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

3.2 Method of calculation of surface tension 25
The kinetic part can be expressed as a generalization of the ideal gas contribution
P K (r) = kBT ρ(r)^1 (3.9)
where ρ(r) is density at position r and ^1 is the 3x3 unit matrix. This kinetic part is a
single particle property and it is well localized in space. Conversely, there is no unambiguous
way of expressing the potential part of the pressure tensor. P U (r) can be
defined as the force acting across a microscopic element of area located at r. Because
the force depends on the position of two particles (for pair additive potentials), there
is no unique way to determine which pairs of particles should contribute to the pressure
across the microscopic element of area at r [71], and to reduce the non local
two-particles force to a local force at r. Irving and Kirkwood [72] derived the equations
of hydrodynamics by means of classical statistical mechanics and obtained the
expression of the pressure tensor in terms of molecular variables. They required that,
in any definition of the pressure tensor, the local virial should be located near the line
connecting the two interacting particles.
Different methods have been proposed to compute the potential part of the pressure
tensor, like the Irving and Kirkwood method [72] or the Harashima method [73].
The various definitions correspond to different choices of the contour which connects
the position in which the microscopic pressure tensor is calculated with the
particles position. It is important to underline that all methods give the same expression
for the total pressure and interfacial tension when integrated over the whole
system, while the expression of the local pressure depends on the applied method.
For a detailed description of these different methods see [66]. In our simulations we
use the Kirkwood-Buff convention [67, 74], which takes as a contour a straight line.
The simulation box of sizes Lx, Ly and Lz, is divided into Ns slabs parallel to the interface
(xy-plane) and the contribution of each pair of interacting particles to the
local pressure tensor is evenly split through all the slabs which intersect the line the
connects the two particles (line of centers).
The normal and lateral components of the local pressure tensor in slab k, including
the kinetic contribution, are then given by
PL(k) = kBT 〈ρ(k)〉 − 1
2Vs
(i,j)
PN(k) = kBT 〈ρ(k)〉 − 1
Vs
(k) x2 ij + y2 ij
u
rij
′ (rij)
(k) z
(i,j)
2 ij
u
rij
′ (rij)
(3.10)
(3.11)
where ρ(k) is the average density in slab k, Vs = LxLyLz/Ns is the volume of a slab,
u ′ (r) is the derivative of the intramolecular potential, and the brackets denote an ensemble
average. (k)
(i,j) means that the summation runs over all pairs of particles (i, j)
of which the slab k (partially) contains the line of centers. A slab k gets a contribu-