# Multilayer Structures of Graphene and Pt Nanoparticles - a Multiscale
Computational Study

Samaneh Nasiri1, Christian Greff 1, Kai Wang2, Mingjun Yang2, Qianqian Li 3,
Paolo Moretti 1, Michael Zaiser 1,2,4 1WW8-Materials Simulation, Department of
Materials Science, Friedrich-Alexander Universität Erlangen-Nürnberg, 90762
Fürth, Germany  2School of Materials Science and Engineering, Southwest
Petroleum University, Chengdu, Sichuan, PR China  3Department of Aerospace
Engineering, Imperial College, Prince Consort Road, London SW7 2BZ, UK
4School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu,
Sichuan, PR China<EMAIL_ADDRESS>
###### Abstract

We report results of a multiscale simulation study of multilayer structures
consisting of graphene sheets with embedded Pt nanoparticles. Density
functional theory is used to understand the energetics of Pt-graphene
interfaces and provide reference data for the parameterization of a Pt-
graphene interaction potential. Molecular dynamics simulations then provide
the conformation and energetics of graphene sheets with embedded Pt
nanoparticles of varying density, form and size. These results are interpreted
using a continuum mechanical model of sheet deformation, and serve to
parameterize a meso-scale Monte Carlo model to investigate the question under
which conditions the free volume around the Pt nanoparticles forms a
percolating cluster, such that the structures can be used in catalytic
applications. We conclude with a discussion of potential applications of such
multilayer structures.

## Keywords

Graphene, Platinum, Catalyst, Multiscale Simulation, Nanoparticles.

##  1 Introduction

It is well known that $sp^{2}$ bonded carbon nanoparticles (CNP), in
particular graphene (GP) flakes and carbon nanotubes (CNT), possess excellent
mechanical properties [1, 2, 3], high specific surface area [4, 5], good
thermal properties [6, 7] and high electrical conductivity[8, 9]. These
outstanding properties make graphene a suitable candidate to act as catalyst
support for metals and non-metals in electro-catalytic applications such as
use in fuel cells and batteries[10, 11, 12, 13]. Since the discovery of
graphene there has been a significant amount of research on using platinum
decorated graphene sheets as new catalysts in low temperature fuel cells
because of their high stability and electro-catalytic ability and more
importantly their high catalyst loading capabilities [14, 15, 16, 17, 18, 19,
20].

Stability of metal nanoclusters on carbon based catalysts is one major issue
in catalytic applications. Due to the $sp^{2}$ carbon bonds, pristine graphene
is relatively inert. Metal nanoparticles adsorbed onto graphene are found to
be fairly mobile on the surface, indicating weak bonding [21, 22, 23, 24].
Using dispersion-corrected DFT methods, Schneider et al.[22] showed that for
nanoclusters larger than about 1 nm, the interaction between platinum and
graphene is purely of van der Waals type. As the weak bonding affects the
mechanical stability of the catalyst, methods have been sought to cause
stronger attachment of the Pt clusters to the graphene sheets. Such methods
include heat treatment [25], defect engineering [26, 22, 27] as well as
chemical surface modification of graphene [28, 29, 30].

An alternative approach towards enhancing the attachment between Pt
nanoparticles and graphene support is to create sandwich structures in which
Pt nanoparticles are anchored between the two adjacent graphene sheets. Such
structures were found to possess higher activity for methanol electrooxidation
and a higher stability than that of Pt-graphene[19]. Somewhat paradoxically,
while mutually adhering graphene sheets have been argued to hold Pt
nanoparticles in place, it has also been argued that discrete metal
nanoclusters on graphene sheets can act as geometrical nano-spacers to reduce
or prevent graphene agglomeration [31, 32, 33, 34]. For instance, it has been
demonstrated that decoration of exfoliated graphene sheets with Pt
nanoclusters can prevent face-to-face aggregation of the sheets [35]. These
apparently contradictory applications – Pt nanoclusters reduce graphene-
graphene adhesion and prevent graphene agglomeration, mutually adhering
graphene sheets hold Pt nanoclusters in place – indicate a need for a
systematic investigation. The aim is to establish, for Pt nanoclusters of
different sizes and densities embedded between graphene sheets, the mechanical
stability and energetic properties of the resulting multilayer structures in
order to decide whether they are promising for use in catalytic applications.

In the present work, we report results of a multiscale simulation study of
bilayers of graphene with embedded Pt nanoparticles. We first perform density
functional theory (DFT) calculations to establish the energetics of Pt-
graphene interfaces and to obtain reference data for parameterizing
interaction potentials for the Van der Waals - like Pt-graphene interactions.
We then use these potentials in a molecular mechanics (MM) and molecular
dynamics (MD) study to establish sheet conformations of graphene bilayers with
embedded Pt clusters of varying size. These simulations are analyzed using a
continuum model to establish analytical relations between the density and the
size of embedded Pt nanoparticles and the effective adhesion properties of the
bilayers. This analysis is complemented by a Monte-Carlo model to investigate
under which conditions the free volume surrounding the nanoparticles in such
structures forms a percolating cluster as required for their catalytic use.

##  2 Atomic-scale simulations

On the atomic scale, we first perform DFT calculations of planar graphene-Pt
interfaces in order to gain data for parameterizing empirical potentials that
are then used, in larger-scale MD simulations, to describe the Van der Waals-
like interaction between graphene sheets and Pt nanoparticles.

