Biomedical Engineering Reference
In-Depth Information
Π
r
(
c, μ
)=
k
B
T
a
c
1+
3
c
2
c
N
1
−
2
3
S
2
+
k
B
T
a
c
36
D
u
kl
Q
kl
432
D
c
2
5
− S
2
,
+
m
a
α
k
B
T
a
(7.75)
where the last term is new and arises from activity. The filament contribution to
the deviatoric stress tensor is given by
=2
k
B
T
a
c
1
−
Q
ij
+
m
a
α
8
k
B
T
a
c
c
N
σ
r,N
ij
432
D
c
2
Q
ij
24
D
1
3
u
ik
Q
kj
+
u
jk
Q
ki
−
δ
ij
u
kl
Q
kl
]
.
c
2
u
ij
+
2
+
k
B
T
a
(7.76)
Activity modifies the stress tensor of a nematic in two ways. The first term on
the right-hand side of Equation (7.76) is equilibrium-like, in the sense that it can be
obtained from the corresponding term in the stress tensor of passive rods,
σ
r,
passive
ij
=
2
k
B
T
1
−
c
N
Q
ij
by letting
T → T
a
(and replacing the transition density
c
N
by
c
IN
,
when
m
s
= 0). The second term on the right-hand side of Equation (7.76) is a truly
nonequilibrium contribution. It was first proposed phenomenologically by Hatwalne
and collaborators [50] who argued that an active element in solution behaves like a
force dipole. Correlations among the axis of each dipole build up orientational order
and yield active contributions to the stress tensor proportional to the orientational
order parameter,
Q
ij
. Our microscopic derivation [57] yields an estimate for the
coecient of this term (undetermined, even in sign, in the phenomenological theory)
and shows that the active cross-linkers yield contractile stresses (
α>
0). Finally,
the third term on the right-hand side of Equation (7.76) is the viscous contribution
which has the standard form for a solution of rod-like filaments. Finally we note
that active contributions proportional to the parameter
β
given in Equation (7.27)
do not appear in the hydrodynamics of the nematic phase. This is expected as terms
proportional to
β
break the inversion symmetry of the ordered state and can only
appear in a system with polar order.
c
7.7.3 Polarized State
The coarse-grained variables describing the dynamics of an active polarized suspen-
sion are the density and flow velocity of the solution and the concentration and
polarization of the filaments. As shown in Section 7.6, in a polarized state the align-
ment tensor is slaved to the polarization field and it is not an independent continuum
field. On the other hand, because our theory only considers terms that are quadratic
in the fields, a nonzero value for
|
P
|
is only obtained by considering the coupled equa-
tions for
P
to
Q
ij
and eliminating
Q
ij
in favor of
P
in the polarization equation
to generate a term of order (
P
)
3
. To see, consider a filament density well into the
polarized state, with
c>c
IN
and
c>c
IP
so that both coecients
a
1
=1
−
c/c
IP
and
a
2
=1
c/c
IN
in Equations (7.44) and (7.45) satisfy
a
1
<
0and
a
2
<
0. Setting
the left-hand side of Equations (7.44) and (7.45) to zero, we solve Equation (7.45)
for
Q
ij
to obtain
−
a
2
P
i
P
j
−
2
δ
ij
P
2
.
b
2
c
1
Q
ij
=
(7.77)
P
2
P
i
on the right-
hand side of Equation (7.44) which has solution
P
2
=(2
a
1
a
2
)
/
(
b
1
b
2
c
2
).
This solution, substituted in Equation (7.44), yields a term
∼
