#
Computational Study of Quasi-2D Liquid State in Free Standing Platinum, Silver, Gold, and Copper Monolayers^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

_{2}[7]. Los et al. predict a more accurate melting point of T

_{m}= 4510 K for graphene using nucleation theory, which is about 250 K higher than that of graphite, and so far the highest of all materials [8]. A melting point of 3700 K has been predicted for single layer freestanding MoS

_{2}[7]. In 2015, Merino and coworkers predicted the melting of boron 40 molecules in DFT simulations. They consider this system to be a nanobubble, and observed diffusion of individual atoms in the shell [9].

## 2. Computational Methods

^{−6}; Smearing = 0.04; Direct Inversion of the Iterative Subspace method (DIIS) = 20; Nosé–Hoover method [20], Nosé Q = 2, Nosé chain length = 2; Fixed center of mass. For the higher temperatures, the first 12 ps of each run were considered an equilibration period (less time was needed for the lower temperatures). More details on each run are in the Supplementary Information (SI). Data were acquired after the equilibration. Materials Studio was also used to create the initial structures and visualize the results. The Root Mean Square Displacement (RMSD) was calculated using the differences between an initial data frame (after equilibration), and the final frame of each simulation.

_{0}) using a simple spring model with spring constant k

_{spring}at low deviations, as shown in Equation (3). Here, $\ell $ is the bond length, and $\ell $

_{0}is the most probable bond length at a given temperature, and corresponds to zero spring extension. We have also included a hard sphere repulsive term ${\left[\ell -{r}_{0}\right]}^{12}$ with C and r

_{0}determined empirically for each plot.

## 3. Results and Discussion

#### 3.1. Platinum

^{2}/ps for 2300 K, and 0.5 Å

^{2}/ps for 2400 K. The velocity auto correlation function for Pt at 2400 K is shown in the SI. We observe that the correlation dissipates quickly after less than 1 ps. In Figure 5b, we also show the bond orientation order parameter for platinum. We see that this is close to one for short distances and lower temperatures, and is reduced as we raise the temperature. Although the value drops somewhat with distance, we see that bond orientational order is maintained over the longest distance available in the simulation. For 2400 K, where we are in the liquid state, we see an average value of around 0.63–0.67.

_{0}) using a simple spring model with spring constant k

_{spring}at low deviations as shown in Equation (3) (with an additional hard sphere repulsive term). For Pt, we see that the effective spring constant k

_{spring}= 4.3 and 4.1 eV/Å

^{2}at 1200 and 1800 K. The fits are quite good at all temperatures. At 2400 K, the system changes to the liquid state. This state has 4% shorter bond length at the maximum. We attribute the shortened bond lengths to the lower number of bonds in the liquid state. We observe that the effective spring constant decreases to k

_{spring}= 2.8 eV/Å

^{2}. This shrinking as we raise the temperature corresponds to a negative coefficient of thermal expansion as we enter the liquid state. At 2400 K, at long bond lengths, there are significant excess observations from 2.8 to 3.5 Å. This is expected as we approach the van der Waals limit as the atom moves away from the plane of the other atoms (the bonding becomes weaker than the spring model and more atoms will be observed with these long bond lengths).

#### 3.2. Gold

^{2}/ps, respectively. These can be compared to the result of 0.14 Å

^{2}/ps at 1600 K from K & K [2]. At 1600 K, the average number of bonds is 5.5, somewhat reduced compared to the low temperature results, but still larger than the Pt results at 2400 K.

#### 3.3. Silver

#### 3.4. Copper

## 4. Conclusions

^{2}for Pt as it goes into the liquid state. Pair correlation functions and diffusion measurements reveal liquid behavior in specific temperature ranges. The pair correlation function drops in the liquid state, while orientation order is reduced to a lesser degree.

