### Publication Type:

Journal Article### Source:

J. Fluid Mech., Cambridge University Press, Volume 723, p.69-90 (2013)### ISBN:

0022-1120### Keywords:

2013, 2013 and earlier, interfacial flows (free surface), low-Reynolds-number flows, lubrication theory### Abstract:

We examine solidification in thin liquid films produced by annealing amorphous \${\mathrm{Alq} }_{3} \$ (tris-(8-hydroxyquinoline) aluminium)

in methanol vapour. Micrographs acquired during annealing capture the

evolution of the film: the initially-uniform film breaks up into drops

that coarsen, and single crystals of \${\mathrm{Alq} }_{3} \$ nucleate

randomly on the substrate and grow as slender . The growth of these

needles appears to follow power-law behaviour, where the growth exponent,

\$\gamma \$, depends on the thickness of the deposited \${\mathrm{Alq}

}_{3} \$ film. The evolution of the thin film is modelled by a lubrication

equation, and an advection–diffusion equation captures the transport of

\${\mathrm{Alq} }_{3} \$ and methanol within the film. We define a

dimensionless transport parameter, \$\alpha \$, which is analogous to an

inverse Sherwood number and quantifies the relative effects of diffusion-

and coarsening-driven advection. For large \$\alpha \$-values, the model

recovers the theory of one-dimensional, diffusion-driven solidification,

such that \$\gamma \rightarrow 1/ 2\$. For low \$\alpha \$-values, the

collapse of drops, i.e. coarsening, drives flow and regulates the growth

of needles. Within this regime, we identify two relevant limits: needles

that are small compared to the typical drop size, and those that are

large. Both scaling analysis and simulations of the full model reveal that

\$\gamma \rightarrow 2/ 5\$ for small needles and \$\gamma \rightarrow 0.

29\$ for large needles.