REFRACTIVE INDICES OF LIQUID CRYSTALS AND PURE FLUIDS B.A. Sc., University of British Columbia, 1966 M.A. Sc., University of British Columbia, 1969 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES THE DEPARTMENT OF PHYSICS We accept this thesis as conforming NEAR PHASE TRANSITIONS by PETER PALFFY-MUHORAY to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1977 (Peter Palffy-Muhoray In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish 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 representatives. It is understood that copying or-publication of this thesis for financial gain shall not be allowed without my written permission. Department of P h y s i c S The University of Brit ish Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 October 5, 1977 i i ABSTRACT Interferometric measurements to determine the r e f r a c t i v e i n -dices of the nematic l i q u i d c r y s t a l s EBBA and BEPC as a function of temperature are described u t i l i z i n g modified Rayleigh and conoscopic interferometers. Theory i s presented r e l a t i n g the r e f r a c t i v e indices and density to the o r i e n t a t i o n a l order, l o c a l f i e l d parameter and mole-cular properties. The r e s u l t s of simple thermal expansivity measure-ments are also given for EBBA. The Lorentz-Lorenz c o e f f i c i e n t for SF, and GeH, has been 6 4 determined from r e f r a c t i v e index and density measurements. The method u t i l i z e s a prism shaped high pressure c e l l which can be removed from a temperature c o n t r o l l e d holder and weighed on a p r e c i s i o n balance. The r e s u l t s i n d i c a t e a v a r i a t i o n of 0.5% f o r SF, and 0.8% f o r GeH, 6 4 over the density range covered. i i i TABLE OF CONTENTS Page List of Tables. v i List of Figures v i i Acknowledgements x Preface x i CHAPTER 1. Refractive Indices in Anisotropic Media 1 1.1 Introduction 1 1.2 The Propagation of Plane Waves 2 1.3 The Local Field Anisotropy Tensor 6 1.4 The Dielectric Tensor and the Local Field 14 CHAPTER 2. Nematic Liquid Crystals 17 2.1 Molecular Order in Liquid Crystals 17 2.2 Types of Liquid Crystalline Order 22 2.3 The Order Parameter in Nematics 29 2.4 The Landau Theory 34 CHAPTER 3. Mean Field Theory 43 3.1 One-body Pseudopotential in the Mean Field Approximation 43 3.2 Pair-Correlation in a Hard Spheroid Model 48 3.3 Dependence of density on the Order Parameter ... 60 3.4 Dipole-Dipole Interaction 67 3.5 London-van der Waals Interaction 71 iv Page CHAPTER 4. Pure Fluids - Experimental 7 7 4.1 The Lorentz-Lorenz Relation 77 4.2 Details of the Experiment 79 4.3 Temperature Control 89 4.4 Results 94 CHAPTER 5. Nematic Liquid Crystals - Experimental 101 5.1 The Anisotropic Lorentz-Lorenz Relation 101 5.2 The Modified Rayleigh Interferometer 104 5.3 Conoscopic Measurements 5.4 Refractive Indices 128 5.5 Thermal Expansivity 1^ 0 5.6 The Order Parameter and the Local Field Anisotropy 1 ^ CHAPTER 6. Distortions in Nematics 1 5 2 6.1 Polarization Anomaly 1^ 2 6.2 Nematic Alignment 155 6.3 Propagation in a Twisted Medium 160 CHAPTER 7. CONCLUSIONS 1 6 3 7.1 Summary of Results l ^ 3 7.2 Suggestions for Future Work 165 APPENDIX A i 6 7 B 170 C 1 7 4 v i LIST OF TABLES Page TABLE 1. Composition of some common nematic liquid crystals 20 V Page APPENDIX D 177 E , 179 F 180 G 183 H 185 I 189 J , 192 K 197 REFERENCES 1 9 9 v i i i Figure Page 19. Coexistence curve on a temperature-density plot il l u s t r a t i n g procedure for obtaining data. 84 20. Temperature and density region at which data points are ob-tained. 87 21. Circuit diagram of the temperature control system. 91 22. The c e l l containers. 92 23. The Lorentz-Lorenz coefficient for SF. as a function of 6 density. 95 24. The Lorentz-Lorenz coefficient for GeH, as a function of 4 density. 97 25. Coexistence curve of GeH.. 98 4 26. The modified Rayleigh interferometer. 106 27. Interference pattern obtained from the modified Rayleigh interferometer. 107 28. Results of fringe number measurements corresponding to changes in the refractive indices of EBBA. 110 29. Results of fringe number measurements corresponding to changes in the refractive indices of BEPC. 112 30. Analysis of c e l l rotation data: x vs. x 2 + cos 20 for EBBA. 115 31. Conoscopic diffraction pattern. 119 32. Conoscopic fringe number measurements for EBBA. 122 33. Conoscopic fringe number measurements for BEPC. 124 34. Refractive indices of EBBA. 129 ix Figure Page 35. Refractive indices of BEPC. 131 36. Published values for the refractive indices of EBBA. 133 37. Photograph of apparatus used in the refractive index mea- 138 surements. 38. Temperature-controlled c e l l holder used in thermal expan- 141 sivity measurements. 39. Relative volume and thermal expansivity for EBBA. 144 40. The order parameter S obtained from experimental measurements 148 for EBBA. 41. The local f i e l d tensor n obtained from experimental measure- 150 ments for EBBA. B.l The local f i e l d anisotropy tensor eccentricity e . g H. l Prism c e l l geometry. I. 1 Geometry of rotated c e l l . J . l Geometry of a plane wave incident material. n as function of the 172 zz 186 190 on a f l a t slab of sample 194 v i i LIST OF FIGURES Figure Page l a . Regions of integration in the local f i e l d calculation, show-ing the arbitrary surface U. 8 lb. The transformed surfaces. 10 2. Structure of common nematogens. 19 3. Canonic order. 23 4. Hexagonal rod systems. 24 5. Arrangement of molecules in smectics. 25 6. The cholesteric configuration. 26 7. Arrangement of molecules in the nematic mesophase. 27 8. Co-ordinate system used in the polarization calculation. 30 9. Free energy and order parameter R; continuous transition. 37 10. Free energy and order parameter R; discontinuous transition. 39 11. Free energy and order parameter S; discontinuous transition. 41 12. The geometry of closest approach. 51 13. Bounds on the effective hard-core diameter D(r). 54 14. The local f i e l d anisotropy tensor as a function of the order 57 parameter. 15. Approximations to the effective hard-core diameter. 63 16. The effect of the anisotropic hard-core on the order parameter in the mean f i e l d approximation. 75 17. Schematic i l l u s t r a t i o n of optical equipment. 80 18. Drawing of sample c e l l . 81 x i PREFACE In an attempt to make this thesis more easily readable by maintaining continuity of thought throughout, we have derived, from basic principles, certain well known theoretical results. Although the methods of derivation are often somewhat different from those in the literature, we have attempted to indicate in the text when-ever the results are not original. In order to clearly define the o r i -ginal contributions presented in this thesis, however, we wish to make note of the following. The results of Sec. 1.2 regarding the propaga-tion of plane waves in anisotropic media are well known; similarly, the results obtained from applying the Landau theory of phase transi-tions to liquid crystals, presented in Sec. 2.4, are not original. Finally, the topics discussed in Sees. 6.2 and 6.3 appear in liquid crystal literature as mentioned in the text, although our formalism is different. Our original contributions may be summarized as follows. We have generalized the Clausius-Mossotti relation so as to make i t applicable to certain anistropic molecular fluids. We have attempted to include the effects of an anisotropic hard-core repulsive term in the intermolecular interaction potential in the mean-field theory of nematic liquid crystals. Experimentally, we have measured directly the Lorentz-Lorenz coefficient of pure fluids, and, using a new and sensi-tive interferometric technique, measured the refractive indices of nematic liquid crystals. The theory enables the orientational order parameter and the local f i e l d anisotropy parameter to be calculated from experimental results. ACKNOWLEDGMENTS I should like to express my sincere appreciation to Professor D.A. Balzarini for his guidance and assistance throughout this work. I am especially indebted to the late Professor R.E. Burgess for his interest in and encouragement of this research, and also for his patience and tolerance during our chess games. I should also like to thank Professor L. Sobrino for his help with the theoretical aspects of this work, and Professor W. Unruh who helped me avert the polarization catastrophe. To A. Rosenberg, M. Burton and J. Shelton I extend my thanks for their participation in this research. I am indebted to the technical staff of the Physics department for their assistance in manufacturing the experimental apparatus. I wish to thank Professors F.W. Dalby, A.V. Gold, W.N. Hardy, W. Opechowski, B.G. Turrell, B.L. White for interesting and illuminating discussions. I should lik e to thank Darlene Crowe for typing the manu-script, and Mikey Burton for his help with the illustrations. Finally, I am grateful to Eunice for putting up with i t a l l . 1 CHAPTER 1 REFRACTIVE INDICES IN ANISOTROPIC MEDIA 1.1 Introduction Dielectric properties of materials reflect the changes in molecular order that occur at phase transitions. Refractive index measurements provide a simple and sensitive method of studying these changes. In addition to the liquid-vapor transition, certain fluids exhibit phase changes that correspond to loss of orientational order of the molecules; these liquid crystals have been the source of con-siderable interest recently. In this thesis, we wish to report a new and sensitive interferometric method of measuring the refractive indices of nematic liquid crystals, and in addition, a simple, direct method of measuring the refractive ..index, together with the density of pure fluids. In order to relate these measurements to molecular properties, we have derived a general relation between the dielectric permittivity and molecular polarizability for a broad class of ordered fluids. Using this relation, experimental results may be interpreted in terms of simple s t a t i s t i c a l mechanical models. 2 1.2 The Propagation of Plane Waves. Refractive indices of materials may be defined formally in terms of the phase velocities of plane waves propagating in them. Maxwell's equations, in Gaussian units, are and for a region that does not contain currents. If B = H, elimination of H'from Eq.'s (1.1) and (1.2) yields \l \ $ = -V x (V x E). (1.3) The dielectric permittivity tensor is defined by D = £ BE f l , (1.4) where a = x,y,z and summation is implied over repeated greek indices. It is well known (1) that is diagonal in a system of coordinate axes coincident with the principal dielectric axes of the material. A mono-chromatic plane wave of frequency v = o)/2ir propagating in the ^-direction 3 in the material w i l l have a time and space variation proportional to exp[i(tot-k« r) ]; then Eq. ( 1 . 3 ) becomes or - 4-D = - £ x ( £ x l ) = Ut'l)-hi2 ( 1 . 5 ) c z S - ^ V ( 1 . 6 ) to x where " E x denotes the component of 111 perpendicular to %. Clearly, Tj, E; and k a r e coplanar, and D i s perpendicular to k. The phase velocity Vp of the wave is given by v^ = to/k, and the refractive index n is defined by n= — = — ( 1 . 7 ) • V CO p Eq. ( 1 , 6 ) may be re-written in the following form: ? „ „ 2 i i i M ( 1 . 8 ) or n2 = D 2 ( 1 . 9 ) (E« D ) 4 If the components of D in the principal axis system are denoted by D i = Ddi where i = x,y,z and the d_^ are the components of a unit vector along D, then and Dd. E i = (1.10) i i , d 2 d 2 d 2 xx yy z z It i s shown in Appendix A, that, for a given the allowed values of d^ are those for which n is an extremum. It follows immediately that for k along one of the principal axes, say the z-axis, the allowed directions of D are along the x-axis, with n = , and along the y-axis, with n = In general, for a given direction of propagation, two allowed perpendicular directions for D exist, with corresponding refractive indices. Materials in which a l l three principal dielectric constants have different values are termed biaxial, since two different direc-tions of propagation exist i n which the phase velocity i s independent of the direction of D. If two of the principal dielectric constants are equal, say = £ ^ = e , then the phase velocity for wave propagation along the z-axis is independent of the direction of D, and n = n x = ve x . For propagation perpendicular to the z-axis, say in the x-direction, extremal values of n are n = n x = ve x i f D is along the y-axis, and n = n)P = ~ ^ H i f u is along the z-axis. Such materials 5 are termed uniaxial; a l l liquid crystals considered in this thesis belong to this category. For historical reasons, n x is often called the ordinary index, denoted by n^, while n(| is called the extraordinary index, denoted by n &; the subscripts " and x denote directions per-pendicular and parallel to the optic axis. Finally, i f a l l the d i -electric constants are equal, n = regardless of the direction of D, and the material is isotropic. 6 1.3 The Local Field Anisotropy Tensor In order to relate the dielectric tensor of fluids to molecular properties, i t is necessary to know the relation between an applied f i e l d 1" and the local electric f i e l d F~ experienced by a molecule in the medium. , In cubic crystals, the local f i e l d has been evaluated by Lorentz, and is given by F = (e+2)E"/3, where e is the isotropic dielectric permittivity. This result ifrass been extended to isotropic fluids by Hirschfelder et a l . (2). A more complicated expression for the local f i e l d in certain anisotropic - crystals has been derived by Ewald (3) and Born (4); and more recently by Neugebauer (5) and Dunmur (6). For anisotropic fluids, the relation between E and F is not known. For fluids whose pair-correlation function can be made isotropic by scaling in radial directions after.averaging over molecular orientations, the local f i e l d can be easily evaluated in terms of the scaling transformation. We assume that a_."f.luadpp61ar.ize'diby an applied electric f i e l d may be represented as an assembly.of identical dipoles u whose relative spatial distribution is characterized by a pair-correlation function g(r), normalized such that the lim g(rr) = 1 where r = rr is a position r-x» vector originating from a dipole in the fl u i d . Let r' = r/f(r) be the radial -> scaling transformation under which g(r) becomes isotropic; that i s , g(rr) = g(r'f(r)r) = g Q ( r ' ) . Let U be the surface given by r = cf(r) where c is some positive constant; evidently g(r) = gg(c), a constant everywhere on U. The electric f i e l d F^ at the origin due to a l l dipoles within a sphere r = R i s given in cartesian tensor notation by 7 pg(rf) ( 3 r q r 3 y g P q ) r 2 drdfi (1.12) r 3 where and are the components of r and u respectively, p is the average number density of molecules i n the f l u i d , and dfi is the element of solid angle. Letting r* = r/f(r) and P = pu where P is the polarization, and F. = 4-rrn QP Q la a3 3 (1.13) \ 6 tiff R/f(r) g 0(r')dr' (3r r Q - 6 p.dfi a 3 a3 (1.14) The integration is performed over two regions; over the volume bounded-by the sphere r' = c, and over the volume between the sphere r' = c and the surface r 1 = R/f(r). The regions of integration are-shown in Fig. 1. The constant c is chosen such that 0<c<R/f(r) for a l l r. Then /c m S 0 ^ ) d r ' / ( 3 r ^ " 6 a B ) d " R/f(r) g Q ( r ' ) d r ' ( 3 r a r e - 6 a g ) d ^ . (1.15) c r' 8 Fig. la Regions of integration in the local f i e l d calculation, showing the arbitrary surface U. 9 10 F i g . l b The transformed surfaces. 