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Modelling of kinetics of directional coarsening in Ni-superalloys
Acta mater.
Pergamon 0956-7151(95)00349-5
Vol. 44, No.
6, pp. 2557-2565,
1996
ElsevierScienceLtd Copyright 0 1996Acta Metallurgica Inc. Printed in Great Britain. All rights reserved 1359-6454/96 $15.00+ 0.00
MODELLING OF KINETICS OF DIRECTIONAL COARSENING IN Ni-SUPERALLOYS J. SVOBODA
and P. LUKAS
Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zizkova 22, 616 62 Brno, Czech Republic (Received 31 October 1994; in revised form 24 April 1995) Abstract-A model of kinetics of diffusion controlled directional coarsening (rafting) of y'-precipitatesin Ni-superalloys has been developed. The model is based on the global energetic treatment using variational formulation of thermodynamics of irreversible processes. The total energy of the investigated system consists of the elastic term, the potential energy of the loading system and the energy of dislocations stored on the boundaries of y-channels. The morphological changes of the y ‘-particles are conditioned by diffusive fluxes balancing the difference in the chemical composition of the y- and y/-phases. The diffusion is connected with energy dissipation. As the process of rafting has been described by the development of one parameter, the energy balance of the system was a sufficient basis for the determination of the rafting kinetics. The results of the model are in good agreement with the experimental data for the rate of rafting and the magnitude of the creep strain accompanying rafting.
1. INTRODUCTION
Monocrystalline nickel-base superalloys are strengthened by a high volume fraction of coherent y’-partitles regularly distributed in a y-matrix. During creep deformation the initially cuboidal y ‘-particles coarsen directionally, forming rafts. The prerequisite for the rafting process to occur is sufficiently high temperature, typically over 900°C [e.g. 141. This strongly suggests that rafting is basically a diffusionconditioned process. The directional rafting occurs only during the creep process. The driving force for the diffusion is due to differences of local internal stresses. Socrate and Parks [2] proposed a model for the driving force of rafting. They assumed that this driving force was given by the decrease of the total energy, which is the sum of the elastic energy of the crystal plus the potential energy of the loading system. The aim of this paper is to present a model of kinetics of rafting which also takes into accountbesides the elastic energy of the crystal and potential energy of the loading system-the energy terms reflecting the plastic deformation of the loaded crystal connected with rafting. 2. MODEL 2.1. Basic considerations There are two ways how the rafting of y’-particles can take place in (001) single crystals under tensile load. The first way is the dislocational and/or diffusional flow of matter from the vertical channels to the horizontal ones. In this way the vertical channels become narrower and the horizontal channels
become wider. This type of rafting would result in a relatively large plastic deformation of about 209/o (for the fully developed rafts). Moreover, the rafting would be insensitive to the sign of the misfit parameter 6. The second way is rafting due to migration of the y/v’ interface. The migration is conditioned by the diffusive fluxes balancing the different chemical composition of the y- and y/-phases and it need not be associated with any plastic deformation of the specimen. The diffusion in this case leads only to exchanges of the positions of different atoms; this kind of diffusion is not conditioned by the existence of sources and sinks of vacancies at the y/v ’interfaces and thus it does not require the presence of dislocations at all y ly ’ interfaces. Our model assumes the second method of rafting, i.e. the migration of the y/r ’ interfaces. Let us consider the case of a (100) oriented single crystal under tensile load (Fig. 1) with a negative misfit. The experimental data clearly show that at low and intermediate applied stresses the creep deformation takes place in the y-matrix [5]. Even at high applied stresses the process of creep deformation starts in the matrix and only later on does the cutting of y’-precipitates by dislocations occur [4]. The stress deviator in the horizontal channels is considerably higher than that in the vertical channels. This is due to the fact that the negative misfit increases the stress deviator in the horizontal channels and decreases the stress deviator in the vertical channels. Thus, the creep process starts in the horizontal channels, namely the grown-in and newly generated dislocation loops spread in these horizontal channels in the whole crystal. In many cases the creep curve exhibits an
2551
2558
SVOBODA and LUKciS:
DIRECTIONAL
IL
1 11 6 Fig. 1. Scheme of y’-particles in y-matrix. P denotes a pair of dislocations which can annihilate provided they climb to the edge of the y’-particle.
incubation period. Simultaneously, the dislocations are deposited on the horizontal ;l/v’ interfaces, decreasing the stress deviator in the horizontal channels. This is why the creep in the horizontal channels slows down. It seems very plausible that the primary creep stage corresponds to the plastic deformation in the horizontal channels. The experiments show that a certain degree of plastic deformation in horizontal channels is necessary before rafting starts [3,4]. The plastic deformation of the channels generally requires that a certain threshold stress state is attained, for which the energy released by this plastic deformation is equal to the energy of dislocations deposited on the r/v’ interfaces. Later we shall show that the corresponding threshold stress is inversely proportional to the width of the y-channel. The conditions for the plastic deformation in the vertical channels are much less favourable than those in the horizontal channels. Consequently, the plastic deformation in the vertical channels either does not occur at all during the primary creep stage or only to a very limited degree; the corresponding density of dislocations on the vertical yly’ interfaces is then nil or low. It can be assumed that shortly after the specimen is loaded such a stress state is established which is characterized by a stress in the horizontal channels slightly exceeding the threshold value and by a low density of dislocations deposited on the vertical y/v’ interfaces (caused by an almost negligible plastic deformation). This is the starting state in our model. The further development of the system will go along two parallel lines. (i) The recovery of the dislocation structure, i.e. the non-conservative motion and annihilation of dislocations on the y/r ’interfaces, will lead to a softening of the y-channels and release of the dislocation activity in the vertical channels. This, in turn, results
C’OARSEhlNG
Ih XI-SUPERAI_LCI\I S
in generation of new dislocations and consequently to an elongatzon of the spccimcn. (ii) The Battening of the -prcclpitates (perpcndicularll to the stress axis) \~a the migration of the ;“;I’ Interfaces results in widening of the horizontal channel. This increases the volume fraction of the plastically deformable material (;-‘-particles are not plastically deformable and the plastic deformation in the vertical channels is assumed to be negligible) and simultaneously the magnitude of the plastic deformation in the horizontal channel as the threshold stress (inversely proporlional to the channel width) decreases. This more intensive plastic deformation in the horizontal channel results in a higher density of dislocations deposited on the horizontal ;‘/;b’ interfaces and to an increase of the macroscopic creep deformation of the specimen. Thus, the flattening of the ?I’-particles changes, as well as the elastic energy of the specimen, the energy of the deposited dislocations and the potential energy of the system. The sum of all these energies represents the total energy of the system. The dependence of this total energy on the morphology of the y’-particles determines the direction and rate of further morphological evolution. The creep process described in (i) takes place also at temperatures, at which no rafting occurs [4. 51. There is no reason why this creep process should significantly influence the process of rafting. On the other hand, the rafting described in (ii) results in narrower vertical channels and makes the plastic deformation in the vertical channels more and more difficult. 2.2. Thermodynamic
approach
In the following derivation the initial y’-particle size will be denoted as L, the initial channel width as h and the particle flattening as 2A (see Fig. 2). For systems under constant temperature, T and pressure, p the total Gibbs free energy
-
l-
Fig. 2. Schematics of rafting process. Tensile load axis is identical with the y-axis. The original cuboidal y’-particle (full lines) flattens into the shape marked by the dashed lines.
SVOBODA and LUKAS:
DIRECTIONAL
is the characteristic potential. Here E is the energy, V the volume and S the entropy of the system. During the evolution of the closed system the Gibbs free energy G decreases and approaches its minimum value corresponding to the equilibrium state. A general treatment of diffusion controlled kinetics of closed systems is presented in Ref. [6]. For the systems characterized by one internal parameter A the rate of its change, A, is given by [6] -dG/dA
= f dR/dA,
(2)
where R is the rate of the total Gibbs free energy dissipation due to diffusive volumetric fluxes j, of individual components. The relation between R and j, can be written as [6]
R=CkT , ciDiQ
s
of the total energy
As already mentioned, the horizontal growth of the y/-particles widens the horizontal channels. The change of the morphology of the y’-particles leads to a change in the following energy terms: elastic energy E elast,energy of dislocations stored on boundaries of the horizontal channels Edls, and potential energy of the loading system E,,,. Thus the total energy, E, is given by E = Eelas,+ EdIs,+ E,,, + constant.
