Anomalous approach to thermodynamic equilibrium: Structure formation of molecules after vapor deposition

We describe experiments and computer simulations of molecular deposition on a substrate in which the molecules (substituted adenine derivatives) self-assemble into ordered structures. The resulting structures depend strongly on the deposition rate (flux). In particular, there are two competing surface morphologies $(\alpha$ and $\beta )$, which differ by their topology (interdigitated vs lamellar structure). Experimentally, the $\alpha$ phase dominates at both low and high flux, with the $\beta$ phase being most important in the intermediate regime. A similar nonmonotonic behavior is observed on varying the substrate temperature. To understand these effects from a theoretical perspective, a lattice model is devised which reproduces qualitatively the topological features of both phases. Via extensive Monte Carlo studies we can, on the one hand, reproduce the experimental results and, on the other hand, obtain a microscopic understanding of the mechanisms behind this anomalous behavior. The results are discussed in terms of an interplay between kinetic trapping and temporal exploration of configuration space.

Figure 1

The comparison of (a) the substituted adenine molecule A-C22 and (b) a model chain where the head bead is shown in green and the four tail beads in white.

Figure 2

Construction of (a) the interdigitated $\alpha$ and (b) the lamellar $\beta$ phase with the lattice model. The possible bonds are highlighted on top.

Figure 3

The different terms, listed in Table 1: (a) $u_{HH,\alpha }$, (b) $u_{HH,\beta }$, (c) and (d) $u_{TT}$ for different orientations of adjacent chains, (e) and (f) $u_{\widetilde{TT}}$ for different orientations of adjacent chains, and (g) $u_{HT}$.

Figure 4

An example to display the definition of a chain with single bonds [free (red) and nonfree (green)] and two bonds (blue).

Figure 5

Large-scale STM constant-current images of A-C22 physisorbed onto a graphite surface by evaporation ($250 \times$ 250 nm). Fixed evaporation temperature is 383 K. (a) ${\mathrm{T}}_{\mathrm{sub}}=318\mathrm{K},{\mathrm{V}}_{\mathrm{bias}}=-1.170\mathrm{V},{\mathrm{I}}_{\mathrm{set}}=0.100\mathrm{nA}$; (b) ${\mathrm{T}}_{\mathrm{sub}}=333\mathrm{K},{\mathrm{V}}_{\mathrm{bias}}=-0.886\mathrm{V},{\mathrm{I}}_{\mathrm{set}}=0.100\mathrm{nA}$; (c) ${\mathrm{T}}_{\mathrm{sub}}=363\mathrm{K},{\mathrm{V}}_{\mathrm{bias}}=-1.64\mathrm{V},{\mathrm{I}}_{\mathrm{set}}=0.901\mathrm{nA}$.

Figure 6

$N_{\alpha ,\beta }/N_{\mathrm{tot}}$ as a function of substrate temperature and either evaporation temperature (left) or deposition rate (right). Left column: experiment; right column: simulations. (a) Fixed evaporation temperature (i.e., deposition rate) $T_{\mathrm{evap}}=383$ K and (b) fixed deposition rate $F=6.7\times {10}^{-6}$. Fixed substrate temperatures (c) $T=318$ K and (d) $T=0.041$. Somewhat larger substrate temperatures (e) $T=363$ K and (f) $T=0.057$.

Figure 7

$N_{\alpha ,\beta }$ as a function of coverage $\mathrm{\Theta }$ at fixed deposition rate (a) $F=6.7\times {10}^{-6}$ and (b) $F=3.3\times {10}^{-7}$ at temperature $T=0.041$.

Figure 8

Increase of $\mathrm{\Delta }{N}_{2\alpha }$ as a function of $N_{1\alpha }^f$. This demonstrates how the increase of chains with two $\alpha$ bonds depends on the availability of free chains. The solid line represents a least-squares fit of $\mathrm{\Delta }{N}_{2\alpha }\propto {\left[N_{1\alpha }^f\right]}^a$ with $a=0.48$. Furthermore, the running average is displayed.

Figure 9

Two typical configurations during the simulation. Left: $\alpha$-rich domain. The few $\beta$ chains are colored in the darker blue. Right: high-resolution image where the free chains are specifically highlighted (blue: free $\alpha$-chains, green: free $\beta$-chains).

Figure 10

$P_{\alpha ,\beta }^{\mathrm{dis}}$ is shown as a function of coverage at $T=0.041$ and $F=6.7\times {10}^{-6}$.

Figure 11

$N_{\alpha ,\beta }$ is plotted as a function of $\mathrm{\Theta }$ in a double-logarithmic representation. Included is a line of slope 2.

Figure 12

$N_{\alpha }$ is shown as a function of $ln(F\times {\tau }_{2\alpha })$. One can see in a unified way how the fraction of $\alpha$ chains depends on the product of the $\alpha$-cluster lifetime and the flux. Shown are $\mathrm{\Theta }=0.05$ (left) and $\mathrm{\Theta }=0.8$ (right).

Figure 13

(a) ${\mathrm{\Gamma }}_{\alpha }^{\mathrm{eff}}$ and (b) $P_{\alpha }^{\mathrm{dis}}$ as a function of flux at $T=0.041$ and $\mathrm{\Theta }=0.20$. Whereas ${\mathrm{\Gamma }}_{\alpha }^{\mathrm{eff}}/L$ reflects the normalized growth rate of $\alpha$ clusters, $P_{\alpha }^{\mathrm{dis}}$ is the probability that an $\alpha$ chain is free, i.e., can be used as a site for an additional attachment.

Figure 14

$N_{\alpha }$ is shown as a function of $F$ at $T=0.057$ for three different coverages $\mathrm{\Theta }=0.05,\mathrm{\Theta }=0.2$, and $\mathrm{\Theta }=0.8$.

Figure 15

Transition probability from the $\beta$ cluster to the $\alpha$ cluster during the interval of coverage $\mathrm{\Theta }=0.2$ and $\mathrm{\Theta }=0.8$ at two different temperatures $T=0.041$ and $T=0.057$.

Figure 16

Transition probability from $\beta$ to $\alpha$ is shown as a function of $Fexp(-U_{\beta }/T)$.

Figure 17

Sketch of the nonequilibrium behavior in dependence of $T$ and $F$. The solid lines correspond to the two crossover flux values, defined by $Fexp(-U_{\alpha }/T)$ and $Fexp(-U_{\beta }/T)$, respectively, as discussed in the text. The anomalous regime is highlighted. The $T$- and $F$-dependent curves of $N_{\alpha }$, shown on the left and the bottom of the figure, respectively, would have been measured along the broken lines.

Figure 18

Extending the flux range of Fig. 5 to an even higher flux value.