\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.
