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Microstructure characteristics of rapidly solidified alloys
Materials Science and Engineering, A 178 (1994) 129-135
129
Microstructure characteristics of rapidly solidified alloys R. T r i v e d i
Ames Laboratory, US Department of Energy and Department of Materials Science and Engineering, Iowa State University, Ames, 1A 50011 (USA)
Abstract The microstructural characteristics of rapidly solidified materials were examined in terms of the effect of high velocity on the physical processes that govern microstructure formation. Under rapid solidification conditions, non-equilibrium conditions at the interface and a modified diffusional instability condition play critical roles in the selection of morphology and its microstructural scales. These effects were examined in terms of their influence on the physical processes. Appropriate characteristic lengths of physical processes for rapid solidification were developed, and relationships obtained for low and high growth rate conditions which characterize microstructural transition conditions and microstructural scales of cellular, dendritic and eutectic structures under free and constrained growth conditions. It is shown that all the results at low and high growth rates can be represented by unified expression in terms of redefined characteristic lengths.
1. Introduction
The evolution of solidification microstructures at low velocities has been studied extensively in the literature [1], and the conditions for microstructure transitions and the variations in important microstructural length scales with processing conditions have been well characterized. However, when the velocity of the solid-liquid interface becomes very high, significant changes in the modeling of microstructures are required. Interface velocities of the order of 10 m s can be achieved in laser processing [2], whereas velocities of the order of 30 m s- I have been achieved in highly undercooled melts [3]. Under such high growth rates, two new effects become important in influencing the formation and stability of microstructures. ( 1 ) The effect of high velocity causes the thermal and/or solutal fields to become more localized, which alters the condition for the diffusive instability of the interface, and (2) non-equilibrium effects at the interface become significant which strongly influence the phase and microstructure selection criteria. In order to visualize conceptually how the rapid solidification influences microstructure formation, we first examine the simple case of the low velocity condition. In this case, we find that the evolution of the microstructure can be readily visualized by considering how the processing variables influence the physical processes. We then apply these ideas to rapid solidification processing and show that the same basic forms of results obtained for low velocity can be recovered if 0921-5093/94/$7.00 SSDI 0921-5093(93)04525-M
the effect of rapid solidification is examined in terms of its influence on the physical processes. By defining appropriate characteristic lengths for solute and thermal diffusion, and for the capillarity effect under rapid solidification conditions, a unified expression is obtained for microstructure scales which is valid for low as well as high growth rate conditions.
2. Microstructure scales and transitions
We first consider the microstructural scales and the conditions for the transition in morphology for a single-phase material under low-growth rate conditions and under local equilibrium conditions at the interface. In order to obtain an insight into the results of analytical models, we consider the case of dilute solution with linear solidus and liquidus lines. In this case, simple relationships can be obtained by examining the physical processes that control the microstructure [4]. For directional solidification, the microstructure and phase selection is controlled by the growth rate, temperature gradient and composition. In an undercooled melt, the variables are composition and undercooling. In rapidly solidified droplets, the undercooling is controlled by the nucleation process so that appropriate nucleation models need to be developed to correlate the undercooling as a function of droplet size, cooling conditions and heterogeneous nucleation events which are present [5]. In this paper we only examine the growth phenomenon and assume that the proper undercooling is © 1994 - Elsevier Sequoia. All rights reserved
130
R. Trivedi / Microstructure of rapidly solidified alloys
either predicted from the nucleation models or that it has been measured experimentally [6-10]. For low velocities, and for systems in which interface kinetic effects are negligible, the microstructure is controlled by the solute and thermal diffusion processes and by the capillarity effect at the interface. Each of these processes can be represented by a characteristic length. (1) The diffusion lengths are given by ID= D / V for solute diffusion and by lD,= D ' / V for thermal diffusion, where D and D' are the solute and thermal diffusion coefficients respectively, and V is the velocity. (2) The thermal length for constrained growth is given by I T = A T J G , where A T s = - m l ( G - C ~ ) , in which mz is the slope of the liquidus at the interface temperature and G and C s are interface solute concentrations in the liquid and in the solid respectively. G is the conductivity-weighted temperature gradient in the liquid and solid. The thermal length in directional solidification is proportional to the length of the dendrite or to the extent of the mushy zone. (3) The capillarity length for an alloy is given by do = F/A Ts, where F is the capillarity constant which is a ratio of the interface energy to the entropy of melting per unit volume. The capillarity length is proportional to the critical radius of nucleation. For pure materials, the thermal capillarity length is defined as do'= F q / AH, where cj and A H are the specific heat and the enthalpy of melting, respectively, of the pure material. Note that, A Ts = - m l ( C 1- Cs) , is defined for a general case and it has different values for planar and dendritic growth. The general definition of A Ts is shown in Fig. i for an arbitrary phase diagram, and it is a function of interface temperature. For a phase diagram with linear solidus and liquidus lines in a dilute
solution regime, the limiting values are A T~= A TOfor planar front growth at the solidus temperature, and A Ts= k A T o for dendritic growth at low velocities where the dendrite tip temperature is close to the liquidus temperature. A To is the equilibrium freezing range of the alloy and k is the solute distribution coefficient. Although a specific value of A TS for a planar interface, or dendritic interface at low velocities, has been commonly used in the literature [11, 12] to define characteristic lengths, a general definition of AT s is required to consider properly the capillarity and thermal effects which are present at the interface temperature. 2.1. Microstructural transitions Various microstructural transitions can be readily represented in terms of these length scales of physical processes, as shown in Fig. 2. In order to conform with the analytical results given in the literature, we first consider the dilute solution case and assume linear solidus and liquidus lines. Note again that A Ts = A TO for a planar interface, and A Ts = k A T o for dendritic growth at low velocities. For directional solidification, the planar to cellular transition is given by lo = lT, and the cellular to planar transition at high velocities is given by ID= kdo, and the dendrite to cell transition at high velocities is given by lD = ado, with AT s = AT0, and a is a constant. The cell-dendrite transition is g i v e n b y ID = IT, w i t h a L = k A T o o r IT = k A T o / G . 2.2. Microstructural scales There are several important basic microstructural scales such as dendrite tip radius, primary and secondary dendrite spacings, cellular spacing and eutectic spacing. If we represent these scales by ;t i, then it is possible to describe a general expression for 2i, in the following two forms.
