3 Generalized Onsager Principle and It Applications
129
We denote the zeroth, first, and second moments of f by c = =1, p = =m, M =
mm, respectively, where c is the rod number density, n =
1
c
p is the polarity vector,
and ˜
Q =
1
c
M −
I
3
is the orientation tensor [1, 6, 7, 9, 19]. (Note: different authors
normalize c to get a particle, mass, or volume fraction.)
In rodlike molecule systems, individual rods are transported by a superposition of
four effects: (1) macroscopic velocity u of the fluid mixture, (2) diffusive translational
transport in x-space due to spatial gradients of the chemical potential, (3) rotational
velocity induced by the macroscopic flow and modeled by the Jeffery orbit equation
in eqn. (3.140), and (4) diffusive rotational transport in m-space due to rotational
gradients of the chemical potential. A key remaining ingredient entails the nonlocal
interactions of high aspect ratio rods and filaments, which we now discuss.
The nonlocal microstructure interaction potential has a general form,
U =
V
S 2
K (m, m
, x, y) f (m, y, t)dm
dy,
(3.136)
where K (m, m
, x, y) is an interaction kernel [4, 9, 10, 20] and V ∈ R
3 is the
physical domain for rods in R
3 . In practice, U is typically approximated by a local
representation in terms of low moments (shown here up to 2nd moments) and their
derivatives via a truncation of the series expansion of the kernel function:
U ≈ U (c, ∇c, p, ∇p, M, ∇M).
(3.137)
The Onsager and Maier-Saupe excluded-volume potentials for liquid crystals are
two classical examples, while Marrucci and Greco [10, 20] extended the potential
for nematic polymers to incorporate gradient elasticity (the kinetic analog of Frank
elasticity).
Combining the nonlocal potential and the entropy of the system, the free energy
over the material volume V for the system is given by
F[ f ] = k B T
V
S 2
ln f − f +
U
2
f dm dx = k B T
V
ln f − f +
U
2
dx.
(3.138)
The chemical potential is then given by
μ = k B T
δ F
δ f
= k B T [ln f + U ].
(3.139)
The transport equation for the probability density function f is given by the Smoluchowski equation that couples advection, physical and configurational space diffusion, and rotation by the flow:
∂ f
∂t
+ ∇ · (u f ) + R · ((m × ( ˙
m f ) = ∇j x + R · j r ,
˙
m = · m + a [D · m − D : mmm],
129
We denote the zeroth, first, and second moments of f by c = =1, p = =m, M =
mm, respectively, where c is the rod number density, n =
1
c
p is the polarity vector,
and ˜
Q =
1
c
M −
I
3
is the orientation tensor [1, 6, 7, 9, 19]. (Note: different authors
normalize c to get a particle, mass, or volume fraction.)
In rodlike molecule systems, individual rods are transported by a superposition of
four effects: (1) macroscopic velocity u of the fluid mixture, (2) diffusive translational
transport in x-space due to spatial gradients of the chemical potential, (3) rotational
velocity induced by the macroscopic flow and modeled by the Jeffery orbit equation
in eqn. (3.140), and (4) diffusive rotational transport in m-space due to rotational
gradients of the chemical potential. A key remaining ingredient entails the nonlocal
interactions of high aspect ratio rods and filaments, which we now discuss.
The nonlocal microstructure interaction potential has a general form,
U =
V
S 2
K (m, m
, x, y) f (m, y, t)dm
dy,
(3.136)
where K (m, m
, x, y) is an interaction kernel [4, 9, 10, 20] and V ∈ R
3 is the
physical domain for rods in R
3 . In practice, U is typically approximated by a local
representation in terms of low moments (shown here up to 2nd moments) and their
derivatives via a truncation of the series expansion of the kernel function:
U ≈ U (c, ∇c, p, ∇p, M, ∇M).
(3.137)
The Onsager and Maier-Saupe excluded-volume potentials for liquid crystals are
two classical examples, while Marrucci and Greco [10, 20] extended the potential
for nematic polymers to incorporate gradient elasticity (the kinetic analog of Frank
elasticity).
Combining the nonlocal potential and the entropy of the system, the free energy
over the material volume V for the system is given by
F[ f ] = k B T
V
S 2
ln f − f +
U
2
f dm dx = k B T
V
ln f − f +
U
2
dx.
(3.138)
The chemical potential is then given by
μ = k B T
δ F
δ f
= k B T [ln f + U ].
(3.139)
The transport equation for the probability density function f is given by the Smoluchowski equation that couples advection, physical and configurational space diffusion, and rotation by the flow:
∂ f
∂t
+ ∇ · (u f ) + R · ((m × ( ˙
m f ) = ∇j x + R · j r ,
˙
m = · m + a [D · m − D : mmm],