Abstract
We present a phenomenological treatment of diffusiondriven martensitic phase transformations in multicomponent crystalline solids that arise from nonconvex free energies in mechanical and chemical variables. The treatment describes diffusional phase transformations that are accompanied by symmetrybreaking structural changes of the crystal unit cell and reveals the importance of a mechanochemical spinodal, defined as the region in strain–composition space, where the freeenergy density function is nonconvex. The approach is relevant to phase transformations wherein the structural order parameters can be expressed as linear combinations of strains relative to a highsymmetry reference crystal. The governing equations describing mechanochemical spinodal decomposition are variationally derived from a freeenergy density function that accounts for interfacial energy via gradients of the rapidly varying strain and composition fields. A robust computational framework for treating the coupled, higherorder diffusion and nonlinear strain gradient elasticity problems is presented. Because the local strains in an inhomogeneous, transforming microstructure can be finite, the elasticity problem must account for geometric nonlinearity. An evaluation of available experimental phase diagrams and firstprinciples free energies suggests that mechanochemical spinodal decomposition should occur in metal hydrides such as ZrH_{2−2c}. The rich physics that ensues is explored in several numerical examples in two and three dimensions, and the relevance of the mechanism is discussed in the context of important electrode materials for Liion batteries and hightemperature ceramics.
Introduction
Spinodal decomposition is a continuous phase transformation mechanism occurring throughout a solid that is far enough from the equilibrium for its freeenergy density to lose convexity with respect to an internal degree of freedom. The latter could include the local composition as in classical spinodal decomposition described by Cahn and Hilliard,^{1} or a suitable nonconserved order parameter as in the theory by Allen and Cahn^{2} for spinodal ordering. A key requirement for continuous transformations is that order parameters can be formulated to uniquely describe continuous paths connecting the various phases of the transformation. These phases then correspond to local minima on a single, continuous freeenergy density surface in that order parameter space. For classical spinodal decomposition inside a miscibility gap, all phases have the same crystal structure and symmetry, and the order parameter is simply the local composition. The existence of a single, continuous freeenergy density surface for all phases participating in a transformation implies, by geometric necessity, the presence of domains in order parameter space, where the freeenergy density is nonconvex. Reaching those domains through supersaturation (by externally varying temperature or composition) makes the solid susceptible to a generalised spinodal decomposition.
Many important multicomponent solids undergo phase transformations that couple diffusional redistribution of their components with a structural change of the crystallographic unit cell. One prominent example is the decomposition that occurs when cubic yttriastabilised zirconia Zr_{1−c}Y_{c}O_{2−c/2} is quenched into a twophase equilibrium region between tetragonal Zr_{1−c}Y_{c}O_{2−c/2} having a lowY composition and cubic Zr_{1−c}Y_{c}O_{2−c/2} having a highY composition. Another occurs in Libattery electrodes made of spinel Li_{c}Mn_{2}O_{4}. Discharging to low voltages causes the compound to transform from cubic LiMn_{2}O_{4} to tetragonal Li_{2}Mn_{2}O_{4} through a twophase diffusional phase transformation mechanism. As with simple diffusional phase transformations, these coupled diffusional/martensitic phase transformations can occur either through a nucleation and growth mechanism or, if certain symmetry requirements are met, through a continuous mechanism due to an onset of an instability with respect to the composition and/or a structural order parameter.
Here we present a treatment of coupled diffusional/martensitic phase transformations triggered by instabilities with respect to both strain and composition. These phase transformations are characterised by a mechanochemical spinodal that is defined as a nonconvex region of the freeenergy density function in the strain–composition space. The possibility of a mechanochemical spinodal decomposition is motivated by the recent firstprinciples studies of martensitic transformations in transition metal hydrides, where a hightemperature cubic phase is predicted to display negative curvatures with respect to strain, thus making strain a natural order parameter to distinguish the cubic parent phase from its lowtemperature tetragonal daughter phases.^{3} In addition, the coupling with composition degrees of freedom (e.g., through the introduction of hydrogen vacancies in the metal hydrides) allows for the possibility that the free energy also exhibits a negative curvature with respect to the composition. Mechanochemical spinodal decomposition, therefore, is a phenomenon that is likely present in many multicomponent materials but has to date been overlooked as a mechanism by which a highsymmetry phase can decompose martensitically and through diffusional redistribution upon quenching into a twophase regime. Structural phase transformations in solids driven by instability with respect to an internal shuffle of the atoms within the unit cell have been treated rigorously in the literature with coupled Cahn–Hilliard and Allen–Cahn approaches,^{4–7} but mechanochemical spinodal decomposition is fundamentally different and necessitates a coupled treatment of both the strain and composition instabilities.
Our treatment is based on a generalised, Landautype freeenergy density function that couples strain and composition instabilities. The governing equations of mechanochemical spinodal decomposition, obtained by variational principles, generalise the classical equations of the Cahn–Hilliard formulation,^{1} and of nonlinear gradient elasticity^{8,9} by coupling these systems. The ability to solve this complex, nonlinear, strain and compositiongradientdriven, mechanochemical system for sufficiently general initial and boundary value problems in two and three dimensions also has been lacking heretofore. We introduce the computational framework to obtain such solutions in general, threedimensional solids. This newfound capability allows us to then reinforce our discussion of the phenomenology with dynamics predicted by the numerical solutions.
