Parametrisation in Dissipative Particle Dynamics: Applications in Complex Fluids - Selected Topics of Computational and Experimental Fluid Mechanics - page 94

ˁ DPD =

3 (three water molecules per mesoscopic particle in an aqueous solution, for
example). For the mono-component system the virial free energy density f v is given

2

by f v / k B T .

When a mixture of 2 components A and B is considered, the virial pressure is
given by (Maiti and McGrother 2003 )

k B T = ˁ ln ˁ − ˁ + 2 ʱ a ˁ /

a AA ˆ

2

2

ʱ k B T ˁ

2

p = + 2 a AB ˆ( 1 − ˆ) + a BB ( 1 − ˆ) , (18)

r c 3

where ˆ is the volume fraction of component A and ( 1 − ˆ) that of component B .

The virial free energy density for this system is

N A

ln ˆ + (

1 − ˆ)

N B

ʱ( 2 a AB − a AA − a BB )ˁ

f v /ˁ k B T = ln ( 1 − ˆ) + ˆ( 1 − ˆ) + ,

(19)

cte

k B T

with ˁ = ˁ A + ˁ B and a AB = a BA .

The relationship between a ij and the physicochemical characteristics of a real
system may be obtained through the Flory-Huggins (FH) theory, based on occupa-
tions of a lattice where we have exclusively and uniquely a polymer segment or a
solvent molecule per lattice site. In the mean-field approximation this exacting single
occupancy is relaxed to a site occupancy probability, which gives a mean-field free
energy of mixing constituted by a combinatorial entropy and a mean-field energy of
mixing

F MF S MF H MF

MIX . The free energy per unit volume for a mixture
of two polymers A and B could then be written as

ʔ

MIX = ʔ MIX + ʔ

F MF
MIX

ʔ N A

ln ˆ + (

1 − ˆ)

N B Nk B T =

ln ( 1 − ˆ) + ˇ(ˆ)( 1 − ˆ), (20)

with N A and N B the number of monomers of species A and B , respectively, and
N

N B . The first two terms on the right hand side contain the information of
the energy of the pure components and correspond to the entropic contribution

= N A +

S MF ʔ

MIX .

H MF The third one involves the excess energy produced by the mixture (

MIX ). The ˇ -
parameter tells us how alike the two phases are, and is known as the Flory-Huggins
interaction parameter . In the mean-field theory this parameter is written in terms of
the nearest-neighbor interaction energies ij as ˇ 12 =

ʔ

2 k B T , where
z is the lattice coordination number. It is a phenomenological parameter, and correc-
tions considering an ionisation equilibrium between counterions and electrolyte are
needed in the presence of long-range forces. But one can also estimate this quantity
by using the Hildebrand-Scatchard regular solution theory (Hildebrand and Wood
1933 ; Scatchard 1931 ; Hildebrand and Scott 1950 ), in which the entropy of mixing
is given by an ideal expression, but the enthalpy of mixing is non-zero and is the
next simplest approximation to the ideal solution. In this approach one can appropri-
ately consider the Coulombic contribution in the enthalpy of mixing via the activity
coefficients in electrolyte solutions (vide infra).

z ( 11 + 22 − 12 )/