Gel and Glass Transitions in Short-Range Attractive Colloidal Systems - Food Colloids: Self-Assembly and Material Science - page 331

Figure 2 The numerical phase diagram, temperature T versus particle packing fraction f ,
indicating the quench analyzed. The crosses are the initial configurations, and the
squares the final ones. The squares are shaded if the final structure is a percolating
gel. The dashed line and the star are the liquid liquid coexistence line and the
critical point from Miller and Frenkel. 30,42 The continuous lines with symbols are
the equi-diffusivity lines, D/D 0 ¼ 0.01, 0.005 and 0.001, going from left to right.
The bold line (with no symbol) is the extrapolated glass line. 28 The inset shows an
enlargement of the low packing fraction region on a logarithmic scale

vanishingly small range. We shall use this model to estimate the location of the
coexistence curve. This potential V B (r) is defined according to
exp ð b V B ð r ÞÞ ¼ y ð r d Þþ d

12 t d ð r d Þ;

ð 4 Þ

where t is an adhesive parameter that plays the role of an effective temperature, y
and d are the Heaviside and the Dirac functions, respectively, and b ¼ 1/k B T.
The main feature of this model is that it allows us to take the limit of a vanishing
interaction range by keeping the second virial coefficient finite, i.e., keeping fixed
the overall strength of the interaction. The success of this model is due to the fact
that it can be solved analytically within the Percus Yevick approximation, and
that it can be successfully used to interpret experimental data. The adhesive
parameter t is directly related to the virial coefficient by the relation:

B 2 ¼ B 2 ¼ 1 1

4 t :

ð 5 Þ

B HS
2

In this way is it possible to relate the adhesive parameter to the temperature of a
finite square-well model by equating the virial coefficients of the two models.
Doing this we obtain:

1 1

4 t ¼ 1 ð e b u 0 1 Þ½ð 1 e Þ 3 1 :

ð 6 Þ