Y ,
'%.
i
i \~-,4 E i
Transitions
~x i ms
i'
l
i
Ii
cs
co
I I i
x,
........ ~
[K=A [tD]' [lT]b
i
ula~]
~i~i
',~~~~~~
ri
i 1
[do]~j
or
cx Composition
Fig. 1. Definition of A Ts for a general phase diagram. T i is the interface temperature.
] [7.2]"1 -A[I DdO]-I +B[/D'do']'I ] Fig. 2. Microstructural transitions and microstructural scales in terms of characteristic lengths of solute, thermal and capillarity processes.
R. Trivedi / Microstructure of rapidly solidified alloys' (1) When all or two of the physical processes are required to control the scale, one obtains a general expression:
J.i= Ai(lD)a(lT )b(do)c
(1)
in which A~ is a constant and the exponents a, b and c are constants whose values must add up to unity [4]. When the thermal effect is negligible, the microstructural scale is obtained for a = c = 1/2. This gives the relationship J,i=Ai(IDdo) 1/2, which indicates that the microstructural scale is the geometric mean of the destabilizing effect of solute diffusion and the stabilizing effect of the capillarity length. Such a relationship is satisfied by a dendrite tip radius in the pure solute or pure thermal case, eutectic spacing, initial secondary dendrite arm spacing, critical wavelength at absolute stability, and primary dendrite spacing at high velocities. For eutectic spacing a weighted average value of the two phases is used for the definition of the capillary length, and for pure undercooled melt the dendrite tip radius is obtained by replacing lD with lD, and d 0 by d0'. When all three physical processes are required to control the magnitude of 2~, these results can often be represented by the geometric mean of all the three processes, i.e. 2i=Ai(lDlTdo) I/3. Such a relationship is satisfied by the wavelength at the critical velocity and by the final secondary dendrite arm spacings. The primary dendrite spacing at low velocities, however, is found to follow the relationship:
;yA,(lD)'/4(1T)'/2(d,,)l/4
131
Note that both the above equations reduce to the result discussed in paragraph (1) when either the thermal or the solute effect is predominant. The constant a*, however, has different characteristic values for different microstructures. The above results give an insight into how the solute diffusion, thermal diffusion and capillarity effect influence the selection of all important microstructural scales of solidified materials. They also show which of the physical phenomena is influenced by the change in a specific processing condition. We now examine how the high growth rate effects influence these basic processes and how they alter the results discussed so far.
3. Rapid solidification conditions Under rapid solidification conditions, two major effects need to be taken into consideration which alter the characteristic lengths of physical processes. ( 1 ) The rapid solidification rate causes a significant change in the solute field (or the thermal field in undercooled alloy melt) which modifies the stability criterion, and it alters the effective diffusion length for the stability of the interface. (2) At high solidification rates, there is a departure from local equilibrium at the interface which must be taken into account, and this modification influences the compositions at the interface, and thus the A T~ term in the thermal (alloy solidification) and capillarity lengths.
(2a) 3.1. Diffusive instability
or
2i=A,(ITR) '/2
(2b)
where R is the dendrite tip radius. (2) When thermal and solute effects are operating in parallel, the following general relationship can be written:
(3) where ~* is constant and is known as the operating point of the microstructure scale, and fl= 0.511 + (Ks/ K~)]. The above relationship is satisfied by the dendrite tip radius in an undercooled melt. The cellular spacing, the marginal wavelengths of an unstable planar interface, and the dendrite tip radius also follow a similar relationship for constrained growth when an appropriate thermal length is used, and this result is
,o,,
(4)
For a sinusoidal perturbation, the stability of the interface is related to the difference in velocities at the peak and valley of the interface. At low velocities, the velocity-dependent term VV C in the diffusion equation is small so that the difference in velocities that controls the stability is proportional to the diffusion lengths lD=D/V or ID,=D'/V, for solute or thermal field respectively. At high velocities, the velocitydependent term in the diffusion equation becomes prominent and influences the stability result. In this case, the difference in the velocities at the peak and the valley is proportional to D/V~c and D'/V~, respectively, for the solute and the thermal fields, where the functions ~i vary from unity to zero as the velocity is increased [13]. From the conceptual viewpoint, we may thus define the effective diffusion stability lengths as IDv = D~ V~c and ID.v= D'/V~,.
3.Z Non-equilibrium effects Under rapid solidification conditions, departure from local equilibrium occurs at the interface so that the values of compositions in the solid and liquid must be determined through appropriate considerations of
132
R. Trivedi / Microstructureof rapidlysolidifiedalloys
non-equilibrium processes. This is achieved by considering three different non-equilibrium effects at the interface, which are (a) the effect of attachment kinetics at the interface, (b) the constraint on interface compositions due to thermodynamic considerations [14, 15], and (c) the variation in the solute distribution coefficient as a function of velocity [16, 17]. These aspects have been considered in detail in the literature [15], and they give rise to the following two equations that describe the interface compositions and interface temperature, for dilute solution and for linear solidus and liquidus lines:
Ti= Tm + mvCL-(2F/R)-(V//~k)
(5)
and
TM'
TI
.
