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Surface reactivity by gas introduction in Knudsen cell mass spectrometry
Journal of Physics and Chemistry of Solids 66 (2005) 488–493 www.elsevier.com/locate/jpcs
Surface reactivity by gas introduction in Knudsen cell mass spectrometry C. Chatillon*, M. Heyrman Laboratoire de Thermodynamique et de Physico-Chimie Me´tallurgiques, (UMR 5614, CNRS-INPG/UJF)—ENSEEG, Domaine Universitaire BP.75, Saint Martin d’He`res F-38402, France Accepted 5 June 2004
Abstract Conditions for controlled gas flow introduction in an effusion cell are calculated according to different flow regimes that can be encountered in the gas line and its connection to a Knudsen cell. Molecular, transition, viscous and chocked flow regimes are analyzed in order to propose a general structure of the connection with the use of a small nozzle at the entrance of the Knudsen cell. Tests are then performed when introducing N2 gas and monitoring by mass spectrometry its pressure in the cell. An example of gas surface reaction—C gasification by O2 gas—is also presented. q 2004 Elsevier Ltd. All rights reserved. Keywords: D. Surface properties
1. Introduction Gas–surface reactions were studied in Knudsen cells mainly in view of new gaseous species thermodynamic data determinations that cannot be obtained by conventional vaporization of a condensed sample. Indeed, introducing a gas in a cell containing a solid or liquid sample can produce extreme values for component activities that favor the production of some intermediate oxidation degrees. This was used, for example, in the determinations of subhalides, oxyhalides [1,2] and more recently for oxy-hydroxides [3,4]. By introduction of a gas or a gas mixture in a Knudsen cell coupled with a mass spectrometer, surface reactions can be studied [5], not only at equilibrium but also in order to relate kinetic steady-state flows to equilibrium conditions. The aim of this kind of study can be the determination of the so-called ‘gross evaporation and condensation’ coefficients [6], these coefficients being obtained by reference to known—calculated or measured—equilibrium conditions. The main advantage of mass spectrometry is to be able to determine the whole set of evaporation and condensation coefficients for each gaseous species due to the analytical * Corresponding author. Tel.: C33 4 76 82 65 11; fax: C33 4 76 82 67 67. E-mail address: [email protected] (C. Chatillon). 0022-3697/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2004.06.024
capabilities of the method. Moreover, the use of effusion cells conversely to free surface vaporization facilitates thermal equilibrium of the surface as well as accurate temperature measurements.
2. Gas introduction principle and related equations One problem arising when introducing gases in a Knudsen cell is first of all the ability to monitor the incident flow in order to obtain a steady-state pressure in the cell available in a range as wide as the mass spectrometric detection, i.e. about 106. The second problem is to calibrate the mass spectrometer in direct relation with the input flow in order to circumvent the usual estimates of some parameters that contribute to the uncertainties attributed to the method. So, special efforts have to be done to build a gas introduction device in such a manner that the flow can be calculated with accuracy. As the molecular, isentropic and viscous flow equations are known, the best solution is probably to go from viscous to molecular regime in the cell using a chocked ideal orifice, the main condition being that the ratio of the input pressure p2 to the cell pressure p0 being R0.525 (value calculated for air at 298 K). But manufacturing such an ideal orifice is not possible because a thickness limit of the walls leads to a nozzle shape.
C. Chatillon, M. Heyrman / Journal of Physics and Chemistry of Solids 66 (2005) 488–493
489
Fig. 1. Our Knudsen cell (inner diameter 36 mm) connected to the regulated pressure gas tank, through a nozzle (50!50 mm2). Regulated pressure at the entrance of the nozzle, p2; at the exit of the nozzle, p1; in the Knudsen cell, p0; as measured by the mass spectrometer from the effused beam (effusion orifice 2!2 mm2).
