UBC Theses and Dissertations

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UBC Theses and Dissertations

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 University of Alberta, 1985 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1988 ® James Theodore Gleeson, 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date March 7 1988 Abstract The s u r f a c e a n c h o r i n g o f the n e m a t i c l i q u i d c r y s t a l 5CB a l i g n e d t a n g e n t i a l l y on two d i f f e r e n t g l a s s s u b s t r a t e t r e a t m e n t s i s s t u d i e d . These two t r e a t m e n t s a re o b l i q u e l y e v a p o r a t e d S iO and b u f f e d p o l y ( v i n y l f o r m a l ) d e p o s i t e d from s o l u t i o n . The l a t t e r i s b e l i e v e d to be a new t r e a t m e n t and o f p o s s i b l e i m p o r t a n c e to the t w i s t e d n e m a t i c l i q u i d c r y s t a l d i s p l a y i n d u s t r y . Two methods a r e employed to measure the s t r e n g t h o f the s u r f a c e a n c h o r i n g p o t e n t i a l . The f i r s t was o r i g i n a l l y d e v e l o p e d by Yokoyama and v a n S p r a n g , and y i e l d s the c o e f f i c i e n t o f the R a p i n i - P a p o u l a r a n c h o r i n g p o t e n t i a l . The s e c o n d i s a new method b a s e d on the F r a n k - O s e e n e l a s t i c i t y t h e o r y . I t i s an e x t e n s i o n o f D e u l i n g ' s 1972 s o l u t i o n o f t he s p l a y F r e e d e r i c k s z t r a n s i t i o n . The s e c o n d y i e l d s the f u n c t i o n a l fo rm o f the a n c h o r i n g p o t e n t i a l . B o t h methods a r e b a s e d on measurements o f the b u l k d i e l e c t r i c c o n s t a n t and the e f f e c t i v e r e f r a c t i v e i n d e x o f a n e m a t i c sample t h a t has undergone a s p l a y F r e e d e r i c k s z t r a n s i t i o n . The f i r s t method r e q u i r e s b o t h t h e s e measurements , and the s e c o n d r e q u i r e s one o r the o t h e r ; b o t h c a n be employed as a v e r i f i c a t i o n o f r e s u l t s . The l a t t e r method i s found to be v e r y s e n s i t i v e to the v a l u e o f t he d i e l e c t r i c a n i s o t r o p y ; t h i s i s b e l i e v e d to be due to n o t a l l o w i n g f o r s p a t i a l v a r i a t i o n s i n the s c a l a r o r d e r p a r a m e t e r . i i T a b l e o f C o n t e n t s A b s t r a c t i i L i s t o f F i g u r e s i v Acknowledgement v i C h a p t e r I INTRODUCTION TO NEMATIC LIQUID CRYSTALS AND SURFACE ANCHORING 1 L i q u i d C r y s t a l s and Nema t i c O r d e r 1 E l a s t i c i t y T h e o r y 4 E l e c t r i c F i e l d E f f e c t s 5 B i r e f r i n g e n c e 6 S u r f a c e A n c h o r i n g 7 C h a p t e r I I THEORETICAL CONSIDERATIONS 8 T h e o r y 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 8 F l e x o e l e c t r i c i t y 13 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 T r a n s i t i o n 17 E f f e c t o f F i n i t e S u r f a c e A n c h o r i n g 20 The Y o k o y a m a - - v a n Sp rang Method 21 E x t e n d e d D e u l i n g Method 26 C h a p t e r Summary 31 C h a p t e r I I I EXPERIMENTAL APPARATUS 32 Sample P r e p a r a t i o n 32 Tempera tu re C o n t r o l 37 E l e c t r o n i c s 39 O p t i c a l Measurements 41 Summary o f E x p e r i m e n t 43 C h a p t e r IV DATA ANALYSIS AND EXPERIMENTAL RESULTS 44 R e d u c t i o n 44 M a t e r i a l C o n s t a n t s 47 R e s u l t s : M a t e r i a l C o n s t a n t s 49 S u r f a c e A n c h o r i n g R e s u l t s 55 Yokoyama-van Sp rang Method 55 E x t e n d e d D e u l i n g Method 61 C h a p t e r V DISCUSSION AND CONCLUSION 75 C o m p a r i s o n w i t h Yokoyama et al's work 75 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 76 C o n c l u s i o n 79 A p p e n d i x EFFECT OF F I N I T E SURFACE ANCHORING ON THE 81 THRESHOLD F I E L D R e f e r e n c e s 85 i i i L i s t of Figures 1. A schematic representation of a nematic l i q u i d with various degrees of order 2. A schematic representation of the splay Freedericksz transition. 3. Director angle vs c e l l position for various 14 reduced displacements. 4. Plot of reduced capacitance vs reduced 18 displacement based on the Deuling calculation. 5. Plot of reduced phase vs reduced displacement 19 based on the Deuling calculation. 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. D i e l e c t r i c constants as a function of 51 temperature. 12. E l a s t i c constant anisotropy as a function of 52 temperature. 13. Splay e l a s t i c temperature. constant as function of 53 14. Bend e l a s t i c temperature. constant as function of 54 15. Extrapolation length as a function of 56 temperature for SiO treated surface. 16. Out of plane anchoring strength as a function 57 of temperature for SiO treated surface. 17. Extrapolation length as a function of 59 temperature for buffed PVF surface. 18. Out of plane anchoring strength as a function 60 of temperature for buffed PVF surface. 19. T i l t angle vs reduced voltage for SiO treated 62 surface. iv 20. S u r f a c e torque as a f u n c t i o n o f t i l t angle f o r 63 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 o f t i l t angle 64 f o r SiO t r e a t e d s u r f a c e . 22. T i l t angle as a f u n c t i o n o f reduced v o l t a g e f o r 65 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 torque as a f u n c t i o n o f t i l t angle f o r 66 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 SiO t r e a t e d 67 s u r f a c e : f i t to phase d a t a . 25. S u r f a c e torque as a f u n c t i o n o f t i l t angle f o r 68 SiO t r e a t e d s u r f a c e : f i t to phase data. 26. A n c h o r i n g s t r e n g t h vs temperature f o r SiO 70 t r e a t e d s u r f a c e : r e s u l t s o f ED method. 27. T i l t angle as a f u n c t i o n o f reduced v o l t a g e f o r 71 rubbed PVf s u r f a c e . 28. S u r f a c e torque as a f u n c t i o n o f t i l t angle f o r 72 rubbed 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 o f t i l t angle 73 f o r rubbed 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 f o r rubbed 74 PVF s u r f a c e : r e s u l t s o f ED method. 31. S u r f a c e torque as f u n c t i o n o f t i l t angle 77 showing the dependence on the d i e l e c t r i c a n i s o t r o p y used i n f i t t i n g the data. v Acknowledgment T h i s work c o u l d n o t have been a t t e m p t e d w i t h o u t the c o n t i n u e d gu i dance and s u p p o r t o f my t h e s i s s u p e r v i s o r , Dr. P e t e r P a l f f y - M u h o r a y . Much o f what i s o r i g i n a l i n t h i s t h e s i s stemmed f rom h i s s u g g e s t i o n s . Most o f the e x p e r i m e n t s d e s c r i b e d h e r e i n were p e r f o r m e d a t the L i q u i d C r y s t a l I n s t i t u t e a t Kent S t a t e U n i v e r s i t y . I am d e e p l y i n d e b t e d t o the members o f t he I n s t i t u t e , e s p e c i a l l y the D i r e c t o r , P r o f . J.W. Doane and the A s s o c i a t e D i r e c t o r , P r o f . M.A. Lee . The I n s t i t u t e p r o v i d e d me w i t h space t o s e t up my e x p e r i m e n t s , equ ipment w i t h w h i c h t o p e r f o r m them. The members o f the I n s t i t u t e p r o v i d e d me w i t h c o u n t l e s s i d e a s and s u g g e s t i o n s i n a d d i t i o n t o unend i ng h o s p i t a l i t y t h r o u g h o u t my s t a y . I am i n d e b t e d t o the N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l o f Canada and the Department o f P h y s i c s a t Kent S t a t e U n i v e r s i t y f o r p r o v i d i n g me w i t h f u n d i n g . Much o f t he cus tom equ ipment r e q u i r e d f o r the e x p e r i m e n t was wrought t h r o u g h the e f f o r t s o f my p r e d e c e s s o r , S.W. M o r r i s . My e f f o r t s i n the l a b o r a t o r y were g r e a t l y a i d e d by the p r e s e n c e o f B . J . F r i s k e n whose i d e a s , a s s i s t a n c e and p a t i e n c e were n e v e r p r o p e r l y a cknow ledged . I s h a l l a lway s be i n d e b t e d t o Becky , who so many t ime s w i l l i n g l y t ook an u n d e s e r v e d b a c k s e a t t o t h i s t h e s i s . v i Chapter I I n t r o d u c t i o n to Nematic L i q u i d C r y s t a l s and S u r f a c e A n c h o r i n g T h i s t h e s i s i s concerned w i t h s u r f a c e a n c h o r i n g of nematic l i q u i d c r y s t a l s . S u r f a c e a n c h o r i n g d e s c r i b e s the i n t e r a c t i o n o f l i q u i d c r y s t a l s w i t h s u r f a c e s s u b j e c t to v a r i o u s treatments. The s u b j e c t was d i s c u s s e d by R a p i n i and Papoular i n 1969 , 1 and i s an a r e a o f i n t e r e s t f o r many l i q u i d c r y s t a l r e s e a r c h e r s today. The mechanism o f s u r f a c e a n c h o r i n g i s not w e l l understood and t h e r e i s an ongoing 2 3 debate over the models o f s u r f a c e a n c h o r i n g ; ' as w e l l t h e r e i s no w i d e l y a c c e p t e d t e c h n i q u e o f measuring s u r f a c e a n c h o r i n g p r o p e r t i e s . Furthermore, c e r t a i n types o f l i q u i d c r y s t a l s d i s p l a y s (LCD's) r e l y on s u r f a c e treatments f o r t h e i r o p e r a t i o n , i n which s u r f a c e a n c h o r i n g p l a y s a major r o l e . The aim o f t h i s t h e s i s i s f i r s t l y to employ a known tec h n i q u e o f measuring s u r f a c e a n c h o r i n g p r o p e r t i e s on a n o v e l s u r f a c e treatment and s e c o n d l y to p r e s e n t a new method of measuring the s u r f a c e a n c h o r i n g p o t e n t i a l . D e s c r i b i n g s u r f a c e a n c h o r i n g n e c e s s i t a t e s f i r s t d i s c u s s i n g the b a s i c p r o p e r t i e s o f nematic l i q u i d c r y s t a l s ; t h i s i s done i n the next s e c t i o n o f t h i s c h a p t e r . The s u r f a c e a n c h o r i n g p o t e n t i a l i s d e f i n e d a t the end o f t h i s c h a p t e r . Liquid Crystals and Nematic Order L i q u i d c r y s t a l s are s t a t e s o f matter i n t e r m e d i a t e between l i q u i d s and c r y s t a l l i n e s o l i d s . For the most p a r t , they e x h i b i t the f l u i d i t y o f l i q u i d s ; they adopt the shape of whatever v e s s e l c o n t a i n s them. 1 F i g u r e 1. A schematic r e p r e s e n t a t i o n o f a nematic l i q u i d c r y s t a l w i t h v a r i o u s degrees o f o r d e r . 2 However they do not show the i s o t r o p y o f o r d i n a r y l i q u i d s . There are many d i f f e r e n t l i q u i d c r y s t a l phases and they are c h a r a c t e r i z e d by the symmetry they do or do not e x h i b i t . The m a j o r i t y o f l i q u i d c r y s t a l phases e x h i b i t o r i e n t a t i o n a l o r d e r to some degree. T h i s type o f o r d e r i s c h a r a c t e r i z e d by long range c o r r e l a t i o n s o f m o l e c u l a r d i r e c t i o n s . That i s to say, on the average the molecules " p o i n t " i n a common d i r e c t i o n . The most common phase o f t h i s type, which shows the l e a s t amount o f o r d e r o f a l l l i q u i d c r y s t a l phases, i s the u n i a x i a l nematic phase. I n t h i s nematic phase, the lowest energy s t a t e i s one i n which on the average the o r i e n t a t i o n o f one a x i s i n the m o l e c u l e ' s r e f e r e n c e frames i s i n a common d i r e c t i o n . Note t h a t u n i a x i a l nematics are observed to be symmetric w i t h r e s p e c t to i n v e r s i o n i n the pl a n e p e r p e n d i c u l a r to t h a t common d i r e c t i o n . Thus the p r o p e r q u a n t i t y to d e s c r i b e the amount o f o r i e n t a t i o n a l o r d e r i n 2 a u n i a x i a l nematic i s m o l e c u l a r average o f cos (6) where 6 i s the angle between the m o l e c u l a r a x i s and the p r e f e r r e d d i r e c t i o n . One u s u a l l y d e f i n e s the o r d e r parameter S(T) as <?^(cos(9))> where i s the second Legendre p o l y n o m i a l . T h i s c h o i c e r e s u l t s i n S b e i n g between 1 and 0, S = 0 c o r r e s p o n d i n g to the unordered ( i s o t r o p i c ) phase and S = 1 c o r r e s p o n d i n g t o a p e r f e c t l y o r d e r e d nematic phase. F i g u r e 1 d e p i c t s s c h e m a t i c a l l y a nematic l i q u i d c r y s t a l w i t h v a r i o u s degrees o f o r d e r . More g e n e r a l l y , the degree o f o r d e r p r e s e n t i n a u n i a x i a l nematic i s d e s c r i b e d by a s p a t i a l l y v a r y i n g t e n s o r o r d e r parameter Q „, where 1.1 and 3 n n = 1 1.2 S(T) i s r e f e r r e d to as the s c a l a r o r d e r parameter and n i s r e f e r r e d to as the director. n ( r ) s h o u l d be thought of as the d i r e c t i o n i n which molecules are p o i n t i n g averaged over an i n f i n i t e s i m a l volume 3 d r c e n t e r e d a t r . The i n f i n i t e s i m a l volume must s t i l l c o n t a i n many mol e c u l e s . Elasticity Theory A In the b u l k nematic, the u n d i s t o r t e d s t a t e , where n ( r ) i s uniform, has the lowest energy d e n s i t y . 5 T h i s i s not t r u e f o r a l l l i q u i d c r y s t a l phases. I f i n one r e g i o n o f a nematic the p r e f e r r e d d i r e c t i o n i s d i f f e r e n t than i n another, t h e r e must be a r e g i o n i n which t h i s d i r e c t i o n changes. Somewhere i n t h a t r e g i o n , t h e r e must be r e g i o n s where the " d i r e c t o r s " are not as p a r a l l e l as i n a nematic w i t h o n l y one p r e f e r r e d d i r e c t i o n . Since t h i s i s d i f f e r e n t from the nematic ground s t a t e , i t must be s t a t e o f h i g h e r energy. The amount by which the energy has i n c r e a s e d i s the s u b j e c t of Frank-Oseen e l a s t i c i t y t h e o r y . E l a s t i c i t y t h e ory says t h a t i n a d i s t o r t e d nematic, t h e r e i s an i n c r e a s e i n the f r e e energy d e n s i t y t h a t depends A on the s p a t i a l change i n the d i r e c t o r n. T h i s i n c r e a s e i s the e l a s t i c f r e e energy d e n s i t y 5 . S i n c e the symmetry of the nematic d A A d i c t a t e s t h a t n must be the same as -n, 5^  must have t h a t symmetry as w e l l . The e l a s t i c f r e e energy d e n s i t y must t h e r e f o r e be expandable i n i n v a r i a n t s formed from d e r i v a t i v e s o f n t h a t are A unchanged when n changes s i g n . I t can be shown t h a t any c ombination o f these i n v a r i a n t s which c o n t a i n s o n l y f i r s t d e r i v a t i v e s can be 4 5 - - K ( V « n ) 2 + K (n«Vxn) 2 + K (nxVxn) 2 d 2 1 2 3 expressed i n terms o f three c a n o n i c a l d e f o r m a t i o n s . These are s p l a y : V»n, t w i s t : n«Vxn, and bend: nxVxn. To f i r s t o r d er, the e l a s t i c f r e e energy i s q u a d r a t i c i n these d e f o r m a t i o n s , w i t h c o e f f i c i e n t s r e f e r r e d to as e l a s t i c c o n s t a n t s . ( i = l - 3 ) These are p r o p o r t i o n a l to S and have u n i t s o f f o r c e . 