section 4.8
We have established a solid-fluid coexistence region for a system of
polydisperse hard spheres with near Gaussian diameter distributions,
as a function of polydispersity. Our approach employs simulation in
the isobaric semi-grand ensemble with a Gaussian activity
distribution. Gibbs-Duhem integration is used to trace the coexistence
pressure as a function of the variance of the imposed activity
distribution. The Gibbs-Duhem integration is initiated with a
monodisperse hard sphere fluid and fcc solid, and throughout the
integration process the solid remains in an fcc structure. We do not
explore the possibility of a polymorphic transition in the solid.
Both the fluid-solid coexistence densities and volume fractions are
monotonically increasing functions of the polydispersity s, which is
given in terms of the standard deviation in the particle diameter distribution
function. The volume change at the freezing transition decreases as a
function of s and eventually takes on negative values, which implies
that the number density of the fluid phase is greater than that of the
solid. However, the packing fraction of the fluid remains always less
than that of the coexisting solid phase. Connected to this is the
observation of significant fractionation between the two phases, which
permits the fluid phase to comprise particles of a smaller average
diameter.
Significantly, we observe a terminal polydispersity, i.e., a
polydispersity above which there can be no fluid-solid
coexistence. This terminus arises quite naturally as the Gibbs-Duhem
integration path leads the pressure to infinity. The existence of
this terminus only at infinite pressure precludes the construction of
a continuous path from the solid to the fluid. While it was
anticipated in previous studies that such a continuous path could not
be constructed, the issue was not addressed as fully as we are able to
here.
At the terminus the polydispersity is 5.7% for the solid and 11.8%
for the fluid while the volume fractions are 0.588 and 0.547 for the
solid and fluid respectively. Large fractionation observed at moderate values
of s (>0.05)
implies that the constrained eutectic assumption implicit in previous studies
is not valid over a very large range of polydispersity. The constrained eutectic
approximation is perhaps the reason that McRae and Haymet [83] obtained
the smaller value of 6% for the terminus. Our results for
the terminal polydispersity are consistent with experiments performed on
polydisperse colloidal suspensions.
We feel that the qualitative conclusion that a terminal polydispersity
exists is generally correct and that it is of the order of 5% in
the solid and 12% in the fluid. However, we have not examined the
sensitivity of the terminal polydispersity to variation in the
chemical potential distribution function (and thus the composition).
It seems likely that the terminal polydispersity would not be very
sensitive to details of composition, and as our distributions are
near-Gaussian we expect our conclusions to be generally valid.
section 4.8