Moreover, as explained by Santeler [7], the pressure decrease from the input pressure—regulated in an on line buffer tank—to the cell at low pressure corresponds to different steps as presented in Fig. 1. An important feature is to consider that the chocked exit—the orifice of the nozzle in the Knudsen chamber—has to be calculated independently of the tubing part of the nozzle that works in the viscous regime. For these two nozzle parts, and due to the large pressure range needed in mass spectrometry, the possibility of working in the transition regime has to be taken into account. The different steady-state pressures, p2 at the nozzle entrance, p1 at the inner face of the nozzle exit are calculated as a function of the desired p0 pressure in our Knudsen cell fixed arbitrarily between 10K4 and 10K7 bar for mass spectrometric detection (lower pressures correspond to molecular flow in the nozzle). The pressure p0 determines the effusion flow Qeff (see Fig. 1 for the symbols) which becomes the steady-state flow in the whole system. The procedure for calculation is the one proposed by Santeler [7,8]: starting from a fixed p0 pressure in the Knudsen cell, with a chocked orifice (50 mm diameter), the p1 pressure is calculated as presented in Fig. 2 (chocked orifice, pressure p1), and a linear relation is attended according to the chocked flow equation QchocZC. p1 (CZ200pr2/RT in MKSA units (mol/s)—some parenthesis are missing in the Santeler formula) with the steady-state condition QchocZ Qeff. Then, calculation of the mean free path of molecules in
the gas at the nozzle orifice (pressure p1) showed that the flow can be partly in the transition regime, and a correction is done using the q function from Santeler, leading to a more accurate value for p1—‘chocked orifice in transition flow’— showed in Fig. 2 for the ideal exit orifice of the nozzle. Then, the viscous flow is calculated in the short tubing of the nozzle (50 mm long) leading to a parabolic relation in Fig. 2 for the pressure p2 (viscous nozzleCchocked orifice in transition flow), and corrections are necessary for the transition regime, leading to the final relation for the pressure p2 (crosses in Fig. 2). A linear fit is given for further comparison with experiments, although the calculated curves let appear a slight S shape corresponding to the transition flow. The calculation of the transition regime may be controversy, and different weighting equations between the molecular and the viscous flow contributions have been proposed in literature. For this reason we calculated also with the DeMuth and Watson [9] formula that is based on the Knudsen number, as presented in Fig. 3. A slight difference with calculations from the Santeler formula is observed on the slopes that are not significant compared to the fit approximations.
3. Mass spectrometric tests Mass spectrometric experiments were performed first with an empty Knudsen cell by introduction of pure N2 gas
490
C. Chatillon, M. Heyrman / Journal of Physics and Chemistry of Solids 66 (2005) 488–493
Fig. 2. Calculated input regulated pressures p1 at the nozzle exit (diamonds and squares) or p2 in the gas tank (triangles and crosses) as a function of the Knudsen cell pressure p0 according to Santeler [7,8] formula and taking into account the different flow regimes in the nozzle. Marks are for calculated values and equations are obtained from least square fits.
in order to test our gas introduction device. The effusion orifice of the Knudsen cell (Fig. 1) has a 2 mm diameter and 2 mm length and works in the molecular flow regime for the chosen p0 pressures. At the bottom of the cell a large orifice (5 mm diameter) let appear the nozzle which is machined in
a stainless steel wall according to the zoom presented in Fig. 1. In this study, the measurement on gaseous species that exist also in the ion source background—namely N2, O2 and H2O—were performed with our special restricted collimation sampling and shutter device [10] in order to
Fig. 3. Calculated input regulated pressures p1 at the nozzle exit (diamonds and squares) or p2 in the gas tank (triangles and crosses) as a function of the Knudsen cell pressure p0 according to Santeler [7,8] formulae and DeMuth and Watson [9] formula for the transition regime.
