Formation of Nanopillar Arrays in Ultrathin Viscous Films: The Critical Role of Thermocapillary Stresses

Experiments by several groups during the past decade have shown that a molten polymer nanofilm subject to a large transverse thermal gradient undergoes spontaneous formation of periodic nanopillar arrays. The prevailing explanation is that coherent reflections of acoustic phonons within the film cause a periodic modulation of the radiation pressure which enhances pillar growth. By exploring a deformational instability of particular relevance to nanofilms, we demonstrate that thermocapillary forces play a crucial role in the formation process. Analytic and numerical predictions show good agreement with the pillar spacings obtained in experiment. Simulations of the interface equation further determine the rate of pillar growth of importance to technological applications.

Figure 1

(a) Experimental setup for formation of nanopillar arrays. (b) AFM image of PMMA pillars [5]: $d=260\mathrm{nm}$, $h_0=95\mathrm{nm}$, ${\lambda }_{\max }=3.4\mu \mathrm{m}$, $\Delta T$ unknown. (c) Optical micrograph of PS pillars [23]: $d=345\mathrm{nm}$, $h_0=100\mathrm{nm}$, ${\lambda }_{\max }=4.5\mu \mathrm{m}$, $\Delta T=46°\mathrm{C}$. (d) AFM image of PMMA pillars [9]: $d=163\mathrm{nm}$, $h_0=100\mathrm{nm}$, ${\lambda }_{\max }=6.5\mu \mathrm{m}$, $\Delta T=10°\mathrm{C}$.

Figure 2

(a) Experimental data from Refs. 6, 7, 8, 23. Material constants, evaluated at the temperature $T_2=170°\mathrm{C}$, were obtained from Refs. 27, 28, 29, 30. (b) Fitting coefficients are of the form [experiments A–D, $C_1({10}^3\mu \mathrm{m}{)}^{0.5}$, $C_2(0.1\mu \mathrm{m}{)}^{1.5}$]: [A, $0.353,-34.7$], [B, $0.650,-64.6$], [C, $0.379,-46.0$], [D, $0.340,-30.7$]. For the AP model, $Q=6.2$ and $u_p=1850\mathrm{m}/\mathrm{s}$.

Figure 3

Solutions of Eq. (3) for $\Delta T=11°\mathrm{C}$ and $46°\mathrm{C}$.

Figure 4

(a) Evolution of the Lyapunov free energy for the case $h_0=100\mathrm{nm}$, $d=285\mathrm{nm}$, $\Delta T=46°\mathrm{C}$. Real time conversion $t=2.28\tau$ hours. Inset: Plot of $H(X,Y,\tau =1.1025)$ from finite element simulations of Eq. (1). (b) Time required, $t_{\mathrm{top}}$, for fastest growing pillars to contact upper wafer for parameter values given in (a). For variation with $h_0$, $d=285\mathrm{nm}$; for variation with $d$, $h_0=100\mathrm{nm}$.