Analysis of Kinetic Curves of Mass Change During the Interaction of Active Gases with Certain Metals and Compounds with Simultaneous Evaporation of the Products of Reaction

1. Introduction

This work is a continuation of [1], where the role of reduction of the reaction surface and evaporation of scale in the process of oxidation of FeCr and FeCrAl alloys is considered. Here we will consider in more detail the influence of the secondary process of evaporation (sublimation) on the formal kinetics (mass change - time) of the interaction of gases (O₂, Cl₂, NH₃, H₂O and N₂H₄ vapors) with some metals and compounds. We used data from various works collected in [2], the results of the indicated author, as well as your data.

The process of scale formation with simultaneous evaporation is called the Tedmon’s process [3] (althoug such cases were discussed a little earlier [4,5]). This model is used when the overall kinetics is determined by the diffusion of ions in the scale (volume diffusion) rather than by the rate of the chemical reaction itself. In this case, the kinetics is parabolic [6–13]. With short-circuit diffusion, cubic kinetics takes place [6–11], and in the case of local electric fields and space charges, the kinetic law of the fourth degree is implemented. Kinetic data with the latter mechanism are also presented in number of works [6,7,17,18].

All of the above is clearly shown in the kinetic dependences of the mass gain, when the following happens: the mass of the sample first monotonically increases with deceleration, reaches a maximum, and then, when the rates of growth and evaporation are equalized, begins to decrease (Figure 1). Here, curve 1 shows the total mass change (M) of the metal-scale system, and curve 2, plotted graphically on its basis (excluding the contribution of evaporation), corresponds to the increase of mass (m) due to the reacted gas: m=M+v_mt, where v_m is the evaporation rate to the metal component of the scale, t is the time. The section OA (OA₁) corresponds to the gradual growth of scale, and the straight section AB (A₁B₁) corresponds to the stationary regime, when the amounts of the formed and evaporating products are equalized (the first gradually decreases due to diffusion limitation, while the second is constant in an isothermal process). In this case, the layer reaches its maximum mass (thickness). The slope of the straight line AB corresponds to the rate of the mass decrease of the sample: v_m=$-$tgβ. (Here we consider the coordinate systems t - m and t - M; the coordinate system t_W – W is considered further.)

The rate of mass increase due to the reacted gas in the general case is:

$$\frac{\mathrm{d}\mathrm{m}}{\mathrm{d}\mathrm{t}}=\frac{{\mathrm{k}}_{\mathrm{n}}/\mathrm{n}}{{\mathrm{m}}^{\mathrm{n}-1}+({\mathrm{k}}_{\mathrm{n}}/\mathrm{n}{\mathrm{r}}_{\mathrm{r}})}-{\mathrm{v}}_{\mathrm{g}},$$

(1)

where n=2, 3 or 4; ${\mathrm{k}}_{\mathrm{r}}\equiv$dm/dt (at point t=0,m=0)=tgα is rectilinear constant; ${\mathrm{k}}_{\mathrm{n}}$ is the power-law constant; v_g is the speed of the system mass reduction due to the gas component of the evaporating part of the scale. Its integral form is as follows:

$$\mathrm{t}=(1+\mathrm{k})\frac{{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{n}-1}}{{\mathrm{v}}_{\mathrm{g}}}\int \frac{\mathrm{d}\mathrm{m}}{{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{n}-1}-{\mathrm{m}}^{\mathrm{n}-1}}-\frac{\mathrm{m}}{{\mathrm{v}}_{\mathrm{g}}},$$

(2)

where k=${\mathrm{v}}_{\mathrm{g}}$/$({\mathrm{k}}_{\mathrm{r}}-{\mathrm{v}}_{\mathrm{g}})$, ${\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}={({\mathrm{k}}_{\mathrm{n}}/\mathrm{n}{\mathrm{k}}_{\mathrm{r}}\mathrm{k})}^{1/(\mathrm{n}-1)}$ is the maximum mass gain of the system at the expense of active gas.

Boundary condition for solving of Eq.(2) for different n: t=0,m=0.

The Tedmon-Wajsel equation (n=2) in our notation will be:

$$\mathrm{t}=-(1+\mathrm{k})\frac{{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{{\mathrm{v}}_{\mathrm{g}}}\ln \frac{{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-\mathrm{m}}{{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}-\frac{\mathrm{m}}{{\mathrm{v}}_{\mathrm{g}}}.$$

(3)

For n=3 and 4 we will have:

$$\mathrm{t}=(1+\mathrm{k})\frac{{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{2{\mathrm{v}}_{\mathrm{g}}}\ln \frac{{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}+\mathrm{m}}{{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-\mathrm{m}}-\frac{\mathrm{m}}{{\mathrm{v}}_{\mathrm{g}}},$$

(4)

$$\mathrm{t}=(1+\mathrm{k})\frac{{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{{\mathrm{v}}_{\mathrm{g}}}\left[\frac{1}{\sqrt{3}}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{g}\frac{\sqrt{3}\mathrm{m}}{\mathrm{m}+2{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}-\frac{1}{6}\mathrm{l}\mathrm{n}\frac{{({\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-\mathrm{m})}^2}{{\mathrm{m}}^2+\mathrm{m}{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}+{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}^2}\right]-\frac{\mathrm{m}}{{\mathrm{v}}_{\mathrm{g}}},$$

(5)

respectively [17]. For the total mass change will be:

M=m-v_mt.