###  2.1 Density-functional calculations: Methods and results

In order to calculate the Pt-graphene interaction using DFT we consider five-
layer Pt(111) slabs adhering to monolayer graphene. Both Pt slabs and graphene
sheet are oriented parallel to the $xy$ plane of a Cartesian coordinate
systems. Periodic boundary conditions are used in all three spatial directions
with a 10 Å vacuum gap between the periodic replicas in the $z$ direction. Two
variants of this structure (graphene on Pt (111) surface) are constructed as
shown in Figure 1; for model a, a larger supercell consisting of p $(7\times
7)$ graphene and p $(6\times 6)$ Pt slabs were used in the calculations to
mimic an incoherent interface. The structure for model b consists of p
$(2\times 2)$ graphene and three atoms per Pt layer and represents a coherent
interface as considered in previous studies [36, 21]. For both models, the Pt
lattice parameters were set to experimental values for bulk Pt.

All the calculations were carried out by the Quantum Espresso package version
6.1 [37], employing PBE-based projected augmented wave(PAW) potentials [38].
Previous studies have shown that the optB88-vdw [40] exchange-correlation
functional gives a reasonably accurate prediction of both interlayer distance
and binding energy for graphite [39] , so here we choose the optB88-vdw
functional in all our DFT calculations. In our calculations, the kinetic
energy cutoffs of 40 Ry and 450 Ry were used for wave function and charge
density calculations respectively, and a Methfessel-Paxton smearing of 0.01 Ry
was used for the electronic convergence. The convergence for self-consistency
calculations was less than 0.0001 Ry. In the calculations of model a, only the
gamma k-point was used due to the large system we chose, while an $8\times
8\times 1$ k-points grid was used in the calculations of model b.

Figure 1: Top views ((a) and (b)) and side views ((c) and (d)) of the geometry
for model a and model b respectively. (e) Pt-graphene binding energy for both
models obtained by DFT calculations.

Since our aim is to obtain data for parameterizing an interaction potential,
rather than performing an accurate analysis of the interface structure and
energy using DFT, we do not carry out any structural relaxation. Instead we
simply separate the graphene sheet rigidly from the Pt(111) slab without
geometry relaxation of either the Pt slab or the graphene. We evaluate the
binding energy per carbon atom as

$\displaystyle E_{\rm b}=[E_{\rm tot}-(E_{\rm pt}+E_{\rm gr})]/n$ (1)

$E_{\rm tot}$ is the total energy of the graphene-Pt(111) system, $E_{\rm pt}$
represents the energy of the five-layer Pt(111) slab, and $E_{\rm gr}$
represents the energy of the free-standing graphene sheet. Results are
depicted in Figure 1 for both models.

Figure 2: Top: Interface energy surface obtained by sliding graphene rigidly
on top of a Pt slab, using model (b) and keeping the distance at the position
of lowest energy; bottom: Interface energy profiles and interface shear stress
profiles taken along the displacement paths $\rm P_{\rm 1}$ and $\rm P_{\rm
2}$.

To obtain an estimate of the stress needed to shear the graphene-Pt(111)
interface, we also evaluate the ’interface energy surface’ for model (b) which
is obtained by sliding the graphene sheet rigidly on the Pt surface. Results
are shown in Figure 2 in conjunction with energy profiles taken along three
typical paths. It can be seen that the interface energy variations are quite
small. Accordingly, the interface shear stresses (IFSS) – derivatives of the
energy profiles – required to slide the graphene over the Pt surface are
comparatively low, of the order of 20 MPa or less. The same is expected to
hold for the typical shear stresses required to slide a Pt cluster on a
graphene sheet.

###  2.2 Molecular simulation: Validation of interaction potentials

We use the LAMMPS simulation package [41] to perform molecular mechanics (MM)
and molecular dynamics (MD) simulations of binary systems consisting of
platinum nanoparticles which are anchored between two layers of graphene
sheets. For describing such a system we need to define Pt-Pt, Pt-C and C-C
interaction potentials, where the latter have to account for both short and
long range interactions between carbon atoms within the graphene sheets and
between both sheets.

In a first step, we examine different sets of interaction potentials to
establish which set of potentials is most appropriate for the problem at hand.
The first potential we examined is the reactive force field (ReaxFF) which has
been developed by Sanz-Navarro et al. [42, 43]. ReaxFF is based on a bond-
order model in conjunction with a charge-equilibration scheme[44]. This
potential is in principle able to describe all the above mentioned
interactions in the Pt-C system, however, it is optimized to correctly
represent reactive processes rather than mechanical properties. The second
potential we considered is a Brenner force field (BrennerFF) parametrized by
Albe et al. [45] to describe short-range interatomic interactions in the Pt-C
systems, with a Lennard-Jones potential added to mimic the weak van der Waals
interaction between the carbon atoms of the two graphene sheets [46]. A
further possibility is to combine potentials of different type that are
separately optimized to describe the energetics and mechanical properties of
covalently bonded carbon, and of platinum. A similar approach was used by [50,
51] for simulating Ni-graphene interactions: These authors use a standard
AIREBO potential [46, 47] for describing covalent C-C interactions in
conjunction with an EAM potential for Ni, and a Lennard-Jones potential for
Ni-carbon interactions at the Ni-graphene interface. We emphasize that such
addition of structurally dissimilar potentials is, in general, problematic.
However, in the present context the approach may be feasible because the
components of the system (metal, graphene) maintain their structural
integrity, and the van der Waals type interaction between these components can
be considered as additive to their internal (metallic, covalent) interactions.
Accordingly we consider a Hybrid combination comprising the standard AIREBO
potential [46, 47] for covalent C-C interactions and the EAM potential of Zhou
and coworkers [48] for Pt-Pt interactions. Pt-C interactions at the interface
are described using either a LJ potential taken from the literature [49]; this
combination is denoted as HybridFF-LJ. Alternatively we consider a Morse
potential of the form $E_{\rm M}=D(\exp[-2\alpha(r-r_{\rm
0})]-2\exp[-\alpha(r-r_{\rm 0})])$ where $r$ is the Pt-C distance, $r_{0}$ an
equilibrium bond distance, $D$ is the well depth, and $\alpha$ controls the
stiffness of the potential (the smaller $\alpha$ is, the smaller the
attractive/repulsive forces). The combination of AIREBO (C-C) and EAM (Pt-Pt)
with this Morse potential, whose parameters we determine by matching with our
own DFT calculations, is denoted as HybridFF-M.

