# Phase Behavior of Rod-Like Viruses and Virus–Sphere Mixtures

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## Summary

(all images from paper) The reviewers have been studying the behavior of viral colloids. In this 78-page article, they describe the various properties they have observed in the colloids and compare different theoretical descriptions with simulated models and experimental results. Here, I will just describe the background of this work and some of the more basic descriptions they give.

This chapter mainly concerns the filamentous phage fd. fd is a rod-shaped virus that infects E. Coli. Previous work concentrated mainly on the Tobacco Mosaic Virus (TMV), a famous virus that infects and damages tobacco plants. TMV was the first virus discovered and was used widely in research because it only infects plants and not animals. Bacteriophage like fd are easier to study, however, because they can easily be mass-produced (using bacteria instead of tobacco plants). Rods-shaped colloids have been less studied than spherical colloids, but their behavior is better known due studies by Onsager that accurately explain liquid crystal phase transitions characterizing rod colloids. TMV accurately follows this model, but fd is semi-flexible and displays different behaviors, most of which are still not satisfactorily explained by theory. This article first presents Onsager's theory, then proceeds to present results of models for fd solutions, followed by mixtures of fd and spherical colloids.

### Onsager's Theory

With rigid rods, there is a lot of wasted space in the nematic phase (a), but with flexible rods, some of this space can be recovered without having a phase transition to the smetic phase (b).

Onsager showed that a second-order virial expansion becomes exact as the aspect ratio of cylindrical colloids approaches infinity. He used this model to characterize the isotropic-nematic (I-N) phase transition in TMV colloids. The article extends this theory to charged rod systems, then to semi-flexible rods characterized by a characteristic persistence length. They concluded that as the colloid becomes more flexible, the transition from the isotropic to nematic phase is pushed back to higher volume fractions. (If you view the link on liquid crystal phases, the nematic phase has orientational order, so it makes sense that it becomes destabilized when the colloid is more flexible. Also see figure below.) They then further extended this analysis to studying the nematic-smetic (N-S) phase transition. In the case of flexible rods, the N-S phase transition is suppressed because flexible rods are able to bend around one another as shown in the figure to the left.

y-axis is concentration where isotropic and nematic phases coexist (proportional to volume fraction). x-axis is flexibility of the polymer, P = length/persistence length.

### fd Virus

fd phase diagram relating the surface charge strength to the isotropic-cholesteric coexistence concentration. Lines indicate different models. Right vertical axis is effective diameter in calculations.

The next section of the review article discusses the properties of a fd virus solution. fd actually forms a cholesteric (chiral nematic, see page2 again) instead of nematic phase. However, it takes a long time for this phase to equilibrate, so in terms of free energy, it is not much different than the nematic phase. The cholesteric phase is about 30% denser than the isotropic phase, so given a couple days, a coexisting solution will phase separate, as shown in the bottom figure below (actually this figure is TMV, but the same phenomenon occurs for fd). The figure on the left shows a phase diagram for fd. fd may have surface charges, and these affect the concentration for coexistence of the isotropic and cholesteric phases.

Electron microscopy image of fd virus, length ~880nm, diameter ~6.6nm, aspect ratio 130.
TMV phase separation of isotropic and nematic phase. fd shows the same phenomenon.

### fd Virus + Polysytrene Spheres

Phase diagram of fd virus with polystyrene spheres of diameter 100nm.
Images of the phases. a)columnar phase, b)transition between a and c, c)lamellar phase). Scale bar is 10 $\mu$m.

The last part of the review discusses systems of rods and spheres, in particular of fd viruses (rods) and polystyrene spheres of varying sizes. In sum, the figure on the right shows the phase diagram of the fd virus with 100nm polystyrene spheres (recall that the fd virus has length ~880nm and diameter ~6.6 nm). The phase diagram predictably changes with the diameter of spheres used. The lamellar phase is the manifestation of 1-D layers of spheres alternating with layers of rods. Although at first this phase may seem strange because the spheres are, well, spherical, meaning that they have no directional preferences, the image below illustrates that the excluded volume of a sphere in the system is actually larger than you would initially think because of the rods, and asymmetric. This excluded volume gives rise to the lamellar phase. At the left, we see images of the columnar and lamellar phases.

Excluded volume of spheres and rods arising to the lamellar phase.
Larger spheres of 300nm or 400nm would change the dynamics of the system.

## Relevance to Soft Matter

This review covered much of the current theory on systems of rods and spheres with short-range interaction. Furthermore, it asserted that many phenomena that can be observed experimentally (such as lamellar structure, columnar structure, ribbon-like and smetic structures, etc) have no current solid theoretical framework, but are very likely to be observed in many similar systems of rods and spheres. This work in rod-like and spherical colloids can be further extended to simple liquids, liquid crystals, and other systems. Moreover, viral colloids themselves have been the object of increasing attention lately with work in phage displays, colloid stars, and viral engineering, such as the tiny lithium batteries produced by the Belcher Lab using virus backbones. But perhaps in the end, the most striking point of this article is that there are so many novel phases that arise once we mix spherical and rod-like colloids, and there is very little known about these so far. In order to establish a theoretical framework for describing more practical and complex systems, we will have to extend our models beyond simple, theoretical constructs.

## References

G. Gompper and M. Schick, eds. Soft Matter, vol. 2: Complex Colloidal Suspensions, Chap. 1: “Phase Behavior of Rod-Like Viruses and Virus–Sphere Mixtures”, Z. Dogic and S. Fraden, 1st ed., Wiley-VCH, Weinheim, 2006.