(6)

Such containing maxima curves were obtained in a number of works [19–29] and works collected in [2]. Here we will look at graphs in which this maximum is clearly expressed and from which reliable information can be obtained (some graphs, which are not considered here, give unrealistic values of kinetic parameters).

2. Discussion

2.1. Kinetic Curves of the Total Mass Change, Having a Maximum

As mentioned above, kinetic curves containing a maximum are presented in many works. Most of them are curves corresponding to parabolic kinetics (n=2); there is little data for n=4; but for n=3 we did not find such data (if we do not consider the curve presented in Figure 4a), although cubic processes (with curves without maxima) are considered in a fairly large number of works.

Figure 2 shows the data of [20,21]. To determine n, we can use the formula [17]:

$$\mathrm{n}=\frac{\mathrm{l}\mathrm{g}\left[\left(1-\mathrm{q}\mathrm{k}\right)(\overline{\mathrm{M}}+{\mathrm{v}}_{\mathrm{m}}\overline{\mathrm{t}})/\mathrm{p}{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right]}{\mathrm{l}\mathrm{g}\left[\right(\overline{\mathrm{M}}+{\mathrm{v}}_{\mathrm{m}}\overline{\mathrm{t}})/{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}]},$$

(7)

where q=v_m/v_g, p=(v_m+v_g)/v_g=q+1. The tangents of the curves in Figure 2 virtually coincide with ordinate axis at the origin of the coordinates: ${\mathrm{k}}_{\mathrm{r}}\to \mathrm{\infty }\Rightarrow \mathrm{k}\to 0$. In this case formula (7) is simplified as follows:

$$\mathrm{n}=\frac{\mathrm{l}\mathrm{g}\left[(\overline{\mathrm{M}}+{\mathrm{v}}_{\mathrm{m}}\mathrm{t})/\mathrm{p}{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right]}{\mathrm{l}\mathrm{g}\left[\right(\overline{\mathrm{M}}+{\mathrm{v}}_{\mathrm{m}}\mathrm{t})/{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}]}.$$

(8)

.

According to formula (8) for [21] it turns out n$\cong$2.02, and for [20] n$\cong$3.75, that are approaching to 2 and 4. Corresponding empirical expressions are: t$\cong -$15.707ln(1-0.554m)$-$8.693m and t$\cong$28.885{[0.577arctg((1.424m)/(0.822m+2)$]-[$0.167ln((1-0.822m)²$/$(${\mathrm{m}}^2$+0.822m+1))]}--23.75m, where m is in mg/cm² and t is in hours. The curves constructed using these equations on the scale used in Figure 2 practically coincide with the experimental curves.

Figure 3 shows the dependence in coordinates (k_p/v_g, $\overline{\mathrm{M}}$) for different samples from [2]. The following reactions are considered here: 2Cr+3O₂=Cr₂O₃, Pb+Cl₂ = PbCl₂, Si+O₂ = SiO₂, SiC+2O₂ = SiO₂ + CO₂, Si₃N₄+3O₂ = 3SiO₂+2N₂ and 4BN+3O₂ = 2B₂O₃ +2N₂ (for all reactions the kinetics are parabolic: k_p - power-law constant at n=2). We have added data for reactions Cr+2HCl=CrCl₂+H₂ [21] and 3Ge+4NH₃=Ge₃N₄+6H₂ [29] also with parabolic kinetics. This data fit well into this dependence (see figure).

2.2. Consideration of Preliminary Mass Reduction

In some cases the growth of scale is preceded by other processes, for example, gas etching of the surface of the metal or alloy (initial section of curve 1 in coordinate system t-W (Figure 1)). In this case, to describe the m – t dependence, it is necessary to solve the differential equation (1) with the boundary condition t = 0, m = m₀. For n=2, 3 and 4 these solutions have the form:

$$\mathrm{t}=-(1+\mathrm{k})\frac{{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{{\mathrm{v}}_{\mathrm{g}}}\ln \frac{{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-\mathrm{m}}{{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}-{\mathrm{m}}_0}}-\frac{\mathrm{m}-{\mathrm{m}}_0}{{\mathrm{v}}_{\mathrm{g}}},$$

(9)

$$\mathrm{t}=(1+\mathrm{k})\frac{{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{{2\mathrm{v}}_{\mathrm{g}}}\ln \frac{\left({\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}+\mathrm{m}\right)({\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\mathrm{m}}_0)}{({\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-\mathrm{m})({\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}+{\mathrm{m}}_0)}-\frac{\mathrm{m}-{\mathrm{m}}_0}{{\mathrm{v}}_{\mathrm{g}}},$$

