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An EPR study of order and molecular orientation in liquid crystals MacKay, Alexander Lloyd 1971

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AN EPR STUDY OF ORDER AND MOLECULAR ORIENTATION IN LIQUID CRYSTALS  "  by ALEXANDER LLOYD MACKAY  B.Sc,  Dalhousie  U n i v e r s i t y > 1969  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS MASTER in  FOR THE DEGREE OF  OF SCIENCE  t h e department of Physics  We  accept  required  THE  this  thesis  as c o n f o r m i n g  to the  standard  UNIVERSITY OF B R I T I S H COLUMBIA April,  1971  In presenting t h i s thesis in partial fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make i t freely available for reference and  study.  I further agree that permission for extensive copying of this thesis for scholarly purposes may by his representatives.  be granted by the.Head of my  Department or  It is understood that copying or publication  of this thesis for financial gain shall not be allowed without my written permission.  Department of  •  P HVS/C S  The University of B r i t i s h Columbia Vancouver 8, Canada  Date  APftlL  10,  f 3-7 1  ii  ABSTRACT  Using ive in  order  in  parameter  the nematic  aniline  EPR,  was  the temperature of the molecule  liquid  crystal  studied.  type  of motion  obtained The  found,  vanadyl  4-methoxy  In the r e s u l t i n g  slope' o f t h e curve  e r a t u r e was  dependence o f t h e e f f e c t -  of e f f e c t i v e  indicating  discontinuity  T  benzylidene-n-butyl data,  order  a  parameter versus  liquid  c a n n o t be i n t e r p r e t e d s o l e l y  temp-  restricted  Similar results  m e a s u r e m e n t s on t h e p u r e  x  discontinuity  a c h a n g e t o a more  at lower temperatures.  f r o m NMR  acetylacetonate  as a  were  crystal. viscosity  effect. Two vanadyl (~20  acetylacetonate  found  of cholesteryl to orient  a.2 p e r c e n t  nematic tended  liquid  crystal  were s u b j e c t e d  to a high  contalnine magnetic  c h l o r i d e and c h o l e s t e r y l m y r i s t a t e  i t shelix by w e i g h t  axis p a r a l l e l mixture  t o arrange  their  long  i  .  ! I  V-  aniline,  axes p a r a l l e l  axis perpendicular  I  to the f i e l d  of cholesteryl  4-methoxy b e n z y l i d e n e - n - b u t y l  making t h e h e l i x  Tiixtin^es  Prom EPR m e a s u r e m e n t s , a 1 . 7 5 : 1 m i x t u r e  kilogauss).  weight  In  chclesteric  to the  -  field by was  direction.  chloride i n the  molecules  t o the f i e l d field.  thus  iii  TABLE OF CONTENTS page Abstract Table  i i  o f Contents  List  of Figures  List  of Tables  i i i v . . . . . .  Acknowledgements  v i i  .  viii  . . . . . . . . . . . .  CHAPTER 1 Liquid  . .  ' Crystals  '  -  i. ' 1  . :  CHAPTER 2 Choice  o f Vanadyl  EPR T e c h n i q u e Preparation  Acetylacetonate  ...............  5 0  o .  .  8 13  o f Samples  CHAPTER 3 The S p i n H a m i l t o n i a n Interpretation  ...  18 21  o f The EPR S p e c t r a  CHAPTER 4 Order  32  i n t h e N e m a t i c Phase  'CHAPTER 5 |  Orientation of Cholesteric a Magnetic F i e l d  Liquid  Crystals  with ^3  iv  page CHAPTER 6  . . . .  Discussion  APPENDIX  53  . . . . . .  1  Transformation of the Spin Hamiltonian f o r A x i a l l y Symmetric M o l e c u l e s APPENDIX  55  2  The E n e r g y E i g e n v a l u e s o f t h e S p i n f o r A x i a l l y Symmetric M o l e c u l e s References  Hamiltonian  58 .  6l  V  L I S T OF FIGURES page crystals  4  o f VAAC  7  Figure  1 .1  The t h r e e  types  Figure  2 .1  Schematic  of the structure  Figure  2 .2  Block  Figure  2 .3  The t e m p e r a t u r e  Figure  3 .1  The m o l e c u l a r  Figure  3 .2  Absorption f o r VAAC  ( a ) and d e r i v a t i v e (b) l i n e s i n a n e m a t i c sample when Y 0  of liquid  d i a g r a m o f t h e EPR control  coordinate  11  spectrometer  12  apparatus s y s t e m f o r VAAC  =  o  25 26  Figure  3 .3  Absorption f o r VAAC  (a) and d e r i v a t i v e ( b ) l i n e s i n a n e m a t i c sample when y=90°  26  Figure  3  Absorption f o r VAAC  (a.) and d e r i v a t i v e ( b ) l i n e s i n a ramdom sample  27  ~>  A h ^.r.-r^  '"  •»-T  - --, ~  '^  *~.  f o r VAAC Figure  3 .6  ^  ^  « - - > • ? • < - - - - - o  i n an i s o t r o p i c  27  sample  S p e c t r u m f o r VAAC i n a n e m a t i c c r y s t a l when y - 0  liquid  S p e c t r u m f o r VAAC i n a n e m a t i c c r y s t a l when y 9 0 °  liquid  o  Figure  3 .7  28 29  =  30  Figure  3 .8  S p e c t r u m f o r VAAC  i n a random  Figure  3•9  S p e c t r u m f o r VAAC  i n an i s o t r o p i c  Figure  -4. 1  Hyperfine s p l i t t i n g versus kinematic v i s c o s i t y f o r VAAC i n O c t o i l  36  Figure  4 .2  E f f e c t i v e order parameter versus reduced t e m p e r a t u r e f o r VAAC i n t h e n e m a t i c l i q u i d c r y s t a l s : ( a ) b i s (-J ' - n - o c t y l oxybental)-2-chloro-l,4-phenyle n e d i a m i n e , ( b ) MBBA, ( c ) 4-methoxy benzylidene-4-amino-a-methyl cinnamic acid-n-propyl ester  37  sample sample  31  6  8  Figure  4.3  Kinematic v i s c o s i t y versus temperature f o r ( a ) O c t o i l , X , ( b ) 4-methoxy b e n z y l idene-4-amino-a-methy1 cinnamic a c i d - n p r o p y l e s t e r , o , and ( c ) MBBA,• 8  Figure  4.4  Luckhursts t h e o r e t i c a l c u r v e i s compared w i t h c u r v e s o f S f f and S f f - 1 / 3 (Syis ) v e r s u s t e m p e r a t u r e i n MBBA 7  e  +  e  D  Figure  4.5  P r o t o n r e l a x a t i o n time T i v e r s u s r e c i p r o c a l o f t h e t e m p e r a t u r e a t 1 8 . 2 MHz f o r MBBA  Figure  5.1 "~  Peak i n t e n s i t i e s f o r t h e C-MBBA m i x t u r e versus o r i e n t i n g magnetic field  Figure  5.2  Peak i n t e n s i t i e s f o r t h e CM m i x t u r e o r i e n t i n g magnetic field  versus  vii  L I S T OF  TABLES page  Table  3.1  P a r a m e t e r s f o r VAAC i n a n e m a t i c liquid crystal  20  viii  ACKNOWLEDGMENTS  I would the C.  like  t o e x p r e s s my  h e l p f u l guidance provided F. S c h w e r d t f e g e r ,  writing students and  appreciation of  by my r e s e a r c h  s u p e r v i s o r , Dr.  i n a l l stages  of this -thesis. i n the Solid  of research  I am a l s o v e r y  -  assistance  sincere  State  Physics  discussions  with  of research.  i n the achievement  m i u i -Ui  f K O O V J T  uu.m..^ ^ _ ~  4- c: ^ *3  y.  The Council  ^  4- l->  scholarships  o f Canada d u r i n g  J. —  o f my p r e s e n t  _J - _  ~J-?^fiiT(S  w _  As w e l l , t h e  Mr. M. M a r u s i c h a v e  extremely b e n e f i c i a l ^3 ^  g r a t e f u l t o a l l the  Group f o r much i n s t r u c t i o n  t h r o u g h o u t my p e r i o d  many e n l i g h t e n i n g  and i n t h e  ^ v n r q f - a  1  q  J i  awarded by t h e N a t i o n a l  the period  of research  r. J_ i  been level of  i  Research  are g r a t e f u l l y  acknowledged. The National  research  Research  f o r this  Council,  grant  t h e s i s was s u p p o r t e d by t h e number  A-2228.  CHAPTER 1 Introduction  to Liquid  Liquid  crystals  Austrian botanist, compound  it  were f i r s t  Frederich  cholesteryl  l45°C i t c h a n g e d  Crystals  Reinitzer,  liquid.  compounds, d e f i n e d  coined  the At  l i q u i d and a t 179°C significant  0. Lehmann who  some o f t h e i r  liquid crystal  roesophase)  of  a l i q u i d and a c r y s t a l l i n e s o l i d . properties  with  i s a state  as a  properties  found  work  other  characteristics,  As w e l l  anisotropic  of matter  i n t P T m p d i a t e  common t o l i q u i d s . o f  form d r o p l e t s .  Liquid  optical  between those  fluidity  characteristics.  Further,  to external  influences,  and  magnetic  than  crystalline solids  through a t r a n s i t i o n  or heating  T h e s e a r e c l a s s i f i e d as t h e r m o t r o p i c .  to  they  s u c h as e l e c t r i c or l i q u i d s .  a r e f o r m e d by c o o l i n g  temperature  to  of crystals  more s u s c e p t i b l e  either  possess  and a b i l i t y  are  fields,  (referred  crystals  they have t h e p r o p e r t y  Some l i q u i d c r y s t a l s  '  The n e x t  to  exhibit  that  by a n  t h e word l i q u i d c r y s t a l . The  the  who. f o u n d  from a s o l i d t o a t u r b i d  was done by a German p h y s i c i s t  and  i n 1888  b e n z o a t e h a d two m e l t i n g p o i n t s .  became a t r a n s p a r e n t  similar  discovered  a liquid  a solid  crystal.  A l l t h e work  consider-  j  |  ed i n t h i s  study  i s on t h e r m o t r o p i c  mesophases a r e f o r m e d by a d d i t i o n T h e s e a r e known  as l y o t r o p i c  -  liquid crystals.  of a solvent  liquid crystals.  1  Other  t o a compound. The  existence  2  of this  type  ture but  by  of l i q u i d  in  i s determined  n o t by  tempera-  solute concentration.  Liquid Slight  crystal  crystals  variations  i n the  have p r e d o m i n a n t l y  long  shape o f t h e s e m o l e c u l e s  molecules. and  changes  the molecular order are r e s p o n s i b l e f o r the three types:  smectic, nematic, "vl.l,  these  and  As  illustrated  t h r e e types have v e r y d i f f e r e n t  In smectic parallel  cholesteric.  