A Unified Model for I-N-S A -S C Phase Transitions for Liquid - PowerPoint PPT Presentation

A Unified Model for I-N-S A -S C Phase Transitions for Liquid Crystal Song Mei School of Mathematical Science, Peking University November 1, 2013 Joint work with Jiequn Han, Wei Wang, Pingwen Zhang . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei 1 / 29

Contents . . Introduction 1 . . Classical models in three levels 2 . . A mechanistic Q Tensor Model 3 . . Vector Model 4 . . Numerical Results 5 . . Conclusion 6 . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei 2 / 29

Contents . . Introduction 1 . . Classical models in three levels 2 . . A mechanistic Q Tensor Model 3 . . Vector Model 4 . . Numerical Results 5 . . Conclusion 6 . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei Introduction 3 / 29

Molecules . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei Introduction 4 / 29

Phases . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei Introduction 5 / 29

Contents . . Introduction 1 . . Classical models in three levels 2 . . A mechanistic Q Tensor Model 3 . . Vector Model 4 . . Numerical Results 5 . . Conclusion 6 . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei Classical models in three levels 6 / 29

Onsager’s Molecular Theory The energy functional: ∫ ∫ β F [ ρ ] = S 2 ρ ( x , m ) ln ρ ( x , m ) d m d x Ω + α ∫ ∫ ∫ ∫ S 2 ρ ( x , m ) G ( m , m ′ , x − x ′ ) ρ ( x ′ , m ′ ) d x d m d x ′ d m ′ . 2 S 2 Ω Ω where ∫ ∫ S 2 ρ ( x , m ) d x d m = N Ω and G represents the interaction potential. For example, G may represent volume exclusion potential: { 0 , molecule ( x , m ) is disjoint with molecule( x ′ , m ′ ) G ( m , m ′ , x − x ′ ) = 1 , joint with each other . or G may represent the Gay-Berne potential: ( r / r 0 − σ ( m , m ′ , r ) + 1) − 6 − ( r / r 0 − σ ( m , m ′ , r ) + 1) − 12 ) ( G ( m , m ′ , r ) = ε ( m , m ′ , r ) . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei Classical models in three levels 7 / 29

Q-Tensor Theory for Liquid Crystal Define a symmetric traceless second order tensor Q ( x ) to describe the orientation of molecule. ∫ S 2 ( mm − 1 Q ( x ) = 3 I ) f ( x , m ) d m , where ∫ f ( x , m ) d m = 1 . Ω Q = 0 : ⇒ isotropic; If Q has two equal eigenvalues: Q = S ( nn − 1 3 I ) , n ∈ S 2 . ⇒ uniaxial; If S = const ̸ = 0: nematic phase; The Landau-de Gennes model: ( A ( T − T ∗ ) ∫ tr ( Q 2 ) − B 3 tr ( Q 3 ) + C ) F LG [ Q ] = 4 ( tr ( Q 2 )) 2 d x 2 Ω ∫ L 1 ∂ j Q ik ∂ k Q ij + L 2 ∂ j Q ij ∂ k Q ik + L 3 |∇ Q | 2 + L 4 Q lk ∂ k Q ij ∂ l Q ij ( ) + d x Ω . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei Classical models in three levels 8 / 29

Vector Model and Oseen-Frank Energy Assume Q ( x ) = S ( nn − 1 3 I ), define n ( x ) as the director. ∫ Define c ( x ) as the relative density, where Ω c ( x ) = 1. Vector Model describes N-S A -S C transitions, with order parameter c ( x ) and n ( x ). The energy functional: β F [ c , n ] = β F 1 + β F 2 , where F 1 contains up to second order derivative terms of c ( x ) C 2 ∫ ( ac 2 + D || [( n · ∇ ) 2 c ] 2 − C || ( n · ∇ c ) 2 + || c 2 β F 1 = 4 D || Ω ij ∇ i c ∇ j c + D ⊥ ( ∇ 2 ⊥ c ) 2 ) d x . + C ⊥ δ T F 2 is the Oseen-Frank free energy for distortions in the nematic director: β F 2 = β ∫ ( K 1 ( ∇ · n ) 2 + K 2 [ n · ( ∇ × n )] 2 + K 3 [ n × ( ∇ × n )] 2 ) d x . 2 . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei Classical models in three levels 9 / 29

Contents . . Introduction 1 . . Classical models in three levels 2 . . A mechanistic Q Tensor Model 3 . . Vector Model 4 . . Numerical Results 5 . . Conclusion 6 . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei A mechanistic Q Tensor Model 10 / 29