2559
IN Ni-SUPERALLOYS
Let us assume that the stress tensor within the y’-particles is constant. The same holds for the horizontal channels and also for the vertical channels. Due to symmetry the shear stress components are zero. The normal stress components within the particle will be denoted as u,,~, the normal stress components in the horizontal channels as g,,n and the normal stress components in the vertical channels as ~,,v (i = x, y, z). These stresses can also be understood as the average stresses in the respective regions. The equilibrium condition on the plane xz (see Fig. 2) cutting the particle is 2c+(h
- A)(L + A) + a,,,(L = a,,.,,[2(h
+ A)’
- A)(L + A) + (L + A)‘],
(5)
where gl.,app is the applied stress. The equilibrium condition on the plane xz going through the channel is
jf dV, Y
where ci is the concentration, Di the coefficient of diffusion, R, the atomic volume of the ith component and k is the Boltzmann constant. As the fluxes j, are proportional to A, R is a quadratic function of A and equation (2) represents a linear equation for A. During the evolution of the system the change of the volume V is negligible. In fact, the whole term p V in equation (1) is a constant. Further, the total volume of the y’-precipitates is not changed and the entropy borne by dislocations is negligible. Thus also the entropy S of the system can be considered to be a constant during the system evolution. Then G in equation (2) can be replaced by E, and R is the rate of heat production in the system due to stress driven diffusion of components. The equation for the balance of energy-as follows from the first law of thermodynamics-is -dE/dt = R. Applying the identities dE/dt = (dE/dA)A and R = i(dR/dA)A to the equation -dE/ dt = R we find our equation (2). Thus, equation (2) can be interpreted as the equation for the balance of energy. This treatment is analogous to the solution of the problems of mechanical systems with friction and one degree of freedom. The equation for the balance of energy is sufficient for the complete description of the evolution of the system. 2.3. Calculation
COARSENING
(4)
The constant on the right-hand side involves the terms independent of the y’-particles morphology.
%,H
The equilibrium particle is a,,,(L
=
condition
((9
oyy.ap~’
on the plane xy cutting the
+ A)(h + 28) + cZ,v(L - 2A)(h -A) +a,,@
+ A)(L -2A)
= 0.
(7)
The equations (5) and (7) implicitly involve the conservation of the volume of y’-particles within 0.7%. The equilibrium condition on the plane xy going through the channel is OX,”= 0.
(8)
Plastic deformation of the horizontal channels will start after the stress deviator 01 = 2(a,,, - a,,,)/3 reaches its critical value CS$= K/(h + 2A), i.e. the value which is inversely proportional to the length of the shortest possible dislocation segments in the horizontal channels, i.e. to the thickness of the horizontal channel. The estimation of the constant K leading to the value K < 17 N/m is presented in Appendix 1. During the plastic deformation of the horizontal channels the dislocations are deposited on the horizontal y/y’ interfaces relaxing the misfit stresses. Let us denote the stored energy per unit y/y’ horizontal interface area as p. The work done by the internal stresses in the horizontal channels during plastic deformation is partly transformed into this stored energy and partly dissipated. For the case that there is no energy dissipation the relation between the stored energy on one side and the work done by the internal stresses on the other side is given by 2P = (h + 2A)(2~a,,pi
where strain; which stored
=
(h
+
2~)%,,,(~y,.
=
3(h + 2A)~,,,,aJ/2
+ -
ov,HE,,p,)
ox,H)
= 3KaJ2,
(9)
of the plastic E,,~I and a.v,plare components they are related by E,,~~= --:E~,~,. The case in only part of the work is transformed into the energy is treated in Appendix 2.
SVOBODA
2560
and LUK.@i:
DIRECTIONAL
As already mentioned, the plastic deformation of the horizontal channels requires that the stress deviator C: reaches its critical value dependent on A, and that this value is kept during the morphological changes; it holds K@ + 2A) = 2(0,.,, - 0,,,),‘3.
(10)
We do not assume any stored dislocations vertical ;J/v’ interfaces. Then it holds E .,.V
-
on the
=6
6.P
(11)
and Ei,v - E,p = 6,
(12)
where S is the misfit. On the other hand, there are stored dislocations on the horizontal rkj’ interfaces, which results in -
&,H
E,,p =
6 +
(13)
&+.
The components of the elastic strain tensor are related to the stress components via the elastic A and B in constants. Denoting these constants the y-matrix and A ’and B ’in the y ‘-precipitates, the Hook law has the form O7.H =
&,H
+
B(E,,H
CI>,H=
A$,H
+
2%
01.~ =
A
‘6,~
+
o,,,~ =
A
'Ey,p+
+
(14)
E.Y,H)
IN NI-SUPERALt.OYS
Here tt was assumed that the redistrtbution of the internal elastic strains contributes only insignificantly to the specimen elongation. The driving force for rafting, F, is then given by F’= -dE/dA.
B ‘(5
,P +
(16)
E,,P)
2B ‘E,,~
(17) (18)
$v = A+
(19)
+ B(Ex.v + s:,v)
oz,v = As:,v + &,v
+ s,,v).
(20)
Sixteen linear equations (5))(20) can be solved with respect to 16 variables, namely cx,Hr cr!,H, o,,~, o,,~, a x.V > Oy,V 1 O2.V > %,H, EJ.H 1 %,P 1 E>.P > %,v 9 %v > %,v > &y,pl and p. Let us now come back to the energy terms presented in equation (4). The part of the specimen consisting of one y’-cuboid and corresponding parts of y-channels will be taken as the system. The elastic energy can be expressed as (density of elastic energy multiplied by volume) A)*(~,H
Ox,H +
To find the kinetics of rafting it is necessary to express the rate of energy dissipation R as a quadratic function, of the rate of rafting b. Let us assume for the sake of simplicity that the chemical composition of the y’-particles is Ni,Al. Then roughly every fourth atom on the y+’ interface is an Al atom. The horizontal growth of the initially cuboidal 7 ‘-particles requires diffusion fluxes, namely diffusion of Ni atoms from the vertical interfaces to the horizontal interfaces and the diffusion of the Al atoms from the horizontal interfaces to the vertical interfaces. This diffusion is assumed to take place in the channels, as the diffusion coefficient in the ordered y’-particle can be expected to be much lower than that in the y-matrix. Let us first treat the diffusion in the vertical channels, The flux j of Al and Ni atoms (in opposite directions) is given by ,j =
Ayi8.
-
24
(L
+
A)2(2~x,,~x,p
R A1.V --8
L?mb
kT
c,,D,, ;(h - A)RA, (’ + ‘) s ”
After integration R
we obtain kT
A’.V= 12c,, D,, (h - A)nA,
(L + A) (L/2 - A)3A?.
+
The equation for the dissipation due to the diffusion of Ni atoms in the vertical channels, RNi,V, can be obtained from equation (27) by replacing cA,, D,, and a,,
by cNi, Diffusion cally shown the marked
b,
and Q,,
in the horizontal channels is schematiin Fig. 3. Let us consider the flux through element. It holds that
E,,, = -(L
(28)
j, = AX/~. (21)
KllS1= 2(L + h)2P of the
r,
E,,P~,..P 1
The stored energy of the dislocations on the boundary of the horizontal channels is given by
energy
(27)
E,..~ nv:.H)
X(%,v~~x,v+ %.v~~,v + E:.v~;,v).
potential as
j2 d_r. (26)
+2(h - A)(L - 2A)(L - A)
and the expressed
(25)
The corresponding dissipation due to diffusion of Al atoms in the vertical channels, RA,,“, can be written as
,j =,j,/cos + V
(24)
2.4. C’ukulution of’the energ.y dissipation rate
(15)
0 r.v = A%v + B(s>,v + %v)
2-G,,, = (h + 2A) (L +
COARSENING
The dissipation then given by r(x) =
(22) system
+ h)2(h + 2A)%,,,, E,,,,,
can
Y(X) dx, is
m4 (jjcos cc)’da cD(h/2 + A)Q s ,,
= 24cD (h /2 + A)Q This equation the constants
element,
2kTx
kT
be
(23)
in the hatched
(29)
A2x’.
(30)
holds for both Al and Ni, which is why c. D and R are not marked by indices.
SVOBODA and LUKAS:
DIRECTIONAL
COARSENING
IN Ni-SUPERALLOYS
x
E-10 cl
0.5
0.0
RAFTING
PARAMETER
1.0
A/h
Fig. 4. Computed dependence of driving force on rafting parameter.
Fig. 3. Schematics of diffusion flux in horizontal channel. The dissipation in the whole horizontal channel, R,, is then given by integration of r(x), namely by R,=8
(L+ A)/2 r(x) dx s cl kT
= 192cD(h/2
+ A)0
A2(,c+ A)“.
