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Diffusional creep of two-phase materials
Mw)I-6160/8Z~081655-10fo3.00;0 Copyright Q 1982 Pcrgamon Press Ltd
ACM mmlt. Vol. 30. pp. 1655 to 1664. 1982 Printed in Great Britain. .A11rights resrrved
DIFFUSIONAL
CREEP OF TWO-PHASE
MATERIALS
I.-W. CHENt Massachusetts Institute of Technology. Cambridge, MA 02139, U.S.A. (Receiced 9 Zlocember 1981; in recisedform
15 &far& 1982)
Abstract-A theory of diffusional creep in two-phase binary alloys is developed. The thermodynamic requirement of phase equilibrium along phase boundaries severely restricts the possible extent of compositional redistribution within each phase domain; meanwhile, the chess atoms of the higher mobility may jump across the phase boundaries. In the absence of long-range connectivity of either phase, it is found that diffusional creep may be controlled by the fastest diffusing species along its fastest path (phase) and is necessarily accompanied by migration of phase boundaries. Since diffusional creep in a single phase binary alloy is controlled by the slowest diffusing species along its fastest path, the twophase alloy may creep much fdster than any corresponding single phase alloy containing one of its constituent phases. At low stresses. a threshold behavior for boundary migration. either due to interfacial energy or impurity adsorption, is found to drastically suppress the diffusional creep rate. This type of migration-assisted diffusional creep is substantially suppressed on a macroscopic scale by long-range phase connectivity, although it may still operate locally. The application of these results to superplasticity is outlined.
R&urn&&e thiorie de Ruagc par diffusion dans des atliages binaires B deux phases est pr&cntte. La contrainte thermodynamique de l’equilibre des phases le long des joints interphases restreint shv&ement I’amplitude possible pour la redistribution en composition dans chaque domaine de phase; pendant ce temps. les atomes de plus grande modilitt en excis peuvent franchir les joints interphases. Dans I’absence de connectivitC B tongue distance des phases. on trouve que le fluage par diffusion peut &re contr6lC par l’esp&ce diffusant le plus rapidement le long de son chemin le plus rapide et qu’il s’accompagne nkessairement d’une migration des joints interphases. Puisque le Ruage par diffusion est cont&li dans un alliage & phase unique par I’esp&ce diffusant le plus lenrement le long de son chemin le plus rapide, l’alliage B deux phases peut fluer beaucoup plus rapidement que les aliiages contenant une de ses phases constituantes. Aux faibles contraintes un seuil pour la migration des joints, dont l’existence se dtpend de l’bnergie interfaciale ou de l’absorption des impuritts, reduit considerablement la vitesse de Ruage par diffusion. Ce type de fluage par diffusion assist6 par la migration est largement supprimt B I’tchelle macroscopique par la connectivitt des phases B longue distance, bien qu’il puisse optret localement. L’application de ces r&sultats il la superplasticitt? est don&e dans ses lignes g&&ales. Zusammenfass~-Eine Theorie wird entwickelt fiir das Diffusionskriechen in einer zweiphasigen bin& ren Legierung. Die thermodynamische Forderung nach Phasengleichgewicht entlang den Phasengrenzen bestimmt, in wclchen Grenzen die Zusammensetzung in jeder Phasendomlne sich umverteilen kann. Die UberschuBatome mit der hBheren Beweglichkeit kBnnen dabei durch die Phasengrenze wandern. Hiingen die Phasengrenzen nicht weitreichend zusammen, dann ergibt sich, dai3 das Diffusionskriechen von der am schnellsten entlang dem schnellsten Weg (Phase) diffundierenden Atomsorte bestimmt sein kann und daher mit der Wanderung von Phasengrenzen verbunden ist. Da das Diffusionsk~echen in einer einphasigen bin&en Legierung durch die am langsamsten entlang dem schnellsten Weg diffundierende Atomsorte kontroltiert ist, kann die zweiphasige Legierung vie1 schneller als jede einphasige Logierung aus einer ihrer Komponenten kriechen. Bei nicdrigen Spannungen ergibt sich ein S&w&nverhalten der Komgrenzwanderung. harvorgerufen entweder durch die Gr&fliichenenergie oder durch Anlagerung von Verunreinigungsatomen, welches die Rate des Diffusionskriechens drastisch absenkt. Dieses Diffusionskriechen mit Komgrenzwanderung wird makroskopisch durch weit zusammenhHngende Phasen sehr stark unterdriickt. wenngleich es Iokal auftreten kann. Die Anwendung der Ergebnisse auf die Su~plastizit~t wird kurz dargestellt.
1. INTRODUmON
The classical theory of diffusional creep due to Nabarro [l], Herring [2] and Coble [3] concerns mainly with single phase polycrystals. Many important metallic and ceramic systems in which diffusional transport is important are actually two-phase oi even multi-phase materials. In the past, little attention, in the diffusional creep literature and in related fields such as sintering and grain growth, has been paid to t Assistant Professor of Nuclear Engineering Materials Science and Engineering.
and
the problems associated with the two-Dhase microstru&ures. This is not surprising, since-problems of this type are rather complicated. In general, diffusivities of various species are usually quite diierent and they vary from phase to phase. Furthermore, the thermodynamic condition of phase equilibrium at the interphase boundaries must be satisfied in addition to the mechanical condition of stress equilibrium and displacement compatibility. Lastly, microstructural features such as volume fraction and phase connectivity also play an important role in deformation. For these reasons, a comprehensive treatment of the subject of the entire scope seems to be formidable.
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to yield two Poisson equations VI (&I _ y(i)) v -- 0 pz (pig) - fl’$) = 0
(3)
within each phase domain. To the first order in stresses, the driving forces are well defined on the phase boundaries and grain-boundaries C2). They are
Fig. 1. A two-phase polycrystal under external stress.
In this paper, we shall attempt to deal with the above aspects, starting from a greatly-simplified model in which phase geometry and distribution are idealized. A full solution of this model problem will be presented. The intent of the exercise is to highlight the unique features of two-phase materials emphasizing the important consequences of the thermodynamic condition of phase equilibrium. On such basis, further observations on the effects of phase geometry and distribution will be made. Finally, these perspectives will be considered in the context of superplasticity to which the present treatment becomes most relevant. 2. BASIC CONCEPTS Consider a stressed polycrystal of a binary A-B alloy containing two equilibrium phases denoted by I and II in which phase-boundaries and grain-boundaries are perfect sources and sinks for atoms and vacancies (Fig. I). The diffusional fluxes of atoms A and B in each phase domain are [1,2]
Jy
where pAo and pso are compositional chemical potentials of atoms A and B at the given local compositions but at zero stress and gn the normal stress on the phase boundaries and the grain boundaries. The boundary value problem is completely specified once the values of P.,~, pl,o and Q, are known on all the boundaries. There is an important and indeed fundamental difference between single phase materials and multiphase materiais in the determination of plo and fill0 on the boundaries. In the single phase materials, Herring [2] pointed out that the faster diffusing species, say A, will accumulate near the sinks and deplete near the sources to oppose further excessive build-up of the compositional gradient. A steady state will soon be reached in which JJJH is exactly X$X”, to aliow net material plating (removal) at the sinks (sources) at the bulk composition, Following the analysis of Herring [2], we can verify that this consideration completely determines pAo and p(HOon the grain boundaries of a single phase material, namely
=
(1) In the above equations, the superscript (i) is omitted J\;'
=
where the superscript (i) stands for the phase labels I or II, 5., and DE are tracer diffusivities of atoms A and B in the lattice, X, and XH the atomic fractions, R the atomic volume, g,%,pa and pv the chemical potentials of atoms and vacancies. We shall also assume that the contribution of short circuit diffusion along interfaces to the total strain is negiigibIe. It should be noted however that this does not preclude short circuit diffusion crossing interface from operating. The parallel theory of Cable creep is trivial. At steady state, there should be no change of compositions with time within each phase. Following Herring [2], we linearize the field equations, which are V.J$)=
V.Jfj’=O
(2)
since only a single phase material is considered here, and Xi and X”, are initial compositions in the stressfree solid. In a multiphase material, the overriding consideration at the interphase boundaries should be phase equilibrium. Specifically, no discontinuity is allowed for pi\ - pclvand in13- pv across the phase boundaries at temperatures when diffusional creep operates. From equation (4) this implies = a, (Q(‘)- Qc”)) g?o - I1’Ab’ &3 - &j = 0” (@I) - Q(“))*
(6)
Referring to the free energy diagram in Fig. 2 in which the equilibrium compositions in the absence of the stresses are denoted by (Xi’“. Xi”‘) and (X!$“‘, X$“‘), we find the above condition to be satisfied by a set of parallel tangents at (X2’), X2’)) and (Xx’? Xjt’)) with a vertical spacing of c,, (R@ - #In). In the special case of $2(t)= #it), (X2”, Xi’)) and (X’$? X$‘o)
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ation of connectivity and its consequences will be more fruitful, however, if the consequences of thermodynamics and geometry can be separately assessed, at least in some idealized cases initially. Such an approach will be taken here. In section 3, we exactly solve a model in which both phases are unconnected so that the finding can be entirely attributed to twophase thermodynamics and its consequences are fully explored in the three subsequent sections. The issue of phase connectivity is then taken up again to present a &ore realistic picture of diffusi&tai creep in twophase materials. Fig. 2. Diagram of compositional chemical potential of a two-phase system. A/J* is taken to be CT,(Q”’ - flu’) in section 2 and r~.(@” - @I”) - I-K(Q’” + Qcs) in section 4.
