“The Princess and the Pea” at the Nanoscale: Wrinkling and Delamination of Graphene on Nanoparticles

Thin membranes exhibit complex responses to external forces or geometrical constraints. A familiar example is the wrinkling, exhibited by human skin, plant leaves, and fabrics, that results from the relative ease of bending versus stretching. Here, we study the wrinkling of graphene, the thinnest and stiffest known membrane, deposited on a silica substrate decorated with silica nanoparticles. At small nanoparticle density, monolayer graphene adheres to the substrate, detached only in small regions around the nanoparticles. With increasing nanoparticle density, we observe the formation of wrinkles which connect nanoparticles. Above a critical nanoparticle density, the wrinkles form a percolating network through the sample. As the graphene membrane is made thicker, global delamination from the substrate is observed. The observations can be well understood within a continuum-elastic model and have important implications for strain-engineering the electronic properties of graphene.

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In Hans Christian Andersen’s fairy tale, “The Princess and the Pea,” multiple mattresses were used to hide the presence of a tiny pea in a test to tell apart a true princess in a commoner’s appearance from a possible impostor, but the exquisitely discerning back of the true princess was indeed not fooled by the multiple mattresses. In this experimental paper we reenact “The Princess and the Pea” tale with props and players from the world of atomic and nanoscale materials: sheets of graphene—an amazing layer of single carbon atoms—for mattresses, ${\mathrm{SiO}}_2$ nanoparticles for peas, and an atomic force microscope for the true princess. Is the princess able to tell the presence of peas hidden under multiple mattresses?

Our interest in this particular line of investigations is of course not theatrical. Beside its unique electronic and thermal properties, graphene also has a mechanical strength that is surprising for the thinnest possible mechanical membrane. Studying how a single sheet, or multiple sheets, of graphene behave when laid on a rough “bed board” formed by nanoparticles randomly placed on a flat solid substrate provides a way to a deeper understanding of graphene’s material properties and extends the realm of thin-sheet mechanics to the nanoscale limit.

Graphene placed over a nanoparticle-decorated “bed board” does not lie flat, showing a degree of mechanical flexibility. But, does it bend, or does it stretch, or does it do both, to conform to the substrate? When the nanoparticles are spaced far apart from each other, graphene forms a bump over each nanoparticle and stretches in the bump to accommodate the particle. But this stretching costs significant energy. As the nanoparticles become closer, therefore, graphene gives up this bump-stretching strategy and instead opts for a more energy-saving one by bending to form a wrinkle that connects two particles next to each other. As the number of graphene sheets becomes larger, bending becomes more difficult. Instead of wrinkling, the whole multisheet film becomes unstuck from the bed board altogether, forming a surface above the rough particles, whose smoothness could have fooled the most discerning “princess” about the hidden “peas.” What is also remarkable is that a mechanical theory that treats graphene as a thin-sheet continuum with stretching and bending elasticity, i.e., ignores its underlying atomic structure, provides a good description of what we have observed, in particular, the critical particle separation for wrinkle formation at the nanoscales.

The tale of “The Princess and the Pea” here adds insights to the research effort of manipulating graphene’s electronic properties by mechanical deformation. It should also apply to other nanoscale thin films adhered to rough surfaces and may have a broad appeal to materials physicists and engineers.

Figure 1

AFM images ($1\times 1\mu {\mathrm{m}}^2$) of graphene on ${\mathrm{SiO}}_2$ nanoparticle/${\mathrm{SiO}}_2$ substrates for a nanoparticle density of (a) 11, (b) 22, (c) 49, (d) 90, and (e) $170\mu {\mathrm{m}}^{-2}$, respectively.

Figure 2

Schematics of (a) a wrinkle formed between two nanoparticles with diameters $d$ and (b) the wrinkle profile along the transverse direction as represented by shaded area in (a).

Figure 3

(a) Profile of the wrinkle along the white dotted line in the AFM image shown in the inset. The scale bar in the inset is 50 nm. The solid red lines are theoretical expectations. (b) The maximum deflection ${\zeta }_0$ as a function of the wrinkle length. The error bar indicates the uncertainty of ${\zeta }_0$ due to the height difference between the protrusions. We choose wrinkles formed between protrusions where the height differences are less than 10%. The area shaded in red is the theoretical prediction for the scaling of ${\zeta }_0$ with $X$ for $\Gamma =0.6–2.8\mathrm{eV}/{\mathrm{nm}}^2$. The inset is a typical AFM image of the wrinkles formed between the two protrusions. The scale bar is 20 nm.

Figure 4

The density of wrinkles ${\rho }_w$ and the mean number of wrinkles per protrusion ${\rho }_w/{\rho }_{\mathrm{np}}$ (inset) as functions of nanoparticle density ${\rho }_{\mathrm{np}}$. Each arrow corresponds to the AFM images shown in Figs. 1a, 1b, 1c, 1d, 1e. The solid red lines are fits described in text.

Figure 5

(a) $P$ as a function of nanoparticle density for $L=1$, 2, and $3\mu \mathrm{m}$. The inset is an AFM image ($1\times 1\mu {\mathrm{m}}^2$) of graphene on the ${\mathrm{SiO}}_2$ nanoparticles with a density of $57\mu {\mathrm{m}}^2$, showing the percolating cluster highlighted by the dashed curve. (b) $\Pi$ as a function of the density of nanoparticles for $L=0.5$, 1, 2, and $3\mu \mathrm{m}$. Points for $L=0.5$, 1, and $2\mu \mathrm{m}$ represent averages in a bin of $10\mu {\mathrm{m}}^{-2}$. The inset is a plot of $\log \Delta$ as a function of $\log L$; the red line indicates a best-fit power exponent of $-1.0$. (c) The mean finite cluster size $S$ as a function of the density of nanoparticles (points represent averages in a bin of $2\mu {\mathrm{m}}^{-2}$). The red dashed line is the theoretical expectation (described in text).

Figure 6

Typical AFM images of (a) monolayer, (b) trilayer, (c) 7-layer, (d) 10-layer, (e) 14-layer, and (f) 18-layer graphene on ${\mathrm{SiO}}_2$ with nanoparticle density $160\pm 24\mu {\mathrm{m}}^{-2}$. The scale bar in each image is 400 nm. The insets in (a), (d), and (f) are corresponding schematics of graphene films on nanoparticles to the AFM images. (g) The fractional area $\varphi$ where graphene conforms to the substrate and the characteristic length $l$ of the delaminated domains, as functions of the number $n$ of graphene layers. The dashed curves are the theoretical expectations for $\varphi$ (orange) and $l$ (blue) (described in text).

Figure 7

Typical AFM (a) height and (b) phase images ($1\times 1\mu {\mathrm{m}}^2$) of 4-layer graphene supported on the nanoparticles. The scale bar is 200 nm. (c) Line profiles of the height and phase along the dashed red and blue lines shown in (a) and (b), respectively. The arrows correspond to those in (b), showing the locations of nanoparticles beneath graphene.

Figure 8

Schematic of graphene supported on a single nanoparticle. The detachment length is $R$.