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Surface anchoring of nematic liquid crystals Gleeson, James Theodore 1988

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SURFACE ANCHORING OF NEMATIC LIQUID CRYSTALS By JAMES THEODORE GLEESON B.Sc,  The U n i v e r s i t y o f A l b e r t a ,  A THESIS SUBMITTED  1985  IN PARTIAL FULFILLMENT OF  THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS  We accept t h i s t h e s i s as conforming to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA March 1988 ® James Theodore Gleeson,  1988  In  presenting  degree  this  at the  thesis  in  partial  fulfilment  University of  British  Columbia,  freely available for reference and study. copying  of  department publication  this or of  thesis by  this  for scholarly  his thesis  or  her  I further agree  purposes  may It  be is  requirements  for  an  Physics  March 7  1988  advanced  that the Library shall make it that permission  for extensive  granted  head  by the  understood  that  for financial gain shall not be allowed without  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  the  I agree  representatives.  permission.  Department of  of  of  my  copying  or  my written  Abstract  The  surface  tangentially These  two  poly(vinyl to  be  a  nematic  anchoring  treatments formal)  new  Sprang,  employed The  and  The  Frank-Oseen e l a s t i c i t y s o l u t i o n o f the  Both  methods  constant has  are  and the  undergone  to  a  splay  second It  can  anisotropy;  spatial variations  employed  found  to  this  i n the  strength  a  be  of  new  the  is  believed  to  the  twisted  of  the  surface  based  on  of Deuling's  The s e c o n d y i e l d s  a  very  the  order  ii  bulk  index of a nematic transition.  and the  as  of  second  the 1972 the  The  dielectric sample  first  requires  one  verification  of  results.  sensitive  the  value  to  i s b e l i e v e d t o be due  scalar  by Yokoyama  Rapini-Papoular  method  an e x t e n s i o n  transition.  Freedericksz  both  dielectric  is  refractive  other;  is  is  measurements  these measurements,  method  latter  buffed  potential.  both  latter  the  coefficient  requires  be  The  studied.  and  was o r i g i n a l l y d e v e l o p e d  anchoring  effective  SiO  importance  measure  the  on  is  industry.  theory.  based  5CB a l i g n e d  treatments  solution.  splay Freedericksz  form o f the  crystal  evaporated  possible  first  yields  potential.  functional  of  crystal display  anchoring p o t e n t i a l .  anchoring  and  from  liquid  substrate  obliquely  deposited  are  nematic  glass  are  treatment  liquid  van  the  on two d i f f e r e n t  Two m e t h o d s  and  of  parameter.  to not  that  method or  the The  of  the  allowing for  Table  of  Contents  Abstract L i s t of Figures Acknowledgement  i i iv vi  Chapter I INTRODUCTION TO NEMATIC LIQUID CRYSTALS AND SURFACE ANCHORING L i q u i d C r y s t a l s and N e m a t i c O r d e r E l a s t i c i t y Theory E l e c t r i c Field Effects Birefringence Surface Anchoring Chapter I I THEORETICAL CONSIDERATIONS Theory o f the F r e e d e r i c k s z T r a n s i t i o n Flexoelectricity E x p e r i m e n t a l O b s e r v a t i o n o f the F r e e d e r i c k s z E f f e c t of F i n i t e Surface Anchoring The Y o k o y a m a - - v a n S p r a n g M e t h o d Extended D e u l i n g Method C h a p t e r Summary  1 1 4 5 6 7  Transition  8 8 13 17 20 21 26 31  Chapter I I I EXPERIMENTAL APPARATUS Sample P r e p a r a t i o n Temperature C o n t r o l Electronics O p t i c a l Measurements Summary o f E x p e r i m e n t  32 32 37 39 41 43  C h a p t e r IV DATA A N A L Y S I S AND EXPERIMENTAL RESULTS Reduction M a t e r i a l Constants R e s u l t s : M a t e r i a l Constants Surface Anchoring Results Yokoyama-van Sprang Method Extended D e u l i n g Method  44 44 47 49 55 55 61  Chapter V DISCUSSION AND CONCLUSION C o m p a r i s o n w i t h Yokoyama et al's work E x t e n d e d D e u l i n g method o f M e a s u r i n g A n c h o r i n g P o t e n t i a l Conclusion  75 75 76 79  Appendix EFFECT OF F I N I T E SURFACE ANCHORING ON THE THRESHOLD F I E L D  81  References  85  iii  L i s t of Figures 1.  A schematic representation of a nematic with various degrees of order  2.  A schematic representation Freedericksz t r a n s i t i o n .  3.  Director angle vs c e l l reduced displacements.  various  14  4.  P l o t of reduced capacitance vs reduced displacement based on the Deuling c a l c u l a t i o n .  18  5.  P l o t of reduced phase vs reduced based on the Deuling c a l c u l a t i o n .  19  6.  Scale drawing of the sample c e l l .  36  7.  Schematic of phase measurement setup.  42  8.  An example of phase vs voltage data.  45  9.  An example of capacitance vs voltage data.  46  10.  Birefringence as a function of temperature  50  11.  Dielectric constants temperature.  12.  E l a s t i c constant anisotropy as temperature.  13.  Splay e l a s t i c temperature.  14.  Bend elastic temperature.  15.  Extrapolation length as a function temperature f o r SiO treated surface.  of  position  for  splay  displacement  51  of  52  of  53  of  54  of  56  16.  Out of plane anchoring strength as a function of temperature for SiO treated surface.  57  17.  Extrapolation length as a function temperature for buffed PVF surface.  of  59  18.  Out of plane anchoring strength as a function of temperature for buffed PVF surface.  60  19.  T i l t angle vs reduced voltage for surface.  62  constant  a  the  of  constant  as  liquid  function a  function function  as as  iv  function  SiO  treated  20.  S u r f a c e t o r q u e as a f u n c t i o n SiO t r e a t e d s u r f a c e .  21.  A n c h o r i n g p o t e n t i a l as a f u n c t i o n for SiO t r e a t e d s u r f a c e .  22.  T i l t a n g l e as a f u n c t i o n o f r e d u c e d SiO t r e a t e d s u r f a c e : worst case.  23.  S u r f a c e t o r q u e as a f u n c t i o n o f t i l t SiO t r e a t e d s u r f a c e : worst case.  24.  T i l t angle vs reduced v o l t a g e f o r s u r f a c e : f i t to phase d a t a .  25.  S u r f a c e t o r q u e as a f u n c t i o n o f t i l t a n g l e SiO t r e a t e d s u r f a c e : f i t to phase data.  for  68  26.  Anchoring strength vs treated surface: results  temperature for o f ED method.  SiO  70  27.  T i l t angle r u b b e d PVf  of reduced  voltage for  71  28.  S u r f a c e t o r q u e as a f u n c t i o n o f t i l t r u b b e d PVF s u r f a c e .  29.  A n c h o r i n g p o t e n t i a l as a f u n c t i o n f o r r u b b e d PVF s u r f a c e .  30.  A n c h o r i n g s t r e n g t h vs temperature PVF s u r f a c e : r e s u l t s o f ED method.  31.  Surface torque as function showing the dependence on a n i s o t r o p y used i n f i t t i n g the  as a f u n c t i o n surface.  of t i l t  v  angle  for  63  angle  64  voltage for  65  of t i l t  angle  SiO  for  66  treated  67  angle  for  72  angle  73  rubbed  74  of tilt angle the dielectric data.  77  of t i l t  for  Acknowledgment  This  work  guidance  could  and  have  support  Palffy-Muhoray. his  not  been  of  attempted  my  thesis  Much o f what i s  without  the  supervisor,  original in this  continued Peter  Dr.  thesis  stemmed  from  suggestions. Most  Liquid  of  experiments  Crystal  indebted Prof.  the  to  J.W.  Institute  Institute  the  members  Doane  and  described at  of  the  countless  am  Council  of  wrought  of  and  throughout  indebted  University Much  ideas  to  Canada  my the  and  through  custom  to set  performed I  especially  Director,  Prof.  the  M.A.  of the I n s t i t u t e  suggestions  in  addition  at  am  the  deeply  Director, Lee.  up my e x p e r i m e n t s ,  The members  The  equipment  p r o v i d e d me to  unending  stay. Natural the  Sciences  Department  f o r p r o v i d i n g me w i t h the  were  University.  Institute,  p r o v i d e d me w i t h s p a c e  hospitality I  the  State  Associate  w i t h w h i c h t o p e r f o r m them. with  Kent  herein  and  of  Engineering  Physics  at  Research  Kent  State  funding.  equipment  required  for  t h e e f f o r t s o f my p r e d e c e s s o r ,  the  S.W.  experiment  was  Morris.  My e f f o r t s i n t h e l a b o r a t o r y w e r e g r e a t l y a i d e d by t h e p r e s e n c e B.J.  Frisken  properly I  whose  ideas,  assistance  and  patience  were  of  never  acknowledged.  shall  always  be  indebted to Becky,  t o o k an u n d e s e r v e d b a c k s e a t  to t h i s  vi  who  thesis.  so  many  times  willingly  Chapter  I  Introduction Surface  Nematic L i q u i d  Crystals  and  Anchoring  This  thesis  crystals.  i s concerned  Surface  crystals was  to  with  interest  by  f o r many  surface  anchoring  surfaces  discussed  anchoring  with  subject  to  and  liquid  crystal  not  anchoring  describes  Rapini  is  surface  the  various  Papoular  in  interaction  1969 ,  understood  nematic  treatments.  researchers  well  of  and  1  today.  and  of  liquid  The is  The  there  liquid  subject  an  area  of  mechanism  of  is  an  ongoing  23 debate  over  the  widely  accepted  Furthermore, on  surface  known  technique  the  major  treatment  surface  necessitates crystals; surface  role. of  The  is  anchoring  anchoring;  measuring  of  liquid  aim  measuring  this  surface to  d i s c u s s i n g the done  in  the  potential  crystals  as  well  thesis  is  surface  section  i s defined at  the  to  method  Describing  end  of  surface  of  of  rely  employ a  novel  anchoring liquid  chapter.  this  a  measuring  nematic  this  no  anchoring  p r o p e r t i e s on  basic properties of next  (LCD's)  firstly  new  is  properties.  displays  anchoring a  there  anchoring  i n which  present  potential.  '  surface  operation,  of  secondly  anchoring  this  of  for their  and  first  surface  types  treatments  a  of  technique  certain  plays  surface  models  The  chapter.  Liquid Crystals and Nematic Order  Liquid and of  crystals  crystalline liquids;  they  are  solids. adopt  states of For the  matter  intermediate  the most p a r t ,  they  between  exhibit  the  shape o f whatever v e s s e l c o n t a i n s  1  liquids fluidity them.  Figure  1.  A  schematic  with  various  representation degrees  of  of  order.  2  a  nematic  liquid  crystal  However many  they  different  the  symmetry  crystal of  do n o t show t h e i s o t r o p y o f o r d i n a r y  they  phases  order  common  the  uniaxial  the  o r do  exhibit  That  long  correlations  are  i n the plane quantity  frames  nematic  crystal phase,  be  t h e amount  "point" i n which  phases,  shows  i s the energy  o f one a x i s i n  direction.  common  type  molecular  the lowest  symmetric  to that  liquid  This  of  type,  the o r i e n t a t i o n  to  perpendicular  to describe  of this  i s i n a common  observed  of  the molecules  a l lliquid  on t h e a v e r a g e  majority  t o some d e g r e e .  common p h a s e of  There are  a r e c h a r a c t e r i z e d by  The  order range  In this  reference  nematics  proper  order  phase.  i s one i n w h i c h  inversion  and they  not exhibit.  orientational  T h e most of  nematic  molecule's  phases  i s t o s a y , on t h e a v e r a g e  amount  uniaxial  the  do  direction.  least  state  crystal  i s c h a r a c t e r i z e d by  directions. a  liquid  liquids.  with  Note  that  respect  direction.  of orientational  to Thus  order i n  2 a  uniaxial  angle  between  usually the  phase  1  i s molecular  the molecular  defines  second  between  the order  Legendre a n d 0,  degrees More  1 depicts  axis  average  polynomial.  S = 0  o f c o s (6) where  and the p r e f e r r e d  parameter  This  choice  where  is  in S  being  results  s c h e m a t i c a l l y a nematic  ordered  liquid  i s the One  to the unordered  to a perfectly  6  direction.  S ( T ) as <?^(cos(9))>  corresponding  and S = 1 c o r r e s p o n d i n g  Figure  is  nematic  (isotropic)  nematic  crystal  phase.  with  various  i n a uniaxial  nematic  o f order. g e n e r a l l y , the degree  described  by  a  spatially  o f order varying  present tensor  order  parameter  Q „,  where  1.1 and  3  n n S(T) to  is referred  as  the  which  to  as  director.  molecules  the  n(r)  are  scalar should  pointing  1  =  1.2  order be  parameter  thought  averaged  over  of  and  as  an  n  the  is  referred  direction in  infinitesimal  volume  3 d r  centered  at  r.  The  infinitesimal  volume  must  still  contain  many  molecules.  E l a s t i c i t y Theory  A  In has  the  the  bulk lowest  crystal  be  this  regions  with  only  nematic by  is  where  in  one  than  the  region in  state,  theory.  there  i s an  a l l  a  in as  Since of is  for  nematic  that  be  a  as  region there  in a  is different  higher  energy.  subject  says  that  f r e e energy  must  from  the  amount  Frank-Oseen a  density  in  nematic  The  of in  liquid  preferred  region,  parallel  the  the  must  this  theory  i n the  true  there  not  state  Elasticity  not  of  increased  increase  is  Somewhere  direction.  has  i s uniform,  another,  i t must be  energy  s t a t e , where n ( r )  This  5  " d i r e c t o r s " are  preferred  the  nematic,  If  d i r e c t i o n changes.  ground  elasticity  undistorted  density.  different  one  which  the  energy  phases.  direction which  nematic,  distorted  that  depends  A  on  the  spatial  elastic  free  change  energy  in  the  density  5  director .  