.
.
.
.
.
.
.
.
.
.
i
i
'i"'",,,, N
i
Cse Cs
CO
CL
mlrtk/(k-l) CLe
Composition
kv = +k ~ P i l+Pi
(6)
Fig. 3. The effect of non-equilibriumconditions at the interface on the kinetic solidus and liquidus lines [18].
where
[ k-k~+kvln(kv/k)] mv=m, 1~ ~
(7)
and, k= Cse/CLe, kv = Cs/CL, ilk = goASm/egT i and V 0 is a constant which is of the order of the velocity of sound for pure metals. The subscript e refers to equilibrium values. Pi = ao V/Di is the interface Peclet number of solute redistribution in which Di/V = li is the interface diffusion length, and a0 is of the order of the interatomic distance. The effect of composition under non-equilibrium conditions has exactly the same mathematical form as that for the local equilibrium case, with m v replacing m~. The physical interpretation of rnv is that it represents the slope of the line in the phase diagram that connects the melting point of pure solvent with the non-equilibrium interface composition in the liquid at the interface temperature for a given velocity in the absence of interfacial energy and interface attachment kinetic effects, as shown in Fig. 3. For an equilibrium diagram with constant m and k, rn~ is a function of V through kv. The effective freezing range of the alloy is now dependent on velocity, and equal to
A ToV= mvCo( kv-1)/k~. The non-equilibrium effects also modify the thermal and capillarity lengths for alloys since the parameter A 72v= -mvC~( 1 - kv). For high velocities, we can now rescale length scales as follows: (1) diffusion length, ID~= D/V~c and ID,v= D' / V~; (2) capillarity length for alloys is now given by
Note that the diffusion, thermal and capillarity lengths, redefined for high velocities and for nonequilibrium conditions, reduce to the corresponding definitions used at low velocities when local equilibrium is present and the functions ~i are unity. We may now examine how rapid solidification influences different microstructural scales. We illustrate this by considering the example of rapid dendrite growth in an undercooled melt and eutectic growth. The same procedure can then also be applied to other microstructural scales and microstructural transitions at high velocities.
4. Rapid dendritic growth in an undercooled melt
The basic dendrite growth model for high growth rate conditions is still given by the tip undercooling expression and the dendrite tip selection criterion. The dendrite tip undercooling is obtained from the modified Ivantsov solution in which the solutal undercooling part is modified to take into account the non-equilibrium conditions at the interface. The relationship between the liquid undercooling A T, velocity and tip radius, for linear solidus and liquidus lines, is obtained as[18]
I[. k_~AToIv(P) + [[1 - ( 1 - kv)Iv(P)
d0 = VIA Lv; (3) the thermal length for directional solidification is given by ITv= A TsV/(G~l).
+[(m-my)Co]
(8)
R. Trivedi I Microstructure of rapidly solidified alloys The second term on the right-hand side contains non-equifibrium effects through kv, m v and A Tov. Note that the solutal undercooling is the difference between the liquidus temperature of the equilibrium phase diagram and the interface temperature, so that an additional term, given by A Tn~ = (m -mv)C0 has to be included in the total undercooling expression, as shown in Fig. 3. Since our aim is to examine the microstructure scale, we now consider the dendrite tip selection criterion which (for ~k = co ) is now given by [18]
/
ge2 k
~c + gR2
~1 =
(')
,.,+
/i)Vd0 v
=
2 , inO /1 t e> I tv~ol
L,
f~m,"AC,,Vl]
The terms in the three brackets correspond to l/o*,/D" and d0", so that one obtains
2= \oS]
(12)
(ID"do") ti2
where Po=O.335(faf~) k, = k/~= k,. is
i~s, and the function ~ , for
5:rDIV2 ~" [1 +(5:rDIV2)2] 1/2- 1 +2k,.
_1 lD,Vdo '
sequently, the eutectic spacing is controlled by the interaction of solute diffusion and capillarity lengths. The general expression for lamellar eutectic growth under rapid solidification condition is given by [19]
(9)
This criterion can now be written in terms of characteristic lengths as
__
133
(13)
(10)
where do'= F/A T0~. The above criterion, in terms of rescaled length scales, has precisely the same form as that for low velocities. Note that 2fl = 1 when K~ = K~. In the regime of solute diffusion controlled growth, in the high growth rate limit, one obtains R=(1/o*) (lDVdo~)1/2. This relationship is similar to that obtained for low velocities, and shows that the fundamental ideas of geometric mean are also valid at high velocities when the effect of high growth rates is included in the definition of characteristic lengths. Note that the velocity effect is quite complicated since it enters through the terms D and ~ in the effective diffusion length and the term A Tov in the capillarity length. At high growth rates, ~ approaches zero so that ID~, and thus the dendrite tip radius, becomes very large. When IDv becomes large, the solute term in eqn. (10) becomes negligible, so that thermal effects become important and one obtains a pure thermal dendrite growth at high velocities. Under this condition the dendrite tip radius is given by R =(2fl/o*)ll2(llyVdo,) 1/2. This expression is the same as that for low velocities, except that the value of the thermal diffusion length is now a function of velocity through V as well as through the function ~. Note that all the characteristic lengths, redefined for high velocities and for non-equilibrium conditions, reduce the corresponding definitions used at low velocities when local equilibrium is present.
