# When The Dealloying Process Is Started English Language Essay

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When the dealloying process is started, we assume that the alloy is in a quasi-equilibrium state. However, the corresponding atomic arrangement is not known a priori. Therefore, in the equilibration process, we take as the initial arrangement a random distribution of particles over the FCC lattice and use the MMC method to find the equilibrium state for the current set of input parameters. Evidently, in this phase of the simulation we do not allow for chemical reaction processes. The relaxation process was performed by gradual cooling down (annealing) to the desired dealloying temperature. This procedure forces the system to changes gradually from the initial random rearrangement of atoms to an equilibrated at certain temperature state. A typical example of cooling a system from infinite temperature down to 300 K is presented in Fig.~\ref{fig5.1} where we plot the total energy $E$ as a function of the number of MMC cycles per particle $n$. Three plateaus indicate that three quasi-equilibrium states have been reached, corresponding to the specified temperatures 400 K, 350 K and 300 K respectively. Thus, the initial configuration of the system, which corresponds to the infinite temperature, equilibrated to three configurations corresponding to these three temperatures, at which the dealloing processes are going to take place.

\begin{figure}[h]

\begin{center}

\includegraphics[width=13cm]{./6/relax.pdf}

\caption{Relaxation of the total energy $E$ of the system as a function of $n$, the number of MMC steps per particle. For $n<3 \times 10^4$, the temperature $T=400~$K, for $3 \times 10^4<n<6 \times 10^5$ , the temperature $T=350$~K and for $n>6 \times 10^5$, the temperature $T=300$~K. The plateaus indicate that quasi-equilibrium states were reached. System interaction constants are $J_{11}=0.047~$eV,~$J_{22}=0.043~$eV,~$J_{12}=0.037~$eV. The system size is $100\times 100\times 50$ ($29.1\times 29.1\times 14.4~$nm$^3$).}

\label{fig5.1}

\end{center}

\end{figure}

\begin{figure}[h]

\begin{center}

\includegraphics[width=11cm]{./6/phase1BW.pdf}

\caption{"Phase diagram" of equilibrated structures. Snapshots of a cross sections of the Au$_{25}$Ag$_{75}$ (wt.~\%) systems were taken after equilibration. Black color corresponds to the gold particles and light-gray color corresponds to silver atoms. The gold-gold interaction $J_{11}$ increases from the left column to the right column in the range of $[0.045, 0.049]~$eV in steps of $0.001~$eV. The silver-silver interaction $J_{22}$ increases from the top row to the bottom row in the range of $[0.035, 0.049]~$eV in steps of $0.002~$eV. The gold-silver interaction $J_{12}$ decreases from the left column to the right column in the range of $[0.039, 0.035]~$eV in steps of $0.001~$eV. $T=300$~K. The systems sizes are $100\times 100\times 50$ ($29.1\times 29.1\times 14.4~$nm$^3$).}

\label{fig5.2}

\end{center}

\end{figure}

\begin{figure}[ht]

\begin{minipage}[b]{0.45\linewidth}

\centering

\includegraphics[width=6.25cm]{./6/mixedBW.pdf}

Mixed

\end{minipage}

\hspace{0.5cm}

\begin{minipage}[b]{0.45\linewidth}

\centering

\includegraphics[width=6.25cm]{./6/clusteredBW.pdf}

Clustered

\end{minipage}

\caption{Snapshots of the atomically mixed and clustered configurations at the temperature $T=300~$K. Only particles of gold are shown. No cluster formation is observed in the case of the mixed configuration. The clustered configuration yields agglomerations of the gold particles. System interaction constants are $J_{11}=0.041~$eV,~$J_{22}=0.047~$eV,~$J_{12}=0.037~$eV for the mixed configuration and $J_{11}=0.049$~eV,$~J_{22}=0.044~$eV,$~J_{12}=0.035~$eV for the clustered configuration. The systems sizes are $100\times 100\times 50$ ($29.1\times 29.1\times 14.4~$nm$^3$).}