C. Chatillon, M. Heyrman / Journal of Physics and Chemistry of Solids 66 (2005) 488–493
491
Fig. 4. Products IT (of ionic intensity (in A)!cell temperature (K)) measured by the mass spectrometer-proportional to the Knudsen cell pressure—as a function of the measured N2 gas input regulated pressure in the tank by a Baratron capacitance. The H2O pressure is imposed by a dew point at 10 8C, and needs some time for obtaining a steady-state value of IT at about 10K14 A.K. (the first measured values start at the bottom left side according to the arrows).
avoid any ion source/oven/vacuum background interactions that can introduce some systematic errors in measurements. In some case, the input N2 gas was flowed over a thermal regulated water bath in order to impose a constant partial water pressure in the effusion cell. Results are shown in Figs. 4 and 5. The experimental slope of N2 effused pressure—or proportional quantity IT—measured versus the input regulated one is very close to the expected one from flow calculations (see Figs. 2–5).
In Fig. 4, the two regimes observed for the H2O pressure—as well as the sequence of observation—show that time is needed to reach a constant value that corresponds probably to the dew point as imposed by thermal regulation of the gas flow after flushing the water tank as well as time for saturation of the thermal regulated input line. In Fig. 5, without water, simultaneous monitoring of NC and OC ions (masses 28 and 32) 2 2 shows that there was some leak in our introduction device,
Fig. 5. Products IT measured by the mass spectrometer as a function of the N2 input regulated pressure in the gas tank. We observe a small leak in the introduction device as revealed by the measure of O2(g) at mass 32 that becomes more important when the total pressure p2 in the on line tank decreases.
492
C. Chatillon, M. Heyrman / Journal of Physics and Chemistry of Solids 66 (2005) 488–493
Fig. 6. Evolution of the products IT for different gaseous species as a function of the O2(g) input regulated pressure at constant temperature for a grinded graphite sample disposed in the Knudsen cell at 683 K. The N2(g) species accounts for the background peak in the ion source housing contrarily to other species that are detected in the effused beam using the shutter. Arrows indicate the sequence of measurements.
more easily observed on the O2 impurity. This feature means that the line has to be built according to ultra-high vacuum techniques (changes are in process).
4. Application to gas surface reactions as studied by mass spectrometry The aim of the gas introduction device is to study reactions of gases with a solid surface that we use as sample in the Knudsen cell (Fig. 1). As an example we studied the gasification of carbon—a graphite sample ground in a mill during 72 h under fore vacuum and under pure O2 during 12 h in order to favor the saturation of the sites by surface complexes—as shown in Fig. 6. The temperature is fixed at 683 K and we analyze by mass spectrometry the gaseous reaction products—i.e. CO, CO2 and by-products due to leaks and impurities—as a function of the flow of O2 which is pressure regulated in the input tank. We observe a production of CO larger than the CO2 one, and a slight relative increase of CO2 proportion versus CO when the total pressure in the cell increases. Beyond this classical thermodynamic effect, a more marked effect occurs when pressure is maintained a long time at high values (see arrows in Fig. 6), probably because of a reaction between the CO(g) and the surface complexes can give CO2(g) as postulated in carbon gasification literature [11,12]. This reaction is postulated to be a slow process depending on total pressure as we observe in this work.
5. Conclusion and perspectives The ability to control an inlet gas flow in an effusion cell over a wide pressure range has been proved when using a small nozzle at the bottom of an effusion cell associated with an input pressure regulated tank. Flow calculations of such a device are clearly possible and show that a wide pressure range values may be reached when using suitable input pressure gauges—at least two different gauges since their usual working range is within 1!10K3. One supplementary evaluation of such a device will be done by calibration of the mass spectrometer with an inert and suitable sample (relatively to the introduced gas) vaporizing in the same temperature range, the ionization cross-section of which being well known.
References [1] D.L. Hildenbrand, Attainment of chemical equilibrium in effusive beam sources of the heterogeneous type, in: Characterization of High Temperature Vapors and Gases, NBS Special Publication 561, vol. 1, NIST, Maryland, 1979; pp. 171–179. [2] D.L. Hildenbrand, K.H. Lau, Thermochemistry of gaseous manganese oxides and hydroxides, J. Chem. Phys. 100 (1994) 8377–8380. [3] D.L. Hildenbrand, K.H. Lau, Thermochemistry of gaseous SiO(OH), SiO(OH)2 and SiO2, J. Chem. Phys. 101 (1994) 6076–6079. [4] J.W. Hastie, D.W. Bonnell, E.R. Plante, Slag and metal oxide vaporization in reactive atmospheres, High Temp. Sci. 13 (1980) 257–279. [5] J. Drowart, A. Pattoret, S. Smoes, F. Degre`ve, D. Detry, Etudes thermodynamiques et cine´tiques en chimie des hautes temperatures in: W.L. Mead (Ed.), Advanced Mass Spectrometry, vol. 3, The Institut of Petroleum, London, 1966, pp. 931–945.
