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 (1993)] 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 DI/V is comparable to the diffuse interface thickness. Here V is the interface velocity and DI characterizes the solute diffusivity in the interfacial region. We characterize the dependence of the critical speed for solute trapping on the equilibrium partition coefficient kE that shows good agreement with experiments by Aziz and co-workers [see M. J. Aziz, Metall. Mater. Trans. A 27, 671 (1996)]. 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.

https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=150730

Solute trapping and solute drag in a phase-field model of rapid solidification

In this paper we discuss some applications of the classical Gibbs adsorption equation to specific diffuse interface models that are based on conserved and non–conserved order parameters. Such models are natural examples of the general methodology developed by J. W. Gibbs in his treatment of the thermodynamics of surfaces. We employ the methodology of J. W. Cahn, which avoids the use of conventional dividing surfaces to define surface excess quantities. We show that the Gibbs adsorption equation holds for systems with gradient energy coefficients, provided the appropriate definitions of surface excess quantities are used. We consider, in particular, the phase–field model of a binary alloy with gradient energy coefficients for solute and the phase field. We derive a solute surface excess quantity that is independent of a dividing surface convention, and find that the adsorption in this model is influenced by the surface free energies of the pure components of the binary alloy as well as the solute gradient energy coefficient. We present one–dimensional numerical solutions for this model corresponding to a stationary planar interface and show the consistency of the numerical results with the Gibbs adsorption equation. We also discuss the Gibbs adsorption equation in the context of other diffuse interface models that arise in spinodal decomposition and order–disorder transitions.

On the Gibbs adsorption equation and diffuse interface models

In this paper we discuss some applications of the classical Gibbs adsorption equation to speci› c di¬use interface models that are based on conserved and non-conserved order parameters. Such models are natural examples of the general methodology developed by J. W. Gibbs in his treatment of the thermodynamics of surfaces. We employ the methodology of J. W. Cahn, which avoids the use of conventional dividing surfaces to de› ne surface excess quantities. We show that the Gibbs adsorption

On the Gibbs adsorption equation and di® use interface models

In this work we assume that the total stored energy functional for a body $\B$ depends not only on the local strain field, but also on the spatial average of the strain field over the body weighted with an influence kernel. We investigate the problem of minimizing the total stored energy subject to a given bulk displacement $\Delta\geq 0$. After the general setup for this problem is reviewed, we give sufficient conditions for an energy minimizing strain field $e(\cdot)$ to satisfy an integro-differential Euler--Lagrange equation. The result is general and applies to material energies that display a wide variety of singular behavior. Through analysis of this Euler--Lagrange equation for a special class of influence kernels, we arrive at a regularity theorem which ensures that energy minimizing strain fields must be periodic, piecewise smooth, and possess a finite number of simple discontinuities. We then combine this with a well-known existence result for relaxed minimization problems to arrive at a genera...

On a model of nonlocal continuum mechanics part I: existence and regularity

(ABSTRACT) We examine a model of solid-solid phase transitions that includes thermo-elastic effects and an order parameter. The model is derived as a special case of the Gurtin-Fried model posed in one space dimension with a symmetric triple-well free energy in which the relative heights of the wells vary with temperature. We examine the temperature independent case, showing existence of a unique classical solution of a regularized system of partial differential equations using semigroup theory. This is followed by numerical study of a finite element algorithm for the temperature independent model. Finally, we present computational material concerning the temperature dependent model. Acknowledgments There are many people who deserve my gratitude in connection with the completion of this dissertation. I wish first to thank my advisor, Dr. Robert C. Rogers, for his continued encouragement, availability and patience. I especially appreciate his professional insight and warmth of character. I also thank Dr. Tao Lin for his continual support and friendship. The many discussions in his office of a numerical nature were essential for this dissertation. In addition, I acknowledge the support of the remaining committee members, Drs. Special thanks to Dr. Paul C. Fife of the University of Utah for generously supplying me with several articles. I wish to extend my undying gratitude to my husband, David Schenck, for his love, support and willingness to take on so many roles this past year. Finally, I thank my daughter, Hannah, for supplying me with the incentive to finish in a timely manner and for providing so much joy.

On an Order-Parameter Model of Solid-Solid Phase Transitions

On an order-parameter model for a binary liquid

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On an order-parameter model for a binary liquid