Kinetics of dissociative congruent evaporation based on the transition state theory Open Access

Schematic of the energy profile of evaporation and the relation with the geometry of the reactive system. The horizontal axis for the potential energy curve represents the reaction coordinate that is perpendicular to the surface of the condensed phase. The dividing surface is located between the reactant condensed phase and the product gas phase, separating them perpendicularly to the reaction coordinate (i.e., in parallel to the surface of the condensed phase). The transition state defined with this dividing surface thus occurs in the middle of the reaction path of the decomposition from the condensed phase to the product molecules.

III. RESULTS AND DISCUSSION

A. Application to evaporation of MgO solid

To test the validity of the theory, we apply it to the evaporation of MgO solid [MgO (s) → Mg (g) + O (g),^2,3 Fig. 2(a)]. Since the properties of the transition state are unknown at present, here we calculate the evaporation rate for the following two extreme cases:

\begin{matrix} MgO (s) \to M g^{‡} + O^{‡} \to Mg (g) + O (g), \end{matrix}

(17)

\begin{matrix} MgO (s) \to Mg O^{‡} \to Mg (g) + O (g), \end{matrix}

(18)

where the species with the superscript ‡ are those at the transition state. So far, we have regarded the reaction coordinate at the transition state as the collective retreat coordinates between the molecules and the condensed phase surface. On the other hand, the decomposition should also proceed with the progress of evaporation. This implies that the location of the potential energy barrier along the reaction coordinate correlates with the degree of decomposition at the transition state.

FIG. 2.

(a) Arrhenius plot of the experimental evaporation rates of MgO solid.2 The linear fitting of the data (shown by the solid line) yields Ea = 542 ± 9 kJ mol−1. (b) Arrhenius plot of the calculated evaporation rate of MgO solid for the extremely “late” barrier case (the blue dashed line) and the extremely “early” barrier case (the red dashed line), shown with the experimental evaporation rates.2

(a) Arrhenius plot of the experimental evaporation rates of MgO solid.² The linear fitting of the data (shown by the solid line) yields E_a = 542 ± 9 kJ mol⁻¹. (b) Arrhenius plot of the calculated evaporation rate of MgO solid for the extremely “late” barrier case (the blue dashed line) and the extremely “early” barrier case (the red dashed line), shown with the experimental evaporation rates.²

In reaction (17), MgO solid is completely dissociated into the same species as the products (atomic Mg and O) at the transition state. This situation is realized at least when the evaporation of MgO solid is a barrierless reaction, in which the transition state locates distant from the reactant along the reaction coordinate,³⁷ and thus, is virtually identical to the products. On the other hand, in reaction (18), Mg and O atoms are not yet separated and exist as a MgO molecule at the transition state. This indicates that the potential energy barrier occurs at an early stage of the reaction. Thus, reactions (17) and (18) can be contrasted as extremely “late” and “early” barrier reactions, where the locations of the transition state are very close (or even identical for the former case) to the product and the reactant, respectively.^34,38

To calculate the evaporation rate for these cases, we assume that the rotations and the surface-parallel translations of the transition-state molecules are completely free just as gas-phase molecules without any hindrance originating from interactions with the surface of the condensed phase (two-dimensional ideal gas³⁹). This approximation is valid when the energy k_BT (∼17 kJ mol⁻¹ at 2000 K) exceeds the barrier height for the molecules to rotate or move parallel to the surface but otherwise overestimates the corresponding partition functions.^40,41 For the energy barrier E₀, we used the internal energy difference between MgO (s) and Mg (g) + O (g), 949 kJ mol⁻¹ (=ΔH⁰ − 2RT at 2000 K).¹⁹ This is correct for the barrierless (i.e., the extremely “late” barrier) case but could be smaller than the actual E₀ of the “early” barrier.

