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NMR of solutes in nematic liquid crystals : an investigation of the mechanisms of orientational ordering Van der Est, Arthur James 1987

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NMR OF SOLUTES IN NEMATIC LIQUID CRYSTALS: AN INVESTIGATION OF T H E MECHANISMS OF ORIENTATIONAL ORDERING by ARTHUR J. VAN DER EST B.Sc. (U.B.C.) A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Chemistry We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1987 ® Arthur J . van der Est, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of (2Ht=:ns<.l s T / The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date (O iT-lsl-ABSTRACT Dipolar and quadrupolar couplings measured from NMR experiments, and order parameters obtained from these couplings, are reported for a number of small solutes dissolved in several nematic liquid crystals. These results are discussed in terms of the solute-solvent interactions. It has been shown that the interaction between the solute molecular quadrupole moment and a mean external electric field gradient due to the liquid crystal accounts for most but not all of the ordering of molecular hydrogen. The remaining contribution to ordering is discussed in terms of possible mechanisms. The anisotropic couplings observed for methane and acetylene are discussed in terms of a model which takes into account the interaction between the vibrations and rotations of the solute. Excellent agreement between the observed and calculated dipolar couplings is obtained. Evidence is given that these two solutes experience the same field gradient as molecular hydrogen. In a mixture of 55wt per cent 1132 (Merck ZLI 1132) and 45wt per cent EBBA (N-(4-ethoxybenzylidene)-4'-n-butylaniline at 301.4K the deuterons in T>2 experience no external electric field gradient. The order parameters of a series of solutes in this mixture are calculated in terms of a simple model for the short range interactions which depend on the size and shape of the solute. These calculated order parameters are in very good agreement with the experimental results. In liquid crystals where the field gradient is not zero it is shown that the combination of the short range interaction model and the electric field gradient - molecular quadrupole moment mechanism predicts the order parameters very well. ii T A B L E OF CONTENTS Abstract ii Table of Contents iii List of Figures v List of Tables vii List of Abbreviations ix Acknowledgements x I. Introduction 1 A. Liquid Crystals 3 B. NMR of Partially Oriented Molecules 6 1. The Order Matrix 6 2. The Nuclear Spin Hamiltonian 8 a. The Zeeman Interaction 8 b. Indirect or J Coupling 9 c. Direct Dipole-Dipole Coupling 10 d. Quadrupolar Coupling 11 3. Spectral Analysis 12 4. NMR Using Liquid Crystalline Solvents 14 C. Outline of Thesis 14 II. Experimental 19 A. Preparation of liquid crystals 20 1. General 20 2. Preparation of EBBA-d2 20 a. Syntheses 20 3. Preparation of 1132/EBBA Mixtures 22 B. Preparation of solutes 23 1. Hydrogens 23 2. Acetylene / Acetylene-di / Acetylene-d2 23 C. Preparation of NMR samples 24 1. Hydrogens 24 2. Methanes 24 3. Acetylenes 25 4. Other Solutes 25 D. NMR spectra 26 ITI. Results and Discussion 28 A. Hydrogen 28 1. Spectra of Partially Ordered H 2 , HD and D2 28 2. Dipolar and Quadrupolar Coupling Constants 30 3. External Electric Field Gradients 33 iii 4. Electric Field Gradient Molecular Quadrupole Moment Interaction 36 5. D2 in 1132/EBBA Mixtures 37 6. Isotope and Temperature Effects 41 a. 60 wt per cent 1132/EBBA 41 b. 50 and 55wt per cent 1132/EBBA 46 7. Summary 49 B. Methane 52 1. Introduction 52 2. Theory 53 a. Solvent-Solute Interaction Potential 53 b. Calculation of Observables 54 3. Methanes in Liquid Crystal Mixtures 59 4. Summary 63 C. Acetylene 64 1. Introduction 64 2. Experimental Results 66 3. Theoretical Considerations 69 4. Calculation of Order Parameters 70 5. Summary 79 D. Size and Shape Effects 80 1. Introduction 80 2. Short Range Interaction Model 83 a. The Interaction Potential 83 b. The Short Range Potential 84 c. Calculation of Order Parameters 88 3. Ordering of solutes in 55wt per cent 1132/EBBA 89 a. Solutes with C3 or Higher Symmetry 89 b. Solutes of Lower Symmetry 96 c. Liquid Crystal Spectrum as an Internal Standard .... I l l d. Summary 120 4. Ordering of solutes in other liquid crystals 120 a. 1132 and EBBA 121 b. 5CB-a,0-d4 1 2 6 IV. Conclusions 137 V. Appendix I ... 139 VI. Appendix U 145 A. Calculation of Maximum and Minimum Circumferences 145 1. Projection of solute 145 2. Maximum circumference 146 3. Minimum circumference 146 VTI. Appendix HI 148 A. Source Code for the Calculation of Expression (51) 148 Vffl. Bibliography 159 iv List of Figures FIGURE 1: Representation of a Nematic Phase 4 FIGURE 2: l H Spectrum of 2,4-Hexadiyne Dissolved in 55wt per cent 1132/EBBA 15 FIGURE 3: Synthesis of EBBA-d2 21 FIGURE 4: l H Spectrum of H2 and HD in a Nematic Phase 29 FIGURE 5A: 2R Spectrum of D2 and HD in a Nematic Phase 30 FIGURE 5B: 2H Spectrum of D2 in a Nematic Phase 31 FIGURE 7: Spectra of D2 in 1132/EBBA Mixtures 38 FIGURE 8: Ordering of Molecular Hydrogen in Nematic Liquid Crystals. I 40 FIGURE 9: Ordering of Molecular Hydrogen in Nematic Liquid Crystals. H .... 42 FIGURE 10: Dipolar Couplings for H2, HD and D2 in 60wt per cent 1132/EBBA 44 FIGURE 11: Quadrupolar Couplings for HD and D2 in 60wt per cent 1132/EBBA 45 FIGURE 12: Dipolar Couplings for H2, HD and D2 in 50wt per cent 1132/EBBA 47 FIGURE 13: Dipolar Couplings for H2, HD and D2 in 55wt per cent 1132/EBBA 48 FIGURE 14: Elastic Tube Model for the Short Range Interactions 85 FIGURE 15: Projection of Solute in XY Plane 86 FIGURE 16A: Short Range interaction Model (Minimum Circumference) 92 FIGURE 16B: Short Range interaction Model (Maximum Circumference) 93 FIGURE 17: Polarizability - Electric Field Mechanism 98 FIGURE 18: Short Range Interaction Model; Experimental vs Calculated Order Parameters in 55wt per cent 1132/EBBA-d2 103 FIGURE 19: Polarizability - Electric Field Interaction in 55wt per cent 1132/EBBA-d2 105 v FIGURE 20A: Asymmetry Parameters for Solutes of C2v and D2h Symmetry : Short Range Interaction Model 109 FIGURE 20B: Asymmetry Parameters for Solutes of C2v and D2h Symmetry Polarizability - Electric Field Interaction 110 FIGURE 21: Structure and Molecule Fixed Axes; EBBA-d2 H2 FIGURE 22: Force Constant in 55wt per cent 1132/EBBA-d2 117 FIGURE 23: Electric Field Gradients in 55wt per cent 1132/EBBA-d2 118 FIGURE 24: Solutes in EBBA-d2 Short Range Interaction Model 124 FIGURE 25: Solutes in 1132 Short Range Interaction Model 125 FIGURE 26: Solutes in 5CB-a,/3-d4: Short Range Interaction Model 129 FIGURE 27: Field Gadients in 5CB-a,/3-d4 131 FIGURE 28: Force Constant in 5CB-a,/3-d4 132 FIGURE 29: Solutes in 5CB-a,/3-d4: Short Range Interaction Model: 1,3,5-trichlorobenzene as Reference 134 FIGURE 30: 2,4-hexadiyne in 5CB-a,j3-d4 135 vi List of Tables TABLE 1: Electric Field Gradients in 1132 and EBBA-d2 37 TABLE 2: D2 in Mixtures of 1132 and EBBA 39 TABLE 3: H2, HD and D2 Dipolar Couplings in Various Liquid Crystals 50 TABLE 4: Methane in 1132, EBBA and 61wt per cent 1132/EBBA at 31 OK: Experimental and Calculated Dipolar Couplings and G/3 Parameters 60 TABLE 5A: Methane in 1132, EBBA and 61wt per cent 1132/EBBA at 31 OK: Experimental and Calculated Quadrupolar Couplings (Hz) ...62 TABLE 5B: Parameters qeq and 9q/9SF 2 With and Without External Field Gradient Corrections 63 TABLE 6: Acetylene in 1132, EBBA and 55wt per cent 1132/EBBA at 3OIK: Experimental Dipolar Couplings (Hz) 67 TABLE 7: Acetylene in 1132, EBBA and 55wt per cent 1132/EBBA at 30 IK: Calculated and Experimental Dipolar Couplings (Hz) 71 TABLE 8: Acetylene in 1132, EBBA and 55wt per cent 1132/EBBA at 30IK: Calculated and Experimental Dipolar Couplings and the Anisotropy in the Indirect C-C Coupling (Hz) 76 TABLE 9: Spectral Parameters for Solutes with C3 or Higher Symmetry 90 TABLE 10: Experimental and Calculated Order Parameters for Solutes with C3 or Higher Symmetry Dissolved in 55wt per cent 1132/EBBA at 301.4K 94 TABLE 11: Solute Polarizabilities and References 97 TABLE 12: Experimental an Calculated Solute Order Parameters in 55wt per cent 1132 at 301.4K 99 TABLE 13: Force Constants, Electric Fields, Liquid Crystal Deuteron Splittings and Order Matrix Elements(55wt per cent 1132/EBBA-d2 at 301.4K) 107 TABLE 14: EBBA-d2 AT 301.4K: Liquid Crystal Order Parameters, Deuteron Quadrupolar Splittings and Dipolar Couplings 114 TABLE 15A: Furan and Thiophene in 55wt per cent 1132/EBBA-d2; Experimental Solute Order Parameters 115 vii TABLE 15B: Furan and Thiophene in 55wt per cent 1132/EBBA-d2:Liquid Crystal Order Parameters, Deuteron Quadrupolar Splittings and Dipolar Coupling Constants 116 TABLE 16: D2 in 55wt per cent 1132/EBBA-d2 119 TABLE 17: Molecular Quadrupole Moments 122 TABLE 18: Solute Order Parameters: 1132 and EBBA-d2 at 301.4K 123 TABLE 19: D2 in 5CB 127 TABLE 20: Order Parameters of Solutes in 5CB at 294K 127 viii LIST OF ABBREVIATIONS NMR Nuclear Magnetic Resonance S order matrix S j^ kl element of order matrix J ajj indirect spin-spin coupling between spins a and b A J a b anisotropy in J & b direct dipole-dipole coupling between nuclei a and b B f l quadrupolar coupling of nucleus a QJJ nuclear quadrupole moment of deuteron q internal electric field gradient tensor at the site of the deuteron EBBA N-(4-ethoxybenzylidene)-4'-n-butylaniline 1132 Merck ZLI 1132: A mixture of three phenylcyclohexanes and one biphenylcyclohexane (see chapter III) 5CB-a,/3-d4 4-(a,/3-d4-rc-pentyl)-4'-cyanobiphenyl. TTF tetrathiofulvalene U solute-solvent interaction potential F^2 external electric field gradient Q 2 Z zz component of the molecular quadrupole moment tensor 7H, 7 D gyromagnetic ratio of proton (deuteron) G&kl parameter describing so vent-solute interaction strength Q M mth normal mode of vibration of solute k elastic force constant anisotropy in mean square electric field ix ACKNOWLEDGEMENTS I wish to thank Dr. E.E. Burnell for his guidance and support. His positive attitude and the many hours of discussions we have had, have been a tremendous help to me. It has been an especially great pleasure to have worked with J.C.T. Rendell and E.J. Delikatny. I am grateful for their help and friendship. During the course of this work I was fortunate enough to spend a year at the Vrije Universiteit in Amsterdam and I am indebted to Dr. C.A. de Lange and Dr. J.G. Snijders for their hospitality and support during my stay. I especially wish to thank Drs. J.B.S. Barnhoorn, with whom I worked quite closely, for making it a most rewarding experience. My thanks are also extended to Dr. P.B. Barker, Dr. G.S. Bates, Dr. G.L. Hoatson, M.Y. Kok, Dr. G.N. Patey, and A. Weaver for their collaboration in various aspects of this work. I also wish to thank Dr. F.G Herring and Dr. D.C. Walker for their helpful suggestions I must reserve my deepest gratitude for my wife Karin, whose patience and understanding I am very thankful for. I am also very appreciative of her help in typing this thesis and preparing some of the figures. x I. INTRODUCTION When a molecule is placed in an anisotropic environment its reorientation may also become anisotropic. That is, not all orientations of the molecule relative to an external frame of reference are equally probable. This is a result of the fact that most molecular properties are anisotropic so that their interaction with an anisotropic environment is orientation dependent. Nematic liquid crystals are anisotropic fluids and as solvents they provide anisotropic environments for solutes. The measurement of any property which depends on the orientation of the solute relative to a space fixed axes system can provide information about the solvent-solute interactions in such systems [1-3]. NMR is ideally suited for this purpose because quantities such as the dipolar and quadrupolar couplings and the anisotropy in the chemical shifts, which depend on the orientation of the solute and which average to zero in isotropic solvents, become observable. The orientation dependent part of the dipolar and quadrupolar couplings is often written as a 3X3 matrix called the order matrix [1] and the individual elements of the matrix are called the order parameters of the solute. The order parameters of a solute are a measure of its average orientation and its interaction with the liquid crystalline environment. Despite the large number of solutes and wide range of liquid crystalline solvents which have been studied [2-5], the exact nature of the intermolecular forces which lead to solute ordering is still not understood. Past attempts at understanding the ordering of solutes have generally centred around correlating the solute order parameters with molecular properties such as the polarizability [6-10], moments of inertia [11, 12] and molecular shape [6, 13]. In a number of other studies the solute ordering has been described in terms of an unspecified interaction potential [10, 14, 15, 16-20]. The 1 Introduction / 2 success of such studies has, however, been limited and special mechanisms are invoked to explain the ordering of solutes which do not correlate well with the chosen molecular property [21]. In this thesis we will reexamine this problem using a slightly different approach [22]. We will begin by studying in detail the NMR spectra of isotopically substituted modifications of some very simple solutes such as molecular hydrogen and methane. Because of the simplicity of these solutes and because a great deal is known about their physical properties, it should be possible to obtain specific information about the mechanisms which lead to their orientational ordering in the liquid crystals used. We will then use the knowledge gained from these studies in order to vary the liquid crystal environment in a controlled fashion. This should allow a more systematic study of the ordering of larger solutes whose molecular properties are not as well known. For these studies we will also use partially deuterated liquid crystals which will allow us to examine the relationship between solute and solvent ordering. A complete description of the ordering of the solvent is complicated by the internal motions of the liquid crystal molecules [23] and we shall restrict ourselves to using the liquid crystal spectrum as a measure of the anisotropic environment experienced by the solutes. As will be seen this approach is quite successful and we will be able to make some useful conclusions concerning the relative importance of various mechanisms to the orientational ordering of a number of solutes in several liquid crystalline solvents. We begin by presenting a brief description of liquid crystalline solvents and the NMR spectra of solutes in such solvents. Introduction / 3 A. LIQUID CRYSTALS The first reported observation of liquid crytalline behaviour was made by Reinitzer [24] who found that cholesteryl benzoate melted at 145.5 °C to form an opaque fluid which cleared suddenly at 178.5 °C. He brought this compound, as well as cholesteryl acetate, to the attention of Lehman [25] who studied the opaque fluid under the polarizing microscope and discovered that it had the optical properties of an uniaxial crystal. He proposed the term "fliissige Kristalle" (liquid crystal) to describe this fluid since it displayed properties of both liquids and crystals. Many compounds have since been discovered which show similar behaviour and it is now known that a wide range of intermediate phases exist between the solid and liquid. These various phases were first described systematically by Friedel [26] and despite his strong objection, the term liquid crystal has come to mean those states of matter whose degree of molecular ordering lies between that of a crystalline solid and an isotropic liquid [27]. In this thesis we will be concerned with the least ordered of these phases. Friedel proposed the name nematic from the Greek word vqua (thread) because of the thread-like discontinuities observed under the polarizing microscope for this phase. The molecules which form this phase are, in general, long and rod-like and are arranged such that,as in an isotropic liquid, there is no long range positional ordering of their centres of mass. Unlike an isotropic liquid, however, all molecules in the phase have the same average orientation with respect to a unique direction referred to as the director. This direction usually lies approximately parallel to the long axes of the molecules. A "snap shot" representation of the nematic phase is shown in figure 1. In the absence of any Introduction / 4 FIGURE 1 Representation of a Nematic Phase A "snap shot" representation of a nematic phase. The long rod-like molecules, represented by the rods in the figure, are aligned so that their long axes are on average parallel to the director, Z. There is no positional ordering of the centres of mass of the molecules. Introduction / 5 external fields the director varies randomly throughout the sample, whereas, in the presence of a strong magnetic field, such as is used in NMR experiments, it normally aligns parallel to the field direction. Alignment of the director may also occur in the presence of an electric field and at the surface of prepared glass plates. A second more ordered liquid crystalline phase, was named smectic by Friedel from the Greek word oprjyqa (soap) because it was first discovered in potassium and ammonium oleate. In this phase the molecules are aligned parallel to the director as in the nematic phase but they also have some degree of positional ordering of their centres of mass. Several modifications of this phase exist and they are labeled smectic A, smectic B, etc. depending on the particular arrangement of the molecules. Both the nematic and smectic phases are examples of thermotropic liquid crystals, so named because the liquid crystallinity is primarily temperature induced. There exists a second class of liquid crystals formed by amphiphilic compounds and whose liquid crystalline behaviour is induced by the presence of water. Such systems are usually referred to as lyotropic liquid crystals and they have attracted a great deal of attention because of their similarity to lipid membranes. Because of its simplicity, the nematic phase is much more attractive as a system for studying intermolecular interactions than either the smectic phase or lytropic liquid crystals. The ordering of the molecules in the nematic phase is completely described by the time or ensemble average of the orientation of a molecule fixed axis system with respect to a single space fixed axis. This is also Introduction / 6 true of solutes dissolved in this phase. We will now describe the effect of such ordering on the NMR spectrum of a molecule and discuss how