5. Eutectic growth The basic eutectic growth model considers the solute diffusion process, and the stability of the structure is controlled by the capillarity effect. Con-
The function ~e is unity at low velocities and approaches zero at very high velocities. This function is analogous to the function ~ for dendritic growth. The value of o* varies from 0.0713 to 0.1386 when the volume fraction f.~ varies from 0.1 to 0.5, and this value is of the same order of magnitude as the o* value for dendritic growth. Consequently, eutectic spacings tend to be of the same order of magnitude. The rapid solidification effects enter through the terms ~e, k v and mi~. The kinetic slopes of the liquidus will be governed by the non-equilibrium effect in the two phases. In general these effects will not be identical in the two phases, so that the kinetic eutectic point will vary with velocity, as shown schematically in Fig. 4. Note that, if one starts with an alloy of equilibrium eutectic composition, this alloy will behave as an off-
F/)x'o..N'..>,
!
-.
I,'/,2//9x \ . "...'..-. \ ¢ . . ~
F4 \ " /,"
I :o/
I
I
,
t
...--~o°%-q~'..;~ ,..'..',, ;;/ ~
/ I¢ i , ill C -----i~
Fig. 4. The effect of non-equilibrium conditions on the kinetic eutectic point [ 18].
R. Trivedi
134
/
Microstructure of rapidly solidified alloys
eutectic alloy at high growth rates, and the deviation from the kinetic eutectic point may increase as the velocity is increased. The shift in the kinetic eutectic point under rapid solidification conditions can influence transitions from lamellar eutectic to fibrous or oscillating eutectic and also alter the coupled zone field. In the undercooled melt, where the temperature gradient in the liquid is negative, it may also be possible to obtain a cellular or dendritic eutectic, particularly at high growth rates where thermal effects become significant, as discussed for the dendritic .growth model. The stability of a eutectic front in undercooled melt when both the solute and the thermal effects are important needs to be investigated.
6. Conclusions The development of microstructural scales under rapid solidification is examined conceptually in term of physical processes that are operative during growth. This approach allows one to visualize clearly how the change in growth conditions influences specific physical phenomena and how this change alters the microstructure. By developing appropriate characteristic lengths for high velocities, which reduce to the corresponding values at low velocities, a unified result is obtained for microstructural scales of different morphologies under low and rapid solidification conditions. Furthermore, high velocity cellular to planar transition is also characterized by the condition /v =
vd0 v .
In order to bring out clearly the role of characteristic lengths of physical processes in the development of microstructural scales, we have used the results which are based on dilute solution approximations and linear solidus and liquidus lines. At large undercoolings, this approximation will not be valid, and one must modify appropriately the expression to take into account the variation in k, m~ and m s with temperature. For non-linear solidus and liquidus and under local equilibrium condition, the parameter AT s = - m l ( C l Q) is valid if ml, C 1and Q are evaluated at the interface temperature, as shown in Fig. 1. The variation in k also influences the interface stability condition through the term ~c for a single phase since the temperature along the perturbed interface varies which alters the value of k along the interface and modifies the stability condition [20, 21]. In this case, one must consider k as well as derivatives of k with the temperature. The function ~c is now given by 2k* ~c = 1 - [1 + Iz)/vK)]'4yg ~'~n'211/2 - 1 +2k*
(14)
where k* is the appropriate solute distribution parameter which contains k as well as derivatives of k with temperature. If solidus and liquidus lines can be approximated as piecewise linear, then for the linear region, k* = k + k(d T/dC)= ms/m ,
(15)
where m s and ml are the local constant slopes of the solidus and the liquidus respectively [21]. For highly non-linear liquidus and solidus lines, the second and possibly higher order effects should also be considered for the stability condition as well as for the capillarity effect. For rapid solidification, under non-equilibrium conditions, the non-equilibrium compositions as a function of velocity can be obtained by considering the thermodynamic treatment of concentrated alloys, and by using the solute trapping model for concentrated solutions [17]. In this case, numerical calculations are required to generate the kinetic liquidus and solidus lines as a function of velocity.
Acknowledgments The author would like to acknowledge many valuable discussions with Professor W. Kurz. This work was carried out at Ames Laboratory which is operated for the US Department of Energy by Iowa State University under contract no. W-7405-ENG-82. This work was supported by the Office of Basic Energy Sciences, Division of Materials Sciences.
References 1 W. Kurz and D. J. Fisher, Fundamentals of Solidification, Trans. Tech., Publications, Aadermansdorf, 1992. 2 M. Carrard, M. Gremaud, M. Zimmermann and W. Kurz, A cta Metall. Mater., 40 (1992) 983. 3 K. Eckler, R. E Cochrane, D. M. Herlach and B. Feuerbacher, Phys. Rev. B, 45 (1992) 5019. 4 R. Trivedi and W. Kurz, Acta Metall. Mater., 42 (1994) 15. 5 J.H. Perepezko, Mater. Sci. Eng., 178 (1994) 105-112. 6 M. Barth, K. Eckler, D. M. Herlach and H. Alexander, Mater. Sci. Eng., A133 (1991) 790. 7 W. L6ser and D. M. Herlach, Metall. Trans. A, 23 (1992) 1585. 8 E. Schleip, R. Willnecker, D. M. Herlach and G. P. G6rler, Mater. Sci. Eng., 98 (1988) 39. 9 Y. Wu, T. J. Piccone, Y. Shiohara and M. C. Flemings, Metall. Trans. A, 18 (1987) 915,925. 10 Y. Wu, T. J. Piccone, Y. Shiohara and M. C. Flemings, Metall. Trans. A, 19 (1988) 1109. 11 J.S. Langer, Rev. Mod. Phys., 52 (1980) 1. 12 R. Trivedi, J. Cryst. Growth, 49 (1980) 219. 13 R. Trivedi and W. Kurz, Acta Metall., 34 (1986) 1663.
