Kinetic Arrest, Not Equilibrium: Why Indium Nanoinclusions Survive in Sphalerite

Version 2.0. This version reports the outcome of a test that version 1 proposed but had not carried out. Version 1 noted that four determinations of the coarsening exponent span a factor of three, and suggested this might reflect a crossover from interface-limited coarsening at the atomic scale towards diffusion-limited coarsening as domains grow, adding that the suggestion was testable within a single simulation. Phase-field runs at four interface widths have since been made and the hypothesis is NOT supported: rescaling the abscissa by interface width reduces the scatter between curves only from 0.011 to 0.008, where collapse would require a ratio well below one; at a common L/w of 4.4 the four runs agree to 0.004 while their absolute domain size differs by a factor of 3.4, so the exponent tracks neither variable; and the exponent rises through every run, from 0.207 to 0.267, where the hypothesis requires a fall from about 1/2 towards 1/3. What the exponent does track is how far each run has progressed, correlating with the logarithm of the step count at 0.99, which points to a common asymptote approached from below. No other conclusion of version 1 is altered, and the discrepancy with the natural end members returns to the unpaired sampling as the remaining explanation. Phase-field and lattice Monte Carlo study of why indium nanoinclusions survive in sphalerite when equilibrium thermodynamics demands a single condensed domain. This record contains the complete source code, the numerical results underlying every figure and table, and the manuscript in three versions. The preceding paper of this series established that the equilibrium state of indium in sphalerite is a single condensed domain, at every composition from 0.10 to 2.00 atomic per cent. The present work adds the interfacial energy in closed form, gamma = lambda/3 - J(Cu-In)/3 - J(like)/6, in which the bulk terms cancel identically, and shows that dividing one domain into two costs between 300 and 1800 k_B T over the size range examined. Bulk configurational thermodynamics therefore opposes a population of separate domains by an enormous margin. Natural sphalerite carries such populations anyway, in the form of nanometric and micrometric inclusions, so the observed configurations lie hundreds of k_B T above the state the free energy selects. We argue that this is kinetic arrest rather than a failure of the thermodynamics. Coarsening is limited by solute transport, which is thermally activated and therefore slows exponentially as temperature falls; the cooling timescale falls only as the square of temperature. Two such functions must cross, and they cross once. Below the crossing the microstructure ceases to evolve, and what survives to be observed is not the equilibrium state but the configuration the system occupied when transport stopped. The arrest condition is strongly buffered. Every quantity entering it, including the diffusion prefactor this work cannot determine, appears inside a logarithm, so an error of an order of magnitude in any of them shifts the arrest temperature by less than a quarter. The existence of arrest therefore does not depend on the parameters that remain uncertain; only its calibration does. The frozen domain size follows a power law in cooling rate, which inverts, so that inclusion size records thermal history. Two calculations support this, with roles that are not interchangeable, and the division of labour is stated in the manuscript before any result is presented. The bulk free energy is measured by thermodynamic integration rather than assumed, because a regular-solution form has no driving force whatever in this system: under random mixing the heterovalent and homovalent contributions cancel identically, so a mean-field free energy would describe a different system rather than approximate this one. The gradient energy coefficient is then fixed by the interfacial energy. A phase-field calculation on that footing reproduces the sequence from homogeneous solution through spinodal decomposition to coarsening and arrest, and returns a coarsening exponent of 0.259; it cannot supply an absolute length, because the interface must be numerically widened for any solver to resolve it, so the domain sizes in those figures are set by the grid rather than predicted. Direct lattice coarsening, with the interface at its physical width of one bond and nothing inflated, gives domains of one to two nanometres in a box of thirteen, and an exponent of 0.150. Set against deposit classes whose cooling rates span eight orders of magnitude, from quenched seafloor massive sulphide systems to basinal brines, the model reproduces the observed ordering of inclusion size but implies an exponent near 0.45, larger than either simulation. We report the disagreement rather than the agreement. Four determinations of the exponent span a factor of three, and it would be dishonest to present the two simulation values as mutual corroboration when they differ by seventy per cent. The spread may reflect a crossover from interface-limited coarsening at the atomic scale towards diffusion-limited coarsening as domains grow, in which case no single exponent describes the whole range; that reading is untested. Until it is resolved the inversion supports ranking deposits by cooling rate, not calibrating one. Two consequences follow regardless of the calibration. A smooth LA-ICP-MS depth profile, often taken as evidence for solid solution, is on this account evidence for rapid cooling: the indium is aggregated, as the preceding paper requires, but into domains too fine for the ablation volume to resolve. And indium deportment, and hence its metallurgical recovery, should differ systematically between deposit classes along an axis predictable in advance from thermal history rather than discoverable only by testing. Several methodological findings are recorded because they cost us time and are easy to repeat. A polynomial fitted to a narrow composition window cannot serve as a global free energy: a quartic through the eight measured points has a negative leading coefficient, so the free energy runs to minus infinity and the field diverges within a few hundred steps. In a semi-implicit spectral scheme the timestep should be set from the target amplification of the unstable mode rather than from an explicit CFL condition, which gives a per-step growth of three parts in ten thousand and produces diffusive smoothing that looks like coarsening. The analytic derivatives of the free energy must be checked against finite differences, since dropping one term made the first derivative inconsistent with the second by a factor of one hundred and fifty while the solver continued to run and to produce plausible pictures. And the coarsening exponent does not identify the mechanism: two calculations in which decomposition never occurred returned exponents within seven per cent of the correct value, and what exposed them was the composition contrast, which must grow by orders of magnitude and did not. Limitations are stated rather than obscured. The phase-field interface is artificially widened, so absolute domain sizes come only from the lattice route. The lattice dynamics moves solute by direct exchange and has no vacancy mechanism, which is not how atoms move in a real crystal and is a plausible contributor to its low exponent. The simulated compositions are far above natural indium tenors, because at natural dilution the solute budget is exhausted into a single domain before any coarsening can be measured. Monte Carlo sweeps are not physical time. The end-member comparison pairs deposit-class cooling rates with textures reported from different samples rather than the same ones, and paired measurements would test the power law properly. Iron is absent, and the chemical term J remains unconstrained by first-principles calculation. Contents. The src directory holds the Cahn-Hilliard solver with Arrhenius mobility, the isothermal and cooling drivers, the direct lattice coarsening, the thermodynamic integration for the bulk free energy, the core lattice module and its verification suite, and the structure generation and fitting harness for the deferred first-principles parameterisation. The data directory holds the numerical results underlying every figure. The figures directory holds them in vector and raster form with the scripts that regenerate them. The paper directory holds the manuscript in plain format, in Elsevier format for journal submission, and in Chinese translation, with LaTeX source and compiled PDF for each. Licensing. Source code is released under the MIT License. Data and figures are released under CC BY 4.0. The manuscript remains the copyright of the authors. Related work. This is the third paper of the Computable Mineral Deposit Chemistry series. The Hamiltonian and its analytic ground state are established in doi:10.5281/zenodo.21880502, and the equilibrium result whose absence in nature this paper explains is doi:10.5281/zenodo.21911898. This study was funded by the Hezhou Municipal Scientific Research and Development Program, Project Numbers 2024143, 2024141 and 2024104.