5 Thus the f r e e energy d e n s i t y due to i n h o m o g e n e i t i e s i n the d i r e c t o r f i e l d i s g i v e n by: A 1.3 Note t h a t to be s t r i c t l y c o r r e c t , the theory of d e f o r m a t i o n s s h o u l d a l l o w f o r s p a t i a l v a r i a t i o n s i n S; and the e l a s t i c energy d e n s i t y s h o u l d be r e p l a c e d by an e l a s t i c free energy d e n s i t y , i n which the A entropy change due to v a r i a t i o n s i n S and n i s c o n s i d e r e d . The l a t t e r i s a s u b t l e p o i n t , because one c o u l d c l a i m t h a t any e n t r o p i c e f f e c t s are i n h e r e n t l y c o n t a i n e d i n the e l a s t i c c o n s t a n t s . These a d d i t i o n s to d e f o r m a t i o n t h e o r y are beyond the scope of t h i s t h e s i s . Electric Field Effects Another a s p e c t r e l e v a n t to t h i s work o f nematic l i q u i d c r y s t a l s i s t h e i r response to an a p p l i e d e l e c t r i c f i e l d . In an i s o t r o p i c medium, the p o l a r i z a t i o n c r e a t e d by an a p p l i e d f i e l d i s n e c e s s a r i l y i n the same d i r e c t i o n as t h a t f i e l d . In a n i s o t r o p i c media t h a t i s not the c a s e . In g e n e r a l , the e l e c t r i c d i s p l a c e m e n t B i s r e l a t e d to an a p p l i e d f i e l d ! by a d i e l e c t r i c t e n s o r e. In u n i a x i a l nematics t h i s t e n s o r has two p r i n c i p a l v a l u e s e and e ; the d i e l e c t r i c r r 1 2 A c o n s t a n t s p a r a l l e l and p e r p e n d i c u l a r to n r e s p e c t i v e l y . Ae = -i s d e f i n e d as the d i e l e c t r i c a n i s o t r o p y . In most nematics ( i n c l u d i n g those used i n the experiments d e s c r i b e d i n t h i s t h e s i s ) Ae i s 5 p o s i t i v e . I f one n e g l e c t s l o c a l f i e l d c o r r e c t i o n s , then Ae i s p r o p o r t i o n a l to the s c a l a r o r d e r parameter. The i n t e r e s t i n g s i t u a t i o n i s where the d i r e c t o r i s not everywhere p a r a l l e l to an a p p l i e d f i e l d . Then the f i e l d can apply a torque to the d i r e c t o r and perhaps induce a d e f o r m a t i o n . T h i s can be d e s c r i b e d f o r m a l l y by i n c l u d i n g the p a r t o f the e l e c t r i c energy d e n s i t y t h a t depends on the A d i r e c t o r n as a term the e l a s t i c energy d e n s i t y . B i r e f r i n g e n c e Nematics, b e i n g a n i s o t r o p i c by d e f i n i t i o n , are a l s o b i r e f r i n g e n t . In g e n e r a l the v e l o c i t y o f l i g h t p a s s i n g through a nematic w i l l depend on the angle between the e l e c t r i c f i e l d v e c t o r o f the l i g h t wave and the d i r e c t o r . L i g h t p o l a r i z e d p a r a l l e l to the d i r e c t o r w i l l t r a v e l a t a speed c/n , and l i g h t p o l a r i z e d p e r p e n d i c u l a r to the e d i r e c t o r w i l l t r a v e l a t a speed c/n . n and n are the o e o e x t r a o r d i n a r y and o r d i n a r y r e f r a c t i v e i n d i c e s r e s p e c t i v e l y . The d i f f e r e n c e An = n - n i s the b i r e f r i n g e n c e and i s p r o p o r t i o n a l to e o 4 the s c a l a r o r d e r parameter. L i g h t p o l a r i z e d a t some angle i> to the d i r e c t o r w i l l t r a v e l a t a speed c/n(V>) . n(V>) may be found by c o n s i d e r i n g the magnitude o f the v e l o c i t y v e c t o r w i t h component c/n e i n the d i r e c t i o n p a r a l l e l to n and component c/n i n the d i r e c t i o n o A p e r p e n d i c u l a r to n: n n n(tf) - 6 ° 1.4 / 2 2 2 2 n s i n (V>) + n cos (V>) 6 S u r f a c e A n c h o r i n g O f t e n , a l i q u i d c r y s t a l experiment or a p p l i c a t i o n w i l l demand a u n i f o r m l y a l i g n e d sample. Since t h i s was s t a t e d to be the nematic ground s t a t e above, a u n i f o r m l y a l i g n e d sample s h o u l d not be d i f f i c u l t to a t t a i n . However, the d i r e c t o r f i e l d p r e s e n t i n a sample o f a nematic depends on the boundary c o n d i t i o n s a t the s u r f a c e and on any e x t e r n a l l y a p p l i e d f i e l d s . The s u r f a c e c o n d i t i o n s are c l a s s i f i e d by the type o f alignment e x p e c t e d i n a nematic i n c o n t a c t w i t h the s u r f a c e . Namely, one can c r e a t e t a n g e n t i a l (homogeneous), normal (homeotropic) or t i l t e d a l ignment. On such a s u r f a c e , one d e f i n e s the "easy a n g l e " as the angle between the d i r e c t o r and the s u r f a c e i n the u n d i s t o r t e d s t a t e . The s i t u a t i o n o f i n t e r e s t f o r t h i s t h e s i s i s the f i r s t , t a n g e n t i a l a lignment i n which the d i r e c t o r i s p e r p e n d i c u l a r to the s u r f a c e normal. The methods o f i n d u c i n g t a n g e n t i a l alignment are d i s c u s s e d i n c h a p t e r I I I . The q u e s t i o n o f t h i s t h e s i s i s how s t r o n g i s the anchoring? More s p e c i f i c a l l y , i f a torque i s a p p l i e d to the d i r e c t o r a t the s u r f a c e , how w i l l the o r i e n t a t i o n change? I t i s assumed t h a t the energy o f i n t e r a c t i o n o f the d i r e c t o r w i t h the s u r f a c e i s a f u n c t i o n o f the a ngle <f> between the d i r e c t o r and the s u r f a c e . By the d e f i n i t i o n o f the easy a n g l e , t h a t f u n c t i o n must be a minimum when 4> i s the easy a n g l e . T h i s f u n c t i o n i s r e f e r r e d to as the s u r f a c e a n c h o r i n g p o t e n t i a l W ^ ) . The t h e o r e t i c a l f o u n d a t i o n f o r measuring p r o p e r t i e s o f W(<£ ) i s the s u b j e c t o f the next c h a p t e r . 7 C h a p t e r I I T h e o r e t i c a l C o n s i d e r a t i o n s 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 The F r e e d r i c k s z t r a n s i t i o n i s a r e o r i e n t a t i o n o f the d i r e c t o r f i e l d 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 the i n f l u e n c e o f an e x t e r n a l magne t i c o r e l e c t r i c f i e l d . The ca se o f i n t e r e s t f o r t h i s t h e s i s i s t he s p l a y F r e e d e r i c k s z t r a n s i t i o n i n d u c e d by an e l e c t r i c f i e l d . The t r a n s i t i o n i s d e p i c t e d s c h e m a t i c a l l y i n f i g u r e 2. Note t h a t t h i s p i c t u r e i m p l i e s the " r i g i d a n c h o r i n g a s s u m p t i o n " . R i g i d a n c h o r i n g i n t h i s c a se means t h a t the d i r e c t o r on the s u r f a c e i s f i x e d p a r a l l e l t o t h a t s u r f a c e r e g a r d l e s s o f s t r e n g t h o f the a p p l i e d f i e l d . One w o u l d l i k e t o c o n s i d e r the t r a n s i t i o n d e p i c t e d i n f i g u r e 2 m a t h e m a t i c a l l y . C o n s i d e r a s y s tem w i t h two i n f i n i t e p a r a l l e l p l a t e s , s e p a r a t e d by a d i s t a n c e d. The n e m a t i c l i q u i d c r y s t a l i s homogeneous ly a l i g n e d a t the p l a t e s and i s c o n s t r a i n e d t o l i e p a r a l l e l t o the p l a t e s . L e t the p l a t e no rma l s be the k d i r e c t i o n . Thus the f i r s t p l a t e i s t he p l a n e z = 0 and the second i s t he p l a n e z = d . The p l a t e s u r f a c e s a r e such t h a t the d i r e c t o r n on the p l a t e s A i s p a r a l l e l t o i . The e l e c t r i c f i e l d w i l l be a p p l i e d by c r e a t i n g a p o t e n t i a l d i f f e r e n c e between the . p l a t e s . The r e s u l t i n g e l e c t r i c A f i e l d c an have no component o t h e r t han the k d i r e c t i o n by symmetry. A A l s o f rom symmetry, n and ^ can be f u n c t i o n s o f z o n l y . A s suming A A t h a t t h e r e w i l l no be component o f n i n t he j d i r e c t i o n , the d i r e c t o r A n can be w r i t t e n a s : A A A n - c o s ( $ ) i + sin(<^)k. 2.1 8 X igure 2. A Schematic representation of the splay Freedericksz transition. 9... where <$> = <j>(z) . In o r d e r to determine the f u n c t i o n 4>(z) and hence the d i r e c t o r f i e l d n i n e q u i l i b r i u m , one must f i n d the thermodynamic p o t e n t i a l F which i s a minimum at c o n s t a n t e l e c t r i c d i s p l a c e m e n t and temperature. The reason the e l e c t r i c d i s placement i s used i s t h a t by Gauss' law, the d i s p l a c e m e n t i s homogeneous throughout. So then which thermodynamic p o t e n t i a l to use? When a p o l a r i z a b l e medium i s i n t r o d u c e d i n t o an e l e c t r i c f i e l d , the f i e l d does work dW = J dB« .^ Hence the d i f f e r e n t i a l change i n energy i s d§ = TdS + JdB«^. I t i s then c o n v e n i e n t to c o n s t r u c t a thermodynamic p o t e n t i a l F = & - TS; a s m a l l change i n t h i s p o t e n t i a l , dF, i s zero i f T and B are c o n s t a n t . Thus t h i s i s the " f r e e energy" to be minimized i n the p r e s e n t problem. As s t a t e d i n Chapter I, one can t h i n k of e n t r o p i c terms i n t h i s a n a l y s i s as b e i n g c o n t a i n e d i n the e l a s t i c c o n s t a n t s , and as w e l l as i n the d i e l e c t r i c a n i s o t r o p y Ae. Thus the p o r t i o n o f the A f r e e energy t h a t depends on the d i r e c t o r f i e l d n i s : F - J £ ? d + i $.t j dz 2.2 Where 5 i s the the energy d e n s i t y o f a deformed nematic l i q u i d d c r y s t a l due to deformations as g i v e n i n e q u a t i o n 1.3. In the geometry o f t h i s problem: ? = ; ( K C O S 2 ( * ) + K s i n 2 ( * ) ] [ f | d 2 I 1 3 J L° z 2.3 As d i s c u s s e d i n c h a p t e r I, the d i e l e c t r i c c o n s t a n t i s a t e n s o r whose p r i n c i p a l a x i s i s p a r a l l e l to the d i r e c t o r . A g a i n by Gauss' law the e l e c t r i c d i s p l a c e m e n t i s the a r e a l charge d e n s i t y a on the A p l a t e s and i s p a r a l l e l to the p l a t e normal, i . e . 3 = Dk. By symmetry the o n l y non-zero component of \ i s a l o n g the p l a t e normal as w e l l ; one can w r i t e \ = Ek. T h e r e f o r e the p r o d u c t D^ E* can be w r i t t e n as 10 DE, and, s i n c e D - e t E, the energy d e n s i t y c o n t r i b u t i o n can be o 33 w r i t t e n as 1 D z e e 0 33 2.4 £ i s the 3-3 component o f the d i e l e c t r i c t e n s o r a f t e r i t has been r o t a t e d so t h a t i t s p r i n c i p a l a x i s l i e s a l o n g n; 2 2 e e s i n (<j>) + e cos (6). E q u a t i o n 2.2 can then be w r i t t e n as 33 l Y 2 M 2 2 -\ cos (<£)+K s i n (0)J dz 2 2 e (e s i n (<^ ) + e cos ($)) J o 1 2 dz 2.5 The e q u i l i b r i u m v a l u e o f the f u n c t i o n 4>(z) t h a t s p e c i f i e s the d i r e c t o r everywhere i s t h a t f u n c t i o n which makes F a minimum a t c o n s t a n t D and g i v e n boundary c o n d i t i o n s . A l t e r n a t i v e l y , one may c o n s i d e r a s i t u a t i o n on which the v o l t a g e between the p l a t e s i s h e l d f i x e d i n s t e a d o f the d i s p l a c e m e n t . In t h i s case, the a p p r o p r i a t e f r e e energy to use i s : „d 2.6 The p o t e n t i a l d i f f e r e n c e V i s g i v e n by: d V j* E dz = \ D dz € £ 0 o 33 dz £ (e s i n 2 ( ^ ) + £ cos 2(<£)) 0 o 1 2 2.7 and the f r e e energy becomes: 11 - ] f [ Cn i =~ , <««, . in*(«) V 2 dz 2.8 2 2 « (c sin (^ ) + e cos (<b)) o o l 2 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. 6 It is more readily obtained from eq. 2.5 . In both cases i t is found that ^(z) is everywhere zero for displacements less than a threshold D given by: th D - 1 / ° 2 1 2.9 th d v 7 or for voltages less than V : th D d /T V . J±- _ * / — L - 2.10 th € t V € € 7 o 2 o 2 Where 7 - e /e - 1. At displacements larger than D (or voltages 1 2 th larger than V ), the director bends in the direction of the applied th f i e ld at the center of the c e l l and s t i l l remains aligned paral le l to the plates at the boundaries. Note that this threshold voltage is for the case of inf ini te anchoring strength; i f the anchoring is f in i t e , the threshold voltage w i l l depend on the strength of the anchoring. This dependence is discussed in the appendix. The change in threshold voltage due to this effect however is not expected to be signif icant; reasons for this are also given in the appendix. Figure 3 shows the angle <f> as a function of z for various reduced displacements. Note that while D may be a more convenient quantity to 12 work w i t h on paper, V i s much e a s i e r to measure and c o n t r o l e x p e r i m e n t a l l y . The important r e l a t i o n s h i p between D and V i s not eq. 2.7 but i s d e r i v e d from the f a c t t h a t the d i s p l a c e m e n t i s g i v e n by the charge d e n s i t y on the p l a t e s . T h i s c o u p l e d w i t h the d e f i n i t i o n o f c a p a c i t a n c e C: C = Q/V, and o f s u r f a c e charge d e n s i t y Q = CTA (where A i s the p l a t e area) enables one to w r i t e 2.11 The i m p l i c a t i o n i s obvious, i f one knows C and V, one knows the e l e c t r i c d i s p l a c e m e n t D, which i s the more n a t u r a l q u a n t i t y to work w i t h m a t h e m a t i c a l l y . Flexoelectricity C e r t a i n terms i n the f r e e energy t h a t depend on the d i r e c t o r c o n f i g u r a t i o n were absent from D e u l i n g ' s 1972 a n a l y s i s , 6 however he r e c t i f i e d t h a t s i t u a t i o n i n 1978. 7 The m i s s i n g terms are due to g f l e x o e l e c t r i c i t y . Meyer f i r s t d e s c r i b e d t h i s phenomenon i n 1969, then c a l l i n g i t p i e z o e l e c t r i c e f f e c t s . F l e x o e l e c t r i c i t y has s i n c e become the a c c e p t e d term to d e s c r i b e t h i s phenomenon. Simply put, i f a nematic i s s u b j e c t to a bend or s p l a y (not a t w i s t ) d e f o r m a t i o n , and i f i t s molecules have a c e r t a i n shape a n i s o t r o p y (the c l a s s i c examples are pear shaped molecules f o r s p l a y and banana shaped m o l e c u l e s f o r bend) as w e l l as a permanent d i p o l e moment, then t h e r e can be a non-zero f l e x o e l e c t r i c p o l a r i z a t i o n . T h i s i s because when the nematic i s deformed, molecules w i t h these p r o p e r t i e s can lower the d e f o r m a t i o n energy i f they a l i g n p a r a l l e l l o c a l l y r a t h e r than a n t i - p a r a l l e l . T h i s produces a net l o c a l p o l a r i z a t i o n . In g e n e r a l , 13 z/d F i g u r e 3. D i r e c t o r angle vs c e l l p o s i t i o n f o r v a r i o u s reduced d i s p l a c e m e n t s . 