles and so according to the present model ao will increase with strain in a manner described by the equation • rrU2 ( 3d ) ( a o ) - 55bkT 1 + ~ e Vo, (20) where V0 is the original volume fraction. The dependence of creep rate on creep strain, again taking the lattice diffuaion case, is then given by :
=~
cr
55bkr
1+ ~
Vo
(21)
and the rate of decrease of ~ with e is :
Oi ~3e --
(~_~) 2 DVo drp '
A \'~,/
where A = 3~BO/220b .~ 7 × 10-19 m 2.
(22)
342
B. BURTON
Thus in order for O~/~3e to be large Vo must be large or d and rp small (for a fixed temperature). However, the parameter which is more important in choosing alloys for technological applications is the limiting creep strain eL. This is the strain at which ~ 0 as a0 approaches a and can be calculated by putting rrU2
55bkT
(1
3d) + 4 r p e V°=a
case of the magnesium alloy where large particlefree zones were formed during creep on boundaries perpendicular to the stress axis, the faster creep of these "pure" zones may have contributed to overall extension. Since this contribution will increase with zone size and thus with strain this may well explain the non-linear behaviour in Fig. 6.
5. CONCLUSIONS
to be:
~55bkTa
4rp =
,- 6 V00
) 1
.
(23)
Now eL oc rp/d and thus the limiting creep strain is smaller for larger grain size material containing a constant volume fraction of small particles. U
~2 x uJ a:
~/
Cu/AI203
z
~
Z
r
2
H
2
4 6 8 CREEP STRAIN x I02
I0
Fig. 6. Variation of strain rate with creep strain for two alloys.
In order t o compare these predictions with experimental results the creep rate of Cu/l~/o A 1 2 0 3 1 ° and Mg/0.5~ ZrH2 s is plotted versus strain in Fig. 6. The values of eL are 0.03 and 0.11 respectively and the line energy U can be calculated from these values using eqn. (23). For the copper alloy putting rv=0.3 pm, d = 1 3 #m, a = l . 1 6 MN/m 2 and T = 1173°K gives U = 6 5 pN, a value close to that calculated earlier in subsection 4.2 from absolute measurements of go. For the magnesium alloy taking V=0.005, rp~-1 #m (by inspection from ref. 8, Fig. 1), T=673°K, d=100 /an and a = 1.1 MN/m 2 gives U - 3 0 pN again a value of the correct order. Thus in this respect behaviour is consistent with the accumulation of particles on longitudinal boundaries. However, Fig. 6 shows that 0~/ae is not constant as predicted but varies by a factor of ~ 2 for the magnesium alloy and ~ 3 for the copper alloy. Such curvature is not easily quantifiable but several possibilities exist. For instance, any primary creep contribution or "settling in" at the beginning of test tends to over-exaggerate the creep rate in the initial stages. Alternatively, in the
Vacancy condensation on grain boundaries requires the movement of line defects through the boundary. If no particles are present this movement is relatively easy and boundaries can act as vacancy sinks (and sources) at small supersaturations. When particles are present, movement of the line defect can be impeded and the defect must bow out in order to by-pass these particles. If the particle/ matrix interface is not a good vacancy sink then any continuing condensation on particle-free regions of the boundary rapidly leads to stress concentration at the particles. This stress concentration can be relaxed if defect loops nucleate and grow on the partiele/matrix interface. For elastically soft particles this process should be easy and these particles have a negligible effect on diffusional creep. For elastically hard particles the nucleation process may be difficult and a critical stress must then be exceeded before boundaries can absorb vacancies at a significant rate. Thus a corresponding threshold stress for diffusional creep exists. When the particle size is relatively large this threshold stress is independent of particle size but below a certain particles size it drops rapidly to zero, indicating that nucleation on very small particles is easy. The collection of particles on longitudinal boundaries during creep leads to a progressive increase in the threshold stress and causes creep rate to diminish with strain. Creep eventually stops when the stress required for condensation approaches the applied stress.
ACKNOWLEDGEMENTS
The author is grateful to Mr. I. G. Crossland, Dr. J. E. Harris, Mr. G. F. Hines, Dr. R. B. Jones of Berkeley Nuclear Laboratories and Mr. R. E. Lewis
INHIBITION OF DIFFUSIONAL CREEP
and Prof. G. W. Greenwood of Sheffield University for permission to quote their unpublished work and to Mr. M. V. Speight for useful discussion. This paper is published by permission of the Central Electricity Generating Board.
