Spinodal decomposition is a mechanism by which a single thermodynamic phase spontaneously separates into two phases (without nucleation). Decomposition occurs when there is no thermodynamic barrier to phase separation. As a result, phase separation via decomposition does not require the nucleation events resulting from thermodynamic fluctuations, which normally trigger phase separation. 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. 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). The dynamics of spinodal decomposition is commonly modeled using the Cahn–Hilliard equation. Spinodal decomposition is fundamentally different from nucleation and growth. When there is a nucleation barrier to the formation of a second phase, time is taken by the system to overcome that barrier. As there is no barrier (by definition) to spinodal decomposition, some fluctuations (in the order parameter that characterizes the phase) start growing instantly. Furthermore, in spinodal decomposition, the two distinct phases start growing in any location uniformly throughout the volume, whereas a nucleated phase change begins at a discrete number of points. Spinodal decomposition occurs when a homogenous phase becomes thermodynamically unstable. An unstable phase lies at a maximum in free energy. In contrast, nucleation and growth occur when a homogenous phase becomes metastable. That is, another biphasic system becomes lower in free energy, but the homogenous phase remains at a local minimum in free energy, and so is resistant to small fluctuations. J. Willard Gibbs described two criteria for a metastable phase: that it must remain stable against a small change over a large area.
History 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. From the spacing of the sidebands, they were able to determine the wavelength of the modulation, which was of the order of 100 angstroms (10 nm). The growth of a composition modulation in an initially homogeneous alloy implies uphill diffusion or a negative diffusion coefficient. 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. The first explanation of the periodicity was given by Mats Hillert in his 1955 Doctoral Dissertation at MIT. Starting with a regular solution model, he derived a flux equation for one-dimensional diffusion on a discrete lattice. This equation differed from the usual one by the inclusion of a term, which allowed for the effect of the interfacial energy on the driving force of adjacent interatomic planes that differed in composition. Hillert solved the flux equation numerically and found that inside the spinodal it yielded a periodic variation of composition with distance. Furthermore, the wavelength of the modulation was of the same order as that observed in the Cu-Ni-Fe alloys. Building on Hillert's work, a more flexible continuum model was subsequently developed by John W. Cahn and John Hilliard, who included the effects of coherency strains as well as the gradient energy term. The strains are significant in that they dictate the ultimate morphology of the decomposition in anisotropic materials.
Cahn–Hilliard model for spinodal decomposition 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. This approach allows one to approximate the free energy as an expansion in terms of the concentration gradient ∇ c {\displaystyle \nabla c} , a vector. Since free energy is a scalar and we are probing near its minima, the term proportional to ∇ c {\displaystyle \nabla c} is negligible. The lowest order term is the quadratic expression κ ( ∇ c ) 2 {\displaystyle \kappa (\nabla c)^{2}} , a scalar. Here κ {\displaystyle \kappa } is a parameter that controls the free energy cost of variations in concentration c {\displaystyle c} . The Cahn–Hilliard free energy is then
F = ∫ v [ f b + κ ( ∇ c ) 2 ] d V {\displaystyle F=\int _{v}[f_{b}+\kappa (\nabla c)^{2}]~dV}
where f b {\displaystyle f_{b}} is the bulk free energy per unit volume of the homogeneous solution, and the integral is over the volume of the system. We now want to study the stability of the system with respect to small fluctuations in the concentration c {\displaystyle c} , for example a sine wave of amplitude a {\displaystyle a} and wavevector q = 2 π / λ {\displaystyle q=2\pi /\lambda } , for λ {\displaystyle \lambda } the wavelength of the concentration wave. To be thermodynamically stable, the free energy change δ F {\displaystyle \delta F} due to any small amplitude concentration fluctuation δ c = a sin ( q → . r → ) {\displaystyle \delta c=a\sin({\vec {q}}.{\vec {r}})} , must be positive. We may expand f b {\displaystyle f_{b}} about the average composition co as follows:
f b ( c ) = f b ( c 0 ) + ( c − c 0 ) ( ∂ f ∂ c ) c = c 0 + 1 2 ( c − c 0 ) 2 ( ∂ 2 f ∂ c 2 ) c = c 0 + ⋯ {\displaystyle f_{b}(c)=f_{b}(c_{0})+\left(c-c_{0}\right)\left({\frac {\partial f}{\partial c}}\right)_{c\,=\,c_{0}}+{\frac {1}{2}}\,\left(c-c_{0}\right)^{2}\left({\frac {\partial ^{2}f}{\partial c^{2}}}\right)_{c\,=\,c_{0}}+\cdots }