reduce to (Xi”‘, Xi(“) and (X k)““.X0,““), hence, there is no compositional redistribution within each phase domain. This conclusion for interphase boundaries in the multiphase materials contrasts with Herring’s observation of the compositional gradient which generally occurs in single-phase materials in which D.4 + DR. From the standard procedure [4], the compositional chemical potentials are
(7)
on the phase boundaries. Again at Q”’ = Q”“, the second term vanishes as expected. Although this simplification does not hold in a more general case, unlike equation (5), boundary values of pA,, and pus0in equation (7) are always independent of D, and D,, for a multi-phase binary alloy. The decoupling of the driving force from kinetics in the latter case, being a new feature which distinguishes a two-phase material from a single-phase material, leads to rather unexpected results in diffusional deformation, which will be demonstrated in the next two sections. The above finding is mostly thermodynamic in nature.t The other important element in deformation of two-phase alloys is phase distribution concerning, in particular, the phase connectivity. Generally, the connected phase must deform while the unconnected phase need not (though it may), although the breaking of full connectivity into a partial one will substantially modify the situation. Further consider-
t Strictly speaking, it also bears on mechanical equilibrium, which requires that the same normal stress be supported on either side of a phase boundary. This, of course, is always satisfied.
3. DIFFUSIONAL
CREEP RATE
An exact calculation is now performed for a square grain array of two phases, as shown in Fig. 3. Being highly idealized, this configuration provides a limiting case of no phase connectivity of an eutectic or duplex polycrystal and highlights a unique feature of twophase materials by allowing only interphase boundaries to exist. The analytic evaluation of all the essential quantities of interest can be effected using this model. For the symmetric configuration studied here we expect that the average normal stress on each of the phase boundaries of a square grain to be the same as the applied normal stress. Furthermore from the nature of the linear field equations and their boundary conditions, equations (2, 4, 6, 7), which are all of first order in o,,, we conclude that the deposition (removal) of matter due to J$‘, Jg’, J:“’ and J’d”, respectively, must be uniform along all the boundaries to avoid back stresses.
-(u)
I
-2
Fig. 3. A square grain, two-phase array under stress creeps by lattice diffusional transport.
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DIFFUSIONAL CREEP OF TWO-PHASE MATERIALS
Recall that the solution of the boundary problem of the following type
value
v’fcx y) = 0
(8)
where f = -b, at Ix/ or lyl = d/2, with (a,) = (a) at 1x1= d/2 and (a,) = -(a) at lyl = d/2, is cr, = &a>(1 - (2y/d)‘) u, = -+(a>(1
at
1x1= d/2
due to the stress bias their signs on the tensile boundary and the compressive boundary are opposite to each other. The overall creep strain rate is defined as 2 = (3’) + j”“)/&j.
(12)
Solving the above equation, we find
(9)
- (2x/d)*) at IyI = d/2
and f=
Pa)
- x’),‘d’.
6(a)(y2
+
We can verify that for our problem exactly the same stress distribution as shown above applies, which in turn drives the atomic fluxes. At the phase boundaries,
-(opx\; + -
Dpxy’
Q(ll
q-y,,y1 X
J”’ = + D$‘X;(” 6(a) A -kTR’“d
t
@‘)X;(“l _ @nX$l) X0,“” _
x;‘l)
> (10)
In the above equations the “+” sign applies on the tensile boundary while the “-‘I sign applies on the compressive boundary. The net material deposition on the interphase boundaries can be evaluated from the following equations
J’:)
+
J’“) A
=
X0”’ A
_ *,,,
+
XY’)
g
(11) Jg)
+
Jf)
=
X;(‘)
$1) nc’)
+
X;(“‘$
xy
_ _
(yll)
fplxy, xyl
> z 11.
(13)
The resultant grain shape is shown in Fig. 4 for the cases of D$Df’) >>DI(‘f# and Dg)Dg” >>D$)D$“, assuming X~“‘X$“’ > X~(“X~“‘) > 0. The most interesting feature of the figure is that it demonstrates that the deformation of two phases are not identical, indeed they do not even elongate in the same direction. The overat diffusional creep in this case, according to equation (13), is controlled by the fastest diffusing species along its fastest path (phase), i.e. 02) or Of’ in Fig. 4a and DC’ or D$” in Fig. 4b. It is noted that interphase boundary sliding, and more significantly, interphase boundary migration must have taken place in the deformation of the two-phase materials modeled here. The quantification of these observations and the effects of interface energy will be studied in the next section.
.
4. MIGRATION OF IYTERPHASE BOUNDARIES In the above equations Xi”) and Xi(‘) were used in lieu of Xx’ and Xje in the spirit of linearization. Here .@ is the deposition rate at the interface on the side of phase i at the composition of Xiti’ and X$i’. Again
4.1 Rare of migration. Migration of interphase boundaries is effected by the net transport of atoms of either species across the (bl
D$‘DBm)>>
DC” A DA(=’
Fig. 4. The resultant grain shape of the two-phase system of Fig. 3 under diffusional creep for which the deformation of the two phases are not identical. The center voids are artifacts of the small strain formulation used here.
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interface in question. In the problem considered here, its occurence is a necessary result of the continuity of driving forces across the interface and the conservation of matter. The flux of A atoms, M, crossing the
or, after symmetrization IMA= -;
.
(Ji’ _ J’{“) _ !
i
(15) Similarly, the flux of B atoms, Ms, crossing the interface from phase I towards phase II is (J$’ _ Jlf’)
OF TWO-PHASE
MATERIALS
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r. Below a certain threshold stress, migration will be restrained by the surface tension which contributes to the driving forces at the phase boundaries in the following way (41
where K(~)is the curvature of the phase boundary defined to be positive if the phase boundary bows out and it always satisfies the relationship, K(‘)= -I&‘~. We thus define K = K(‘)= -#‘) for simplicity. We can now go through the same procedure which leads to equations (4,6. 7). referring to the free energy diagram shown in Fig. 2. to find, on the interphase boundaries,
.
-
(16)
The net flux of either atoms, &!, crossing the interface from phase I toward phase II. corresponding to the net ~~g~~rio~ of the interface in the reverse direction from phase II toward phase I, is the sum of the above two quantities, M.4 and MI3 M = (Xi”’ - x.;““)-‘(x:“‘J\;’ -
Xp:l’J’:’
-
ow (Ill x,1 J,* ).
-t x.4OIIIIJ,,III1
(17)
In the special case considered in Fig. 3, M is reduced to the following form M = f !?.?&X~~r~ _ X0,““)-2
In the above equation, the ‘+’ sign applies on the tensile boundary while the ‘-’ sign applies on the compressive boundary. As in the case of the overall creep rate, the migration rate is controlled by the fastest diffusing species, while its direction is determined by the consideration that the phase which is richer in the fastest diljEtsingspecies must grow ut the expense of the other phase at the sink, and cite versa at the source. 4.2 Eflect of interfacial energy
Results for the overall creep rate and the migration rate, equation (13) and flS), were obtained negtecting any effects of the interfacial energy of the phase boundaries. This assumption is justified if the driving force, CL?, substantially exceeds the surface tension term which is of the order of IX/d for an interface energy
f TK ~(1)~~‘~~~ ‘_ ~~~(1)) * (
*
As expected, the above equation reduces to equation (7) if frc = 0. At the steady state, we expect K to assume the magnitude which renders M = 0 and to vary finearly with a,, namely, TK/CY,to be a constant. The constant is determined beiow. The solution of the boundary value problem now yields
(2l) @‘Xi”’ J&l’ = z!zkTR”’
&3) d
~(l)X~{lI~_ ~Ii)X~~l) C --
‘ycp
_
/-K fpxyl~
0”
ow1 X.4
Xcyh
+' ~(lilx~~ll _
xy
>
on the interphase boundaries, with the same sign notation used earlier. Meanwhile equation (17) for the rate of migration still holds. Therefore, the value of ~K/u” which renders M = 0 can be calculated from
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equations (9) for the stress distribution. we find
equation (17) and (21). i.e.
