2.1 The construction of 2·1/4 Cr1Mo micro model

The first-principles calculations were performed using DFT as implemented in VASP code and using the projector augmented wave (PAW) pseudo-potential ^[19] to describe the electron-ion interaction. The electron exchange and correlation energy is mostly treated in the generalized gradient approximation (GGA) parameterized by Perdew, Burke and Enzelhof ^[20].

To perform the first-principle calculation, the micro-structure of 2·1/4Cr1Mo was determined in the first step. There are many kinds of alloy steel based on different dopants. These alloys have many different lattice structures. The 2·1/4Cr1Mo used in HTR-PM is a face-centered cubic structure. The strong interaction between the two adjacent layers of metal crystal and the multi-layer structure should be considered. According to the manufacturing data, the chrome-molybdenum steel is composed of Fe, Cr and Mo with a weight fraction of 95.81%, 2.24% and 0.95%, respectively ^[21]. The numerical ratio of these three elements is about 171:4.3:1. The Cr and Mo have the similar chemical properties ^[22], and they have nearly the same effect as the doping element in our study. These two elements could be unified and simplified with a ratio of iron and chromium of 33:1. This ratio is similar to the replacement of one iron atom with a chromium atom at each 3 × 3 × 1 ideal primitive cells of face-centered cubic iron. This significantly simplifies the calculation. Finally, the super-cells consisting of an adatom on the 4 layers of Cr doped body centered cubic iron with 3 × 3 × 2 primitive cells (Fig. 1) are employed to calculate the adsorption energy using a κ-point of 4 × 4 × 1. The vacuum thickness is set to 10.0 Å, and the kinetic energy cut-off is set to 500 eV ^[19].

Fig. 1.Full-screen View

The crystal structure of Cr doped body centered cubic iron: (a) ideal primitive cell, (b) ideal 3 × 3 × 1 primitive cells, and (c) one Cr doped 3 × 3 × 1 primitive cells.

In contrast to pure substrates, an adatom on the Cr doped face-centered cubic iron has more adsorptive sites. As shown in Fig. 2, there are five and three adsorptive sites for Cr atoms in the top and bottom layers, respectively. In this work, all eight adsorptive sites are considered to obtain more comprehensive information about the different adsorption geometries that affect the adsorption energy and adsorption properties.

Fig. 2.Full-screen View

The adsorptive sites: (a) Cr at the top layer and (b) Cr at the bottom layer.

2.2 The theory of adsorption energy, Fermi energy, CDD and DOS in the system of fission products and 2·1/4 Cr1Mo

The adsorption energy is defined as

where $E_{\text{ad}}$, $E_{\text{metal}}$, $E_{\text{atom}}$, and $E_{\text{atom-metal}}$ are the energies of the adsorption, metal, a single atom, and the adatom-metal system, respectively. The numerical values of $E_{\text{metal}}$, $E_{\text{atom}}$, and $E_{\text{atom-metal}}$ can be determined by the first-principle calculations directly, and $E_{\text{ad}}$ is obtained from Eq. (1). This is one of the most important parameters in determining the adsorption rate.

The Fermi level $E_{\text{F}}$ is usually defined as the position of highest energy that is filled with the free electrons at 0 K. For example, for the isolated I atom, the position of the $5p$ orbit will be the Fermi level. When the atom is adsorbed on the metal surface, the $E_{\text{F}}$ will have a shift for both adatom and substrate.

The charge density difference (CDD) is one of the most important parameters in the calculation because it can display the visual image of electron transfer that enables us to see the interaction between nuclide and substrate intuitively. Another important parameter is density of state (DOS), which is used to calculate the distributions of electrons in each energy level. Detailed information of electron transfer will be seen in the DOS analysis. Both of these methods are used here.

Of note, even the energies in Eq. (1) are calculated at 0 K because the energy of an electron is insensitive to temperature < 10³ K ^[23], and the 2·1/4 Cr1Mo is mainly used for preheating, evaporating and low overheating section. This ensures that the temperature is below 10³ K ^[24], and the results can be used for real scenarios in HTR-PM.

2.3 The calculation methods of adsorption rate with equilibrium statistical physics

In the primary circuit, the radioactive fission products used a coolant gas from the fuel elements. Some of them will be adsorbed on the metal. For a normal operating reactor, the adsorbed and separate nuclide atoms will have a dynamic equilibrium system. In following the literature approach ^{[24, 25]}, the grand canonical ensemble model has described this system and evaluated the balanced adsorption rate of the nuclide on the metal. In this model, the metal-adsorbing nuclide atoms are considered to be an open system, and the separate nuclide atoms in the coolant gas are treated as a particle and heat source. We define $\overline{N}$ and $N_0$ to represent the number of adsorbed nuclide atoms and the total adsorptive centers, respectively, and the relationship between$\overline{N}$ and $N_0$ is given by the theory of the statistic physics as follows:

Here,$\mu$ is the chemical potential of nuclide atoms, $K_{\text{b}}$ is the Boltzmann’s constant and $T$ is the temperature of the circumstance. The adsorption rate $\theta$ is determined by $\overline{N}/N_0$.

