Solid-State Nanopores: Methods of Fabrication and Integration, and Feasibility Issues in DNA Sequencing - Nanopores: Sensing and Fundamental Biological Interactions - page 180

Fig. 8.2 Amodel for pore shrinking and expansion based on Laplace's surface tension mechanism:
(a) A simplified sketch of a pore inside a SiO 2 membrane. (b) Free energy of the pore [ 6 ]. Used with
permission. Copyright Nature 2004

behaves like a fluid. As such, if a hole was formed inside the SiO 2 membrane prior
to fluidization, it will shrink or expand responding to the unbalanced Laplace
pressures due to the finite radius of the hole and the finite thickness of the
membrane. A small hole below the critical radius will shrink while a large one
will expand without bound. When the electron beam is switched off, the material
cools off (by black-body radiation) and freezes, and the pore should retain its shape.
In the model shown in Fig. 8.2 , the free energy change of forming a pore inside
a membrane is,

r 2

d F= g (2 p rh --- 2 p Þ; (8.1)

where is surface tension, r is the radius of the pore, h the thickness of the

membrane.
If the starting pore radius is less than h /2, the pore will shrink spontaneously
under the surface tension; on the other hand, if the starting pore diameter is greater
than h /2, the pore expands without bound, causing the membrane to rupture.
The “critical” diameter is the thickness of the sheet. This scaling argument is
valid at any scale. The key prediction that a pore with a starting diameter larger
than the thickness of the membrane will expand upon heating was verified in the
subsequent experiments of Storm et al. [ 6 ].
We must point out that the exact mechanism of the pore shrinking induced by
an e-beam is likely to be more complicated than the simple picture above.
For example, the inner rim of the pore during shrinking is not circular, as would
be expected from a simple surface-tension mechanism, see Fig. 8.1 (top). It is quite
possible that the SiO 2 material is no longer uniform during the exposure to the
e-beam. Later studies indeed revealed significant loss of oxygen especially after
prolonged exposure to the high energy e-beam (see Kim et al. [ 11 ]).
The surface-tension model in Fig. 8.2 predicts that the physics of a critical
diameter for pore shrinking or expanding is scale independent. Indeed this was
confirmed by Wu et al. [ 8 ] in a laser-heating experiment on plastic pores (holes).

g