Condensed and soft matter systems often contain thin structures such as
(i) domain boundaries, vortex lines,
polymers, and filaments, or (ii) interfaces, monolayers, and membranes etc which behave as
1-dimensional lines or 2-dimensional surfaces. These
low-dimensional manifolds undergo thermally excited shape fluctuations
in order to increase their configurational entropy. These fluctuations are scale-invariant
and can be characterized by a corresponding critical exponent.
One particularly interesting case are domain boundaries and interfaces in
Because of their molecular architecture, 1-dimensional lines and 2-dimensional surfaces
interact via electrostatic and van der Waals forces or other constraints.
The shape fluctuations renormalize
these latter interactions and lead to
between bound and unbound states of the manifolds.
These latter transitions represent wetting
and adsorption transitions
for interfaces, membranes, and polymers, respectively.
In all cases, one encounters a strong fluctuation regime
as well as an
intermediate fluctuation regime
with more complex behavior
In the latter regime, the unbinding transitions are discontinuous and
characterized by a
kink in the internal energy but the shape fluctuations exhibit unusual
properties and are
governed by a power law distribution
The scaling properties of different systems are found to be related in unexpected ways.
The continuous unbinding of two membranes in three dimensions
, for example,
exhibits the same scaling properties as the unbinding of two domain boundaries or "strings"
in two dimensions
For 1-dimensional lines, there is even an intimate relationship between the
scaling properties of bound and unbound states
Both bundles of N strings in two dimensions
bunches of N membranes in three dimensions
undergo continuous unbinding transitions
whereas bundles of N filaments unbind discontinuously in
In the continuous case, extensive numerical studies
and analytical solutions based on a Bethe Ansatz
have led to somewhat different conclusions about the N-dependence
of the critical exponents which remains an open issue
Recent studies have focused on the critical behavior of filaments and semiflexible polymers:
- For more information, see
on lines and surfaces.