(25) where 9 is the quantity on the RHS of equation (22). The critical condition occurs when dy’d.r approaches infinity, at 1.~1= d/2 at the grain corner. which defines a threshold stress +
(,Y~“‘X;“‘D’,”
x
(~mx~111’
+ +
X~111’X,o”11~~11)
~1lllxy1)~~
(22)
Finally, SC’)and s(“) can be obtained as before to yield the creep strain rate. For brevity, we only present the result in the case of Qc” = a(“’ = R
Below(~hhrlrrllold,migration is suppressed by the bow-out of the interphase boundary and the overall creep rate is given by equation (23). Above (Q)~,_,,~~~, migration becomes possible and the overall creep rate approaches that described by equation (13). It is also noted that even below the threshold, migration is still possible during the transient state before the steady state bow-out is fully developed. Indeed, the steady state can only be reached after certain creep strain. This conclusion holds for both the subcritical case and the supercritical case. A schematic diagram of bow-out is shown in Fig. 5.
5. THE ROLE OF THE INTERFACE + X;““XpI”“[(X~C” + X;(“‘)Djf” + (X?
+ XoH’“‘)Df”] 1
(23)
which in the case of a single-phase system reduces to the result of Herring [2] as it should, namely k
_
6Q kTd2
441 D.4X; + D,,Xo,’
(234
In the context of the simplified notion of two-phase materials in which both phases are dispersed and the interphase boundaries are the only type of boundary that exist, major observations made in section 3 and 4 apply, not only to diffusional creep, but also to other kinetic processes which involve incoherent phase boundaries as sources and sinks. In the following, a physical interpretation is afforded for a better under-
It can be verified that the diffusional creep rate in the present case, equation (23), in which migration is suppressed by the bow-out of the phase boundaries, is controlled by the slowest diffusing along its fastest path (phase). 4.3 Threshold stress The profile of the interphase boundary can be determined completely from its curvature. A critical condition is reached when the bow-out at the grain corners extends an angle of rr/2 (Fig. 5), a condition similar to the Orowan’s condition in the dislocation theoiy [YJ. From the geometric consideration [6]
where X, y are the plane coordinates of the phase boundary, Solving dy1d.x as a function of x and noting
c
Fig. 5. The antisymmetric bow-out of phase boundaries can be superposed on the diffusional creep when the interfacial energy restrains boundary migration at the steady state.
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standing which eventually focuses on the role of interface in kinetic processes. Kinetic processes in multi-component systems have often been thought to be controlled by the slowest diffusing species along its fastest path. Essentially, this concept follows Herring’s analysis of the compositional gradient which is set up on application of a driving force. Kinetically, this corresponds to a build-up of the fast diffusing species at the sinks in relation to the sources, hence, a situation of up-hill diffusion for the fast-diffusing species, and the opposite one of down-hill diffusion for the slow-diffusing species. At the steady-state, the overall diffusion of all species due to the applied driving force and the compositional gradient are properly adjusted to be in proportion to their respective atomic fractions. The validity of the above statement depends on the ability of the system, at the sources and sinks in particular, to sustain a compositional redistribution which, by definition, deviates from the equilibrium composition (at zero stress) without incurring any phase changes or microstructural evolution. A counter-example was provided by Chen [7] in Coble creep of multi-component single-phase materials. Migration of grain boundaries, once initiated, will leave alloyed and de-alloyed zones behind to alleviate the excessive composition build-up along grain boundaries. The diffusional creep rate in this case is controlled by the fastest diffusing species and the deformation is considerably faster than Herring’s prediction. The results of sections 3 and 4 demonstrate that more counter-examples exist in mult-component multi-phase materials. In fact, the ‘anomalous’ fast diffusional creep [7] which is facilitated by grain boundary migration in the single-phase material becomes entirely normal in the present case. It was shown earlier that the compositions of either phases are locked in by the requirement of phase equilibrium at the phase boundaries to be those of equilibrium compositions at zero stress. Physically, the existence of two phases which are intermixed at the scale of the grain size assures the presence of an adjacent thermodynamic reservoir for each grain to prohibit and indeed adsorb, any build-up of composition within neighboring grains. Thus, diffusion responds solely to the applied driving force and is not affected by the compositional redistribution which, if present, would have slowed down (fastened) the fast (slow) diffusing species. The overall creep rate consequently is controlled by the fastest diffusing species along its fastest path, which in the present case turns out to be the path along the phase where diffusion is fastest. In general, diffusional fluxes set up in this way have at the sinks an excess of the fast diffusing species, and vice versa at the sources. Atoms on either side of the phase boundary must be allowed to jump across the boundary, to maintain thermodynamic equilibrium of the two-phase system. Migration of the phase boundary is hence necessitated. As we found in section 4.1, the migration rate is controlled by the fastest diffusing
I\.Y.30.a---r
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species, while its direction is determined by the consideration that the phase which is richer in the fastest diffusing species must grow at the expense of the other phase at the sink, and vice versa at the source. Migration of this sort alleviates any potential compositional build-up at the boundary and makes possible the fast diffusional flow at the steady state. In other words, in maintaining the phase equilibrium across itself, the phase boundary also facilitates the faster kinetics which would be impossible if boundary migration were inhibited. Hence, as in Ref. [7], it is again a case of migration-assisted diffusional creep, only to be effected by the thermodynamic consideration applicable to phase equilibrium. The connection between migration and enhanced overall kinetics is made even clearer when we consider the other case in which migration is suppressed at low driving forces. Below the thresholds of migration, the exchange of atoms at the phase boundary with the reservoir (the neighboring phase) is not permitted. Only under such circumstances does compositional gradient again become pronounced and the creep rate controlled by the slowest diffusing species along its fastest path. It would be interesting to examine the more general case beyond the binary system treated here. The result of such study will be reported shortly later [14]. It is interesting to note that the specific model of Fig. 3 implies that the two-phase alloy may creep via diffusional flow at a rate faster than the single-phase alloy made of either phase I or phase II. This is because diffusional creep of a single-phase alloy is controlled by the slowest species, while that of a twophase alloy is controlled by the fastest species. This remarkable result, however, had a direct bearing on the assumption that only interphase boundaries exist in the microstructure and obviously needs modification when phase connectivity is taken into account. This is the subject of the next section. 6. PHASE CONNECI’IVITY ANSTRESS REDISTRIBUTION 6.1 Phase connecticity The model developed in sections 3-4 is unrealistic in that phase connectivity is entirely avoided in the configuration of Fig. 3. Generally speaking shortrange and long-range connectivity may exist and their extent depends on the volume fractions of each phase, as well as other factors, such as interfacial energies and transformation kinetics. Here, the range of phase connectivity is defined to be the average length of continuous paths along the same phase in the aggregate. If each phase domain is equiaxed and the phase distribution is random, the range of connectivity is only a function of volume fractions and dimensionality. In a two-dimensional system, long-range connectivity is realized when the volume fraction of the phase in question exceeds approximately 0.5, i.e. when
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the two phases are roughly of equal volume fractions. In a thr~-dimensional system. the critical volume fraction is reduced to 0.4, due to the additional degree of freedom in a three-dimensional network. The above statement is substantiated by the theory of random alloys in physics, in which percolation is investigated, and a mechanical analogy in terms of deformation was analyzed by a number of authors(S-IO] using the technique of self-consistent method. It is deduced from these considerations that spatial connectivity can be minimized if the volume fractions of the constituent phases are roughly the same, and the majority phase always possesses long-range connectivity. Although in a two-phase spatial aggregate it is difEcuIt to avoid fang-range connectivity, even with equal volume fractions for both phases, it becomes definitely possible to eliminate long-range connectivity entirely with three phases of equal volume fractions, 4.2 Stress re~isfr~~t~tion Turning to the point of stress redistribution among phases, we shall examine the consequences of phase connectivity with the aid of Fig. 6a and 6b. The configuration of Fig. 6b contains only short-range connectivity at equal volume fractions for both phases, while that of Fig. 6a contains some long-range connectivity for phase I which is also the majority phase. For simplicity, the latter case is chosen to possess certain symmetry which makes analytic solution possible. As before, diffusivities of different species are different and dependent on the phases, The most interesting case, however, is the one in which phase I is the ‘harder’ phase due to lower diffusivities therein and its stress distribution will be compared with that of Fig. 3. 6.2,l Long-range connectirity. Consider first Fig. 6a. On the side of compression, no stress redistribution is necessary due to symmetry. On the side of tension, the second and the fifth row along which the harder phase is continuous should carry higher
(a)
LONG -RANGE
CONNECTIVITY
MATERLALS
stresses, while all other rows should carry the same stress, again due to symmetry. The detailed analytic solution of diffusional creep of this con~guration demonstrates that the major material flows in a unit cell, i.e.. cdefgh in Fig. 6a. always originate from inter. phase boundaries, though they terminate both at interphase-boundaries and grain-boundaries. As before, interphase boundaries migrate, and certain rows (namely. rows 1, 3.1 and 6) deform almost entirely by fast diffusion in phase II and by attendant migration of phase boundaries. In contrast with the previous case. Fig. 3, those vertical grain-boundaries in rows 2 and 5 of the harder phase develop segregation which slows down diffusion drastically. The net result is that the slower diffusing species in domains d and g controls the overall kinetics. It is nevertheless correct to’state that the preservation of phase equilibrium at phase boundaries has already facilitated the overall kinetics considerably, since many sluggish diffusion paths were by-passed. It is also confirmed by the analytic solution that the ‘local’ deformation resistance aiong rows 1, 3,4 and 6, taken to be the ratio of the average normal stress to the average normal strain rate, is indeed lower than that of a single-phase alloy of either phase I or phase II. In other words, the two-phase alloy still flows easier locally wherever long-range phase connectivity is lost, but the harder phase controls the macroscopic strain rate if it is connected over a long-range in the microstructure. 6.22 Short-range connectivity. The configuration in Fig. 6b contains no long-range phase connectivity. It is then possible to isolate regions of each phase which are completely surrounded by the other phase. Two such regions, i.e. mnpq and ijkl, are shown in Fig. 6b. Due to the lack of symmetry, the problem is not analytically tractable. Mechanically, however, no excessive stress redistribution. by more than a factor of three or so [9], is possible in this case of equiaxed grains, for the simple reason of mechanical equilibrium. It should be noted that despite its frequent use in this field 111. 121, the upper bound analysis which
fb)
SHORT-RANGE
CONNECTIVITY
Fig. 6. Two-phase systems with long-range connectivity in phase I. (a) and with oniy short-range connectivity in both phases, (b). The diffusional flows of major importance are shown in a unit ceil cdefgh in (a) and in a representative domain in (b), assuming fast diffusion in phase 11.
