Coarse-grained models of tethers for fast self-assembly simulations

Long molecular ligands or “tethers” play an important role in the self-assembly of many nanoscale systems. These tethers, whose only interaction may be a hard-core repulsion, contribute significantly to the free energy of the system because of their large conformational entropy. Here, we investigate how simple approximate models can be developed and used to quickly determine the configurations into which tethers will self assemble in nanoscale systems. We derive criteria that determine when these models are expected to be accurate. Finally, we propose a generalized two-body approximation that can be used as a toy model for the self-assembly of tethers in systems of arbitrary geometry and apply this to the self-assembly of self-assembled monolayers on a planar surface. We compare our results to those in the literature obtained via atomistic and dissipative particle dynamics simulations.

Figure 1

(Color online) Three tethers $i$, $j$, and $k$ with $L_i=L_j=L_k=6$ located along a line at positions $x$, $y$, and $z$, respectively.

Figure 2

(Color online) Plots of $p\left(i\otimes j\mid k\right)$ vs $z$ for $x=0$ and $y=1$ (red, circle), 2 (green, square), and 3 (blue, triangle). This data was obtained from $10000$ trials with tethers of length $L_i=6$ beads each of radius ${\sigma }_i=0.5$.

Figure 3

A plot of $p\left(i\otimes \left\{k\right\}\right)$ vs the range of tethers included in the simulation. This data were obtained from $10000$ trials with tethers of length $L_i=5$ beads each of radius ${\sigma }_i=0.25$.

Figure 4

(Color online) A schematic of the free energy $F$ vs configuration $\psi$ calculated exactly (black) and using some approximate method (red). While there is a large net shift, the relative values of $F$ are constant so that the dynamics obtained by a MC algorithm will be identical in both systems.

Figure 5

(Color online) Two tethers $i$ and $j$ separated by some distance $r_{ij}$.

Figure 6

(Color online) A plot of $p\left(i\otimes j\right)$ vs $r_{ij}$ using both data from MC integration (red circles) and the fitted functional form (black line). This data was obtained from $10000$ trials with tethers of length $L_i=6$ beads each of radius ${\sigma }_i=0.5$. The data were fit to the function form given in Eq. (16) with $m=0.56442$ and $\varphi =0.97332$.

Figure 7

(Color online) A plot of $p\left(i\otimes \left\{k\right\}\right)$ vs $L_i$ calculated numerically using MC integration (black, square) and using the two-body approximation (red, circle). This data were obtained from ${10}^4$ trials for tether beads of radius ${\sigma }_i=0.25$ with $R=5$.

Figure 8

(Color online) A plot of $p\left(i\otimes \left\{k\right\}\right)$ vs ${\sigma }_i$ calculated using the $N$-body MC integration method (black, square) and using the two-body approximation (red, circle). This data were obtained from ${10}^4$ trials for tethers with $L_i=5$ beads with $R=5$.

Figure 9

(Color online) A chart showing tether configuration and energetic contributions from the D (left), S (middle), and I (right) macrostates. The local configuration around either long (red) or short (yellow) tethers determines the energy per particle $E_{M,i}$ based on the average number of nearest neighbors. For the demixed state, this is computed in the limit that $N\to \infty$ so that the interface and boundaries can be neglected. For simplicity, we only consider stripes of width 2 since they exhibit only one type of local configuration for both tethers. In principle, one could calculate the average energy and entropy per particle for wider stripes, but one must average over multiple local configurations to do so. The $N$-body and two-body methods are used to compute $S_{M,i}$ and $S_{M,i}^{\prime }$, respectively. Numerical values of both entropies are given in units of ${10}^{-4}\text{eV}/\text{K}$. For the isotropic state, the entropy is taken as the average of 100 randomly generated configurations. The error $\epsilon$ is given as the difference between the exact and approximate entropy.

Figure 10

(Color online) Plots of the free energy $\Delta F_i/\left(T\Delta {\epsilon }_i\right)$ for macrostate transitions $\text{D}\to \text{I}$ (red, dashed), $\text{D}\to \text{S}$ (blue, dotted), and $\text{S}\to \text{I}$ (black, solid) vs $\xi$ measured in eV.

Figure 11

(Color online) Plots of the free energy $F_i$ vs $\xi$ (measured in eV) for macrostates I (blue, dotted), S (red, dashed), and D (black, solid) computed (a) using $N$-body MC integration and (b) using the two-body approximation.

Figure 12

(Color online) System configurations obtained as a function of tether length difference $dL$ and strength of attraction $\xi$. As observed in DPD and atomistic simulations [5], stripes are observed for sufficiently large $dL$ provided $\xi$ is not too large.

Figure 13

A plot of the effective interaction $V\left(r_{ij}\right)$ vs separation distance $r_{ij}$ obtained for $\xi =0.2$, $m=0.660$, and $\varphi =0.998$.