On the cubic instability in the Q-tensor theory of nematic liquid - PowerPoint PPT Presentation

On the cubic instability in the Q-tensor theory of nematic liquid crystals Arghir Zarnescu University of Sussex Joint work with Gautam Iyer (Carnegie Mellon) , Xiang Xu (Carnegie Mellon) Diffuse Interface Models Workshop Levico Terme, September 2013

Liquid crystals modelling: physics A measure µ such that 0 ≤ µ ( A ) ≤ 1 ∀ A ⊂ S 2

Liquid crystals modelling: physics A measure µ such that 0 ≤ µ ( A ) ≤ 1 ∀ A ⊂ S 2 The probability that the molecules are pointing in a direction contained in the surface A ⊂ S 2 is µ ( A )

Liquid crystals modelling: physics A measure µ such that 0 ≤ µ ( A ) ≤ 1 ∀ A ⊂ S 2 The probability that the molecules are pointing in a direction contained in the surface A ⊂ S 2 is µ ( A ) Physical requirement µ ( A ) = µ ( − A ) ∀ A ⊂ S 2

Liquid crystals modelling: physics A measure µ such that 0 ≤ µ ( A ) ≤ 1 ∀ A ⊂ S 2 The probability that the molecules are pointing in a direction contained in the surface A ⊂ S 2 is µ ( A ) Physical requirement µ ( A ) = µ ( − A ) ∀ A ⊂ S 2

Q-tensors-statistical mechanics interpretation and constraints � S 2 p ⊗ p d µ ( p ) − 1 Q = 3 Id

Q-tensors-statistical mechanics interpretation and constraints � S 2 p ⊗ p d µ ( p ) − 1 Q = 3 Id Q is a 3 × 3 symmetric, traceless matrix - a Q-tensor

Q-tensors-statistical mechanics interpretation and constraints � S 2 p ⊗ p d µ ( p ) − 1 Q = 3 Id Q is a 3 × 3 symmetric, traceless matrix - a Q-tensor If e i , i = 1 , 2 , 3 are eigenvectors of Q , with corresponding eigenvalues λ i = 1 , 2 , 3, we have � − 1 S 2 ( p · e i ) 2 d µ ( p ) − 1 3 ≤ 2 3 ≤ λ i = 3 � for i = 1 , 2 , 3, since S 2 d µ ( p ) = 1.

Landau-de Gennes Q-tensor reduction and earlier theories � S 2 p ⊗ p d µ ( p ) − 1 Q = 3 Id

Landau-de Gennes Q-tensor reduction and earlier theories � S 2 p ⊗ p d µ ( p ) − 1 Q = 3 Id The Q -tensor is: isotropic is Q = 0 uniaxial if it has two equal eigenvalues biaxial otherwise

Landau-de Gennes Q-tensor reduction and earlier theories � S 2 p ⊗ p d µ ( p ) − 1 Q = 3 Id The Q -tensor is: isotropic is Q = 0 uniaxial if it has two equal eigenvalues biaxial otherwise Ericksen’s theory (1991) for uniaxial Q -tensors which can be written as � � n ( x ) ⊗ n ( x ) − 1 s ∈ R , n ∈ S 2 Q ( x ) = s ( x ) 3 Id ,

Landau-de Gennes Q-tensor reduction and earlier theories � S 2 p ⊗ p d µ ( p ) − 1 Q = 3 Id The Q -tensor is: isotropic is Q = 0 uniaxial if it has two equal eigenvalues biaxial otherwise Ericksen’s theory (1991) for uniaxial Q -tensors which can be written as � � n ( x ) ⊗ n ( x ) − 1 s ∈ R , n ∈ S 2 Q ( x ) = s ( x ) 3 Id , Oseen-Frank theory (1958) take s in the uniaxial representation to be a fixed constant s +

The Q-tensor energy functionals I The simplest way to obtain physically relevant configurations is by minimizing an energy functional: � F [ Q , D ] = ψ ( Q ( x ) , D ( x )) dx Ω where ( Q ij ( x )) i , j = 1 ,..., d is a Q -tensor, i.e. symmetric and traceless d × d matrix ( d = 2 , 3) and ‘ D ∼ ∇ Q ‘ , is a third order tensor.

