Size-dependent shape characteristics of 2D crystal blisters

Abstract

Micro- and nano-sized blisters can form spontaneously when two-dimensional (2D) crystals are transferred onto substrates because liquid molecules that are initially adsorbed on 2D material and substrate surfaces can be squeezed and trapped by interfacial forces. In this work, we use a combination of experiments, continuum theories, and coarse-grained molecular dynamics (CGMD) simulations to investigate the shape characteristics of spontaneously formed blisters under 2D crystals with heights ranging from a few ångströms to tens of nanometers. We show three distinct regimes in which the height-to-radius ratios (i.e., aspect ratios) of liquid-filled 2D crystal blisters are size-independent, rough linearly proportional, and inversely proportional to the blister radius. We reveal that the blister shape characteristics are governed by three factors: the 2D crystal elasticity, the interfacial interactions, and the phases of confined substances. The characteristic length scales (to which comparing the blister height or radius can define the boundary between these different regimes) are also discussed. We also provide approximate analytical solutions to the blister aspect ratios, which, together with complementary CGMD simulation results, agree with our experimental measurements.

Introduction

Two-dimensional (2D) crystals refer to a single layer or a few layers of covalently bonded atoms, including graphene, hexagonal boron-nitride (h-BN), transition metal-dichalcogenides (TMDCs, e.g., ${\mathrm{MoS}}_2$), among others (Novoselov et al., 2004, Novoselov et al., 2016, Gibertini et al., 2019). The atomic-level thickness and exceptional mechanical and physical properties of 2D materials make them promising for a wide range of applications, such as flexible and stretchable electronics and photonics (Mak and Shan, 2016, Akinwande et al., 2017, Jang et al., 2022). In most applications, 2D crystals need to be placed on supporting substrates for further fabrication and integration (Geim and Grigorieva, 2013). When 2D crystals interface with another surface, the interfacial van der Waals (vdW) forces squeeze interfacial substances into a blister-shaped volume to lower the system-level energy (Stolyarova et al., 2009, Levy et al., 2010, Sanchez et al., 2021). On the one hand, blisters are undesirable in 2D material devices as they impede charge/photon/phonon transport across the interface, so various means were developed to eliminate interfacial blisters (Uwanno et al., 2015, Pizzocchero et al., 2016, Jain et al., 2018, Purdie et al., 2018, Rosenberger et al., 2018). On the other hand, mechanics analysis has elucidated that the blister morphology is a good indicator of the interfacial properties of 2D crystals, such as adhesion (Khestanova et al., 2016, Sanchez et al., 2018). Furthermore, the inhomogeneous strain distributions and the rich surface topographies of 2D crystal blisters can be leveraged for optimizing luminescence and exciton transport in 2D crystals (De Palma et al., 2020, Brennan et al., 2020) as well as designing mechanical sensors, microlenses, and more (Lin et al., 2020, Yang et al., 2022). Additionally, 2D crystal blisters can form high-pressure liquid cells for the observation of chemistry under nanoconfinement within transmission electron microscopy (TEM) due to their atomic thinness (Vasu et al., 2016). Therefore, fundamental mechanistic understandings of the spontaneously formed 2D crystal blisters can provide necessary and useful insights for 2D crystal applications.

The 2D crystal blister is a notable example of blistering at thin-film/rigid-substrate interfaces, where crack growth is controlled by balancing the reduction in potential energy with the increase in surface energy in accordance with the Griffith criterion (Griffith, 1921). As a result, self-similarity, i.e., a constant aspect ratio, of blisters at monolayer 2D crystal/substrate interfaces has been widely observed (Khestanova et al., 2016, Sanchez et al., 2018). Their profiles have been quantitatively predicted by considering the competition between the stretching energy of the bulged 2D crystal and the change of interfacial energy. In this scenario, the blister profile features an apparent kink across the blister edge, forming the “contact angle”, which is captured by membrane theory (Dai et al., 2018, Blundo et al., 2021, Fang et al., 2022). As the number of 2D crystal layers increases, the bending energy of 2D crystals starts to play a role in the blister morphology (Wang et al., 2022). The blister aspect ratio becomes size-dependent, and the blister edge features a smooth transition to a zero slope, following plate theory (Yue et al., 2012, Wang et al., 2013). The slippage of 2D crystals on substrates also affects the blister morphology (Wang et al., 2017, Dai et al., 2018, Sanchez et al., 2018, Rao et al., 2021). In particular, the hoop compression caused by inward sliding may induce radial wrinkling instabilities (Dai et al., 2020, Dai and Lu, 2021, Ares et al., 2021, Dai et al., 2022).

Although 2D crystal blisters have been investigated from different aspects, the mechanics of 2D crystal blisters with limiting sizes (e.g., a few nm and below) are rather complex (Ares et al., 2021, Dai et al., 2022). There exists a transition between stretching- and bending-dominated blisters (Ma et al., 2022). Moreover, instead of a discontinuity in interfacial energy across the blister edge, continuous vdW interactions have to be considered for these ultra-shallow blisters. The vdW interactions can be viewed as nonlinear elastic springs acting between 2D crystals and substrates (Aitken and Huang, 2010, Gao and Huang, 2011, Xue et al., 2022). It is therefore natural to ask: how large is the vdW process zone, and when should it be incorporated into 2D crystal blister models? Unfortunately, quantitative answers are lacking in existing work. Furthermore, recent studies have indicated that as the blister height approaches a few ångströms, a minimum blister height corresponding to a single molecular layer of interfacial substances emerges, as observed in both experimental studies and molecular dynamics (MD) simulations of 2D crystal blisters (Lee et al., 2012, Iakovlev et al., 2017, Bampoulis et al., 2018, Villarreal et al., 2021). It is therefore interesting to investigate the breakdown of continuum mechanics for ultra-shallow blisters.

Driven by the aforementioned curiosities, we carry out experimental, analytical, and MD investigations on micro- and nano-sized blisters formed by 2D crystals. This paper is organized as follows. In Section 2, we report the experimental procedures for creating parent-satellite blisters at the interface between multilayer graphene (MLG) and silicon-dioxide (${\mathrm{SiO}}_2$) substrate. We then establish a Griffith-type model (GTM) in Section 3 to elucidate the spontaneous formation of blisters governed by the competition among the stretching and bending energies of the bulged 2D crystal, the change of interfacial energy, and the interfacial sliding. The GTM predicted aspect ratios agree well with the experimental measurements on the parent blisters but not the satellite ones. Therefore, we establish a cohesive zone model (CZM) to account for the continuous vdW interactions between the bulged 2D crystal and the substrate in Section 4. When the CZM hits a limit of minimum blister size, we leverage coarse-grained molecular dynamics (CGMD) simulations in Section 4 to illustrate a monolayer lattice of confined liquids trapped in ultra-shallow blisters with heights of only a few ångströms. We conclude with a summary of the main findings in Section 5.