The mechanochemical spinodal in two dimensions
For accessibility of the arguments, we first consider the twodimensional analogue of the cubictotetragonal transformation: the squaretorectangle transformation. The highsymmetry square lattice will serve as the reference crystal relative to which strains are measured. Lowersymmetry lattices that can be derived from the square lattice by homogeneous strain include the rectangle, the diamond and lattices, where there are no constraints on the cell lengths and their angles.
We use the composition c, varying between 0 and 1, as our order parameter for the chemistry of our binary twodimensional solid. Symmetrybreaking structural changes are naturally described by the strain relative to the highsymmetry square lattice. The elastic freeenergy density is also a function of strain. In both contexts, strain will in general be of finite magnitude. Following Barsch and Krumhansl,^{8} we use the Green–Lagrange strain tensor, E, relative to the square lattice. Rotations are exactly neutralised in this strain measure; for any rigid body motion, E=0 (Supplementary Information). In two dimensions, the relevant strain components are E_{11}, E_{22} and E_{12}=E_{21}. However, it is more convenient to use linear combinations of these components that transform according to the irreducible representations of the point group of the highsymmetry square lattice. In two dimensions, these include ${e}_{1}=\left({E}_{11}+{E}_{22}\right)\sqrt{2}$, ${e}_{2}=\left({E}_{11}{E}_{22}\right)\sqrt{2}$ and ${e}_{6}=\sqrt{2}{E}_{12}$. Here e_{1} and e_{6} reduce to the dilatation and shear strain, respectively, in the infinitesimal strain limit. The strain measure e_{2} uniquely maps the square lattice into the two rectangular variants (Figure 1a): positive and negative e_{2} generate the rectangles elongated along the global X_{1} and X_{2} directions, respectively. The equivalence of the rectangular variants under the point group symmetry of the square lattice (e_{2}=0) restricts the freeenergy density to even functions of e_{2}.
If the crystalline solid has multiple chemical species, its freeenergy density dependence on e_{1}, e_{2} and e_{6} can change with composition, c. Figure 2a illustrates a freeenergy density surface, $\mathcal{F}(c,{e}_{2})$, for a binary solid that, at higher temperature, forms a solid solution having square symmetry. In this case, ℱ is convex, a condition made precise by specifying positive eigenvalues of its Hessian matrix over the {c, e_{2}} space. In addition, $\mathcal{F}(c,0)$ is a minimum with respect to e_{2} for fixed c, making the square phase stable for all c at this temperature. However, at a lower temperature, ℱ may lose convexity, inducing the notion of a mechanochemical spinodal region. We define it as the domain in {c, e_{2}} space over which the Hessian matrix admits negative eigenvalues, as illustrated in Figure 2b. Here we focus on the conditions ${\partial}^{2}\mathcal{F}/\partial {c}^{2}<0$ and ${\partial}^{2}\mathcal{F}/\partial {e}_{2}^{2}<0$. The square phase (e_{2}=0) remains stable at high composition (c~1) with positive eigenvalues of the Hessian (Figure 2b). But it is mechanically unstable at low composition, with ${\partial}^{2}\mathcal{F}/\partial {e}_{2}^{2}<0$ for (c, e_{2})~(0, 0), and has two symmetrically equivalent, stable, rectangular phases, with ${\partial}^{2}\mathcal{F}/\partial {e}_{2}^{2}>0$ at (c, e_{2})=(0,±s_{e}). Although not shown in Figure 2b, we assume that ℱ is convex with respect to e_{1} and e_{6}. Figure 2c illustrates a schematic temperature–composition phase diagram consistent with the freeenergy densities of Figure 2a,b. The square/cubic phase, β, forms at high composition or at high temperature; the rectangle/tetragonal phase, α, forms at low composition and low temperature. A large twophase region separates them.
Zuzek et al.^{10} have obtained phase diagrams experimentally that are topologically equivalent to Figure 2c for the ZrH_{2−2c} system. Recent firstprinciples calculations^{3} have demonstrated the existence of a mechanical instability that exists in this system at low c via nonconvexity with respect to strain. Furthermore, the twophase coexistence at low temperature also implies nonconvexity with respect to the composition, as we have demonstrated in Supplementary Information. Figure 2a,b therefore represents a twodimensional analogue of the free energy for such systems.
Consider our model binary solid, annealed at high temperature T_{h}, to form a solid solution in the square phase, β. Its state is at point A in Figure 2a, with e_{2}=0. It is then quenched into the twophase region (Figure 2c) with freeenergy density at point B in Figure 2b. For a quench at sufficiently high rate, the dimensions of the square lattice, controlled by strains e_{1}, e_{2} and e_{6}, and the composition remain momentarily unchanged. However, as the state at point B satisfies ${\partial}^{2}\mathcal{F}/\partial {e}_{2}^{2}<0$ and ${\partial}^{2}\mathcal{F}/\partial {c}^{2}<0$, there exist thermodynamic driving forces for segregation by strain and composition within the mechanochemical spinodal.