14 n e m a t i c s a re no t f e r r o e l e c t r i c because they have r e f l e c t i o n symmetry i n the p l a n e p e r p e n d i c u l a r to the d i r e c t o r . I f a d e f o r m a t i o n b r e a k s t h a t symmetry, t han i t becomes p o s s i b l e t o have f e r r o e l e c t r i c i t y i n the r e g i o n o f the d e f o r m a t i o n . F l e x o e l e c t r i c i t y , i f p r e s e n t , r e s u l t s i n two a d d i t i o n a l terms i n the f r e e energy d e n s i t y . The f i r s t i s the e l e c t r o s t a t i c ene rgy o f the f l e x o e l e c t r i c p o l a r i z a t i o n i n the f i e l d i t c r e a t e s . The second i s the i n t e r a c t i o n o f the f l e x o e l e c t r i c p o l a r i z a t i o n w i t h the e x t e r n a l f i e l d . The f l e x o e l e c t r i c 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 , i s : A A A A P" - e (V«n )n - e nxVxn . 2.12 1 3 where e and e a r e the f l e x o e l e c t r i c c o e f f i c i e n t s ; t hey a re r e l a t e d 1 3 J t o the shape a n i s o t r o p y o f the m o l e c u l e as w e l l as i t s permanent g d i p o l e moment and the s p l a y and bend e l a s t i c c o n s t a n t s r e s p e c t i v e l y . I n the geometry o f the p r ob l em a t hand : P = (e +e ) s i n ( 0 ) c o s ( ^ ) | ^ 2.13 z 1 3 dz The f l e x o e l e c t r i c p o l a r i z a t i o n canno t p roduce an e l e c t r i c f i e l d component i n the x d i r e c t i o n because t h e r e i s t r a n s l a t i o n a l symmetry i n t h a t d i r e c t i o n . One t h e n must w r i t e down the f r e e ene rgy anew w i t h t he c o n t r i b u t i o n s o f the " s e l f - e n e r g y " o f the f l e x o e l e c t r i c p o l a r i z a t i o n as w e l l as the ene rgy o f the e x t e r n a l f i e l d c o u p l e d t o t he f l e x o e l e c t r i c p o l a r i z a t i o n . The e l e c t r i c f r e e energy d e n s i t y i s now: ? - - e e E 2 2.14 E 2 o 33 U s i n g a c o n s t i t u t i v e r e l a t i o n : D - e e E + P , the above can be 0 33 z w r i t t e n : l 5 = E 2 £ £ o 33 D 2 - 2PD + P Z 15 2.15 The f i r s t term i s the e l e c t r i c f r e e energy due to l i n e a r p o l a r i z a t i o n i n the medium; the second i s the c o u p l i n g of the e x t e r n a l a p p l i e d f i e l d to the f l e x o e l e c t r i c p o l a r i z a t i o n and the t h i r d i s the s e l f - e n e r g y o f the f l e x o e l e c t r i c p o l a r i z a t i o n . The f r e e energy d e n s i t y per u n i t a r e a i s then the i n t e g r a l o f t h i s e x p r e s s i o n over the c e l l . As D i s homogeneous throughout the c e l l , i t can be taken o u t s i d e the i n t e g r a l ; comparing w i t h eq. 2.7, the f i r s t term can be i n t e g r a t e d immediately. One has then: d F - - DV E 2 _d P2 dz + i 2 £ £ 0 0 33 dz 2.16 £ £ 0 33 Upon c l o s e r e x a mination o f the second i n t e g r a l i n t h i s e x p r e s s i o n , one can i n t e g r a t e i t as w e l l : ( e x + e 3 ) s i n ( ^ ) c o s ( ^ ) H • 2 / A\ , Z r ±\ <$Z £ s i n (0) + £ cos (cp) 0 1 2 dz (e + e ) 1 1 3 2 £ A£ o l o g [ e s i n 2 ( ^ ) + £ 2cos 2(<&)] 2.17 I f we d e f i n e t h i s q u a n t i t y as V , we can w r i t e the e l e c t r i c energy p per u n i t a r e a a g a i n as: F = - D( V - 2V ) + ( e i + e 3 ) e 2 p „d 2 £ £ o 2 2 2 s i n (4>)cos (4>) (I + ysin2(4>) ) r d £ 2 dz dz 2.18 Note t h a t one e f f e c t o f the f l e x o e l e c t r i c i t y i s then to change the p o t e n t i a l d i f f e r e n c e a c r o s s the c e l l . In the r i g i d a n c h o r i n g assumption V v a n i s h e s because ^(0) = 4>(d) , and the o n l y p f l e x o e l e c t r i c term l e f t i s the s e l f - e n e r g y . Thus the energy per u n i t a r e a o f the b u l k nematic which i s expected to be minimum i n the e q u i l i b r i u m c o n f i g u r a t i o n at c o n s t a n t D i s : 16 F = 2 2 U „ _ L A s i n (^)cos (<t>) l+/c s m ( 0 ) + 2 1 + 7 s i n (<£) 0 $ D 2 - 2D (e +e ) s i n ( ^ ) c o s(4>)ir 1 3 dz dz 2.19 Where K = K /K - 1 and A <= (e + e ) /K £ £ . F must be 3' 1 1 3 ' 1 o 2 m i n i m i s e d a t c o n s t a n t D t o d e t e r m i n e the d i r e c t o r eve rywhere i n the s amp le . The boundary c o n d i t i o n s when f i n i t e a n c h o r i n g i s assumed have y e t t o be e s t a b l i s h e d ; t h i s w i l l be examined i n the l a s t two s e c t i o n s o f t h i s c h a p t e r . 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 T r a n s i t i o n The F r e e d e r i c k s z t r a n s i t i o n can be o b s e r v e d as change i n the b u l k p r o p e r t i e s o f a n e m a t i c l a y e r as the v o l t a g e a c r o s s i t i s i n c r e a s e d . The c a p a c i t a n c e o f such a l a y e r i s g i v e n b y : 1 C 1 A dz £ £ o 3 3 2.20 I f D i s b e l o w t he t h r e s h o l d , (£(z) r ema in s c o n s t a n t a t z e r o , and hence the c a p a c i t a n c e w i l l r ema i n c o n s t a n t a t C = f e A/d . A t the r 0 o 2 t h r e s h o l d D, <0(z) w i l l become n o n - z e r o a b r u p t l y and hence C w i l l i n c r e a s e s h a r p l y . U s i n g D e u l i n g ' s 6 r e s u l t s , the r e d u c e d c a p a c i t a n c e c = C / h a s been c a l c u l a t e d as a f u n c t i o n o f r e d u c e d d i s p l a c e m e n t D/D ; t h i s i s shown i n f i g u r e 4. th I n a d d i t i o n , t he o p t i c a l p r o p e r t i e s o f such a l a y e r w i l l change i n an ana l ogou s manner. The phase d i f f e r e n c e between the e x t r a o r d i n a r y and the o r d i n a r y r a y s ( h e r e a f t e r r e f e r r e d t o as the phase) i s w r i t t e n 17 Figure 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 Figure 5. P l o t of reduced phase vs reduced displacement based on the Deuling c a l c u l a t i o n 19 2.21 o where n(4>) i s the q u a n t i t y d e f i n e d i n eq. 1.4 and A i s the vacuum wavelength o f the l i g h t . The phase w i l l be c o n s t a n t at 5 q = 2?rdAn/A below the t h r e s h o l d , and w i l l decrease s h a r p l y a t the F r e e d e r i c s k z t r a n s i t i o n . A p l o t o f 5/6q V S D/ n t h ( a g a i n based on D e u l i n g ' s c a l c u l a t i o n ) i s shown in f i g u r e 5. Effect of Finite Surface Anchoring The s u b j e c t o f t h i s t h e s i s i s to lo o k a t the problem above when th e r e i s no r i g i d a n c h o r i n g on the p l a t e s ; indeed, the main t h r u s t i s to measure j u s t how s t r o n g l y the d i r e c t o r i s c o n s t r a i n e d to l i e p a r a l l e l to the p l a t e s f o r d i f f e r e n t methods o f s u r f a c e treatment. F i n i t e s u r f a c e a n c h o r i n g can be accounted f o r i n the b u l k f r e e energy d e n s i t y (eq. 2.19) simply by the i n c l u s i o n o f the terms: and are thus the a n c h o r i n g e n e r g i e s on p l a t e s z = 0 and z = d r e s p e c t i v e l y . R a p i n i and P a p o u l a r 1 (R-P) proposed i n 1969 t h a t a n c h o r i n g s u r f a c e energy was s p r i n g - l i k e ; i . e . t h a t the d i r e c t o r a t the s u r f a c e was bound by simple harmonic p o t e n t i a l : Here 6 i s the easy angle ( z e r o i n the geometry c o n s i d e r e d ) and <f> i s the d i r e c t o r angle a t z = 0. Indeed, one c o u l d say t h a t t h i s i s t r u e i n g e n e r a l f o r s m a l l enough d e v i a t i o n s , and t h a t the R a p i n i - P a p o u l a r form i s merely the f i r s t term i n the expansion o f any s u r f a c e a n c h o r i n g energy. A few authors have proposed o t h e r W U)6(z) , W (0)5(z-d). 1 2 2.22 W(<£) - - W (0 - 8)Z 2.23 20 forms f o r the s u r f a c e a n c h o r i n g energy and there i s an ongoing debate 2 3 as to which i s the c o r r e c t one. ' However, the R-P form i s a l i m i t i n g case f o r a l l proposed forms i f the d e v i a t i o n angle i s s m a l l . The c o n s t a n t of p r o p o r t i o n a l i t y i n the R-P form i s i n any case a measure o f how s t r o n g l y the d i r e c t o r i s c o n f i n e d to a s u r f a c e . T h i s c o n s t a n t i s r e f e r r e d to as the anchoring strength. The Yokoyama - van Sprang Method g Yokayama and van Sprang (Y-vS) developed a method o f measuring the a n c h o r i n g s t r e n g t h based on the " s u r f a c e e x t r a p o l a t i o n l e n g t h " : d . T h i s parameter may be thought o f as how f a r one must e x t r a p o l a t e e <j>(z) i n t o to the r e g i o n s z. < 0 and z > d to reach the p o i n t 4> - 0. The Y-vS method r e l a t e s the c a p a c i t a n c e o f a c e l l and the v o l t a g e a c r o s s t h a t c e l l to the phase. The a n a l y s i s t h a t r e l a t e s these q u a n t i t i e s to the s u r f a c e e x t r a p o l a t i o n has been devel o p e d twice by 9 10 Yokoyama and h i s c o l l a b o r a t o r s u s i n g d i f f e r e n t approaches. ' The f i r s t ( r e f . 9) i s d i f f i c u l t to f o l l o w i n t u i t i v e l y and f o r t h a t r e a s o n the second ( r e f . 10) i s reviewed h e r e . The r e s u l t s o f the two a n a l y s e s are the same as f a r as the a n c h o r i n g s t r e n g t h i s concerned. Other r e s u l t s o f t h e i r f i r s t c a l c u l a t i o n w i l l be d i s c u s s e d l a t e r i n t h i s c h a p t e r . The second c a l c u l a t i o n has the added a t t r a c t i o n t h a t one i s a b l e to add f l e x o e l e c t r i c terms wi t h o u t changing the essence o f the a n a l y s i s . T h i s was not done i n r e f . 10. The b a s i s o f t h i s a n a l y s i s i s the form of the b u l k f r e e energy o f a s e m i - i n f i n i t e c e l l , t h a t i s e q u a t i o n 2.19 w i t h the upper l i m i t o f the i n t e g r a l changed to i n f i n i t y . I f one then changes the v a r i a b l e 21 of integration 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. This implies that 4>(z) i s determined uniquely by £ = Dz, 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>1 >* 0, and since we expect '<f> to be function of D, the function 6 must have a more complicated dependence on D than indicated above. Furthermore, the additional 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 additional 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, related d i r e c t l y to how strongly the director i s constrained to l i e i n the plane of the plates. Note also that d may be a e function of D. One would l i k e to relate the quantity d to experimentally e measured quantities. Recall the expression for the phase, eq. 2.21, in a semi-infinite geometry, i.e. i n the l i m i t d -» °°. I f one changes the variable of integration to <f> and uses equation 2.25, we can write this as: 2TT AD .17 2 [n(tf) - n ] d* 2.26 d0 l 9 1 i s the inverse function of 6; ^ i s the director angle at z=0, and we have used the fact that far enough away from the region of d i s t o r t i o n , <f> ~ rt/2 (Later in this chapter i t w i l l be shown that this 22 a p p r o x i m a t i o n i s not always d e s i r a b l e ) . The q u a n t i t y we are i n t e r e s t e d i n i s d e x t r a p o l a t e d to D = 0. In or d e r to see how S e v a r i e s w i t h D f o r s m a l l D, we expand the i n t e g r a l i n e q u a t i o n 2.26 about D = 0. In t h i s r e g i o n <f> i s expected to be s m a l l as w e l l . T h i s r e s u l t s i n : •X - 11. «D - ! = ( . < # „ ) - n j g ' 1 ! * . D=*0 D 2.27 D=0 to f i r s t o r d e r i n D. By e q u a t i o n 2.25, we see t h a t t h i s i s j u s t : 5 = — — ° - (n - n ) d ; 2.28 D A e 0 e In a more r e a l i z a b l e s i t u a t i o n , i . e . a f i n i t e geometry, i n which the c e l l i s bounded by p l a t e s a t z=0 and z=d, the above i s s t i l l v a l i d p r o v i d e d the d e f o r m a t i o n i n the c e l l i s l o c a l i z e d near these two p l a t e s and n e g l i g i b l e i n the c e n t e r . I f t h a t i s the case, then a f i n i t e c e l l i s e q u i v a l e n t to two s e m i - i n f i n i t e c e l l s p l a c e d back to back. In the c a l c u l a t i o n to produce f i g u r e 3, when D > 10 D the th angle <f> a t the c e n t e r o f the c e l l i s w i t h i n .001 r a d o f w/2. T h i s c a l c u l a t i o n was f o r the case o f r i g i d a n c h o r i n g . In the case o f f i n i t e a n c h o r i n g s t r e n g t h , the d e f o r m a t i o n i n the c e n t e r w i l l be even s m a l l e r a t these d i s p l a c e m e n t s . The seemingly c o n f l i c t i n g c o n d i t i o n s t h a t D > D and d> = 0 are s a t i s f i e d when the i n e q u a l i t i e s th 1 ^ d < £ < d 2.29 e h o l d , where £ i s the e l e c t r i c coherence l e n g t h d e f i n e d by i / K e e < " BZ-T-1 ' Assuming t h a t one may c o n s i d e r the s e m i - i n f i n i t e case to be c o m p l e t e l y analogous to the f i n i t e case, the o p t i c a l phase d i f f e r e n c e f o r the f i n i t e case, g i v e n a c o m p l e t e l y symmetric c e l l , w i l l be 23 e x a c t l y twice t h a t f o r the semi - i n f i n i t e case, and eq. 2.28 becomes — = — - 2 - e 2 31 S CV d ' o Here we have used e q u a t i o n the r e l a t i o n D = CV/A, and the u n d i s t o r t e d phase, <$o — 27rAnd/A. B i s a c o n s t a n t depending o n l y on m a t e r i a l p r o p e r t i e s and c e l l dimensions. T h i s i s the fundamental e q u a t i o n o f the e x p e r i m e n t a l method. I f the o p t i c a l phase d i f f e r e n c e i s p l o t t e d v e r s u s the i n v e r s e o f the p r o d u c t CV, a s t r a i g h t l i n e w i l l r e s u l t , and the i n t e r c e p t o f t h a t l i n e w i l l be p r o p o r t i o n a l to the s u r f a c e e x t r a p o l a t i o n l e n g t h . One e x p ects the e x t r a p o l a t i o n l e n g t h d to be i n v e r s e l y e p r o p o r t i o n a l to the s t r e n g t h o f the s u r f a c e a n c h o r i n g . The e x a c t r e l a t i o n s h i p i s o b t a i n e d from c o n s i d e r i n g the torque b a l a n c e a t the i n t e r f a c e . The torque a p p l i e d by the s u r f a c e on the i n f i n i t e s i m a l l a y e r s o f l i q u i d c r y s t a l a t the s u r f a c e s must equal the torque a p p l i e d on those l a y e r s by the b u l k l i q u i d c r y s t a l . T h i s b a l a n c e i s c o n t a i n e d i n the c o n d i t i o n s : £1 . • £F = ^ 2 2 32 dd> dd> ' dd> ' dd> l l 2 2 ^ and 0^  a r e the t i l t a n g l e s on the two s u r f a c e s r e s p e c t i v e l y . I t i s n a t u r a l to assume t h a t i n the geometry c o n s i d e r e d , i f the p l a t e s have been p r e p a r e d the same way, t h a t the s u r f a c e i n t e r a c t i o n s a r e the same on b o t h p l a t e s : W = = W. Yokoyama and van Sprang assumed not o n l y t h i s , but a l s o t h a t the angles at the two s u r f a c e s were the same. T h i s i s j u s t i f i e d i f the c e l l i s t r u l y symmetric. However, the term i n the f r e e energy c o r r e s p o n d i n g to the c o u p l i n g o f the e x t e r n a l a p p l i e d e l e c t r i c f i e l d to the f l e x o e l e c t r i c p o l a r i z a t i o n i s a n t i - s y m m e t r i c about the c e n t e r o f the c e l l . A l l the o t h e r terms 24 are symmetric. T h i s broken symmetry then r e s t r i c t s the assumption t h a t the a n g les a t the two s u r f a c e s are e q u a l . Note t h a t the term i n the f r e e energy due s o l e l y to the e x t e r n a l e l e c t r i c f i e l d does not come i n t o the torque b a l a n c e e q u a t i o n because e l e c t r i c f o r c e s cannot ap p l y a torque on an i n f i n i t e l y t h i n s l i c e o f the sample. The e l a s t i c f o r c e s can a p p l y a torque because no matter how t h i n the s l i c e , i t w i l l always have n e i g h b o r s , which w i l l t r a n s m i t the torque to the s l i c e . The g o a l i s to r e l a t e the s u r f a c e a n c h o r i n g energy to the e x t r a p o l a t i o n l e n g t h d which can be measured d i r e c t l y w i t h the e Yokoyama - van Sprang method. R e c a l l t h a t the e x t r a p o l a t i o n l e n g t h i s a f u n c t i o n o f the a p p l i e d e l e c t r i c f i e l d . The q u a n t i t y which i s r e l a t e d to the a n c h o r i n g s t r e n g t h i s l i m d (D) . The torque b a l a n c e D-»0 * a t the f i r s t i n t e r f a c e can be w r i t t e n as: K 1 2 2 , . 