the parameters obtained from the NMR spectrum can be related to the ordering of the molecule. B. NMR OF PARTIALLY ORIENTED MOLECULES 1. The Order Matrix Before discussing the NMR spectrum of partially oriented molecules we will first introduce the order matrix [1] because it will prove useful in much of what follows. Many of the properties of the molecules that we will be concerned with are of second rank tensorial form. Because we can only measure the ensemble or time average of these properties in the laboratory fixed frame of reference, the transformation of the tensor from a molecule fixed axis sj'stem to a space fixed axis system, averaged over the motions of the molecule, is of interest. For a second rank tensor, T, this is given by: <TU> = E <cos0fc, cos0 u> l k l , (1) k.,1 where <TJJ> are the components of T averaged over the molecular motions in the space fixed axis system and are the components in the molecule fixed frame. (We will use upper case labels for space fixed axes and lower case labels for the molecule fixed axes throughout). is the angle between axes k and I and we have assumed that the components of the tensor in the molecule fixed axes are independent of the internal motions of the molecule. For a molecule in a nematic liquid crystal, the orientations 0^ 1 a n d ~&kl a r e e ( i u a u y probable but the individual values of Introduction / 7 dfri are not. Because of this <cos0£j> is zero but <cos 0^j> is not. In most cases, when such a system is placed in a strong magnetic field the director is oriented parallel to the field direction. Because of this, and the fact that there is no positional ordering of the molecules, the lab fixed frame of reference is axially symmetric. This leads to the result < T I J > = 0 for I / J and < T X X > = < T Y Y : > if Z is the symmetry axis of the lab frame. The trace of the tensor must be invariant to rotations so that <T'xx> = < TYY> = ( f T « - <Tzz>)/2 • ( 2 ) Thus we need only specify <r^2,z> a n c ^ ^ e t r a c e < Expression (1) then becomes <TZZ> = -3 E T ^ + I £<f co s0 a cos0 l z - J 6kl>Tkl , (3) where 6^ / is the Kronecker 6. (1/3)Z T -^ is often referred to as the isotropic i part of T because it is the value observed if the molecular motion is isotropic. It is convenient to collect the angular dependent part of expressions (3) as a 3X3 matrix S such that Skl ~ 1 < 3 C O S 0 A 2 C O S 0 H ~ 6kl> . (4) This matrix is called the order matrix. It is real, symmetric and traceless and has the transformation properties of a second rank tensor under rotations of the molecule fixed axis system. In general it has five independent elements. However, through an appropriate transformation of axes it can be diagonalized leaving only two independent non-zero elements. If the molecule possesses a three fold or higher rotation axis then in an axis system with z parallel to the rotation axis Introduction / 8 S j^ = 0, k / 1 and Sxx = S = ~2$zz- Thus S has only one independent element, Szz, which is referred to as the order parameter of such a molecule. In this case Szz is the average of P2(cos#2 )> the second Legendre polynomial of cosfl2Z. 2. The Nuclear Spin Hamiltonian The Hamiltonian for a spin system in a large external magnetic field consists of contributions from (i) the Zeeman interaction of the nuclear spins with the magnetic field, (ii) the indirect or J coupling between the nuclear spins via the electron spins, (iii) the direct dipole-dipole coupling of the nuclear magnetic moments and (iv) for spins greater than 1/2, the interaction of the nuclear quadrupole moment of each spin with the electric field gradient at the site of the nucleus. The total Hamiltonian may be written as a sum of contributions from each of these terms. K = X Z + #J + #D + #Q ( 5 ) c. The Zeeman Interaction The Zeeman Hamiltonian has the form * Z = " f I j . . ( 6 ) where v& is the resonance frequency of nucleus a and l^a is the operator for the component of the spin of nucleus a along the field direction. The resonance frequency depends of the local field at the site of the nucleus and is given by Introduction / 9 7 a n H Z . (7) "a = ~ (1 " ^ZZ,a) » 27T where y& is the gyromagnetic ratio of nucleus a, h is Planck's constant, is the applied magnetic field and O"^ a is the ZZ component of the shielding tensor of nucleus a and is averaged over the motions of the molecules. From expressions (3) and (4) we have _ 1 2 °ZZ,s ~ 3 f a i i * + 3 ? , Skl°kl,a • (8) t kl In an isotropic solution the second term vanishes and <7 Z Z a = \ ^ a iia i b. Indirect or J Coupling The contribution to the Hamiltonian from the indirect spin-spin coupling between spins a and b is given by: #j = S E I a • J a b • I b , a b>a w where J a b is the second rank tensor describing the strength of the indirect coupling between spins a and b. Again using expressions (2-4) this becomes # j = E 2 5 E J « . a b I. • I b + a b>a t I 2 2 ? J w > a b Skl (3 l Z a l Z b - I. • Ib) . (10) a b>a kl The second term is again zero for an isotropically tumbling molecule and (1 /3)Z JJ-J- a b is the scalar coupling observed in a normal high resolution N M R i experiment. For most couplings involving protons the anisotropic part of expression (10) is small and may be ignored [5]. This has important Introduction / 10 consequences because, as we shall see in a moment, the direct dipole-dipole coupling has exactly the same form as this term and thus the two coupling constants cannot be distinguished in the spectrum. c. Direct Dipole-Dipole Coupling The contribution to the Hamiltonian from the dipolar coupling between the spins in a molecule is given by MD = E E D a b (3 l Z a I Z b - I a . ] b) , (ii) a b>a where _ h7 a7b, 3 c o s 20 a b z - 1 Dab = ~ ( — 2 - ) < 3 > • ( 1 2 > 4 7 r 2 r a b 6?abZ is the angle between the laboratory Z axis and the vector joining the two nuclei a and b. If the averaging over the vibrations and rotations can be done separately then expression (12) becomes n _ _ ( h 7 a 7 b \ < r " 3 > S w (13) ^ab _ V . 2 / ^ r a b ^ 0ab » 47T where S a b is the order parameter of the vector between nuclei a and b and may be related to the elements of the order matrix, S of the molecule fixed axis system. This separation of the vibrational and rotational averages is by no means always valid and we shall see evidence of this in the spectrum of methane. For most solutes the effect of separating the two averages is quite small and the dipolar couplings can be used to determine the order matrix. Moreover, if one distance in the molecule is fixed, then the average structure Introduction / 11 may be determined. It is also apparent that for an isotropically tumbling molecule the dipolar couplings average to zero. d. Quadrupolar Coupling In anisotropic environments there is also a contribution to the Hamiltonian from the coupling between the quadrupole moment of nuclei with spin > 1/2 and the electric field gradient at the site of the nucleus. The Hamiltonian for this interaction has the form #Q = £ I. ' B a • 1. . (14) The tensor B a , which describes the interaction strength of the nuclear quadrupole moment of nucleus a with the electric field gradient, is symmetric and traceless so that from expressions (3) and (4) we have *Q = 5 J BZZ.a (3IZa " I.*) . (15) where a is the ZZ component of tensor B & and is given by 4hI a [2I a - l ] kl Q a is the quadrupole moment of nucleus a, e is the charge of the electron, I a is the spin of nucleus a and q j^ are the elements of the electric field gradient tensor at the site of nucleus a, averaged over the internal motions of the molecule. We have again made the assumtion that the averages over vibrations and rotations are separable. B ^ a is, of course, zero for isotropic motion and its value may be used to determine the elements of the order matrix for anisotropic Introduction / 12 motion if Q & and are known. Unfortunately, the electric field gradient tensor is very difficult to determine accurately. In this thesis is the only nucleus with I > 1/2 which will be studied. For many compounds the electric field gradient tensor at the site of the deuteron may be taken as axially symmetric which allows a considerable simplification of expression (16). Using values of the field gradient tensor from single crystal or molecular beam studies, the order matrix at the site of the deuteron may then be determined fairly accurately from the spectrum. These, then, are the various contributions to the total Hamiltonian which is given by a 8 b>a z + I g,. (».b + -31 J«,b s „ ) (3 iz. i Z b - i . • ib) + i ? B. (31* - I.2) . 3. Spectral Analysis The analysis of the spectrum of a given spin system inolves calculation of the matrix elements of H with the various spin states of the system. The transition frequencies may then be calculated from the eigenvalues of the Hamiltonian matrix and are functions of the various coupling constants and chemical shifts. These values are adjusted until agreement between the calculated and experimental spectra is obtained. This procedure is referred to as fitting a Introduction / 13 spectrum and is usually done with the aid of a computer. For many simple highly symmetric molecules the analysis of the spectrum is trivial and the coupling constants may be obtained directly from the line positions. In this thesis all spectra, whose analysis is not trivial, have been fit using the computer program LEQUOR [28]. The elements of the order matrix in an appropriate molecule fixed axis system may be calculated from the values of the dipolar and quadrupolar couplings. If the dipolar couplings are used, the average geometry of the molecule must be known. However, in general, the average geometry of a molecule, obtained by electron diffraction or microwave spectroscopy, is not consistent with the observed dipolar couplings. This is a result of either changes in the vibrational and conformational averaging of the solute in the liquid crystalline phase or contributions to the dipolar couplings due to the interaction between the vibrations and rotations of the molecule. A proper treatment of the vibrational problem requires some knowledge of the solvent-solute interaction mechanism and, except for the simplest of solutes, is rather complicated [14-16, 29]. Because of this, it is usually assumed that the averages over the vibrations and rotations may be done separately and a least squares fit of the average geometry to the dipolar couplings is done. The order matrix for this fitted geometry is then obtained. The computer program SHAPE [30] has been developed for this purpose and will be used where necessary to obtain the order matrix of the solutes studied here. Introduction / 14 4. NMR Using Liquid Crystalline Solvents In this thesis we will use the procedure just described to obtain the dipolar and quadrupolar coupling constants of solutes dissolved in a variety of nematic liquid crystals and we will attempt to explain these results (or order parameters obtained from them) in terms of the solute-solvent interactions. We will ignore contributions to the measured dipolar coupling due to the anisotropy in the indirect coupling and we will not be concerned with the anisotropy in the chemical shifts. Figure 2 shows the spectrum of 2,4-hexadiyne in the nematic liquid crystal mixture 55wt per cent 1132/EBBA-d2- The sharp lines are due to the solute and the broad featureless peaks in the baseline are due to the nematic solvent. This difference in linewidths arises because the enormous number of lines in the spectrum of the liquid crystal cannot be resolved and because of the slower tumbling of the solvent molecules due to their large size. This is convenient because it eliminates the need to remove the solvent spectrum as is done when using isotropic solvents. C. OUTLINE OF THESIS Our goal is to interpret the dipolar and quadrupolar couplings and order parameters obtained from spectra such as shown in figure 2, in terms of the intermolecular interactions. To do this, it is important to choose the system carefully because of the problems associated with the interpretation of both the dipolar and quadrupolar couplings. The dipolar couplings depend on the average over all of the motions of the molecule so that we need to study systems in which the internal molecular motions may be either safely ignored or described Introduction / 15 FIGURE 2 l H Spectrum of 2,4-Hexadiyne Dissolved in 55wt per cent 1132/EBBA i 1 1 1 1 1 1 1 1 1 1 -6.0 -2.5 OJO ZJ6 SJO Frequency (kHz) The narrow peaks are due to the solute and the broad peaks in the baseline are due to the liquid crystalline solvent. Solute concentration = ~1 mole per cent, T = 301.4K. very simply. Quadrupolar couplings can only be observed in molecules which contain deuterons or some other quadrupolar nucleus. The interpretation of the couplings requires a knowledge of the nuclear quadrupole moment and the electric field gradient tensor at the site of the nucleus. These requirements can only be met for the simplest of molecules. Thus we are restricted, initially at least, to studying very simple solutes. We will begin by presenting results obtained for molecular hydrogen and its deuterated analogues [10, 16, 31, 32], For this solute both the dipolar and quadrupolar couplings can be related quite accurately to the Introduction / 16 molecular ordering. Moreover, the electronic properties of this solute are very well known so that it should be possible to evaluate the relative importance of mechanisms involving these properties. It has been shown that this solute experiences an external electric field gradient due to its liquid crystalline environment and that the interaction between the molecular quadrupole moment and the anisotropy in this field gradient accounts for most of the solute ordering [31, 32]. Quantum mechanical effects must be taken into account in order to explain the dipolar couplings of H^, HD and D2 [10, 16]. By mixing two liquid crystals where the measured electric field gradients have opposite sign it will be shown that a system may be produced where the field gradient has values intermediate between those in the two component liquid crystals (see also reference [32]). In these liquid crystal mixtures some very unusual isotope effects are observed in the ordering of molecular hydrogen. These results will be discussed in terms of some posible contributions to the mechanisms which lead to orientational ordering. Because of the small number of couplings which can be measured in hydrogen and because of its extremely simple structure, this solute does not provide a very good system for the study of the influence of internal motion on the observed spectrum. Methane and its deuterated analogues are ideally suited for this purpose [33]. Because these molecules are tetrahedral in their equilibrium geometry, any observed spectrum can only arise because of the coupling between the molecular vibrations and rotations and, in the case of the partially deuterated species, deviations of their average geometries from tetrahedral symmetry. These two effects may be calculated [14-16, 34] in terms of the well known vibrational Introduction / 17 force field [35] and the strength of the solvent solute interaction. If the latter interaction is taken to be of second rank tensorial form the observed dipolar couplings [33] are predicted very accurately [14, 15, 36]. It will be shown that this solute experiences the same external electric field gradient as hydrogen, but that the interaction with the molecular quadrupole moment does not account for all of the observed dipolar couplings (see also reference [36]). Methane and hydrogen are unusual solutes because of their small size and, in the case of methane, very high symmetry. Thus we will consider a solute where vibrational effects are expected to play a smaller but still important role. For this purpose we will choose acetylene. We will see that for this solute the electric field gradient - molecular quadrupole moment mechanism plays an important role and that there is a second mechanism present which is roughly proportional to the size and shape of the solute. The results for hydrogen, methane and acetylene provide a basis for the study of other solutes [37]. For all three of these solutes the electric field gradient -molecular quadropole moment mechanism plays an important role, although it is not the only mechanism. From the results on hydrogen it has been shown [32] that a liquid crystal mixture where the field gradient is zero may be produced. This mixture, then, provides a system for studying other mechanisms [38, 39]. For most solutes vibrational effects are small and will be ignored. Because the short range interactions are expected to play an important role for larger solutes, we will develop a simple model for this mechanism and it will be shown that this model can be used to successfully predict the order parameters of a large number of solutes in the zero field gradient mixture (see also reference [38]). Introduction / 18 Finally, we will see that the short range interaction model combined with the electric field gradient - molecular quadrupole moment mechanism predicts the ordering of solutes quite well in liquid crystals where the electric field gradient is non-zero [38-40]. The results discussed above and the relevant theory will be presented following chapter II which deals with the experimental details. Chapter III is divided up into sections dealing with hydrogen, methane and acetylene respectively, as well as a discussion of the short range interaction model. II. EXPERIMENTAL The liquid crystals used were; EBBA: N-(4-ethoxybenzylidene)-4'-M-butylaniline, EBBA-d2: N-(4-ethoxybenzylidene)-2,6-dideutero-4'-n-butylaniline. D. 1132: An eutectic mixture of frans-4-n-alkyl-(4-cyanophenyl)-cyclohexane (alkyl = propyl, pentyl and heptyl) and frans-4-n-pentyl-(4'-cyanobiphenyl-4)-cyclohexane [5]. 3y R = C 4 H g (24 per cent) C-H.,., (36 per cent) o n C _ H , C (25 per cent) / 15 C H 3 ( C H 2 ) 4 - i 5CB-a,/3-d4: 4-(a,p,-d4-n-pentyl)-4'-cyanobiphenyl. C H 3 C H 2 C H 2 C D 2 C D 2 19 Experimental / 20 A. PREPARATION OF LIQUID CRYSTALS 1. General All transition temperatures reported, were measured using a Gallenkamp melting point apparatus and are uncorrected. Flash evaporation of all solutions was performed using a Buchi rotary evaporator under reduced pressure. 1132 and non deuterated EBBA were obtained from Merck and K&K respectively. Both compounds were used without further purification. Measured transition temperatures: EBBA (solid-nematic 308 ± IK, nematic-isotropic 351 ± IK) 1132 (nematic-isotropic 347 ± IK). 5CB-a,/3-d^ synthesized in high purity was provided by Dr. G. S. Bates [40, 41]. 