On applying Eq. to reactions and , the evaporation rate is given by

\begin{matrix} J_{late} = \frac{k_{B} T}{h} {\{\frac{{(q_{Mg}^{T} / L)}^{2} q_{Mg}^{E} {(q_{O}^{T} / L)}^{2} q_{O}^{E}}{{q_{MgO (s)}^{V}}^{6}}\}}^{\frac{1}{2}} \exp (- \frac{1}{2} \frac{E_{0}}{k_{B} T}), \end{matrix}

(19)

\begin{matrix} J_{early} = \frac{k_{B} T}{h} \frac{{(q_{MgO}^{T} / L)}^{2} q_{MgO}^{R^{2}} q_{MgO}^{V} q_{MgO}^{E}}{{q_{MgO (s)}^{V}}^{6}} \exp (- \frac{E_{0}}{k_{B} T}), \end{matrix}

(20)

where J_late and J_early designate the evaporation rates for the extremely “late” barrier [reaction ] and the extremely “early” barrier [reaction ], respectively. We calculated them using the parameters presented in Table I as functions of temperature, which are shown in Fig. 2(b) with the experimental data.² In addition, we evaluated the activation energies of J_late and J_early by Eq. and obtained E_a = 466 kJ mol⁻¹ and 927 kJ mol⁻¹ at 2000 K, respectively. These values are similar to E₀/l, as we discussed in Sec. II.

TABLE I.

Partition functions per degree of freedom and parameters used for the calculations of the evaporation rates.

General formula	Partition function	Parameter	Value (T = 2000 K)
Translation (per unit length)
$\frac{q^{T}}{L} = \frac{{(2 π m k_{B} T)}^{\frac{1}{2}}}{h}$ ⁠, L: length of the system and m: molecular mass	$q_{Mg}^{T} / L$	m = 24.3 amu	1.26 × 10¹¹ m⁻¹
	$q_{O}^{T} / L$	m = 16.0 amu	1.02 × 10¹¹ m⁻¹
	$q_{MgO}^{T} / L$	m = 32.3 amu	1.46 × 10¹¹ m⁻¹
Rotation
$q^{R} = \frac{{(8 π^{2} I k_{B} T)}^{\frac{1}{2}}}{h}$ ⁠, I: moment of inertial
	$q_{MgO}^{R}$	I = 29.3 amu Å² a	49.2
Vibration
$q^{V} = \frac{1}{1 - \exp (- \frac{h ν}{k_{B} T})}$ ⁠, ν: vibrational frequency	$q_{MgO}^{V}$	ν = 785 cm⁻¹ a	2.32
	$q_{MgO (s)}^{V}$	ν = 443 cm⁻¹ b	3.66
Electron
$q^{E} = \sum_{i} g_{i} \exp (- \frac{E_{i}}{k_{B} T})$ ⁠, g_i: degeneracy of the ith state and E_i: energy of the ith state
	$q_{Mg}^{E}$	¹S₀: g = 1	1
	$q_{O}^{E}$	³P₂: g = 5 (ground state)	8.53
		³P₁: g = 3, E = 158.3 cm⁻¹ c
		³P₀: g = 1, E = 227.0 cm⁻¹ c
	$q_{MgO}^{E}$	¹Σ⁺: g = 1	1

General formula	Partition function	Parameter	Value (T = 2000 K)
Translation (per unit length)
$\frac{q^{T}}{L} = \frac{{(2 π m k_{B} T)}^{\frac{1}{2}}}{h}$ ⁠, L: length of the system and m: molecular mass	$q_{Mg}^{T} / L$	m = 24.3 amu	1.26 × 10¹¹ m⁻¹
	$q_{O}^{T} / L$	m = 16.0 amu	1.02 × 10¹¹ m⁻¹
	$q_{MgO}^{T} / L$	m = 32.3 amu	1.46 × 10¹¹ m⁻¹
Rotation
$q^{R} = \frac{{(8 π^{2} I k_{B} T)}^{\frac{1}{2}}}{h}$ ⁠, I: moment of inertial
	$q_{MgO}^{R}$	I = 29.3 amu Å² a	49.2
Vibration
$q^{V} = \frac{1}{1 - \exp (- \frac{h ν}{k_{B} T})}$ ⁠, ν: vibrational frequency	$q_{MgO}^{V}$	ν = 785 cm⁻¹ a	2.32
	$q_{MgO (s)}^{V}$	ν = 443 cm⁻¹ b	3.66
Electron
$q^{E} = \sum_{i} g_{i} \exp (- \frac{E_{i}}{k_{B} T})$ ⁠, g_i: degeneracy of the ith state and E_i: energy of the ith state
	$q_{Mg}^{E}$	¹S₀: g = 1	1
	$q_{O}^{E}$	³P₂: g = 5 (ground state)	8.53
		³P₁: g = 3, E = 158.3 cm⁻¹ c
		³P₀: g = 1, E = 227.0 cm⁻¹ c
	$q_{MgO}^{E}$	¹Σ⁺: g = 1	1

^a

Values for the gas-phase MgO molecule.^43,44

^b

Peak frequency of the infrared absorption spectra of a MgO crystal.⁴⁵

^c

Reference 46.