Since  n.  This  the  increase  symmetry  of  the  is  the  nematic  d A  dictates as  that  well.  expandable  A  n  must  The  be  the  elastic  in  same  free  invariants  as  -n,  energy  formed  from  5^  must  density  have must  derivatives  that  symmetry  therefore  of  n  that  be are  A  u n c h a n g e d when n of  these  changes  invariants  sign.  which  I t can  contains  4  be  shown  only  first  that  any  combination  derivatives  can  be  expressed V»n,  twist:  energy to  i n terms o f  as  n«Vxn,  have  and  is quadratic elastic  three  of  i n these  i n the  To  Thus  5  first  These  the  the  are  splay:  elastic  free  coefficients referred  are  free  director field  These  order,  deformations, with  . (i=l-3)  force.  inhomogeneities  deformations.  b e n d : nxVxn.  constants  units  canonical  proportional  energy  i s given  to  density  S  and  due  to  by: A  5  - d  Note  that  allow  to  be  2  spatial  be  replaced  + K  2  2  correct,  variations by  + K (n«Vxn)  2  1  strictly  for  should  K (V«n)  an  in  2  1.3  3  the S;  elastic  (nxVxn)  theory  and  free  of  the  deformations  elastic  energy  should  energy  density,  in  density  which  the  A  entropy latter  change is a  effects  to  subtle  are  additions  due  variations  point,  inherently  to  in  because  one  contained  deformation theory  S  in  and  could the  n  is  claim  elastic  are  beyond  this  work  considered.  the  that  any  entropic  constants.  scope  of  The  These  this  thesis.  E l e c t r i c F i e l d Effects  Another is  their  medium, in  the  not an  aspect response  the  an  applied  d i r e c t i o n as  case.  applied  this  to  to  p o l a r i z a t i o n created  same  the  relevant  In  field  tensor  general, !  has  that  by two  a  the  electric by  nematic field.  applied In  electric  dielectric  r  an  field.  principal  of  In  field  tensor  e.  e  and  1  r  In  crystals  an is  anisotropic  displacement  values  liquid  isotropic necessarily  media  B  that  is  is related  to  uniaxial  e ;  the  nematics  dielectric  2  A  constants is  parallel  defined  those  used  as  the in  and  perpendicular  dielectric the  to  anisotropy.  experiments  n  respectively.  I n most n e m a t i c s  described  5  Ae  in  this  =  -  (including  thesis)  Ae  is  positive.  I f one  proportional situation applied  to  the  i s where  field.  perhaps  neglects scalar  a  the d i r e c t o r  corrections,  parameter.  i s not  can apply  deformation.  the part  field  order  Then the f i e l d  induce  including  local  This  The  everywhere  a torque  c a n be  o f the e l e c t r i c energy  Ae  then  is  interesting  parallel  to  to the d i r e c t o r  described  density  that  an and  formally  by  d e p e n d s on t h e  A  director  n as a term  the e l a s t i c energy  density.  Birefringence Nematics, In  general  depend  on  being  anisotropic  the v e l o c i t y the angle  travel  at  a  director  speed  will  of  light  between  wave a n d t h e d i r e c t o r .  by d e f i n i t i o n ,  the e l e c t r i c  Light  c/n , e  travel  passing  polarized  and  at  light  a  are also through  field  difference  and  ordinary  An = n  - n e  c/n .  the  scalar  director  order will  considering in  refractive  at  the magnitude  the d i r e c t i o n  vector  o f the  light  n  and  indices  n  will  to the are  the  o  respectively.  and  The  i s proportional  to  4  parameter.  travel  will  e  i s the birefringence o  nematic  perpendicular  o  extraordinary  a  p a r a l l e l to the d i r e c t o r  polarized  speed  birefringent.  Light a  polarized  c/n(V>)  speed  o f the v e l o c i t y  parallel  to n  .  a t some  n(V>)  vector  a n d component  may with  c/n  o  angle be  i> t o t h e found  component  i n the  by  c/n e  direction  A  perpendicular  t o n: n n  n(tf)  -  6  /  2  2  n sin  6  (V>)  °  2  2  + n cos  (V>)  1.4  Surface  Anchoring  Often,  a  uniformly ground  aligned  state  difficult of any  liquid  attain.  angle  On  between  s i t u a t i o n of  alignment normal.  in The  chapter The  the  are  conditions  classified  i n contact  with  be  the  a  nematic  should  present at  demand  the  not  in a  be  sample  surface  by  the  the  type  surface.  of  and  on  normal  (homeotropic)  a  defines  the  surface,  one  this  the  surface  director  of  inducing  in  thesis is  "easy  the  is  one  or  the  angle"  first, to  alignment  can  tilted as  undistorted  perpendicular  tangential  alignment  Namely,  (homogeneous),  interest for  the  of  i f a  this  thesis  torque  i s how  is applied  o r i e n t a t i o n change? of  the  <f> b e t w e e n  easy  boundary  to  sample  director field  will  the  state.  tangential the  are  surface discussed  III.  interaction angle  the  stated  aligned  the  methods  question  will  was  d i r e c t o r and  which  specifically,  the  such the  Since  this  application  fields.  nematic  tangential  alignment.  or  uniformly  conditions  in a  experiment  However,  d e p e n d s on  surface  create  how  a  externally applied  expected  in  sample.  above,  to  a nematic  The  The  crystal  angle,  angle.  This  potential  W^).  o f W(<£ ) i s t h e  the that  d i r e c t o r and function  function The  is  to  It  d i r e c t o r with  the the  must  strong  be  referred  the  is  surface  to  of  the  next  7  chapter.  a  By  minimum as  that  is  surface. a  anchoring?  d i r e c t o r at  assumed  t h e o r e t i c a l foundation  subject  i s the  the  the the  surface, energy  function the  when  More  of  the  definition 4>  is  the  of  of  easy  surface  anchoring  f o r measuring  properties  Chapter  II  Theoretical  Theory  of  The field  the F r e e d e r i c k s z  Freedricksz  or  splay  transition  electric field.  Freedericksz  transition picture this  Transition  is  a  reorientation  of  the  director  i n a n e m a t i c l i q u i d c r y s t a l due t o t h e i n f l u e n c e o f an e x t e r n a l  magnetic the  Considerations  is  of  interest  t r a n s i t i o n i n d u c e d by  depicted  implies  The c a s e  schematically  the " r i g i d anchoring  in  One  regardless  would  like  mathematically. separated  to  Thus  first  z = d.  a  figure  2.  assumption".  the  at  plates. plate  d.  The  the  plates  Let  is  transition  Note Rigid  is  the  the plane  The p l a t e s u r f a c e s  and  that  in  z = 0 and  be  the  in  figure  the  second  2  plates,  crystal  constrained  normals  this  anchoring  liquid  is  The  field.  depicted  nematic  plate  is  f i x e d p a r a l l e l to  of the a p p l i e d  the  thesis  a s y s t e m w i t h two i n f i n i t e p a r a l l e l  distance  aligned  parallel the  consider  Consider  by  homogeneously  to  of strength  this  an e l e c t r i c f i e l d .  c a s e means t h a t t h e d i r e c t o r o n t h e s u r f a c e  that surface  for  k  to  is lie  direction.  is  the  a r e s u c h t h a t t h e d i r e c t o r n on t h e  plane plates  A  is  p a r a l l e l to  potential  i .  The  difference  electric field  between  the  will  be  .plates.  a p p l i e d by  The  creating  resulting  a  electric  A  field  can have  no  component  other  than  the  k  d i r e c t i o n by  symmetry.  A  Also  from  symmetry,  n and ^  can  be A  that there w i l l  no be component  functions  of  z  only.  Assuming  A  of n i n the j  d i r e c t i o n , the  director  A  n c a n be w r i t t e n  as: A  n -  A  cos($)i  8  A  + sin(<^)k.  2.1  X  igure 2.  A Schematic  representation  Freedericksz transition. 9...  of  the  splay  where the  <$> =  <j>(z) .  director  potential  field  Hence then  the  change  Thus  this  i n this  is  problem.  As  as  potential, "free  stated  analysis  as w e l l  the  as  i n the  When  the  dF,  field  to  I,  one  So  polarizable work  potential i f T  are  can  think of entropic  a n i s o t r o p y Ae.  the  dB«^. It is  - TS;  the  a  present terms  constants,  Thus  is  constant.  minimized  elastic  in  then  = J  dW  F = &  B  and  by  medium  JdB«^.  +  and  i s that  be  c o n t a i n e d i n the  dielectric  thermodynamic  i s used  does  hence  displacement  i s d§ = TdS  i s zero  and  throughout.  a  thermodynamic  energy"  i n Chapter  being  use?  4>(z)  the  electric  displacement  i n energy  construct a  find  homogeneous  field,  change  function  must  constant  is to  the  one  electric  potential  differential  small  the  electric  to  determine  at  displacement  an  convenient  this  reason  the  into  to  in equilibrium,  thermodynamic  introduced  order  i s a minimum  The  law,  which  n  F which  temperature. Gauss'  In  in  and  portion  of  the  A  free  energy  that  d e p e n d s on F  Where  5  is  the  the  the  J  -  energy  director  field  £ ? +  i $.t  density  of  d  j  n is:  dz  2.2  a  deformed  in  equation  nematic  liquid  d crystal  due  to  deformations  geometry o f t h i s  =  d  whose law  discussed principal  the  given  1.3.  In  the  problem:  ?  As  as  in  (K C O S ( * ) + K 2  I  chapter  axis  electric  ; 2  1  I,  the  2  J L°  is parallel  displacement  is  dielectric  to the  the  2.3  [f|  sin (*)]  3  z  constant  director.  areal  charge  is  a  tensor  Again  by  Gauss'  density  a  on  the  A  plates  and  the  only  one  can  is parallel  non-zero write  to the p l a t e  component  \ = Ek.  of \  Therefore  normal,  i s along the 10  i.e. 3 the  product  = Dk.  plate D^E*  By  normal  can  be  symmetry as  well;  written  as  DE,  and,  D - e t  since  o  written  E,  the  energy  density  1  D 0  i s t h e 3-3  rotated e  component  so  that  its  2  2.4 33  o f the d i e l e c t r i c principal  2  e s i n Y(<j>) + e c o s (6). l 2  33  Equation  equilibrium  value  o  of  1  2  2  the  constant  D and g i v e n  boundary  conditions.  this  case,  consider  i s held  the appropriate  4>(z)  function  function  the p l a t e s  along  n; as  dz  J  dz  2.5  2  i s that  between  been  -\  s i n (<^) + e c o s ( $ ) )  everywhere  one may  lies  (0)J  (<£)+K s i n  director  Alternatively,  i t has  c a n t h e n be w r i t t e n  2  (e  after  axis  2  e  tensor  2.2  M  cos  The  c a n be  as  z e e  £  contribution  33  which  that makes  specifies F  a  a s i t u a t i o n on w h i c h  fixed  instead  f r e e energy  of  to use  minimum  the  the at  voltage  the displacement.  In  is:  „d 2.6  The  p o t e n t i a l difference V i s given  by:  d V  j* E dz =  \ 0  €  o  £  2.7 £ (e s i n ( ^ ) + £ cos (<£)) 2  33 0  and  the f r e e  energy  dz  D dz o  becomes:  11  1  2  2  -  ] f  [ Cn =~ <««,.in*(«) ,  i  V  2  2.8 dz 2  2  « (c s i n (^) + e cos (<b)) o l 2  o  The Euler-Lagrange equation,  whose solution  gives  the equilibrium  configuration of the director f i e l d has been obtained from eq. 2.8 by Deuling.  It  6  is more readily obtained from eq. 2.5  i t is found that ^(z)  . In both cases  is everywhere zero for displacements  less than  a threshold D given by: th D  -1 /  th  d  °  v  2  2.9  1  7  or for voltages less than V : th D d . J±€ t o 2  V th Where 7 - e  /e 1  - 1.  2  larger than V ), th  _  /T  * / — L V € € 7 o2  2.10  At displacements  larger than D (or voltages th the director bends i n the direction of the applied  f i e l d at the center of the c e l l and s t i l l remains aligned p a r a l l e l to the plates at  the boundaries.  for  of  the  finite,  case the  anchoring.  infinite  Note that this  anchoring strength;  threshold voltage  will  threshold voltage  is  if  is  the  depend on the  This dependence is discussed  anchoring  strength  of  the  in the appendix. The change  in threshold voltage due to this effect however is not expected to be significant; 3  shows  reasons for this are also given i n the appendix.  the  displacements.  angle  <f>  as  a  function  of  z  for  various  Figure reduced  Note that while D may be a more convenient quantity to  12  work  with  on  paper,  experimentally. eq.  2.7  by  the  but  The  is  much  important  i s d e r i v e d from  charge  definition  V  of  density  Q = CTA (where A  fact  the  c a p a c i t a n c e C:  C  i s the p l a t e  to  relationship  the  on  easier  that  the  Q/V,  area)  of  enables  one  control  and  V  displacement  This  and  and  D  between  plates.  =  measure  coupled  is  is  not  given  with  s u r f a c e charge  the  density  to w r i t e 2.11  The  implication  electric with  is  obvious,  displacement  D,  i f one  which  knows  i s the  C  and  V,  one  knows  more n a t u r a l  q u a n t i t y to  that  on  the work  mathematically.  Flexoelectricity  Certain  terms  configuration rectified  in  were  that  the  free  absent  energy  from  situation  in  Deuling's 1978.  1972  The  7  depend  the  analysis,  missing  director  however  6  terms  are  due  he to  g flexoelectricity. then  calling  become a  the  nematic  and  Meyer  i t piezoelectric  accepted  term  molecules a  are  pear  can  be  non-zero  the  nematic  the  deformation  is  anti-parallel.  a  effects.  bend  have  shaped  f o r b e n d ) as  a  as  T h i s produces  (not  shape for  with  a net  13  twist)  dipole  moment,  these  since  put, i f  (the  and  This  has  classic  banana then  i s because  properties locally  polarization.  1969,  deformation,  splay  parallel  local  a  in  Simply  anisotropy  polarization.  align  phenomenon  phenomenon.  