2. Preparation of E B B A - d 2 Partially deuterated EBBA was prepared according to the following reaction sequence shown in figure 3. c. Syntheses N,N-2,6-tetradeutero-n-butylaniline (reference [42]): An aqueous solution of 4-n-butylaniline hydrochloride (124 g, 0.66 moles) was prepared by adding freshly distilled 4-n-butylaniline (100 g, 0.66 moles) to 2M HCI (350 ml). The water was then removed by flash evaporation. The hydrochloride salt was then dissolved in D2O (100 ml) and refluxed for several hours. The resulting solution was neutralized by the dropwise addition of a saturated solution of KOH. The partially deuterated 4-n-butylaniline was then extracted with ether which was subsequently removed by flash evaporation leaving a brown oily liquid. The crude Experimental / 21 FIGURE 3 Synthesis of EBBA-02 product was then purified by vacuum distillation. Experimental / 22 EBBA-d 2 (reference [43]): A solution of N,N-2,6-d4-n-butylaniline (60 g, 0.4 moles), 4-ethoxybenzaldehyde (60 g, 0.4 moles) and 4-toluenesulfonic acid (0.75 g, 4.4 mmoles) in toluene (600 mis) was refluxed for 4 hours. The water formed during the reaction was removed by means of a Dean-Stark apparatus. The toluene was then removed by flash evaporation and the crude EBBA-d2 was recrystallized from methanol and dried under vacuum (yield 61 g, 55 per cent, degree of deuteration: NMR 63 per cent, mass spectroscopy 61 per cent). The measured solid-nematic and nematic-isotropic transition temperatures (308 ± IK; 351 ± IK) are in good agreement with literature values [44] for the non-deuterated compound. 3. Preparation of 1132/EBBA Mixtures Several mixtures of EBBA or EBBA-d2 and 1132 were prepared by adding appropriate weights of each of the liquid crystals to a vial or Erlenmeyer flask, heating the mixture to the isotropic phase, and stirring it vigorously using a Vortex stirrer. B. PREPARATION OF SOLUTES Experimental / 23 1. Hydrogens [10] High purity H 2 was purchased from Matheson Ltd. and D 2 was produced by electrolyzing 99 per cent pure D 2 0 purchased from Merck, Sharp & Dohme. Statistical mixtures of H 2 / HD / D 2 were prepared by mixing varying amounts of H 2 and D 2 in a 100 ml round bottom flask attached to a vacuum line and fitted with a platinum wire which could be heated to ~1300K by passing a current through it. The wire was then heated for approximately 1 minute to produce the H 2 / HD / D 2 mixture. 2. Acetylene / Acetylene-d^ / Acetylene-d2 Solid calcium carbide (5 g, 0.1 moles) was placed in a 100 ml round bottom flask equipped with a dropping funnel and a magnetic stirrer. 4M H 2 S 0 4 (50 mis) in a 50 per cent mixture of D 2 0 and H 2 0 was added slowly to the calcium carbide. The resulting mixture was stirred at room temperature for several hours. The gas evolved during the reaction was driven off with a stream of nitrogen, passed through a H 2 S 0 4 / CuS0 4 solution, and collected in a cold trap cooled to liquid nitrogen temperature. The cold trap was then attached to a vacuum rack and the temperature was raised to approx. —10 °C with a salt/ice bath. The acetylene evolved was then condensed in a clean, dry flask cooled to liquid nitrogen temperature. The proton NMR spectrum of the product dissolved in EBBA showed no impurities and an approximately 2:1 ratio of C 2 H D to C 2 H 2 . No further analysis of the purity of the product was done. Experimental / 24 C. PREPARATION OF NMR SAMPLES For all solutes, a sufficient volume of one of the liquid crystals to fill the NMR coils was placed in either a standard or medium walled 5 mm NMR tube or a 9 mm o.d. standard pyrex glass tube. The sample tubes were then attached to a vacuum line via a piece of rubber tubing and the liquid crystal was thoroughly degassed by performing repeated freeze-pump-thaw cycles until no more gas was evolved. Solutes were then dissolved in the degassed liquid crystals as follows: 1. Hydrogens [10, 32] 9 mm o.d. sample tubes containing degassed liquid crystals were placed in a dewar containing liquid helium and a sufficient volume of hydrogen (H2, D2 or a mixture of H2 / HD / D2) to produce a final pressure of 15 to 20 atm in the head volume of the tube at room temperature, was condensed into each tube. The tubes were then flame sealed and pressure tested in an oven at approximately 42 5K for 10-15 minutes. 2. Methanes [14, 15, 33, 36] 1 3 C H 4 (90 per cent 1 3 C ) , 1 3 C H g D (90 per cent 1 3 C , 98 per cent D), C H 2 D 2 (98 per cent D), CHDg (98 per cent D) were obtained from Merck, Sharp and Dohme (Canada) and used without purification. Samples of * 3 C H 4 / CD4, 1 3 C H 3 D , C H 2 D 2 and CH3D were prepared in 1132, EBBA, and 61.3wt per cent 1132/EBBA, in a similar manner to the hydrogen samples except that (i) for the samples in 1132, 5 mm medium walled NMR tubes were used, (ii) liquid nitrogen was used to condense the gasses and (iii) the final pressures were Experimental / 25 between 5-15 atm. 3. Acetylenes 1 3 C 2 H 2 (90 per cent 1 3 C ) and 1 3 C 2 D 2 (98 per cent 1 3 C , 99 per cent D) were obtained from Merck, Sharp and Dohme (Canada) and used without purification. A mixture of C 2 H 2 , C 2 H D and C 2 D 2 was prepared by the hydrolysis of calcium carbide (see above). Samples containing a mixture of all of the isotopically substituted acetylenes were prepared in 1132, EBBA and 55wt per cent 1132/EBBA. All samples were prepared in 9 mm o.d. tubes in the same manner as the methane samples. The volume of gas condensed was sufficient to produce a final pressure of 5 atm in the head volume of the sample tube at room temperature. 4. Other Solutes [38-40] The 28 solutes listed in table 12 (chapter III, section D) were obtained from a variety of chemical suppliers and used without purification. Samples were prepared in one of EBBA, EBBA-d 2 1132, 55wt per cent 1132/EBBA, 55wt per cent 1132/EBBA-d2, or 5CB-a,/3-d4. For the solutes which are solids or liquids at room temperature a sufficient amount of the solute to produce a final concentration of 1-2 mole per cent was added to the degassed liquid crystal in a standard 5 mm NMR tube. The tubes were then capped and sealed with paraffin film, heated until the liquid crystal became isotropic, and stirred vigorously using a Vortex stirrer. All samples containing EBBA or EBBA-d 2 and CHgl or CHgBr became orange over a period of several hours if allowed to stand at room temperature. In one sample (CH^Br / 55wt per cent 1132/EBBA) Experimental / 26 a considerable amount of another compound was apparent in the NMR spectrum after the sample was allowed to stand at room temperature overnight. No attempt was made to identify this compound. To avoid these problems all iodo-and bromomethane samples were stored at —15 °C until their spectra were measured. For the solutes which are gasses at room temperature samples were prepared in the same manner as for the methanes and acetylenes. For two solutes, allene and propyne, a pressure of approximately 5 atm caused a noticeable drop in the nematic to isotropic phase transition temperatures in 1132, EBB A and 55wt per cent 1132 indicating a solute concentration of considerably more than 1-2 mole per cent. New samples of these two solutes were prepared using a pressure of approximately 0.25 atm. At this pressure no depression of the nematic to isotropic transition temperatures was observed and the orientational order parameter of the molecular symmetry axis of each solute was larger than observed at higher solute pressure. D. NMR SPECTRA H, H and C free induction decays for the samples were measured on a Bruker WH400 spectrometer operating at 400.1 MHz for protons 61.4 MHz for deuterons and 100.6 MHz for carbon. For the samples containing ^ 9 F nuclei either, (i) ^ 9 F and ^H free induction decays were measured on a Bruker CXP 200 spectrometer operating at 188.2 MHz for 1 9 F and 200.0 MHz for *H or iq o (ii) F and H free induction decays were measured on a Varian XL300 spectrometer operating at 282.3 MHz for 1 9 F and 46.07 MHz for 2 H . Where more than one nucleus was measured, one of the nuclei was measured through the decoupling channel ( J H on the WH400 and CXP200, 1 9 F on the XL300) or Experimental / 27 through the lock channel ( H) without removing the sample from the magnet. Thus all spectra were recorded at the same temperature. The temperature was controlled by means of a variable temperature gas flow unit and was calibrated to ±0.3K using the proton chemical shift differences in a sample of ethylene glycol. All samples were heated until the liquid crystal became isotropic and were shaken vigorously before being placed in the probe. Approximately thirty minutes was allowed for temperature equilibration before the spectra were measured. The free induction decays obtained on the Bruker WH400 and CXP200 spectrometers were transferred for processing and plotting to a Nicolet 1280 computer equipped with a Zeta 8 digital plotter. DI. RESULTS AND DISCUSSION A. HYDROGEN In this section we present results obtained for H2, HD and D 2 in a number of nematic liquid crystals [10, 31, 32] and will begin with a discussion the spectra obtained. 1. Spectra of Partially Ordered HD and D 2 As a result of the Pauli exclusion principle [46] both H 2 and D 2 may exist in one of two forms. In H2 the symmetric combinations of nuclear spin states (total spin 1 = 1) may combine only with the antisymmetric, odd numbered rotational states. The antisymmetric combinations of nuclear spin states (1=0) may combine only with the even numbered rotational states. The first form (1=1) is referred to as ortho-H2 and the second (1 = 0) is para-O-2- The para form with 1 = 0 has no observable NMR spectrum. For D2 the orfAo form has total spin 1=2 or 0 and J even whereas para-D2 has 1=1 and J odd. In contrast to H 2 both the ortho and para forms of D 2 have an observable NMR spectrum. For HD, only one form exists because the two nuclei are different and there are no restrictions on which rotational states may combine with which nuclear spin states. Thus in a mixture of H2, HD and D2 the NMR spectra of orf/io-H2, HD, ortho-T)2 para-D2 may be observed. The *H spectrum of such a mixture dissolved in a nematic phase is shown in figure 4. For ortho-H2 the spectrum is a doublet of splitting |3D H H | and for HD it is a 1:1:1 triplet of splitting |2DJJJJ + J^\- The ^H spectrum of the mixture is shown in figure 5. For HD the spectrum is a doublet of doublets with splittings |2BD| and |2DgQ+Jgp|. For para-D2 it is a 28 Results and Discussion / 29 FIGURE 4 l H Spectrum of H£ and HD in a Nematic Phase 3D H H i i - 0 0 0 i i i i i i i i i - 4 0 0 0.0 400 BOO Frequency (Hz) Proton spectrum of HD and H2 in 50wt per cent 1132/EBBA: The doublet due to H£ has a splitting |3Djj[j| and the 1:1:1 triplet due to the HD has a splitting |2DHD + JHDI (see text). T=298.9K doublet of splitting |2Bp—6DpD| and for ortho-T>2 six lines are observed. The frequencies and intensities of the transitions for both ortho- and para-^2 are given in reference [10] and the relative positions of some of the lines are shown in figure 6. From these spectra the values of the scalar, dipolar and quadrupolar couplings may be obtained. The sign of the J coupling is known to be positive [47] and the signs of B D , D i m and D D D may also be determined [10]. Results and Discussion / 30 FIGURE 5 2H Spectrum of D2 and HD in a Nematic Phase — 1 — -400 1— -200 —I 0.0 I 200 400 Frequency (Hz) Deuteron spectrum of D2 and HD in 55wt per cent 1132/EBBA the doublet of doublets due to HD is labeled with *'s in the figure and has splittings |2DJJD + J|jrj| and 2Br> The remaining lines are due to D2- T=309.4K 2. Dipolar and Quadrupolar Coupling Constants Using expressions (13) and (16) the dipolar and quadrupolar couplings may be related to the order parameter of the internuclear axis z: Dab = - ( 2~) < r 8 b > s z z 4n (18) Results and Discussion / 31 FIGURE 6 2H Spectrum of D2 in a Nematic Phase Frequency (Hz) Deuteron spectrum of D2 in EBBA-d.2- B is the quadrupolar coupling, D is the dipolar coupling between the deuterons of the solute, and J is the indirect coupling between the deuterons of the solute. The centre of the spectrum, which contains no peaks, has been removed. T=301.4K 3 ^ q Q c B D = 4 <—7 > S " (19) n In expression (19) q=qzz, Q n is the deuteron nuclear quadrupole moment and the angled brackets imply the average over the internal molecular motions. Using these expressions the spectra of H 2 , HD and D 2 dissolved in several nematic liquid crystals [10] yield interesting results. First, the signs of the order parameters are solvent dependent. Second, the longer the average internuclear Results and Discussion / 32 distance the smaller the magnitude of Szz. Third, ortho- and para-D2 have the same order parameters to within experimental error. Finally the ratio B/D for HD and D 2 is solvent dependent and differs from the gas phase value [49]. In order to understand these results it is necessary to describe the solute-solvent interactions which lead to solute ordering. Without specifying the exact nature of the interactions involved, Burnell, de Lange and Snijders [10, 14-16, 50] have proposed a second rank tensorial form for the interaction potential. For hydrogen it may be written where A is the strength of the solvent-solute interaction. This potential may then be treated as a perturbation on the freely rotating, and vibrating molecule. For the unperturbed wave functions, products of the harmonic oscillator and rigid rotor functions are used. Then by standard perturbation theory expressions for the observables such as the dipolar and quadrupolar couplings may be obtained. For hydrogen the effects of vibrations will be ignored. The order parameter is a function of the matrix elements of the direction cosines for the rigid rotor wave functions summed over all J states and a parameter describing the solvent-solute interaction and is given by [10, 14-16, 50]: 3 2 1 U = -A ( g COS 6Zz - 2 ) f (20) S 2 2 = | r S P j <J|P2(cos0Z2)|J>2 -2A E E P <JlP 2(cos0Z z)lJ'XJ'|P 2(cos6Z 2)lJ> Ej - Ey (21) where (2J+1) exp(-BJ(J+l)/kT) E(2J+1) exp(-BJ(J+l)/kT) ' (22) J Results and Discussion / 33 and P 2(cos0 Z 2) = lcos\z - i . (23) A is the interaction parameter and J is the rotational quantum number. Using these expressions along with <r > averaged over the rotational states in the ground vibrational state, the dipolar couplings may be calculated as a function of A. The experimental couplings are predicted quite well by adjusting the value of A in a least squares fit. From this fit, it is immediately apparent that the isotope dependence of the order parameters is largely a quantum mechanical effect since in the classical limit the same order parameter is predicted for all three molecules. The variation in the sign of the order parameter between liquid crystals is explained in terms of a change in the sign of the parameter describing the solvent-solute interaction. 3. External Electric Field Gradients The liquid crystal dependence of the ratio B/D also provides some insight into the ordering of hydrogen [31]. This ratio should be a molecular property to a good approximation. Thus the liquid crystal dependence must be a result of changes in the electronic properties of the molecule or due to an external contribution to either B or D. It is unlikely that the electronic structure will change much with environment. If the separation of the vibrational and rotational averages is valid then the dipolar coupling depends only on the order parameter and <r"°> (where r is the internuclear distance). Thus the variation in B/D most likely arises because of variations in B. This can be explained very simply by proposing an external electric field gradient due to the liquid crystal Results and Discussion / 34 environment. The total electric field gradient at the site of the deuteron, is then the sum of an internal contribution due to the electrons and the other nucleus within the molecule, and a term due to the liquid crystal environment. The observed quadrupolar coupling is due to the interaction between the total electric field gradient and the deuteron quadrupole moment Bobs = " 4 < ~ 7 ~ > <FZZ - e<*Szz) • (24) n where is the mean electric field gradient due to the liquid crystal and q is the internal electric field gradient averaged over the internal motions of the molecule. If the internal contribution to can be calculated then an estimate of the external electric field gradient may be obtained. This requires a knowledge of the internal field gradient, q, at the site of the deuteron and the order parameter, S 2 2 . The value of S 2 2 can be calculated from the observed dipolar coupling using expression (18) and a quantum mechanically averaged value of q q <r"°> . Here, we expand r"° in a Taylor series in terms of the dimensionless q parameter £ about the equilibrium position, r'g <r"3> = r"e3 (1 - 3 < £ > + 6 < £ 2 > + . . . ) . ( 2 5 ) where £ = ( r - r . ) / r , . (26) < £ > and < £ > are given by [51]: <f> = - 3(1/ + ^)a[B ew c !] + 4<J(J + l)>[B ew 6 1] 2 (27) and <iZ>v = 2(1/ + {)[BeJel] , Results and Discussion / 35 (28) where a, uQ and B g are the Dunham coefficient, equilibrium vibrational frequency and equilibrium rotational constant. The values of these constants have been taken [10] from the literature [52]. For the internal electric field gradient we have [53] q ( r ) = q ( r e ) r c V 3 (1 + a ^ > + a 2 < * > + ... ) (29) The coefficients and a 2 have been obtained by fitting the calculated values of Reid and Vaida [54], The value of <B>/<D> in the absence of the external field gradient is then <B> (1 + (oc,-3)<£> + (a 2+6-30. ,)<{2>) ^ = ( B / D ) " (. -3<?> + e < e » • ( 3 0 ) The value of (B/D) has been obtained by inserting the experimental value [49] eq of <B>/<D> for the 01 state of D 2 . This value may then be used to calculate <B>/<D> for the 02 state of D 2 and the 01 state of HD. These two ratios have been measured in molecular beam experiments and our calculated values agree essentially exactly with the experimental values [49]. The internal contribution to B is then <B>/<D> from expression (30) multiplied by the experimental dipolar coupling. The difference between this value and the experimental value of B is the contribution due to the external field gradient. The value of F r , z may then be obtained from expression (24) using the known value of the deuteron nuclear quadrupole moment Q D [54]. In table 1 experimental couplings and values of Results and Discussion / 36 F z z for D 2 in 1132 and EBBA-d2 a r e s n o w n - It 1 S interesting to note that the value of F z z is of opposite sign in these two liquid crystals. 