2.. . A s i n (d> )cos (<f> ) 1 + K s i n (<f>J + V V V y i 1 + 7 sin2(<f> dz 2=0 1 T h i s d e f i n e s the s l o p e ^ a t z — 0. Under the assumption t h a t <f>2^ i s s m a l l compared to <j>^, one may e x t r a p o l a t e <f>(z) l i n e a r l y to <f> = 0. T h i s i s by d e f i n i t i o n the p o i n t z = -d . That w i t h the s l o p e enables e one to w r i t e : + x d -— 2.34 dz z=0 2 Note t h a t the assumption t h a t <f>^ i s s m a l l compared to <f>^ i s the c o n d i t i o n t h a t one i s i n the regime where the R-P form o f W(<^) i s 2 v a l i d . D i s c a r d i n g terms o f 0(# ) i n eq. 2.33 above r e s u l t s i n : ^ . B 2 35 25 R e c a l l i n g eq. 2.23, the R-P s u r f a c e a n c h o r i n g p o t e n t i a l , the above reduces to W = K /d . T h i s i s the fundamental r e l a t i o n t h a t r e l a t e s o 1 e the measured d to the an c h o r i n g s t r e n g t h . T h i s i s a l s o the 6 d e f i n i t i o n o f d used by deGennes. 5 e Extended D e u l i n g Method The D e u l i n g 5 a n a l y s i s has been extended to a l l o w the t i l t angle $ to v a r y . T h i s e x t e n s i o n enables one to c a l c u l a t e the torque a t the s u r f a c e due to the p l a t e s as a f u n c t i o n o f the t i l t a n g le from measurements on the bu l k . T h i s method w i l l be r e f e r r e d to as the Extended Deuling (ED) method. C o n s i d e r 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 l e f t out. ,d r K j l + «sin 2(*)] [§§] e ,e„(l + 7 s i n 2 ( ^ ) ) o 2 ' dz 2.36 The j u s t i f i c a t i o n o f l e a v i n g out these terms w i l l be d i s c u s s e d i n the next c h a p t e r . As d i s c u s s e d e a r l i e r , the d i r e c t o r f i e l d i s s p e c i f i e d by t h a t f u n c t i o n <£(z) which minimizes F a t c o n s t a n t D. P e r f o r m i n g a v a r i a t i o n o f F under these c o n d i t i o n s y i e l d s the f o l l o w i n g n o n - l i n e a r d i f f e r e n t i a l e q u a t i o n : d_ dz K ( l + K sin(4>)) r d £ 2 dz e e f l + 7 s i n (4>)) o 2 *• J = 0 2.37 d<t> T h i s may be i n t e g r a t e d immediately, u s i n g the c o n d i t i o n t h a t ^ — 0 at z = d/2, and 4>{d/2) - <j> . T h i s r e s u l t s i n : d± dz £ £ K o 2 1 2 2 s i n (.</>) - s i n (cj>) (l+Ksin2(<j>)) ( l + 7 s i n 2 (0)) (l+ 7 s i n 2 ( < £ ) 1/2 2.38 26 T h i s can a l s o be i n t e g r a t e d f o r m a l l y r e s u l t i n g i n : D V o 2 1 s i n (<f> ) (1+Ksin ( ^ ) ) ( l + 7 s i n (<?))" s i n (<j> ) s i n (4>) d<j> 2.39 T h i s e q u a t i o n r e l a t e s the measurable q u a n t i t y D to the two angles <j> m and 4>^ which d e s c r i b e the c e l l d e f o r m a t i o n u n i q u e l y . In t h i s s i t u a t i o n one may not make the ap p r o x i m a t i o n <j> = n/2 because the m i n t e g r a n d i s s i n g u l a r a t the upper l i m i t i n t h a t case. The c e l l v o l t a g e , eq, 2.7 w i t h the change i n v a r i a b l e o f i n t e g r a t i o n from z to <f> ( u s i n g eq. 2.38 above) i s : K V - 2 £ A£ (1 + /csin (0))(1+ 7 s i n (<j> )) m ( s i n 2 ( 0 ) - s i n 2 ( ^ ) ) ( l + 7 s i n 2 ( < / 0 ) 1/2 d<j> 2.40 Shimoda et al used a s i m i l a r method to show that p r e t i l t e d c e l l s have ii no F r e e d e r i c k s z t h r e s h o l d v o l t a g e . With the s u b s t i t u t i o n s rj — s i n (4> ) and s in (^ ) = sin(<£ ) sin(V>) , the two i n t e g r a l s above in ro become, r e s p e c t i v e l y : V o 2 1 ,*72 r rj, L (1+ K f;sin (VO ) (l+7r?sin (VO) 1 - T] s i n (VO 1/ 2 dV> 2.41 and - T / 2 v - 2/rh A™ (1 + KT) s i n (V1)) (1 - rjsin 2 ( V 0 ) (1 + 7f?sin 2(V>)) 1/2 d<f> 2.hi Where ib = s i n l sin(4> ) l sin(<£ ) With the d e f i n i t i o n s o f V and C these o o, can i n t e g r a l s can be r e w r i t t e n as: _7T / 2 C V C V C V 0 0 (l+/cr7sin2(i/>)) (l+7r7sin 2(VO ) " | 1 / 2 dV-1 - rj sin(V') 2.43 27 and V 2 / -1T/Z (1 + KT] s i n (VO) ( l - r ? s i n 2 ( V ' ) ) d+7'/sin 2(V ')) n 1/ 2 d<£ 2 .44 r\ i s then a f i n e i n d i c a t o r o f the t o t a l d e f o r m a t i o n i n the c e l l ; r? s h o u l d go smoothly from 0 to 1 as D goes from D to i n f i n i t y . Indeed th r\ can be thought o f as the o r d e r parameter f o r the F r e e d e r i c k s z t r a n s i t i o n . The essence o f the ED method o f d e t e r m i n i n g the a n c h o r i n g p o t e n t i a l i s c o n t a i n e d i n the two i n t e g r a l s above: The c a p a c i t a n c e and v o l t a g e , as w e l l as the m a t e r i a l c o n s t a n t s determine rj and <f> u n i q u e l y . <f>^ may be used i n the torque balance eq. 2.33 ( i n which ^ i s known from the f i r s t i n t e g r a l o f the E u l e r - L a g r a n g e e q u a t i o n , eq. 2.38) and thus the s u r f a c e torque SW/d^ determined d i r e c t l y . The s u r f a c e torque can be i n t e g r a t e d to y i e l d W(^) w i t h i n an a d d i t i v e c o n s t a n t . A c o m p l e t e l y analogous c a l c u l a t i o n can be done w i t h the phase. D e f i n i n g the reduced phase as 6 = 1 - &/6q, e q u a t i o n 2.21 becomes r 1 . ^ « ) d , f — ^ J J / l + u s i n 2 ( 0 ) 2.45 2 2 Where i/ — n /n - 1. Making the change o f v a r i a b l e from z to 4> a g a i n e o r e s u l t s i n : l A n J / V 2 K i 2 A + 7 s i n 2 ( * ) " f ( l ^ S i n 2 ( ^ ) ) ( l + 7 s i n 2 W ) V>2^ m J . l-(sin 2 ( 0 ) - s i n 2 ( ^ ) ) (1 + i / s i n 2 ( 0 ) J 2.46 or i n the reduced n o t a t i o n used p r e v i o u s l y : 28 n ( l + w g s i n 2 (x(>)) (l+7>?sin 2 (i/>))' -,1/2 cv - \ A ™ '-(l-rysin (V>)) (1 + vrjsxxi (ip)J dip 2.47 G iven the phase da t a i n a d d i t i o n to the c a p a c i t a n c e , t h i s e q u a t i o n and e i t h e r of eqs. 2.43,2.44 may be s o l v e d s i m u l t a n e o u s l y f o r r\ and and then the s u r f a c e torque may be determined i n the same manner as o u t l i n e d above. Thus i f one has both phase and c a p a c i t a n c e data, b oth can be used i n d e p e n d e n t l y to determine WC^) ; t h i s i s o b v i o u s l y a u s e f u l method o f v e r i f y i n g r e s u l t s . T h i s i s a s i g n i f i c a n t d e p a r t u r e from the method of Yokoyama and van Sprang. The method they d e v i s e d , which i s o u t l i n e d p r e v i o u s l y i n t h i s c h a p t e r may be extended to determine W(0 ) as w e l l as d . The 1 e p r i n c i p a l d i f f e r e n c e between t h a t e x t e n s i o n and the ED method i s t h a t the Y-vS method r e q u i r e s phase data as w e l l as c and v. I t i s n o t e d t h a t r\, even though i t i s the n a t u r a l q u a n t i t y to d e s c r i b e the d e f o r m a t i o n i s not p a r t i c u l a r l y u s e f u l a t l a r g e r v o l t a g e s . T h i s i s because i t becomes too c l o s e to u n i t y to be 18 d i s t i n g u i s h a b l e from i t . I t i s f o r t h i s reason the s u b s t i t u t i o n where or becomes the parameter o f i n t e r e s t i s u s e f u l . When a n a l y s i n g the d a t a f o r these experiments, a became as l a r g e as 174 a t v o l t a g e s o f l e s s than 100 V . The a n a l y s i s was stopped then because the th computer c o u l d not e a s i l y s t o r e numbers as l a r g e or as s m a l l as exp(±174). T h i s was not a s i g n i f i c a n t problem because there was much i n f o r m a t i o n c o n t a i n e d i n the data t h a t c o rresponded to a < 174. Another problem i s t h a t the i n t e g r a l s i n eqs. 2.43,2.44 do not r e a d i l y l e n d themselves to n u m e r i c a l i n t e g r a t i o n because t h e i r denominators i n c r e a s e s h a r p l y near the upper l i m i t . In l i g h t o f t h i s V 1 - exp( - a ) 2.48 29 i t i s u s e f u l to r e w r i t e i n t e g r a l s o f t h i s form as: . T / 2 f(0)d<6 V 1-nsm (4>) ,7T / 2 -ff/2 f(«)-f(w/2) ^ / 1-rjsin'(4>) d<j> + f (JT/2) V1 l - r7s in 2 (0) 2.49 The f i r s t term on the r i g h t i s a w e l l behaved i n t e g r a l ; i t i s e v a l u a t e d by an a d a p t i v e Simpson's method a l g o r i t h i m . The second term can be w r i t t e n as a d i f f e r e n c e o f complete and incomplete e l l i p t i c i n t e g r a l s o f the f i r s t k i n d ; these are e a s i l y e v a l u a t e d u s i n g e x i s t i n g a l g o r i t h i m s . 1 3 As the e q u a t i o n s 2.43, 2.44 cannot be s o l v e d d i r e c t l y f o r rj and <t> , a f i x e d p o i n t i t e r a t i v e scheme was adopted. R e w r i t i n g these e q u a t i o n s (note t h a t the s u b s t i t u t i o n i n eq. 2.48 has been performed) as: and cv = f (a,6 ) l l 2.50 v = f (a,4> ) 2 1 2.51 I n i t i a l guesses are made a t both a and 6 , these are a and d> ° 1 o 10 r e s p e c t i v e l y . L i n e a r i z i n g these e q u a t i o n s i n the unknowns g i v e s r i s e to the f o l l o w i n g r e c u r s i o n : Q i+1 a i + L li+l J df /da df /8<f> I r I df /da df /8<f> 1 ~ l _ cv- f 1 V - f 2 J 2.52 1 l i A FORTRAN program was w r i t t e n to perform t h i s r e c u r s i o n as w e l l as e v a l u a t e the f u n c t i o n s f^ and and t h e i r d e r i v a t i v e s . The program performed t h i s r e c u r s i o n u n t i l the convergence c r i t e r i a were met. These c r i t e r i a were t h a t the d i f f e r e n c e between s u c c e s s i v e l y i t e r a t e d v a l u e s was l e s s than some c o n s t a n t , t y p i c a l l y 10 The convergence was r e a s o n a b l y f a s t f o r the lower v o l t a g e data but became slow a t 30 h i g h e r v o l t a g e s . I t i s f o r t h i s r e a s o n t h a t the program was made a d a p t i v e ; the s t e p between s u c c e s i v e Q ' S was a d j u s t e d to speed c o n v e r g e n c e . To t e s t the r o u t i n e s , a r t i f i c i a l d a t a was g e n e r a t e d u s i n g p u b l i s h e d m a t e r i a l c o n s t a n t s and v a r i o u s v a l u e s o f a and <j> . The program t h e n u sed t h i s d a t a as i n p u t and i n a l l c a se s e x c e p t one s m a l l r e g i o n i t c onve r ge s e x a c t l y t o the c o r r e c t v a l u e s o f a and \f>^. The one ca se was l o c a t e d a t a - 19.5 ± . 3 . T h i s v a l u e d e f i n e d the bounda ry be tween the r e g i o n o f f a s t conve rgence and t h a t o f s l ow c o n v e r g e n c e . The r e a s o n f o r t h i s b e h a v i o r i s n o t y e t u n d e r s t o o d ; the same l a c k o f conve rgence was 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 was u sed as i n p u t as w e l l . As a consequence o f t h i s , on some s e t s o f d a t a , one o r two p o i n t s were d i s c a r d e d . C h a p t e r Summary The aims o f t h i s t h e s i s a r e t h e n : t o measure the c a p a c i t a n c e and the o p t i c a l phase 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 f rom as s m a l l a v o l t a g e as p r a c t i c a l t o a v e r y l a r g e v o l t a g e (> 100 V ) . The method th o f Yokoyama and v a n Sprang can t h e n be a p p l i e d t o d e t e r m i n e the e x t r a p o l a t i o n l e n g t h . I n a d d i t i o n , a new method ( t h e ED method) ba sed s o l e l y on the con t i nuum t h e o r y o f n e m a t i c s can be a p p l i e d i n d e p e n d e n t l y t o b o t h the c a p a c i t a n c e and the phase i n f o r m a t i o n t o y i e l d the a n c h o r i n g p o t e n t i a l . The n e x t c h a p t e r d e s c r i b e s how t h i s i n f o r m a t i o n i s o b t a i n e d . 