Iron Corrosion in the Radiative Plasma of Moist Air

Abstract

The rate of iron corrosion in a low-temperature plasma of moist air formed under the action of a fast electron beam, depending on the relative air humidity, was measured. It is shown that a beam of fast electrons (radioactive β-radiation) significantly intensifies the corrosion of iron in a plasma-forming gas medium in which oxygen and water vapour are simultaneously present. It has been established that the rate of corrosion under conditions of radioactive irradiation increases sharply when the relative air humidity exceeds 10%. Numerical simulation of the ionic composition of the plasma is carried out taking into account 12 types of positive and 12 types of negative hydrated ions at different intensities of the external ionization source and at different values of relative humidity from 10^–6 to 100%. A hypothesis has been put forward about the determining role of cluster hydrated ions, which are formed in the plasma of moist air at atmospheric pressure, in heterogeneous processes of iron oxidation.

1 INTRODUCTION

The problem of long-term and reliable storage of radioactive waste (RW) is an urgent task of the nuclear industry, which has a significant impact on the development of the entire nuclear industry, and above all nuclear energy, worldwide. A radiation chemically active low-temperature plasma is formed during the processing and storage of radioactive waste in installations and containers under the influence of radiation arising from the decay of radioactive elements of radioactive waste. This leads to the initiation and intensification of ion–molecular processes of plasma chemical corrosion of metal walls of equipment and storage facilities and a significant reduction in their safe operation. The current level of understanding of complex processes and phenomena occurring in closed volumes of RW storage facilities is very limited and far from meeting the increasing needs of practice [1, 2]. In this regard, the study of the mechanisms and kinetics of physics–chemical corrosion processes under increased radioactive radiation conditions is important for the development methods for protecting the walls of equipment and containers from destruction and, accordingly, preventing radiation leakage into the environment and ensuring the maximum level of safety of radioactive waste storage. This paper presents the results of experimental studies of the corrosion rate of iron, which is the main component of most structural alloys of container walls, in a moist air plasma excited by a stationary fast electron beam as a high-energy ionizing beta radiation simulator for decay products of radioactive elements of spent nuclear fuel. The effect of the number density of water vapor in the air on the rate of iron corrosion has been studied. The results of numerical simulation of the plasma composition in moist air irradiated by a fast electron beam at different values of relative humidity are also presented.

It should be noted that the type of ionizing radiation that creates a plasma has practically no effect on the physicochemical processes in such a plasma, therefore, all of the following is also true for the plasma generated in humid air by γ-quanta, α-particles, or other high-energy charged particles at equality of the production rate of electron-ion pairs per unit volume.

2 EXPERIMENTAL

Figure 1 shows a schematic diagram of an experimental setup designed to study the rate of iron corrosion in a radiation plasma of moist air formed by fast electron beam at different number densities of water vapor. The setup allows experiments to be carried out at different number densities of water vapor in air or other gases and at different gas ionization rates.

Fig. 1.

Schematic diagram of an experimental setup for studying the rate of iron corrosion in a radiation plasma of moist air generated by fast electron beam.

The main module of the experimental setup is an electron accelerator (electron gun) with an incandescent cathode, which generates a continuous (stationary) fast electron beam. The electron energy and the beam current could be varied in the range W = 80–140 keV and I = 0.1–1.5 mA, respectively. A fast electron beam formed in the high-vacuum chamber of the accelerator is extracted into the plasma chemical reactor (high-pressure region) through an aluminum foil with a thickness of 14 microns. The cylindrical plasma-chemical reactor was made of a glass tube with an internal diameter of 175 mm and a height of 70 mm, with one end closed with a metal flange, and the other vacuum tightly docked to the output of the electronic accelerator. The reactor is equipped with vacuum ports for pumping the internal volume to a pre-vacuum pressure of P = 10^–3 Torr and subsequent injection of a working plasma-forming gas of a given composition and pressure. In this paper, all experiments were carried out at a pressure of P = 1 bar, weak pumping of gas with a volume flow rate of v_f = 100 cm³/s and a processing time of t = 20 min. A massive steel cylinder with a diameter of 100 mm was placed on the bottom of the plasma chemical reactor, which together with the flange of the chamber served as a refrigerator for the iron sample being processed. A chromel— copel thermocouple was inserted into the upper end of the steel cylinder, with the help of which the temperature of the processed metal sample was measured. The processed samples were plates of technically pure ARMCO iron with transverse dimensions of 10 × 10 mm², 20 × 70 mm² and 70 × 70 mm² and a thickness of 0.5 mm, which were placed on a massive thermostat that allows maintaining a constant temperature of iron samples equal to approximately 300 K. The choice of several geometric dimensions for the processed samples is due to the requirements of the diagnostic methods used, as well as the spatially heterogeneous nature of the radiation corrosion phenomenon under study. Chemical composition of iron (wt %), according to the manufacturer’s passport, was as follows: Fe 99.92, C 0.010, Mn 0, 010, P 0.010, S 0.008, N 0.006, Cu 0.03, Sn 0.010.

Before installation into the chamber, iron samples were mechanically ground using fine sandpaper and washed with acetone and isopropyl alcohol. Experimental studies of the rate of iron corrosion in a moist air plasma excited by a stationary fast electron beam were carried out at an absolute air humidity of ε, varying within the range of ε = 0.01–2.5%. When the air temperature in the laboratory room in which the experimental installation is located is equal to T = 295 K, the specified absolute humidity corresponds to the relative humidity RH = 0.4–94%. Obtaining a given level of humidity in the air was achieved by passing it through a humidifying system consisting of several sequentially connected containers with distilled water. In the case of a high level of relative humidity RH ≥ 60%, dust-free indoor air was used as the pumped gas. At RH <60%, dry synthetic air was pumped through the humidification system, obtained by mixing nitrogen and oxygen in a ratio of 0.79 : 0.21. The water vapor content in the air was measured at the inlet and outlet of the plasma chemical reactor. At a low humidity level (ε ≤ 0.1% = 1000 ppm) measurements were carried out using a Baikal MK coulometric hygrometer at ε > 0.1% used a thermometer – hygrometer “Testo 623” with a humidity sensor of the capacitive type. In order to form a complete picture of the studied phenomenon of radiation corrosion of iron, a number of experiments were carried out in the inert gas argon, as well as in nitrogen with a water vapor content of ε = 0.005 and 1.2%.