4. Electric Field Gradient Molecular Quadrupole Moment Interaction The presence of this electric field gradient immediately suggests a possible orienting mechanism through the interaction with the molecular quadrupole moment [31]. This interaction is of second rank tensorial form and the interaction potential is given by [56]: U Q = " J 2 F u Qmol,IJ • (31) Because both the nematic liquid crystal, and hence F, and the hydrogen molecule have cylindrical symmetry, expression (31) becomes U Q = " 2 F"ZZ QmoUz (f C O S 2 0 2 Z - { ) . (32) The contribution to the dipolar couplings and order parameters due to this mechanism maj' be calculated using expression (21) with A = — $F Z Z Q £ 2 . The value of Qzz has been obtained from reference [55]. The results of this calculation for D 2 in 1132 and EBBA-d2 are also shown in table 1. It is obvious from the good agreement between the experimental and calculated order parameters that this mechanism dominates the ordering of D 2 in these two liquid crystals. The negative order parameter in EBBA is also explained very simply. The discrepancies between the calculated and experimental order parameters are, in part, due to errors in the various physical constants required in the calculation of F Z 2 . However, these errors do not account for all of discrepancy Results and Discussion / 37 T A B L E 1 Electric Field Gradients in 1132 and EBBA-d2 Solvent T (K) FZZ* (esu XIOH) (exp) S(D2) X10-3 t (calc) 1132 298 6 .33±0 .13 9.28 8 .57±0 .65 EBBA-d2 301 - 7 . 2 3 ± 0 . 2 0 -11.07 - 9 . 6 9 ± 0 . 2 7 * F z z I S t n e external electric field gradient calculated from the ratio of BID for D2 (see text and [10, 15, 32, 36]) t S(D£) is the order parameter for D£. The experimental value is obtained from the dipolar coupling between the two deuterons. (The dipolar coupling for D2 in 1132 has been taken from [10].) The calculated values are obtained using expression (20) with A = ~±FzzQzz- Qzz has been calculated from the values given in reference [55]. and it is likely that other mechanisms play a role. 5. D 2 in 1132/EBBA Mixtures The change in sign of F z z in EBBA and 1132 raises the interesting possibility of producing a mixture of the two liquid crystals in which F z z may have values intermediate between those in the two components [32]. Of particular interest would be a mixture in which F z z = 0 since this could provide a system for studying other solutes where mechanisms involving F z z could be ignored. Experimental dipolar and quadrupolar couplings [32] for D 2 in several mixtures of 1132 and EBBA are shown in table 2 and the spectra obtained in three of these mixtures are shown in figure 7. As can be seen the spectral width, which is largely determined by the solute order parameters, depends very strongly on the liquid crystal composition. In 61.3wt per cent 1132/EBBA at 310K the spectrum collapses to nearly a single line of width 2.4 Hz. In fact, there are Results and Discussion / 38 FIGURE 7 Spectra of D2 in 1132/EBBA Mixtures 76j0wt per rant 61.3w I per cent 2&9wt per ent -1000 -«00 Frequency (Hz) 000 WOO 61.3 MHz deuterium NMR spectra of D2 dissolved in nematic mixtures of EBB A and 1132 at 31 OK. two shoulders with splitting 6|D| + 2|B| = 4.7 Hz. By interpolating plots of B and D versus temperature the values D = —0.45 ± 0.1 Hz and B = 1.0 ± 0.3 Hz are obtained. From table 2 it can be seen that F j , z varies continuously with the liquid crystal composition and that the order parameter follows the field gradient very closely. This is an important result since it provides us with a liquid crystal system in which one orienting mechanism can be controlled. From table 2 it can be seen that F z z and S do not pass through zero at exactly the same composition. Some of this discrepancy can be attributed to the experimental error in F ^ . However, it is likely that more than one mechanism is involved in the ordering of hydrogen. To investigate this, experimental and calculated order Results and Discussion / 39 T A B L E 2 D 2 in Mixtures of 1132 and EBBA* wt per cent 1132 D** (Hz) (Hz) S(exp)t (X10-3) F Z Z * (X10 1 1 esu) 0.0 23.9 49.3 55.0 61.3 66.1 76.0 100.0 65.95 + 0.50 33 .67±0 .14 11 .15±0 .18 5 .59±0.07 - 0 . 4 5 ± 0 . 1 0 -5.38 + 0.14 - 1 6 . 4 6 ± 0 . 0 6 - 4 9 . 7 6 ± 0 . 6 0 - 1 5 3 5 . 2 2 ± 1 . 5 0 - 7 8 8 . 2 6 ± 0 . 4 3 -267.6310.55 - 1 3 8 . 0 2 ± 0 . 2 9 116.06 + 0.42 373 .07±0 .20 1151.86+1.70 1.0±0.30 9.65 4.94 1.64 0.82 0.06 0.79 2.42 7.31 5.42 + 0.8 2.6910.2 0.4710.3 0.0110.1 0.6410.2 1.0610.2 2.1010.1 4.8210.9 * All results are from reference [32] and are at 31 OK except for 55.0 wt per cent 1132 which is at 301.4K ** D is the dipolar coupling between the deuterons in D2 *** B is the quadrupolar coupling of the deuterons in D2 f Calculated from the experimental dipolar coupling $ Fzz I S the external electric field gradient due to the liquid crystal environment and is calculated from the solute couplings (see text) parameters for D 2 and HD in 11 different liquid crystals have been obtained [10, 32, 40]. The experimental values have been calculated from the dipolar couplings using expressions (18) and (25). (Experimental couplings and order parameters for H 2 , HD and D 2 in. these liquid crystals at various temperatures are tabulated in in appendix I.) The calculated values have been obtained using expression (21) with A = ~^zz^zzm ^ e v a ^ u e s °f Qzz have been obtained from reference [55]. In figure 8 the calculated values have been plotted against the experimental values. There are two striking features of this plot. First, the agreement between the experimental and calculated order parameters is very good especially since there are no adjustable parameters involved. This strongly Results and Discussion / 40 FIGURE 8 Ordering of Molecular Hydrogen in Nematic Liquid Crystals. -12 - 6 0 6 12 S (exp) xlO 3 Experimental values vs calculated values of Szz, the order parameter of the molecular symmetry axis, of HD and D2 in 11 liquid crystals. The experimental values are obtained from the dipolar coupling and the quantum mechanically averaged value of <r~3>. The calculated values are obtained using expression (28) with A = — i^zzQzz- FzZ *s obtained from the deuteron spectrum (see text). Triangles = HD, Circles = D2. a = EBBA, b = phase V, c = 1167, d = 23.9wt per cent 1132/EBBA, e = 49.3wt per cent 1132/EBBA, f = 55wt per cent 1132/EBBA, g = 61.3wt per cent 1132/EBBA, h = 66.1wt per cent 1132/EBBA, i = 76.4wt per cent j = 5CB-a,$-d4, k = 1132. For composition of phase V and 1167 see ref [10]. Results and Discussion / 41 supports the conclusion that the electric Field gradient - molecular quadrupole moment mechanism dominates the ordering of HD and D 2 in all of these liquid crystals and suggests that this may be true for most, if not all, nematics. Second, the points lie on a straight line whose slope is slightly different from one and which does not pass through the origin. There are several reasons why such a plot may be obtained. If we arbitrarily make the value of (BfD)„n in expression (30), 0.8 per cent larger the plot shown in figure 9 is obtained. As can be seen this small adjustment can account for most of the deviation in both the slope and intercept., Since we are quite confident of the calculated gas phase value of (B/D) , this implies a change in the electronic structure of the molecule eq between the gas and liquid crystal phases. Although it is quite small, it is unlikely that a change of this magnitude in the electronic structure would occur. A more reasonable explanation of these results is that the field gradient at the site of the deuterons is slightly different from the one which causes the orientation of the molecule. It is also interesting to note in figure 9 that all of the points lie very slightly to the positive side of the line of slope 1. 6. Isotope and Temperature Effects a. 60wt per cent 1132/EBBA In order to investigate the possibility of a second mechanism it is useful to consider the ordering of H2, HD and D 2 in a mixture where S z z for all three solutes passes through zero. Let us assume that the ordering is the result of the interaction between a mean field due to the liquid crystal and some property of the solutes. If all three solutes experience the same mean field, then the Results and Discussion / 42 F I G U R E 9 Ordering of Molecular Hydrogen in Nematic Liquid Crystals - 6 0 6 12 S (exp) x l O 3 The data are the same as figure 8 except that the equilibrium value of BID is 0.8 per cent larger (see text). Results and Discussion / 43 order parameters of the solutes should all pass though zero at the point where the mean field is zero. However, if more than one mechanism contributes to the ordering, then the order parameter may be zero as a result of the sum of the various contributions being zero. In this case it is likely that the order parameters of the three solutes will pass through zero at different temperatures and liquid crystal compositions. Dipolar couplings obtained for H2, HD and D 2 as a function of temperature in 60wt per cent 1132 are shown in figure 10. The couplings for HD and D 2 have been scaled by the factors (y^y^) a n ^ ^ H ^ D ^ respectively. Unfortunately over the temperature range studied, the spectrum of D 2 is very poorly resolved and it was only possible to obtain dipolar couplings for this solute at three temperatures. It is very surprising to observe that all of the dipolar couplings obtained for D 2 lie between the H 2 and HD values. The quadrupolar couplings for D 2 are more easily determined and they are plotted in figure 11 along with the values obtained for HD as a function of temperature. From the figure we see that Bp(D2) is more positive BD(HD) for all temperatures. This implies that S 2 2(D 2) is more positive than S2 2(HD) which is consistent with Dp D more negative than D ^ . From the two figures it is also apparent that the dipolar and quadrupolar couplings do not pass through zero at exactly the same temperature. This is a direct result of the external electric field gradient. When the order parameter of D 2 or HD is zero then the dipolar coupling is zero and the quadrupolar coupling is entirely due to the external field gradient. If the electric field gradient - molecular quadrupole moment mechanism accounts for all of the ordering then the quadrupolar and dipolar couplings will pass through zero at the same point. The fact that they do not suggests that there is a second mechanism. It is also clear that the order parameters of the Results and Discussion / 44 FIGURE 10 Dipolar Couplings for H2, HD and D2 in 60wt per cent 1132/EBBA 290 294 298 T e m p e r a t u r e ( K ) 302 The dipolar couplings for HD have been scaled by the factor TTH^TD^ and those for D2 are scaled by (yft/yD)2. The temperature has been calibrated using the chemical shift differences in a sample of ethylene glycol. Circles = H2, triangles = HD and crosses = D2-three solutes do not pass through zero at the same temperature. At higher temperatures where S is negative |SJJJJ| > |S n | > |S^ | but at lower temperatures where S is postive I S ^ I < |S n | < |SH j. This also indicates the presence of a second mechanism. Because of the quantum mechanical averaging, second rank tensorial mechanisms such as the electric field gradient - molecular quadrupole moment interaction have | S D J > I S ^ I > | S H J . Since the contribution to the order parameters from a second mechanism must cancel the contribution Results and Discussion / 45 FIGURE 11 Quadrupolar Couplings for HD and D2 in 60wt per cent 1132/EBBA 290 294 298 Temperature (K) 302 The temperature has been calibrated as indicated in the caption to figure 10. Triangles = HD and crosses = D2. due to the electric field gradient - molecular quadrupole moment mechanism when S = 0, the second mechanism would have to have and S D more negative than S H . Moreover, for this mechanism SJJJJ would have to l i e closer to Sr, than to S U . This is an unusual isotope dependence and it is worthwhile investigating it somewhat further. Results and Discussion / 46 b. 50 and 55wt per cent 1132/EBBA In 50 and 55wt per cent 1132/EBBA the order parameters of the hydrogens are considerably larger than in 60wt per cent 1132/EBBA but are still much smaller than in the component liquid crystals where the electric field gradient - molecular quadrupole moment mechanism dominates the ordering. Thus results in these two mixtures should provide some information about the dependence of the contributions to the solute ordering on the composition of the liquid crystal. In particular we are interested in the isotope dependence of the couplings. In figures 12 and 13 the observed couplings are plotted as a function of temperature. The most striking feature of these two plots is the fact that the dipolar couplings go through a maximum. This is not too surprising, given the positive slope at lower temperatures and the fact that the dipolar couplings must go to zero at the isotropic phase transition temperature. It should be noted that the plots shown in figures 10-13 are quite different from those usually obtained for the liquid crystal solvent and which are predicted quite well by the mean field theory of Maier and Saupe [57]. This shows quite clearly that the intermolecular forces responsible for the ordering of the liquid crystal molecules are different from those responsible for the ordering of the solutes. It is also apparent from figures 12 and 13 that although the values of the scaled dipolar couplings for HD lie between those for H 2 and D2, they lie much closer the D 2 values. The experimental dipolar couplings for H 2 , HD and D 2 in 50, 55 and 60wt per cent 1132/EBBA (as well as several other liquid crystals [10]) are shown in table 3. The difference between the observed HD dipolar coupling and the average of the H 2 and D 2 couplings is also given in table 3. Both the HD and D 2 have been scaled to the proton value in calculating this difference. As can be seen the Results and Discussion / 47 F I G U R E 12 Dipolar Couplings for H2, HD and D2 in 50wt per cent 1132/EBBA 295 300 305 T e m p e r a t u r e ( K ) 310 The couplings have been scaled and labeled in the same manner as for figure 10. values obtained are remarkably constant in the three mixtures. In the other liquid crystals the values are again always positive and although there is some dependence on temperature and the solvent used, they are also approximately the same as obtained in the mixtures. Thus it appears that there is a roughly constant extra positive contribution to the dipolar couplings of HD, which implies an extra negative contribution to its order parameter. The mass dependence of electronic properties such as the quadrupole moment and polarizability is very Results and Discussion / 48 FIGURE 13 Dipolar Couplings for H2, HD and D2 in 55wt per cent 1132/EBBA N 8 bXI a a, o o V i cd 1-4 o T 3 co o co 260 — + + + + + + + + A A A A A 240 + A A A A 220 — O (D O G) O O O O 200 1 1 1 1 1 1 1 1 300 303 306 309 Temperature (K) 312 315 The couplings have been scaled and labeled in the same manner as for figure 10. nearly linear and cannot account for this effect. HD does have a small dipole moment which could account for this effect through its interaction with a mean electric field. However, the magnitude of the electric field required is quite large (~10^ V cm"l) and a field of this magnitude would produce a very large order parameter through the interaction with the polarizability. Moreover, it is well known [17] that there is no correlation between the dipole moment of a solute and its ordering, thus this explanation is rather unlikely. The main difference Results and Discussion / 49 between HD and the other two isotopes is that its centre of mass does not lie half way between the two nuclei. This means that as the molecule tumbles about its centre of mass it sweeps out a larger area making it effectively larger that H 2 and D 2 - If steric interactions are important this would lead to a more positive order parameter for HD. Exactly the opposite is observed. This is an intriguing problem and its solution would lead to a better understanding of the intermolecular forces involved in the ordering of hydrogen. At present we are unable to explain the results adequately. 7. Summary For molecular hydrogen we have gained a great deal of insight into its ordering, although the picture is not yet complete. We have seen that the ratio of the quadrupolar to dipolar couplings is solvent dependent and that this solvent dependence can be explained in terms of an external electric field gradient due to the liquid crystalline environment. The ordering of the solute can then be explained largely as a result of the interaction between this electric field gradient and the molecular quadrupole moment. We have also seen that by mixing 1132 and EBBA the value of the field gradient may be varied continuously between the values found in the two component liquid crystals. In a 55wt per cent 1132/EBBA mixture at 301.4K there is no measureable external electric field gradient. In principle, this allows us to determine what role, if any, the electric field gradient plays in the orientation of other solutes and also allows us to investigate other mechanisms. In this mixture the order parameter of hydrogen is not zero which indicates the presence of a second mechanism. In mixtures where the electric field gradient is small a very unusual isotope dependence is observed Results and Discussion / 50 T A B L E 3 H2, HD and D2 Dipolar Couplings in Various Liquid Crystals D H D -Temp. L>HH D H D L>DD * ( D H H + D D D ) * (K) (Hz) (Hz) (Hz) (Hz) 50wt per cent 1132/EBBA 290.14 376.8110.50 65.3310.14 10.5010.02 14.3910.51 293.67 373.5610.50 65.8710.14 10.6110.02 17.1910.51 295.42 383.8110.50 66.4210.14 10.6610.02 14.5910.51 297.17 s.n 66.96 + 0.14 10.7810.02 298.92 s.n 67.3910.14 10.8110.02 301.65 394.2010.50 67.7110.14 10.9510.07 11.6510.51 303.34 394.9210.50 68.0410.14 10.8610.07 15.3510.51 305.03 399.8310.50 67.8110.14 10.8910.07 10.7610.51 306.72 396.8010.50 67.7110.14 10.8910.07 11.6210.51 55wt per cent 1132/EBBA 300.68 200.3710.09 35.7810.14 5.5910.14 14.2910.74 302.43 203.6910.09 36.2810.14 5.7710.13 12.0710.70 304.18 207.0610.09 36.7310.14 5.8810.10 10.9810.59 305.93 209.6410.09 37.1510.14 5.9910.12 10.0910.66 307.69 212.1710.09 37.5010.14 6.1210.05 8.3510.43 309.44 214.5010.09 37.7910.14 6.1510.05 8.4410.43 311.19 216.4310.09 38.0210.14 6.1610.05 8.7610.43 312.94 217.8310.09 38.2010.14 6.1310.05 9.8710.43 314.69 218.3510.09 38.1510.14 6.0810.05 10.3410.43 60wt per cent 1132/EBBA 291.34 -42.6110.23 -3.8210.28 n.m. 293.29 -34.2510.12 -2.7910.14 -0.7210.04 14.2310.41 294.26 -29.7510.12 -1.4410.11 n.r. 295.23 -27.4210.24 -1.3010.11 n.r. 296.20 -24.0010.19 -0.9310.11 n.r. 297.17 -19.3910.47 -0.0410.11 n.r. 298.14 -16.0110.21 0.4610.14 n.r. 299.11 -13.5910.14 0.91 + 0.14 n.r. 300.08 -7.3410.19 1.6210.14 n.r. 301.05 -2.7610.38 2.1010.18 n.r. 302.03 0.6010.23 2.8010.11 0.2110.04 13.4810.37 303.00 3.9610.14 3.4910.11 0.4510.04 11.2110.35 304.94 10.2510.16 n.m. n.m. Results and Discussion / 51 Table 3 cont. Temp. (K) (Hz) °HD (Hz) ODD (Hz) DHD-i ( D H H + D D D )* (Hz) 310 1141.2±0.30 1167T 185.85±0.15 30.6310.02 -9.8210.45 (19.6410.90)* 298 320 3059 .3±0 .50 2632.0 + 0.30 EBB At 506 .6±0 .60 434.35 + 0.30 83.110.20 71.2110.10 7.2811.82 2.5610.92 298 2670.5 + 0.30 Phase Vt 442.710.60 72.510.30 10.3312.0 298 320 - 2 3 4 0 . 