A Stable and Efficient Regression Approach for Determination of Coefficients in Linear Multicomponent Diffusion Alonso V. Jaques and Jeffrey C. LaCombe (Submitted November 30, 2011; in revised form February 14, 2012) Existing methods for determining constant multicomponent diffusion coefficients are typically based on generalizations of the established Boltzmann-Matano method. They often require performing numerical integration and differentiation operations on individual experimental concentration profiles. Scatter in these data, present due to uncertainties and variations in the measured concentrations, often necessitates the use of smoothing procedures for noise removal. A regression approach is presented here to determine constant multicomponent diffusion coef- ficients that eliminates the need to perform smoothing operations on the concentration profile data. This approach simultaneously fits the data from multiple diffusion couples to a functional form of the mathematical expression for the concentration profile, and allows us to determine the diffusivity matrix directly from the fitted parameters. Reformulation of the equation for the analytical solution is done in order to reduce the size of the problem and increase the conver- gence rate. The objective function for the regression can incorporate point estimations for uncertainty in the concentration, improving the statistical confidence in the estimated diffusivity matrix. Case studies are presented to demonstrate the reliability and the stability of the method. Keywords concentration profiles, diffusivity coefficient, diffusiv- ity measurements, estimation of data, experimental techniques, multicomponent diffusion, square root diffusivity 1. Introduction In measurement of diffusion coefficients in high-order alloy systems n  3 ð Þ, there are challenges in the experi- mental design and the analysis of results. It has often been proven difficult to obtain reliable results using limited sets of experimental data. The case of linear multicomponent diffusion (i.e., a constant diffusivity matrix, ½D) is a strong simplification of the true nature of diffusion. However, when this simplification is adequately valid, it permits a direct implementation of established analytical methods for the measurement and prediction of diffusion phenomena. A number of techniques currently exist for the extraction of constant and variable diffusion coefficients. [1-5] One feature common to these methods, with the exception of Ref 1, is the requirement to perform numerical integrations and differentiations of experimentally-derived concentration profile data, Cx; t ð Þ, because these methods extend from a generalization of the Boltzmann-Matano method (summa- rized in Ref 6), applied to multicomponent systems. Usually the Cx; t ð Þ experimental data include scattering (noise) from the instrumental measurement of the compositions, which affects the numerical differentiation and integration pro- cesses. These methods suffer particularly when the point-to- point slopes between adjacent data points jump from positive to negative (common in noisy data). In such cases, significant errors can be introduced into the extracted ½D values. To deal with this experimental reality (scattered data), smoothing or filtering of the Cx; t ð Þ data is often employed. [7,8] However, the smoothing process itself adds additional influences, uncertainties, or errors (which are not well-characterized) into the estimation of ½D. Thus, it is desirable (when possible) to develop alternative data analysis methods that use the data as measured, i.e., without smoothing or filtering prior to the analytical estimation of ½D. We present here, a regression methodology that avoids the need for smoothing of the Cx; t ð Þ data. This method also features an ability to incorporate measurement uncertainties in the experimental data more fully than other existing methods for determining ½D in multicomponent systems. For example, each Cx; t ð Þ data point typically has an uncertainty associated with it due to the accuracy and resolution limits of the instrument used to measure it. These uncertainty values are relevant for the best estimation of the diffusivity values, and in the method described here, can be incorporated into the regression scheme as a weighting parameter. To date, such a weighted regression scheme has not been incorporated into other methods for measuring ½D from diffusion couples. In recent years, several improved and experimentally efficient methods have been developed for full character- ization of the diffusivity matrix using a minimum number of Alonso V. Jaques, Departamento de Ingenierı ´a Quı ´mica y Ambiental, Universidad Te ´cnica Federico Santa Marı ´a, Casilla 110-V, Valparaiso, Chile; and Jeffrey C. LaCombe, Chemical and Materials Engineering Department, University of Nevada, Reno, Reno, NV 89557. Contact e-mail: lacomj@unr.edu. JPEDAV (2012) 33:181–188 DOI: 10.1007/s11669-012-0028-x 1547-7037 ÓASM International Basic and Applied Research: Section I Journal of Phase Equilibria and Diffusion Vol. 33 No. 3 2012 181