(31)
The total dissipation, both in vertical and horizontal channels, can be written as
R = RAI,V + RN,, + hi,” + RNJI. Substitution
from equations
x
(L + A)(L/2 h-A
(27) and (31) leads to
- A)3 +
x
(32)
(L + A)’ 16(h + 2A)
A2= u(A)A2.
The whole mathematical treatment was incorporated into a computer program. The derivative dE/dA was computed numerically using finite differences; equation (34) was integrated in time by means of the Euler method. 3. RESULTS For the numerical simulation of the rafting process we have inserted into the program the elastic constants and misfit parameter used by Pollock and Argon [3] for T = 1050°C. In terms of our nomenclature, the numerical values are A = 102 GPa, B = 29 GPa, A’=95GPa, B’=27GPa and 6 = -0.38%. The dimensions of the structure were chosen to be L = 5 x lo-‘m and h = 5 x 10m8m. Three different values of the applied stress and two values of the constant K were used; they are listed in Table 1. The numbering of the curves in Figs 4-7 corresponds to the numbering of the simulations. Figure 4 displays the dependence of the driving force on the rafting parameter A/h. Figure 5 displays
1 (33)
2.5. Kinetics of rafting In previous sections the dependences E = E(A) and R = R(A, A) were derived. Equation (2) can be rewritten and solved with respect to A; we obtain A =
_dEldA
(34)
u
Table
1. Values of op,app and K used for numerical
Simulation oJ.app WPa) K (N/m)
No.
simulation
1
2
3
4
5
6
200 12
100 12
50 12
200 17
100 17
50 17
0.0
1 .C)
0.5
RAFTING
PARAMETER
A/h
Fig. 5. Computed dependence of creep strain on rafting parameter.
2562
SVOBODA
0.0
0
and LUKAS:
DIRECTIONAL
I
I
I
10000
20000
30000
TIME
[s]
Fig. 6. Computed dependence of rafting parameter on time. the dependence of the creep strain E,,,_ on the rafting parameter. The value of E,,,,~ was calculated from the value of E>,~,using the relation h +2A %,,,p = %P1 L + h and corresponds thus to the plastic deformation of the specimen due to the deformation of the horizontal channels which changes due to rafting. The model has a physical sense only for E,,,,~ > 0 (E,,~,> 0). Consequently, the simulation No. 6 has a sense starting with the value A/h N 0.08 and the simulation No. 5 starting with the value A/h N 0.01. From Fig. 4 it follows for simulations Nos 5 and 6 that the driving force is negative for these and even higher values of A/h. The model yields positive driving force only for E,,,~ > 0. Thus, for A = 0 in the case of simulations Nos 5 and 6 there is no plastic deformation of the horizontal channels and no rafting is predicted by the model. Besides the driving force the kinetics of rafting is crucially affected by the magnitude of the diffusion coefficients in equation (33). For temperature T = 105O”C, the effective diffusion coefficient DeE= 6 x 10-‘6m2/s was calculated for material of a comparable composition [7]. The term kT[l/(c,,D,,R,,) + I/(c,,D,~Q,~)] in equation (33) is then of the order of 1O25J s/m’. This value was used for the computation of rafting kinetics. The results are presented in Figs 6 and 7. In Fig. 6 it can be seen that rafting (for the given temperature of T = 1050°C) is accomplished within several hours or, at most, within several tens of hours. This is in full agreement with the experimental observation [3]. The increase of temperature by 50°C would result in shorter times, roughly by a factor of five. Figure 7 displays the time dependence of the creep deformation related to the rafting. It can be seen that the minimum plastic deformation accompanying rafting is about 0.0024.003. Comparison of the computed
COARSENING
Ih
NI-SUPERALLOYS
creep curves presented in Fig. 7 wtth the cur\c determined experimentally by Mughrabi (‘t al. [4] for monocrystalline nickel-base superalloy CMSX-4 UI 1100 C shows a reasonable agreement (i) in the time needed for the rafting. (ii) in the time dependence oi the creep strain rate and (iii) in the values of J,~,~~,’ corresponding to the maximum o~E,,,,~ and to the end of rafting. It is useful to note that the results presented in Figs 4-7 do not take into account the diffusion creep accompanying rafting which is due to the replacement of the horizontal atomic layers of the y’-particle (with smaller lattice parameter) by the :)-layers (with larger lattice parameter). The calculation taking into account this effect leads to a slightly higher driving force and to a slightly higher creep strain rate accompanying rafting. In both cases the increase is about 10%. 4. DISCUSSION Let us first compare the results of the presented model with the experimentally established facts. (i) Rafting occurs at temperatures roughly above 900°C. This is due to the strong temperature dependence of the diffusion coefficients. It can be shown that the application of the diffusion coefficient estimated for temperatures below 900°C results in unrealistically long times needed to accomplish rafting. (ii) Rafting starts in the primary creep stage and is accomplished after a relatively small creep strain. Figure 5 shows that there is no rafting without creep deformation and that the process of rafting (i.e. the attainment of A/h = I) is a matter of a small creep strain of about 0.2%. (iii) Raising the test temperature and increasing the applied stress accelerates rafting. The effect of the temperature is due to the already mentioned temperature dependence of the diffusion coefficient. The effect of the applied stress can be seen in Fig. 6 to be the expected one, i.e. for otherwise identical parameters, 1 .o
0.0 0
10000
20000
30000
TIME [s] Fig. 7. Computed
dependence
of creep strain
on time.
SVOBODA and LUKAS:
DIRECTIONAL
rafting is accomplished within a shorter time for a higher stress. (iv) A characteristic feature of the rafted structure is the presence of networks of dislocations at the r/v’ interfaces [2]. The presented model takes into account the dislocations deposited during the creep process at the horizontal y/v’ interfaces; the corresponding stored energy [equation (22)] is part of the total energy used for the calculation of the driving force. It can be shown that theoretical spacing of the deposited dislocations is of the same order of magnitude as the experimentally observed one. The tool for the solution of the problem of rafting is the thermodynamics of irreversible processes. The classical thermodynamics of irreversible processes yields phenomenological equations for diffusive fluxes of the individual components of the system; they are given at every point of the system by the respective diffusion coefficient and by the gradient of the chemical potential. In turn, the chemical potential is determined by the concentration of the components and by the stress state at the given point. An explicit formulation of the chemical potential itself is a complex problem. This kind of modelling of the rafting process must be accompanied by calculations of the development of the chemical and stress state in the system. The results of the modelling must be then presented without unimportant details in the form of evolution of several parameters of interest. All this complicated procedure can be overcome by the method used in the present paper. The basis is the proper choice of parameters describing the changes in the system. Then it is possible to derive relatively simply the equations of motion for these parameters [6]. The procedure is thus automatically free of unnecessary details. Several models of rafting have been recently published [2, 8-101. The model of Arrel and Valles [8] is based on the assumption that in the case of negative misfit single crystals under tensile load, dense dislocation networks are formed on the boundaries of the vertical channels, while the dislocation networks on the boundaries of the horizontal channels are loose. The driving force for rafting then stems from the difference of the interface energies of the vertical and horizontal channels. The basic assumption of Arrell and Valles implicitly means that the vertical channels are the main bearers of the plastic deformation. This contradicts all the experimental findings. The paper by Mishin et al. [9] deals with the coarsening of the already rafted structure. In this respect its results are not directly relevant to our paper. The paper by V&on et al. [lo] is based on the assumption that the material from the vertical channels flows into the horizontal channels by diffusion and dislocation creep. In their model the rafting is related to large creep strains exceeding by two orders of magnitude the creep strains found experimentally to be sufficient for the rafting [4].