\label{3dconf}

\end{figure}

\begin{figure}[h]

\begin{center}

\includegraphics[width=10cm]{./6/mixedEA.pdf}

\caption{The Euler characteristic $\chi$ (squares) and the specific surface area $A$ (circles) of gold particles as a function of a temperature $T$. The negative values of the $\chi$ indicate that gold particles mix with silver. The sharp drop in $\chi$ signals the transition into mixed configurations. The largest value of the specific surface area $A$ corresponds to the most mixed system. Near the temperature $T=250$~K, a transition from a clustered to a mixed configurations is observed. System interaction constants are $J_{11}=0.041~$eV,~$J_{22}=0.047~$eV,~$J_{12}=0.037~$eV. The systems sizes are $100\times 100\times 50$ ($29.1\times 29.1\times 14.4~$nm$^3$).}

\label{fig5.3}

\end{center}

\end{figure}

\begin{figure}[h]

\begin{center}

\includegraphics[width=10cm]{./6/clusteredEA.pdf}

\caption{The Euler characteristic $\chi$ (squares) and the specific surface area $A$ (circles) of gold particles as a function of a temperature $T$. The largest value of $\chi$ corresponds to the configuration with the largest amount of disconnected gold clusters. The largest value of a specific surface area $A$ corresponds to the most mixed system. Near the temperature $T=400$~K, a transition from a clustered to a mixed configurations is observed. System interaction constants are $J_{11}=0.049$~eV,$~J_{22}=0.044~$eV,$~J_{12}=0.035~$eV. The systems sizes are $100\times 100\times 50 (29.1\times 29.1\times 14.4~$nm$^3$).}

\label{fig5.4}

\end{center}

\end{figure}

\hspace{10 mm} Repeating the equilibration simulation for different choices of the interaction strength between gold and silver yields the "phase diagram" presented in Fig.~\ref{fig5.2}. In the Fig.~\ref{fig5.2} presented cross sections of the Au$_{25}$Ag$_{75}$ (wt.~\%) systems after equilibration process. Here, the black color corresponds to the gold atoms and the light-gray color corresponds to the silver atoms. The gold-gold interaction $J_{11}$ increases from the left column to the right column in the range of $[0.045, 0.049]~$eV in steps of $0.001~$eV. The silver-silver interaction $J_{22}$ increases from the top row to the bottom row in the range of $[0.035, 0.049]~$eV in steps of $0.002~$eV. The gold-silver interaction $J_{12}$ decreases from the left column to the right column in the range of $[0.039, 0.035]~$eV in steps of $0.001~$eV.

\hspace{10 mm} In this diagram one can clearly see two distinct configurations - the mixed configuration in the top-left corner and the clustered configuration at the bottom-right corner. The distinction between these two types of configuration was done by eye using visualizing program (see section~\ref{visual}).

\hspace{10 mm} Fig.~\ref{3dconf} represents 3D pictures of mixed and clustered configurations. Only particles of gold are shown here. The snapshots was taken for the different systems at the temperature $T=300~$K after the equilibration process. No cluster formation is observed in the case of mixed configuration. for the case of the clustered configuration one can clearly see some agglomerates (clusters) of gold particles which lead us to the conclusion that formation of clusters started already during the relaxation stage. The interaction constants of the system that yields mixed configuration are $J_{11}=0.041~$eV,~$J_{22}=0.047~$eV,~$J_{12}=0.037~$eV. The interaction constants of the system that yields clustered configuration are $J_{11}=0.049$~eV,$~J_{22}=0.044~$eV,$~J_{12}=0.035~$eV.

\hspace{10 mm} Atomically mixed configurations yield a negative value of Euler characteristic which is of the order of $N_{Au}$, number of gold particles in the system. In the case of a clustered configuration the Euler characteristic $\chi \sim N_{Cl}$, where $N_{Cl}$ is a number of disconnected gold clusters in the system. In simple words the Euler characteristic is equal to the number of disconnected gold clusters minus the number of tunnels between gold particles in the system.

\hspace{10 mm} The evolution of the Euler characteristic and specific surface area of gold structures (disregarding silver) during the relaxation process is shown in Fig.~\ref{fig5.3} and Fig.~\ref{fig5.4}. Two different sets of interaction constants were chosen for these simulations, corresponding to the two different types of equilibrium configurations. Relaxation with one set of interaction constants leaves the system in an atomically mixed configuration and relaxation with another set produces gold clusters immersed in silver (Fig.~\ref{3dconf}).

\hspace{10 mm} Prior to dissolution the alloy must be homogeneous with no phase separation \cite{ERL03+}. Porosity evolution thus forms dynamically during dissolution and is not due to one phase simply being excavated out of the two-phase material \cite{ERL03+}. Therefore, interaction constants which lead to clustered configurations may be excluded from further considerations.