C. Chatillon, M. Heyrman / Journal of Physics and Chemistry of Solids 66 (2005) 488–493 [6] G.M. Rosenblatt, Evaporation from Solids, in: N.B. Hannay (Ed.), Treatise on Solid-State Chemistry VI. Surfaces, Plenum Press, New York, 1976, pp. 165–239. Chapter 3. [7] D.J. Santeler, Exit loss in viscous tube flow, J. Vac. Sci. Technol. A4 (1986) 348–353. [8] D.J. Santeler, New concepts in molecular gas flow, J. Vac. Sci. Technol. A4 (1986) 338–343. [9] S.F. DeMuth, J.S. Watson, Prediction of flow rates through an orifice at pressures corresponding to the transition between molecular and isentropic flow, J. Vac. Sci. Technol. A4 (1986) 344–347.
493
[10] P. Morland, C. Chatillon, P. Rocabois, High temperature mass spectrometry using the Knudsen effusion cell. I. Optimization of sampling constraints on the molecular beam, High Temp. Mat. Sci. 37 (1997) 167–187. [11] N.M. Laurendeau, Heterogeneous kinetics of coal char gasification and combustion, Prog. Energy Combust. Sci. 4 (1978) 221–270. [12] Z. Du, A.F. Sorofin, J.P. Longwell, L. Tognotti, The CO/CO2 ratio in the products of the carbon oxygen reaction in fundamental issues, in: J. Lahaye, P. Ehrburger (Eds.), Control of Carbon Gasification Reactivity, Kluwer Academic Publishers, Netherland, 1991, pp. 91–106.
Surface reactivity by gas introduction in Knudsen cell mass spectrometry C. Chatillon*, M. Heyrman Laboratoire de Thermodynamique et de Physico-Chimie Me´tallurgiques, (UMR 5614, CNRS-INPG/UJF)—ENSEEG, Domaine Universitaire BP.75, Saint Martin d’He`res F-38402, France Accepted 5 June 2004
Abstract Conditions for controlled gas flow introduction in an effusion cell are calculated according to different flow regimes that can be encountered in the gas line and its connection to a Knudsen cell. Molecular, transition, viscous and chocked flow regimes are analyzed in order to propose a general structure of the connection with the use of a small nozzle at the entrance of the Knudsen cell. Tests are then performed when introducing N2 gas and monitoring by mass spectrometry its pressure in the cell. An example of gas surface reaction—C gasification by O2 gas—is also presented. q 2004 Elsevier Ltd. All rights reserved. Keywords: D. Surface properties
1. Introduction Gas–surface reactions were studied in Knudsen cells mainly in view of new gaseous species thermodynamic data determinations that cannot be obtained by conventional vaporization of a condensed sample. Indeed, introducing a gas in a cell containing a solid or liquid sample can produce extreme values for component activities that favor the production of some intermediate oxidation degrees. This was used, for example, in the determinations of subhalides, oxyhalides [1,2] and more recently for oxy-hydroxides [3,4]. By introduction of a gas or a gas mixture in a Knudsen cell coupled with a mass spectrometer, surface reactions can be studied [5], not only at equilibrium but also in order to relate kinetic steady-state flows to equilibrium conditions. The aim of this kind of study can be the determination of the so-called ‘gross evaporation and condensation’ coefficients [6], these coefficients being obtained by reference to known—calculated or measured—equilibrium conditions. The main advantage of mass spectrometry is to be able to determine the whole set of evaporation and condensation coefficients for each gaseous species due to the analytical * Corresponding author. Tel.: C33 4 76 82 65 11; fax: C33 4 76 82 67 67. E-mail address: [email protected] (C. Chatillon). 0022-3697/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2004.06.024
capabilities of the method. Moreover, the use of effusion cells conversely to free surface vaporization facilitates thermal equilibrium of the surface as well as accurate temperature measurements.
