Polar nematic elasticity

dp = ndµ

F (N,V, T ) = V f(n, T )p = �f + µn G = F + pV = µN

Gibbs-Duhem (at T=const):

Simple Fluid:

Polar nematic:

df = µ dn� ~h · d~p Polarity field (director) : “Molecular field” :

~p~h = �@f

@~p

Torque: ~� = ~p⇥ ~h

Equilibrium: ~p · ~h = 0

Splay Bend Twist

Polar term, surface contribution

Usual simplifications in biological applications

2d + “isotropic” : K1 = K2 (K3 = 0)

+ constant modulus: |~p| = 1

Fnem =1

2

Zd

2x

⇣K(~r✓)2 � ~p

2h

0k

⌘

Fnem =1

2

Zd

3x

✓K1(~r · ~p)2 +K2

⇣~p⇥ (~r⇥ ~p)

⌘2+K3

⇣~p · (~r⇥ ~p)

⌘2◆

“isotropic” + non-constant |~p|

+Kd~r · ~p

Coarse-grained Frank free energy:

Fnem =

Zd

3x

✓V(|p|) + 1

2K(@↵p�)(@↵p�)

◆

Fnem =

Zd

3x f(p↵, @↵p�)

h↵ = � �F

�p↵= �

✓@f

@p↵� @�

@f

@(@�p↵)

◆

Gibbs-Duhem (at T=const):

n@↵µ = @↵p+@f

@p�@↵p�

ndµ = dp+@f

@~p· d~p

n@↵µ = �h�@↵p� � @��e↵�

Ericksen tensor: �e↵� = �p�↵� � @f

@(@�p�)@↵p�

For isotropic F: �e↵� = �p�↵� �K(@↵p�)(@�p�)

(symmetric)