After penetrating through the aluminum foil of the accelerator and entering the plasma chemical reactor, the fast electron beam becomes divergent due to the elastic and inelastic scattering of electrons on foil neutral atoms and air molecules [3, 4]. In this paper, the spatial distribution of the current density of fast electrons in the plane of the processed samples was measured using a multi-element electrical probe. The measuring sections of the probe were isolated from each other by a dielectric gap 0.5 mm wide, they had transverse dimensions of 5 × 5 mm² and were arranged in one row (line) and connected through an individual measuring resistance to a grounded bus. An electrical signal from each measuring resistance was applied to the input of a multichannel recorder. The entire line of probes was surrounded by a protective grounded copper electrode, which prevents the influence of external electric fields on the data of the probes. The probe module was placed on a two-coordinate moving system, with the help of which it moved in a plane in two mutually perpendicular directions during measurements. In turn, the displacement system could be installed at any given distance from the entrance of the electron beam to the plasma-chemical reactor.

Figure 2a shows, for example, the measured distributions of the beam current density at the total current of the gun I = 0.5 mA and I = 1.0 mA. It can be seen that the beam current density has a maximum value on the beam axis (x = 5.5 cm) and monotonically decreases with increasing distance from the axis (with increasing radius). The obtained electron beam current density distributions were taken into account when choosing the sizes of the processed samples and their location in the processing zone in order to obtain various scenarios and the most complete picture of radiation corrosion. Large samples (70 × 70 mm²) fell into areas with significant differences in radiation intensity, while small samples (10 × 10 mm²) were in areas with fairly uniform, but different in magnitude, radiation.

Fig. 2.

Radial distribution of the current density of fast electrons in the plane of processing iron samples (a) at different values of the total beam current I = 1 mA (circles and curve 1) and 0.5 mA (triangles and curve 2) and the corresponding gas ionization rate distributions (b).

The rate of iron corrosion in the plasma of moist air, depending on the number density of water vapor, was estimated using three independent diagnostic methods:

1. Change in the mass of the corroding sample (gravimetric method).

2. Qualitative and quantitative changes in the elemental chemical composition of the upper surface layer of the sample.

3. Evolution of the spatial structure (morphology) of the sample surface in the corrosion process.

To measure the weight gain of the processed sample, digital analytical scales of the CCS Services A‑250A brand were used with a measurement accuracy of 10^–4 g, which have automatic calibration and control of zero readings. Qualitative and quantitative changes in the elemental chemical composition of the upper surface layer of the sample, occurring during corrosion, were investigated by X-ray microanalysis using a high-resolution scanning electron microscope of the brand Axia ChemiSEM (Thermo Scientific), equipped with a detector for elemental analysis Everhart–Thornley SE. The evolution of the spatial structure (morphology) of the corroding surface of the samples was also studied using this microscope. The change in the spatial structure of the sample surface during its processing was also studied using an Altami MET 5T digital optical microscope. This microscope allows analysis of the surface in polarized light using various light filters and adjustable aperture and field diaphragms.

3 RESULTS AND DISCUSSION

Figures 3a–3h show photographs of the surface of the initial (untreated) iron sample and samples exposed to an electron beam and radiation plasma in air with different humidity, as well as photographs of samples treated in an atmosphere of nitrogen and argon (Fig. 3i) at relative humidity RH = 50%. It is clearly seen from the presented photos that relative humidity has a significant effect on the processes of radiation corrosion of iron in the air. At low relative humidity RH ≤ 10%, no visible traces of corrosion are detected on iron samples (Fig. 3b). Noticeable traces of corrosion processes begin at a relative humidity of RH = 15%, while, as can be seen from Fig. 3c, the distribution of corrosion products over the surface of the sample is significantly heterogeneous. From the comparison of Figs. 3 and 2, it follows that with the selected total beam current I = 1 mA, the maximum current density corresponds to the minimum corrosion of the sample. Note that the heterogeneous nature of corrosion is observed only when the total beam current exceeds a certain critical value of I_cr ≈ 0.5 mA. With an increase in relative air humidity, the heterogeneous nature of corrosion persists, but the size of the area of heterogeneous corrosion decreases. At the same time, even at a high level of RH = 90%, the area of heterogeneous corrosion does not completely disappear. A more detailed study shows that corrosion actually occurs in the central part of the sample, but unlike its peripheral areas, where corrosion is fairly uniform (but with different intensity), in the center it has a point (pitting) character. The most probable reason for the decrease in the corrosion rate in the central region of the electron beam, where the maximum current density and accordingly the maximum gas ionization rate (see Fig. 2) is observed, is the appearance of a passive state of iron [5], in which an increase in the corrosive properties of the medium leads to reducing the rate of corrosion. Also note that in the absence of oxygen, the presence of water vapor in the plasma-forming mixture (nitrogen, argon) does not lead to the appearance of visible corrosion products (Fig. 3i).

Fig. 3.

Photos of iron samples processed in moist air plasma. (a) Initial (untreated) sample; relative air humidity RH = (b) 0–10, (c) 15, (d) 23, (e) 37, (f) 63, (g) 80, and (h) 90%; (i) Ar, N₂, RH = 50%. The sample sizes are 70 × 70 mm².

Figures 4a–4e show images of the surface of the initial (untreated) iron sample and samples exposed to an electron beam and radiation plasma in air with different humidity, as well as in nitrogen and argon, obtained using a scanning electron microscope at magnification k = 5000. From the presented photographs it can be seen that under the influence of radiation in a humid oxygen-containing environment (air), a yellow-brown loose flaky layer of rust forms very quickly (compared to atmospheric conditions) on the surface of the samples. With other similar processing conditions in wet nitrogen and argon, such a layer of rust does not form on the samples.

