Kinetics and nucleation dynamics in ion-seeded atomic clusters

The time-dependent kinetics of formation and evolution of nanosize atomic clusters is investigated and illustrated with the nucleation dynamics of ion-seed ${\mathrm{Ar}}_n{\mathrm{H}}^{+}$ particles. The rates of growth and degradation of Ar-atomic shells around the seed ion are inferred from molecular dynamics (MD) simulations. Simulations of cluster formation have been performed with accurate quantum-mechanical binary interaction potentials. Both the nonequilibrium and equilibrium growth of ${\mathrm{Ar}}_n{\mathrm{H}}^{+}$ are investigated at different temperature and densities of the atomic gas and seed ions. Formation of ${\mathrm{Ar}}_{n\le 40}$ shells is the main mechanism which regulates the kinetics of nanocluster growth and the diffusive fluctuations of the cluster size distribution. The time evolution of the cluster intrinsic energy and cluster size distributions are analyzed at the nonthermal, quasiequilibrium, and thermal equilibrium stages of ${\mathrm{Ar}}_n{\mathrm{H}}^{+}$ formation. We have determined the self-consistent model parameters for the temporal fluctuations of the cluster size and found coefficients of the diffusive growth mechanism describing the equilibrium distribution of nanoclusters. Nucleation of haze and nanodust particles in astrophysical and atmospheric ionized gases is discussed.

Figure 1

Potential energy curves for the MD simulations. The screened ${\mathrm{H}}^{+}\text{−}{\mathrm{H}}^{+}$ interaction is Debye shielded and modeled as a Yukawa-type potential, due to the electron and proton fields. The Ar-Ar interaction is modeled by a LJ potential and describes the short-range vdW interaction between Ar atoms. The Ar-${\mathrm{H}}^{+}$ interaction asymptotically leads to a $r^{-4}$ polarization potential and is the longest range attraction in our simulation. The repulsion and attraction energies of the Ar-Ar and Ar-${\mathrm{H}}^{+}$ interactions are approximately equalized at $\sim 2.5\AA$.

Figure 2

The energies $u\left(n\right)$ of an Ar particle in the cluster shells, $u\left(n\right)=U\left(n\right)-U(n-1)=dU/dn$, as a function of the number of Ar atoms $n$ in the cluster. The minimal energy required at $T=0$ to remove an Ar atom from the ${\mathrm{Ar}}_n{\mathrm{H}}^{+}$ cluster is $\left|u\right(n\left)\right|$. The sharp increase of the $u\left(n\right)$ function at $n\le 4$ corresponds to construction of the deepest 1T Ar shell. The plateau at $n\ge 5$ shows that shell energies $u\left(n\right)$ of larger clusters are weakly sensitive to the value of cluster size $n$. The energy $\left|u\right(n\left)\right|$ is the maximal energy which can be transferred to the thermostat in ${\mathrm{Ar}}_{n-1}{\mathrm{H}}^{+} +\mathrm{Ar} \to {\mathrm{Ar}}_n{\mathrm{H}}^{+}$ transitions. The irregular oscillations of $u\left(n\right)$, shown in the insert, reflect energy $n$ alternation of Ar atoms in the open shells. The value of these oscillations is reduced at $n\gg 1$.

Figure 3

The time evolution of Ar gas velocity distributions (a) and ${\mathrm{H}}^{+}$ ions (b) during growth of ${\mathrm{Ar}}_n{\mathrm{H}}^{+}$ clusters at 200 K. The Ar atom density is ${10}^{20}{\mathrm{cm}}^{-3}$ with 1000 Ar atoms and 60 ${\mathrm{H}}^{+}$ ions in the simulation box. The small fraction of Ar atoms and ${\mathrm{H}}^{+}$ ions in the high-velocity tail are omitted. The distributions are normalized to the unity. The theoretical Maxwell-Boltzmann velocity distributions are shown for 100 K (dashed line), and 200 K (solid line) together with results of simulations at 3 ns. These figures illustrate the timescale required for the relaxation of velocity distribution functions.

Figure 4

The abundances $N_c(n,t)$ of ${\mathrm{Ar}}_n{\mathrm{H}}^{+}$ clusters with different number ($n$) of Ar atoms ($n\le 5$ in (a), $6\le n\le 9$ and the cumulative abundance of $n\ge 10$ in (b)] as functions of time $t$. Clusters are formed by an ensemble of 200 ${\mathrm{H}}^{+}$ ions embedded into the equilibrium Ar gas at the temperature $T=200$ K and the atom density of ${10}^{20}{\mathrm{cm}}^{-3}$. H atoms have been ionized at the time $t=0$.The inset axes in (a) show the abundances $N_c(n,t)$ of ${\mathrm{Ar}}_n{\mathrm{H}}^{+}$ ($2\le n\le 5$) averaged over the interval $\mathrm{\Delta }t=10\text{ns}$.

