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Thermodynamic driving forces and growth rates in rapid solidification are analysed. Taking into account the relaxation time of the solute diffusion flux in the model equations, the present theory uses, in a first case, the deviation from local chemical equilibrium, and ergodicity breaking. The second case of ergodicity breaking may exist in crystal growth kinetics of rapidly solidifying glass-forming metals and alloys. In this case, a theoretical analysis of dendritic solidification is given for congruently melting alloys in which chemical segregation does not occur. Within this theory, a deviation from thermodynamic equilibrium is introduced for high undercoolings via gradient flow relaxation of the phase field. A comparison of the present derivations with previously verified theoretical predictions and experimental data is given. This article is part of the theme issue 'Heterogeneous materials: Metastable and nonergodic internal structures'. ©2019 The Author(s) Published by the Royal Society.

A mathematical model for reactive-transport processes in porous media is presented. The modeled system includes diffusion, electromigration and electroosmosis as the most relevant transport mechanism and water electrolysis at the electrodes, aqueous species complexation, precipitation and dissolution as the chemical reactions taken place during the treatment time. The model is based on the local chemical equilibrium for most of the reversible chemical reactions occurring in the process. As a novel enhancement of previous models, the local chemical equilibrium reactive-transport model is combined with the solution of the transient equations for the kinetics of those chemical reactions that have representative rates in the same order than the transport mechanisms. The model is validated by comparison of simulation and experimental results for an acid- enhanced electrokinetic treatment of a real Pb-contaminated calcareous soil. The kinetics of the main pH buffering process, the calcite dissolution, was defined by a simplified empirical kinetic law. Results show that the evaluation of kinetic rate entails a significant improvement of the model prediction capability. This work has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 778045. Part of this work was supported financially by the European Commission within the project LIFE12 ENV/IT/442 SEKRET “Sediment electrokinetic remediation technology for heavy metal pollution removal”. Paz-Garcia acknowledges the financial support from the “Proyecto Puente - Plan Propio de Investigación y Transferencia de la Universidad de Málaga”, code: PPIT.UMA.B5.2018/17. Villen-Guzman acknowledges the financial support from the University of Malaga through a postdoctoral contract.

A numerical model describing transport of multiple species and chemical reactions during electrokinetic treatment is presented. The transport mechanisms included in the model were electromigration and electroosmosis. The chemical reactions taken into account were water electrolysis at the electrodes, aqueous species complexation, precipitation, and dissolution. The model was applied to simulate experimental data from an acid-enhanced electrokinetic treatment of a Pb-contaminated calcareous soil. The kinetics of the main pH buffering process (i.e., calcite dissolution) was taken into account and its time-dependent behavior was described by a rate law. The influence of kinetics was evaluated by comparing the results from a set of simulations in which calcite dissolution was implemented considering thermodynamic equilibrium and another set in which the same reaction was described by the rate law. The results show that the prediction capability of the model significantly improves when the kinetic rate is taken into account. Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech.