During rapid solidification, solute may be incorporated into the solid phase at a concentration significantly different from that predicted by equilibrium thermodynamics. This process, known as solute trapping, leads to a progressive reduction in the concentration change across the interface as the solidification rate increases. Theoretical treatments of rapid solidification using traditional sharp-interface descriptions require the introduction of separately derived nonequilibrium models for the behavior of the interfacial temperature and solute concentrations. In contrast, phase-field models employ a diffuse-interface description and eliminate the need to specify interfacial conditions separately. While at low solidification rates equilibrium behavior is recovered, at high solidification rates nonequilibrium effects naturally emerge from these models. In particular, in a previous study we proposed a phase-field model of a binary alloy ͓A. A. Wheeler et al., Phys. Rev. E 47, 1893 ͑1993͔͒ in which we demonstrated solute trapping. Here we show that solute trapping is also possible in a simpler diffuse interface model. We show that solute trapping occurs when the solute diffusion length D I /V is comparable to the diffuse interface thickness. Here V is the interface velocity and D I characterizes the solute diffusivity in the interfacial region. We characterize the dependence of the critical speed for solute trapping on the equilibrium partition coefficient k E that shows good agreement with experiments by Aziz and co-workers ͓see M. J. Aziz, Metall. Mater. Trans. A 27, 671 ͑1996͔͒. We also show that in the phase-field model, there is a dissipation of energy in the interface region resulting in a solute drag, which we quantify by determining the relationship between the interface temperature and velocity.

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Partial least squares discriminant analysis (PLS-DA) is a well-known technique for feature extraction and discriminant analysis in chemometrics. Despite its popularity, it has been observed that PLS-DA does not automatically lead to extraction of relevant features. Feature learning and extraction depends on how well the discriminant subspace is captured. In this paper, discriminant subspace learning of chemical data is discussed from the perspective of PLS-DA and a recent extension of PLS-DA, which is known as the locality preserving partial least squares discriminant analysis (LPPLS-DA). The objective is twofold: (a) to introduce the LPPLS-DA algorithm to the chemometrics community and (b) to demonstrate the superior discrimination capabilities of LPPLS-DA and how it can be a powerful alternative to PLS-DA. Four chemical data sets are used: three spectroscopic data sets and one that contains compositional data. Comparative performances are measured based on discrimination and classification of these data sets. To compare the classification performances, the data samples are projected onto the PLS-DA and LPPLS-DA subspaces, and classification of the projected samples into one of the different groups (classes) is done using the nearest-neighbor classifier. We also compare the two techniques in data visualization (discrimination) task. The ability of LPPLS-DA to group samples from the same class while at the same time maximizing the between-class separation is clearly shown in our results. In comparison with PLS-DA, separation of data in the projected LPPLS-DA subspace is more well defined.