COARSENING
IN Ni-SUPERALLOYS
2563
The most sophisticated of the quoted models of rafting seems to be the 2-D model by Socrate and Parks [2], which is based on FEM computation. Their model yields the driving force for rafting at its very beginning and shows-in agreement with the experimental findings-that the prerequisite for rafting in the negative misfit crystals under tension is plastic deformation in the horizontal channels. The magnitude of the driving force can be characterized by the stress At, which is of the order of 1 MPa. In this respect the results of Socrate and Parks and our results are in agreement, as the magnitude of the driving force in our case, expressed in terms of stress as -(dE/dA)/L*, is approximately the same. In our opinion, some of the results of Socrate and Parks require a more detailed analysis. They obtained their results using FEM in the frame of small-strain approximation. This approximation assumes that the net of the finite elements is not deformed during the development of the system. Thus, under constant external conditions the system brought-after a certain time-into a steady state, in which some state parameters, such as stresses, stabilize at certain values. The effective normal force per unit area of the interface T” and the driving stress AZ, defined by Socrate and Parks, belong to such parameters. However, their Fig. 12 shows that the value of r” steadily decreases with continuing creep; their Fig. 13 does not indicate any saturation of the value Ar,. The graphs showing the dependence of r, on the normalized arc length also do not show any tendency to stabilization in time. Miiller et al. [ 1 l] used, as well as Socrate and Parks, the Norton relation for the description of the creep strain in the y-channels. Miiller et al. [ll] clearly showed that the misfit stress is quickly relaxed out and that the stress state in the steady-state creep is independent of the initial value of the misfit parameter. As the sign of AT, is identical to the sign of 6 [2], the value of the AZ, in the steady state must be then equal to zero. This would mean that after annealing (no rafting can occur during this treatment) the cuboidal morphology would be forever stabilized. The heat treatment of the superalloy single crystals does involve high temperature annealing and there is rafting during subsequent creep. The information on misfit is preserved in the material in the form of the dislocations deposited on the r/y’ interfaces. That is why the models which do not take into account the presence of the interface dislocations cannot, in principle, physically explain the rafting process correctly. The model of rafting presented in this paper is requiring certainly a first-approximation-model further refinement. It is necessary to combine the rafting model with the creep model, taking into account the discrete nature of the slip process in the y-channels and the recovery of the dislocation structure.
2564
SVOBODA
and LUKAS:
DIRECTIONAL
COARSENING
1N NI-SUPERALLOYS
5. CONCLUSIONS
The model of kinetics of directional coarsening of the y’-particles in Ni-superalloys under load has been developed. The state of rafting was described by one parameter, namely by the flattening of the particles, 2A. Basically, the determination of the kinetics of rafting required:
(9 calculation
of the total energy of the system E = E(A); and (ii) calculation of the rate of the energy dissipation R = R(A, A) due to the diffusive processes connected with rafting. Then the condition of the balance of energy -dE/dt = R represents the
equation
of
total
energy
consists
potential energy
energy of
deformation) horizontal
motion
of
of the
the elastic
of the loading
dislocations
The
energy,
system
(produced
deposited
system.
by
on the boundary
the
and
T/T’
the
INTERFACE
plastic of the
T’
channels.
The computed results on the rate of rafting and creep behaviour are in good agreement with the experimental findings. Specifically, the results show that a certain plastic deformation of the horizontal channels is a necessary prerequisite for the rafting process to start and that the time needed to accomplish rafting is of the order of hours to tens of hours for typical conditions under which rafting occurs. REFERENCES M. Feller-Kniepmeier and T. Link, Metall. Trans. 20A, 1233 (1989). S. Socrate and D. M. Parks, Acta metal/. muter. 41,2185 (1993). T. M. Pollock and A. S. Argon, Artu metall. mater. 42, 1859 (1994). H. Mughrabi, W. Schneider, V. Sass and C. Lang, in Strength of Materials (edited by H. Oikawa et al.), p. 705. The Japan Institute of Metals (1994). T. M. Pollock and A. S. Argon, Actu metall. muter. 40, 1 (1992). J. Svoboda and I. Turek, Phil. Mug. B 64, 749 (1991). J. R6iiEkova and B. Million, Muter. Sci. Engng 50, 59 (1981). D. J. Arrell and J. L. Valles, Scripta metall. muter. 30, 149 (1994); J. L. Vales and D. J. Arrell, Actu metull. muter. 42, 2999 (1994). Y. Mishin, N. Orekhov, I. Razumovskii, G. Alyoshin and P. Noat, Muter. Sci. Engng A171, 163 (1993). M. Veron, Y. Brechet and F. Louchet, Key Engng Muter. 97-98, 213 (1994). L. Miiller, U. Glatzel and M. Feller-Kniepmeier, Actu metall. muter. 41, 3401 (1993).
Fig. Al. Section (101) of the crystal. The activity of the slip system (11 l)[OTl] in the horizontal y-channel results in deposition of dislocations on the )JF~’ interface; their distance along the [loll direction is 1. remaining eight slip systems, i.e. (11 l)[Oil], (11 I)[i IO], (Tll)[oTll, (~ll)[llO],(lTl)[Oll],(lTl)[llO],(llT)[Oll]and (llT)rTlol, have the same Schmid factor and thus contribute equally to the plastic deformation. It is then sufficient to treat one of the slip systems. Let us consider the slip system (11 l)[OTl], i.e. dislocations with Burgers vector b = f[OTl] lying on the plane (111). Figure Al represents the section (TOl) of the crystal. The slip planes (111) are perpendicular to the plane of the paper. The distance of active slip planes measured along the [OlO] direction is H. The vector B denotes the projection of the Burgers vector b onto the plane of the paper. It can be shown that B = a[1211 and, consequently, B = fJ3b. Every elementary slip by b causes plastic deformation (in the direction of the applied stress) B sin c1 %cm= ____ H
Bcoscc I
’
I is the distance between the active slip planes measured along the horizontal rh’ interface. This quantity can be then understood as the distance between the dislocations stored on the y/y’ interface. As there are altogether eight equivalent slip systems, the total plastic deformation is (A.2)
%,pl= 8sdcm and the total density interface is
of the dislocations
stored
p * = 8/i.
Relation Between the Plastic Deformation E,,,~,and the Energ.y Density p of Dislocations Stored on the Horizontal y/y’ Interfaces There are altogether 12 slip type on which the dislocations case the direction of the external to be [OlO]. The Schmid factor the slip direction either [loll
systems of the { 11 l}( 110) generate and move. In our load was chosen (see Fig. 1) of the four systems having or [TOl] is zero. All the
The energy of dislocation estimated using the relation
per
on the y:y’ (A.3)
Combining the last three equations and taking that B = f J3b and cos a = l/,/3 we find p * = 2Ey,p,ib.
APPENDIX
(A.11
unit
Gb’ M,= ~- ln(R :r). 4n where r is the radius of the dislocation is the distance between the dislocations estimations we obtain ln(R/r) x 5 and sity of energy stored per unit area of p = M.&J * = Gbr:, p,.
into account
(A.4) length,
M?, can
be
(A.5) core (r 5 2b) and R (R z h). Using these w = fGb’. The denthe yh’ interface is (A.6)
SVOBODA
and LUKAS:
From the comparison of this equation the main text it follows that
DIRECTIONAL
with equation
K=4Gb/3=17N/m.
(9) in
(A.7)
Due to the splitting of dislocations their energy decreases and K assumes some lower values. The expression for K can be inserted into equation (10) to obtain the threshold stress for the plastic deformation in the y-channels.
COARSENING
2565
IN Ni-SUPERALLOYS
to diffusive processes connected with the rafting. Replacing equation (9) with equation (A.8) and using equation (A.lO) together with the reasoning at the end of Section 2.4, we find for the kinetics of the system -dE*/dA
= f dR/d&
where E* = E’ + Q. In analogy
(A.1 1)
to equation
(4) we can write
E* = EL,,,, + E;,,, + I$,, + constant 2. ModiJcation of the Model for Nonconservative Deformation in Horizontal Channels
Let us assume that only a part f of the work done by the internal stresses in the horizontal channels during plastic deformation is transformed into the stored energy. The rest of the work is apparently dissipated and transformed into the heat. Then the relation between the stored energy per unit interface area, p, and the plastic deformation in the horizontal channels, E,,~,, is [see also equation (9) in the main text] (A.81 2~ = 3fKe,,p, and the energy
transformed
into the heat is
Q = 3(1 -f Energy
conservation
)(L + h)‘Ke,,,,.
= dQ/dt
The elastic energy and the potential off. On the other hand, the stored J It holds that K:,,,, = E,,,,,
> E;o,= f&t
Ed,,, =fE,,,, = 3fG
(A.13) (A. 14)
+ h )zKe,,p,.
Let us now calculate the sum ,??A,,,+ Q. Substitution equations (A.9) and (A.14) yields E;,,, + Q = 3f(L + h)2K&,,,, + 3(1 -f
(A. 10)
where E’ is the total energy of the system in which only part of the work done by the internal stresses is transformed into the stored energy and R is the rate of energy dissipation due
from
)(L + h)*Kq,,
It means that the energy E* does not depend E*=E.
f R,
(A.12)
energy are independent energy does depend on
= 3(L + h)*KE,,,, = Edls,. (A.9)
requires -dE’/dt
+ Q.
Plastic
(A.15)
on f and that (A.16)
In other words, the kinetics of rafting does not depend on the fraction of the energy dissipated during plastic deformation of the horizontal channels. Thus, the calculation presented in the main text is, in this sense, valid generally.
