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\begin{document}
\title{How structure-directing agents control nanocrystal shape: PVP-mediated growth of Ag nanocubes}
\author{Tonnam Balankura}
\affil{Affiliation not available}
\date{\today}
\maketitle
\section{Kinetic Wulff Plot}
Away from equilibrium, the NC shape is governed by the kinetics of inter- and intrafacet atom diffusion, as well as by the kinetics of deposition to various facets. At nonequilibrium growth conditions, the resulting shapes are expected to be different from the thermodynamic shapes. Examples of well-known kinetic shapes include nanowires and highly branched (bi- and tripods) structures \cite{Xiong_2007}. When NCs grow beyond a critical size, the relative atom deposition rate to various facets becomes a major influence in the NC shape. In this kinetically-controlled growth regime, the kinetic Wulff construction can predict the shape evolution of faceted crystal growth based on the surface kinetics \cite{Du_2005,frank1958growth,osher1997wulff}. Using 3-dimensional shape evolution calculation method \cite{Zhang_2006}, we correlate the relative flux of Ag atom deposition to \{111\} and \{100\} facets $\frac{F_{111}}{F_{100}}$ and the resulting kinetic Wulff shape in the reversible octahedron-to-cube transformation. This transformation is observed in the seed-mediated growth of Ag NCs \cite{Xia_2012}, in which the shape-controlling parameter is the concentration of poly(vinylpyrrolidone) (PVP) in the solution. The constructed kinetic Wulff plot is shown in Fig. \ref{fig:kinetic-wulff}. The construction of the kinetic Wulff plot is described in the supporting information. When the relative flux to \{111\} facets is less than half of the flux to \{100\} facets, the octahedra is predicted as the kinetic Wulff shape. As $\frac{F_{111}}{F_{100}}$ increases, we observe a shape progression from octahedra to cubo-octahedra, then to truncated cubes, and eventually to cubes at $\frac{F_{111}}{F_{100}} \geq \sqrt{3}$.
To study the mechanism by which SDAs impart shape selectivity, we use the seed-mediated Ag polyol synthesis in the presence of PVP \cite{Xia_2012} as our model. We utilize large-scale MD simulations to quantify $F_{100}$ and $F_{111}$ using \textit{in-silico} deposition and potential of mean force calculation.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/phase-composed-rotated/phase-composed-rotated}
\caption{{\label{fig:kinetic-wulff} The correlation between the fraction of surface covered by {100} facets and the required relative
flux in a reversible octahedron-to-cube transformation. Examples of predicted shapes are given, with the {100} facets are colored in green and the {111} facets are colored in orange.%
}}
\end{center}
\end{figure}
\section{Findings from Potential of Mean Force Calculations}
<<<<<<< HEAD
We explored the potential of mean force (PMF) along the absorption path of Ag atoms from solution phase to the Ag NC surface, with the goal of gaining quantitative insight of the influence of the adsorbed PVP layer. To calculate the PMF of the Ag atom, we use umbrella sampling \cite{K_stner_2011} with harmonic bias potential on the canonical molecular dynamics simulation of the previously described system for the \textit{in-silico} deposition shown in Fig \ref{fig:sim-setup}. Umbrella sampling is used to enhance the sampling because the free energy barrier of absorption is greater than $k_B T$. Umbrella integration \cite{Ka_stner_2005} is used to combine data from individual windows sampled, also yielding a statistical error of the PMF calculated \cite{Ka_stner_2006}. The reaction coordinate of the PMF is the orthogonal axis of the Ag slab, with the origin at the surface layer of the bottom slab. Further description of the PMF calculation methods can be found in the supporting information. In this section, we will present our result of the PMF profile of the Ag atom and calculate the relative atom flux to \{111\} and \{100\} facets $\frac{F_{111}}{F_{100}}$ using the framework of transition-state theory \cite{H_nggi_1990}.
The calculated PMF profile of the Ag atom along the orthogonal axis of the Ag slab with Ag100 and Ag111 surfaces is shown in Fig. \ref{fig:pmf}. The Ag atom approaching the surface goes through the PMF profile from the right to left. On the far right, the PMF is a flat maxima, which is where the Ag atom is in bulk solvent. As the Ag atom move closer to the surface, it interacts with the PVP monolayer which causes the PMF to decline from the flat maxima. The PMF declines until it reaches a local basin trapped by an energy barrier, which is caused by the hindering effect of the network of PVP anchored on the surface as observed in the \textit{in-silico} deposition trajectories. Once the Ag atom overcomes the energy barrier, it reaches an energy minimum where the Ag atom is absorbed onto the surface.