R. Trivedi
/
Microstructure of rapidly solidified alloys"
14 J. C. Baker and J. W. Cahn, in Solidification, ASM, Metals Park, OH, 1971, p. 23. 15 W.J. Boettinger and S. R. Coriell, in Science and Technology of Undercooled Melt, Martinus Nijhoff, Dordrecht, 1985, p. 81. 16 M.J. Aziz, J. Appl. Phys., 53(1982) 1158.
135
M.J. Aziz and T. Kaplan, Acta Metall., 36 (1988) 2335. R. Trivedi and W. Kurz, Int. Mater. Rev., in press. W. Kurz and R. Trivedi, Metall. Trans. A, 22 ( 1991 ) 3(151. S. R. Coriell and R. F. Sekerka, J. Cryst. Growth, 61 (1983) 499. 21 R. Trivedi and W. Kurz, Metall. Trans. A, 21 (1990) 1311. 17 18 19 20
129
Microstructure characteristics of rapidly solidified alloys R. T r i v e d i
Ames Laboratory, US Department of Energy and Department of Materials Science and Engineering, Iowa State University, Ames, 1A 50011 (USA)
Abstract The microstructural characteristics of rapidly solidified materials were examined in terms of the effect of high velocity on the physical processes that govern microstructure formation. Under rapid solidification conditions, non-equilibrium conditions at the interface and a modified diffusional instability condition play critical roles in the selection of morphology and its microstructural scales. These effects were examined in terms of their influence on the physical processes. Appropriate characteristic lengths of physical processes for rapid solidification were developed, and relationships obtained for low and high growth rate conditions which characterize microstructural transition conditions and microstructural scales of cellular, dendritic and eutectic structures under free and constrained growth conditions. It is shown that all the results at low and high growth rates can be represented by unified expression in terms of redefined characteristic lengths.
1. Introduction
The evolution of solidification microstructures at low velocities has been studied extensively in the literature [1], and the conditions for microstructure transitions and the variations in important microstructural length scales with processing conditions have been well characterized. However, when the velocity of the solid-liquid interface becomes very high, significant changes in the modeling of microstructures are required. Interface velocities of the order of 10 m s can be achieved in laser processing [2], whereas velocities of the order of 30 m s- I have been achieved in highly undercooled melts [3]. Under such high growth rates, two new effects become important in influencing the formation and stability of microstructures. ( 1 ) The effect of high velocity causes the thermal and/or solutal fields to become more localized, which alters the condition for the diffusive instability of the interface, and (2) non-equilibrium effects at the interface become significant which strongly influence the phase and microstructure selection criteria. In order to visualize conceptually how the rapid solidification influences microstructure formation, we first examine the simple case of the low velocity condition. In this case, we find that the evolution of the microstructure can be readily visualized by considering how the processing variables influence the physical processes. We then apply these ideas to rapid solidification processing and show that the same basic forms of results obtained for low velocity can be recovered if 0921-5093/94/$7.00 SSDI 0921-5093(93)04525-M
the effect of rapid solidification is examined in terms of its influence on the physical processes. By defining appropriate characteristic lengths for solute and thermal diffusion, and for the capillarity effect under rapid solidification conditions, a unified expression is obtained for microstructure scales which is valid for low as well as high growth rate conditions.
2. Microstructure scales and transitions
We first consider the microstructural scales and the conditions for the transition in morphology for a single-phase material under low-growth rate conditions and under local equilibrium conditions at the interface. In order to obtain an insight into the results of analytical models, we consider the case of dilute solution with linear solidus and liquidus lines. In this case, simple relationships can be obtained by examining the physical processes that control the microstructure [4]. For directional solidification, the microstructure and phase selection is controlled by the growth rate, temperature gradient and composition. In an undercooled melt, the variables are composition and undercooling. In rapidly solidified droplets, the undercooling is controlled by the nucleation process so that appropriate nucleation models need to be developed to correlate the undercooling as a function of droplet size, cooling conditions and heterogeneous nucleation events which are present [5]. In this paper we only examine the growth phenomenon and assume that the proper undercooling is © 1994 - Elsevier Sequoia. All rights reserved
130
R. Trivedi / Microstructure of rapidly solidified alloys
either predicted from the nucleation models or that it has been measured experimentally [6-10]. For low velocities, and for systems in which interface kinetic effects are negligible, the microstructure is controlled by the solute and thermal diffusion processes and by the capillarity effect at the interface. Each of these processes can be represented by a characteristic length. (1) The diffusion lengths are given by ID= D / V for solute diffusion and by lD,= D ' / V for thermal diffusion, where D and D' are the solute and thermal diffusion coefficients respectively, and V is the velocity. (2) The thermal length for constrained growth is given by I T = A T J G , where A T s = - m l ( G - C ~ ) , in which mz is the slope of the liquidus at the interface temperature and G and C s are interface solute concentrations in the liquid and in the solid respectively. G is the conductivity-weighted temperature gradient in the liquid and solid. The thermal length in directional solidification is proportional to the length of the dendrite or to the extent of the mushy zone. (3) The capillarity length for an alloy is given by do = F/A Ts, where F is the capillarity constant which is a ratio of the interface energy to the entropy of melting per unit volume. The capillarity length is proportional to the critical radius of nucleation. For pure materials, the thermal capillarity length is defined as do'= F q / AH, where cj and A H are the specific heat and the enthalpy of melting, respectively, of the pure material. Note that, A Ts = - m l ( C 1- Cs) , is defined for a general case and it has different values for planar and dendritic growth. The general definition of A Ts is shown in Fig. i for an arbitrary phase diagram, and it is a function of interface temperature. For a phase diagram with linear solidus and liquidus lines in a dilute