Pergamon 0956-7151(95)00349-5
Vol. 44, No.
6, pp. 2557-2565,
1996
ElsevierScienceLtd Copyright 0 1996Acta Metallurgica Inc. Printed in Great Britain. All rights reserved 1359-6454/96 $15.00+ 0.00
MODELLING OF KINETICS OF DIRECTIONAL COARSENING IN Ni-SUPERALLOYS J. SVOBODA
and P. LUKAS
Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zizkova 22, 616 62 Brno, Czech Republic (Received 31 October 1994; in revised form 24 April 1995) Abstract-A model of kinetics of diffusion controlled directional coarsening (rafting) of y'-precipitatesin Ni-superalloys has been developed. The model is based on the global energetic treatment using variational formulation of thermodynamics of irreversible processes. The total energy of the investigated system consists of the elastic term, the potential energy of the loading system and the energy of dislocations stored on the boundaries of y-channels. The morphological changes of the y ‘-particles are conditioned by diffusive fluxes balancing the difference in the chemical composition of the y- and y/-phases. The diffusion is connected with energy dissipation. As the process of rafting has been described by the development of one parameter, the energy balance of the system was a sufficient basis for the determination of the rafting kinetics. The results of the model are in good agreement with the experimental data for the rate of rafting and the magnitude of the creep strain accompanying rafting.
1. INTRODUCTION
Monocrystalline nickel-base superalloys are strengthened by a high volume fraction of coherent y’-partitles regularly distributed in a y-matrix. During creep deformation the initially cuboidal y ‘-particles coarsen directionally, forming rafts. The prerequisite for the rafting process to occur is sufficiently high temperature, typically over 900°C [e.g. 141. This strongly suggests that rafting is basically a diffusionconditioned process. The directional rafting occurs only during the creep process. The driving force for the diffusion is due to differences of local internal stresses. Socrate and Parks [2] proposed a model for the driving force of rafting. They assumed that this driving force was given by the decrease of the total energy, which is the sum of the elastic energy of the crystal plus the potential energy of the loading system. The aim of this paper is to present a model of kinetics of rafting which also takes into accountbesides the elastic energy of the crystal and potential energy of the loading system-the energy terms reflecting the plastic deformation of the loaded crystal connected with rafting. 2. MODEL 2.1. Basic considerations There are two ways how the rafting of y’-particles can take place in (001) single crystals under tensile load. The first way is the dislocational and/or diffusional flow of matter from the vertical channels to the horizontal ones. In this way the vertical channels become narrower and the horizontal channels
become wider. This type of rafting would result in a relatively large plastic deformation of about 209/o (for the fully developed rafts). Moreover, the rafting would be insensitive to the sign of the misfit parameter 6. The second way is rafting due to migration of the y/v’ interface. The migration is conditioned by the diffusive fluxes balancing the different chemical composition of the y- and y/-phases and it need not be associated with any plastic deformation of the specimen. The diffusion in this case leads only to exchanges of the positions of different atoms; this kind of diffusion is not conditioned by the existence of sources and sinks of vacancies at the y/v ’interfaces and thus it does not require the presence of dislocations at all y ly ’ interfaces. Our model assumes the second method of rafting, i.e. the migration of the y/r ’ interfaces. Let us consider the case of a (100) oriented single crystal under tensile load (Fig. 1) with a negative misfit. The experimental data clearly show that at low and intermediate applied stresses the creep deformation takes place in the y-matrix [5]. Even at high applied stresses the process of creep deformation starts in the matrix and only later on does the cutting of y’-precipitates by dislocations occur [4]. The stress deviator in the horizontal channels is considerably higher than that in the vertical channels. This is due to the fact that the negative misfit increases the stress deviator in the horizontal channels and decreases the stress deviator in the vertical channels. Thus, the creep process starts in the horizontal channels, namely the grown-in and newly generated dislocation loops spread in these horizontal channels in the whole crystal. In many cases the creep curve exhibits an
2551
2558
SVOBODA and LUKciS:
DIRECTIONAL
IL
1 11 6 Fig. 1. Scheme of y’-particles in y-matrix. P denotes a pair of dislocations which can annihilate provided they climb to the edge of the y’-particle.
incubation period. Simultaneously, the dislocations are deposited on the horizontal ;l/v’ interfaces, decreasing the stress deviator in the horizontal channels. This is why the creep in the horizontal channels slows down. It seems very plausible that the primary creep stage corresponds to the plastic deformation in the horizontal channels. The experiments show that a certain degree of plastic deformation in horizontal channels is necessary before rafting starts [3,4]. The plastic deformation of the channels generally requires that a certain threshold stress state is attained, for which the energy released by this plastic deformation is equal to the energy of dislocations deposited on the r/v’ interfaces. Later we shall show that the corresponding threshold stress is inversely proportional to the width of the y-channel. The conditions for the plastic deformation in the vertical channels are much less favourable than those in the horizontal channels. Consequently, the plastic deformation in the vertical channels either does not occur at all during the primary creep stage or only to a very limited degree; the corresponding density of dislocations on the vertical yly’ interfaces is then nil or low. It can be assumed that shortly after the specimen is loaded such a stress state is established which is characterized by a stress in the horizontal channels slightly exceeding the threshold value and by a low density of dislocations deposited on the vertical y/v’ interfaces (caused by an almost negligible plastic deformation). This is the starting state in our model. The further development of the system will go along two parallel lines. (i) The recovery of the dislocation structure, i.e. the non-conservative motion and annihilation of dislocations on the y/r ’interfaces, will lead to a softening of the y-channels and release of the dislocation activity in the vertical channels. This, in turn, results
C’OARSEhlNG
Ih XI-SUPERAI_LCI\I S
in generation of new dislocations and consequently to an elongatzon of the spccimcn. (ii) The Battening of the -prcclpitates (perpcndicularll to the stress axis) \~a the migration of the ;“;I’ Interfaces results in widening of the horizontal channel. This increases the volume fraction of the plastically deformable material (;-‘-particles are not plastically deformable and the plastic deformation in the vertical channels is assumed to be negligible) and simultaneously the magnitude of the plastic deformation in the horizontal channel as the threshold stress (inversely proporlional to the channel width) decreases. This more intensive plastic deformation in the horizontal channel results in a higher density of dislocations deposited on the horizontal ;‘/;b’ interfaces and to an increase of the macroscopic creep deformation of the specimen. Thus, the flattening of the ?I’-particles changes, as well as the elastic energy of the specimen, the energy of the deposited dislocations and the potential energy of the system. The sum of all these energies represents the total energy of the system. The dependence of this total energy on the morphology of the y’-particles determines the direction and rate of further morphological evolution. The creep process described in (i) takes place also at temperatures, at which no rafting occurs [4. 51. There is no reason why this creep process should significantly influence the process of rafting. On the other hand, the rafting described in (ii) results in narrower vertical channels and makes the plastic deformation in the vertical channels more and more difficult. 2.2. Thermodynamic
approach
In the following derivation the initial y’-particle size will be denoted as L, the initial channel width as h and the particle flattening as 2A (see Fig. 2). For systems under constant temperature, T and pressure, p the total Gibbs free energy
-
l-
Fig. 2. Schematics of rafting process. Tensile load axis is identical with the y-axis. The original cuboidal y’-particle (full lines) flattens into the shape marked by the dashed lines.
SVOBODA and LUKAS:
DIRECTIONAL
is the characteristic potential. Here E is the energy, V the volume and S the entropy of the system. During the evolution of the closed system the Gibbs free energy G decreases and approaches its minimum value corresponding to the equilibrium state. A general treatment of diffusion controlled kinetics of closed systems is presented in Ref. [6]. For the systems characterized by one internal parameter A the rate of its change, A, is given by [6] -dG/dA
= f dR/dA,
(2)
where R is the rate of the total Gibbs free energy dissipation due to diffusive volumetric fluxes j, of individual components. The relation between R and j, can be written as [6]
R=CkT , ciDiQ
s
of the total energy
As already mentioned, the horizontal growth of the y/-particles widens the horizontal channels. The change of the morphology of the y’-particles leads to a change in the following energy terms: elastic energy E elast,energy of dislocations stored on boundaries of the horizontal channels Edls, and potential energy of the loading system E,,,. Thus the total energy, E, is given by E = Eelas,+ EdIs,+ E,,, + constant.
