Mesoscopic models of lipid bilayers and bilayers with embedded ...

24 Surface tension in lipid bilayers
with
From equations 3.1 and 3.2 we have
Pxx(r) = Pyy(r). (3.3)
∂Pxx
∂x ex + ∂Pyy
∂y ey + ∂Pzz
∂z ez = 0 (3.4)
where eα (α = x, y, z) are the orthogonal basis vectors in the Cartesian space. From
equations 3.4 and 3.3 results that the lateral (also called tangential) components of
the pressure tensor are function of z only: PL(z) = Pxx(z) = Pyy(z). The normal
component is constant throughout the system and equal to the external pressure:
PN(z) = Pzz(z) = Pext. The lateral components are also equal to the external pressure
in the bulk phases.
The surface tension is defined as the integral over the interface of the difference
between the normal and lateral components of the pressure tensor [67, 68]
γ =
z2
z1
dz [PN(z) − PL(z)] =
z2
z1
dγ(z) (3.5)
where z1 and z2 are positions in the bulk phases and γ(z) is the local surface tension
at position z.
In molecular simulations, statistical mechanics is used to relate thermodynamic
quantities to ensemble averages over microscopic degrees of freedom. For an homogeneous
system the pressure is a scalar and it can be expressed by the virial equa-
tion [69, 70]
P = ρkBT + 1
N
ri · Fi
3V
i=1
(3.6)
where Fi is the total internal force on particle i and the brackets indicate an ensemble
average. If the intermolecular forces are pairwise additive, the above may be written
as
P = ρkBT + 1
3V
i
j>i
where Fij is the force on particle i due to particle j.
rij · Fij
(3.7)
For an inhomogeneous system the pressure tensor at position r can still be expressed
in a tensorial form of the virial equation and it can be split in a kinetic P K
and a potential part P U [66],
P(r) = P K (r) + P U (r) (3.8)