Graphoepitaxy for Pattern Multiplication of Nanoparticle Monolayers

We compute the free energy minimizing structures of particle monolayers in the presence of enthalpic barriers of a finite height $\beta V_{\mathrm{ext}}$ using classical density functional theory and Monte Carlo simulations. We show that a periodic square template with dimensions up to at least 10 times the particle diameter disrupts the formation of the entropically favored hexagonally close-packed 2D lattice in favor of a square lattice. The results illustrate how graphoepitaxy can successfully order nanoparticulate films into desired patterns many times smaller than those of the prepatterned template.

Figure 1

Monolayers assembled via deposition onto a substrate with repulsive barrier templates separated by a distance commensurate with the square lattice unit cell. The template in (a) trivially fixes each particle in its ideal location. The template in (b) uses sparser templating to direct assembly into the same structure.

Figure 2

Equilibrium density profiles with $\beta (\mu -{\mu }_{\text{hcp}})=2.40$ predicted using FMT. The external field is as illustrated in Fig. 1, with barriers of height $\beta V_{\mathrm{ext}}=(\mathrm{a})$ 0, (b) 2, (c) 5, and (d) 10, each separated by a distance of $10R$.

Figure 3

Equilibrium density profiles predicted by FMT as a function of template feature spacing for $\beta (\mu -{\mu }_{\text{hcp}})=2.40$. Enthalpic barriers of height $\beta V_{\mathrm{ext}}=10$ are placed at each border of the displayed region. Template feature separation for each profile is (a) $8R$, (b) $12R$, (c) $16R$, (d) $20R$, (e) $24R$, and (f) $28R$.

Figure 4

(a) Fraction of microstates with $N_s$ particles as a function of chemical potential $\beta (\mu -{\mu }_{\text{hcp}})$. Here, a microstate is defined by a single periodic $10R\times 10R$ square bounded by the external field from Fig. 2. Configurations shown in (b) and (c) are snapshots taken from the GCMC simulations where $\beta (\mu -{\mu }_{\text{hcp}})=1.87$ and 5.87, respectively. Lines in plots (b) and (c) represent thin barriers of height $\beta V_{\mathrm{ext}}=10$.