31 Chapter III: Ex p e r i m e n t a l Apparatus The experiment to be performed i s to a l i g n a l i q u i d c r y s t a l p a r a l l e l to c o n d u c t i n g g l a s s p l a t e s apply a v o l t a g e a c r o s s these p l a t e s and measure the c a p a c i t a n c e and the phase. The method o f alignment s h o u l d have a c h a r a c t e r i s t i c a n c h o r i n g s t r e n g t h which i s what one wishes to i n v e s t i g a t e . Sample Preparation The l i q u i d c r y s t a l employed was 4- n - 4 - n ' - p e n t y l - c y a n o b i p h e n y l , commonly r e f e r r e d to as 5CB; i t was o b t a i n e d from BDH and used w i t h o u t f u r t h e r p u r i f i c a t i o n . The s t r u c t u r e o f 5CB i s : In c h a p t e r I I , the magnitude o f f l e x o e l e c t r i c e f f e c t s were d e s c r i b e d by two parameters: V and A ( c f . eqs. 2.17, 2.19). In p 5CB one expects these parameters to bo t h be s m a l l . F o l l o w i n g g H e l f r i c h ' s procedure f o r c a l c u l a t i n g f l e x o e l e c t r i c c o e f f i c i e n t s , i t i s e s t i m a t e d t h a t V < 4. x 10~3 V and A ~ 3. x 10" 3. T h i s r e s u l t i s p d e r i v e d from the f a c t t h a t the d i p o l e moment o f the cyano group on the 5CB molecule dominates a l l o t h e r c o n t r i b u t i o n s to the t o t a l m o l e c u l a r d i p o l e moment. T h i s i m p l i e s e^ > e^. These upper l i m i t s on the s t r e n g t h o f f l e x o e l e c t r i c e f f e c t s j u s t i f y the assumption o f i g n o r i n g the e f f e c t s i n the ED method. The 5CB i s sandwiched between g l a s s p l a t e s . The p l a t e s used were 1/4" t h i c k and were purchased w i t h a c o n d u c t i n g , t r a n s p a r e n t 32 Indium-Tin-Oxide (ITO) c o a t i n g on them. A r e g i o n o f t h i s c o a t i n g was masked and the remainder etched o f f w i t h an a c i d s o l u t i o n to produce c i r c u l a r e l e c t r o d e s w i t h c o n t a c t tabs. The e t c h i n g was done such t h a t d u r i n g the pr o c e s s the p l a t e s u r f a c e s were not rubbed or s c r a t c h e d The c o n d u c t i n g area c o n t r i b u t i n g to the c a p a c i t a n c e was -I* 2 1.3 i .1 x 10 m , determined by measuring the diameter o f the area w i t h a v e r n i e r c a l i p e r . C l e a n i n g the p l a t e s i s an important s t e p , and th e r e were two proc e d u r e s employed. In the f i r s t , the p l a t e s were c l e a n e d by s o n i c a t i o n i n f o u r s t a g e s ; d u r i n g a l l stages the s o n i c a t i n g b a t h was at - 70°C and the d u r a t i o n was about t h i r t y minutes. The s o n i c a t i n g agents used were r e s p e c t i v e l y : a d e t e r g e n t s o l u t i o n to remove gross i m p u r i t i e s , a d e g r e a s i n g s o l v e n t , u s u a l l y x y l e n e s or t r i c h l o r o e t h y l e n e , e l e c t r o n i c s grade methanol and e l e c t r o n i c s grade acetone. The p r o c e s s was completed by r i n s i n g i n e l e c t r o n i c s grade acetone and d r y i n g i n a stream o f hot a i r . Cleanness was c o n f i r m e d by o b s e r v i n g w e t t i n g o f the s u r f a c e by d e i o n i z e d water; b e a d i n g o c c u r s i f the g l a s s i s not c l e a n . The o t h e r c l e a n i n g procedure employed, which gave e q u a l l y good r e s u l t s , was to s o n i c a t e i n 5% d e t e r g e n t s o l u t i o n and then i n ammonium h y d r o x i d e to which 30% hydrogen p e r o x i d e had been added. Both s o n i c a t i n g baths were a g a i n h e l d a t - 70°C. The d r y i n g p r o c e s s was the same as the i n the f i r s t p r o c e d u r e . The l a t t e r p r o c e s s i s l e s s e x p ensive. Two s u r f a c e alignment t e c h n i q u e s were employed; q u a l i t a t i v e l y the u n i f o r m i t y o f alignment was about e q u a l between them. The f i r s t t e c h n i q u e , d i s c o v e r e d by J a n n i n g 1 5 was s i l i c o n oxide (SiO) evap o r a t e d a t an o b l i q u e angle onto the s u b s t r a t e . S u b s t r a t e s p r e p a r e d i n t h i s 33 way are s a i d to produce " s t r o n g " p l a n a r alignment w i t h a low t i l t a n g l e . 1 6 , 1 7 The e v a p o r a t o r used r o u t i n e l y a c h i e v e d 10 5 T o r r . The SiO was Union C a r b i d e S e l e c t Grade. The p l a t e s were h e l d i n p o s i t i o n 20 cm above the boat, and the p l a t e normal was at an angle o f 60° to the l i n e between the p l a t e c e n t e r and the e v a p o r a t i o n boat. The boat c u r r e n t was 240 A, a t which the d e p o s i t i o n r a t e was - 12 A/sec The p l a t e s were exposed f o r - 25 sec, to produce a l a y e r - 300 A t h i c k . The second method employed a t h i n , b u f f e d f i l m o f p o l y ( v i n y l f o rmal) (PVF). Rubbed polymer f i l m s are used e x t e n s i v e l y i n the manufacture o f t w i s t e d nematic d i s p l a y s . The PVF was i n a v e r y d i l u t e (-.05% by weight) s o l u t i o n o f c h l o r o f o r m , and the s o l u t i o n was a p p l i e d l i b e r a l l y to the p l a t e s . Sometimes i t was n e c e s s a r y to use a s p i n c o a t e r when a p p l y i n g the s o l u t i o n to the p l a t e s ( t h i s w i l l be d i s c u s s e d l a t e r ) . A f t e r the s o l u t i o n had evaporated, a t h i n f i l m o f the polymer remains. T y p i c a l d e n s i t i e s -4 2 were - 10 g/cm ( c a l c u l a t e d from the c o n c e n t r a t i o n of the s o l u t i o n and the volume of s o l u t i o n p l a c e d on the p l a t e s ) . The f i l m was then g e n t l y rubbed i n one d i r e c t i o n a few times w i t h a c o t t o n c l o t h . The p r o c e s s i s s a i d to be i n s e n s i t i v e to the c o m p o s i t i o n o f the f i b r e s i n the r u b b i n g m a t e r i a l . 1 6 Uniform homogeneous alignment was produced r o u t i n e l y w i t h t h i s method; t h i s i s b e l i e v e d to be the f i r s t r e p o r t o f PVF used to a l i g n l i q u i d c r y s t a l s homogeneously. PVF alignment c o u l d be d e s t r o y e d by h e a t i n g the sample f a r i n t o the i s o t r o p i c . T h i s was p r o b a b l y due to h e a t i n g above the g l a s s t r a n s i t i o n o f the PVF and d e s t r o y i n g any e f f e c t s the r u b b i n g had on the polymer f i l m . The g l a s s t r a n s i t i o n o f PVF i s around 100°C, 1 8 and p r e c a u t i o n s had to 34 be taken whereby f i l m s d i d not reach t h a t temperature. 19 Geary e t al r e c e n t l y d e s c r i b e d a p o s s i b l e mechanism o f rubbed polymer alignment. They propose t h a t b u f f i n g o r i e n t s the polymer c h a i n s , and t h a t t h i s o r i e n t a t i o n , not s c r a t c h i n g or g r o o v i n g o f the s u r f a c e , t h a t i s r e s p o n s i b l e f o r p r o d u c i n g alignment o f the l i q u i d c r y s t a l . They found t h a t alignment o c c u r s when the polymer i s both o r i e n t e d and c r y s t a l l i n e . I t i s important to be sure t h a t whatever the p l a t e treatment i s , i t i s indeed p r o d u c i n g u n i f o r m homogeneous alignment. T h i s was done once the c e l l i s f i l l e d by m i c r o s c o p i c examination under c r o s s e d p o l a r i z e r s . A w e l l a l i g n e d sample i s u n i f o r m l y dark when the alignment i s p a r a l l e l to e i t h e r the p o l a r i z e r or the a n a l y z e r axes and u n i f o r m l y b r i g h t when the alignment i s a t 45°. The u n i f o r m i t y of alignment o f two samples may be compared by o b s e r v i n g the amount o f l i g h t they s c a t t e r . Note t h a t t h i s i s o n l y u s e f u l i f they are the same t h i c k n e s s and a t the same temperature. A f t e r t r e a t i n g the s u r f a c e s the p l a t e s were clamped t o g e t h e r w i t h 40/J Mylar s p a c e r s between them. The l i q u i d c r y s t a l i s i n t r o d u c e d i n t o the c e l l by c a p i l l a r y a c t i o n a t one o f the edges. Note t h a t i f the l i q u i d c r y s t a l i s i n the nematic phase when i n t r o d u c e d t h i s way, 20 then s u r f a c e a d s o r p t i o n can a l s o produce alignment. One must v e r i f y t h a t any observed alignment i s due to the s u r f a c e p r o p e r t i e s and not due to a d s o r p t i o n e f f e c t s . T h i s was checked by h e a t i n g a t l e a s t f i v e degrees above the c l e a r i n g p o i n t where the a d s o r p t i o n 20 e f f e c t s d i s a p p e a r and then c h e c k i n g the alignment a g a i n a f t e r the c e l l has c o o l e d to the nematic. I f the c e l l showed good alignment, i t was then g l u e d a t the c o r n e r s w i t h M i l l e r Stephenson Epoxy 907. 35 Electrode F i g u r e 6 S c a l e drawing o f the sample c e l l used. 36 F i g u r e 6 shows the sample c e l l to s c a l e with the e l e c t r o d e areas o u t l i n e d . C r u c i a l to measurements of the a n c h o r i n g s t r e n g t h by t h i s method i s an a c c u r a t e d e t e r m i n a t i o n of nematic l a y e r t h i c k n e s s . The method employed was the i n t e r f e r e n c e o f o p t i c a l rays r e f l e c t i n g from the two 21 s u r f a c e s o f the l a y e r . The angle of i n c i d e n c e was v a r i e d u n t i l i n t e r f e r e n c e was observed. Angles (tj)) a t which d e s t r u c t i v e i n t e r f e r e n c e was observed are g i v e n by: n^ i s the r e f r a c t i v e index o f the 5CB i n the i s o t r o p i c phase, m i s the o r d e r o f the i n t e r f e r e n c e and A i s the vacuum wavelength o f the l i g h t used. The sample c e l l was housed i n a b l o c k w i t h c i r c u l a t i n g temperature c o n t r o l l e d water around i t to keep the temperature c o n s t a n t and to keep the l i q u i d c r y s t a l i n the i s o t r o p i c phase. The same water c o n t r o l l e d the temperature o f an Abbe r e f r a c t o m e t e r where the r e f r a c t i v e index was measured. One c o u l d observe f i v e o r d e r s of d e s t r u c t i v e i n t e r f e r e n c e . The t u r n t a b l e on which the c e l l was p l a c e d had a maximum r e s o l u t i o n o f -.007°. The c e l l t h i c k n e s s was determined to w i t h i n -1.8% Temperature C o n t r o l The temperature c o n t r o l o f t h i s experiment was the one used by 13 S. M o r r i s . The c o n t r o l l e r i s capable o f sub m i l l i K e l v i n s t a b i l i t y . For the purposes o f t h i s experiment, the c o n t r o l l e r was s e t f o r 1-2 mK s t a b i l i t y . The sample c e l l was h e l d i n a.copper b l o c k , which a l s o s e r v e d as a grounded s h i e l d f o r the c a p a c i t a n c e measurements. T h i s m 3.1 b l o c k was s e t i n s i d e a l a r g e r copper c y l i n d e r and copper end-pieces were f i t t e d at e i t h e r end to h o l d the b l o c k i n p l a c e . The c y l i n d e r had keyways on the i n s i d e and the c e l l b l o c k had tabs matching tabs on e i t h e r s i d e to a l i g n the b l o c k i n s i d e the c y l i n d e r . The c y l i n d e r was wound n o n - i n d u c t i v e l y w i t h h e a t e r wire. The c y l i n d e r was p l a c e d i n s i d e a c y l i n d e r o f s tyrofoam i n s u l a t i o n , which was c o n t a i n e d i n a c y l i n d r i c a l b r a s s can. The b r a s s can had copper t u b i n g s o l d e r e d around i t f o r c i r c u l a t i n g water. Four Fenwal t h e r m i s t o r s were mounted i n the ends o f the c y l i n d e r ; each of these had been c a l i b r a t e d w i t h a Hewlett Packard 2804A q u a r t z thermometer, which was c a l i b r a t e d w i t h a t r i p l e p o i n t c e l l . The t h e r m i s t o r temperature c o u l d be a c c u r a t e l y determined to l e s s than 10 mK. One o f these t h e r m i s t o r s s e r v e d as one arm o f a dc Wheatstone b r i d g e ; i t was b a l a n c e d a g a i n s t a Time E l e c t r o n i c s 1051 decade r e s i s t o r h a v i n g a r e s o l u t i o n o f . Ol f i . The b r i d g e i s powered by a mercury b a t t e r y . The e r r o r s i g n a l o f the b r i d g e was f e d i n t o a H e w lett-Packard 419A n u l l v o l t m e t e r ; t h i s v o l t m e t e r s e r v e d as the f i r s t a m p l i f i c a t i o n stage o f the feedback c i r c u i t ; the a m p l i f i e d e r r o r s i g n a l then c o n t r o l l e d a Kepco OPS 40-0.5B 20 W power a m p l i f i e r which d r i v e s the copper c y l i n d e r h e a t e r . The Kepco has a v a r i a b l e r e s i s t o r i n i t s feedback loop, as w e l l as an o p t i o n a l l a r g e c a p a c i t o r f o r i n t e g r a l c o n t r o l . S t a b i l i t y i s o p t i m i z e d by i n c r e a s i n g the g a i n o f the feedback network to j u s t below the p o i n t where o s c i l l a t i o n s o c c u r . The o u t e r b r a s s can i s h e l d t y p i c a l l y t h r e e degrees below the sample temperature by c i r c u l a t i n g temperature c o n t r o l l e d water through the aforementioned copper t u b i n g . The water was pumped and temperature c o n t r o l l e d to w i t h i n 500mK by a Haake FE2 h e a t i n g b a t h 38 and c i r c u l a t o r . In t h i s arrangement the temperature s t a b i l i t y was 1-2 mK. Another t h e r m i s t o r ' s temperature was monitored by a K e i t h l e y 175 d i g i t a l m u l t i m e t e r ; t h i s was taken to be the sample temperature. E l e c t r o n i c s For the most p a r t the experiment was c o n t r o l l e d w i t h an IBM XT c o m p a t i b l e microcomputer, the NOVA XT. The computer c o n t r o l l e d the c e l l v o l t a g e as w e l l as a i d e d i n the the data c o l l e c t i o n . The e l e c t r i c f i e l d r e q u i r e d to induce the deformations i s produced by a p p l y i n g a known p o t e n t i a l d i f f e r e n c e between the ITO e l e c t r o d e s on the g l a s s p l a t e s . An ac s i g n a l i s n e c e s s a r y because o f the presence o f i o n i c i m p u r i t i e s i n the sample. In a dc f i e l d these i m p u r i t i e s would have time to migrate to the e l e c t r o d e s and p a r t i c i p a t e i n e l e c t r o c h e m i c a l r e a c t i o n s as w e l l as s c r e e n the a p p l i e d e l e c t r i c f i e l d . S i n c e l i q u i d c r y s t a l r e o r i e n t a t i o n s are r e l a t i v e l y slow, on the o r d e r o f m i l l i s e c o n d s , (the t y p i c a l t u r n - o f f time o f a t w i s t e d nematic d i s p l a y ) , they w i l l see e s s e n t i a l l y the rms v a l u e o f any f i e l d whose fre q u e n c y i s g r e a t e r than a k i l o H e r t z . The a p p l i e d s i n u s o i d a l v o l t a g e i s produced by a m p l i f y i n g the output o f a Hewlett Packard 3312A f u n c t i o n g e n e r a t o r w i t h a Kepco BOP 72-5M a m p l i f i e r . The a m p l i f i e d s i g n a l i s then stepped up by a 16:1 t r a n s f o r m e r and a p p l i e d to the sample c e l l through the r a t i o t r a n s f o r m e r i n the c a p a c i t a n c e b r i d g e . T h i s source can s u p p l y over 200 V rms a t 6 kHz (the frequency used) a c r o s s the sample. The s i g n a l amplitude was computer c o n t r o l l e d v i a the amplitude m o d u l a t i o n 39 i n p u t on the 3312A; the modulation s i g n a l was the output o f a Data T r a n s l a t i o n DT2814 12 b i t d/a c o n v e r t e r on the bus of the NOVA. Thus the c e l l v o l t a g e was c o n t r o l l a b l e to 1/4096 o f the maximum. A K e i t h l e y 197 d i g i t a l multimeter monitors the v o l t a g e a c r o s s the pr i m a r y o f the b r i d g e r a t i o t r a n s f o r m e r . T h i s m u l t i m e t e r i s i n t e r f a c e d to the NOVA v i a the IEEE p a r a l l e l bus. The c a p a c i t a n c e o f the c e l l i s measured w i t h a GenRad 1615-A b r i d g e . The unknown c a p a c i t a n c e i s on one arm o f the b r i d g e , and s i x decades o f p r e c i s i o n c a p a c i t o r s are on the opposing arm. On a t h i r d arm are f i v e decades o f p r e c i s i o n r e s i s t o r s . The b r i d g e i s manually nulled; i . e . the decade c a p a c i t o r s and r e s i s t o r s are a d j u s t e d so th e r e i s ze r o e r r o r v o l t a g e a t the d e t e c t o r . When t h i s i s the case the c a p a c i t a n c e a p p e a r i n g on the c a p a c i t a n c e decades equals the unknown c a p a c i t a n c e ( i n t h i s case the c e l l c a p a c i t a n c e ) and the r e s i s t a n c e decades c o r r e s p o n d to the l o s s f a c t o r o f the unknown c a p a c i t a n c e . Thus the b r i d g e has a maximum r e s o l u t i o n o f 1 ppm f o r c a p a c i t a n c e . T y p i c a l l y i n t h i s experiment the a c c u r a c y was .003% . The b r i d g e was used i n the thr e e t e r m i n a l mode i n which c a p a c i t a n c e i n the l e a d s i s e x c l u d e d and the o n l y c a p a c i t a n c e measured was t h a t o f the sample p l a t e s . The c e l l h o l d e r s e r v e d as the grounded s h i e l d f o r t h i s arrangement; s h i e l d e d c o a x i a l c a b l e s made the c o n n e c t i o n between the c e l l and the b r i d g e . When a G e n e r a l Radio 500 pF s t a n d a r d c a p a c i t o r was p l a c e d a c r o s s the t e s t l e a d s used i n the experiment, the c a p a c i t a n c e measured by the b r i d g e agreed w i t h i n l e s s than .2 pF. The e r r o r v o l t a g e was monitored by a P r i n c e t o n A p p l i e d Research 5102 phase i n s e n s i t i v e l o c k - i n a m p l i f i e r . T h i s instrument has a f u l l s c a l e s e n s i t i v i t y o f 100 ^V a t maximum g a i n . 