343
1 R. B. Jones, Nature, 207 (1965) 70. 2 H. Jones, Mater. Sei. Eng., 4 (1969) 106. 3 F. R. N. Nabarro, Rept. of Conf. on the Strength of Solids, 1948, The Physical Society, London, p. 75. 4 C. Herring, J. Appl. Phys., 21 (1950) 437. 5 R. L. Coble, J. Appl. Phys., 34 (1963) 1679. 6 P. Greenfield and W. Vickers, J. Nucl. Mater., 22 (1967) 77. 7 W. Vickers and P. Greenfield, J. Nucl. Mater., 27 (1968) 73. 8 J. E. Harris, R. B. Jones, G. W. Greenwood and M. J. Ward, J. Nuel. Mater., 29 (1969) 154.
9 F. K. Sautter and E. S. Chen, in C. S. Ansell (ed.), Oxide Dispersion Strengthening, Gordon and Breach, New York, 1968, p. 495. 10 B. Burton, Metal Sci. J., 5 (1971) 11. 11 M. F. Ashby, Scripta Met., 3 (1969) 837. 12 G. W. Greenwood, Scripta Met., 4 (1970) 171. 13 R. L. Squires, R. T. Weiner and M. Phillips, J. Nucl. Mater., 8 (1963) 77. 14 B. Burton, Mater. Sci. Eng., 10 (1972) 9. 15 J. E. Harris, to be published. 16 I. G. Crossland and R. B. Jones, personal communication. 17 R. B. Jones, in D. G. Brandon and A. Rosen (eds.), Quantitative Relation between Properties and Micro-Structure, Israel Universities Press, 1969, p. 343. 18 R. B. Jones, personal communication. 19 G. F. Hines, personal communication. 20 B. Burton, Ph. D. Thesis, University of Sheffield, 1969. 21 M. F. Ashby, Proc. 2nd Intern. Conf. on Strength of Metals and Alloys, Vol. II, Am. Soc. Metals, 1970. 22 B. Ya Pines and Ye Ye Badyan, Fiz. Metal. i Metalloved., 29 (1970) 847. 23 R. E. Lewis and G. W. Greenwood, personal communication.
MOcanisme d'inhibition du fluage-diffusion par des particules arune seconde phase
Zum Mechanismus der Behinderuny des Diffusionskriechens durch Teilchen einer zweiten Phase
Le fluage-diffusion peut ~tre inhib~ par la pr6sence de particules d'une seconde phase darts les joints de grains. L'auteur 6tudie ce ph6nom6ne en consid6rant les limitations apport6es par les particules au r61e de sources et de puits de lacunes jou6 par les joints de grains. Si les lacunes se condensent dans des zones du joint d6pourvues de particules, il en r6sulte une concentration de contrainte sur les particules. Cette concentration de contrainte peut &re relax6e par la cr6ation et la croissance de boucles de d6fauts ~t l'interface particule-matrice. Comme la facilit6 de formation de ces boucles d6pend de l'6nergie de ligne des d6fauts et que celle-ci d6pend du module de cisaillement de la particule, on peut pr6voir que les particules dures d'un point de vue 61astique seront les plus efficaces pour inhiber le fluage-diffusion. Cette id6e est d6velopp6 et les pr6dictions du module sont compar6es avec des observations exp6rimentales publi6es.
Diffusionskriechen kann durch die Gegenwart von Teilchen einer zweiten Phase an Korngrenzen behindert werden. Diese Behinderung wird unter dem Gesichtspunkt einer Einschr~nkung der Funktion der Korngrenzen als Leerstellenquelle und -senke durch die Teilchen diskutiert. Kondensieren Leerstellen in teilchenfreien Bereichen einer Korngrenze, so entsteht an den Teilchen eine Spannungskonzentration, die durch die Bildung und das Wachstum von Versetzungsringen in der TeilchenMatrix-Grenzfl~iche gemildert werden kann. Da die Keimbildungswahrscheinlichkeit von der Linienenergie des Defektes und diese wiederum vom Schermodul des Teilchens abh~ingt kann man vorhersagen, dab Diffusionskriechen durch elastisch harte Teilchen besonders wirkungsvoll behindert wird. Dieses Modell wird entwickelt und seine Vorhersagen werden mit ver6ffentlichten experimentellen Beobachtungen verglichen.
REFERENCES