Liquid Crystals Some aspects Martin Oettel May 14 1
The phases in Physics I gas / vapor change in density (scalar order parameter) no change in continuous translation liquid change in density change from continuous to discrete translation in all 3 directions solid / crystalline
The phases in Advanced Soft Condensed Matter There is a lot of room between the liquid and the crystalline phase. Key necessity: anisotropic building blocks liquid continuous translational continuous orientational liquid crystal (nematic) continuous translational discontinuous orientational ( mm: 1 rotation axis, mirror planes) molecules may be oriented up or down, so it is not a polarized material! 5CB - a working horse 3
The phases in Advanced Soft Condensed Matter liquid continuous translational continuous orientational liquid crystal (discotic) continuous translational discontinuous orientational ( mm: 1 rotation axis, mirror planes) benzene-hexa-n-alkanoate derivatives 4
The phases in Advanced Soft Condensed Matter n n s liquid crystal (nematic) liquid crystal (biaxial) continuous translational discontinuous orientational continuous translational discontinuous orientational ( mm: 1 rotation axis, mirror planes) (/m m: -fold axis with reflection, mirror plane) banana bend in and out of plane banana bend in the plane 5
The phases in Advanced Soft Condensed Matter liquid crystal (nematic) liquid crystal (cholesteric) continuous translational discontinuous orientational one discrete translational discontinuous orientational screw axis one needs chiral molecules! 6 MBBA: nematic methylized MBBA: cholesteric
The phases in Advanced Soft Condensed Matter liquid crystal (nematic) liquid crystal (smectic) continuous translational discontinuous orientational one discrete translational ( 1d crystal ) discontinuous orientational molecules need to be long enough 5CB 8CB 7
The phases in Advanced Soft Condensed Matter liquid crystal (discotic) continuous translational discontinuous orientational ( mm: 1 rotation axis, mirror planes) liquid crystal (columnar) two discrete translational ( d crystal ) discontinuous orientational 8
Order parameters James Sethna Order parameters, broken, and topology Loosely: important variables in the respective phases More precisely: different phases different broken symmetries order parameter M: labels which broken state the material has order parameters may vary in space: M(x,y,z) is a field Major exception: gas liquid transition: no broken order parameter field: density ρ(r) Example 1: Magnet - M is the locally averaged direction of magnetic dipoles, can take values which lie on a sphere 9
Order parameters: d crystal u 1 u u 1, u must be periodic! A crystal consists atoms arranged in regular, repeating rows and columns. At high temperatures, or when the crystal is deformed or defective, the atoms will be displaced from their lattice positions. The displacements u are shown. Even better, one can think of u(x) as the local translation needed to bring the ideal lattice into registry with atoms in the local neighborhood of x. Also shown is the ambiguity in the definition of u. Which ``ideal'' atom should we identify with a given ``real'' one? This ambiguity makes the order parameter u equivalent to u + (m a,n a). Instead of a vector in two dimensional space, the order parameter space is a square with periodic boundary conditions. Example : Two-dimensional crystal - M is the average displacement field of atoms from an ideal lattice. M can take values which lie on a two-dimensional torus. Part of M is the basis a of the unit cell! 1
Order parameters: uniaxial nematic Nematic liquid crystals are made up of long, thin molecules that prefer to align with one another. Since they don't care much which end is up, their order parameter isn't precisely the vector n' along the axis of the molecules. Rather, it is a unit vector up to the equivalence n' = -n'. The order parameter space is a half-sphere, with antipodal points on the equator identified. Thus, for example, the path shown over the top of the hemisphere is a closed loop: the two intersections with the equator correspond to the same orientations of the nematic molecules in space. Example 3: Nematic liquid crystal - M is the average orientation field of molecules. The order parameter space is a hemisphere, with opposing points along the equator identified. (RP ) 11
Order parameters: uniaxial nematic Example 3: Nematic liquid crystal - M is the average orientation field of molecules. The order parameter space is a hemisphere, with opposing points along the equator identified. (RP) M n( r ) : director: unit vector along preferred direction with n= n M n ' (r ) Practical solution for the n' = -n' : Tensor! M αβ = 1 3 n α ' (r) nβ ' (r ) δα β (quadrupole tensor) 1
Order parameters: uniaxial nematic M αβ = Properties: 1 3 n α ' (r) nβ ' (r ) δα β Tr M = M αα =, M αβ = M β α S / n e z : M = S / S ( ) (rotational!) n e z : direction obtainable from diagonalization T MD = OMO (O : rotation matrix)) and n = O e z M M 3 αβ β α 1 agrees, of course, with: S = 3 cos θ 1, where cos θ = n n ' strength of nematic order: S = director representation of M : M αβ (r ) = 1 S (r ) 3 nα (r) nβ (r ) δα β 13
Order parameters: biaxial nematic c three orthogonal main axes b a We have to worry about the orientation of all three axes. Generalization of M, now with spatial and axis indices : M ijαβ = 1 3i α (r ) j β (r ) δα β δij α,β = {x, y, z} i, j = {a, b,c} diagonal form n e z : M aa ( S / +T = S / T S ) (breaking of rotational!) strength of biaxiality: T 14
Order parameters: measurement high resolution NMR I1 I proton spins H (magnetic field) a interaction of molecule with field and dipole interaction within molecule H = ℏ H (γ 1 I 1 + γ I ) ℏ γ1 γ [3( I 1 e a )( I e a ) I 1 I ] 3 a H int H int = ℏ γ1 γ a 3 M aa αβ I 1, α I,β expectation value H int detectable as a line shift 15
Order parameters: susceptibilities Magnetic susceptibilities isotropic material: M = χ H nematic material: magnetic susceptibility χ is a scalar M = χh magnetic susceptibility χ is a tensor χ ( χ n e z : χ = n e z : χαβ = χ χ ) (rotational!) δαβ + (χ χ )n α nβ molecular susceptibility χ ' αβ = χ δαβ + (χ ' χ ' )n ' α n ' β and weak correlation between magnetization of neighboring molecules (since χ 1 ): χαβ = N (χ ' χ ' 3 ) M αβ + 1 N (χ ' + χ ' 3 )δαβ in particular χ χ = N S (χ ' χ ' ) Anisotropy in magnetic susceptibility directly reflects nematic order! 16
Inhomogeneous phases: free energy response of material to external fields primarily: order parameter changes (in directions of broken )! ( Goldstone-modes, low-energy excitations ) secondary: all other values nematic: director changes (but not strength of nematic order) l a : locally, the material is a bulk nematic director variation on scale l averaging box size a strength of nematic order: S (r) S = const. (degree of nematic order is not influenced by inhomogeneities in the director!) 17
degennes and Prost, The physics of liquid crystals Inhomogeneous phases: free energy local gradient expansion: F = d r f (n α( r ), n α ( r )) condition: gradient small on molecular [averaging] scale: a n α a /l 1 symmetries: local ideality at every point r the symmetries of the idela bulk nematic should be present in the free energy density f (no more, no less!) f should be invariant under: n n r r rotations around n need at least two n 's and two 's solution to lowest order in f (nα ( r ), nα ( r )) = K1 K K3 (div n) + (n rot n) + (n rot n) Frank-Oseen free energy Ki - Frank free energy constants (material parameters!) 18
Inhomogeneous phases: principle modes of director deformation splay (d) electric field of wires and point charges splay (3d) div n rot n = torsion div n = n (rot n) n x (rot n) = bending magnetic field of wires div n = n (rot n) = n x (rot n) Kleman and Lavrentovich, Soft matter physics 19
Inhomogeneous phases: accessing the different modes via the Frederiks transition splay torsion bending