Fig. 4.

Images of the surface of iron samples obtained using a scanning electron microscope. (a) The original (unprocessed) sample; (b–e) wet air plasma treatment, relative air humidity RH = (b) 10, (c) 23, (d) 40, and (e) 85%; (f) Ar, N₂, RH = 50%. Sample sizes are 10 × 10 mm², magnification k = 5000. Electron beam current I = 1 mA, electron energy W = 115 keV, processing time t = 20 min.

Figure 5 shows experimental data on the increase in the mass of iron samples treated in moist air plasma depending on the relative humidity of the air, obtained with a constant current of a fast electron beam I = 1 mA and a constant processing time t = 20 min. The dimensions of the processed samples were 70 × 70 mm². The samples were weighed 10 minutes after their extraction from the plasma chemical reactor. From the data presented, it can be seen that at a relative humidity of the plasma-forming air RH <10%, the increase in the ΔM mass of the treated sample is below the sensitivity limit of the ΔM weights <0.1 mg. The results obtained by the gravimetric method are in good agreement with the results of microscopic and optical studies (Figs. 3b and 4b). Also, Fig. 5 clearly shows that there is a “critical” relative humidity RH ≈10%, above which the rate of mass gain of an iron sample treated in a radiation plasma of moist air increases significantly.

Fig. 5.

The specific increase in the mass of iron samples treated in moist air plasma, depending on the relative air humidity. Electron beam current I = 1 mA, electron energy W = 115 keV, processing time t = 20 min. The size of the samples is 70 × 70 mm².

Figure 6 shows experimental data on the oxygen content in the near-surface oxidized layer of iron samples treated in moist air plasma, depending on the relative humidity of the air. The results of measurements for two samples with a size of 10 × 10 mm², which were placed in different areas of the irradiation zone during processing, are presented. Sample 1 was located on the axis of the electron beam, where the minimum corrosion rate is observed, and sample 2 was located at a distance of 3 cm from the beam axis, where the corrosion rate is higher.

Fig. 6.

The oxygen content in the oxidized near-surface layer of iron samples treated in moist air plasma, depending on the relative air humidity. 1—the sample was located on the axis of the electron beam, 2—the sample was located on the periphery of the beam, the size of the samples is 10 × 10 mm². Electron beam current I = 1 mA, electron energy W = 115 keV, processing time t = 20 min.

At first glance, these data do not agree with the results presented in Figs. 3b and 4b, since an increase in oxygen content is detected already at relative humidity RH = 5%. The contradiction is resolved if we estimate the depth of the Δh layer analyzed by X‑ray microanalysis using a scanning electron microscope. With the electron energy in the diagnostic beam equal to W = 1.5 keV, the depth of analysis is Δx = 2 nm. Estimates show that the increase in the mass of the sample due to the introduction of oxygen atoms into the surface layer with a thickness of 2 nm is at the level of 5 × 10^–6 g/cm², which is below the measurement limit. At the same time, the thickness of the oxidized layer of iron (rust) at high levels of relative air humidity, measured using an optical profilometer, turns out to be 10–200 times greater.

4 NUMERICAL SIMULATION OF THE PLASMA COMPOSITION AND ESTIMATION OF THE CORROSION RATE

As follows from the data presented in Fig. 5, the rate of radiation corrosion of iron increases sharply at a relative humidity RH ≥ 10%. At such low humidity values, continuous water films do not form on the metal surface [6] and, accordingly, the electrochemical mechanism of corrosion does not work under these conditions. In [2], it was suggested that hydrated positive ions can play a significant role in the mechanism of radiation corrosion, while no specific processes of ion participation in the mechanism of corrosion were proposed. To clarify the role of cluster hydrated ions in heterogeneous processes of iron oxidation, numerical modeling of plasma composition in moist air irradiated by a fast electron beam at different values of relative humidity and different electron beam ionization rates was carried out in this paper.

The calculation of the rate of gas ionization by a beam of fast electrons Q_ion was carried out according to the model described in detail in [3]. It is similar to the model from [4] and the ionization rate was calculated by the relation:

$${{Q}_{{{\text{ion}}}}} = \frac{{{{j}_{b}}}}{e}\frac{\rho }{{{{\varepsilon }_{{{\text{ion}}}}}}}\frac{{dE}}{{dz}}\phi \left( {r,z} \right)F\left( {{{E}_{0}}} \right),$$

(1)

where j_b is the beam current density, e is the absolute value of the electron charge, ε_ion is the energy cost of the ion-electron pair production, which is close to two ionization potentials of atoms and molecules of gas (for humid air ε_ion ≈ 33 eV [7]), dE/dz is the specific energy loss by a fast electron per unit path length, ϕ is the function that takes into account diffusion spreading of the beam, z, r are the coordinates in a cylindrical coordinate system with the origin at the center of the foil, F is the quantity that takes into account the effect of the electric field E₀ applied to the discharge gap. In [4], to determine the specific energy loss and electron range, approximate relations were proposed, obtained as a result of approximation of the data calculated by the Monte Carlo method:

$$\begin{gathered} \frac{{dE}}{{dZ}}\left( {{{E}_{b}},Z} \right) = \left[ {0.78 - 0.0005{{E}_{b}}} \right] \\ \, \times \frac{{{{E}_{b}}}}{{{{R}_{e}}}}\left( {1 + \sin \left\{ {1.29\pi \frac{{{{Z}_{f}} + Z}}{{{{R}_{e}}}}} \right\}} \right), \\ \end{gathered} $$

(2)

$${{R}_{e}} = 0.00753E_{b}^{{1.661}},$$

(3)

where R_e is the beam electron range in mg/cm², E_b is the beam electron energy in keV, dE/dZ is the specific energy loss in keV cm²/mg, Z_f is the foil thickness in mg/cm², Z = ρz is the electron range in gas in mg/cm² (foil is at z = 0), ρ is the gas density.