A mechanstic Q Tensor Model With order parameter c ( x ) , Q 2 ( x ), the energy functional is: ∫ β F [ c , Q ] = c ( x )(ln c ( x ) + B Q : Q − ln Z ( x )) d x + θ ( A 1 + 3 2 A 2 | Q 2 | 2 + 35 ∫ 8 A 3 | Q 4 | 2 ) c 2 d x 2 ( ∫ { G 1 |∇ c | 2 + G 2 |∇ ( cQ 2 ) | 2 + G 3 |∇ ( cQ 4 ) | 2 + G 4 ∂ i ( cQ 2 ij ) ∂ j ( c ) − + G 5 ∂ i ( cQ ik ) ∂ j ( cQ jk ) + G 6 ∂ i ( cQ 4 iklm ) ∂ j ( cQ 4 jklm ) + G 7 ∂ i ( cQ 4 ijkl ) ∂ j ( cQ 2 kl ) } d x ∫ + { H 1 ∂ ij ( cQ 4 ijpq ) ∂ kl ( cQ 4 klpq ) + H 2 ∂ ij ( cQ 2 ij ) ∂ kl ( cQ 2 kl ) + H 3 ∂ ik ( cQ 2 ip ) ∂ jk ( cQ 2 jp ) + H 4 ∂ ij ( cQ 2 pq ) ∂ ij ( cQ 2 pq ) + H 5 ∂ ij ( c ) ∂ ij ( c ) + H 6 ∂ ij ( cQ 4 ijkp ) ∂ kl ( cQ 2 lp ) + H 7 ∂ ij ( cQ 4 ijpq ) ∂ kk ( cQ 2 pq ) + H 8 ∂ ij ( cQ 4 ijkl ) ∂ kl ( c ) + H 9 ∂ ij ( cQ 2 ij ) ∂ kk ( c ) } d x ) Here A i , G i , H i are coefficients depend on molecular parameters. θ : the average concentration; β = 1 / ( k B T ). c ( x ): the normalized local concentration. Q 2 ( x ): second-order tensor; Q 4 is a fourth-order tensor depending on Q 2 . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei A mechanistic Q Tensor Model 11 / 29

Taylor expansion Onsager’s molecular theory: ∫ F [ ρ ] = ρ ( x , m ) ln ρ ( x , m ) d m d x ⇒ Entropy term + α ∫ ρ ( x , m ) G ( m , m ′ , x − x ′ ) ρ ( x ′ , m ′ ) d m ′ d x ′ d m d x ⇒ Interaction 2 = F entropy + α ∫ ρ ( x , m ) G ( m , m ′ , r ) ρ ( x + r , m ′ ) d m ′ d m d x d r 2 where r = x ′ − x . Using Taylor expansion: ρ ( x + r , m ) = ρ ( x , m ) + r · ∇ ρ ( x , m ) + 1 2 rr : ∇ 2 ρ ( x , m ) + · · · . The energy functional becomes: F interaction = α ∫ ρ ( x , m ) G ( m , m ′ , r ) ρ ( x , m ′ ) d r d m ′ d m d x ⇒ Bulk energy 2 + α ∫ ρ ( x , m ) G ( m , m ′ , r ) rr : ∇ 2 ρ ( x , m ′ ) d r d m ′ d m d x 4 + α ∫ ρ ( x , m ) G ( m , m ′ , r ) rrrr : ∇ 4 ρ ( x , m ′ ) d r d m ′ d m d x + · · · ⇒ Elastic energy 48 . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei A mechanistic Q Tensor Model 12 / 29

Spherical Invariants Expansion Expand the interaction potential in terms of spherical invariants: ∑ J lmk ( | r | ) T lmk ( m , m ′ , ˆ G ( m , m ′ , r ) = r ) . lmk Thus, the moments of G can be write down as summation of interaction of Legendre polynomials: ∫ G ( m , m , r ) d r = a 1 + a 2 P 2 ( m ) : P 2 ( m ′ ) + a 3 P 4 ijkl ( m ) P 4 ijkl ( m ′ ) + · · · , Ω ∫ G ( m , m , r ) rr d r = g 1 I + g 2 ( P 2 ( m ) + P 2 ( m ′ )) + g 3 P 2 ( m ) : P 2 ( m ′ ) I + g 4 P 2 ( m ) · P 2 ( m ′ ) · · · , Ω ∫ G ( m , m , r ) rrrr d r = h 1 I 4 + h 2 ( P 2 ( m ) + P 2 ( m ′ )) I 2 + h 3 P 2 ( m ) P 2 ( m ′ ) + h 4 P 2 ( m ) · P 2 ( m ) I 2 Ω + h 5 P 2 ( m ) : P 2 ( m ) I 4 + h 6 ( P 4 ( m ) + P 4 ( m ′ )) + · · · . where P n is the n-th order Legendre polynomial and thus P n ( m ) is an n-th order tensor. For example P 2 ( m ) = 3 2 mm − 1 2 I . Here a i , g i , h i can be calculated by integration, and are expressed by molecular parameters. We truncated to 4-th order Legendre polynomials terms. . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Song Mei A mechanistic Q Tensor Model 13 / 29