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[MZZ45-P07] Kinetics of dissociative congruent evaporation based on transition state theory
Keywords:evaporation, kinetics, transition state theory
Introduction:
Non-transition metal oxides (including some minerals) and group 13-15 compounds are known to evaporate decomposing into multiple gas molecules with maintaining their stoichiometric compositions under non-equilibrium (Langmuir) conditions [1,2]. Such dissociative congruent evaporation is characterized by smaller activation energies than the enthalpy of sublimation or vaporization per composition formula. To theoretically describe the kinetics of this evaporation, the Hertz-Knudsen equation has been commonly used. In this equation, the evaporation rate is equated to the condensation rate in equilibrium based on the detailed balance. Because the maximum condensation rate in equilibrium can be calculated only with thermodynamic data as the collision flux of gas molecules with the equilibrium vapor pressure onto a surface, the Hertz-Knudsen equation conveniently provides predictions of ideal evaporation rates, which are often in good agreement with experiments (e.g., within 1-2 orders of magnitude for silicate and oxide minerals). However, since this equation is essentially derived as the evaporation rate in equilibrium, it is insufficient for interpretations of the experimental kinetic data of non-equilibrium evaporation. In this study, we derive the absolute rate of dissociative congruent evaporation in non-equilibrium based on transition state theory and apply it to evaporation of MgO solid to discuss its dynamics [3].
Theory:
We applied transition state theory to dissociative congruent evaporation by describing it as an elementary reaction from the reactant condensed phase to the product multiple gas molecules. The transition state locates in the middle of the reaction path, where the decomposition is in progress. Regarding that the reaction coordinate is orthogonal to the surface of the condensed phase, we set the dividing surface in parallel to the condensed phase surface as a thin layer separating the two phases. The molecules at the transition state were assumed to be mobile within the dividing surface and thus follow Boltzmann statistics for indistinguishable particles. In the derivation, we used the hypothesis that the concentrations of the molecules at the transition state that are becoming products can be calculated using equilibrium theory even when the whole system is not in equilibrium, which follows from the basic “no-recrossing assumption” of transition state theory [4]. Consequently, the absolute evaporation rate was expressed with the statistical properties of the reactant and the transition state.
Results and Discussion:
The temperature dependence of the derived evaporation rate indicated that the activation energy closely corresponds to the average energy of the molecules at the transition state, reflecting the degree of decomposition at the potential energy barrier along the reaction coordinate of evaporation. By comparing the theoretical and experimental evaporation rates for the reaction MgO (s) → Mg (g) + O (g) [1,5], we found that there is an activation barrier close to the product side (i.e., “late” barrier) where the decomposition is almost achieved (Fig. 1). We also compared the present theory with the Hertz-Knudsen equation by reformulating it using statistical mechanics and found that the Hertz-Knudsen equation provides a good prediction when the stabilization from the transition state to the equilibrated products associated with the decrease of one molecule is comparable to the activation energy, which is the case for evaporation of MgO solid. The present theory is advantageous to the Hertz-Knudsen equation in that it describes dissociative congruent evaporation as a non-equilibrium process and thus provides insights into the reaction dynamics.
[1] Hashimoto (1990) Nature 347, 53. [2] Goldstein et al. (1976) Surf. Sci. 57, 733. [3] Inada et al. JCP, under review. [4] Steinfeld et al. (1989) Chemical Kinetics and Dynamics. [5] Wu et al. (1991) CPL 182, 472.
Non-transition metal oxides (including some minerals) and group 13-15 compounds are known to evaporate decomposing into multiple gas molecules with maintaining their stoichiometric compositions under non-equilibrium (Langmuir) conditions [1,2]. Such dissociative congruent evaporation is characterized by smaller activation energies than the enthalpy of sublimation or vaporization per composition formula. To theoretically describe the kinetics of this evaporation, the Hertz-Knudsen equation has been commonly used. In this equation, the evaporation rate is equated to the condensation rate in equilibrium based on the detailed balance. Because the maximum condensation rate in equilibrium can be calculated only with thermodynamic data as the collision flux of gas molecules with the equilibrium vapor pressure onto a surface, the Hertz-Knudsen equation conveniently provides predictions of ideal evaporation rates, which are often in good agreement with experiments (e.g., within 1-2 orders of magnitude for silicate and oxide minerals). However, since this equation is essentially derived as the evaporation rate in equilibrium, it is insufficient for interpretations of the experimental kinetic data of non-equilibrium evaporation. In this study, we derive the absolute rate of dissociative congruent evaporation in non-equilibrium based on transition state theory and apply it to evaporation of MgO solid to discuss its dynamics [3].
Theory:
We applied transition state theory to dissociative congruent evaporation by describing it as an elementary reaction from the reactant condensed phase to the product multiple gas molecules. The transition state locates in the middle of the reaction path, where the decomposition is in progress. Regarding that the reaction coordinate is orthogonal to the surface of the condensed phase, we set the dividing surface in parallel to the condensed phase surface as a thin layer separating the two phases. The molecules at the transition state were assumed to be mobile within the dividing surface and thus follow Boltzmann statistics for indistinguishable particles. In the derivation, we used the hypothesis that the concentrations of the molecules at the transition state that are becoming products can be calculated using equilibrium theory even when the whole system is not in equilibrium, which follows from the basic “no-recrossing assumption” of transition state theory [4]. Consequently, the absolute evaporation rate was expressed with the statistical properties of the reactant and the transition state.
Results and Discussion:
The temperature dependence of the derived evaporation rate indicated that the activation energy closely corresponds to the average energy of the molecules at the transition state, reflecting the degree of decomposition at the potential energy barrier along the reaction coordinate of evaporation. By comparing the theoretical and experimental evaporation rates for the reaction MgO (s) → Mg (g) + O (g) [1,5], we found that there is an activation barrier close to the product side (i.e., “late” barrier) where the decomposition is almost achieved (Fig. 1). We also compared the present theory with the Hertz-Knudsen equation by reformulating it using statistical mechanics and found that the Hertz-Knudsen equation provides a good prediction when the stabilization from the transition state to the equilibrated products associated with the decrease of one molecule is comparable to the activation energy, which is the case for evaporation of MgO solid. The present theory is advantageous to the Hertz-Knudsen equation in that it describes dissociative congruent evaporation as a non-equilibrium process and thus provides insights into the reaction dynamics.
[1] Hashimoto (1990) Nature 347, 53. [2] Goldstein et al. (1976) Surf. Sci. 57, 733. [3] Inada et al. JCP, under review. [4] Steinfeld et al. (1989) Chemical Kinetics and Dynamics. [5] Wu et al. (1991) CPL 182, 472.