After a fast electron has passed a certain path in a decelerating medium (according to [8], approximately equal to half the electron range in gas), the spreading function ϕ is described by the Laplace equation:

$$\Delta \phi = 0$$

(4)

with the following boundary conditions [4]:

$$\phi \left( {r,z = 0} \right) = \left[ {\begin{array}{*{20}{c}} {1,\quad r \leqslant a,} \\ {0,\quad r > a,} \end{array}\quad \mathop {\lim }\limits_{z \to \infty } \phi \left( {z,r} \right) \to 0,} \right.$$

(5)

where a is the radius (half-width in the case of a window in the form of an infinite strip) of the exit window of the electron gun.

In our experiments, the exit window of the electron gun had a shape close to round; therefore, the solution of the Laplace equation (4) can be represented in the following form [3] (see also [9]):

$$\begin{gathered} \phi = \frac{z}{\pi }\int\limits_0^\pi {\frac{1}{{[{{z}^{2}} + {{r}^{2}}{{{\sin }}^{2}}\psi ]}}} \\ \, \times \left[ {{{{({{r}^{2}} + {{z}^{2}})}}^{{1{\text{/}}2}}} - \frac{{{{r}^{2}} + {{z}^{2}} - ar\cos \psi }}{{{{{({{r}^{2}} + {{z}^{2}} + {{a}^{2}} - 2ar\cos \psi )}}^{{1{\text{/}}2}}}}}} \right]d\psi . \\ \end{gathered} $$

(6)

By integrating the spreading function, the current density per individual section of the probe was determined in the form of squares 0.5×0.5 cm² in size:

$${{j}_{p}}\left( {x,y,z} \right) = {{j}_{b}}\int\limits_{x - {{{{a}_{p}}} \mathord{\left/ {\vphantom {{{{a}_{p}}} 2}} \right. \kern-0em} 2}}^{x + {{{{a}_{p}}} \mathord{\left/ {\vphantom {{{{a}_{p}}} 2}} \right. \kern-0em} 2}} {\int\limits_{y - {{{{a}_{p}}} \mathord{\left/ {\vphantom {{{{a}_{p}}} 2}} \right. \kern-0em} 2}}^{y + {{{{a}_{p}}} \mathord{\left/ {\vphantom {{{{a}_{p}}} 2}} \right. \kern-0em} 2}} \phi } \left( {x{\kern 1pt} ',y{\kern 1pt} ',z{\kern 1pt} '} \right)dx{\kern 1pt} 'dy{\kern 1pt} ',$$

(7)

where h = ap/2, a_p is the section size: a_p = 0.5 cm. Figure 2 shows the beam current density distributions calculated by Eq. (7). One can see a good agreement between the calculated and experimental data. To achieve the best agreement, two parameters were varied in the calculations: the value of the beam current density j_b passing through the foil, since a part of the total beam current I fell on the structural elements of the electron gun supporting and cooling the foil, and also the coordinate z, since the spreading function, which has already been noted above, is described by the Laplace equation only from a certain distance after the passage of the foil [8]. It turned out that a good agreement between the beam current densities at the maximum in the calculation and experiment is achieved if we assume that 46.5% of the beam current is released at I = 1 mA, and it is 40.7% at I = 0.5 mA. And the form of the dependence on the x coordinate in the experiment and in the calculation practically coincide with each other, as can be seen in Fig. 2, at z' = 3 cm, i.e., the shift was 2.5 cm: z – z' = 2.5 cm. Therefore, the air ionization rate was calculated using formula (1) with this shift and with the found values of the beam current density. The calculation results are shown in Figs. 2b. It can be seen that the rate of gas ionization on the axis of the working chamber reaches 2 × 10¹⁶ cm^–3 s^–1.

The kinetic model of ion-molecular processes described in detail in [7] was used to calculate the ionic composition of moist air plasma. The main calculations were performed at the ionization rate Q_ion=10¹⁶ cm^–3 s^–1. Figure 7 shows the relative number densities of non-hydrated positive and negative ions in the radiative plasma of humid air as a function of relative humidity. It can be seen from the presented results that the fraction of non-hydrated ions is noticeable only at low relative humidity RH ≤ 0.1%, and already at RH ≥ 1%, the total fraction of non-hydrated both positive and negative ions becomes less than 10%.

Fig. 7.

The number densities of ${\text{O}}_{4}^{ + }$ ions (curve 1), ${\text{NO}}_{2}^{ + }$ (2), NO⁺O₂ (3), H₃O⁺(OH) (4), NO⁺(H₂O) (5), H₃O⁺ (6), ${\text{O}}_{4}^{ - }$ (7), ${\text{O}}_{2}^{ - }$ (8) at Q_ion = 10¹⁶ cm^–3 s^–1.

Figure 8 shows the relative number densities of positive H⁺(H₂O)_n (a) and negative ${\text{O}}_{2}^{ - }$(H₂O)_n (b) ions in radiative plasma of humid air as a function of relative humidity at a constant rate of gas ionization by an electron beam. It can be seen that as the relative humidity increases, the number n in the cluster complex increases (i.e., the cluster ions “get heavier”), and for negative ions this process begins much earlier (at lower RH) than for positive ions and accordingly at a fixed RH value, the fraction of negative ions with high n is greater than the fraction of positive ions. For example, at relative humidity RH = 80%, the fraction of ${\text{O}}_{2}^{ - }$(H₂O)₁₂ ions is 11–14%, depending on the gas ionization rate, while the proportion of positive H⁺(H₂O)₉ is at the level of 1%.

Fig. 8.

Relative number densities of positive H⁺(H₂O)_n (a) and negative ${\text{O}}_{2}^{ - }$(H₂O)_n (b) ions with different numbers of water molecules in the radiation plasma of moist air, depending on relative humidity (the numbers on the right are equal to the value of the number of water molecules in the complex n). The ionization rate of Q_ion = 10¹⁶ cm^–3 s^–1.

