Design Rule for Colloidal Crystals of DNA-Functionalized Particles

We report a Monte Carlo simulation study of the phase behavior of colloids coated with long, flexible DNA chains. We find that an important change occurs in the phase diagram when the number of DNAs per colloid is decreased below a critical value. In this case, the triple point disappears and the condensed phase that coexists with the vapor is always liquid. Our simulations thus explain why, in the dilute solutions typically used in experiments, colloids coated with a small number of DNA strands cannot crystallize. We understand this behavior in terms of the discrete nature of DNA binding.

Figure 1

Simulations of colloids coated with $\kappa =16$ DNA chains. (a) The total Helmholtz free energy ($\beta A$) as a function of $\eta$ for different binding strengths ($\beta f_{\mathrm{hyb}}$). The two minima correspond to the liquid (left) and CsCl crystal (right) phases. (b) Pair distribution function $g(r)$ and snapshot of the simulated crystal phase. We show the total $g(r)$, the “opposite-type” $A\mathrm{\text{−}}B$ $g(r)$, and the “same-type” $A\mathrm{\text{−}}A$ (or $B\mathrm{\text{−}}B$) $g(r)$. The peaks are located in the positions consistent with the bcc structure.

Figure 2

The average number of colloids ($\tau$) to which each colloid is directly connected as a function of $\eta$, for different values of $\kappa$ and in the limit of strong hybridization. The location of the minimum in free energy is indicated with a circle for each value of $\kappa$.

Figure 3

Phase diagram of the DNA-coated colloids in the plane $\beta f_{\mathrm{hyb}}$ vs $\eta$ for different values of $\kappa$. Dashed lines only represent a qualitative description of the expected phase behavior. Since a single fluid phase is observed for the weak-binding regime (i.e., $0<\beta f_{\mathrm{hyb}}<+\infty$) we limit our plots to $\beta f_{\mathrm{hyb}}=0$. Also, other phases may be stable for $\eta >0.30$ but they will not coexist directly with the vapor phase. (a) $\kappa =16$, the system displays a triple point (TP) where the vapor phase ($V$), the liquid phase ($L$), and the bcc solid phase ($S$) coexist. (b) $\kappa =11$, the system no longer presents a TP. The vapor-liquid coexistence envelope ($V+L$) and the liquid-crystal envelope ($L+S$) are always separated by a narrow region where the pure $L$ phase is stable. (c) $\kappa =10$, the $L+S$ envelope shifts to increasing values of $\eta$.