Spinodal decomposition
Spinodal decomposition is observed when mixtures of metals or polymers separate into two co-existing phases, each rich in one species and poor in the other.
[2] When the two phases emerge in approximately equal proportion (each occupying about the same volume or area), characteristic intertwined structures are formed that gradually coarsen (see animation).
As there is no barrier (by definition) to spinodal decomposition, some fluctuations (in the order parameter that characterizes the phase) start growing instantly.
J. Willard Gibbs described two criteria for a metastable phase: that it must remain stable against a small change over a large area.
[3] In the early 1940s, Bradley reported the observation of sidebands around the Bragg peaks in the X-ray diffraction pattern of a Cu-Ni-Fe alloy that had been quenched and then annealed inside the miscibility gap.
Further observations on the same alloy were made by Daniel and Lipson, who demonstrated that the sidebands could be explained by a periodic modulation of composition in the <100> directions.
Becker and Dehlinger had already predicted a negative diffusivity inside the spinodal region of a binary system, but their treatments could not account for the growth of a modulation of a particular wavelength, such as was observed in the Cu-Ni-Fe alloy.
In fact, any model based on Fick's law yields a physically unacceptable solution when the diffusion coefficient is negative.
Starting with a regular solution model, he derived a flux equation for one-dimensional diffusion on a discrete lattice.
Hillert solved the flux equation numerically and found that inside the spinodal it yielded a periodic variation of composition with distance.
[6][7][8] Free energies in the presence of small amplitude fluctuations, e.g. in concentration, can be evaluated using an approximation introduced by Ginzburg and Landau to describe magnetic field gradients in superconductors.
is given by: which corresponds to a fluctuations above a critical wavelength Spinodal decomposition can be modeled using a generalized diffusion equation:[10][11][12] for
The boundary of the unstable region sometimes referred to as the binodal or coexistence curve, is found by performing a common tangent construction of the free-energy diagram.
Inside the binodal is a region called the spinodal, which is found by determining where the curvature of the free-energy curve is negative.
Regions of negative curvature (∂2f/∂c2 < 0 ) lie within the inflection points of the curve (∂2f/∂c2 = 0 ) which are called the spinodes.
In some systems, ordering of the material leads to a compositional instability and this is known as a conditional spinodal, e.g. in the feldspars.
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.
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 c1'1' in terms of the constants referred to the standard axes.
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 an isotropic material: so that: This equation can also be written in terms of Young's modulus E and Poisson'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.
In the study of a Fourier series, complicated periodic functions are written as the sum of simple waves mathematically represented by sines and cosines.
If θ were measured in seconds then the waves e2πiθ and e−2πiθ would both complete one cycle per second—but they represent different frequencies in the Fourier transform.
is obtained by substituting this solution back into the diffusion equation as follows: For solids, the elastic strains resulting from coherency add terms to the amplification factor
as follows: where, for isotropic solids: where E is Young's modulus of elasticity, ν is Poisson's ratio, and η is the linear strain per unit composition difference.
Thus, the amplitude of composition fluctuations should grow continuously until a metastable equilibrium is reached with preferential amplification of components of particular wavelengths.
The kinetic amplification factor R is negative when the solution is stable to the fluctuation, zero at the critical wavelength, and positive for longer wavelengths—exhibiting a maximum at exactly
Limitations of this theory would appear to arise from this assumption and the absence of an expression formulated to account for irreversible processes during phase separation which may be associated with internal friction and entropy production.
Very small regions will shrink away due to the energy cost of maintaining an interface between two dissimilar component materials.
-type materials, the Landau free-energy is a good approximation of the free energy near the critical point and is often used to study homogeneous quenches.
[30] Another interesting property of spinodal materials is the capability to seamlessly transition between different classes, orientations, and densities,[30] thereby enabling the manufacturing of effectively multi-material structures.