The Q-tensor energy functionals I The simplest way to obtain physically relevant configurations is by minimizing an energy functional: � F [ Q , D ] = ψ ( Q ( x ) , D ( x )) dx Ω where ( Q ij ( x )) i , j = 1 ,..., d is a Q -tensor, i.e. symmetric and traceless d × d matrix ( d = 2 , 3) and ‘ D ∼ ∇ Q ‘ , is a third order tensor. Physical invariances require that: ψ ( Q , D ) = ψ ( Q ∗ , D ∗ ) where Q ∗ = RQR t and D ∗ ijk = R il R jm R kn D lmn for any R ∈ O ( 3 ) .

The Q-tensor energy functionals II We can decompose: ψ ( Q , D ) = ψ ( Q , 0 )+ ψ ( Q , D ) − ψ ( Q , 0 ) = ψ B ( Q ) + ψ E ( Q , D ) � �� � � �� � bulk elastic Then ψ B ( Q ) = ψ B ( RQR t ) for R ∈ O ( 3 ) implies that there exists ¯ ψ B so that ψ B ( Q ) = ¯ ψ B ( tr ( Q 2 ) , tr ( Q 3 )) .

The Q-tensor energy functionals II We can decompose: ψ ( Q , D ) = ψ ( Q , 0 )+ ψ ( Q , D ) − ψ ( Q , 0 ) = ψ B ( Q ) + ψ E ( Q , D ) � �� � � �� � bulk elastic Then ψ B ( Q ) = ψ B ( RQR t ) for R ∈ O ( 3 ) implies that there exists ¯ ψ B so that ψ B ( Q ) = ¯ ψ B ( tr ( Q 2 ) , tr ( Q 3 )) . Example of elastic terms that respect the physical invariances: I 1 = Q ij , k Q ij , k , I 2 = Q ij , j Q ik , k , I 3 = Q ij , k Q ik , j , I 4 = Q ij , l Q ij , k Q kl Note that I 2 − I 3 = ( Q ij Q ik , k ) , j − ( Q ij Q ik , j ) , k .

The cubic instability Take an elastic energy: 4 � ψ E ( Q , D ) = L j I j j = 1 where L j , j = 1 , . . . , 4 are elastic constants. If L 4 � = 0 (corresponding to the cubic term) then � F [ Q , D ] = Ω ψ B ( Q ( x )) + ψ E ( Q ( x ) , D ( x )) dx is seen to be unbounded from below by taking (J.M. Ball): � � | x | ⊗ x x | x | − 1 Q = s ( x ) 3 Id for suitable s ( x ) (then 9 s ( s ′ 2 − 3 I 4 = 4 r 2 s 2 ) ).

The cubic instability Take an elastic energy: 4 � ψ E ( Q , D ) = L j I j j = 1 where L j , j = 1 , . . . , 4 are elastic constants. If L 4 � = 0 (corresponding to the cubic term) then � F [ Q , D ] = Ω ψ B ( Q ( x )) + ψ E ( Q ( x ) , D ( x )) dx is seen to be unbounded from below by taking (J.M. Ball): � � | x | ⊗ x x | x | − 1 Q = s ( x ) 3 Id for suitable s ( x ) (then 9 s ( s ′ 2 − 3 I 4 = 4 r 2 s 2 ) ). If L 4 = 0 then for ψ B ≡ 0 we have: F [ Q ] ≥ µ �∇ Q � 2 L 2 for some µ > 0 if and only if (Longa, Monselesan, and Trebin, 1987): L 3 > 0 , − L 3 < L 2 < 2 L 3 , − 3 5 L 3 − 1 10 L 2 < L 1

Q-tensors versus directors energies Take Q ( x ) = s ( n ( x ) ⊗ n ( x ) − 1 3 Id ) with s fixed and n ( x ) ∈ S 2 . Let K 1 := 2 L 1 s 2 + L 2 s 2 + L 3 s 2 − 2 3 L 4 s 3 , K 2 := 2 L 1 s 2 − 2 3 L 4 s 3 K 3 = 2 L 1 s 2 + L 2 s 2 + L 3 s 2 + 4 3 L 4 s 3 , K 4 = L 3 s 2 Then F [ Q , D ] = G [ n , ∇ n ] with � K 1 ( div n ) 2 + K 2 ( n · curl n ) 2 + K 3 ( n × curl n ) 2 dx G [ n , ∇ n ] = Ω � ( K 2 + K 4 )( tr ( ∇ n ) 2 − ( div n ) 2 )] dx + Ω the Oseen-Frank energy functional.

