Biomedical Engineering Reference
In-Depth Information
Table 23.1
Representative relative permittivities
Substance
ε
r
Free space
1
Air
1.0006
Glass
6
Water
81
B in the molecule of Figure 23.1 will not have much exposure to a solvent and so a pure
solvent value of ε
r
is not appropriate.
A
Solvent
B
Figure 23.1
Atoms A and B in a molecule surrounded by solvent
23.2 Langevin Dynamics
Molecules in solution undergo collisions with other molecules and the solvent, and they
can also be thought to experience frictional forces as they move through the solvent. The
Langevin technique allows for both these effects. The collisions aremodelled by the addition
of a random force
R
, and the frictional effects are modelled by adding a frictional drag that is
directly proportional to the velocity of a given particle. Solvent molecules are not explicitly
included.
We saw in Chapter 10 that the key equation for a MD simulation is Newton's second law:
m
d
2
r
d
t
2
=
F
(23.1)
In Langevin dynamics, the force is modified according to the ideas discussed above:
m
d
2
r
d
t
2
m
γ
d
r
=
F
−
d
t
+
R
(23.2)
The frictional term introduces energy and the random force removes it as kinetic energy.
The quantity γ is the collision frequency, and 1/γ is often called the
velocity relaxation
time
; it can be thought of as the time taken for a particle to forget its initial velocity. There
is an equation due to Einstein that relates γ to the diffusion constant
D
of the solvent
k
B
T
mD
=
γ
