# 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 (PMMA). 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

- Entropically driving force due to polymers:

Fig. 1A shows the depletion effect. The centers of nonadsorbing polymer coils (small spheres with radius <math>R_g</math>) are excluded from a depletion zone (hashed regions) outside the large colloids spheres with radius <math>a</math> 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 <math>\Pi \Delta V</math> as spheres approach each other. Here <math>\Pi</math> is the polymer osmotic pressure, and <math>\Delta V</math> is the overlap volume, shown in black in Fig. 1A. The free energy reduction at contact at temperature <math>T</math> is <math>F_0 \approx - 2 \pi a R_g^2 n_p k_B T</math>, <math>n_p</math> is the number density of the dilute polymer coils. the template creates effectively a periodic surface potential for colloidal self-assembly.

Fig. 1B and 1C show two different PMMA templates used in the experiments.

- 1D colloidal liquids:

Fig. 2 shows the formation of 1D colloidal liquids in the grooves of the 1D grating templates (Fig. 1B). Fig. 2A shows the 1D liquid phase (i.e., stripe phase) arising when the spheres are large enough to fill the groove, but not large enough to interact with spheres in adjacent grooves. Pair correlation function along the groove, <math>g(r)</math>, is plotted in Fig. 3A for three different combinations of bulk particle volume fraction,<math>\Phi</math> , and polymer concentration, <math>C_p</math>. At low volume fraction <math>\Phi</math>, the measured pair correlation function <math>g(r)</math> exhibited peaks whose positions were identical to and whose asymmetric shapes were similar to those of a classical hard-core gas. At higher concentrations the mean nearest-neighbor spacing, d, derived from the first peak in <math>g(r)</math>, shifted to smaller values, but the typical spacing was larger than the depletion interaction range, about 1.1 diameter. These observations suggest that the surface density of spheres is determined by the competition of the depletion attractions driving the spheres to the surface and the osmotic pressure of the spheres already there.

When the sphere diameter increases relative to the grating period <math>p</math>, the 1D colloidal liquids in adjacent rows interact more strongly. The most important parameter characterizing the 2D phase behavior is the commensurability ratio, <math>\chi = d/p</math>, where <math>d</math> is derived from the pair correlation function along the groove. Hexagonal symmetry emerges in 2D through the interlacing of spheres in different grooves for <math>1 < \chi < 2/ \sqrt{3}</math> (see Fig. 2B).