2. Gas introduction principle and related equations One problem arising when introducing gases in a Knudsen cell is first of all the ability to monitor the incident flow in order to obtain a steady-state pressure in the cell available in a range as wide as the mass spectrometric detection, i.e. about 106. The second problem is to calibrate the mass spectrometer in direct relation with the input flow in order to circumvent the usual estimates of some parameters that contribute to the uncertainties attributed to the method. So, special efforts have to be done to build a gas introduction device in such a manner that the flow can be calculated with accuracy. As the molecular, isentropic and viscous flow equations are known, the best solution is probably to go from viscous to molecular regime in the cell using a chocked ideal orifice, the main condition being that the ratio of the input pressure p2 to the cell pressure p0 being R0.525 (value calculated for air at 298 K). But manufacturing such an ideal orifice is not possible because a thickness limit of the walls leads to a nozzle shape.
C. Chatillon, M. Heyrman / Journal of Physics and Chemistry of Solids 66 (2005) 488–493
489
Fig. 1. Our Knudsen cell (inner diameter 36 mm) connected to the regulated pressure gas tank, through a nozzle (50!50 mm2). Regulated pressure at the entrance of the nozzle, p2; at the exit of the nozzle, p1; in the Knudsen cell, p0; as measured by the mass spectrometer from the effused beam (effusion orifice 2!2 mm2).
Moreover, as explained by Santeler [7], the pressure decrease from the input pressure—regulated in an on line buffer tank—to the cell at low pressure corresponds to different steps as presented in Fig. 1. An important feature is to consider that the chocked exit—the orifice of the nozzle in the Knudsen chamber—has to be calculated independently of the tubing part of the nozzle that works in the viscous regime. For these two nozzle parts, and due to the large pressure range needed in mass spectrometry, the possibility of working in the transition regime has to be taken into account. The different steady-state pressures, p2 at the nozzle entrance, p1 at the inner face of the nozzle exit are calculated as a function of the desired p0 pressure in our Knudsen cell fixed arbitrarily between 10K4 and 10K7 bar for mass spectrometric detection (lower pressures correspond to molecular flow in the nozzle). The pressure p0 determines the effusion flow Qeff (see Fig. 1 for the symbols) which becomes the steady-state flow in the whole system. The procedure for calculation is the one proposed by Santeler [7,8]: starting from a fixed p0 pressure in the Knudsen cell, with a chocked orifice (50 mm diameter), the p1 pressure is calculated as presented in Fig. 2 (chocked orifice, pressure p1), and a linear relation is attended according to the chocked flow equation QchocZC. p1 (CZ200pr2/RT in MKSA units (mol/s)—some parenthesis are missing in the Santeler formula) with the steady-state condition QchocZ Qeff. Then, calculation of the mean free path of molecules in
the gas at the nozzle orifice (pressure p1) showed that the flow can be partly in the transition regime, and a correction is done using the q function from Santeler, leading to a more accurate value for p1—‘chocked orifice in transition flow’— showed in Fig. 2 for the ideal exit orifice of the nozzle. Then, the viscous flow is calculated in the short tubing of the nozzle (50 mm long) leading to a parabolic relation in Fig. 2 for the pressure p2 (viscous nozzleCchocked orifice in transition flow), and corrections are necessary for the transition regime, leading to the final relation for the pressure p2 (crosses in Fig. 2). A linear fit is given for further comparison with experiments, although the calculated curves let appear a slight S shape corresponding to the transition flow. The calculation of the transition regime may be controversy, and different weighting equations between the molecular and the viscous flow contributions have been proposed in literature. For this reason we calculated also with the DeMuth and Watson [9] formula that is based on the Knudsen number, as presented in Fig. 3. A slight difference with calculations from the Santeler formula is observed on the slopes that are not significant compared to the fit approximations.