Figure 9 shows the total number density of positive ions and the total number density of negative ions equal to each other, as well as the total number densities of hydrated positive and negative ions in the radiative plasma of humid air depending on relative humidity at three gas ionization rates of 10¹⁴, 10¹⁵, and 10¹⁶ cm^–3 s^–1. As can be seen from Fig. 9, at a constant rate of gas ionization by an electron beam, the total number densities of positive and negative ions practically do not change with changes in air humidity, and hydrated complex ions become the predominant ions at RH > 10%.

Fig. 9.

Dependences of (solid curves) the total concentration of all ions of the same sign, (dashed curves) the total concentration of positive hydrated ions, and (dash-dotted curves) the total concentration of negative hydrated ions on the relative air humidity at an ionization rate Q_ion = (1) 10¹⁶, (2) 10¹⁵, and (3) 10¹⁴ cm^–3 s^–1.

In the range of RH > 20%, the composition of the predominant ions, as can be seen from Fig. 8 changes insignificantly, while the number of molecules in the complex changes by 1–2 units. Also in this area, according to Fig. 9, the total number densities of hydrated complex ions do not change. Therefore, we can conclude that the change in the corrosion rate and its maximum at RH ~ 45%, established in [2], are not associated with a change in the type of hydrated ions.

To establish the reasons for the appearance of the noted features of corrosion under the action of ionizing radiation, the process of chemical corrosion is considered in more detail. The analysis will be carried out on the basis of the Langmuir–Hinshelwood model [2, 10]. The process of corrosion by neutral molecules of oxygen and water vapor in the Langmuir-Hinshelwood model is described as a bimolecular process on the surface:

$${{{\text{O}}}_{2}} + {\text{ S}} \leftrightarrow {{{\text{O}}}_{2}} - {\text{S}},$$

(8)

$${{{\text{H}}}_{2}}{\text{O}} + {\text{S}} \leftrightarrow {{{\text{H}}}_{2}}{\text{O}} - {\text{S}},$$

(9)

where “S” means the surface, followed by a two-step process

$$2{\text{Fe}} \to 2{\text{F}}{{{\text{e}}}^{{2 + }}} + 4{{e}^{ - }},$$

(10)

$${{{\text{O}}}_{2}} + 4{{e}^{ - }} + 2{{{\text{H}}}_{2}}{\text{O}} \to 4{\text{O}}{{{\text{H}}}^{ - }},$$

(11)

which can be written as one process

$$2{\text{Fe}} + {{{\text{O}}}_{2}} + 2{{{\text{H}}}_{2}}{\text{O}} \leftrightarrow 2{\text{F}}{{{\text{e}}}^{{2 + }}} + 4{\text{O}}{{{\text{H}}}^{ - }}.$$

(12)

The overall corrosion equation in the first stage is:

$$\begin{gathered} 2{\text{Fe}} + {{{\text{O}}}_{2}} + 2{{{\text{H}}}_{2}}{\text{O}} \to 2{\text{Fe}}{{\left( {{\text{OH}}} \right)}_{2}} \\ {\text{(white rust)}}. \\ \end{gathered} $$

(13)

Under the influence of atmospheric oxygen, the further oxidation of iron will occur with the formation of iron (III) hydroxide (the second stage) according to the reaction:

$$4{\text{Fe}}{{\left( {{\text{OH}}} \right)}_{2}} + {{{\text{O}}}_{2}} + 2{{{\text{H}}}_{2}}{\text{O}} = 4{\text{Fe}}{{\left( {{\text{OH}}} \right)}_{3}}.$$

(14)

Iron (III) hydroxide is unstable and decomposes with the release of water according to the following reaction:

$${\text{Fe}}{{({\text{OH}})}_{3}} \to {\text{FeOOH}} + {{{\text{H}}}_{2}}{\text{O}},$$

(15)

where FeOOH is the leaf rust, the end product of iron corrosion in a neutral environment (рН 7) [11].

The rate of absorption of oxygen and water molecules by the surface is described by the balance equations [2]:

$$\begin{array}{*{20}{c}} {\frac{{d{{\Theta }_{{{{{\text{O}}}_{2}}}}}}}{{dt}} = {{k}_{{a1}}}\left( {1 - {{\Theta }_{{{{{\text{O}}}_{2}}}}} - {{\Theta }_{{{{{\text{H}}}_{2}}{\text{O}}}}}} \right){{n}_{{{{{\text{O}}}_{2}}}}} - {{k}_{{d1}}}{{\Theta }_{{{{{\text{O}}}_{2}}}}},} \\ {\frac{{d{{\Theta }_{{{{{\text{H}}}_{2}}{\text{O}}}}}}}{{dt}} = {{k}_{{a2}}}\left( {1 - {{\Theta }_{{{{{\text{O}}}_{2}}}}} - {{\Theta }_{{{{{\text{H}}}_{2}}{\text{O}}}}}} \right){{n}_{{{{{\text{H}}}_{2}}{\text{O}}}}} - {{k}_{{d2}}}{{\Theta }_{{{{{\text{H}}}_{2}}{\text{O}}}}},} \end{array}$$

(16)

where k_a1, k_d1 are the rate constants of adhesion and evaporation of oxygen and k_a2, k_d2 of water molecules, respectively, ${{n}_{{{{{\text{O}}}_{2}}}}}$ and ${{n}_{{{{{\text{H}}}_{2}}{\text{O}}}}}$ are the volume number densities of oxygen and water molecules in air, respectively, ${{\Theta }_{{{{{\text{O}}}_{2}}}}}$ and ${{\Theta }_{{{{{\text{H}}}_{2}}{\text{O}}}}}$ are the fraction of surface sites occupied by oxygen and water molecules, respectively. The iron density under normal conditions is ρ_Fe = 7.874 g/cm³, the iron lattice structure is body-centered cubic with a lattice parameter a = 2.866 Å. The density of surface sites n_s depends on the type of lattice face; for estimates, it was assumed that it is determined by the expression:

$${{n}_{s}} = \frac{{{{\rho }_{{{\text{Fe}}}}}a}}{{2{{m}_{{{\text{Fe}}}}}}} = 1.2 \times {{10}^{{15}}}\;{\text{c}}{{{\text{m}}}^{{ - 2}}},$$