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maintains a uniform strain rate everywhere is often unsatisfactory, for both mechanical reasons in general [8, lo] and geometric reasons in particular for diffusional creep [13,14]. For the case of two-phase flow by diffusion, the disparity between the inherent deformation resistance of different phases is so significant that we expect an almost complete breakdown of the upper bound analysis for the estimation of stress redistribution. Hence, isolated regions, e.g. mnpq of phase I and ijkl of phase II, should not be stressed very differently and should creep by the same mechanism of fast diffusional creep described earlier. More specifically, this will take place via diffusion along the softer phase, i.e. phase II, with only phaseboundaries as the source and the sink, and is assisted by migration of phase-boundaries. Although the length of diffusion becomes somewhat longer, it still has the advantage of being controlled by the fastest diffusing species along its fastest path (phase). For this reason, we expect that the two-phase alloy with only short-range phase connectivity still creeps relatively easily by diffusional flow, in a manner very similar to the one described in sections 35. The above discussion, of course, is valid only if the connectivity is truly a short-range one. The intermediate case between short-range connectivity and long-range connectivity is difficult to assess, but should lie between the two limiting cases treated above. Operationally, more ease of diffusional creep should be possible to realize if a third phase is added to the alloy and the volume fractions of all three phases are roughly the same. This, however, is only possible with ternary systems on which we will report later [14]. It is worth noting that the isolated regions of phase I, e.g. mnpq, also undergoes shape changes and has an apparent deformation strain. This statement does not negate our previous assumption that phase I is a much harder phase than phase II. The shape changes happen by the migration of phase boundaries which can be effected without necessarily the participation of diffusional flow through phase I, since an alternative fast diffusion path is available through the surrounding phase II. Indeed, this is perhaps the most significant feature of diffusional creep which differs from continuum deformation. In the latter case, a hard inclusion may remain rigid while the soft matrix flows around. In diffusional creep, a hard inclusion must deform, even though the diffusional flow is restricted to the soft matrix. The consequence of phase equilibrium and phase boundary migration is again observed here. 7. APPLICATIONS The results obtained here have some interesting applications in suck phenomenon as superplasticity [15]. Some of these will be briefly discussed below, although a detailed analysis and review of the experimental evidence is beyond the scope of the present paper.
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It is well known that superplasticity occurs more often in two-phase materials. This feature is usually attributed to the fact that grain growth can be more feasibly suppressed in a two-phase microstructure at a fine grain size which is a prerequisite for superplasticity. Our results, however, suggest that perhaps it is the unique thermodynamic and kinetic conditions which are realized in two-phase materials. especially at a one-to-one ratio of phase volume fractions, that bring about superplasticity. In particular, a twophase microstructure may creep via diffusional flow locally and sometimes even globally at a rate much faster than either constituent phase in the form of a single-phase polycrystal, due to the different rate-controlling mechanisms in question. This remarkable and yet entirely overlooked feature of two-phase materials certainly is a very important, if not the most important, cause of superplasticity which has been most often reported in two-phase alloys. The introduction of a third phase should strengthen this effect. The migration of phase-boundaries in the manner described by Figs -1 and 5 is also pertinent. We note that in Fig. 5 each phase domain is being ‘pinched off by inward hexing of the phase-boundaries, leading at the same time to the eventual approach of two initially separated neighboring grains. Such morphological evolution is reminiscent of the thermally activated grain switching model proposed by Ashby and Verrall [16]. Indeed. the form of the threshold stress calculated by these authors using the absolute rate theory is very similar to our result of equation (26). The analysis in this paper in our opinion provides a somewhat more natural and general basis for such evolution. It is more natural since equilibrium is maintained everywhere in the evolution for which a kinetic path is explicitly described in terms of diffusion and migration. It is more general since the possible cause for the threshold in our model is not limited to the interfacial energy and may include such well-known phenomenon as Langmuir adsorption at the boundary. In particular, if the non-linear dependence on driving force of boundary migration [17] is included in our analysis, a sigmoidal ( - 0 behavior is expected near the threshold. It should be noted that in reality processes described by Figs 4 and 5 are likely to operate simultaneously due to grain size distributions, internal stresses, and complex transient behavior of boundary sliding and boundary migration [ 181. One interesting feature of migration of phase boundary due to unbalanced atomic fluxes at the boundary, equation (18), is that migration becomes very fast if X0:” approaches X0,““. Although a detailed analysis taking into account the energetics of the phases and the interface and the mobility of the boundary is still required for any conclusive prediction, this observation raises some interesting questions concerning the possible role of diffusion in a nominally diffusionless allotropic transformation. In particular, if diffusion is active at the transformation
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temperature, applied
stress,
both
DIFFUSIONAL
the internal
which
stresses
can cause diffusion
and
CREEP OF TWO-PHASE
the
of atoms,
might have important effects on the stability of the phases and the rate of transformation. It is interesting to note that this phenomenon of diffusion-induced phase-boundary migration may explain some observations [19] which were, to this date, broadly associated with the so-called transformation-induced superplasticity [20]. Acknowledgements-This research is supported by the National Science Foundation Grant No. DMR-7824185. Helpful comments of Professor M. F. Ashby are gratefully appreciated.
REFERENCES 1. F. R. N. Nabarro, Report of a Conference on rhe Strength of Solids. Phys. Sot. London, p. 75 (1948). 2. C. Herring, J. appl. Phys. 21, 437 (1950). 3. R. L. Coble, J. appl. Phys. 34, 1679 (1963). 4. J. W. Christian, in The Theory of Transformations on Memls and Alloys, 2nd edn, Chap. 6, Part I. Pergamon Press, Oxford (1975). 5. F. A. McClintock and A. S. Argon, in Mechanical Behavior of Materials, Chap. 4. Addison-Wesley, Reading (1966).
6. J. Mathews
MATERIALS and
.bfethodsofPhysics,
R. L. Walker, in Mathemarical 2nd edn. Chap. 15. Benjamin, New
York (1970). 7. I.-W. Chen, Acra meroll. (1982). In press. 8. J. W. Hutchinson, Proc. R. Sot., Lond. 319A, 247 (1970). 9. I.-W. Chen and A. S. Argon, Acta merall. 27, 785 (1979). 1”. en M. P. Cleary, I.-W. Chen and S.-M. Lee, J. Engng Me&. A.S.C.E. 106. 861 11980). 11. A. K. Ghosh and R.. Raj. kn’metall. 29, 607 (1981). 12. R. Raj and A. K. Ghosh. Acra metall. 29, 283 (1981). 13. J. H. Schneibel, R. L. Coble and R. M. Cannon, Acta metall. 29. 1285 (1981). 14. I.-W. Chen, To be presented at Annual Meeting, Am. Ceram. Sot. (1982).
15. J. W. Edington. K. N. Melton and C. P. Cutler. Prog. Mater. Sci. 21, 2 (1976). 16. M. F. Ashby and R. A. Verrall, Acta merall. 21, 149 (1973). 17. K. Liicke, K. R. Rixen and F. W. Rosenbaum, in The Nature and Behavior of Grain-boundaries (edited by H. Hu), p. 245. Plenum Press, New York (1972). 18. H. C. Chann and N. J. Grant. Trans. metoll. Sec. A.I.M.E. 206: 544 (1956). 19. L. F. Porter and P. C. Rosenthal, Acta me&all. 1. 504 (1959). 20. G. W. Greenwood and R. H. Johnson. Proc. R. Sot. 283A. 403 (1965).
