Electro-Opto-Mechanical Effects in Swollen Nematic Elastomers - Liquid Crystal Elastomers: Materials and Applications - page 139

The free energy of nematic gels under electric fields may be written as the sum of
the Frank energy ( F F ), the electrostatic energy ( F el ), and the gel elasticity energy ( F g ):

F ¼ F F þ F el þ F g (13)

0 in the present case where the director rotates
uniformly around the y -axis, i.e., the director is always confined to the x - z plane.
The electrostatic energy is:

It should be noted that F F

ð ð

e 0
2

e 0
2

2 2 2

F el ¼ ð e a ðr'ðxÞ n ðxÞÞ þ e ? r'ðxÞ j j Þ d x ð e s r'ðxÞ j j Þ d x (14)

A B

where ' is the electric potential, n ¼ (cosy, 0, siny) is the nematic director, and
A and B are the spaces occupied by gel and silicone oil, respectively. The dielectric
constant e s is for silicone oil, whereas e || and e ⊥ are those parallel and normal to n
for the gel, respectively, and e a ¼ e || e ⊥ .
The elastic energy of the gel is:

ð G
2

d x

2

F g ¼ j e u ðxÞe 0 ð n ðxÞÞ j þ f an ð n ðxÞÞ (15)

where G is the shear modulus and e u is the linear strain with components ( e u ) ij ¼
( ∂x i )/2. The quantity u i ( x ) is the i th component of the displacement

at point x of the gel with respect to its reference configuration (chosen as the
configuration the specimen would take in a high-temperature isotropic state), and
x j is the j th coordinate of x . The strain e 0 (n) corresponds to that in the stress-free
state associated with n, which we assume to be a uniaxial stretching along n . This is
expressed by the following matrix as a function of the angle y:

∂u i / ∂x j þ ∂u j /

2
4

#

3

cos 2

1 3

2 gð

y

3 Þ

0

2 g sin y cos y

2

½e 0 ð n Þ ¼ 0 0 ¼½e 0 ð y Þ (16)

3 3

sin 2

1

2 g sin y cos y

0

2 gð

y

3 Þ

where g is a measure of the chain anisotropy in the gel and related to k as g ¼ 2

( k 2). The dimensional variation in the x direction driven by the rotation

(y) of the director in the x - z plane is given by ( e u ) 11 ¼

1)/( k þ

[ e 0 (y)] 11 . The corresponding

strain g x is given by:

l x ð y Þ l x 0
l x 0

¼ ½e 0 ð y Þ 11 ½e 0 ð

0 Þ 11 3 g

21

sin 2

g x ð y Þ¼

Þ 11 ¼

y (17)

1 þ½e 0 ð 0 ð þ g Þ

This relation is substantially equivalent to ( 11a ) since the parameter k is given by
k ¼ l x (0)/ l x (90 ).
In ( 15 ) , f an is the anisotropic energy density for the memory effect of the director,
driving the director n towards the initial director n 0 ¼ (1,0,0) at cross-linking.