11 12 The f i r s t term corresponds to the contribution of dipoles within U to the local f i e l d ; i t vanishes for a l l c since the integral over the solid angle is zero. The second term, corresponding to the contribution of dipoles between the surface U and the sphere r = R to the local f i e l d , can be integrated over r' i f c is sufficiently large that g 0(r') = 1 for r' > c. Then T1ae="47y ( l n f ( r y ^ 3 r a r B - 6 a e ) d f i ( 1 ' 1 6 ) independent of R, c and the detailed structure of g(r). The symmetric zero-trace tensor defined by Eq. (1.16), yields a measure of the anisotropy of f ( f ) , or equivalently of U. The off-diagonal elements of n can be made to vanish i f f(r) i s a quadratic form, and n ^ = 0 i f f(r) is spherically symmetric. The simplest case of anisotropy occurs when the pair-correlation function g(r) can be made isotropic by a scaling in one direction only. The surface U upon which g(r) is a constant i s , . i n this case, a spheroid; and the scaling transformation which renders g(r) isotropic is given by r' = r/f(r) = r(r A _ r _ ) 2 . If the spheroid is prolate, a ap g 1 0 0 0 1 0 0 0 1-e2 g (1.17) where e is the eccentricity of the pair correlation function. The g diagonal elements of the anisotropy tensor n are given by 13 /*2TT /•IT ^ n = -r- I I (ln(l-e 2cos 26) 2)(3cos2e-l)sin6ded((.; (1.18) ZZ 4lT / / g 0 0 n =-n = - h n and n „ = 0 i f a ^ g . The integration i s carried xx yy - 2 zz ap out in Appendix B, with the result that 2 1 1 n 1 M ( 1 + e ^ " d- 1 9) \ z = 3 " ^ " 2e~ ( 1 " ^ ^ T i ^ O ' 8 - 8 8 - 8 Furthermore, i t can be shown that i f U is a general ellipsoid, 1 N i n.. = - -rr + -,— where N. is the demagnetizing factor of the ellipsoid, i i 3 4ir l 14 1 . 4 The Dielectric Tensor and the Local Field The local electric f i e l d F , experienced by a molecule in the fluid, i s given by F = E +F-. +F. = E +4TTL 0 P D ( 1 . 2 0 ) a a 0 a l a a ap g 4TTP where E is the applied f i e l d , F_ =•—— is the Lorentz cavity f i e l d , a r 0 a 3 and L . is the Lorentz-factor tensor introduced by Dunmur (7). Substitu-ag tion of Eq. ( 1 . 1 3 ) into Eq. ( 1 . 2 0 ) yields L = ± 8 +n . ( 1 . 2 1 ) ag 3 ag ag The dielectric permittivity. tensor is obtained from Eq. ( 1 . 4 ) by recalling that D = E +4irP ; then a a a e Q E Q = E +4TTP . ( 1 . 2 2 ) ag g a a The expression for the local electric f i e l d F in terms of the dielectric a permittivity is obtained by substituting Eqs. ( 1 . 2 1 ) and ( 1 . 2 2 ) into Eq. ( 1 . 2 0 ) ; then F = ( - k e Q +26 0 ) + n (e -fi Q ) ) E Q . ( 1 . 2 3 ) a 3 ag ag ay y3 YP 3 If the assumption is made that the polarization P^ is related to the local f i e l d F by a 15 P = pa Q F C a ag {. (1.24) where is the effective molecular polarizability, substitution of Eqs. (1.20) and (1.24) into Eq. (1.22) yields the anisotropic Clausius-Mossotti equation e -6 D = 4irp((e s-& .)L„ +6 )a 0 ag ag ao ao 6y ay Y3 (1.25) or, as expressed in terms of the anisotropy tensor n ag» (e .-6 Q ) ( 6 D - 4 T r p n D Xa„ ) = (e „+26 Q ) a Q ag ag gy go Sy -3- ag ag gy (1.26) If the permittivity tensor and the local f i e l d anisotropy tensor are both diagonal, Eq. (1.26) becomes (e . .-1) 4iTpa. . (l - 4 i r p T i. .a. .) = (ei;L+2) 1 1 1 1 3 (1.27) and the local f i e l d i s given by ± (e..+2) + n..(e..-D 3 xi xi xx -E, (1.28) These results have been published in our recent,paper (24). If the assumptions made in deriving ^these^relations, are valid r )at,.optical fre-quencies, then for a uniaxial f l u i d , Eq. (1.27) becomes 1 6 ( n l _ 1 ) ATT (l-4TTpn„a 0) = -5-pa (1.29) (n2+2) £ £ 3 £ since = /e^ where £ = " or j-K Since is traceless, n M = ~2r\j_ . and i t follows from Eq. (1.29) that 2(n 2 +2) (n2,;+2) + (n, -1) (n,,,-l) 1, + 2 (1.30) A l l refractive index measurements w i l l be interpreted on the basis of Eq. (1.29); in the case of isotropic fluids, e = 0, n = 0 § 0t p and Eq. (1.29) reduces to the well known Lorentz-Lorenz relation. In general, however, the refractive indices of a f l u i d depend on the effective molecular polarizability a., the pair-correlation determined anisotropy tensor and the number density p. whereas an exact determination is d i f f i c u l t , an approximate determination of the temperature dependence of these quantities is possible from simple s t a t i s t i c a l mechanical models. 17 CHAPTER 2 NEMATIC LIQUID CRYSTALS 2.1 Molecular Order in Liquid Crystals In crystalline solids, long-range order exists in the three positional degrees of freedom of the constituent molecules. If the molecules are anisotropic, long-range orientational order can also exist. Loss of order in one or more degrees of freedom constitutes a change of phase, and in an isotropic liquid no long-range order remains. The term "mesophase" is applied to the intermediate phases that occur between the phase that has long-range order i n the largest number of degrees of freedom and the unordered phase, the isotropic liquid. Two broad categories of mesophases exist. "Plastic crystals" are characterised by loss of orientational order, while positional order of the crystal lattice i s maintained. Commonly occuring examples are solid hydrogen, ammonium halides, and the ferroelectrics barium titanate and rochelle salts. In the strongly ordered phase, the molecules in the crystal lattice have a well—defined orientation, whereas in the less ordered "plastic crystal" mesophase, they commute between several equivalent orientations. In "liquid crystals" the converse is true, orientational order is maintained while positional order is diminished. It follows from the definitions that the constituent molecules of these mesophases must be anisotropic, since one cannot speak of orientation of spherically symmetric molecules, and symmetry 18 prevents the loss of positional order in fewer than three dimensions. In fact, plastic crystal molecules are nearly spherical in shape, whereas liq u i d crystal molecules are strongly elongated. There exists usually a parameter such that a mesophase exists for some range of i t s values; the end-points of the range corresponding to transitions to more or less ordered phases. If this parameter is temperature, the material is thermotropic, i f i t is composition (i.e. concentration in a solvent), i t is lyotropic. Three categories of liquid crystals are distinguished. In canonic liquid crystals, long range positional order is lost in one dimension, in smectics in two dimensions, and in cholesterics in three. Canonics are regularly packed long rod-like molecules, where the centers of molecules are randomly distributed in one dimension (i.e. one dimensional liquids). Smectics form uniformly separated layers of two-dimensional liquids with oriented molecules; while cholesterics form three-dimensional liquids with well-defined (but not necessarily uniform) orientation. Compounds capable of forming the liquid crystalline phase; are made up of long molecules that are f a i r l y r i g i d along their long axes. These can be small organic molecules, giving rise to thermo-tropic phases, long helical rods, giving rise to lyotropic phases or associated structures of molecules and ions, which can be thermo- and/or lyotropic. The general pattern of small organic molecules i s two nearly coplanar para-substituted aromatic rings rigidly linked by a double or triple bond (A-B) and short, partly flexible chains, R shown in Fig. 2. The composition of central link A-B and the chains R are Fig. 2 Structure of common nematogens. 20 TABLE I. Composition of some common nematic liquid crystals. A-B R R' Mol. Wt. PAA -N=N-1 0 •CH3-0- -0-CH3 258.28 MBBA -CH=N- CH3-0- -(CH 2) 3-CH 3 267.37 EBBA -CH=N- C2H50- -(CH 2) 3-CH 3 281.40 BEPC -0-C-II CH3-CH2-0- -(C0 3)-(CH 2) 3- 358.39 21 given in Table I for some common materials. Examples of long h e l i c a l rods are DNA, tobacco mosaic virus and certain synthetic polypeptides; The typical lengths of these are in the hundreds of angstroms, with widths an order of magnitude smaller. Associated structures consist of long apolar aliphatic chains (10-20 CIL, groups) having an ionizable polar group at one end, occuring naturally in soaps and phospholipids. 22 2.2 Types of Liquid Crystalline Order The different types of liquid crystalline order can best be illustrated graphically. The position and orientation of the molecules i s shown by a dash. In canonics, long rods are hexagonally packed i n two dimensions, as shown in Figure 3. The long molecules can be h e l i c a l rods or groups of associated structures. The probability density of finding a molecule at point (x,y,z) is P(x,y,z) = P(x)S(y-dn- ^ - a)6(z - ^ ,'dm-g) , where 6 i s the Dirac delta function, d i s the repeat distance in the y-direction, m and n are integers and a and g are arbitrary constants. The average value of the orientation of the long axes of the rods i s — TT — 0 = - j - and <j) = 0. Long rods can be made up of associated structures by the following mechanism. The ionic head dissociates in a solvent, making the head hydrophilic and the long t a i l hydrophobic. End views of the two stable concentration dependent rod-like configurations are shown in Figure 4. In sm'e"Sti'csj, long range positional order i s retained in one direction, giving rise to a layered liquid structure. In addition, the long axes of the molecules have the same average orientation. The variations smectic A and smectic C are shown in Figure 5. The probability density of finding a molecule at point (x,y,z) i s P(x,y,z) - : P(x,y)6(z-md-a). The average orientation is 8 = 6fi and n i s a unit Fig. 3 Canonic order. ^ 3l zte. /p. •f* vr* \V \J> v$£ ^ VfT a) Excess solvent 4 Hexagonal rod systems . b) Excess solute . 111} mm M\\\\\\\I\\W lAIII/IUIUW '/Mil Mil WW ro X a) Smectic A. Fig. 5 Arrangement of molecules in smetics, b) Smectic C. 26 z X Fig. 6 The cholesteric configuration. 27 Fig. 7 Arrangement of molecules in the nematic mesophase. 28 vector pointing i n the direction of average orientation. Smectic A i s a special case of smectic C, where 6=0. The arrangement of molecules in the cholesteric phase is shown in Figure 6. In moving through the liquid in the z-direction, n (averaged in the x-y plane) is periodic in a distance X^. Here n = cosfr^-z+aj, n = sin ( 7^ z+a) and n = 0 . Note that there i s no x \Xh /' y \ \ ) z long-range positional order, and that n and -n are equivalent i f the molecules are simple rods. Arrangement of molecules in the nematic phase is shown in Figure 7. This i s a special case of the cholesteric phase, obtained when X^ 00. The unit vector n pointing in the direction of average orientation i s called the nematic director. In the cholesteric phase, the helical deformation is caused by chiral ( l e f t - or right-handed) molecules. The nematic phase i s a special case of the cholesteric where the constituents are either achiral or racemic. Because of their relative simplicity, nematics were chosen as the subject of this study. A l l subsequent discussion i s therefore restricted to the nematic mesophase. 29 2.3 The Order Parameter in Nematics In order to quantitatively characterize order in a liquid cry-stalline mesophase, i t i s desirable to have an order parameter whose value reflects the amount of order present. The molecular order manifests i t s e l f as the anisotropy of macroscopic properties of the mesophase. Ideally, then, one would like to relate the anisotropy of measurable physical properties to the molecular order through an order parameter. To gain this end, we calculate the polarization P for an array of spheroidal molecules, carrying permanent dipole moments along -> their long axes in an electric f i e l d F. Let a and a be the m„ m i molecular polarizabilities parallel and perpendicular to the long axis th respectively, and let be a unit vector along the long axis of the i molecule, shown in Figure 8. The contribution of the i molecule to the polarization i s p. = u n. + T *a T F (2.1) l p i ap mRv YO o a a where T „ transforms F into components parallel and perpendicular to the ag a long axis of the molecule. If 6 and <j> are the usual colatitude and longitude, -sincfi^ -cos<))^ 0 T „ = I cos9.cos(i>. -cos8 . sin* . -sin6. ag I I l i i i I sin0 . cosd>. -sinO . sin*. cos0.J l Y i I l i (2.2) 30 Fig. 8 Co-ordinate system used in the polarization calculation. 31 m a g m, 0 0 ma 0 0 0 x (2.3) and sin9. cos*, l l sin8. sin*. I l cos9. (2.4) T ! = T. and multiplication yields ag 3a T Q a T ag D Y<5 mgy a +3Asin 20.cos 2*. mA l l -3Asin20 . sin*. cos*. l l l 3Asin0 . cos0 . cos*. 1 X 1 -3Asin20 . sin*. cos*. 3Asin6 . cos0 . coscb. i i i i l l a +3 sin 26.sin 2*. -3 sin0.cos6.sin*. mx i i i i i -3Acos0.sin0.sin*. a +3Acos20. l l l m, l (2.5) where A = (a - a )/3. Letting a = (a +2a )/3, Eq. (2.5) can be m m . m mj_ re-written as T *a T = aS ,+A ap m. YO ao 3Y 3n. n..-6 . la 10 ao (2.6) and P i a = u n. +a6 0F +A p i a ap g 3n. n -6 1 ia lg ag (2.7) 32 The polarization P i s given by P = apF + u p < n. > + ApF 2<%(3n n -<T ) a a p xa p xa xp ap (2.8) where p is the number density, and < > indicates averaging over a l l mole-cules. The second and third terms in the r.h.s. of Eq. (2.8) constitute the anisotropic part of P; they vanish i f the material i s isotropic and a l l molecular orientations are equiprobable. We choose, therefore, the vector as the dipolar and quadrupolar order-parameters respectively. In a uni-axial material the orientational probability density function does not depend on the angle 9 i f the symmetry axis of the material coincides with the z-axis. Then, dropping the subscript i , n = <n. > a xa (2.9) and the symmetric traceless tensor (2.10) 0 n = 0 (2.11) R where R = <cos8>, and 33 ex 6 0 0 (2.12) where S = <Jg(3cos20-l)>. If R = 0, the bulk polarizability obtained from Eq. (2.8) is in agreement with published results (21). The two scalars R and S are thus sufficient to describe order in uniaxial materials; they have the value of unity i f a l l the molecules are aligned, and they vanish in the completely disordered isotropic phase. R distinguishes between molecules flipped end for end, whereas S does not. Note that R and S are not independent, the condition S > %(3R2-1) must be satisfied, (i.e. S = 0 -fa R = 0) and that negative and positive values of R describe the same physical configuration, whereas for S they do not. In fact, negative values of S indicate molecules perpendicularly aligned to the symmetry axis, randomly distributed otherwise. 34 2.4 The Landau Theory In the thermotropic nematic phase, as the temperature is i n -creased, the angular distribution of the symmetry axes of the molecules becomes more random, until the value of the order parameter(s) drops to zero catastrophically at some temperature T^. Experimentally, the diminishing order can be seen by measuring the anisotropy of such macro-scopic properties as the magnetic susceptibility, the dielectric constant and the refractive index. If the nature of the intermolecular interaction i s known, i t is possible in principle to calculate the temperature dependence of the order parameters. In practice, exact solutions for three-dimensional systems have not yet been obtained. A great deal of qualitative informa-tion can, however be obtained from the elegant Landau theory of phase-transitions (8). We start with the partition function Q, given by " 8 E " r 1 e - e « ( C > - k T i n « a ) ) d c -B r -BE(c) r = I e 1 =J n(e)e d? =J where B = ^ , ? is the order parameter, and JKc) is the density of states. The Landau free energy density F(£,T) i s defined by F(C,T) = |[E(c)-kTlnfi(c)] (2.14) where V is the volume. The basic assumption of the Landau theory is that F has a minimum value for some t,, say £ , and that F may be expanded near t, in a power series -^ c in c. Then 35 F(C,T) = F(X,T) + a 2 ( ? - C c ) 2 + (2.15) = 0 since 7^-c Eq. (2.13) yields = 0. Substitution of Eq. (2.15) into Eq. (2.14) and -evF(c , T ) r -eva (c-o2 Q = e / e d ? ' ( 2 < 1 6 ) and the average value of t, is given by -BVF( VT) - BVa ?(C-C r ) 2 < c>=^ / c e 2 C d? (2.17) As V o>, clearly <C> •> C , and the Helmholtz free energy density F is given by ,T r -eva (c-c ) 2 F = - ^ In Q = F( ? c,T) - ^ l n / e dc -> F(C C >T) (2.18) Eq. (2.15) can therefore be regarded as an expansion of the Helmholtz free energy about i t s equilibrium value; the expected value of the order parameter i s that which minimizes F(t,T). It is convenient to define a quantity AF = F(t,T)-F(t c,T); since Z,^ < £ < 1, we assume that AF can be expanded about C = 0. Then AF = a 2 ? 