The performances of the different potentials or combinations of potentials are
compared in view of two main criteria. First, we ask how well the potentials
reproduce theoretical predictions for the mechanical properties of the system
components. Second, we study how well they describe the adhesion between Pt
and graphene. Regarding mechanical properties, the data in Table 1 show that
all potentials correctly represent the lattice constant and cohesive energy of
Pt, however, the ReaxFF produces isotropic elastic behavior for Pt, completely
disregarding the significant cubic anisotropy of the material. The other
potentials yield acceptable results regarding the elastic properties of Pt.
Lattice properties of monolayer graphene as well as the interlayer distance
$d_{\rm 0}$ and interlayer adhesion energy $\gamma_{\rm GG}$ of bilayer
graphene are reported in Table 2.

Table 1: Lattice properties of platinum. | EAM | ReaxFF | BrennerFF | Experiment[52, 53]
---|---|---|---|---
$a_{\rm 0}$(Å) | 3.92 | 3.94 | 3.92 | 3.91
$E_{\rm 0}$(eV) | -5.77 | -5.72 | -5.77 | -5.77
$C_{11}$(GPa) | 357.11 | 332.02 | 351.91 | 373.42
$C_{12}$(GPa) | 260.13 | 194.63 | 248.45 | 241.74
$C_{44}$(GPa) | 77.97 | 194.63 | 89.94 | 77.65
Table 2: Lattice properties of graphene. | ReaxFF | BrennerFF | AIREBO | DFT[54]
---|---|---|---|---
$a_{\rm 0}$(Å) | 2.46 | 2.46 | 2.46 | 2.45
$E_{\rm 0}$(eV) | -7.46 | -7.35 | -7.39 | -7.46
$d_{\rm 0}$(Å) | 3.29 | – | 3.41 | 3.35
$\gamma_{\rm GG}$$(\rm J/m^{2})$ | -0.42 | – | -0.29 | -0.25 – -0.36

Concerning the binding between Pt and graphene, we use our DFT results for the
interaction between Pt(111) and graphene as a reference. We consider the same
configuration as in the DFT simulations, i.e., we separate the graphene sheet
rigidly from the Pt(111) surface, which allows us to directly compare the MD
and DFT energies. First, we observe that the BrennerFF completely fails to
reproduce the interfacial bonding between graphene and Pt(111) – the
interaction reduces to a short-range repulsion and the binding energy is thus
zero – which renders this potential unsuitable. The ReaxFF significantly over-
estimates the bonding between Pt and graphene while it under-estimates the
interfacial separation. Therefore, and also in view of its poor performance in
modeling the elastic properties of Pt, this potential is also ruled out, which
leaves us with the two hybrid combinations of potentials (HybridFF-LJ and
HybridFF-M).

Figure 3: Binding force and energy for interface model a (left) and model b
(right); comparison of DFT data with MD results for different interaction
potentials.

The two hybrid potentials offer the advantage that the parameters of the LJ or
Morse potential, which describes Pt-C interactions at the interface between
graphene sheets and Pt, can be fitted to correctly reproduce the interface
separation and binding energy. At variance with previous studies [49], we find
that the LJ potential has clear deficiencies as it is too stiff and thus
significantly over-estimates the forces acting across the Pt-graphene
interface (see Figure 3 (c,d)). The Morse potential is the only one that can
be fitted very well to the DFT data because the well depth, well width and
equilibrium distance parameters can be optimized to reproduce the quantum
mechanical interaction energy curve from the density functional calculations.
The resulting parameters for the Pt-C Morse potential are $D$ = 0.0071 eV,
$r_{\rm 0}$ = 4.18 Å and $\alpha$ = 1.05 $\mathrm{\AA}^{\rm-1}$.

We therefore use in the following the HybridFF-M combination of potentials:
The AIREBO potential [46, 47] for covalent C-C interactions, the EAM potential
of Zhou and coworkers [48] for Pt-Pt, and the DFT parameterized Morse
potential for C-Pt interactions. Van der Waals interactions between different
graphene sheets are described by a standard LJ potential.

