LogoSmall An introduction to the Defect Free Q -Tensor Approximation K.R. Daly, G. D’Alessandro & M. Kaczmarek School of Mathematics University of Southampton DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 1 / 41
LogoSmall Outline Introduction to modelling nematic liquid crystals 1 A mathematical detour 2 Defect Free Q -tensor approximation in 2D 3 Defect Free Q -tensor approximation in 3D 4 Extensions 5 Conclusions 6 DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 2 / 41
LogoSmall Outline Introduction to modelling nematic liquid crystals 1 A mathematical detour 2 Defect Free Q -tensor approximation in 2D 3 Defect Free Q -tensor approximation in 3D 4 Extensions 5 Conclusions 6 DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 3 / 41
LogoSmall What are liquid crystals? Liquid crystals exists between the isotropic liquid and crystalline solid states. Nematic liquid crystals posses some orientational order due to the elastic interactions of the molecules but no positional order. The local average direction of the molecules is the director field . It is possible to move into and out of the liquid crystalline phase simply through changing the temperature. DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 4 / 41
LogoSmall An example application: the electro-optic response From a device point of view one of the most interesting properties of nematic liquid crystals is their interaction with an electric field. DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 5 / 41
LogoSmall An example application: the electro-optic response From a device point of view one of the most interesting properties of nematic liquid crystals is their interaction with an electric field. Specifically the liquid crystals will align in the presence of external electric fields. DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 5 / 41
LogoSmall An example application: the electro-optic response From a device point of view one of the most interesting properties of nematic liquid crystals is their interaction with an electric field. Specifically the liquid crystals will align in the presence of external electric fields. The optical properties, which related to the liquid crystal alignment, can be altered by applying an electric field. DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 5 / 41
LogoSmall An example application: the electro-optic response From a device point of view one of the most interesting properties of nematic liquid crystals is their interaction with an electric field. Specifically the liquid crystals will align in the presence of external electric fields. The optical properties, which related to the liquid crystal alignment, can be altered by applying an electric field. DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 5 / 41
LogoSmall A simple model of liquid crystals: a vector The orientation of the liquid crystal can be modelled in terms of a unit vector (Frank–Oseen and Ericksen–Leslie). sin ϑ cos ψ . n = ˆ sin ϑ sin ψ cos ϑ DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 6 / 41
LogoSmall A simple model of liquid crystals: a vector The orientation of the liquid crystal can be modelled in terms of a unit vector (Frank–Oseen and Ericksen–Leslie). sin ϑ cos ψ . n = ˆ sin ϑ sin ψ cos ϑ Models based on the vector representation have the advantage that they are physically intuitive. DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 6 / 41
LogoSmall A simple model of liquid crystals: a vector The orientation of the liquid crystal can be modelled in terms of a unit vector (Frank–Oseen and Ericksen–Leslie). sin ϑ cos ψ . n = ˆ sin ϑ sin ψ cos ϑ Models based on the vector representation have the advantage that they are physically intuitive. However, they can only model uniaxial liquid crystals. DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 6 / 41
LogoSmall A simple model of liquid crystals: a vector The orientation of the liquid crystal can be modelled in terms of a unit vector (Frank–Oseen and Ericksen–Leslie). sin ϑ cos ψ . n = ˆ sin ϑ sin ψ cos ϑ Models based on the vector representation have the advantage that they are physically intuitive. However, they can only model uniaxial liquid crystals. Furthermore, the inversion symmetry of liquid crystals would require “ ˆ n = − ˆ n ”, which is not possible for a non-zero vector. DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 6 / 41
LogoSmall Limitations of the vector representation To illustrate this problem we consider a rectangular domain of liquid crystal. DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 7 / 41
LogoSmall Limitations of the vector representation To illustrate this problem we consider a rectangular domain of liquid crystal. Under the application of an external force the liquid crystal is distorted. DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 7 / 41
LogoSmall Limitations of the vector representation To illustrate this problem we consider a rectangular domain of liquid crystal. Under the application of an external force the liquid crystal is distorted. If the external force is strong enough it can force defects to appear. DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 7 / 41
LogoSmall Limitations of the vector representation To illustrate this problem we consider a rectangular domain of liquid crystal. Under the application of an external force the liquid crystal is distorted. If the external force is strong enough it can force defects to appear. DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 7 / 41
LogoSmall Limitations of the vector representation To illustrate this problem we consider a rectangular domain of liquid crystal. Under the application of an external force the liquid crystal is distorted. If the external force is strong enough it can force defects to appear. This can cause conflicts in the vector representation of the liquid crystal alignment. DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 7 / 41
The Q -tensor representation of the director field LogoSmall We use a traceless symmetric tensor, Q , to describe the liquid crystal orientation. DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 8 / 41
The Q -tensor representation of the director field LogoSmall We use a traceless symmetric tensor, Q , to describe the liquid crystal orientation. The direction of alignment of the liquid crystal is the eigenvector n of Q with largest eigenvalue. DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 8 / 41
The Q -tensor representation of the director field LogoSmall We use a traceless symmetric tensor, Q , to describe the liquid crystal orientation. The direction of alignment of the liquid crystal is the eigenvector n of Q with largest eigenvalue. The scalar order parameter, i.e. the number that identifies whether the liquid crystal is in the isotropic ( S = 0) or fully nematic ( S = 1) phase, is given by S 2 = Tr ( Q 2 ) . DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 8 / 41
The Q -tensor representation of the director field LogoSmall We use a traceless symmetric tensor, Q , to describe the liquid crystal orientation. The direction of alignment of the liquid crystal is the eigenvector n of Q with largest eigenvalue. The scalar order parameter, i.e. the number that identifies whether the liquid crystal is in the isotropic ( S = 0) or fully nematic ( S = 1) phase, is given by S 2 = Tr ( Q 2 ) . The Q -tensor naturally respects the inversion symmetry of the liquid crystal ( n and − n are both eigenvectors with the same eigenvalue). DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 8 / 41
The Q -tensor representation of the director field LogoSmall We use a traceless symmetric tensor, Q , to describe the liquid crystal orientation. The direction of alignment of the liquid crystal is the eigenvector n of Q with largest eigenvalue. The scalar order parameter, i.e. the number that identifies whether the liquid crystal is in the isotropic ( S = 0) or fully nematic ( S = 1) phase, is given by S 2 = Tr ( Q 2 ) . The Q -tensor naturally respects the inversion symmetry of the liquid crystal ( n and − n are both eigenvectors with the same eigenvalue). The Q -tensor describes also biaxial liquid crystals and defects (where S = 0 and n is not defined). DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 8 / 41
Q -tensor and vector director field LogoSmall It is possible to express the Q tensor in terms of the vector director field n as � � � 3 n − 1 Q = 2 S ˆ n ⊗ ˆ 3 I . DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 9 / 41
Q -tensor and vector director field LogoSmall It is possible to express the Q tensor in terms of the vector director field n as � � � 3 n − 1 Q = 2 S ˆ n ⊗ ˆ 3 I . This representation shows clearly that the Q -tensor is invariant with respect to sign reversal of n , n ⊗ ˆ n = ( − ˆ n ) ⊗ ( − ˆ n ) . ˆ DDK (Univ. of Southampton) Intro to the DFQTA 31 March 2012 9 / 41
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