2 + a 3C 3 + a 4 ^ (2.19) 36 and a stable ordered phase w i l l occur i f AF is negative and is a minimum. Since for a low temperature ordered phase to exist a^ must change sign near the transition temperature, i t is assumed to have the form a^ = qCT-T^), where q and TQ are positive constants. The other a's are assumed to be constants. If the interaction i s predominantly the permanent dipole-dipole type, that i s , the pair interaction potential is not invariant i f one molecule i s turned end for end, the configurational free energy AF can be expanded in terms of R, and is given by AF = a„R 2 + a.Rh + a,R6 (2.20) 2 4 6 where we have chosen somewhat arbi t r a r i l y to retain only the f i r s t three terms. There are no odd powers of R, since +R and -R describe physically equivalent orientations. At the phase transition, AF = 0, R = Rc and T = T . There are two solutions which minimize AF; i f a. > 0 c 4 and a, < 0, then R = 0 and 6 c R2 = (^j (T c-T) (2.21) where T = T_.. The details of the calculations in this section are given c 0 • b in Appendix C. The free energy AF and R are shown in Figure 9. If and a. < 0 and a, > 0, then R = / a4 4 6 c v - — 2 a6 R2 = I R2 + k ' /T +-T (2.22) 1 v c V \ I I a) the Landau free energy b) the order parameter R as a function of temperature Fig. 9 Free energy and order parameter R; continuous transition. 38 a 2 where k' 2 = and T = T_ + -: . Furthermore — = 0 for 1 3a^ c 0 ^fJEig 8R + a4 T < T = T„ + ; a metastable ordered phase can therefore persist, - c 0 3qa, v r 5 o -„+ = -y R . Similarly, T o e + 9 on heating, up to a temperature T = T with R Z 9AF ~ c since for small R, — — = 2a0R, a metastable unordered (R = 0) phase can exist, on cooling, down to a temperature T = TQ with R 2 - 1 R2 T 0 2 E c The free energy AF and R are shown in Figure 10. If the interaction i s predominantly induced dipole-dipole type, that i s , the pair interaction potential is invariant i f one mole-cule i s turned end for end, AF can be expanded in terms of S and is given by AF = b 2 S 2 + b 3 S 3 + (2.23) where, as before b^ = q(T-Tg). The salient feature here is the inclusion of odd powers of S in the expansion; b^ cannot be zero since negative and positive values of S correspond to physically different configurations. At the transition, AF = 0, S = S and T = T . There are two c c solutions which minimize AF; i f b~ > 0 and b, < 0, then S =0 and 3 4 c 3b„ (T -T) (2.24) where T^ = T^. This solution corresponds to a local minimum only, however; energetically an ordered phase with S = -1 w i l l always be favorable. Be-haviour of the type given by Eq. (2.24) is therefore not expected to occur. Molecular symmetry thus forces the second solution, where " b3 b 3 2 b Q < 0 and b. > 0, then S = r r ^ , T = T„ + -j^- and 3 4 c 2b. c 0 4qb. Fig. 10 Free energy and order parameter R; discontinuous transition. 40 S = T S + k! / T + - T (2.25) 4 c 2 V c + qb? where T = T N + ,,0 , and k i 2 = . c 0 32qb^ 2 2b^ As previously, a metastable ordered phase can p e r s i s t , on heating, up to a temperature T*, with S 3 + = -r S ; and a metastable unordered (S = 0) T 4 c c phase can e x i s t , on cooling, down to a temperature T = TQ with 3 S = T S . The free energy AF and S are shown i n Figure 11. 0 For most nematic l i q u i d c r y s t a l s , the predominant i n t e r a c t i o n i s assumed to be the anisotropic Van der Waals i n t e r a c t i o n , and the constant b^ i s known to be negative. Thus, the temperature dependence of the order parameter i s expected to be s i m i l a r to that of Eq. (2.25); i n f a c t , experimental measurements (9) i n d i c a t e extremely good agree-ment with Eq. (2.25). The presence of metastable phases near the c r i t i c a l point whose s t a b i l i t y i s enhanced by sample impurities (10), however, make accurate comparison between theory and experiment d i f f i c u l t . In a d d i t i o n to the nematic-isotropic t r a n s i t i o n , i t i s i n t e r e s t -ing to consider the liquid-vapor t r a n s i t i o n i n a f l u i d . Using the s o - c a l l e d l a t t i c e gas model, we consider a volume V, divided into 2N i d e n t i c a l c e l l s , containing N molecules. If a l i q u i d and a vapor phase coexist i n V, then we denote the number of occupied and unoccupied c e l l s i n the l i q u i d by and ^ r e s p e c t i v e l y ; we assume that each c e l l can e i t h e r be empty, or contain only one molecule. If the law of r e c t i l i n e a r diameters holds, i . e . N^+N^ = N, then the number of occupied and unoccupied c e l l s i n the vapor phase i s given ^ and r e s p e c t i v e l y . The p r o b a b i l i t y of a c e l l being occupied i n the l i q u i d or a c e l l being Fig. 11 Free energy and order parameter S; discontinuous transition. 42 N l empty in the vapor is ; whereas the probability of a c e l l , any-1 2 ± where in V, being either empty or f u l l i s The deviation of the probability in either phase from i t s average value of is N N -N 1 1 1 2 Y = „ ,>T - v = — ™ — • If there is only a single homogeneous phase ' Nj+N2 2 2N existing, then y = 0; thus y may be regarded as the order parameter. P L " P V 2NX 2N2 N In terms of densities, y = —; , where p = — ~ , p = — —and p = — . P J-i V v '.V *— • c Since +y and - y describe the same physical configuration, the expansion of the Landau free energy in even powers of y is appropriate. If the coefficient of the second term in the expression is positive, then Eq. (2.21) yields immediately P 2 - p v = C ] L ( T c - T ) P (2.26) where 3 = 2" a n c * = p ^ ("2a ) ' 1 1 1 v i e w °^ t^ i e simplicity of the model, Eq. (2.26) gives a remarkably accurate description of the liquid-vapor transition. 43 CHAPTER 3 MEAN FIELD THEORY 3.1 One-body Pseudopotential i n the Mean F i e l d Approximation For a c l a s s i c a l f l u i d of N i d e n t i c a l molecules contained i n a volume V, the configurational p a r t i t i o n function Q^, n - p a r t i c l e d i s -t r i b u t i o n function p^ n^ and n - p a r t i c l e c o r r e l a t i o n function can be written j y*exp (-3UM) dy,... dy„ Z. <N T T N! (4ir) 'N N N N! (4TT) N (3.1) P ( " > ( v r . . ^ ) -e x p ( - e U N ) d Y n + 1 ^ / Y K (N-n)'.Z N (3.2) i S i ' a ) < ? i > (3.3) where U N i s the N- p a r t i c l e conf i g u r a t i o n a l p o t e n t i a l energy, and 9,^ denote, r e s p e c t i v e l y , the p o s i t i o n and o r i e n t a t i o n of molecule i , 6 = 1/kT and dy. = dr.dfi. = dr. sine ,d9 .deb. . If the molecules are i n t e r a c t i n g pair-wise, then UN = , N N z 1=1 j= 1 1 , J 1 3 (3.4) 44 where W. .(y.»Y.) is the interaction energy of the pair i , j , and the 1 • 3 1 J potential energy of say, molecule 1, e1 is given by Eq.(3.5)by i t s average value; a l l thermodynamic properties of the system can then be obtained from the resulting one-body pseudopotential E ^ ( Y ^ ) . — — Since the system is a f l u i d , e i / ^ l ^ = e x ^ x ^ a n c^ N uAT = y E.(a.) (3.6: 1=1 From Eq.(3.2), (3.5) The mean-field approximation consists of replacing the sum in (3.7) where p = N/V, exp(-ge^ (n)) (3.8) and self-consistency requires that the average value of U N (3.9) 45 In most mean-field theories of the nematic state, molecular interactions consist exp l i c i t l y only of anisotropic long-range attrac-tive forces (13),(15),(25). Repulsive interactions are taken into account implicitly through assumptions made about the positional distribution.of par-ticl e s in the calculation of the one-body pseudopotential. e.(^). These assumptions, (the most frequent one being that of spherical symmetry), are usually not well j u s t i f i e d . It i s possible, however, to obtain an exact expression for e(fi) in the mean-field approximation, since the molecular distribution is consistently determined by the form of the interaction potential Id „. 1 ) » The average value of e(fi) i s given by <i<*i> - T / V I P<i)flJ d T i • < 3 - 1 0 ) Substitution from Eqs. (3.3) and (3.7) yields ^ i ( V = i p / * w i , j g ( 2 )(vY\)faydY. . (3.ii) The pair-correlation function can be evaluated by considering pairs of particles; the potential energy of the pair of molecules, say 1 and 2 is given by £ 1 , 2 ( V - - V = W 1 , 2 + I j . W i f j + l J V j • ( 3 ' 1 2 ) 46 As before, the mean f i e l d approximation consists of replacing the sums i n (3.12) by t h e i r average value; then, as N -> °°, ^ t f i . f y " W l,2 + 5 1 ( V + W (3.13) (2) . - -g (Y-L>Y2) c a n n o w be evaluated, since UJJ = E X 2 + E3 4 + 'N-1,N 2 " 1 N! exp ( -3e 1 2 ) / exp(-3e. . )dYi,dy N (N-2): y e x p ( - f - ->- ->• ie. JdY.,dy. i»3 i J_ pf(n 1) Pf(n 2) N(N-l)exp(-g e i 2) '-£(tt1)f(a2)J'exV(-£e1 2)dy1,dY2 (3.14) In p r i n c i p l e , f o r a given i n t e r a c t i o n p o t e n t i a l W 9 , the one-body pseu-dopotential e^(fi^) can be obtained from Eqs. (11) and (14). As the r e s u l t i n g i n t e g r a l equation appears i n t r a c t a b l e , the assumption i s made that g^CY-^Y^) may be approximated i n Eq. (11) by i t s average value g ^ ( r , r 2 ) where denotes averaging over molecular or i e n t a t i o n s . L e t t i n g r = r r r 2 , i < 2 ) ( ? ) -= y g ( 2 ) ( Y l 5 Y 2 ) f (n x)f (n2)dn1dn, ^ 2 x p ( - 3 e 1 2 ) d ^ 1 d f i 2 (3.15) 47 -(2) where K is the normalization factor such that as r + <», g ->• 1 given by _ p 2/exp (-ge ?)dY,,dY9 / * _ / * _ K = — - ±*=- = / exp(-Be-.)dfi / exp(-Be,)dft N(N-l) J 1 V (3.16) as N -*- 0 0 48 3.2 Pair-Correlation in a Hard Spheroid Model In order to take steric repulsion of the molecules into consideration ex p l i c i t l y , the interaction potential W „ is assumed to consist of a short-range repulsive part U(Y^,Y2) a n d a l° n§ r a n § e attrac-tive part V(Y^,Y2) # ^ E N N J=l D 1 . J ( ? 1 . ? J > + V 1 . J ( ? 1 ' V (3.17) and E1,2^ Y1' Y2' ) = U l 2^ Y1' Y2^ + e i ^ i ^ + E2^2' ) (3.18) where 2 ^ \ ^ 2 ^ i s a s s u m e < l t o ^ e contained in and e^. If the mole-cules are represented by cylindrically symmetric "hard" surfaces, whose orientation i s given by Q, a convenient form for IL. _ i s x, z U (r,n ,fl ) = lim -ln(y + - tan 1a[r-d. r)]) (3.19) x,z x z z TT l,z l j where r = r r i s the intermolecular vector and d „ i s the distance between x, z molecular centers when the surfaces are i n point contact e x t e r n a l l y . ( 2 ) The anisotropy of the p a i r - c o r r e l a t i o n function g ~ (rr) may be obtained from D(r), defined to be the e f f e c t i v e hard core diameter given by D(r) = .,/ ( l - g ( 2 ) ( r r ) ) d r ; (3.20) 49 — ( 2 ^ * in order to simplify the notation, henceforth we shall write g (rr) as g(rr). We assume that there exists a radial scaling transformation r = r'f(r) which renders g(rr) isotropic; then, as in Section 1.3, g(rr) = g(r'f(r)r) = S Q ( r ' ) . From Eq. (3.20), i t follows that = / (l-g(rr))dr = f(r) / J 0 J Q D(r) = / ( l - g ( ) ( r ' ) ) d r ' (3.21) -1 and f(r) = ( 2 R Q ) D(r) where R ^ , the effective hard-sphere radius, is assumed constant. Thus, i f D(r) is known, the average values of expressions involving two-particle interactions can be evaluated by simple scaling. Substitution of Eqs. (3.15), (3.18) and (3.19) into Eq. (3.20) yields d * x (3.22) D(r) = lira K / / tan a [ r - d C f i ^ n ^ r ) ] ) exp (-B [e 1 (S\)+e2 ( ^ > d ^ ± d ^ 2 and integration over r yields D(r) = K J d 1 2 ( ^ , ^ 2 , ^ )exp(-B[i 1(n i)+i 2(^ 2))dfi,df2 2 = d(r) (3.23) 50 In this model, the molecules are represented by hard spheroids, whose equation i s r 2 ( l - e 2 ( r -n)' 2) = R2 (3.24) m mm U A th where r = r r is a position vector originating at the center of the m m m m ° ° molecule, n^ is a unit vector along i t s symmetry axis and e is the eccen-t r i c i t y . Unfortunately, i t has not been possible to obtain a closed-form expression for the distance of closest approach d^ 2 l n spite of considerable effort; consequently bounds on i t must be considered. If d^ +(r,fi^) is the length of the normal projection of surface 1 onto a line through i t s center in the direction r, and d^_(r,fi^) i s the length of the intercepted segment of the same line, as shown in Figure 12, then clearly ^ ^ ( r , ^ ) + |d 2_(r,Q 2) < d < | d 1+(r,^ 1) + \ d 2 + ( r , f i 2 ) (3.25) Letting d +(r) =jf d +(r,^)f(fi)dft , d_(r) < D(r) < d +(r) (3.26) from elementary geometry, as shown in Appendix D, 2R 2 d (r,n) = A T-pr and d, (r,fi) = 2R (1 + ( r - n ) 2 ) 1 / 2 . - 2/ \2\1 £ + U 1-e^ (l-e z(r.n)^) (3.27) 51 Fig. 12 The geometry of closest approach. 52 Since d_(r) > d_((r«n)2) and d +(r) < d +((r-n) 2) , d_((r-n) 2) < D(r)<< d +((r-n) 2) (3.28) Eq. (3.28) can be expressed in terms of the order parameter S^g as follows: ( r ' n ) 2 = r a ( n a n B ) r £ 2 "T R z a -T (3n n -6 ) 2 a g a3 rB + 3 (3.29) and (3.30) Substitution of Eq. (3.30) into Eq. (3.27) f i n a l l y yields R Q ( l - f 2 ( l + 2 r a S a 6 r g ) y i / 2 <D(r)<2R n[l + 3 T i = 5 Z 5 - ( l + 2 r o S a e r B ) 11/2 (3.31) It i s worth noting that there have been no specific assumptions made regarding the attractive part of the pair-interaction potential in obtaining Eq. (3.31), and that in a condensed phase, D(r) may be thought of as the average repeat distance of molecules in the direction r. 53 The bounds d ^ r ^ S ^ r ^ ) on D(r), given in Eq. (3.31) are shown in Figurel3 for r in the z and x (or y) directions as a function of the order para-meter S = Sz, for the case when the length to width ratio of the hard core is 2:1, (i.e. e 2 = — ) , and the symmetry axis of the material is in the 4 z-direction. As can be seen from Fig. 13, for this model of hard spheroids, an essential feature of the ordered phase is the anisotropy of the mole-cular distribution. In order to avoid the necessity of performing two sets of calculations, one for each bound, we assume D(r) = | ( d _ ( r a S a e r g ) + d + ( r o S r ) ) . (3.32) This choice of D(r) preserves the essential features of the molecular distribution; in fact, i t i s plausible that i t is the exact effective hard core diameter for some elongated hard core potential. To simplify the notation somewhat, in analogy with tetragonal crystals we shall denote D(r) by "c" i f r is in the z-direction, and by "a" i f r is in the x or y directions for the case when the symmetry axis of the material i s in the z-direction. Eq. (3.31) suggests that a more general treat-ment might be to expand D(r) in a power series in r S »r_; in order to a ag g avoid introducing additional parameters, however, we continue with the spheroidal model. The local f i e l d anisotropy tensor n „ can now be evaluated; ag the eccentricity of D(r) and hence of f g ( r ) is 9 a 2 eg = 1--H2 (3.33) 54 Fig. 13 Bounds on the effective hard-core diameter D(r). 56 where, from Eqs. (3.32) and (3.31) c = R„ e2 1/2 ( 1 - | - (1+2S)) ± / Z + (1 + 3(l-e 2) (1+2S)) -1/2 (3.34) and a = R„ (1 - y- ( 1 - S ) ) 1 / 2 + (1 + 6 2 -1/2' (1-S)) 1 U 3(l-e 2) (3.35) The value of n i s shown in Figure 14 as a function of S for several zz values of the molecular eccentricity e. Values of n obtained from zz refractive index measurements of PAA(26) are also shown; giving reasonable agreement with theory. The small deviation of the experimental values from the linear S-dependence predicted ,by the theory may in part be caused by a decrease in the molecular eccentricity e as a function of temperature due to molecular vibrations. Once f (r) = (2R„) 1D(r) is known, the one particle pseudopotential s u e^(fi^) can be evaluated from Eq. (3.11) as follows. By noting that, for a general hard-core repulsive potential U, U(Y1>Y2)g (vV d Y2 = ° (3.36) (2) where g is given by Eq. (3.14), Eq. (3.11) yields ^1 (V = \ P J v ^ C ^ ^ g ^ ^ V Y ^ f W ^ d ^ . (3.37) 57 Fig. 14 The local f i e l d anisotropy tensor as a function of the order parameter. 