ACM mmlt. Vol. 30. pp. 1655 to 1664. 1982 Printed in Great Britain. .A11rights resrrved
DIFFUSIONAL
CREEP OF TWO-PHASE
MATERIALS
I.-W. CHENt Massachusetts Institute of Technology. Cambridge, MA 02139, U.S.A. (Receiced 9 Zlocember 1981; in recisedform
15 &far& 1982)
Abstract-A theory of diffusional creep in two-phase binary alloys is developed. The thermodynamic requirement of phase equilibrium along phase boundaries severely restricts the possible extent of compositional redistribution within each phase domain; meanwhile, the chess atoms of the higher mobility may jump across the phase boundaries. In the absence of long-range connectivity of either phase, it is found that diffusional creep may be controlled by the fastest diffusing species along its fastest path (phase) and is necessarily accompanied by migration of phase boundaries. Since diffusional creep in a single phase binary alloy is controlled by the slowest diffusing species along its fastest path, the twophase alloy may creep much fdster than any corresponding single phase alloy containing one of its constituent phases. At low stresses. a threshold behavior for boundary migration. either due to interfacial energy or impurity adsorption, is found to drastically suppress the diffusional creep rate. This type of migration-assisted diffusional creep is substantially suppressed on a macroscopic scale by long-range phase connectivity, although it may still operate locally. The application of these results to superplasticity is outlined.
R&urn&&e thiorie de Ruagc par diffusion dans des atliages binaires B deux phases est pr&cntte. La contrainte thermodynamique de l’equilibre des phases le long des joints interphases restreint shv&ement I’amplitude possible pour la redistribution en composition dans chaque domaine de phase; pendant ce temps. les atomes de plus grande modilitt en excis peuvent franchir les joints interphases. Dans I’absence de connectivitC B tongue distance des phases. on trouve que le fluage par diffusion peut &re contr6lC par l’esp&ce diffusant le plus rapidement le long de son chemin le plus rapide et qu’il s’accompagne nkessairement d’une migration des joints interphases. Puisque le Ruage par diffusion est cont&li dans un alliage & phase unique par I’esp&ce diffusant le plus lenrement le long de son chemin le plus rapide, l’alliage B deux phases peut fluer beaucoup plus rapidement que les aliiages contenant une de ses phases constituantes. Aux faibles contraintes un seuil pour la migration des joints, dont l’existence se dtpend de l’bnergie interfaciale ou de l’absorption des impuritts, reduit considerablement la vitesse de Ruage par diffusion. Ce type de fluage par diffusion assist6 par la migration est largement supprimt B I’tchelle macroscopique par la connectivitt des phases B longue distance, bien qu’il puisse optret localement. L’application de ces r&sultats il la superplasticitt? est don&e dans ses lignes g&&ales. Zusammenfass~-Eine Theorie wird entwickelt fiir das Diffusionskriechen in einer zweiphasigen bin& ren Legierung. Die thermodynamische Forderung nach Phasengleichgewicht entlang den Phasengrenzen bestimmt, in wclchen Grenzen die Zusammensetzung in jeder Phasendomlne sich umverteilen kann. Die UberschuBatome mit der hBheren Beweglichkeit kBnnen dabei durch die Phasengrenze wandern. Hiingen die Phasengrenzen nicht weitreichend zusammen, dann ergibt sich, dai3 das Diffusionskriechen von der am schnellsten entlang dem schnellsten Weg (Phase) diffundierenden Atomsorte bestimmt sein kann und daher mit der Wanderung von Phasengrenzen verbunden ist. Da das Diffusionsk~echen in einer einphasigen bin&en Legierung durch die am langsamsten entlang dem schnellsten Weg diffundierende Atomsorte kontroltiert ist, kann die zweiphasige Legierung vie1 schneller als jede einphasige Logierung aus einer ihrer Komponenten kriechen. Bei nicdrigen Spannungen ergibt sich ein S&w&nverhalten der Komgrenzwanderung. harvorgerufen entweder durch die Gr&fliichenenergie oder durch Anlagerung von Verunreinigungsatomen, welches die Rate des Diffusionskriechens drastisch absenkt. Dieses Diffusionskriechen mit Komgrenzwanderung wird makroskopisch durch weit zusammenhHngende Phasen sehr stark unterdriickt. wenngleich es Iokal auftreten kann. Die Anwendung der Ergebnisse auf die Su~plastizit~t wird kurz dargestellt.
1. INTRODUmON
The classical theory of diffusional creep due to Nabarro [l], Herring [2] and Coble [3] concerns mainly with single phase polycrystals. Many important metallic and ceramic systems in which diffusional transport is important are actually two-phase oi even multi-phase materials. In the past, little attention, in the diffusional creep literature and in related fields such as sintering and grain growth, has been paid to t Assistant Professor of Nuclear Engineering Materials Science and Engineering.
and
the problems associated with the two-Dhase microstru&ures. This is not surprising, since-problems of this type are rather complicated. In general, diffusivities of various species are usually quite diierent and they vary from phase to phase. Furthermore, the thermodynamic condition of phase equilibrium at the interphase boundaries must be satisfied in addition to the mechanical condition of stress equilibrium and displacement compatibility. Lastly, microstructural features such as volume fraction and phase connectivity also play an important role in deformation. For these reasons, a comprehensive treatment of the subject of the entire scope seems to be formidable.
1655
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to yield two Poisson equations VI (&I _ y(i)) v -- 0 pz (pig) - fl’$) = 0
(3)
within each phase domain. To the first order in stresses, the driving forces are well defined on the phase boundaries and grain-boundaries C2). They are
Fig. 1. A two-phase polycrystal under external stress.
In this paper, we shall attempt to deal with the above aspects, starting from a greatly-simplified model in which phase geometry and distribution are idealized. A full solution of this model problem will be presented. The intent of the exercise is to highlight the unique features of two-phase materials emphasizing the important consequences of the thermodynamic condition of phase equilibrium. On such basis, further observations on the effects of phase geometry and distribution will be made. Finally, these perspectives will be considered in the context of superplasticity to which the present treatment becomes most relevant. 2. BASIC CONCEPTS Consider a stressed polycrystal of a binary A-B alloy containing two equilibrium phases denoted by I and II in which phase-boundaries and grain-boundaries are perfect sources and sinks for atoms and vacancies (Fig. I). The diffusional fluxes of atoms A and B in each phase domain are [1,2]
Jy
where pAo and pso are compositional chemical potentials of atoms A and B at the given local compositions but at zero stress and gn the normal stress on the phase boundaries and the grain boundaries. The boundary value problem is completely specified once the values of P.,~, pl,o and Q, are known on all the boundaries. There is an important and indeed fundamental difference between single phase materials and multiphase materiais in the determination of plo and fill0 on the boundaries. In the single phase materials, Herring [2] pointed out that the faster diffusing species, say A, will accumulate near the sinks and deplete near the sources to oppose further excessive build-up of the compositional gradient. A steady state will soon be reached in which JJJH is exactly X$X”, to aliow net material plating (removal) at the sinks (sources) at the bulk composition, Following the analysis of Herring [2], we can verify that this consideration completely determines pAo and p(HOon the grain boundaries of a single phase material, namely
=
(1) In the above equations, the superscript (i) is omitted J\;'
=
where the superscript (i) stands for the phase labels I or II, 5., and DE are tracer diffusivities of atoms A and B in the lattice, X, and XH the atomic fractions, R the atomic volume, g,%,pa and pv the chemical potentials of atoms and vacancies. We shall also assume that the contribution of short circuit diffusion along interfaces to the total strain is negiigibIe. It should be noted however that this does not preclude short circuit diffusion crossing interface from operating. The parallel theory of Cable creep is trivial. At steady state, there should be no change of compositions with time within each phase. Following Herring [2], we linearize the field equations, which are V.J$)=
V.Jfj’=O
(2)
since only a single phase material is considered here, and Xi and X”, are initial compositions in the stressfree solid. In a multiphase material, the overriding consideration at the interphase boundaries should be phase equilibrium. Specifically, no discontinuity is allowed for pi\ - pclvand in13- pv across the phase boundaries at temperatures when diffusional creep operates. From equation (4) this implies = a, (Q(‘)- Qc”)) g?o - I1’Ab’ &3 - &j = 0” (@I) - Q(“))*
(6)
Referring to the free energy diagram in Fig. 2 in which the equilibrium compositions in the absence of the stresses are denoted by (Xi’“. Xi”‘) and (X!$“‘, X$“‘), we find the above condition to be satisfied by a set of parallel tangents at (X2’), X2’)) and (Xx’? Xjt’)) with a vertical spacing of c,, (R@ - #In). In the special case of $2(t)= #it), (X2”, Xi’)) and (X’$? X$‘o)
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ation of connectivity and its consequences will be more fruitful, however, if the consequences of thermodynamics and geometry can be separately assessed, at least in some idealized cases initially. Such an approach will be taken here. In section 3, we exactly solve a model in which both phases are unconnected so that the finding can be entirely attributed to twophase thermodynamics and its consequences are fully explored in the three subsequent sections. The issue of phase connectivity is then taken up again to present a &ore realistic picture of diffusi&tai creep in twophase materials. Fig. 2. Diagram of compositional chemical potential of a two-phase system. A/J* is taken to be CT,(Q”’ - flu’) in section 2 and r~.(@” - @I”) - I-K(Q’” + Qcs) in section 4.