3. Mass spectrometric tests Mass spectrometric experiments were performed first with an empty Knudsen cell by introduction of pure N2 gas
490
C. Chatillon, M. Heyrman / Journal of Physics and Chemistry of Solids 66 (2005) 488–493
Fig. 2. Calculated input regulated pressures p1 at the nozzle exit (diamonds and squares) or p2 in the gas tank (triangles and crosses) as a function of the Knudsen cell pressure p0 according to Santeler [7,8] formula and taking into account the different flow regimes in the nozzle. Marks are for calculated values and equations are obtained from least square fits.
in order to test our gas introduction device. The effusion orifice of the Knudsen cell (Fig. 1) has a 2 mm diameter and 2 mm length and works in the molecular flow regime for the chosen p0 pressures. At the bottom of the cell a large orifice (5 mm diameter) let appear the nozzle which is machined in
a stainless steel wall according to the zoom presented in Fig. 1. In this study, the measurement on gaseous species that exist also in the ion source background—namely N2, O2 and H2O—were performed with our special restricted collimation sampling and shutter device [10] in order to
Fig. 3. Calculated input regulated pressures p1 at the nozzle exit (diamonds and squares) or p2 in the gas tank (triangles and crosses) as a function of the Knudsen cell pressure p0 according to Santeler [7,8] formulae and DeMuth and Watson [9] formula for the transition regime.
C. Chatillon, M. Heyrman / Journal of Physics and Chemistry of Solids 66 (2005) 488–493
491
Fig. 4. Products IT (of ionic intensity (in A)!cell temperature (K)) measured by the mass spectrometer-proportional to the Knudsen cell pressure—as a function of the measured N2 gas input regulated pressure in the tank by a Baratron capacitance. The H2O pressure is imposed by a dew point at 10 8C, and needs some time for obtaining a steady-state value of IT at about 10K14 A.K. (the first measured values start at the bottom left side according to the arrows).
avoid any ion source/oven/vacuum background interactions that can introduce some systematic errors in measurements. In some case, the input N2 gas was flowed over a thermal regulated water bath in order to impose a constant partial water pressure in the effusion cell. Results are shown in Figs. 4 and 5. The experimental slope of N2 effused pressure—or proportional quantity IT—measured versus the input regulated one is very close to the expected one from flow calculations (see Figs. 2–5).
In Fig. 4, the two regimes observed for the H2O pressure—as well as the sequence of observation—show that time is needed to reach a constant value that corresponds probably to the dew point as imposed by thermal regulation of the gas flow after flushing the water tank as well as time for saturation of the thermal regulated input line. In Fig. 5, without water, simultaneous monitoring of NC and OC ions (masses 28 and 32) 2 2 shows that there was some leak in our introduction device,
Fig. 5. Products IT measured by the mass spectrometer as a function of the N2 input regulated pressure in the gas tank. We observe a small leak in the introduction device as revealed by the measure of O2(g) at mass 32 that becomes more important when the total pressure p2 in the on line tank decreases.
492
C. Chatillon, M. Heyrman / Journal of Physics and Chemistry of Solids 66 (2005) 488–493
Fig. 6. Evolution of the products IT for different gaseous species as a function of the O2(g) input regulated pressure at constant temperature for a grinded graphite sample disposed in the Knudsen cell at 683 K. The N2(g) species accounts for the background peak in the ion source housing contrarily to other species that are detected in the effused beam using the shutter. Arrows indicate the sequence of measurements.
more easily observed on the O2 impurity. This feature means that the line has to be built according to ultra-high vacuum techniques (changes are in process).
