# Pair Potential of Charged Colloidal Stars.

(in progress)

## Contents

## Keywords

## Summary

## Relevance for Soft Matter

## References

F. Huang, K. Addas, A. Ward, N.T. Flynn, E. Velasco, M.F. Hagan, Z. Dogic, and S. Fraden. "Pair Potential of Charged Colloidal Stars". *Physical Review Letters* 102, 108302 (2009).

## Pair Potential of Charged Colloidal Stars (by Hsin-I Lu)

## Summary

This paper reported the construction of colloidal stars: 1 <math>\mu</math>m polystyrene beads grafted with a dense brush of 1 <math>\mu</math>m long and 10 nm wide charged semiflexible filamentous viruses. The pair interaction potential of colloidal stars are studied under optical traps and measured using an implementation of umbrella sampling. The influence of ionic strength and grafting density on the interaction is measured and in good agreement with theoretical predictions.

## Soft matter keywords

osmotic pressure, laser tweezers, colloid, TEM

## Soft Matter

- Colloid stars:

Colloid stars are constructed by grafting M13 viruses to polystyrene spheres. The viruses are rodlike, semiflexible charged polymers of length <math>L=880</math> nm, diameter <math>D=6.6</math> nm, and persistence length ~ 2 <math>\mu</math>m with a linear charge density of ~<math>7 e^{-1}</math>/nm. The M13 are rigid enough to form liquid crystals, but when grafted to a sphere remain flexible enough to be distorted by the director field.

Fig.1 (a) and (b) show TEM images of 10 nm Au-bound M13 viruses of different nanoarchitectures and (c)–(e) show TEM (right panel) and fluorescence (left panel) images of labeled phage grafted to unlabeled 1<math>\mu</math>m PS beads with varying grafting densities. (c) 3 phages/bead. (d) 38 phages/bead. (e) 135 phages/bead. At the grafting density of 135 phages/bead [Fig. 1(e)], the anchored dye-labeled rods form a spherically symmetric corona around the bead.

- Measurement of interaciton potential:

The potential of mean force as a function of separation between two colloidal particles <math>W_{int} (r)</math>can be determined up to an additive offset by the Boltzmann relation, <math>P(r) \sim exp(-W_{int} (r) /k_B T)</math>. Therefore, measuring the probability <math>P(r)</math> of finding the particles at a separation r can determine <math>W_{int} (r)</math>. Typically for states of even moderate repulsive interaction energies (few <math>k_B T</math>) <math>P(r)</math> becomes very small. Using optical field to displace colloidal stars in various distance, the authors can construct single interparticle pair-potentialcan from seperate measuremnts.

Fig. 2 shows two colloidal stars in separate laser traps and the histogram of separation distances between the colloids. Since there is repulsion interaction between stars, the seperation would be larger than two bare beads (Fig. 2(a)). In Fig. 2(c), separation histograms of bare beads (A) and phage-grafted beads (B) confirmed this point. To meausre the full range of <math>W_{int} (r)</math>, the authors measured <math>P(r)</math> for 30 different trap positions with 50 nm increments in separation. In each seperation the stars fluctuate about the minimum of a total potential resulting from a combination of the dual traps and interparticle star potential. Only 6kBT of each of the total potentials is sampled and each minimum has a different energy. The overlap between windows can be combined to produce a single interparticle pair-potential.

To extrapolate <math>W_{int} (r)</math> between to stars under the infulence of external optical field, the authors performed two experiments for any given trap seperation. In one experiment, two colloidal stars were placed in two separate laser traps and in the other experiment two bare colloids were placed in the same two traps. For both experiments, the separation histogram of the colloids was measured. The potential of mean force, <math>W_{sub}</math>, is then obtained by subtracting the results from each experiment.

<math>W_{sub}(r) /k_B T = -log(f_f(r)) + log(f_{nf} (r))</math>,

with <math>f_f(r)</math> and <math>f_{nf}(r)</math> the fraction of measured displacements that fall within the histogram bin associated with the displacement value <math>r</math> for functionalized and nonfunctionalized beads, respectively.

Fig. 3 (b) shows the interaction potentials of colloidal stars at varying solution ionic strengths with 135 phages/bead.

- Discussion:

By attaching viruses to polystyrene spheres and trapping polystyrene spheres in optical fields, the interaction of colloidal stars due to repulsion between charged viruses can be measured. This technique can be used to studied the interactions between hybrid colloidal materials. As mentioned in the paper, this method allows measurement of strong repulsive potentials of the order of 100 <math>k_B T</math> given the laser power (which determines the optical trap depth) and optical resolution in the experiment.