Anisotropy of interfaces in an ordered HCP binary alloy

Cahn, J W; Han, S C; McFadden, G B

doi:10.6028/nist.ir.6217

Cited by 4 publications

(12 citation statements)

References 6 publications

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“…But we now have found two surprises: firstly, transverse isotropy in γ and in the Wulff shape for the A1-L1 0 IPB with only tetragonal symmetry is greater symmetry than expected. Secondly, we found less symmetry in the hcp investigation; there was no isotropy transverse to the six-fold axis [13]. Here the gradient energy coefficient bore no resemblance to elements in a tensor of rank 4, which not only confirms that these coefficients are not tensors, but forms a strong counterexample to any conjecture that the orientation dependence of γ of a diffuse interface might reflect that of a low rank tensor.…”

Section: Summary and Discussionmentioning

confidence: 55%

“…The model can describe a variety of phase diagrams; a series of diagrams topologically similar to the Au-Cu system was presented in [8]. It can be used for other ordered crystal structures that occur in fcc and other systems [13], though more order parameters may be required. This method allows the computation of interfaces for all orientations at a wide variety of conditions.…”

Section: Summary and Discussionmentioning

confidence: 99%

“…For the hcp crystal, the disordered state where either type of atom is equally likely to occur on any of the four sublattices, the structure is designated A3 in Strukturbericht notation. When one of the sublattices is occupied (on average) by a different atom than the other three, the crystal structure is denoted DO 19 ; this ordered state is exactly analogous to the L1 2 structure on an fcc lattice [13].…”

Section: Summary and Discussionmentioning

confidence: 99%

“…The C 12 term is absent because of the fcc symmetry. We emphasize, however, that the non-conserved order parameters do not constitute the components of a tensor, nor do the gradient energy coefficients transform as a fourth-rank tensor, as discussed in Appendix B of [7,13].…”

Section: Governing Equations For Interfacesmentioning

confidence: 99%

“…The multiple-order-parameter formulation leads to a natural anisotropy, i.e. without introducing ad hoc parameters in the model, in fcc [7] and even transverse to a six-fold axis in hcp [13]. Furthermore, the orientation dependence of interfacial properties (such as the interfacial energy and mobility [6]) is continuous and easily allows computation of the properties for all orientations.…”

Section: Introductionmentioning

confidence: 99%

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A1-L$1_0$ phase boundaries and anisotropy via multiple-order-parameter theory for an fcc alloy

Tanoğlu¹,

Braun²,

Cahn³

et al. 2003

Interfaces Free Bound.

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The dependence of thermodynamic properties of planar interphase boundaries (IPBs) and antiphase boundaries (APBs) in a binary alloy on an fcc lattice is studied as a function of their orientation. Using a recently developed diffuse interface model based on three non-conserved order parameters and the concentration, and a free energy density that gives a realistic phase diagram with one disordered phase (A1) and two ordered phases (L1 2 and L1 0 ) such as occur in the Cu-Au system, we are able to find IPBs and APBs between any pair of phases and domains, and for all orientations. The model includes bulk and gradient terms in a free energy functional, and assumes that there is no mismatch in the lattice parameters for the disordered and ordered phases. We catalog the appropriate boundary conditions for all IPBs and APBs. We then focus on the IPB between the disordered A1 phase and the L1 0 ordered phase. For this IPB we compute the numerical solution of the boundary value problem to find its interfacial energy, γ , as a function of orientation, temperature, and chemical potential (or composition). We determine the equilibrium shape for a precipitate of one phase within the other using the Cahn-Hoffman "ξ -vector" formalism. We find that the profile of the interface is determined only by one conserved and one non-conserved order parameter, which leads to a surface energy which, as a function of orientation, is "transversely isotropic" with respect to the tetragonal axis of the L1 0 phase. We verify the model's consistency with the Gibbs adsorption equation. † description of the fcc disordered phase (A1, in Strukturbericht notation) and two ordered phases, both of which have wide ranges of composition away from stoichiometry: the Cu 3 Au phase, typified by the Strukturbericht L1 2 structure, and the CuAu(I) phase, typified by the Strukturbericht L1 0 structure, in the Cu-Au system. The fcc lattice can be viewed as four interpenetrating simple cubic lattices (see Figure 1). In the disordered A1 structure, the four sublattices have equal probabilities of being occupied by either type of atom. In the L1 2 phase, one of the sublattices has a different occupation probability than the other three sublattices; for the L1 0 phase, two of the sublattices are occupied differently than the other two.In previous work by the authors [5, 7, 6], a free energy density was employed which provided a useful description of A1-L1 2 IPBs and L1 2 APBs. In that model, however, the resulting phase diagram featured a multicritical point for all three phases [41], rather than the separate congruent disordering points with first order A1-L1 2 and A1-L1 0 transitions, as commonly observed in fcc systems such as Cu-Au. There was no co-existence of the A1 and L1 0 phases; the A1-L1 0 transition occurred only at the multicritical point and was second order. A more realistic phase diagram can be obtained with the improved free energy that we employ here (see Figure 2); there is A1-L1 2 and A1-L1 0 coexistence and the transitions at the congruent po...