Using the framework of transition-state theory \cite{H_nggi_1990}, we can obtain the rate constant of atom flux from the calculated PMF profile. Methods are described in the supporting information. The rate constant of atom flux towards Ag111 and Ag100 is calculated to be 25.5 ns^{-1} and 12.2 ns^{-1}, respectively. From the rate constants calculate, the ratio of rate constants $\frac{k_{111}}{k_{100}}$ is 2.10. The atom flux calculated by transition-state theory is one order-of-magnitude larger than the atom flux calculated by \textit{in-silico} deposition. This is likely to be a consequence from the neglecting recrossings, which causes the over-estimation of the atom flux by the transition state theory. The ratio of recrossings to successful crossings as high as 10 has been shown in the literature \cite{Pritchard_2005}, which is possible for our system where the energy barrier is only 2 to 4 $k_B T$. We focus more on the accuracy of the relative flux $\frac{F_{111}}{F_{100}}$ by sufficient sampling of the domain space because it can be used to define the kinetic Wulff shape of the grown NCs.
To calculate the relative flux $\frac{F_{111}}{F_{100}}$, we also need to obtain the ratio of trapping coefficient $\frac{P_{111}}{P_{100}}$. Higher mean fluffiness of the PVP layer adsorbed on the Ag111 surface associates with higher trapping coefficient, which is reflected by the further distance for the PMF to reach the maxima plateau. We use a linear absorbing Markov chain \cite{kemeny1976finite} to model how the difference in PVP layer fluffiness affects the ratio of trapping coefficient. The length from the top PVP layer to the bulk solution is divided into discrete Markov states. The absorbing states are at the top PVP layer and at the bulk solution. The difference in PVP layer fluffiness is reflected by a longer Markov chain for the Ag100 system than the Ag111 system. Further description of the absorbing Markov chain is in the supporting information. We calculate the ratio of trapping coefficient $\frac{P_{111}}{P_{100}}$ to be 1.21 from our Markov chain model of the PMF profile. The obtained relative flux $\frac{F_{111}}{F_{100}}$ from $\frac{k_{111}}{k_{100}} \times \frac{P_{111}}{P_{100}}$ is 2.541, which predicts that the kinetic Wulff shape is a cube as shown in Fig. \ref{fig:kinetic-wulff}. Slight discrepancy from the relative flux calculated by \textit{in-silico} deposition is likely a consequence of recrossing frequency difference for the Ag100 and Ag111 surface because the smaller energy barrier permits more recrossing possibilities.
=======
We explored the potential of mean force (PMF) along the absorption path of an Ag atom from solution phase to the Ag NC surface, with the goal of gaining quantitative insight of the influence of the adsorbed PVP layer. To calculate the PMF profile of the Ag atom, we use umbrella sampling \cite{K_stner_2011} with a harmonic bias potential on the canonical molecular dynamics simulation. We use the same system as for the \textit{in-silico} deposition, which is shown in Fig \ref{fig:sim-setup}. Umbrella sampling is used to enhance the sampling because the free energy barrier of absorption is greater than $k_B T$. We use the umbrella integration method \cite{Ka_stner_2005} to combine data from individual windows sampled. The advantage of the umbrella integration method over the conventional weight-histogram analysis method (WHAM) is the independence of the number of grid points and one can obtain the statistical error directly through umbrella integration \cite{Ka_stner_2006}. The reaction coordinate of the PMF is the orthogonal axis of the Ag slab, with the origin at the surface layer of the bottom slab. Further description of the PMF calculation methods can be found in the supporting information. In this section, we present our result of the PMF profile of the Ag atom and calculate the relative atom flux to \{111\} and \{100\} facets $\frac{F_{111}}{F_{100}}$ using transition-state theory.
>>>>>>> parent of 55b9e94... edited section_Findings_from_PMF.tex\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Xin-Simulation-Setup/Xin-Simulation-Setup}
\caption{{\label{fig:sim-setup} Snapshots from MD simulations. (a) An example initial configuration of deposition on Ag(100) facet. A free Ag atom is inserted at the bulk-PVP interphase (dashed blue line) with an initial velocity facing the surface. A successful deposition is counted when the atom reaches the surface, and the atom fails to grow the surface if it passes the dashed red line. (b) A demonstration of how an Ag atom accomplish deposition: when an Ag atom (yellow) arrives at the bulk-PVP interphase (A), it can be attracted into the PVP film by interacting with O atoms (red) on the pyrrolidone ring (B); the Ag atom is retained in the PVP layer but the direct deposition is hindered by the structure of PVP (C). The deposition finished when a "hole" of suitable size opens (D).%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/PMF2/PMF2}
\caption{{\label{fig:pmf} Potential of mean force of a Ag atom along the orthogonal axis of the Ag slab with Ag100 and Ag111 surfaces. The potential of mean force of each profile have been adjusted so that the potential of mean forces at distance far away from the Ag surfaces are equal.%
}}
\end{center}
\end{figure}
\section{Supporting Information}
\subsection{Kinetic Wulff Plot}
Test
\subsection{Potential of Mean Force Calculation}
To calculate the potential of mean force (PMF) profile, we use umbrella sampling \cite{K_stner_2011} with harmonic bias potential on the canonical molecular dynamics simulation and we combine data from different windows using umbrella integration \cite{Ka_stner_2005}. The harmonic bias $\omega_i (\xi)$ used has the form
\begin{equation}
\label{eqn:harmonic}
\omega_i (\xi) = K/2 (\xi - \xi^{ref}_i)^2
\end{equation}
where $K$ is harmonic bias constant, $\xi$ is the reaction coordinate, and $\xi^{ref}_i$ is the reference point of each window $i$. For our model, the harmonic bias constant of 16.14 kcal mol^{-1} Angstrom^{-2} is sufficient to enhance the sampling. We sampled 100 windows, with window spacing of 0.38 Angstrom.