Comparing these results to the experimental data, the experimental activation energy [E_a = 542 ± 9 kJ mol⁻¹, Fig. 2(a)] is slightly larger than that of the extremely “late” barrier case and much smaller than that of the extremely “early” barrier case. Because the pre-exponential terms do not largely differ between J_late and J_early, the absolute rates also show the same tendency: the experimental rate is smaller than the rate of the extremely “late” barrier case only by 1–2 orders of magnitude, while it is larger by ∼9 orders of magnitude than the rate of the extremely “early” barrier case. This implies that the evaporation of MgO solid is likely to be the “late” barrier reaction rather than the “early” barrier reaction, which is consistent with the expectation from its endothermicity.⁴² On the other hand, it should not be completely barrierless (extremely “late” barrier), considering the larger experimental activation energy than the theoretical value calculated by assuming the extremely “late” barrier case. Therefore, we infer that there is a potential energy barrier at the location slightly before the products along the reaction coordinate, and the MgO solid is almost, but not completely, converted into the product atomic gas at the energy barrier (Fig. 3).

FIG. 3.

Schematic of the potential energy of the evaporation of MgO solid along the reaction coordinate. The blue and red dotted curves show the energy profiles of the extremely “late” barrier (barrierless) and extremely “early” barrier cases, respectively. The experimental data of evaporation of MgO solid suggest that there is an energy barrier closer to the product as shown by the black line. The barrier height E₀ is equal to (for barrierless case) or larger than (for earlier barrier case) the potential energy difference between the reactant and the product, which is designated as ΔE.

Note that the above-mentioned discussion is not largely affected by the simplicity of the approximation that we have made here about the properties of the transition state. Since both the possible larger internal energy E₀ and the hindered rotations and translations of the MgO molecules at the “early” barrier result in the decrease in J_early, the discrepancy between the extremely “early” barrier case and the experiments could be even larger. On the other hand, the difference in the activation energy between the extremely “late” barrier case and the experiments is not likely to be ascribed only to the free translation approximation because the temperature dependence of the partition functions is not significant even when the hindered translators are considered.⁴⁰ This means that based on the present theory, even the simplest model can provide the implication about the reaction dynamics of evaporation of MgO.

B. Comparison with the Hertz–Knudsen equation

Here, we reformulate the Hertz–Knudsen equation for dissociative congruent evaporation as a statistical mechanical expression to directly compare to the present theory. Let us consider the following case, where the evaporation products in equilibrium with the condensed phase A consist of l^eq molecules, C_i (i = 1, 2, … , l^eq):

\begin{matrix} A (C P) ⇄ C_{1} (g) + C_{2} (g) + \dots + C_{l^{e q}} (g) . \end{matrix}

(21)

According to the Hertz–Knudsen equation, the ideal evaporation rate of C_i is given by

\begin{matrix} J_{ideal} = \frac{P_{C_{i}}^{e q}}{{(2 π m_{C_{i}} k_{B} T)}^{\frac{1}{2}}}, \end{matrix}

(22)

where

P_{C_{i}}^{e q}

and

m_{C_{i}}

are the equilibrium vapor pressure and the mass of C_i, respectively. Since J_ideal should be equal for all values of C_i to maintain the stoichiometry of the condensed phase,^13,22

\begin{matrix} J_{ideal} = {\{\prod_{i = 1}^{l^{e q}} \frac{P_{C_{i}}^{e q}}{{(2 π m_{C_{i}} k_{B} T)}^{\frac{1}{2}}}\}}^{\frac{1}{l^{e q}}} = \frac{{\{\prod_{i = 1}^{l^{e q}} P_{C_{i}}^{e q}\}}^{\frac{1}{l^{e q}}}}{{(2 π {\bar{m}}_{C} k_{B} T)}^{\frac{1}{2}}}, \end{matrix}

