Ward Identities and Semisoft Elasticity

T.C. Lubensky 

Fangfu Ye

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Semisoft Elastic Response

Λ- and Λ+ signal phase transitions at which uxz, respectively, becomes nonzero and vanishes

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Differential Elastic Response

Iij ijkl kl

K σδ δσ′Λ =

Displacement of the top plate relative to the bottom one produces xy and zy stress in (a) and (b) and xz and yz stresses in (c).   

( )

0zx zy xy

yx

c δ δ δδ

′ ′ ′Λ = Λ = Λ′= Λ =

( )

0yx yz xz

zx

a δ δ δδ

′ ′ ′Λ = Λ = Λ′= Λ =

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Forms of elastic tensors independent of detailed model: follow from Ward identities. 

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Deformations and Strains( ) ( )= +R x x u x

ii i i

Rxα α αα

δ η∂

Λ = = +∂

i iuα αη = ∂

( )

1 12 2

12

( ) ( )T T

k k

u

u u u uαβ α β β α α β

δ η η= Λ Λ− ≈ +

= ∂ +∂ +∂ ∂ u

2 2 2dR dx u dx dxαβ α β− =uαβ is invariant under rotations in target space – used to describe systems without aligning stresses

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Elastic Energy

iso ani( ) ( ) ( )f u f u f u= +

ani( )

zzf u hu=−

( ) ( )ij ij

g u f u uσ= −

Decompose free energy into an isotropic part invariant under rotations in SR (i.e. of x) and an anisotropic part (Assume Maier Saupe order parameter Qij has been integrated out).

Add “internal” or second PK stress: still invariant w.r.t. to rotations in ST

New equilibrium deformation and strain

0 0

0 0

0

0

0 0

0 0

xx xz

ij yy

zz

⎛ ⎞Λ Λ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜Λ = Λ ⎟⎜ ⎟⎜ ⎟⎜ ⎟Λ ⎟⎜⎝ ⎠( )0 0 012ij ki kj ij

u δ= Λ Λ −

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Elastic Tensor

12 ijkl ij kl

g C u uδ δ δ=

0 0;i ij j ik ik kj

x x′ ′= Λ Λ = Λ Λ

0Tu uδ ′= Λ ⋅ ⋅ Λ

12 ijkl ij kl

g C u uδ δ δ′ ′ ′ ′=

0 0 0 0

0

1

detT T

ijkl ip kr pqrs qj slC C′ = Λ Λ Λ Λ

Λ

Energy of “harmonic”deviations from equilibrium

New reference position

Strain w.r.t. to new reference pos.

Elastic tensor w.r.t to new reference space –symmetric under interchange of i and j and k and l

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Elastic Tensor for Deformations

( )I Iij ij

g f u σ= − Λ

Iij ik

ij kj

f fu

σ∂ ∂

= = Λ∂Λ ∂

0

2

0 0

ijklij kl

ik jl ip kr pjrl

fK

Cδ σΛ=Λ

∂=

∂Λ ∂Λ

= +Λ Λ

External 1st PK stress induces deformation: Defines a direction in ST: σI

xx is the force along x in the target space per unit area of the reference space

First PK stress tensor (in equilibrium)

0 1( ) Iij ik kj

σ σ−= Λ

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Deformation Tensor Relative to New State

1;

2I

ijkl ij kl ijkl ik jl ijklg K K Cδ δ δ δ σ′ ′ ′ ′ ′ ′ ′= Λ Λ = +

0 0

0

1

detT

jl jp qs slσ σ′ = Λ Λ

Λ

2 21[ ( ) 4 ]

21 1( ); ( )

2 2

Ixx y xz xzxz xz

y xz zx xz xz zx

g Cδ σ δθ ε ε

δθ ε

′ ′ ′= − +

′ ′ ′ ′= Λ −Λ = Λ +Λ

K’ijklhas a symmetric part C’ijkl and an anti‐symmetric part, which defines preferred direction in ST

Small Deformations:

0ik ik kj

′Λ = Λ Λ

Cauchy Stress Tensor

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Elastic Moduli at “Constant Stress”

1; 2; 3;

4; 5; 6.

xx yy zz

xz zy xy

= = == = =

1;

Iab b a

Ia ab b

K

S S K

δ δσ

δ δσ −

′ ′ ′Λ =′ ′ ′ ′ ′Λ = =

1Ia

aab aa

KS

σ σ′∂= =

′ ′∂Λ

0

or 0; and

0

yx yz xz zx

xy zy

xx yy zz

δ δ δ δδσ δσδσ δσ δσ

′ ′ ′ ′Λ = Λ = Λ = Λ =≠

= = =

0

or 0; and

0

zx zy xy yx

xz yz

xx yy zz

δ δ δ δδσ δσδσ δσ δσ

′ ′ ′ ′Λ = Λ = Λ = Λ =≠

= = =

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Ward Identities for C’ijkliso

( ) ( )zz

f u f u hu= −−

aij ij a ajk

U δ θ ε= − ( )ij ij a aki kj akj ik

u u u uθ ε ε→ + +

aniij ijij

a ij a a a

du dudfdf fh

d u d d dθ θ θ θ∂

= = =−∂

( )( ) 0ari rj arj ir ij ij

u u hε ε σ+ + =

2 0, ;

2 ( ) 0, ;

2 0, .

yz

xz xx

xy xx

u h a x

u h a y

u a z

σσ

= =− = =

= =

1iso

( ) f u u UuU −→is invariant under

uxz=0 or h = σxx

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These conditions must hold:

Henceelectr

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Ward Identities for Cijkl

1;

2

1;

2

1.