a permanent  molecules  i f they  this  Flexoelectricity  splay  molecules  well  deformed,  or  certain  flexoelectric  energy  described  to d e s c r i b e t h i s  i s s u b j e c t to  i f i t s molecules  examples  first  can  rather  shaped there when lower than  In g e n e r a l ,  z/d  Figure  3.  Director  angle  vs  cell  displacements.  14  position  for  various  reduced  nematics in  are  the p l a n e  that  not  perpendicular  symmetry,  the r e g i o n  f e r r o e l e c t r i c because  of  than  becomes  electrostatic  terms  energy  creates.  The  have  reflection  If  a deformation  the d i r e c t o r . possible  the d e f o r m a t i o n .  i n two a d d i t i o n a l  it  it  to  they  to  have  the  second  Flexoelectricity,  i f present,  the  The  the  interaction  polarization  with  external  p o l a r i z a t i o n as  f i r s t c a l c u l a t e d by Meyer, A  of  the  field.  A  in  is  the  the  field  flexoelectric  The  flexoelectric  is:  A  A  P" - e ( V « n ) n - e n x V x n . 1  where  e  and e 1  to  the  the  flexoelectric coefficients;  moment  anisotropy  of  flexoelectric  component that  with  the  in  the  the  z  molecule  =  are  related  as  well  as  as  constants  its  permanent  respectively.  then  cannot  must  of  the  the  energy  there  is  write  of  The  2.13  produce  down  the  an  electric  translational  "self-energy"  flexoelectric polarization.  g  hand:  the x d i r e c t i o n because  contributions  well  (e +e ) s i n ( 0 ) c o s ( ^ ) | ^ 1 3 dz  polarization  One  as  and b e n d e l a s t i c  the problem a t  direction.  polarization  of  and the s p l a y  P  in  they J  the geometry  The  2.12  3  3  shape  dipole In  are  in  results  first  flexoelectric polarization  is  breaks  ferroelectricity  i n the f r e e energy d e n s i t y .  of  symmetry  the  free  of  the  external  e l e c t r i c free  field  symmetry  energy  anew  flexoelectric  f i e l d coupled energy  density  to is  now: ?  E  Using  a  constitutive  - e e 2  relation:  o  D -  E  2.14  2  33  e e E + P , 0 33 z  the  above  can  be  written:  5E  =  l 2£  D  £ o  33  15  2  - 2PD  +  P  Z  2.15  The in  first  term  i s the e l e c t r i c  t h e medium;  field  to  the  self-energy density the  the  of  the  outside  As  integrated  comparing  with  One h a s  of  e q . 2.7,  -  one  closer  can integrate  (e  dz  2  2  examination  of  this  free  the  energy  expression i t c a n be  the f i r s t  term  over taken  c a n be  _d P  £  0  Upon  The  is  then:  - DV  E  applied  third  the c e l l ,  d F  the  polarization.  throughout  polarization  the e x t e r n a l  and  the i n t e g r a l  i s homogeneous  immediately.  of  polarization  i s then  the i n t e g r a l ;  due t o l i n e a r  coupling  flexoelectric  area  D  energy  i s the  flexoelectric  per unit  cell.  second  free  £  0  i  +  dz  2  £  33  the second  integral  2.16  £  0  33  i n this  expression,  i t as w e l l :  x  +  e  )  3  H  sin(^)cos(^) •  / A\  2  ,  £ s i n (0) + £ c o s 0  1  (e + e  1  1  2  dz  <$Z  r ±\  Z  (cp)  2  ) 3  log[  £ A£  e sin (^)  +  2  £ cos (<&)]  2.17  2  2  o  If  we  define  this  q u a n t i t y a s V , we  can w r i t e  the e l e c t r i c  energy  p  per  unit  area again as:  „d F  = e  - D(  V  - 2V  2  ) + p  (  e  i  +  2 £ £ o  Note  that  one  potential  flexoelectric  expected  the  V  p  across  vanishes  term energy  3  2  )  2  2  left  (I + ysin (4>) 2  because  cell.  In  ^ ( 0 ) = 4>(d) ,  2  dz  2.18  )  i s then the  d £ dz  t o change  rigid and  the  anchoring the  only  i s the s e l f - e n e r g y .  per  t o be minimum  the  r  s i n (4>)cos (4>)  o f the f l e x o e l e c t r i c i t y  difference  assumption  Thus  effect  e  unit  area  of  i n the e q u i l i b r i u m  is:  16  the  bulk  nematic  configuration  which  is  at constant D  F  „ l+/c s m  U  =  _ L A s i2 n ( ^ ) c o2 s (<t>) 0 $  (0)  +  2  1+7  D  Where  =  K /K  minimised  at  constant  sample.  The  have  to  K  yet  sections  3'  - 1  D to  of  <=  determine  the  director  finite  will  of  be  capacitance  increase  D/D  anchoring  layer  as  such a l a y e r  the v o l t a g e is  will  given  1  <0(z)  sharply.  ; this  remain  will  remains  constant  become  Using  been is  must  in  is  be  in  the  assumed  the  last  two  change  across  non-zero  Deuling's  calculated  as  shown i n f i g u r e  4.  6  a  it  in  the  bulk  is  increased.  by:  dz £ £ o 33  A (£(z)  as  2.20  constant at  C  at zero,  o  abruptly  results, function  the of  and  = f e A/d.  0  c = C / h a s  F  Transition  r  D,  2.19  everywhere  examined  the F r e e d e r i c k s z  below the t h r e s h o l d ,  threshold  dz  e ) /K £ £ . 3 ' 1 o 2  the  when  this  1 C D is  +  1  t r a n s i t i o n c a n be o b s e r v e d  a nematic  The c a p a c i t a n c e o f  If  (e  dz  chapter.  Freedericksz  properties  )sin(^)cos(4>)ir  1 3  A  established;  this  (e +e  conditions  Experimental Observation  The  2D  and  boundary be  of  1  -  2  s i n (<£)  hence  At  the  C  will  2  and  hence  reduced reduced  capacitance displacement  th In  addition,  the o p t i c a l p r o p e r t i e s  an a n a l o g o u s m a n n e r .  The  and the o r d i n a r y  ( h e r e a f t e r r e f e r r e d t o as 17  rays  phase  of such a l a y e r w i l l  d i f f e r e n c e between  the  change  in  extraordinary  t h e phase)  is  written  F i g u r e 4.  P l o t o f reduced c a p a c i t a n c e vs reduced based on the D e u l i n g c a l c u l a t i o n .  18  displacement  F i g u r e 5.  Plot  o f reduced phase vs reduced displacement  on the D e u l i n g c a l c u l a t i o n  19  based  2.21  o where  n(4>) i s t h e q u a n t i t y  wavelength below  o f the l i g h t .  the t h r e s h o l d ,  transition.  A  and w i l l  plot  i n e q . 1.4  The p h a s e  5/6  of  i s shown in  calculation)  defined  will  be c o n s t a n t  decrease VS  q  D/  and A  sharply  t h  at 5  a t the  (again  n  i s t h e vacuum  based  q  = 2?rdAn/A  Freedericskz on  Deuling's  5.  figure  Effect of F i n i t e Surface Anchoring  The there to  subject  of this  i s no r i g i d  measure  parallel Finite  just  thesis  anchoring how  i s to look  on t h e p l a t e s ; i n d e e d ,  strongly  energy d e n s i t y  anchoring  ( e q . 2.19) s i m p l y  a r e thus  respectively. anchoring the  surface  and  the true  6 i s the easy  in  general  Rapini-Papoular any  surface  energies 1  angle  (zero  at z = for  form  anchoring  -  0.  small  i s merely energy.  - W  treatment.  20  few  i n 1969  that  the d i r e c t o r a t  potential: 8)  one  2.23  Z  could  term  authors  a n d <f>  considered) say that  deviations,  the f i r s t A  z = 0 and z = d  i . e . that  i n t h e geometry  enough  free  2.22  on p l a t e s  (0 -  Indeed,  l i e  o f the terms:  (R-P) p r o p o s e d  harmonic  when  to  f o r i n the bulk  by the i n c l u s i o n  e n e r g y was s p r i n g - l i k e ;  s u r f a c e was b o u n d b y s i m p l e  d i r e c t o r angle  accounted  Papoular  W(<£) Here  i s constrained  , W (0)5(z-d). 2  the anchoring  Rapini  above  the main t h r u s t i s  methods o f s u r f a c e  c a n be  W U)6(z) 1 and  the d i r e c t o r  to the p l a t e s f o r d i f f e r e n t surface  a t the problem  and  this i s  that  the  i n the expansion have  is  proposed  of  other  forms  for  the  surface  anchoring  energy 2  as  to  which  limiting The  is  case  of  constant  The  correct  of  how  strongly  the  i s r e f e r r e d to  Yokoyama  - van  as  i s an  However, i f the  in  the  director the  Sprang  '  forms  proportionality  there  ongoing  debate  3  one.  f o r a l l proposed  constant  measure  the  and  the  R-P  deviation  R-P  form  i s confined  anchoring  angle  is to  form  in  a  is  is  a  small.  any  case  surface.  a  This  strength.  Method  g Yokayama the d  and  anchoring  .  This  van  Sprang  strength  (Y-vS)  based  p a r a m e t e r may  be  on  developed  the  thought  a  "surface  of  as  how  method  of  measuring  extrapolation f a r one  must  length":  extrapolate  e  <j>(z) i n t o The  to  Y-vS  across  the  regions  method that  quantities  relates  cell to  z. <  to  the  the  the  0  and  d  capacitance  phase.  surface  z >  The  to  reach  of  a  cell  analysis  extrapolation  has  the  point  and  that  been  4> -  the  voltage  relates  developed  first the  second  Other this  a  are  is difficult  the  their  reviewed f a r as  first  the  flexoelectric  analysis.  The  basis  This this  i n t e g r a l changed  was  to  The  anchoring  c a l c u l a t i o n has  not  analysis that  here.  terms  i s the  form  infinity.  If 21  results  be  one  is  The reason  the  concerned. later  attraction  changing  two  the  in  that  essence  10.  of  2.19  of  discussed  added  without  in ref.  for that  strength  the  done  i s equation  and  by  10  approaches. '  intuitively  calculation will  add  the  different  follow  second  semi-infinite cell,  the  is  to  using  The to  of  10)  same as  of  chapter. able  collaborators  (ref.  results  is  of  his  ( r e f . 9)  analyses  one  and  these  twice 9  Yokoyama  0.  the  bulk  free  energy  of  with  the  upper  limit  of  then  changes  the  variable  of i n t e g r a t i o n to £ = Dz , the energy can be written as F = D J [<f>] , where J i s a functional of <j> without e x p l i c i t D dependence. that 4>(z)  implies  i s determined uniquely  by  £ = Dz,  This  i . e . one  can  write =  4>(z)  6 (Dz).  2.24  However, since we are interested i n the f i n i t e anchoring case, i . e . the case where 4>(0) = 4> >* 0, and since we expect '<f>  to be  1  of D,  function  the function 6 must have a more complicated dependence on D  than indicated above.  Furthermore, the a d d i t i o n a l D dependence must  be such that when D = 0, we  recover the undistorted  configuration.  One can see that on these bases, and also on dimensional grounds, the a d d i t i o n a l D dependence must be of the form <(>(z)  =  9  (D(z  +  d  ))  .  2.25  e  I t i s then obvious that the quantity d  i s indeed an  extrapolation  6  length, r e l a t e d d i r e c t l y to how to l i e i n the  plane of the  strongly the d i r e c t o r i s constrained  plates.  Note also  that d  function of D. One  would  may  be  a  e  like  to  relate  the  quantity  d  to  experimentally  e  measured q u a n t i t i e s .  R e c a l l the expression  for the phase, eq.  i n a s e m i - i n f i n i t e geometry, i . e . i n the l i m i t d -» °°.  2.21,  I f one changes  the v a r i a b l e of i n t e g r a t i o n to <f> and uses equation 2.25,  we can write  t h i s as: 2  .17  2TT  [n(tf) - n ]  AD  d0  d*  2.26  l  9  1  i s the inverse function of 6; ^  i s the d i r e c t o r angle at  and we have used the f a c t that f a r enough away from the region  z=0, of  d i s t o r t i o n , <f> ~ rt/2 (Later i n t h i s chapter i t w i l l be shown that t h i s 22  approximation interested  is  not  in is d  always  desirable).  extrapolated  The  t o D = 0.  In  quantity  order  to  we  are  s e e how  S  e varies  with  D  f o r small  a b o u t D = 0. In  this  D,  we  r e g i o n <f>  expand  the i n t e g r a l  i s expected  i n equation  2.26  t o be s m a l l a s w e l l .  This  results i n :  •X  1 1 . «D - ! = ( . < # „ )  -  -  n j g '  1  ! * .  D=*0 to  first  order  i n D. By e q u a t i o n  5  =  — — °  2.25, we  -  In  a more r e a l i z a b l e  cell  i s bounded  provided  by p l a t e s  the deformation  plates  and n e g l i g i b l e  finite  cell  back.  situation,  In the c a l c u l a t i o n  0  e  a n d z=d,  i n the c e l l  2.28  e geometry,  t h e above  I f that  t o two s e m i - i n f i n i t e to produce  figure  i s just:  ;  i n which the  is still  i s localized  i n the center.  i s equivalent  - n ) d  i.e. a finite  a t z=0  2.27  see t h a t t h i s  (n A  D  D  D=0  near  these  i s the case, cells  3,  placed  when  valid two  then  a  back to  D > 10 D  the  th <f> a t t h e c e n t e r  angle  calculation finite  anchoring  smaller that  was  a t these  o f the c e l l  f o r the case  of r i g i d  displacements.  In  The s e e m i n g l y  This  the case  i n the c e n t e r w i l l conflicting  of  be e v e n  conditions  when t h e i n e q u a l i t i e s  1  ^ d  where £ i s t h e e l e c t r i c  <  £ <  d  2.29  e c o h ie r e n/ cK e el ee n g t h d e f i n e d b y 1  < " BZ-T- '  Assuming  that  completely for  .001 r a d o f w/2.  anchoring.  s t r e n g t h , the d e f o r m a t i o n  a n d d> = 0 a r e s a t i s f i e d  D > D  th hold,  i s within  the  one  analogous  finite  may  consider  to the f i n i t e  case,  given  a  the case,  semi-infinite the o p t i c a l  completely  23  symmetric  case  phase  to  be  difference  cell,  will  be  exactly  twice  that  for  the  semi - i n f i n i t e  —  H e r e we  have used  —  <$  phase,  o  properties the  equation  27rAnd/A.  and  cell  experimental  versus and  the  the  One  is  a  of of  If  the  - 2 d  relation  This  the  that  line  will  CV/A,  and  depending  is  CV,  eq.  the  a  the  becomes  only  on  line  proportional  31  material  equation  difference  straight  '  undistorted  fundamental  phase  be  2.28  2  D =  optical  product  and  e  constant  dimensions.  inverse  extrapolation  B  — CV  o the  method.  intercept  =  S  case,  is  of  plotted  will  result,  the  surface  to  length.  expects  the  extrapolation  length  d  to  be  inversely  e  proportional  to  relationship  i s obtained  interface.  