When The Dealloying Process Is Started English Language Essay

When the dealloying procedure is started, we assume that the metal is in a quasi-equilibrium province. However, the corresponding atomic agreement is non known a priori. Therefore, in the equilibration procedure, we take as the initial agreement a random distribution of atoms over the FCC lattice and utilize the MMC method to happen the equilibrium province for the current set of input parametric quantities. Obviously, in this stage of the simulation we do non let for chemical reaction processes. The relaxation procedure was performed by gradual chilling down ( tempering ) to the desired dealloying temperature. This process forces the system to alterations bit by bit from the initial random rearrangement of atoms to an equilibrated at certain temperature province. A typical illustration of chilling a system from infinite temperature down to 300 K is presented in Fig.~
ef { fig5.1 } where we plot the entire energy $ E $ as a map of the figure of MMC rhythms per atom $ n $ . Three tableland indicate that three quasi-equilibrium provinces have been reached, matching to the specified temperatures 400 K, 350 K and 300 K severally. Therefore, the initial constellation of the system, which corresponds to the infinite temperature, equilibrated to three constellations matching to these three temperatures, at which the dealloing procedures are traveling to take topographic point.

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

caption { Relaxation of the entire energy $ E $ of the system as a map of $ n $ , the figure of MMC stairss per atom. For $ n & lt ; 3 imes 10^4 $ , the temperature $ T=400~ $ K, for $ 3 imes 10^4 & lt ; n & lt ; 6 imes 10^5 $ , the temperature $ T=350 $ ~K and for $ n & gt ; 6 imes 10^5 $ , the temperature $ T=300 $ ~K. The tableland indicate that quasi-equilibrium provinces were reached. System interaction invariables are $ J_ { 11 } =0.047~ $ electron volt, ~ $ J_ { 22 } =0.043~ $ electron volt, ~ $ J_ { 12 } =0.037~ $ electron volt. The system size is $ 100 imes 100 imes 50 $ ( $ 29.1 imes 29.1 imes 14.4~ $ nm $ ^3 $ ) . }

label { fig5.1 }

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

caption { “ Phase diagram ” of equilibrated constructions. Snapshots of a cross subdivisions of the Au $ _ { 25 } $ Ag $ _ { 75 } $ ( wt.~ % ) systems were taken after equilibration. Black colour corresponds to the gold atoms and light-gray colour corresponds to silver atoms. The gold-gold interaction $ J_ { 11 } $ additions from the left column to the right column in the scope of $ [ 0.045, 0.049 ] ~ $ electron volt in stairss of $ 0.001~ $ electron volt. The silver-silver interaction $ J_ { 22 } $ additions from the top row to the bottom row in the scope of $ [ 0.035, 0.049 ] ~ $ electron volt in stairss of $ 0.002~ $ electron volt. The gold-silver interaction $ J_ { 12 } $ lessenings from the left column to the right column in the scope of $ [ 0.039, 0.035 ] ~ $ electron volt in stairss of $ 0.001~ $ electron volt. $ T=300 $ ~K. The systems sizes are $ 100 imes 100 imes 50 $ ( $ 29.1 imes 29.1 imes 14.4~ $ nm $ ^3 $ ) . }

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Assorted

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

Clustered

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caption { Snapshots of the atomically assorted and clustered constellations at the temperature $ T=300~ $ K. Merely particles of gold are shown. No bunch formation is observed in the instance of the assorted constellation. The clustered constellation outputs agglomerations of the gold atoms. System interaction invariables are $ J_ { 11 } =0.041~ $ electron volt, ~ $ J_ { 22 } =0.047~ $ electron volt, ~ $ J_ { 12 } =0.037~ $ electron volt for the assorted constellation and $ J_ { 11 } =0.049 $ ~eV, $ ~J_ { 22 } =0.044~ $ electron volt, $ ~J_ { 12 } =0.035~ $ electron volt for the clustered constellation. The systems sizes are $ 100 imes 100 imes 50 $ ( $ 29.1 imes 29.1 imes 14.4~ $ nm $ ^3 $ ) . }

label { 3dconf }

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

caption { The Euler characteristic $ chi $ ( squares ) and the specific surface country $ A $ ( circles ) of gold atoms as a map of a temperature $ T $ . The negative values of the $ chi $ indicate that gold atoms mix with Ag. The crisp bead in $ chi $ signals the passage into assorted constellations. The largest value of the specific surface country $ A $ corresponds to the most assorted system. Near the temperature $ T=250 $ ~K, a passage from a clustered to a assorted constellations is observed. System interaction invariables are $ J_ { 11 } =0.041~ $ electron volt, ~ $ J_ { 22 } =0.047~ $ electron volt, ~ $ J_ { 12 } =0.037~ $ electron volt. The systems sizes are $ 100 imes 100 imes 50 $ ( $ 29.1 imes 29.1 imes 14.4~ $ nm $ ^3 $ ) . }

