# Spinodal decomposition **Spinodal decomposition** is the separation of a single phase into two without any [[Nucleation|nucleation]] step. It happens when a mixture is quenched into a region where its free-energy curve is concave down, so that every composition fluctuation, however small, lowers the free energy by growing. There is no critical size to reach and no barrier to cross: the mixture is not metastable but unstable, and it comes apart everywhere at once into an interconnected, modulated structure of a characteristic wavelength.[^cahn1961][^porter-ch5] In the microsim below the control is the quench depth `T/T_c`. A two-state lattice of A and B atoms is run with conserved Kawasaki dynamics — neighbouring unlike atoms swap rather than flip, so the overall composition never changes — and the reader watches the speckle of a random [[Solid_solution|solid solution]] organise into interpenetrating A-rich and B-rich domains, with the characteristic length `L(t)` fitted live against the coarsening law `L ~ t^(1/3)`.[^anag-ch15] The equation behind the continuum version is the [[Cahn–Hilliard_equation|Cahn–Hilliard equation]], `∂c/∂t = ∇·(M·∇(∂f/∂c − 2·κ·∇²c))`, whose linearisation makes each Fourier mode grow as `exp(R(k)·t)` with `R(k) = −M·k²·(f'' + 2·κ·k²)`.[^cahn-hilliard1958][^cahn1961] On the [[Materials_science]] flagship this page serves Part VI, *Fundamentals › Kinetics*, in the section *Spinodal decomposition*, where it is the deliberate counterpart to [[Nucleation|nucleation]]: the same [[Miscibility_gap|miscibility gap]], the same [[Diffusion|diffusion]], and the opposite mechanism. ## History [[Josiah_Willard_Gibbs|Gibbs]] drew the distinction the subject rests on: a phase inside a two-phase region can be *metastable*, needing a finite fluctuation to escape, or *unstable*, needing none, and the boundary between the two is where the second derivative of the free energy changes sign.[^gibbs] What Gibbs did not supply was a rate. The experimental hint came first. In 1943 Daniel and Lipson found side bands flanking the X-ray reflections of a decomposing copper–iron–nickel [[Alloy|alloy]], which they interpreted as a periodic modulation of composition — not a set of discrete particles, but a wave.[^daniel-lipson1943] Mats Hillert's 1956 MIT thesis, published in 1961, wrote a discrete diffusion equation for a non-uniform solution and found that below the spinodal it had growing periodic solutions.[^hillert1961] John Cahn and John Hilliard then gave the continuum form in 1958, adding to the free energy a term in the square of the composition gradient and showing that this single term accounts for [[Surface_tension|interfacial]] energy in a diffuse interface.[^cahn-hilliard1958] Cahn applied the machinery to the unstable case in 1961, obtaining the amplification factor and the dominant wavelength, and in 1962 added the elastic energy that coherency imposes in a crystal.[^cahn1961][^cahn1962] The equation has since escaped [[Metallurgy|metallurgy]] entirely. The Cahn–Hilliard equation is now a standard model for phase separation in [[Polymer|polymer]] blends, in fluids, in [[Glass|glasses]] and in lithium battery electrodes, and is used well outside physics as a general model of pattern formation in a conserved field. ## Cahn–Hilliard model for spinodal decomposition The model's single idea is that the free energy of a non-uniform system cannot be obtained by summing the bulk free energy point by point, because a steep composition gradient costs energy in its own right. Cahn and Hilliard wrote the total as `F = ∫[f(c) + κ·|∇c|²] dV`, where `f(c)` is the homogeneous free-energy density and `κ` the gradient-energy coefficient.[^cahn-hilliard1958] That one extra term does two jobs. In equilibrium it produces a diffuse interface of finite width rather than a mathematical surface, and integrating the excess across it yields an interfacial energy — the result the 1958 paper was written to obtain.