Chemistry Reference
In-Depth Information
Figure 1 Snapshot from an MC simulation of two proteins. The black and grey spheres
illustrate mobile cations and anions. Amino acids are depicted as white spheres,
clustered to form the two proteins. In the simulation, the proteins are displaced
along a line and rotated independently. Ions are displaced in all three directions
and the whole system is enclosed in a sphere of appropriate radius
where Z
i
is the ion valency, e the elementary charge,
e
0
the dielectric permit-
tivity of vacuum and d
hc
the hard sphere diameter of an ion. For simplicity, we
will in this communication mostly assume d
hc
to be the same for all ionic species
and equal to 4 A
˚
.
While these charges can be the small mobile ions in a salt solution, they can
also be the charged groups on a protein or some other macromolecule. The
model is schematically depicted in Figure 1 in terms of two macromolecules in a
salt solution. We will solve this model exactly using Monte Carlo (MC)
simulations, or in an approximate way with either the Poisson-Boltzmann
(PB) equation or its linearized version, the Debye-Hu¨ ckel (DH) equation. For
an introduction to the DH theory, the reader is recommended to consult the
excellent textbook of Hill.
1
Engstro¨ m and Wennerstro¨ m
2
have solved the PB
equation for a charged surface with neutralizing counter-ions, and their paper
is a good starting point on this subject. Monte Carlo and other simulations are
well described in the textbooks by Allen and Tildesley
3
and Frenkel and Smit.
4
The MC simulations allow us to emphasize where the simple theory is appli-
cable and where a more accurate treatment is needed. The simulations also give
an opportunity to clarify certain physical mechanisms, providing a deeper
understanding of the system at hand.
9.2 The Simple Electrolyte Solution
An important property of an electrolyte solution is the activity factor (activity
coefficient)
g
, or excess chemical potential
m
ex
, which is a part of the total
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