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Section: Summary and Discussionmentioning

confidence: 55%

Section: Summary and Discussionmentioning

confidence: 99%

Section: Summary and Discussionmentioning

confidence: 99%

Section: Governing Equations For Interfacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A1-L$1_0$ phase boundaries and anisotropy via multiple-order-parameter theory for an fcc alloy

Tanoğlu¹,

Braun²,

Cahn³

et al. 2003

Interfaces Free Bound.

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On the Gibbs adsorption equation and diffuse interface models

McFadden

Wheeler

2002

Proc. R. Soc. Lond. A

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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 equation holds for systems with gradient energy coe¯cients, provided the appropriate de nitions of surface excess quantities are used. We consider, in particular, the phaseeld model of a binary alloy with gradient energy coe¯cients for solute and the phase eld. We derive a solute surface excess quantity that is independent of a dividing surface convention, and nd that the adsorption in this model is in®uenced by the surface free energies of the pure components of the binary alloy as well as the solute gradient energy coe¯cient. 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 di¬use interface models that arise in spinodal decomposition and order{disorder transitions.

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Development of Phase Field Simulation for the Growth of Dendrite Structure of Al-Si Cast Alloy

Ramdan

Takaki

Djuansjah

et al. 2013

MSF

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Micro to nano scale study of dendrite structure is important in order to have better properties control of casting product. The present study concerns on the morphological study of dendrite structure by phase-field simulation, in order to obtain the morphological growth of this structure that close its real morphology. Focus was given on the morphological growth of dendrite structure of Al-Si cast alloys, therefore thermodynamic data were taken for this type of materials. Anisotropy noise, strength of anisotropy and different undercooled conditions were applied as the variable parameters in the present works. It was observed that by introducing higher anisotropy noise, higher degree fragmentation of dendrite structure was obtained. Similar condition was obtained by introducing higher strength of anisotropy value, that higher degree of fragmentation was obtained. Both of these phenomena was also supported by the heat flux rate features of these variations that higher heat flux rate to almost all direction was obtained with the higher value of anisotropy noise and strength of anisotropy. In addition it was also observed that higher degree fragmentation of dendrite only possible to occur if sufficient undercooled condition established.

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Anisotropy of interfaces in an ordered HCP binary alloy

Cited by 4 publications

References 6 publications

A1-L$1_0$ phase boundaries and anisotropy via multiple-order-parameter theory for an fcc alloy

A1-L$1_0$ phase boundaries and anisotropy via multiple-order-parameter theory for an fcc alloy

On the Gibbs adsorption equation and diffuse interface models

Development of Phase Field Simulation for the Growth of Dendrite Structure of Al-Si Cast Alloy

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