Minimization of Gibbs Free Energy

An alternative approach to calculate thermodynamic equilibrium is the minimization of the Gibbs fi ee energy. In this energy-based considerations, no reactions need to be assigned. It is assumed that a mixture of defined compounds experiences a minimum of the Gibbs fi ee energy in equilibrium. 

In Equation (5.14), G is the Gibbs fi ee energy, is the chemical potential and Hi is the molar amount of species i. Equation (5.15) describes the side condition where bj is the quantity of chemical element /, and is the elemental matrix assigning the elements j to the species i. Hence, it represents the conservation of material. The solution of the constrained optimization problem is rather complex and has been described elsewhere [3]. Furthermore, each calculation is carried out for constant pressure and temperature and requires an iteration with the heat balance to calculate the system equilibrium temperature. The advantages include correct prediction of trace compounds and inclusion of non-ideal 

As is well known from thermodynamics,6 a system will be in equilibrium when the Gibbs Free Energy is at a minimum. Therefore, all that is necessary is to express the Gibbs Free Energy in terms of the degree of completion of the reaction, and then minimize that function. The Gibbs Free Energy can be expressed as follows,7 

Then the solution to the problem of determining the equilibrium state at thermodynamic equilibrium reduces to one of finding the minimum in the function G subject to the constraints of Equation (24). There are a number of numerical techniques for the solution of this minimization problem, 7/8but rather than review the details, we will simply describe some of the results of such calculations. 

Each formula is written to contain one niobium, except for pure Ge. 

Other systems can be treated in a similar manner such as the Ti-B-CI-H system,10 and of course the Si-H-CI system has been exhaustively studied because of the commercial importance of silicon epitaxial films used for integrated circuit fabrication.11 

Experimentation shows that the best, fully dense, and homogeneous carbon deposits are produced at an optimum negative value of AG. For smaller negative values, the reaction rate is very low and, for higher negative values, vapor-phase precipitation and the formation of soot can occur. Such factors are not revealed in the simple free-energy change calculation. A more complete analysis is often necessary. 

A method of analysis is the minimization of the Gibbs free energy, a calculation based on the rule of thermodynamics which states that a system will be in equilibrium when the Gibbs free energy is at a minimum. The objective then is the minimization of the total free energy of the system and the calculation of equilibria at constant temperature and volume or constant pressure. It is a complicated and lengthy operation but, fortunately, computer programs are now available that simplify the task considerably.  

All of this is valuable information which can be of great help. Yet, it must be treated with caution since, in spite of all the progress in thermodynamic analysis, the complexity of many reactions in the CVD of carbon, and the fact that these calculations are based on chemical equilibrium which is rarely attsuned in CVD reactions, make predictions relying on thermody-neunic calculations alone still questionable. 

It follows that, in orderto provide a reliable and balanced investigation, it is preferable to combine the theoretical calculations with an experimental program. Fortunately, carbon deposition experiments are relatively easy to design and carry out without the need for expensive equipment, and results can usually be obtained quickly and reliably. 

The CVD reactions to deposit pyrolytic graphite are based on the thermed decomposition (pyrolysis) of hydrocarbons. The most common precursor is methane (CH4), which is generally pyrolyzed at 1100 C or above, over a wide range of pressure from about 100 Pa (0.001 atm) to 10 Pa (1 atm). The reaction in a simplified form is as follows  

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He showed that positive dX give d that reduce Gibbs free energy. This method is analogous to that of steepest descent, a first-order method for minimization of Gibbs free energy. Ma and Ship-man (11) used Naphtali s method to estimate compositions at equilibrium and the Newton-Raphson method to achieve convergence. 

For these systems, mass balances must account for these restrictions and replace the atom balances (2) as constraints for minimization of Gibbs free energy. These mass balances are given by Schott (37) ... 

One unresolved question concerns whether it is possible to use volatility parameters as iteration variables in a nonlinear programming algorithm, with an approach similar to that introduced by Boston and Britt (13) for solution of nonlinear algebraic equations involving K-values. Our conclusion is that volatility parameters apply where K-values are used, and would be awkward to use in minimization of Gibbs free energy. 