reduce to (Xi”‘, Xi(“) and (X k)““.X0,““), hence, there is no compositional redistribution within each phase domain. This conclusion for interphase boundaries in the multiphase materials contrasts with Herring’s observation of the compositional gradient which generally occurs in single-phase materials in which D.4 + DR. From the standard procedure [4], the compositional chemical potentials are
(7)
on the phase boundaries. Again at Q”’ = Q”“, the second term vanishes as expected. Although this simplification does not hold in a more general case, unlike equation (5), boundary values of pA,, and pus0in equation (7) are always independent of D, and D,, for a multi-phase binary alloy. The decoupling of the driving force from kinetics in the latter case, being a new feature which distinguishes a two-phase material from a single-phase material, leads to rather unexpected results in diffusional deformation, which will be demonstrated in the next two sections. The above finding is mostly thermodynamic in nature.t The other important element in deformation of two-phase alloys is phase distribution concerning, in particular, the phase connectivity. Generally, the connected phase must deform while the unconnected phase need not (though it may), although the breaking of full connectivity into a partial one will substantially modify the situation. Further consider-
t Strictly speaking, it also bears on mechanical equilibrium, which requires that the same normal stress be supported on either side of a phase boundary. This, of course, is always satisfied.
3. DIFFUSIONAL
CREEP RATE
An exact calculation is now performed for a square grain array of two phases, as shown in Fig. 3. Being highly idealized, this configuration provides a limiting case of no phase connectivity of an eutectic or duplex polycrystal and highlights a unique feature of twophase materials by allowing only interphase boundaries to exist. The analytic evaluation of all the essential quantities of interest can be effected using this model. For the symmetric configuration studied here we expect that the average normal stress on each of the phase boundaries of a square grain to be the same as the applied normal stress. Furthermore from the nature of the linear field equations and their boundary conditions, equations (2, 4, 6, 7), which are all of first order in o,,, we conclude that the deposition (removal) of matter due to J$‘, Jg’, J:“’ and J’d”, respectively, must be uniform along all the boundaries to avoid back stresses.
-(u)
I
-2
Fig. 3. A square grain, two-phase array under stress creeps by lattice diffusional transport.
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DIFFUSIONAL CREEP OF TWO-PHASE MATERIALS
Recall that the solution of the boundary problem of the following type
value
v’fcx y) = 0
(8)
where f = -b, at Ix/ or lyl = d/2, with (a,) = (a) at 1x1= d/2 and (a,) = -(a) at lyl = d/2, is cr, = &a>(1 - (2y/d)‘) u, = -+(a>(1
at
1x1= d/2
due to the stress bias their signs on the tensile boundary and the compressive boundary are opposite to each other. The overall creep strain rate is defined as 2 = (3’) + j”“)/&j.
(12)
Solving the above equation, we find
(9)
- (2x/d)*) at IyI = d/2
and f=
Pa)
- x’),‘d’.
6(a)(y2
+
We can verify that for our problem exactly the same stress distribution as shown above applies, which in turn drives the atomic fluxes. At the phase boundaries,
-(opx\; + -
Dpxy’
Q(ll
q-y,,y1 X
J”’ = + D$‘X;(” 6(a) A -kTR’“d
t
@‘)X;(“l _ @nX$l) X0,“” _
x;‘l)
> (10)
In the above equations the “+” sign applies on the tensile boundary while the “-‘I sign applies on the compressive boundary. The net material deposition on the interphase boundaries can be evaluated from the following equations
J’:)
+
J’“) A
=
X0”’ A
_ *,,,
+
XY’)
g
(11) Jg)
+
Jf)
=
X;(‘)
$1) nc’)
+
X;(“‘$
xy
_ _
(yll)
fplxy, xyl
> z 11.
(13)
The resultant grain shape is shown in Fig. 4 for the cases of D$Df’) >>DI(‘f# and Dg)Dg” >>D$)D$“, assuming X~“‘X$“’ > X~(“X~“‘) > 0. The most interesting feature of the figure is that it demonstrates that the deformation of two phases are not identical, indeed they do not even elongate in the same direction. The overat diffusional creep in this case, according to equation (13), is controlled by the fastest diffusing species along its fastest path (phase), i.e. 02) or Of’ in Fig. 4a and DC’ or D$” in Fig. 4b. It is noted that interphase boundary sliding, and more significantly, interphase boundary migration must have taken place in the deformation of the two-phase materials modeled here. The quantification of these observations and the effects of interface energy will be studied in the next section.
.
4. MIGRATION OF IYTERPHASE BOUNDARIES In the above equations Xi”) and Xi(‘) were used in lieu of Xx’ and Xje in the spirit of linearization. Here .@ is the deposition rate at the interface on the side of phase i at the composition of Xiti’ and X$i’. Again
4.1 Rare of migration. Migration of interphase boundaries is effected by the net transport of atoms of either species across the (bl
D$‘DBm)>>
DC” A DA(=’
Fig. 4. The resultant grain shape of the two-phase system of Fig. 3 under diffusional creep for which the deformation of the two phases are not identical. The center voids are artifacts of the small strain formulation used here.
CHEN:
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interface in question. In the problem considered here, its occurence is a necessary result of the continuity of driving forces across the interface and the conservation of matter. The flux of A atoms, M, crossing the
or, after symmetrization IMA= -;
.
(Ji’ _ J’{“) _ !
i
(15) Similarly, the flux of B atoms, Ms, crossing the interface from phase I towards phase II is (J$’ _ Jlf’)
OF TWO-PHASE
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1659
r. Below a certain threshold stress, migration will be restrained by the surface tension which contributes to the driving forces at the phase boundaries in the following way (41
where K(~)is the curvature of the phase boundary defined to be positive if the phase boundary bows out and it always satisfies the relationship, K(‘)= -I&‘~. We thus define K = K(‘)= -#‘) for simplicity. We can now go through the same procedure which leads to equations (4,6. 7). referring to the free energy diagram shown in Fig. 2. to find, on the interphase boundaries,
.
-
(16)
The net flux of either atoms, &!, crossing the interface from phase I toward phase II. corresponding to the net ~~g~~rio~ of the interface in the reverse direction from phase II toward phase I, is the sum of the above two quantities, M.4 and MI3 M = (Xi”’ - x.;““)-‘(x:“‘J\;’ -
Xp:l’J’:’
-
ow (Ill x,1 J,* ).
-t x.4OIIIIJ,,III1
(17)
In the special case considered in Fig. 3, M is reduced to the following form M = f !?.?&X~~r~ _ X0,““)-2
In the above equation, the ‘+’ sign applies on the tensile boundary while the ‘-’ sign applies on the compressive boundary. As in the case of the overall creep rate, the migration rate is controlled by the fastest diffusing species, while its direction is determined by the consideration that the phase which is richer in the fastest diljEtsingspecies must grow ut the expense of the other phase at the sink, and cite versa at the source. 4.2 Eflect of interfacial energy
Results for the overall creep rate and the migration rate, equation (13) and flS), were obtained negtecting any effects of the interfacial energy of the phase boundaries. This assumption is justified if the driving force, CL?, substantially exceeds the surface tension term which is of the order of IX/d for an interface energy
f TK ~(1)~~‘~~~ ‘_ ~~~(1)) * (
*
As expected, the above equation reduces to equation (7) if frc = 0. At the steady state, we expect K to assume the magnitude which renders M = 0 and to vary finearly with a,, namely, TK/CY,to be a constant. The constant is determined beiow. The solution of the boundary value problem now yields
(2l) @‘Xi”’ J&l’ = z!zkTR”’
&3) d
~(l)X~{lI~_ ~Ii)X~~l) C --
‘ycp
_
/-K fpxyl~
0”
ow1 X.4
Xcyh
+' ~(lilx~~ll _
xy
>
on the interphase boundaries, with the same sign notation used earlier. Meanwhile equation (17) for the rate of migration still holds. Therefore, the value of ~K/u” which renders M = 0 can be calculated from
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equations (9) for the stress distribution. we find
equation (17) and (21). i.e.