t o each  o t h e r and  of a l l the molecules another  and  Smectic  liquid  liquid  crystals arranged  i n layers.  are  The  are p a r a l l e l  o f t e n p e r p e n d i c u l a r to the plane are o p t i c a l l y  figure  structures.  the molecules  i n a given layer  crystals  in  to  of the  positive  l o n g axes  and  one  layer. usually  unaxial. Nematic materials.  The  into  and  optically  determined. which The is  planes.  of nematic  have  are p a r a l l e l  substances  Some s c i e n t i s t s  The has  not  are arranged  as  "continuum  at every point  i n the  liquid  This  preferred  position.  orientation  direction  yet been  support  i s the  f o r the  order than  t o each  l o n g range  view  direction  less  the  not  unaxial  molecular  stru-  satisfactorily  "swarm t h e o r y " i n  i n f e r r o m a g n e t i c domains. t h e o r y " which  crystal  of the  smectic  o t h e r but  Nematic m a t e r i a l s are u s u a l l y  positive.  the molecules  opposing  crystals  molecules  divided  ture  liquid  claims there  a definite  long molecular  i s assumed t o v a r y  preferred axes.  continuously with  3  In  the t h i r d type  the molecules parallel in  a r e grouped  to the layer.  of liquid  i n layers  I n each  crystal,  w i t h t h e i r l o n g axes  layer  the molecular  t h e same d i r e c t i o n b u t due t o t h e s l i g h t l y  shape o f t h e c h o l e s t e r i c a x e s i n one l a y e r direction  i s slightly  i n the neighbouring  this  direction will  line  cutting  liquid optical  through  rotary  power e x i s t s  layers..  from  the corresponding  T h r o u g h many  the planes  about  axis. this  of a  liquid  36O . 0  The  A very  strong  axis.  c r ? t ; p i p. v  layers  cholesteric  research described i n this thesis  and c h o l e s t e r i c  point  unsymmetric  change c o n t i n u o u s l y t h r o u g h  normally  axes  the d i r e c t i o n of the long  different  c r y s t a l i s c a l l e d the h e l i x  The nematic  molecules  cholesteric,  deals  with  :  >  helix  axis  -  Cholesteric Figure  1.1  The  three  liquid  crystal  structures.  CHAPTER 2 Choice of Vanadyl  The  Acetylacetonate  liquid  c r y s t a l s used  t h e m s e l v e s , e x h i b i t EPR s p e c t r a . ation in  about  the l i q u i d  i n t h i s study d i d n o t , In order  to extract  c r y s t a l s i t was n e c e s s a r y  inform-  to dissolve  them a p a r a m a g n e t i c p r o b e w h i c h d i d have an EPR s i g n a l . For  t h i s study, vanadyl acetylacetonate  c h o s e n as t h e p a r a m a g n e t i c rated VAAC  i n figure loses  2.1.  probe.  -Being e s s e n t i a l l y p l a n a r  of a liquid  in  to useful  the solute  leads  crystal.  about  o f freedom  the l i q u i d  was c h o s e n f o r s e v e r a l  It i s readily available.  temperature  liquid  splitting. This  loss  itself.  reasons.  most  i n shape,  This  information  Vanadyl acetylacetonate  quired  is illust-  some m o t i o n a l f r e e d o m when i n s o l u t i o n w i t h t h e  elongated molecules  crystal  I t s structure  (VAAC) was  range  crystals. Only  I  i n the r e -  (0°C-100°C) and c a n be d i s s o l v e d  VAAC h a s a l a r g e  one n u c l e a r  moment  i s t h e vanadium n u c l e u s w i t h  hyperfine  I t i s stable  anisotropic  contributes  i t s spin  in "  hyperfine  to the spectra.  7/2 p r o d u c i n g 8  lines. Before discussing  t h e u s e o f VAAC  i t i s necessary  i ;to d e t e r m i n e how w e l l of  the l i q u i d  treatments  i t s spectrum represents  crystal.  In the l i t e r a t u r e there  o f t h i s problem.  .  -  the o r i e n t a t i o n  G l a r u m and M a r s h a l  5  are several 1  h a v e shown  that  the degree  observed ponds  f r o m t h e u s e o f VAAC  closely  refractive that  to that  o f the pure  i n d e x measurements.  plot  by c h a n g e s  liquid crystal  curve  that  of solutes  the ordered structure  ed by t h e p r e s e n c e  of*a  not  i t appears  yet conclusive  paramagnetic  probe  second  i s at least  corres-  as f o u n d by have  2  a liquid crystal of a second  reduced temperature  f o r a wide range  probe  Chen and L u c k h u r s t  i n the concentration  of order against  dependence  as a p a r a m a g n e t i c  t h e d e g r e e .of o r d e r o f V A A C . i n  altered  ing  o f o r d e r and i t s t e m p e r a t u r e  produced  found  i s not solute.  A  a common  and c o n c e n t r a t i o n s s u g g e s t -  o f t h e mesophase was n o t a l t e r solute. that  While  the evidence i s  t h e u s e o f VAAC  qualitatively  as a  justified.  Figure  2.1  Schematic  of the  structure  of  VAAC.  8  EPR  Technique  The an  spectra  X-band EPR  used  spectrometer  wave power s o u r c e was V-153/6315. to  operate  the  Packard  k l y s t r o n the  microwaves  by  the  k l y s t r o n was and  The  Varian  directly  cavity  gain  of  the  The  output  40db  i n the  was  the  used  traced  the  of the  micro-  was  used  range  one  output.  side  into  After  side the  Power a  slide sample  a f t e r one  made  were mounted  detector  of  1KC  to  magnetic  an  c a r r i e d out  Communications Packard Moseley  Inc 680  coupled  AC  current  IMC.  A  100KC  field  and  to  s i g n a l f o r phase s e n s i t i v e was  then  cavity.  to modulate the  and  and  bridge.  patterned coils  passed  50db.  other  1N23B c r y s t a l  frequency  A Hewlett the  the  modulation  phase s e n s i t i v e d e t e c t i o n  Amplifier.  on  on  a p r e a m p l i f i e r w h i c h had  reference  t r o n i c s , .Missies,  er  of  and  arms -  c a v i t y was  The  isolator  a magic t e e  i n t o two  load  first  of range 0 to  entered  side walls  s i g n a l to  oscillator provide  a dummy  TE102 sample  The the  fed  Associates. on  The  reflex klystron  microwaves ferrite  attenuation  cavity.  KMHz.  with  klystron.  type attenuator  tuner  9.1  HP716B power s u p p l y  a Starrett flap  scre'v  at  Associates  a Microwave A s s o c i a t e s  from the  t h e s i s were o b t a i n e d  operating  a Varian  A Hewlett  Prom t h e through  in this  detection.  with  m o d e l RJB strip  an  Elec-  Lock-In  chart  record-  9  To ation  stabilize  was i m p o s e d  Frequency  the k l y s t r o n  on I t s r e f l e c t o r v o l t a g e  Controller  modulation produced  (AFC).  was  on t h e c a v i t y  klystron  was  on e i t h e r  appropiate  side  when a m p l i f i e d  tection  i n t h e AFC, p r o v i d e d  For Varian  u s e d . Tc a p p l y ation to  This The to  magnet  of cholesteric  pole  tips  an e r r o r  liquid  o f range magnetic  liquid  capable  M a g n i o n magnet  was  capable  I f the  a 10KC s i g n a l o f This  s e n s i t i v e deto restore the  c r y s t a l s :.a.  fields  necessary  c r y s t a l molecules  of producing  a glycerine  NMR  from  field  i t was  magnet probe.  0 to 23  was  for orient-  Electromagnet  s y s t e m was a r o t a t i n g  of providing  small  0 to 4 kilogauss  from  of the Varian Associates  ± 0 . 1 gauss w i t h  than  signal  magnet was p o w e r e d by a H a r v e y W e l l s field  when t h e  resonance.  the high  the  necessary with  high  kilogauss.  FFC-4  power  could  be m e a s u r e d  supply.  Incorporated i n coil  gaussmeter  measurements w i t h  errors  which less  one p e r c e n t . The  temperatures is  by p h a s e  use a Magnion model L 2 6 L a b o r a t o r y  field  frequency  at the preamplifier.  and r e c t i f i e d  t h e work on n e m a t i c  Associates  frequency.  of the resonance  p h a s e was m e a s u r e d  to cavity  by an A u t o m a t i c  at the preamplifier  resonance  signal,  klystron  a 10KC m o d u l -  The r e s u l t i n g m i c r o w a v e  a 20KC s i g n a l  klystron  the  frequency  experiments  i n this  between 0° and 1 0 0 ° C .  technically  divided  into  thesis This  two. r e g i o n s :  were  carried  temperature  out a t region  one f r o m .C°C t o room  10  temperature  The ly  temperature  of a dry nitrogen  To m i n i m i z e  cavity  this  i n liquid  less  t h e sample  inside  tube.  e i t h e r by r e g u l a t i n g  100°C.  from  about  I t was p l a c e d  ''I i s t o r was c a l i b r a t e d icury  thermometer.  were  coil  des-  immer-  c o n t r o l was  t h a n room t e m p e r a t u r e t h e nitrogen  dewar j u s t  coil  b u t went  before reach-  case.temperature  c o n t r o l was  t h e gas f l o w r a t e  o r by v a r y i n g  coil.  t o measure t e m p e r a t u r e . varied  between  rate.  the glass  In this  i n the heater  gas p a s s e d  a copper  Temperature  t h e "liquid  A Fenwall Electronics  istor  through  greater  path bypassed  along a heater c o i l  used  to the enviro-  t h a n room t e m p e r a t u r e  t h e gas f l o w  temperatures  tlcvi  the current  tube.  tube.  a t 77°K.  nitrogen  For  acheived  loss  The n i t r o g e n  g a s was p a s s e d  a c h e i v e d by r e g u l a t i n g  ing  a l o n g t h e sample  due t o h e a t  t h e sample.  