Figure 5

MD simulations of number of Ar atoms $n\left(t\right)$ in a single ${\mathrm{Ar}}_{n\left(t\right)}{\mathrm{H}}^{+}$ cluster. The simulation time is $t=100$ ns, the Ar gas temperature is $T=90$ K, and the number of Ar atoms in the simulation box $N={10}^3$. The dashed line is the theoretical curve for an average cluster size $\overline{n}\left(t\right)$ as a function of time. The time delay parameter $t_d=3.2\mathrm{ns}$, and the scaling relaxation time is $t_T=1.4\mathrm{ns}$.

Figure 6

The number of Ar atoms $n\left(t\right)$ in the single $H^{+}$ ion cluster as a function of the time in the nonequilibrium stage of nucleation: (a) Formation of the deep shells in ${\mathrm{Ar}}_n{\mathrm{H}}^{+}$ clusters for the time interval $t=0–2$ ns after creation of the ${\mathrm{H}}^{+}$ ion in the argon bath gas. (b) The intermediate stage of ${\mathrm{Ar}}_n{\mathrm{H}}^{+}$ growth and development of the thermal fluctuations of $n\left(t\right)$ inside the time interval $0<t<$ 10 ns. The gas temperature and density are $T=90$ K and ${\mathrm{n}}_g={10}^{20}{\mathrm{cm}}^{-3}$, respectively. Theoretical curve is computed using thermodynamic formula for the two-state approximation given by Eq. (2) with the kinetic parameters $t_d\simeq 3.2\mathrm{ns}, t_T=1.4\mathrm{ns}$, and $n_c=21.6$.

Figure 7

(a) The number of Ar atoms $n_1\left(t\right)$ and $n_2\left(t\right)$ in two independent clusters, 1 and 2, are shown as functions of the time $t$ under the thermal equilibrium conditions inside the time interval $t=40–44$ ns. The gas temperature and density are $T=90$ K and ${\mathrm{n}}_g={10}^{20}{\mathrm{cm}}^{-3}$, respectively. (b) The timescale filtering of fluctuations are shown with the smoothed curves shown for cluster 2 from (a). The averaging time for the Gaussian filters are $2\sigma =40\mathrm{ps}$, 0.2 ns, and 1 ns for solid, dot-dashed, and dashed curves, respectively.

Figure 8

The empirical probability $P(n,t,\tau ,T)$ to find the ${\mathrm{Ar}}_n{\mathrm{H}}^{+}$ clusters with $n$ Ar atoms in the cluster shells. The total simulation time $t=100$ ns for a single ${\mathrm{H}}^{+}$ ion at $T=90$ K. The curves A, B, and C are shown at different times: curve A, the earlier stage of cluster formation $0\le t\le 20\mathrm{ns}$; curve B, just after thermalization $20\le t\le 60\mathrm{ns}$; and curve C, $60\le t\le 100\mathrm{ns}$ at the end of the MD simulation process. The averaging interval for the later distributions is $\tau =40\mathrm{ns}$, and for the early stage A is 20 ns. The 0–20 ns time interval includes essentially nonequilibrium stage of the cluster nucleation and this causes a visible fraction of small clusters in the empirical probability given by curve A.

Figure 9

The probability distribution function $P(n,N_{{\text{H}}^{+}})$ to find ${\mathrm{Ar}}_n{\mathrm{H}}^{+}$ clusters with $n$ Ar atoms in the cluster shells as a function the number of argon atoms $n$. ${\mathrm{Ar}}_n{\mathrm{H}}^{+}$ clusters are formed by the initial ensemble of ${\mathrm{N}}_{{\text{H}}^{+}}$ ions. The total simulation time is $t=100$ ns. The gas temperature $T=90$ K, with either ${\mathrm{N}}_{{\text{H}}^{+}}=1$ (circles) or ${\mathrm{N}}_{{\text{H}}^{+}}=20$ (triangles) ${\mathrm{H}}^{+}$ ions, all at Ar density ${10}^{20}{\mathrm{cm}}^{-3}$. Data are averaged over the time interval 60–100 ns; error bars for the 20 ${\mathrm{H}}^{+}$ ion curve are on the order of the marker size. The introduction of more ions leads to a reduction of the mean cluster size compare with the independent proton growth.

Figure 10

The thermal equilibrium probability $P(n,N_{{\text{H}}^{+}},T)$ to detect the ${\mathrm{Ar}}_n{\mathrm{H}}^{+}$ clusters with the $n$ Ar atoms for the ensembles of ${\mathrm{N}}_{{\text{H}}^{+}}=1$ and 200 ${\mathrm{H}}^{+}$ ions. The results of the independent ${\mathrm{H}}^{+}$ ion growth are show for $T=200$ K (hexagons, left vertical axis) and $T=90$ K (circles, right vertical axis). The cluster size distribution $P(n,N_{{\text{H}}^{+}}=200,T=200\text{K})$ for the averaged value over the ensemble of 200 ${\mathrm{H}}^{+}$ ions at the single snapshot at $t=80\mathrm{ns}$ are shown (squares, left vertical axis) with corresponding error bars. The triangles (left vertical axis) show results of $P(n,N_{{\text{H}}^{+}}=200,T=200\text{K})$ obtained with time and ensemble averaging. The error bars for this case is smaller than the marker size.