Extracting thermodynamic information from local composition fluctuations in solids : extended theory and its application to simulated and experimental atom probe data

Abstract

In case of liquids, thermodynamic fluctuation theory has been applied for decades to obtain direct thermodynamic information (e.g. miscibility gap, mixing/demixing tendencies, critical solution temperature) from local composition fluctuations. Recently, this theory has been extended to solids by introducing an additional elastic work term between the evaluated sub-system and the entire system, which does not arise in liquids. This extended theory has been verified via atomistic simulations in an exemplary Cu-Ni embedded-atom system using Monte Carlo simulations at a fixed temperature over the entire composition range. Composition fluctuations in the system that are represented by the relative variance of the composition histogram are tracked in various-sized subvolumes over time, revealing a systematic dependence on the size of the evaluation volume due to interface effects. Nonetheless, these surface effects can be excluded by extrapolation to an infinitely large subvolume, leading to perfect agreement with the prediction by the extended theory. Thus, the recovery of the Gibbs free energy of mixing from evaluation of the fluctuations is possible also in the case of solids. Atom Probe Tomography (APT) delivers combined high-resolution chemical and sub nanometric three-dimensional (3D) spatial information, and is therefore the perfect technique to determine local composition fluctuations by using spatial frequency distribution analysis in practical applications. In this work, the applicability of the extended theory is tested on the Cu-Ni alloy and ionic CuO systems via frequency distribution analysis on simulated and experimental atom probe data, and eventually compared to available phase diagram data, thereby proving the validity of extracting the Gibbs free energy from local composition fluctuations in solids. In the first part of this work, the spatial frequency distribution analysis is applied to simulated crystals of long-range ordered L12 and monoclinic structures numerically modeled disregarding thermodynamic interaction between atoms. The relative variance displays an evaluation size dependence, but goes to zero (i.e. no composition fluctuations) if extrapolated to sufficiently large evaluation size. This result meets the expectation as no composition fluctuations should be found in perfectly ordered materials. In the second part, this approach is applied to simulated alloys including thermodynamic interactions. Cu-Ni alloys of various compositions are firstly equilibrated using a Monte Carlo simulation with an embedded-atom potential. Afterwards, the alloys are numerically field-evaporated by the evaporation simulation package TAPSim and the 3D coordinates of the field-evaporated sample are recovered through the usual reconstruction algorithm. Throughout this process, two practical considerations related to the atom probe technique have been effectively addressed: i) The newly developed model tackles the challenges associated with the limited detection efficiency and allows the reconstruction of the relative variance for the bulk system from limited atom probe data scaled by detection efficiency; ii) An additional correction term which is proportional to the evaluation size and magnitude of composition inhomogeneity is introduced. It enables the separation of thermodynamic fluctuations from artificial composition variations inherent in the experimental method based on their different size dependence, so that the extrapolation still recovers the intrinsic thermodynamic composition fluctuations. In the third part, this approach is finally applied to experimental atom probe data. The Cu-Ni alloys are prepared by induction melting of pure Cu and Ni and CuO thin films are prepared via ion beam sputtering. After sufficient equilibration by heat treatment, Cu-Ni and CuO specimens for the APT measurement are fabricated via focused ion beam cutting. By experimentally conducting the same approach as developed theoretically, local composition fluctuations are obtained for both Cu-Ni and CuO systems. After the elastic work term correction, the CALPHAD-style parametrization of the Gibbs free energy is obtained by linking it to the measured local composition fluctuations. In this way, the Cu-Ni miscibility gap is successfully reconstructed from data measured at elevated temperature (800 K), and the resulting phase diagram is in agreement with the CALPHAD results in literature. The frequency distribution analysis of the reconstructed CuO tends to approach the binomial distribution (i.e. behavior of random alloys), since field evaporation of molecules (e.g. CuO, Cu2O) but not only single ions destroys the long-range order structure and deteriorates the resolution in the reconstruction. This effect indicates the partial limitation of this method on ionic compounds. In summary, the present work has systematically extended and proven the application of the composition fluctuation theory to metallic alloys, and makes it possible to directly access thermodynamic information from local composition fluctuations. APT is demonstrated as a new technique to extract direct thermodynamic information, and a general route from the APT measurement to the Gibbs free energy is presented. Given that the composition fluctuation is a local property and only a substantially short diffusion length for equilibration is required, this represents an efficient methodology especially for systems where slow diffusion hinders the establishment of large scale thermodynamic equilibrium. APT, as a sub-nanometric resolution technique, promises to extract more accurate thermodynamic information in a wider temperature composition range. Besides, this study advances our understanding of the size dependence in the traditional frequency distribution analysis. It is pointed out that potential misinterpretation could happen and is presented in literature, if a sample evaluation size in the frequency distribution analysis is arbitrarily chosen. Only the bulk relative variance obtained via extrapolation to infinitely large sub-system is thermodynamically meaningful.