  Several groups have reported spontaneous formation of periodic pillar-like
arrays in molten polymer nanofilms confined within closely spaced substrates
maintained at different temperatures. These formations have been attributed to
a radiation pressure instability caused by acoustic phonons. In this work, we
demonstrate how variations in the thermocapillary stress along the nanofilm
interface can produce significant periodic protrusions in any viscous film no
matter how small the initial transverse thermal gradient. The linear stability
analysis of the interface evolution equation explores an extreme limit of
B\'{e}nard-Marangoni flow peculiar to films of nanoscale dimensions in which
hydrostatic forces are altogether absent and deformation amplitudes are small
in comparison to the pillar spacing. Finite element simulations of the full
nonlinear equation are also used to examine the array pitch and growth rates
beyond the linear regime. Inspection of the Lyapunov free energy as a function
of time confirms that in contrast to typical cellular instabilities in
macroscopically thick films, pillar-like elongations are energetically
preferred in nanofilms. Provided there occurs no dewetting during film
deformation, it is shown that fluid elongations continue to grow until contact
with the cooler substrate is achieved. Identification of the mechanism
responsible for this phenomenon may facilitate fabrication of extended arrays
for nanoscale optical, photonic and biological applications.
