One of the most important assumption of the closed-form model derived above, is that the silica layer does not rate limit the elimination of SiO(g). Indeed, for reaction (R5) to occur, the SiO(g) formed at the Si(s)-SiO2(s) interface must be evacuated.
If the silica layer were perfectly continuous, this would occur by solid state diffusion through this layer. According to Sasse and König [SK90], the diffusion of the SiO molecular entity would control the kinetics. The diffusion coefficient in vitreous silica, DSiOl,SiO2, is estimated as 3.2×10-20 and 2.7×10-19 m² s-1 at 1000 and 1100 °C respectively. These values are considerably lower than the molecular diffusion coefficient of silicon monoxide in the gas phase, DSiOmol (# 8×10-4 m² s-1, Figure B.6 in Appendix B.4.1).
Assuming a serial diffusion of silicon monoxide in the silica layer (thickness,
SiO2
e ) and in the gas phase (distance, zf), an effective diffusion coefficient, DSiOeff , can be overestimated from Equation (III.33).
mol SiO 9 nm
1 1 mol SiO f SiO l, SiO
SiO SiO
f eff
SiO 1 10
SiO2 2
2
2 D
D z D
e e z
D e (III.33)
Even when considering the diffusion of silicon monoxide in a silica layer of nanometric thickness, the effective diffusion coefficient is much lower than the molecular diffusion coefficient, which is used in the model to describe the reduction kinetics in section III.3.3.
Accordingly, if the silica layer were perfectly continuous, it would entirely preclude silica reduction through reaction (R5). It is then fair to question the continuity of this layer.
From interfacial energy considerations, the silica layer should be stable and continuous. The surface energy of amorphous silica, SiO2V T 0.307 0.0002 T 1720 C Jm 2 [END99], is much lower than the surface energy of silicon, SiVT 1.08 0.0001 T 937 C Jm 2 [END99], as usually observed when comparing oxide to metal surface energies. The silicon- silica interfacial energy, SiSiO2, can be estimated from wettability measurements of silicon onto silica. The wetting is then unfavorable with an equilibrium contact angle, ~ 90° ± 10°
[WTT99]. Accordingly, the silicon-silica interfacial energy, SiSiO2, should be of the same order as the surface energy of amorphous silica, SiO2V. As shown in Figure III.12, as the temperature increases, the global energy of a silica layer covering a silicon surface becomes even less than the energy of a free silicon surface. Accordingly, dewetting of silica on silicon particles is unlikely to occur.
Figure III.12: Surface energy of a free silicon surface compared with the estimated surface energy of a silicon surface covered with a silica layer.
The inconsistency is removed by considering the presence of defects, such as oxygen vacancies or holes, in the silica layer. These defects could be initially present inside the native oxide or created during heating. However, during heating below the passive to active transition, continuous growth of the silica layer occurs through reaction (R1). This growth should fill the holes and decrease the vacancy concentration in the layer.
Si(s) + 2H2O(g) = SiO2(s) + 2H2(g) (R1)
But, once the passive to active transition temperature is overpassed, the overall vacancy concentration should increase and accelerate the diffusion of silicon monoxide molecules (Figure III.13 (a)).
Considering the thermal expansion coefficients of silicon (4.26×10-6 °C-1 [Hul99]) and vitreous silica (0.55×10-6 °C-1 [Smi83]) at 1000 °C, tensile stresses should also appear during heating and result in an increment of oxygen vacancies (Figure III.13 (b)) or even cracks or holes (Figure III.13 (c)).
Nevertheless, according to the Pilling-Bedworth ratio, which is defined as the ratio of the molar volume of silica to the molar volume of silicon, the silica layer is initially under
800 1000 1200 1400
0,5 0,6 0,7 0,8 0,9 1,0 1,1 1,2 1,3 1,4
Surface energy (J m-2 )
Temperature (°C) Free silicon surface, Si V
Silicon surface covered with a silica layer, SiO2 V + Si SiO2
compressive stresses. Indeed, this ratio is close to 2.1 in the case of amorphous silica, tridymite or cristoballite, which are the most common crystalline strcutres proposed for thin native silica layers. The silica layer should be then initially continuous and passive, as observed expiremtally since silicon is usually protected against further oxidation once a native silica layer is formed.3
Another possible reason for the acceleration of the diffusion in the silica layer after the passive to active transition is the generation of vacancies by the reaction (R5), itself, at the silicon-silica interface.
Whatever the origin of the defects, considering the thickness of the silica layer (0.5 nm, two or three silica tetrahedrons), these should rapidly propagates in non-healable holes (Figure III.13 (d)) as observed by Tromp et al. [TRB+85] under vacuum.
Figure III.13: Defects in a silica layer covering a silicon surface. (a) Generation of vacancies, (b and c) Generation of vacancies, cracks or holes under the effect of thermal tensile stresses, (d) Non-healable holes.
From the gas kinetic theory (Appendix B.4), the evaporation flux of silicon monoxide through the holes is given in Equation (III.34), where is the surface area fraction of holes at the particle surface.
T M j P
R 2 SiO
R v SiO
SiO
5
(III.34)
3 Paragraph added to the manuscript following a discussion with the comitee during the PhD defence.
SiO2 Si
SiO2 Si
SiO(g) SiO(g) SiO(g)
Vacancies
(a) (b) (c)
(d)
The diffusion flux of silicon monoxide from the sample surface is roughly estimated in Equation (III.35) from discussion of section III.3.3.
f R SiO mol d SiO
SiO
5
R z
P T
j D (III.35)
According to a mass balance, at the passive to active transition, the evaporation does not limit the silicon monoxide departure if Equation (III.36) is verified.
d SiO v
SiO j
j (III.36)
Using Equations (III.34) to (III.36), this is verified for surface area fraction of holes of approximately 4×10-5 as shown in Equation (III.37).
T z
M D
R 2
f
SiO mol
SiO (III.37)
From our results on silica reduction kinetics, it is fair to assume that the holes surface area fraction overpasses this limit as soon as the passive to active transition is overpassed. Cracks or holes then rapidly propagate in non-healable holes and silica reduction kinetics is limited by molecular diffusion in the gas phase.