(17)

where m_Fe is the mass of iron atoms. When the fluxes of molecules to the surface and from the surface are not very different (i.e., for small values of the accommodation coefficient), the sticking rate constants can be determined using the kinetic flux of molecules to the surface:

$${{k}_{{a1}}} = {{\gamma }_{{{{{\text{O}}}_{2}}}}}\frac{{{{v}_{{th,{{{\text{O}}}_{2}}}}}}}{{4{{n}_{s}}}},\quad {{k}_{{a2}}} = {{\gamma }_{{{{{\text{H}}}_{2}}{\text{O}}}}}\frac{{{{v}_{{th,{{{\text{H}}}_{2}}{\text{O}}}}}}}{{4{{n}_{s}}}},$$

(18)

where ${{\gamma }_{{{{{\text{O}}}_{2}}}}}$ and ${{\gamma }_{{{{{\text{H}}}_{2}}{\text{O}}}}}$ are the accommodation coefficients (probabilities of adhesion) to the iron surface of oxygen molecules and water molecules, respectively, ${{{\text{v}}}_{{th{\text{,}}{{{\text{O}}}_{2}}}}}$ and ${{{\text{v}}}_{{th{\text{,}}{{{\text{H}}}_{2}}{\text{O}}}}}$ are the directed thermal velocities of oxygen and water molecules:

$${{v}_{{th,{{{\text{O}}}_{2}}}}} = \sqrt {\frac{{8{{k}_{B}}T}}{{\pi {{m}_{{{{{\text{O}}}_{2}}}}}}}} ,\quad {{v}_{{th,{{{\text{H}}}_{2}}{\text{O}}}}} = \sqrt {\frac{{8{{k}_{B}}T}}{{\pi {{m}_{{{{{\text{H}}}_{2}}{\text{O}}}}}}}} .$$

(19)

The accommodation coefficient is usually on the order of 10^–3−10^–4. In the stationary state, from (16) we find:

$$\begin{gathered} {{\Theta }_{{{{{\text{O}}}_{2}}}}} = \frac{{{{{{k}_{{a1}}}{{n}_{{{{{\text{O}}}_{2}}}}}} \mathord{\left/ {\vphantom {{{{k}_{{a1}}}{{n}_{{{{{\text{O}}}_{2}}}}}} {{{k}_{{d1}}}}}} \right. \kern-0em} {{{k}_{{d1}}}}}}}{{1 + {{{{k}_{{a1}}}{{n}_{{{{{\text{O}}}_{2}}}}}} \mathord{\left/ {\vphantom {{{{k}_{{a1}}}{{n}_{{{{{\text{O}}}_{2}}}}}} {{{k}_{{d1}}}}}} \right. \kern-0em} {{{k}_{{d1}}}}} + {{{{k}_{{a2}}}{{n}_{{{{{\text{H}}}_{2}}{\text{O}}}}}} \mathord{\left/ {\vphantom {{{{k}_{{a2}}}{{n}_{{{{{\text{H}}}_{2}}{\text{O}}}}}} {{{k}_{{d2}}}}}} \right. \kern-0em} {{{k}_{{d2}}}}}}}, \\ {{\Theta }_{{{{{\text{H}}}_{2}}{\text{O}}}}} = \frac{{{{{{k}_{{a2}}}{{n}_{{{{{\text{H}}}_{2}}{\text{O}}}}}} \mathord{\left/ {\vphantom {{{{k}_{{a2}}}{{n}_{{{{{\text{H}}}_{2}}{\text{O}}}}}} {{{k}_{{d2}}}}}} \right. \kern-0em} {{{k}_{{d2}}}}}}}{{1 + {{{{k}_{{a1}}}{{n}_{{{{{\text{O}}}_{2}}}}}} \mathord{\left/ {\vphantom {{{{k}_{{a1}}}{{n}_{{{{{\text{O}}}_{2}}}}}} {{{k}_{{d1}}}}}} \right. \kern-0em} {{{k}_{{d1}}}}} + {{{{k}_{{a2}}}{{n}_{{{{{\text{H}}}_{2}}{\text{O}}}}}} \mathord{\left/ {\vphantom {{{{k}_{{a2}}}{{n}_{{{{{\text{H}}}_{2}}{\text{O}}}}}} {{{k}_{{d2}}}}}} \right. \kern-0em} {{{k}_{{d2}}}}}}}. \\ \end{gathered} $$

(20)

The rust growth rate R in the process of chemical corrosion is determined by the expression [2, 10]

$$R = k{{\Theta }_{{{{{\text{O}}}_{2}}}}}{{\Theta }_{{{{{\text{H}}}_{2}}{\text{O}}}}} \equiv k\frac{{{{K}_{1}}{{K}_{2}}{{n}_{{{{{\text{O}}}_{2}}}}}{{n}_{{{{{\text{H}}}_{2}}{\text{O}}}}}}}{{{{{\left( {1 + {{K}_{1}}{{n}_{{{{{\text{O}}}_{2}}}}} + {{K}_{2}}{{n}_{{{{{\text{H}}}_{2}}{\text{O}}}}}} \right)}}^{2}}}},$$