temperatures  the nitrogen  nitrogen  basical-  dewar made by V a r i a n A s s o c i a t e s was p l a c e d i n  around  If  sed  gas f l o w d i r e c t e d  dewar a n d t h e sample  ired  c o n t r o l mechanism c o n s i s t e d  thermal gradients  ment, a g l a s s the  up t o 1 0 0 ° C .  a n d t h e o t h e r f r o m room t e m p e r a t u r e  Inc  GA-J3P21  t h e r m i s t o r was  The r e s i s t a n c e  70 k i l o o h m s  of this  therm-  a t 5°C t o 2 k i l o o h m s a t  i n a simple bridge c i r c u i t . w i t h a G. H. Z e a l  (-10°C  The t h e r m -  t o 1 1 0 ° C ) mer-  dummy load  slide screw tuner 7K  klystron TpC—  3  isolator  attenuator  magic trip II  crystal 'detector .  5 preamplifier  ;  cavity  AFC  100KC  modulation  phase $ sensitive detector  magnet klystron power supply  recorder error  Figure  2.2 B l o c k d i a g r a m  signal  o f t h e EPR s p e c t r o m e t e r ,  sample  tube  dewar -\,  cavity heater  ^PS^s _  ir d r y N2 gas  liquid nitrogen  heater power supply  Figure  2.3  The t e m p e r a t u r e  control  apparatus.  13  Preparation  o f Samples  For these experiments used.  These  were:  aniline  (MBBA),  esteryl  chloride  ( a ) VAAC  ( b ) VAAC  only with  radicals  i n a 1 . 7 5 : 1 by w e i g h t  of chol-  of cholesteryl  chloride  crystal., molecules  ina  i n MBBA. (VAAC)  and n o t w i t h  i t was n e c e s s a r y t o k e e p t h e VAAC c o n c e n t r a t i o n  than 1 0 " molar.  VAAC  this  Owing t o t h e i n h e r e n t low s o l u b i l i t y o f  3  c o n d i t i o n was e a s i l y f o i ' o b s e r v a t i o n oy pyrex  3 millimeters, wall 16 c e n t i m e t e r s . from  mixture  the organic r a d i c a l molecules  liquid  less  were p u t i n t o  were  i n 4-methoxy b e n z y l i d e n e - n - b u t y l  mixture  In order that  other  samples  and c h o l e s t e r y l m y r i s t a t e , and ( c ) VAAC  2 p e r c e n t by w e i g h t  interacted  three different  satisfied.  J-JPR  the l i q u i d  t u b e s . ' These  crystal  t u b e s were o f i n s i d e  thickness 1 millimeter,  They  2 t o -J k i l o g a u s s  produced  samples  and l e n g t h  no E P R s i g n a l  diameter about  i n t h e range  and d i d n o t s u b s t a n t i a l l y  decrease the  Q of the cavity. (a) F o r t h e i n v e s t i g a t i o n the nematic  liquid  crystal  o f MBBA h a v e b e e n f o u n d their  isotropic  function crystal  of time.  A fresh  t o an i s o t r o p i c  MBBA was u s e d .  t o possess  transition  sample  liquid  been proposed  this  samples  property that  are a decreasing  changes  a t about  parameter  Unevacuated  the peculiar  temperatures  MBBA i n t h e l a b h a d a n i s o t r o p i c has  o f the order  from  47°C.  transition  a nematic  liquid  Some 5 month o l d  point  at 3 7 ° C I t  d i s c r e p a n c y i s due t o c h e m i c a l  absorp-  14  tion  o f o x y g e n by t h e l i q u i d  crystal.  was c i r c u m v e n t e d  by u s i n g  used  as o b t a i n e d  from V a r i L i g h t  that  i n the presence  erature  decreased  only  f r e s h samples.  strength  The MBBA was  Corporation.  o f VAAC t h e i s o t r o p i c  I t was f o u n d  transition  temp-  t o 44°C.  Because i t i s d i f f i c u l t the  I n any c a s e t h e p r o b l e m  t o d i s s o l v e VAAC  i n MBBA and  o f t h e EPR s i g n a l i s p r o p o r t i o n a l t o t h e number  of solute molecules,  a s p e c i a l p r o c e d u r e was d e v e l o p e d  sample.  m i x i n g was c a r r i e d  The i n i t i a l  l o n g pyrex tube which had a t i n y a t fche o t h e r  end.  First,  hole  a small  out i n a 1 0  for this  centimeter  a t one end a n d was open  amount  ( a few m i l l i g r a m s ) o f  VAAC was g r o u n d up i n t o a " f i n e powder and p u t i n t o t h e b o t t o m of the tube. the  The MBBA was t h e n p o u r e d  i n t o the tube.  open end o f t h e sample t u b e was a t t a c h e d  small  the mixture.  was a d j u s t e d tube.  such t h a t  The s u c t i o n  exerted  A f t e r s e v e r a l minutes  of this,  remained a great was removed  deal  i n two s t e p s .  until  a l l undissolved  using  a drawn-out  transferred  of undissolved First,  out o f t h e  a sufficient  VAAC  vigorously  by t h e vacuum pump  t h e MBBA d i d n o t b u b b l e  VAAC was d i s s o l v e d f o r s u c c e s s f u l EPR work.  air  t o a vacuum pump,  a i r b u b b l e s were p u l l e d t h r o u g h t h e b o t t o m ,  stirring  When  amount o f  However,  there  i n t h e sample.  This  t h e s a m p l e was c e n t r i f u g e d  VAAC h a d s e t t l e d  t o t h e bottom.  Then,  e y e d r o p p e r , t h e r e q u i r e d amount o f sample was  t o a clean pyrex tube.  To e l i m i n a t e  f r o m t h e s a m p l e , t h e vacuum pump was a g a i n  dissolved  applied f o r  15  s e v e r a l minutes.  At t h i s  point  t h e . s a m p l e was r e a d y  f o r exp-  eriment . If sential  that  minimized. the  an order  any t h e r m a l g r a d i e n t s  To  The b e s t  volume o f l i q u i d  such  amount that  i n the l i q u i d that  way t o r e d u c e  crystal  o f MBBA was u s e d  this  smaller.  gradient  For this  this  i t was. n e c e s s a r y  h a v e a n EPR s i g n a l i n t h e 2 t o 4 k i l o g a u s s  Fortunately,  (b) c r y s t a l with myristate,  mixed t o g e t h e r situation  molecules  there  i s similar  which d e v i a t e  the  crystal  optical  arranged  inside  the thermistor d i d range -  should  t o a nematic  I s weakly  forces  liquid  cholesteric.  optical  liquid  optical  from t h e proper  fields  Cholesteryl  a right-handed  of these  be z e r o  a r e more r e s p o n s i v e  intermolecular  r o t a r y power.  proportions  slightly  and m a g n e t i c  The  that  hand, possesses  I f the appropiate  liquid  the  only a  Cholesteryl chloride i s a cholesteric liquid  on t h e o t h e r  electric  i t was f o u n d  a left-handed  the  as  reason  t o put t h e t h e r m i s t o r  not  tures  was t o make  and t h e sample t u b e was  cavity.  This  along  o f the experimental  the  are  c r y s t a l be  t h e e n t i r e volume o f sample was i n s i d e t h e c a v i t y .  achieve  power.  i t i s es-  a thermal gradient  t u b e was i n e v i t a b l e b e c a u s e  arrangement.  small  i s t o be a c c u r a t e  F o r MBBA i t was f o u n d  sample  the  measurement  crystals  rotary  power ;  crystal. nematic  In this  I n mix-  conditions  condition  to external influences  whereas  9  such  i n a pure c h o l e s t e r i c  a r e dominant.  c h o l e s t e r y l c h l o r i d e and c h o l e s t e r y l m y r i s t a t e  16  liquid  c r y s t a l s were o b t a i n e d  Corporation. the  i n powder f o r m f r o m V a r i  To p r o d u c e a mesophase i t was n e c e s s a r y  powders t o t h e i s o t r o p i c t r a n s i t i o n p o i n t .  they  formed  cholesteric  In  preparing  amounts o f c h o l e s t e r y l VAAC were w e i g h e d while  still  liquid  chloride,  c r y s t a l mixture cholesteryl  and t h e n s t i r r e d  into  After  eral  minutes.  m i x e d by b e i n g  Since  the l i q u i d  v i s c o u s when i n t h e c h o l e s t e r i c out  In the i s o t r o p i c s t a t e .  solved until  the correct  a homogeneous into  Relatively were u s e d  since  the  presence  The  sample  VAAC large  state,  t h e m i x i n g was  samples  the 'cavity.  sufficient percent  liquid  while  of cholesteryl  the bulk  mixture  on them  acceptable. VAAC  i n the  of the l i q u i d  amount o f a n o p t i c a l l y  c r y s t a l i s i n some  t o produce a c h o l e s t e r i c  by weight  was  the undissolved  The p r e s e n c e o f a s m a l l i n ,a n e m a t i c  dis-  centrifuged  of the cholesteric  so t h a t  crystal  solute  carried  had s e t t l e d t o t h e b o t t o m .  t h e EPR c a v i t y  active  f o r sev-  When enough VAAC h a d b e e n  b o t t o m was o u t s i d e  (c)  point.  c r y s t a l m i x t u r e was r e l a t i v e l y  o f a moderate t h e r m a l g r a d i e n t  was i n s i d e  a sample  shaken v i g o r o u s l y  f o r the experiments being performed  t u b e was p l a c e d  mixture  above i t s t r a n s i t i o n  and t h e sample a p p e a r e d homogeneous i t was a l l undissolved  cooling  m y r i s t a t e , and  b e i n g poured  t u b e , t h e powder m i x t u r e was h e a t e d T h e n i t was f u r t h e r  t o heat  crystals.  the l i q u i d  i n powder f o r m .  Upon  Light  mesophase .  chloride  1 1  cases  I f about  2  i s added t o MBBA t h e  r e s u l t i n g mixture fields  the h e l i c a l The  The  In fact,  untwist  i n strong  t o a nematic  p r e p a r a t i o n o f t h e above sample was v e r y c h l o r i d e was added  The m i x t u r e  c h l o r i d e became i s o t r o p i c  was h e a t e d  and t h e n  one  1 1  .  simple to a  u n t i l the  shaken  until  sample was homogeneous. It above  For t h i s within with  structure w i l l  o f MBBA and" VAAC.  cholesteryl  the  cholesteric.  r e q u i r e d amount o f c h o l e s t e r y l  solution  the  i s weakly  was s u g g e s t e d  samples. reason  VAAC  reacted chemically with  T h i s was e s p e c i a l l y  evident with  sample p r e p a r a t i o n was a l w a y s  a few h o u r s  b e f o r e a n EPR r u n .  