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

caption { The Euler characteristic $ chi $ ( squares ) and the specific surface country $ A $ ( circles ) of gold atoms as a map of a temperature $ T $ . The largest value of $ chi $ corresponds to the constellation with the largest sum of staccato gold bunchs. The largest value of a specific surface country $ A $ corresponds to the most assorted system. Near the temperature $ T=400 $ ~K, a passage from a clustered to a assorted constellations is observed. System interaction invariables are $ J_ { 11 } =0.049 $ ~eV, $ ~J_ { 22 } =0.044~ $ electron volt, $ ~J_ { 12 } =0.035~ $ electron volt. The systems sizes are $ 100 imes 100 imes 50 ( 29.1 imes 29.1 imes 14.4~ $ nm $ ^3 $ ) . }

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hspace { 10 millimeter } Repeating the equilibration simulation for different picks of the interaction strength between gold and Ag outputs the “ stage diagram ” presented in Fig.~
ef { fig5.2 } . In the Fig.~
ef { fig5.2 } presented transverse subdivisions of the Au $ _ { 25 } $ Ag $ _ { 75 } $ ( wt.~ % ) systems after equilibration procedure. Here, the black colour corresponds to the gold atoms and the light-gray colour corresponds to the Ag atoms. The gold-gold interaction $ J_ { 11 } $ additions from the left column to the right column in the scope of $ [ 0.045, 0.049 ] ~ $ electron volt in stairss of $ 0.001~ $ electron volt. The silver-silver interaction $ J_ { 22 } $ additions from the top row to the bottom row in the scope of $ [ 0.035, 0.049 ] ~ $ electron volt in stairss of $ 0.002~ $ electron volt. The gold-silver interaction $ J_ { 12 } $ lessenings from the left column to the right column in the scope of $ [ 0.039, 0.035 ] ~ $ electron volt in stairss of $ 0.001~ $ electron volt.

hspace { 10 millimeter } In this diagram one can clearly see two distinguishable constellations – the assorted constellation in the top-left corner and the clustered constellation at the bottom-right corner. The differentiation between these two types of constellation was done by oculus utilizing visualising plan ( see section~
ef { ocular } ) .

hspace { 10 millimeter } Fig.~
ef { 3dconf } represents 3D images of assorted and clustered constellations. Merely particles of gold are shown here. The snapshots was taken for the different systems at the temperature $ T=300~ $ K after the equilibration procedure. No bunch formation is observed in the instance of assorted constellation. for the instance of the clustered constellation one can clearly see some agglomerates ( bunchs ) of gold atoms which lead us to the decision that formation of bunchs started already during the relaxation phase. The interaction invariables of the system that yields assorted constellation are $ J_ { 11 } =0.041~ $ electron volt, ~ $ J_ { 22 } =0.047~ $ electron volt, ~ $ J_ { 12 } =0.037~ $ electron volt. The interaction invariables of the system that yields clustered constellation are $ J_ { 11 } =0.049 $ ~eV, $ ~J_ { 22 } =0.044~ $ electron volt, $ ~J_ { 12 } =0.035~ $ electron volt.

hspace { 10 millimeter } Atomically assorted constellations yield a negative value of Euler feature which is of the order of $ N_ { Au } $ , figure of gilded atoms in the system. In the instance of a clustered constellation the Euler characteristic $ chi sim N_ { Cl } $ , where $ N_ { Cl } $ is a figure of staccato gold bunchs in the system. In simple words the Euler feature is equal to the figure of staccato gold bunchs minus the figure of tunnels between gold atoms in the system.

hspace { 10 millimeter } The development of the Euler feature and specific surface country of gold constructions ( ignoring Ag ) during the relaxation procedure is shown in Fig.~
ef { fig5.3 } and Fig.~
ef { fig5.4 } . Two different sets of interaction invariables were chosen for these simulations, matching to the two different types of equilibrium constellations. Relaxation with one set of interaction invariables leaves the system in an atomically assorted constellation and relaxation with another set produces gilded bunchs immersed in Ag ( Fig.~
ef { 3dconf } ) .