and for the perturbation δ c = a sin ( q → . r → ) {\displaystyle \delta c=a\sin({\vec {q}}.{\vec {r}})} the free energy change is
f b + κ ( ∇ c ) 2 = f b ( c 0 ) + a sin ( q → . r → ) ( ∂ f ∂ c ) c = c 0 + 1 2 a 2 sin 2 ( q → . r → ) ( ∂ 2 f ∂ c 2 ) c = c 0 + a 2 κ q 2 cos 2 ( q → . r → ) {\displaystyle f_{b}+\kappa (\nabla c)^{2}=f_{b}(c_{0})+a\sin({\vec {q}}.{\vec {r}})\left({\frac {\partial f}{\partial c}}\right)_{c\,=\,c_{0}}+{\frac {1}{2}}\,a^{2}\sin ^{2}({\vec {q}}.{\vec {r}})\left({\frac {\partial ^{2}f}{\partial c^{2}}}\right)_{c\,=\,c_{0}}+a^{2}\kappa q^{2}\cos ^{2}({\vec {q}}.{\vec {r}})}
When this is integrated over the volume V {\displaystyle V} , the sin ( q → . r → ) {\displaystyle \sin({\vec {q}}.{\vec {r}})} gives zero, while sin 2 ( q → . r → ) {\displaystyle \sin ^{2}({\vec {q}}.{\vec {r}})} and cos 2 ( q → . r → ) {\displaystyle \cos ^{2}({\vec {q}}.{\vec {r}})} integrate to give V / 2 {\displaystyle V/2} . So, then
δ F V = a 2 4 [ ( ∂ 2 f ∂ c 2 ) c = c 0 + 2 κ q 2 ] {\displaystyle {\frac {\delta F}{V}}={\frac {a^{2}}{4}}\left[\left({\frac {\partial ^{2}f}{\partial c^{2}}}\right)_{c=c_{0}}+2\,\kappa \,q^{2}\right]}
As a 2 > 0 {\displaystyle a^{2}>0} , thermodynamic stability requires that the term in brackets be positive. The 2 κ q 2 {\displaystyle 2\kappa q^{2}} is always positive but tends to zero at small wavevectors, large wavelengths. Since we are interested in macroscopic fluctuations, q → 0 {\displaystyle q\to 0} , stability requires that the second derivative of the free energy be positive. When it is, there is no spinodal decomposition, but when it is negative, spinodal decomposition will occur. Then fluctuations with wavevectors q < q c {\displaystyle q<q_{c}} become spontaneously unstable, where the critical wave number q c {\displaystyle q_{c}} is given by:
q c = − 1 2 κ ( ∂ 2 f ∂ c 2 ) c = c 0 {\displaystyle q_{c}={\sqrt {{\frac {-1}{2\kappa }}\left({\frac {\partial ^{2}f}{\partial c^{2}}}\right)_{c=c_{0}}}}}
which corresponds to a fluctuations above a critical wavelength
λ c = − 8 π 2 κ / ( ∂ 2 f ∂ c 2 ) c = c 0 {\displaystyle \lambda _{c}={\sqrt {-8\pi ^{2}\kappa /\left({\frac {\partial ^{2}f}{\partial c^{2}}}\right)_{c=c_{0}}}}}
Dynamics of spinodal decomposition when molecules move via diffusion Spinodal decomposition can be modeled using a generalized diffusion equation:
∂ c ∂ t = M ∇ 2 μ {\displaystyle {\frac {\partial c}{\partial t}}=M\nabla ^{2}\mu }
for μ {\displaystyle \mu } the chemical potential and M {\displaystyle M} the mobility. As pointed out by Cahn, this equation can be considered as a phenomenological definition of the mobility M, which must by definition be positive. It consists of the ratio of the flux to the local gradient in chemical potential. The chemical potential is a variation of the free energy and when this is the Cahn–Hilliard free energy this is
μ = δ F δ c = ( ∂ f ∂ c ) c = c 0 − 2 κ ∇ 2 c {\displaystyle \mu ={\frac {\delta F}{\delta c}}=\left({\frac {\partial f}{\partial c}}\right)_{c=c_{0}}-2\kappa \nabla ^{2}c}
and so
∂ c ∂ t = M ∇ 2 μ = M [ ( ∂ 2 f ∂ c 2 ) c = c 0 ∇ 2 c − 2 κ ∇ 4 c ] {\displaystyle {\frac {\partial c}{\partial t}}=M\nabla ^{2}\mu =M\left[\left({\frac {\partial ^{2}f}{\partial c^{2}}}\right)_{c=c_{0}}\nabla ^{2}c-2\kappa \nabla ^{4}c\right]}
and now we want to see what happens to a small concentration fluctuation δ c = a exp ( ω t ) sin ( q → . r → ) {\displaystyle \delta c=a\exp(\omega t)\sin({\vec {q}}.{\vec {r}})} - note that now it has time dependence as a wavevector dependence. Here ω {\displaystyle \omega } is a growth rate. If ω < 0 {\displaystyle \omega <0} then the perturbation shrinks to nothing, the system is stable with respect to small perturbations or fluctuations, and there is no spinodal decomposition. However, if ω > 0 {\displaystyle \omega >0} then the perturbation grows and the system is unstable with respect to small perturbations or fluctuations: There is spinodal decomposition. Substituting in this concentration fluctuation, we get
ω δ c = M [ − ( ∂ 2 f ∂ c 2 ) c = c 0 q 2 − 2 κ q 4 ] δ c {\displaystyle \omega \delta c=M\left[-\left({\frac {\partial ^{2}f}{\partial c^{2}}}\right)_{c=c_{0}}q^{2}-2\kappa q^{4}\right]\delta c}
This gives the same expressions for the stability as above, but it also gives an expression for the growth rate of concentration perturbations
ω = M q 2 [ − ( ∂ 2 f ∂ c 2 ) c = c 0 − 2 κ q 2 ] {\displaystyle \omega =Mq^{2}\left[-\left({\frac {\partial ^{2}f}{\partial c^{2}}}\right)_{c=c_{0}}-2\kappa q^{2}\right]}
which has a maximum at a wavevector
q m a x = − ( ∂ 2 f ∂ c 2 ) c = c 0 / ( 4 κ ) {\displaystyle q_{\rm {max}}={\sqrt {-\left({\frac {\partial ^{2}f}{\partial c^{2}}}\right)_{c=c_{0}}/(4\kappa )}}}
So, at least at the beginning of spinodal decomposition, we expect the growing concentrations to mostly have this wavevector.