s a m p l e s more t h a n  contamination  that  2k h o u r s  or d e t e r i o r a t i o n .  c a r r i e d out  No r u n s  o l dor after  MBBA.  v/ere made  any s u s p i c i o n o f  CHAPTER 3 The  Spin  Hamiltonian  An transitions magnetic by  EPR s p e c t r u m  i s d e r i v e d from  between t h e e l e c t r o n i c  ion.  The e n e r g i e s  microwave  energy  o f these  levels  levels  of a para-  are determined  the p r o p e r t i e s o f the i o n , the a p p l i e d magnetic  local  field  about  the i o n , the i n t e r a c t i o n  Hamiltonian ation  data maybe  data  appropiate  erimental  i  the spin  a l l relavent  R  f ^1"^ o ^ ur  Then t h i s  informThe p r o -  "First - the e x p -  data  i s fitted  t o an  From.the magnitude o f t h e terms  c o n c l u s i o n s a r e made a b o u t  i s an a l m o s t  axially  symmetric  " g " and "A" v a l u e s  g i s the gyromagnetic  t h e exp-  ratio  paramagnetic  are l i s t e d  i n table  and A i s t h e h y p e r f i n e  tensor. By  j f o r VAAC  a  that  system.  Its principle  coupling  Q  i s accumulated.  Hamiltonian  VAAC probe.  In order  a s m a l l number o f p a r a m e t e r s .  spin Hamiltonian.  of the spin  3.1.  with  c f a n E^R e x p e ^ i r ^ e ^ t  erimental  theory,  has been c r e a t e d t o express  concisely  cedure  compared w i t h  f i e l d , the  o f e l e c t r i c and  n u c l e a r moments, and s e v e r a l o t h e r m e c h a n i s m s . experimental  induced  f a r the largest  in a liquid  crystal  term  i n the spin  i s the e l e c t r o n i c  This  term  describes the i n t e r a c t i o n  with  the a p p l i e d e x t e r n a l f i e l d -  -  --  18  Zeeman t e r m .  of the electron  H and h a s t h e f o r m :  -H =3H.£.S Z  Hamiltonian  spin S  19  The be w r i t t e n in  g - t e n s o r f o r an a x i a l l y  i n terms  symmetric  o f two components, g  molecule can  and g , r e s u l t i n g  ((  x  the following expression: H =3(g i  The electron  H S  ( (  z  A  x  + HySy))  x  i n t e r a c t i o n between t h e e f f e c t i v e s p i n  S and t h e a c t u a l  hyperfine  + g (H S  z  spin  of the  o f t h e n u c l e u s I_, known as t h e  interaction i s written: H„=S.A.I Again  the  hyperfine  onents  A  (l  i n t h e case o f an a x i a l l y  splitting  and AJL .  onian describing  c o n s t a n t c a n be d i v i d e d  The s i g n i f i c a n t terms  Ai  of the spin  Hamilt-  L  H  the  I n t o two comp-  ' e-i-^•'•x^x • " y - ' y ' '  ,  + A S I  of  molecule  VAAC a r e :  ^ 5 - M \.&n z'- 2  By  symmetric  Z  + A (S I  Z  X  X  X  + Syly)  l i m i t i n g H t o t h e x - z p l a n e and c h a n g i n g t h e axes  q u a n t l t i z a t i o n of the e l e c t r o n H a m i l t o n i a n may b e w r i t t e n  spin  and t h e n u c l e a r  i n the following  form  spin,  (Appendix  1 ) :  H =g3HS ' + 5  AS -I '  z  Z  Z  + (A,,A /A)S 'I ' A  X  X  +  AxSy'Iy'  + 1(A„2 - A 2)g„g sin(2e)S "I " 2 A x  x  where g = g „ c o s 0 2  and  2  A g =A 2  2  2  2 1 1  3.1  x  + gx sin 8 2  g„ cos 0 2  x  2  +  2  A  2 x  gx sin 6 2  2  0 i s t h e a n g l e b e t w e e n H and t h e V-0 bond  of the  20  VAAC m o l e c u l e .  The primes denote t r a n s f o r m e d  A c a l c u l a t i o n c a r r i e d t o second yields  t h e energy  difference  (M-l,m) f o r an e l e c t r o n the  following  electron  order perturbation  theory  f o r t h e t r a n s i t i o n (M,m)-  of spin  equation.  quantities.  1/2.  This  i s given  by  M and m d e n o t e t h e components o f t h e  s p i n and t h e n u c l e a r  spin  along the axis  of quantit-  ization. AE=gBH + Am + (An  2  +  "Wgp  A )Ax (I(I+D-m ) 2  2  2  -  + ( A , , - A^ ) (g„gJ" 2  2  2  " sin  2  (2G)m  3*.'2  2  8A^g3H  This  r e l a t i o n i s derived  TABLE Parameters  f o r VAAC  Sxx  = 1  -985  . g =1.979 y y  g =1.943 z z  i n Appendix  3.1  i n a Nematic L i q u i d A =525  gauss  95  gauss  x x  A  yy  = 4  A =1257 z z  gauss  Crystal  2.  21  Interpretation  o f t h e EPR S p e c t r a  Since, Is  from  t h e VAAC m o l e c u l e  crystal VAAC  e x p l a i n e d , t h e EPR  and n o t d i r e c t l y  t h e s h a p e o f t h e VAAC m o l e c u l e  i s p l a n a r and a p p r o x i m a t e l y  jects the  as p r e v i o u s l y  n o r m a l l y from  magnetic  is. called Since  the Z  arrange  the X  their  neighbouring around  through  parallel  defines  liquid  has a x i a l and Y  square.  and Y  r  symmetry  axes.  r  r  The V=0 bond  pro-  the l i q u i d  axes  direction  are i n the plane.  VAAC m o l e c u l e s  parallel  crystal  The V=0  i t i s not n e c e s s a r y t o are forced to  t o t h e l o n g axes  c r y s t a l molecules. molecule  of the  They a r e f r e e o r about  toro-  an a x i s  bond.  a nematic  mesophase t h e VAAC m o l e c u l e s  to the l i q u i d  have used  must be c o n s i d e r e d .  t h e a n g l e 6.  and t h e X  planar sides  t h e V=0 In  r  from t h e l i q u i d  t h e c e n t e r o f t h e p l a n e and a l o n g w i t h  axis  r  the molecule  distinguish  tate  field  signal  crystal  molecules.  an a n g u l a r d i s t r i b u t i o n  function  align  "Meier and S a u p e  3  of the following  form: f  (a)=exp(-Ccos a) 2  where C i s a c o n s t a n t t o be d e t e r m i n e d  and a i s t h e a n g l e  b e t w e e n t h e l o n g m o l e c u l a r axes  and t h e o r i e n t a t i o n  (e.  by i m p o s i n g  g. magnetic  field * 1  crystal  field).  or s u p e r c o o l i n g  Either 5  i t is. possible  sample whose m o l e c u l e s  a strong  axis electric  t o have a n e m a t i c  are oriented  about  a  liquid  direction  22  Y,<J> f r o m of  t h e magnetic  field  the angular d i s t r i b u t i o n  liquid  crystals  ferent  y's *' . 1  y=90° w i l l  of long molecular  understanding axes i n n e m a t i c  study  o n l y t h e two e n d p o i n t s  y=0°  difand  be d i s c u s s e d . When Y 0 °  the l i q u i d  =  about  Some  c a n b e o b t a i n e d by t a k i n g EPR s p e c t r a . f o r In this  5  3.1).  (figure  the magnetic  field  c r y s t a l molecules  are oriented  i n a distribution:  f(a)=exp(-Ccos a). 2  F o r t h e VAAC  molecule  t h i s may be w r i t t e n :  f (-6)=exp(-Csin e) . 2  The  preferred  pendicular  direction  i s t h a t w h i c h makes  t o the magnetic  field.  and e q u a l number o f Xp a n d Y  of  axes a l o n g t h e magnetic  for  r  Y  = U  °  solving Z  r  of  for H .  X  r  and Y  absorptions occur the p r i n c i p l e  energy  arated by has 3.'1.  A single  i n figure  absorption  3.2(a).  absorptions occur  When t h e f i e l d  a x e s o f t h e VAAC  f o r Y=0°  a distance approximately  b e e n c a l c u l a t e d by computer  line  H H and  molecule  f o rY °° =  consists  a t H o . and  i s n o t a l o n g one (6?-0°  or 9 0 ° ) , the  a t an i n t e r m e d i a t e f i e l d  of the absorption l i n e  The EPR s p e c t r u m  per-  and 6 = 9 0 ° i n e q u a t i o n 3.2 and  energy  at H„.  absorption occurs  derivative '"'3.2(b).  r  9 = 0°  axis  a x e s and a s m a l l number  r  field.  h a s t h e shape i l l u s t r a t e d  H j . a r e o b t a i n e d by s e t t i n g  r  T h e n , t h e r e s h o u l d be a  large Z  the Z  value.  The  i s shown i n f i g u r e o f 8 o f these  equal t o A . A  This  and i s i l l u s t r a t e d  sep-  spectrum  i n figure  23  A procedure ferent the  y has  f o r c a l c u l a t i n g EPR  been developed  writing  i n the  are d i s t r i b u t e d (Y=90°), equal  the  around  sum  more Z  in  the  Y 0°  absorptions case.  single  line  are  r  =  T h e n , i n an EPR  ivative and  Z  R  curves  3.6(b).  It  consists  by  A„. In  and  Y  EPR  Y  i n the time  will than  curves  3.3(b). Xrnes  are  for a  The  EPR  in. w b i c b  s e p a r a t e d by  c o n s i d e r e d i s where  An.  the  f i x e d randomly i n a l l d i r e c t i o n s .  there w i l l  line  be  The  an e q u a l  separated  number  a b s o r p t i o n and  by  Aj_ and  i n figure  8 peaks  examples d e s c r i b e d a b o v e , t h e VAAC  corresponding  t o one  of  der-  3.6(a)  a r e shown i n f i g u r e  were assumed t o have u n d e r g o n e a n e g l i g i b l e ion  There  derivative  spectrum- i s i l l u s t r a t e d  of 8 l i n e s  field  absorptions  r  absorptions  absorptions.  for a single  the  r  are  experiment  The  and  3 . 3 ( a ) and  s i t u a t i o n t o be  energy  r  of 8 d e r i v a t i v e  to Z  molecules  approximately  absorptions.  r  fewer X  consists  c r y s t a l molecules  and  R  undertaking.  to the magnetic be  only  A re-  crystal  absorptions w i l l  r  enabled 90°.  and  liquid  a b s o r p t i o n and  corresponding Another  r  90°  shown i n f i g u r e s  —  the peaks  axis  and  The  fcr Y 90°  spectrum  an  of the X  be  Y ,  study  0°  endpoints:  However,  5  for this  c a s e where t h e  number o f Z  to the  liquid  two  dif-  Luckhurst .  o f t h e p r o g r a m w o u l d have b e e n a m a j o r Now  R  James and  computing r o u t i n e a v a i l a b l e  the r e p r o d u c t i o n of the  X ,  by  spectra for  3.8.  