CVD normally involves a multi-component and a multi-phase system. There are various ways to calculate thermodynamic equilibrium in multicomponent systems. The following is a brief discussion of the optimization method where the minimization of Gibbs free energy can be achieved. The free energy G of a system consisting of m gaseous species and s solid phases can be described by. 

An alternative way of relating concentrations (mass fractions) of individual species to/ is the assumption of chemical equilibrium. An algorithm based on minimization of Gibbs free energy to compute mole fractions of individual species from / has been discussed by Kuo (1986). The equilibrium model is useful for predicting the formation of intermediate species. If such knowledge of intermediate species is not needed, the much simpler approximation of mixed-is-burnt can be used to relate individual species concentrations with/. In order to calculate the time-averaged values of species concentrations the probability density function (PDF) approach is used. 

Chemical equilibrium model Most reactive transport formulations use the mass action law to solve the chemical equilibrium equations. In this formulation an alternative (though thermodynamically equivalent) approach is used, based on the minimization of Gibbs Free Energy. This approach has a wider application range extending to highly non-ideal brine systems. 

To compute the concentrations of the species in equilibrium, a Newton-Raphson algorithm is applied to the direct minimization of Gibbs Free Energy. Lagrange multipliers are used to incorporate the restrictions of the problem. Its implementation is adapted to problems with high dynamic minerals appearance/disappearance as in the case presented in next section. 

It can be shown that two sets of equations arises when employing minimization of Gibbs free energy (Jones and Rigopoulos, 2005). One set becomes algebraic equations defining the... 

Direct minimization of Gibbs free energy in muitiphase fiash... 

Pan, H and A. Firoozabadi Complex Multiphase Equilibrium Calculations by Direct Minimization of Gibbs Free Energy Using Simulated Annealing, SPE Res. Eval. and Engl., p. 36, Feb. 1998. 

The minimization of Gibbs free energy of the total system with respect to the independent variables subject to the constraints of Eqs. (5.85) and (5.86) provides the amount and composition of each phase. In general there are eight variables for the calculation of an equilibrium state at constant Tand P. Those are, n y D, n j., n , na and n -Once these variables are obtained, the micellar size can be readily determined. Pan and Firoozabadi (1997a) have used the feasible sequential quadratic programming (FSQP) to minimize G (Zhou, Tits, and Lawrence, 1996). 

Direct minimization of Gibbs free energy. The Gibbs free energy of the system in Fig. 5.3 is given by... 

The current list of 26 chemical elements treated by VICTORIA includes not only the volatiles that are the primary concern in the event of an accident, but also those that may interact with the volatile species and inhibit their release, those that are easily measurable experimentally and so are beneficial for validation, and those that are important because of their quantity within the RCS. Chemical interaction of a set of 288 chemical species are analyzed by VICTORIA (condensed and vapor forms are regarded as separate species), which is given in Table 1. Both chemical equilibria and phase behavior are determined by minimization of Gibbs free energies. Chemical equilibria are imposed within the fuel grains, in the porosity of the fuel, within the fuel/clad gap, in the bulk gas, and in structural films. Within the fuel grains, either equilibrium chemistry using a small subset of the full species set (primarily UO2, oxides, and uranates) is allowed or else no chemistry at all is performed. 

The equilibrium state to which a system tends, whether by chemical reaction or by chemical transfer between phases, is governed by the minimization of Gibbs free energy. In a chemical reaction at equilibrium, the change in free energy per additional mole reacting, as expressed in Eq. (1.9), must be zero ... 

Equation (1.12), the expression of the equilibrium state of a reaction, is known as the mass action law and is a direct consequence of the minimization of Gibbs free energy. Table 1.1 presents equilibrium constants and standard free energy changes for some common environmental... 

A combustion chamber can be ealeulated either based on kinetic theory (reaction rate equations) or a ehemical equilibrium constant. Knowledge about the exact value of partial pressures of the reactants is unnecessary this time, and simple calculations based on minimize of Gibbs free energy can be used. 

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