Flexoelectricity: Symmetry and Geometry

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FlexoelectricitySymmetry and Geometry

• Robert B. Meyer
• Brandeis University
• Oxford, September, 2006

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F. C. Frank
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Origins

• Discuss. Faraday Soc. vol.25, p.19 (1958) F.C.
  Frank pointed out the intrinsic coupling between
  polar symmetry (parallel to the director) and
  splay, noting that since nematics do not exhibit
  spontaneous splay, they are also not
  spontaneously polar. Analogous to the coupling
  of molecular chirality to spontaneous twist.
• PRL, vol.22, p.918 (1969) Meyer generalizes
  Franks argument to induced polar order electric
  field orients molecular dipoles, and thereby
  induces splay or, vice versa, splay induces polar
  molecular ordering and, through molecular
  dipoles, electrical polarization. Also points
  out the coupling of bend and polar order
  perpendicular to the director.

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Basic Symmetry
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Free Energy
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Sign convention
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Surface Effect

• Flexoelectric terms do not appear in the Euler
  Lagrange equations.
• Volume integral of flexoelectric terms in the
  free energy can be converted to a surface
  integral.

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Meaning of surface effectOrientation of the
director at boundaries determines the energy
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Elastic energy controls structure within the
fixed boundaries
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Some comparisons

• The spontaneous twist term does not show up in
  the Euler Lagrange equation for chiral nematics.
• In the analogous 1D problem for a chiral nematic,
  the spontaneous twist term in the free energy
  depends only on the boundary values.

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Some Warnings!

• In problems with variable boundary conditions,
  including the appearance, disappearance or
  movement of defects, be wary of throwing away
  surface terms.
• Example structure and energy of blue phases of
  cholesterics, which depend on arrays of defects
  for their stability.

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Only one flexoelectric coefficient

• Corollary of the surface effect point of view,
  for certain problems.
• More a mathematical than a physical statement.
• For intuitive understanding of effects, the
  simple splay and bend pictures are most helpful.

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Molecular models
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Molecular models

• J. Prost and J.P. Marcerou, Journal de Physique
  vol.38, p.315 (1977).

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Polarization of a circular sample
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Same argument for quadrupolar molecules
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Cholesteric flexoelectro-optic effect

• Patel and Meyer, PRL vol.58, p.1538 (1987).
• Jay Patel discovered the effect experi-mentally,
  and concluded it must be related to
  flexoelectricity since the measured rotation of
  the optical axis was linear in electric field
  (sign and magnitude). Meyer unraveled the
  geometry.

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• The work of Bouligand was crucial to
  understanding the geometry. It connected the
  arches pattern of Meyers first paper to
  oblique sections of the chosteric helical
  structure seem in electron micrographs of crab
  cuticle and other biological structures. (J. de
  Physique, colloq. 30, C4-90 (1969).
• A note to grad students read widely in your
  field. Completely unrelated pieces of knowledge
  may come together in an important way!

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Modern times!

• Bimesogens are shown to have very large
  flexoelectric coefficients.
• Coles group JAP vol.99 p.034104 (2006).
• Zenithally bistable nematic devices various
  conference references, and U.S. Patent No.
  6,249,332, (June 19, 2001) by ZBD inventors.
• Development of the chiral nematic
  flexoelectro-optical device, including polymer
  stabilization.
• Molecular and device modeling.

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Bimesogens splay
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Bimesogens bend
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Bimesogens large effect

• Argument based on long range dipole dipole
  correlation functions created by the molecular
  pairing ???

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Flexoelectro-optic effect

• Using bimesogens, the combination of positive
  splay and bend flexoelectric coefficients is most
  favorable.

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ZBND modeling switching

• L.A Parry-Jones and S.J. Elston, JAP vol. 97,
  p.093515 (2005).
• A.J. Davidson and N.J. Mottram, PRE vol.65
  p.051710 (2002).
• This conference
• A qualitative sketch for the case of equal
  positive splay and bend flexoelectric
  coefficients.

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ZBND Top electrode -
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Relaxation to defect state
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ZBND Top electrode
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Relaxation to vertical state
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Switching defective to vertical state
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Switching vertical to defective state