59 Replacing g^Cy^Y..) by g(r), where r = = | v 1 ^ j ( n 1 , n j ; r ) g ( r ) f ( n j ) d n j d r . (3.38) Letting dr = r 2drdft^ and r = r ' f s ( r ) , ^ ( f i ) = | p y V l j j ( n i , n j , r ' f § ( r ) r ) g 0 ( r , ) f 0^ )^ .0 r) r' 2dr'dft dtt. r J (3.39) o 60 3.3 Dependence of Density on the Order Parameter The equation of state can be obtained from the partition function, which, in the mean f i e l d approximation is given by IN e x p ( - B e ( f t ) ) d y N! (4TT) N (3.39) The free volume seen by one particle i s V,. = / dr = V-NV , where V , f J m m the volume effectively occupied by one molecule is V m = — / (l-g(r m 23 h J ))r2drdft. (3.40) where h is the packing fraction. Since, from Eq. (3.11) e(fi) is propor-tional to the number density, we may write e(ft) = -pz(£2). The pressure P is given by P Then and the free energy F is given by F = - — InQ. P = 3 V InQ N 2 z ( n ) e x p ( 3 ^ z ( Q ) ) d Q V-NV V m exp(B f z(fi ) ) d f i (3.41) and l V l - Pvm) = ^ \ p 2 z / p z (3.42) 61 Far below the liquid-vapor transition (kT << pz) in the condensed phase at low pressure (P << p 2z) Eq. (3.42) yields p = Y~ - + (3.43) m z We suppose most of the volume dependence at T ~ to reside in V^; although i t i s clear from Eq. (3.41) that z may change considerably across the transition. As a f i r s t order approximation we let p * = V , then making the transformation r = r ' f ( r ) , Eq. (3.40) yields -1 ^ J ( l - g 0 ( r ' ) ) r ' 2 d r ^ f 3 ( r ) d ^ (3.44) Since f (r) = D(r)/2Rn s U j D3(r) P 1 = 2^h / D i ( i ) d f i r • ( 3 * 4 5 ) If D(r) is a spheroid with eccentricity e 2 = 1 r- , then D(r) = g a(l-e 2cos 2G and g r q I S i n D Q H o -1 ira 3 I r r ira^c /s 6 d9(l-e 2cos 2G 12h / 2 2Q ^3/2 6h ' (l ^cos^e ) g r (3.46) where a and c are given by Eqs. (3.34) and (3.35). In order to further simplify calculations, we replace a and c in Eq. (3.46) by a linear app roximat ion, 62 a = A - SS (3.47) and c - A + 26S (3.48) 2 R 0 2 R o where A = —^— (K+2) and <5 = — ( K - 1 ) and K is the molecular length to width ratio K = 1 . This i s equivalent to letting v^e~ 2 D(r) = 2R 0 1 + (1 + 2r S .r.) 3 a ag g (3.49) The exact values of a and c and the linear approximations are shown in Figure 15. The volume V(S) i s given by 4TTR! H n T "l r 1 V(S) = •^zr~ N (K+2) - (K-1)S 2 (K+2) + 2(K-1)S (3.50) o l n |_ or V(S) = -V(0) (3.51) The thermal expansivity i s obtained from Eq. (3.51), and 63 Fig. 15 Approximations to the effective hard-core diameter. 65 3 '~+— From Eq. (2.25), near the transition S = S £ + k 2 / Tc-T , and 1 3V = J K-lV V 8T \K+2/ ,»2 3S 1 + 4k! ( t;- t )" (3.53) Arrott and Press (10) have shown that experimentally obtained values of the expansion coefficients of MBBA obey such a relation. Their experimental values were fitt e d to (3.54) - 1/2 with the result that G = 0.56 ± .1 and Z = 0.435 x 10 3/°C .We can estimate Z from Eq. (3.42) where we take K = 1.6 as suggested by local f i e l d results, and = 0.08, then M 2 k-^K+2 j k2 3/o 1/2 „S = 1.512 x 10 3/°C 2 c The value of k^ was obtained from published values of Landau expansion coefficients for MBBA (11)• In view of the approximations made in obtaining Eq. (3.53), the agreement is considered satisfactory. The discontinuous volume change across the nematic-isotopic transition is given approximately by 66 For K = 1.6 and S c - 0.3, Eq. (3.55) yields a volume change of 0.75%. If a and c given by Eqs. (3.34) and (3.35) are used instead of the linear approximation, then the volume change is further reduced to 0.25%. Experimental results (10) for MBBA suggest 0.13% as a minimum estimate, the existence of a two-phase region in the neighborhood of the transition makes accurate determination of the discontinuous volume change d i f f i c u l t . 67 3.4 Dipole-Dipole Interaction The interaction energy between two point dipoles un_^ and un^ separated by r = rr is V.. = - 4f3(n. ,r)(n.'r) - n.-n.l , (3.56) iJ r 3 |_ 1 J i j j substitution into Eq. (3.38) yields, for a spherical sample of radius R i f ( 3 R R R - < 5 ft)N-R £ 1 ( V = " I p y 2 n l a l " 3 3 g(r)f(fi^dfljdr . (3.57) Performing the transformation r = r ' f g ( r ) yields R / f s ( r ) g 0(r')dr' 0 x (3r r -6 Q)dfi . . a 3 a3 r (3.58) where the f i r s t integral i s just the dipolar (vector) order parameter n^ of Eq. (2.9), and the second integral i s the local f i e l d anisotropy tensor n of Eqs. (1.14) and (1.16). Then ^1 ( V = - 2 l ^ 2 n l a V a 3 * ( 3 ' 5 9 ) If the symmetry axis of the f l u i d i s in the z-direction, then Eq. (3.59) becomes, on dropping the subscript 1, 68 e(fi) = -2Trpy 2cos0Rn (3.60) zz and the quantity 2irpyRrizz= H^ . may be thought of as the effective molecular f i e l d . The orientational probability density function £(Q) is obtained by substituting Eq. (3.60) into Eq. (3.8), and from Eq. (3.9), the self-consistent equation for R is I r n cos6exp(gyn' cos8)sin8d9 R= £ (3.61) J 0 exp(pyrLcos6)sin6d6 The quadrupolar order parameter S is similarly given by I. L 7r(3cos 26-l)exp(gyH,cos6)sined8 S = (3.62) J 0 exp(ByH„cose)sinede E The integrals are evaluated in Appendix E with the results that R=-L(guH E) (3.63) and S - 1 " ByfT ^ « E > E (3.64) 69 where L is Langevin's function, i-(x) = coth(x) - — . The sensitivity of the solution to the anisotropy of the molecular distribution i s apparent. If, as is frequently assumed, the distribution is isotropic, then T) = fL = 0, and R = S = 0. If, as assumed by Born (12) and later zz E • by Chandrasekhar (13) the distribution i s anisotropic but independent of the temperature, then H = uc.R where c, is a constant and the solu-E I 1 tion for R is identical to the magnetization in the Weiss theory of ferromagnetism. i-(x) may be expanded about x = 0, and letting (3.65) and, for small R, while (3.66) (3.67) We have shown, however, that the anisotropy of the distribution is temperature dependent; as can be seen from Figure 14, n - c^S where is a constant i f changes in the density are neglected. Since, for prolate molecules c „ < 0 , R = S = 0 ; from Eq. (3.41) i t is clear that 70 for a given R a n t i p a r a l l e l alignment i s e n e r g e t i c a l l y favorable. Oblate molecules may be considered by replacing the hard-core (prolate) e c c e n t r i -- e 2 9 0 9 c i t y +e^ by ^_ 2 where e^ i s the e c c e n t r i c i t y of an oblate spheroid. 6 0 y 2 c 2 Then n > 0 hence c„ > 0, and l e t t i n g H = y„c 9RS and T = .. „ y i e l d s , from Eq. (3.64) 2 ~ 2 / 1 - f (3.68) c where only + gives a p h y s i c a l l y meaningful s o l u t i o n , and R = L L T T "1 1/2 12_c RS = S ' (3.69) where the r e s u l t R 2 - S has been obtained by numerical means. 71 3.5 London-van der Waals Interaction If molecule i has a dipole moment V^11^ caused by a spontaneous charge fluctuation, the resulting electric f i e l d of a point r = rr is given by y o r /*. s\ s\ I = 7J [3(n.-r)r-n.J. (3.70) If molecule j at r has a polarizability a„ along i t s symmetry axis and a x perpendicular to i t , and i f a„ >> a x, then the induced dipole is ( ^ " n j ) a n n j • The interaction energy of the two dipoles is o Ct || u V. . = -a„(E-n.) 2 = - -^p- (3(n.-r)(n.rr) - n.-n.) 2 . (3.71) The same form of is obtained from quantum mechanical calculation of the dipole-dipole contribution to the dispersion energy (14). Substitution into Eq. (3.38) yields for the one-body pseudopotential ± r ( 3 V g - % ) ( 3 r Y V V e l ( f i ) = " 2 P Ps a" nla nl Yy n j e n j 6 r 6 (3.72) ;(r)f (^)dft.dr Performing the transformation r = r ' f g ( r ) yields 72 n / n n f (Q. lYJ jg j5 J )<K2. R/f (r) g . C r ^ d r ' ( 3 r r - 6 J ( 3 r r - 6 ) S a e a? y 6 ^ 6 dft ( 3 . 7 3 ) 0 r , l t fs ( r> The f i r s t i n t e g r a l can be expressed i n terms of S „, and in t e g r a t i o n over r' y i e l d s , f o r large R, P y s a " n l a n l Y 48R3 L 3 ( 2 S B 6 + f i 3 « ) / ( 9 V g V 5 - 3 r a r g 5 Y 6 - 3 r Y r 6 6 a g + 6 a B 6 Y 6 ) d ^ f3 (;) ( 3 . 7 4 ) If there exists rotational symmetry about the z-axis, then, as shown in Appendix F, Eq. (3.58) becomes e 1(^ 1) = *pvl f\ 2 2 / 3 c o s 2 6 -1 \ / (^3cos2e i-5 ) ( 3 . 7 5 ) (3cos 26.,-l) + S „ (9cos49 -8cos29 +1) 2 r r sin6 d9 r r f3(r) 73 If a spherically symmetric molecular distribution i s supposed, then f Q ( r ) = 1 and the usual Maier-Saupe pseudopotential i s obtained; dropping the subscript 1, e(fi) = S(3cos 26-l) x constant . (3.76) If the molecular distribution i s anisotropic, then, as before D(r) a 2 2* N~1/2 , 2 . a1 f ( r ) = = U-el*0*2^ w h e n e g = 1 " c 2 • Furthermore, the density p is given by p = ^\ , and Eq. (3.75) becomes, 7 T 3 . C on omitting the terms not containing 6^ from the integrand 2y2a„h (3cos 26-l) . ^ --"ife —J d-eWer)3/2 0 2 (3cos 26 r-l) + S(9cos 1 + e r-8cos 2 e r+l) s i n e d e . (3.77) The dependence of e(fi) on S may be condensed into a function <j>(S), then e(«) = - y(3cos 20-l)*(S) (3.78) and S i s obtained self-consistently from Eq. (3.9) Jo \ ( 3 c o s 2 e- 1) e xp[f <t'(S)(3cos 2e-l)] sine J exp ^ | <(((S) (3cos 20-l)] sinGdO dO — . (3.79) 74 Eq.(3.79) cannot be solved a n a l y t i c a l l y ; the temperature dependence of S has been obtained using numerical methods. The method i s described i n Appendix G, and the r e s u l t s are shown i n Figure 16. The e f f e c t of the elongated hard core i s to make the t r a n s i t i o n more abrupt; the behavior of the system v a r i e s continuously from the Maier-Saupe s o l u t i o n (15) to Onsager's r e s u l t for hard rods (23) as the length to width r a t i o of the repulsive p o t e n t i a l i s increased. Since t h i s behavior i s not i n agreement with experimental r e s u l t s , a more r e a l i s t i c i n t e r a c t i o n p o t e n t i a l should be considered; a model i n which the molecular p o l a r i z -a b i l i t y perpendicular to the symmetry axis i s not neglected may predict more ph y s i c a l behavior. 75 Fig. 16 The effect of the anisotropic hard-core on the order parameter in the mean f i e l d approximation. 76 CO tD vfr CM 77 CHAPTER 4 PURE FLUIDS ~ EXPERIMENTAL 4.1 The Lorentz-Lorenz Relation In an isotropic f l u i d , the pair-correlation function g(r) is spherically symmetric; the local f i e l d anisotropy tensor therefore vanishes.and Eq. (1.29) reduces to the well-known Lorentz-Lorenz relation ^ = i p M • (4.1) n2+2 The Lorentz-Lorenz coefficient L i s given by 4TT M where a is the effective molecular polarizability, M is the gram molecular weight"-TandnAv ils Avogadro's -number''and p M isttfte mas S= T density. It is of interest to investigate:the. validity^O'fisEqt'(4 .l).qacr.osspthetliquid4-vapor_transition, since optical techniques are commonly used (16,17)-to measure the order para-p. -p > £ c v meter — near the c r i t i c a l point. The Lorentz-Lorenz coefficients p c for sulfur hexafluoride and germane have been measured together with their c r i t i c a l constants. Refractive indices and density are both measured in the same experiment, yielding values of L accurate to 0.05 per cent for densities near c r i t i c a l . The method u t i l i z e s a prism-78 shaped high-pressure c e l l which can be removed from a temperature con-t r o l l e d holder and weighed on a p r e c i s i o n balance. The c e l l i s equipped with a needle valve which allows the high pressure gas to be bled out i n steps, r e f r a c t i v e indices are thus measured as a function of weight and hence density. 79 4.2 Details of the Experiment Optical equipment used in this experiment i s schematically illustrated in Figure (17). A laser beam, rendered uniphase and c o l l i -mated by a beam expanding telescope and pinhole f i l t e r , traverses a prism-shaped sample vessel and is reflected by a mirror into a telescope. The prism is oriented with i t s axis vertical; the beam is refracted horizontally through an angle which depends on the index of refraction of the flu i d in the^prism^vessel. The beam is reflected into an auto-collimating telescope (Davison model D275) by a differential-micrometer driven mirror (Lansing Research Corp. Model 10.253). The high pressure sample vessel is shown in Figure 18. The body i s made of aluminum with a brass needle valve at one end. Two sapphire windows, clamped in place by flanges, form a prism-shaped region at the end of the vessel. The space between the windows is kept to a minimum i n order to keep the light path length small. The long cylindrical portion of the vessel provides a reasonably large volume of f l u i d . Experimental error decreases with increasing volume of the vessel, but the precision balance available for this experiment had a limit of 200 gm. Therefore, the sample vessel was designed to ob-tain a large volume of f l u i d but under the restriction that the mass of the vessel and contents not exceed the limit of the balance. The prism angle between the sapphire windows was measured to be 20.088°. The relationship between the refractive index, n, of the fl u i d and the angle of deviation, 9, through the prism vessel for a ray incident normally to the front face of the prism i s derived in Appendix H and i s HE-NE LASER IRIS Fig. 17 Schematic illustration of optical equipment'. THERMOSTATIC HOUSING ADJUSTABLE MIRROR TELESCOPE 81 PRISM -ALIGNMENT PIN CELL BODY V'ALVL St AT •FILLING HOLE VALVE 18 Drawing of sample c e l l . 82 n = n + (sinO-sine )[B+C(sine+sin6 )] a a a where n is the refractive index of air, 6 is the angle of deviation a a for air in the sample c e l l , and B and C depend on the c e l l geometry. The angle of deviation through the prism is measured by adjust-ment of the micrometer screw on the mirror mount. The mirror is ad-justed until the beam is centered on the cross hairs of the auto-collimator, the reading of the micrometer is then recorded. The procedure is repeated ten times and the readings are averaged. Five readings are taken with ,the micrometer screw turned clockwise and five C.C.W. This is done in order to eliminate backlash in the screw and in order to eliminate the effect that would occur i f the amount of backlash were not a constant amount, but depended on the particular orientation of the screw. A reference angle is obtained when the c e l l is removed for weighing. This reference angle is measured on the micrometer screw each time the c e l l is removed in order to avoid any problems due to alterations in the alignment of the optical system during the course of the experiment. The zero angle used for calculation of the deviation angle is obtained with the c e l l containing air at N.T.P. The relationship between the angle of deviation and the read-ings of the micrometer screw is established by calibrating using a 50 line per inch Ronchi ruling in place of the sample vessel. The micro-meter readings for sixty-five orders were measured, and a relationship between sine and micrometer screw reading was established. The relationship i s linear except for a small correction which i s nearly 83 negligible except for larger angles. (It should be noted that the maximum angle of deviation encountered in the experiment is only 3 degrees.) The measurement of the refractive index described above yields two values, n^ and n^, when both liquid and vapour are present in the c e l l . The density and temperature of interest i s in the single phase region at,the coexistence curve boundary. The sample vessel was i n i t i a l l y f i l l e d to an average density p' such that i t was nearly a l l l iquid at room temperature with only a small amount of vapour present. The sample vessel was then placed in the thermostatic housing at temperature T^. If i s below T'^ both liquid and vapour are present, and n^(T^) and n^(T^) can both be measured (see Fig. 19). The temperature was then increased to resulting in an increase in the fraction of liquid present. The values of n^Cl^) and n C ^ ) were then measured. The temperature was then increased to T^ and measure-ments of n repeated. As the temperature approaches T', the meniscus rises i n the vessel and i t becomes impossible to measure n . For v temperatures above T', measurements of n result in obtaining n(p',T). n(p',T) is almost independent of T for T > T' and i t is easy to extra-polate to obtain h(p',T') which is the value sought in this work. Following the carrying out of the above procedure for obtaining n(p',T') the sample vessel was removed from the thermostatic housing and weighed. The density was obtained using the known volume and the mass of the vessel when evacuated. 84 Fig. 19 Coexistence curve on a temperature-density plot il l u s t r a t i n g procedure for obtaining data. 85 86 The density was then decreased by bleeding out a small amount of f l u i d with the needle valve. The procedure was then repeated to obtain a value corresponding to n(p",T"), etc. As the c r i t i c a l point is approached at the top of the co-existence curve, the large compressibility of the f l u i d results in large density gradients. Therefore, i t i s d i f f i c u l t to obtain n at the coexistence curve boundary near the c r i t i c a l point. The values used for calculating the Lorentz-Lorenz coefficient, L, are those obtained at temperatures shown in the shaded area in Figure 20\ The measurements on the vapour side of.the coexistence curve are made in a similar way. except the meniscus level decreases with increasing temperature and only vapour is present for temperatures above the coexistence curve. The values used in obtaining L are ob-tained in the shaded region. The measurements were continued for densities for which the coexistence curve is well below room temperature and L for these points correspond to n(p,T ). c r room The mass of the • evacuated vessel was obtained by weighing i t after, evacuating i t . The volume of the vessel was obtained by f i l l i n g i t with d i s t i l l e d water and weighing i t . Small corrections were made for the change in volume of ,tfte>cell with temperature. Small corrections were also made for the change in volume of the c e l l with pressure a l -though the accuracy.of this correction is less reliable because of the di f f i c u l t y of estimating the change in volume with pressure for the oddly shaped vessel. 