section 4.6
plane starting
with an fcc solid and a fluid at the monodisperse hard sphere freezing
point. Both systems consisted of 256 particles and were well
equilibrated. Using the ghost-growing method of section 4.4 in
combination with eqn.
4.18 we found an initial slope of
=1400 (
50) (in units such that 
and 
are unity).
This value was used in the first predictor
step to go from 
=0 to finite 
, where the slope could be
directly measured. We evaluated the coexistence line by gradually
increasing 
from one simulation to the next while integrating eqn.
4.10 to determine the pressure. In figure 4.1 the
equilibrium pressure is shown as a function of the polydispersity
parameter 
. The slope starts off at a value of 
=1400 as
we obtained from eqn.
4.18 and increases when 
is
increased. At a value of 
=0.0056 the slope becomes infinite, so
we had to invert the integration, taking P as the independent
variable. In this case, we can calculate 
for increasing
values of P by applying the same integration scheme to the
reciprocal of eqn.
4.10. Surprisingly, the equilibrium curve
continues to bend back, approaching 
=0 at infinite pressure. In
the inset of figure 4.1 we rescale the pressure to 
, which remains finite and follows a straight
line as 
. The fact that
is finite in this limit actually allows us to perform
simulations at infinite pressure as will be discussed in the next
section. The divergence of the pressure is somewhat misleading. The pressure indeed goes to infinity on a scale characterized by
, but
when reduced instead by the volume or an average diameter it remains
bounded. In fact, in this limit 
loses its relevance as a
length scale because all particle diameters are going to values much
smaller than 
. Scaling by an average diameter (or the
volume) makes the interpretation of the results more intuitive because
in experiments the important microscopic length scale is the
characteristic particle diameter. Consequently, in most of what
follows we present our results in terms of 
(where the angle brackets indicate an isobaric semigrand ensemble
average). We choose arbitrarily to use 
of the
solid to perform the reduction; we could just as well have used the
fluid value. We will continue to refer to the limit of infinite
pressure because this represents a limiting behavior of our isobaric
semigrand system. It should be understood that in this case the
pressure is infinite on the 
scale but not on, say, the
scale.Although the parameter
shows a maximum as a function of P, the
real polydispersity---given in terms of the width of the composition
distribution 
---does not. This polydispersity s is
defined as
is plotted against the polydispersity
s in both the fluid and the solid. The equilibrium pressure
increases monotonically until at infinite pressure a limiting value of
s and 
is reached. From this plot it
becomes immediately clear that the polydispersity of the fluid and the
solid at equilibrium can be very different. As reviewed in the
introduction of this chapter,
this fact, although anticipated, was discounted in previous studies.
, i.e., the terminal
polydispersity.
for fluid-solid equilibrium at different pressures. In the figures, the leftmost solid curve represents the fluid
composition, the rightmost one the solid phase composition.
a) 
for 
=15. The dotted curve is the imposed (Gaussian) activity distribution.
b) 
for 
=100. The dotted curve is the imposed (Gaussian) activity distribution.
c) 
for 
=6400. The dotted
line denotes the imposed chemical potential difference function, which is becoming straight at high 
.
d) 
for 
.In this limit the diameters are pushed to zero. By dividing them by 
they remain finite. The dotted
line denotes the imposed chemical potential difference function.
given in terms of
the real volume fraction, because this is the quantity one measures in
experiments. We define 
;
note that 
=0.7405 for monodisperse close packed spheres. The phase
diagram in the 
plane is shown in figure
term in eqn.
seen in figure 
. As the equilibrium
pressure (or equivalently the value of 
) is increased, the
composition distributions depart from the ideal activity. Although
still almost Gaussian, they are shifted considerably to lower values
of 
. The average diameter is smaller in the fluid, whereas the
solid composition is located at larger diameters and is more narrowly
distributed. The difference between the phases becomes more pronounced
at higher pressures. At infinite pressure, all diameters go to zero on
the scale of 
. Interesting distributions can be recovered by
proper scaling of the diameters by their average, as presented
in figure 
has decreased to zero in this limit
(and of course has the same value in the two phases), the (rescaled) composition
distributions are still near-Gaussian with a finite width (on a scale
of 
). This curious outcome is a result of the
limiting process in which 
while
. 
, as discussed above, becomes irrelevant. In
figure 
.