Wrinkle motifs in thin films

On length scales from nanometres to metres, partial adhesion of thin films with substrates generates a fascinating variety of patterns, such as ‘telephone cord’ buckles, wrinkles, and labyrinth domains. Although these patterns are part of everyday experience and are important in industry, they are not completely understood. Here, we report simulation studies of a previously-overlooked phenomenon in which pairs of wrinkles form avoiding pairs, focusing on the case of graphene over patterned substrates. By nucleating and growing wrinkles in a controlled way, we characterize how their morphology is determined by stress fields in the sheet and friction with the substrate. Our simulations uncover the generic behaviour of avoiding wrinkle pairs that should be valid at all scales.

The strength of friction is quantified by the strain energy per area of the sheet once it has relaxed in this potential, without any external forces applied.
We drain energy from the system with a viscous fix with damping parameter 5 × 10 −4 eV/ps. We run our simulations until no change in sheet configuration can be seen in two snapshots separated by 10 4 time steps. We use a time step of 0.0005 ps.
The stress on particle i is given by the tensor (S1) The first term is a contribution from the kinetic energy, which is zero at the end of our simulations, when we measure the stress. The second term gives contributions from each of the N (i) B bonds particle i is a member of, and the final term gives contributions from the N (i) I impropers particle i is a member of. r (i) is the position of particle i and F (i) refers to the force on i as a result of the relevant interaction. The quantity S given by Eq. (S1) is not strictly a stress, but is multiplied by a characteristic volume and is returned with units barsÅ 2 . We obtain the stress σ on the sheet by taking the particle volume to be the area of a triangle (here ∼ 10.8Å 2 ), multiplied by the sheet thickness, which is approximately 3n A. We focus on the stress trace in part because it is invariant and therefore independent of the local orientation of the sheet.

Nature of substrate radial interactions
As discussed in the main text, in our simulations the substrate exerts a harmonic repulsion on the graphene sheet, with a potential energy of the form where r is the radial distance from the substrate, k = 1.0 eV/Å 2 is a spring constant and r c = 1Å. In addition to this interaction, in order to push the sheet towards the substrate a constant downward force of f down = 0.001 eV/Å is applied to each particle in the sheet throughout the simulation. Figure S1 shows the potential energy as a function of height above the substrate for a flat sheet. Minimization of the total potential energy gives an equilibrium height above the substrate of r c − f down /(2k) = 0.9995Å.
We have also tested a Lennard-Jones substrate interaction. In that case, the interaction with the substrate has potential energy with = 0.04 eV and σ = 1Å. In those simulations, a position-independent downward force of 0.001 eV/Å is applied for the first 5 × 10 5 simulation steps (out of typically 10 6 required for full equilibration), and is then turned off. The potential energy as a function of height above a substrate for these simulations is shown in Fig. S1.
Although the energy minimum near a height of 1Å is more pronounced for the LJ potential, as seen in Fig. S1, we find this makes little difference to simulations. Figure S2 shows height maps for simulations of a monolayer sheet deposited on a substrate with four particles arranged in a square with side length 140 nm, for the two types of substrate interactions.
The differences between the two configurations are minimal, and importantly the wrinkle wavelengths are unaffected.

Friction between sheet and substrate
As discussed in the main text, we have studied a substrate with random pinning to model friction. This is implemented as randomly-positioned Gaussian potentials whose centers are on a plane at a distance z 0 below the minimum of a Lennard-Jones potential. The LJ potential is the dominant contribution so that in the absence of external forces the graphene sheet sits near the minimum of the LJ potential. Varying z 0 enables us to tune the strength of the random contributions to the sheet's energy.
We make two characterizations of the potential. (i) The mean strain energy of a sheet that has relaxed in the potential, ε. This is exponentially distributed, so we also give the maximum values measured, ε max (the minimum values are always ∼ 0). (ii) The mean adhesion energy of the relaxed sheet, Γ. Table S1 reports these characteristics for the three substrates we study in the main text. The graphene layer is thermalized using a Berendsen [6] thermostat set at 300K, with a characteristic relaxation time set to 0.1 ps; the simulation timestep is set at 1 fs to ensure a correct time integration of the atom dynamics. All simulations were performed using the LAMMPS molecular dynamics toolbox [1]. The simulation protocol consists of four steps.
(i) the graphene layer is generated, and the system is brought to the energy minimum using conjugate gradient minimization; (ii) the graphene layer is placed above the substrate, just beyond the cutoff of the potential, and the atoms are given a small initial velocity towards the substrate; (iii) the graphene interacts with the substrate and becomes attached to it; the simulation is run until the whole patch has come in contact; in some cases (PBC), periodic oscillations around an equilibrium configuration can be observed; the thermostat induces only minimal damping, and the oscillations can be observed for long times; (iv) we drive the system to its equilibrium configuration by removing the thermostat and by adding an artificial viscous damping term. The simulation is stopped once a lower threshold in atom motion is reached.
As seen in Fig. S3, periodic boundary conditions result in a very smooth deformation of the sheet, which remains flat and attached to the substrate away from the particle pair. We The graphene adheres strongly on contact and once adhered, can neither detach nor move laterally. As a result, although wrinkles are present in the graphene, they are lower than those observed experimentally and are atomically narrow. We believe they correspond to boundaries between regions of graphene that have adhered to the substrate in an incompatible way, and are therefore fundamentally different in origin to the wrinkles observed in experiments [2] and our model.

