Ab initio thermodynamics study of ambient gases reacting with amorphous carbon

Amorphous carbon (a-C) occurs as a tribologically induced phase in diamond or diamond-like carbon coatings. The interaction of ambient gases (${\mathrm{H}}_2$, ${\mathrm{N}}_2$, ${\mathrm{O}}_2$, ${\mathrm{H}}_2\mathrm{O}$, ${\mathrm{CO}}_2$) with a-C of varying mass density is studied by means of ab initio thermodynamics simulations. Different scenarios such as moist air or pure gases are investigated under different pressure and temperature conditions. Equilibrium concentrations of chemisorbed and fragmented final states are found to only exhibit a minor dependence on the specific conditions of the gas phase reservoir. The differences in local structure of the a-C samples with varying mass density such as pore size and coordination and dangling bonds, as well as the competition among different gas molecules for a-C atoms as reactants, affect the equilibrium concentrations to a greater degree. The availability and reactivity of a-C atoms are thus found to mainly control the chemical composition of a-C interacting with ambient gases in thermodynamic equilibrium. Trends found in the analysis of chemical groups occurring in equilibrium can possibly be transferred to a more realistic nonequilibrium tribological scenario.

Figure 1

Mass density for the a-${\mathrm{C}}_{128}$ samples after quenching and relaxing with the Rebo2Scr potential and further relaxing with DFT as a function of initial mass density. ${\rho }_{\text{in}}$ denotes the initial and $\rho$ the final mass density after quenching and relaxation with the classical force field Rebo2Scr (orange squares) as well as after further relaxation using DFT (green circles). Error bars (black vertical lines) indicate the absolute error of $\rho \left(\left\{V_i\right\}\right)$ assuming that the absolute error of $V_i$ is the sample standard deviation of $\left\{V_i\right\}$.

Figure 2

Coordination ratio for 2, 3, and 4 nearest neighbors (NN) of the Rebo2Scr-relaxed and DFT-relaxed samples as a function of the final mass density $\rho$ for different particle numbers. Error bars for $\rho$ as in Fig. 1. Error bars for ${\alpha }_{\text{c}}$ as explained in the main text.

Figure 3

Radius of the largest test sphere fitting into the a-C as a function of mass density. Horizontal lines show the radii $R_{\text{M}}$ of the molecules.

Figure 4

Histogram of the absolute values of the local magnetic moments of the carbon atoms in the a-C samples from the DFT simulations. Shaded region: Carbon atoms qualified as having a dangling bond. White arrow: Count exceeds visible region of plot.

Figure 5

Schematic illustration of the transfer of a molecule M from the gas phase into an a-C pore undergoing fragmentation and adsorption resulting in the product X in the $NpT$ ensemble. Here, (g) denotes gas phase and (s) solid state species.

Figure 6

(a) Histogram of absolute contribution $C\left(\mathrm{\Delta }{E}_{\text{DFT}}\right)$ to the Gibbs free energy (see main text). (b) Same for $C\left(\mathrm{\Delta }{G}_{\text{r}}\right)$. (c) Same for $C\left(p\mathrm{\Delta }V\right)$. (d) Histogram of Gibbs free energy values.

Figure 7

(a) Histogram of DFT energy difference $\mathrm{\Delta }{E}_{\text{DFT}}^{\text{initial}}$ before optimizing the X structures. (b) Histogram of DFT energy difference $\mathrm{\Delta }{E}_{\text{DFT}}$ after local geometry optimization.

Figure 8

Equilibrium number concentrations $c_{\text{X}}^{\text{eq}}\left(\text{M}\right)$ of the chemisorbed or fragmented molecules for moist air at $p={10}^5$ Pa as a function of temperature. (a)–(d) Representative samples 1 to 4 with different mass densities.

Figure 9

Same as Figs. 8–8 except that concentrations are given as number per a-C atom. 100% corresponds to one molecule per a-C atom in the sample.

Figure 10

Same as Figs. 9–9 except that the partial pressures are all set to $\frac{p}{5}$.

Figure 11

Same as Figs. 9–9 except for pure gases (partial pressures are set subsequently to $p_{\text{M}}=p$ for all molecules).

Figure 12

Upper limits ($\frac{\mathrm{\Delta }{G}_i}{k_{\text{B}}T}=-\infty$) for the dimensionless concentrations of pure gases as a function of a-C mass density.

Figure 13

Dimensionless equilibrium concentrations of the chemisorbed or fragmented molecules for moist air and pure water at $T=400$ K as a function of total pressure $p$ (logarithmic pressure scale). (a)–(d) Representative samples 1 to 4 with different mass densities. Dashed vertical line: $p={10}^5$ Pa.

Figure 14

Occurrence $[{\tilde{c}}_{\text{C}}\left(K\right)$, hollow bars] and occupation $[{\tilde{c}}_{\text{C}\text{occ}}^{\text{eq}}\left(K\right)$, left solid bars: air, right solid bars: pure water] of the a-C atoms in the four representative samples. $\mathrm{c}n$: $n$-fold-coordinated a-C atom. ${}^{*}$: with dangling bond. 100% corresponds to all atoms in the sample. Coordination and dangling bonds refer to pristine a-C before the reaction with gas molecules.

Figure 15

Same as Fig. 14 except that the relative occupations ${\tilde{c}}_{\text{C}\text{occ}}^{\text{eq}}\left(K\right)/{\tilde{c}}_{\text{C}}\left(K\right)$ are plotted. Data points with ${\tilde{c}}_{\text{C}}\left(K\right)=0$ are omitted (no label for the atom kind is drawn). Hollow bars are just a guide to the eye to discriminate air and pure water data points. 100% corresponds to all available atoms of a certain kind being occupied in equilibrium.

Figure 16

Hollow bars: Dimensionless equilibrium concentrations of chemical groups for the four representative a-C samples in contact with air at $T=300$ K and $p={10}^5$ Pa. Filled bars: Contributions of different reactant combinations leading to the same group. Color by molecule: Gray: ${\mathrm{H}}_2$, red: ${\mathrm{O}}_2$, blue: ${\mathrm{N}}_2$. 100% correspond to one group per a-C atom.

Figure 17

Same as Fig. 16 but for pure water.