the  The  torque  layers  of  liquid  applied  on  those  contained  i n the  strength from  crystal layers  by  0^  It  assumed the  However, the is  tilt  same not  on only  same. the  external  term  to  assume  both  the  the  the  bulk  This  .  on that  the  but is  i n the  anchoring.  the  torque  surface  surfaces liquid  • £F =  l  the  2  two  in  W  on  must  The  balance  the  exact at  the  infinitesimal  equal  crystal.  the  This  torque  balance  the  is  24  32  2  that  the  center  2  respectively.  geometry  that  field  2  dd>  surfaces  =  free energy  ^  '  =  justified  the  dd>  '  same way,  also  electric  about  dd>  l  plates:  this,  applied  anti-symmetric  by  at  angles  have been p r e p a r e d  the  were  the  is natural  plates are  are  surface  conditions: dd>  and  the  considering  applied  £1 ^  of  W.  i f the  the  considered, surface  Yokoyama angles cell  and  van  the  is  truly  to  the  the  interactions  at  corresponding  i f  two  Sprang surfaces  symmetric. coupling  of  to  the  flexoelectric polarization  of  the  cell.  A l l the  other  terms  are  symmetric.  that the  the  This  angles  free  at  energy  come  into  apply  a  the  the  due  forces  two  to  balance  on  an  can  symmetry  surfaces  solely  torque  torque  elastic  broken  are  the  equation  a  torque  i t will  always have n e i g h b o r s ,  to  slice.  The  the  goal  extrapolation  i s to  length  d  equal.  of no  which  will  can  be  that  assumption the  field  electric  slice  the  the  electric  because  relate  which  Note  because  thin  slice, the  restricts  external  infinitely  apply  then  the  does  forces  not  cannot The  how  the  transmit  thin the  anchoring  measured  in  sample.  matter  surface  term  torque  energy  to  with  the  directly  e  Yokoyama is  a  function  related at  the  This  to  method.  Recall  applied  anchoring  . 2..  .  be  Asin  2  compared i s by  slope to  ^  <j>^, one  definition  the  is  (d> ) c o s  V V  V  z  at  may point  y  the  field.  The  The  which  torque  is  balance  2 (<f> )  i  dz  2=0  sin (<f> — 0.  Under  extrapolate =  quantity  length  *  2  z  extrapolation  l i m d (D) . D-»0 as:  written  1 + 7 the  that  electric  strength  can  1 + K s i n (<f>J +  defines  small  the  interface  , 1  Sprang of  the  first  K  This  - van  -d  the  1  assumption  that  <f>(z) l i n e a r l y  . That  with  the  to  slope  <f> ^ 2  <f> = enables  e  one  to  write:  +x  -—  d  dz  2.34 z=0  2 Note  that  condition valid.  the that  assumption one  Discarding  is  in  that  <f>^ i s  small  the  r e g i m e where 2 t e r m s o f 0(# ) i n e q . 2.33  ^  .  25  B  compared the  R-P  above  to  form results  <f>^ i s of  the  W(<^)  is  in: 2  35  is 0.  Recalling reduces the  e q . 2.23,  to W  = K /d . 1 e  o  measured  definition  t h e R-P s u r f a c e  d  to  6  of d  used  This  anchoring  i s the fundamental  the anchoring by deGennes.  potential, relation  strength.  t h e above  that  This  relates  i s a l s o the  5  e  Extended  The  Deuling  Deuling  vary.  a n a l y s i s has been extended  5  This  surface  Method  extension  enables  due t o t h e p l a t e s  measurements  Extended  on the bulk.  Deuling  Consider  ,d  (ED)  to allow  the t i l t  one t o c a l c u l a t e  as a This  function method  angle  the torque  o f the t i l t  will  $ to  at  angle  be r e f e r r e d  the from  t o as  the  method.  a g a i n eq.  2.19 w i t h  the f l e x o e l e c t r i c  terms  left  out.  r  Kjl  2.36  dz  + « s i n ( * ) ] [§§] 2  e ,e„(l + 7 s i n ( ^ ) ) 2  o 2'  The  justification  next by  chapter.  that  o f l e a v i n g o u t these  As d i s c u s s e d e a r l i e r ,  terms w i l l the director  f u n c t i o n <£(z) w h i c h m i n i m i z e s  variation  o f F under  differential  d_ dz  these  be d i s c u s s e d field  F a t constant  conditions yields  D.  i n the  i s specified Performing  a  the f o l l o w i n g non-linear  equation:  K  ( l + K sin(4>)) d £ r  2  = 0  dz  2.37  e e f l + 7 s i n (4>)) o 2 *• J  This at  may b e i n t e g r a t e d i m m e d i a t e l y ,  z = d/2,  a n d 4>{d/2) -  <j> .  using  This results  2 d± dz  sin £ £K o 2 1  (l+Ksin (<j>)) 2  d<t> the condition that ^ — 0  (.</>) (l+7sin  26  2  in:  2 s i n (cj>)  1/2 2.38  (0)) ( l + 7 s i n ( < £ ) 2  This  c a n a l s o be i n t e g r a t e d  D  sin V  This  formally  ( ^ ) ) ( l + 7 s i n (<?))"  (1+Ksin  (<f> )  2 1  o  resulting in:  sin  (<j> )  e q u a t i o n r e l a t e s the measurable  sin  2.39  d<j>  (4>)  D t o t h e two a n g l e s <j>  quantity  m  4>^ w h i c h  and  situation  describe  one may  the  n o t make  cell  deformation  uniquely. <j> = n/2  the approximation  In  this  because  the  m  integrand The  i s s i n g u l a r a t the upper  cell  integration  voltage,  eq,  2.7  limit  with  i n that  the  change  f r o m z t o <f> ( u s i n g e q . 2.38 a b o v e ) K  V - 2  case. variable  1/2  (<j> )) m  £ A£  of  i s :  (0))(1+ 7 s i n  (1 + /csin  in  d<j>  2.40  )-sin (^))(l+7sin (</0)  (sin (0 2  2  2  Shimoda e t al used a s i m i l a r method t o show t h a t p r e t i l t e d c e l l s have no  Freedericksz  rj — s i n (4> )  threshold  voltage.  With  and s i n ( ^ ) = sin(<£ ) sin(V>) ,  in  become,  ii  the  substitutions  the two i n t e g r a l s  above  ro  respectively: ,*72  V  o  2 1  rj,  r  (1+Kf;sin 1  L  (VO ) ( l + 7 r ? s i n -  T]  sin  (VO)  1/ 2  dV>  (VO  2.41  and  v  Where can  -/rh A™  -T/2  (1 +  2  ib = s i n l  sin(4>  KT)  sin  1/2  (V )) 1  d<f>  (1 )  l sin(<£ )  2.hi  - r j s i n ( V 0 ) (1 + 7f?sin (V>)) 2  With  2  the d e f i n i t i o n s o f V o  and C these o,  i n t e g r a l s c a n be r e w r i t t e n a s :  (l+/cr7sin (i/>)) _7T / 2 2  CV  C V C V 0 0  (l+7r7sin (VO ) "  1 - rj sin(V') 27  2  |  1  /  2  dV-  2.43  and -1T/Z  V  2  (1  /  +  n 1/ 2  (VO)  sin  KT]  2 .44  d<£  ( l - r s i n ( V ' ) ) d+7'/sin (V')) 2  2  ?  r\  i s then  should  go  a  fine  indicator  smoothly  from  of  the  0 t o 1 as  total  deformation  D goes  from  D  to  in  the  cell;  infinity.  r?  Indeed  th r\  can  be  thought  of  as  the  order  parameter  for  the  Freedericksz  transition. The  essence  potential and  voltage,  as  and  surface  well  <f>^ may  known f r o m  2.38)  the  i s contained  uniquely. is  of  be  the  thus  method  the the  used  first  can  in  as  the  torque  ED  determining  integrals  material  i n the  of  constants  torque  integrated  the  capacitance rj  determine  eq.  2.33  W(^)  and <f>  ( i n which  determined  yield  anchoring  The  Euler-Lagrange  SW/d^  to  the  above:  torque balance  integral  surface be  two  of  ^  equation,  eq.  directly.  The  within  an  additive  constant. A  completely  Defining  analogous  calculation  t h e r e d u c e d p h a s e as 6 =  r  . ^ « )  1  2 2 Where i / — n / n - 1. e  Making  o  done  with  the  e q u a t i o n 2.21  q  l + u  /  phase.  becomes  , f — ^ J  the  be  1 - &/6 , d  J  can  2.45  sin (0) 2  change o f v a r i a b l e  from  z t o 4> a g a i n  results in:  l  2  A  sin (*  )  2  +  7  m  or  A  "f ( l ^  S  i n  J . -(sin (0 l  2  / V  J  n  2  ( ^ ) )  (l  )-sin (^))  i n the reduced n o t a t i o n used  2  K 2  +  7  s i n  (1 +  previously:  28  i  2 W  V> ^  )  2  i/sin (0) 2  J  2.46  n  Given and  phase  either  of  then  can  eqs.  be  in  addition  surface  used  torque  Thus  van  is  a  Sprang.  this  independently  significant The  chapter  may  method  may  be  be  to  vrjsxxi  (ip)  this  simultaneously  determined  both  J  capacitance,  solved  has  2.47  dip  the  be  i f one  a u s e f u l method o f v e r i f y i n g This  to  2.43,2.44 may  above.  2  '-(l-rysin (V>)) (1 +  data  the  outlined  both  A ™  - \  the  and as  -,1/2 ( l + 7 > ? s i n (i/>))'  ( l + w g s i n (x(>)) 2  cv  phase  in  and  WC^)  determine  the  equation  the  Y-vS  manner  capacitance  data,  ; this  is  obviously  results.  departure  they  from  devised,  extended  It  d i f f e r e n c e between method r e q u i r e s  is  noted  describe  the  voltages.  r\,  that  to  the  which  method  of  Yokoyama  W(0  even  This  is  as  though  is  because  extension  data  )  not  it  and  well i t  as  is  as  well  as  d .  the  ED  c and  the  too  in The  e  method  is  that  v.  natural  particularly  becomes  and  is outlined previously  determine  that  phase  deformation  and  same  1  principal  r\  for  quantity  useful  close  to  at  to  larger  unity  to  be  18  distinguishable  from  i t . It is for this  where the of  or becomes  data less  the  for  these  than  100  parameter  of  experiments, V  .  The  the  substitution  - exp(-a)  1  V  reason  2.48  interest is useful. a  became  analysis  was  as  large  When  as  stopped  174  then  analysing  at  voltages  because  the  th computer  could  exp(±174).  This  information Another readily  not  denominators  not  contained  problem lend  was  easily  is  a in  that  themselves increase  store  numbers  significant the the to  sharply  data  that  numerical  29  the  large  or  as  problem because  integrals  near  as  there  corresponded in  eqs.  limit.  to  much  a <  174.  do  not  because In  light  as  was  2.43,2.44  integration  upper  small  their of  this  it  i s useful  .T/  2  to rewrite  form as: -ff/2  ,7T / 2  f(0)d<6  V  f(«)-f(w/2)  1-nsm  The  first  ^  (4>)  term  /  on  by  an  term  be  written  can  elliptic using As  existing  fixed  equations  a  the  point  (note  2.43,  is  a  well  l-r7sin (0)  of  kind;  2  behaved  method  difference  integral;  algorithim. complete  these  are  i t is  The  and  second  incomplete  easily  evaluated  directly  f o r rj a n d  1 3  2.44  iterative  that  V  first  algorithims.  2.49  1  Simpson's  as  of  the equations  a  right  adaptive  integrals  d<j> + f (JT/2)  1-rjsin'(4>)  the  evaluated  <t> ,  integrals of this  cannot  scheme  be  was  solved adopted.  Rewriting  t h e s u b s t i t u t i o n i n e q . 2.48 h a s b e e n  these  performed)  as: cv  = f (a,6 ) l l  2.50  v  = f (a,4> ) 2 1  2.51  and  Initial  guesses °  respectively. to  are  made  both  L i n e a r i z i n g these  the f o l l o w i n g  a  and  equations  6 , 1  these  are  i n t h e unknowns  a  o  a i  i+1  FORTRAN  evaluate  li+l  df /da  df /8<f> 1 ~ r I  df /da  df /8<f>  I  +  J  program  was  written  the f u n c t i o n s  f^ a n d  performed These values was  this  criteria was  recursion were  less  reasonably  10  gives  l  cv- f  rise  V-  f  _  1 2  2.52 J  1 A  a n d d>  recursion:  Q L  at  that  than fast  to perform and t h e i r  until  the  this  recursion  derivatives.  convergence  criteria  as w e l l The  constant,  f o r the  lower  30  typically voltage  10  data  The  as  program  were  the d i f f e r e n c e between s u c c e s s i v e l y  some  li  met.  iterated  convergence  b u t became  slow  at  higher  voltages.  adaptive;  the  step  convergence. using  To  published  The p r o g r a m small The  for  test  the  as  between  data,  as  The  to  data  was  generated  values  of  i n a l l cases  a -  19.5  ±  the  of  fast  convergence  region  reason  for  well.  this was  As  .3.  This  behavior  is  made  not  and  <j> .  except  one  o f a a n d \f>^.  value and  a  speed  defined that  yet  of  the slow  understood;  o b s e r v e d when e x p e r i m e n t a l d a t a  a consequence  of  this,  on  some  was  sets  of  discarded.  Summary  aims  of  this  the o p t i c a l phase voltage  i n p u t and  was  adjusted  artificial various  program  e x a c t l y to the c o r r e c t v a l u e s  one o r two p o i n t s w e r e  Chapter  the  Q ' S was  and  d a t a as  that  located at  The  input  succesive routines,  same l a c k o f c o n v e r g e n c e  used  reason  material constants  was  convergence.  this  between  i t converges  case  boundary  is  then used t h i s  region  one  the  It  thesis  are  then:  t o measure  the  capacitance  d i f f e r e n c e as a f u n c t i o n o f v o l t a g e  as p r a c t i c a l  to a very  large voltage  f r o m as  (> 100 V  ) .  and  small a  The  method  th of  Yokoyama  and  van  Sprang  extrapolation  length.  based  on  the  to  both  solely  independently yield  the  anchoring  information is  In  can  addition,  continuum the  then a  theory  capacitance  potential.  be  The  obtained.  31  applied  new of and  next  to  determine  the  method  (the  ED  method)  nematics  can  be  applied  the  phase  chapter  information  describes  how  to  this  Chapter III: Experimental  The  Apparatus  experiment  parallel plates  to  be  to conducting and measure  alignment  should  what one w i s h e s  performed  glass  is  plates  a  align  apply  the capacitance  have  to  and  characteristic  a  a  liquid  voltage  the phase. anchoring  crystal  across The  these  method  strength  of  which i s  to investigate.  Sample Preparation  The  liquid  commonly without  In  crystal  referred further  by  as  5CB;  II,  the  was  obtained  from  BDH  and  used  T h e s t r u c t u r e o f 5CB i s :  magnitude  two p a r a m e t e r s :  4-n-4-n'-pentyl-cyanobiphenyl,  i t was  purification.  chapter  described  to  employed  V  of  and A  flexoelectric ( c f .  effects  were  e q s . 2.17, 2 . 1 9 ) .  In  p  5CB  one  expects  these  parameters  to  both  be  small.  Following  g Helfrich's is  procedure  estimated  that V  for calculating < 4. x 10~  3  flexoelectric  coefficients, i t  V a n d A ~ 3. x 1 0 " . 