Precipitation from Solid Solution J. C.F I S H E R , J. H. H O L L O M O N , A N D J. 0.L E S C H E N GENERAL ELECTRIC RESEARCH LABORATORY, SCHENECTADY, N . Y.
Elastic strains can be present in particles t h a t precipitate from solid solution, and in the surrounding parent phase as well. When t h e nucleation theory is modified to include the effects of elastic distortion, most of t h e characteristics of precipitation from solid solution can be accounted for.
P
RECIPITATION from solid solution proceeds in very much the same way as does precipitation from vapor or liquid sohtion: Nuclei form here and there in the solution and then grow. The three tvpes of precipitation are not identical, however, for in solids there is the possibility of altering the interfacial energy by changing the relative orientation of the precipitate and parent phases, and by straining one or both phases. The factors that control the rate of nucleation in solids are assumed to be (1) the interfacial free energy, which sometimes can be decreased by straining the precipitate and/or parent phase SO that they are coherent, ( 2 ) the free-energy change per unit volume of precipitate, and (3) the strain energy that is associated with coherency. At any temperature and state of stress, and for every precipitate that can form in a given parent phase, the precipitates that do in fact nucleate them should be those with structure, composition, orientation, and state of strain such that the product of nucleation rate and growth rate is a maximum. Usually this means that the free energy of the critical size nucleus is a minimum. The growth rate of precipitated particles usually is controlled by diffusion. If the precipitate and parent phase differ in composition, diffusion of matter will limit the growth rate. If there is no composition change, diffusion of heat can be the limiting factor. The free-energy change per unit volume of precipitate provides the driving force for diffusion. Several types of precipitation can occur in a single system. Consider, for example, a solid solution, CY, in which p precipitates upon cooling. At the highest transformation temperatures, noncoherent p of equilibrium compoeition will form, for insufficient free energy is available for producing elastic deformation. St somewhat lower temperatures, coherent p of substantially the same composition would be expected to form, for the nucleation rate of coherent precipitates, with their reduced interfacial free energies, is very great when sufficient volume free energy is available to provide the necessary strain. At still lower temperatures a coherent precipitate with the same composition as the parent CY might form. Essentially, such a precipitate is a metastable allotropic modification of the solid solution, which may or may not have a stable counterpart. If the crystal structures of p and 01 are different, this precipitate could be p with the composition of CY. No diffusion, save perhaps of heat, is required for the growth of such precipitates, and when sufficient volume free energy is available their rapid rates of nucleation and growth assure their formation. This paper is concerned primarily with the nucleation of precipitates, and only secondarily with their growth. The possibility of coherent nucleation is perhaps the most striking feature of solid-solution precipitation, and is discussed in some detail. COHERENT NUCLEATION
Consider the formation of a spherical precipitate of radius T . If the particle is strain-free, the free energy associated with its formation is A F = ( 4 d )y
+ (4rr3/3) Afv
(1) where y is the interfacial free energy and A& is the free-energy change per unit volume of precipitate. The free energy of forming a particle is a maximum for particles of critical radius r*
1324
=
-2-,/Afw
(2)
and is given by the expression
(3)
AF* = 167ry3/3Afu2
The particle of radius r* is the critical size nucleus, and its energy of formation AF* is the free energy of nucleation. Since the rate of nucleation is proportional to exp ( - A F " / k T ) , the precipjtrltes that nucleate most rapidly are those for which AF* is a minimum.
