Quantitative Prediction of the Phase Diagram of DNA-Functionalized Nanosized Colloids

We present a coarse-grained model of DNA-functionalized colloids that is computationally tractable. Importantly, the model parameters are solely based on experimental data. Using this highly simplified model, we can predict the phase behavior of DNA-functionalized nanocolloids without assuming pairwise additivity of the intercolloidal interactions. Our simulations show that, for nanocolloids, the assumption of pairwise additivity leads to substantial errors in the estimate of the free energy of the crystal phase. We compare our results with available experimental data and find that the simulations predict the correct structure of the solid phase and yield a very good estimate of the melting temperature. Current experimental estimates for the contour length and persistence length of single-stranded (ss) DNA sequences are subject to relatively large uncertainties. Using the best available estimates, we obtain predictions for the crystal lattice constants that are off by a few percent: this indicates that more accurate experimental data on ssDNA are needed to exploit the full power of our coarse-grained approach.

Figure 1

Left: The pair interaction ${\Phi }_2^{AB}(r,T)$ (bold lines) is the sum of the $T$-independent steric repulsion ${\Phi }_{2,\mathrm{rep}}(r)$ and the strongly $T$-dependent attractive hybridization energy ${\Phi }_{2,\mathrm{hyb}}(r,T)$ (both as labeled). Right: The three-body interaction $\beta {\Phi }_3^{ABA}(r)$ (bold dashed lines) of two $A$ and one $B$ colloids arranged in an equilateral triangle (inset) compared with the sum of the two-body contributions $3\beta {\Phi }_{2,\mathrm{rep}}+2\beta {\Phi }_{2,\mathrm{hyb}}$ (bold solid lines). The repulsive and the attractive hybridization contributions to the interactions are shown explicitly in thin lines (two-body, dashed lines; three-body, solid lines). Both: Results are shown at $56.9°\mathrm{C}$, $59.1°\mathrm{C}$, $61.4°\mathrm{C}$, $63.2°\mathrm{C}$, and $65.1°\mathrm{C}$. Inset: A simulation snapshot of one $B$ colloid (top) interacting with two $A$ (bottom) at $59.1°\mathrm{C}$ and $r=5.75R_C$. Free blobs are drawn as small spheres, hybridized sticky ends as rods. Translucent spheres indicate the extension of the core.

Figure 2

The dimensionless free energy $\beta \eta F/N$ at $T=56.9°\mathrm{C}$ as function of the colloidal volume fraction $\eta$ shows that the CsCl structure is the most stable one. Pair potential approach, dashed lines; core-blob approach, solid lines. CsCl, ▴; s-hcp, ●; CuAu, ▪; NaTl, $◆$; disordered CsCl, *; disordered CuAu, ×; and liquid, ◂. The common tangents between the equilibrium CsCl crystal and the dilute vapor are shown as dotted lines for both models. Right inset: the lattice constant $a$ as function of $T$ in $°\mathrm{C}$ as measured in the experiments (heating, ▴; cooling, ▾; data from Ref. 6) and as obtained from simulations with the core-blob model (●) and using the pair potential approach (▪). The data of $a$ have been scaled to the values of $a_{\mathrm{ref}}=a(56.9°\mathrm{C})$, as measured with the respective approaches: $a_{\mathrm{ref}}^{\exp }=45.3\mathrm{nm}$, $a_{\mathrm{ref}}^{\mathrm{core}\mathrm{\text{−}}\mathrm{blob}}=39.7\mathrm{nm}$, and $a_{\mathrm{ref}}^{{\Phi }_2}=36.9\mathrm{nm}$. In the shaded region, crystals are not stable in experiments. Left inset: a cut through a simulated CsCl structure at $\eta =0.024$ and $T=56.9°\mathrm{C}$. For a better visualization, only the hybridized links are drawn.