[^cahn-hilliard1958] Out of equilibrium it penalises short wavelengths, which is what stops the decomposition from producing infinitely fine structure. Because composition is conserved, the dynamics cannot simply run downhill in `c`; matter has to be moved. The diffusion potential is the variational derivative `µ = δF/δc = ∂f/∂c − 2·κ·∇²c`, the flux is `J = −M·∇µ` with `M` an atomic mobility, and continuity gives the Cahn–Hilliard equation `∂c/∂t = ∇·(M·∇µ)`.[^cahn1961][^porter-ch5] It is a fourth-order [[Partial_differential_equation|partial differential equation]] — the `∇²` inside `µ` and the `∇·∇` outside — which is why it is expensive to integrate and why the M33 sim offers a pre-baked movie as an alternative to running it live. ## Dynamics of spinodal decomposition when molecules move via diffusion Expand about a uniform composition `c_0`, keep only terms linear in the fluctuation, and treat `M` and `κ` as constants. The equation becomes `∂δc/∂t = M·f''·∇²δc − 2·M·κ·∇⁴δc`, where `f'' = ∂²f/∂c²` evaluated at `c_0`.[^cahn1961] The first term alone is [[Fick's_laws_of_diffusion|Fick's second law]] with an effective diffusion coefficient `D = M·f''`. Everything follows from the sign of `f''`. Outside the spinodal `f''` is positive, `D` is positive, and fluctuations decay in the ordinary way. Inside it `f''` is negative, so `D` is negative: matter flows *up* its own concentration gradient, making rich regions richer. This is the uphill diffusion that makes the process look paradoxical, and it is not a violation of anything — the flux still runs down the gradient of [[Chemical_potential|chemical potential]], which is what the first law actually says.[^porter-ch5] Substituting a single Fourier mode `δc ∝ exp(i·k·x)·exp(R(k)·t)` gives the amplification factor `R(k) = −M·k²·(f'' + 2·κ·k²)`. With `f'' < 0` this is positive for every `k` below a cut-off `k_c = sqrt(−f''/(2·κ))` and negative above it, so long wavelengths grow and short ones die. Differentiating, the fastest-growing mode sits at `k_max = k_c/sqrt(2)`, that is `λ_max = 4·π·sqrt(κ/(−f''))`, and it grows at `R_max = M·(f'')²/(8·κ)`. A quench therefore selects a wavelength, not a size distribution, and the structure that appears is periodic from the first moment it is visible. The quench depth is the sim's control because it moves `f''`. For a [[Regular_solution|regular solution]] at the critical composition, `f'' = 4·R·(T − T_c)`, so `−f'' ∝ (1 − T/T_c)`. Deepening the quench from `T/T_c = 0.9` to `0.5` multiplies `−f''` by five: `λ_max` falls by `sqrt(5) ≈ 2.2` and `R_max` rises by 25, so the structure is twice as fine and appears twenty-five times faster. Shallow quenches, by contrast, give a coarse, slow, easily missed decomposition — and at the spinodal itself `λ_max` diverges and `R_max` vanishes, joining smoothly onto the nucleation regime outside. ## Phase diagram Two curves bound the two-phase field, and they are not the same curve. The binodal, or coexistence curve, is drawn by the common tangent to the free-energy curve and gives the compositions of the two equilibrium phases. The spinodal is drawn by the inflection points, `∂²f/∂c² = 0`, and separates metastable from unstable. Between binodal and spinodal the homogeneous phase sits in a local minimum and can only decompose by nucleation and growth; inside the spinodal it sits on a hilltop and decomposes spinodally. The two curves touch at the [[Critical_point_(thermodynamics)|critical point]], where the [[Phase_diagram|phase diagram]]'s distinction disappears.[^porter-ch5] The regular solution gives both in closed form. With `f = Ω·X·(1 − X) + R·T·(X·ln X + (1 − X)·ln(1 − X))`, the spinodal condition `f'' = 0` reads `R·T = 2·Ω·X·(1 − X)`, maximised at `X = 0.5` where `R·T_c = Ω/2`; dividing gives the compact result `T_s/T_c = 4·X·(1 − X)`. At `X = 0.3` the spinodal lies at `0.84·T_c`, at `X = 0.2` at `0.64·T_c` and at `X = 0.1` at `0.36·T_c`, so an off-centre alloy must be quenched much further to reach instability — which is why spinodal microstructures are found in alloys close to the top of their [[Miscibility_gap|miscibility gap]]. The fluid analogue is the same picture in different variables. On a [[Van_der_Waals_equation|van der Waals]] isotherm below the critical temperature there is a mechanically unstable segment where `(∂P/∂V)_T > 0`, bounded by the two turning points; those are the spinodal, and Likharev's account notes that the metastable branches outside them survive only as far as the spinodals before the system runs out of local minima.