In nanoscale clusters, the configuration of Pt atoms differs from the planar
slab used in our reference DFT calculation because many atoms have reduced
coordination number and/or local environments of reduced symmetry. To verify
that the HybridFF-M combination of potentials correctly describes the adhesion
of small Pt clusters to graphene, we consider the extreme case of a very small
$\mathrm{Pt}_{\rm 13}$ nanocluster adhering to a pristine graphene sheet. DFT
calculations for this system were performed by Fampiou and Ramasubramanian
[26] who report an adhesion energy of -0.84 eV for a minimum energy
configuration of non icosahedral symmetry. Repeating the calculation with the
same structure file but using the HybridFF-M potential for energy evaluation
gives an adhesion energy of -0.94 eV, i.e., the deviation from the DFT result
is only 11%. (We note that calculations using HybridFF-LJ, Tersoff and ReaxFF
yield discrepancies of over 300 % of the DFT adhesion energy.) This gives us
confidence that the HybridFF-M combination used in the following MM and MD
simulations adequately captures the adhesion of larger Pt clusters on
graphene.

us

Figure 4: Modeling the conformation of graphene bilayer around an annealed Pt
particle using MM simulation.

###  2.3 MM and MD simulations of Pt-Graphene structures: Set-up and results

For simulating the energy and geometrical properties of graphene-encapsulated
Pt nanoparticles, we consider two types of particles: On the one hand, we use
Pt particles constructed with cylindrical geometry (radius $R_{\rm p}$,
particle height $2h$) to produce a prescribed aspect ratio of $q_{\rm
p}=h/R_{\rm p}$. On the other hand, we consider annealed particles where an
initially cylindrical particle is, before encapsulation, heated above the
melting point of Pt and slowly cooled down to near 0 K using a Nosé-Hoover-
thermostat in order to minimize the particle energy. Such annealed particles
generally exhibit a spheroidal particle shape. In order to encapsulate the
particles, we start from two graphene sheets distanced significantly beyond
the cut-off radius in order to avoid van der Waals interaction between the
layers. One platinum particle is positioned on the lower graphene sheet within
the cut-off radius of the Pt-C interaction (Figure 4: step 1). To prevent edge
effects, periodic boundary conditions are used in all directions. To form a
system where the platinum particle is acting as a spacer between the two
graphene sheets, the upper graphene sheet is rigidly moved right above the
platinum particle (Figure 4: step 2). Then, the outer areas of the two sheets
are brought into contact with each other with 3.4 $\mathrm{\AA}$ distance
between them. A minimization step brings the system into mechanical
equilibrium, where the platinum particle functions as a mechanical obstacle
that prevents the two graphene sheets from adhering to each other over a
’detachment area’ surrounding the particle (Figure 4: step 3). In all
simulations, we determine the total energy of the relaxed structure as well as
the elastic energy of the graphene sheet and the energies of graphene-
graphene, Pt-Pt, and graphene-Pt interaction. In addition, we determine the
detachment area $A_{\rm d}$ as the area over which the spacing between the
graphene layers exceeds the value for pure bilayer graphene by more than 1 Å,
and for single encapsulated particles we define the detachment radius via
$A_{\rm d}=:\pi R_{\rm d}^{2}$.

Figure 5: (a) Detachment area, energy of GG detachment, energy of PtG
attachment, and graphene elastic energy as functions of Pt nanoparticle
radius; data points: numerical values obtained from our MM simulations; solid
red line: $E_{\rm d}$ after Eq. (6) with $q_{\rm d}=4.0$, solid black line:
$E_{\rm el}$ after Eq. (7) with $q_{\rm d}=4.0$; solid blue line: $E_{\rm a}$
after Eq. (8) with $q_{\rm c}=0.87$; (b) Detachment radius as function of
particle radius, solid line: $R_{\rm d}=q_{\rm d}R_{\rm p}$ with $q_{\rm
d}=4.0$; (c) Elastic energy as a function of detachment energy, solid line:
Eq. (7) with $q_{\rm d}=4.0$.

MM simulations of single embedded Pt nanoparticles of varying radius but
constant aspect ratio indicate that the detachment radius surrounding a
particle is proportional to the particle radius, i.e. $R_{\rm d}=4.2R_{\rm p}$
(Figure 5b). The energy of the graphene bilayer is raised by embedding a Pt
nanoparticle. This energy increase (’GG excess energy’) is due to the energy
of adhesion which must be provided in order to detach the two graphene sheets
over the detachment area (detachment energy: $E_{\rm d}$), diminished by the
adhesion between the two graphene sheets and the nanoparticle (attachment
energy: $E_{\rm a}$), and due to the elastic distortion of the sheets (elastic
energy: $E_{\rm el}$). All three energy contributions, which we evaluate from
the MM/MD simulations, are found to increase in mutual proportion as the Pt
particle radius increases. A simplified continuum model formulated in section
3 allows to understand this behavior.

Figure 6: Molecular mechanics simulation of a two-nanoparticle system; the
coordinates of the Pt atoms were kept fixed during relaxation of the graphene
sheets.

In order to establish the dependence of the GG excess energy on the distance
between Pt particles, we perform calculations on nanoparticle pairs as shown
in Figure 6. To control the inter-particle separation distance $d$, in these
MM simulations we keep the coordinates of the Pt atoms fixed during the final
relaxation step. If the detachment areas surrounding two Pt nanoparticles
overlap, this leads to a reduction in GG excess energy since the overall
detachment area is reduced and also the elastic deformation of the graphene
sheet decreases. The energy reduction becomes more pronounced as the distance
between the two particles decreases, which leads to an attractive
configurational force acting on the particles. Curves showing the GG excess
energy of two-nanoparticle systems ($E^{\rm 2p}$) with different nanoparticle
size vs the nanoparticle separation $d$ are shown in Figure 7. A data collapse
can be obtained when the GG excess energy is scaled by its value for a single
particle ($E^{\rm 1p}$), and the inter-particle separation is measured in
units of the detachment radius $R_{\rm d}$ of the graphene sheet around that
particle. The energy vs. distance curve can be well approximated by a linear
dependency, $E^{\rm 2p}=E^{\rm 1p}(1+d/d_{\rm c})$ where $d\approx 2.4R_{\rm
d}$ is the critical distance where the particles first interact with each
other. Above that distance, we are dealing with two separate single particles
(see Figure 7 inset).