Delamination of graphene multilayers on particle-decorated substrates
We have studied the role of wrinkle coalescence in the transition from conformation to delamination of a sheet deposited on a substrate decorated with nanoparticles. Previous simulations of deposition on small clusters of nanoparticles indicate interesting pseudomagnetic field distributions [7], which provide technological motivation for these studies. We simulate a 600 × 600 nm 2 graphene sheet, on a substrate with four nanoparticles of diameter 8 nm, arranged in a rectangle with aspect ratio close to unity and a range of side lengths s, with centre close to the sheet centre. (We use this configuration, rather than a square centred exactly, in order to avoid spurious effects arising from symmetry.) Example movies are provided (four particles spacing160nm 1layer.avi and four particles spacing160nm 5layers.avi). Figure S5(a-d) shows typical configurations for close-spaced and distant particles, for a monolayer (panels a and b) and a 5-layer sheet (panels c and d). When particles are close together, the sheet detaches in the centre of the rectangle. As particle spacing is increased, the sheet conforms to the substrate except at wrinkles that radiate from the particles.
Increasing the sheet thickness 'smooths' the sheet and increases the size of features, as seen by comparing panels (a,b) and (c,d) of Fig. S5. Thicker sheets also have lower stresses, as seen in Fig. S5(e-h). Near the particles, stresses are tensile (Tr(σ) > 0), but compressive regions exist inside wrinkles and especially at the wrinkle tips.
The delamination transition is quantified in Fig. S5(i), which shows the height of the centre of the sheet as a function of particle spacing s, for 1-, 2-, 5-and 10-layer sheets. In all cases, a sharp transition is observed at a particle spacing s * that depends on sheet thickness.
A first estimate of this dependence is given by assuming bending rigidity does not enter the problem and the sheet deforms purely through stretching. Then the delaminated region of an n-layer sheet around a protrusion has radius R ≈ r(4nE 2D /3Γ) 1/4 [2, 8], where r is the particle radius, E 2D is the tensile rigidity of the sheet and Γ characterizes the adhesion energy. Under the assumption that the sheet detaches when delaminated regions meet, the critical particle spacing scales as s * ∼ n 1/4 . As seen in Fig. S5(j), this scaling collapses the 1and 2-layer curves well, but already breaks down for the 5-layer sheet. This is unsurprising, because the argument does not take into account the existence of stress fields and wrinkles connecting the particles. [5] Tersoff, J. New empirical approach for the structure and energy of covalent systems. Phys.

List of supplementary movies
Included are five video files. They are: 1. four particles spacing160nm 1layer.mov Wrinkles form during deposition before the sheet is in full contact with the substrate. The video shows deposition of a monolayer sheet on four particles of diameter 8nm, with interparticle spacing 160 nm. monolayer sheet, particle spacing 60 nm, (b) monolayer sheet, particle spacing 160 nm, (c) 5-layer sheet, particle spacing 60 nm, (d) 5-layer sheet, particle spacing 160 nm. Panels (e-h) show the corresponding trace of the stress tensor. Where the stress is above 15 GPa, the color scale has been clipped (black regions on top of the particles). All eight panels have the same scale; the scale bar in panel (a) represents 100 nm. (i) Height of the center of the graphene sheet as a function of particle spacing, for 1, 2, 5 and 10 layer sheets. (j) As a first approximation, the delamination transition occurs at a particle spacing s * ∼ n 1/4 .