Diffusion being substantially slower than elastic relaxation, the solid immediately deforms to either positive or negative e_{2}, driven towards a local minimum at constant, c. These deformations due to the mechanical instability will happen like many martensitic transformations, where a mix of symmetrically equivalent rectangular variants coexist to minimise macroscopic strain energy. For finite but moderate strain, the transformation could proceed coherently,^{11} even if the two symmetrically equivalent rectangular variants coexist. We neglect nonessential complexities of this process and assume that, instantly upon quenching, finitely sized neighbourhoods of the solid deform homogeneously into one of these rectangular variants at the original composition.
The solid also becomes susceptible to uphill diffusion because ${\partial}^{2}\mathcal{F}/\partial {c}^{2}<0$ implies a negative diffusion coefficient. However, it does not occur at constant e_{2}, as the valleys traversing the local minima, $\partial \mathcal{F}/\partial {e}_{2}=0$, between the square lattice at c=1 and the rectangular lattices at c=0 span intervals of negative and positive e_{2}. Mechanochemical spinodal decomposition sets in. Composition modulations are amplified: highc regions strive to be more square (point C), whereas lowc regions strive to be more rectangular (points D or D′). However, as coherency is maintained, some neighbourhoods in the solid will be frustrated from attaining strains that ensure minima, $\partial \mathcal{F}/\partial {e}_{2}=0$, for local values of c. Coherency straininduced freeenergy penalties arise to alter the driving forces for purely chemical spinodal decomposition. Supplementary Movie S5 in the Supplementary Information shows the evolution of the state (c, e_{2}) of the material points on the freeenergy manifold ℱ.
Mathematical formulation: three dimensions
Armed with the insight conveyed by the twodimensional study, we next lay out the threedimensional treatment, in which setting the considered phase transformations proceed via lattice deformation and diffusion. The arguments are made more concrete by considering the general mathematical form of the freeenergy density.
Strain order parameters
The strain measures e_{1}–e_{6} are first redefined using the full Green–Lagrange strain tensor components in three dimensions:
In the limit of infinitesimal strains, e_{1} describes the dilatation, whereas e_{4}, e_{5} and e_{6} reduce to shears. The point group operations of the cubic crystal leave e_{1} invariant, as it is the trace of E, while collecting each of the subsets {e_{2}, e_{3}} and {e_{4}, e_{5}, e_{6}} into a symmetryinvariant subspace whose elements transform into each other. The measures e_{2} and e_{3} are especially suited as order parameters to describe cubictotetragonal distortions. All three tetragonal variants that emerge from the cubic reference crystal can be uniquely represented by these two measures. See Figure 1b, where tetragonal distortions of the cubic crystal along the X_{1}, X_{2} and X_{3} axes have been labelled as 1, 2 and 3, respectively. Deviations from the dashed lines in the e_{2}–e_{3} space of Figure 1b correspond to orthorhombic distortions of the cubic reference crystal. This role of e_{2} and e_{3} as structural order parameters to denote the degree of tetragonality, and to distinguish between the three tetragonal variants, complements their fundamental purpose as arguments of the elastic freeenergy density.
The mechanochemical spinodal in three dimensions
For brevity, we write the strains as e={e_{1},…,e_{6}}. We introduce further phenomenology by specifying that $\mathcal{F}(c,\mathit{e})$ is only one part of the total freeenergy density. It is a homogeneous contribution whose composition and strain dependence cannot be additively separated. Figure 3, for example, illustrates contour plots of $\mathcal{F}(c,\mathit{e})$ on the e_{2}–e_{3} and e_{3}–c planes for a binary solid having a temperature–composition phase diagram similar to that of Figure 2c. To generate the tetragonal α phase as illustrated in that diagram, the homogeneous freeenergy density as a function of strain must qualitatively follow the contours of the e_{2}–e_{3} plane at c=0 in Figure 3. Here the tetragonal variants are local minimisers of ℱ in e_{2}–e_{3} space, with equal minima. In turn, to obtain the cubic β phase, $\mathcal{F}(c,\mathit{e})$ must follow the contours of the e_{2}–e_{3} plane at c=1. Thus, ℱ changes smoothly from convex with respect to e_{2} and e_{3} on the c=1 plane to nonconvex at c=0 to form the three variants of the tetragonal α phase. Importantly, the planes at c=0 and c=1 themselves must be minimum energy surfaces to represent the tetragonal α and cubic β phases, respectively. Supplementary Movie S8 in the Supplementary Information shows the evolution of the state (c, e_{2}, e_{3}) of the material points on the freeenergy manifold ℱ.
Gradient regularisation of the freeenergy density
Following van der Waals,^{12} and Cahn and Hilliard^{1} in their treatment of nonuniform composition fields, we can extend the total freeenergy density beyond ℱ by writing it as a Taylor series, retaining terms that depend on the composition gradient, ∇c. We extend this gradient dependence to the strain measures ∇e, as did Barsch and Krumhansl^{8} following Toupin.^{9} (Also see Karatha et al.,^{13,14} for treatments using infinitesimal and finite strain, respectively.) These gradient dependences appear in a nonuniform freeenergy density $\mathcal{G}(c,\mathit{e},\nabla c,\nabla \mathit{e})$. Frame invariance of ℱ and $\mathcal{G}$ is guaranteed, as the members of e are linear combinations of the tensor components of E. They also must be invariant under point group operations of the cubic reference crystal.