40 The 1615 b r i d g e i s such t h a t i n the n u l l c o n d i t i o n , the v o l t a g e a c r o s s the primary o f the r a t i o t r a n s f o r m e r i s the same as t h a t a c r o s s the sample c e l l . Thus i t i s a p p r o p r i a t e to measure the c e l l v o l t a g e a t the ge n e r a t o r t e r m i n a l s . O p t i c a l Measurements I t was o r i g i n a l l y thought t h a t the phase c o u l d be measured by m o n i t o r i n g the i n t e n s i t y o f t r a n s m i t t e d l i g h t through c r o s s e d p o l a r i z e r s w i t h the sample i n between a l i g n e d a t 45° to the a x i s o f the p o l a r i z e r . However, the envelope o f the r e s u l t i n g f r i n g e p a t t e r n was not c o n s t a n t as the v o l t a g e was v a r i e d , making the phase 13 measurement d i f f i c u l t . T h i s e f f e c t was a l s o seen by M o r r i s . T h i s was l i k e l y due to both v o l t a g e dependent d e p o l a r i z e d s c a t t e r i n g from the c e l l , as w e l l as i n t e r f e r e n c e e f f e c t s from m u l t i p l e r e f l e c t i o n s from the g l a s s p l a t e s . 22 The method employed was s i m i l a r to t h a t used by van Sprang. When l i g h t o r i g i n a l l y p o l a r i z e d a t 45° to the o p t i c a x i s o f a p o s i t i v e b i r e f r i n g e n t medium e x i t s t h a t medium i t i s e l l i p t i c a l l y p o l a r i z e d ; the major a x i s o f i t s e l l i p s e o f p o l a r i z a t i o n i s a l o n g 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 i n c i d e n t l i g h t . The i n f o r m a t i o n about the phase 6 i s c o n t a i n e d e n t i r e l y i n the e c c e n t r i c i t y which i s g i v e n by e - t a n 2 [ - | - ] . 3.2 I f a q u a r t e r wave p l a t e i s p l a c e d b e h i n d the medium w i t h i t s f a s t a x i s p a r a l l e l 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 i n c i d e n t on the sample, the l i g h t upon e x i t w i l l be l i n e a r l y p o l a r i z e d w i t h 41 I Figure 7. Schematic of phase measurement setup. 42 i t s p l ane o f p o l a r i z a t i o n r o t a t e d through an angle 5/2 from the plane o f the i n c i d e n t beam. An a n a l y z e r can be p l a c e d a f t e r the q u a r t e r wave p l a t e and r o t a t e d u n t i l i t e x t i n g u i s h e s the l i g h t ; the angle at which t h i s happens i s d i r e c t l y p r o p o r t i o n a l to 6. T h i s i s d e p i c t e d s c h e m a t i c a l l y i n f i g u r e 7. The l i g h t source was a . 5mW HeNe l a s e r and O r i e l f i l m p o l a r i z e r s were used f o r p o l a r i z e r and a n a l y z e r . D u r i n g t h i s experiment, the a n a l y z e r was r o t a t e d by hand u n t i l e x t i n c t i o n . The e x t i n c t i o n i s d e t e c t e d by m o n i t o r i n g the i n t e n s i t y o f t r a n s m i t t e d l i g h t w i t h a S i l i c o n D e t e c t o r photodiode. Summary of Experiment The experiment proceeds as f o l l o w s . The sample c e l l i s p l a c e d i n the temperature c o n t r o l l e d h o u s i n g where i t i s a l l o w e d to come to thermal e q u i l i b r i u m . An ac s i g n a l i s a p p l i e d v i a the b r i d g e r a t i o t r a n s f o r m e r to the sample c e l l . T h i s s i g n a l i s ramped by the microcomputer from -50 mV to the maximum v o l t a g e d e s i r e d (between 50 and 100V) i n v a r i a b l e s i z e s t e p s . To take a d a t a p o i n t , the ramping st o p s and the a p p l i e d v o l t a g e i s h e l d c o n s t a n t . A f t e r the b r i d g e e r r o r v o l t a g e has stopped changing, ( t h i s may take twenty minutes or more near the F r e e d e r i c k s z t r a n s i t i o n ) , the c a p a c i t a n c e b r i d g e i s n u l l e d and the 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 l i g h t . Both the angle o f e x t i n c t i o n and the c a p a c i t a n c e are r e c o r d e d manually. The c e l l v o l t a g e and the temperature are r e c o r d e d a u t o m a t i c a l l y by the microcomputer. The v o l t a g e ramp then c o n t i n u e s , and the the above p r o c e s s i s r e p e a t e d . The d a t a i s then a n a l y s e d to y i e l d the e x t r a p o l a t i o n l e n g t h and the a n c h o r i n g p o t e n t i a l . The a n a l y s i s o f the d a t a and the r e s u l t s comprise the next c h a p t e r . 43 Chapter IV Data A n a l y s i s and E x p e r i m e n t a l R e s u l t s The experiment d e s c r i b e d i n the p r e v i o u s c h a p t e r y i e l d s the both the c a p a c i t a n c e and the phase o f the sample as f u n c t i o n s o f v o l t a g e a t c o n s t a n t temperature. T h i s , i n a d d i t i o n to knowledge o f the c e l l dimensions, i s s u f f i c i e n t to determine s e v e r a l m a t e r i a l c o n s t a n t s o f the l i q u i d c r y s t a l sample. Some o f these c o n s t a n t s are important i n f u r t h e r a n a l y s i s o f t h a t data. The data i s a n a l y s e d u s i n g the Yokoyama - van Sprang method and a l s o u s i n g the extended D e u l i n g method r e s u l t i n g i n the a n c h o r i n g p o t e n t i a l W(c^) . Reduction The raw o p t i c a l phase da t a i s modulo 2n. Since the angle o f the a n a l y z e r can always be exp r e s s e d as an angle d where 0 < 8 < it, the o p t i c a l r e t a r d a t i o n i n f o r m a t i o n 6 w i l l always be such t h a t 0 < 5 < 2n. Based on the t h e o r e t i c a l p r e d i c t i o n o f the phase ( c f . f i g . 5 ) , one expects the phase to always decrease as the c e l l v o l t a g e i s i n c r e a s e d above the t h r e s h o l d . One t h e r e f o r e expects the phase a t v e r y h i g h v o l t a g e to be c l o s e to z e r o . The procedure was t h e r e f o r e to examine the phase da t a s t a r t i n g from the p o i n t c o r r e s p o n d i n g to the h i g h e s t v o l t a g e ; the phase angle i n c r e a s e d m o n o t o n i c a l l y u n t i l i t reaches 2n a t which p o i n t i t drops d r a m a t i c a l l y to a v a l u e near z e r o . A t t h i s p o i n t 2TT was added to a l l the - phase p o i n t s a t a lower v o l t a g e . The c o u n t i n g i s then c o n t i n u e d from the p o i n t where the drop o c c u r r e d and the p r o c e s s r e p e a t e d u n t i l a l l the dat a has been 44 7 - i • 6-5-CM * 3-2-1-0 - T -20 0.4 0.6 0.8 1.0 — i ^ — T - — 40 60 r m s V o l t a g e T-80 100 Figure 8. An example of phase vs voltage data. 45 900-1 800- • • • • 700H S 600 H £ 500-J! «3 4004 o 300-' • i — | — i — r .4 0.6 0.8 1.0 200 20 —I ' 1 — 40 60 r m s V o l t a g e —r* 80 100 F i g u r e 9. An example o f c a p a c i t a n c e vs v o l t a g e data. 4 6 c o u n t e d . An example o f phase vs v o l t a g e d a t a i s d e p i c t e d i n f i g u r e 8. O b t a i n i n g the c a p a c i t a n c e d a t a i s s t r a i g h t f o r w a r d ; i t i s s i m p l y r e a d o f f the b r i d g e . An example o f c a p a c i t a n c e vs v o l t a g e d a t a i s shown i n f i g u r e 9. M a t e r i a l C o n s t a n t s The b i r e f r i n g e n c e was o b t a i n e d from the z e r o - f i e l d phase : 5q " 27rAnd/A i n w h i c h SQ was o b t a i n e d by f i t t i n g the phase d a t a b e l o w the F r e e d e r i c k s z t r a n s i t i o n to a s t r a i g h t l i n e and e x t r a p o l a t i n g to z e r o v o l t a g e . was e v a l u a t e d w i t h i n a s i m i l a r manner by e x t r a p o l a t i n g the c a p a c i t a n c e v s v o l t a g e c u r v e b e l o w the t r a n s i t i o n a l o n g a s t r a i g h t l i n e t o V = 0 , t h i s i s by d e f i n i t i o n C = e e A / d . F u r t h e r m o r e , e J 0 o 2 ' 1 c o u l d be e v a l u a t e d by e x t r a p o l a t i n g the C-V c u r v e t o V • » . T h i s i s e a s i e s t done by n o t i n g t h a t f o r l a r g e v o l t a g e , = T T / 2 , and t h a t the m c a p a c i t a n c e s h o u l d be a p p r o x i m a t e l y l i n e a r i n 1 / V . 1 3 Hence C v s 1/V was f i t t o a s t r a i g h t l i n e and the i n f i n i t e f i e l d c a p a c i t a n c e C^ was o b t a i n e d . A f l a w 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 i s t h a t one needs t o know the r a t i o o f the c e l l a r e a to the t h i c k n e s s A / d . T h i s p r o b l e m was " s o l v e d " by f i t t i n g C q t o p u b l i s h e d v a l u e s f o r 23 and t h u s c a l c u l a t i n g A / d . There i s a n o t h e r p r o b l e m i n d e t e r m i n i n g f rom the e x t r a p o l a t i o n to i n f i n i t e v o l t a g e because i n the c a s e o f f i n i t e a n c h o r i n g e f f e c t s the c e l l c o u l d be c o m p l e t e l y a l i g n e d ( 4>(z) = n/2 ) a t a f i n i t e v o l t a g e , and hence e x t r a p o l a t i n g the d a t a t o V — °° w i l l o v e r e s t i m a t e e . A t b e s t one o b t a i n s two l bounds on e : the e x t r a p o l a t i o n to V = » w h i c h o v e r e s t i m a t e s , and the 47 l a r g e s t v a l u e of C measured, which un d e r e s t i m a t e s . The d i f f e r e n c e i n these e s t i m a t e s i s -.5%. The importance of such a seemingly s m a l l d i f f e r e n c e i s d i s c u s s e d i n Chapter V. The s p l a y e l a s t i c c o n s t a n t was determined from the t h r e s h o l d v o l t a g e f o r the F r e e d e r i c k s z t r a n s i t i o n ; c. f. eq. 2.10: The t h r e s h o l d can be seen q u i t e c l e a r l y i n f i g u r e s 8 and 9. S i n c e the t r a n s i t i o n i s never p e r f e c t l y sharp, nor can one guarantee t h a t a d a t a p o i n t would f a l l on the t r a n s i t i o n even i f i t were, a scheme i s r e q u i r e d to deduce V more a c c u r a t e l y than merely p e r u s i n g the data. th G i v e n t h a t one has a l r e a d y f i t the data below the t r a n s i t i o n w i t h a s t r a i g h t l i n e , i t i s n a t u r a l to f i t the data above i t to something and then f i n d the i n t e r s e c t i o n o f the two f i t s . One cannot e a s i l y d e s c r i b e a l l the d a t a above the t r a n s i t i o n w i t h o u t f i t t i n g to eqs. 13 2.43 and 2.44, a f o r m i d a b l e t a s k . The method employed was to f i t a s m a l l number o f p o i n t s immediately a f t e r the t r a n s i t i o n to a q u a d r a t i c f u n c t i o n o f the v o l t a g e , u s i n g a l i n e a r l e a s t squares a n a l y s i s . There i s an a r b i t r a r i n e s s i n how many p o i n t s one s h o u l d use i n t h i s f i t ; i n o r d e r to be c o n s i s t e n t , the data was always f i t up u n t i l V = 2V . V i s u a l examination v e r i f i e d t h a t the q u a d r a t i c th was a r e a s o n a b l e f i t i n t h i s r e g i o n . A s m a l l number o f p o i n t s i n the immediate v i c i n i t y o f the t r a n s i t i o n were d i s c a r d e d because they showed a "rounding" o f what s h o u l d be a sharp knee i n the curve. The i n t e r s e c t i o n o f the two f i t s , d i s r e g a r d i n g "rounded" data p o i n t s y i e l d e d the t h r e s h o l d v o l t a g e . T h i s was done f o r both the phase and the c a p a c i t a n c e data. The s p l a y c o n s t a n t i s then determined from the V th 0 2 4.1 48 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 i n the e l a s t i c constants, K = K/K - 1, was determined from the slope of the C-V curve in the region immediately 1 3 above the transition. This slope M i s given by: M " 2( « +\ + 1) 4 - 2 It was determined using the quadratic f i t described previously; 7 was determined from C q and C^. A completely analogous analysis was used to determine K from the phase data as well. In this case the 2 constant u - (n /n ) - 1 i s required. I t was determined from the e o measured birefringence An and n — n + 2n , the l a t t e r being taken e 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 this chapter, results corresponding to two different samples are quoted. The two samples differed i n the technique of surface alignment used. Sample 1 employed tangentially evaporated SiO and sample 2 employed rubbed PVF. Figure 10 shows results for birefringence vs temperature. Plotted 2 4 on the same figure are Karat's and Madhusudana's results, 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 results obtained by Dunmur et al are also plotted for comparison. The systematic deviation in t has been observed before. 1 3 I t i s not 4 9 0.18-, 0.15H 0.12H 0.09J 0.06H 0.03A 0.00 ° o • O o • Sample 1 • Sample 2 o Ref. 25 - i 1 • 1 1 1 « 1 1 j • 1 1 1 1 1 • 1 26 27 28 29 30 31 32 33 34 35 Temperature (°C) F i g u r e 10. B i r e f r i n g e n c e as a f u n c t i o n o f temperature 50 22.0T J ° ° o oo o o g o o ^ 18.0H d cP o © a c O CO a H.OH d o o o £ 10.0H a a> Q) Q o o • 0 a o a t » 6.0H 2.0-• Sample 2 • Sample 1 o Ref. 23 27 28 T — 1 — r T ' 1 1 — I — I — I — 1 — I — 1 — I — I 29 30 31 32 33 34 35 Temperature (°C) F i g u r e 11. D i e l e c t r i c c o n s t a n t s as a f u n c t i o n of temperature 51 0.6-. 