separated  molecules  change o f  direct-  Larmour p r e c e s s i o n o f  2k  their  spins.  With t h i s  to the l i q u i d ection.  crystal  condition  as b e i n g  The r e m a i n i n g  liquid  usually  crystal  during  satisfied.  AE=g3H + am +  lines  where t h e VAAC  where  g=l(g« 3  and  a=l(A« 3  molecules  one Larmoui* p r e c e s s i o n .  For the i s o t r o p i c case,  3.2 i s r e p l a c e d  The  i n a particular dir-  i s i n i t s isotropic state,  averaging of the angular parts Equation  oriented  case i s t h a t  u n d e r g o many r e v o l u t i o n s a  i t i s meaningful to r e f e r  of the spin  this there  When  condition i s i s an  hamiltonian.  by : 5  la (I(I+l)-m ) 2 g3H 2  +  2  2  Sx)  + 2A ). X  i s o t r o p i c EPR s p e c t r u m  s e p a r a t e d by t h e d i s t a n c e  a.  consists  o f 8 symmetric  Figure  3.1  The  molecular  coordinate  system  f o r VAAC  26  Figure  3'..3 A b s o r p t i o n ( a ) a n d d e r i v a t i v e ( b ) l i n e s in n e m a t i c s a m p l e when y = 90° .  a  f o r VAAC  Figure  3.5  A b s o r p t i o n ( a ) and d e r i v a t i v e i n an i s o t r o p i c s a m p l e .  (b)  lines  f o r VAAC  -^2.5 67.9 -J  Figure  3.6  84.2  S p e c t r u m f o r VAAC m a n e m a t i c l i q u i d c r y s t a l when Y 0 . On t h e x - a x i s f i e l d v a l u e s - ( i n g a u s s ) a r e given r e l a t i v e to the resonant f i e l d of the electron 7  ro co  Figure  3.8  S p e c t r u m f o r VAAC i n a: r a n d o m - s a m p l e . The x - a x i s i s r e p r e s e n t e d as i n f i g u r e 3.6. The numbered p e a k s and l i n e s a r e r e f e r r e d t o i n f i g u r e s 5 - 1 and 5.2. -  uo o  Figure  3.9  S p e c t r u m f o r VAAC i n an i s o t r o p i c s a m p l e . i s r e p r e s e n t e d as i n f i g u r e 3-6.  The  x-axis U)  CHAPTER 4 Order  i n t h e Nematic  Phase  F o r t h e n e m a t i c mesophase t h e m o l e c u l a r o r d e r p a r ameter  expression 2 o .  S i s d e f i n e d by t h e f o l l o w i n g S = l < 3 c o s 6 - 1> 2 0 i s t h e a n g l e between t h e l o n g 2  tfhere  crystal  m o l e c u l e s and t h e m a g n e t i c  thermodynamic and S = l .  average.  F o r complete  VAAC h a s i t s symmetry molecular it  axis,  I f there disorder axis  disorder.  liquid  I n nematic  b e t w e e n t h e s e two v a l u e s .  i s a universal  to the l i q u i d  ordering  calculated for  crystals,  and S=0 d e n o t e s  total  S i s intermediate  due_to  thermal 7  function  fluctuations.  have i n t r o d u c e d  of order of a solute  i n a nematic  a model meso-  o f t h e r e d u c e d t e m p e r a t u r e T..  have used t h e f o l l o w i n g  the temperature dependence  crystal  c r y s t a l by t h e f a c t o r - 1 / 2 .  ( e T / T k where Tk i s t h e i s o t r o p i c - n e m a t i c t u r e ) . They  Because  2  C h e n , James, and L u c k h u r s t  phase  2  S measures t h e e x t e n t t o which t h e .  i s disordered  t h e degree  a  order <cos 6>=l  values o f the order parameter  perfect  i n which  and < > i n d i c a t e s  < c o s 6 > = l / 3 and S=0.  perpendicular  Then S=-l/2 i n d i c a t e s  crystal  field  of the l i q u i d  i s complete  d i f f e r , from those o f the l i q u i d  liquid  axes  transition  expression  tempera-  to calculate  o f S. exp(U/KT)dcos6  - 1 2  33  U i s the o r i e n t a t i o n a l molecules.  potential  energy  of the solute  I f i t i s assumed t h e n e m a t i c  to dispersion forces (forces of t h e m o l e c u l e s ) , U w i l l  order  i s mainly  due  b e t w e e n i n s t a n t e o u s d i p o l e moments  have t h e f o l l o w i n g f o r m :  U=-lAS(3cos G-l) 2V 2  2  V i s t h e m o l a r volume o f t h e l i q u i d characteristic  o f the l i q u i d  ence o f S c a n be found An the  concept  chapter  EPR s t u d y  the they  of liquid  t h e two e x t r e m e s  time  crystal.  t h e above  crystals  tumbling  time.  i n tumbling  make a n e g l i g i b l y  fixed  depend-  expressions.  i s c o m p l i c a t e d by In the previous  time  were  investigated.  small angular  o f one L a r m o u r p r e c e s s i o n o f t h e i r  are e s s e n t i a l l y  and A i s a c o n s t a n t  The t h e o r e t i c a l  n u m e r i c a l l y from  o f molecular  I f the molecules  crystal  and d e p e n d i n g  spins  change i n  (10~  10  sec),  on t h e e x p e r i m e n t a l  conditions  they . y i e l d  an o r i e n t e d o r a random EPR s p e c t r u m .  There w i l l  be 8 l i n e s  s e p a r a t e d by A4, and u s u a l l y  separated  by A . u  8 peaks  On t h e o t h e r h a n d , i f t h e m o l e c u l e s  undergo  many r e v o l u t i o n s d u r i n g a L a r m o u r p r e c e s s i o n an i s o t r o p i c spectrum  of 8 lines The  by  calculation First, ally  s e p a r a t e d by " a " i s p r o d u c e d .  slow t u m b l i n g  Schwerdtfeger  and D i e h l * 1  of molecular  an a n g u l a r  c o n d i t i o n has been  investigated  and James and L u c k h u r s t .  order  distribution  f o r the l i q u i d  EPR  5  i s carried  o u t i n two s t e p s .  f u n c t i o n i s found  c r y s t a l molecules.  The  experiment-  As d i s c u s s e d i n t h e  34  previous this  chapter t h i s  distribution  has  the  function  <3cos 6 - 1>.  In p r a c t i c e  cedure  i n order  2  because  function  magnetic  f(8)=exp(-Ccos 6).  this  to f i n d  Then  2  i s used  to c a l c u l a t e  i s a l o n g and an  at a g i v e n temperature,  examined i n a range  angular  the  of orientations  the  average  tedious  pro-  distribution  liquid  c r y s t a l must  b e t w e e n 0°  and  90°  be  to  the  field. The  and  form  fast  Gelerinter  p a r a m e t e r may  tumbling  and  6  be  condition  C h e n , James and  d e r i v e d from  i s d e s c r i b e d by  Luckhurst .  The  7  the H a m i l t o n i a n  for  Pryburg order  isotropic  liquids.  S=l<3cos 6 2  2  <a>  - l>=l(<a> 2 (a -  a) Ai.)  i s the measured h y p e r f i n e s p l i t t i n g .  easy  to f i n d  the temperature  It is  relatively  dependence o f S w i t h  this  relation. The in  tumbling  discussed  preceeding paragraphs  time.  i n this  U n f o r t u n a t e l y , the thesis  of v i s c o s i t y . nematic large the  molecular In the  experimental  above methods a r e t r u l y tumbling  temperature  mesophase e x i s t s ,  changes  two  extremes  situation  i n v o l v e s intermediate tumbling  f o r which n e i t h e r of the The  d e s c r i b e the  liquid  in viscosity.  o r d e r parameter i s not  time  applicable.  i s largely  range  through  crystals  times  a  function  which  usually  the  undergo  Hence a r i g o r o u s c a l c u l a t i o n straightforward.  For  this  of  study,  35  the  assumption  expression ations.  of fast  tumbling  (<a> - a ) / 2 * ( a  Further  was made i n s o f a r as t h e  - Ax) was u s e d  f o r order  calcul-  c o n s i d e r a t i o n was made f o r c h a n g e s i n  viscosity. The  hyperfine  s p l i t t i n g o f VAAC i n " O c t o i l " , a  vacuum pump o i l , was o b t a i n e d viscosity  (figure  viscosity  covers  crystals The to  used  hyperfine  4.1).  i n this  study  s  order  (<a>  +  s  vis*  - a)/2*(a  appropiate  crystals  a  v  b  effective  4.2.  and t h e q u a n t i t i e s  t o be a d d i t i v e .  estimated  e  by  S i s an  of viscous Then  order S ff= e  calculating curve  order  parameter  calculated  for several l i q u i d crystals  The k i n e m a t i c  are plotted  interesting ities  m  t h e measured  - Aj.) f r o m t h e VAAC i n O c t o i l  o f temperature figure  expected  at the  viscosity. The  in  has suggested  8  parameter  ^v±s  was  range.  d e p e n d e n c e on v i s c o s i t y .  t r u e o r d e r may be c o n s i d e r e d  true  ion  i n a similar.temperature  s p l i t t i n g i n the l i q u i d c r y s t a l s  Schwerdtfeger  and  T h i s o i l was c h o s e n b e c a u s e i t s  t h e range o f t h e v i s c o s i t i e s o f t h e l i q u i d  have a s i m i l a r  effective  as a f u n c t i o n o f k i n e m a t i c  against  f e a t u r e o f these  are continuous crystals  the e f f e c t i v e  inuity  at an i n t e r m e d i a t e  i sillustrated  v i s c o s i t i e s f o r t h e same l i q u i d temperature curves  throughout  liquid  as a f u n c t -  i n figure  i s that while  the nematic range,  order  4.3.  The  the viscosf o r a l l the  parameter has a d i s c o n t -  temperature  i n the nematic  range.  