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\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 theAg100 system and the  Ag111 system iscompared, and the distance between its location is  2.81 Angstrom. 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 Angstromaway  from the top PVP layer is treated as if the Ag atom is absorbed  inbulk solution because it will take a very long time for  the Ag atom to diffuse back to the PVP layer. 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 
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\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 lead propagate  to a particular absorbing state from 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.