Young’s modulus and breaking strength

In plain words

How hard it is to stretch a material, and how far it can be stretched before it breaks. Young’s modulus is the stiffness: the force needed to stretch a material by a given fraction of its length. Breaking strength is the stress at which it fails. For a sheet one atom thick both are measured by pressing an atomic force microscope tip into a stretched over a hole like a drum skin. Graphene came out as the strongest material ever measured, with a Young’s modulus of about a terapascal and a breaking strength of about 130 gigapascals.

Going deeper

Young’s modulus is the slope of stress against strain; breaking strength is where the curve ends. For one-atom-thick sheets both are measured by pressing an AFM tip into a flake suspended over a hole. Graphene is the strongest material measured; MoS₂ is about as stiff as steel.

Stiffness and strength

Stretch a bar and it resists in proportion: the stress needed per unit of is its Young’s modulus, the measure of stiffness. Pull harder and eventually it breaks; the stress at that point is its breaking strength. Steel has a modulus of about 200 gigapascals, rubber a few megapascals.

For a sheet one atom thick, stress – force per area of cross-section – needs a thickness, and the thickness of a is not obvious. What is measured is a two-dimensional stiffness, in newtons per metre, and converting it to a Young’s modulus means assuming a thickness, for graphene usually the 0.335 nm spacing of graphite layers. Comparisons between sheets are cleanest in the two-dimensional units.

Drums pressed by a tip

The standard measurement suspends a flake over a small hole, like the skin of a drum, and presses the tip of an atomic force microscope into its centre while recording force against displacement, up to the point where the sheet breaks. For defect-free graphene this gave a two-dimensional stiffness of 340 N/m and a breaking strength of 42 N/m – a Young’s modulus of 1.0 TPa and an intrinsic strength of 130 GPa, the strongest material ever measured (Lee and colleagues, 2008).

Monolayer MoS₂ is softer but still strong: 180 ± 60 N/m, a Young’s modulus of about 270 GPa, comparable to steel. It breaks at 6–11% strain, and its best reach a strength of about a tenth of their modulus – the theoretical upper limit, a sign of nearly defect-free crystals (Bertolazzi and colleagues, 2011).

Layers that slide

Stacks are not simply stronger by the number of layers. Under an indenter, layers can slide over one another: few-layer graphene loses more than 30% of its strength between one and eight layers, while boron nitride sheets keep theirs, because their layers grip each other better – one reason boron nitride nanosheets are attractive as mechanical reinforcement (Falin and colleagues, 2017).

Defects, and lower both stiffness and strength, so the record values belong to small, perfect membranes rather than to the flakes in a .

For specialists

The linear elastic modulus E, stress per unit strain, and the stress at fracture. For a monolayer the measured quantity is a two-dimensional stiffness in N/m, converted to E with an assumed thickness: nanoindentation of suspended graphene gives 340 N m⁻¹ and an intrinsic strength of 42 N m⁻¹, E ≈ 1.0 TPa and σ ≈ 130 GPa with the of graphite; monolayer MoS₂ gives 180 ± 60 N m⁻¹ (E ≈ 270 GPa, comparable to steel) and breaks at 6–11% strain, its best membranes reaching about a tenth of E, the theoretical limit. In few layers, sliding matters: graphene’s strength falls by more than 30% from one layer to eight, while boron nitride’s barely changes.

Where this comes from

Measurement of the elastic properties and intrinsic strength of monolayer graphene Lee et al. · Science 321, 385 (2008) cited by 20,851
Stretching and breaking of ultrathin MoS₂ Bertolazzi et al. · ACS Nano 5, 9703 (2011) cited by 2,682
Mechanical properties of atomically thin boron nitride and the role of interlayer interactions Falin et al. · Nature Communications 8, 15815 (2017)

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The newest items in the site’s news feed that use the term, one from each source.

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2D membrane

A sheet that lets some molecules through and holds others back. A perfect layer of graphene is so tight that not even helium gets through, so useful membranes need openings: tiny holes made in a single layer, or the narrow gaps between flakes stacked like a pack of cards. Because such a membrane is only atoms thick, water or gas can pass far faster than through today’s plastic membranes, which could make filtering, purifying water or separating gases cheaper.

Separation membranes made from 2D materials, in two designs: nanoporous single layers (graphene, hBN, MoS₂ with pores opened by ion irradiation, oxidative etching or defects), whose pore size sets the selectivity, and laminates of stacked flakes (graphene oxide, MXenes, COF and MOF nanosheets), whose interlayer channels – often swelling in water – act as sieves. Pristine graphene is impermeable to gases including helium, yet transmits protons; atomic thickness promises high permeance, and restraining the laminate spacing can exclude hydrated ions. Against mature polyamide reverse-osmosis membranes the obstacles are large defect-free areas, swelling and stability in water, fouling and cost. The word also names any freestanding thin film, as in oxide membranes lifted off a sacrificial layer.

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Bubbles and wrinkles

Flaws that appear when a thin layer is laid on a surface or on another layer. Water and dirt trapped underneath gather into small bubbles that push the layer up, and a layer with more length than the surface beneath it folds into wrinkles. Both stretch or bend the layer where they sit and change how it conducts and glows there, so careful stacking keeps them away from the part that is measured.

