Chemistry Reference
In-Depth Information
The above set of equations is sufficient to describe the adsorption behaviour
of mixed protein + surfactant solutions. Therefore, the theoretical description
of such a mixture can be formulated in the following way. Once the values of T,
o
0
,
o
min
,
o
max
, a
S
, a
P
, a
PS
, a
SPS
, m, b
P
, b
PS
, c
P
, and b
S
are known, the
dependencies of
o
, Y
P
, Y
S
, Y
PS,
and P as a function of the surfactant
concentration c
S
can be calculated. Assuming that the approximations a
PS
¼
a
P
, a
SPS
¼
0 (or a
SPS
¼
(a
S
+ a
P
)/2)
29
and b
PS
¼
b
P
are all valid, it is possible to
calculate the adsorption behaviour of mixtures using only the characteristics of
the individual components.
Among all the parameters necessary for the calculations, there are five
parameters corresponding to the individual protein solution. The values of
o
min
and
o
max
can be determined from the geometrical dimensions of the
protein molecule in the completely folded and unfolded states, respectively. The
value of
o
0
is almost the same for all proteins, being in the range 2
3
10
5
m
2
mol
1
. All the rest of the parameters can be estimated from a fitting procedure,
using some of the experimental dependencies of P versus c
P
, P versus G, and P
versus c
P
, or all simultaneously.
25
The remaining three parameters related to the
individual surfactant solution (
o
S
, a
S
, and b
S
) can be determined by fitting the
experimental dependency P versus c
S
. The parameter m is either known from
independent experimental data, or it can be estimated by fitting the model to
the surface tension data of the mixture. We note that this would be the only
additional parameter obtained by the fitting, with all others coming from the
pure compounds.
The assumption a
SPS
¼
0 is more realistic than a
SPS
¼
(a
S
+ a
P
)/2 when the
surface layer is strongly inhomogeneous, i.e., when protein and surfactant
molecules do not mix in the surface layer but form domains containing
essentially one of the components.
19-20,22-23
The assumption b
PS
¼
b
P
is based
on the fact that m is small (10-100 times lower than the number of amino-acid
groups in the protein molecule), and therefore the adsorption activity of the
protein
surfactant complex only changes slightly.
14.2.2 Adsorption Kinetics
The analysis of adsorption kinetics and dynamic surface tension under quasi-
equilibrium conditions is based on the equation of state for the surface layer
and the adsorption isotherm. The most general relation between dynamic
adsorption G(t) and sub-surface concentration c(0,t) is described by the Ward
and Tordai equation.
30
For a freshly formed non-deformed surface this equa-
tion has the following form:
2
3
p
Z
p
r
D
p
p
t
0
t
4
5
;
c 0
;
t
t
0
G
ðÞ¼
2
c
0
ð
Þ
d
ð
6
Þ
0
where c
0
is the protein bulk concentration, D the diffusion coefficient, t time,
and t
0
the integration variable. The dependence of G(t) on the lifetime of the
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