37  <d-  C V J A T  ro  o  o"  o"  ro  d  d  CM  d o "  <T  co  ro  d  d  ~ eff s  Figure  k ,2  E f f e c t i v e order parameter versus reduced temperature f o r VAAC i n t h e n e m a t i c l i q u i d c r y s t a l s : (a) b i s (4'-n-octyloxybenzal)-2-chloro-l,4-phenylenediam i n e , (b) MBBA, and ( c ) 4-methoxy b e n z y l i d e n e - 4 amino-a-methyl cinnamic a c i d - n - p r o p y l e s t e r 6  8  38  X  TEMPERATURE (°C) Figure  4.3  K i n e m a t i c v i s c o s i t y versus temperature f o r (a) O c t o i l , X, (b) 4-methoxy b e n z y l i d e n e - 4 - a m i n o - a - m e t h y 1 c i n - • namic a c i d - n - p r o p y l e s t e r , 6 , and ( c ) MBBA,* 8  Figure  Luckhursts t h e o r e t i c a l c u r v e i s compared w i t h c u r v e s o f S f f and S e f f - l / 3 ( S v i s + D) v e r s u s t e m p e r a t u r e i n MBBA. The d a s h e d c u r v e i s S f f - l / 3 ( S y i s + D) and t h e b o t t o m c u r v e i s L u c k h u r s t s curve. e  e  vo  40  l/T'10  F i g u r e 4.5  3  (°K)  _ 1  P r o t o n r e l a x a t i o n time T t h e t e m p e r a t u r e a t 18.2  versus r e c i p r o c a l of MHz f o r MBBA  41  The universal  theoretical  with  effective  S j_s  may  are  restricted  i n only  curve may  be  That  to v i s c o s i t y  curve  that  curves.  as  This  theoretical  in  so  dependence  -  that  introducing  l/3(S  three  the  difference  v i s  function + D)  VAAC  the  slope  are  constant.  molecules  p o s s i b l e d i r e c t i o n s of  is parallel  to the e  theoretical  from  St  r  u  e  alone. on  two  liquid  d i s c o n t i n u i t y i n the  suggests the  order  U.  where  D is a  crystals  d o m i n a n t r e l a x a t i o n mechanism a t  d i s c o n t i n u i t y may  liquid  smoothly  i t i s u n l i k e l y they  crystal. varying  could  effect  Both  the  effective be  due  to  viscosity  functions the  8  of  observed  tempchange  slope. .. I n  are  f o r by  deviation in S f f  effects  some p h y s i c a l c h a n g e i n t h e  erature  the  the  i n MBBA i s compared  p r e v i o u s l y and  spin relaxation studies  same t e m p e r a t u r e  than  suggested the  e  of the  this  due  show a c h a n g e i n t h e  and  has  indication  one  indication  NMR  order  an  i s an  be  f o r VAAC  the 7  temperature  is S f f  described  f a c t o r 1/3  motion.  the  accounted  i n f i g u r e 4.4  The  angular  be  i s greater  to  Luckhurst .  o r i e n t a t i o n a l p o t e n t i a l energy  i s c a l c u l a t e d as  v  slope  Luckhurst may  correspond  c a l c u l a t e d by  parameter  curves  curve  curve  I n f i g u r e 4.4  order  terms i n the  dashed  order  curve.  two  i n f i g u r e 4.2  experimental  slope.  between the  The  the  Luckhursts  other  lines  theoretical  In each case  of the  dashed  conclusion,  considerably  i t appears  a f f e c t e d by  the  the  effective  viscosity  of the  order liquid  curves  42  crystal.  But t h e s l o p e d i s c o n t i n u i t y  temperature fore  r a n g e c a n n o t be due t o v i s c o s i t y  indicate  itself. data  The mechanism o f t h i s  available  ideal  Firstly,  tool  at the present  liquid times for  f o r order  used  Thirdly,  or fast  study  hyperfine  with  Secondly,  the molecular  tumbling  case f o r the  tumbling  was a c c u r a t e l y a p p l i c a b l e . where  precision.  t h e measurements  broadening  w h i c h made i t d i f f i c u l t  splitting  crystals.  r a n g e where n e i t h e r t h e t h e o r y  t h e r e was c o n s i d e r a b l e l i n e lines  i n liquid  crystal.  i n the range o f v i s c o s i t y  of adjacent  t h a t EPR i s n o t  i s of the solute, i n this  were i n a n i n t e r m e d i a t e spins  be n o t e d  determination  i n this  from t h e  time.  i t should  obtained  there-  crystal  change i s n o t apparent  that of the l i q u i d  crystals  fixed  taken,  stage  the order  VAAC, a n d . n o t  and must  some p h y s i c a l c h a n g e i n t h e l i q u i d  • At t h i s an  i n the intermediate  were  and o v e r l a p p i n g  t o determine the  CHAPTER 5 of Cholesteric L i q u i d Crystals  Orientation  The  liquid  approximately  a x i a l symmetry.  susceptibility of  may  be  The  difference  determines  how  a free  magnetic  i n t e r a c t i o n with following  a  n  (  ( (  Xx •  external  Two  ascribed  axis, X J  the molecular  tion,  c r y s t a l molecules  values  i one  field.  a magnetic  free  field  In order to minimize x will  i t s symmetry  structure the  field.  behavior cules  and  H may  v a l u e s , X Xt» =  behaves  involved  be  Xx  -  i n an  in„the  expressed  by  the  i n an  tend  2  axis  i n the  with  of a l i q u i d  t e n d i n g to a l i g n the  local  a nematic  external  field  t o a l i g n i n the magnetic  a s i t u a t i o n with  a l l the molecules  43  parallel  to orient  field. with  exerts a torque symmetry  mesophase w i t h  magnetic  with  c r y s t a l sample  field  a  a molecule  axis  p e r p e n d i c u l a r to the case  molecular  negative x w i l l  magnetic  For  long  energy  i t s symmetry  a molecule  o f p o s i t i v e XJ  molecules  to  two  energy  i t s free  orient with  field  Now,  the  i n the d i r e c t i o n  i n the t r a n s v e r s e d i r e c -  v e c t o r i n the d i r e c t i o n of the  the magnetic with  have  proportionality.  n i s a unit  positive  Field  diamagnetic  c r y s t a l molecule  The  a Magnetic  i n t h i s study  of  t o them: one  Fmag'-xCn-H)  axis.  used  between t h e s e  liquid  by  axis  x positive  i s simple.  The  on  parallel the mole-  field  direction.  Since  aligned  or n e a r l y  aligned  44  is  compatible  arrangement  with  of  As  the  the  liquid  described  mesophase i s made up crystal  molecules  parallel is  to  the  c a l l e d the  aligned  the  axes  netic  structure ection. the  one  a  the  along  icular  to  verse  to  axis,  they  plane.  axis.  helix  axis  the  helix  axis.  helix  are  axis  pointing  liquid  Meyer  1 0  9 1 1 0  perpendicular  1 1  and  the  The  axes  normal to the  a small  that  planes are  planes  the  a n g l e making  breaks  pointing  up  that  a x  a magnetic  same  directions: plane  through would  t h e s e two  field.  orient His  sym-  perpend-  arranged  cholesteric should  of  two  field of  dir-  behavior  Perpendicular  a magnetic  nematic  helical  are  0°  a  mag-  The  i n the  i n a l l angles,  one  down t o  the  trivial.  other  external  i n the  fields  direction.  reasons  these  liquid  molecules  i n high  molecules  c r y s t a l along  to  symmetry  structure  whose m o l e c u l e s h a v e p o s i t i v e axis  '  i s not  I t i s assumed t h a t  cholesteric  their  mesophase d e f i n e s  the  the  the  i n successive  intermediate  structure  cholesteric  i n which  each p l a n e  molecules  cholesteric  final  structure.  been shown  a l l the  i s the  planes  direction  In  helical  However, f o r  of  The  cholesteric  with  this  c h a p t e r , the  with  c h a n g e d i r e c t i o n by  cholesteric  metry  situated  same d i r e c t i o n b u t  has  fields  first  successive  planes.  characteristic It  structure,  crystal.  i n the  of  are  helix  i n the  symmetry  nematic  to  transthe  helix  360°,in orient  the a  directions.  liquid  crystal  with  i t s helix  qualitative  45  argument  i s based  on  a unit  with  b e h a v e s as axis for  (x„  and  + Xx)/2 XJ  positive  (X«  s h o u l d have l e s s icular  to  the  increased,  causing  the  axis.  an At  the  cholesteric  s u s c e p t i b i l i t i e s of  Xx)/2  +  to  the  Xx  than x  i s greater  In  t h i s c a s e as  more m o l e c u l e s untwisting higher  of  fields  along  helix  e n e r g y when i t s h e l i x  field.  direction  assumption  perpendicular  free  more and  helix  the  the  t  mesophase the  axis. h  Since,  structure  e  x  axis  field  helix  is  perpend-  intensity  are  turned  toward  the  liquid  c r y s t a l about  the  structure  the  is field  becomes  nematic. A crystal  to have i t s h e l i x x  negative In  second p o s s i b i l i t y  the  structure  and  Labes  that a  the  due  to  liquid  the  axis  the  torque  to  increases  this  structure  begins  4°  1 2  ,  follow  in field  If x  be  liquid  field.  the  As  the  larger  unwind  the  a small  and  out  When t h e  cholesteric  structure  well angle  cause  Baessler case,  direction i f  of  their  magnetic  magnetic as  For  perturbation  angle to  .  1 3  not  field  field  conical  them by  i s at  angle gets to  on  should  electric  i n the  This  field.  is positive,  c r y s t a l molecules  exerted  the  aligned  the  liquid  f r o m Meyer* "s t h e o r y  strength  analogous  occurs.  c r y s t a l molecule  perpendicular  reaches  can  perturbation of  would  the  cholesteric  p a r a l l e l to  t o become n e m a t i c .  helix  a tilting  increase  suggest, for  1 2  conical  axis  this orientation  t h i s c a s e an  i s f o r the  field  of  planes  field.  the  the  is  Each  direction strength  helical perturbation  b r e a k s down and  a  46  nematic  structure The  crystals orient  fields  orientation  liquid  planes  achievable).  cholesteric  i s large In this  If  a small  added  t o a nematic  is  often  weakly  to  a magnetic  mixture  preparations  i s more  suc-  readily  study, t h e magnetic behavior  o f two  i s examined.  liquid  crystal,  active  material  the r e s u l t i n g  When 2 p e r c e n t  mixture  by w e i g h t o f  (C) i s a d d e d t o MBBA, a c h o l e s t e r i c i t s molecules  i t must have a p o s i t i v e  x-  above i s an example o f a weak  parallel  Then t h e cholesteric  p o s i t i v e x«  with  It that  was b e l i e v e d  on t h e b a s i s  t h e C-MBBA m i x t u r e w o u l d  perpendicular  t o a magnetic  orient  field.  