\subsection{Transition-State Theory}
Using the framework of transition-state theory \cite{H_nggi_1990}, we can calculate the rate constant of atom flux from the PMF profile by
\begin{equation}
\label{eqn:tst}
k^{TST}_{A \rightarrow B}=\frac{1}{2}(\frac{2}{\pi \beta m})^{1/2} \frac{\int d\textbf{x} \Theta_A \delta^{\dagger}_{AB} \exp(-\beta V)}{\int d\textbf{x} \Theta_A \exp(-\beta V)},
\end{equation}
where $\textbf{x}$ is the 3\textit{N}-dimensional configuration of the \textit{N}-particle system, $\beta = 1/k_B T$, $m$ is the effective mass, $V$ is the potential energies, $\Theta_A$ has a value of one if the system is in state $\textbf{A}$ and is zero otherwise, $\delta^{\dagger}_{AB}$ has a value of one if the system is at the dividing hypersurface and is zero otherwise, and the factor of $1/2$ limits the flux to trajectories that are exiting from $\textbf{A}$. We neglect the recrossings of the dividing surface once the trajectories exits from $\textbf{A}$.
For our model, the dividing hypersurface is at the energy barrier, state $\textbf{A}$ is the local basin on the right side of the energy barrier, state $\textbf{B}$ is the energy minimum on the left side of the energy barrier, $\textbf{x}$ is the reaction coordinate, $m$ is the mass of one Ag atom, and $V$ is the potential of mean force. The domain space of state $A$ is the region within one $k_B T$ from the local basin.
\subsection{Absorbing Markov chain}
The length from the top PVP layer to the bulk solution is divided into discrete Markov states. The absorbing states are at the bulk solution and at the top PVP layer where the PMF starts to decline. The difference in PVP layer fluffiness is reflected by a longer Markov chain for the Ag100 system than the Ag111 system. The exact location of the declining point of the PMF is obtained from the first derivative of the PMF profile. Moving from right to left, the first maxima of the PMF profile's first derivative is defined as the declining point. The declining point for the Ag111 system is 2.81 Angstrom farther away from the Ag surface than for the Ag111 system. This distance is divided into 50 discrete Markov states, defining the distance between consecutive Markov states to be 0.0562 Angstrom. From the \textit{in-silico} deposition, trajectories with the Ag atom moving further than 13.0 Angstrom from the top PVP layer is treated as if the Ag atom is absorbed in the bulk solution. Therefore we used 13.0 Angstrom as the length of Markov chain of the Ag111 system, which represents the distance from the top PVP layer absorbed to Ag111 surface to the bulk solution.
The transition matrix $P$ which holds the transition probabilities between all Markov states is constructed as following; all states except for the absorbing states have a transition probability of 0.5 to each of their adjacent state, and absorbing states have a transition probability to itself of unity. The transition matrix have the form
\begin{equation}
\label{eqn:markov-p}
P =
\begin{pmatrix}
I_m & 0 \\
R & Q
\end{pmatrix},
\end{equation}
where $I_m$ is an identity matrix of size $m$, which $m$ is the number of absorbing states. The fundamental matrix $F$ describing the number of visits of each state can be calculated as
\begin{equation}
\label{eqn:markov-f}
F = (I_n - Q)^{-1},
\end{equation}
where $I_n$ is an identity matrix of size $n$, which is has the same dimensions as matrix $Q$. The probabilities that a particular initial state will propagate to a particular absorbing state is calculated by taking the matrix product $F.R$. We compute the probability that the Ag atom will be trapped by the PVP for Ag100 and Ag111 systems, taking the same initial conditions defined by the distance from the bulk solution absorbing state. The ratio of trapping coefficient $\frac{P_{111}}{P_{100}}$ is calculated from these probabilities.
\section{Supporting Info}
\subsection{Kinetic Wulff Plot}
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