# Entropically driven colloidal crystallization on patterned surfaces

Original entry: Hsin-I Lu, APPHY 225, Fall 2009

"Entropically driven colloidal crystallization on patterned surfaces"

Keng-hui Lin, John C. Crocker, Vikram Prasad, Andrew Schofield, D. A. Weitz, T. C. Lubensky, and A. G. Yodh, PRL 85, 1770 (2000)

## Summary

This paper studies the self-assembly of colloidal spheres on periodically patterned templates which are made from polymethylmethacrylate. Sefl-assembly of colloids in this system is induced entropically by the presence of dissolved, nonadsorbing polymers. Either colloids adsorbing on the templates or two colloids overlaping with each other can create more space for polymers in the system. Therefore, free energy reduction due to these two process can provide attractive forces between colloids and the templates. The authors observed two-dimensional fluidlike and solidlike phases form on templates with both one- and two-dimensional symmetry. The same methodology was then used to nucleate an oriented single fcc crystal more than 30 layers thick.

## Soft Matter Keywords

Colloids, colloidal self-assembly, polymer, osmotic pressure, pair correlation function

## Soft Matter

Fig. 1
• Entropically driving force due to polymers:

Fig. 1A shows the depletion effect. The centers of nonadsorbing polymer coils (small spheres with radius $R_g$) are excluded from a depletion zone (hashed regions) outside the large colloids spheres with radius $a$ and corrugated surface. When these depletion zones overlap (dark shading), the volume accessible to the polymer increases, increasing polymer-coil entropy and inducing an attractive force between the surfaces. Similarly, spheres are preferentially drawn to interior corners.

The Helmholtz free energy of a colloid/polymer mixture decreases by $\Pi \Delta V$ as spheres approach each other. Here $\Pi$ is the polymer osmotic pressure, and $\Delta V$ is the overlap volume, shown in black in Fig. 1A. The free energy reduction at contact at temperature $T$ is $F_0 - 2 \pi a R_g^2 n_p k_B T$, $n_p$ is the number density of the dilute polymer coils.