9 ± 0 . 3 0 - 1 7 9 6 . 1 ± 0 . 3 0 1132t -382.310.35 -292.0210.80 -63.27 -48.010.80 22.4910.94 14.2014.2 * D{JD and DrjD scaled to proton value. t Couplings in 1167, EBB A, Phase V, and 1132 are from reference [10] t Scaled by the order parameter of the liquid crystal director with rvspect to the magnetic field direction. n.m. not measured. n.r. D2 spectrum not resolved. s.n coupling not measured because of poor signal to noise. and it appears that there is an extra negative contribution to the ordering of HD. In other liquid crystals this effect is also observed, however, most of the isotope dependence is a result of differences in the quantum mechanical averaging for the three isotopes. At present, there is no adequate explanation of the extra contribution to the ordering of HD, although it is possible that it is related to the shift in the centre of mass. Results and Discussion / 52 B. METHANE 1. Introduction In the preceding section we have discovered that molecular hydrogen experiences an average electric field gradient due to its liquid crystalline environment and that the interaction between this field gradient and the molecular quadrupole moment apparently accounts for most of the observed orientational ordering. We have also seen that in nematic mixtures of 1132 and EBBA the field gradient varies continuously as a function of composition between the values observed for the two component liquid crystals. Throughout the discussion of the hydrogen results we have made the assumption that the averages over the vibrations and rotations may be done separately. Given the excellent agreement between theory and experiment, this assumption seems well justified. If this assumption is strictly valid then for molecules with full tedrahedral symmetry no anisotropic couplings should be observed since the order matrix is traceless and invariant to rotations of the molecule fixed axis for such symmetry. Experimentally, anisotropic splittings are observed in nematic solvents for such solutes as methane [33, 58], silane [59-61], tetramethylenetetramine [62, 63], adamantane [63], tetramethyl tin [64] and neopentane [59, 65] and it is clear that the separation of vibrations and rotations is not valid for these molecules. An adequate description of the effects of the coupling between the molecular vibrations and rotations on the observed anisotropic couplings, is complicated by the fact that such effects occur because of the dependence of the solute-solvent interactions on the vibrations and rotations of the solute. Thus the crux of the problem is the nature of the solute-solvent interactions. In this section we will Results and Discussion / 53 outline a general theory developed by Snijders, de Lange and Burnell [14-16, 50] for the orientation of solutes with internal motions in nematic liquid crystals. A summary of the application of the theory to methane and its deuterated analogues dissolved in 1132, EBBA and a 61.3wt per cent 1132/EBBA mixture will be given in order to determine what role, if any, the electric field gradient plays in determining the observed couplings. 2. Theory The details of the theory presented here are given in references [14-16] and we will only outline the main points. a. Solvent-Solute Interaction Potential In order to describe the anisotropic couplings observed for a solute in a liquid crystalline environment, we require the solvent-solute interaction potential. This potential is determined by the intermolecular interactions between the solvent and solute molecules. The observed couplings result from an average over all molecular motions in the presence of the potential. An exact treatment of this problem is prohibitively complex and instead we consider a simple model for the interaction. We will assume that the solute experiences a mean field due to the liquid crystal and that this mean field may be described by a second rank tensor F. The solute then interacts with this field via some second rank tensorial property p\ Thus we have specified the form of the interaction but not its exact nature. If we assume that the liquid crystal environment has axial symmetry about the Z direction, defined by the magnetic field, then the tensor F should also have this symmetry. The tensor may be written as the sum of an isotropic Results and Discussion / 54 part (the trace) and an anisotropic, symmetric, traceless part. The isotropic part cannot lead to anisotropic couplings and, because of the axial symmetry, the anisotropic part has only one independent non zero element which we will call G. The tensor j3 is a function of the positions of the nuclei of the solute which may be written in terms of the normal coordinates Q m . For the interaction potential we have U = — - E Fjj /SJJ j 3 1 = ~ 3 G ?, M Q m ) (acosflfczcosfljz - -26kl) , Kl,m (oo) where I, J and Z are space fixed axes and k, I are molecule fixed axis. The potential is a function of both the vibrations and the orientation of the solute and will couple these two types of motion. b. Calculation of Observables This potential may then be treated as a perturbation on the freely rotating and vibrating molecule. Using first order perturbation theory, the perturbed wavefunctions may be obtained from the zeroth order functions for which simple products of the harmonic oscillator and rigid rotor functions are used. Expressions for the expectation values of observables such as the dipolar and quadrupolar couplings in terms of the various Gj3^;(Qm) may then be obtained. Because we have only specified the form of the potential the values of these parameters and their dependences on the normal coordinates are not known. If we assume that only small displacements occur, the 0 tensor may be expanded in a Taylor series about the equilibrium in geometry in terms of the normal coordinates and truncated after the linear term. Thus the unknown parameters are reduced to Results and Discussion / 55 G(tu and G 9 p V 3 Q m | = Dipolar Couplings: In terms of these parameters the following expressions are obtained for the dipolar couplings Dab = Vii*" + D a n b ° n _ r i g i d , ( 3 4 ) a b kl a b'* f a (35) n non rigid_ where . 88 v,- 9d a b k, h G E E E (-^) ( — < A A i > • 3 m V a Q m Qm=0 9 Q m Qm=0 2 7 T Q m V (36) j _ , h7a7b, cos6>abJfc c o s 0 a b > i a a b * Z ~ ~ I 2 / 3 . 4n r, ab (37) i _ 3 1 *i - ^ o s f l ^ c o s f l ^ - -26kl (38) and <&kl> - Skl (39) Results and Discussion / 56 U ) m is the vibrational frequency of normal mode m. The rigid term arises from the ordering of the molecule in the absence of any coupling of the vibrations and rotations. The non rigid term arises from the vibration rotation interaction. The average over the orientations in (35) and (36) must be done quantum mechanically for hydrogen but for other solutes this is not necessary. In the absence of quantum effects this is given by where fl represents a particular orientation of the molecule fixed axes with respect to the space fixed axes. For methane the interaction energy, U, (expression (33)) is sufficiently small that the exponential in expression (40) can be expanded in a Taylor series and truncated after the linear term. The average over the internal motions in expression (35) is obtained by expanding expression (37) in terms of the normal coordinates and retaining only those terms which make a significant contribution. In a purely harmonic potential only the even terms will contribute. For methane the series may be truncated after the linear term and symmetry greatly reduces the number of non-zero parameters. Because the equilibrium geometry is tetrahedral the parameter Gt$kieq does not contribute to the coupling. Similarly, the derivative of 0 with respect to the symmetric breathing mode, which retains the tetrahedral symmetry, does not contribute. This leaves only the derivatives with respect to the E (bending), F 2 (predominantly stretching) and F 2 (predominantly bending) normal modes. Because the derivatives are evaluated in the equilibrium geometry which has tetrahedral symmetry, they are isotope independent. Thus the dipolar couplings of the series of isotopically substituted methanes may be calculated in terms of these three parameters. For / \j exp(-U(n)/k B T) dO / e x P(-u(n)/k B T ) dn ' (40) Results and Discussion / 57 the symmetrically substituted methanes (CD^ and CH^) the average geometry is also tetrahedral so that only the non rigid term in expression (34) contributes. For the asymmetrically substituted species (CHgD, CH2D2 and CHDg) there is also a rigid contribution which arises because the average geometry, and hence the /3 tensor averaged over the molecular vibrations, does not have tetrahedral symmetry. A fit of 17 experimental dipolar couplings [33] for the various partially deuterated species dissolved in 1132 and EBBA in terms of the derivatives of the 0 tensor with the respect to the E , F2 (stretch) and F2 (bend) modes has been done [14] and excellent agreement between the experimental and calculated values is obtained. It is clear from these results that the form of the interaction potential is essentially correct. From the parameters obtained, it is found that the effects due to the F2 stretching mode dominate the observed couplings. Quadrupolar Couplings: The quadrupolar couplings have essentially the same form as the dipolar couplings and again two terms are obtained: p _ R " g i d . R n o n - r i g i d BD ~ BD + BD (41) e2Q B ° g d = ~Zh E < q k l > S k l ( 4 2 ) Bnon r i g i d = !<, 2 Q D Results and Discussion / 58 2 h ^ (43) x E E E ( — ) ( ^ ~ ) <<&»AM> • Thus to calculate the quadrupolar couplings we require the same parameters which determine the dipolar couplings, the average value of the electric field gradient tensor at the site of the deuteron <q>, and the derivatives of this tensor with respect to the various normal modes. The G/3 parameters may be obtained from the fit to the dipolar couplings and some simplifying assumptions may be made about the electric field gradient tensor. For the rigid part deviations of the electric field gradient tensor elements from the equilibrium values due to vibrations will be ignored. Then, because of the local C g y symmetry about the C-D bonds in the equilibrium geometry, there is only one independent element qon. Electronic structure calculations [66] have shown that for small distortions the local C g y symmetry is retained and that the derivatives of the field gradient with respect to the E and F 2 bending modes may be neglected. Thus the quadrupolar couplings are determined by two unknown parameters q g y and 9(le(y'3QF2(str)" ^ ^ to ^ e experimental couplings in 1132 and EBBA [33] in terms of these two parameters [15] also produces excellent agreement between experiment and theory. Both parameters here should be electronic properties and independent of the liquid crystalline environment. While this is true of q ^ there is a large difference in the values of 3<le^9QF2 (str) obtained [15] which suggests that this description is not complete. Results and Discussion / 59 3. Methanes in Liquid Crystal Mixtures In section A we saw that the ordering of hydrogen is dominated by the molecular quadrupole moment electric field gradient mechanism and that in a 55wt per cent 1132/EBBA mixture the external electric field gradient is zero. It is important to point out that at this composition the order parameter is not exactly zero which suggests the presence of a second orienting mechanism. In a 60wt per cent mixture the order parameter of hydrogen goes through zero as a function of temperature but there is a small but appreciable field gradient present. It seems likely that the electric field gradient also plays a role in determining the coupling in methane. To investigate this, the spectra of the methanes were measured [36] in 1132, EBBA and a 61.3wt per cent 1132/EBBA mixture at 310K. At this temperature the order parameter of as measured from the dipolar coupling, is zero in the mixture. An analysis of the dipolar couplings of the isotopically substituted methanes as described above was done and these results are given in table 4. Initial fits to the EBBA results gave relatively large deviations. The measurements were repeated several times and it was discovered the ^^CHgD sample was slowly changing over time probably because of a slow degradation of the liquid crystal. The cause of this was not investigated and with the 1 3 C H g D couplings removed a satisfactory fit was obtained. It is immediately apparent that the dipolar couplings of methane do not go through zero in the mixture, although they do he between the values in the two component liquid crystals. This suggests that the external electric field gradient which is responsible for the ordering of D 2 is at least partly responsible for the observed dipolar couplings in methane. If methane does experience an external field gradient there will be an extra contribution to the quadrupolar Results and Discussion / 60 T A B L E 4 Methane in 1132, EBBA and 61wt per cent 1132/EBBA at 310K: Experimental and Calculated Dipolar Couplings (Hz) and G/3 Parameters*1' (10-5 erg cm-1) Molecule Dipolar coupling Experimental Calculated EBBA 1 3 C H , HH 0.68 ± 0.05 0.69 CH 10.87 ± 0.1 10.87 1 3 C H 3 D HH t HD t CH t CD t C H 2 D 2 HH -9.15 ± 0.05 -9.15 HD 0.104 ± 0.10 0.096 DD 0.26 ± 0.02 0.25 C H D 3 HD -0.59 ± 0.07 -0.66 DD 0.15 ± 0.10 0.13 C D , DD 0.015 ± 0.010 0.016 G/32 = -0.197 G/3 3 = 1.498 G/3„ = -0.078 1 3 C H , 1 3 C H 3 D C H 2 D 2 C H D 3 C D , 61.3wt per cent 1132/EBBA HH 0.0 ± 0.18 0.13 CH . 1.17 ± 0.10 1.17 HH -3.476 ± 0.18 -3.78 HD 0.60 ± 0.03 0.62 CH -1.67 + 0.08 -1.68 CD 1.425 ± 0.02 1.49 HH -7.174 ± 0.02 -6.99 HD 0.022 ± 0.003 0.015 DD 0.17 ± 0.01 0.017 HD -0.51 ± 0.02 -0.53 DD 0.09 ± 0.01 0.09 DD 0.010 ± 0.003 0.003 G 0 2 = G/33 = GpY = -0.197 0.919 -0.109 Results and Discussion / 61 Table 4 cont. Molecule Dipolar Experimental Calculated coupling 1132 1 3 C H 4  1 3 C H 3 D C H 2 D 2 C H D 3 CD„ HH <, 0.18 0.15 CH -5.17 ± 0.14 -5.23 HH -3.83 ± 0.05 -3.99 HD 0.57 ± 0.07 0.66 CH -8.27 ± 0.7 -8.26 CD 0.31 ± 0.4 0.59 HH -7.60 ± 0.05 -7.46 HD -0.04 ± 0.07 0.02 DD HD -0.59 ± 0.07 -0.55 DD ~ 0.09 0.09 DD ~ 0.0 0.004 G/32 = -0.243 G/3 3 = 0.824 G/3, = -0.182 * G/3 2,G/3 3 and G/3q are the derivatives of the /3 tensor with respect to the E, F2 (stretch) and F2 (bend) symmetry modes (see text) t Unreliable results, see text couplings given by: B e x t = _ 3 ( ^ Q D F Z _ Z ) ( 4 4 ) In table 5 the experimental quadrupolar couplings are given along with the values calculated as described in section 2 above and the uncorrected values of q ^ and ^(leq^^^F2 (str) a r e obtained. If the contribution to the quadrupolar couplings given in expression (44) is taken into account, only the value of 3qg^/3Qp2( s t r) is affected [15, 16]. The corrected parameters shown in table 5 have been obtained using the value of from D 2 experiments. As can be Results and Discussion / 62 T A B L E 5A Methane in 1132, EBBA and 61wt per cent 1132/EBBA at 310K: Experimental and Calculated Quadrupolar Couplings (Hz) Molecule Experimental Calculated EBBA 1 3 C H 3 D t C H 2 D 2 -26.15 ± 0 . 1 0 -25.93 C H D 3 -3.20 ± 0 . 1 0 -3.64 CD„ 18.40 ± 0 . 0 1 0 18.40 1.3wt per cent 1132/EBBA 1 3 C H 3 D -65.26 ± 0.02 -66.45 C H 2 D 2 -49.00 ± 0.03 -46.79 C H D 3 -30.5 ± 0 . 0 3 -31.06 CD, -15.17 ± 0 . 0 3 -15.63 1132 1 3 C H 3 C H 2 D 2 C H D 3 CD„ D •90.08 ± 0.35 •74.95 ± 0.10 -55.20 ± 0.10 •37.76 ± 0.10 -92.16 -71.62 -55.07 -39.14 t Unreliable results, see text seen the value of 3Qe^9QF2(str) * s n o w independent of the liquid crystalline environment as one would expect for a molecular property. This result provides very strong evidence that the deuterons in both D 2 and methane experience the same external electric field gradient. However, it is clear that for methane, the interaction with the molecular quadrupole moment does not account for all of the observed dipolar couplings. Results and Discussion / 63 T A B L E 5B Parameters' qeq (kHz) and 9q/9SF, (kHz A " 1 ) With and Without External Field Gradient Corrections Parameter EBBA 61.3wt per 1132 cent 1132/EBBA Uncorrected 193 187 184 9q/9SF 2 -61.4 -293 582 Corrected 193 187 184 9q/9SF 2 BD(ext)t -286 -252 -243 86 -10.1 -72 *qeq is the principle component of the intramolecular field gradient tensor in the equilibrium geometry and 9q/9SF 2 is the derivative with respect to the F 2 stretching symmetry mode, evaluated in the equilibrium geometry, t Obtained from D2 experiments [10]. 