Self-assembly of a drop pattern from a two-dimensional grid of nanometric metallic filaments

We report experiments, modeling, and numerical simulations of the self–assembly of particle patterns obtained from a nanometric metallic square grid. Initially, nickel filaments of rectangular cross section are patterned on a ${\mathrm{SiO}}_2$ flat surface, and then they are melted by laser irradiation with $\sim 18$-ns pulses. During this time, the liquefied metal dewets the substrate, leading to a linear array of drops along each side of the squares. The experimental data provide a series of SEM images of the resultant morphology as a function of the number of laser pulses or cumulative liquid lifetime. These data are analyzed in terms of fluid mechanical models that account for mass conservation and consider flow evolution with the aim to predict the final number of drops resulting from each side of the square. The aspect ratio, $\delta$, between the square sides' lengths and their widths is an essential parameter of the problem. Our models allow us to predict the $\delta$ intervals within which a certain final number of drops are expected. The comparison with experimental data shows a good agreement with the model that explicitly considers the Stokes flow developed in the filaments neck region that lead to breakup points. Also, numerical simulations that solve the Navier-Stokes equations along with slip boundary condition at the contact lines are implemented to describe the dynamics of the problem.

Figure 1

(a) Initial square grid of Ni strips with rectangular cross section. The as-deposited metal thickness is $h_g=(10\pm 1)$ nm. The inner square sides are $L_g=1587$ nm long. (b) After the first laser pulse, the strips decrease their width by dewetting and the ends detach from the vertices, leaving drops there. (c) Drops pattern resulting from the breakup of the filaments after five pulses.

Figure 2

Drop patterns for $h_g=10$ nm and $w_g=162$ nm after five pulses for different initial lengths: (a) $L_g=1387$ nm ($\delta =17.21$), (b) $L_g=987$ nm ($\delta =12.25$), and (c) $L_g=606$ nm ($\delta =7.52$).

Figure 3

Number of filaments (count) that yield a certain number of drops as the thickness, $h_g$, and the aspect ratio, $\delta =L_g/w$ are varied: (a) $h_g=5$ nm, (b) $h_g=10$ nm, and (c) $h_g=20$ nm.

Figure 4

Symbols correspond to experiments carried out with different values of thickness, $h_g$, and aspect ratio, $\delta =L_g/w$. The experiments are grouped according to the modal number of drops, $n$, as determined from the histograms in Fig. 3. The error bars correspond to the uncertainty in $w$, see Eq. (2). The horizontal dashed lines are the predictions obtained from LSA, showing (dimensionless) wavelength of maximum growth, ${\lambda }_m/w\left(h_g\right)$, see Eq. (5). The dot-dashed curves are obtained by the MCM, see Eq. (13). The solid horizontal lines correspond to the FDM, see Eq. (17).

Figure 5

Scheme illustrating the time evolution of a portion of square grid. See the text for the discussion of the various stages of breakup process.

Figure 6

(a) Sketch of the head and bridge regions showing the parameters used in the model. (b) Sketch of the longitudinal section showing the interpretation of both roots for $L_b$, namely $L_s$ and $L_l$.

Figure 7

Time evolution of the fluid thickness for the parameters of Fig. 1 using $\ell =20$ nm: (a) $t=40$ ($t_c=8$ ns), (b) $t=80$ ($t_c=16$ ns), and (c) $t=105$ ($t_c=21$ ns). Here we have $w=80$ nm and $L_{\mathrm{cyl}}=1749$ nm. The lengths are in units of $w$.

Figure 8

Thickness, $h$, along a filament ($x$ coordinate is defined along the symmetry line of a filament), at different times for the grid shown in Fig. 7 with $\ell =20$ nm: (a) $t=0$ and 1 ns, (b) $t=8$ ns, (c) $t=16$ ns, (d) $t=21$ ns. Note the corner drops at $x=0$ and $x/w=L_{\mathrm{cyl}}=21.81$. The arrow indicates the position of maximum thickness at the bulk, $x_h\left(t\right)$, further discussed in the text.

Figure 9

Final numerical drop patterns for the experimental cases shown in Fig. 2 using $\ell =20$ nm: (a) $L_g=1387$ nm ($\delta =17.21$), (b) $L_g=987$ nm ($\delta =12.25$), and (c) $L_g=606$ nm ($\delta =7.52$). The results are shown at the late times such that no further evolution is expected.

Figure 10

Time evolution of the position of the point of maximum height in the bulge, $x_h$, as a function of time for several values of $\ell$. The $x$ coordinate is measured from the filament intersection and $n$ stands for the number of drops formed along each side of the square. The horizontal dashed line stands for $x_h^{\max }$ as predicted by FDM.

Figure 11

Closeup of a SEM for a grid with $h_g=5$ nm and $L_g=1556$ nm ($\delta =24.4$). Note that the lower right corner drop is missing.

Figure 12

(a) Sketch showing the intersection region for the original grid (dashed lines) of width $w_g$ with rectangular cross section of thickness $h_g$ and the (liquefied) cylindrical cap filaments of width $w$. $V_0$ corresponds to the square with thick lines, and $V_{\mathrm{cross}}$ to the colored cross region. (b) Relative variation of volume in the intersection region as a function of $h_g$ [see Eq. (22)] when comparing the original volume $V_0$ with the assumed cylindrical cap arms as depicted in (a).