Figure 7: GG excess energy of two-nanoparticle systems as function of
nanoparticle separation distance; red symbols: $R_{\rm p}=2$nm, blue symbols:
$R_{\rm p}=1$nm; inset: normalized excess energy, measured in units of the GG
excess energy around a single particle, vs. inter-particle distance in units
of the single-particle detachment radius.

##  3 Continuum mechanical model

In the following we use the results of our MM/MD simulations to parameterize a
simplified continuum mechanical model which transfers the atomic interactions
into continuum-level concepts such as elastic energy, work of adhesion, and
interfacial shear stress acting between continuous bodies. Such a model is
necessarily inadequate for very small Pt clusters (an extreme case being
$\mathrm{Pt}_{\rm 13}$) where atomic structure is crucial. However, we expect
the continuum model to become more accurate as we move to larger and larger Pt
nanoparticles. In fact, the scaling relations between energies and geometrical
parameters we expect from the continuum model extend down to clusters with a
diameter as low as 1nm. This is demonstrated in Figure 5 by the comparison of
the discrete data points (simulation data) and straight lines (continuum
mechanical scaling relations). A continuum mechanical treatment is useful for
several reasons: (i) through simplified calculations it provides analytical
estimates of energies and interactions in the system at hand; (ii) by their
nature, continuum mechanical models are scalable and can be applied to larger
nanoparticles without adding additional computational cost; (iii) numerical
implementation of such models (not studied here) can be easily generalized to
more complex multilayer architectures.

###  3.1  Model of a single embedded particle

To obtain semi-analytical relations between the size of the Pt nanoparticles
encapsulated between the graphene sheets, the detachment between the two
sheets, and the energy of the system, we use a simplified continuum model as
illustrated in Figure 8, top. The particle is idealized as a cylindrical body
with a radius of $R_{\rm p}$ and a height of $2h$, with its axis perpendicular
to the adhering graphene sheets. Accordingly, a cylindrical coordinate system
is used with its origin in the particle center. The $z$-axis runs along the
particle axis, and the radial coordinate $r$ and angular coordinate $\theta$
denote the position parallel to the the plane $z=0$, which corresponds to the
plane of adhesion between the two graphene sheets. The deformation of the
graphene sheets is described by the displacements $u_{r},u_{\theta{}},u_{z}$.
Pt-graphene binding length ($h_{\rm b}^{*}$) and graphene-graphene inter-layer
distance ($2h_{\rm b}$) are assumed to be equal. A matching atomistic model is
provided by a graphene sheet indented by a cylindrical indenter of radius
$R_{\rm p}$. In this model all atoms within $R_{\rm p}$ from the center of the
bilayer sheet are displaced rigidly in the respective up- and downward
direction by a distance $h$ while the rest of the sheet is allowed to relax
(Figure 8, bottom).

Figure 8: Schematic of the continuum mechanical model (top) and reference MD
model (indentation model) (bottom).

The graphene sheets are modeled as 2D elastic membranes with a bending
stiffness that is negligible compared to their in-plane stiffness. The latter
is characterized by the planar elastic modulus $E_{\rm 2D}$ and Poisson number
$\nu$. The radius of the detached area surrounding the particles is denoted as
$R_{\rm d}>R_{\rm p}$. Because of the mirror symmetry of the problem with
respect to the plane $z=0$, it is sufficient to consider a single membrane
deformed by the half particle, hence, the problem is similar to the problem of
indentation of a 3D membrane of finite thickness as described by Begley [55],
whose treatment we adapt to a 2D membrane sheet. We thus consider a membrane
that is displaced, along the radius $r=R_{\rm p}$, by $u_{z}=h$ whereas for
$r\geq R_{\rm d}$ the displacement is fixed at $u_{z}=0$. A solution of the
elastic problem with these boundary conditions is sought for $R_{\rm p}\leq
r\leq R_{\rm d}$. The approximate solution of this elastic boundary value
problem is discussed in supplementary information. Here we only summarize the
main results:

  * •

The displacement profile of the membrane is approximately given by

$u_{z}(r)=h\frac{q_{\rm d}^{a}-(r/R_{\rm p})^{a}}{q_{\rm d}^{a}-1}\quad,q_{\rm
d}=\frac{R_{\rm d}}{R_{\rm p}},\quad a\approx 2/3$ (2)

  * •

The elastic energy of the system is given by

$E_{\rm el}=\pi R_{\rm p}^{2}\frac{2E_{2D}}{27}\frac{q_{\rm p}^{4}}{(q_{\rm
d}^{2/3}-1)^{3}}$ (3)

The parameter $q_{\rm p}$ is the aspect ratio of the sandwiched particle. To
determine the parameter $q_{\rm d}$ and hence the radius of the detached area
surrounding the particle, we use a virtual work argument. The virtual work
incurred upon expanding the detached area consists of the work released by the
change in elastic energy and the work of adhesion that must be expended to
detach the adhering graphene sheets. This detachment energy is simply given by
${\rm d}E_{\rm d}=2\pi(\gamma_{\rm GG}/2)R_{\rm d}{\rm d}R_{\rm d}$ where
$\gamma_{\rm GG}$ is the adhesion energy per unit area of the graphene
bilayer, and $\gamma_{\rm GG}/2$ the adhesion energy per sheet. We thus set
${\rm d}W={\rm d}E_{\rm el}+{\rm d}E_{\rm d}=0$ which leads to the equation