2

xxxyxy

xx yy

yzyzzz yy

xxxzxz

zz xx

Cu u

hC

u uh

Cu u

σ

σ

=−

=−−

=−

2 ( ) ( ) 0ari rj ijkl aki il il ali ik ik

u C h hε ε σ ε σ+ + + + =

2

2

1;

2

( 2 ) ;( )

( ) ;

( )( )

zz yyxyxy yzyz

xx yy

xzxzxz zzzz xxzz xxxx

zz xx

xzxxxz xxzz xxxx

zz xx

xx yy zz yy xz

u u hC C

u u

uC C C C

u uu

C C Cu u

u u u u u

−= =

− Δ

= − +−

= −−

Δ = − − −

0xz

u = 0xz

u ≠

( )( ) 0ari rj arj ir ij ij

kl

u u hu

ε ε σ∂

+ + =∂

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Ward Identities for C’ijkl

2 2

2 2

2 2

2 2

2 2

2 2

1;

det

1;

det

( )1det

xx yy xxxyxy xyxy

xx yy

yy zzyzyz yzyz

zz yy

zz xx xxxzxz xzxz

zz xx

C K

hC K

hC K

σ

σ

Λ Λ′ ′= =

Λ Λ −ΛΛ Λ

′ ′= =Λ Λ −Λ

Λ Λ −′ ′= =

Λ Λ −Λ

0xz

u =

When uxz is nonzero, expressions are complicated

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Minimal Model2

21o

2

s 2

3 2

iTr r

12

TTr w u v uf Bu r uαα

⎛ ⎞⎟⎜− + ⎟⎝= ⎜ ⎟⎜ ⎠

+

13

iso

0

u u u

B u

f

αβ αβ αβ γγ

αα

δ= −→ ⇒→∞ =Maier - Saupe - de Gennes

σ= − −iso

( )zz xx xx

g f u hu u

• Not full incompressibility• Nematic in crossed electric and magnetic fields

= =Tr 0 not det(1+2 ) 1 u u

σ ∂=∂

nd: 2 ij

ij

fu

Piola - Kirchhoff Stress tensor

Return to Eng. and Cauchy stress later

Ye, et al. PRL 98, 147801 (2007)

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Phase diagram at σxx=0

−=

: ;

c Z

Z

N c

c

nematic - paranematic coexistencemechanical critcal point

1 2: ( , ) (cos2 , sin2 )

: -

Z

Z Z Z

S

t N t

Biaxial coexistence -attains equilibrium value

Tricritical point; First - order line

η η η η θ θη

= =

−

η η

η η

⎛ ⎞− +⎜ ⎟⎜ ⎟⎜ ⎟= − −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

1 2

2 1

03

03

20 0

3

S

Su

S

0 : 0

0 : 0

h S

h S

> >< <

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Global Phase Digram, , :

, , :

X Y Z

X Y Z

D D D

S S S

Coexistence ofdiscrete biaxial phases

Coexistence ofcontinuous biaxial phases

η η

η η

⎛ ⎞− +⎜ ⎟⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟

− −⎜ ⎟⎝ ⎠

1 2

2 1

03

20 0

3

03

S

Su

S

( )

( , , ) ( , , )

0, 0

zz xx xx yy xx xx

xx xx

xx xx

hu u hu h u

g r h g r h h

h hPlane Planes

σ σσ σ

σ σ

+ = − + −= − −

= ≡ = = 0

YS S <Order parameter near

withR.G. Priest, Phys. Lett. A 47 475 (1974);Frisken, Bergersen, Palffy-Muhoray, Mol. Cryst. Liq. Cryst. 148, 45 (1987).

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Modulus at “Constant Stress”11 13 14

13 33 34

14 34 44

55

0

0

0

0 0 0

xxC C C

C C CK

C C C

C

σ⎛ ⎞′ ′ ′ ′+ ⎟⎜ ⎟⎜ ⎟⎜ ′ ′ ′ ⎟⎜ ⎟⎜ ⎟′ = ⎜ ⎟′ ′ ′⎜ ⎟⎟⎜ ⎟⎜ ⎟⎜ ′ ⎟⎟⎜⎝ ⎠

1; 2; 3;

4; 5; 6.

xx yy zz

xz zy xy

= = == = =

144

0 for ;ij xzxz

ij xz yy K Cσδ ′ ′Λ = ≠ =1

550 for ;

ij yzyzij yz yy K Cσδ ′ ′Λ = ≠ =

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||2

233 11 13

det0 for , , ,

( ) ( )ij xzxz

xx

Kij xx yy zz xz K

C C Cσδ

σ

′′Λ = ≠ =

′ ′ ′ ′+ −2

550 for , , ,

ij yzyzij xx yy zz yz K Cσδ ′ ′Λ = ≠ =

0yy

σ =

0yy zz xx

σ δσ δσ= = =electr

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Moduli Again

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The general behavior of Elastic moduli shown below is independent of model details.

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