Blisters of trapped adsorbates – mainly hydrocarbons and water – between transferred or stacked layers, from nanometres to micrometres across, and ridges where compressive strain from transfer or thermal-expansion mismatch is relieved out of plane. Van der Waals adhesion squeezes contamination into bubbles, leaving atomically clean interfaces between them, and the bubbles take a universal shape set by adhesion and in-plane stiffness. Both strain the layer locally, shifting band edges, Raman modes and emission, and both scatter carriers.

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Composites and coatings

The biggest market for 2D materials so far: flakes of graphene or other layered materials mixed in small amounts into plastics, rubber, concrete, paints or battery electrodes, to make them stiffer, tougher, more conductive or less permeable. Sold by the tonne, these products use cheap flakes made in bulk rather than perfect single layers. Whether a few percent of flakes really helps depends on how well they are spread out, how they line up and how firmly they hold on to the material around them.

Bulk uses in which exfoliated or chemically derived flakes – graphene nanoplatelets, reduced graphene oxide, hBN, MXenes, clays – are dispersed in a matrix or applied as a film. In polymer composites a few weight percent can raise stiffness and toughness, cross the percolation threshold for electrical conduction – as low as about 0.1 percent by volume for thin, wide flakes – and raise thermal conductivity; aligned flakes in barrier coatings lengthen the diffusion path of water, oxygen and ions and slow corrosion; flakes used as conductive additives replace part of the carbon black in battery electrodes; MXene films give strong electromagnetic shielding. Performance depends on flake size and thickness, dispersion without restacking, alignment and bonding to the matrix, and on powders whose properties vary widely between suppliers.

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Atomic force microscopy (AFM)

A tip on a tiny springy arm is moved across a surface and its bending is measured, mapping height point by point – much as a fingertip reads Braille. It works on anything, conducting or not, which is why it is the routine way to check how thick a flake is. With a conducting tip it can map electric current instead (conductive AFM).

Force microscopy in contact, tapping or non-contact mode, used for thickness, roughness, bubbles and polymer residue, with variants for surface potential (KPFM) and local conductivity (c-AFM). Apparent height is not the crystallographic thickness: the tip interacts differently with flake and substrate, so a monolayer commonly reads between 0.4 and 1 nm. Quote the mode, and calibrate against a step of known height.

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Photocatalysis and electrocatalysis

Using light, or an applied voltage, to drive a chemical reaction at a surface – most often splitting water to make hydrogen. Thin materials are natural candidates because nearly every atom is a surface atom, and because their edges are usually far more active than their flat faces.

Surface reactions driven by photons (photocatalysis) or by electrode potential (electrocatalysis), judged by overpotential, Tafel slope and turnover per site rather than by raw current. In MoS₂ the basal plane is close to inert while the edge sites carry the hydrogen-evolution activity – the result that made layered chalcogenides catalyst candidates – so activity scales with edge length, defect density and phase, and any comparison needs an active-site count rather than a geometric area.

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Chemical bond

The glue that holds atoms together. In a covalent bond two atoms share electrons, as within a sheet of graphene. In an ionic bond the atom that pulls harder on electrons – the one with the higher electronegativity – takes them from the other, and the resulting charges attract, as in table salt. In a metal the outer electrons are shared by all the atoms at once. Much weaker van der Waals forces hold molecules and the layers of a layered crystal together. A 2D material is a crystal with strong bonds within its layers and only weak ones between them – which is why it can be peeled.

The attractive interactions that hold atoms in molecules and solids, classified by how electrons are shared: covalent bonds, from overlapping orbitals holding shared electron pairs, strong and directional (sp² carbon, Mo–S in TMDCs); ionic bonds, from electron transfer and electrostatic attraction (halides, oxides); metallic bonding, with electrons delocalised in a partly filled band; and the weak, non-directional van der Waals interaction from correlated charge fluctuations, besides hydrogen bonds. Real bonds are mixtures, characterised by electronegativity differences and charge analyses. Layered crystals combine in-plane bonds of several electronvolts with interlayer binding of only about 20 meV per Å² (some 0.3 J/m²), the criterion high-throughput screens use to identify exfoliable compounds.

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Battery electrode

A battery stores energy by moving ions – in phones and cars, lithium ions – between two electrodes, the anode and the cathode. Charging pushes the ions into the anode, where they slip between the layers of a material such as graphite; using the battery lets them flow back. Layered and 2D materials matter because ions can enter and leave the gaps between layers, and because thin conductive flakes can help electrons reach every grain of an electrode.

A lithium- or sodium-ion cell shuttles ions through an electrolyte between a negative electrode (the anode on discharge) and a positive one (the cathode), while electrons pass through the external circuit. Graphite, the standard anode, intercalates lithium between its layers up to LiC₆, about 372 mAh/g; layered oxides such as LiCoO₂ and NMC dominate cathodes. 2D materials enter as conductive additives (graphene flakes replacing part of the carbon black), as higher-capacity anodes that alloy or convert (phosphorus, antimony, SnS₂, SnO), as hosts for larger ions such as sodium, which graphite takes up poorly, and as MXene electrodes. Their weaknesses are large volume change, irreversible first-cycle loss from the high surface area, low packing density and cost, so gains at cell level are usually far smaller than those claimed for the material.

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