$\frac{{\rm d}E_{\rm el}}{dR_{\rm d}}+\pi\gamma_{\rm GG}R_{\rm d}=0$ (4)

From this virtual work balance we derive an implicit relation connecting
$q_{\rm p}$ and $q_{\rm d}$:

$\frac{q_{\rm p}}{q_{\rm d}^{1/3}(q_{\rm
d}^{2/3}-1)}=\left(\frac{27\gamma_{\rm GG}}{4E_{2D}}\right)^{1/4}=0.283$ (5)

where we used the adhesion energy $\gamma_{\rm GG}=0.288$ J/m2 for bilayer
graphene and a planar elastic modulus of $E_{2D}=300$ N/m. Taking the
detachment ratio for a given particle aspect ratio from this relation, we can
write the detachment energy as a function of particle radius

$E_{\rm d}=\pi R_{\rm d}^{2}\gamma_{\rm GG}=\pi\gamma_{\rm GG}q_{\rm
d}^{2}R_{\rm p}^{2}$ (6)

From this relation and the detachment energies calculated by MM simulation in
Figure 5, we find $q_{\rm d}\approx 4.0$ (red datapoints and red line in
Figure 5a). This is consistent with direct determination of the detachment
area as a function of nanoparticle radius, which gives $q_{\rm d}\approx 4.2$
(Figure 5b). By using Eq. (5) elastic energy of one graphene sheet can be
obtained as

$E_{\rm el}=\pi R_{\rm d}^{2}\frac{\gamma_{\rm GG}}{2}(1-q_{\rm d}^{-2/3})=\pi
R_{\rm p}^{2}\gamma_{\rm GG}\frac{q_{\rm d}^{2}}{2}(1-q_{\rm d}^{-2/3})$ (7)

which, with $q_{\rm d}=4.0$, gives us the black lines in Figure 5a and 5c.
Finally, geometrical similarity requires that, also in case of non cylindrical
particles, the effective area of contact between the graphene sheet and the Pt
nanoparticle is proportional to the square of the particle radius. This gives
us for the Pt-graphene adhesion energy

$E_{\rm a}=\pi R_{\rm p}^{2}\gamma_{\rm PG}q_{\rm c}^{2}$ (8)

By matching this with MM simulation data, we find the geometrical factor of
$q_{\rm c}=0.87$ for annealed particles (blue data points and blue line in
Figure 5a). Where $\gamma_{\rm PG}=0.476$ J/m2 represents Pt-graphene adhesion
energy per unit area. For cylindrical particles $q_{\rm c}=1$.

The elastic energy and the detachment radius are shown in Figure 9 as
functions of the nanoparticle aspect ratio. By comparing the continuum results
with the results of our simulations of annealed particles we can show that,
for those particles, the elastic energy amounts to about $0.62E_{\rm d}$,
hence $q_{\rm d}\approx 4$ according to the mechanical model. This is in good
agreement with the observed proportionality $R_{\rm d}\approx 4.2R_{\rm p}$
found in the MM simulations in Figure 5c and Figure 9. These values correspond
to an effective aspect ratio of $q_{\rm p}\approx 0.67$.

Figure 9: Left: detachment radius and elastic energy of the detached sheet as
functions of the particle aspect ratio; right: detachment radius as a function
of particle radius; solid symbols: MD simulation data, solid line: continuum
prediction for cylindrical particle of aspect ratio $q_{\rm p}=1$.

###  3.2 Two-particle interaction and mechanical stability of a two-particle
system

As noted in the MD section, if two particles get close and the respective
detachment radii overlap, they experience an attractive mutual interaction. A
semi-quantitative model can be formulated by approximating the conformation of
the graphene sheets around a pair of interacting particles as two detached
semicircles of radius $R_{d}$ connected by a detached ribbon of constant width
$w\approx 2R_{\rm d}$ and area $2R_{\rm d}d$ (see Figure 6). The ribbon forms
as soon as the particles approach below a critical distance $d_{\rm c}$, and
the interaction energy then changes linearly with distance $d$.

Since the elastic distortion of the graphene sheet over the two detached
semicircles is very similar to that around a single particle, we approximate
the detachment energy in these regions by that of a single particle. On the
other hand, across the detached ribbon the elastic energy is significantly
reduced, and we therefore approximate the energy per unit area of the ribbon
by $\gamma_{\rm GG}$. The total GG excess energy of the interacting two-
particle system is then

$E^{\rm 2p}(d,R_{\rm d})=1.6\pi\gamma_{\rm GG}R_{\rm d}^{2}+2\gamma_{\rm
GG}dR_{\rm d}.$ (9)

The detached ribbon forms as soon as this energy falls below the GG excess
energy of two non interacting particles, $2E^{\rm 1p}=2\times
1.6\pi\gamma_{\rm GG}R_{\rm d}^{2}$. Based on this argument the critical
separation follows via $1.6\pi R_{\rm d}=2d_{\rm c}$ as $d_{\rm c}\approx
2.5R_{\rm d}$ which is in good agreement with the MD data shown in Figure 7.
The GG excess energy can then be written as

