Gel and Glass Transitions in Short-Range Attractive Colloidal Systems - Food Colloids: Self-Assembly and Material Science - page 332

While the coexistence curve can be calculated analytically with Percus-Yevick
theory, the outcome is not reliable. Indeed, the predicted behaviour is thermo-
dynamically inconsistent since the two equations of state obtained by the
energy and compressibility route give different results. A more sensible route is
to use the results of numerical simulations of Miller and Frenkel, 30,31 who
performed accurate Monte Carlo simulations to evaluate the correct Baxter
coexistence curve. These data provide an accurate estimate of the coexistence
curve of all short-range interacting potentials, via an appropriate law of
corresponding states. Indeed, Noro and Frenkel 25 noticed that, when the
attraction range is much smaller than a few percent of the particle diameter,
the value of the first virial coefficient at the critical temperature is essentially
independent of the details of the potential, down to the Baxter limit. Following
this prescription we report in Figure 2 the coexistence curve calculated by
Miller and Frenkel 30,31 in the f - t plane after transforming it to the T plane
according to Equation (5). To give independent support to the mapping, we
calculated explicitly the location of the spinodal line, by bracketing it with the
lowest T value at which no phase separation takes place and the highest T value
at which we observe a growing spinodal decomposition peak. The calculated
spinodal line, also reported in Figure 2, is in very good agreement with Miller
and Frenkel estimates.
Figure 2 also illustrates the protocol of our numerical experiment. 23 We
prepared several independent configurations equilibrated at T ¼ 1.0 at packing
fractions ranging from 0.01 to 0.50. These configurations were then quenched
at temperature T f ¼ 0.05, deep down in the liquid-liquid phase separating
region. The temperature was kept fixed by coupling the system to a thermostat,
and the evolution was then followed as a function of time. This same procedure
has been followed for both Newtonian and Brownian dynamics.
The first observation is that, if the packing fraction is lower than a certain
threshold (the open symbol in Figure 2), the system does not form any
percolating structure; whereas above this threshold value (the filled symbols),
the system gets arrested in an out-of-equilibrium state. This is an indication
that the percolation threshold has been passed. Indeed, it is possible to evaluate
the percolation line for this model: it passes close to the critical point, intersects
the bimodal, and moves to lower density. 30 Indeed, the transition from a
percolating to a non-percolating structure is seen better following the time
evolution of the potential energy per particle, U/N.
An important property of the square-well system is the fact that the bonds
between the particles are defined with no ambiguity, i.e., two particles are
bonded if they are within each other's attractive well. In this way the number of
bonds per particle is directly proportional to the energy, i.e., 2U/(Nu 0 ). In
Figure 3, we present the evolution of the energy per particle versus time at the
various packing fractions for both ND and BD. This evolution presents three
characteristic regimes. At the beginning the energy remains constant for a time
that gets longer the lower the packing fraction. This is a reflection of the typical
free path for the particles, which increases on lowering the density and regulates
the formation of large aggregates. After this initial stage the aggregation