Oseen-Frank energy: interpretation and its darkest secret...the K 13 term Recall the Oseen-Frank energy functional: � K 1 ( div n ) 2 + K 2 ( n · curl n ) 2 + K 3 ( n × curl n ) 2 dx G [ n , ∇ n ] = Ω � ( K 2 + K 4 )( tr ( ∇ n ) 2 − ( div n ) 2 )] dx + Ω A term allowed by symmetries � � K 13 ∇ · ( n · ∇ n ) dx = K 13 ( n · ν )( ∇ · n ) d σ Ω ∂ Ω would make the energy unbounded from below...Thus it is never included in mathematical studies...

Π αντα ρǫι H ǫρακλǫιτ o ζ

Π αντα ρǫι H ǫρακλǫιτ o ζ Everything flows Heraclitus

The L 2 gradient flow A gradient flow: dQ dt = − grad F [ Q , D ( Q )]

The L 2 gradient flow A gradient flow: dQ dt = − grad F [ Q , D ( Q )] We consider the L 2 gradient, taking into account the constraints � δ F � ∂ Q ij = − + λδ ij + µ ij − µ ji ∂ t δ Q ij where λ is a Lagrange multiplier corresponding to the constraint tr ( Q ) = 0 and µ ij , i , j = 1 , 2 , 3 are Lagrange multipliers corresponding to the constraint Q ij = Q ji .

The L 2 gradient flow A gradient flow: dQ dt = − grad F [ Q , D ( Q )] We consider the L 2 gradient, taking into account the constraints � δ F � ∂ Q ij = − + λδ ij + µ ij − µ ji ∂ t δ Q ij where λ is a Lagrange multiplier corresponding to the constraint tr ( Q ) = 0 and µ ij , i , j = 1 , 2 , 3 are Lagrange multipliers corresponding to the constraint Q ij = Q ji . More explicitly and for the standard potential ψ B ( Q ) = a 2 tr ( Q 2 ) + b 3 tr ( Q 3 ) + c 4 tr 2 ( Q 2 ) this becomes: Q ik Q kj − tr ( Q 2 ) ∂ Q ij � � − c tr ( Q 2 ) Q ij = 2 L 1 ∆ Q ij − aQ ij + b δ ij ∂ t 3 +( L 2 + L 3 ) ( ∇ j ∇ k Q ik + ∇ i ∇ k Q jk ) − 2 3 ( L 2 + L 3 ) ∇ l ∇ k Q lk δ ij + 2 L 4 ∇ l Q ij ∇ k Q lk + 2 L 4 ∇ l ∇ k Q ij Q lk − L 4 ∇ i Q kl ∇ j Q kl + L 4 3 |∇ Q | 2 δ ij .

“Energy conservation” and its use-the L ∞ bound Recall the abstract equation: � δ F � ∂ Q ij = − + λδ ij + µ ij − µ ji ∂ t δ Q ij Multiplying by − RHS and integrating over Ω we get: �� δ F � 2 3 � � d � dt F [ Q ] = − + λδ ij + µ ij − µ ji ≤ 0 (1) δ Q Ω ij i , j = 1 so the energy is apriori bounded, i.e. � L 1 |∇ Q | 2 + L 2 ∇ j Q ik ∇ k Q ij + L 3 ∇ j Q ij ∇ k Q ik F [ Q ( t )] = + L 4 Q lk ∇ k Q ij ∇ l Q ij dx � �� � � �� � Ω small? ≥ µ �∇ Q � 2 L 2 � a 2 tr ( Q 2 ) + b 3 tr ( Q 3 ) + c 4 tr 2 ( Q 2 ) + dx ≤ F [ Q 0 ] Ω � �� � ≥− c 2 0