2559
IN Ni-SUPERALLOYS
Let us assume that the stress tensor within the y’-particles is constant. The same holds for the horizontal channels and also for the vertical channels. Due to symmetry the shear stress components are zero. The normal stress components within the particle will be denoted as u,,~, the normal stress components in the horizontal channels as g,,n and the normal stress components in the vertical channels as ~,,v (i = x, y, z). These stresses can also be understood as the average stresses in the respective regions. The equilibrium condition on the plane xz (see Fig. 2) cutting the particle is 2c+(h
- A)(L + A) + a,,,(L = a,,.,,[2(h
+ A)’
- A)(L + A) + (L + A)‘],
(5)
where gl.,app is the applied stress. The equilibrium condition on the plane xz going through the channel is
jf dV, Y
where ci is the concentration, Di the coefficient of diffusion, R, the atomic volume of the ith component and k is the Boltzmann constant. As the fluxes j, are proportional to A, R is a quadratic function of A and equation (2) represents a linear equation for A. During the evolution of the system the change of the volume V is negligible. In fact, the whole term p V in equation (1) is a constant. Further, the total volume of the y’-precipitates is not changed and the entropy borne by dislocations is negligible. Thus also the entropy S of the system can be considered to be a constant during the system evolution. Then G in equation (2) can be replaced by E, and R is the rate of heat production in the system due to stress driven diffusion of components. The equation for the balance of energy-as follows from the first law of thermodynamics-is -dE/dt = R. Applying the identities dE/dt = (dE/dA)A and R = i(dR/dA)A to the equation -dE/ dt = R we find our equation (2). Thus, equation (2) can be interpreted as the equation for the balance of energy. This treatment is analogous to the solution of the problems of mechanical systems with friction and one degree of freedom. The equation for the balance of energy is sufficient for the complete description of the evolution of the system. 2.3. Calculation
COARSENING
(4)
The constant on the right-hand side involves the terms independent of the y’-particles morphology.
%,H
The equilibrium particle is a,,,(L
=
condition
((9
oyy.ap~’
on the plane xy cutting the
+ A)(h + 28) + cZ,v(L - 2A)(h -A) +a,,@
+ A)(L -2A)
= 0.
(7)
The equations (5) and (7) implicitly involve the conservation of the volume of y’-particles within 0.7%. The equilibrium condition on the plane xy going through the channel is OX,”= 0.
(8)
Plastic deformation of the horizontal channels will start after the stress deviator 01 = 2(a,,, - a,,,)/3 reaches its critical value CS$= K/(h + 2A), i.e. the value which is inversely proportional to the length of the shortest possible dislocation segments in the horizontal channels, i.e. to the thickness of the horizontal channel. The estimation of the constant K leading to the value K < 17 N/m is presented in Appendix 1. During the plastic deformation of the horizontal channels the dislocations are deposited on the horizontal y/y’ interfaces relaxing the misfit stresses. Let us denote the stored energy per unit y/y’ horizontal interface area as p. The work done by the internal stresses in the horizontal channels during plastic deformation is partly transformed into this stored energy and partly dissipated. For the case that there is no energy dissipation the relation between the stored energy on one side and the work done by the internal stresses on the other side is given by 2P = (h + 2A)(2~a,,pi
where strain; which stored
=
(h
+
2~)%,,,(~y,.
=
3(h + 2A)~,,,,aJ/2
+ -
ov,HE,,p,)
ox,H)
= 3KaJ2,
(9)
of the plastic E,,~I and a.v,plare components they are related by E,,~~= --:E~,~,. The case in only part of the work is transformed into the energy is treated in Appendix 2.
SVOBODA
2560
and LUK.@i:
DIRECTIONAL
As already mentioned, the plastic deformation of the horizontal channels requires that the stress deviator C: reaches its critical value dependent on A, and that this value is kept during the morphological changes; it holds K@ + 2A) = 2(0,.,, - 0,,,),‘3.
(10)
We do not assume any stored dislocations vertical ;J/v’ interfaces. Then it holds E .,.V
-
on the
=6
6.P
(11)
and Ei,v - E,p = 6,
(12)
where S is the misfit. On the other hand, there are stored dislocations on the horizontal rkj’ interfaces, which results in -
&,H
E,,p =
6 +
(13)
&+.
The components of the elastic strain tensor are related to the stress components via the elastic A and B in constants. Denoting these constants the y-matrix and A ’and B ’in the y ‘-precipitates, the Hook law has the form O7.H =
&,H
+
B(E,,H
CI>,H=
A$,H
+
2%
01.~ =
A
‘6,~
+
o,,,~ =
A
'Ey,p+
+
(14)
E.Y,H)
IN NI-SUPERALt.OYS
Here tt was assumed that the redistrtbution of the internal elastic strains contributes only insignificantly to the specimen elongation. The driving force for rafting, F, is then given by F’= -dE/dA.
B ‘(5
,P +
(16)
E,,P)
2B ‘E,,~
(17) (18)
$v = A+
(19)
+ B(Ex.v + s:,v)
oz,v = As:,v + &,v
+ s,,v).
(20)
Sixteen linear equations (5))(20) can be solved with respect to 16 variables, namely cx,Hr cr!,H, o,,~, o,,~, a x.V > Oy,V 1 O2.V > %,H, EJ.H 1 %,P 1 E>.P > %,v 9 %v > %,v > &y,pl and p. Let us now come back to the energy terms presented in equation (4). The part of the specimen consisting of one y’-cuboid and corresponding parts of y-channels will be taken as the system. The elastic energy can be expressed as (density of elastic energy multiplied by volume) A)*(~,H
Ox,H +
To find the kinetics of rafting it is necessary to express the rate of energy dissipation R as a quadratic function, of the rate of rafting b. Let us assume for the sake of simplicity that the chemical composition of the y’-particles is Ni,Al. Then roughly every fourth atom on the y+’ interface is an Al atom. The horizontal growth of the initially cuboidal 7 ‘-particles requires diffusion fluxes, namely diffusion of Ni atoms from the vertical interfaces to the horizontal interfaces and the diffusion of the Al atoms from the horizontal interfaces to the vertical interfaces. This diffusion is assumed to take place in the channels, as the diffusion coefficient in the ordered y’-particle can be expected to be much lower than that in the y-matrix. Let us first treat the diffusion in the vertical channels, The flux j of Al and Ni atoms (in opposite directions) is given by ,j =
Ayi8.
-
24
(L
+
A)2(2~x,,~x,p
R A1.V --8
L?mb
kT
c,,D,, ;(h - A)RA, (’ + ‘) s ”
After integration R
we obtain kT
A’.V= 12c,, D,, (h - A)nA,
(L + A) (L/2 - A)3A?.
+
The equation for the dissipation due to the diffusion of Ni atoms in the vertical channels, RNi,V, can be obtained from equation (27) by replacing cA,, D,, and a,,
by cNi, Diffusion cally shown the marked
b,
and Q,,
in the horizontal channels is schematiin Fig. 3. Let us consider the flux through element. It holds that
E,,, = -(L
(28)
j, = AX/~. (21)
KllS1= 2(L + h)2P of the
r,
E,,P~,..P 1
The stored energy of the dislocations on the boundary of the horizontal channels is given by
energy
(27)
E,..~ nv:.H)
X(%,v~~x,v+ %.v~~,v + E:.v~;,v).
potential as
j2 d_r. (26)
+2(h - A)(L - 2A)(L - A)
and the expressed
(25)
The corresponding dissipation due to diffusion of Al atoms in the vertical channels, RA,,“, can be written as
,j =,j,/cos + V
(24)
2.4. C’ukulution of’the energ.y dissipation rate
(15)
0 r.v = A%v + B(s>,v + %v)
2-G,,, = (h + 2A) (L +
COARSENING
The dissipation then given by r(x) =
(22) system
+ h)2(h + 2A)%,,,, E,,,,,
can
Y(X) dx, is
m4 (jjcos cc)’da cD(h/2 + A)Q s ,,
= 24cD (h /2 + A)Q This equation the constants
element,
2kTx
kT
be
(23)
in the hatched
(29)
A2x’.
(30)
holds for both Al and Ni, which is why c. D and R are not marked by indices.
SVOBODA and LUKAS:
DIRECTIONAL
COARSENING
IN Ni-SUPERALLOYS
x
E-10 cl
0.5
0.0
RAFTING
PARAMETER
1.0
A/h
Fig. 4. Computed dependence of driving force on rafting parameter.
Fig. 3. Schematics of diffusion flux in horizontal channel. The dissipation in the whole horizontal channel, R,, is then given by integration of r(x), namely by R,=8
(L+ A)/2 r(x) dx s cl kT
= 192cD(h/2
+ A)0
A2(,c+ A)“.