hspace { 10 millimeter } Prior to dissolution the metal must be homogenous with no stage separation cite { ERL03+ } . Porosity development therefore forms dynamically during disintegration and is non due to one stage merely being excavated out of the two-phase stuff cite { ERL03+ } . Therefore, interaction invariables which lead to clustered constellations may be excluded from farther considerations.

section { Dealloying }

label { deall }

hspace { 10 millimeter } The relaxation procedure changes the infinite temperature ( random ) constellation of the gold and Ag atoms into a typical equilibrium constellation at the temperature at which the dealloying is traveling to take topographic point. To get down the dealloying, the lone alteration to the Monte Carlo algorithm is the inclusion of the chemical reaction processes. The simulation of the dealloying is performed utilizing the same set of theoretical account parametric quantities $ J_ { 11 } , J_ { 12 } $ and $ J_ { 22 } $ as in the equilibration procedure. The fact that in the simplified theoretical account Eq.~
ef { energy } , there is no nearest-neighbor interaction, between acid and metal atoms does non forestall a chemical reaction between these atoms to take topographic point. Indeed, by building, in the Monte Carlo algorithm such reactions can take topographic point whenever a Ag and acid atoms are close neighbours.

hspace { 10 millimeter } A “ stage diagram ” of dealloyed constructions is presented in Fig.~
ef { fig5.5 } . The diagram is made of snapshots of a cross subdivisions of the Au $ _ { 25 } $ Ag $ _ { 75 } $ ( wt.~ % ) dealloyed systems. Black colour corresponds to the gold atoms and light-gray colour corresponds to silver atoms. The gold-gold interaction $ J_ { 11 } $ additions from the left column to the right column in the scope of $ [ 0.045, 0.049 ] ~ $ electron volt in stairss of $ 0.001~ $ electron volt. The silver-silver interaction $ J_ { 22 } $ additions from the top row to the bottom row in the scope of $ [ 0.035, 0.049 ] ~ $ electron volt in stairss of $ 0.002~ $ electron volt. The gold-silver interaction $ J_ { 12 } $ lessenings from the left column to the right column in the scope of $ [ 0.039, 0.035 ] ~ $ electron volt in stairss of $ 0.001~ $ electron volt.

hspace { 10 millimeter } In the diagram Fig.~
ef { fig5.5 } one can separate three distinguishable stages – constellations with gold bunchs that accumulated largely in the underside of the sample ( e.g. the system with $ J_ { 11 } =0.045~ $ electron volt, ~ $ J_ { 22 } =0.035~ $ electron volt, ~ $ J_ { 12 } =0.039~ $ electron volt, ~ $ T=300~ $ K ) , constellations with broken NPG ( e.g. the system with $ J_ { 11 } =0.047~ $ electron volt, ~ $ J_ { 22 } =0.043~ $ electron volt, ~ $ J_ { 12 } =0.037~ $ electron volt, ~ $ T=300~ $ K ) and constellations which correspond to constellate equilibrated constellations ( e.g. the system with $ J_ { 11 } =0.049~ $ electron volt, ~ $ J_ { 22 } =0.049~ $ electron volt, ~ $ J_ { 12 } =0.035~ $ electron volt, ~ $ T=300~ $ K ) . Clearly, we are non interested in the first stage because it is non porous constructions. The 3rd stage corresponds to the clustered constellations in the “ stage diagram ” of equilibrated constructions ( see Fig.~
ef { fig5.2 } ) . Therefore, the 3rd stage besides is non of involvement for the present intent.

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

caption { “ Phase diagram ” of dealloyed constructions. Snapshots of a cross subdivisions of the Au $ _ { 25 } $ Ag $ _ { 75 } $ ( wt.~ % ) dealloyed systems. Black colour corresponds to the gold atoms. The gold-gold interaction $ J_ { 11 } $ additions from the left column to the right column in the scope of $ [ 0.045, 0.049 ] ~ $ electron volt in stairss of $ 0.001~ $ electron volt. The silver-silver interaction $ J_ { 22 } $ additions from the top row to the bottom row in the scope of $ [ 0.035, 0.049 ] ~ $ electron volt in stairss of $ 0.002~ $ electron volt. The gold-silver interaction $ J_ { 12 } $ lessenings from the left column to the right column in the scope of $ [ 0.039, 0.035 ] ~ $ electron volt in stairss of $ 0.001~ $ electron volt. $ T=300~ $ K. The systems sizes are $ 100 imes 100 imes 50 ( 29.1 imes 29.1 imes 14.4~ $ nm $ ^3 $ ) . }

label { fig5.5 }

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hspace { 10 millimeter } Comparing the two “ stage diagrams ” ( Fig.~
ef { fig5.2 } and Fig.~
ef { fig5.5 } ) , we may reason that the NPG constructions can merely be obtained within a narrow scope of interaction parametric quantities. Within this scope, these constructions are robust to altering of interaction parametric quantities.

