Thermally induced interfacial instabilities and pattern formation in confined liquid nanofilms

The dynamics, instability, and pattern formation of thermally triggered thin liquid films are investigated numerically under a long-wave limit approximation. To determine the mechanisms responsible for instability growth and pattern formation in confined heated nanofilms, acoustic phonon (AP) and thermocapillary (TC) models are examined using both linear and nonlinear analyses. Under uniform heating conditions, both AP and TC models predict the formation of raised columnar structures (pillars) and bicontinuous structures for very low and high filling ratios defined as the ratio of the initial film thickness to the plate separation distance, $D^{-1}$. A transition threshold is observed when $D\approx 2.5$. However, the TC model predicts smaller features for larger $D$, whereas the opposite prediction applies for the AP model. Under spatially variable cooling conditions involving a patterned top plate, both TC and AP models exhibit similar predictions: pillars form under the top plate protrusions. When the heating is spatially variable, the lower plate is patterned; the AP model predicts pillar formation above ridges, whereas the TC model predicts pillar formation above the valleys.

Figure 1

A 2D schematic of the ultrathin liquid film sandwiched between top (cold) and bottom (hot) plates.

Figure 2

TC model, interface height profile versus time, and the 3D snapshots of the interface. Initial film thickness, $h_0=25\mathrm{nm}$, plate separation distance, $d=100\mathrm{nm}$, temperature difference, $\mathrm{\Delta }T=46{}^{\circ }\mathrm{C}$, and nondimensional times, $\tau =$ (a) 0.225, (b) 0.306, and (c) 0.528.

Figure 3

TC mode, interface height profile versus time, and the 3D snapshots of the interface. Initial film thickness, $h_0=55\mathrm{nm}$, plate separation distance, $d=100\mathrm{nm}$, temperature difference, $\mathrm{\Delta }T=46{}^{\circ }\mathrm{C}$ and nondimensional times, $\tau =$ (a) 0.17, (b) 0.252 and (c) 0.49.

Figure 4

TC model, (i–v) 3D snapshots of the interface profile. Initial film thickness, $h_0=$ (i) 65 nm, (ii) 55 nm, (iii) 45 nm, (iv) 35 nm, and (v) 25 nm. $\text{Ma}=$ (i) 1.56, (ii) 2.31, (iii) 3.64, (iv) 5.73, and (v) 9.69. Plates distance, $d=$ 100 nm, temperature difference, $\mathrm{\Delta }T=46{}^{\circ }\mathrm{C}$, and nondimensional times, $\tau =$ (i) 0.49, (ii) 0.30, (iii) 0.26, (iv) 0.41, and (v) 0.53.

Figure 5

AP model, (i–vi) 3D snapshots of the interface profile. Initial film thickness, $h_0=$, (i) 65 nm, (ii) 55 nm, (iii) 45 nm, (iv) 35 nm, (v) 25 nm, and (vi) 25 nm. Plates distance, $d=$ 100 nm, temperature difference, $\mathrm{\Delta }T=46{}^{\circ }\mathrm{C}$ and nondimensional times, $\tau =$ (i) 0.10, (ii) 0.10, (iii) 0.07, (iv) 0.10, and (v) 0.10.

Figure 6

(a) 2D schematic view of patterned top plate (PTP) and patterned lower plate (PLP) setup. (b) Temperature distribution versus interface height for flat, PTP and PLP configurations. (b) $k_r$ = 3.61, $d=100\mathrm{nm}$, $d_p=l_p=20\mathrm{nm}$.

Figure 7

Patterned top plate (PTP) and TC model, interface height profile at (a) early stages of deformation and (b) quasisteady stage of pattern formation. Protrusion height $d_p=0.1d$ and protrusion width $p_w=$ (i) ${\lambda }_{max}$ and (ii) $0.5{\lambda }_{max}$, initial film thickness, $h_0=35\mathrm{nm},$ and plates distance, $d=100\mathrm{nm}$.

Figure 8

Patterned top plate (PTP) and AP model, interface height profile at (a) early stages of deformation and (b) quasisteady stage of pattern formation. Protrusion height $d_p=0.1d$ and protrusion width $p_w=$ (i) ${\lambda }_{max}$ and (ii) $0.5{\lambda }_{max}$, initial film thickness, $h_0=35\mathrm{nm},$ and plates distance, $d=100\mathrm{nm}$.

Figure 9

Patterned lower plate (PLP) and TC model, interface height profile at (a) early stages of deformation and (b) quasisteady stage of pattern formation. Protrusion height $l_p=0.1d$, and protrusion width $p_w=$ (i) ${\lambda }_{max}$ and (ii) $0.5{\lambda }_{max}$, initial film thickness, $h_0=35\mathrm{nm},$ and plates distance, $d=100\mathrm{nm}$.

Figure 10

Patterned lower plate (PLP) and AP model, interface height profile at (a) early stages of deformation and (b) quasisteady stage of pattern formation. Protrusion height $l_p=0.1d$ and protrusion width $p_w=$ (i) ${\lambda }_{max}$ and (ii) $0.5{\lambda }_{max}$, initial film thickness, $h_0=35\mathrm{nm}$, and plates distance, $d=100\mathrm{nm}$.