3  This  result i s  p  derived the  from  5CB  molecular on  the fact  molecule  The 1/4"  t h e d i p o l e moment  dominates  d i p o l e moment.  the strength  ignoring  that  a l l other  This  implies  of flexoelectric  the e f f e c t s  and  contributions e^ > e^.  effects  These  justify  to  group the  upper  on  total limits  the assumption  of  i n t h e ED m e t h o d .  5CB i s s a n d w i c h e d b e t w e e n thick  o f the cyano  were  glass plates.  purchased  with  32  a  The p l a t e s u s e d  conducting,  were  transparent  Indium-Tin-Oxide masked  and the remainder  circular during The  ( I T O ) c o a t i n g on them.  electrodes with  the process conducting  with  at  - 70°C  agents  the  plates  i n four  impurities,  a  first Two  was  of  a  capacitance  was  o f the area  completed  good  then  had been  The l a t t e r  process  onto  and e l e c t r o n i c s  grade  i n electronics  grade  deionized other  was  i s less  was  between  the s u b s t r a t e .  33  to  beading procedure  sonicate  i n 5%  to  which  baths  were  as t h e i n t h e  expensive.  equal  silicon  cleaning  t h e same  was  was  confirmed  water;  sonicating  employed;  1 5  was  hydroxide  were  d i s c o v e r e d by J a n n i n g angle  or  techniques about  sonicating  xylenes  Both  The d r y i n g p r o c e s s  was  usually  ammonium  added.  bath  by  gross  results,  in  cleaned  The  Cleanness  The  two  t o remove  by r i n s i n g  by  were  solution  methanol  surface  were  minutes.  of hot a i r .  the  there  the s o n i c a t i n g  thirty  grade  and  plates  solvent,  equally  alignment  step,  detergent  clean.  and  a t - 70°C.  an o b l i q u e  that  or scratched  the diameter  the  about  i s not  surface alignment  technique, at  gave  peroxide  of  the  during a l l stages  i n a stream  solution  procedure.  uniformity  first,  electronics  glass  which  hydrogen  again held  the  degreasing  wetting  i f the  detergent  to  important  respectively:  and d r y i n g  employed,  30%  In  The p r o c e s s  observing  occurs  i s an  stages;  trichloroethylene,  by  n o t rubbed  by measuring  a n d t h e d u r a t i o n was were  acetone  to produce  caliper.  used  acetone.  were  contributing  employed.  sonication  solution  t a b s . The e t c h i n g was done s u c h  surfaces  m , determined  a vernier  procedures  the p l a t e  an a c i d  c o a t i n g was  2  .1 x 10  Cleaning  o f f with  contact  area  -I*  1.3 i  etched  A region of this  oxide  Substrates  qualitatively them.  The  the first  (SiO) evaporated prepared  i n this  way  are  angle. SiO 20  said  cm  the  above  the  between  plates  and  the  A,  The  plate  center  for  alignment  routinely  at which  exposed  planar  Grade.  the  plate  240  were  used  Select  boat,  c u r r e n t was  The  "strong"  evaporator  Union Carbide  line  boat  produce  The  1 6 , 1 7  was  to  -  10  a  was  the  at  an  in  angle  evaporation  to  produce  a  tilt The  position of  60°  boat.  d e p o s i t i o n r a t e was  sec,  low  Torr.  5  p l a t e s were h e l d  and  25  achieved  normal  the  with  to The  -  12  A/sec  layer  -  300  A  thick. The  second  poly(vinyl in  the  very  method  formal)  solution  plates  use  (this  and  10  the  gently  thin  film  a  the  rubbed  PVF  could This  be was  solution  i n one  the  of  PVF  was and  a  the  i t  was  solution  to  the  solution  had  the  remains.  in  Sometimes the  After  of  extensively  chloroform,  applying  polymer  used The  plates.  later).  t o be  to  this  probably  method; liquid  by due  PVF  and  The  glass transition  a  Typical  d e s t r o y i n g any  plates).  times  to the  with  The  a  densities  the  above  the  rubbing  i s around  34  cotton cloth.  The  of  the  alignment  was  far  the  be  the  into  glass had  100°C,  1 8  film  on and  solution then  homogeneously.  sample  the  was  composition  i s b e l i e v e d to  heating  o f PVF  the  crystals  effects  concentration of  homogeneous  this  heating to  on few  Uniform  1 6  the  placed  insensitive  align  destroyed  from  direction  material.  with  used  of  displays.  the  when  discussed  (calculated of  i s said  routinely of  g/cm  rubbing  coater  are  film  2  volume  process  spin  to  buffed  films  solution  liberally  be  thin,  nematic  weight)  will  -4 -  a  a  Rubbed p o l y m e r  twisted  by  applied  to  evaporated,  of  (-.05%  was  necessary  were  (PVF).  manufacture  dilute  employed  fibres  produced  first PVF  the  report  alignment isotropic.  transition the  in  polymer  of  the  film.  p r e c a u t i o n s had  to  be  t a k e n whereby  films  d i d not reach  that  temperature.  19  e t al  Geary  recently  polymer  alignment.  chains,  and t h a t  surface,  that  crystal.  It it  They  this  propose  a  possible  that  orientation,  found  that  not scratching  alignment  when  the  cell  polarizers. alignment  t o be s u r e  is filled A  well  is parallel  uniformly bright  alignment light  that  whatever  After  scatter.  treating  Mylar  into the  of  rubbed polymer o f the  the  the polymer  sample  to either  liquid i s both  Note  the c e l l liquid  between  by c a p i l l a r y  crystal  or  dark  i s only  crossed the  the a n a l y z e r  axes  The u n i f o r m i t y o f  by o b s e r v i n g useful  t h e amount o f  i f they  are the  temperature.  the surfaces the p l a t e s  spacers  under  done  when  i s a t 45°.  be c o m p a r e d this  T h i s was  uniformly  the p o l a r i z e r  that  treatment i s ,  examination  is  when t h e a l i g n m e n t may  the p l a t e  alignment.  microscopic  aligned  o f two s a m p l e s  they  by  same t h i c k n e s s a n d a t t h e same  40/J  the  or grooving  alignment  occurs  i s i n d e e d p r o d u c i n g u n i f o r m homogeneous  and  orients  of  and c r y s t a l l i n e .  i s important  once  mechanism  buffing  i s responsible f o r producing  They  oriented  described  them.  The  action  were  liquid  clamped  together  crystal  i s introduced  a t one o f t h e e d g e s .  i s i n the nematic  p h a s e when  with  Note  that i f  introduced this  way,  20 then  surface  verify and  n o t due  least  five  effects cell it  that  adsorption  any o b s e r v e d  also  alignment  produce i s due  to adsorption effects. degrees 20  disappear  has c o o l e d  was  can  above  the  and then  clearing  checking  to the nematic.  This  alignment. to the s u r f a c e  was  point  then glued a t the corners with M i l l e r  must  properties  checked  by h e a t i n g a t  where  the a d s o r p t i o n  the alignment  I f the c e l l  35  One  showed  again good  Stephenson  after  the  alignment,  E p o x y 907.  Electrode  Figure  6  Scale  drawing of  the  sample  36  cell  used.  Figure  6  shows  the  sample  cell  to  scale  with  the  electrode  areas  outlined. Crucial is  an  to  measurements  of  the  anchoring strength  a c c u r a t e d e t e r m i n a t i o n of nematic  employed  was  the  interference  layer  of optical  by  this  method  thickness.  The  method  rays r e f l e c t i n g  from  the  two  21  surfaces  of  the  interference  layer.  was  interference  was  The  angle  observed.  of  (tj))  Angles  observed are given  incidence at  was  varied  which  until  destructive  by: 3.1  m n^ the  i s the  refractive  order of  light  used.  index  the  interference  The  sample  temperature  controlled  constant  to keep  and  same w a t e r the  refractive  had  a  Temperature  Control  temperature  was  5CB A  housed  around  of  isotropic  to  i n the  o f an  One  turntable  keep  -.007°.  The  of  the  the  temperature phase.  refractometer five  the c e l l cell  is  circulating  observe  on w h i c h  m  with  isotropic  Abbe  could  phase,  wavelength  i n a block  i t  crystal  measured. The  i n the  i s t h e vacuum  temperature  resolution  to w i t h i n  and  liquid  interference.  maximum  the  water  the  i n d e x was  determined  The  cell  the  controlled  destructive  of  The where  orders of was  placed  thickness  was  -1.8%  control  of  this  experiment  was  the  one  used  by  13  S. M o r r i s . For mK  the  The  purposes  stability.  served  as  a  controller of  The  this  sample  grounded  i s c a p a b l e o f sub  experiment, cell  shield  was  the  held  f o r the  milliKelvin  controller  i n a.copper  was block,  stability. set  for  which  c a p a c i t a n c e measurements.  1-2 also This  block were had on was  was  set  inside  fitted  at  either  keyways  on  the  either  side  to  a  end  to  inside align  cylinder  cylindrical around  for  in  could  be  One  with  the  it  resistor  having  mercury  was  first  balanced  The 419A  null  amplification  stage  error  signal  which  drives  resistor  i n i t s feedback  integral the  occur.  copper  control.  feedback The  outer  temperature  through  the  temperature  . Olfi.  signal  of  of  the  well  as  to  can  is  just  is held  aforementioned to  copper within  was  voltmeter  500mK  38  by  The  W  Kepco  optional by  fed  by  into as  a a  the  the  amplified  power  amplifier  has  a  large  variable capacitor the  below  controlled was  FE2  gain  oscillations  degrees  water Haake  decade  powered  where  three  The  Wheatstone  increasing  point  a  was  temperature  served  circuit;  temperature  tubing.  been  1051  bridge  the  were  had  dc  the  typically  circulating  a  is  optimized  below  of  bridge  an  soldered  these  The  as  a  mK.  40-0.5B 20  loop,  in  thermistors  Electronics  feedback  placed  thermometer, which  arm  this  a K e p c o OPS  tubing  of  10  Time  cylinder  was  thermistor  one  heater.  by  controlled  as  cylinder  network  sample  than  a  of  Stability  brass  less  tabs  contained  copper  The  voltmeter;  then c o n t r o l l e d the  to  The  was  Fenwal  cylinder  matching  cylinder  each  cell.  against  error  had  end-pieces  The  tabs  which  Four  served  resolution  The  2804A q u a r t z  point  place.  cylinder.  wire.  can  copper  had  the  cylinder;  Packard  thermistors  battery.  Hewlett-Packard  brass  determined  in  insulation,  the  triple  a  heater  and  block  inside  water.  of  a Hewlett  these  bridge;  The  block  cell  block  with  cylinder  the  the  styrofoam  ends  accurately of  hold  circulating  a  copper  and  can.  the  calibrated with calibrated  of  brass  it  mounted  of  larger  wound n o n - i n d u c t i v e l y  inside  for  a  water  pumped heating  the  and bath  and  circulator. In  this  arrangement  thermistor's  the  temperature  multimeter;  this  was  temperature was  taken  stability  monitored  t o be  by  a  the sample  was  1-2  mK.  Keithley  Another  175  digital  temperature.  Electronics  For  the  most  compatible cell  electric  applying  on  the  a  glass  presence  of  impurities  time  plates.  in  slow,  An  have  o f any  applied Hewlett  field  the  amplifier.  transformer V rms  signal  and  3312A  the  necessary In the  reactions  as  well  liquid  crystal  display),  they w i l l  dc  function  produced  electrodes of  field  was  the  is  (the  frequency  computer  these  as  screen  the  reorientations  are  typical  turn-off  essentially  the  a  the  output  Kepco  BOP  stepped  cell  This  used)  controlled  39  then  sample  capacitance bridge.  with  up  through  source  across  the  and  amplifying  generator  signal to  by  the  electrodes  (the  see  ITO  is  because  a  to  order of milliseconds,  applied  6 kHz  amplitude  between  migrate  Since  amplified  i n the at  to  XT  collection.  sample.  v o l t a g e i s produced  The  transformer  the  IBM  controlled  the d e f o r m a t i o n s  is  an  whose f r e q u e n c y i s g r e a t e r t h a n a k i l o H e r t z .  sinusoidal Packard  in  with  computer  the d a t a  signal  electrochemical  on  The  difference  time  field.  controlled  to induce  ac  impurities  would  was  NOVA XT.  required  of a t w i s t e d nematic  value  200  the  known p o t e n t i a l  electric  relatively  experiment  as a i d e d i n t h e  field  ionic  participate applied  the  microcomputer,  v o l t a g e as w e l l  The by  part  can  the  by  The of  a  72-5M a  the supply  sample.  v i a the a m p l i t u d e  rms  16:1 ratio over The  modulation  input  on t h e 3312A;  Translation the  cell  197  primary  of  interfaced  decades  there the  multimeter  of  of precision  the c e l l  error voltage appearing  on  ( i n this  Thus  the bridge  capacitance.  Typically  bridge  was u s e d  in  the leads  of  t h e sample  has  plates.  was  shielded  with  Thus A  across  the  multimeter  is  a  placed  scale sensitivity  1615-A  the c e l l the  experiment terminal  When  across  a  the  was m o n i t o r e d lock-in  unknown  o f 1 ppm f o r was  .003% .  capacitance  measured  was  as t h e grounded made  the  General  test  used  agreed  gain.  shield  500  pF  i n the  within  by a P r i n c e t o n This  that  connection  Radio  leads  amplifier.  the  and the  the  i n which  o f 100 ^V a t maximum  40  equals  the accuracy  cables  so  i s the case  of  resolution  served  coaxial  this  factor  mode  third  i s manually  decades  capacitance  holder  On a  capacitance)  loss  and s i x  are adjusted  When  measured by the b r i d g e  insensitive  GenRad  arm.  resistors  capacitance  the bridge.  The e r r o r v o l t a g e  5102 p h a s e  and  and the only  the capacitance  .2 p F .  