4. Summary In summary, then, we have seen that the observed dipolar couplings for the series of deuterated methanes dissolved in nematic liquid crystals is a result of two contributions. For all of the isotopically substituted species there is a non-rigid contribution which is a result of the fact that the orientational averaging is dependent on the vibrational state of the molecule. For the asymmetrically substituted species there is also a rigid contribution from the deviations of the average geometry from tetrahedral symmetry. The dipolar couplings are predicted very well by assuming that the solvent-solute intraction is of second rank tensorial form and by fitting the calculated couplings to the Results and Discussion / 64 experimental values in terms of three parameters, G/32, a n d which describe the interaction between the liquid crystal mean field, G, and the derivatives of the second rank tensorial molecular property, p\ with respect to the E (bend), F 2 (stretch) and F 2 (bend) normal modes. The quadrupolar couplings may be calculated in terms of these same three parameters along with qgg. the principle component of the intramolecular field gradient tensor in the equilibrium geometry and 9q e^/8Qp 2 (g t r) the derivative with respect to the F 2 stretching mode. Both of these parameters should be liquid crystal independent. However, this is true only when one takes into account the external electric field gradient as measured from D 2 experiments. Finally, it is apparent from the correlation between the G/J parameters and the external electric field gradient that it also plays some role in determining the dipolar couplings. However, unlike hydrogen where the electric field gradient quadrupole moment mechanism dominates, for methane it makes a relatively small contribution. C. A C E T Y L E N E 1. Introduction In the previous two sections we have seen that the observed dipolar couplings for methane and hydrogen can be explained very well in terms of a simple model in which the solvent-solute interaction is of second rank tensorial form. In hydrogen we have also seen the electric field gradient - molecular quadrupole moment mechanism plays an important role. For methane, the interaction mechanism has not been specified, but there is strong evidence suggesting that this solute also experiences the same electric field gradient as hydrogen. These Results and Discussion / 65 two solutes have thus provided some insight into the solvent-solute interactions responsible for the observed spectra. They are, however, unusual solutes: hydrogen, because of its extremely small size and methane, because of its very high symmetry. In order to see whether the conclusions we have drawn here are generally applicable it is worthwhile considering a somewhat more typical solute. In choosing such a solute there are some important considerations. First, in order to perform the vibrational analysis in a manner similar to that used to analyse the methane spectra the number of normal modes of vibration should be as small as possible and the number of dipolar couplings as large as possible. Second, both the harmonic and anharmonic force fields must be known. A solute which fits these criteria is acetylene. It has 5 normal modes of vibration but by symmetry the derivatives of the /3 tensor, which describes some unspecified molecular property, will be zero with respect to the ungerade (asymmetric) modes. There are two of these which leaves 3 non zero derivatives. A total of 29 dipolar couplings may be measured if all of the possible C and H isotopically substituted species are available. Thus the ratio of unknown parameters to measured couplings is quite favourable for this solute. Diehl et al. [21] have studied the spectra of acetylehe-l-^C and acetylene-1,2- C£ in EBBA and Merck phase IV. The observed dipolar couplings where analysed in terms of the vibrationally averaged structure, but no account of the vibration-rotation interaction was taken. The ratios of the C-H and the C-C bond lengths, as determined from the dipolar couplings were found to be strongly solvent and temperature dependent. This result was explained in terms Results and Discussion / 66 of a two site model. In the model, the acetylene molecules were envisaged as occupying two different environments. In these two environments, the geometry and orientation of the molecules were allowed to vary. The ratio of the order parameters, bond lengths and enthalpies for the two sites were then taken as adjustable parameters and the experimental results were predicted. Although this model correctly predicts the observed behaviour, no evidence was given that the two sites actually exist. In light of the results for methane it seems likely that the solvent and temperature dependence of the dipolar coupling ratios is a result of the coupling between vibrations and rotations. In this section results for the analysis of various isotopically substituted species of acetylene in terms of the model used for methane and hydrogen will be presented. As will be seen very good agreement with the experimental dipolar couplings is obtained. However, there are some small remaining discrepancies. These discrepancies will be discussed in terms of a possible contribution to the C-C dipolar couplings from the anisotropy in the indirect coupling, as well as anharmonic corrections to the calculated rigid dipolar couplings. 2. Experimental Results The experimental dipolar couplings for the various isotopically substituted species in 1132, EBBA and 55wt per cent 1132/EBBA at 301.4K are shown in table 6. These couplings have been obtained by fitting the spectra using the program LEQUOR where necessary. Unfortunately, the quadrupolar coupling constants of the deuterons could not be determined accurately. Because of the large magnitude of the observed quadrupolar splittings (eg. ~120 kHz in 1132 at 30IK), the Results and Discussion / 67 TABLE 6 Acetylene in 1132, EBBA and 55wt per cent 1132/EBBA at 301.4K: Experimental Dipolar Couplings (Hz) Nucleus Measured  Molecule Coupling ' H 1 3 C H - 1 3 C =1 3 C - H H - 1 3 C = 1 2 C - H D - 1 3 C = 1 3 C - D H - 1 2 C = 1 2 C - H H - 1 2 C =1 2 C - D D C H L > C H ' L>HH D C C D C H D C H ' D H D EBBA 1386 .471±0.008 147 .652±0.008 189 .415±0.008 236 .929±0 .018 1386 .4±0 .2 147 .6±0 .2 189.410.2 189.410.2 29.210.2 1386.3910.04 147.6510.04 189.3510.09 236.6110.05 1386.3610.04 147.5010.04 DCD D C D ' DDD 218.7110.05 22.9710.05 4.4010.11 239.0210.12 55wt per cent 1132/EBBA H _ 1 3 C _ 1 3 C _ H D c H -2412.9710.07 -2413.5410.11 D C H . -265.6810.07 -265.6810.11 D H H -337.7810.07 -337.8210.26 D C C -442.3810.16 -440.5510.12 H - 1 3 C e 1 2 C - H D C H -2412.9110.16 D C H ' -265.7110.16 D H H -337.8210.11 Results and Discussion / 68 Table 6 cont. Nucleus Measured Molecule Coupling 1 H 1 3 C 55wt per cent 1132/EBBA D - 1 3 C _1 3 C - D -380.5710.06 D C D < -41.1910.06 D DD -7.9110.09 DCC -444.0810.10 H - 1 2 C _ 1 2 C - H DHH -337.8 + 0.7 H - 1 2 C _ 1 2 C - D DHD - 5 2 . 1 ± 0 . 7 1132 H - 1 3 C _ 1 3 C - H DCH - 5 7 7 0 . 9 6 ± 0 . 1 8 -5770.7710.04 D c H 1 -632.5810.18 -632.3010.04 DHH -804.9810.19 -804.2110.10 Dec -1043.3310.42 -1043.4710.05 H - i 3 C _1 2 C - H • D C H -5770.7210.40 DCH' -632.3110.40 DHH -805.1310.27 D - 1 3 C _ 1 3 C - D D CD -902.2810.10 DCD' -97.3710.10 DDD -20.1410.16 D C C -1042.9910.16 H - 1 2 C _1 2 C - H DHH -804.610.3 H - i 2 C _1 2 C - D DHD -124.210.3 Results and Discussion / 69 contributions to the linewidths due to, e.g. temperature inhomogeneity, will also be large. In the best deuteron spectrum obtained the linewidths in the solute spectrum were ~100 Hz and very little fine structure could be resolved. As can be seen from the results in table 6, the signs of the dipolar couplings change between 1132 and EBBA in the same manner as those for hydrogen. However, the composition where the order parameter of acetylene is zero is quite different from that for hydrogen. In a mixture of approximately 16wt per cent 1132/EBBA the proton spectrum of 1 2 C 2 H 2 is a single line at 340K (uncalibrated). Unfortunately this mixture has a fairly narrow nematic range and at lower temperatures a more ordered phase (probably smectic) is formed. Because of problems with temperature stability and homogeneity in the nematic range, a good set of couplings has not yet been obtained in this mixture. 3. Theoretical Considerations Before calculating the dipolar couplings for acetylene in terms of our second rank tensorial interaction model, there are some points which should be emphasized. First, unlike methane, the B tensor will not be isotropic and the largest contribution to the couplings will be the rigid effect which will be dominated by the molecule in its equilibrium geometry. Second, we need to consider the isotope dependence of the derivatives of the B tensor with respect to the various normal coordinates [14, 16, 67]. The B tensor is entirely determined by the electronic structure of the molecule. Thus, the derivatives of B with respect to the cartesian displacements are isotope independent. However, because of the Eckart conditions of zero angular momentum [68], the relationship between the cartesian Results and Discussion / 70 coordinates and internal coordinates is mass dependent. Thus, in general, the parameters 9/3/3Qm | _ will be isotope dependent. Crawford [69] has shown Q m - 0 that the derivatives with respect to the internal coordinates of a second rank tensorial property, such as the polarizability, for the various isotopic species can be related to one another through and appropriate rotation of the molecule fixed axes. For methane the |3 tensor is invariant to such a rotation and thus the derivatives are isotope independent. This is not the case for acetylene. It is convenient, here, to relate the derivatives for the isotopically substituted species to a hypothetical molecule with atoms of unit mass [70]. The dipolar couplings may then be calculated in terms of parameters for this hypothetical molecule. Finally, we also note the order parameters for acetylene will be considerably larger than for the asymmetrically substituted methanes and therefore more terms in the expansion of expression (37) in terms of the normal coordinates will be required. 4. Calculation of Order Parameters Initial attempts at fitting the experimental couplings from table 6 did not converge and the values of the derivatives of the |3 tensor with respect to the two stretching modes were unreasonably large. This is a result of the fact that the dependence of the dipolar coupling on these two modes is effectively identical and of opposite sign. Removal of the derivative of the /J tensor with respect to either stretching mode yielded much better results which were independent of which parameter was removed. In table 7, the results where the derivative of the P tensor with respect to the C-C stretch has been set to zero, are presented. The value obtained for the derivative with respect to the symmetric Results and Discussion / 71 TABLE 7 Acetylene in 1132, EBBA and 55wt per cent 1132/EBBA at 30IK: Calculated and Experimental Dipolar Couplings (Hz) Molecule Coupling Rigid Non-rigid Total * Experiment EBBA H - 1 3 C _ 1 3 C - H D C H 1289.88 96.81 1386.69 1386.43 + 0.04 D o r 144.12 2.90 147.03 147 .65±0.04 D H H 181.09 6.47 187.56 189 .38±0.09 D e c 244.20 -5.54 238.66 236.7710.05 H - 1 3 C _1 2 C - H D C H 1291.04 95.62 1386.66 1386.4010.22 D C H ' 144.25 2.79 147.04 147.5710.16 D H H 181.25 6.33 187.58 189.4010.22 D - 1 3 C _ 1 3 C - D D CD 207.37 11.33 218.70 218.7110.05 D C D ' 22.59 0.19 22.78 22.9710.05 D DD 4.37 0.10 4.46 4.4010.11 Dee 246.48 -7.31 239.17 239.0210.12 H - 1 2 C _ 1 2 C - H D HH 181.41 6.19 187.59 189.410.2 H - 1 2 C _ 1 2 C - D DHD 28.10 0.75 28.85 29.210.2 G/3 parameters 1" GBeq = -0.2546X10" 1 3 erg G/3, = -1.318X10" 5 erg c m ' 1 G/3 2 = 0.0 G/33 =-0.2254X10' 5 erg cm" 1 rms error = 1.05 Hz* Results and Discussion / 72 Table 7 cont. Molecule Coupling Rigid Non-rigid Total * Experiment 55wt per cent 1132/EBBA H - 1 3 C _1 3 C - H DCH -2361.63 -51.71 -2413.34 - 2 4 1 3 . 2 6 ± 0 . 1 3 D C H ' -263.72 -1.69 -265.41 -265.6910.13 D H H -331.55 -3.44 -334.99 -337.8110.27 D e c -447.09 -3.01 -444.08 -441.4610.20 H - 1 3 C _1 2 C - H D C H -2363.03 -50.61 -2413.65 -2412.9110.16 D C H ' -264.02 -1.44 -265.46 -265.7110.16 D H H -331.75 -3.32 -335.06 -337.8210.11 D - 1 3 C _ 1 3 C - D PCD -376.26 -4.43 -380.69 -380.5710.06 DCD ' -40.99 -0.07 -41.06 -41.1910.06 D D D -7.92 -0.04 -7.96 -7.9110.09 D e c -447.23 3.04 -444.19 -444.0810.10 H - 1 2 C _ 1 2 C - H D H H -331.94 -3.20 -335.14 -337.810.7 H - 1 2 C _ 1 2 C - D D H D -51.27 -0.33 -51.60 -52.110.7 G/3 parameterst G/3e<? = 0.4116X10" 1 3 erg G/3, = 0.7219X10"5 erg cm" 1 G/32 = 0.0 G/33 = 0.0109X10" 5 erg cm" 1 rms error = 1.55 Hz* Results and Discussion / 73 Table 7 cont. Molecule H - 1 3 C = 1 3 C - H H - 1 3 C = 1 2 C - H D - 1 3 C =1 3 C - D H _ i 2 C =1 2 C _ H H - 1 2 C = 1 2 C - D Coupling D C H DCH' D H H D C C D CH D C H ' D HH D CD DCD' D DD Dec D H H D H D Rigid 1132 -5598.89 -625.59 -786.03 -1059.97 -5600.90 -625.79 -786.30 -885.62 -96.48 -18.65 -1052.67 -786.68 -121.19 Non-rigid Total Experiment -173.20 -5772.09 - 5 7 7 0 . 8 7 ± 0 . 1 8 -5.48 -631.07 - 6 3 2 . 4 4 ± 0 . 1 8 -11.82 -797.85 - 8 0 4 . 5 9 ± 0 . 2 1 -1043 .40±0 .42 9.00 -1050.97 170.32 -5771.22 -5.24 -631.03 -11.5 -797.80 -17.58 -0.36 -0.16 9.97 -903.20 -96.84 -18.81 -1042.70 -11.18 -797.76 -1.30 -122.49 - 5 7 7 0 . 7 2 ± 0 . 4 0 - 6 3 2 . 3 1 ± 0 . 4 0 - 8 0 5 . 1 3 ± 0 . 2 7 - 9 0 2 . 2 8 ± 0 . 1 0 -97.37 + 0.10 - 2 0 . 1 4 ± 0 . 1 6 - 1 0 4 2 . 9 9 ± 0 . 1 6 - 8 0 4 . 6 ± 0 . 3 -124.210.3 G/3 parameterst GBeq = 0.922X10" 1 3 erg G/3, = -0.622X10" 5 erg cm" 1 G/32 = 0.0 G/33 = 0.1214X10" 5 erg cm" 1 rms error = 4.05 Hz* * Average of 1 ff and 1 3 C results t G is the anisotropy in the second rank tensor describing the mean field due to the liquid crystal. B is a second rank tensorial property of the solute. By, B2 and B3 are the derivatives of B with respect to the symmetric C-H stretch, C-C stretch and symmetric bend symmetry modes. t rms error between calcuated and experimental couplings. Results and Discussion / 74 C-H stretch is thus the sum of the derivatives with respect to both stretching modes. The calculated dipolar couplings have been obtained using the force field of Strey and Mills [71] and the equilibrium geometry and definitions of the internal coordinates are also from this reference. For the rigid dipolar couplings we have ^ . _ . eq ^ v i d ^ k l ] Qm=0 ^ m (45) _ _ < Q „ Q „ > + . . . -m n 3Q m 3Qn Qm-Qn=0 <d a b W > = d a b e q „ + E ( — — ) n n <Qm> sb.hl ab,*i m 7 Q m = 0 m ™ m i d2d a b i i "2 2 2 ( _ ahfJ)n _ <QmQ n> + • • • For a purety harmonic potential terms of order < Q m > , < Q m » <^m » e t c -are zero. The rigid couplings reported in table 7 have been calculated with only the harmonic terms (even powers of Q) included and the series has been truncated after the sixth0 order term. The non-rigid contribution to the couplings has been calculated from expression (36) in the section on methane. The angular dependent parts of the expressions for the rigid and non-rigid couplings are calculated classically from expression (40) (also in the section on methane). For acetylene the interaction energy is sufficiently large that the expansion of the exponential in expression (40) can no longer be truncated after the linear term. Because of this, the integrals in expression (40) have been calculated numerically. As can be seen from table 7, quite good agreement between the total calculated couplings and experimental values is obtained. The rigid effect clearly dominates the observed couplings, but it is also apparent that non-rigid effects also make a considerabale contribution. The change in sign of the order parameter is explained very simply as a change in the sign of the anisotropy in the mean field, G. In 55wt per cent 1132 the order parameter is positive and the change in the order Results and Discussion / 75 parameter between the three liquid crystals is qualitatively consistent with the electric field gradient - molecular quadrupole moment mechanism. Thus, the ordering of acetylene can be explained as resulting from two mechanisms; (i) the electric field gradient molecular quadrupole moment mechanism and (ii) a second mechanism which makes a positive contribution to the order parameter. Although the fit obtained is quite good, the discrepancies lie well outside the error in the experimental dipolar couplings. It is also clear that the discrepancy between theory and experiment increases as the magnitude of the order parameter increases. One reason for this could be the contribution to the carbon-carbon couplings due to the anisotropy in the J coupling. Thus, the calculations were repeated with these couplings removed. These results are shown in table 8. As is obvious, better agreement with the experimental couplings is obtained. The values of the anisotropy in J , obtained from the difference between the calculated and experimental carbon-carbon dipolar couplings, are roughly the same in all 3 liquid crystals. However, the values obtained for the deuterated compound are consistently lower than those for the protonated species and all of the values are considerably larger than the theoretical values of 99.3 Hz [5, 72] and 85.0 Hz [5, 73]. A second possible reason for the discrepancies in the initial fit is the neglect of the anharmonic terms in the calculation of < d a D £ / > . The linear term can be calculated using the anharmonic force field [14, 16, 74] and for acetylene it is of the order of one per cent of the total coupling. Some initial fits with this term included were slightly worse than those shown in table 7. However, it is Results and Discussion / 76 TABLE 8 Acetylene in 1132, EBBA and 55wt per cent 1132/EBBA at 301K: Calculated and Experimental Dipolar Couplings and the Anisotropy in the Indirect C-C Coupling (Hz) Molecule Coupling Calculated Experiment A JQC EBBA H - 1 3 C _ 1 3 C - H DCH 1386.41 1386.4310.04 DCH* 148.40 147.6510.04 D H H 188.78 189.3810.09 D e c 242.84 236.7710.05 H - 1 3 C _1 2 C - H D C H 1386.44 1386.4010.22 D C H * 148.42 147.5710.16 D H H 188.81 189.4010.22 D - 1 3 C _ 1 3 C - D DCD 218.74 218.7110.05 D C D ' 22.99 22.9710.05 D D D 4.49 4.4010.11 D e c 248.33 239.0210.12 H - 1 2 C _ 1 2 C - H D H H 188.83 189.410.2 H - 1 2 C _1 2 C - D D H D 29.06 29.210.2 Gj3 parameters t GBeq = -0.2570X10- 1 3 erg G/3, = -1.149X10- 5 erg cm" 1 GB2 = 0.0 GB3 =-0.1696X10" 5 erg cm" 1 rms error = 0.47 Hz* Results and Discussion / 77 Table 8 cont. Molecule Coupling Calculated Experiment AJrx 55wt per cent 1132/EBBA H - 1 3 C = 1 3 C - H H - 1 3 C = 1 2 C - H D - 1 3 C =' 3 C - D H - 1 2 C = 1 2 C - H H _ 1 2 C s i 2 C _ D D C H D o r D H H D e c D C H D C H * D H H DCD DcD 1 D D D DCC D H H D H D -2412.96 - 2 4 1 3 . 2 6 ± 0 . 1 3 -267.26 -265.69 + 0.13 -336.64 -337.8110.27 -452.07 -441.4610.20 305.53 -2413.37 -2412.9110.16 -267.32 -265.7110.16 -336.72 -337.8210.11 -380.51 -380.5710.06 -41.32 -41.1910.06 -7.99 -7.9110.09 -449.50 -444.0810.10 156.47 -336.79 -337.810.7 -51.87 -52.110.7 G/3 parameters'!' GBeq = 0.4137X10" 1 3 erg G/3, = 0.4396X10- 5 erg cm" 1 G/3 2 = 0.0 G/3 3 = -0.0578X10" 5 erg cm" 1 rms error = 0.92 Hz* Results and Discussion / 78 Table 8 cont. * * Molecule Coupling Calculated Experiment AJcc 1132 H - 1 3 C = 1 3 C - H D C H -5771.23 - 5 7 7 0 . 8 7 ± 0 . 1 8 D C H- -635.97 -632.4410.18 D H H -802.23 -804.5910.21 D C c -1065.89 -1043.4010.42 274.17 H - 1 3 C = 1 2 C - H D C H -5770.49 -5770.7210.40 D C H . -635.94 -632.3110.40 D H H -802.19 -805.1310.27 D - 1 3 C = 1 3 C - D D C D -902.72 -902.2810.10 D C D< -97.53 -97.3710.10 D D D -18.90 -20.1410.16 D C C -1056.76 -1042.9910.16 168.66 H - 1 2 C = 1 2 C - H D H H -802.16 -804.610.3 H - 1 2 C = 1 2 C - D D H D -123.18 -124.210.3 G/3 parameters t Gpeq = 0.928X10" 1 3 erg G/3 , = -1.354X10- 5 erg cm" 1 G/3 2 = 0.0 G/33 = -0.0548X10" 5 erg cm" 1 rms error = 2.11 Hz* *C-C coulpings not included in fit (see text). ** Average of 1 H and 1 3 C results *** AJcc I S the anisotropy in the indirect C-C coupling tensor, t G is the anisotropy in the second rank tensor describing the mean field due to the liquid crystal. /3 is a second rank tensorial property of the solute. /3,, /3 2 and /33 are the derivatives of /3 with respect to the symmetric C-H stretch, C-C stretch and symmetric bend symmetry modes. t rms error between calcuated and experimental couplings. Results and Discussion / 79 clear that the contribution from the cubic term, which will partially cancel the linear term, must also be included. This work is not yet complete and it remains to be seen whether including these terms will result in better agreement with the experimental couplings. 