(31)
The total dissipation, both in vertical and horizontal channels, can be written as
R = RAI,V + RN,, + hi,” + RNJI. Substitution
from equations
x
(L + A)(L/2 h-A
(27) and (31) leads to
- A)3 +
x
(32)
(L + A)’ 16(h + 2A)
A2= u(A)A2.
The whole mathematical treatment was incorporated into a computer program. The derivative dE/dA was computed numerically using finite differences; equation (34) was integrated in time by means of the Euler method. 3. RESULTS For the numerical simulation of the rafting process we have inserted into the program the elastic constants and misfit parameter used by Pollock and Argon [3] for T = 1050°C. In terms of our nomenclature, the numerical values are A = 102 GPa, B = 29 GPa, A’=95GPa, B’=27GPa and 6 = -0.38%. The dimensions of the structure were chosen to be L = 5 x lo-‘m and h = 5 x 10m8m. Three different values of the applied stress and two values of the constant K were used; they are listed in Table 1. The numbering of the curves in Figs 4-7 corresponds to the numbering of the simulations. Figure 4 displays the dependence of the driving force on the rafting parameter A/h. Figure 5 displays
1 (33)
2.5. Kinetics of rafting In previous sections the dependences E = E(A) and R = R(A, A) were derived. Equation (2) can be rewritten and solved with respect to A; we obtain A =
_dEldA
(34)
u
Table
1. Values of op,app and K used for numerical
Simulation oJ.app WPa) K (N/m)
No.
simulation
1
2
3
4
5
6
200 12
100 12
50 12
200 17
100 17
50 17
0.0
1 .C)
0.5
RAFTING
PARAMETER
A/h
Fig. 5. Computed dependence of creep strain on rafting parameter.
2562
SVOBODA
0.0
0
and LUKAS:
DIRECTIONAL
I
I
I
10000
20000
30000
TIME
[s]
Fig. 6. Computed dependence of rafting parameter on time. the dependence of the creep strain E,,,_ on the rafting parameter. The value of E,,,,~ was calculated from the value of E>,~,using the relation h +2A %,,,p = %P1 L + h and corresponds thus to the plastic deformation of the specimen due to the deformation of the horizontal channels which changes due to rafting. The model has a physical sense only for E,,,,~ > 0 (E,,~,> 0). Consequently, the simulation No. 6 has a sense starting with the value A/h N 0.08 and the simulation No. 5 starting with the value A/h N 0.01. From Fig. 4 it follows for simulations Nos 5 and 6 that the driving force is negative for these and even higher values of A/h. The model yields positive driving force only for E,,,~ > 0. Thus, for A = 0 in the case of simulations Nos 5 and 6 there is no plastic deformation of the horizontal channels and no rafting is predicted by the model. Besides the driving force the kinetics of rafting is crucially affected by the magnitude of the diffusion coefficients in equation (33). For temperature T = 105O”C, the effective diffusion coefficient DeE= 6 x 10-‘6m2/s was calculated for material of a comparable composition [7]. The term kT[l/(c,,D,,R,,) + I/(c,,D,~Q,~)] in equation (33) is then of the order of 1O25J s/m’. This value was used for the computation of rafting kinetics. The results are presented in Figs 6 and 7. In Fig. 6 it can be seen that rafting (for the given temperature of T = 1050°C) is accomplished within several hours or, at most, within several tens of hours. This is in full agreement with the experimental observation [3]. The increase of temperature by 50°C would result in shorter times, roughly by a factor of five. Figure 7 displays the time dependence of the creep deformation related to the rafting. It can be seen that the minimum plastic deformation accompanying rafting is about 0.0024.003. Comparison of the computed
COARSENING
Ih
NI-SUPERALLOYS
creep curves presented in Fig. 7 wtth the cur\c determined experimentally by Mughrabi (‘t al. [4] for monocrystalline nickel-base superalloy CMSX-4 UI 1100 C shows a reasonable agreement (i) in the time needed for the rafting. (ii) in the time dependence oi the creep strain rate and (iii) in the values of J,~,~~,’ corresponding to the maximum o~E,,,,~ and to the end of rafting. It is useful to note that the results presented in Figs 4-7 do not take into account the diffusion creep accompanying rafting which is due to the replacement of the horizontal atomic layers of the y’-particle (with smaller lattice parameter) by the :)-layers (with larger lattice parameter). The calculation taking into account this effect leads to a slightly higher driving force and to a slightly higher creep strain rate accompanying rafting. In both cases the increase is about 10%. 4. DISCUSSION Let us first compare the results of the presented model with the experimentally established facts. (i) Rafting occurs at temperatures roughly above 900°C. This is due to the strong temperature dependence of the diffusion coefficients. It can be shown that the application of the diffusion coefficient estimated for temperatures below 900°C results in unrealistically long times needed to accomplish rafting. (ii) Rafting starts in the primary creep stage and is accomplished after a relatively small creep strain. Figure 5 shows that there is no rafting without creep deformation and that the process of rafting (i.e. the attainment of A/h = I) is a matter of a small creep strain of about 0.2%. (iii) Raising the test temperature and increasing the applied stress accelerates rafting. The effect of the temperature is due to the already mentioned temperature dependence of the diffusion coefficient. The effect of the applied stress can be seen in Fig. 6 to be the expected one, i.e. for otherwise identical parameters, 1 .o
0.0 0
10000
20000
30000
TIME [s] Fig. 7. Computed
dependence
of creep strain
on time.
SVOBODA and LUKAS:
DIRECTIONAL
rafting is accomplished within a shorter time for a higher stress. (iv) A characteristic feature of the rafted structure is the presence of networks of dislocations at the r/v’ interfaces [2]. The presented model takes into account the dislocations deposited during the creep process at the horizontal y/v’ interfaces; the corresponding stored energy [equation (22)] is part of the total energy used for the calculation of the driving force. It can be shown that theoretical spacing of the deposited dislocations is of the same order of magnitude as the experimentally observed one. The tool for the solution of the problem of rafting is the thermodynamics of irreversible processes. The classical thermodynamics of irreversible processes yields phenomenological equations for diffusive fluxes of the individual components of the system; they are given at every point of the system by the respective diffusion coefficient and by the gradient of the chemical potential. In turn, the chemical potential is determined by the concentration of the components and by the stress state at the given point. An explicit formulation of the chemical potential itself is a complex problem. This kind of modelling of the rafting process must be accompanied by calculations of the development of the chemical and stress state in the system. The results of the modelling must be then presented without unimportant details in the form of evolution of several parameters of interest. All this complicated procedure can be overcome by the method used in the present paper. The basis is the proper choice of parameters describing the changes in the system. Then it is possible to derive relatively simply the equations of motion for these parameters [6]. The procedure is thus automatically free of unnecessary details. Several models of rafting have been recently published [2, 8-101. The model of Arrel and Valles [8] is based on the assumption that in the case of negative misfit single crystals under tensile load, dense dislocation networks are formed on the boundaries of the vertical channels, while the dislocation networks on the boundaries of the horizontal channels are loose. The driving force for rafting then stems from the difference of the interface energies of the vertical and horizontal channels. The basic assumption of Arrell and Valles implicitly means that the vertical channels are the main bearers of the plastic deformation. This contradicts all the experimental findings. The paper by Mishin et al. [9] deals with the coarsening of the already rafted structure. In this respect its results are not directly relevant to our paper. The paper by V&on et al. [lo] is based on the assumption that the material from the vertical channels flows into the horizontal channels by diffusion and dislocation creep. In their model the rafting is related to large creep strains exceeding by two orders of magnitude the creep strains found experimentally to be sufficient for the rafting [4].
