Ligand structure and mechanical properties of single-nanoparticle-thick membranes

The high mechanical stiffness of single-nanoparticle-thick membranes is believed to result from the local structure of ligand coatings that mediate interactions between nanoparticles. These ligand structures are not directly observable experimentally. We use molecular dynamics simulations to observe variations in ligand structure and simultaneously measure variations in membrane mechanical properties. We have shown previously that ligand end group has a large impact on ligand structure and membrane mechanical properties. Here we introduce and apply quantitative molecular structure measures to these membranes and extend analysis to multiple nanoparticle core sizes and ligand lengths. Simulations of nanoparticle membranes with a nanoparticle core diameter of 4 or 6 nm, a ligand length of 11 or 17 methylenes, and either carboxyl (COOH) or methyl $\left({\mathrm{CH}}_3\right)$ ligand end groups are presented. In carboxyl-terminated ligand systems, structure and interactions are dominated by an end-to-end orientation of ligands. In methyl-terminated ligand systems large ordered ligand structures form, but nanoparticle interactions are dominated by disordered, partially interdigitated ligands. Core size and ligand length also affect both ligand arrangement within the membrane and the membrane's macroscopic mechanical response, but are secondary to the role of the ligand end group. Moreover, the particular end group (COOH or ${\mathrm{CH}}_3$) alters the nature of how ligand length, in turn, affects the membrane properties. The effect of core size does not depend on the ligand end group, with larger cores always leading to stiffer membranes. Asymmetry in the stress and ligand density is observed in membranes during preparation at a water-vapor interface, with the stress asymmetry persisting in all membranes after drying.

Figure 1

Top row: Configurations of NPs with 6 nm diameter cores and $n=12$ COOH-terminated ligands sampled at 10 and 50 ns during a free-particle simulation. Nanoparticles are placed at the liquid-vapor interface and are allowed to diffuse freely. Bottom row: The same NPs at the liquid-vapor interface undergoing isotropic compression. Water molecules are not shown.

Figure 2

A $5\times 5$ array of NPs with open boundary conditions for 6 nm diameter NPs with $n=12 {\mathrm{CH}}_3$-terminated ligands viewed from the vapor side. Dashed red lines show interior NPs used to determine lattice spacings.

Figure 3

Stress $S_{zz}\left(z\right)$ for dry membrane samples with (a) 6 nm diameter NP cores and (b) 4 nm diameter NP cores with $n=12$ or 18, COOH- or ${\mathrm{CH}}_3$-terminated ligands. The top of each membrane sample is oriented away from the water, in the positive $z$ direction.

Figure 4

The ligand carbon number density as a function of height $\rho \left(z\right)$ through dry membrane samples with (a) 6 nm diameter NP cores and (b) 4 nm diameter cores.

Figure 5

The ligand carbon number density as a function of height $\rho \left(z\right)$ through NPs with (a) 6-nm-diameter core and $n=12{\mathrm{CH}}_3$-terminated ligands and (b) 4 nm diameter core and $n=18 {\mathrm{CH}}_3$-terminated ligands. In the dilute case NPs are not touching and axes for these data are on the right. Blue dotted and dashed density curves indicate the water density in the dilute and wet case, respectively.

Figure 6

Typical stress-strain curves for samples undergoing pure shear deformation. Data for 6 nm diameter cores from Ref. [13] are shown to contrast with the 4 nm core diameter data.

Figure 7

The radial density function $\rho \left(r\right)$ for end-group carbon atoms on differing NPs for different chain length and end group. Curves for different NP diameters are similar. Data for $n=12$ from Ref. [13] are shown to contrast with the $n=18$ case.

Figure 8

A snapshot of ligands on adjacent NPs at 300 K for 6 nm diameter NPs with $n=12 {\mathrm{CH}}_3$-terminated ligands taken from a $6\times 6$ membrane on water. Ligand carbon atoms are colored according to the $p_2$ measure of local order. The left snapshot shows the atoms from the membrane top view. On the right the axes have been rotated by ${90}^{\circ }$ to show the band of disordered ligands corresponding to the membrane plane.

Figure 9

(a) The normalized number of neighbors $q_{ij}$ with $i=1$ for dry membranes with different NP diameter and ligand chain length $n.$ (b) The normalized number of neighbors with $i=n$ for the same NP configurations.

Figure 10

(a) $q_{ij}$ for $i=j=1$ for the 6 nm diameter $n=12$ COOH end-group system and (b) ${\sum }_j{q}_{ij}$ for $i=12$ for the 6 nm $n=12{\mathrm{CH}}_3$ end-group system as a function of time since membrane formation and drying. Insets are the full histograms of $q_{ij}$ for (a) $i=1$ and (b) $i=12$ for the 6 nm diameter $n=12$ NP systems taken at 1, 8, and 22 ns.

Figure 11

A snapshot of ligands on adjacent NPs at 300 K before annealing (left), 400 K (middle), and 300 K after annealing (right) for 6 nm diameter NPs with $n=12 {\mathrm{CH}}_3$-terminated ligands taken from a $6\times 6$ membrane on water. Ligand carbon atoms are colored according to the $p_2$ measure of local order.

Figure 12

The radial density function $\rho \left(r\right)$ for 6 nm diameter $n=12$ COOH-terminated ligand membranes that have or have not undergone thermal annealing. In both cases membranes are dry and have been equilibrated for at least 8 ns.

Figure 13

(a) The normalized number of neighbors $q_{ij}$ for backbone index $i=1$ for 6 nm diameter, $n=12$ dry membrane samples with and without thermal annealing. (b) The same membrane samples for backbone index $i=12.$