(10)

$$\mathrm{t}=(1+\mathrm{k})\frac{{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{{\mathrm{v}}_{\mathrm{g}}}\left[\frac{1}{\sqrt{3}}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{g}\frac{\sqrt{3}\mathrm{m}(\mathrm{m}-{\mathrm{m}}_0)}{\mathrm{m}+2{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}-\frac{1}{6}\mathrm{l}\mathrm{n}\frac{{\left({\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-\mathrm{m}\right)}^2({\mathrm{m}}_0^2+{\mathrm{m}}_0{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}+{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}^2)}{{\left({\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\mathrm{m}}_0\right)}^2{(\mathrm{m}}^2+\mathrm{m}{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}+{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}^2)}\right]-\frac{\mathrm{m}-{\mathrm{m}}_0}{{\mathrm{v}}_{\mathrm{g}}},$$

(11)

respectively. To demonstrate, we present data on the interaction of single-crystalline Ge with NH₃+H₂O and N₂H₄ vapors (Figure 4).

P_H2O/P_NH3); and (c) with N₂H₄ (P_N2H4=2$\bullet$10³Pa) at 720^oC – (1) dependences W – t, (2) – dependences m – t; 2’ – calculated curves (in the scale of the figure (c), the experimental and calculated curves practically coincide with each other). .

Figure 4. Kinetic dependences of interaction of Ge: with NH₃+H₂O at (a) P=2%, 820^oC, (b) P=5%, 800^oC, (P$\equiv$ P_H2O/P_NH3); and (c) with N₂H₄ (P_N2H4=2∙103Pa) at 720oC – (1) dependences W – t, (2) – dependences m − t; 2’ – calculated curves (in the scale of the figure (c), the experimental and calculated curves practically coincide with each other). .

Figure 4. Kinetic dependences of interaction of Ge: with NH₃+H₂O at (a) P=2%, 820^oC, (b) P=5%, 800^oC, (P$\equiv$ P_H2O/P_NH3); and (c) with N₂H₄ (P_N2H4=2∙103Pa) at 720oC – (1) dependences W – t, (2) – dependences m − t; 2’ – calculated curves (in the scale of the figure (c), the experimental and calculated curves practically coincide with each other). .

On these curves, the initial decrease of mass is due to the etching of the germanium surface by water vapor, which is contained in small quantities also in concentrated hydrazine (volatile GeO is formed here: GeO+H₂O=GeO+H₂) [30]. Also it is obvious that the formation of nitride on the germanium surface will begin before the zero point in the t - m coordinate system. But from the presented model it follows that the m - t dependences are convex in the positive direction. Time shifts betveen equations (3) and (9), (4) and (10), (5) and 11) are:

$${\mathrm{t}}_0=(1+\mathrm{k})\frac{{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{{\mathrm{v}}_{\mathrm{g}}}\ln \frac{{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\mathrm{m}}_0}{{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}+\frac{{\mathrm{m}}_0}{{\mathrm{v}}_{\mathrm{g}}},$$

(9’)

$${\mathrm{t}}_0=(1+\mathrm{k})\frac{{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{2{\mathrm{v}}_{\mathrm{g}}}\ln \frac{{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}+{\mathrm{m}}_0}{{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\mathrm{m}}_0}+\frac{{\mathrm{m}}_0}{{\mathrm{v}}_{\mathrm{g}}},$$

(10’)

and

$${\mathrm{t}}_0=(1+\mathrm{k})\frac{{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{{\mathrm{v}}_{\mathrm{g}}}\left[\frac{1}{6}\mathrm{l}\mathrm{n}\frac{{\left({\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\mathrm{m}}_0\right)}^2}{{{\mathrm{m}}_0}^2+{\mathrm{m}}_0{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}+{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}^2}-\frac{1}{\sqrt{3}}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{g}\frac{\sqrt{3}{\mathrm{m}}_0}{{\mathrm{m}}_0+2{\mathrm{m}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}\right]+\frac{{\mathrm{m}}_0}{{\mathrm{v}}_{\mathrm{g}}},$$

(11’)

respectively. Thus, the values of m₀ can be estimated by solving of transcendental equations (9’)-(11’) by substituting the values of k, m_max, v_g, and t₀ determined from experimental data. The main difficulty is the accurate determination of t₀ in the initial section of the curves - conducting an additional experiment of short duration would lead to even larger errors.

According to the experimental data presented in Figure 4, one can estimate t₀≈$-$3 min, $-$0.14 h, and $-$0.33 h, respectively with Figures (a), (b) and (c). Then the values ​​of m₀ will be ≈0.3, 0.05 and 0.03 mg/cm². As you can see, m₀ makes up (20-34)% of corresponding m_max (0.145, 1.42, 0.092 mg/cm², respectively) and this cannot be ignored when conducting an experiment using the gravimetric method.

3. Conclusions

A general equation is given that describes the scale growth-evaporation kinetic (sample mass change - time) curves during the interaction of gases with the surface of metals and compounds. Special cases of parabolic, cubic and fourth degree processes are discussed. Equations are also given for the case when scale formation is preceded by the process of gas etching of the metal surface.