Phase diagram This type of phase transformation is known as spinodal decomposition, and can be illustrated on a phase diagram exhibiting a miscibility gap. Thus, phase separation occurs whenever a material transition into the unstable region of the phase diagram. 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. The binodal and spinodal meet at the critical point. It is when a material is moved into the spinodal region of the phase diagram that spinodal decomposition can occur. The free energy curve is plotted as a function of composition for a temperature below the convolute temperature, T. Equilibrium phase compositions are those corresponding to the free energy minima. Regions of negative curvature (∂2f/∂c2 < 0 ) lie within the inflection points of the curve (∂2f/∂c2 = 0 ) which are called the spinodes. Their locus as a function of temperature defines the spinodal curve. For compositions within the spinodal, a homogeneous solution is unstable against infinitesimal fluctuations in density or composition, and there is no thermodynamic barrier to the growth of a new phase. Thus, the spinodal represents the limit of physical and chemical stability. To reach the spinodal region of the phase diagram, a transition must take the material through the binodal region or the critical point. Often phase separation will occur via nucleation during this transition, and spinodal decomposition will not be observed. To observe spinodal decomposition, a very fast transition, often called a quench, is required to move from the stable to the spinodal unstable region of the phase diagram. 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.
Coherency strains For most crystalline solid solutions, there is a variation of lattice parameters with the 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 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 ao and that of the undeformed slice a. If the slice is to be coherent after the addition of the slab, it must be subjected to a strain ε in the z' and y' directions which is given by:
ϵ = a − a 0 a 0 {\displaystyle \epsilon ={\frac {a-a_{0}}{a_{0}}}}
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:
σ x ′ = σ y ′ = σ z ′ {\displaystyle \sigma _{x'}=\sigma _{y'}=\sigma _{z'}}
The elastic work per unit volume is given by:
W E = 1 2 ∑ i σ i ϵ i {\displaystyle W_{E}={\frac {1}{2}}\displaystyle \sum _{i}\sigma _{i}\epsilon _{i}}
where the ε's are the strains. The work performed per unit volume of the slice during the first step is therefore given by:
W E ( 1 ) = 3 2 ( c 11 + 2 c 12 ) ϵ 2 {\displaystyle W_{E}(1)={\frac {3}{2}}(c_{11}+2c_{12})\epsilon ^{2}}
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, εz' = εy' = 0. The result is that:
W E ( 2 ) = ϵ 2 ( c 11 + 2 c 22 ) 2 c 11 {\displaystyle W_{E}(2)={\frac {\epsilon ^{2}(c_{11}+2c_{22})}{2c_{11}}}}
The net work performed on the slice in order to achieve coherency is given by:
W E = W E ( 1 ) − W E ( 2 ) {\displaystyle W_{E}=W_{E}(1)-W_{E}(2)}
or
W E = ϵ 2 2 ( c 11 + 2 c 12 ) ( 3 − [ c 11 − 2 c 12 c 1 ′ 1 ′ ] ) {\displaystyle W_{E}={\frac {\epsilon ^{2}}{2}}(c_{11}+2c_{12})\left(3-\left[{\frac {c_{11}-2c_{12}}{c_{1'1'}}}\right]\right)}
The final step is to express c1'1' in terms of the constants referred to the standard axes. From the rotation of axes, we obtain the following:
c 1 ′ 1 ′ = c 11 + 2 ( 2 c 44 − c 11 + c 12 ) ( l 2 m 2 + m 2 n 2 + l 2 n 2 ) {\displaystyle c_{1'1'}=c_{11}+2(2c_{44}-c_{11}+c_{12})(l^{2}m^{2}+m^{2}n^{2}+l^{2}n^{2})}
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:
W E = Y ϵ 2 {\displaystyle W_{E}=Y\epsilon ^{2}}
Y = 1 2 ( c 11 + 2 c 12 ) [ 3 − c 11 + 2 c 12 c 11 + 2 ( 2 c 44 − c 11 + c 12 ) ( l 2 m 2 + m 2 n 2 + l