(25) where 9 is the quantity on the RHS of equation (22). The critical condition occurs when dy’d.r approaches infinity, at 1.~1= d/2 at the grain corner. which defines a threshold stress +
(,Y~“‘X;“‘D’,”
x
(~mx~111’
+ +
X~111’X,o”11~~11)
~1lllxy1)~~
(22)
Finally, SC’)and s(“) can be obtained as before to yield the creep strain rate. For brevity, we only present the result in the case of Qc” = a(“’ = R
Below(~hhrlrrllold,migration is suppressed by the bow-out of the interphase boundary and the overall creep rate is given by equation (23). Above (Q)~,_,,~~~, migration becomes possible and the overall creep rate approaches that described by equation (13). It is also noted that even below the threshold, migration is still possible during the transient state before the steady state bow-out is fully developed. Indeed, the steady state can only be reached after certain creep strain. This conclusion holds for both the subcritical case and the supercritical case. A schematic diagram of bow-out is shown in Fig. 5.
5. THE ROLE OF THE INTERFACE + X;““XpI”“[(X~C” + X;(“‘)Djf” + (X?
+ XoH’“‘)Df”] 1
(23)
which in the case of a single-phase system reduces to the result of Herring [2] as it should, namely k
_
6Q kTd2
441 D.4X; + D,,Xo,’
(234
In the context of the simplified notion of two-phase materials in which both phases are dispersed and the interphase boundaries are the only type of boundary that exist, major observations made in section 3 and 4 apply, not only to diffusional creep, but also to other kinetic processes which involve incoherent phase boundaries as sources and sinks. In the following, a physical interpretation is afforded for a better under-
It can be verified that the diffusional creep rate in the present case, equation (23), in which migration is suppressed by the bow-out of the phase boundaries, is controlled by the slowest diffusing along its fastest path (phase). 4.3 Threshold stress The profile of the interphase boundary can be determined completely from its curvature. A critical condition is reached when the bow-out at the grain corners extends an angle of rr/2 (Fig. 5), a condition similar to the Orowan’s condition in the dislocation theoiy [YJ. From the geometric consideration [6]
where X, y are the plane coordinates of the phase boundary, Solving dy1d.x as a function of x and noting
c
Fig. 5. The antisymmetric bow-out of phase boundaries can be superposed on the diffusional creep when the interfacial energy restrains boundary migration at the steady state.
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standing which eventually focuses on the role of interface in kinetic processes. Kinetic processes in multi-component systems have often been thought to be controlled by the slowest diffusing species along its fastest path. Essentially, this concept follows Herring’s analysis of the compositional gradient which is set up on application of a driving force. Kinetically, this corresponds to a build-up of the fast diffusing species at the sinks in relation to the sources, hence, a situation of up-hill diffusion for the fast-diffusing species, and the opposite one of down-hill diffusion for the slow-diffusing species. At the steady-state, the overall diffusion of all species due to the applied driving force and the compositional gradient are properly adjusted to be in proportion to their respective atomic fractions. The validity of the above statement depends on the ability of the system, at the sources and sinks in particular, to sustain a compositional redistribution which, by definition, deviates from the equilibrium composition (at zero stress) without incurring any phase changes or microstructural evolution. A counter-example was provided by Chen [7] in Coble creep of multi-component single-phase materials. Migration of grain boundaries, once initiated, will leave alloyed and de-alloyed zones behind to alleviate the excessive composition build-up along grain boundaries. The diffusional creep rate in this case is controlled by the fastest diffusing species and the deformation is considerably faster than Herring’s prediction. The results of sections 3 and 4 demonstrate that more counter-examples exist in mult-component multi-phase materials. In fact, the ‘anomalous’ fast diffusional creep [7] which is facilitated by grain boundary migration in the single-phase material becomes entirely normal in the present case. It was shown earlier that the compositions of either phases are locked in by the requirement of phase equilibrium at the phase boundaries to be those of equilibrium compositions at zero stress. Physically, the existence of two phases which are intermixed at the scale of the grain size assures the presence of an adjacent thermodynamic reservoir for each grain to prohibit and indeed adsorb, any build-up of composition within neighboring grains. Thus, diffusion responds solely to the applied driving force and is not affected by the compositional redistribution which, if present, would have slowed down (fastened) the fast (slow) diffusing species. The overall creep rate consequently is controlled by the fastest diffusing species along its fastest path, which in the present case turns out to be the path along the phase where diffusion is fastest. In general, diffusional fluxes set up in this way have at the sinks an excess of the fast diffusing species, and vice versa at the sources. Atoms on either side of the phase boundary must be allowed to jump across the boundary, to maintain thermodynamic equilibrium of the two-phase system. Migration of the phase boundary is hence necessitated. As we found in section 4.1, the migration rate is controlled by the fastest diffusing
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species, while its direction is determined by the consideration that the phase which is richer in the fastest diffusing species must grow at the expense of the other phase at the sink, and vice versa at the source. Migration of this sort alleviates any potential compositional build-up at the boundary and makes possible the fast diffusional flow at the steady state. In other words, in maintaining the phase equilibrium across itself, the phase boundary also facilitates the faster kinetics which would be impossible if boundary migration were inhibited. Hence, as in Ref. [7], it is again a case of migration-assisted diffusional creep, only to be effected by the thermodynamic consideration applicable to phase equilibrium. The connection between migration and enhanced overall kinetics is made even clearer when we consider the other case in which migration is suppressed at low driving forces. Below the thresholds of migration, the exchange of atoms at the phase boundary with the reservoir (the neighboring phase) is not permitted. Only under such circumstances does compositional gradient again become pronounced and the creep rate controlled by the slowest diffusing species along its fastest path. It would be interesting to examine the more general case beyond the binary system treated here. The result of such study will be reported shortly later [14]. It is interesting to note that the specific model of Fig. 3 implies that the two-phase alloy may creep via diffusional flow at a rate faster than the single-phase alloy made of either phase I or phase II. This is because diffusional creep of a single-phase alloy is controlled by the slowest species, while that of a twophase alloy is controlled by the fastest species. This remarkable result, however, had a direct bearing on the assumption that only interphase boundaries exist in the microstructure and obviously needs modification when phase connectivity is taken into account. This is the subject of the next section. 6. PHASE CONNECI’IVITY ANSTRESS REDISTRIBUTION 6.1 Phase connecticity The model developed in sections 3-4 is unrealistic in that phase connectivity is entirely avoided in the configuration of Fig. 3. Generally speaking shortrange and long-range connectivity may exist and their extent depends on the volume fractions of each phase, as well as other factors, such as interfacial energies and transformation kinetics. Here, the range of phase connectivity is defined to be the average length of continuous paths along the same phase in the aggregate. If each phase domain is equiaxed and the phase distribution is random, the range of connectivity is only a function of volume fractions and dimensionality. In a two-dimensional system, long-range connectivity is realized when the volume fraction of the phase in question exceeds approximately 0.5, i.e. when
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the two phases are roughly of equal volume fractions. In a thr~-dimensional system. the critical volume fraction is reduced to 0.4, due to the additional degree of freedom in a three-dimensional network. The above statement is substantiated by the theory of random alloys in physics, in which percolation is investigated, and a mechanical analogy in terms of deformation was analyzed by a number of authors(S-IO] using the technique of self-consistent method. It is deduced from these considerations that spatial connectivity can be minimized if the volume fractions of the constituent phases are roughly the same, and the majority phase always possesses long-range connectivity. Although in a two-phase spatial aggregate it is difEcuIt to avoid fang-range connectivity, even with equal volume fractions for both phases, it becomes definitely possible to eliminate long-range connectivity entirely with three phases of equal volume fractions, 4.2 Stress re~isfr~~t~tion Turning to the point of stress redistribution among phases, we shall examine the consequences of phase connectivity with the aid of Fig. 6a and 6b. The configuration of Fig. 6b contains only short-range connectivity at equal volume fractions for both phases, while that of Fig. 6a contains some long-range connectivity for phase I which is also the majority phase. For simplicity, the latter case is chosen to possess certain symmetry which makes analytic solution possible. As before, diffusivities of different species are different and dependent on the phases, The most interesting case, however, is the one in which phase I is the ‘harder’ phase due to lower diffusivities therein and its stress distribution will be compared with that of Fig. 3. 6.2,l Long-range connectirity. Consider first Fig. 6a. On the side of compression, no stress redistribution is necessary due to symmetry. On the side of tension, the second and the fifth row along which the harder phase is continuous should carry higher
(a)
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stresses, while all other rows should carry the same stress, again due to symmetry. The detailed analytic solution of diffusional creep of this con~guration demonstrates that the major material flows in a unit cell, i.e.. cdefgh in Fig. 6a. always originate from inter. phase boundaries, though they terminate both at interphase-boundaries and grain-boundaries. As before, interphase boundaries migrate, and certain rows (namely. rows 1, 3.1 and 6) deform almost entirely by fast diffusion in phase II and by attendant migration of phase boundaries. In contrast with the previous case. Fig. 3, those vertical grain-boundaries in rows 2 and 5 of the harder phase develop segregation which slows down diffusion drastically. The net result is that the slower diffusing species in domains d and g controls the overall kinetics. It is nevertheless correct to’state that the preservation of phase equilibrium at phase boundaries has already facilitated the overall kinetics considerably, since many sluggish diffusion paths were by-passed. It is also confirmed by the analytic solution that the ‘local’ deformation resistance aiong rows 1, 3,4 and 6, taken to be the ratio of the average normal stress to the average normal strain rate, is indeed lower than that of a single-phase alloy of either phase I or phase II. In other words, the two-phase alloy still flows easier locally wherever long-range phase connectivity is lost, but the harder phase controls the macroscopic strain rate if it is connected over a long-range in the microstructure. 6.22 Short-range connectivity. The configuration in Fig. 6b contains no long-range phase connectivity. It is then possible to isolate regions of each phase which are completely surrounded by the other phase. Two such regions, i.e. mnpq and ijkl, are shown in Fig. 6b. Due to the lack of symmetry, the problem is not analytically tractable. Mechanically, however, no excessive stress redistribution. by more than a factor of three or so [9], is possible in this case of equiaxed grains, for the simple reason of mechanical equilibrium. It should be noted that despite its frequent use in this field 111. 121, the upper bound analysis which
fb)
SHORT-RANGE
CONNECTIVITY
Fig. 6. Two-phase systems with long-range connectivity in phase I. (a) and with oniy short-range connectivity in both phases, (b). The diffusional flows of major importance are shown in a unit ceil cdefgh in (a) and in a representative domain in (b), assuming fast diffusion in phase 11.