solution regime, the limiting values are A T~= A TOfor planar front growth at the solidus temperature, and A Ts= k A T o for dendritic growth at low velocities where the dendrite tip temperature is close to the liquidus temperature. A To is the equilibrium freezing range of the alloy and k is the solute distribution coefficient. Although a specific value of A TS for a planar interface, or dendritic interface at low velocities, has been commonly used in the literature [11, 12] to define characteristic lengths, a general definition of AT s is required to consider properly the capillarity and thermal effects which are present at the interface temperature. 2.1. Microstructural transitions Various microstructural transitions can be readily represented in terms of these length scales of physical processes, as shown in Fig. 2. In order to conform with the analytical results given in the literature, we first consider the dilute solution case and assume linear solidus and liquidus lines. Note again that A Ts = A TO for a planar interface, and A Ts = k A T o for dendritic growth at low velocities. For directional solidification, the planar to cellular transition is given by lo = lT, and the cellular to planar transition at high velocities is given by ID= kdo, and the dendrite to cell transition at high velocities is given by lD = ado, with AT s = AT0, and a is a constant. The cell-dendrite transition is g i v e n b y ID = IT, w i t h a L = k A T o o r IT = k A T o / G . 2.2. Microstructural scales There are several important basic microstructural scales such as dendrite tip radius, primary and secondary dendrite spacings, cellular spacing and eutectic spacing. If we represent these scales by ;t i, then it is possible to describe a general expression for 2i, in the following two forms.
Y ,
'%.
i
i \~-,4 E i
Transitions
~x i ms
i'
l
i
Ii
cs
co
I I i
x,
........ ~
[K=A [tD]' [lT]b
i
ula~]
~i~i
',~~~~~~
ri
i 1
[do]~j
or
cx Composition
Fig. 1. Definition of A Ts for a general phase diagram. T i is the interface temperature.
] [7.2]"1 -A[I DdO]-I +B[/D'do']'I ] Fig. 2. Microstructural transitions and microstructural scales in terms of characteristic lengths of solute, thermal and capillarity processes.
R. Trivedi / Microstructure of rapidly solidified alloys' (1) When all or two of the physical processes are required to control the scale, one obtains a general expression:
J.i= Ai(lD)a(lT )b(do)c
(1)
in which A~ is a constant and the exponents a, b and c are constants whose values must add up to unity [4]. When the thermal effect is negligible, the microstructural scale is obtained for a = c = 1/2. This gives the relationship J,i=Ai(IDdo) 1/2, which indicates that the microstructural scale is the geometric mean of the destabilizing effect of solute diffusion and the stabilizing effect of the capillarity length. Such a relationship is satisfied by a dendrite tip radius in the pure solute or pure thermal case, eutectic spacing, initial secondary dendrite arm spacing, critical wavelength at absolute stability, and primary dendrite spacing at high velocities. For eutectic spacing a weighted average value of the two phases is used for the definition of the capillary length, and for pure undercooled melt the dendrite tip radius is obtained by replacing lD with lD, and d 0 by d0'. When all three physical processes are required to control the magnitude of 2~, these results can often be represented by the geometric mean of all the three processes, i.e. 2i=Ai(lDlTdo) I/3. Such a relationship is satisfied by the wavelength at the critical velocity and by the final secondary dendrite arm spacings. The primary dendrite spacing at low velocities, however, is found to follow the relationship:
;yA,(lD)'/4(1T)'/2(d,,)l/4
131
Note that both the above equations reduce to the result discussed in paragraph (1) when either the thermal or the solute effect is predominant. The constant a*, however, has different characteristic values for different microstructures. The above results give an insight into how the solute diffusion, thermal diffusion and capillarity effect influence the selection of all important microstructural scales of solidified materials. They also show which of the physical phenomena is influenced by the change in a specific processing condition. We now examine how the high growth rate effects influence these basic processes and how they alter the results discussed so far.
3. Rapid solidification conditions Under rapid solidification conditions, two major effects need to be taken into consideration which alter the characteristic lengths of physical processes. ( 1 ) The rapid solidification rate causes a significant change in the solute field (or the thermal field in undercooled alloy melt) which modifies the stability criterion, and it alters the effective diffusion length for the stability of the interface. (2) At high solidification rates, there is a departure from local equilibrium at the interface which must be taken into account, and this modification influences the compositions at the interface, and thus the A T~ term in the thermal (alloy solidification) and capillarity lengths.
(2a) 3.1. Diffusive instability
or
2i=A,(ITR) '/2
(2b)
where R is the dendrite tip radius. (2) When thermal and solute effects are operating in parallel, the following general relationship can be written:
(3) where ~* is constant and is known as the operating point of the microstructure scale, and fl= 0.511 + (Ks/ K~)]. The above relationship is satisfied by the dendrite tip radius in an undercooled melt. The cellular spacing, the marginal wavelengths of an unstable planar interface, and the dendrite tip radius also follow a similar relationship for constrained growth when an appropriate thermal length is used, and this result is
,o,,
(4)
For a sinusoidal perturbation, the stability of the interface is related to the difference in velocities at the peak and valley of the interface. At low velocities, the velocity-dependent term VV C in the diffusion equation is small so that the difference in velocities that controls the stability is proportional to the diffusion lengths lD=D/V or ID,=D'/V, for solute or thermal field respectively. At high velocities, the velocitydependent term in the diffusion equation becomes prominent and influences the stability result. In this case, the difference in the velocities at the peak and the valley is proportional to D/V~c and D'/V~, respectively, for the solute and the thermal fields, where the functions ~i vary from unity to zero as the velocity is increased [13]. From the conceptual viewpoint, we may thus define the effective diffusion stability lengths as IDv = D~ V~c and ID.v= D'/V~,.
