  We study the mechanics of pressurized graphene membranes using an
experimental configuration that allows the determination of the elasticity of
graphene and the adhesion energy between a substrate and a graphene (or other
two-dimensional solid) membrane. The test consists of a monolayer graphene
membrane adhered to a substrate by surface forces. The substrate is patterned
with etched microcavities of a prescribed volume and when they are covered with
the graphene monolayer it traps a fixed number (N) of gas molecules in the
microchamber. By lowering the ambient pressure, and thus changing the pressure
difference across the graphene membrane, the membrane can be made to bulge and
delaminate in a stable manner from the substrate. Here we describe the analysis
of the membrane/substrate as a thermodynamic system and explore the behavior of
the system over representative experimentally-accessible geometry and loading
parameters. We carry out companion experiments and compare them to the
theoretical predictions and then use the theory and experiments together to
determine the adhesion energy of graphene/SiO2 interfaces. We find an average
adhesion energy of 0.24 J/m2 which is lower, but in line with our previously
reported values. We assert that this test, which we call the constant N blister
test, is a valuable approach to determine the adhesion energy between
two-dimensional solid membranes and a substrate, which is an important, but not
well-understood aspect of behavior. The test also provides valuable information
that can serve as the basis for subsequent research to understand the
mechanisms contributing to the observed adhesion energy. Finally, we show how
in the limit of a large microcavity, the constant N test approaches the
behavior observed in a constant pressure blister test and we provide an
experimental observation that suggests this behavior.