COARSENING
IN Ni-SUPERALLOYS
2563
The most sophisticated of the quoted models of rafting seems to be the 2-D model by Socrate and Parks [2], which is based on FEM computation. Their model yields the driving force for rafting at its very beginning and shows-in agreement with the experimental findings-that the prerequisite for rafting in the negative misfit crystals under tension is plastic deformation in the horizontal channels. The magnitude of the driving force can be characterized by the stress At, which is of the order of 1 MPa. In this respect the results of Socrate and Parks and our results are in agreement, as the magnitude of the driving force in our case, expressed in terms of stress as -(dE/dA)/L*, is approximately the same. In our opinion, some of the results of Socrate and Parks require a more detailed analysis. They obtained their results using FEM in the frame of small-strain approximation. This approximation assumes that the net of the finite elements is not deformed during the development of the system. Thus, under constant external conditions the system brought-after a certain time-into a steady state, in which some state parameters, such as stresses, stabilize at certain values. The effective normal force per unit area of the interface T” and the driving stress AZ, defined by Socrate and Parks, belong to such parameters. However, their Fig. 12 shows that the value of r” steadily decreases with continuing creep; their Fig. 13 does not indicate any saturation of the value Ar,. The graphs showing the dependence of r, on the normalized arc length also do not show any tendency to stabilization in time. Miiller et al. [ 1 l] used, as well as Socrate and Parks, the Norton relation for the description of the creep strain in the y-channels. Miiller et al. [ll] clearly showed that the misfit stress is quickly relaxed out and that the stress state in the steady-state creep is independent of the initial value of the misfit parameter. As the sign of AT, is identical to the sign of 6 [2], the value of the AZ, in the steady state must be then equal to zero. This would mean that after annealing (no rafting can occur during this treatment) the cuboidal morphology would be forever stabilized. The heat treatment of the superalloy single crystals does involve high temperature annealing and there is rafting during subsequent creep. The information on misfit is preserved in the material in the form of the dislocations deposited on the r/y’ interfaces. That is why the models which do not take into account the presence of the interface dislocations cannot, in principle, physically explain the rafting process correctly. The model of rafting presented in this paper is requiring certainly a first-approximation-model further refinement. It is necessary to combine the rafting model with the creep model, taking into account the discrete nature of the slip process in the y-channels and the recovery of the dislocation structure.
2564
SVOBODA
and LUKAS:
DIRECTIONAL
COARSENING
1N NI-SUPERALLOYS
5. CONCLUSIONS
The model of kinetics of directional coarsening of the y’-particles in Ni-superalloys under load has been developed. The state of rafting was described by one parameter, namely by the flattening of the particles, 2A. Basically, the determination of the kinetics of rafting required:
(9 calculation
of the total energy of the system E = E(A); and (ii) calculation of the rate of the energy dissipation R = R(A, A) due to the diffusive processes connected with rafting. Then the condition of the balance of energy -dE/dt = R represents the
equation
of
total
energy
consists
potential energy
energy of
deformation) horizontal
motion
of
of the
the elastic
of the loading
dislocations
The
energy,
system
(produced
deposited
system.
by
on the boundary
the
and
T/T’
the
INTERFACE
plastic of the
T’
channels.
The computed results on the rate of rafting and creep behaviour are in good agreement with the experimental findings. Specifically, the results show that a certain plastic deformation of the horizontal channels is a necessary prerequisite for the rafting process to start and that the time needed to accomplish rafting is of the order of hours to tens of hours for typical conditions under which rafting occurs. REFERENCES M. Feller-Kniepmeier and T. Link, Metall. Trans. 20A, 1233 (1989). S. Socrate and D. M. Parks, Acta metal/. muter. 41,2185 (1993). T. M. Pollock and A. S. Argon, Artu metall. mater. 42, 1859 (1994). H. Mughrabi, W. Schneider, V. Sass and C. Lang, in Strength of Materials (edited by H. Oikawa et al.), p. 705. The Japan Institute of Metals (1994). T. M. Pollock and A. S. Argon, Actu metall. muter. 40, 1 (1992). J. Svoboda and I. Turek, Phil. Mug. B 64, 749 (1991). J. R6iiEkova and B. Million, Muter. Sci. Engng 50, 59 (1981). D. J. Arrell and J. L. Valles, Scripta metall. muter. 30, 149 (1994); J. L. Vales and D. J. Arrell, Actu metull. muter. 42, 2999 (1994). Y. Mishin, N. Orekhov, I. Razumovskii, G. Alyoshin and P. Noat, Muter. Sci. Engng A171, 163 (1993). M. Veron, Y. Brechet and F. Louchet, Key Engng Muter. 97-98, 213 (1994). L. Miiller, U. Glatzel and M. Feller-Kniepmeier, Actu metall. muter. 41, 3401 (1993).
Fig. Al. Section (101) of the crystal. The activity of the slip system (11 l)[OTl] in the horizontal y-channel results in deposition of dislocations on the )JF~’ interface; their distance along the [loll direction is 1. remaining eight slip systems, i.e. (11 l)[Oil], (11 I)[i IO], (Tll)[oTll, (~ll)[llO],(lTl)[Oll],(lTl)[llO],(llT)[Oll]and (llT)rTlol, have the same Schmid factor and thus contribute equally to the plastic deformation. It is then sufficient to treat one of the slip systems. Let us consider the slip system (11 l)[OTl], i.e. dislocations with Burgers vector b = f[OTl] lying on the plane (111). Figure Al represents the section (TOl) of the crystal. The slip planes (111) are perpendicular to the plane of the paper. The distance of active slip planes measured along the [OlO] direction is H. The vector B denotes the projection of the Burgers vector b onto the plane of the paper. It can be shown that B = a[1211 and, consequently, B = fJ3b. Every elementary slip by b causes plastic deformation (in the direction of the applied stress) B sin c1 %cm= ____ H
Bcoscc I
’
I is the distance between the active slip planes measured along the horizontal rh’ interface. This quantity can be then understood as the distance between the dislocations stored on the y/y’ interface. As there are altogether eight equivalent slip systems, the total plastic deformation is (A.2)
%,pl= 8sdcm and the total density interface is
of the dislocations
stored
p * = 8/i.
Relation Between the Plastic Deformation E,,,~,and the Energ.y Density p of Dislocations Stored on the Horizontal y/y’ Interfaces There are altogether 12 slip type on which the dislocations case the direction of the external to be [OlO]. The Schmid factor the slip direction either [loll
systems of the { 11 l}( 110) generate and move. In our load was chosen (see Fig. 1) of the four systems having or [TOl] is zero. All the
The energy of dislocation estimated using the relation
per
on the y:y’ (A.3)
Combining the last three equations and taking that B = f J3b and cos a = l/,/3 we find p * = 2Ey,p,ib.
APPENDIX
(A.11
unit
Gb’ M,= ~- ln(R :r). 4n where r is the radius of the dislocation is the distance between the dislocations estimations we obtain ln(R/r) x 5 and sity of energy stored per unit area of p = M.&J * = Gbr:, p,.
into account
(A.4) length,
M?, can
be
(A.5) core (r 5 2b) and R (R z h). Using these w = fGb’. The denthe yh’ interface is (A.6)
SVOBODA
and LUKAS:
From the comparison of this equation the main text it follows that
DIRECTIONAL
with equation
K=4Gb/3=17N/m.
(9) in
(A.7)
Due to the splitting of dislocations their energy decreases and K assumes some lower values. The expression for K can be inserted into equation (10) to obtain the threshold stress for the plastic deformation in the y-channels.
COARSENING
2565
IN Ni-SUPERALLOYS
to diffusive processes connected with the rafting. Replacing equation (9) with equation (A.8) and using equation (A.lO) together with the reasoning at the end of Section 2.4, we find for the kinetics of the system -dE*/dA
= f dR/d&
where E* = E’ + Q. In analogy
(A.1 1)
to equation
(4) we can write
E* = EL,,,, + E;,,, + I$,, + constant 2. ModiJcation of the Model for Nonconservative Deformation in Horizontal Channels
Let us assume that only a part f of the work done by the internal stresses in the horizontal channels during plastic deformation is transformed into the stored energy. The rest of the work is apparently dissipated and transformed into the heat. Then the relation between the stored energy per unit interface area, p, and the plastic deformation in the horizontal channels, E,,~,, is [see also equation (9) in the main text] (A.81 2~ = 3fKe,,p, and the energy
transformed
into the heat is
Q = 3(1 -f Energy
conservation
)(L + h)‘Ke,,,,.
= dQ/dt
The elastic energy and the potential off. On the other hand, the stored J It holds that K:,,,, = E,,,,,
> E;o,= f&t
Ed,,, =fE,,,, = 3fG
(A.13) (A. 14)
+ h )zKe,,p,.
Let us now calculate the sum ,??A,,,+ Q. Substitution equations (A.9) and (A.14) yields E;,,, + Q = 3f(L + h)2K&,,,, + 3(1 -f
(A. 10)
where E’ is the total energy of the system in which only part of the work done by the internal stresses is transformed into the stored energy and R is the rate of energy dissipation due
from
)(L + h)*Kq,,
It means that the energy E* does not depend E*=E.
f R,
(A.12)
energy are independent energy does depend on
= 3(L + h)*KE,,,, = Edls,. (A.9)
requires -dE’/dt
+ Q.
Plastic
(A.15)
on f and that (A.16)
In other words, the kinetics of rafting does not depend on the fraction of the energy dissipated during plastic deformation of the horizontal channels. Thus, the calculation presented in the main text is, in this sense, valid generally.