Freeenergy functional
The crystal occupies a reference (undeformed) configuration Ω with boundary Γ. The total freeenergy, Π, is the integral of the freeenergy density $\mathcal{F}+\mathcal{G}$ over the solid with boundary contributions included. Thus, Π is a functional of the composition c and the displacement vector field u, from which the strains are derived (Supplementary Information):
where traction vector component T_{i} is specified on the boundary subset ${\Gamma}_{{T}_{i}}\subset \Gamma $. Following the above authors, we include only quadratic terms in the gradients, but in a generalisation we also allow cross terms between ∇c and ∇e in the nonuniform contribution $\mathcal{G}$, which can therefore be written as
Here κ is a symmetric tensor of compositiongradient energy coefficients, each γ^{αβ} (α, β=1,…,6) is a tensor of strain gradient energy coefficients, and each θ^{α} is a tensor of the mixed, composition–strain gradient energy coefficients. Note that, in general, these coefficients will be functions of the local composition and strain. The point group symmetry of the cubic reference crystal imposes constraints on the tensor components of κ, γ^{αβ} and θ^{α} as well as on the form of ℱ.
Although the gradient energies bestow greater accuracy upon the freeenergy description of solids with nonuniform composition and strain fields, they are essential at a more fundamental level if the homogeneous freeenergy density is nonconvex. At compositions that render ℱ nonconvex, the absence of a gradient energy term will allow spinodal decomposition characterised by composition fluctuations of arbitrary fineness, thus leading to nonunique microstructures—a fundamentally unphysical result.^{15} With κ≠0, the compositiongradient energy penalises the interfaces wherein composition varies rapidly between high and low limits. This ensures physically realistic results, manifesting in unique microstructures with a mathematically wellposed formulation.
An essentially analogous situation exists with respect to the negative curvatures of ℱ in the e_{2}–e_{3} plane at low c, which drive the cubic lattice to distort into the tetragonal variants corresponding to the three freeenergy wells. Consider a solid with a homogeneous freeenergy density as in Figure 3 and a strain state lying between the valleys in the e_{2}–e_{3} plane at c=0. Absent the strain gradient energy, mechanochemical spinodal decomposition would allow tetragonal variants of arbitrary fineness—an unphysical result, reflecting further mathematical illposedness. Retention of the strain gradient energy, (γ^{αβ}≠0) penalises interfaces of sharply varying strain between tetragonal variants to ensure physically realistic results and unique microstructures from a mathematically wellposed formulation. This is well understood in the literature that studies the formation of martensitic microstructures from nonconvex freeenergy density functions.^{16–18}
Governing equations of nonequilibrium chemistry
The free energy for nonhomogeneous composition and strain fields (equation 2), must be a minimum at equilibrium. The state of a solid out of equilibrium will evolve to reduce the free energy ∏ [c, u]. In formulating a kinetic equation for the redistribution of atomic species through diffusion, we are guided by variational extremisation of the free energy to identify the chemical potential, μ. Details of this calculation appear as Supplementary Information. The result follows:
For solids where c tracks the composition of an interstitial element within a chemically inert host, such as Li in Li_{c}Mn_{2}O_{4}, μ in equation (4) corresponds to the chemical potential of the interstitial element. If c tracks the composition of a substitutional species, such as in alloys or on sublattices of complex compounds (e.g., the cation sublattice of yttriastabilised zirconia), μ is equal to the chemical potential difference between the substitutional species.
The common phenomenological relation for the flux is J=−L (c, e)∇μ, where L is the Onsager transport tensor (see de Groot and Mazur^{19}). For an interstitial species, L is related to a mobility,^{20} whereas it is a kinetic, intermixing coefficient for a binary substitutional solid.^{21} Inserting the flux in a mass conservation equation yields the strong form of the governing partial differential equation for timedependent mass transport. It is of fourth order in space due to the compositiongradient dependence of μ in equation (4). See Supplementary Information for strong and weak forms of this partial differential equation.
Governing equations of mechanical equilibrium: strain gradient elasticity
Mechanical equilibrium is assumed as elastic wave propagation typically is a much faster process than diffusional relaxation in crystalline solids. Equilibrium is imposed by extremising the freeenergy functional with respect to the displacement field. Standard variational techniques lead to the weak and strong forms of strain gradient elasticity. The treatment is technical, for which reason we restrict ourselves to the constitutive relations that are counterparts to the chemical potential equation (4) for chemistry. Coordinate notation is used for transparency of the tensor algebra, and summation is implied over repeated spatial index, I. Details appear in Supplementary Information. The final form of the equations is complementary to that mentioned in Toupin,^{9} as our derivation is relative to the reference crystal, Ω.