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maintains a uniform strain rate everywhere is often unsatisfactory, for both mechanical reasons in general [8, lo] and geometric reasons in particular for diffusional creep [13,14]. For the case of two-phase flow by diffusion, the disparity between the inherent deformation resistance of different phases is so significant that we expect an almost complete breakdown of the upper bound analysis for the estimation of stress redistribution. Hence, isolated regions, e.g. mnpq of phase I and ijkl of phase II, should not be stressed very differently and should creep by the same mechanism of fast diffusional creep described earlier. More specifically, this will take place via diffusion along the softer phase, i.e. phase II, with only phaseboundaries as the source and the sink, and is assisted by migration of phase-boundaries. Although the length of diffusion becomes somewhat longer, it still has the advantage of being controlled by the fastest diffusing species along its fastest path (phase). For this reason, we expect that the two-phase alloy with only short-range phase connectivity still creeps relatively easily by diffusional flow, in a manner very similar to the one described in sections 35. The above discussion, of course, is valid only if the connectivity is truly a short-range one. The intermediate case between short-range connectivity and long-range connectivity is difficult to assess, but should lie between the two limiting cases treated above. Operationally, more ease of diffusional creep should be possible to realize if a third phase is added to the alloy and the volume fractions of all three phases are roughly the same. This, however, is only possible with ternary systems on which we will report later [14]. It is worth noting that the isolated regions of phase I, e.g. mnpq, also undergoes shape changes and has an apparent deformation strain. This statement does not negate our previous assumption that phase I is a much harder phase than phase II. The shape changes happen by the migration of phase boundaries which can be effected without necessarily the participation of diffusional flow through phase I, since an alternative fast diffusion path is available through the surrounding phase II. Indeed, this is perhaps the most significant feature of diffusional creep which differs from continuum deformation. In the latter case, a hard inclusion may remain rigid while the soft matrix flows around. In diffusional creep, a hard inclusion must deform, even though the diffusional flow is restricted to the soft matrix. The consequence of phase equilibrium and phase boundary migration is again observed here. 7. APPLICATIONS The results obtained here have some interesting applications in suck phenomenon as superplasticity [15]. Some of these will be briefly discussed below, although a detailed analysis and review of the experimental evidence is beyond the scope of the present paper.
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It is well known that superplasticity occurs more often in two-phase materials. This feature is usually attributed to the fact that grain growth can be more feasibly suppressed in a two-phase microstructure at a fine grain size which is a prerequisite for superplasticity. Our results, however, suggest that perhaps it is the unique thermodynamic and kinetic conditions which are realized in two-phase materials. especially at a one-to-one ratio of phase volume fractions, that bring about superplasticity. In particular, a twophase microstructure may creep via diffusional flow locally and sometimes even globally at a rate much faster than either constituent phase in the form of a single-phase polycrystal, due to the different rate-controlling mechanisms in question. This remarkable and yet entirely overlooked feature of two-phase materials certainly is a very important, if not the most important, cause of superplasticity which has been most often reported in two-phase alloys. The introduction of a third phase should strengthen this effect. The migration of phase-boundaries in the manner described by Figs -1 and 5 is also pertinent. We note that in Fig. 5 each phase domain is being ‘pinched off by inward hexing of the phase-boundaries, leading at the same time to the eventual approach of two initially separated neighboring grains. Such morphological evolution is reminiscent of the thermally activated grain switching model proposed by Ashby and Verrall [16]. Indeed. the form of the threshold stress calculated by these authors using the absolute rate theory is very similar to our result of equation (26). The analysis in this paper in our opinion provides a somewhat more natural and general basis for such evolution. It is more natural since equilibrium is maintained everywhere in the evolution for which a kinetic path is explicitly described in terms of diffusion and migration. It is more general since the possible cause for the threshold in our model is not limited to the interfacial energy and may include such well-known phenomenon as Langmuir adsorption at the boundary. In particular, if the non-linear dependence on driving force of boundary migration [17] is included in our analysis, a sigmoidal ( - 0 behavior is expected near the threshold. It should be noted that in reality processes described by Figs 4 and 5 are likely to operate simultaneously due to grain size distributions, internal stresses, and complex transient behavior of boundary sliding and boundary migration [ 181. One interesting feature of migration of phase boundary due to unbalanced atomic fluxes at the boundary, equation (18), is that migration becomes very fast if X0:” approaches X0,““. Although a detailed analysis taking into account the energetics of the phases and the interface and the mobility of the boundary is still required for any conclusive prediction, this observation raises some interesting questions concerning the possible role of diffusion in a nominally diffusionless allotropic transformation. In particular, if diffusion is active at the transformation
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temperature, applied
stress,
both
DIFFUSIONAL
the internal
which
stresses
can cause diffusion
and
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the
of atoms,
might have important effects on the stability of the phases and the rate of transformation. It is interesting to note that this phenomenon of diffusion-induced phase-boundary migration may explain some observations [19] which were, to this date, broadly associated with the so-called transformation-induced superplasticity [20]. Acknowledgements-This research is supported by the National Science Foundation Grant No. DMR-7824185. Helpful comments of Professor M. F. Ashby are gratefully appreciated.
REFERENCES 1. F. R. N. Nabarro, Report of a Conference on rhe Strength of Solids. Phys. Sot. London, p. 75 (1948). 2. C. Herring, J. appl. Phys. 21, 437 (1950). 3. R. L. Coble, J. appl. Phys. 34, 1679 (1963). 4. J. W. Christian, in The Theory of Transformations on Memls and Alloys, 2nd edn, Chap. 6, Part I. Pergamon Press, Oxford (1975). 5. F. A. McClintock and A. S. Argon, in Mechanical Behavior of Materials, Chap. 4. Addison-Wesley, Reading (1966).
6. J. Mathews
MATERIALS and
.bfethodsofPhysics,
R. L. Walker, in Mathemarical 2nd edn. Chap. 15. Benjamin, New
York (1970). 7. I.-W. Chen, Acra meroll. (1982). In press. 8. J. W. Hutchinson, Proc. R. Sot., Lond. 319A, 247 (1970). 9. I.-W. Chen and A. S. Argon, Acta merall. 27, 785 (1979). 1”. en M. P. Cleary, I.-W. Chen and S.-M. Lee, J. Engng Me&. A.S.C.E. 106. 861 11980). 11. A. K. Ghosh and R.. Raj. kn’metall. 29, 607 (1981). 12. R. Raj and A. K. Ghosh. Acra metall. 29, 283 (1981). 13. J. H. Schneibel, R. L. Coble and R. M. Cannon, Acta metall. 29. 1285 (1981). 14. I.-W. Chen, To be presented at Annual Meeting, Am. Ceram. Sot. (1982).
15. J. W. Edington. K. N. Melton and C. P. Cutler. Prog. Mater. Sci. 21, 2 (1976). 16. M. F. Ashby and R. A. Verrall, Acta merall. 21, 149 (1973). 17. K. Liicke, K. R. Rixen and F. W. Rosenbaum, in The Nature and Behavior of Grain-boundaries (edited by H. Hu), p. 245. Plenum Press, New York (1972). 18. H. C. Chann and N. J. Grant. Trans. metoll. Sec. A.I.M.E. 206: 544 (1956). 19. L. F. Porter and P. C. Rosenthal, Acta me&all. 1. 504 (1959). 20. G. W. Greenwood and R. H. Johnson. Proc. R. Sot. 283A. 403 (1965).