Consider the formation of SiO2 as O2 diffuses through the Si substrate as shown in figure 1 below.

T (C) 1 100 1000 900 800 1000 100 Kp (nm?/min) 10 K (nm/min) 0.1 P = 1 atm 0.01 7.0 7.5 B.0 9.0 104 /T (K-' ) Arrhenius plot of the linear and parabolic rate constants for silicon oxidation at one atmosphere of oxidant. Data from Irene and Dong [9, Fig. 2]: reprinted by permission of the publisher, The Electrochemical Society, Inc.UEIUVV. O L" J I Figure 1: Three-phase oxidation system and excess 0; distribution in the oxide. In summary, the governing equations for the system are: 07.20 (15562 A 0 C(zi: = O) = 00 MIL + D (0005' m) = 0 6:12 pm) GIL g - (2160;. 15(0) : 0 where the last equation is the initial condition for the layer thickness. (14) (15) (16) (17) (18) \fC(t,x) = concentration of O2 in excess of the amount contained in the glass network. Therefore the diffusion equation should govern the transport (diffusive) of the oxygen in the silica network but we shall see that a simplifying approximation is appropriate. C(t,L(t) ) = CL is the concentration at the interface, L(t) = position of the Si/SiO2 interface, r = k cz - is the reaction rate at the interface assuming that the reaction is first order in the concentration of O2 at the Si/SiO2 interface, k = the rate constant, C(t,0) = Co is the (constant) concentration of Oz at the exterior Si02 interface , D = diffusion coefficient of O2 in Si, 2 = specific volume of SiO2