Biomedical Engineering Reference
In-Depth Information
Polar head
Hydrophobic core
A spherical micelle of amphiphilic molecules.
Figure 6.1
The fraction of molecules in micelles of size N is
N
∞
N
1
μ
μ
X
N
¼
NX
1
exp
;
X
N
¼
C
:
ð
6
:
4
Þ
kT
N
¼
1
Here C is the molar fraction of amphiphilic molecules in solution. When micelles
form, the difference in the standard free energies has to be negative:
0
N
0
1
5
0
μ
μ
:
The structure of a micelle is shown in
Figure 6.1
. In most currently used surfactants
there is an optimum aggregate size or aggregation number N = N
agg
for a spherical
micelle made of N
agg
molecules. The value of N
agg
is obtained by minimizing the free
energy of the micelle versus the surface area of a molecule, taking into account two
opposing effects:
*
the attractive energy arising from the interfacial tension between the liquid hydro-
carbon and water
*
the repulsion between the hydrophilic heads due to steric hindrance or to electrostatic
repulsion.
An equilibrium is established between single molecules and aggregates of size N
agg
;
when the concentration c increases, single molecules dissolve until the so-called critical
micellar concentration (CMC) is reached. At this concentration, single molecules coexist
with the
first micelle:
1
N
agg
μ
μ
X
ð
crit
¼
½
CMC
≈
exp
ð
Þ;
X
N
agg
¼
N
agg
:
ð
6
:
5
Þ
k
B
T
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