87 Fig. 20 Temperature and density region of which data points are obtained. The existence of large density gradients creates d i f f i c u l t i e s in obtaining data at the coexistence curve near the c r i t i c a l point. 88 89 4.3 Temperature Control The. temperature co n t r o l systems used both i n the pure f l u i d and the nematic l i q u i d c r y s t a l experiments consist of two b a s i c parts; an outer housing, whose temperature i s c o n t r o l l e d by a f l u i d c i r c u l a t i n g through i t , and an inner c e l l - h o l d e r , carrying heating c o i l s , embedded i n a thermally i n s u l a t i n g foam. The c i r c u l a t i n g f l u i d i n t h i s experi-ment was water pumped by a Forma S c i e n t i f i c Model 2095 whose tempera-ture was constant to within ±0.05°C. In some of the l i q u i d c r y s t a l experiments at- higher temperatures, a Haake Model E12 c i r c u l a t o r of s i m i l a r thermostatic accuracy was used with S h e l l V i t r e a O i l 21. Temperature sensing was.done with high-resistance Fenwal thermistors, whose resistance was greater than 2000 Q near the operating temperature. The thermistors were epoxied into 1/4" copper b o l t s , which i n ; t u r n were screwed into tapped holes i n the c e l l - h o l d e r Thermistor.^resistance was measured by means of s p e c i a l l y constructed Wheatstone bridges u t i l i z i n g matched r e s i s t o r s and i n d i v i d u a l l y c a l i b r a t e d to each thermistor. The output of each Wheatstone bridge was measured by a Hewlett-Packard Model 419A DC n u l l voltmeter. Temperature control was e f f e c t e d by feeding the proportional output of the d.c. n u l l meter into a low output impedance operational a m p l i f i e r , Kepco Operational Power Supply 7-2B, whose gain was adjustable. The operational a m p l i f i e r then supplied current to the non-inductive windings on the c e l l holder. The Wheat-stone bridges used i n temperature control had, i n addition to the usual balancing decade box, a 25-turn motor-driven h e l i - p o t , which could be used to sweep temperature continuously. The c i r c u i t diagram f or the 90 temperature control system is shown in Figure (21). The thermistors, their associated bridges and decade boxes, were calibrated using a Hewlett-Packard Model DY-2801A quartz thermometer. Deviations from 8/T the theoretically predicted resistance R = R^eJ' where RQ and B are characteristic constants of each thermistor were fi t t e d by poly-nomials in T, thus reducing the uncertainty in the measured temperature to ±0.0001°C. The decade boxes used.in temperature monitoring were General Radio Type 1433, with a specified temperature coefficient of resistance less than ±20 ppm/°C; in addition, every attempt was made to regulate room temperature during the course of the experiment. The temperature control was accurate to ±0.0005°C over an eight hour period. In some of the liquid crystal experiments, the quartz thermo-meter was used in addition to thermistor resistance measurements; the thermistor characteristics appeared to remain constant over the course of the experiments. The c e l l containers showing the location of the heating c o i l and thermistors is shown in Figure (22f),. Fig. 21 Circuit diagram of the temperature control system. Fig. 22 The c e l l containers. threaded cap heating wire hole for light aluminum cylinder thermistor in a screw Side View hole for alignment pin hole for c e l l "body hole for c e l l windows End View copper tubing copper cylinder hole for light beam styrofoam insulation plywood cover L external tubing for connection to water bath 94 4.4 Results The experimental procedure described above was carried out for SF, and GH.. Figures (23) and (24) show the results for the O H O Lorentz-Lorenz coefficient for X = 6328 A as a function of density, the results for SF^ have appeared in the literature (27). The coefficient is essentially constant within +0.5% over the range covered, although a small decrease i s observed with increasing density. The decrease in L with increasing density is opposite to the prediction of Yaris and Kurtman (18) who predicted an increase with density. The decrease with density agrees with the conclusions of Chapman, Finnimore, and Smith (22) for xenon. The precision of the measurements presented in Figures (23) and(24) is almost an order of magnitude better than Smith's xenon data because density and refractive index are both measured in this experiment whereas analysis of the xenon refractive index data required use of published PVT data for interpretation. The c r i t i c a l density and temperature for GeH^ were measured for the f i r s t time in this experiment; the obtained values are p = 0.5029 gm/cm3 and T = 38.925°C. c c 95 Fig. 23 The Lorentz-Lorenz coefficient for SF^ as a function density. 0.0790 h r x cm 3/g 0.0780 o SF C 1% o O Y o h o J 9* ° o * ° cp~ o 0,0 o cr-CO o OO Q O 0 U Q OOQOO 00~. 0,5 g/cnv 1.0 97 Fig. 24 The Lorentz-Lorenz coefficient for GeH^ as a function of density. 0.184 cm3/g <p (g/cc) V£5 00 99 F i g . 25 Coexistence curve of GeH G e H , U J o -tt« I O O i CC ce LU ' CUO 1 : 1 1 1 ! 1 1 1 1 I 1 1 1 D 0 0 .08 0 .16 0.24 0 .32 0 . 4 0.43 0.5S 0 .64 0 .72 0 .B 0.BB 0.SS . 1.04 DENSITY g/cc 101 CHAPTER 5 NEMATIC LIQUID CRYSTALS - EXPERIMENTAL 5.1 The Anisotropic Lorentz-Lorenz Relation For nematic liquid crystals, the relation between refractive indices, number density, polarizability and local f i e l d anisotropy i s given by Eq. (1.29). In a principal axis system, for a uniaxial material, Eq. (1.29) yields (n 2-l) (n2+2) (l -4Trph l la 1 1) = — pa II (5.1) and (n 2-l) (n2+2) (l-4TTpnJ_aJ_) = — pa x . (5.2) The polarizabilities a for the medium are given by Eq. (2.8); since for a system of prolate molecules R = <cos6> = 0, a II 1 2S = -r- (a +2a ) + — (a -a ) J mM m, j m,, m " j . it j _ (5.3) and 102 a, = (a +2a ) - -f- (a -a ) (5.4) -1- 3 m„ m 2 m„ m II j . II x where the a m denote molecular polarizability. The order parameter S may be eliminated from Eqs. (5.3) and (5.4) to yield a+2a = a +2a . (5.5) Since the anisotropy tensor n „ is traceless and the material is uniaxial a3 n + 2n =0. (5.6) II X If the refractive indices and the number density are known, then the four equations Eq. (5.1), (5.2), (5.5) and (5.6) may be solved for the four unknowns a n , a^, TiM> 1 x at a given temperature. The order parameter S can then be obtained from a„-°<a. S = a — • < 5' 7> m.. m, In this experiment, the refractive indices n and n x are measured as a function of temperature using a modified Rayleigh interferometer. The difference between the refractive indices is separately measured by use of a previously reported (20) conoscopic techniques. Thermal expansivity measurements were obtained by f i l l i n g conventional mercury and alcohol thermometers with liquid crystal samples and measuring the height of the meniscus in the capillary as a function of temperature. The nematic 103 liquid crystals used in the measurements reported here were EBBA (p-Ethoxy Benzylidene-p-n-Butylaniline) and BEPC (Butyl p-(p-Ethoxy-phenoxycarbonyl) phenyl Carbonate) obtained from Eastman Kodak Co. 104 5.2 The Modified Raylelgh Interferometer The liquid crystal samples were contained in rectangular fused quartz cells manufactured by Hellma Ltd. The inside dimensions of the cells are 1 cm x 5 cm x 0.2 cm; with a window thickness of 0.12 cm. The cells were connected to a vacuum system and pumped on for a period of 24 hours prior to f i l l i n g . They were then approximately one-half f i l l e d with the liquid crystal sample in the isotropic phase, and were then immediately reconnected to the vacuum system and were pumped on again in order to remove air dissolved in the sample and from the rest of the c e l l . The cells were then vacuum sealed, and kept at a s u f f i -ciently high temperature to prevent sample recrystallization. The c e l l containing the sample to be measured was then placed inside the tempera-ture controlled c e l l holder. Sample alignment was effected by placing the c e l l holder between the poles of a conventional electromagnet. The pole-piece separation was 8", providing a B-field of 1.8 kG at a 5 A supply current with 99.8% homogeneity over the volume occupied by the c e l l . The light source used was a Hughes Model 3178H 0.5 mW He-Ne laser. The beam was collimated and rendered uniphase by a beam expand-ing telescope and a lOu pinhole f i l t e r , i t was then passed through a 1 cm diameter i r i s and a Spindl.er-Hqyer polarizer. The beam was normally incident on the c e l l in such a way that approximately one half of the beam passed through the liquid crystal sample in the lower portion of the c e l l , while the rest of the beam traversed the upper portion of the c e l l containing only vapor from the sample. The two beams were mixed using a 1.5 cm x 1.5 cm x 1.5 cm beam cube and the 105 resulting interference pattern was enlarged using a simple f/2.8 f = 4.5 cm lens. Due to imperfections in the alignment of the two halves of the beam cube, the interference pattern consisted of approxi-mately 15 vertical fringes, spaced a distance apart. Using a 0.5 mm s l i t , a 2.5 cm x 0.5 mm horizontal portion was continuously recorded on film. The optical system and the interference pattern i s shown in Figures 26 and 27. As the temperature of the sample was varied, the optical path length difference of the beam through the liquid and of the beam through the vapor varied, resulting in horizontal movement of the fringes. A displacement of the fringes through a distance \^ corresponds to a change in the phase difference of the two beams of 2ir radians. Since the vapor pressure of the samples i s much less than one atmosphere, changes in the optical path length due to vapor density varia-tions are neglected. For light polarized parallel or perpendicular to the optic axis, the change in the corresponding refractive index An^ is thus An a = — (5.8) where £ = 2 mm is the sample thickness, A = 6328 A is the free-space wavelength of the laser and AN is the number of fringes that move past any given point within the interference pattern. In addition to photographing the interference pattern, the movement of fringes was monitored by a sili c o n photocell, whose amplified output was recorded on a strip-chart recorder. Sample temperature was -cell housing i 1 He-Ne polarizer sample o camera photocell Fig. 26 The modified Rayleigh interferometer. 107 Fig. 27 Interference pattern obtained from the modified Rayleigh interferometer. 108 109 recorded by photographing the display of the quartz thermometer every 5 minutes, and in addition, by manually balancing the monitor thermistor bridge and recording the thermistor resistance. Temperature was con-tinuously increased from approximately 20°C below the transition tempera-ture to 10°C above i t ; with sweep rates of .5°C/hr far from the transi-tion, and .002°C/hr near i t . Two separate runs were made for each sample; one with light polarized along the optic axis, and one with light polarized perpendicular to i t . The direction of motion of the fringes was opposite in the two cases. The refractive index in the isotropic phase i s known to decrease with increasing temperature due to decreasing density. Since the direction of fringe movement when the polarization was along the optic axis and the sample was in the nematic phase was in the same direction as that in the isotropic phase, i t was concluded that the extraordinary refractive index decreases with increasing tempera-ture, while the ordinary index increases. The number of fringes moving past the center of the interference pattern corresponding to changes in the refractive indices is shown in Figures (28) and (29). Since fringe displacements of corresponding to AN = -|" are easily detected, the accuracy in measuring the change in the refractive indices i s better than one part in three thousand; an order of magnitude better than pre-viously reported (21) refractive index measurements. The absolute value of the ordinary index at a given temperature was obtained by slowly rotating the c e l l about i t s vertical axis, and counting the fringes as they moved past a given point near the center of the pattern. The c e l l rotation was accomplished by using a 1/240 r.p.m. 110 Fig. 28 Results of fringe number measurements corresponding to changes in the refractive indices of EBBA. I l l r» •z. . <rUJ ZD I— ui o i l -- L r M I i n | : 1 ; — I — B'09C O'Oflt 1 1 " D'Ot-t O'DOC U3QWnN 30NIyj 0'09t O'OJt 112 Fig. 29 Results of fringe number measurements corresponding to changes in the refractive indices of BEPC. en 114 A.C. motor manufactured by Graham Canada Ltd., Model 267777. Care was taken to mount the motor i n such a way as to minimize v i b r a t i o n of the sample. A 10-tooth spur gear on the motor output shaft was used to drive a 96-tooth spur gear mounted on the v e r t i c a l axis of the c y l i n -d r i c a l c e l l holder; the angular v e l o c i t y of c e l l r o t a t i o n was 9.375°/hr. C e l l r o t a t i o n was i n i t i a t e d with the c e l l positioned at +30° from i t s normal p o s i t i o n perpendicular to the beam, and was rotated through 0° to a f i n a l o r i e n t a t i o n of -30°. In order to correct for errors due to the gear being s l i g h t l y o f f axis, the c e l l was rotated 180° and the above procedure was repeated. As shown i n Appendix I, the r e l a t i o n between n x and AN i s given by x 2 + s i n 2 0 = ax + b (5.9) where AMX . x = — j 1- cosG - 1 d a = - 2 ( ^ + n x ) b = n 2 - ( f ) 2 and e i s a constant between zero and unity. The fringes p l o t t e d on the chart paper were symmetric about 8 = 0 ; the angle corresponding to each i n t e n s i t y maximum was calculated from the known r o t a t i o n rate. The quantity x was evaluated for each 115 Fig. 30 Analysis of c e l l rotation data; x vs. x 2 + cos 20 for EBBA. x2+sin26 1 00 116 75 -3 40 50 25 BEPC T= 74.430 0 -10 ' • t ' • I I L. <10 r3 -20 -30 117 fringe (AM = integer) and x 2 + sin 20 was then plotted vs. x. The re-fractive index n^ was calculated from the slope and intercept of the resulting straight line, since n x = / ( f ) 2 + b . (5.10) A typical graph i s shown in Figure 30; the error in n^ is estimated to be ±0.005. In principle, a similar procedure could have been used to determine the absolute value of n . Rotation of the c e l l , however, II * ' necessitates the continuous re-alignment of the nematic director along the applied B-field. Although the observed time constant for this re-laxation process was less than one second, i t was f e l t that more reliable results could be obtained by using the conoscopic method described in the next section. The advantages of the modified Rayleigh interferometer des-cribed in this section that merit mention are i t s simplicity, i t s relative ease of alignment, i t s inherent insensitivity to the optical path length of the sample container (i.e. c e l l windows) and i t s insen-s i t i v i t y to building vibrations. Although the steel frame table support-ing the optical bench and magnet was floated on 14 automobile tire inner-tubes, an identical experiment using a Mach-Zender interferometer could not be made to yield reliable results due to building vibrations. No such problems were encountered during the course of this experiment. 