5. Summary Although, the analysis presented in this section is not complete, it is clear that the vibration-rotation interaction plaj's an important role in determining the couplings observed for acetylene. When this interaction is included there is no need to invoke any specific interactions between solute and solvent. The negative order parameter in EBBA, which had been explained in terms of specific interactions [21], is accounted for quite naturally by assuming that acetylene experiences the same external electric field gradient as molecular hydrogen. Finally, there is some evidence of a contribution to the carbon-carbon dipolar couplings due to the anisotropy in the indirect spin-spin coupling. However, until all significant contributions to the calculated couplings have been taken into account, this result remains somewhat speculative. D. SIZE AND SHAPE EFFECTS Results and Discussion / 80 1. Introduction In the preceding sections we have been able to make some significant conclusions about the nature of the interactions which lead to the orientation of hydrogen, methane and acetylene in nematic solvents. For these small solutes excellent agreement with the observed dipolar couplings is obtained by considering the solvent-solute interaction as being purely electrostatic and of second rank tensorial form and, in the cases of methane and acetylene, by taking into account the effects of the interaction between vibrations and rotations. For hydrogen we have also seen that the deuterium nuclei in D 2 and HD experience an external electric field gradient due to their liquid crystal environment. Moreover, the interaction between this external field gradient and the molecular quadrupole moment accounts for most of the orientational ordering of the hydrogens. For the deuterated methanes the intramolecular quadrupolar couplings depend upon two parameters: (i) the principal component of the electric field gradient tensor along the CD bond when the molecule is in its equilibrium geometry and (ii) the derivative of the electric field gradient with respect to the F 2 stretching mode. Both of these parameters should be molecular properties which depend only upon the electronic structure of the molecule. We have found that if the effect of the external electric field gradient, as measured from the hydrogen experiments, is taken into account, the values of the parameters obtained by fitting to the experimental couplings become liquid crystal independent as one would expect for a molecular property. This result suggests that deuterium nuclei in both hydrogen and methane experience the same electric field gradient. Results and Discussion / 81 For acetylene we have shown that although vibrations play an important role in determining the dipolar couplings, the interaction between the liquid crystal and the solute in its equilibrium geometry accounts for most of the observed orientational ordering. Moreover, a qualitative examination of the molecular order parameters in 1132 and EBBA suggests that the interaction between the external electric field gradient (as measured from the D 2 experiments) and the molecular quadrupole moment also plays an important role in determining the ordering of this solute. Finally, we have shown that in a 55wt per cent mixture of 1132/EBBA at 301.4 ± 0.3K, the deuterons in D 2 and HD experience no external electric field gradient. It is important to point out, however, that both hydrogen and acetylene are appreciably ordered in this mixture and anisotropic couplings are observed for methane. This implies that although the interaction between the electric field gradient and the molecular quadrupole moment is an important orienting mechanism, it is not the only one. In this section we will use these results as a basis for studying a series of small solutes in an attempt to investigate the contributions to their ordering. We shall begin by making some assumptions based on the above results. First, we shall assume that all solutes experience the same electric field gradient as the deuterons in D 2 - Thus, in the 55wt per cent 1132 mixture we assume that the electric field gradient - molecular quadrupole moment mechanism does not contribute to the ordering. Second, we shall assume that the averaging over internal molecular motions may be done separately from the averaging over Results and Discussion / 82 molecular reorientation. This means that the dipolar coupling between two spins a and b with the solute may be written D a b = - ( * ^ T - b ) <ra"b> S a b, (46) where S a b =<lcos20abz - |> (^ ) is the order parameter for the a,b direction in the molecule. If we consider only the interaction between the solute and its average environment and neglect quantum mechanical effects the order parameters of the molecule are given by / (3cos0iZcos0jZ - (5^ ) exp(-U(n)/k BT) dfl (48) i j ~ 2f exp(-U(n)/k BT) df} where the integrations are over all orientations St=U{8^,8^2) and U(S2) is the mean potential describing the interaction between the solute and its environment. The problem, then, is to define U(fi). Results and Discussion / 83 2. Short Range Interaction Model c. The Interaction Potential In the limit of low solute concentration, the potential, U(fl), is determined by the intermolecular interactions between solute and solvent averaged over all motions of the molecules. In general, these interactions may be divided into two types: (i) long range interactions where the intermolecular distances are much larger than the molecular dimensions and (ii) short range interactions where the intermolecular distances are shorter than (or comparable to) the molecular dimensions. The mean potential may then be written as u(0)=uSR(n)+uLR(a). For the long range contribution to the potential, the liquid crystal provides an average electric field, electric field gradient, etc. which interacts with some electronic property of the solute. Two examples of this are the interaction between the permanent dipole moment or molecular polarizability of the solute and a mean electic field due to the liquid crystal. The interaction between the molecular quadrupole moment of the solute and the mean electric field gradient is also of this type. As we have seen this last mechanism plays a dominant role in the orientation of hydrogen. Hydrogen is an extremely small solute, however, and for larger solutes the short range interactions will become increasingly important. Results and Discussion / 84 b. The Short Range Potential The short range potential, which is a result of the coulombic repulsion between the electron clouds of the solvent and solute molecules, is effectively the energy required to displace the solvent molecules upon introduction of the solute. It is the angular dependence of this energy with respect to the liquid crystal director which will lead to orientation of the solute. We will assume that the energy required to displace the liquid crystal molecules in a direction parallel to the director is negligible compared to the energy required for displacements perpendicular to the director. Then, to avoid the complexities inherent in averaging over the relative positions of the solute and solvent molecules, we model the liquid crystal as an elastic tube aligned parallel to the external Z direction. A schematic representation of this is shown in figure 14. If the liquid crystal molecules are assumed to be rigid then the walls of the tube may be displaced in the XY direction but will remain parallel to the Z direction. The displacement caused by the presence of the solute then leads to a restoring force on the solute which we assume to be proportional to the deformation of the tube in the XY direction: F(Q) = - k / r(B,Q) dB = -kC(fi), (49) where r(B,Sl) is the vector in the XY plane from the origin to the tube surface, B is the angle between the laboratory X axis and r(B,Sl) (see figure 15), C(Q) is the circumference of the deformed tube, and k is the Hooke's law force constant Results and Discussion / 85 FIGURE 14 Elastic Tube Model for the Short Range Interactions T liquid / crystal ,;, , J- . H 11 M t::::H:::::::H::::*::::HHHn:H::::n:!:!::^  M n l l I l l l l I I I I I I I N I I I I f l l l l l I I I I I I I I = l | i ^ ^ i i l l l l l l l l I l ^ p J t l l l l = l l i l l f l f i i " _ _ =_ The solute, represented by an array of van der Waals spheres, displaces the liquid crystal which in turn exerts a restoring force on the solute proportional to the displacement. The liquid crystal, shown here only in the plane of the paper for clarity, is displaced in such a way that the displaced volume is a tube whose walls are parallel to the laboratory fixed Z axis. fl is the angle between the molecule and liquid crystal axes. The notation fl is meant to describe the orientation of a molecule of any symmetry. For molecules with Csv or greater symmetry the single angle 8 is used to describe their orientation. for the restoring force. The mean potential energy arising from the interaction of the solute with the elastic walls is then u^n) = -k / F(n) dc(Q) = | kc2(n). In general, the circumference of the deformed tube, C(O), will depend on how the interaction between the surface of the molecule and the surface of the tube is modeled. If we assume that both surfaces are hard then C(fl) is the Results and Discussion / 86 A. Definition of symbols used in expression (49) r is a vector from the origin to the surface of the projected molecule and (} is the angle between r and the X axis. B. Maximum circumference: The maximum circumference of the elastic tube representing the deformation of the liquid crystal solvent by a solute at some orientation is shown in bold face. C. Minimum circumference: The minimum circumference of the elastic tube representing the deformation of the liquid crystal solvent by a solute at some orientation is shown in bold face. Results and Discussion / 87 cirumference of the projection of the solute molecule onto the XY plane as shown in figure 15. This circumference depends on how the solute molecule is described and how its outer surface is defined. Here, we model the solute as a rigid array of nuclei with fixed positions in the molecule fixed axis system. Each nucleus then has a sphere of radius r associated with it, where r is the van der Waals radius of the atom. The electron distribution around an atom in a molecule is, of course, not spherical and it could be represented by some other shape. For example, in aromatic systems the 7T orbitals extend much further than the a orbitals so that a better representation of the electron distribution would be an elipsoid or some complicated shape which more closely resembles the molecular orbitals. However, in the interest of keeping the model as simple as possible we shall describe the solute as a collection of overlapping spheres. The projection of the molecule in the XY plane is thus an array of overlapping circles. From this array of circles we want to calculate the circumference of the projection of the outer surface of the molecule. This surface is also the surface of the elastic tube and represents the boundary of the region of space which is inaccessible to the solvent molecules. This region is, however, not easily defined and its circumference should be calculated somewhat differently for each solute. In order to avoid this, we approximate the circumference of the projection of the outer surface of the molecule by either the maximum (figure 15b) or minimum (figure 15c) circumference of the array of overlapping circles, both of which are defined in the same way for each solute molecule. A detailed description of the calculation of the maximum and minimum circumferences in terms of the angles 6 and <f> is given in appendix II. The definitions of the angles 8 and <j> are also given in the appendix. Results and Discussion / 88 c. Calculation of Order Parameters Having calculated C(J2) and thus U(£2), the order parameters of the solute may then be calculated by integrating numerically over all orientations of the solute. In terms of our model S^, for example, is given by: / / ( 3 cosz0 - l)exp(-kC2(6\0)/2kBT) sin0d0d0 o 0—0 8=0  ^zz _ T T T T / 2 " • (51) 2 f f exp(-kC (0,0)/2kBT) sin0d0d0 0=0 0=0 It is often assumed that the interaction potential may be expanded in terms of the Legendre polynomials and then truncated after the first term. In such a case the potential becomes independent of <f> for a solute with a Cg or higher symmetry axis. We have not made this assumption and symmetry requires only that the potential be invariant to the symmetry operations of the point groups of the solute and its environment. Thus our potential is independent of 0 only for solutes with a C ,^ axis. If we calculate the integrals in expression (51) using a trapezoidal rule and a step size of 7T/36, the values of the resulting order parameters are accurate to 4 figures. This model is appealing because it should be applicable to a large number of solutes and involves only one adjustable parameter, the force constant k, which should be same for all solutes. Results and Discussion / 89 3. Ordering of solutes in 55wt per cent 1132/EBBA a. Solutes with Cg or Higher Symmetry' In order to test the model described above, we will begin by measuring the order parameters of some simple solutes with Cg or higher symmetry in 55wt per cent 1132/EBBA where we assume that they experience no external electric field gradient due to the liquid crystal. Experimental dipolar couplings for these solutes dissolved in 55wt per cent 1132/EBBA and the two component liquid crystals, EBBA and 1132, are given in table 9. Where necessary the couplings have been obtained by fitting the experimental spectrum using the computer program LEQUOR [28]. All of these solutes have Cg or higher symmetry and it can be shown [1] that for such molecules there is only one independent order parameter, Szz, where z is the symmetry axis. Experimental values of Szz for the solutes dissolved in 55wt per cent 1132/EBBA have been obtained from the measured dipolar couplings using expression (46) and a structure from electron diffraction or microwave data. These results are shown in table 10. As a test of the model for the short range interactions the order parameters have also been calculated using expression (51). The results of this calculation are also given in table 10 and are plotted in figures 16a and 16b, against the experimental values. In figure 16a the minimum circumference has been, used and in figure 16b the maximum circumference. The points have been labeled in the same manner as the results in tables 9 and 10. The van der Waals radii have been taken from reference [75] and the values of k=5.0 and 4.0 dyne cm** for the minimum and maximum circumferences have been determined by a least squares fit to the lines of slope 1. As can be seen the correlations are quite good. This Results and Discussion / 90 TABLE 9 Spectral Parameters (Hz)* for Solutes with C3 or Higher Symmetry at 301.4K Solute Parameter 1132 55wt per cent 1132 EBBA a) 1,3,5-tribromobenzene DRH -213.56(3) -159.2(3) -111.90(9) b) 1,3,5-trichlorobenzene D H H -213.23(3) -156.60(3) -148.12(3) c) Benzene Dortho -987.99(2) -708.47(2) -461.48(1) D meta -191.73(2) -137.15(3) -88.46(1) Dpara -125.05(2) -89.32(4) -57.34(1) d) Hydrogen D H H -2340.9(3) 203.69(9) 3059.3(5) -382.5(6) 36.19(14) 506.5(5) ODD -63.3(3) 5.77(13) 83.1(5) 1461.5(5) -140.4(4) -1943.6(5) B H D 1351.5(5) -137.22(14) -1817.2(5) e) Propyne*"" D C H 3 2853.10(14) 1656.75(3) 595.23(1) D H C H 3 -401.43(14) -232.89(3) -82.47(1) C. Shift 2.395ppm 1.319ppm 0.338ppm f) Acetylene D C H -4763.2(3) -2131.23(8) 1075.1(5) -4607.4(8) 1060.4(3) D C H- -506.3(3) -234.73(8) 114.9(5) -490.5(8) 112.0(3) D C C -834.6(7) -389.4(2) 180.2(12) -832.8(6) 178.6(2) D H H -644.6(3) -298.56(7) 145.8(4) -643.81(8) 140.0(7) g) 2-butyne 2761.16(3) 1493.03(4) 1200.06(3) D H H ' -219.31(1) -118.74(4) -95.68(3) J H H ' 2.75(2) 2.69(7) 2.77(5) Results and Discussion / 91 Table 9 cont. Solute Parameter 1132 55wt per cent 1132 EBBA h) 2,4-hexadiyne Dgem 4547.59(14) 3824.06(11) 2795.38(12) D H H ' -120.51(9) -101.06(9) • -74.36(8) J H H ' 2.75(2) 2.69(7) 2.77(5) i) Allene D c i s 1886.50(3) 1227.34(2) 878.36(1) D t rans -334.44(2) -223.68(1) -158.59(1) j) 1,3,5-trifluorobenzene DHF , o r t h o -837.97(6) -646.08(4) -545.56(8) DHF , p a r a -105.84(9) -81.81(6) -69.64(12) D F F -132.50(5) -102.48(3) -87.12(9) D HH -195.74(15) -151.084(10) -127.4(2) k) Hexafluorobenzene Dortho -571.06(3) -583.42(3) -812.73(2) Dnieta -106.90(4) -109.18(3) -151.86(3) Dpara -76.26(4) -77.78(4) -108.09(4) Jortho -22.47(6) -22.40(6) -22.40(5) Jmeta -3.14(8) -3.07(8) -3.03(6) Jpara 3.78(10) 3.73(10) 3.75(8) 1) Bromomethane D HH 988.41(9) 326.59(13) m) Iodomethane D H H 960.00(13) 426.5(4) 223.02(9) n) Nitrous Oxide D N N -63.21(10) -7641(10) o) Nitrogen**** DNN -36.3 -48.7 * J couplings not reported here have been taken from the literature and the references are given in table 11. ** C. Shift is the proton chemical shift between CH and CH3. The CH3 resonance is to low field in all cases. *** 77ie first set of numbers refers to the proton spectrum and the second set to the 13C spectrum. **** DipQiar C0UpHngS from reference [134]. Results and Discussion / 92 FIGURE 16a Short Range Interaction Model (Minimum Circumference) S (exp) Experimental vs calculated order parameters in 55wt per cent 1132. The calculated order parameters have been fit by the least squares method to the solid lines of slope 1. The labeling of the points refers to the molecules indicated in tables 9 and 10. T = 301.4 K, k = 5.0 dyne/cm, F^z = 0.0 esu, correlation coefficient = 0.985. Results and Discussion / 93 FIGURE 16B Short Range Interaction Model (Maximum Circumference) - 0 .2 0.0 0.2 0.4 S (exp) Experimental vs calculated order parameters in 55wt per cent 1132. The calculated order parameters have been fit by the least squares method to the solid lines of slope 1. The labeling of the points refers to the molecules indicated in tables 9 and 10. T = 301.4 K, k = 4.0 dyne/cm, Fzz ~ 0.0 esu, correlation coefficient = 0.982. Results and Discussion / 94 T A B L E 10 Experimental and Calculated Order Parameters for Solutes with C3 or Higher Symmetry Dissolved in 55wt per cent 1132/EBBA at 301.4K Solute S(exp)* S(calc)**  minimum maximum polarizability circumference circumference a 1,3,5-tribromobenzene -0.215 -0.210 -0.237 -0.259 b 1,3,5-trichlorobenzene -0.211 -0.196 -0.225 -0.229 c Benzene -0.181 -0.112 -0.121 -0.175 d Hydrogen H 2 -0.0007 0.014 0.011 0.012 HD -0.0008 0.014 0.011 0.012 D 2 -0.0009 0.014 0.011 0.012 e Propyne 0.160 0.118 0.106 0.132 f Acetylene 0.089 0.074 0.062 0.074 g 2-butyne 0.140 0.186 0.170 0.187 h 2,4-hexadiyne 0.359 0.400 0.358 0.207 i Allene 0.129 0.107 0.092 0.202 j 