Gel and Glass Transitions in Short-Range Attractive Colloidal Systems - Food Colloids: Self-Assembly and Material Science - page 330

We choose k B ¼ 1 and set the depth of the potential to u 0 ¼ 1. Hence T ¼ 1
corresponds to a thermal energy equal to the attractive well depth. The
diameter of the smaller species is chosen to be the unit of length, i.e., d 2 ¼ 1.
Density is expressed in terms of packing fraction f ¼ ( r 1 d 1 3 + r 2 d 2 3 )( p /6)
where r i ¼ N i /L 3 , L being the box size and N the number of particles of
component i. Time is measured in units of d 2 (m/u 0 ) 1/2 .
Newtonian dynamics (ND) has been coded via a standard event-driven
algorithm, commonly used for particles interacting with step-wise potentials. 26
Between collisions, the particles move along straight lines with constant veloc-
ities. When the distance between the particles becomes equal to the distance
where the potential has a discontinuity, the velocities of the interacting particles
instantaneously change. The algorithm calculates the shortest collision time in
the system and propagates the trajectory from one collision to the next one.
Calculations of the next collision time are optimized by dividing the system in
small sub-systems, so that collision times are computed only between particles
in the neighbouring sub-systems.
Brownian dynamics (BD) has been implemented via the position Langevin
equation,

! k ¼ D 0

!

k ð t Þþ ! k ð t Þ;

k B T f

ð 2 Þ

using the algorithm coding developed by Scala et al. 27 In Equation (2), r k (t)is
the position of particle k, f k (t) the total force acting on the particle k, D 0 the
short-time (bare) diffusion coefficient, and !

k ð t Þ a random thermal noise

satisfying

D E ¼ k B T d ð t Þ:

! k ð 0 Þ ! k ð t Þ ð 3 Þ

In this algorithm, a random velocity (extracted from a Gaussian distribution of
variance

k B T = p Þ is assigned to each particle and the system is propagated for
a finite time-step, 2mD 0 /k B T, according to event-driven dynamics. We chose D 0
such that short-time motion is diffusive over distances smaller than the well
width.
The phase diagram for the binary mixture is shown in Figure 2. A clear way
to indicate the position and shape of the glass line is represented by the
equi-diffusivity lines, 14 defined as the loci where D/D 0 is constant. Here D is
the diffusion coefficient calculated from the equilibrium mean square
displacement (MSD) using the Einstein relation normalized by the ND bare
diffusivity, D 0 ð k B Td 2 = m Þ 1 = 2 ¼ v th d 2 , to take into account the differences in
the microscopic time because of the different thermal velocity v th . The
equi-diffusivity lines already show signs of the re-entrant behaviour that, at
high density, characterizes the liquid pocket between the repulsive and the
attractive glass; moreover, the extrapolation to the expected glass line is also
plotted. 28
One of the most used models to describe interaction in the SRACS is the
Baxter potential 29 that describes the short-attractive interaction in the limit of