hspace { 10 millimeter } Tuning the temperature for dealloying we could besides bring forth all of these three stages ( see Fig.~
ef { fig5.6 } ) . As the illustration, the system with the interaction constants $ J_ { 11 } =0.047~ $ electron volt, ~ $ J_ { 22 } =0.041~ $ electron volt and ~ $ J_ { 12 } =0.037~ $ electron volt was chosen. The temperature scope is $ [ 200, 400 ] ~ $ K. The first column in the Fig.~
ef { fig5.6 } corresponds to the 3rd stage where equilibrated construction has clustered constellation. Second and 3rd columns could be considered as 2nd stage which leads to the nanoporous constructions. The last two columns represent foremost phase where gold bunchs are accumulated largely in the underside. Therefore, simulations with different temperatures produce distinguishable constructions that exhibit all of these three “ stages ” .

hspace { 10 millimeter } The development of the Euler feature and the specific surface country of gold atoms during the dealloying procedure are shown in Fig.~
ef { ea1 } and Fig.~
ef { ea2 } . In Fig.~
ef { ea1 } and Fig.~
ef { ea2 } the plateau-like parts correspond to the phase when acid reaches the underside of the sample. The changeless Euler feature during this phase is due to the fact that the system exhibits infiltration. In this phase, making extra holes in the system is really hard. The slow lessening of a specific surface country $ A $ with the increasing figure of MMC stairss is due to tempering of the sample.

hspace { 10 millimeter } The first constellation ( see Fig.~
ef { ea1 } ) with interaction invariables $ J_ { 11 } =0.047~ $ electron volt, ~ $ J_ { 22 } =0.043~ $ electron volt and ~ $ J_ { 12 } =0.037~ $ eV represents the construction which is on the edge between two clustered and assorted constellations in equilibrated “ stage diagram ” ( see Fig.~
ef { fig5.2 } ) . The same constellation corresponds to the spongy-like construction in the “ stage diagram ” of dealloyed constructions ( see Fig.~
ef { fig5.5 } ) .

hspace { 10 millimeter } The 2nd constellation ( see Fig.~
ef { ea2 } ) with interaction invariables $ J_ { 11 } =0.046~ $ electron volt, ~ $ J_ { 22 } =0.035~ $ electron volt and ~ $ J_ { 12 } =0.038~ $ eV represents the construction which is far off ( top-left corner ) from the edge between two clustered and assorted constellations in equilibrated “ stage diagram ” ( see Fig.~
ef { fig5.2 } ) . The same constellation corresponds to a dealloyed construction with gold bunchs which are accumulated largely in the underside of the “ stage diagram ” of dealloyed constructions ( see Fig.~
ef { fig5.5 } ) .

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

caption { Snapshots of cross subdivisions of equilibrated stuctures of the Au $ _ { 25 } $ Ag $ _ { 75 } $ ( wt.~ % ) sample ( underside ) and matching dealloyed constructions ( top ) for different temperatures. Temperature increases from the left column to the right column in the scope of $ [ 200, 400 ] ~ $ K in stairss of $ 50~ $ K. Black colour corresponds to the gold atoms and light-gray colour corresponds to silver atoms. System interaction invariables are $ J_ { 11 } =0.047~ $ electron volt, ~ $ J_ { 22 } =0.041~ $ electron volt, ~ $ J_ { 12 } =0.037~ $ electron volt. The systems sizes are $ 100 imes 100 imes 50 ( 29.1 imes 29.1 imes 14.4~ $ nm $ ^3 $ ) . }

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

caption { The Euler characteristic $ chi $ ( squares ) and the specific surface country $ A $ ( circles ) of gold atoms as a map of $ n $ , the figure of MMC stairss per atom. The changeless Euler features $ chi $ during the concluding phase indicate that the systems are leaching. The slow lessening of $ A $ at the concluding phase of the simulation is due to farther equilibration of the samples. System interaction invariables are $ J_ { 11 } =0.047~ $ electron volt, ~ $ J_ { 22 } =0.043~ $ electron volt, ~ $ J_ { 12 } =0.037~ $ electron volt, ~ $ T=300~ $ K. The systems sizes are $ 100 imes 100 imes 50 ( 29.1 imes 29.1 imes 14.4~ $ nm $ ^3 $ ) . }