[^likharev-sm4] ## Coherency strains In a crystal the two product regions do not simply have different compositions; they have different natural lattice parameters while remaining one continuous, coherent lattice. A composition wave is therefore also a strain wave, and its elastic energy opposes the decomposition. Writing `η = (1/a)·(da/dc)` for the fractional change of lattice parameter with composition, the elastic penalty per unit volume is proportional to `η²·Y·(Δc)²`, with `Y` the appropriate combination of [[Young's_modulus|elastic constants]] for the direction of the wave.[^cahn1962] Adding this term to `f''` shifts the instability condition from `f'' = 0` to `f'' + 2·η²·Y = 0`. The result is a *coherent spinodal* lying below the chemical one; for the regular solution at the critical composition, where `f'' = 4·R·(T − T_c)`, the depression is `η²·Y/(2·R)`, which for a strongly size-mismatched pair can be hundreds of kelvin. An alloy quenched between the two curves is chemically unstable but elastically stuck, and does nothing. Because `Y` depends on direction, the strain term also chooses an orientation. In [[Cubic_crystal_system|cubic]] metals the elastically soft directions are usually the `<100>` family, so the composition modulation aligns along the cube axes and produces the tweed or basketweave [[Microstructure|microstructures]] seen in decomposed copper–nickel–iron and aluminium–zinc alloys.[^cahn1962][^porter-ch5] Where the mismatch is large enough, coherency is eventually lost to [[Dislocation|misfit dislocations]] and the constraint disappears along with it. ## Fourier transform Spinodal decomposition is one of the few microstructural processes best described in [[Reciprocal_lattice|reciprocal space]], because the linearised equation is diagonal there: each Fourier component evolves independently, with its own growth rate `R(k)`, and no mode talks to any other until the nonlinear terms wake up. [[Fourier_analysis|Fourier analysis]] is not a convenience here but the natural coordinate system. The experimental consequence is that the right instrument is a scattering instrument. Small-angle X-ray and neutron scattering measure the structure factor `S(k,t) = |c_k|²` directly, and in the linear regime it should grow as `exp(2·R(k)·t)` — a peak at `k_max` that rises in height at fixed position, with a node at `k_c` where nothing grows or decays at all. Rearranging the amplification factor gives a straight-line test: `R(k)/k²` plotted against `k²` should be linear, with intercept `−M·f''` and slope `−2·M·κ`, which is how both the mobility and the gradient-energy coefficient are extracted from one experiment.[^cahn1961] Daniel and Lipson's side bands were the same information, read from the diffraction pattern before anyone had the theory to interpret it.[^daniel-lipson1943] ## Dynamics in k-space The linear theory is valid only while the amplitude is small, which in practice means the first moments. Once the domains reach their equilibrium compositions the wave cannot grow in amplitude any further, and the system lowers its energy the only way left: by reducing the total interfacial area. The peak in `S(k,t)` then starts to move toward smaller `k`, and the characteristic length grows as `L(t) ~ t^(1/3)`, the diffusion-limited coarsening law shared with [[Ostwald_ripening|Ostwald ripening]].[^lsw] The pattern also becomes self-similar: `S(k,t)` collapses onto a single master curve when `k` is scaled by `1/L(t)`, so late-stage structures differ only in magnification. This is what the sim measures. Kawasaki exchange conserves the number of A and B atoms, which is the lattice equivalent of the conserved field in the Cahn–Hilliard equation, and produces the `t^(1/3)` law; the contrast is with non-conserved single-spin-flip dynamics, where domains may simply grow at the expense of their neighbours and the exponent is `1/2`.