\section{Dealloying}

\label{deall}

\hspace{10 mm} The relaxation process changes the infinite temperature (random) configuration of the gold and silver atoms into a typical equilibrium configuration at the temperature at which the dealloying is going to take place. To start the dealloying, the only change to the Monte Carlo algorithm is the inclusion of the chemical reaction processes. The simulation of the dealloying is performed using the same set of model parameters $J_{11}, J_{12}$ and $J_{22}$ as in the equilibration process. The fact that in the simplified model Eq.~\ref{energy}, there is no nearest-neighbor interaction, between acid and metal particles does not prevent a chemical reaction between these particles to take place. Indeed, by construction, in the Monte Carlo algorithm such reactions can take place whenever a silver and acid particles are nearest neighbors.

\hspace{10 mm} A "phase diagram" of dealloyed structures is presented in Fig.~\ref{fig5.5}. The diagram is made of snapshots of a cross sections of the Au$_{25}$Ag$_{75}$ (wt.~\%) dealloyed systems. Black color corresponds to the gold particles and light-gray color corresponds to silver atoms. The gold-gold interaction $J_{11}$ increases from the left column to the right column in the range of $[0.045, 0.049]~$eV in steps of $0.001~$eV. The silver-silver interaction $J_{22}$ increases from the top row to the bottom row in the range of $[0.035, 0.049]~$eV in steps of $0.002~$eV. The gold-silver interaction $J_{12}$ decreases from the left column to the right column in the range of $[0.039, 0.035]~$eV in steps of $0.001~$eV.

\hspace{10 mm} In the diagram Fig.~\ref{fig5.5} one can distinguish three distinct phases - configurations with gold clusters that accumulated mostly in the bottom of the sample (e.g. the system with $J_{11}=0.045~$eV,~$J_{22}=0.035~$eV,~$J_{12}=0.039~$eV,~$T=300~$K), configurations with disordered NPG (e.g. the system with $J_{11}=0.047~$eV,~$J_{22}=0.043~$eV,~$J_{12}=0.037~$eV,~$T=300~$K) and configurations which correspond to clustered equilibrated configurations (e.g. the system with $J_{11}=0.049~$eV,~$J_{22}=0.049~$eV,~$J_{12}=0.035~$eV,~$T=300~$K). Clearly, we are not interested in the first phase because it is not porous structures. The third phase corresponds to the clustered configurations in the "phase diagram" of equilibrated structures (see Fig.~\ref{fig5.2}). Thus, the third phase also is not of interest for the present purpose.

\begin{figure}[h]

\begin{center}

\includegraphics[width=11cm]{./6/phase2BW2.pdf}

\caption{"Phase diagram" of dealloyed structures. Snapshots of a cross sections of the Au$_{25}$Ag$_{75}$ (wt.~\%) dealloyed systems. Black color corresponds to the gold particles. The gold-gold interaction $J_{11}$ increases from the left column to the right column in the range of $[0.045, 0.049]~$eV in steps of $0.001~$eV. The silver-silver interaction $J_{22}$ increases from the top row to the bottom row in the range of $[0.035, 0.049]~$eV in steps of $0.002~$eV. The gold-silver interaction $J_{12}$ decreases from the left column to the right column in the range of $[0.039, 0.035]~$eV in steps of $0.001~$eV. $T=300~$K. The systems sizes are $100\times 100\times 50 (29.1\times 29.1\times 14.4~$nm$^3$).}

\label{fig5.5}

\end{center}

\end{figure}

\hspace{10 mm} Comparing the two "phase diagrams" (Fig.~\ref{fig5.2} and Fig.~\ref{fig5.5}), we may conclude that the NPG structures can only be obtained within a narrow range of interaction parameters. Within this range, these structures are robust to changing of interaction parameters.

\hspace{10 mm} Tuning the temperature for dealloying we could also produce all of these three phases (see Fig.~\ref{fig5.6}). As the example, the system with the interaction constants $J_{11}=0.047~$eV,~$J_{22}=0.041~$eV and ~$J_{12}=0.037~$eV was chosen. The temperature range is $[200, 400]~$K. The first column in the Fig.~\ref{fig5.6} corresponds to the third phase where equilibrated structure has clustered configuration. Second and third columns could be considered as second phase which leads to the nanoporous structures. The last two columns represent first phase where gold clusters are accumulated mostly in the bottom. Thus, simulations with different temperatures produce distinct structures that exhibit all of these three "phases".

