Statistical mechanics of fullerene coalescence growth

Among the different carbon allotropes fullerenes are exceptionally intriguing for their spheroidal topology out of pentagons and hexagons. However, the dominant formation mode is still ambiguous. Here, we analyze the fullerene formation process by the statistical analysis of fullerene sizes produced in a laser-induced microplasma finding that a simple two-parameter lognormal distribution describes impressively well the cluster frequencies under various conditions. Our findings clearly reveal coalescent growth following a classical collision dynamics and disagree with several earlier assumptions.

Figure 1

Mass spectrum of positive fullerene ions. The mass spectrum shows a bimodal structure with a gap around $400u$ which separates smallest carbon clusters from the fullerenes. Above the gap the “magic” fullerenes ${\mathrm{C}}_{50}$, ${\mathrm{C}}_{60}$, and ${\mathrm{C}}_{70}$ (Ref. 3) stand out by their overabundance compared to the residual fullerene distribution. The intersection in the gap was chosen for a better comparability of the different ion yields in both mass ranges.

Figure 2

(Color online) Statistical analysis of a fullerene mass spectrum. The normalized histogram in (a) shows the frequency of each fullerene size $\tilde{N}$ in Fig. 1 (see text for definition). The excess abundance of the “magic” fullerenes ${\mathrm{C}}_{50}$, ${\mathrm{C}}_{60}$, and ${\mathrm{C}}_{70}$ is thereby removed as indicated by the vertical intersections. A lognormal fitting curve is applied to the histogram data, and the values of 16%, 50%, and 84% of the integral area are marked by dotted lines. The 50% value directly translates into the median fullerene size $⟨\tilde{N}⟩$, which is connected to the alternative parameter $\mu$. The lognormal statistics is emphasized by the linearization given in (b). Here, the logarithm of the cluster size $\tilde{N}$ is shown against the cumulative fullerene frequency. The slope in this representation directly provides the shape parameter $\sigma$. We obtain $\mu =4.18$ (corresponding to $⟨\tilde{N}⟩=65.5$), respectively, and a shape parameter of $\sigma =0.43$.

Figure 3

(Color online) Correlation between fullerene ion yields and lognormal distribution parameters. (a) Fullerene ion yields obtained in the laser fluence regime below $1\mathrm{J}∕{\mathrm{cm}}^2$. Beyond $0.2\mathrm{J}∕{\mathrm{cm}}^2$ first a moderate fullerene ion yield is observed in the mass spectrum (1), which abruptly increases above a threshold near $0.8\mathrm{J}∕{\mathrm{cm}}^2$ (2). The appearance of the threshold is also reflected within the cumulative probability plots in (b) by a rotation of the straight line around the lowest fullerene size.

Figure 4

(Color online) Temporal evolution of the fullerene growth process as observed from the size statistics. The histograms show fullerene size distributions obtained (a) $300\mathrm{ns}$ and (c) $900\mathrm{ns}$ after the ablation laser pulses. With increasing growth duration the distributions are shifted to higher cluster sizes. Simultaneously, the distribution asymmetry decreases as it is recognized from the shape of the lognormal fitting curves and from the corresponding straight lines in the cumulative plots (insets) that rotate about the highest cluster size.