0.4 0.2-1 0.0H o Sample 1 (C) • Sample 1 (<5) o Ref. 24 A Sample 2 (C) A Sample 2 (6) ° * O A A a O A A A -.4- "T 1 T -31 32 35 27 28 29 30 33 34 Temperature (*C) F i g u r e 12. E l a s t i c c o n s t a n t a n i s o t r o p y as f u n c t i o n o f temperature. 52 6 5A 55 9* 3-J 1 o Sample • Sample D Sample • Sample A Ref. 24 2.C 2.o l.C Lo-ft B — i — i • 1 1 1 • 1— 27 28 29 30 31 —I r 32 33 Temperature (*C) e o •n r— 34 —I 35 F i g u r e 13. Splay 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 o f temperature. 53 7-, A A O A O ^ 5-A a 2 H • o A o 2 J A O D B A " o Sample l.C A 2 J • Sample 1.6 o Sample 2.C • Sample 2,6 A Ref. 24 1 I ' 1 ' 1 1 1 1 1 • 1 1 1 « 1 27 26 29 30 31 32 33 34 35 Temperature (°C) F i g u r e 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 o f temperature. 54 unreasonable to expect a d e v i a t i o n i n t h i s q u a n t i t y when one c o n s i d e r s t h a t was measured i n a system with extremely h i g h d e f o r m a t i o n , w h i l e was measured on a homogeneous system; the r e s u l t s o f r e f . 23 were a l l measured with a homogeneous system. T h i s i s d i s c u s s e d a t g r e a t e r l e n g t h i n the next c h a p t e r . F i g u r e s 12 through 14 show r e s u l t s r e l a t e d to the e l a s t i c c o n s t a n t s . In a l l cases the r e s u l t s o f Bunning et al are shown f o r comparison. F i g u r e 12 shows the e l a s t i c c o n s t a n t a n i s o t r o p y K. vs temperature f o r both samples. The temperature dependence of « has 13 been seen b e f o r e i n measurements made i n the same way. F i g u r e 13 shows the s p l a y e l a s t i c c o n s t a n t s vs temperature and f i g u r e 14 shows the bend e l a s t i c c o n s t a n t vs temperature. S u r f a c e A n c h o r i n g R e s u l t s : Yokoyama's Method Yokoyama and van S p r a n g ' s 1 0 method f o r d e t e r m i n i n g the e x t r a p o l a t i o n l e n g t h d i s s t r a i g h t f o r w a r d . The r e l a t i v e phase 5/6 e 0 i s p l o t t e d vs 1/CV; t h i s i s f i t to a s t r a i g h t l i n e and the i n t e r c e p t i s 2d /d. F i g u r e 15 d e p i c t s the e x t r a p o l a t i o n l e n g t h d as a e e f u n c t i o n o f temperature f o r sample 1, the SiO c o a t e d s u r f a c e . A l s o shown are Yokoyama, Kobayashi and and Kamei's 1 1 r e s u l t s o f u s i n g the same method. The e r r o r b a r s on my d a t a p o i n t s are c a l c u l a t e d from: 2 a) the x °f t n e f^- c t o a s t r a i g h t l i n e , b) the s t a n d a r d d e v i a t i o n from r e p e a t i n g the run and c) the e r r o r i n the c e l l t h i c k n e s s measurement. Of these, the r e p r o d u c i b l i t y o f the experiment was by 55 s 190-j 170-150-130-110-90-x This work a R«f. 11 60 C a o 50-1 30 A 10 29 I 30 - i r 31 —T~" 32 33 34 35 Temperature (*C) F i g u r e 15. E x t r a p o l a t i o n l e n g t h as a f u n c t i o n o f temperature f o r SiO t r e a t e d s u r f a c e . 56 30-t 25-4 w 20 Xi ti 15. l-l CO ti Xi 5' o 5 0-o This work o Ref. 11 29 30 31 32 33 34 Temperature (*C) 8 o o a o o  3 35 F i g u r e 16. Out o f plane a n c h o r i n g s t r e n g t h as a f u n c t i o n o f temperature f o r SiO t r e a t e d s u r f a c e . 57 f a r the l a r g e s t ; the r e s u l t s f o r d from subsequent runs d i f f e r e d by e as much as 2 0 % . The e r r o r bars used i n r e f . 1 1 are i n c l u d e d f o r comparison; t h e i r o r i g i n i s not known. I t i s not u n l i k e l y t h a t t h e i r experiment had the same problem w i t h r e p r o d u c i n g d as t h i s one. e F i g u r e 1 6 shows the o u t - o f - p l a n e a n c h o r i n g s t r e n g t h W q , based on the R a p i n i - P a p o u l a r model o f the s u r f a c e i n t e r a c t i o n , f o r t a n g e n t i a l l y e v a p o r a t e d SiO as determined from the e x t r a p o l a t i o n l e n g t h . The r e s u l t s quoted i n r e f . 1 1 are a g a i n i n c l u d e d f o r comparison. The e r r o r b a r s have been l e f t o f f f o r c l a r i t y . F i g u r e 1 7 shows the e x t r a p o l a t i o n l e n g t h f o r sample 2 , the b u f fed-PVF c e l l . Note t h a t the f i r s t d a t a p o i n t on the r i g h t shows a n e g a t i v e e x t r a p o l a t i o n l e n g t h ! T h i s data p o i n t c orresponds to a d i f f e r e n t sample than the o t h e r s . T h i s sample was made w i t h a PVF l a y e r t h a t was not s p i n c o a t e d on. Note t h a t t h i s was a c o m p l e t e l y r e p r o d u c i b l e r e s u l t . A l l the o t h e r d a t a p o i n t s on the p l o t c o r r e s p o n d to a l a y e r t h a t was s p i n c o a t e d . The sample t h a t was not s p i n c o a t e d tended to " p i n " domain w a l l s a f t e r the f i e l d was removed, and hence was not u s e f u l f o r r e p e a t i n g measurements w i t h o u t a n n e a l i n g i t i n t o the i s o t r o p i c a f t e r each run; t h i s sample was not used f u r t h e r . The i m p l i c a t i o n s o f t h i s r e s u l t are d i s c u s s e d i n the l a s t c h a p t e r . The o t h e r d a t a p o i n t s show e x t r a p o l a t i o n l e n g t h s t h a t are comparable to those o f the SiO t r e a t e d c e l l . F i g u r e 1 8 shows the a n c h o r i n g s t r e n g t h o f 5 C B on rubbed PVF as a f u n c t i o n o f temperature. T h i s i s the f i r s t r e p o r t o f the a n c h o r i n g s t r e n g t h o f a rubbed polymer f i l m measured by t h i s method. Note t h a t t h e r e i s no data p o i n t c o r r e s p o n d i n g to the n e g a t i v e e x t r a p o l a t i o n l e n g t h d e s c r i b e d e a r l i e r ; w h i l e one may imagine a n e g a t i v e e x t r a p o l a t i o n l e n g t h 5 8 I 100-. 3 8 ( H A "So 60 a 0) « 40-^ 20-1 2 0 -20 " > — [ — 28 ~1 ' r-31 32 —I • I 33 34 35 27 29 30 Temperature (°C) F i g u r e 17. E x t r a p o l a t i o n l e n g t h as a f u n c t i o n o f temperature f o r a b u f f e d PVF s u r f a c e . 59 30-i 25-1 20-1 o o 60 g 15-1 CO 00 ( 4 o o 10-5-0 29 30 o o - T -31 32 3  34 35 Temperature (*C) F i g u r e 18. Out o f plane a n c h o r i n g s t r e n g t h vs temperature f o r a b u f f e d PVF s u r f a c e . 60 p h y s i c a l l y , a n e g a t i v e a n c h o r i n g s t r e n g t h i s i n c o n s i s t e n t w i t h the f a c t t h a t u n i f o r m t a n g e n t i a l alignment was observed i n t h i s c e l l . I t can be seen t h a t the a n c h o r i n g s t r e n g t h o f rubbed PVF i s comparable to t h a t o f SiO t r e a t e d s u r f a c e s . R e s u l t s o f the Extended D e u l i n g Method F i g u r e s 19 through 21 c o r r e s p o n d to sample 1 a t about 300 mK below the c l e a r i n g p o i n t . F i g u r e 19 shows the t i l t angle <f>^ vs reduced v o l t a g e . F i g u r e 20 shows the d e r i v a t i v e o f the a n c h o r i n g p o t e n t i a l SW/d^ as f u n c t i o n o f <j>^ and l a s t l y f i g u r e 21 shows W(<^) as a f u n c t i o n o f ^ , o b t a i n e d by n u m e r i c a l l y i n t e g r a t i n g the da t a i n f i g u r e 20. These t h r e e r e s u l t s were c a l c u l a t e d from the c a p a c i t a n c e vs v o l t a g e d a t a u s i n g the extended D e u l i n g method d e s c r i b e d a t the end o f Chapter II. Yokoyama and van S p r a n g ' s 1 0 d a t a ( o b t a i n e d u s i n g a d i f f e r e n t method) are a l s o i n c l u d e d on these t h r e e graphs f o r c o m p a r i s i o n . A d i s c u s s i o n o f the i m p l i c a t i o n s o f these r e s u l t s i s p r e s e n t e d i n the c o n c l u s i o n . Many o f the r e s u l t s o f the ED method f i t showed f e a t u r e s t h a t were d i s t u r b i n g . F i g u r e 22 shows the worst r e s u l t s from f i t t i n g w i t h the same program. These c o r r e s p o n d to a temperature about 6 K below the c l e a r i n g p o i n t . The most p u z z l i n g f e a t u r e i s the f a c t t h a t the t i l t a n g l e ^ became n e g a t i v e over an a p p r e c i a b l e range o f v o l t a g e . The d e r i v a t i v e SW/S^ ( f i g u r e 23) a l s o shows some unusual f e a t u r e s , f o r example i t i s not a s i n g l e v a l u e d f u n c t i o n . F i t s to the phase data were q u a l i t a t i v e l y the same as the the f i t s to the c a p a c i t a n c e , but showed much more s c a t t e r . An example i s shown i n f i g u r e 24. While 61 0 . 4 n 0 . 3 H 0 . 2 A 0.1 A 0 . 0 - o R e l . 10 1 0 — T " 2 0 —J— 3 0 —T" 4 0 i SO V/V th — I 6 0 F i g u r e 19. T i l t angle vs reduced v o l t a g e f o r SiO t r e a t e d s u r f a c e 62 1.6-J I - 0.8H *° 0.4H 0.0- • Ref. 10 I ' — 0.3 I 0.4 0.0 0.1 I 0.2 F i g u r e 20. S u r f a c e torque as a f u n c t i o n of t i l t angle f o r SiO t r e a t e d s u r f a c e . 63 0.3-, 0.2H 0.0-0.0 I 0.1 — I — 0.2 • Ref. 10 I ' 1 0.3 0.4 F i g u r 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 o f t i l t angle f o r SiO t r e a t e d s u r f a c e . 64 0.14-, O.HH 0.06-1 0.02 A -.02H -.06-* i 10 20 - r -30 V/V —r~ 40 i 50 —j 60 th F i g u r e 22. T i l t a n gle as a f u n c t i o n o f reduced v o l t a g e f o r SiO t r e a t e d s u r f a c e : worst case. 65 4-. a 3A o S H 06 i i i i i i i -.02 0.02 T 1 1 1 1 1 > 1 1 0.06 0.10 0.14 F i g u r e 23. S u r f a c e torque as a f u n c t i o n o f t i l t angle f o r SiO t r e a t e d s u r f a c e : worst case. 66 F i g u r e 24. T i l t a n gle vs reduced v o l t a g e f o r SiO t r e a t e d s u r f a c e : f i t to phase d a t a . 67 1.8-1 1.5-I 1.2-O to 0.6-1 0.3-1 0.0--.10 0.00 0.10 0.20 0.30 0.40 F i g u r e 25. S u r f a c e torque as a f u n c t i o n o f t i l t angle f o r SiO t r e a t e d s u r f a c e : f i t to phase d a t a . 68 these r e s u l t s were not encouraging, they were s t i l l u s e f u l i n one r e s p e c t : i n a l l cases the graph o f s l o p e 3W/30 vs 4>1 c o u l d be f i t t e d to a s t r a i g h t l i n e i n the r e g i o n o f l a r g e r </> The s l o p e o f t h i s l i n e i s the c o n s t a n t o f the R a p i n i - Papoular a n c h o r i n g p o t e n t i a l . The r e s u l t s from t h i s a n a l y s i s are p l o t t e d a l o n g s i d e the r e s u l t s from u s i n g the Y-vS method i n f i g u r e 26. Note t h a t t h i s i s f o r sample 1. F i g u r e s 27-29 r e l a t e the same i n f o r m a t i o n as f i g u r e s 19-21 r e s p e c t i v e l y but are the r e s u l t s f o r the PVF a l i g n e d s u r f a c e . F i g u r e 29 i s the f i r s t r e p o r t e d measurement of the a n c h o r i n g p o t e n t i a l of rubbed PVF o r any o t h e r rubbed polymer. The same problems i n the ED method were observed f o r the PVF s u r f a c e as are o u t l i n e d above f o r the SiO s u r f a c e . The i m p l i c a t i o n s o f these r e s u l t s are d i s c u s s e d i n the c o n c l u s i o n . F i g u r e 30 shows the r e s u l t s o f e x t r a c t i n g the c o e f f i c i e n t W from the r e s u l t s o f the ED method f o r the PVF s u r f a c e o as o u t l i n e d above; a g a i n these r e s u l t s are p l o t t e d w i t h the r e s u l t s o f the Y-vS method f o r comparison. 69 tD a Q> -i-> CO a o Xi o d 30-1 25-^  20-1 i s - e ID-S' 0-29 — r -30 o ED Method. C • ED Method. 6 A Y-vS Method o Ref. 11 o • o9. —i 1 1— 31 32 —T— 33 ~~r~ 34 35 Temperature (°C) F i g u r e 26. A n c h o r i n g s t r e n g t h vs temperature f o r SiO t r e a t e d s u r f a c e : r e s u l t s o f ED method. 70 0.8H 0.6-1 0.4H 0 10 20 30 40 50 60 V/V th F i g u r e 27. T i l t angle as f u n c t i o n o f reduced v o l t a g e f o r rubbed PVF s u r f a c e . 71 F i g u r e 28. S u r f a c e torque as a f u n c t i o n o f t i l t angle f o r rubbed PVF s u r f a c e . 72 30 T 2 5 H 20H 00 g WH 10-5-0-29 A o • 30 o • e i 31 I 32 a ED Method, C o ED Method. 6 A Y-vS Method - T -33 Temperature (°C) A S —r-34 i 35 F i g u r e 30. A n c h o r i n g s t r e n g t h vs temperature f o r rubbed PVF s u r f a c e : r e s u l t s o f ED method. 0.30-t 0.25-*E 0.20-\ 0 2 0.15-3~ 0.10-0.05-0.00 0.0 0.2 0.4 0.6 o.a 1.0 F i g u r 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 o f t i l t angle f o r rubbed PVF s u r f a c e . 73 Chapter V D i s c u s s i o n and C o n c l u s i o n s Comparison w i t h Yokoyama et a l ' s Work The s u r f a c e a n c h o r i n g s t r e n g t h of the nematic l i q u i d c r y s t a l 5CB when a l i g n e d homogeneously on two d i f f e r e n t s u r f a c e s was measured u s i n g the method developed by Yokoyama and van S p r a n g . 1 0 The f i r s t s u r f a c e , t a n g e n t i a l l y evaporated SiO, was employed i n o r d e r to v e r i f y t h a t the method was b e i n g employed c o r r e c t l y and t h a t the r e s u l t s o f r e f . 