∆SCM(c) = ∆S

D Onsager theory “redux”
Hard rods:
D
⇤
= ⇡D LcD⇤ /4 ⇡ D/L
= for⇡(D/2)
⇤Estimate
isotropic-nematic
2
Lc ⇡
D⇥ Rod-rod ! S
⇡
1
⇥
rot
(N
1)
1
coexistence:
arrangements
(N 1)
1
Vtot
6
Vex ⇤ VVex
cVex cVex
cVex
tot⇤1Vtot 1
⇤
⇤) ⇡
2
2
S
(c
== Srot
2
2
S
(c)
CM
CM
cVex
S
(c
)
S
CM
rot
2
L
SCM (c) ⇡
2
L = ⇡(D/2)2 Lc⇤ ⇡ D/L
critical density:
⇤
side!
c =- N/V
# density of rods
c = N/V
⇤
⇤
2
view
S
⇡
1
1
rot
D
c
⇡
1/V
⇡
1/(DL
)
ex
1⇥
CM
c
⇤
2
2
c ⇥
“VolumeSExcluded”
rot ⇡ 1to center of
DL
2
c ⇡ 1/V ⇡ 1/(DL )
Center of mass entropy decrease !
from inter-rod exclusions:
S
mass of rod by neighbor rod:
Vex
2
2
V
⇤
DL
ex | sin || sin |
⇤ DL
cVex
S⇤CM
(c) 2⇡
= ⇡D
Lc ⇤ /42 ⇡ D/L
Rotational entropy diff. (per rod)
between isotropic and aligned states:
⇤
SSCM
(c
rot ⇡)1=
freely rotating!
(isotropic)
D
aligned!
(nematic)
L
(c ) =
S
ex
SCM (c⇤ ) critical
= volume
Srot fraction:
2
⇤
=
⇡(D/2)
Lc
⇡ D/L
⇤
top!
view
c⇤ ⇡ 1/Vex ⇡ ⇤
1/(DL2 )
SCM (c⇤ ) =exSrot
c ⇡ 1/V
Srot
D
⇤ 2
2
c
⇡
1/V
⇡
1/(DL
= ⇡D Lc ⇤ex/4 ⇡ D/L )
L
DL
⇤
orientational!
entropy dominates:!
Isotropic Phase
⇡ 1/(D
positional!
2
c⇤entropy
⇡ 1/Vdominates:!
⇡
1/(DL
)
ex
Nematic Phase
⇠ D/L
ICAMP 2014 (Grason - Lecture I)
- scaled vol. frac.
p
L
⌧
`
⇤
p
⇠ D/`
p
Isotropic-Nematic for
semi-flexible
filaments:
⇤
L⇠
⌧D/L
`p , rigid filament⇠ `p ⇤ ⇠ D/L
⇠ `p
`
⇤L
p
Onsager rod limit
⇠ D/L
For
For
L
`p ,
( c⇤ ⇠ 1/L2 )
⇤
c ⇠ 1/L
coiled filament
chain flexible beyond !
persistence length
⇤
⇠ D/`p
2
L
`p
⇤⇤
D/`
c⇠
⇠ 1/L
p
⇤
L
`p ⇠ D/`p
⇤
( c ⇠ 1/L )
⇤
⇠ `p
⇠ D/`p
nematic
isotropic
- scaled length
Dijkstra & Frenkel, PRE (1995).
continuous !
filament
chain uncorrelated,
rigid links
ICAMP 2014 (Grason - Lecture I)
2
Condensed filaments: orientational
& positional order
⇤
c ⇠ 1/L
Orientational elasticity (nematic):
n(x) - nematic director!
(chain orientation)
Z
⇢(x)
o
⇥
⇤
⇥
⇤
1
2
2
FF rank =
d3 x K1 (r · n)2 + K2 n · (r ⇥ n) + K3 (n · r)n
2
Z
Z
twist
splay
Fliquid
Fsolid
1
=
2
Z
n(x)
1
1
3
2
=Fliquidd =
x B(⇢(x)
d3 x ⇢B(⇢(x)
⇢20 )2 + C(r⇢)2
0 ) + C(r⇢)
2
2
bend
3
d x
2
(ukk ) + 2µuij uij
⇢(x)
⇢(x)
7
ZPositional elasticity (columnar):
“Density elasticity” (line liquid):
2D
1
3
2
2
2
Z
1
F
=
d
x
(u
)
+ 2µuij uij
Z
solid
kk
0
2 i uj +1 @j u3i )
1
u
=
(@
3
2
2
ij
Fliquid =
d x B(⇢(x) ⇢0 ) + C(r⇢) 2 Fsolid =
d x (ukk )2 + 2µuij uij
2
2
B(⇢(x)
⇢(x)
Fsolid
⇢ ) + C(r⇢)
⇢(x)
- local chain density
1
=
2
Z
preferred !
density
1
uij = (@i uj + @j ui ) - 2D elastic strain
2
1
uij =
(@i uj + @j ui )
displacement
u(x) - chain
2
from equilibrium
u(x)
d3 x
(ukk )2 + 2µuij uij
u(x)
ICAMP 2014 (Grason - Lecture II)
n · (r ⇥ n) / @z xy
k
?
Structure factor of polymer nematic: “Butterfly” profile
n · (r ⇥ n) / @zn · xy
(r ⇥ n) / @z
xy
kB T
S(k) = h| ⇢(k)| i =
2 2
2 4
2
2
B + Ck + K1 ⇢0 kz + K3 ⇢0 kz /k?
kB T
2
S(k) = h| ⇢(k)| i =
2
2
B + Ck + K1 ⇢0 k
z
2
k
2
= h| ⇢(k)| i =
kB T
Density fluctuations are
2
2
2
extinguished
along
k
B + Ck + K1 ⇢0 k? z= 0+! K3
WHY???
kz
k?
ICAMP 2014 (Grason - Lecture II)

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