deletions | additions       diff --git a/section_Findings_from_PMF.tex b/section_Findings_from_PMF.tex  index 2e3a787..da45801 100644  --- a/section_Findings_from_PMF.tex  +++ b/section_Findings_from_PMF.tex 
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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 where which causes  the PMF declines 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.