## Supplementary Materials

**Figure S1.**Platinum 2400 K side view, and ball and stick model;

**Figure S2.**Side views of Au models at 500, 800, 1200, 1400, and 1600 K;

**Figure S3.**Last frame of gold simulation at 2000 K, top view and ball and stick model of gold at 1600 K;

**Figure S4.**Silver 1050 K angle view;

**Figure S5.**Copper side view at 1400 K;

**Figure S6.**Plot of copper root mean square displacement versus time;

**Figure S7.**Plot of Au root mean square displacement versus time;

**Figure S8.**Plot of Ag root mean square displacement versus time;

**Figure S9.**Plot of Ag velocity auto correlation function at 1050 K;

**Figure S10.**Plot of Pt velocity auto correlation function at 2400 K;

**Figure S11.**Orientation correlation function g6(r) for Au at various temperatures;

**Figure S12.**Orientation correlation function g6(r) for Ag at various temperatures;

**Figure S13.**Bond length distribution for silver at 800 K and 1050 K;

**Table S1.**Simulation times and skip times for MD runs;

**Table S2.**RMSD and Diffusion Constant D Values after Equilibration; and

**Table S3.**Comparison of 2D Liquid Range to Bulk Melting Points for all materials. Supplementary videos are available at https://zenodo.org/record/:

**Video S1.**5 ps Molecular Dynamics Movie of Pt Freestanding Monolayer at 2400 K.

**Video S2.**6 ps Molecular Dynamics Movie of Ag Freestanding Monolayer at 1050 K.

**Video S3.**4 ps Molecular Dynamics Movie of Au Freestanding Monolayer at 1600 K.

**Video S3.**3 ps Molecular Dynamics Movie of Cu Freestanding Monolayer at 1400 K.

## Acknowledgments

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**Figure 2.**Molecular dynamics snapshots of quasi-2D Pt layer. We show two snapshots at 2300 K, separated by 0.2 ps and one snapshot at 2400 K. Atoms are mobile and holes come and go. This is quasi-2D liquid behavior.

**Figure 3.**Pair correlation function for freestanding platinum quasi-2D monolayer at temperatures ranging from 0 K to 2400 K. At 2300 K and 2400 K the pair correlation function drops off quickly, and the freestanding quasi-2D Pt monolayer is a quasi-2D liquid. Higher temperatures are offset for visibility.

**Figure 4.**Bond length distribution for freestanding quasi-2D Pt monolayer at 1200 K, 1800 K, and 2400 K. The 0 K bond length of 2.63 Å is shown as a short vertical green line below the curves. At 1200 K and 1800 K, almost all of the bond lengths are well above the 0 K value. The most likely length shrinks by 4% at 2400 K in the liquid state, compared to 1200 and 1800 K. Solid lines are fitted to the Boltzmann distribution as discussed in the text.

**Figure 5.**(

**a**) Root mean square displacement (RMSD, in Å) for platinum simulations at 1200–2400 K. At lower temperatures, the RMSD is constant around 1 Å due to oscillation primarily in the Z direction. At 2300 and 2400 K, we see extended diffusion. (

**b**) Bond orientation correlation function.

**Figure 6.**Molecular dynamics snapshots of quasi-2D Au layers at 1400 K and 1600 K. At 1600 K, we show two frames separated by 0.5 ps. quasi-2D liquid behavior is observed at these temperatures.

**Figure 7.**Pair correlation function for gold quasi-2D layer at temperatures ranging from 0 K to 1600 K. Higher temperatures are offset for visibility.

**Figure 9.**Pair correlation function for silver quasi-2D monolayers at temperatures ranging from 0 K to 1050 K. Higher temperatures are offset for visibility.

**Figure 10.**Molecular dynamics snapshot of quasi-2D Cu layer. Two snapshots separated by 0.2 ps are shown at 1320 K and 1400 K. Angle view shown at 1400 K.

**Figure 11.**Pair correlation function for freestanding copper quasi-2D monolayers at temperatures ranging from 0 K to 1400 K. Higher temperatures are offset for visibility.

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**MDPI and ACS Style**

Yang, L.-M.; Ganz, A.B.; Dornfeld, M.; Ganz, E.
Computational Study of Quasi-2D Liquid State in Free Standing Platinum, Silver, Gold, and Copper Monolayers. *Condens. Matter* **2016**, *1*, 1.
https://doi.org/10.3390/condmat1010001

**AMA Style**

Yang L-M, Ganz AB, Dornfeld M, Ganz E.
Computational Study of Quasi-2D Liquid State in Free Standing Platinum, Silver, Gold, and Copper Monolayers. *Condensed Matter*. 2016; 1(1):1.
https://doi.org/10.3390/condmat1010001

**Chicago/Turabian Style**

Yang, Li-Ming, Ariel B. Ganz, Matthew Dornfeld, and Eric Ganz.
2016. "Computational Study of Quasi-2D Liquid State in Free Standing Platinum, Silver, Gold, and Copper Monolayers" *Condensed Matter* 1, no. 1: 1.
https://doi.org/10.3390/condmat1010001