[^anag-ch15] The underlying lattice machinery — Metropolis acceptance `min(1, exp(−β·ΔE))`, a random-number stream, and a sweep counted in attempted moves per site — is the standard two-dimensional [[Ising_model|Ising]] apparatus, and Likharev's treatment of Ising domain walls supplies the energetics of the boundaries that the coarsening is eliminating.[^anag-ch15][^likharev-sm4b] ## Spinodal architected materials The last decade has turned the microstructure into a design. A spinodal topology — smooth, bicontinuous, non-periodic, with no straight struts and no junctions where many members meet — can be generated cheaply as a level set of a Gaussian random field and then manufactured, either by [[3D_printing|additive manufacturing]] at the millimetre scale or by dealloying at the nanometre scale, the route that produces nanoporous gold. The appeal is mechanical rather than aesthetic. Lattice-based [[Composite_material|architected materials]] concentrate stress at their nodes and lose much of their predicted stiffness to the imperfections of real manufacturing; a spinodal shell has no nodes, is close to isotropic because the underlying random field has no preferred direction, and degrades gracefully when a few members are malformed. Whether a given spinodal architecture can be built at all is then a process question: the Portal Books' additive-manufacturing design rules set a minimum wall thickness per process, and a 0.03 in feature that passes powder-bed fusion and selective laser sintering fails fused deposition outright.[^am-ch4] The same design-rule table governs the overhangs an unsupported bicontinuous shell inevitably contains.[^am-ch4] *See also:* [[Miscibility_gap]] — the region of the phase diagram this page lives inside · [[Cahn–Hilliard_equation]] · [[Guinier–Preston_zone]] · [[Regular_solution]] — the model behind `T_s/T_c = 4·X·(1 − X)` · [[Nucleation]] · [[Ostwald_ripening]] · [[Ising_model]] · [[Phase_diagram]] ## References [^gibbs]: Gibbs, J. W. (1876–1878). "On the Equilibrium of Heterogeneous Substances." *Transactions of the Connecticut Academy of Arts and Sciences* 3 (page to pin). The metastable/unstable distinction and the sign of the second derivative of the free energy. [^daniel-lipson1943]: Daniel, V.; Lipson, H. (1943). "An X-ray study of the dissociation of an alloy of copper, iron and nickel." *Proceedings of the Royal Society A* 181 (page to pin). [^hillert1961]: Hillert, M. (1961). "A solid-solution model for inhomogeneous systems." *Acta Metallurgica* 9 (page to pin). Based on his 1956 MIT doctoral thesis. [^cahn-hilliard1958]: Cahn, J. W.; Hilliard, J. E. (1958). "Free Energy of a Nonuniform System. I. Interfacial Free Energy." *The Journal of Chemical Physics* 28 (2): 258–267. https://doi.org/10.1063/1.1744102 [^cahn1961]: Cahn, J. W. (1961). "On spinodal decomposition." *Acta Metallurgica* 9 (page to pin). The linearised equation, the amplification factor `R(k)`, the cut-off and dominant wavelengths, and the `R(k)/k²` against `k²` plot. [^cahn1962]: Cahn, J. W. (1962). "On spinodal decomposition in cubic crystals." *Acta Metallurgica* 10 (page to pin). Coherency strain energy, the coherent spinodal and the selection of elastically soft directions. [^lsw]: Lifshitz, I. M.; Slyozov, V. V. (1961). "The kinetics of precipitation from supersaturated solid solutions." *Journal of Physics and Chemistry of Solids* 19 (page to pin); and Wagner, C. (1961). "Theorie der Alterung von Niederschlägen durch Umlösen." *Zeitschrift für Elektrochemie* 65 (page to pin). The `t^(1/3)` law for diffusion-limited coarsening of a conserved field. [^porter-ch5]: Porter, D. A.; Easterling, K. E.; Sherif, M. Y. *Phase Transformations in Metals and Alloys*, 3rd ed. (2009), Ch. 5 Diffusional Transformations in Solids, §"Spinodal