\hspace{10 mm} The evolution of the Euler characteristic and the specific surface area of gold particles during the dealloying process are shown in Fig.~\ref{ea1} and Fig.~\ref{ea2}. In Fig.~\ref{ea1} and Fig.~\ref{ea2} the plateau-like regions correspond to the stage when acid reaches the bottom of the sample. The constant Euler characteristic during this stage is due to the fact that the system exhibits percolation. In this stage, creating additional holes in the system is very difficult. The slow decrease of a specific surface area $A$ with the increasing number of MMC steps is due to annealing of the sample.

\hspace{10 mm} The first configuration (see Fig.~\ref{ea1}) with interaction constants $J_{11}=0.047~$eV, ~$J_{22}=0.043~$eV and ~$J_{12}=0.037~$eV represents the structure which is on the bound between two clustered and mixed configurations in equilibrated "phase diagram" (see Fig.~\ref{fig5.2}). The same configuration corresponds to the spongy-like structure in the "phase diagram" of dealloyed structures (see Fig.~\ref{fig5.5}).

\hspace{10 mm} The second configuration (see Fig.~\ref{ea2}) with interaction constants $J_{11}=0.046~$eV,~$J_{22}=0.035~$eV and ~$J_{12}=0.038~$eV represents the structure which is far away (top-left corner) from the bound between two clustered and mixed configurations in equilibrated "phase diagram" (see Fig.~\ref{fig5.2}). The same configuration corresponds to a dealloyed structure with gold clusters which are accumulated mostly in the bottom of the "phase diagram" of dealloyed structures (see Fig.~\ref{fig5.5}).

\begin{figure}[h]

\begin{center}

\bigskip

\includegraphics[width=11cm]{./6/phaseTBW.pdf}

\caption{Snapshots of cross sections of equilibrated stuctures of the Au$_{25}$Ag$_{75}$ (wt.~\%) sample (bottom) and corresponding dealloyed structures (top) for different temperatures. Temperature increases from the left column to the right column in the range of $[200, 400]~$K in steps of $50~$K. Black color corresponds to the gold particles and light-gray color corresponds to silver atoms. System interaction constants are $J_{11}=0.047~$eV,~$J_{22}=0.041~$eV,~$J_{12}=0.037~$eV. The systems sizes are $100\times 100\times 50 (29.1\times 29.1\times 14.4~$nm$^3$).}

\label{fig5.6}

\end{center}

\end{figure}

\begin{figure}[h]

\begin{center}

\includegraphics[width=10cm]{./6/deal1EA.pdf}

\caption{The Euler characteristic $\chi$ (squares) and the specific surface area $A$ (circles) of gold particles as a function of $n$, the number of MMC steps per particle. The constant Euler characteristics $\chi$ during the final stage indicate that the systems are percolating. The slow decrease of $A$ at the final stage of the simulation is due to further equilibration of the samples. System interaction constants are $J_{11}=0.047~$eV,~$J_{22}=0.043~$eV,~$J_{12}=0.037~$eV,~$T=300~$K. The systems sizes are $100\times 100\times 50 (29.1\times 29.1\times 14.4~$nm$^3$).}

\label{ea1}

\end{center}

\end{figure}

\begin{figure}[h]

\begin{center}

\includegraphics[width=10cm]{./6/deal2EA.pdf}

\caption{The Euler characteristic $\chi$ (squares) and the specific surface area $A$ (circles) of gold particles as a function of $n$, the number of MMC steps per particle. The constant Euler characteristics $\chi$ during the final stage indicate that the systems are percolating. The slow decrease of $A$ at the final stage of the simulation is due to further equilibration of the samples. System interaction constants are $J_{11}=0.046~$eV,~$J_{22}=0.035~$eV,~$J_{12}=0.038~$eV,~$T=300~$K. The systems sizes are $100\times 100\times 50 (29.1\times 29.1\times 14.4~$nm$^3$).}

