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where C
t
(g L
1
) represents the concentration of protein at a given reaction
time t, the parameter n the reaction order and k
n
((g L
1
)
1
n
s
1
) the reaction
rate constant. Using the integrated form
1
n
C
t
C
0
¼
1
þ
n
1
ð
Þ
kt
ð
2
Þ
the reaction rate k can be derived for a given reaction order n. The quantity
C
t
/C
0
is the protein concentration at a reaction time t divided by the initial
protein concentration.
23,24
12.2.4 Determination of Stability Ratio
The aggregation mechanism of Brownian particles can be depicted as belonging
to two limiting regimes.
25
Diffusion-limited cluster aggregation (DLCA) is
defined by the power-law dependence
D
h
ð
t
Þ¼
D
h
ð
0
Þð
1
þ
a
c
s
t
Þ
z
=
d
f
;
c
s
¼
4k
B
T
3
Z
N
0
ð
3
Þ
In Equation (3), D
h
(0) is described as the cluster or particle diameter extrap-
olated to time zero,
a
is the sticking probability, and d
f
the fractal dimension of
the particles. The dynamic exponent z denotes the reactivity between the
particles and is close to unity for a pure DLCA mechanism. The quantity c
s
is the Smoluchowski rate constant with k
B
T as the thermal energy,
Z
the
viscosity of the solvent and N
0
the number of monomeric particles at t
¼
0.
Several authors have suggested that the aggregation kinetics of colloids can
be described using a so-called stability ratio W.
26
It represents the ratio of the
rate constant for rapid coagulation kinetics, where every collision is effective
(DLCA), and slow aggregation kinetics, where not every collision is effective
(RLCA). The parameter W is linked to the sticking probability
a
by
W
¼
1
a
:
ð
4
Þ
Thus, the DLCA aggregation regime is reached for W values close to unity, i.e.,
where the sticking probability is highest. As the calculation of the stability ratio
W relates to the disappearance of monomers at early reaction times, we choose
to calculate it from the available reaction rates determined by RP-HPLC.
12.2.5 Determination of Protein Aggregate Molecular Weight and
Second Virial Coefficient
The weight-averaged molecular weight M
w
and the second virial coefficient A
2
of the aggregates formed during heat treatment were determined using the
Nanosizer ZS (Malvern Instruments, UK). The measurements were done using
static light scattering (SLS), which measures the time-averaged intensity of the
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