Depletion forces and confinement effects in vibrofluidized granular materials

Ranging from nano- to granular-scales, control of particle assembly can be achieved by limiting the available free space, for example by increasing the concentration of particles (crowding) or through their restriction to 2D environments. It is unclear, however, if self-assembly principles governing thermally equilibrated molecules can also apply to mechanically excited macroscopic particles in nonequilibrium steady-state. 

We have shown that low densities of vibrofluidized steel rods, when crowded by high densities of spheres and confined to quasi-2D planes, can self-assemble into linear polymer-like structures. Our 2D Monte Carlo simulations show similar finite sized aggregates in thermally equilibrated binary mixtures. Using theory and simulations, we have been able to demonstrate how depletion interactions create oriented ‘‘binding’’ forces between rigid rods to form these ‘‘living polymers.’’ Unlike rod–sphere mixtures in 3D that can demonstrate well-defined equilibrium phases in coexistence, our mixtures confined to 2D lack these transitions because lower dimensionality favors the formation of linear aggregates, thus suppressing a true phase transition. 

The qualitative and quantitative agreement between equilibrium and granular patterning for these mixtures suggests that entropy maximization is the determining driving force for bundling. Furthermore, this study uncovers a previously unknown patterning behavior at both the granular and nanoscales, and may provide insights into the role of crowding at interfaces in molecular assemblyMoreover, we have found that rods assemble and disassemble to form nematic phases. Interestingly, we have been able to find elastic constants for these systems that would correspond to the appropriate continuum “Frank free energy” that describes molecular nematics. The method relies on analyzing defects in the container to infer the cost of creating such defects in terms of splay and bend contribution to the free-energy.

References:

• J. Galanis, R. Nossal, D. Harries, Depletion Forces Drive Polymer-like Self-Assembly in Vibrofluidized Granular Materials, Soft Matter 6, 1026-1034 (2010).
• J. Galanis, R. Nossal, W. Losert, D. Harries, Nematic order in small systems: measuring the elastic and wall-anchoring constants in vibrofluidized granular rods, Phys. Rev. Lett. 105, 168001 (2010).
• J. Galanis, D. Harries, D. L. Sackett, W. Losert, R. Nossal, Spontaneous Patterning of confined granular rods, Phys. Rev. Lett. 96, 028002 (2006).

This diagram depicts the area that rods exclude from spheres (dashed regions) and excluded area overlap (dark shaded regions) when two rods are less than one sphere diameter apart, shown for rod "binding".

The free energy gain is greater when there is more overlap that “releases” excluded volume. The experimental realization of this depletion force is seen as clustering of rods when spheres are introduced to rod mixtures. The same patterning is achieved in 2D Monte Carlo simulations of hard rods and spheres (shown in the bottom panels).

Patterning in a confined geometry. a) The free energy minimum derived by mean field theory. Fluctuations to the solution seen in (b) equilibrium Monte Carlo simulations of hard rods and (c) granular experiments of vibrated steel needles. In (b,c), individual rods are shown in black, while the color maps the local orientational ordering and highlights the location of the two defects (blue).