3.Z Non-equilibrium effects Under rapid solidification conditions, departure from local equilibrium occurs at the interface so that the values of compositions in the solid and liquid must be determined through appropriate considerations of
132
R. Trivedi / Microstructureof rapidlysolidifiedalloys
non-equilibrium processes. This is achieved by considering three different non-equilibrium effects at the interface, which are (a) the effect of attachment kinetics at the interface, (b) the constraint on interface compositions due to thermodynamic considerations [14, 15], and (c) the variation in the solute distribution coefficient as a function of velocity [16, 17]. These aspects have been considered in detail in the literature [15], and they give rise to the following two equations that describe the interface compositions and interface temperature, for dilute solution and for linear solidus and liquidus lines:
Ti= Tm + mvCL-(2F/R)-(V//~k)
(5)
and
TM'
TI
.
.
.
.
.
.
.
.
.
.
.
i
i
'i"'",,,, N
i
Cse Cs
CO
CL
mlrtk/(k-l) CLe
Composition
kv = +k ~ P i l+Pi
(6)
Fig. 3. The effect of non-equilibriumconditions at the interface on the kinetic solidus and liquidus lines [18].
where
[ k-k~+kvln(kv/k)] mv=m, 1~ ~
(7)
and, k= Cse/CLe, kv = Cs/CL, ilk = goASm/egT i and V 0 is a constant which is of the order of the velocity of sound for pure metals. The subscript e refers to equilibrium values. Pi = ao V/Di is the interface Peclet number of solute redistribution in which Di/V = li is the interface diffusion length, and a0 is of the order of the interatomic distance. The effect of composition under non-equilibrium conditions has exactly the same mathematical form as that for the local equilibrium case, with m v replacing m~. The physical interpretation of rnv is that it represents the slope of the line in the phase diagram that connects the melting point of pure solvent with the non-equilibrium interface composition in the liquid at the interface temperature for a given velocity in the absence of interfacial energy and interface attachment kinetic effects, as shown in Fig. 3. For an equilibrium diagram with constant m and k, rn~ is a function of V through kv. The effective freezing range of the alloy is now dependent on velocity, and equal to
A ToV= mvCo( kv-1)/k~. The non-equilibrium effects also modify the thermal and capillarity lengths for alloys since the parameter A 72v= -mvC~( 1 - kv). For high velocities, we can now rescale length scales as follows: (1) diffusion length, ID~= D/V~c and ID,v= D' / V~; (2) capillarity length for alloys is now given by
Note that the diffusion, thermal and capillarity lengths, redefined for high velocities and for nonequilibrium conditions, reduce to the corresponding definitions used at low velocities when local equilibrium is present and the functions ~i are unity. We may now examine how rapid solidification influences different microstructural scales. We illustrate this by considering the example of rapid dendrite growth in an undercooled melt and eutectic growth. The same procedure can then also be applied to other microstructural scales and microstructural transitions at high velocities.
4. Rapid dendritic growth in an undercooled melt
The basic dendrite growth model for high growth rate conditions is still given by the tip undercooling expression and the dendrite tip selection criterion. The dendrite tip undercooling is obtained from the modified Ivantsov solution in which the solutal undercooling part is modified to take into account the non-equilibrium conditions at the interface. The relationship between the liquid undercooling A T, velocity and tip radius, for linear solidus and liquidus lines, is obtained as[18]
I[. k_~AToIv(P) + [[1 - ( 1 - kv)Iv(P)
d0 = VIA Lv; (3) the thermal length for directional solidification is given by ITv= A TsV/(G~l).
+[(m-my)Co]
(8)
R. Trivedi I Microstructure of rapidly solidified alloys The second term on the right-hand side contains non-equifibrium effects through kv, m v and A Tov. Note that the solutal undercooling is the difference between the liquidus temperature of the equilibrium phase diagram and the interface temperature, so that an additional term, given by A Tn~ = (m -mv)C0 has to be included in the total undercooling expression, as shown in Fig. 3. Since our aim is to examine the microstructure scale, we now consider the dendrite tip selection criterion which (for ~k = co ) is now given by [18]
/
ge2 k
~c + gR2
~1 =
(')
,.,+
/i)Vd0 v
=
2 , inO /1 t e> I tv~ol
L,
f~m,"AC,,Vl]
The terms in the three brackets correspond to l/o*,/D" and d0", so that one obtains
2= \oS]
(12)
(ID"do") ti2
where Po=O.335(faf~) k, = k/~= k,. is
i~s, and the function ~ , for
5:rDIV2 ~" [1 +(5:rDIV2)2] 1/2- 1 +2k,.