With the deformation gradient F being related to the Green–Lagrange strain as ${E}_{KL}=\frac{1}{2}({F}_{iK}{F}_{iL}{\delta}_{KL})$, the first Piola–Kirchhoff stress tensor and the higherorder stress tensor, respectively, are given by
The higherorder stress B, which is absent in classical, nongradient elasticity^{22} (and in earlier treatments of mechanochemistry^{23,24}), makes the strong form of gradient elasticity a fourth order, nonlinear partial differential equation in space (Supplementary Information). The first threedimensional solutions to general boundary value problems of Toupin’s strain gradient elasticity theory at finite strains were recently presented by the authors.^{25}
Results
Twodimensional examples
We first consider a twodimensional solid to better visualise the microstructures that can emerge from mechanochemical spinodal decomposition. Plane strain elasticity is assumed, for which E_{13}, E_{23} and E_{33}=0, giving e_{4} and e_{5}=0, and ${e}_{3}={e}_{1}/\sqrt{2}$, reducing equations (1, 2, 3, 4, 5 and 6) to two dimensions. The discussion on twodimensional mechanochemical spinodal decomposition (Figure 2) holds: as a particular parameterisation of $\mathcal{F}(c,{e}_{1},{e}_{2},{e}_{6})$, we consider a regular solution model as a function of c at zero strain. At c=0, $\mathcal{F}(c,{e}_{1},{e}_{2},{e}_{6})$ is doublewelled in e_{2}, corresponding to the two rectangular variants. The gradient energy is $\mathcal{G}(\nabla c,\nabla {e}_{2})$ with a constant isotropic κ≠0 and constant γ^{22}≠0, all other gradient coefficients being zero. Although the fullest possible complexity of the coupling is not revealed by these simplifications, the aim here is to present the essential physics that is universal to mechanochemical spinodal decomposition, postponing details of the more complex couplings and tensorial forms to future communications, where they will be derived for specific materials systems. See Supplementary Information for specific forms of $\mathcal{F}\left(c,{e}_{1},{e}_{2},{e}_{6}\right)$ and $\mathcal{G}(\nabla c,\nabla {e}_{2})$.
Figure 4 shows the evolution of microstructure over a 0.01×0.01 domain whose reference (initial) state has the square crystal structure and c=1. The displacement component u_{1}=10^{−5} on the right boundary (X_{1}=0.01), with the remaining boundaries fixed (u=0). An outward flux is imposed on the top and bottom boundaries causing a decrease in composition, c, starting from the boundaries. As the composition falls, the homogeneous freeenergy density ℱ loses convexity and the state of the material (c,e_{2}) enters the mechanochemical spinodal along e_{2}=0 (Figure 2b). The continuing outward flux first drives material near the top and bottom boundaries fully into the regime, where the rectangular crystal structure is stable. As explained in Figure 2b, the negative curvature ${\partial}^{2}\mathcal{F}/\partial {e}_{2}^{2}<0$ creates thermodynamic driving forces that distort the square structure, shown in green, into rectangular variants, which form as red/blue laminae (all e_{2} values). The coexistence of the parent, square structure with the daughter, rectangular variants also can give rise to crosshatched microstructures that parallel the tweed microstructures described in the work of Karatha et al.^{13,14} Supplementary Movie S9 in the Supplementary Information shows the formation of such a microstructure.
A laminar, microtwinned microstructure develops as the two rectangular variants form, distinguished by the sign of e_{2} (see legend). Note that e_{2} remains continuous because of the penalisation of strain gradients ∇e_{2}, although discontinuities can develop in ∇e_{2} itself. Because our numerical framework uses basis functions that are continuous up to their first derivatives (see Methods and Supplementary Information), ∇e_{2} is indeed discontinuous in the computations. The lamination accommodates the strain difference between the two rectangular variants to minimise the free energy. If strains were infinitesimal (e_{1}, e_{2}, e_{6}≳0) e_{2} would correspond to shear along the directions that are rotated π/4 radians from the crystal axes. But the strains in these microstructures can be finite (reflected in the range −0.1⩽e_{2}⩽0.1 in Figure 4) and involve large rotations to accommodate the rectangular variants; this necessitates a finitestrain formulation, as shown in the recent studies by Alphonse et al.^{26} comparing infinitesimalstrain and finitestrain formulations for polytwinned microstructure evolution in martensitic alloys, and by Hildebrand and Miehe.^{7}
The microtwinning and coherency strains are seen in the distorted mesh of discretisation (inset of top centre panel). The undeformed mesh cells are squares, and hence the numerical discretisation strikingly delineates the kinematics of the cubictorectangular transformation, and highlights the rectangular twins as well as the concentrated distortions along the twin boundaries. Supplementary Movie S6 in the Supplementary Information shows such mesh distortion and formation of the rectangular twins with a different form of the function ℱ.
In some cases, the longrange nature of elasticity forces like rectangular variants to align even when separated by an untransformed square phase. This is first seen in the fingerlike extensions of strain contours from the rectangular variants into the square phase in Figure 4, followed by their alignment and eventual incorporation into laminae of the same variant (top right panel and its evolution shown as an inset). In this instance, although the microtwins end at an invariant habit plane that is common with the untransformed square phase, the lattice parameters of the rectangular variants differ from those of the square phase, inducing elastic strain energy in the latter. The alignment lowers this strain energy, and also is seen in Figure 5a. Note, however, that this is not a universal feature. Supplementary Movie S2 shows instances where unlike variants come close to alignment.
