Elsevier

Calphad

Volume 50, September 2015, Pages 23-31
Calphad

Relationship between surface tension and Gibbs energy, and application of Constrained Gibbs Energy Minimization

https://doi.org/10.1016/j.calphad.2015.04.008Get rights and content

Highlights

  • A new interpretation for the surface tension using graphical representation.

  • Surface tension being equivalent to chemical potential of “area” element.

  • Numerical complexity in solving Butler's equation in multicomponent system is avoided.

  • Role of various properties on the surface tension of solutions is discussed.

Abstract

In the context of a boundary phase model, surface tension (σ) of a solution can be regarded as a system property of an equilibrium between a bulk phase and a surface phase. In the present article, a geometric relationship is shown among molar Gibbs energy of the bulk phase (g), that of the surface phase (gs), and corresponding surface tension of the system. The geometric relationship is based on a phase equilibrium between the bulk phase and the surface phase, under a constraint: constant surface area (A). The relationship is consistent with the proposal of Butler, Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci., 135 (1932) 348 [1], and is mathematically equivalent to the Constrained Gibbs Energy Minimization (CGEM) for the surface tension calculation by Pajarre et al., Calphad 30 (2006) 196 [7]. The geometric relationship can be simply utilized by available CALPHAD type code, in order to calculate surface tension of a solution composed of any number of components. Role of various properties (surface tension (σi°) and molar surface area (Ai°) of pure components, excess Gibbs energy of the bulk phase and that of the surface phase) in the surface tension and surface concentration is examined using the CGEM.

Introduction

In the theoretical calculation of surface tension, one of widely applied concepts is that a surface of a condensed material may be considered as a thin separated phase, so to speak, a surface phase. An assumption is made that the surface phase is supposed to be in equilibrium with a corresponding bulk phase. As a result, constitution of the surface phase may be known, and the surface tension of the bulk phase may be derived. This approach has been widely used in order to calculate surface tension of liquid alloys, mostly utilizing the proposal of Butler [1] and subsequent several modifications [2], [3], [4], [5], [6]. While this approach is, in principle, valid generally in multicomponent system, many of previous researches utilizing this approach have been limited to binary systems because of complexity for solving equations. A new approach has been proposed by Pajarre et al. [7] where the surface tension can be calculated in a simple manner utilizing commercially available CALPHAD type code [8], even in multicomponent system. However, although mathematical aspect of the surface tension calculation was given, the principle for the calculation of surface tension is to be more explicitly described in the view of thermodynamic equilibrium.

In the present article, it is shown that how the CALPHAD approach, which has generally been used to calculate phase diagram and thermodynamic properties of multicomponent–multiphase system, can be utilized in the calculation of surface tension as well as the surface constitution (surface concentration). In this regard, a relationship between the surface tension and the Gibbs energy of solution is first described for a binary system. The binary system is treated as a pseudo-ternary system taking into account a new system component, “area”. A graphical representation is provided in order for better understanding the relationship. Based on this relationship, a series of phase equilbria are calculated using the CALPHAD code, and surface tension and surface concentration are obtained as the results of the phase equilibrium calculations. With the aid of such simple method, surface tension and surface concentration of binary systems of various conditions are predicted. Role of various properties including surface tension, molar surface area of pure components, excess Gibbs energy in bulk and in surface phases is discussed. Finally, results of example calculations are shown for a ternary system and a quaternary system, not only for the surface tension, but also for the surface concentration.

Section snippets

Surface tension calculation

According the Butler [1], a surface of a solution is considered as a separate phase, distinguished from the corresponding bulk phase. Moreover, the surface phase is assumed to be confined to a monolayer of atoms. In the following, superscript s refers to “surface”, throughout the present article.

General representation of Gibbs energy in a binary A–B system

Molar Gibbs energy of the A–B binary bulk phase (g) is shown as solid lines in Fig. 1, as a function of XB. Molar Gibbs energies of pure components A and B in the bulk phase are gA° and gB°, respectively. According to Eq. (9), molar Gibbs energies of those in the surface phase are increased as much as Ai°σi°. Molar Gibbs energy of the surface phase (gs) is thus shown as dashed lines.

Fig. 1(a) represents the relationship when the molar surface area of A and that of B are the same (AA°=AB°=A). At

Surface tension calculation using the Constrained Gibbs Energy Minimization

Surface tension of example solutions of binary, ternary, and quaternary systems was calculated using the CGEM of which basic relationship between the surface tension and the Gibbs energy has been described in the previous section. Surface composition was also obtained as a result of the calculations. Gibbs energies of the bulk and the surface phases were described using a regular solution model:g=XgAA°+XBgB°++RTXAlnXA+XBlnXB++LABXAXB+gs=XAsgAs,°+XBsgBs,°++RTXAslnXAs+XBslnXBs++LABsXAsXBs+

Conclusions

After a concept for calculation of surface tension has been proposed by Butler [1], considerable progress has been made by Tanaka et al. [4], [5], [6] and Pajarre et al. [7]. In particular, the concept of CGEM is found to be very useful for the calculation in the multicomponent solution. Already available Gibbs energy minimization code can be effectively used for the calculation of surface tension in multicomponent solution. A practical application of this concept to the liquid steel has been

Acknowledgments

This study was supported by a Grant (NRF-2013K2A2A2000634) funded by the National Research Foundation of Korea, Republic of Korea.

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