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caption { The Euler characteristic $ chi $ ( squares ) and the specific surface country $ A $ ( circles ) of gold atoms as a map of $ n $ , the figure of MMC stairss per atom. The changeless Euler features $ chi $ during the concluding phase indicate that the systems are leaching. The slow lessening of $ A $ at the concluding phase of the simulation is due to farther equilibration of the samples. System interaction invariables are $ J_ { 11 } =0.046~ $ electron volt, ~ $ J_ { 22 } =0.035~ $ electron volt, ~ $ J_ { 12 } =0.038~ $ electron volt, ~ $ T=300~ $ K. The systems sizes are $ 100 imes 100 imes 50 ( 29.1 imes 29.1 imes 14.4~ $ nm $ ^3 $ ) . }

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section { Morphologic analysis }

hspace { 10 millimeter } First, we present some illustrations of 2D pieces of the system during the dealloying procedure, see Fig.~
ef { fig5.8 } . In the figure the black colour corresponds to the gold atoms and light-gray colour corresponds to silver atoms. In this figure we can track the kineticss of the typical dealloying procedure. Note that the lone 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.~
ef { fig5.8 } clearly show a kineticss of the rearrangement of gold atoms from assorted stage to the clustered stage. The 3D image of the dealloyed sample, the consequences of the analysis of the geometry of dealloyed construction and simulation parametric quantities are shown in Fig.~
ef { fig5.9 } . The system has heterogeneousness in $ z $ way due to the fact that acid penetrates the system from top to bottom ( see Fig.~
ef { underside } ) . Several “ underside ” beds of the dealloyed system contain enhanced sums of gold atoms due to the boundary status in this way ( Fig.~
ef { fig5.8 } vitamin D, Fig.~
ef { fig5.9 } ) . Qualitatively, the form of the fake system ( see Fig.~
ef { fig5.10 } ) is similar to the microscopy image of the dealloyed sample ( see Fig.~
ef { fig1 } B ) . From the 2D snapshot ( see Fig.~
ef { fig5.8 } a ) and the 3D snapshot ( see Fig.~
ef { 3dconf } ) of the equilibrated system and the behaviour of the Euler feature ( see Fig.~
ef { fig5.3 } ) we may reason that gold ( black ) and Ag ( light-gray ) are wholly assorted. Vacancies in the sample are represented by nothingnesss that are indiscriminately located in the initial constellation. During the equilibration, the vacancies merge into “ bunchs ” ( see bunchs of white pels in Fig.~
ef { fig5.8 } a ) ) . During the dealloying, vacancies can merely look as the consequence of the chemical reactions Eq.~
ef { chem1 } . In this reaction, three atoms are converted into two atoms and one gas molecule. The latter is non explicitly taken into history in the simulation. Phrased otherwise, our simulation does non separate between true vacancies and sites occupied by a gas molecule. The dealloying procedure is stopped when 99~ % of the initial sum of Ag was dissolved. The dealloyed construction is wholly connected.

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

caption { Snapshots of cross subdivisions of the Au $ _ { 25 } $ Ag $ _ { 75 } $ ( wt.~ % ) sample taken during the MMC simulation after n MMC stairss per atom. Black colour corresponds to the gold atoms and light-gray colour corresponds to silver atoms. extbf { a } : $ n = 0 $ ; extbf { B } : $ n = 5 imes 10^2 $ ; extbf { degree Celsius } : $ n = 2 imes 10^3 $ ; extbf { vitamin D } : $ n = 6 imes 10^3 $ . System interaction invariables are $ J_ { 11 } =0.047~ $ electron volt, ~ $ J_ { 22 } =0.043~ $ electron volt, ~ $ J_ { 12 } =0.037~ $ electron volt. The systems size is $ 100 imes 100 imes 50 ( 29.1 imes 29.1 imes 14.4~ $ nm $ ^3 $ ) . }

label { fig5.8 }

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

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

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

caption { Bottom position of the dealloyed system. Heterogeneity in $ z $ way is due to the fact that acid penetrates the system from top to bottom. Several beds in the underside of the system contain an enhanced concentration of gold atoms due to the boundary status at the bottom bed. The system parametric quantities are the same as in Fig.~
ef { fig5.9 } . }

label { underside }

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

caption { Plane cut taken from the system shown in Fig.~
ef { fig5.9 } , demoing the typical distribution of gold atoms as obtained by imitating chemical dealloying of an Au/Ag sample. }

label { fig5.10 }

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