ing  e x p e r i m e n t was t o d e m o n s t r a t e  ing  field  7 minutes. field and  cholesteric  i n special  S i n c e MBBA a r r a n g e s  field  described  Thus f a r ,  only  liquid  ( i . e., t h e a n g l e between  cholesteric.  chloride  results.  .  amount o f an o p t i c a l l y  is  cholesteryl  1 1  cholesteric  than those necessary t o  and o r i e n t a t i o n  o f weak c h o l e s t e r i c s  mixture  to orient  crystals  has been produced  which a r e weakly  types  required  a r e i n g e n e r a l much h i g h e r  nematic  cessive  i s produced.  was i m p o s e d  of other  with  experiments ' 1 0  i t s helix  The o b j e c t  of the follow-  t h i s u s i n g EPR.  o n t h e C-MBBA sample  axis  An o r i e n t -  f o ra period  of  T h e n an EPR s p e c t r u m was t a k e n by s w e e p i n g t h e  between  2 and 4 k i l o g a u s s .  a n o t h e r s p e c t r u m was t a k e n .  The sample was r o t a t e d T h e n t h e sample was  90°  rotated  1 1  47  back t o 0 ° , the o r i e n t i n g f i e l d kilogauss,  was i n c r e m e n t e d by 5  and t h e p r o c e d u r e r e p e a t e d .  was c a r r i e d o u t  for  orienting field  All  m e a s u r e m e n t s were made a t room t e m p e r a t u r e .  ation the a  time o f t h e l i q u i d  crystal  o f t i m e , t o be a b o u t  were r e q u i r e d  from  predict  to obtain  30 m i n u t e s .  EPR s p e c t r a  order  to analyse  state to  Since  between each  8 minutes  orientation  t h e EPR d a t a i t i s u s e f u l t o  the c h a r a c t e r i s t i c s of the spectra  90° t o t h e f i e l d  as a f u n c t i o n  obtained  of f i e l d  t o p r o d u c e a random EPR s p e c t r u m . increases field  the h e l i x axis w i l l  d i r e c t i o n . Since  plane perpendicular still  be random.  spectra w i l l creases  will  However, a l o n g  the h e l i c a l  high  t o the f i e l d ,  fields be t h a t  t o the  c a n be anywhere i n a  the f i e l d  unwind  will  have t h e f o l l o w i n g p r o p e r t i e s .  will  be s i m i l a r and w i l l  intensity i n -  such t h a t  more and  In the limit  t h e s t r u c t u r e becomes n e m a t i c a t 0°.  should  d i r e c t i o n , the  As t h e f i e l d  to the f i e l d .  o f a nematic  field  t h e s p e c t r u m a t 90°  structure w i l l  are p a r a l l e l  A  c a n be e x p e c t e d  As t h e o r i e n t i n g  the h e l i x axis  a t 0°  strength.  move p e r p e n d i c u l a r  n o t r e m a i n random.  more m o l e c u l e s  rum  The r e l a x -  the oriented  sample w h i c h h a s n o t u n d e r g o n e any o r d e r i n g  very  kilogauss.  e x p e r i m e n t was e s s e n t i a l l y q u a l i t a t i v e . In  and  b e t w e e n 0 and 20  random s t a t e was f o u n d , by m e a s u r i n g p e a k i n t e n s i t i e s as  function  the  intensities  This  of  and t h e s p e c t -  T h e n , t h e EPR  spectra  A l l the spectra  a t 90°  be r e p r e s e n t a t i v e  o f randomly  48  arranged zero  molecules.  field  T h e s p e c t r a , a t 0° w i l l  b u t as f i e l d  strength  increases  o n e n t s become l a r g e r and t h e p e r p e n d i c u l a r Figure  5.1 shows t h e h e i g h t s  peaks and t h e h e i g h t s function 90°.  o f magnetic  increase  decrease  strength  strength  constant  Meyer's t h e o r y  oriented along increases  t o the f i e l d .  f o rliquid  effect  a nematic  crystals  I f these  This with  myristate  A  of this  orients with i t s i s i n agreement positive a left  x  values.  handed  (M) h a s a r i g h t  two a r e m i x e d i n t h e  t h e r o t a r y powers have a  a weak c h o l e s t e r i c  structure.  cholesteryl room  r o t a r y power.  producing  cor-  to the f i e l d  On t h e b a s i s  Cholesteryl myristate  proportions  do  i n c r e a s e s . " The peak h e i g h t s a r e  o p t i c a l r o t a r y power.  appropiate  the lines  the f i e l d  oriented transverse  c h l o r i d e (C) p o s s e s s e s  optical  as a  and t h e l i n e s  Cholesteryl  handed  lines  i n t h e 0° d a t a  f o r t h e 90° d a t a .  axis perpendicular  smaller.  o r i e n t a t i o n s o f 0° and  i t i s a s s e r t e d t h a t t h e C-MBBA m i x t u r e  helix with  t o note that  t o VAAC m o l e c u l e s  as f i e l d  components  comp-  o f some o f t h e p e r p e n d i c u l a r  f o r sample  t o VAAC m o l e c u l e s  essentially data  field  as t h e f i e l d  responding  the p a r a l l e l  o f one o f t h e p a r a l l e l  I t i s interesting  corresponding  be random a t  o r i n some  cancelling circumstances.  1.75:1 by w e i g h t c h o l e s t e r y l c h l o r i d e -  mixture  (CM) i s w e a k l y  cholesteric at  temperature. The  experiment with  1.75:1 CM m i x t u r e .  C-MBBA was r e p e a t e d  On t h e b a s i s  o f other work ' 9  12  with the this  mixture  49  Figure  5.1  Peak i n t e n s i t i e s f o r t h e C-MBBA m i x t u r e v e r s u s o r i e n t i n g magnetic f i e l d . The numbers r e f e r t o t h e c o r r e s p o n d i n g p e a k s and l i n e s i n f i g u r e  3.8.  50 was  expected  magnetic  to orient  field.  any  ordering  Now,  as t h e  magnetic  crease. be the it  expected  orienting  field  Then,  i s increased  and  i n the p a r a l l e l  p e r p e n d i c u l a r components. e x i s t e d , w o u l d be  less  spectra  20  i n figure  k i l o g a u s s , was  5.2.  not  t h e 90°  spectra  components A conical  than  4°  and  12  reinforce helix  and  parallel The  i n two  negative x  that  t o a magnetic  above e v i d e n c e ways.  v a l u e s , the h e l i x  to the f i e l d " .  feasible  because  crystal  1  t h e CM  The  parallel  i s that  data i s  field  used,  helical to  orients with i t s  m i x t u r e may i n t h e CM  a x i s w o u l d be  9  that  to the f i e l d mixture  be  expected  under  inter-  mixture  possibility  be made n e m a t i c  t h e CM  obser-  field.  on t h e CM  been shown  m i x t u r e may  molecules  interpretation  i t has  not  data i s beleived  However, t h i s  1  de-  perturbation, i f  mixture  I f the molecules  parallel  fields  t h e CM  the  there should  therefore  This  crystal  a decrease i n  h i g h e s t magnetic  structure.  the a s s e r t i o n  axis  preted  The  spectrum.  components  i n t e n s e enough t o p r o d u c e  breakdown t o a nematic  undergone  the p e r p e n d i c u l a r  vable . w i t h i n the accuracy of the experiment. presented  not  the  transverse to  the p a r a l l e l  the other hand,.for  an i n c r e a s e  has  to  the l i q u i d  themselves  f o r t h e 0°  should increase  On  parallel  t o g i v e a random EPR  tend to o r i e n t  field.  components  axis  As b e f o r e , a s a m p l e w h i c h  c a n be  molecules w i l l  with i t s h e l i x  high  with the  direction.  t a k e s on a  to  have orient  i s not magnetic liquid The  conical  second  I  I  0  5 . ORIENTING  Figure  5.2  I  10 FIELD  I  15  :  1  20  (KILOGAUSS)  Peak i n t e n s i t i e s f o r t h e CM m i x t u r e v e r s u s o r i e n t i n g magnetic f i e l d . The numbers r e f e r t o t h e c o r r e s p o n d i n g p e a k s and l i n e s i n f i g u r e 3-8.  52  perturbation existence  1 2  '  1 5  .  of this  interpretation  Although  p e r t u r b a t i o n was  not p o s s i b l e ,  the  of the second  seems t o be t h e most r e a s o n a b l e mechanism.  However,.more e x p e r i m e n t a l complete  experimental^ v e r i f i c a t i o n  description  evidence  o f the problem  i s necessary c a n be made.  before a  CHAPTER 6 Discussion  This techniques  to  thesis two  determination  The order  order  unately, slope  application  problems order  i s t h a t no  f r o m EPR  i n nematics  o f the One  order  o f one  occurs  of p r i n c i p l e diamagnetic  compare them w i t h The is  order  information directly  a s o l u t e i n the  relatively  mol-  principle  r e v o l u t i o n of the  part  of the  in this  liquid  another  parameter  1 6  .  and  are  Since  Unfortthe the Infrared  scattering,  indices,  liquid  crystal,  technique.  time crystal  curve,  Possibly  X-ray  susceptibilities  the  liquid  order  region.  refractive  on  calculate  Larmour p r e c e s s i o n .  d i c h r o i s m , NMR,  on  of the  of  tumbling  dichroism, u l t r a v i o l e t  give  theory  intermediate  f o r one  the  behavior  s p e c t r a i n the  order with  the  the  to  answer i s t o d e t e r m i n e  of c a l c u l a t i n g  and  been d e v i s e d  discontinuity,  of p r i n c i p l e  crystals:  field.  complete.  t h e most i n t e r e s t i n g  measurement  EPR  method has  time  i s of the  of  in liquid  state of understanding  r e g i o n where t h e molecules  the  i n a high magnetic  i s f a r from  difficulties true  important  of molecular  of c h o l e s t e r i c s  ecular  presents  measurement  a l t e r n a t e methods these  crystal  i t w o u l d be  techniques  itself fruitful  and to  EPR  knowlege o f magnetic b e h a v i o r c o n s i s t e n t but  of  more e x p e r i m e n t a l  53  cholesterics evidence  and  not  54  theoretical data  c o u l d be o b t a i n e d  computer This the  background  verify  CM m i x t u r e .  