(23)

where

{\bar{m}}_{C}

is the geometric mean of the mass of the equilibrated product molecules. Using the equation of state for an ideal gas, Eq. can also be written as

\begin{matrix} J_{ideal} = {(\frac{k_{B} T}{2 π {\bar{m}}_{C}})}^{\frac{1}{2}} {\{\prod_{i = 1}^{l^{e q}} \frac{N_{C_{i}}}{V}\}}^{\frac{1}{l^{e q}}}, \end{matrix}

(24)

where

N_{C_{i}} / V

is the number density (m⁻³) of C_i in equilibrium. Based on statistical mechanics, the equilibrium between A and the product molecules is expressed by

\begin{matrix} \prod_{i = 1}^{l^{e q}} \frac{N_{C_{i}}}{V} = \frac{\prod_{i = 1}^{l^{e q}} (q_{C_{i}} / V)}{q_{A}} \exp (- \frac{E_{0}^{e q}}{k_{B} T}), \end{matrix}

(25)

where

q_{C_{i}} / V

is the molecular partition function of C_i per unit volume (m⁻³), and

E_{0}^{e q}

(J) is the energy difference of the minimum quantum level between A and the products in equilibrium with A. Substituting Eqs. into , we obtain

\begin{matrix} J_{ideal} = {(\frac{k_{B} T}{2 π {\bar{m}}_{C}})}^{\frac{1}{2}} {\{\frac{\prod_{i = 1}^{l^{e q}} (q_{C_{i}} / V)}{q_{A}}\}}^{\frac{1}{l^{e q}}} \exp (- \frac{1}{l^{e q}} \frac{E_{0}^{e q}}{k_{B} T}) . \end{matrix}

(26)

The partition function

q_{C_{i}}

includes three translational degrees of freedom of the molecule C_i. Separating a one-dimensional partition function for one of these translational motions,

q_{C_{i}}

is written as

\begin{matrix} q_{C_{i}} = \frac{{(2 π m_{C_{i}} k_{B} T)}^{\frac{1}{2}} L}{h} q_{C_{i}}^{*}, \end{matrix}

(27)

where L is the length of the system, and

q_{C_{i}}^{*}

is the partition function of C_i for all the other degrees of freedom. Substituting Eq. into , J_ideal is given by

\begin{matrix} J_{ideal} = \frac{k_{B} T}{h} {\{\frac{\prod_{i = 1}^{l^{e q}} (q_{C_{i}}^{*} / S)}{q_{A}}\}}^{\frac{1}{l^{e q}}} \exp (- \frac{1}{l^{e q}} \frac{E_{0}^{e q}}{k_{B} T}), \end{matrix}

(28)

where

q_{C_{i}}^{*} / S

is

q_{C_{i}}^{*}

per unit area (m⁻²).

One should notice that the statistical mechanical expression of the ideal rate of the Hertz–Knudsen equation [Eq. (28)] has a similar form to Eq. (13) derived from the transition state theory. They become equal when the transition state is identical to the equilibrated products in terms of the internal energy and the molecular structures and can be regarded as two-dimensional ideal gas with free rotations and surface-parallel translations³⁵ (i.e., $E_{0} = E_{0}^{e q}$ ⁠, l = l^eq, and $q_{i}^{‡} / S = q_{C_{i}}^{*} / S$ ⁠). This means that in non-equilibrium conditions, the ideal rate of the Hertz–Knudsen equation can be interpreted as the rate of evaporation with the “ideal transition state” having the above-mentioned properties.

