Thermodynamics of interfacial energy in binary metallic systems: influence of adsorption on dihedral angles
Introduction
Dihedral angles in binary eutectic and monotectic systems of metals-liquid alloys commonly exhibit systematic change with the liquidus temperature [1], [2], [3], [4], [5], [6] (Fig. 1). The decrease of the equilibrium dihedral angle (θ) with increasing temperature (T) observed in most metals has been explained by temperature dependence of solid–liquid interfacial tension (equivalent to the solid–liquid interfacial energy per unit area, σSL) and solid–solid interfacial tension (equivalent to the grain boundary energy per unit area, σSS), since θ is controlled by the ratio of σSL and σSS [7]:where Θ is a positive function of θ. The change of σSL and σSS have been evaluated from temperature derivatives of θ by employing the ‘dihedral angle method’, in which σSL and σSS are assumed to be linearly dependent on T [4], [8], [9]. However, this assumption has no physical basis and is not generally assured.
In binary systems, the liquid composition simultaneously changes with the liquidus temperature. Hence, in terms of compositional change, the above tendency of the dihedral angle can be stated as follows: “When the atoms of a solid component (abbreviated as A) increase in the liquid phase (i.e., T increases), the dihedral angles decrease” (Fig. 2). Although the influence of liquid composition on dihedral angles is indistinguishable from the temperature effect in binary systems, the compositional effects are confirmed in experiments on ternary systems under fixed temperature conditions [3], [4]. The role of liquid composition on the interfacial energy is rather easily understood by the following: if A is enriched in the liquid phase, A–A bonds across the solid–liquid interface increase. Then σSL is reduced because of the affinity between the solid phase and its solutes A dissolved in the liquid phase. The grain boundary energy σSS is, on the other hand, not directly relevant to the liquid properties. Therefore, increasing A atoms in the liquid will cause the reduction of Θ and θ.
The aim of this paper is to formulate dihedral angles in eutectic and monotectic systems, with special attention to the compositional change of σSL. After the establishment of the classical thermodynamic theory of interface by Gibbs [10], two kinds of approaches have been introduced to this field: one is based on the lattice-liquid statistical model, where interfacial energy is calculated by summing up inter-atomic bonding energy [11], [12], [13], [14]. The other is based on the macroscopic thermodynamic theory of interface proposed by Cahn and Hilliard [15], [16]. In this paper, we describe interfacial energy in binary metallic systems by using both models of the lattice statistical theory and of the Cahn–Hilliard theory.
Section snippets
Speculation from the Gibbs theory
In the Gibbs theory of interface, the interfacial energy σSL is defined for an arbitrarily taken dividing surface (abbreviated as Ω) by the following equation:where is the area of Ω, GΩ is the interfacial excess Gibbs free energy, Γi is the amount (in moles) of adsorbed atoms, and μi is the equilibrium chemical potential of component i [10], [17]. By definition, there is no interfacial excess volume. GΩ is therefore written aswhere UΩ and SΩ are the interfacial excess
A lattice-liquid model
Now we carry out statistical calculation of the solid–liquid interfacial energy, following the formulation of Ono [11]. A lattice-like configuration is used for both solid and liquid phases (Fig. 4(a)). For simplicity, the lattice shape and the molar volume v are taken to be the same for both phases and both end-member components of A and B. Furthermore, we assume that only the first layer of the liquid, which is directly in contact with the solid surface, is affected by interfacial adsorption,
Hard interface models
We have assumed a liquid-like nature for the adsorption layers (Fig. 3, Fig. 4(a)), but actual chemical properties of the solid–liquid interfaces may be different from those of the bulk phases. If bonding between the solid and the first adsorption layer is stronger than bonding in the liquid (eij), as illustrated in Fig. 4(b), then σSL in the lattice-liquid model is changed through W′ and W* terms in (27). For example, if we consider that the bonding energy between the solid and the
A continuum multilayer model
In the previous sections, we simply assumed that interfacial adsorption is limited to a monoatomic layer. The lattice-liquid model can also be used for multilayer adsorption [11], [12], but the expressions become much more complicated and a numerical method is required to solve a set of difference equations. In this section, we employ an analytical method proposed by Cahn and Hilliard [15].
The Cahn–Hilliard theory deals with a gradual change of composition c over interfacial layers, as
Eutectic and monotectic systems
In a binary system under fixed pressure conditions, the liquid composition x and temperature T cannot be changed independently because of the equilibrium condition (5) between the bulk solid and liquid phases. Equating (38), (39), and using (16), (18), we obtain the expression of liquidus curves:Here, non-ideality of the system is indicated by a non-dimensional parameter Tc/Tm; low- and high-Tc/Tm values correspond to eutectic and monotectic systems, respectively (
Applications
Now we evaluate the dihedral angles in binary metallic systems shown in Fig. 1, Fig. 2, by using thermodynamic data of pure metals (Table 1) [23]. We apply the empirical law (41) for the interfacial energy of pure metals. For eutectic systems, the critical temperature Tc is determined by fitting the liquidus curves with (54). Tc values of monotectic systems are known from binodal curves [24]. The Zn–Bi system is excluded from our quantitative analysis because its binodal deviates considerably
Acknowledgement
We are grateful to Prof. Mineo Kumazawa for drawing our attention to this subject. We thank Susumu Ikeda and Takehiko Hiraga for helpful comments.
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