_1 lD,Vdo '
sequently, the eutectic spacing is controlled by the interaction of solute diffusion and capillarity lengths. The general expression for lamellar eutectic growth under rapid solidification condition is given by [19]
(9)
This criterion can now be written in terms of characteristic lengths as
__
133
(13)
(10)
where do'= F/A T0~. The above criterion, in terms of rescaled length scales, has precisely the same form as that for low velocities. Note that 2fl = 1 when K~ = K~. In the regime of solute diffusion controlled growth, in the high growth rate limit, one obtains R=(1/o*) (lDVdo~)1/2. This relationship is similar to that obtained for low velocities, and shows that the fundamental ideas of geometric mean are also valid at high velocities when the effect of high growth rates is included in the definition of characteristic lengths. Note that the velocity effect is quite complicated since it enters through the terms D and ~ in the effective diffusion length and the term A Tov in the capillarity length. At high growth rates, ~ approaches zero so that ID~, and thus the dendrite tip radius, becomes very large. When IDv becomes large, the solute term in eqn. (10) becomes negligible, so that thermal effects become important and one obtains a pure thermal dendrite growth at high velocities. Under this condition the dendrite tip radius is given by R =(2fl/o*)ll2(llyVdo,) 1/2. This expression is the same as that for low velocities, except that the value of the thermal diffusion length is now a function of velocity through V as well as through the function ~. Note that all the characteristic lengths, redefined for high velocities and for non-equilibrium conditions, reduce the corresponding definitions used at low velocities when local equilibrium is present.
5. Eutectic growth The basic eutectic growth model considers the solute diffusion process, and the stability of the structure is controlled by the capillarity effect. Con-
The function ~e is unity at low velocities and approaches zero at very high velocities. This function is analogous to the function ~ for dendritic growth. The value of o* varies from 0.0713 to 0.1386 when the volume fraction f.~ varies from 0.1 to 0.5, and this value is of the same order of magnitude as the o* value for dendritic growth. Consequently, eutectic spacings tend to be of the same order of magnitude. The rapid solidification effects enter through the terms ~e, k v and mi~. The kinetic slopes of the liquidus will be governed by the non-equilibrium effect in the two phases. In general these effects will not be identical in the two phases, so that the kinetic eutectic point will vary with velocity, as shown schematically in Fig. 4. Note that, if one starts with an alloy of equilibrium eutectic composition, this alloy will behave as an off-
F/)x'o..N'..>,
!
-.
I,'/,2//9x \ . "...'..-. \ ¢ . . ~
F4 \ " /,"
I :o/
I
I
,
t
...--~o°%-q~'..;~ ,..'..',, ;;/ ~
/ I¢ i , ill C -----i~
Fig. 4. The effect of non-equilibrium conditions on the kinetic eutectic point [ 18].
R. Trivedi
134
/
Microstructure of rapidly solidified alloys
eutectic alloy at high growth rates, and the deviation from the kinetic eutectic point may increase as the velocity is increased. The shift in the kinetic eutectic point under rapid solidification conditions can influence transitions from lamellar eutectic to fibrous or oscillating eutectic and also alter the coupled zone field. In the undercooled melt, where the temperature gradient in the liquid is negative, it may also be possible to obtain a cellular or dendritic eutectic, particularly at high growth rates where thermal effects become significant, as discussed for the dendritic .growth model. The stability of a eutectic front in undercooled melt when both the solute and the thermal effects are important needs to be investigated.
6. Conclusions The development of microstructural scales under rapid solidification is examined conceptually in term of physical processes that are operative during growth. This approach allows one to visualize clearly how the change in growth conditions influences specific physical phenomena and how this change alters the microstructure. By developing appropriate characteristic lengths for high velocities, which reduce to the corresponding values at low velocities, a unified result is obtained for microstructural scales of different morphologies under low and rapid solidification conditions. Furthermore, high velocity cellular to planar transition is also characterized by the condition /v =
vd0 v .
In order to bring out clearly the role of characteristic lengths of physical processes in the development of microstructural scales, we have used the results which are based on dilute solution approximations and linear solidus and liquidus lines. At large undercoolings, this approximation will not be valid, and one must modify appropriately the expression to take into account the variation in k, m~ and m s with temperature. For non-linear solidus and liquidus and under local equilibrium condition, the parameter AT s = - m l ( C l Q) is valid if ml, C 1and Q are evaluated at the interface temperature, as shown in Fig. 1. The variation in k also influences the interface stability condition through the term ~c for a single phase since the temperature along the perturbed interface varies which alters the value of k along the interface and modifies the stability condition [20, 21]. In this case, one must consider k as well as derivatives of k with the temperature. The function ~c is now given by 2k* ~c = 1 - [1 + Iz)/vK)]'4yg ~'~n'211/2 - 1 +2k*
(14)
where k* is the appropriate solute distribution parameter which contains k as well as derivatives of k with temperature. If solidus and liquidus lines can be approximated as piecewise linear, then for the linear region, k* = k + k(d T/dC)= ms/m ,
(15)
where m s and ml are the local constant slopes of the solidus and the liquidus respectively [21]. For highly non-linear liquidus and solidus lines, the second and possibly higher order effects should also be considered for the stability condition as well as for the capillarity effect. For rapid solidification, under non-equilibrium conditions, the non-equilibrium compositions as a function of velocity can be obtained by considering the thermodynamic treatment of concentrated alloys, and by using the solute trapping model for concentrated solutions [17]. In this case, numerical calculations are required to generate the kinetic liquidus and solidus lines as a function of velocity.
Acknowledgments The author would like to acknowledge many valuable discussions with Professor W. Kurz. This work was carried out at Ames Laboratory which is operated for the US Department of Energy by Iowa State University under contract no. W-7405-ENG-82. This work was supported by the Office of Basic Energy Sciences, Division of Materials Sciences.
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R. Trivedi
/
Microstructure of rapidly solidified alloys"
14 J. C. Baker and J. W. Cahn, in Solidification, ASM, Metals Park, OH, 1971, p. 23. 15 W.J. Boettinger and S. R. Coriell, in Science and Technology of Undercooled Melt, Martinus Nijhoff, Dordrecht, 1985, p. 81. 16 M.J. Aziz, J. Appl. Phys., 53(1982) 1158.
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