| 529 |
|
dramatically inward ($c \rightarrow$ 1). |
| 530 |
|
|
| 531 |
|
The computed corrugation factors are shown in Figure |
| 532 |
< |
\ref{fig:NPthiols_combo} for bare nanoparticles and for |
| 532 |
> |
\ref{fig:NPthiols_corrugation} for bare nanoparticles and for |
| 533 |
|
ligand-protected particles as a function of ligand chain length. The |
| 534 |
|
largest nanoparticles are only slightly restructured by the presence |
| 535 |
|
of ligands on the surface, while the smallest particle ($r$ = 10 \AA) |
| 544 |
|
\AA ) particles show significant disruption to their crystal |
| 545 |
|
structures, and the length and stiffness of the ligands is a |
| 546 |
|
contributing factor to the surface disruption.} |
| 547 |
< |
\label{fig:NPthiols_combo} |
| 548 |
< |
\end{figure} |
| 549 |
< |
|
| 550 |
< |
\begin{figure} |
| 551 |
< |
\includegraphics[width=\linewidth]{figures/P2_3.pdf} |
| 552 |
< |
\caption{Computed ligand and interfacial solvent orientational $P_2$ |
| 553 |
< |
values for 4 sizes of solvated nanoparticles that are bare or |
| 554 |
< |
protected with a 50\% coverage of C$_{4}$, C$_{8}$, or C$_{12}$ |
| 555 |
< |
alkanethiolate ligands. Increasing stiffness of the ligand orients |
| 556 |
< |
these molecules normal to the particle surface, while the length |
| 557 |
< |
of the ligand chains works to prevent solvent from lying flat on |
| 558 |
< |
the surface.} |
| 559 |
< |
\label{fig:NPthiols_combo} |
| 547 |
> |
\label{fig:NPthiols_corrugation} |
| 548 |
|
\end{figure} |
| 549 |
|
|
| 550 |
|
Because the thiolate ligands do not significantly alter the larger |
| 601 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 602 |
|
\subsection{Orientation of Ligand Chains} |
| 603 |
|
|
| 604 |
< |
As the ligand chain length increases in length, it exhibits |
| 604 |
> |
As the saturated ligand chain length increases in length, it exhibits |
| 605 |
|
significantly more conformational flexibility. Thus, different lengths |
| 606 |
|
of ligands should favor different chain orientations on the surface of |
| 607 |
|
the nanoparticle. To determine the distribution of ligand orientations |
| 608 |
< |
relative to the particle surface we examine the probability of |
| 609 |
< |
finding a ligand with a particular orientation relative to the surface |
| 610 |
< |
normal of the nanoparticle, |
| 608 |
> |
relative to the particle surface we examine the probability of finding |
| 609 |
> |
a ligand with a particular orientation relative to the surface normal |
| 610 |
> |
of the nanoparticle, |
| 611 |
|
\begin{equation} |
| 612 |
|
\cos{(\theta)}=\frac{\vec{r}_i\cdot\hat{u}_i}{|\vec{r}_i||\hat{u}_i|} |
| 613 |
|
\end{equation} |
| 625 |
|
|
| 626 |
|
\begin{figure} |
| 627 |
|
\includegraphics[width=\linewidth]{figures/NP_pAngle} |
| 628 |
< |
\caption{{\bf The two extreme cases of ligand orientation relative |
| 629 |
< |
to the nanoparticle surface: the ligand completely |
| 630 |
< |
outstretched ($\cos{(\theta)} = -1$) and the ligand fully |
| 631 |
< |
lying down on the particle surface ($\cos{(\theta)} = 0$).}} |
| 628 |
> |
\caption{The two extreme cases of ligand orientation relative to the |
| 629 |
> |
nanoparticle surface: the ligand completely outstretched |
| 630 |
> |
($\cos{(\theta)} = -1$) and the ligand fully lying down on the |
| 631 |
> |
particle surface ($\cos{(\theta)} = 0$).} |
| 632 |
|
\label{fig:NP_pAngle} |
| 633 |
|
\end{figure} |
| 634 |
|
|
| 647 |
– |
|
| 648 |
– |
|
| 649 |
– |
% \begin{figure} |
| 650 |
– |
% \includegraphics[width=\linewidth]{figures/thiol_pAngle} |
| 651 |
– |
% \caption{} |
| 652 |
– |
% \label{fig:thiol_pAngle} |
| 653 |
– |
% \end{figure} |
| 654 |
– |
|
| 635 |
|
An order parameter describing the average ligand chain orientation relative to |
| 636 |
|
the nanoparticle surface is available using the second order Legendre |
| 637 |
|
parameter, |
| 643 |
|
$P_2$ values of 1, while ligand populations lying flat on the |
| 644 |
|
nanoparticle surface have $P_2$ values of $-0.5$. Disordered ligand |
| 645 |
|
layers will exhibit mean $P_2$ values of 0. As shown in Figure |
| 646 |
< |
\ref{fig:NPthiols_combo} the ligand $P_2$ values approaches 0 as |
| 646 |
> |
\ref{fig:NPthiols_P2} the ligand $P_2$ values approaches 0 as |
| 647 |
|
ligand chain length -- and ligand flexibility -- increases. |
| 648 |
|
|
| 649 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 681 |
|
bare particles, but are not as randomly oriented as the longer ligand |
| 682 |
|
lengths. |
| 683 |
|
|
| 684 |
+ |
\begin{figure} |
| 685 |
+ |
\includegraphics[width=\linewidth]{figures/P2_3.pdf} |
| 686 |
+ |
\caption{Computed ligand and interfacial solvent orientational $P_2$ |
| 687 |
+ |
values for 4 sizes of solvated nanoparticles that are bare or |
| 688 |
+ |
protected with a 50\% coverage of C$_{4}$, C$_{8}$, or C$_{12}$ |
| 689 |
+ |
alkanethiolate ligands. Increasing stiffness of the ligand orients |
| 690 |
+ |
these molecules normal to the particle surface, while the length |
| 691 |
+ |
of the ligand chains works to prevent solvent from lying flat on |
| 692 |
+ |
the surface.} |
| 693 |
+ |
\label{fig:NPthiols_P2} |
| 694 |
+ |
\end{figure} |
| 695 |
+ |
|
| 696 |
|
These results are particularly interesting in light of our previous |
| 697 |
|
results\cite{Stocker:2013cl}, where solvent molecules readily filled |
| 698 |
|
the vertical gaps between neighboring ligand chains and there was a |
| 790 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 791 |
|
% **ACKNOWLEDGMENTS** |
| 792 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 793 |
< |
\begin{acknowledgements} |
| 793 |
> |
\begin{acknowledgments} |
| 794 |
|
Support for this project was provided by the National Science Foundation |
| 795 |
|
under grant CHE-1362211. Computational time was provided by the |
| 796 |
|
Center for Research Computing (CRC) at the University of Notre Dame. |
| 797 |
< |
\end{acknowledgements} |
| 797 |
> |
\end{acknowledgments} |
| 798 |
|
|
| 799 |
|
\newpage |
| 800 |
|
\bibliographystyle{aip} |