Because the density of fission products is very small (less than 10¹² m^-3 in HTR-PM^[24]), the ideal gas model can be used to deal with the nuclide gas source. When the reactor achieves an equilibrium state, the chemical potentials of the nuclide atoms in the open system are identical to those of the gas source. According to the theory of the ideal gas model, the chemical potential of the nuclide can be written as follows,

Here, $n$ is the concentration of the nuclide in the primary circuit, $h$ is the Planck constant, and $m$ is the atomic mass of the nuclide atom.

By multiplying Eq. (2) and Eq. (3), one can easily obtain the adsorption rate,

Eq.(4) shows that the adsorption rate depends on tree parameters of $n$,$T$ and $E_{ad}$. The value of $n$ and $T$ in HTR-PM can be determined by either experimental measurements or reactor physics calculations, while $E_{\text{ad}}$ can be obtained through the first-principle calculations using Eq. (1). This grand canonical ensemble model provides a new way to understand the microscopic level behavior of nuclides adsorbed on the substrate.

3.1 Adsorption energy

The variations in the adsorption energy with different adsorptive sites were first studied. This was used to obtain the most stable sites. The adsorption energies at different adsorptive sites are listed in Table 1 and Table 2. This shows that the adsorption energies of the hollow site are significantly higher than other sites regardless of the Cr atom on the top or bottom layer. Therefore the most stable adsorption sites are a hollow site for all four adatoms.

In this result, it is necessary to highlight that the negative values appear in the adsorption energy indicating that the adsorption behavior for this site is physical adsorption or non-absorption. This point will be more clearly seen in the discussion of adsorption rate below. The appearance of the large negative numbers on the top site is likely because these adsorptive sites are quite unstable for the adatom-metal system ^[26]. Table 1 and Table 2 indicate that the adsorption behavior of cesium and strontium tends to be physical adsorption or non-adsorption, and the adsorption of silver and iodine is chemisorption, qualitatively.

3.2 Analysis of electronic structure

Next, we studied the electronic structure of the adatom-metal system to identify the underlying physical nature of the adsorptive behavior. The charge density difference of the adatom-metal system is shown in Fig. 3 and Fig. 4 using the VESTA software ^[27]. Note that the yellow areas represent the increases in charge density, and the blue areas represent the decreases in charge density. With Cr at the top layer, there is no change in electronic cloud between Cs and substrate. The same image appears for Sr. This indicates that Cs and Sr cannot be directly adsorbed on the metal.

Fig. 3.Full-screen View

The charge density difference with Cr at the top layer: (a) Cs, (b) Sr, (c) Ag, and (d) I.

Fig. 4.Full-screen View

The charge density difference with Cr at the bottom layer: (a) Cs, (b) Sr, (c) Ag, and (d) I.

For Ag, there is slight electron transfer between the adatom and substrate, and this is considered to be physical adsorption. I has a strong electron transfer from the metal to the iodine atom, which indicated that the adsorption between I and the metal is chemisorption. For the bottom layer of Cr (Fig. 4), the results are somewhat different from the previous data. Sr has physical adsorption, and the case three other nuclides are the same.

The DOS should be calculated to discuss details of electron transfer. Figures 5, 6, 7, 8 and 9 show the DOS for single adatom, metal substrate, and the adatom-metal system. Because there is no adsorption interaction between Cs and metal substrate for both top and bottom Cr, the DOS analysis will not contain Cs. The abscissa values in the figures are represented by the energies of states relative to the Fermi level, and this is applied to all Figs. 5, 6, 7, 8 and 9.

Fig. 5.Full-screen View

A comparison of DOS with top site Cr: (a) for the I adsorption system, the isolated metal substrate and the I adatom on the metal, and (b) between the isolated I atom and I adatom on the metal. The energy is relative to the Fermi level E_F, which is indicated by the dashed line at E=0 eV.

Fig. 6.Full-screen View

A comparison of DOS with bottom site Cr: (a) for the I adsorption system, the isolated metal substrate and the isolated I atom, and (b) between the isolated I atom and I adatom on the metal.

Fig. 7.Full-screen View

A comparison of DOS with top site Cr: (a) for the Ag adsorption system, the isolated metal substrate and the Ag adatom on the metal, and (b) between the isolated Ag atom and Ag adatom on the metal.

Fig. 8.Full-screen View

A comparison of DOS with bottom site Cr: (c) for the Ag adsorption system, the isolated metal substrate and the isolated Ag atom, and (d) between the isolated Ag atom and Ag adatom on the metal.

Fig. 9.Full-screen View

A comparison of DOS with bottom site Cr: (a) for the Sr adsorption system, the isolated metal substrate and the Sr adatom on the metal, and (b) between the isolated Sr atom and Sr adatom on the metal.