(21)

where k is the growth rate constant of the corroded layer, K₁ = k_a1/k_d1, K₂ = k_a2/k_d2. Expression (21) can be reduced to the form:

$$R = \frac{{k{{k}_{{a1}}}{{k}_{{a2}}}{{n}_{{{{{\text{O}}}_{2}}}}}{{n}_{{{{{\text{H}}}_{2}}{\text{O}}}}}}}{{\left( {{{k}_{{d1}}} + {{k}_{{a1}}}{{n}_{{{{{\text{O}}}_{2}}}}} + {{k}_{{a2}}}{{n}_{{{{{\text{H}}}_{2}}{\text{O}}}}}\frac{{{{k}_{{d1}}}}}{{{{k}_{{d2}}}}}} \right)\left( {{{k}_{{d2}}} + {{k}_{{a1}}}{{n}_{{{{{\text{O}}}_{2}}}}}\frac{{{{k}_{{d2}}}}}{{{{k}_{{d1}}}}} + {{k}_{{a2}}}{{n}_{{{{{\text{H}}}_{2}}{\text{O}}}}}} \right)}}.$$

(22)

It can be expected that, under normal conditions, all sites on the iron surface will be occupied by water and oxygen molecules: ${{\Theta }_{{{{{\text{O}}}_{2}}}}} + {{\Theta }_{{{{{\text{H}}}_{2}}{\text{O}}}}} \approx 1$. In this case, the first terms in brackets in the denominator (22) (terms k_d1 and k_d2) can be neglected. Also, for estimates, we can assume that k_a1 ≈ k_a2. Then, to estimate the growth rate of the corrosion layer from (22), using (18) and (19), we obtain:

$$R \approx k\frac{{\gamma {{n}_{{{{{\text{O}}}_{2}}}}}{{n}_{{{{{\text{H}}}_{2}}{\text{O}}}}}}}{{{{{\left( {{{n}_{{{{{\text{O}}}_{2}}}}} + \gamma {{n}_{{{{{\text{H}}}_{2}}{\text{O}}}}}} \right)}}^{2}}}},\quad \gamma = \frac{{{{\gamma }_{{{{{\text{H}}}_{2}}{\text{O}}}}}}}{{{{\gamma }_{{{{{\text{O}}}_{2}}}}}}}\sqrt {\frac{{{{m}_{{{{{\text{O}}}_{2}}}}}}}{{{{m}_{{{{{\text{H}}}_{2}}{\text{O}}}}}}}} .$$

(23)

The corrosion rate under normal conditions varies within 50–200 µm/year [12]. For estimates, we assume that at a relative humidity of 95% (the corrosion growth rate at this humidity is maximum), the growth rate is 200 µm/year. Estimates show that the rate of growth of the corrosion layer in the atmosphere on the relative humidity of the air depends on the value of the parameter γ and this dependence has a maximum at γ ≈ 8, which is quite explainable by the higher accommodation coefficient of water molecules, which have a constant dipole moment, than the accommodation coefficient of oxygen molecules, having only a constant quadrupole moment. In this case, the rate constant turned out to be equal to k = 800 μm/year.

Let us now consider the corrosion in plasma. In this case, the process of cathodic depolarization during electrochemical corrosion of metals in humid air under the influence of an external ionization source can be carried out not only by oxygen molecules, but also by positively charged H₃O⁺(H₂O)_n ions [2]

$${{{\text{H}}}_{3}}{{{\text{O}}}^{ + }}{{({{{\text{H}}}_{2}}{\text{O}})}_{n}} + {{e}^{ - }} \to 1{\text{/}}2{{{\text{H}}}_{2}} + (n + 1){{{\text{H}}}_{2}}{\text{O}}.$$

(24)

If the efficiency of oxygen as a depolarizer is sufficiently low, then the ion will neutralize the negative charge of the electrode with almost unity probability. The flux of positive ions to the surface in the kinetic region is given by

$${{J}_{{k + }}} = \frac{1}{4}{{n}_{ + }}\sqrt {\frac{{8{{k}_{B}}T}}{{\pi {{m}_{ + }}}}} ,$$

(25)

where n₊, m₊ are the number density and mass of positive ions. And in the diffusion region, the ion flux is determined by the expression

$${{J}_{{d + }}} = {{n}_{ + }}\frac{{{{\pi }^{2}}{{D}_{ + }}}}{{2{{R}_{{sh}}}}},$$

(26)

where D₊ is the diffusion coefficient of positive ions, which can be determined from the mobilities given in Table 1 using the Einstein relation: D₊/μ₊ = k_BT/e, R_sh is the characteristic screening length in nonequilibrium plasma [7]. According to the equally accessible surface method [13], in the general case, the ion flux will be determined by the expression:

$${{J}_{ + }} = {{\left( {J_{{k + }}^{{ - 1}} + J_{{d + }}^{{ - 1}}} \right)}^{{ - 1}}} = {{n}_{ + }}{{\left( {\sqrt {\frac{{2\pi {{m}_{ + }}}}{{{{k}_{B}}T}}} + \frac{{2{{R}_{{sh}}}}}{{{{\pi }^{2}}{{D}_{ + }}}}} \right)}^{{ - 1}}}.$$

(27)

Table 1. Parameters for calculating the growth rate of the corrosion layer under the action of positive hydrated ions at the gas ionization rate Q_ion = 2 × 10¹⁶ cm^–3 s^–1

This expression determines the rate of heterogeneous processes involving ions.

The screening length in Eqs. (26) and (27) is 1/k_s (see [7]), the expression for which is generally given in [7]. In the case of one predominant sort of both positive and negative ions, it goes into the following expression:

$${{R}_{{sh}}} = {{\left[ {{{\beta }_{{ii}}}{{n}_{ + }}\left( {\frac{1}{{{{D}_{ + }}}} + \frac{1}{{{{D}_{ - }}}}} \right)} \right]}^{{ - 1/2}}},$$

(28)

where β_ii is the coefficient of ion-ion recombination, D_– is the diffusion coefficient of negative ions.

