Biomedical Engineering Reference
In-Depth Information
σ
ij
=2
k
B
T
a
c
1
−
c
c
IN
Q
ij
− k
B
T
a
c
2
c
IP
P
i
P
j
−
1
2
δ
ij
P
2
c
2
4
2
δ
ij
P
2
+
m
a
b
2
α
k
B
T
a
l
3
72
D
1
3
Q
ij
+
P
i
P
j
−
c
2
∂
j
P
i
−
4
∂
i
P
j
− ∂
j
P
i
,
+
m
a
b
2
β
k
B
T
a
l
4
216
D
1
2
δ
ij
∇
·
P
−
1
(7.38)
where
c
IP
=
D
r
/
(
m
a
b
2
γ
P
l
2
)and
c
IN
=
c
N
/
[1 +
c
N
l
2
m
s
b
2
γ
NP
/
(4
D
r
)] are the den-
sities for the isotropic-polarized (IP) and isotropic-nematic (IN) transition, respec-
tively, at finite density of active polar motor clusters (
m
a
) and stationary nonpolar
cross-linkers (
m
s
)[36]. Finally,
c
N
=3
π/
(2
l
2
) is the density of the IN transition
in passive systems. There are three types of contributions to the active part of the
stress tensor. The first consists of the first two terms on the right-hand side of Equa-
tion (7.38). These are equilibrium-like terms, in the sense that they have the same
structure one would obtain in a nematic and polar passive fluid, respectively, with
the transition densities replaced by their active values. In particular, the first term
on the right-hand side of Equation (7.38) should be compared to the corresponding
contribution for isotropic (
c<c
N
) passive solutions,
σ
ij
=2
k
B
Tc
1
−
Q
ij
.
The third term is a homogeneous nonequilibrium contribution that remains nonzero
even for
κ
ij
= 0. This “spontaneous stress” arises from activity and is proportional
to the ATP consumption rate that acts as an additional driving force and can build
up stresses even in the absence of mechanical deformations. This term is generated
by motor-induced filament bundling and is proportional to the bundling rate
α
.
It would therefore vanish in the absence of spatial inhomogeneities in the motor
stepping rate. Finally, the fourth term contains active contributions proportional
to gradients of the polarization (we have omitted here terms of linear order in the
gradients containing both polarization and alignment tensor. The full expression for
the stress tensor can be found in the Appendix). These stresses are generated by
motor-induced filament sorting and are proportional to
β
.Theyareimportantonly
in the polarized phase, where we expect they will play an important role in enhanc-
ing the relaxation of longitudinal fluctuations of the filaments and the corresponding
relaxation of shear via reptation.
Finally, the viscous contribution to the stress is
c
c
N
48
1
2
u
ij
−
2
κ
kk
δ
ij
+
1
3
Q
ij
κ
kk
−
δ
ij
κ
kq
Q
qk
lcζ
⊥
σ
ij
=
1
3
u
ik
Q
kj
+
u
jk
Q
ki
,
+
2
(7.39)
7.6 Homogeneous States of a Quiescent Solution
We first examine the case of a quiescent suspension, with
v
= 0. We consider a
system with a concentration
m
a
of active, polar motor clusters and a concentration
m
s
of stationary nonpolar cross-linkers. For convenience, we define a dimensionless
parameter
μ
a
measuring activity as
μ
a
=
m
a
b
2
γ
P
D
r
∼ m
a
R
AT P
,
(7.40)
