  Liquid phase exfoliation is a commonly used method to produce 2D nanosheets
from a range of layered crystals. However, such nanosheets display broad size
and thickness distributions and correlations between area and thickness, issues
that limit nanosheet application potential. To understand the factors
controlling the exfoliation process, we have liquid-exfoliated 11 different
layered materials, size-selecting each into fractions before using AFM to
measure the nanosheet length, width, and thickness distributions for each
fraction. The resultant data show a clear power-law scaling of nanosheet area
with thickness for each material. We have developed a simple nonequilibrium
thermodynamics-based model predicting that the power-law prefactor is
proportional to both the ratios of in-plane-tearing/out-of-plane-peeling
energies and in-plane/out-of-plane moduli. By comparing the experimental data
with the modulus ratio calculated from first-principles, we find close
agreement between experiment and theory. This supports our hypothesis that
energy equipartition holds between nanosheet tearing and peeling during
sonication-assisted exfoliation.