4. Application to gas surface reactions as studied by mass spectrometry The aim of the gas introduction device is to study reactions of gases with a solid surface that we use as sample in the Knudsen cell (Fig. 1). As an example we studied the gasification of carbon—a graphite sample ground in a mill during 72 h under fore vacuum and under pure O2 during 12 h in order to favor the saturation of the sites by surface complexes—as shown in Fig. 6. The temperature is fixed at 683 K and we analyze by mass spectrometry the gaseous reaction products—i.e. CO, CO2 and by-products due to leaks and impurities—as a function of the flow of O2 which is pressure regulated in the input tank. We observe a production of CO larger than the CO2 one, and a slight relative increase of CO2 proportion versus CO when the total pressure in the cell increases. Beyond this classical thermodynamic effect, a more marked effect occurs when pressure is maintained a long time at high values (see arrows in Fig. 6), probably because of a reaction between the CO(g) and the surface complexes can give CO2(g) as postulated in carbon gasification literature [11,12]. This reaction is postulated to be a slow process depending on total pressure as we observe in this work.
5. Conclusion and perspectives The ability to control an inlet gas flow in an effusion cell over a wide pressure range has been proved when using a small nozzle at the bottom of an effusion cell associated with an input pressure regulated tank. Flow calculations of such a device are clearly possible and show that a wide pressure range values may be reached when using suitable input pressure gauges—at least two different gauges since their usual working range is within 1!10K3. One supplementary evaluation of such a device will be done by calibration of the mass spectrometer with an inert and suitable sample (relatively to the introduced gas) vaporizing in the same temperature range, the ionization cross-section of which being well known.
References [1] D.L. Hildenbrand, Attainment of chemical equilibrium in effusive beam sources of the heterogeneous type, in: Characterization of High Temperature Vapors and Gases, NBS Special Publication 561, vol. 1, NIST, Maryland, 1979; pp. 171–179. [2] D.L. Hildenbrand, K.H. Lau, Thermochemistry of gaseous manganese oxides and hydroxides, J. Chem. Phys. 100 (1994) 8377–8380. [3] D.L. Hildenbrand, K.H. Lau, Thermochemistry of gaseous SiO(OH), SiO(OH)2 and SiO2, J. Chem. Phys. 101 (1994) 6076–6079. [4] J.W. Hastie, D.W. Bonnell, E.R. Plante, Slag and metal oxide vaporization in reactive atmospheres, High Temp. Sci. 13 (1980) 257–279. [5] J. Drowart, A. Pattoret, S. Smoes, F. Degre`ve, D. Detry, Etudes thermodynamiques et cine´tiques en chimie des hautes temperatures in: W.L. Mead (Ed.), Advanced Mass Spectrometry, vol. 3, The Institut of Petroleum, London, 1966, pp. 931–945.
C. Chatillon, M. Heyrman / Journal of Physics and Chemistry of Solids 66 (2005) 488–493 [6] G.M. Rosenblatt, Evaporation from Solids, in: N.B. Hannay (Ed.), Treatise on Solid-State Chemistry VI. Surfaces, Plenum Press, New York, 1976, pp. 165–239. Chapter 3. [7] D.J. Santeler, Exit loss in viscous tube flow, J. Vac. Sci. Technol. A4 (1986) 348–353. [8] D.J. Santeler, New concepts in molecular gas flow, J. Vac. Sci. Technol. A4 (1986) 338–343. [9] S.F. DeMuth, J.S. Watson, Prediction of flow rates through an orifice at pressures corresponding to the transition between molecular and isentropic flow, J. Vac. Sci. Technol. A4 (1986) 344–347.
493
[10] P. Morland, C. Chatillon, P. Rocabois, High temperature mass spectrometry using the Knudsen effusion cell. I. Optimization of sampling constraints on the molecular beam, High Temp. Mat. Sci. 37 (1997) 167–187. [11] N.M. Laurendeau, Heterogeneous kinetics of coal char gasification and combustion, Prog. Energy Combust. Sci. 4 (1978) 221–270. [12] Z. Du, A.F. Sorofin, J.P. Longwell, L. Tognotti, The CO/CO2 ratio in the products of the carbon oxygen reaction in fundamental issues, in: J. Lahaye, P. Ehrburger (Eds.), Control of Carbon Gasification Reactivity, Kluwer Academic Publishers, Netherland, 1991, pp. 91–106.