If a precipitate is forming by homogeneous nucleation-Le., if nucleation is not catalyzed by foreign particles-in a liquid or gaseous solution, it is not possible to decrease the interfacial free energy significantly by elastic strain in the precipitate. For precipitates forming in a solid solution, however, where a considerable fraction of the interfacial free energy may be associated with the presence of lattice misfits that can be eliminated by strain, y often can be decreased considerably by straining the precipitate or parent phase. At the same time, strain energy is introduced. However, if the decrease in y is large enough and the required strain small enough, an over-all decrease in A F * will result, and coherent nucleation will take place more rapidly than noncoherent. Coherent precipitation has been discussed by RIott and Xabarro (9-11) and more recently by Barrett ( I ) and by Leschen and Fisher ( 7 ) . Precipitation a t the surfaces of solids has been investigated in more detail by Frank and Van der Merwe (3)for monolayers and by Turnbull and Vonnegut ( 1 2 ) for three-dimensional precipitates. The analysis given by Turnbull and Vonnegut, which is based upon that of Frank and Van der Merwe, is immediately applicable to precipitation in solids. The essential feature of a coherent interface is that it shall contain no dislocations, whereas a noncoherent interface does contain dislocations as shown in Figure 1. [For those unfamiliar with dislocations, the paper by Cottrell ( 2 ) contains a good review.] According to this definition, some precipitates may be partially coherent, the precipitate-matrix interface containing dislocations, but fewer than for a strain-free precipitate. The austenite-bainite and austenite-martensite interfaces in steel, for example, probably are partially coherent. The interfacial free energy of a coherent interface is assumed to be relatively small, and to be comparable with the coherent twin boundary free energy in copper which is reported to be about 20 ergs per sq. em. (4). On the other hand, the energy of a noncoherent interface should be considerably larger, because of the presence of dislocations; it should be more nearly equal to the grain-boundary energy of copper, which is reported to be about 600 ergs per sq. cm. (4). The energy of partially coherent interfaces should be intermediate. For simplicity, we shall assume that precipitating particles are spherical. Let the precipitate and the parent phase have the same structure, one with lattice parameter a and the other with Aa. Let the energy of dislocation be 0 lattice parameter a per unit length. Consider now the free energy associated with the formation of a particle of radius r , strained equally in all directions by an amount e in such a way that the misfit is decreased. The free energy of forming the particle is
+
AF
( 4 i ~ ~?*a )
+ (4sr3/3) (Afu + AS.)
INDUSTRIAL AND ENGINEERING CHEMISTRY
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Vol. 44, No. 6
NUCLEATI ON-From where y. is the surface free energy, Afu the volume free-energy change in the absence of strain, and Af. the strain energy per unit volume of precipitate. The surface free energy depends upon strain,
+
ye = YO (O/a),8 - el (5) where 6 = / A a / a / , since ( 6 - .)/a is the total length of dislocation per unit area of interface. When the strain just equals the misfit, y e has its minimum value YO. The strain energy is a simpler function of strain,
Af. = me2 (6) where m is a constant that can be calculated from elastic theory.
Sol ids
the curve in Figure 2 will not change when the assumption of a spherical particle, adopted for the purposes of illustration, is dropped. SHAPE A N D H A B I T OF COHERENT PRECIPITATES
When the particles of a coherent precipitate have grown to a sufficient size, their equilibrium (subject to the constraint that coherency is not lost) shapes and habits are determined only by elastic energy considerations. The surface free energy of large particles is negligible in comparison with the strain energy. (The habit of a platelike particle refers t o the plane of the parent phase to which the plane of the plate is parallel.) If the particle has lower elastic constants than the parent phase, its equilibrium shape will be such that strain is concentrated in the particle and relieved in the parent phase. In other words, relatively nonrigid precipitates will be plates. There is no strain in the parent phase when a coherent precipitate is in the form of a thin plate, and no stress in the plate normal to its thickness. Because the stress normal to the plate is zero, the elastic energy in a plate with anisotropic elastic constants will vary as the habit varies, and a particular habit will have the lowest energy, giving the equilibrium habit. The precipitate can be more rigid than the parent phase in one direction, and still be platelike. The rigid direction of the plate will be normal to the habit plane. Barrett (1) has calculated the habit planes of several cubic metals in this way, and finds no contradiction with experiment. If the particle is more rigid than the parent phase, it will tend to be spherical, thereby minimizing its own strain energy and maximizing that in the matrix. If it is more rigid than the parent phase in two directions and less rigid in the third, it will tend t o be rodlike, with the axis of the rod lying in the less rigid direction.
COHERENT CUBIC PRECIPITATE
Figure 1.
