Spinodal Decomposition - Coherency Strains

Coherency Strains

For most crystalline solid solutions, there is a variation of lattice parameter with composition. If the lattice of such a solution is to remain coherent in the presence of a composition modulation, mechanical work has to be done in order to strain the rigid lattice structure. The maintenance of coherency thus affects the driving force for diffusion.

Consider a crystalline solid containing a one-dimensional composition modulation along the x-direction. We calculate the elastic strain energy for a cubic crystal by estimating the work required to deform a slice of material so that it can be added coherently to an existing slab of cross-sectional area. We will assume that the composition modulation is along the x' direction and, as indicated, a prime will be used to distinguish the reference axes from the standard axes of a cubic system (that is, along the <100>).

Let the lattice spacing in the plane of the slab be a_o and that of the undeformed slice a. If the slice is to be coherent after addition of the slab, it must be subjected to a strain δ in the z' and y' directions which is given by:

In the first step, the slice is deformed hydrostatically in order to produce the required strains to the z' and y' directions. We use the linear compressibility of a cubic system 1 / ( c₁₁ + 2 c₁₂ ) where the c's are the elastic constants. The stresses required to produce a hydrostatic strain of δ are therefore given by:

The elastic work per unit volume is given by:

where the ε's are the strains. The work performed per unit volume of the slice during the first step is therefore given by:

In the second step, the sides of the slice parallel to the x' direction are clamped and the stress in this direction is relaxed reversibly. Thus, ε_z' = ε_y' = 0. The result is that:

The net work performed on the slice in order to achieve coherency is given by:

or

The final step is to express c_1'1' in terms of the constants referred to the standard axes. From the rotation of axes, we obtain the following:

where l, m, n are the direction cosines of the x' axis and, therefore the direction cosines of the composition modulation. Combining these, we obtain the following:

The existence of any shear strain has not been accounted for. Cahn considered this problem, and concluded that shear would be absent for modulations along <100>, <110>, <111> and that for other directions the effect of shear strains would be small. It then follows that the total elastic strain energy of a slab of cross-sectional area A is given by:

We next have to relate the strain δ to the composition variation. Let a_o be the lattice parameter of the unstrained solid of the average composition c_o. Using a Taylor's series expansion about c_o yields the following:

in which

where the derivatives are evaluated at c_o. Thus, neglecting higher order terms, we have:

Substituting, we obtain:

This simple result indicates that the strain energy of a composition modulation depends only on the amplitude and is independent of the wavelength. For a given amplitude, the strain energy W_E is proportional to Y. Let us consider a few special cases.

For an isotropic material:

so that:

Ths equation can also be written in terms of Young's modulus E and Poissons's ratio υ using the standard relationships:

Substituting, we obtain the following:

For most metals, the left hand side of this equation

is positive, so that the elastic energy will be a minimum for those directions that minimize the term: l2m2 + m2n2 + l2n2. By inspection, those are seen to be <100>. For this case:

the same as for an isotropic material. At least one metal (molybdenum) has an anisotropy of opposite sign. In this case, the directions for minimum W_E will be those that maximize the directional cosine function. These directions are <111>, and

As we will see, the growth rate of the modulations will be a maximum in the directions that minimize Y. These directions therefore determine the morphology and structural characteristics of the decomposition in cubic solid solutions.

Rewriting the diffusion equation and including the term derived for the elastic energy yields the following:

or

which can alternatively be written in terms of the diffusion coefficient D as:

The simplest way of solving this equation is by using the method of Fourier transforms.