Figure 5 gives the total DOS for the isolated I atom, the isolated metal substrate, the I adatom-metal system, and a comparison of the partial DOS of the I atom between an isolated atom and I on the metal surface. Fig. 5 (b) that for an isolated I atom, the peak of the $5s$ state is below$E_{\text{F}}$, and the $5p$ state lies in$E_{\text{F}}$, which is the same as the electron configuration of $[\text{Kr}]4d^{10}5s^{2}5p^7$. When the I atom is adsorbed on the metal surface, the densities of the $5s$ and $5p$ states have been broadened due to adsorption interaction, and the $5p$ state shifts left toward less energy than$E_F$. This indicates that I atoms gained partial electrons from the metal and their energy could be reduced. These partial metal electrons occupy the $5p$ state of I, and this reflects the resonance between the $5p$ state of I atom and metal surface. In other words, the I and the metal formed a chemical bond through orbital hybridization, and thus the adsorptive behavior between I and metal is chemisorption.

Figure 5 (a) also shows that the total DOS of the I-metal system shifts right versus the isolated metal substrate. The appearance of more states above $E_{\text{F}}$ suggests that the metal loses its partial electrons to form chemical bonds. On the other hand, an electrochemical view is that that the electronegativity of I is much larger than Fe and Cr. This causes I to gain partial electrons from metal when they interact with each other. For the bottom layer of Cr, the results are similar to Fig. 5 (a) and Fig. 5 (b) but with a more significant right shift of total I-metal system. Fig. 6(a) and Fig. 6(b) present the case for bottom Cr, and the results are similar to Fig. 5(a) and Fig. 5(b) with a more significant right shift of total I-metal system DOS.

The DOS for Ag and Sr are also shown in Fig. 7, Fig. 8 and Fig. 9. There is no obvious change in total DOS of adatom-metal system, and we conclude that the adsorptive behavior of these two nuclides is both physisorption. However, Fig. 6 (b) shows that the $5p$ state of Ag broadens toward low energy. For Sr, Fig. 7 (b) indicates that few of new states appear above the Fermi level when adatom due to adsorptive interactions between adatoms and metal substrate. These results reveal that there is a weak interaction between adatoms and metal. We also analyzed these phenomena concerning different electronegativity values. As we know, the Ag atom with an electron configuration of $[\text{Kr}]4d^{10}5s^1$ is stable, and its electronegativity is larger than Fe and Cr. This leads the Ag atoms to gain partial electrons from metal when they interact. In contract, the electronegativity of Sr is smaller than Fe and Cr, and the Sr atoms will share their outermost electrons during the interaction.

We note that the electronic structure of Cs, Sr, Ag and I adsorbed on the metal substrate are quite different from when these nuclides are adsorbed on graphite. The Cs and Sr have positive adsorption energies that are larger than Ag and I on graphite ^[25,28-30]. These results for the two substrates illustrates that in HTR-PM, the main solid radioactive fission productions Cs, Sr, Ag and I have different adsorption and deposition behavior. Elements with strong metallic character like Cs and Sr tend to be adsorbed on the graphite first, this is then deposited on the metal together. However, elements with strong nonmetallic character like Ag and I prefer to be adsorbed on the metal directly.

3.3 Adsorption rate

The adsorption rate is determined by $n$, $T$ and $E_{\text{ad}}$ as shown in Eq. (4). As an example, the density of nuclide has been set to be $n=1\times {10}^{12}{\text{m}}^{-3}$^[24]. The adsorption energy $E_{ad}$ is determined by the above first-principle calculations with the bottom site chromium shown in Table 2 to investigate the variation trend of adsorption rate with $T$. It is clear that if the value of $E_{ad}$ is positive, then the adsorption rate of the nuclide will decreases exponentially with increasing temperature. On the other hand, when the value of $E_{\text{ad}}$ is negative, the adsorption rate will always close to zero regardless of temperature. This is seen in the fact that the nuclide atom could not be adsorbed on the substrate with negative adsorption energy. Therefore only I, Ag and Sr’s curves appear on Fig. 10.

Fig. 10.Full-screen View

Variation of adsorption rate with temperature for I, Ag and Sr nuclides.

Figure 10 shows that for the nuclide density of $n=1\times {10}^{12}{\text{m}}^{-3}$, the adsorption rate decreases dramatically when the temperature is close to 800 K, 300 K and 70 K for I, Ag and Sr, respectively. This indicates that when the temperature is higher than the corresponding value, the state of nuclide on the metal surface will have a transition from adsorption to desorption.

In addition, the real scenarios of HTR-PM are also been calculated here. The parameters and results are shown in Table 3, and the densities of the four nuclides are all less than $1\times {10}^{12}{\text{m}}^{-3}$. The temperature range is about 500-1000 K. Table 3 shows that only I has a significant adsorption rate for the normal operating temperature in HTR-PM; Ag has a small adsorption rate at the cold end of HTR-PM, and Cs and Sr cannot be adsorbed on the metal directly.

The total amount of nuclides adsorbed on the metal in the first primary circuit should be calculated to evaluate the release amount of radioactive nuclides in the supposed accident of HTR-PM for the nuclear safety evaluation ^[24]. This can be processed by calculating the total amount $\overline{N}$of adsorbed radioactive nuclides through the sum of contributions from all parts in the first primary circuit according to temperature profile, nuclide concentration and the metal surface area. When$\overline{N}=N_0\times \theta$, the present results of adsorption rate can be used to determine the amount of adsorbed radioactive nuclides for nuclear safety evaluation of HTR-PM.