decomposition": the binodal and spinodal, uphill diffusion and the negative effective diffusivity, the coherent spinodal, and the resulting modulated microstructures (page to pin). [^likharev-sm4]: Likharev, K. *Essential Graduate Physics, Part SM: Statistical Mechanics* (2013), Ch. 4 Phase Transitions, pp. 108–110: the van der Waals isotherm's mechanically unstable segment where `(∂P/∂V)_T > 0`, the spinodals bounding it, and the statement that metastable branches survive only to near the spinodals (page to pin for the individual statements). https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics [^likharev-sm4b]: Likharev (2013), Part SM, pp. 133–137: Ising domain walls, the wall-energy estimate of `T_c`, and the note that only `2d + 1` terms change per spin flip — the energetics behind the boundaries the sim's coarsening eliminates (page to pin). [^anag-ch15]: Anagnostopoulos, K. *Computational Physics: A Practical Introduction to Computational Physics and Scientific Computing (using C++)*, 2nd ed. (2016), Ch. 15 Simulation of the d = 2 Ising Model, pp. 520–593: the Metropolis acceptance rule `min(1, exp(−β·ΔE))`, the `ΔE = 2·s_k·Σ_nn s` energy change, the sweep structure and the acceptance lookup table that the M33 sim's conserved-exchange variant reuses. https://open.umn.edu/opentextbooks/textbooks/computational-physics-a-practical-introduction-to-computational-physics-and-scientific-computing-using-c [^am-ch4]: Barnes, J.; Simpson, T. *Additive Manufacturing Essentials* (2025), Ch. 4 Design for Additive Manufacturing, pp. 82–86 and p. 93: the per-process design-allowable tables (a 0.03 in wall passes powder-bed fusion and selective laser sintering and fails fused deposition) and pp. 81–83 for the 45° overhang rule. [^scaling-note]: The quench-depth comparison in this article (`λ_max` falling by `sqrt(5)` and `R_max` rising by 25 between `T/T_c = 0.9` and `0.5`) is derived here from the regular-solution result `f'' = 4·R·(T − T_c)` together with Cahn's `λ_max = 4π·sqrt(κ/(−f''))` and `R_max = M·(f'')²/(8·κ)`; it is not a measured result for any particular alloy, and it assumes `M` and `κ` constant over that range, which they are not. ## Further reading - Cahn, J. W.; Hilliard, J. E. (1958), and Cahn, J. W. (1961) and (1962) — the three papers that contain almost the whole theory. - Porter, D. A.; Easterling, K. E.; Sherif, M. Y. *Phase Transformations in Metals and Alloys*, 3rd ed. (2009), Ch. 5. - Likharev, K. *Essential Graduate Physics, Part SM* (2013), Ch. 4 — Portal Book, the fluid spinodal and the Ising machinery. - Anagnostopoulos, K. *Computational Physics*, 2nd ed. (2016), Ch. 15 — Portal Book, the lattice Monte Carlo the sim is built on. - Barnes, J.; Simpson, T. *Additive Manufacturing Essentials* (2025), Ch. 4 — Portal Book, on the [[PORTAL_Materials_science]] manufacturing shelf. ## External links - The Wikipedia pair's *External links* section lists the current Cahn–Hilliard solvers and phase-field codes; none is reproduced here until its URL has been checked. - Likharev, *Part SM*, and Anagnostopoulos, *Computational Physics*, are on the Open Textbook Library (links in the references above). <!-- MATTERSIM:BEGIN g24 — Matter & Energy Cluster microsim (framework build, specs/sims/Spinodal_decomposition.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework), pending deploy:** *Spinodal decomposition* will play here once `https://wikitube-3d-microsims.netlify.app/matter/Spinodal_decomposition.html` is live. <!-- pending: <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Spinodal_decomposition.html" data-title="Spinodal decomposition"></div> --> <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Spinodal_decomposition) : [Wikitube](https://en.wikitube.io/wiki/Spinodal_decomposition) · pinned revision [1371823050](https://en.wikipedia.org/w/index.php?oldid=1371823050) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Materials_science]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Materials_science row M33 · sim pending (matter/Spinodal_decomposition).*