Noncoherent and Coherent Cubic Precipitates
Planes of atoms are continuous across coherent interface. Some are discontinuous across noncoherent interface, giving rise t o dislocations
Substituting for
yf
A F = (474)[yo
+ (O/a)l6 - ell + ( 4 7 ~ * / 3(Afv ) + me2)
Holding
E
and Af. in the equation for AF,
(7)
constant, the free energy of nucleation is
AFe* = ( 1 6 ~ / 3 )[ y o
+ (O/a)Is -
el]*
(AS.
+ me2)-'
(8) 0
A F , * is a minimum for = 2(6
+
-
[4(6
+ yoa/e)2+ 3 ~ f ~ / m l ~ / 2
(9) as can be shown by differentiating, equating to zero, and solving for E. As a consequence of the absolute value sign in Equation 8, this expression for e is correct only up to e = 6. The conditions for coherency are just met when e = 6. Equating 6 and the expression for e in Equation 9, and assuming yoa/08 0, a somewhat larger (numerically) free energy per unit volume is required.
A schematic plot of 6 - E (the mismatch between parent phase and precipitate) and Afv (the free-energy decrease per unit volume of unstrained precipitate) is shown in Figure 2, the value of 6 - E corresponding at all times to the minimum free energy of critical nucleus formation. The curve is similar to that given by Turnbull and Vonnegut, showing that when the available free energy is small the precipitate is noncoherent and relatively unstrained, and when the available free energy reaches or exceeds a critical value the precipitate is coherent. The general shape of June 1952
1
-
1.0
-Af,/a8 *
-
Figure 2. Mismatch (6 E) between Parent Phase and Precipitate as a Function of Volume Free Energy (At,,), for M i n i m u m Free-Energy Nucleus Schematic
These considerations of the equilibrium shapes of coherent particles are qualitative only, but they serve t o illustrate the point that equilibrium shapes do exist, as determined by considerations of elastic energy. The shapes are independent of particle size, so long as the particles are large enough to produce negligible surface free-energy effects. T H E R M O D Y N A M I C S OF COHERENT PRECIPITATES
It has been shown that a large coherent particle has an equilibrium shape (subject, of course, t o the condition that coherency be maintained) such that the strain energy is minimized. All large particles are geometrically similar, and the strain energy is proportional to the particle volume; under these conditions it is proper to lump the,volume free energy in the absence of strain and the strain energy per unit volume of precipitate, whether or
INDUSTRIAL AND ENGINEERING CHEMISTRY
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NUCLEATION-From
Solids
not all the strain energy resides in the precipitate, and to call the sum the free energy per unit volume for coherent precipitates. Let us consider the precipitation of a phase from a solid solution 01. Coherent P will be denoted by Pc. Figure 3 gives the usual equilibrium diagram (solid lines) for the A - B system, where 01 is a solid solution of B in A , and 0 a solid solution of A in B. Superposed thereon is an equilibrium diagram for the Q
z
a W
3 W W
U K.
I
I
I
I
I
W
a
2
g
E
a I W
c
Nodule or dendrite. V N DP/z A41thoughdendritic growth occurs most rapidly, fairly large particles in the shape of spheres, rods, or plates are observed under certain conditions. Coherency, for example, should lead to such shapes where there is too little volume free energy to supply the strain energy associated with a dendrite. Surface free energy combined with rapid grain-boundary diffusion may lead to a roughly spherical shape for fairly large particles. The amount of material that must be diffused to allow the growth of a precipitate is proportional to l/AC, where AC is the composition difference between precipitate and parent phase. At a given particle size and tamperature therefore the growth rate is proportional to - Afv/AC, where Afv is the free-energy change per unit volume of part,icle. This expression is valid for noncoherent or coherent precipitates, although for coherent precipitates Afv must include the strain energy. If the composition of the precipitate departs from its equilibrium value and approaches more closely that of the parent phase, AC decreases and the growth rate tends to increase. On the other hand l A l v l decreases too, so that an increase in growth velocity can be realized only when AC decreases more rapidly than lAfvl. For some precipitates, it may happen that, Aie is still negativei.e., the precipitate is still stable-when AC = 0. Mart