118 5.3 Conoscopic Measurements The difference between the principal refractive indices of the liquid crystal samples were measured using a conoscopic technique. A converging beam focussed approximately 0.5 cm below the meniscus of the sample was obtained by placing compound f/1.8 f = 85 mm lenses directly in front of and behind the sample. The beam cube of Figure 24 was replaced by an analyzer. The polarizer and analyzer were crossed in such a way that the polarization of the incident light was at an angle of 45° to the optic axis. The resulting interference pattern is shown in Figure 31. As the difference between the refractive indices decreased due to increasing temperature, fringes were observed to move towards the center of the pattern in the horizontal direction and away from the center in the vertical direction. A si l i c o n photocell was used to monitor the intensity varia-tions due to fringes sweeping through the center of the pattern. The number of fringes AN that move past the center as the difference between the refractive indices changes is given by AN = f (n -n) - N N (5.11) Where NQ is a constant. Changes in the refractive index difference can therefore be measured with an accuracy greater than one part in six thousand; AN is shown in Figures 32 and 3 3 as a function of temperature, These results may be compared with changes in the refractive index difference calculated from the modified Rayleigh interferometer results of the previous section; the two sets of results agree to within ± 1 fringe. 119 Fig. 31 Conoscopic diffraction pattern. 121 122 Fig. 32 Conoscopic fringe number measurements for EBBA. 124 Fi g . 33 Conoscopic f r i n g e number measurements for BEPC. Z3 A N 53.0 60.S8S 62.375 64.063 6S.7S 67.438 63.125 70.813 7Z.S TEMPERATURE 75.875 77.SS3 73.2S 80.338 az.szs B4.313 — I £ 5 . 0 126 The absolute value of n,,-^ can be obtained from the density of fringes in the diffraction pattern shown in Figure 31 i f n is known. The diffraction pattern was photographed approximately ten times dur-ing the course of each run. The value of the ordinary index n^ was obtained from the modified Rayleigh interferometer results of the pre-vious section for temperatures corresponding to each photograph. If til 9 is the angle subtended by the N fringe moving in a vertical direc-tion from the center of the pattern, then the relation between 9 , N and the refractive indices i s , as shown in Appendix J, for small angles, where 0 < 6 < 1. Eq. (5.12) may be rewritten as sin 2 9 = mN + c (5.13) 2n(ln^A where m = —,—: N and c is a constant. The refractive index difference £(n„-njL) then becomes, in terms of m n 2 n.-n » -<- ml 2X " n a (5.14) For each conoscopic photograph, N was plotted vs. s i n 2 9 , and m was obtained. The refractive index difference was then obtained from I X Eq. (5.14). The quantity — (n„""n ) w a s then subtracted from AN of 127 Eq. (5. 1) for each temperature to yield NQ. The accuracy of the measurement of the absolute value of n.,-n i s determined by the scatter of points about N^; for EBBA i t was found to be ±0.005, whereas for BEPC i t was slightly worse due to existence of fewer fringes in the diffraction pattern. This accuracy could be improved considerably by photographing the pattern more frequently and by using lenses with smaller f-number. However, the existing thermostatic cell-holder necessitates the use of lenses with focal lengths of at least 85 mm. 128 5.4 Refractive Indices The refractive indices for each sample are calculated in a number of distinct steps. The absolute values of n^ in the nematic phase is obtained from the results of c e l l rotation. The absolute value of n -n^ in the nematic phase i s obtained from analysis of the conoscopic diffraction pattern, yielding n n since n,_ is now known. The temperature variation of n n , n + and n_^ i s accurately'determined from measurements using the modified Rayleigh Interferometer. The consistency of the results i s established by the conoscopic measure-ments, since the number of fringes traversing the center of the cono-scopic diffraction pattern corresponds to variations in the difference between n„ and n,. The refractive indices n„, n, and n. are shown in ll - l - II * - L . ^ Figures (34) and (35) for EBBA and BEPC. The meaning of the error bars i s that the whole series of points can be shifted as a unit. In addition to the refractive indices, the quantity n = -j(nn+2n_j_) is also plotted. The expected behavior of n is obtained as follows. Eq. (5.1) may be re-written as n 2 - l Y0 (5.15) 4TT -1 where = - j p o ^ (1-4'n-pn^a^) and I = „ , x'•: Then n„ = '1 + 2 ^ \ 1 / 2 3 3 =1 + T Y „ +4 Y 2 + .... (5.16) I Vl-Y / 2 8 'I 1 2 9 Fig. 34 Refractive indices of EBBA. 1 cr Cera ce - 1 35.0 5 ? . 0 50.0 51.0 63.0 TEMPERATURE 100.0 136.0 131 Fig. 35 Refractive indices of BEPC. 9 EPC X UJ a CE CHID -1 116.0 5B.Q 60.3 64.0 68.0 76.0 80.0 1 1 84.0 8S.0 TEMPERATURE S2.0 95.0 100.0 104.0 108.0 U2.0 120.1 133 Fig. 36 Published values for the refractive indices of EBBA. h-CD *~ re << £ o CD o t*. 1— LO II • K o CO o o o C o o 135 ~ 4 since < !• Since Y^ = "J i rP a£ » substitution for yields and n = 1 + 2iTp(a+2AS) + (5.17) n = 1+ 2irp(a-AS) + (5.18) Then, to a f i r s t order _ n M + 2 n 4 . 2 n = 5 = 1 + 2 i r p a + 0 ( S ) (5.19) and thus n varies as the density i f the average molecular polarizability a i s a constant. Eq. (55.1'9)) yields for the thermal expansivity n-1 From Figure.28* i t is clear that n decreases with temperature and that ^ is very nearly constant, giving evidence of the predicted increase of the thermal expansivity as the transition is approached from below. For EBBA at 67°C the above expression yields, since n.= 1.599 and = -6.33 x 10"4/°C - % = -1.06 x 10~3/°C. This i s in good agree-dT p dx ment with measured results, as discussed in the next section. The theory further predicts that 136 2(n 2+2) (n2+2) + (ni-1) ( i i 2 - l ) 4iTp + ^ a a, = A, (5.21) as given by Eq. (1.30). Expanding the p o l a r i z a b i l i t i e s a i n terms of S y i e l d s and A = 4irpa |_" 1-26S+462S2-8<53S3.. . + 2 + 26S•+ 2S 2S 2 + 26 3S 3 + (5.22) A = -7—— | l +'26 2S 2-26 3S 3 4irp a i J (5.23) where 5 = A/a. Le t t i n g p = N/V and using Eq. (3.51) f o r the volume V y i e l d s A = ( l - 3 k 2 S 2 + 2k 3S 3)(l+2 2<5S 2-2<5 3S 3) (5.24) where k = , and A = A(0)(^l+(26 2-3k 2)S 2-2(6 3-k 3)S 3+6k 26 2S l t+...^ . (5.25) For EBBA, the values of 6 and k are extremely close. From recent l i t e r a -ture (19) a = 38.4 x 10~ 2^ cm3, A =10.5 x l O " 2 ^ cm3 and 6 = 0.27 at X = 5790. I f the hard core length to width r a t i o K i s assumed to be ^2, then k = 0.25 and t h e ' c o e f f i c i e n t s of S 2 and S 3 i n Eq. (5.15) are 137 very nearly equal to zero. The quantity A i s thus expected to be nearly independent of temperature; values calculated from the r e f r a c t i v e indices i n d i c a t e that A/A(0) i s constant to within ±0.0015 throughout the covered 20°C temperature range. 138 Fig. 37 Photograph of apparatus used in the refractive index measure-ments . 139 140 5.5 Thermal Expansivity Experimental determination of the temperature dependence of the density is of interest because i t enables comparison with theory and makes possible the calculation of the order parameter S and the local f i e l d tensor n from refractive index measurements. The experimental method consists of f i l l i n g a conventional mercury or alcohol thermometer with a liquid crystal sample and mea-suring the height of the meniscus in the capillary as a function of temperature. The thermometers were evacuated by the following pro-cedure. The top sealed portion of the thermometer stem was broken off. The thermometer was then turned upside down and heated un t i l a portion of the substance in the bulb vaporized, forcing some of the liquid out through the capillary. The thermometer was allowed to cool so that a i r was drawn into the bulb. The bulb was again heated and the procedure was repeated until a l l liquid has been*expelled; a glass tube was then attached to the opened end. The c e l l thus formed was flushed with alcohol and was attached to a vacuum pump and jpumped on for several days. Sample material was then placed in the -glass tube • above the thermometer stem, and the c e l l was re-connected via valves to the vacuum pump and to a regulated helium bottle. After pumping on the sample in the isotropic phase for one hour, repeated applications of helium at approximately 0.5 atm. forced the. sample into the thermo-meter bulb. After further pumping the c e l l was sealed and placed in the temperature-controlled c e l l holder shown in Figure 38.- The results obtained using a c e l l made from an,alcohol thermometer (range: 40-120°F) 1 4 1 Fig. 38 Temperature-controlled c e l l holder used in thermal expansivity measurements. 1 4 2 143 are shown i n Figure 39; the points shown were obtained from three d i f -ferent runs. An expression of the form = 3 0 + z ( T t - T ) - 1 / 2 (5.26) suggested by Eq. (3.53) was used to f i t the data with the r e s u l t that &0;.= 0.81 ± .01 x 10~ 3/°C and z = 0.77 ± .02 x l O " 3 / 0 ^ ^ 2 . Due to the existence of an apparently stable two-phase region extending over 0.15°C, i t was d i f f i c u l t to determine the t r a n s i t i o n temperature Tfc. In t h i s temperature fin.feerv.a'l; the sample in equilibrium consisted of a nematic phase at the bottom and an i s o t r o p i c phase on-top separated by a w e l l -defined meniscus. This e f f e c t was observed in both the thermometer-cells and i n the quartz c e l l s used i n interferometry. I f , as suggested by conoscopic measurements, T = 77.75°C, then from Eq. (5.16) the thermal expansivity — at 67°C i s -lC.04xx 10 3/°C in good agreement p dx with the r e s u l t obtained i n the previous se c t i o n . The main problem encountered i n these measurements was the segmentation of the l i q u i d c r y s t a l filament i n the c a p i l l a r y ; t h i s could be overcome however by using a stem with l a r g e r bore. A new c e l l with increased capacity u t i l i z i n g an overflow trap has been made using 0.2 mm i . d . pyrex c a p i l l a r y ; r e s u l t s from measurements using this c e l l are not yet ava i l a b l e . 144 F i g . 39 Relative volume and thermal expansivity for EBBA. 1.03 145 THERMAL EXPANSIVITY 12x10"' 1x10" -20 -10 T-Tc 0 10 •0 146 5.6 The Order Parameter and the Local Field Anisotropy The order parameter S and the local f i e l d anisotropy tensor n = r i z z can be obtained as a function of temperature by simultaneously solving Eqs. (5.1), (5.2), (5.5) and (5.6). As shown in Appendix K, this results in a " = 2b 3(ab-l) + /9-30ab+9azbz (5.27) where a = (a +2a )/3 and m.. m, 4TT b = - J P (n2+2) 2(n2+2) + (n 2-l) (n 2-l) (5.28) S and r\ are then given by S = a +a, a -a (5.29) and n = 4irpa 3 (n 2-D (5.30) In order to evaluate a , i t is necessary to know a = (a +2a„ )/3 11' J x mn mj_' and the mass density p^. We have no direct way of measuring a or A, hence we have extrapolated published values (19) of the molecular polarizabilities of EBBA using Cauchy' s formula to obtain a = 37.1x10 2 4 cm3 _ o and A = 9.90x10 2 3 cm3 at X = 6328 A. To obtain an expression for the 147 density consistent with our thermal expansivity measurements, we integrate Eq. (5.26) to obtain P M = p Q exp [B 0(T t-T) + 2 Z ( T t - T ) 1 / 2 ] ' (5.31) where £Q and z are as given previously. The constant p ^ was determined by f i t t i n g Eq. (5.31) to published data (19) with the result that PQ = 0.977 gm/cc; the values of p obtained from Eq. (5.31) were within 0.1/% to the published data over the entire 40°C temperature range. Using these values, the order parameter S and the anisotropy tensor n were calculated for EBBA as a function of temperature. The results are shown in Figures (40) and (41). Both S and n are very sensitive to density variations. In view of the poor accuracy of the available density data, we have not attempted to f i t the calculated values of S to a theoretical expression. Calculated values of the local f i e l d anisotropy tensor n are in reasonable agreement with the pre-dictions of theory; we have no explanation for the deviations from linearity. 148 Fig. 40. The order parameter S obtained from experimental measurements for EBBA. 149 , o oo cn cn CO t— e o I •fr in O o o i o-CD o to 00 LO O CO o O -4" O CO 150 F i g . 41 The l o c a l f i e l d t e n s o r n o b t a i n e d f r o m e x p e r i m e n t a l m e a -s u r e m e n t s f o r EBBA. 152 CHAPTER 6 DISTORTIONS IN NEMATICS 6.1 Polarization Anomaly The experiments described in this thesis were performed using Spindier-Hoyer Model 03-6320 polarizing f i l t e r s . The 80 mm diameter f i l t e r elements were mounted on a rotatable ring with 1° divisions. Since an arrow and the letter P were inscribed at 0° on the polarizer ring; we had assumed that the electric f i e l d vector of a transmitted plane wave was co-linear with the direction of the arrow. This assump-tion yielded consistent results i n a l l respects; the extraordinary (fast-decreasing) index was observed when P was along the applied B-field and hence the optic axis, the ordinary (slowly increasing) index was observed when P was perpendicular to B, and the conoscopic pattern of Figure 3fli was observed when P was inclined at an angle of 45° to B. In addition, the conoscopic pattern was seen to disappear due to lack of contrast when P was either parallel or perpendicular to B; the polarization of the emerging wave was identical to that of the incident wave. It was felt then that the polarization and optic axis alignment were well understood. Near the completion of the experiments, however, while observing light reflected from a plane surface through a polarizer, we noticed that the electric vector E of the transmitted wave was not co-linear with P, but was instead per-pendicular to i t . This observation could only be reconciled with a l l 153 of the experimental results i f the polarization of a wave incident on the sample was rotated an angle 8 on entry, and an angle - 6 on exit, where 0 is the angle between the long axis of the fused quartz c e l l and the applied B-field. Usually the angle 0 was 90° since in i t s normal position the c e l l is vertical while B is horizontal, but conoscopic measurements were performed with the c e l l t i l t e d in order to gain a better understanding of this effect. We concluded that there exists a transition layer i n the nematic sample near the c e l l windows which i s responsible for the rotation of polarization. The transition layer i s a result of competition between field-induced alignment and strong * ^ n anchoring of the nematic director n (h. = ———) at the c e l l windows in |n| the direction of the long axis of the c e l l . The anchoring i s thought to be caused by striae on''the window surfaces due to polishing during manufacture; the alignment of nematics by rubbed glass surfaces has long been known. The effect of the twisted transition layer i s clearly seen if.the c e l l Is t i l t e d so that i t s long axis makes an angle 6 with the B-field and the conoscopic diffraction pattern is observed. The pattern does not rotate as 6 is changed, since the direction n in the bulk of the sample i s always along B. Rotation of the crossed polarizer-analyzer pair however causes the pattern to vanish i f the incident polarization is either parallel or perpendicular to the c e l l long axis; in this case the polarization of the emerging, wave is identical to that of the incident wave. The transition region may be imagined as a set of thin uniaxial plates cut parallel to the optic axis with each plate rotated 154 through a small angle with respect to the preceeding one. The only s i g n i f i c a n t e f f e c t of such a system i s to change the p o l a r i z a t i o n and phase of a wave transmitted through i t ; we assume therefore that a l l of the previously used analysis i s v a l i d i f the t r a n s i t i o n l a y e r i s small compared to the bulk sample. A quantitative j u s t i f i c a t i o n of t h i s assumption i s given i n the next two sections. 