The difference between Eq. (28) and the actual rate can arise both from the pre-exponential and exponential terms. If rotations and/or translations of the transition-state molecules are hindered, Eq. (28) overestimates the pre-exponential term, as previously pointed out for evaporation without decomposition.^23–25 At higher temperatures where those hindered motions approach the free rotor and translator limits,^40,41 the effect of the exponential term becomes more important. When the average energy of the transition state per molecule E₀/l is larger (smaller) than that of the equilibrated products $E_{0}^{e q} / l^{e q}$ ⁠, J is expected to be smaller (larger) than the ideal rate of the Hertz–Knudsen equation with γ exponentially dependent on the temperature. However, as discussed earlier, experimental evaporation rates often agree with the ideal rate to some extent with small temperature dependence of γ, suggesting that E₀/l and $E_{0}^{e q} / l^{e q}$ are not largely different. Such agreement is expected when the stabilization from the transition state to the equilibrated products associated with the decrease in one molecule (i.e., $(E_{0} - E_{0}^{e q}) / (l - l^{e q})$ ⁠) is comparable to the activation energy ∼E₀/l. For the case of MgO and presumably for other oxides, this situation is likely to be achieved reflecting the similarity between the activation energy (542 ± 9 kJ mol⁻¹ for MgO) and the internal energy change in the reaction 2O (g) → O₂ (g), 494 kJ mol⁻¹,¹⁹ in which the number of molecules decreases by 1 (Fig. 4). Further verification of this condition for the agreement between experiments and the Hertz–Knudsen equation requires speciation of the transition state for a wide range of substances that evaporate by dissociative congruent evaporation.

FIG. 4.

Internal energy difference (in kJ mol−1) among 2MgO (s), the experimental products 2Mg (g) + 2O (g), and the equilibrium species 2Mg (g) + O2 (g) at 2000 K. As discussed in Sec. III A, the experimental activation energy 542 ± 9 kJ mol−1 is close to the average energy of the experimental products, 1897/4 = 474 kJ mol−1. Due to the similarity between these values and the energy difference between 2O (g) and O2 (g), 494 kJ mol−1, the average energy of the equilibrium species is also similar (1403/3 = 468 kJ mol−1).

Internal energy difference (in kJ mol⁻¹) among 2MgO (s), the experimental products 2Mg (g) + 2O (g), and the equilibrium species 2Mg (g) + O₂ (g) at 2000 K. As discussed in Sec. III A, the experimental activation energy 542 ± 9 kJ mol⁻¹ is close to the average energy of the experimental products, 1897/4 = 474 kJ mol⁻¹. Due to the similarity between these values and the energy difference between 2O (g) and O₂ (g), 494 kJ mol⁻¹, the average energy of the equilibrium species is also similar (1403/3 = 468 kJ mol⁻¹).

IV. CONCLUSION

In this study, we formulated the absolute rate of dissociative congruent evaporation based on the transition state theory and applied it to the evaporation of MgO solid. In the derived expression of the rate of dissociative congruent evaporation, 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. Consequently, by comparing the theoretical and the experimental evaporation rates, we were able to discuss the location of the activation barrier along the reaction coordinate of the evaporation of MgO. As a result, it was suggested that the barrier occurs at a late stage of the reaction, where the decomposition into product atoms (Mg and O) is almost achieved.

Although the present theory is expected to be generally applicable to dissociative congruent evaporation, the evaporation products are not always experimentally confirmed, as in the case of MgO solid. A wider application of this theory will be possible through experimental speciation of evaporation products from a wider variety of compounds. It is also expected that the properties of the transition state of dissociative congruent evaporation inferred from the present theory will be compared to precise quantum mechanical computations. Another application of this theory will be the evaluation of kinetic isotope effects in evaporation, which is currently in progress.

As our understanding of the heterogeneous reaction dynamics of dissociative congruent evaporation is still limited, the statistical rate theory is an essential approach to interpret the experimental kinetic data of evaporation under non-equilibrium conditions. On the other hand, it is also desirable for the molecular-level evaporation dynamics to be uncovered through future experimental studies and to be theoretically connected to the macroscopic evaporation kinetics.

ACKNOWLEDGMENTS

This work was supported by JSPS Kakenhi (Grant No. 20H05846). The authors thank Professor Naoki Watanabe for the helpful discussions and the two anonymous reviewers for the constructive comments.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Shiori Inada: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Methodology (lead); Validation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Tetsuya Hama: Conceptualization (equal); Validation (equal); Writing – review & editing (equal). Shogo Tachibana: Conceptualization (equal); Funding acquisition (lead); Project administration (lead); Supervision (lead); Validation (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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