The Equilibrium Fate of Indium in Sphalerite: Large-Scale Replica-Exchange Simulation and the Absence of a Dilute Solid Solution

Replica-exchange Monte Carlo evidence that indium in sphalerite does not form a dilute solid solution: across every composition examined, from 0.10 to 2.00 atomic per cent, the equilibrium state is condensed. This record contains the complete source code, the numerical results underlying every figure and table, and the manuscript. Whether indium in sphalerite resides in true solid solution or in nanoscale inclusions has resisted empirical resolution because the two states differ at length scales near the limit of detection. The question is not academic. Indium is essentially never mined for its own sake, the overwhelming majority of primary production being recovered as a by-product of zinc refining, so the global indium endowment is controlled not by a distinct ore-forming process but by the trace-element chemistry of sphalerite itself. Whether that indium is dispersed or aggregated bears on whether bulk assay measures a representative concentration, on how the metal partitions during metallurgical recovery, and on whether micro-analytical mapping refines an assay or measures a different quantity altogether. The preceding paper of this series established a configurational Hamiltonian for the sphalerite cation sublattice in which the charge-compensation term reduces exactly, by a topological property of the zinc-blende structure, to a nearest-neighbour pair interaction whose coefficient is fixed by the measured dielectric constant of ZnS rather than fitted. It could not settle the question of state, and said so: at four thousand cation sites and four atomic per cent solute, the solute budget is exhausted by a single domain, so complete condensation was consistent with, but not diagnostic of, a thermodynamic preference. This work removes that limitation in both directions at once. The box is eight times larger, and the composition is varied over a twentyfold range down to 0.10 atomic per cent, at which each solute has some five hundred lattice sites of room. At every composition the solute population condenses completely. The monomer fraction is zero throughout, against a random-solution expectation of 98 per cent monomers at the most dilute point, and the largest cluster differs from the random-solution value by four orders of magnitude. Attaching a physical scale, the box is 10.8 by 10.8 by 10.8 nanometres: indium declines to disperse even when given ten nanometres in which to do so. The result does not depend on convergence at ore-forming temperatures, which the sampler cannot guarantee and which the preceding paper wrongly believed it had demonstrated. Each state point is instead run twice, from a dispersed and from a pre-condensed configuration. These approach equilibrium from opposite sides, so it lies between them whether or not either run has converged. At 1800 kelvin, where the acceptance ratio is near ten per cent and the chain is demonstrably mobile, dispersed solutes spontaneously aggregate to between 64 and 95 per cent in one cluster while condensed solutes do not disperse; the bracket narrows to 100 per cent by 1000 kelvin. Since the ordering preference strengthens as temperature falls, condensation throughout the hydrothermal window follows without further assumption. Two subsidiary results bear on how trace-element data are read. The pair enrichment factor rises sixteenfold across the composition scan while the quantity it is meant to describe, the copper content of the first coordination shell around indium, falls slightly; the apparent variation lies almost entirely in a normalisation whose ceiling is the reciprocal of the copper fraction. Any measure of association normalised by bulk concentration will appear to strengthen on dilution whether or not the underlying association changes, and datasets spanning orders of magnitude in tenor are where this is most available to be misread. Separately, the internal structure of the condensed domain is independent of bulk composition to within a surface correction, so the object that forms is the same at every concentration examined, differing only in size. The conclusion that the equilibrium state is condensed does not depend on the one class of parameter left unconstrained by first-principles calculation. Electrostatics alone gives a sixteenfold enrichment of copper-indium nearest-neighbour bonds and removes 96 per cent of the local charge imbalance; the pair binding energy that drives condensation is fixed by the dielectric constant and contains no adjustable parameter; and the electrostatics-only model condenses more readily than the full model, not less, because its energy landscape is smoother. Removing the chemical term strengthens the central claim rather than weakening it. It follows that the ideal dilute-mixing entropy assumed in conventional partition-coefficient models is not available at ore-forming temperatures, so indium solubilities inferred from such models are too high. Three methodological results are reported that transfer beyond this system. Replica round-trip count, not swap acceptance, is the diagnostic that matters for parallel tempering: reducing the interval between exchange attempts from five sweeps to one takes the round-trip count from zero to seven at identical computational cost while the median swap acceptance moves by less than 0.01, so a study reporting acceptance alone would have seen nothing. In a system whose acceptance ratio is of order ten to the minus five, the convergence of two trajectories towards each other is not a test of equilibration, since two chains launched into the same basin agree with each other whether or not that basin is the equilibrium one; the correction this forces on the preceding paper is documented in an appendix. And the ladder length required scales as the energy fluctuation width, measured here as proportional to composition to the power 0.85 times the number of sites to the power 0.51, which is what makes dilute compositions affordable. The limitations are stated rather than obscured. The compositions examined run an order of magnitude above natural tenors; an entropic estimate places the composition boundary between roughly three parts per million and ten to the minus five parts per million, below crustal abundance, but locating it requires boxes of order five hundred thousand sites. The implementation is verified correct at that scale and the scaling laws are established, so the obstacle is computational rather than conceptual. The chemical term is not determined from first principles: the parameterisation campaign is specified and its structure generation and fitting harness are validated against synthetic data, but a 216-atom supercell requires 18.7 gigabytes by the code's own estimate against roughly 12 available on the workstation used throughout. Iron is absent from the model. And Monte Carlo sweeps are not physical time, so whether a natural sphalerite grain attains this equilibrium state is a question of kinetics, treated in a companion study. All calculations were performed on a single workstation, which we regard as a feature rather than an apology: every result in this work can be reproduced on ordinary hardware. Runs use fixed seeds, long calculations checkpoint and resume, and a verification suite checks lattice topology, energy bookkeeping, every analytic limit to machine precision, and the random-solution limit of the order parameters at box sizes up to five hundred thousand cation sites. Contents. The src directory holds the core lattice module, the replica-exchange sampler and adaptive ladder construction, the production scripts for the composition scan, the bracketing test and the coarsening test, the scaling and verification scripts, and the structure generation and fitting harness for the deferred parameterisation. The data directory holds the numerical results as JSON at full precision and as CSV, one file per manuscript table. The figures directory holds the figures in vector and raster form together with the script that regenerates them. The paper directory holds the manuscript in plain and Elsevier formats. The dft directory holds twelve supercell structures with input files and a manifest. 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. The lattice model and its analytic results are established in the preceding paper of this series, archived at doi:10.5281/zenodo.21880502 with code at github.com/Ruqing1963/sphalerite-lattice-mc This study was funded by the Hezhou Municipal Scientific Research and Development Program, Project Numbers 2024143, 2024141 and 2024104.