155 6.2 Nematic Alignment If a sample of nematic liquid crystal oriented in the direction - > - - > . n is placed in a uniform magnetic f i e l d H, the magnetization M is given by M = x„(H-n)n + x_jH-(H«n)n] (6.1) where x is t n e bulk diamagnetic susceptibility. The associated free energy per unit volume E„ is given by M E M = - / M-dH = - |( X l l-X +)(H-n) 2 - \ X j H 2 . (6.2) Thus, i f Y > x . » E„, is minimized i f n i s along H. Since for a l l known nematics the diamagnetic susceptibility is positive, the director n and hence the optic axis i s expected to orient parallel to an externally applied B-field. The free energy per unit volume associated with spatial varia-tions of the nematic director n is given, from (23) by E D = j ^ ( v n ) 2 + | K 2(n-(vxn)) 2 + | K 3(nx(vxn)) 2 (6.3) where K^ , and are the elastic constants associated with the deforma-tions splay, twist and bend respectively. Clearly E^ is a minimum i f n is a constant everywhere. In the absence of external fields, then, i f n is constrained to l i e in a given direction at a sample boundary, surface 156 anchoring serves to align the bulk sample. If an external f i e l d is also present, then competition exists between the two effects; the ex-pected alignment corresponds to spatial variations of n which minimize the total free energy. Two distinct deformations w i l l be considered; pure twist, corresponding to the rotation of n in the plane of the c e l l windows from i t s anchored vertical direction at the c e l l windows to a direction parallel to B in the bulk of the sample, and a combina-tion of bend and splay, corresponding to the rotation of n from i t s (nearly) vertical orientation of the liquid-vapor meniscus to a direc-tion parallel to B in the bulk of the sample. If i , j and k are unit vectors along the x, y and z axes, ys A. ^ then simple twist may be described by letting n = cosOi + sin0j where 0 is a function of z only. Then n*(Vxn) = - -— and contribution of o Z twist to the free energy is The combination of bend and splay may be described by letting n = cos0coscj> i + cos0sin<j)j + sin0k, where 0 is a function of z only, and <j> is a constant. " 30 Then V«n = cos6 — and the contribution of splay to the free energy is 3z E s = Y K l « Z ! ( ! f • ( 6 - 5 ) * * 13 0 i Since [nx(Vxn)] 2 = sin 20l —I , the contribution of bend to the free energy is 157 ^ - K ^ M H - i • (6-6) In the one-constant approximation = = = K; formally the two distortions become identical and E = f K ( f f ) 2 . (6.7) To obtain 0(z) which minimizes the total energy, we consider a small slab of sample of area A and thickness t. The total energy E associated with r the magnetization and the distortion is E F = t A [ - y( X l l-xjH2cos 20 + \ K ^ J d z . (6.8) " 0 Proceeding somewhat differently than (23), we note that the integral may be minimized by using Lagrange's equation; letting K Y 2 = ( x _ x ) H2 ' 6 m u s t satisfy Y 2 |~T " sin0cos0 = 0. (6.9) a Z Eq. (6.9) may be written as which yields upon integration 158 2 I £2.1 = r . 2 = C 1-cos 26 . (6.11) Since ^- , 8 + 0 as z + <», C = 1. Then a Z J-i S e = T l - t a n f + C 2 . (6.12) Since e = | i f z = 0, C 2 = 0 and f i n a l l y = 2 tan 1 exp(-z/y) (6.13) or 2 (6.14) To determine which solution i s energetically favorable for a sample of length ZQ, = 0 i f 0 = I , the non-trivial solution w i l l occur i f < 0, and at the Frederiksz transition 2 E T = / " I Y Z ( ^ " cosMdz = 0. ( 6 ' 1 5 ) Substitution from Eq. (6.11) gives 0 (l-2cos 26)dz = / ° (l-2tanh 2z/ Y)dz = 0 (6.16) U ^ 0 159 and integration yields z z ^ = tanh — . (6.17) 2y Y The solution of Eq. (6.17) is ZQ = 1.915 y. This is in good agreement with published results (23) for the case when anchoring occurs at both extremities of the twisted region; then 2z^ = iry. Thus, a f i e l d greater than H c is required to overcome surface-induced alignment in a sample of width " -1/2 d = 2z_ where H = 3 * ^ 0 c d K (x,rxj.) The anisotropy in the diamagnetic susceptibility is typically ^10 7, while the elastic constant is ^ 3x10 7 dyne (23). For the d = 0.2 cm c e l l used in this experiment, the c r i t i c a l f i e l d H c is only 33 oerstedts. The magnetic coherence length y for our applied 1.8 kG f i e l d is calculated to be 9.6u. At three coherence lengths from the c e l l walls the director n is only at an angle of 6° from the applied f i e l d direction, thus the twisted transition region is small compared to the bulk sample width. 160 6.3 Propagation in a Twisted Medium In order to determine the effect of the twisted transition region, we solve Maxwell's equations for light propagating along the twist (z) axis. (A more general derivation is given in (28)). From Eq.(1.3) 1 82D "2 f # = V ^ " V ( V * 6 ) * ( 6 ' 1 8 ) If i , j and k are unit vectors along the x, y and z axes, then the vectors n = cos6i + sinBj (6.19) /\ /\ <j> = sin9i - cos6j (6.20) and k define the directions of the principal axes of the dielectric tensor; 0 i s a function of z. For a wave propagating along the z-axis, we l e t E = E(z)exp[i(tjt-kz)]; then V(V»E) = and Eq. (6.17) yields ->- " ~ z 2 -> D = E =0. We write E(z) = E., n + E„d>, then D = e.,E-,ri + e„E0d> since z z 1 2 1 1 2 2 is now diagonal. Eq. (6.17) then yields \l ( ^ n + e2E2h = f i > ( E ^ .+ E 2 i ) . (6.21) For simplicity, we assume that 0 is proportional to z instead of the 2 TT Z relation given by Eq. (6.18), that i s , 9 = —— where \ u is the helix A, u d e l pitch. Since from Eq. (6.11) the maximum value of — = — , a reasonable dz y 161 3 ^ A 9 ^ ^ choice of pitch i s A^ = 2-ny. Noting that — n = -ct and — <j) = n, Eq. (6.21) yields after some manipulation ( e | - k , 2 - l ) E 1 = 2ikE 2 (6.22) and ( e ^ - k , 3 - l ) E 2 = -2ikE 1 (6.23) \ 2 k A h where we have let e 1 = e( T— ) and k' = —z— where A is the free-space A 2TT wavelength of the wave. We obtain for the wave-vector k' k' 2 = 1 + e' ± /6,2+4e (6.24) and (6.25) VE 2/ 2 + / S ^ e ' - 6' where e' = (ej+e^)I'2 and 6' = (e^-ep/2. The two solutions for E cor-responding to the roots of Eq. (6.25) are E± = ( E n + E 1 2<j ))exp[i(a)t-k 1z)] (6.26) and E2 = ( E21 n + E 2 2 < f , ) e x p [ 1 ( t J t " k 2 z ) : i • (6.27) The normal modes are therefore two e l l i p t i c a l l y polarized waves, whose components are polarized along the principal axes of the dielectric tensor. 162 In our experiment, the magnetic coherence length y was cal-culated to be ~ lOy; then -r^- = = ^—^-^-z-~-r - 100. From our 6.3 x 10"'m refractive index measurements we estimate that e' = 2.5 x 10^ and 6' = 6.2 x 10 3. Eq. (6.23) yields for 6' > 2e' .i2 = 1 + e' ± (<5 + 2e' (6.28) or k!2 = e| within an error of 0.1%, and since k. = / eT , n. x i x A 1 1 as before. Intensity ratios are obtained from Eq. (6.25), -for k« 2 = el /FT I J l l J12 6' = 1000 (6.29) I + 6' and for k'2 = '21 l22 1 - £ 1 6* . 1 -6 2000 (6.30) In our experiment therefore the normal modes are essentially plane polarized waves tracking the principal axes of the dielectric tensor with an associated refractive Index n equal to v^eT to within 0.1%. 163 CHAPTER 7 CONCLUSIONS 7.1 Summary of Results We have obtained a simple relation between the refractive indices, density and local f i e l d anisotropy in ordered fluids whose pair-correlation function can be made isotropic by radial scaling. Using mean f i e l d theory, we have shown that in a system of molecules with elon-gated hard core, the molecular distribution is anisotropic in the ordered phase. The corresponding local f i e l d anisotropy tensor i s shown to be proportional to the order parameter. We have obtained an expression for the specific volume as a function of the hard-core eccentricity and the degree of orientation order. Using the order parameter obtained from the Landau theory, an expression was obtained for the density as a function of temperature. Using a self-consistent formulation for the single particle pseudopotential, we have shown that ferroelectric order is not expected to occur for a fl u i d of hard rods, but may do so for a f l u i d of hard disks. In the latter case, the transition is expected to be of the f i r s t order. Using a single oscillator model, the London-van der Waals interaction predicts an ordered phase, but inclusion of the anisotropic hard core causes the order to vanish more abruptly than is experimentally observed. Experimentally, we have shown that the Lorentz-Lorenz coefficient i s constant to within ± 0.8% for GeH, and to within ±0.5 % for SF A along 164 the coexistence curve. We have determined the c r i t i c a l density and tem-perature of GeH. for the f i r s t time with the result that p = 0.504 ± ^ 4 c .005 g/cc and T = 38.925 ± .05°C. c We have measured the refractive indices of the liquid crystals EBBA and BEPC using a sensitive interferometer technique. Changes in the refractive indices were measured with an accuracy better than ±0.02%, whereas the absolute values were determined to within ±0.5%. The temperature dependence of the refractive indices was in qualitative agreement with the predictions of the theory. We have measured the thermal expansivity of EBBA across the nematic isotropic transition, the observed behavior i s in good agree-ment with the predictions of theory. The discontinuous volume change across the transition was M3.3%; the accuracy of this measurement is d i f f i c u l t to assess due to the existence of an apparently stable two-phase region near the transition. Using these measurements, the order parameter and local f i e l d anisotropy were calculated. Results for the order parameter are in good agreement with existing published results, while the anisotropy tensor i s nearly proportional to the degree of order, as predicted by the theory. 165 7.2 Suggestions for Future Work In order to exploit the sensitive refractive index measure-ments in the order parameter and local f i e l d tensor determination, den-sity data of comparable accuracy is needed. To gain this end, we have constructed a special c e l l capable of holding 10 cc of sample; time, however, has not been available to make extensive measurements. Pre-liminary results using this c e l l indicate greatly improved accuracy. By the use of smaller f-number lenses and by photographing the diffrac-tion pattern more frequently, the accuracy of the conoscopic measure-ments could be further increased. An independent scheme of measuring the c e l l orientation would similarly increase the accuracy of the c e l l rotation results, while a combination of the conoscopic and Mach-Zender interferometry could furnish additional refractive index information. We have noticed during the course of the experiments that near the transi-tion in the nematic phase that increasing the applied B-field caused the birefringence to increase. This effect, together with the field-induced re-alignment of the nematic director merits further investigations. Exploration of the two phase region in the neighborhood of T^ may also yield information about the nature of the transition. It would be of interest to see i f a relation similar to the law of corresponding states for pure fluids holds for order in nematic liquid crystals scaled to molecular eccentricity; measurements on more samples are thus required. It is believed that the mean-field theory could be further improved i f a closed-form solution for the distance of closest approach between two ar b i t r a r i l y oriented elongated surfaces could be obtained. 166 A more r e a l i s t i c attractive potential consisting of coupling between two sets of three mutually orthogonal oscillators is currently being considered. Fluctuations of the nematic director have not been con-sidered in this thesis, although known to be of considerable importance. Recently acquired photon correlation equipment is currently being set up to study light scattered from nematic materials. 167 APPENDIX A Letting x = nd^ , y = nd^ and z = nd^ Eq. (1.11) may be re-written as 2 v 2 z 2 + 1 — + - — = 1 (A.l) £ £ £ xx yy zz Since D i s perpendicular to k, D'k = 0 and xk + yk + zk =0 (A.2) x y z Furthermore, n 2 .= x 2 + y 2 + z 2. We wish to consider extrema of n 2 subject to the constraints of Eq. (A.l) and Eq. (A.2). Using the method of Lagrange's undertermined multipliers, we wish to find the extrema of 2 2 2 F = x2+y2+z2+2X, (xk +yk +zk ) + X„(-— + 1 — + 1) (A. 3) l x y z 2 e e e xx yy zz 9F 3F 9F This demands that — = — = — = 0. Then 9x 3y 9z x + A,k + X0 - — = 0 (A.4) 1 x 2 e xx y y + X.,k + A. ^ . J 1 y 2 E =0 (A.5) yy z + x.k + x. - — = 0 (A.6) 1 z 2 e zz 168 Combining Eq.'s (A.4), (A.5) and (A.6) with (A.l) and (A.2) obtains and n 2 + X 2 = 0 (A.7) k x k y k z k 2 ^ + K I — + + — 1= 0 <A-8> 1 2 \ e e e xx yy zz y Solving for X^ and X^ Xn = -n 2 (A.9) and 2 / k x k y k z , *i " ? | — + " ^ - + — <A'10> 1 k z \ e e e xx yy zz Substitution into Eq. (A.4) yields z 9 x n 2k /k x k y k z xxx 7 \ xx yy zz ' similarly for y and z. Letting x = nd , etc., in Eq. (A.11) and multi-plying by D we obtain 169 D I 1- )+4k ( E k + E k + E k ) = 0 x y to 2 e x x / w z x x y y y z z or 2 _ _ _ ^ D - k 2 E + k(E«k) = 0 which i s Eq. (1.5). Thus the permitted d i r e c t i o n s of D are those for which n 2 = x 2+y 2+z 2 i s an extremum. 170 APPENDIX B From Eq. (1.16), integration over <f> yields zz - h f ( l n ( l -e2cos 26) 2)(3cos 29-l)sined6 g (B.l) If x = cos0, integration by parts yields |1 / * ! V 3 n = %ln(l-e 2x 2)(x 3-x) +/ ( f - - x)dx, (B.2) Z Z g lo l - e 2 x 3 3 where the f i r s t term equals zero. Since e 2 < 1, we let e x = sinE g " g Then ^ a s i n ^ e o sin20cos0d0 1 / sm H0 1 / Tl = I cos0d0 I z z e 3 J o C O S 9 %J0 cos 2e (B.3) Furthermore, ' U " " ~ s i n l e d0 • . e 3 L = r s i n eq-cos 20) 2d0 _ r 1 J 0 cos0 J n cos0 6 3 (B.4) and 171 s i n *e (l-cos 29) COS0 dO •f. sm 'e d9 COS0 - e (B.5) I f y = In ("*"+S""?^ ) then dy = d6/cos0, and COS0 . - 1 s i n e d0 . , 1+sine In COS0 COS0 -1 s i n e 1+e = lnl £_ /1+e = In »Vl-ez/ v ^ e (B.6) and f i n a l l y zz 2 3 1 1 e z 2e g g g-l n 1+e 1-e g J (B.7) I t i s i n t e r e s t i n g to consider the l i m i t s of n as e -* 0 and as e -> 1, zz g g In the f i r s t case, since 1+e E 1-e e 3 e 5 e + ^ + + .. g 3 5 (B.8) for small e , g zz 2e'+ _2 e2 g_ 15 g: 35 (B.9) and n = 0 i f e =0. For e ~ 1, l e t t i n g x = 1-e y i e l d s zz g g g 1 (2-x)x . f2-x" 2 - TTTZZ\ L N 'zz 3 ( l - x ) z 2(l-x) (B.10) Since limxlnx = 0, n ,•=--=• i f e =1. In Figure B l , n i s shown as a zz 3 B & zz x+0 function of e g 172 Fig. B.l The local f i e l d anisotropy tensor T i z z as function of the eccentricity e . g 173 174 APPENDIX C From Eq. (2.20) AF = a 2R 2 + a 4R 4 + a 6R 6 (C.l) and the conditions that AF be negative and a minimum, i.e. AF < 0, 3AF 92AF . - 0 and ^ 0 yield, respectively a 2R 2 + a 4R 4 + a 6R 6 < 0 (C.2) 2a„R + 4a,R3 + 6a^R5 = 0 (C.3) 2 4 6 and 2a 2 + 12a4R2 + 30a6R4 > 0. (C.4) At the transition, AF = 0, R = Rc and T = T . Eliminating a 2 between -a, Eqs. (C.l) and (C.2) and solving for R yields R = 0 or R2 = -=— . C C C /SL 6 If R =0, then R is small near the transition, and Eq. (C.3) yields o ~ C _ a3 Rz = -x— . Then, i f a. > 0, 2a. 4 r 2 • i t <V T ) • ( c- 5 ) 4 ~a2 Sub0*"* n^ r 2 = stitution of Rz = into Eq. (C.l) yields 4 —SL SL AF = R 2 ( a 2 + a 4 ( 2 ^ ) ) =R 2 / < 0 (C.6) for T < T . 175 " a2 Substitution of R2 = -=— into Eq. (C.4) yields 2 a4 Isr = 2 a 2 + 1 2 a 4 ( § ) " - S > 0 ( c - 7 ) for T < TQ. -a 9 4 If Rz = - r — , then, solving Eq. (C.3) directly yields, i f a, < 0 c -^ a^ ^ and a, > 0, o R2 = 4 R2 + k' / T + - T (C.8) 3 c IV c a 2 a4 where k' 2 = and = Tn + 0 _ . To obtain T^, substitution of 1 3a^ c 0 3qa^ c -a a' R2 = into Eq. (C.8) yields T £ = T Q + ^ & . Straightforward 6 6 substitutions show that Eq. (C.4) is satisfied for T < T C , while Eq. (C.l) is satisfied for T < T . - c From Eq. (2.23) A F = b 0S 2 + b„S 3 + b.S 4 (C.9) 2 3 4 and the conditions that A F be negative and a minimum, i.e. A F < 0, 9AF . . , 8 2 A F ^ . . , t_. , - 5 ^ — = 0 and 9 > yield, respectively b 0S 2 + b_S 3 + b.S4 < 0 (CIO) 2 3 4 -2b2S + 3b 3S 2 + 4b 4S 3 = 0 (C.ll) 176 and 2b 2 + 6b3S + 12b 4S 2 > 0 (C.12) At the transition, AF = 0, S = S and T = T . Eliminating between -b 2 Eqs. (C.9) and (C.ll) and solving for S yields S = 0 or S = . 4 If S =0, then S i s small near the transition, and Eq. (C.ll) yields c s = ~ 2 b2 . Then, i f b, > 0, 3b 3 S = 3 ^ ( V T ) (C.13) -2b3 Substitution of S = TCT— into Eq. (C.9) yields 3 h + b 3 ft))' AF = S z(b 0 + b 0 1 ^ J J . Sz -f 1 0 (C-14) -2b for T < T.. Substitution of S = - T T — - into Eq. (C.12) yields - U jb„ = 2 b2 + 6 b 3 ( 3^) = " 2 b2 > ° ( C ' 1 5 ) for T < T Q. The absolute minimum of AF is attained however i f S = -1; then, near TQ AF = -b 3-|b 4| (C16) since b. < 0. 