The growth rate of the corrosion layer under the action of a positive ion flux is ultimately determined by the expression

$${{R}_{ + }} = \frac{1}{2}\frac{{{{\gamma }_{ + }}{{J}_{ + }}a}}{{{{n}_{s}}}} \equiv \frac{1}{2}\frac{{{{\gamma }_{ + }}{{n}_{ + }}a}}{{{{n}_{s}}}}{{\left( {\sqrt {\frac{{2\pi {{m}_{ + }}}}{{{{k}_{B}}T}}} + \frac{{2{{R}_{{sh}}}}}{{{{\pi }^{2}}{{D}_{ + }}}}} \right)}^{{ - 1}}}.$$

(29)

Table 1 shows the parameters used in the calculation of the growth rate of the corrosion layer under the action of negative hydrated ions from expression (29). According to the calculations performed in this section, at a gas ionization rate Q_ion ~ 10¹⁶ cm^–3 s^–1 and a relative humidity RH > 20%, the main positive ions are H₃O⁺(H₂O)₆, H₃O⁺(H₂O)₇ and H₃O⁺(H₂O)₈, and the dominant sort of negative ions are ${\text{O}}_{2}^{ - }$(H₂O)₅ and ${\text{O}}_{2}^{ - }$(H₂O)₉ ions (see Fig. 8). Therefore, the calculations were carried out for H₃O⁺(H₂O)₇ and ${\text{O}}_{2}^{ - }$(H₂O)₉ ions, since both the mobilities (diffusion coefficients) and the ion–ion recombination coefficients for all the named ions are very close.

As can be seen from Table 1, the increase in the mass of the iron plate due to the flow of positive ions in 20 minutes turns out to be an order of magnitude less than those observed in the experiment (see Fig. 5). It should be noted that the above estimates did not take into account the formation of an ambipolar diffusion layer near the plate, since the mobilities of negative and positive ions are close to each other. Near the plate, the rate of electron loss due to escape to it can exceed the rate of loss in the process of three-body attachment to oxygen molecules. According to [14], the condition for the dominance of the electron flux over the flux of negative ions to the wall (this condition is the condition for the formation of an ambipolar diffusion layer, in which the diffusion and drift components of the flux of more mobile charge carriers in plasma are approximately equal to each other) has the form:

$${{Q}_{{{\text{ion}}}}} \gg {{Q}_{{{\text{ion,cr}}}}} = \frac{1}{{{{\beta }_{{ii}}}}}{{\left( {\frac{{{{\mu }_{ - }}{{\nu }_{a}}}}{{{{\mu }_{e}}}}} \right)}^{2}},$$

(30)

where ν_a is the frequency of three-body electron attachment, β_ii, μ₋, μ_e are the mobilities of negative ions and electrons. Calculations with the use of the B-OLSIG+ program [15] with cross sections from [16, 17] showed that the diffusion coefficient of electrons in air at atmospheric pressure and at a reduced electric field strength E/n = 0.1 Td (n is the total number density of air molecules) is equal to D_e = 460 cm²/s (mobility μ_e = 1.43 × 10⁴ cm²/V s). During the characteristic time of the process of three-body attachment of electrons, the main channel of their loss in the bulk plasma, equal to τ_a =1/ν_a = 18 ns (ν_a ≈ 5.5 × 10⁷ s^–1) at atmospheric pressure in air, they manage to cover a distance of ${{R}_{a}} = \sqrt {2D{}_{e}{{\tau }_{a}}}$ ≈ 40 μm, which significantly exceeds both the Debye screening length and the screening length (24). From (30) we find that Q_ion,cr ≈ 10¹³ cm^–3 s^–1, therefore, at Q_ion ≈ 10¹⁶ cm^–3 s^–1 in our experiments, an ambipolar diffusion layer will form near the iron plate under study. In this layer, in which the density of negative charge carriers will be strongly suppressed due to the action of the ambipolar electric field that slows them down, the main channel for the loss of positive ions will be their escape to the wall. Consequently, the positive ion flux will be of the order of Q_ionR_a ≈ 8 × 10¹³ cm^–2 s^–1, which is almost two times higher than the ion flux given in Table 1. This shows the need for more accurate calculations of the ion flux using the balance equations, taking into account the processes of charge exchange and ion conversions.

CONCLUSIONS

In this paper, experimental studies of the corrosion rate of iron samples placed in a chemically active plasma of moist air generated by fast electron beam as a simulator of radiation from the decay radioactive products of spent nuclear fuel, depending on the relative air humidity, are carried out. It was found that radioactive β-radiation (fast electron beam) significantly intensifies iron corrosion in a working gas environment in which oxygen and water vapor is simultaneously present. It has been found that the rate of corrosion under conditions of radioactive irradiation increases sharply when the relative humidity of the air exceeds RH > 10%. The existence of a “critical” air humidity RH ~ 60% for the rate of iron corrosion under atmospheric conditions (in the absence of radiation) is well known [18] and is explained by the change of the corrosion mechanism from chemical to electrochemical during the formation of a continuous film of water on the metal surface. At a relative humidity of RH ~ 10% continuous films of water do not form on the metal surface, therefore, an increase in the corrosion rate under these conditions has a different nature. In [1] it was shown that the effect of fast electrons on metal corrosion processes in a vapor-gas medium occurs through radiolysis of this medium, and not through activation of the metal surface by electrons. The mechanism of this effect remains unclear to date. Numerical calculations of the composition of the radiation plasma in humid atmospheric air performed in this paper indicate that the main ions at a relative humidity RH > 10% are hydrated ion clusters H⁺(H₂O)_n and ${\text{O}}_{2}^{ - }$(H₂O)_n with n = 6−8 for positive and n = 9–12 for negative ions. The close correlation between the behavior of the corrosion rate and the composition of hydrated ions in moist air plasma indicates the decisive role of cluster hydrated ions in heterogeneous iron oxidation processes. Numerical simulation of the corrosion process based on the Langmuir–Hinshelwood model showed that hydrated ions significantly increase the corrosion rate, however, to achieve a quantitative agreement between the simulation results and experimental data, more accurate calculations of ion fluxes are required, taking into account the conversion and charge exchange processes.