Researchers show how blister shapes in atomically thin materials like graphene can be used to map pressure, membrane tension, and surface adhesion.
(Nanowerk Spotlight) Anyone who has ever struggled with smoothing out a screen protector or applying a decal without trapping air bubbles has encountered a simple version of a much deeper physical problem. When a thin, flexible sheet is placed onto a solid surface, even a slight mismatch or trapped fluid can lead to a blister—an enclosed pocket that distorts the film and breaks its contact with the substrate.
In most everyday cases, this is a cosmetic nuisance. But when the film is only one atom thick, as in the case of graphene and other two-dimensional (2D) materials, these blisters are far more than visual imperfections. They are structural features that affect how the material behaves and how it can be used.
In nanoscale systems, blisters are no longer just defects. They have enabled applications as diverse as high-pressure chemistry, where reactions occur under confined conditions that are otherwise difficult to generate, and advanced electron microscopy, where the liquid trapped inside blisters slows down the motion of biomolecules for imaging. But to control or exploit these features, researchers need to understand what’s happening inside them—how much pressure they contain, how much tension stretches across their surface, and how strongly they adhere to the underlying support.
For blisters with circular shapes, previous models have offered some answers. Yet many blisters, especially those formed in single-layer 2D crystals, take on irregular, asymmetric shapes. These cannot be adequately described by circular models, and until now, there has been no reliable way to extract mechanical information from them.
In a new study (Advanced Functional Materials, “Fracture Mechanics of 2D Crystal Blisters with Irregular Geometry”), researchers from Peking University present a method that addresses this gap. By combining atomic force microscopy with a theoretical model rooted in fracture mechanics, they show that the shape of a blister—even when irregular—contains enough information to reconstruct key physical properties of the system. These include not just the strain and stress in the 2D material, but also the pressure inside the blister and the adhesion energy at its interface.
This framework allows researchers to turn what was once seen as an uncontrollable byproduct of fabrication into a precise diagnostic tool for measuring nanoscale mechanical interactions.
Elastic blisters across multiple scales. A) Photo of a sticker “carelessly” placed on window glass. B) Photo of a thin polymeric sheet partially conforming to a sphere with capillary adhesion. C) Optical image of interfacial blisters formed after the deposition of a 60-nm-thick gold film on a thin PMMA layer. D) AFM amplitude images of blisters formed by transferring graphene sheets onto hBN substrates with water molecules trapped at the interface. E) Schematic illustration of the classical droplet problem, featuring Young’s contact angle. F) Schematic illustration of the elastic blister problem, where the liquid is covered by an elastic sheet of Young’s modulus E and thickness t, resulting in an elastic version of contact angle. (Image: Reprinted with permission by Wiley-VCH Verlag)
The authors focused on blisters that form when graphene sheets are transferred onto substrates like hexagonal boron nitride or graphite using a wetting process. This method frequently traps small amounts of water between the sheet and the surface, producing spontaneous blisters with varied and complex shapes.
Atomic force microscopy was used to record the topography of these blisters with nanometer-scale resolution. From these images, the researchers computed the curvature of the surface, which they used as input for a theoretical model based on Föppl–von Kármán plate theory—a framework that captures how thin elastic films bend and stretch.
To solve the equations that connect curvature with internal stress, the researchers used a spectral method involving Chebyshev polynomials. This technique transforms the original differential equations into a set of algebraic equations that can be efficiently solved while preserving accuracy. The model yields a function that describes the internal stress distribution across the blister, which then allows calculation of both strain and internal pressure using well-established physical laws.
They first validated their approach using circular blisters, for which analytical solutions are known. The model’s predictions agreed closely with these known results, confirming its accuracy. The researchers then applied the same method to irregular blisters, such as triangular ones with straight edges and sharp corners. These shapes had previously resisted precise analysis, partly because the residual stresses left in the film from fabrication were difficult to quantify.
To resolve this, the authors looked to features that appear consistently in triangular blisters: small buckled regions at the corners. These buckles result from compressive stresses and offer clues about the underlying stress field. By analyzing the shape of these buckles—specifically, their amplitude and wavelength—the team estimated the compressive forces acting on the material. This step made it possible to incorporate realistic stress conditions into their model, improving its ability to capture the true mechanical behavior of the blister.
With the stress and strain fields calculated, the authors turned to another key parameter: adhesion. In fracture mechanics, the energy release rate describes how much energy is available to propagate a crack or delaminate a film. This value balances against the adhesion energy that resists separation. By measuring the tension along the edge of the blister and the angle at which the sheet lifts off from the substrate, the researchers used a formula adapted from classical peeling theory to calculate this energy balance.
For this calculation to be valid, they had to ensure that the contact angle was measured at a location where the assumptions of the theory hold—specifically, where the sheet is still governed by elastic mechanics rather than dominated by local atomic-scale interactions. They derived a cutoff condition based on the geometry and mechanical properties of the material to identify valid regions for measurement.
The team applied this method to triangular blisters formed by monolayer and few-layer graphene on different substrates. They found that the membrane tension and the local contact angle are related in a predictable way: the tension is approximately inversely proportional to the square of the angle. Using this relationship, they computed the adhesion energy for each blister.
Their measurements showed that graphene adheres more strongly to hexagonal boron nitride than to graphite—a result that agrees with known trends in materials assembly. They also observed that adhesion decreases as the number of graphene layers increases. Thicker films make less intimate contact with the substrate, which reduces the effective bonding strength.
Importantly, this method also highlighted a key limitation of prior approaches. When they applied the same framework to circular blisters under the assumption of zero residual stress—a common simplification in earlier models—they found systematically lower adhesion energies than those derived from triangular blisters. This discrepancy suggests that ignoring residual stress can lead to significant underestimation of adhesion, especially in cases where the stress state is not well characterized. Triangular blisters, by contrast, tend to release stress through localized buckling, making them more reliable for this type of analysis.
The significance of this work lies not only in its technical precision but also in its broad applicability. The model does not depend on the specific fluid inside the blister, only that it exerts pressure. This means it can be used for both liquid- and gas-filled systems. The technique is also not limited to graphene and can be extended to other 2D materials and flexible thin films. Because it relies only on surface shape measurements, it offers a non-invasive way to study mechanical interactions at the nanoscale.
The ability to extract strain, stress, pressure, and adhesion from blister geometry opens up new possibilities for material characterization and device engineering. It provides a practical path to measure quantities that have traditionally been hard to access in thin films, especially when working with atomically thin materials.
In the future, the authors suggest this approach could be combined with automated image analysis and machine learning to enable large-scale mapping of mechanical properties across materials, potentially accelerating the design of devices that rely on the unique behavior of 2D crystals.
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