4 177 APPENDIX D We wish to find the length of the normal projection of the surface R2 - e 2(R-n) 2 = RQ (D.l) onto a line in direction I through the origin. The normal § to the sur-face i s given by N(R) = V[R 2-e 2(R-n) 2] = 2R-2e2(R-n)n (D.2) The point at which N i s colinear with I is given by NCR^) = 2uJt, where u is some constant. The projection d + of the surface onto the line i s then d, = 2R To obtain R , consider + u u R - e 2(R «n)n = u£ (D.3) u u Multiplication by n,£ and Ru yields, respectively, (l-e 2)($ u-n) = u(£-n), (D.4) R •£ - e2(Ru«n)(n-£) = u (D.5) and R2 - e 2(R «n) 2 = u(R •£) = R2 (D.6) 178 Combining Eqs. (D.4), (D.5) and (D.6) yields e 2R 2 \ 2 R2 (D.7) u (1-e2) (R •*)• ( R U , A> and f i n a l l y dj = 4(R 4-£) 2 = 4R2(1 + (^* u) 2)- (D-8) The length of the line in the I direction intercepted by the surface i s simply given by d* = 4R2(1 - e 2(£-u) 2) -1. (D.9) Noting that, for a random variable x, x™ > x™ for a l l positive even m, Eq. (D.8) gives d + < 2R. / 2 ~ ~ \ l / 2 Q ( l (£-n) 2 ) . (D.10) Expanding d_, given by Eq. (D.9), in a Taylor series in (£*n)2) and averaging yields d_ = 2R (1 + j e 2 ( i l - n ) 2 + ) > 2R 0(l-e 2(£-n) 2) (D.ll) 179 APPENDIX E We wish to evaluate the integrals in Eq. (3.42) R = L cos8exp(gyf/ cos9)sin0d0 L) E f (E.l) exp(gyfL,cos0)sin0de E Letting y = 3yH and x = cos0 yields E and Y X , Y X X 1 x e dx e ? Z_ = Y IT -1 (eY+e Y) - - (e Y-e Y) Yx , e dx 1 yx — e Y 1 -1 Y _Y e-y-e (E,2) R = cothy - - = L(3yW„) Y 11 (E.3) Then 3R 1 o S is obtained by noting that, from Eq. (E.2) — = —(2S+1)-R2. dy j I (R2 + w) -1 = f (< 9 2 1 coth^Y _ -cothy H 9 Y Y z sin In l zY Y2/ 2 (E.4) and S = 1 - -(cothy - -) Y Y 180 APPENDIX F Rotational symmetry about the z-axis implies that S ^ i s dia-gonal and i s given by Eq. (2.10), and consequently, from Eq. (3.31), D(r) and f (r) are independent of the longitude $ . Eq.;(3.74) then s i m p l i f i e s as follows: £..(ft) = A n, n. 1 l a l y 3 ( 2 S 3 6 + V (F.l) &2n r TT • I (9r r f l r r.-3r r J .-3r r.6 .+«iD(S .) sine de dd> r r T r "a. 3 y 6 a 3 Y<$ Y 6 ag yS f 3 , 2fl '0 " 0 r S t ' c o s H r ; where A = -p y S a n 48R03 Upon multiplying through, the f i r s t term I, in the integrand becomes I. = 3n. n r r . r r_(2S 0.+6 Q.) = 3n_ n r r (2r.r.S.,+l) 1 l a l y a 3 y 6 36 36 l a 1Y a Y 3 6 36 (F.2) and since S . i s diagonal and S = S zz 2 xx - 7 s 2 yy I, = 3n n n., r r (l+S(3cos 2e-l)). 1 l a l y a y (F.3) Since integration over <j> eliminates a l l the off-diagonal elements of r r to obtain a Y I, = 3n 2 r 2 ( l + S ( 3 c o s 2 6 - l ) ) . 1 l a a (F.4) 181 Proceeding as above, the following terms i n the integrand are I. = -n. n. r r_6 . (2S..+6-.J = -n 2 r 2 ( 2 S +1) (F.5) 2 l a l y a 6 y6 36 86 l a a aa I„ = -n. n. r r.fi Q ( 2 S Q . + 6 Q J =.-n2 r 2 ( 2 S +1) (F. 6) 3 l a l y y 6 a3 36 36 l a a aa I = ^ n. n. 6 .6 .(2Sfl.+60_) = \ n 2 S + ^ (F.7) 4 3 l a l y a3 y6 36 36 3 l a aa 3 Expressing the components of n^ and r i n terms of polar angles and i n t e g r a t i n g over <|>r y i e l d s I (3cos 26 -1) r n (3cos 2G -1) _ i = s h c o s ^ e -6cos 2 e +1 +2S 2TT „ I r r J ( 3 c o s 2 6 1 - l ) (3cos 26 -1) + 2 — + 1 (F. 8) I 2 + I 3 ( 3 c o s 2 9 1 - l ) 4S ( 3 c o s 2 9 r - l ) _ _ _ . _ s (2cos 29 r- 3) - 3 . ( 3 c o s 2 6 1 - l ) (3cos 26 -1) 4_._ 1 r 3 3 J 2 2 J 2 (F.9) and 182 Summing a l l the terms f i n a l l y y i e l d s e 1(n) = 2TT A (3cos 26 -1) / (3cos 20 -1) 1*1 . (S + ~ (3cos 20 -1) + S — Ocosfo -8cos 20 +1) sin0d0 J f|(r) ( F . l l ) 183 APPENDIX G The self-consistent equation for S given by Eq. (3.79) becomes, on letting x = cos0 1 -(3x 2 - l H(S ) B fp f ( 3 * 2 - l ) e 2 < dx S = " : \ (G.l) -(3x2-i)<|)(s)e dx 0 Letting a Q = —<k(S)8, and noting that 2?= (l+2S)/3 2 2 0 r - a. X 2 /"I a Qx 2 ^ - o l l • J / e dx where (G.2) /* ^ a_x2 0 0 n 1 = / e U dx = I. a0 . (G.3) Jn n=0 (2n+l)n'. It is worth noting that the series converges rapidly; since n! >.nne n , the remainder after N terms is bounded by = J . w • 184 Since i s the order of unity, only about 10 terms are needed to get seven place accuracy, thus eliminating the need to use tabulated values of Dawson's integral. Eq. (G.2) is solved as follows. The constant a^ may be written as a Q=|<|) 1(S) (G.5) where <t>-^(S) i s a polynomial in S whose coefficients are known for a given eccentricity e from Eq. (3.77). The constant A incorporates the oscillator strengths, hard sphere radius, packing fraction, etc.; T' = T/A is regarded as a normalized temperature. An arbitrary value of ag is chosen to begin with, the corresponding value of S is obtained from Eq. (G;2). Once S is known, a, c and e can be calculated for a given molecular eccentricity e, and <f>^ (S) can be evaluated. The normlized temperature corresponding to this value of S is simply ^(S) T' = . Thus S vs. T' is obtained simply without resorting to ao root-finding techniques. S decreases monotonically with a^; T' however f i r s t increases to T^ ,, then decreases. Only those values of S are of interest which decrease with increasing T' . Having determined T^ _, the I T.' reduced temperature is obtained from the relation 1 - — =1 - — , . c c 185 APPENDIX H The windows of the c e l l have surfaces which are nearly p a r a l l e l ; the f i r s t window has a wedge angle of less than 0.01° and the exit win-dow has a wedge angle of 0.080°. If the windows'had perfectly parallel surfaces, then the deviation angle for a i r in the c e l l would be zero and would be identical to the reference angle measured with the c e l l removed. The effect of the window wedges can be calculated and i f the deviation angle measured with air in the c e l l is used as a zero deviationaigle, the effect of the window wedge angles can be neglected. The wedge angle.of the f i r s t window is only 0.01° and is neglected in the following calculation. The geometry of the c e l l i s shown in Figure H-3l. Let the prism angle between the windows be denoted by a and the wedge angle of the second window by <|>. The angle <j) is of opposite sense to the prism angle a for the c e l l used in this experiment. Consider a ray entering the c e l l perpendicular to the f i r s t window. Application of Snell's law at the surfaces leads to the following equations: nsina = n sin(\JH-(j>) (H.l) n sinijj = n sin (a-<frt-8) (H.2) o 3. where n, n and n are the refractive indices of the sample f l u i d , o 3-sapphire window and air. Elimination of ty yields 186 F i g . H.l Prism c e l l geometry. 187 nsina = n sin(a-(b+6)cosq)+sin<&(n 2-n 2sin 2 (a-(j>+e)) SL o cL 1/2- (H.3) Expanding i n sinG and r e t a i n i n g the f i r s t three terms gives nsin = a + bsin8 + c s i n 2 f (H.4) Since the c o e f f i c i e n t s a, b and c are derivatives of nsina with respect to sin8 evaluated at 0 = 0, d i f f e r e n t i a t i o n of Eq. (H.3) with respect to sin(a-(|>+0) and!application of the chain r u l e y i e l d s a = sina 1/2 n sin(a-<f>)coscb + sina) (n2-n2sin2'(a-<}))0 * SL o cL 3=0 (HV5) , _ 3 (nsina) _ 3sin0 1 2 n sinctsin(a-d)) 2 A A n^ cos<j> . (n|-n 2 s i n 2 (a-cb) ) cos (a-<J>) and (H.6) _ 1, 3 2 (nsina) ,J, 2 3 ( s i n G ) 2 0=0 n 2n 2sincbcos 2 (a-cb) SL o ( n 2 - n 2 s i n 2 (a-cb)) o 3. 2/2* + n^coscbsin (a-cb) n 2sin<bsin 2 (a-cb) SL ( n 2 - n 2 s i n 2 (a-cb) ) 1 / 2 O 3. (H.7) 188 If the c e l l i s f i l l e d with a i r at normal density and the deviation angle 0 i s measured, Eq. (H.6) y i e l d s n = r~ |n + (sin0-sin0 ) [b + c(sin0+sin0 ) ] | . (H.8) sxna [ a a a JJ The use of 0 rather than the angle measured with the c e l l removed a. eliminates most of the ef f e c t due to the window wedges. If the f i r s t window were also wedged, the above expression could be appropriately modified. 189 APPENDIX I The geometry of the c e l l rotated through an angle 8 is shown in Figure L l . and £^ are the path lengths of rays through the evacuated position of the c e l l , and through that portion which i s f i l l e d with the liquid crystal sample, respectively. For incident light polarized perpendicular to the optic axis, Snell's law is obeyed, and sin9 = n sin0 1 (1.1) where n = n /n. and n. = 1.00029 is the refractive index of a i r . Since - i - A A JLTcos6 = I (1.2) N X,1cos81 = I (1.3) and Z2 = t sine = (Jl Nsine-Jc. 1sine i)sine , (1.4) The optical path length difference L between the two rays is 1 2 N nZ . Jtsin26 Jlsin 2 e I + cos6^ cos6 ncosB^ cost 190 Fi g . 1.1 Geometry of rotated c e l l . J 191 The change in the optical path length difference L i f the c e l l i s rotated from zero angle to an angle 0 is AL = I p n 2 - s i n z 0 - cos0 - n+1 . (1.6) Letting AL = (AN+e)A, Eq. (1.6) yields ANA % + cos0-l 1 + + n = /n z-sin z0 (1.7) Letting ANA ^ Q 1 x = — — + COS0-1 a = -2|f^-+ n b = n 2 - ( f ) 2 and squaring Eq. (1.7) yields c2 + sin 20 = ax + b (1.8) 192 APPENDIX J If a plane wave of angular frequency o> and wave vector is incident on a plane dielectric boundary and k g is the wave vector of the transmitted wave, then continuity of the tangential component of the E-field across the boundary demands that cot-It. »r = cot-It «r (J.l) X L5 everywhere on the boundary. If r is in the plane of the boundary, then k.cos(f-e.) = k scos(f-e s) (J.2) where 6. and 0 are the angles between k. and £ and the normal to the i S i o boundary. Since k = ^ , A n.sin0. = n„sin0„, l l S S where n. and n are the refractive indices in the two media. Thus Snell's law i s valid for wave-normals in anisotropic media. The refractive indices corresponding to the two allowed polari-zations for a wave propagating in the k direction i s obtained from Eq. (A.l). If n 2 = n 2 = n 2 and n 2 =.n2, then, letting e 2 = 1 -193 r 2(l-e 2cos 26) = n 2 (J.4) where 0 is the colatitude and r 2 = n 2. If k makes an angle a with the z-(optic) axis, then the extremal values of r 2 or n 2) are n 2 = n 2 (J.5) i f 0 = TT/2, and n 2 n 2 = (J.6) ^ -i 2 • 2 1-e sm^a i f 0 = Tv/2-a. The geometry for a plane wave incident on a plane slab of sample is shown in Figure J . l . From Snell's law, n^sin© = n^sin©^ and n^sinG = ^ 3 1 ^ 2 . The difference in the optical path lengths of the two waves is In £n„ - d = -NX (J.7) cos0^ cost<2 where we have let n. =1. Furthermore, A d = sin0(£tan61-£tan02) (J.8) and simple algebra yields 194 Fig. J . l Geometry of a plane wave incident on a f l a t slab of sample material. 195 n^cosG^ - n2Cos92 NX SL (J.9) If the wave vector k is perpendicular to the optic axis, cor-responding to the vertical fringes in the conoscopic pattern, then a = TT/2 and n„ = n . Then 2 II h + ( ! - ^ 3in2e) 1 / 2 - „„(l - i j .ln*») 1/2 = _ NX (J.10) and expanding in sin 20 yields (n„-n ) 1 + 2n n„ sin20 NX I (J..11) If the optic axis is in the plane of incidence and is parallel n' to the front surface, then sina = cos02> and n£ = i _ e 2 c o s 2 Solving for cos 20„ yields cos20„ = sin 20-n 2 2 9 9 9 e^sin^G-nf (J.12) and n 2cos0 2 = n ^ l - s i n 2 0 ^ 1 / 2 . (J.13) Eq. (J.9) gives 196 and expanding in sin z0 as before yields (n„-nx) [ l - 2^ 2- s i n 2 e ] = ^ . (J.15) Eqs. ( J . l l ) and (J.15) predict the experimentally observed motion of the fringes in the conoscopic pattern; as n -n x decreases (with increasing temperature) the horizontal fringes more towards the center, while the vertical fringes move away from i t . A commonly occuring mistake in optics text books (see 'Optics' by Rossi for example) i s the statement that the extraordinary wave sur-face in uniaxial crystals i s an ellipsoid. From Eq. (J.6) i t follows that the wave surface i s given by r 2 = -i- ( l - e 2 s i n 2 a ) , (J.16) a fourth degree ovaloid. If the wave surface is assumed to be an e l l i p -soid, the angle of refraction of the extraordinary ray can be easily obtained. If the optical path length difference for the ordinary and the extraordinary wave is then calculated, using this result, the re-sulting equation resembles Eq. (J.15), except the sign of sin 20 i s reversed. The fact that such an equation is incorrect could easily be overlooked in situations where n -n A cannot be varied continuously. 197 and APPENDIX K We wish to solve the following equations (n 2-l) " 4IT — (1 - 4T7pa l l n l l ) = -r- pa (n2+2) J to*-1* 4. (1 - 4irpaJ_nJ_) = ~~rT P a j (n2+2) ; J n„ + 2n =0 a.. + 2a, = 3a Letting (n2-D z = (n2+2) and (ni-1) x = (n2+2) elimination of n from Eqs. (K.l) and (K.2) yields 198 Elimination of a from Eqs. (K.5) and (K.6) gives ba 2 + a„(3-3ab) + 3a = 0 (K.6) Solution of Eq. (K.6) is a„ = [3(ab-l) + /9-30ab+9azbz, (K.7) and then a_,_ = (3a-a,()/2. Since a„ = a+2AS and a_,_ = a-AS, the order para-meter S is given by S = - 3 ^ — . (K.8) The anisotropy tensor n is obtained from Eq. (K.l); n = T2" (K-9> 11 4irpa 3z and n, = - \ n • (K.10) 11 199 REFERENCES (1) Max Born and Emil Wolf, Principles of optics. (Pergamon Press, New York, 1965), 3rd ed. (2) J.O. Hirschfelder, C.F. Curtiss and R.B. Bird, Molecular theory of gases and liquids (Wiley, New York, 1954). (3) P.P. Ewald, Ann. Physik 64 (1921) 253. (4) M. Born and M. Bradburn, Proc. Cambridge Phil. Soc. 39 (1934) 104. (5) H.E.J. Neugebauer, Can. J. Phys. 32 (1954) 1. (6) D.A. Dunmur, Moi. Phys. 23 (1972) 109. (7) D.A. Dunmur, Chem. Phys. Letters 10 (1971) 49. (8) L.D. Landau, Collected Papers of L.D. Landau (Gordon and Breach, New York, 1965). (9) Y. Poggi, P. Atten and J.C. F i l i p p i n i , Moi. Cryst. Liq. Cryst. 37 (1961) 1. (10) M.J. Press and A.S. Arrott, Phys. Rev. A, 8_, 3, 1973. (11) T.W. Stinson and J.D. Litster, Phys. Rev. Lett., 25, p.503 (1970) (12) M. Born, Ann. Physik 4 (1918) 225. (13) S. Chandrasekhar, D. Kirshnamurti and N.V. Madhusana, Moi. Cryst. Liq. Cryst. 8 (1969) 45. (14) A.J. van der Merwe, Z. Physik 196 (1966) 212. (15) W. Maier and A. Saupe, Z. Naturforsch. 14A (1959) 882. (16) D. Balzarini and K. Ohrn, Phys. Rev. Lett. 29 (1972) 840. (17) D. Balzarini, Can. J. Phys. 50 (1972) 2194. (18) R. Yaris and B. Kurtman, J. Chem. Phys. 37 (1962) 1775. 0 200 (19) H.S. Subramhanyam and J. Shashidhara Prasad, Moi. Cryst. Liq. Cryst., 37 (1976) 23. (20) D. Balzarini, Phys. Rev. Lett. 25 (1970) 103. (21) I. Haller, Prog. Sol. Stat. Chem. 10 (1975) 103. (22) J.A. Chapman, CP. Finnimore and B.L. Smith, Phys. Rev. Lett. 21 (1968) 1306. (23) P.G. de Gennes, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1974). (24) P. Palffy-Muhoray, Chem. Phys. Lett. 48 (1977) 315. (25) M.J. Freiser, Liquid Crystals 3, Part 1. ed. G.H. Brown and M.M. Labes (Gordon and Breach Science Publishers, New York, 1972) p. 281. (26) H.S. Subramhanyam and D. Krishnamurti, Moi. Cryst. Liq. Cryst. 22, (1973) 239. (27) D. Balzarini and P. Palffy-Muhoray, Can. J. Phys. 52 (1974) 2007. (28) D.W. Berreman and T.J. Scheffer, Moi. Cryst. Liq. Cryst. 11, (1970) 395.
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Refractive indices of liquid crystals and pure fluids near phase transitions Palffy-Muhoray, Peter 1977
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Title | Refractive indices of liquid crystals and pure fluids near phase transitions |
Creator |
Palffy-Muhoray, Peter |
Publisher | University of British Columbia |
Date Issued | 1977 |
Description | Interferometric measurements to determine the refractive indices of the nematic liquid crystals EBBA and BEPC as a function of temperature are described utilizing modified Rayleigh and conoscopic interferometers. Theory is presented relating the refractive indices and density to the orientational order, local field parameter and molecular properties. The results of simple thermal expansivity measurements are also given for EBBA. The Lorentz-Lorenz coefficient for SF₆, and GeH₄, has been determined from refractive index and density measurements. The method utilizes a prism shaped high pressure cell which can be removed from a temperature controlled holder and weighed on a precision balance. The results indicate a variation of 0.5% for SF₆, and 0.8% for GeH₄, over the density range covered. |
Subject |
Liquid crystals |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-02-25 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085754 |
URI | http://hdl.handle.net/2429/20911 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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