The fineness of laminae depends on the strain gradient elasticity length scale ${l}_{\mathrm{e}}\sim \sqrt{{\gamma}^{22}/\sqrt{{\sum}_{\alpha ,\beta =1,2,6}{\left({\partial}^{2}\mathcal{F}/\partial {e}_{\alpha}\partial {e}_{\beta}\right)}^{2}}}$, as explored in Figure 5. In Figure 5a, the initial and boundary conditions are the same as in the example of Figure 4. Decreasing l_{e} weakens the penalty on the strain gradient ∇e_{2} across neighbouring, unlike rectangular variants and allows more twin boundaries. Notably, selfsimilarity is not maintained between microstructures for different l_{e}, even for the same initial and boundary value problem. We understand this to be the influence of elastic strain accommodation: to minimise the total free energy when the strain gradient penalty changes, the physics optimises twin boundaries via laminations of different sizes as well as different patterns. Importantly, however, crystal symmetry admits nonvanishing strain gradient energy coefficients beyond γ^{22}≠0 used in these simulations (Supplementary Information). Furthermore, the composition dependence of ℱ could be more complex than the simple regular solution model used here. Given the already strong effect of l_{e} alone, we conjecture that varying these forms will have a significant influence on the resulting coherent twin microstructure. The proper form, while guided by crystal symmetry arguments must ultimately be determined experimentally or by firstprinciples statistical mechanical methods.
The example in Figure 5b further explores the influence of l_{e}. In this suite of computations, the unstrained material with an initially square microstructure, convex freeenergy density and composition having random fluctuations about c=0.45 was quenched to a low temperature and into the mechanochemical spinodal. The local composition and strain evolve under the thermodynamic driving forces detailed in the context of Figure 2. We draw attention, once again, to the changing identity of rectangular variants, their shapes and sizes depending on l_{e}. Also, note the progressively finer lamination of rectangular domains with decreasing l_{e}. Such studies suggest how dynamic mechanochemical spinodal decomposition can lead to an atlas of microstructures, which in turn will determine material properties. Supplementary Movies S1–S4 in the Supplementary Information show the evolution of some of these microstructures.
A threedimensional study
This final example displays the full threedimensional complexity of microstructures resulting from the mechanochemical spinodal. We persist with the above simplifications aimed at presenting the essential physics of mechanochemical spinodal decomposition, and postpone a full exploration of the coupling and anisotropies in coefficients to future work on specific materials systems. The freeenergy density function used for the threedimensional study appears as Supplementary Information.
Figure 6 shows the equilibrium microstructure that results in a solid that is initially in the cubic phase, c=1, subject to an outflux on all surfaces. The domain is a unit cube with displacement components u_{2},u_{3}=0.01 on the boundary X_{1}=1, with zero displacement, u=0, on the boundary X_{1}=0. The cubictotetragonal transformation takes place as c→0. Under these conditions, all three tetragonal variants form, with an intricately interleaved microstructure for strain accommodation. Note the finer microstructures and changing pattern for smaller l_{e}. The inset shows the surface straining around a corner, delineated by the distorted mesh lines. Observe the three, oriented, tetragonal variants formed by twinning deformation from the initial cubic structure. See Supplementary Movie S7 in the Supplementary Information for a detailed view of the threedimensional structure of these individual tetragonal variants. Supplementary Movie S10 in Supplementary Information shows other threedimensional microstructures whose crosssectional planes bear closer resemblance to the platelike structures predicted by twodimensional computations. To the best of our knowledge, such computations of a cubictotetragonal transformation with twinned variants whose microstructure is controlled by nonlinear, strain gradient interface energies have not been previously presented.
Discussion
Several kinetic mechanisms have been proposed to describe the decomposition of a solid upon entering a twophase region.^{27,28} A commonly observed mechanism occurs through localised nucleation followed by diffusional growth that is mediated by the migration of interfaces separating the daughter phases from the parent phase. Nucleation and growth mechanisms are common when the phases participating in the transformation have little or no crystallographic relation to each other. The transformation then proceeds reconstructively, where the interphase interfaces are disordered or at best only semicoherent. Numerical treatments of this mechanism rely on sharpinterface methods such as a level set approach.^{29}
Other kinetic mechanisms of decomposition involving some degree of coherency are also possible when the participating phases share sufficient crystallographic commonality that order parameters can be defined describing continuous pathways connecting the parent and daughter phases. The structural changes of the crystal that accompany these transformations can fall in one of two categories. One subclass of structural transformations is driven by an internal shuffle, where the atomic arrangement within the unit cell of the crystal undergoes a symmetrybreaking change. The unit cell vectors of the crystal may also undergo a change and lower the symmetry of the lattice, but this effect is secondary and in response to the atomic shuffle within the unit cell. The order parameters for the structural change therefore describe the atomic shuffles within the unit cell and are nonconserved. The second subclass of structural transformations is driven by a symmetryreducing change of the lattice vectors of the crystal, with any atomic rearrangement of the basis of the crystallographic unit cell occurring in response to the symmetrybreaking changes of the unit cell vectors. The natural order parameters describing these transitions are strains. The free energy in the first subclass will exhibit negative curvatures with respect to the nonconserved shuffle order parameters, whereas the free energy in the second subclass will have negative curvatures with respect to the relevant strain order parameters, as was recently predicted for the cubictotetragonal transition of ZrH_{2.}^{3}
Decomposition reactions that require both diffusional redistribution and a structural change due to an internal shuffle have been treated successfully and rigorously with phase field approaches that combine Cahn–Hilliard with Allen–Cahn. The Cahn–Hilliard description accounts for the composition degrees of freedom, whereas the Allen–Cahn component describes the evolution of the nonconserved shuffle order parameters required to distinguish the various phases of the transformation.^{4–7} These treatments have included strain energy as a secondary effect serving only as a positive contribution to the overall free energy due to coherency strains. The approach is rigorous if the freeenergy density remains convex with respect to strain, with instabilities in the free energy appearing as a function of concentration and the nonconserved shuffle order parameters.