$E^{\rm 2p}(d)\approx 2\gamma_{\rm GG}R_{\rm d}^{2}\frac{d_{\rm c}+d}{R_{\rm
d}}$ (10)

from which the interaction force of the two particles follows as

$F^{\rm 2p}=\frac{\partial E^{\rm 2p}(d)}{\partial d}\approx 2\gamma_{\rm
GG}R_{\rm d}$ (11)

or, expressed in terms of $R_{\rm p}\approx R_{\rm d}/4$

$F^{\rm 2p}=\frac{\partial E^{\rm 2p}(d)}{\partial d}\approx 8\gamma_{\rm
GG}R_{\rm p}$ (12)

The interaction force is, hence, approximately proportional to the Pt
nanoparticle radius. Two nanoparticles are stable if this force is unable to
overcome the shear resistance at the Pt-Graphene interface. The shear force
required to slide a Pt nanoparticle inside the graphene bilayer can be
estimated as $F_{\rm c}\approx 2\pi R_{\rm p}^{2}\tau_{\rm c}$ where
$\tau_{\rm c}$ is the critical shear stress for interface sliding. Equating
this to the inter-particle force we find a critical radius below which the
particles will slide towards each other:

$R_{\rm p,c}=\frac{4\gamma_{\rm GG}}{\pi\tau_{\rm c}}$ (13)

With a typical critical shear stress of $\tau_{\rm c}=15$ MPa (Figure 3) we
find a critical nanoparticle radius $R_{\rm p}\approx 24$nm. Particles below
this radius, if closer than $d_{\rm c}\approx 10R_{\rm p}$, are likely to
spontaneously aggregate.

##  4 Monte Carlo Simulation

In the following we study the properties of assemblies of statistically
distributed nanoparticles, assuming the nanoparticles are above the critical
radius for nanoparticle aggregation and thus on stable locations within the
graphene bilayer. Assuming the particles are deposited at random locations
with the sole constraint that two particles cannot overlap, we can then use
static Monte-Carlo simulation to generate particle assemblies and investigate
their geometrical properties. We use this approach to address two questions:
(1) what is the minimum concentration of particles that can generate a
connected pore space as required for catalytic applications, (2) what is the
concentration of particles that yields the smallest overall adhesion of the
graphene bi-layer?

In the following, nanoparticles are modeled as disks of radius $R_{\rm p}$ in
a two-dimensional continuum, subject to a hard-core repulsion pair potential.
Their configurations are obtained using the particle Monte Carlo method,
employing the rejection-free Geometric Cluster Algorithm in order to
accelerate simulations [56]. The surface area occupied by a nanoparticle is
quantified by counting the number of carbon atoms it covers on the underlying
graphene sheet. The same procedure is used to measure detachment areas. For
simplicity we assume that the detachment area around a nanoparticle has
circular shape with radius $R_{\rm d}=q_{d}R_{\rm p}$, which provides a lower
bound to the actual detachment area in a given multi-particle configuration.
Upon increasing the number of nanoparticles, detachment areas form connected
components. Component sizes are evaluated employing a standard breadth-first
search algorithm on the lattice [57].

Figure 10: (a) Area fraction of the largest connected pore cluster as a
function of particle density, for different particle aspect ratios; (b) Area
fraction of the second largest cluster, aspect ratios as in plot (a); (c) Net
adhesion energy (energy per unit area required to detach one graphene sheet)
as function of particle density, aspect ratios as in plot (a), full line:
upper bound given by Eq. (14), top, dashed line: lower bound given by Eq.
(14), bottom.

We first analyze the question at which nanoparticle density a connected pore
space emerges. This is a standard percolation problem. The critical particle
density (the percolation threshold) is most conveniently identified by
investigating the size $s_{1}$ of the largest cluster in the system and, more
prominently, the size $s_{2}$ of the second largest cluster, which is expected
to be the largest at the percolation point Figure (10). Figure 10 (b) shows
that, almost independent from the aspect ratio, percolation occurs when the
particle concentration $n$, measured in units of $1/R_{\rm d}^{2}$ where
$R_{\rm d}$ is the detachment radius around a single particle, reaches a value
of $n\approx 0.5R_{\rm d}^{-2}$. The area of the largest connected cluster
then continues to increase with increasing particle concentration as remaining
areas of direct contact between the two graphene sheets become detached, see
Figure 10 (a). The detached area reaches a maximum at a concentration that
increases with increasing ratio of detachment radius to particle radius, but
is of the order of $n\approx R_{\rm d}^{-2}$ for all radii. Beyond this
maximum, the graphene sheets are completely detached from each other and
adding more Pt particles now reduces the available pore space.

Looking at the energy of adhesion required to remove one of the graphene
sheets from Pt on one side and separate apart the mutually adhering GG ( going
backward from step 3 to step 1 in Figure 5 ) we can estimate the corresponding
energy per unit area as $E_{\rm ad}=f_{\rm GG}\gamma_{\rm GG}+f_{\rm
PG}\gamma_{\rm PG}-E_{\rm el}$ where $f_{\rm GG}$ is the area fraction of
graphene-graphene contacts, and $f_{\rm PG}$ the area fraction of Pt-graphene
contacts, whereas $E_{\rm el}$ is the energy due to elastic distortion of the
detached graphene. For the respective adhesion energies we use the values
$\gamma_{\rm GG}=0.288$ J/m2 and, from our DFT calculation, $\gamma_{\rm
PG}=0.476$ J/m2. For the energy of elastic distortion we consider the two
limits $f_{\rm PG}<<1$ and $f_{\rm GG}<<1$. In the first case we see mainly
isolated nanoparticles. In that case the elastic energy of the detachment area
around a nanoparticle is, as shown in Section 2.3, about 0.62 times the
detachment energy. The detached area fraction equals $(1-f_{\rm GG}-f_{\rm
PG})$, and the elastic energy is then $E_{\rm el}\approx 0.62(1-f_{\rm
GG}-f_{\rm PG})\gamma_{\rm GG}$. In the second case, $f_{\rm GG}<<1$, both
graphene sheets are no longer in mutual contact and the elastic energy is
negligible. This gives us the following expressions for the net energy of
adhesion between the two sheets:

