26 research outputs found
A lattice model for the line tension of a sessile drop
Within a semi--infinite thre--dimensional lattice gas model describing the
coexistence of two phases on a substrate, we study, by cluster expansion
techniques, the free energy (line tension) associated with the contact line
between the two phases and the substrate. We show that this line tension, is
given at low temperature by a convergent series whose leading term is negative,
and equals 0 at zero temperature
A compact null set containing a differentiability point of every Lipschitz function
We prove that in a Euclidean space of dimension at least two, there exists a
compact set of Lebesgue measure zero such that any real-valued Lipschitz
function defined on the space is differentiable at some point in the set. Such
a set is constructed explicitly.Comment: 28 pages; minor modifications throughout; Lemma 4.2 is proved for
general Banach space rather than for Hilbert spac
SURFACE INDUCED FINITE-SIZE EFFECTS FOR FIRST ORDER PHASE TRANSITIONS
We consider classical lattice models describing first-order phase
transitions, and study the finite-size scaling of the magnetization and
susceptibility. In order to model the effects of an actual surface in systems
like small magnetic clusters, we consider models with free boundary conditions.
For a field driven transition with two coexisting phases at the infinite volume
transition point , we prove that the low temperature finite volume
magnetization m_{\free}(L,h) per site in a cubic volume of size behaves
like
m_\free(L,h)=\frac{m_++m_-}2 + \frac{m_+-m_-}2
\tanh \bigl(\frac{m_+-m_-}2\,L^d\, (h-h_\chi(L))\bigr)+O(1/L),
where is the position of the maximum of the (finite volume)
susceptibility and are the infinite volume magnetizations at
and , respectively. We show that is shifted by an amount
proportional to with respect to the infinite volume transitions point
provided the surface free energies of the two phases at the transition
point are different. This should be compared with the shift for periodic boun\-
dary conditons, which for an asymmetric transition with two coexisting phases
is proportional only to . One also consider the position of
the maximum of the so called Binder cummulant U_\free(L,h). While it is again
shifted by an amount proportional to with respect to the infinite volume
transition point , its shift with respect to is of the much
smaller order . We give explicit formulas for the proportionality
factors, and show that, in the leading term, the relative shift is
the same as that for periodic boundary conditions.Comment: 65 pages, amstex, 1 PostScript figur