Temperature Effects on Threshold Counterion Concentration to Induce Aggregation of fd Virus

We seek to determine the mechanism of like-charge attraction by measuring the temperature dependence of critical divalent counterion concentration ($C_{c}$) for the aggregation of fd viruses. We find that an increase in temperature leads to a decrease in the dielectric constant ($\varepsilon$) of the solvent, thus causing $C_{c}$ to decrease. At a constant $\varepsilon$, $C_{c}$ is found to increase as temperature increases. The effect of $T$ and $\varepsilon$ on $C_{c}$ is combined to that of one parameter: Bjerrum length ($l_{B}$). $ C_{c}$ decreases exponentially as $l_{B}$ increases. The exponential decay of $C_{c}$ suggests that entropic effect of counterions plays an important role at the onset of bundle formation.