Thermodynamic Model of Gibbs Free Energy

Mathematical descriptions of the thermodynamic properties as a function of composition and temperature has been developed in order to estimate thermodynamic properties at different temperatures [Sau98] . The CALPHAD (CALculation of Phase Diagram) method is usually used for describing the Gibbs model for different condition.

Gibbs Free Energy Model of Pure Elements

The Gibbs energy of pure elements i with the structure , refered to the enthalpy of its standard state element reference (SER) at 298.15K, , is described as a function of the temperature by the following equation:

This quantity is denoted GHSER where the structure corresponds to SER. Values of the coefficients, a, b,…, h, are taken from the SGTE database [Din91]. The thermodynamic data for 78 pure elements used by SGTE (Scientific Group Thermodata Europe) were published and has been widely adopted within the international community as a basis for thermodynamic modeling of multi-component systems.

Gibbs Free Energy Model of Solid Solution

Solid solutions are described by a random substitutional model and molar Gibbs energy is given by the general compound energy formalism:

where the is the contribution of the pure components of the phase to the Gibbs energy, is the ideal mixing contribution and is the excess Gibbs energy corresponding to the non-ideal interactions between the components.

The three terms are express by the following equations:

where represents the modal fraction of components i and is a binary interaction parameter:

Assessment of Gibbs Free Energy Model

The optimization of these parameters can be done by means of the Parrot module of the Thermocalc TM program package [Sun85] which consists of sophisticated and generalized software for the calculation of chemical equilibria and phase diagrams [Jon90]. The criteria of the best estimation of the model parameters can be obtained from a statistical treatment of the errors associated with all experimental determinations of equilibrium states in the system. The most commonly used method in parameter evaluation is the least-square method [Jan84]. Figure 1.1 shows the flowchart of optimizing parameters using CALPHAD method [Har01].

Many more energy models other than those described above have been developed in order to describe thermodynamic properties of phases. The energy model has been developed step by step, taking into account various complicating aspects, and is now capable of describing the properties of many different types of phases. It is thus widely used in CALPHAD assessments. The energy model was reviewed by Mats Hillert [Hil01].

Figure 1.1. Flowchart of the CALPHAD Method [Har01]

Thermodynamic Model for Estimating Thermodynamic Property

In order to obtain an accurate phase diagram within a short time, precise measurement of thermodynamic property is necessary. The measurement is not easy if it is a multi-component system and the problem cannot be solved easily due to the complexity of experiments in a multicomponent system. The calculation of thermodynamic properties has been playing an important role in developing phase diagrams.

Different models have been developed to estimate different thermodynamic property, these model could be empirical, semi-empirical or based on first principles. Table 1.3 shows the several models used in recent thermodynamic calculations.

Table 1.3. Thermodynamic Model Developed on Different Basis