11 c o u l d be reproduced; one f e a t u r e however t h a t was not seen i n t h i s experiment was a " c r i t i c a l " i n c r e a s e i n d near the c l e a r i n g e p o i n t . I t i t i s not known i f the i n c r e a s e t h a t was seen i n d would e become more pronounced n e a r e r to the t r a n s i t i o n temperature. The graph of a n c h o r i n g s t r e n g t h of SiO vs temperature ( f i g u r e 16) appears to have a f e a t u r e a t - 32.5°C. T h i s was not commented on r e f 11, but the f a c t t h a t i t has been reproduced i m p l i e s s t r o n g l y t h a t something i n t e r e s t i n g i s happening i n t h a t r e g i o n . C e r t a i n l y no t h e o r y o f s u r f a c e a n c h o r i n g p r e d i c t s t h i s b e h a v i o r . F u r t h e r study i n t h i s r e g i o n i s needed. The a n c h o r i n g s t r e n g t h o f a b u f f e d PVF f i l m was measured u s i n g the same method. T h i s i s the f i r s t r e p o r t o f the measurement o f t h i s q u a n t i t y by t h i s method on any rubbed polymer s u r f a c e . The a n c h o r i n g s t r e n g t h was found to be as l a r g e as t h a t f o r SiO. T h i s i s an important r e s u l t because SiO i s s a i d to produce " s t r o n g " s u r f a c e a l i g n m e n t , 1 6 , 1 7 and thus PVF produces s t r o n g alignment a l s o w i t h a s i m p l e r and cheaper treatment. Furthermore, on some samples i n which 75 Che s u r f a c e treatment used was b u f f e d PVF, a n e g a t i v e e x t r a p o l a t i o n l e n g t h was observed. The i m p l i c a t i o n such r e s u l t s i s t h a t the e f f e c t o f s u r f a c e alignment p e r s i s t s through a l a y e r o f t h i c k n e s s d next to e the p l a t e s , even under the i n f l u e n c e o f a v e r y l a r g e r e o r i e n t i n g f i e l d . Presumably there i s a f i e l d s t r e n g t h at which the e f f e c t o f the s u r f a c e on t h i s l a y e r i s overcome, but t h i s was not seen. In the case o f a n e g a t i v e e x t r a p o l a t i o n l e n g t h the r e l a t i o n s h i p between i t and the a n c h o r i n g s t r e n g t h breaks down. The PVF method o f p r o d u c i n g s u r f a c e alignment c o u l d be o f i n t e r e s t to the t w i s t e d nematic LCD i n d u s t r y , i n which a rubbed p o l y ( v i n y l - a l c o h o l ) f i l m i s used as the a l i g n i n g treatment. My e x p e r i e n c e w i t h PVA has been t h a t i t i s much more d i f f i c u l t to produce u n i f o r m alignment w i t h than PVF, mainly due to i t s poor s o l u b i l i t y i n most s o l v e n t s . The a n c h o r i n g s t r e n g t h o f PVA has not y e t been measured to my knowledge. Extended D e u l i n g Method o f Measuring A n c h o r i n g P o t e n t i a l A new method o f measuring the s u r f a c e a n c h o r i n g p o t e n t i a l , based s o l e l y on the continuum t h e o r y o f nematics, was developed. The new method i s an e x t e n s i o n o f D e u l i n g ' s 6 s o l u t i o n o f the s p l a y F r e e d e r i c k s z t r a n s i t i o n . T h i s method does not assume the R a p i n i - P a p o u l a r 1 form o f the a n c h o r i n g p o t e n t i a l . A s e r i o u s d i f f i c u l t y i n the ED method i s the s t r o n g dependence o f the r e s u l t s on the c o n s t a n t 7. The d i r e c t r e s u l t o f t h i s dependence was t h a t t h e r e i s an u n c e r t a i n t y i n 7 of a t most 1%, and t h i s u n c e r t a i n t y was l a r g e enough to a f f e c t the r e s u l t s c o n s i d e r a b l y as can be seen i n f i g u r e 31. T h i s s t r o n g dependence on 7 may be due to 76 F i g u r e 31. S u r f a c e torque as a f u n c t i o n of t i l t angle showing the dependence on the d i e l e c t r i c a n i s o t r o p y used i n f i t t i n g the data. 77 the assumption of the continuum model t h a t the o r d e r parameter S i s u n i f o r m throughout the c e l l and dependent o n l y on temperature. T h i s i s a poor assumption i n r e g i o n s of v e r y h i g h d e f o r m a t i o n -- i t i s w e l l known t h a t d e f e c t s i n nematics o f t e n have an i s o t r o p i c c o r e . 5 R e c a l l the e l e c t r i c coherence l e n g t h (eq. 2.30): In eq. 2.38, i t can be seen t h a t t h i s l e n g t h i s the a p p r o x i m a t e l y the w i d t h of the r e g i o n where the s l o p e 34>/dz i s the l a r g e s t . In the experiments performed, £ c o u l d become as s h o r t as 50 nm. C e r t a i n l y i t i s p o s s i b l e to have a decrease i n the o r d e r parameter i n a r e g i o n such as t h i s ; a f i r s t o r d e r c a l c u l a t i o n , e q u a t i n g the s p l a y energy (w i t h a c o n s t a n t e l a s t i c c o n s t a n t ) to the m o l e c u l a r f i e l d energy due to nematic o r d e r i n g shows t h a t decreases i n S o f the o r d e r o f 1% may be expected w i t h a coherence l e n g t h o f 50 nm. When one c o n s i d e r s t h a t 2 the e l a s t i c c o n s t a n t s are p r o p o r t i o n a l to S , the change i n S would become even l a r g e r . F i g u r e 31 shows the r e s u l t o f a .6% change i n 7. I t i s thus c o n j e c t u r e d t h a t the continuum model needs to be extended to a l l o w the o r d e r parameter to v a r y s p a t i a l l y b e f o r e the s u r f a c e i n t e r a c t i o n can be w e l l d e s c r i b e d w i t h experiments o f t h i s type. Other methods o f d e t e r m i n i n g the a n c h o r i n g p o t e n t i a l u s i n g t h i s type o f experiment have found g r e a t s e n s i t i v i t y to 7 . 1 0 The r e s u l t s o f the ED method ( f i g u r e 19) i n the r e g i o n o f s m a l l v o l t a g e (v < 2) e x h i b i t e d an unusual shape. T h i s may have been due the f a c t t h a t the F r e e d e r i c k s z t r a n s i t i o n may be thought o f as a second o r d e r phase t r a n s i t i o n , and hence one expects f l u c t u a t i o n s to be important i n the r e g i o n near the t r a n s i t i o n . T h i s a l s o i s not 5.1 78 accounted f o r i n continuum theory. A l s o i n f i g u r e 19, a " j o g " can be seen i n the curve i n the v i c i n i t y o f V ~ 17 V . T h i s was a f e a t u r e of a l l data f i t i n t h i s J th way, f o r both types o f s u r f a c e s . I t s o r i g i n i s unknown, but i t s presence p o i n t s to e i t h e r an unexpected jump i n the s u r f a c e torque 3W/3^i o r a shortcoming o f the continuum theory. Note t h a t no break was v i s i b l e i n the raw data, and hence the e f f e c t must be s u r f a c e e f f e c t and not a b u l k e f f e c t . I t s o r i g i n i s not y e t understood. When the analogous c a l c u l a t i o n was a p p l i e d to the phase data, q u a l i t a t i v e l y e q u i v a l e n t r e s u l t s were o b t a i n e d , but w i t h v e r y much more s c a t t e r : c. f. f i g u r e 24 T h i s c o u l d be due to the f a c t t h a t another m a t e r i a l c o n s t a n t (i/) was r e q u i r e d , however the r e s u l t s d i d not depend so s t r o n g l y on the m a t e r i a l c o n s t a n t s as i n the c a p a c i t a n c e d a t a case. I t i s f o r t h i s r e a s o n t h a t 7 was a d j u s t e d ( w i t h i n the bounds d i s c u s s e d i n Chapter IV) so t h a t the two f i t s a greed i n the r e g i o n o f h i g h v o l t a g e . In b o t h c a s e s , dW/3^ as a f u n c t i o n o f <f> was f i t to a s t r a i g h t l i n e i n the r e g i o n of l a r g e s t ^ and hence the a n c h o r i n g s t r e n g t h W Q was c a l c u l a t e d . These r e s u l t s were comparable those o f the Y-vS method. C o n c l u s i o n Two methods have been used to study s u r f a c e a n c h o r i n g p r o p e r t i e s , each on two d i f f e r e n t s u r f a c e s . The f i r s t method i s s t r a i g h t f o r w a r d , but based on a l i n e a r expansion to f i r s t o r d e r i n the d i s p l a c e m e n t . The r e s u l t s o f t h i s method were d i f f i c u l t to reproduce on subsequent 79 runs. The r e s u l t s from t h i s method f o r the s u r f a c e a n c h o r i n g s t r e n g t h o f t a n g e n t i a l l y evaporated SiO were found to agree r e a s o n a b l y w i t h p r e v i o u s l y p u b l i s h e d r e s u l t s . The ED method develop e d i n t h i s t h e s i s was based s o l e l y on the continuum t h e o r y o f nematic d e f o r m a t i o n s . T h i s method a l l o w s the d e t e r m i n a t i o n o f the s u r f a c e a n c h o r i n g p o t e n t i a l from e i t h e r d i e l e c t r i c ( c a p a c i t a n c e ) or o p t i c a l (phase) d a t a . The ED method was found to be v e r y s e n s i t i v e to v a l u e s o f m a t e r i a l parameters; t h i s i s thought to be due to s p a t i a l v a r i a t i o n s i n the s c a l a r o r d e r parameter caused by l a r g e d i r e c t o r g r a d i e n t s . As a r e s u l t o f t h i s , i t s u s e f u l n e s s i s l i m i t e d ; the cases where i t was most s u c c e s s f u l were those a t h i g h e r temperatures where the s p a t i a l v a r i a t i o n o f the s c a l a r o r d e r parameter i s expected to be s m a l l e r . 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 e q u a t i o n s f o r the angle o f the d i r e c t o r a t the s u r f a c e s i s n u m e r i c a l l y i n t e n s i v e and r e q u i r e s s o p h i s t i c a t e d t e c h n i q u e s o f n u m e r i c a l i n t e g r a t i o n . Both the Yokoyama-van Sprang method and the extended D e u l i n g method were a p p l i e d to a new type o f s u r f a c e f o r a l i g n i n g nematic l i q u i d c r y s t a l s : rubbed PVF. The a n c h o r i n g s t r e n g t h o f t h i s s u r f a c e was found to 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 e v a p o r a t e d SiO. 8 0 Appendix: E f f e c t o f F i n i t e S u r f a c e A n c h o r i n g on the T h r e s h o l d F i e l d The F r e e d e r i c k s z t h r e s h o l d v o l t a g e V i n g e n e r a l depends on the t h s t r e n g t h o f the s u r f a c e a n c h o r i n g . In the experiments d e s c r i b e d i n t h i s t h e s i s t h a t change i s l e s s than the ac c u r a c y o f the v o l t a g e measurements and thus i s not an o b s e r v a b l e e f f e c t . The change i n the t h r e s h o l d magnetic f i e l d was p r e d i c t e d by R a p i n i and P a p o u l a r , 1 and ob s e r v e d by R o s e n b l a t t and Yang. 2 6 I t i s a l s o shown t h a t there e x i s t s a s a t u r a t i o n f i e l d a t which the c e l l becomes homogeneously a l i g n e d a l o n g the a p p l i e d f i e l d . T h i s was p r e d i c t e d by Nehring, Kmetz and S c h e f f e r 2 7 and observed by Yokoyama and van S p r a n g . 1 0 R e c a l l e q u a t i o n 2.44: 7T/2 1 - 2 / V n V 1 + o 1 + K.rj s i n (V>) T 1/2 (1 - n s in 2(i/>))( l + tv sin2(V>) di/> A . l Where V = IT / K /e e which i s the F r e e d e r i c k s z t h r e s h o l d v o l t a g e o v V o 2 6 c a l c u l a t e d by D e u l i n g 6 f o r the case o f i n f i n i t e a n c h o r i n g s t r e n g t h . The t h r e s h o l d v o l t a g e i s d e f i n e d as the l i m i t o f V above when <j> and 4>^ b o t h go to zero; i n e q u a t i o n A . l , t h i s l i m i t i s e v a l u a t e d by l e t t i n g rj = 0. T h i s y i e l d s : V — = I - - rp V * " 1 o A.2 In the i n f i n i t e a n c h o r i n g case the lower l i m i t o f the i n t e g r a l i s z e r o , and the t h r e s h o l d v o l t a g e i s V q as expected. In the f i n i t e a n c h o r i n g case Tp i s d e f i n e d through sin(V> )sin((£ ) = sin ( 0 ) and 1 1 m 1 hence: 81 cos (^ v) = s i n ( ^ ) s i n ( 0 ) A . 3 when both <j> and <f> are s m a l l . The r e l a t i o n s h i p between 4> and 4> i s 1 m 1 m r) th o b t a i n e d from the torque balance e q u a t i o n (2.33) i n which i s o b t a i n e d from the f i r s t i n t e g r a l o f the Euler-Lagrange e q u a t i o n (2.37). The torque balance e q u a t i o n i s thus: 2 , , , . 2 aw 34> - K D / ' — 1 1 v £ e K ( s i n (0 ) - s i n (4 )) (1 + K s i n U ) ) m l 1 ( 1 + 7 s i n 2 ( ^ ) )( 1 + 7 s i n 2 ( ^ ) IS 1 1/2 A.4 o 2 1 I f we i n s e r t the R-P form o f the a n c h o r i n g energy (Eq. 2.23) (which i s g e n e r a l l y v a l i d i n the regime where <f> i s s m a l l ) , and then o n l y keep terms l i n e a r i n 4> and <j> , the r e s u l t i s : m 1 W <j> = K D 0 1 1 £ £ K 0 2 1 <t>2 - s m 1 1/2 A.5 N o t i n g t h a t D q i s g i v e n by e q u a t i o n 2.9, t h i s e q u a t i o n becomes, upon s q u a r i n g b o t h s i d e s : .2 2 A v 4>Z -m 1 A.6 where A = 7r K /W d and D /D = V /V = v. When e q u a t i o n A. 6 i s 1 0 th 0 th 0 i n s e r t e d i n t o e q u a t i o n A.3, a t r a n s c e n d e n t a l e q u a t i o n r e s u l t s f o r v. c o t (— u) = A 1/. 2 A.7 U s i n g t y p i c a l r e s u l t s from c h a p t e r IV, A i s found to range between .0007 and .0015. T h i s r e s u l t s i n 1/ r a n g i n g between .9990 and .9995. That i s the t h r e s h o l d v o l t a g e observed w i l l be reduced from the p r e d i c t e d v a l u e s by l e s s than .1%. T h i s d e v i a t i o n i s a p p r o x i m a t e l y the a c c u r a c y w i t h which v o l t a g e can be measured i n the experiment 82 d e s c r i b e d i n Chapter III. R o s e n b l a t t and Yang, 2 6 u s i n g a v e r y t h i n c e l l (d ~ 3 JJ) observed a decrease i n the t h r e s h o l d magnetic f i e l d f o r the bend F r e e d e r i c k s z t r a n s i t i o n o f ap p r o x i m a t e l y 25%. T h i s decrease was i n e x c e l l e n t agreement w i t h the v a l u e p r e d i c t e d from a the o r y analogous to t h a t above. A s a t u r a t i o n v o l t a g e V i s d e f i n e d as the l i m i t of e q u a t i o n 2.40 s a t when both 4> and 4> go to n/2. That l i m i t may be e v a l u a t e d by making m 1 2 the s u b s t i t u t i o n s | = cos (<f> ), and cos(<£) = cos (4> )cosh(V>) i n the m m i n t e g r a l o f e q u a t i o n 2.40. In the l i m i t where £ goes to ze r o t h i s e q u a t i o n becomes: V s a t 2 / , = - V 1 + K i/> A. 8 V o Where cosh(V> ) = cos($ )/cos(<£ ). D e f i n i n g a as n/2 - <t> and /? as 1 1 tn 1 7r/2 - <f> , and i n the l i m i t where both a and jl are s m a l l one can ra w r i t e : coshC^) = a//3. A. 9 One can a l s o e v a l u a t e the torque b a l a n c e e q u a t i o n i n t h i s l i m i t . However the R a p i n i - P a p o u l a r a n c h o r i n g p o t e n t i a l may not be v a l i d i n the r e g i o n where 4> - w/2. Nonetheless, whatever the a n c h o r i n g energy, i t must be symmetric about n/2. T h e r e f o r e , when the t i l t a n g l e i s c l o s e to n/2, one can approximate the a n c h o r i n g energy as: W(^) - \ W j \ - 0 j 2 A.10 The torque b a l a n c e e q u a t i o n i n the l i m i t where a and (3 are bo t h s m a l l i s : K D 1 sat /~T~ f (a - / T ) ( l + * ) " / £ £ K <- (1 + 7 ) 2 1/2 = W Q A l l l A. 11 o 2 1 83 R e p l a c i n g D /D w i t h v ' t /<r , t h i s becomes: v ° sat 0 1 2 A' 2 i/ ' 2 (1 + K) (a2 - P2) = a2 Where A' = n K /W d. T h i s i s i n s e r t e d i n A.9 and the r e s u l t i s : l l A. 12 c o t h • -J 1 + K • - A'/ 1 + K v' A.13 10 Yokoyama and van Sprang measured the s a t u r a t i o n v o l t a g e and found r e a s o n a b l e agreement w i t h t h i s t h e o r y . 84 References 1. A. R a p i n i and M. Papoular, J . Phys. C o l l o q . , 30. C4-54 (1969). 2. G. Barbero and G. Durand, J . Phys. ( P a r i s ) 47, 2129 (1986). 3. G. Barbero, N.V. Madhusudana and B. Durand, Z. N a t u r f o r s c h . 39a. 1066 (1984). 4. 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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 W.T. V e t t e r l i n g , Numerical R e c i p e s . (Cambridge,1986). 15. J . L . J a n n i n g , A p p l . Phys. L e t t . 21, 173 (1972). 16. J.Cognard, Molec. C r y s t . L i q . C r y s t . Supp., 1. 17. D. R i v i e r e and Y. Levy, J . Phys. L e t t . 40, L-215 (1979). 18. W.A. La and G.J. Knight, i n Polymer Handbook, e d i t e d by J . Brandup and E.H. Immiyat ( I n t e r s c i e n c e , New York, 1966) p 111-61. 85 19. J.M. Geary, J.W. Goodby, A.R. Kmetz and J.S. P a t e l , J . A p p l . Phys. 62, 4100 (1987). 20. H. Yokoyama, S. Kobayashi and H. Kamei, J . A p p l . Phys. 56, 2645 (1984). 21. Stephen A. Casalnuovo, R.C. Mockler and W.J. O ' S u l l i v a n , Phys. Rev. A 29, 257 (1984). 22. H.A. van Sprang, J . Phys. ( P a r i s ) 44, 421 (1983). 23. D.A. Dunraur, M.R. M a n t e r f i e l d , W.H. M i l l e r and J.K. Dunleavy, Mol. C r y s t . L i q . C r y s t . 45, 127 (1978). 24. J.D. Bunning, T.E. 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