a full  voltage  The b r i d g e  h a s a maximum  The c e l l  and  capacitor  to  i n this  i s excluded  the c e l l  the  i n the three  arrangement;  experiment,  Research  Data  t h e maximum.  This  resistors.  case  capacitance.  than  the  a t the detector.  correspond  standard  of a  i s o n one arm o f t h e b r i d g e ,  capacitors  decades  between  of  c a p a c i t o r s a r e on the opposing  capacitance  this  1/4096  i s measured  decades o f p r e c i s i o n  i s zero  resistance  for  to  transformer.  T h e unknown c a p a c i t a n c e  capacitance  The  the output  on t h e b u s o f t h e NOVA.  monitors  ratio  i . e . t h e decade  unknown  was  t o t h e NOVA v i a t h e I E E E p a r a l l e l b u s .  are five  nulled;  controllable  bridge  capacitance  bridge.  arm  was  digital the  signal  12 b i t d/a c o n v e r t e r  voltage  Keithley  The  DT2814  the modulation  less  Applied  instrument  The  1615 b r i d g e  across  the primary  across  the sample  i s such  that  of  ratio  the  cell.  Thus  voltage  a t the g e n e r a t o r  Optical  Measurements  It  was  originally  monitoring  the  polarizers the  with  polarizer.  was  not  i n the n u l l transformer  i t i s appropriate  thought  that  of  t h e sample  the phase  transmitted  i n between  However, t h e e n v e l o p e as  is  the  the  same  t o measure  voltage as  that  the  cell  terminals.  intensity  constant  condition,  the  could  be  light  aligned  measured  through  crossed  a t 45° t o t h e a x i s  o f the r e s u l t i n g f r i n g e  voltage  was  by  varied,  making  of  pattern  the  phase  13  measurement was  likely  the  cell,  from  difficult.  This  due t o b o t h  voltage  as w e l l  the g l a s s  as  e f f e c t was  also  dependent  interference  seen  by M o r r i s .  depolarized  effects  from  This  scattering  multiple  from  reflections  plates. 22  The When  method light  positive  was  originally  birefringent  polarized; direction  employed  polarized medium  the major a x i s  the phase  given  by  to  at  exits  of  6 i s contained  that  entirely  2  a  axis on  quarter parallel  the sample,  wave  plate  to  i s placed  used the  medium  by  van  optic  i t is  Sprang.  axis  light.  The  upon e x i t  41  the  information  i n the e c c e n t r i c i t y which i s  .  behind  will  a  elliptically  3.2 t h e medium  with  i t s fast  to the d i r e c t i o n o f p o l a r i z a t i o n o f the l i g h t the l i g h t  of  o f p o l a r i z a t i o n i s along  the i n c i d e n t  e - tan [-|-] If  that  45°  of i t s ellipse  of polarization  about  similar  be l i n e a r l y  polarized  incident with  I  Figure 7.  Schematic of phase measurement setup.  42  its of  plane the  of p o l a r i z a t i o n  incident  beam.  wave p l a t e  and  which  happens  this  schematically and  Oriel  During  the  to  microcomputer  from  error  v o l t a g e has  cell  the  near  angle  the  and of  the  is  6. a  by  monitoring  angle  at  is depicted  . 5mW  rotated  plane  quarter  the  This  was  the  the  light;  was  Detector  and  housing  ac  mV  stopped  is  cell. the  HeNe  and by  laser  analyzer. hand  the  until  intensity  photodiode.  sample i t  is  take  is held  changing,  a  transition),  the  is  data  the  The  capacitance  temperature  voltage  length the  the  ramp  The and  results  are  the  then  data  is  anchoring  comprise  43  (between  ramping  minutes  or  bridge  is  light.  recorded  bridge  Both  manually.  and  analysed  chapter.  the to The  the  the The  a u t o m a t i c a l l y by  potential.  50  twenty  capacitance  then  the  the  the  continues,  to  ratio  by  the  in  After  take  recorded  the next  ramped  may  are  come  bridge  point,  a n a l y z e r i s r o t a t e d to e x t i n g u i s h the and  to  desired  constant.  (this  i s placed  allowed  signal  maximum v o l t a g e To  cell  applied via  This  steps.  Freedericksz  The  where  signal  to  size  repeated.  extrapolation data  after  polarizer  follows.  applied voltage  and  microcomputer.  the  -50  extinction  voltage  process  the  from  for  analyzer  as  sample  in variable  and  nulled  An  the  stops  more  placed  source  i s detected  controlled  equilibrium.  100V)  used  a Silicon  proceeds  temperature  transformer  and  were  5/2  Experiment  experiment  thermal  light  the  with  be  angle  p r o p o r t i o n a l to  The  extinction  transmitted light  The  7.  experiment,  can  an  i t extinguishes  polarizers  The  Summary o f  analyzer  is directly  figure  film  extinction. of  An  rotated until  in  this  r o t a t e d through  the  above  yield analysis  the of  Chapter  IV  Data A n a l y s i s and E x p e r i m e n t a l  The the at  experiment  capacitance constant  dimensions, the  liquid  further  described  i n the previous  and the phase  This,  i s sufficient  to determine  crystal  sample. of  - van  Sprang  method r e s u l t i n g  that  chapter  o f t h e sample  temperature.  analysis  Yokoyama  Results  The  and  i n the anchoring  to knowledge  o f the c e l l  several material constants  Some o f t h e s e  method  the both  as f u n c t i o n s o f v o l t a g e  i n addition  data.  yields  constants  data  also  are important i n  i s analysed  using  of  using  the extended  the  Deuling  W(c^) .  potential  Reduction  The  raw o p t i c a l  phase  analyzer  c a n always  optical  retardation  0 < 5 < 2n. fig. is  the threshold.  voltage;  reaches  2n a t w h i c h p o i n t i t d r o p s  voltage. drop  The  d where always  prediction  data  starting  the phase  2TT was counting  angle  added  to  i s then  o c c u r r e d and the p r o c e s s  from  the p o i n t  increased  o f the  0 < 8 < it, be  such  as t h e c e l l  voltage  was t h e r e f o r e t o  corresponding  to the  until i t  d r a m a t i c a l l y to a value near  continued  repeated  44  from  until  that  the phase a t  monotonically  a l l t h e - phase  the  o f the phase ( c f .  t o b e c l o s e t o z e r o . The p r o c e d u r e  highest  point  will  the angle  One t h e r e f o r e e x p e c t s  the phase  this  Since  t o always decrease  examine  At  6  information  the phase  2n.  as an a n g l e  on the t h e o r e t i c a l  5 ) , one e x p e c t s  very high voltage  i s modulo  be e x p r e s s e d  Based  i n c r e a s e d above  data  points  at  the p o i n t  zero.  a  where  lower the  a l l the data has been  7-i •  650.4  CM *  0.6  0.8  1.0  3-  210  -T  -  20  Figure 8.  — i  ^  —  40  T  -  —  60  rms Voltage  An example of phase vs voltage data.  45  T-  80  100  900-1 •  800-  •  •  •  700H  S  600 H  £  500-J!  «3 4004  o  • i — | — i —  .4  0.6  r  0.8  1.0  300-' 200  20  —I  '  40  —r*  1 —  80  60  rms Voltage  Figure  9.  An e x a m p l e  of capacitance  46  vs voltage  data.  100  counted.  An e x a m p l e o f p h a s e v s v o l t a g e  Obtaining read  off  the  the  capacitance  bridge.  shown i n f i g u r e  Material  An example  birefringence  was  zero  Freedericksz  vs  it  is  voltage  8.  simply data  is  obtained  from  the  zero-field  transition  to  a  straight  line  phase:  phase data  and  below  extrapolating  to  voltage. was  evaluated  capacitance line  capacitance  was o b t a i n e d b y f i t t i n g t h e  Q  the  of  straightforward;  i n figure  9.  " 27rAnd/A i n w h i c h S  q  is  is depicted  Constants  The 5  data  data  to  vs  V = 0,  with  in  a  voltage  curve  this  by  is  similar  below  manner  the  easiest  definition C  for  the  large  along  = e e A/d.  0  done b y n o t i n g t h a t  extrapolating  transition  J  c o u l d be e v a l u a t e d b y e x t r a p o l a t i n g  by  o  2  a  straight  Furthermore,  '  e 1  C-V curve  voltage,  the  to V • » . = TT/2,  This  is  and t h a t  the  Hence C v s  1/V  m  capacitance was f i t  s h o u l d be  to a s t r a i g h t  obtained. that  one  A/d.  A flaw needs  approximately l i n e and the  linear  in 1/V.  infinite field  1 3  capacitance  C ^ was  i n t h i s method o f m e a s u r i n g d i e l e c t r i c c o n s t a n t s  to  know t h e  ratio  of  the  cell  T h i s p r o b l e m was " s o l v e d " b y f i t t i n g C  q  area  to  the  is  thickness  to p u b l i s h e d v a l u e s  for  23  and  thus  determining the  case  aligned the  from the finite  to  V — °° w i l l  on e : t h e  A/d.  There  extrapolation  anchoring  ( 4>(z) = n/2 )  data  bounds  of  calculating  at  a  effects  finite  to  the  e .  l  another  problem  i n f i n i t e voltage  voltage,  overestimate  is  cell  could  be  and hence At  best  one  because  in  completely  extrapolating obtains  e x t r a p o l a t i o n to V = » which o v e r e s t i m a t e s , 47  in  and  two the  largest these  value  of  C measured, which u n d e r e s t i m a t e s .  estimates  difference The  -.5%.  The  i s discussed  splay  voltage  is  elastic  f o r the  i n Chapter constant  Freedericksz  V  The  threshold  the  transition  data  point  required  can  be  was  fall  on  deduce V  such  determined  transition;  a  seemingly  small  c.  f.  from  eq.  the  threshold  2.10:  4.1  th  seen  of  difference in  V.  0 2  quite  clearly  in  i s never p e r f e c t l y sharp,  would  to  importance  The  the  nor  transition  more a c c u r a t e l y  figures  8  can  guarantee  even  one  and  i f i t were,  than merely  9.  a  Since that  scheme  perusing  the  a is  data.  th  Given  that  one  straight  line,  and  find  then  has  the  a l l the  2.43  2.44,  small  number  quadratic  use up  in this until  data  above  a  formidable  of  points  There  of  is  an  2V  .  the  the  data  two  The  Visual  be  i t to  One  without  using  the a  consistent,  cannot  fitting  examination  many  to  to  data  verified  least  points  that  eqs.  f i t a to  a  squares  one  was  a  easily  transition  linear  the  with  something  m e t h o d e m p l o y e d was  a r b i t r a r i n e s s i n how to  transition  above  after  voltage,  the  fits.  transition 13  immediately  the  below  f i t the  task.  f i t ; i n order  V =  to  data  i n t e r s e c t i o n of  function  analysis.  f i t the  i t is natural  describe  and  already  should  always f i t  the  quadratic  th  was  a reasonable  immediate showed a  vicinity "rounding"  intersection yielded the  f i t in this  the  of  of  the  two  should  fits,  threshold voltage.  capacitance  data.  The  A  transition  o f what  the  region.  splay  be  a  s m a l l number o f p o i n t s were  s h a r p knee  disregarding This  was  constant  48  discarded  done  because  i n the  "rounded" f o r both  in  they  curve. data  the  i s then determined  the  The  points  phase  and  from  the  threshold voltage and the previously determined d i e l e c t r i c constants. 2 4  This i s a standard method of measuring this constant. The  anisotropy  in  the  elastic  K = K/K  constants,  determined from the slope of the C-V  curve i n the region  - 1,  was  immediately  13  above the t r a n s i t i o n .  This slope M i s given by: M  " 2( «  \  +  +  1)  4  -  2  I t was determined using the quadratic f i t described previously; 7 was determined from C  and C^.  q  A completely analogous analysis was used  to determine K from the phase data 2  constant u - (n /n ) e  - 1 i s required.  as w e l l . I t was  In t h i s  case  determined from the  o  measured birefringence An and n — n + 2n , the l a t t e r being e  the  taken  o  2 4  from the l i t e r a t u r e .  Once the splay e l a s t i c constant and K were  evaluated, the bend e l a s t i c constant could be  determined.  Results: Material Constants  In the remainder of t h i s chapter, different  samples are  quoted.  The  technique of surface alignment used. evaporated SiO  r e s u l t s corresponding  to  two  two  in  the  samples  differed  Sample 1 employed t a n g e n t i a l l y  and sample 2 employed rubbed PVF.  Figure 10 shows r e s u l t s for birefringence vs temperature. P l o t t e d 2 4  on the same figure are Karat's and Madhusudana's  r e s u l t s , measured  using Chatelain's wedge technique. Figure 11  shows the d i e l e c t r i c constants vs temperature.  The  2 3  r e s u l t s obtained by Dunmur et al  are also p l o t t e d for comparison.  The systematic d e v i a t i o n i n t has been observed b e f o r e . 49  13  I t i s not  0.18-,  °  o  •  0.15H O  o  0.12H  0.09J 0.06H  0.03A 0.00  -i  1  •  1  1  1  «  1  1  j  •  1  • • o 1  1  1  •  1  26 27 28 29 30 31 32 33 34 35 Temperature (°C)  Figure  1  Sample 1 Sample 2 Ref. 25  10. B i r e f r i n g e n c e  as a f u n c t i o n  50  o f temperature  22.0T J  ^  18.0H  d o o  H.OH  d  ° ° o oo o o  g o o cP o © a  c  O CO  a  o  £ 10.0H a a>  • a  Q  o  t »  o a  0  Q)  o  6.0H • •  2.0-  27  Figure  T —  28  1  — r  11. D i e l e c t r i c  o  29  1  —  I  —  I  —  I  —  1  —  I  —  1  —  I  —  I  Sample 2 Sample 1 Ref. 23  30 31 32 33 Temperature (°C)  constants  as a f u n c t i o n  51  o f temperature  34 35 T  '  1  0.6-.  0.4  o • o A A  Sample Sample Ref. 24 Sample Sample  1 (C) 1 (<5) 2 (C) 2 (6)  ° * O  A  0.2-1 A  O  27  A  A A  0.0H  -.4-  a  28  29  30  "T  31  1  T  -  32  33  34  Temperature (*C)  Figure  12. E l a s t i c  constant  anisotropy  52  as f u n c t i o n o f t e m p e r a t u r e .  35  6  o • D •  1  5A  Sample Sample Sample Sample Ref. 24  A  2.C 2.o l.C Lo-  55 ft  B  e  9*  o  3-J  27  — i — i  28  •  1  1  29  1  30  •  1—  31  —I  •n  r  32  33  r—  34  Temperature (*C)  Figure  13. S p l a y  elastic  constant  as a f u n c t i o n o f  53  temperature.  —I  35  7-,  A A O  A O  5-  ^  2  A  a  •  o  J  A  o  A  O  2 H  2  J  B  A  o • o • A 1  27  D  Sample l.C Sample 1.6 Sample 2.C Sample 2,6 Ref. 24 I ' 1 '  26  29  "  A  1  30  1  1  31  1  1  •  32  1  33  1  1  34  Temperature (°C)  Figure  14. Bend e l a s t i c  c o n s t a n t as a f u n c t i o n  54  of  temperature.  «  1  35  unreasonable considers  to  that  deformation, results is  expect  of  was  was  r e f . 23 at  greater  12  through  constants.  In  a l l cases  temperature  Figure for  in  in a  measured  length  on  12  both  i n the  system;  the  a homogeneous s y s t e m .  This  homogeneous  related  results the  of  to  the  e t al  Bunning  elastic  The  one high  results  the  when  extremely  a  with  chapter.  show  samples.  quantity  next  14  shows  this  system  were a l l m e a s u r e d w i t h  Figures  comparison.  deviation  measured  while  discussed  a  are  constant  temperature  elastic shown  anisotropy  dependence  of  for  K. v s «  has  13  been  seen  shows the  the  before splay  i n measurements elastic  bend e l a s t i c  Surface  constants  constant  Anchoring  made  vs  vs  in  the  same  temperature  way.  and  Figure  13  f i g u r e 14  shows  determining  the  temperature.  Results:  Yokoyama's M e t h o d  Yokoyama  and  extrapolation  van  length  Sprang's  d  method  1 0  for  is straightforward.  The  relative  5/6  phase  0  e  is  p l o t t e d vs  is  2d / d .  1/CV;  this  Figure  15  i s f i t to a  depicts  the  straight  line  and  extrapolation  the  length  e  d  as  a  e  function  of  shown a r e same  intercept  temperature  Yokoyama,  method.  The  for  sample  Kobayashi  e r r o r bars  1,  and on  and my  the  SiO  coated  Kamei's  data  1 1  surface.  results  points  are  of  Also  using  calculated  the  from:  2  a)  the  from  x  °f  t  n  e  repeating  measurement.  Of  f^the  c  t  o  a  run  these,  straight and the  c)  line, the  b)  the  error  in  r e p r o d u c i b l i t y of  55  the  standard the  cell  deviation thickness  e x p e r i m e n t was  by  190-j  s  170-  x a  This work R«f. 11  15060 130C 110-  a o  90-  50-1 30 A 10  I  - i  30  29  —T~"  r  31  32  33  34  Temperature (*C)  Figure  15. E x t r a p o l a t i o n SiO  treated  length  as a f u n c t i o n  surface.  56  o f temperature f o r  35  30-t  o o  This work Ref. 11  25-4  20  w  Xi  ti 15. l-l  CO  8  ti  o Xi  o  5  '  5  029  30  31  32 32  33 33  a  o o  o  34  35  Temperature (*C)  Figure  16. O u t  of  plane  temperature  anchoring  f o r SiO treated  57  strength surface.  as  a  function  of  far  the  as  much  largest;  the  20%.  as  comparison;  The  their  e x p e r i m e n t had  results for error  origin  the  d  bars  i s not  from  e  used  known.  same p r o b l e m w i t h  subsequent  runs  11  in  ref.  It  i s not  reproducing  d i f f e r e d by  are  included  u n l i k e l y that d  as  this  for their  one.  e  16  Figure the  shows  the  out-of-plane  Rapini-Papoular  tangentially length.  SiO  results  quoted  The 17  Figure  shows  layer  the  Note  sample  that  was  result.  correspond  a  coated  the  spin  reproducible  spin  to  not  it  into  isotropic  after  implications  chapter.  The  other  to  those  polymer point  the film  data of  earlier;  the  measured  while  one  other  spin  to  may  by  each this  points SiO  report  corresponding  of  5 C B on  of  first  the  this the  imagine  this  r e s u l t are  corresponds made was  as  a  the  that  Note  a negative  not  removed,  not  in  used  the  last  lengths  that  are  18  shows  the  temperature.  strength  extrapolation  PVF  annealing  was  function of  that  a  plot  was  was  Figure  anchoring  a  a  completely  on  discussed  to  with  a  field  sample  cell.  the  r i g h t shows  sample  extrapolation  method.  58  the  points  a f t e r the  run;  treated  negative  this  The  for  2,  sample  measurements w i t h o u t  show  the  was  data  coated.  r u b b e d PVF of  on  point  that  included  clarity.  for  sample  Note  for repeating  The  is  data  for  extrapolation  again  off for  This  on.  the  are  data point  on  q  interaction,  from  length  This  was  further.  This  first  coated  useful  strength  left  " p i n " domain w a l l s  h e n c e was  anchoring  11  ref.  others.  that  and  comparable  in  the  All  layer  tended  the  determined  length!  than  not  to  as  W , based  strength  surface  extrapolation  that  extrapolation  different  the  e r r o r b a r s have been  buffed-PVF c e l l . negative  of  evaporated  The  comparison.  model  anchoring  of  there length  extrapolation  a is  rubbed no  data  described  length  I  100-.  3  8  (  H  A  "So a  60  0)  «  ^  2  40-  20-1 0  -20  27  ">—[—  28  ~1  30  29  31  '  r  —I  -  33  32  •  I  35  34  Temperature (°C)  Figure  17.  Extrapolation buffed  PVF  length  as  surface.  59  a  function  of  temperature  for  a  o  30-i  o  25-1  20-1 g10-15-1 5029 60  CO ( 4  o 00  o  Figure  30  18. O u t  of  buffed  o o  31  -T  32 33 34 35  -  Temperature (*C)  plane  anchoring  PVF s u r f a c e .  60  strength  vs  temperature  for a  physically, fact can to  be  seen  19 t h r o u g h  clearing  voltage.  a  of  20.  ^ ,  data  different  Figure shows  three  using II.  A  PVF  the  cell.  It  i s comparable  shows  the  the d e r i v a t i v e lastly  by  were  Yokoyama are  tilt  21  Deuling  d i s c u s s i o n o f the  shows  integrating  calculated  included  <f>^ v s  angle  figure  from  on  these  implications  reduced  potential  W(<^) the  as  in  a t the  (obtained  three  of  a  data  described  data  1 0  below  the c a p a c i t a n c e  method  and v a n S p r a n g ' s  also  300 mK  o f the a n c h o r i n g  numerically  results  1 a t about  these  using  graphs results  for is  i n the c o n c l u s i o n .  disturbing.  Figure  same p r o g r a m . clearing  point.  angle  became  derivative  were  in this  o f rubbed  t o sample  the extended  Many o f t h e r e s u l t s  example  19  <j>^ a n d  of  method)  comparision.  ^  observed  with  D e u l i n g Method  obtained  These  o f Chapter  presented  20  function  voltage  end  is inconsistent  was  strength  21 c o r r e s p o n d  point.  Figure  as  figure  alignment  the anchoring  o f the Extended  function  strength  SiO t r e a t e d s u r f a c e s .  Figures  SW/d^  anchoring  tangential  that  that of  the  negative  that uniform  Results  vs  a  22  These  o f t h e ED m e t h o d f i t showed shows  correspond  results  negative  over  (figure  23)  i t i s not a single  qualitatively  scatter.  an  as  shows  function.  with  that the  tilt The  features, for  t o the phase  to the c a p a c i t a n c e , i n figure  the  6 K below the  of voltage.  unusual Fits  A n e x a m p l e i s shown  61  about  range  some  the the f i t s  fitting  i s the f a c t  appreciable  also  valued  t h e same  from  to a temperature  The most p u z z l i n g f e a t u r e  SW/S^  showed much more  the worst  f e a t u r e s t h a t were  24.  data but  While  0.4n  0.3H  0.2  0.1  A  A  0.010  — T "  —J—  20  30  —T"  o  Rel.  40  i  10  SO  V/V th  Figure  19.  Tilt  angle  vs  reduced voltage  62  f o r SiO  treated  surface  —I  60  1.6-J  I  - 08.H *°  0.4H  0.0-  0.1  0.0  Figure  20.  Surface  torque  • I  I  0.2  as  a function  surface.  63  Ref.  10  ' —  0.3  of  tilt  angle  f o r SiO  I  0.4  treated  0.3-,  0.2H  0.00.0  Figure  21.  treated  potential  as  a  surface.  64  function  Ref.  I  0.2  0.1  Anchoring  •  —I—  I  '  0.3  of  tilt  10 1  0.4  angle  for  SiO  0.14-,  O.HH  0.06-1 0.02  A  -.02H  *  -.06-  i  10  30  V/V  Figure  22. T i l t  angle  treated  as  surface:  a  —r~ 40  -r-  20  function  worst  case.  65  i  50  —j 60  th  of  reduced  voltage  for  SiO  4-.  a  3A  o  S  H  06  Figure  i  i  23.  i  i  -.02  Surface treated  0.02  torque  as  i  i  a  i  T  case.  66  1  1  0.06  function  s u r f a c e : worst  1  1  1  >  1  0.10  of  tilt  angle  1  0.14  for  SiO  Figure  24.  Tilt  a n g l e vs  to phase  reduced voltage  data.  67  f o r SiO  treated  surface: f i t  1.8-1  1.5-  I  1.2-  O  0.6-1 to  0.3-1  0.0-.10  Figure  25.  0.00  Surface surface:  torque  0.10  as  a  0.20  function  f i t to phase  data.  68  of  tilt  0.30  angle  f o r SiO  0.40  treated  these  results  respect: to  a  line The  not  i n a l l cases  straight  line  results  from  t h e Y-vS  Figures  i n the  respectively  method  i n f i g u r e 26.  relate  reported  PVF o r a n y o t h e r  method  were  observed  SiO surface.  the  conclusion.  coefficient  W  o  The  from  of  were  still  3W/30  larger  1  </>  The  - Papoular  the  Note t h a t  same  of  f o r t h e PVF  surface  shows  the r e s u l t s  above; a g a i n  these  the  potential.  outlined  results  of  t h e Y-vS m e t h o d f o r c o m p a r i s o n .  69  from  i s f o r s a m p l e 1. as  figures  19-21 Figure  p o t e n t i a l of  The  same p r o b l e m s  i n t h e ED  as  are o u t l i n e d  above f o r  results  results  o f t h e ED m e t h o d  as  this  the r e s u l t s  the anchoring  i m p l i c a t i o n s o f these 30  fitted of  f o r t h e PVF a l i g n e d s u r f a c e .  measurement  i n one  be  slope  anchoring  this  information  rubbed polymer.  Figure  useful  v s 4> c o u l d  analysis are p l o t t e d alongside  rubbed  the  of slope  region  but are the r e s u l t s  i s the f i r s t  they  o f the R a p i n i  this  27-29  encouraging,  the graph  i s the constant  using  29  were  of  are discussed extracting  f o r t h e PVF  are p l o t t e d with  the  in the  surface results  30-1  o • A o  25-^  ED Method. C ED Method. 6 Y - v S Method Ref. 11  20-1 tD  a  is-  e  Q>  -i->  CO  a  o  Xi  o  d  ID-  o •  S' 029  —r-  30  —i  1—  1  31  32  —T—  33  o9.  ~~r~ 34  35  Temperature (°C)  Figure  26. A n c h o r i n g s t r e n g t h results  vs temperature  o f ED method.  70  f o r SiO treated  surface:  0.8H  0.6-1  0.4H  0  10  20  30  V/V  Figure  27.  Tilt  angle  as  function  of  surface.  71  40  50  60  th  reduced voltage  f o r rubbed  PVF  Figure  28.  Surface  torque  as  a  function  surface.  72  of  tilt  angle  f o r rubbed  PVF  30  T  25 H  20H  A  a o A  o  ED Method, C ED Method. 6 Y-vS Method  •  00  g  WH  o  e  •  10-  A  50-  29  30  i  I  31  32  -T  -  33  —r34  S i  35  Temperature (°C)  Figure  30. A n c h o r i n g results  strength  vs temperature  o f ED method.  f o r rubbed  PVF  surface:  0.30-t  0.25-  *E \  0.20-  0  2  0.15-  3~  0.100.05-  0.00  0.2  0.0  Figure  29.  Anchoring PVF  0.4  p o t e n t i a l as  0.6  a function  surface.  73  of t i l t  o.a  angle  1.0  for  rubbed  Chapter  V  Discussion  and  Conclusions  Comparison with  The  Yokoyama e t a l ' s Work  surface  when  aligned  using  the  anchoring  strength  homogeneously  method  on  developed  by  that  the  m e t h o d was  ref.  11  this  experiment  be  was  a  SiO,  employed  reproduced;  one  nematic  different  Yokoyama  t a n g e n t i a l l y evaporated  could  the  two  surface,  being  of  and  was  and  in  1 0  the  t h a t was d  The  first  to  verify  results not  near  5CB  measured  i n order that  however  increase  was  Sprang.  employed  feature  crystal  surfaces  van  correctly  "critical"  liquid  the  seen  of in  clearing  e  point.  It  i t i s not  known i f t h e  become more p r o n o u n c e d n e a r e r The  graph  appears 11,  the  something  this  same  of  interesting surface  method. by  strength  this  was  is  transition  of  SiO  32.5°C. been  vs  This  predicts  in  seen  temperature was  that  this  was  in d  would  e  temperature.  reproduced  happening  strength  alignment,  1 6 , 1 7  is  of  the  m e t h o d on  found  result  and  i t has  anchoring  This  important  simpler  at -  that  not  commented  implies region.  behavior.  (figure  16)  on  strongly  ref that  Certainly Further  no  study  in  i s needed.  anchoring  quantity  that  the  strength  feature  fact  region  The  anchoring  to have a  but  theory  of  to  increase  to  and  cheaper  thus  first any  be  because  a buffed  report  film of  large  SiO  is  said  produces  treatment.  as  that to  75  measured u s i n g measurement  surface. for  produce  strong  Furthermore,  was  the  rubbed polymer  as  PVF  PVF  SiO.  also  some s a m p l e s  this  anchoring  This  "strong"  alignment on  The  of  the  is  an  surface with  a  i n which  Che  surface  treatment  used  l e n g t h was o b s e r v e d . of  surface  alignment  was  buffed  PVF,  The i m p l i c a t i o n s u c h p e r s i s t s through  a negative  results  extrapolation  i s that  the e f f e c t  a layer of thickness  d  next to e  the  plates,  field. the  and  on t h i s  the anchoring  aligning  treatment.  i t s poor  PVA  is a field  could  A  new  solely  be  My  solubility  i n most  at  large which  was n o t s e e n .  The PVF method to  (vinyl-alcohol) film with  PVA h a s b e e n  alignment with  solvents.  1  serious  that  than  results  there  i s an  o f nematics, Deuling's This  7.  was  enough  can  i n f i g u r e 31. T h i s  strong  was  method  i t i s much  strength  of  potential,  developed.  does  of  not  based  The  the  new  splay  assume  the  potential. i s the strong  in 7  to a f f e c t  76  as the  PVF, m a i n l y due  solution  6  The d i r e c t  uncertainty  uncertainty be s e e n  large  anchoring  i n t h e ED m e t h o d  on the c o n s t a n t  LCD  knowledge.  