\label{ea2}

\end{center}

\end{figure}

\section{Morphological analysis}

\hspace{10 mm} First, we present some examples of 2D slices of the system during the dealloying process, see Fig.~\ref{fig5.8}. In the figure the black color corresponds to the gold particles and light-gray color corresponds to silver atoms. In this figure we can track the dynamics of the typical dealloying process. Note that the only difference of the dealloying algorithm from the equilibration algorithm is in turning on the chemical reaction processes. The dealloying was started at the same temperature at which this system was equilibrated. Fig.~\ref{fig5.8} clearly show a dynamics of the rearrangement of gold atoms from mixed phase to the clustered phase. The 3D picture of the dealloyed sample, the results of the analysis of the geometry of dealloyed structure and simulation parameters are shown in Fig.~\ref{fig5.9}. The system has heterogeneity in $z$ direction due to the fact that acid penetrates the system from top to bottom (see Fig.~\ref{bottom}). Several "bottom" layers of the dealloyed system contain enhanced amounts of gold particles due to the boundary condition in this direction (Fig.~\ref{fig5.8}d, Fig.~\ref{fig5.9}). Qualitatively, the pattern of the simulated system (see Fig.~\ref{fig5.10}) is similar to the microscopy picture of the dealloyed sample (see Fig.~\ref{fig1}b). From the 2D snapshot (see Fig.~\ref{fig5.8}a) and the 3D snapshot (see Fig.~\ref{3dconf}) of the equilibrated system and the behavior of the Euler characteristic (see Fig.~\ref{fig5.3}) we may conclude that gold (black) and silver (light-gray) are entirely mixed. Vacancies in the sample are represented by voids that are randomly located in the initial configuration. During the equilibration, the vacancies merge into "clusters" (see clusters of white pixels in Fig.~\ref{fig5.8}a)). During the dealloying, vacancies can only appear as the result of the chemical reactions Eq.~\ref{chem1}. In this reaction, three particles are converted into two particles and one gas molecule. The latter is not explicitly taken into account in the simulation. Phrased differently, our simulation does not distinguish between true vacancies and sites occupied by a gas molecule. The dealloying process is stopped when 99~\% of the initial amount of silver was dissolved. The dealloyed structure is entirely connected.

\begin{figure}[h]

\begin{center}

\includegraphics[width=8cm]{./6/abcdBW.pdf}

\caption{Snapshots of cross sections of the Au$_{25}$Ag$_{75}$ (wt.~\%) sample taken during the MMC simulation after n MMC steps per particle. Black color corresponds to the gold particles and light-gray color corresponds to silver atoms. \textbf{a}: $n = 0$; \textbf{b}: $n = 5\times 10^2$; \textbf{c}: $n = 2\times 10^3$ ; \textbf{d}: $n = 6\times 10^3$. System interaction constants are $J_{11}=0.047~$eV,~$J_{22}=0.043~$eV,~$J_{12}=0.037~$eV. The systems size is $100\times 100\times 50 (29.1\times 29.1\times 14.4~$nm$^3$).}

\label{fig5.8}

\end{center}

\end{figure}

\begin{figure}[h]

\begin{center}

\includegraphics[width=10cm]{./6/iso3d+.pdf}

\caption{Typical distribution of the nanoporous structure as obtained by simulating chemical dealloying of a Au/Ag sample. View from the "top" of the dealloyed system. System interaction constants are $J_{11}=0.047~$eV,~$J_{22}=0.041~$eV,~$J_{12}=0.037~$eV.~$T=300~$K. The systems size is $100\times 100\times 50~ (29.1\times 29.1\times 14.4~$nm$^3$). Volume of the system $V=4524.0~$nm$^3$. Surface area $S=14663.9~$nm$^2$. Specific surface area $A=3.24~$1/nm. The Euler characteristic $\chi=46$. The ligament size is of the order of 2-3~nm.}

\label{fig5.9}

\end{center}

\end{figure}

\begin{figure}[h]

\begin{center}

\includegraphics[width=10cm]{./6/iso3d2+.pdf}

\caption{Bottom view of the dealloyed system. Heterogeneity in $z$ direction is due to the fact that acid penetrates the system from top to bottom. Several layers in the bottom of the system contain an enhanced concentration of gold particles due to the boundary condition at the bottom layer. The system parameters are the same as in Fig.~\ref{fig5.9}.}

\label{bottom}

\end{center}

\end{figure}

\begin{figure}[h]

\begin{center}

\includegraphics[width=7.5cm]{./6/iso+.pdf}

\caption{Plane cut taken from the system shown in Fig.~\ref{fig5.9}, showing the typical distribution of gold particles as obtained by simulating chemical dealloying of an Au/Ag sample.}

\label{fig5.10}

\end{center}

\end{figure}