$E_{\rm ad}=\left\\{\begin{array}[]{ll}f_{\rm GG}\gamma_{\rm GG}+f_{\rm
PG}\gamma_{\rm PG}&,\quad f_{\rm GG}\ll 1\\\ f_{\rm GG}\gamma_{\rm GG}+f_{\rm
PG}\gamma_{\rm PG}-0.62(1-f_{\rm GG}-f_{\rm PG})\gamma_{\rm GG}&,\quad f_{\rm
PG}\ll 1\end{array}\right.$ (14)

Here, the first expression provides an upper bound and the second expression a
lower bound to the actual energy of adhesion. Both bounds are shown in Figure
10 (c) as functions of nanoparticle density. We find a minimum of the adhesion
energy which occurs close to the density where the pore space is maximum. The
depth of this maximum and hence the maximum reduction in adhesion depend on
nanoparticle aspect ratio: pancake-shaped nanoparticles of small aspect ratio
provide better adhesion. As a function of nanoparticle density, we find thus
three regimes: At low density, we find a non connected pore space concomitant
with strong adhesion that decreases with increasing nanoparticle density. At
high density, we find a continuous pore space in between the nanoparticles,
and adhesion increases with nanoparticle density. At an intermediate density,
there is a minimum of adhesion where even spontaneous detachment might occur,
in agreement with the experimental finding that decoration with Pt
nanoparticles may prevent graphene sheets from aggregating.

##  5 Summary and Conclusions

We have performed a multiscale simulation study of graphene bi-layers
containing embedded Pt nanoparticles. Such composite systems have potential
applications for catalysis, but also for the simple purpose of reducing the
mutual adhesion of graphene sheets which is important, for instance, if these
are to be dispersed as fillers in composites. DFT calculations have provided
us with data to parameterize and benchmark potentials for describing the Pt-
graphene interface, and to characterize the resistance of this interface to
shear displacements of Pt on graphene or vice versa. MD calculations were then
performed to determine the energetics and interactions of Pt nanoparticles
embedded into graphene, and the results were used to validate a mechanical
model which envisages the embedded nanoparticles as rigid cylindrical
particles of variable aspect ratio that are inserted between two adherent
elastic sheets. With an effective aspect ratio of about 0.67, excellent
agreement between the MD results and the predictions of the mechanical model
was obtained, both regarding the energetics of the system and the size of the
detachment zone surrounding the embedded nanoparticles. The model was then
extended to describe the interaction of two nanoparticles mediated by the
intermediate detachment zone, and it was shown that this interaction makes
embedded nanoparticles below a critical radius of about $R_{\rm p,c}\approx
24$nm and separation below about 10 $R_{\rm p}$ unstable with respect to
spontaneous aggregation. Such aggregation reduces the overall detached area
and may push a system with initially connected pore space below the
percolation threshold.

Regarding the geometrical and energetic properties of systems of randomly
dispersed nanoparticles above the critical radius, we find that percolation of
the detachment areas occurs above a critical nanoparticle density of
approximately $1/R_{\rm d}^{2}$. Above this concentration, the area fraction
of the largest connected detachment area first continues to increase, then
decreases as the sheets forming the bilayer are completely detached and the
addition of further nanoparticles reduces the free space between the graphene
sheets. At the same time, the energy of adhesion (the energy needed to
separate the bilayer) first decreases, then increases again. In the context of
catalytic applications, this allows to increase both the catalytically active
nanoparticle surface and the stability of the structure, and trading this off
against a reduction in pore space. In the context of using Pt nanoparticles as
graphene ’spacers’ we find an optimum nanoparticle density where the adhesive
energy of the structure is lowest, and thus the separation of the graphene
sheets most easy.

It remains a task for future investigation to extend the present modelling
approach from bilayer to multilayer structures. The interaction potentials
derived in the present study can serve as a foundation for such a model.
However, on larger scales, graphene-mediated interactions between Pt particles
located in different layers of a stacked multilayer structure need to be taken
into account. A systematic investigation of the resulting complex multi-
particle effects and the resulting optimal structures will be a task for
future work.

## Acknowledgments

SN, KW, MY and MZ acknowledge financial support from DFG under grant no.
Za171/11-1. M.Z. also acknowledges support by the Chinese State Administration
of Foreign Expert Affairs under Grant No MS2016XNJT044 for the initial visit
to Chengdu during which this research was planned. The authors gratefully
acknowledge the compute resources and support provided by the Erlangen
Regional Computing Center (RRZE).

## Author contributions

SN and CG performed MD simulations and analyzed MD data using the continuum
mechanical model. KW and MY performed DFT simulations. PM performed MC
simulations. QL provided unpublished experimental data regarding Pt decorating
of graphene which guided the simulations. MZ is responsible for the simulation
design. All authors were involved in the preparation of the manuscript. The
final version has been read and approved by all authors.

## Conflict of Interest

the Authors declare no Competing Financial or Non-Financial Interests.

## Reference

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