Biomedical Engineering Reference
In-Depth Information
The interaction between two model colloidal particles can be represented by the
generalized Lennard
-
Jones (LJ) m-n potential:
h
r
m
r
n
i
ð
Þ¼
ð
;
Þ ε
;
ð
2
:
33
Þ
U
r
A
m
n
where r is the inter-particle centre-to-centre separation,
σ
is the colloidal particle diameter
and A(m, n) is a normalization factor which ensures that
is the potential well depth.
Varying the exponents m and n generates various interaction ranges. The minimum of
the potential occurs at a separation r =2
1/n
ε
, displaced to smaller separations as n or m
increases. Lodge and Heyes (
1997
,
1998
,
1999a
) performed simulations using 12:6,
24:12 and 36:18 potentials (m=2n) and A = 4. Along this series, the attractive part of the
potential becomes shorter-range. The thermodynamic properties are given in units of
σ
σ
for length and
ε
for energy. The reduced temperature T
*
is given by
k
B
T
ε
T
¼
:
ð
2
:
34
Þ
The units of time are a²/D
0
, where a =
σ
/2 is the particle radius, D
0
is the self-diffusion
coef
cient of the colloidal particle in very dilute suspensions and
ζ
0
is the in
nite dilution
friction coef
cient:
k
B
T
ζ
0
D
0
¼
ð
2
:
35
Þ
ζ
0
*
= 1 and then D
0
*
= T
*
.
A typical phase diagram for a colloidal system of attractive particles is given in
Figure 2.19
, plotted as reduced temperature versus density
In reduced units,
ρ
-
for the vapour
liquid
two-phase system (or versus particle volume fraction in the case of a liquid
liquid phase
separation in a suspension). The system can be quenched into various domains within the
two-phase region (phase separated region) with a binodal boundary line. Inside the binodal
line there is a spinodal line marking the transition from metastable to unstable regions.
Inside the spinodal boundary, it is possible to form a rigid but mechanically unstable
network which has a
-
is drawn on
Figure 2.19
,a
line which marks the boundary of states, to the right of which they display a gel-like
structure. These systems are referred to as
finite yield stress (see below): a
'
gel line
'
because the colloidal particles
do not have permanent bonds and therefore the structure tends to collapse.
Experimentally, phase separation can be induced in a number of ways, for example by
adding non-absorbing polymers (depletion
'
transient gels
'
flocculation), changing the pH or salt con-
centration for aqueous systems or changing the temperature. In real experiments, adding
polymers or salts is widely used since the quench is essentially instantaneous, so there is
minimal change within the system during the quench. In computer simulations, changing
the temperature is often the most convenient way of initiating phase separation. Starting
from an equilibrium con
guration, above the critical point, the system is quenched in a
single time step. The change of temperature enters the model through the Brownian force
term. Experimental and simulation procedures are equivalent, as T
*
, proportional to the
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