Here we have presented a mathematical formulation and an accompanying computational framework for decomposition reactions that combine diffusional redistribution with a structural change driven by a symmetrybreaking strain of the crystallographic unit cell as opposed to an internal shuffle within the unit cell. Strains have a primary role, explicitly serving as order parameters to distinguish variants of a daughter phase that has a symmetry subgroup–group relationship with its parent phase due to a structural change of the crystallographic unit cell. Crucial to the description is that the driving force for the formation of the daughter from a supersaturated parent phase emerges from an instability with respect to composition. The treatment therefore combines Cahn–Hilliard for composition with a description of martensitic transformations at fixed concentration introduced by Barsch and Krumhansl.^{8} The existence of simultaneous instabilities with respect to strain and composition has not been considered, as a mechanism to describe decomposition reactions upon quenching into a twophase region.
The possibility of such a mechanochemical spinodal mechanism is motivated by recent firstprinciples studies of the cubictotetragonal phase transformations of ZrH_{2}, where the free energy of the hightemperature cubic form was predicted to become unstable with respect to strain upon cooling below the cubictotetragonal secondorder transition temperature.^{3} The ZrH_{2−2c} hydride can accommodate large concentrations of hydrogen vacancies, c, and has a phase diagram that is topologically identical to that depicted in Figure 2c, with a twophase region separating a hydrogenrich tetragonal form of ZrH_{2−2cα} from a cubic form of ZrH_{2−2cβ} (with c_{α}<c_{β}). See Zuzek et al.^{10} (in their phase diagrams, Zuzek et al. have an inverted composition axis relative to our notation, the tetragonal phase is labelled ε and the cubic phase is δ). To be consistent with the predicted free energies for stoichiometric ZrH_{2} (i.e., c=0) and the experimental T versus c phase diagram with the form of Figure 2c, the free energy of this hydride as a function of composition and strain (i.e., e_{2} and e_{3}) should be similar to those depicted in Figures 2a,b and 3. Decomposition upon quenching cubic ZrH_{2−2c} into the twophase region can therefore proceed through a mechanochemical spinodal decomposition mechanism.
Not only does the phenomenological description introduced in this work describe a new mechanism of decomposition that is qualitatively distinct from previous combined Cahn–Hilliard and Allen–Cahn approaches, its numerical solution also proves to be substantially more complex due to contributions from strain gradient terms. Indeed, even numerical solutions to general, threedimensional boundary value problems of gradient elasticity at finite strain were not available until presented recently by the authors.^{25} Furthermore, as other authors have pointed out before, the use of strain metrics as order parameters makes a reliance on infinitesimal strains untenable due to the rigid rotations that accompany the finite strains characterising most structural transformations.^{7,26} The use of finitestrain metrics introduces geometric nonlinearity into the problem, which although having been treated in past Cahn–Hilliard and Allen–Cahn approaches,^{7} presents additional numerical challenges when also considering strain gradient contributions. These challenges have been overcome in this work as demonstrated by our threedimensional numerical examples.
Mechanochemical spinodal decomposition as described here can be expected in solids forming hightemperature phases that exhibit dynamical instabilities at low temperature. An accumulating body of firstprinciples calculations of the Born–Oppenheimer surfaces have shown that many hightemperature phases are dynamically unstable at low temperature,^{30,31} becoming stable at high temperature through large anharmonic vibrational excitations,^{3,32–35} usually through a secondorder phase transition. Although such instabilities are frequently dominated by phonon modes describing an internal shuffle, a subset of chemistries become dynamically unstable at low temperature with respect to phonon modes that break the symmetry of the crystal unit cell,^{3,34} an instability that can be described phenomenologically with strain order parameters. If, upon alloying such compounds, the highsymmetry phase becomes stable by passing through a twophase region, its freeenergy surface will resemble that of Figures 2b and 3, and thereby make the solid susceptible to the mechanochemical spinodal decomposition mechanism described here.
Although firstprinciples and experimental evidence suggest that mechanochemical spinodal decomposition should occur in ZrH_{2−2c}, we expect it to occur in a wide range of other chemistries as well. One possible example, as described in the Introduction, is the decomposition of cubic yttria stabilised zirconia^{36,37} upon quenching from the hightemperature cubic phase into a twophase region separating a lowY tetragonal phase from a Yrich cubic phase. Past treatments of this transformation^{4–6} relied on nonconserved order parameters to distinguish the different tetragonal variants from each other and from the cubic parent phase. The elastic part of the freeenergy density function was parameterised using linearised elasticity and infinitesimal strains, as other authors have also done.^{38,39} The chemical part of the freeenergy density was assumed to have a negative curvature as a function of the nonconserved order parameter at Zrrich compositions, but made up of convex (and quadratic) potentials with respect to strain. We reiterate that such a treatment rests on the implicit assumption that internal shuffles drive the cubictotetragonal transformation, whereas the free energy as a function of strain remains convex for all relevant values of the nonconserved order parameters.