form o f the anchoring  difficulty  nematic  The a n c h o r i n g  the surface  of  producing  i s used  o f measuring  extension  of  the t w i s t e d  method  an  In the  down.  Method o f M e a s u r i n g A n c h o r i n g P o t e n t i a l  is  of  the r e l a t i o n s h i p between i t  t o my  theory  reorienting  the e f f e c t  Deuling  Rapini-Papoular  that  very  strength  interest  to produce uniform  transition.  A  a  length  experience  Freedericksz  was  of  a rubbed p o l y  on the continuum  method  the  breaks  has n o t y e t been measured  Extended  of  l a y e r i s overcome, b u t t h i s  strength  i n which  more d i f f i c u l t  influence  extrapolation  alignment  industry,  the  there  of a negative  surface  to  under  Presumably  surface  case  even  result of  at  of this most  the r e s u l t s  dependence  dependence o f  1%,  dependence and  this  considerably  o n 7 may b e due t o  as  Figure  31.  Surface  torque  dependence the  on  as the  a  function  dielectric  data.  77  of  tilt  angle  anisotropy  used  showing in  the  fitting  the  assumption  uniform is  a  the  throughout  poor  well  of  the  assumption  known  Recall  continuum  the  that  cell in  defects  electric  model  and  dependent  regions in  of  very  nematics  coherence  that  the  order  only  on  high  deformation  often  have  (eq.  2.30):  length  parameter  S  temperature.  an  is  This  --  i t is  isotropic  core.  5  5.1  I n eq. width  2.38, of  i t can  the  experiments it  region  as  (with  to have  this;  a  nematic  be  expected with  £  shows  as  i n the  to  order  of  approximately  the  largest.  the  the as  50  the  molecular i n S of  50  nm.  nm.  in a  region  splay  energy  field  the  energy  order  When one  In  Certainly  parameter  equating  the  decreases  length  is  short  calculation,  that  i s the  34>/dz  become  constant)  a coherence  length  slope  could  order  elastic  ordering  the  this  a decrease  first  a constant  to  seen that  where  performed,  is possible  such  be  of  due  1%  may  considers  that  2  the  elastic  constants  become e v e n l a r g e r . It  i s thus  to  allow  the  order can  results  voltage the  fact  second be  (v <  that  order  important  of  2)  well  ED  exhibited the  phase in  the  the  vary  transition, region  the  78  would  change  t o be  before  in  the  surface  using  this  region  of  7.  to 19)  one  7.  extended  potential  transition  near  S  of  shape.  hence  .6%  in  experiments  anchoring  and  of a  spatially  (figure  change  model needs  with  unusual  Freedericksz  the  result  sensitivity  method an  S , the  continuum to  found great the  to  shows t h e  described  determining  o f e x p e r i m e n t have The  that  31  parameter  be  methods o f  proportional  Figure  conjectured  interaction Other  are  this  type. type  1 0  in  the  This may  may  be  been  thought  expects  transition.  have  small  of  as  due a  f l u c t u a t i o n s to  This  also  is  not  accounted  f o r i n continuum  Also  in  vicinity  figure V ~  of  19,  17  V  for  both  presence 3W/3^ was  a  and  When  can  T h i s was  types  of  surfaces.  to  either  shortcoming  visible  effect  "jog"  .  points  or  i  a  in not  the  the  an  of  raw  the  data,  a bulk effect.  analogous  material  depend  so  capacitance  In  was  f.  figure  constant  case.  bounds  Its  jump  Its origin  curve  in  the  f i t in  in  This  on is  this  in  c o u l d be  this  the  due  surface  phase  data,  very  much  the  fact  that  the  constants that  so  break  with  to  however  IV)  torque  no  be  its  understood.  but  reason  Chapter  that  must  to  obtained,  but  surface  Note  effect  material  for  the  applied  required,  the  unknown,  i s not y e t  was  were  (i/) was  discussed  is  the  24  It  origin  hence  calculation  strongly  data  the  7  that  results  did  as  the  was the  in  adjusted two  fits  i n the r e g i o n o f h i g h v o l t a g e . both  line  the  a l l data  and  c.  scatter:  agreed  feature of  theory.  more  (within  in  continuum  equivalent results  not  a  seen  unexpected  qualitatively  another  be  th  J  way,  theory.  cases,  i n the  dW/3^  region of  calculated.  as  a  largest  These  function ^  results  and  of  hence  were  <f> was the  f i t to  those  straight  strength W  anchoring  comparable  a  of  the  Q  Y-vS  method.  Conclusion  Two each  on  methods have been two  but  based  on  The  results  different a of  linear this  used  to  surfaces. expansion  method were  study The to  surface anchoring  first first  difficult  79  method order to  properties,  is straightforward, i n the  reproduce  displacement. on  subsequent  runs.  The  strength  results  of  reasonably developed  from  tangentially  with i n this  thesis  deformations.  surface  anchoring  optical  (phase)  values  spatial  where  temperatures parameter  where  for  numerically numerical Both  either  a result  the  this order  t o be angle  of  and  the  found is  to  The  agree  ED  method  theory  determination  t o be  very  thought  to  were  those  of  the  or  sensitive be  caused  i t s usefulness  of  o f the  (capacitance)  parameter  variation  smaller.  found  anchoring  the continuum  successful  spatial  were  dielectric  of this,  most  on  allows  scalar  surface  due  by  to  large  i s limited; at  scalar  higher order  S o l u t i o n o f the t r a n s c e n d e n t a l  the  director  requires  at  the  sophisticated  surfaces  is  techniques  of  integration. the  Yokoyama-van  were  liquid  crystals:  found  the  from  the  results.  ED m e t h o d was  was  intensive  method  was  i t  i s expected  equations  method  the  As  SiO  solely  parameters;  in  gradients.  cases  The  material  for  published based  potential  variations  director the  of  was This  data.  method  evaporated  previously  nematic  to  this  applied  to a  rubbed  Sprang new  PVF.  method  type The  of  and  surface  anchoring  the  Deuling  for aligning  nematic  strength of this  surface  t o be as s t r o n g as t h a t o f t a n g e n t i a l l y  80  extended  evaporated  SiO.  Appendix:  Effect  Anchoring  The  Freedericksz  strength this  thesis  that  measurements and thus threshold  magnetic  observed exists  by  a  aligned  saturation the  Kmetz a n d S c h e f f e r Recall  was and  field  predicted Yang.  -  2  o Where  V  =  o  calculated The  letting  go  was  shown  and  1  that  there  homogeneously  predicted  by  Nehring, 1 0  T  s i n (V>)  K.rj  1/2 di/>  V  e 6  zero;  rj = 0. T h i s  which  2  o  Deuling  to  This  voltage  and P a p o u l a r ,  becomes  2  threshold voltage  4>^ b o t h  the c e l l  the  The c h a n g e i n t h e  i s also  (1 - n s i n ( i / > ) ) ( l + tv  IT / K /e by  effect.  It  1 +  /  v  of  in  2.44:  n V 1 +  V  accuracy  by R a p i n i  2 6  field.  7T/2  1  the  on t h e  described  a n d o b s e r v e d b y Yokoyama a n d v a n S p r a n g .  2 7  equation  than  a t which  applied  depends  In the experiments  i s less  field  Field  i n general  th  i s n o t an o b s e r v a b l e  Rosenblatt  along  V  voltage  anchoring.  change  Surface  on t h e T h r e s h o l d  threshold  o f the s u r f a c e  of Finite  i s the  f o r the case i s defined  as t h e l i m i t  i n equation  2  Freedericksz  of infinite  A . l , this  A.l  sin (V>) threshold  anchoring  voltage 6  strength.  of V  above when <j>  limit  i s evaluated  and by  yields: V  — V  In  the  zero,  infinite  and  anchoring  anchoring  the t h r e s h o l d case  Tp  is  =  *"  o  case  voltage  defined  I - - rp  the lower is V  q  through  1  as  A.2 1  limit  81  the  expected.  In  integral the  m  is  finite  sin(V> )sin((£ ) = s i n ( 0 )  1  hence:  of  1  and  sin(^ cos  v)  (^  )  =  A.  3  sin(0 ) when b o t h  <j> and  <f> are  1  obtained  from  the  torque  -  K D 1  1  /  insert  e  o  K 2  the  is  generally v a l i d  between  (0  m  (2.33)  in  4>  and  .2  ) - s i n (4 ) )  (1 +  l  sin U  K  ) )( 1 + 7 s i n ( ^  sin (^ 2  2  1  form in  of  the  i n 4>  the  anchoring  regime  and  where  <j> , t h e  m  energy  <f> i s  4> i s m  r) th  which  the Euler-Lagrange thus:  IS  R-P  linear  equation  2 ,, ,  (1+7  1  we  terms  balance  '—  v £  If  keep  relationship  first integral of balance equation i s (sin  aw  The  1  obtained from the (2.37). The t o r q u e  34>  small.  m  1  ) )  is  equation 1/2 A.4  )  (Eq.  2.23)  s m a l l ) , and  (which  then  only  result i s :  1 1/2  W  0  <j>  =  1  K D  <t>  1  Noting  that D  i s g i v e n by  q  squaring both  equation  2 1  A = 7r K /W 1  inserted  into  d  and  0  D  v  2  /D th 0  e q u a t i o n A.3,  typical  .0007 a n d That  is  predicted the  results  .0015. the  This  from  by  with  4> m  s  A.5  equation  becomes,  upon  =  less which  A.6  -  Z  V  (—  2  1  /V th 0  u)  chapter  results  threshold  values  accuracy  this  1  v.  =  When  equation  A. 6  is  results  for  v.  a transcendental equation  cot Using  2.9,  -  sides:  .2 A  where  m  £ £ K 0  2  A 1/.  IV,  A  is  A.7  found  to  i n 1/ r a n g i n g b e t w e e n  voltage than  =  observed .1%.  voltage  can  82  This be  will  be  range  .9990 a n d reduced  deviation is measured  between  in  .9995.  from  the  approximately  the  experiment  described cell for  i n Chapter  III.  ( d ~ 3 JJ) o b s e r v e d the bend  decrease theory A  Rosenblatt  a decrease  Freedericksz  was i n e x c e l l e n t  analogous  to that  and Y a n g ,  using  i n the t h r e s h o l d  transition agreement  2 6  a very  magnetic  o f approximately  with  the value  thin field  25%.  This  p r e d i c t e d from  a  above.  saturation voltage V  i s d e f i n e d as t h e l i m i t  o f e q u a t i o n 2.40  sat  4> a n d 4>  when b o t h  go t o n/2.  That  limit  may be e v a l u a t e d b y m a k i n g  1  m  2 the  | = c o s (<f> ) , a n d cos(<£) = c o s (4> )cosh(V>)  substitutions  m  integral equation  Where  o f equation  2.40.  cosh(V> )  V  =  - <f> ,  where  2  / , - V 1 +  =  1  K  £ goes  to zero  this  i/>  A. 8  Defining a  a s n/2  -  <t> a n d /? a s  1  tn  and i n the l i m i t  the  o  V c o s ( $ )/cos(<£ ) .  1 7r/2  In the l i m i t s at  becomes:  in  m  where  both  a  a n d jl a r e s m a l l  one c a n  ra write:  coshC^) = a//3. One  can also  However the  angle  the torque  the Rapini-Papoular  region  energy,  evaluate  where  i t must  4> -  anchoring  w/2.  torque  balance  potential  about  n/2.  one c a n a p p r o x i m a t e W(^)  The  equation  equation  Nonetheless,  be symmetric  i s c l o s e t o n/2,  balance  A. 9  may n o t b e v a l i d i n the  anchoring  Therefore,  when  the  the anchoring  tilt  energy a s :  2  A.10  w h e r e a a n d (3 a r e b o t h  i n the l i m i t  limit.  whatever  \ -0 j  - \ Wj  i n this  small  is:  /~T~  K D  1  f (a - / T ) ( l + * ) "  / £ £K < -  sat  o  21  1/2 =  (1 + ) 7  83  2  W  Q  l  A  l  l  A. 11  Replacing v  D  /D sat  °  with  v't  0 A'  Where A' = n K /W d. l l  2  /<r , t h i s 1 2  i/'  This  2  becomes: (a  (1 + K)  reasonable  van  i n A.9  •  and  2  i s inserted  coth  Yokoyama  - P)  2  Sprang  10  agreement w i t h  -J  1 +  K  a  and  the r e s u l t i s :  A'/ 1 +  2  K v'  12  A.13  •  measured this  A.  =  the  theory.  84  saturation voltage  and  found  References  1.  A. R a p i n i and C4-54 ( 1 9 6 9 ) .  2.  G.  3.  G. B a r b e r o , N.V. Madhusudana and 39a. 1066 (1984).  4.  W.H.. de Jeu, Physical Properties of Liquid M a t e r i a l s . G o r d o n and B r e a c h , New Y o r k ( 1 9 8 2 ) .  5.  P.G. de G e n n e s , The Physics of Clarendon P r e s s , O x f o r d (1974).  6.  H.  7.  H.J. Deuling, i n Liquid Crystals. S u p p l e m e n t 14, e d i t e d b y L. L i e b e r t P r e s s , New Y o r k , 1 9 7 8 ) .  8.  R.B.  9.  W.  10.  H. Yokoyama a n d (1985) .  11.  Barbero  and  M.  G.  Papoular,  Phys.  Helfrich,  Z.  Phys.  Durand, J . Phys.  J . D e u l i n g , Mol.  Meyer,  J.  Cryst.  Rev.  (Paris)  B.  47,  Durand,  Liquid  L i q . Cryst.  Z.  19,  2129  (1986).  Naturforsch.  123  Crystalline  (1972).  Solid State (Academic  918  (1969).  N a t u r f o r s c h . 26a,  833  (1971).  J. Appl.  H. Yokoyama, S. K o b a y a s h i 61, 4501 (1987).  and  Kamei,  12.  S. S h i m o d a , H. Mada a n d P h y s . 17, 1359 (1978).  S.  13.  S.W. Morris, M.Sc. Columbia, 1985.  14.  W.H. P r e s s , B.P. F l a n n e r y , S.A. T e u k o l s k y and V e t t e r l i n g , Numerical Recipes. (Cambridge,1986).  15.  J . L . J a n n i n g , A p p l . Phys.  16.  J.Cognard,  17.  D. R i v i e r e a n d L-215 (1979).  18.  W.A. L a a n d G.J. K n i g h t , i n P o l y m e r Handbook, e d i t e d b y J . B r a n d u p a n d E.H. Immiyat ( I n t e r s c i e n c e , New York, 1966) p 111-61.  Y.  H.  Cryst. Levy,  21,  L i q . Cryst. J.  85  Phys.  173  4520  Phys.  Jpn.  Appl.  University  Lett.  57,  J. Appl.  Kobayashi,  thesis,  Phys.  Physics  Sprang,  Molec.  van  30.  Crystals.  22,  H.A.  Lett.,  Colloq.,  J.  of  British  W.T.  (1972).  Supp., Lett.  1. 40,  19.  J.M. Appl.  Geary, Phys.  J.W. 62,  Goodby, 4100  A.R.  Kmetz  H. Yokoyama, S. K o b a y a s h i 56, 2645 ( 1 9 8 4 ) .  21.  S t e p h e n A. P h y s . Rev.  and  H.  22.  H.A.  23.  D.A. Dunraur, M.R. Manterfield, Dunleavy, Mol. C r y s t . L i q . 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