liquid These  Although  EPR  many o t h e r  1 7  of o p t i c a l  ,  measurement  be i n c r e a s e d  EPR  value  1 8  o f EPR  i f more  determination  one o f t h e most  are a v a i l a b l e .  a b s o r p t i o n by d i s s o l v e d constants  1 2  ,  to  9  i n the study  radical  of l i q u i d  paramagnetic was  of order.  the  liquid  reason, and  The s p e c t r a l i n e s  i t s c i g a r shape t h i s crystal  i t should  crystals  probes  u s e d by F e r r u t t i  This r a d i c a l ,  o r i e n t a t i o n than The  be t o a t t a c h selves.  ideal  a more  way  spin labels  i n the  from  than  i t are w e l l  direct  c a n be et a l  1  9  in  2,2,6-tetra-  "g" t e n s o r  i s cigar and an  separated.  be more  the planar  may  restricted VAAC.  axial Due by  For this  r e p r e s e n t a t i o n of order  VAAC. t o u s e EPR  in liquid  to the l i q u i d  As y e t no c o n v i n c i n g  been found  axial  molecule w i l l  molecules give  measurement  and NMR .  and has an a p p r o x i m a t e l y  A tensor.  con-  u s e f u l techniques  convenient  An i n t e r e s t i n g  molecules.  perturbation i n  methyl-4-(p-actloxy)-benzoylomino-piperdine-l-axyl shaped  extensive  orientation i n  of light  ,  EPR  of determining  of .dielectric  rotary power The  of a conical  i s perhaps  methods  i n c l u d e measurement  radicals  an  accurate  i t w o u l d be u s e f u l t o do an  the existence  and p r o d u c t i v e crystals,  found.  I f more  a n a l y s i s of the o r i e n t a t i o n of c h o l e s t e r i c  might  venient  are required.  literature.  treatment  crystals  crystal of this  would  molecules  them-  p r o b l e m has  APPENDIX 1 Transformation o f the Spin Hamiltonian  for Axially  Symmetric  Ions In t h e case  of axial  choose the x - a x i s such x-z  plane.  angle  symmetry  that the external f i e l d  T h e n H =0 w i t h no l o s s y  0 to the -z-axis. l l  Z  Using  +  z  + A„S I  of generality.  H i s at  g S sin0) x  x  + AX(S I  Z  H i s i n the  The H a m i l t o n i a n may be w r i t t e n a s :  H =3H(g S cos0 s  i t i s convenient to  X  + Syly)  X  the transformation:  S = S ''cos(J) - Sx^sincf) z  z  S =S ^sin<|) + S ""cos(j> x  s  the  electron  about  y  z  = s  x  y"  s p i n s may be c h a n g e d t o a s e t o f a x e s r o t a t e d  t h e y - a x i s by a n a n g l e <J>..  expression  f o r t h e Zeeman Z  the following  term.  H = 3 H [ S ^ (g sin0sin<|> 4  This yields  +  x  g„cos0cost|>)  + S " (g^sinecostj) - g„cos0sin(|>) ] x  The  term  i n S " i s zero i f : x  g^sinOcosc}) or:  - g cos0sincj)=O (l  tan(j>=g tan0 x  Define: g =g„ cos 0 2  2  2  +  g  2 x  sin 0  55  2  56  Then:  sin<j)=g_ sin0 g i  and cos<j)=gnCos8 g  Upon s u b s t i t u t i n g the  these  back i n t o  the Hamiltonian  result i s : H =gSHS ' a  If  the transformed  z  S ^ , S "*, and Sy"* a r e p u t i n t o z  x  the h y p e r f i n e p a r t o f the Hamiltonian,  X  energy  s t a t e s s e p a r a t e d b y g3H, may b e t r e a t e d by s e c o n d  perturbation  Z  theory  X  i n finding  |A j <<g3H f o r VAAC. that way. axes  differ Z  X  t h e energy  However, b e c a u s e S ' I Z  i n energy  But S -*I  t h r e e , which  i n S 'I ,  S 'I , Z  The f i r s t  exist  Sy'ly,  X  and S " I .  terms  o n l y by A i t c a n n o t  connects  .order  since states  be h a n d l e d  may be e l i m i n a t e d by c h o o s i n g  f o r I t h a t a r e r o t a t e d about  connect  eigenvalues X  X  i n this  a new s e t o f  t h e y - a x i s a t an a n g l e \p.  Then:  I = I ' c o s ^ - I ^sinijj z  z  x  I =I ^sini|j + Ix"cos^ x  z  T =1 " With Hamiltonian  this  transformation the hyperfine part  may be w r i t t e n :  H = S ' I ^ . ( Aj.sincj)sini|J + A« cos<J>cosip) Z  z  + . S *I "(AiSin<{>cosif) - A» coscj)sin^) -  z  x  + S x ^ I ^ C A x c o s ^ c o s ^ -. An sincj)sin^) + S ^ I " ( A x c o s t j j s i n ^ - Ansincj>cosip) X  Z  t'Sy'Iy'Ax  ofthe  57  Now  S "*I ^ i s eliminated i f : Z  X  A sincj>cosiJ; - A ,cos<J>sin^=0 A  (  Define: A g =A„ g„ cos 9 2  Then:  2  2  2  + Ax gj. sin 6  2  2  2  sin^=Axgj. s i n 6 and cos^=A»gn c o s 9 Ag By  in  terms  2  Ag  substituting  t h e above  expressions  o f 6 the f o l l o w i n g spin Hamiltonian H =g3HS - + A S ' I ' 5  z  Z  + AMAXS 'I " X  x  Z  + Aj.S "I ' Y  Y  + (A,> - A x ) g „ g s i n ( 2 6 ) S - ' I / 2A 2  2  x  This  i s e x p r e s s i o n 3.1.  Y  7  f o r \p and cf)  i s achieved.  , APPENDIX 2 The  Energy  Symmetric  Eigenvalues  o f the Spin Hamiltonian  for Axially  Ions  It  i s required to find  t h e energy  o f the s t a t e  |M,m> f o r t h e H a m i l t o n i a n 3.1. Hj.=g!3HS - + A S * I ' z  Z  Z  + A„Ai.S 'I " Y  '  +  —^  (An  + AiS'I ' y y  v  X  X  - A i. )g„gxsin(2e)S "I "  2  2  v  gr  2A To  first  f7  order the eigenvalues  ofH are:  E =<M,m|H |M,m>=g3HM + AMm 1  s  A p p l y i n g second order p e r t u r b a t i o n t h e o r y : E =E* + E l<M",m"|H?|M,m>| M*,mVM,m E -E 2  2  u  where E ° i s t h e z e r o t h o r d e r e n e r g y  u  i  ggHM.  Then t h e s o l u t i o n t o t h e problem the  following matrix  lies  i n calculating  elements. <M",m'|Hy|M,m>  The S+ + S_; S y = S  +  values  2  <M-l,m|H |M m>=(A„  2  s  5  J  J  elements  - S_; a n d s i m i l a r  <M+l m|H |M m> = ( A „ i  o f these  are: (using S =  relations  x  for I  x  and l y  - A a , )g„ g ^ s i n ( 29 )m[S (S + l ) -M (M+l) ] ^ 2  - Ai )g„g sin(29)m[S(S+1)-M(M-1)V 4A " i 2  x  2  -  58  59  <M,m+l|Hs|M,m>=0 <M,m-l|H  |M ,m>=0  s  <M+l,m-l| Hs |M,m>= (A„ A <M-l,m+l| H  S  : + 4A  x  A A O [ S ( S + l )- M ( M + 1 ) ] * [ I ( I + 1 )  i  -m(m-l)]*  |M,m>= (An Ao. + AAJ. ) [S(S+1)- M ( M - l ) ] * [ 1 ( 1 + 1 ) -m(m+l) HA  <M+l,m+l| H  S  <M-l,m-l| H  5  |M,m>= ( A  T L  A  AAJL)[S(S+1) -M(M+1)]'[I(I+1) -m(m+l)]*  X  4A |M,m>= ( A „ Ax  AAA)[S(S+1) -M(M-l)]*[I(I+1)-m(m-l)]*  .Then t h e s e c o n d o r d e r e n e r g y e i g e n v a l u e s a r e : E=g$HM + AMm . +  l<M+l,m|HslM,m>|  + [<M-1,m1H 1M,m>|  2  2  4  +g6H  -ggH +  1<M+l,m-l|Hs |M,m> j -g3H  + j <M-1,m+l|Hy [M,m>| +g3H  +  l<M+l,m+llH |M,m>| -g3H  + |<M-l,m-l|H |M m>1 +ggH  2  2  3  5  On m u l t i p l y i n g following result  this  2  f  out and c o l l e c t i n g  terms t h e  i s obtained.  E=g3HM + AMm + A„Ax m[M -S(S+l] + ( A u + A )Ax M[I(I+l)-m ] 2AggH 4A ggH 2  2  2  2  2  2  z  -  (Au  - Ax )  2  2  2  g„gxl s i n 2  (20)Mm  5  8A g3H z  The transition solving the  energy absorbed by t h e microwave  between e l e c t r o n  energy l e v e l s  field  ina  i s o b t a i n e d by  t h e above e q u a t i o n f o r (M,m) a n d (M-l,m) and t a k i n g  difference.  The f o l l o w i n g r e s u l t  i s obtained.  60  AE=g3H + Am +  (A  + A )Ax (I(I+l)-m ) 4A^ggH  +  (A,,  2  z  -  2  Ax )  8A g3H z  This  (  2  2  t >  2  ;  2  I  i s expression  S 3-2.  sin (20)m 2  :  61  REFERENCES 1.  G l a r u m , S. H. a n d M a r s h a l l , 16_» 55, (1967).  2.  Chen, D. H. a n d L u c k h u r s t ,  J . H., J . Chem. P h y s .  G. R., T r a n s . F a r a d a y S o c .  6 5 6 , (1969).  65, 3.  Meier,  4.  S c h w e r d t f e g e r , C. F . and D i e h l , P., M o l . P h y s . 17, 417, (1969). James, P. G. and L u c k h u r s t , G. R., M o l . P h y s . 19, 489, (1970).  5.  G. and Saupe, A., M o l . C r y s t .  1, 515,  (1966).  6.  F r y b u r g , G. C. and G e l e r i n t e r , E . J . , J . Chem P h y s . 52, 3378, (1970).  7.  C h e n , D. H., James, P. G., a n d L u c k h u r s t , 8, 789, (1969).  8.  S c h w e r d t f e g e r , C. F . , M a r u s i c , M., Mackay, A. L , , '.arid Dong, R. Y., t o be p u b l i s h e d i n M o l . C r y s t .  9.  Sachman, E . , Meiboom, S., a n d S n y d e r , L . C , Soc.  8 9 , 5981,  G. R., M o l . C r y s t  J . Am.  Chem.  (1968).  10.  M e y e r , R. B., A p p l i e d  11.  D u r a n d , G., L e g e r , L . , R o n d e l e z , F . , and V e y s s i e , M., P h y s . Rev. L e t t . 2_2, 2 2 7 , (1969). B a e s s l e r , H., L a r o n g e , T. M., a n d L a b e s , M. M., J . Chem. Phys.. 5 1 , 3 2 1 3 , (1969).  12.  P h y s . L e t t . 1_4, 2 0 8 , (1969).  13.  B a e s s l e r , H., L a r o n g e , T. M., P h y s . 5 2 , 6453, (1970).  14.  K e s s l e r , J . 0., i n " L i q u i d C r y s t a l s and O r d e r e d F l u i d s " e d i t e d by J o h n s o n , R. F. and P o r t e r , R. S., Plenum P r e s s , New Y o r k (1970) p . 3 6 1  15.  De G e n n e s , P. G., M o l . C r y s t .  16.  S a u p e , A., Angew. Chem. I n t e r n a t .  17.  Sachman, E . , Chem. P h y s . L e t t . 3, 253,  and L a b e s , M. M., J . Chem.  8_, 531,  (1969).  Edit.  7, 97, (1969).  (1969).  62  18.  W y s o c k i , J . J . , Adams, J . , and H a a s , 2p_, 1 0 2 4 , ( 1 9 6 8 ) .  19.  F e r r u t t i , P., G i l l , D., H a r p o l d , M. A., and K l e i n , J . Chem. P h y s . 5 0 , 4 5 4 5 , ( 1 9 6 9 ) .  20.  Saupe,  A., Z. N a t u r f .  A 19,  l 6 l ,  W.,  (1964).  P h y s . Rev.  Lett. M.  P.,  

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