The possibility also exists that a coarsegrained freeenergy density of cubic ZrO_{2} exhibits negative curvatures with respect to strain below the cubictotetragonal transition temperature making the formalism developed here an accurate description of decomposition reactions of yttriastabilised zirconia. Although the precise mechanism and nature of the cubictotetragonal transition of pure ZrO_{2} remain to be resolved,^{33} firstprinciples calculations predict that the cubic form of ZrO_{2} is dynamically unstable with respect to the transformation to the tetragonal variant.^{40} A rigorous statistical mechanics treatment^{3,32} is required to determine whether the cubictotetragonal transition of pure ZrO_{2} is accompanied by a change in the sign of the curvature of the freeenergy density with respect to e_{2} and e_{3}. If this proves to be the case, yttriastabilised zirconia should also be susceptible to the mechanochemical spinodal decomposition upon quenching, consistent with the coherent spinodal microstructures between tetragonal and cubic phases observed in singlecrystal regions of quenched cubic Zr_{1−c}Y_{x}O_{2−c/2.}^{37}
A mechanochemical spinodal may also have a role in a variety of important electrode materials for Liion batteries, and intercalation compounds considered for twodimensional nanoelectronics. These include cubic LiMn_{2}O_{4} transforming to tetragonal Li_{2}Mn_{2}O_{4.}^{41} Although most mechanochemical spinodal transitions will occur in three dimensions, many of the qualitative features of these transitions are more conveniently illustrated in two dimensions. Our twodimensional studies should also prove relevant to understanding mechanochemical phase transformations in twodimensional layered materials for nanoelectronics. Materials such as TaS_{2}, are susceptible to Peierls instabilities upon variation of the composition of adsorbed or intercalated guest species that donate to or extract electrons from the sheetlike host.^{42} Our phenomenological treatment by introduction of the concept of a mechanochemical spinodal, coupled with gradient stabilisation of the ensuing nonconvex free energy in strain–composition space, thus offers a framework with potential for extension to a wide range of phase transformation phenomena.
With regard to the fundamental thermodynamic underpinnings of analogous processes, the phenomenological model introduced here can also be used to describe temperaturedriven martensitic transformations. The composition, c, would then be analogous to the internal energy density, $\mathcal{U}$, whereas the chemical potential, μ, would be analogous to the temperature, T. Due to the presence of temperature gradients throughout the solid, though, the starting point must be a formulation of the entropy density, $\mathcal{S}$, (as opposed to the Helmholtz free energy) as a functional of spatially varying $\mathcal{U}$ and displacements, u. In analogy with equation (2), the total entropy can be written as a volume integral over a homogeneous entropy density that depends on the local internal energy density and strain $\overline{\mathcal{S}}(\mathcal{U},e)$, as well as a nonuniform entropy density contribution expressed in terms of gradients in internal energy density and strain ($\nabla \mathcal{U}$, ∇e). Variational maximisation of the entropy will yield mechanical equilibrium equations as well as an expression for the temperature T (strictly speaking, for its reciprocal, 1/T), similar to equation (4) for the chemical potential. Due to the presence of gradient terms, $\nabla \mathcal{U}$ and ∇e, the temperature will not only be a function of the local internal energy density and strain, but also gradients in these field variables. As with the diffusion problem treated here, heat flow can be described with an Onsager expression relating the heat flux to a gradient in temperature. The possibility exists that the homogeneous entropy $\overline{\mathcal{S}}$ exhibits instabilities with respect to internal energy density $\mathcal{U}$, allowing for spinodal decomposition with respect to the redistribution of $\mathcal{U}$ in a manner that is similar to the well understood problem of spinodal decomposition with respect to composition. The treatment introduced here can therefore describe (upon replacing c with $\mathcal{U}$ and μ with T) temperaturedriven martensitic transformations for solids that exhibit instabilities with respect to both internal energy density and strain. Past treatments of this phenomenon also relied on a Barsch and Krumhansl approach, solving the mechanical problem together with Fourier’s law of heat conduction.^{38,39} However, these studies were restricted to two dimensions,^{26,38,39} with some neglecting geometric nonlinearity by relying on infinitesimal strains.^{38,39} At a more fundamental level, they did not consider the possibility of spinodal instabilities with respect to the redistribution of internal energy density.
Materials and methods
The governing fourthorder partial differential equations of mass transport and gradient elasticity are solved numerically in weak form, wherein secondorder spatial gradients appear on the trial solutions and their variations. Numerical solutions require basis functions that are continuous up to their first spatial gradients, at least. Our numerical framework (see Rudraraju et al.^{25}) uses isogeometric analysis^{43} and the PetIGA code framework^{44} with spline basis functions, which can be constructed for arbitrary degree of continuity (Supplementary Information). This framework has proven pivotal to the current work. The code used for the numerical examples in this paper can be downloaded at the University of Michigan Computational Physics Group opensource codes webpage: http://www.umich.edu/~compphys/codes.html
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Acknowledgements
The mathematical formulation for this work was carried out under an NSF CDI Type I grant: CHE1027729 ‘MetaCodes for Computational Kinetics’ and NSF DMR 1105672. The numerical formulation and computations have been carried out as part of research supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under award no. DESC0008637 that funds the PRedictive Integrated Structural Materials Science (PRISMS) Center at University of Michigan.
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Rudraraju, S., Van der Ven, A. & Garikipati, K. Mechanochemical spinodal decomposition: a phenomenological theory of phase transformations in multicomponent, crystalline solids. npj Comput Mater 2, 16012 (2016). https://doi.org/10.1038/npjcompumats.2016.12
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