1,3,5-trifluorobenzene -0.200 -0.155 -0.177 -0.180 k Hexafluorobenzene -0.219 -0.184 -0.250 -0.197 1 Iodomethane 0.022 0.070 0.079 0.106 * Order parameters obtained from the experimental dipolar couplings given in table 9. ** Order parameters calculated using the short range interaction model with the minimum and maximum circumferences, and the polarizability electric field mechanism. supports the idea that in this mixture the ordering of solutes is dominated by their size and shape and that the external electric field gradient experienced by all solutes is small. The fact that the correlation is not affected greatly by using the maximum or minimum circumference suggests that both approximations provide a reasonable representation of the surface of the molecule within the limits of the model. As may be seen in table 10 the order parameter which is predicted the least Results and Discussion / 95 accurately and is in fact of the wrong sign, is that of hydrogen. We have seen, in the preceding sections that most of the orientational ordering of this solute is due to the electric field gradient - molecular quadrupole moment mechanism and that quantum mechanical effects must be taken into account in order to describe the isotope dependence of the order parameters. That the contributions to the ordering due to short range effects should be small for this molecule and that this contribution cannot be described using the same force constant as used to describe the orientation of the larger solutes, is not surprising since for small values of the circumference the repulsive forces between the solvent molecules themselves should allow the solute to tumble more freely. Moreover, it may be necessary to consider quantum mechanical effects in the description of the short range interactions since such effects do play a role in the long range interactions. In considering other possible mechanisms for the orientation of the solutes it is useful to consider briefly the nature of the molecular properties involved. In many cases [10, 16, 12, 13, 17] proposed mechanisms involve molecular properties which are approximately equal to the sum of contributions associated with each bond. It is important to note, however, that this is not true of the molecular quadrupole moment. The contribution associated with a given bond usually depends in some way on the size of the atoms involved so that the molecular property is a function of the size and shape of the molecule as a whole. Because of this, mechanisms involving such molecular properties should also predict the order parameters in the 55wt per cent 1132/EBBA mixture reasonably well. To investigate this possibility order parameters for the various Results and Discussion / 96 solutes have been calculated using as an orienting mechanism the interaction between the molecular polarizability of the solute (given in table 11) and the anisotropy of a mean square electric field due to the liquid crystal. The results of this calculation are shown in table 10 and are plotted in figure 17. The value of the anisotropy in the mean square elecric field has been obtained from a least squares fit of the calculated order parameters to the experimental values -i r o o and is found to be 2.27X10 i O cm" . The correlation is again quite good. This suggests that if the contribution to the solute ordering from the molecular quadrupole - electric field gradient mechanism is sufficiently small, then any molecular property which is related to the size and shape of the solute can be used to predict the order parameters. b. Solutes of Lower Symmetry Because solutes with C3 or higher symmetry have only one independent order parameter, it is not possible to distinguish between imperfections in the model and variations in the average environment experienced by different solutes. Solutes of lower symmetry, however, have two or more independent order parameters and because the order parameters must describe orientation in exactly the same environment, such solutes present a more rigorous test of the model. Moreover, deuteration of the liquid crystal allows us to monitor the experimental conditions by using the deuteron spectrum as an internal standard. A series of solutes with and &2h ByTnme^ry are listed in table 12 along with their experimental order parameters in 55wt per cent 1132/EBBA-d2. Order parameters for the solutes with Cg or higher symmetry have also been measured in this mixture and the results are also shown in table 12. The order parameters, Results and Discussion / 97 T A B L E 11 Solute Polarizabilities and References Solute Polarizability (cm-2 4) References axx ayy a-zz a J Structure a) 1,3,5-tribromobenzene 22.39 22.39 13.34 [76P [77] b) 1,3,5-trichlorobenzene 18.88 18.88 11.36 [76]a [78] c) Benzene 11.73 11.73 6.54 [79] [58] [80] d) Hydrogen 0.71 0.71 1.01 [81, 82] [10] e) Propyne 4.54 4.54 7.57 [79] [84] [80] f) Acetylene 2.78 2.78 4.53 [79] [21] [80] g) 2-butyne 5.29 5.29 9.51 [79] [80] h) 2,4-hexadiyne 7.78 7.78 12.42 [80, 85]d [80] i) Allene 4.47 4.47 9.02 [97]b [86] [80] j) 1,3,5-trifluorobenzene 11.56 11.56 6.19 [87]b [88] [80, 78] k) Hexafluorobenzene 12.18 12.18 6.19 [87, [78, 89] 122]b 1) Iodomethane 6.47 6.47 8.92 [87]b [80] m) Bromomethane 4.81 4.81 6.74 [34]b [80] 1) Acetone 7.15 5.16 7.05 [90] [91] [92] 2) Furan 7.75 5.36 7.48 [93]a [94] [95] 3) Thiophene 9.51 7.27 11.02 [93]a [94] [96] 4) Pyridine 11.94 6.21 11.02 [90] [97] [98] 5) Fluorobenzene 11.88 6.32 11.37 [99]b [100] [101] 6) Chlorobenzene 14.52 7.42 15.84 [90] [102] [9 IF 7) T T F f 17.62 4.96 25.74 [104] [83] [83] 8) 2,6-difluoropyridine [105] [106]c 9) Iodobenzene [102] [119]e 10) 1,2-dichlorobenzene [107] [108]c 11) 1,2-dicyanobenzene [109] [109]c 12) 1,3-dichlorobenzene [110] [108F 13) 1,3-dinitrobenzene [111] [112, 113]c 14) 1,4-dichlorobenzene [114] [115] 15) 1,4-dibromobenzene [115] [115] a) Extrapolated to zero frequency using frequency dependence of benzene given in reference [79]. b) Extrapolated to zero frequency. c) The structure has been calculated from data in the given reference(s). d) The polarizability has been estimated using data from the given references. e) Regular hexagon assumed. f) Tetrathiofulvalene Results and Discussion / 98 FIGURE 17 Polarizability - Electric Field Mechanism S (exp) Experimental vs calculated order parameters in 55wt per cent 1132. T = 301.4 K, C<Eu2> - <EXZ>)1/2 = 1.59X105 statvolt cm'l, Fzz = 0.0 esu. The calculated order parameters have been fit by the least squares method to the solid line of slope 1 (correlation coefficient = 0.947). The labeling of the points refers to the molecules indicated in tables 9 and 10. Results and Discussion / 99 T A B L E 12 Experimental and Calculated Solute Order Parameters in 55wt per cent 1132/EBBA-d2 at 301.4K Solute S(exp)£ fit to all results S.Rb S(calc) Pol.c separate fit to each solute S.Rd Pole Acetone syy 0.0711 (2) -0.0720 (1) 0.0009 (1) 0.0754 0.0718 •0.0036 0.0366 0.0669 0.0303 0.0711 0.0696 0.0033 0.0408 0.0736 0.0340 Fur an Syy s 2 2 0.0907 (1) •0.1365 (1) 0.0458 (2) 0.0636 •0.0894 0.0258 0.0469 0.0770 0.0302 0.0974 0.1336 0.0362 0.0841 0.1383 0.0520 Thiophene Jxx Dyy $zz 0.0586 (1) •0.1490 (4) 0.0905 (5) 0.0547 •0.0998 0.0451 0.0057 0.0997 0.0940 0.0810 0.1474 0.0664 0.0061 •0.1235 0.1166 Pyridine Jxx Syy Szz 0.1093 (2) -0.1545 (4) 0.0452 (6) 0.0727 •0.1138 0.0412 0.1129 0.1632 0.0502 0.1016 0.1566 0.0550 0.1067 0.1555 0.0488 Fluorobenzene Jxx Dyy Szz 0.0507 -0.1906 0.1399 (1) (1) (2) 0.0242 •0.1384 0.1142 0.0645 -0.1640 0.0995 0.0274 -0.1804 0.1530 0.0760 0.1947 0.1188 Chlorobenzene TTFi ?xx ^yy Szz ?xx 0.0070 (1) -0.2037 (2) 0.1967 (3) Szz -0.081 (3) 0.2514 (17) 0.3320 (17) 0.0109 -0.1501 0.1610 0.1513 -0.2570 0.4083 0.0622 0.2196 0.1575 0.1299 -0.3520 0.4819 0.0201 0.1856 0.2057 0.1129 •0.2189 0.3318 0.0635 -0.2229 0.1594 0.0499 •0.2828 0.3327 2,6-difluoro-pyridine ^xx syy Szz 0.1529 (2) -0.1908 (2) 0.0379 (4) 0.1231 •0.1503 0.0272 0.1577 0.1868 0.0291 Results and Discussion / 100 Table 12 cont. Solute S(exp)a S(calc)  fit to all separate fit to results each solute S.Rb Pol.c S.R.d Pole Iodobenzene ^xx Szz -0.0243 (5) -0.2146 (6) 0.2389 (11) -0.0594 0.1735 0.2329 -0.0642 0.1811 0.2453 1,2-dichloro-benzene Jxx Syy 0.0647 (1) -0.2248 (3) 0.1601 (2) 0.0254 0.1754 0.1500 0.0245 -0.2040 0.1795 1,2-dicyano-benzene Jxx °yy S 2 2 0.063 (11) -0.247 (13) 0.184 (24) 0.0272 0.2100 0.1828 0.0254 -0.2255 0.2001 1,3-dichloro-benzene ^>xx Syy Szz 0.1958 (4) -0.2243 (3) 0.0285 (7) 0.2029 0.1870 0.0159 0.2092 0.1915 -0.0177 1,3-dinitro-benzene Jxx *yy ^zz 0.1961 (2) -0.2287 (2) 0.0326 (4) 0.2211 •0.2096 •0.0116 0.2090 0.1999 -0.0091 1,4-dichloro-benzeneg ^xx "yy $zz -0.0721 -0.2467 0.3188 (1) (4) (5) 0.1046 0.2003 0.3049 -0.1123 -0.2105 0.3228 1,4-dibromo-benzeneg &XX syy Szz -0.1021 (1) 0.2546 (1) 0.3567 (2) 0.1420 0.2173 0.3594 -0.1420 -0.2170 0.3590 Benzene 1,3,5-tri-fluorbenzene Szz S 'zz -0.1756 -0.1220 -0.1608 -0.2014 -0.1687 -0.1654 1,3,5-tri-chlorobenzene Jzz -0.2380 -0.2121 -0.2141 Results and Discussion / 101 Table 12 cont. Solute S(exp)a S(calc)  fit to all separate fit to results each solute S.Rb Pol.c S.Rd Pole 1,3,5-tri- szz -0.1886 -0.2268 -0.2434 bromobenzene hexafluoro- S 2 2 -0.2280 -0.1990 -0.1804 benzene 2,4-hexadiyne Szz 0.3664 0.4439 0.1846 2-butyne 0.1943 0.2081 0.1670 Allene Szz 0.1302 0.1197 0.1302 Bromomethane Szz 0.0676 0.0664 0.0734 Iodomethane Szz 0.0650 0.0786 0.0942 Hydrogen Szz -0.0008 0.0152 0.0109 Acetylene szz 0.1123 0.0825 0.0663 Propyne Szz 0.1559 0.1314 0.1177 a) Order parameters obtained from the experimental dipolar coupling constants. The bracketed numbers are errors. The axis system is chosen such that: z is the symmetry axis, y is perpendicular to the plane of the molecule, and x is perpendicular to y, and z. b) Order parameters calculated using a model for the short range repulsive interactions. The force constant governing the interaction has been obtained by a least squares fit of SXx> Syy, and Szz for all solutes to the experimental values. c) Order parameters calculated using the interaction between the molecular polarizability of the solute and the mean electric field due to the liquid crystal solvent. The anisotropy in the mean suare electric field has been obtained by a least squares fit of Sxx, Syy, and Szz for all solutes to the experimental values. d) Order parameters calculated in the same manner as b) except that the force constant has been fit for each solute independently. e) Order parameters calculated in the same manner as c) except that the anisotropy in the mean electric field squared has been fit independently for each solute. f) For TTF z is along the C~C double bond joining the two rings, x is parallel to the vector joining the sulfur atoms within one ring, and y is perpendicular to x and z. g) For 1,4-dichlorobenzene and 1,4-dibromobenzene the z axis is chosen such that it passes through both halogen atoms. The y axis is perpendicular to the plane of the molecule and the x axis is perpendicular to y and z. Results and Discussion / 102 which are taken from references [39] and [116], have been calculated from the measured dipolar couplings using the computer program SHAPE [30] and a geometry from one of electron diffraction, microwave spectroscopy or NMR using liquid crystal solvents. References for the J couplings and geometries are given in table 11 and the definitions of the molecule fixed axes for the solutes are given in the footnotes to table 12. Order parameters for the solutes have been calculated using the short range interaction model and the minimum circumference of the projection of the molecule onto the XY plane (figure 15C). The value of the force constant, k, has been obtained by a least squares fit of the calculated order parameters S , S,„, and S to the experimental values and is found to xx yj zz be 5.55 dyne cm"*. The order parameters calculated from this fit are listed in table 12 and are plotted against the experimental values in figure 18 (the solutes with Cg or higher symmetry are also included in this plot). As can be seen the fit is very good and the model is successful in explaining the results for all the solutes. These results may also help to distinguish among various mechanisms which correlate roughly as the size and shape of the solute. With this in mind order parameters for the solutes have been calculated using the polarizability-electric field interaction as the orienting mechanism. Unfortunately, the low symmetry of the solutes makes determination of their polarizability tensors more difficult because one needs to measure three properties of the solute which depend on the three independent elements of the polarizability tensor, aQ, azz and a ^ — a ^ , [99]. In some recent publications [90, 93, 99] the authors have combined measurements of the Kerr and Cotton-Mouton effects with light scattering Results and Discussion / 103 FIGURE 18 Short Range Interaction Model S (exp) Experimental versus calculated order parameters for solutes in 55wt per cent 1132/EBBA-d2- The calculated order parameters, also shown in tables 10 and 11, have been obtained using the minimum circumference. For solutes with C2v or I>2h symmetry the values of Sxx> Syy, and Szz are plotted (circles). For solutes with C3V or higher symmetry only Szz is plotted (trianlges). The definitions of the axes, x, y, and z are given as footnotes to table 12. The force constant describing the short range interactions has been obtained from a least squares fit to the solid line of slope 1. T = 301.4K, k = 5.55 dyne cm-1, F^z = 0.0 esu, correlation coefficient = 0.985. Results and Discussion / 104 measurements to obtain the elements of the polarizability tensors of six of the solutes with C2w and s y m m e t r y used here. These solutes are labeled 1-6 in table 11. Their polarizabilities extrapolated to zero frequency are also given in table 11 along with calculated values for tetrathiofulvalene (TTF). These polarizabilities have been used to calculate order parameters for the solutes and the value of the mean square electric field has been obtained from a least squares fit of S^, S^, and S 2 2 to the experimental values, and is found to be •i r o o 1.64X10 V cm- . The results of this calculation are shown in table 12 and are plotted in figure 19. As can be seen, the agreement between the experimental and calculated values is quite good, although a slightly better fit is obtained with the short range interaction model (figure 18). As with the solutes of C3 or higher symmetry we are unable to clearly distinguish between these two mechanisms because both are functions of the size and shape of the solutes. In the above discussion we have tested our model by comparing order parameters for different solutes with one another. There is no reason to expect that all solutes will be governed by exactly the same mean field, as each may experience a different distribution of local environments. For example, it is well known that various tetrahedral solutes experience different local magnetic susceptibility anisotropies and therefore must he in different average environments [117, 135]. For the solutes of and D 2 n any two of the three non zero order parameters, Sxx, Syy and Szz are independent of each other. Since all three order parameters must depend on the environment in the same way, they should be predicted exactly by the model if the form of the potential is correct. To investigate this, order parameters for these solutes have been calculated using Results and Discussion / 105 0.6 FIGURE 19 Polarizability - Electric Field Interaction O o 0.4 0.2 -^ 0.0 -0.2 --0.4 J I I I L -0.4 -0.2 0.0 0.2 0.4 0.6 S (exp) Experimental versus calculated order parameters for solutes in 55wt per cent 1132IEBBA-d2' The calculated order parameters, also shown in tables 10 and 11, have been obtained using the interaction between the polarizability tensor of the solute, (given in table 12), and the mean electric field squared due to the liquid crystal. For solutes with C2v o r &2h symmetry the values of SXx> Syy, and Szz are plotted (circles). For solutes with C3V or higher symmetry only Szz is plotted (triangles). The anisotropy in the mean electric field squared has been obtained from a least squares fit to the line of slope 1. T = 301.4K, (<E*> - <E±Z>)112 = 1.35X10 & statvolt cm'1, Fzz = 0.0 esu, correlation coefficient — 0.941. Results and Discussion / 106 the size and shape model and fitting the force constant seperately for each solute. The order parameters obtained from this calculation are shown in table 12 and the force constants are shown in table 13. As can be seen the agreement between the experimental and calculated order parameters is excellent. This demonstrates very clearly that the scatter in figure 18 is primarily due to our assumption that each solute experiences exactly the same average environment and that the form of our potential is correct. Order parameters have also been calculated using the polarizability - electric field mechanism and fitting the anisotropy in the mean square electric field separately for each solute. These order parameters are also shown in table 12 and the values obtained for the anisotropy in the mean square electric field are shown in table 13. Although the agreement with experiment is good, in most cases it is not as good as obtained using the short range interaction model. In order to see more clearly how well the relationships between the order parameters for each solute are predicted we define an asymmetry parameter _ S l l ~ S22 (52) such that for the experimental order parameters |Sgg|>|S22l>|Snl- This definition ensures that 17 calculated from the experimental order parameters lies between zero and one. Figures 20a and 20b show the comparison between the experimental values of 77 and those calculated from the separate fits to each solute for the short range and polarizability electric field interactions respectively. Although there are some deviations, the correlation shown in figure 20a is Results and Discussion / 107 T A B L E 13 Force Constants, Electric Fields, Liquid Crystal Deuteron Splittings and Liquid Crystal Order Matrix Elements (55wt per cent 1132/EBBA-d2 at 301.4K) 2 2 Solute k» < E U > ~ < E 1 > AI/QC DHD<* SQ.c.)e (dyne (10 2 (kHz) (Hz) Szz S ^ - S , cm-1) cm" 2) Acetone 5.37 2.26 16.91 738 0.610 0.016 Fur an 8.54 3.74 17.93 778 0.643 0.016 Thiophene 8.47 2.54 17.46 750 0.620 0.014 Pyridine 7.90 1.92 17.18 755 0.624 0.018 Fluorobenzene 7.48 2.46 17.65 765 0.633 0.016 Chlorobenzene 7.03 2.05 17.81 733 0.606 0.006 TTF 4.50 1.29 17.82 750 0.620 0.010 2,6-difluoropyridine 7.12 16.89 743 0.614 0.017 lodobenzene 5.83 17.58 733 0.606 0.009 1,2 -dichlorobenzene 6.64 17.68 750 0.620 0.010 1,2-dicy anobenzene 6.09 17.77 765 0.633 0.014 1,3-dichlorobenzene 5.72 17.75 750 0.620 0.011 1,3-dinitrobenzene 5.24 17.64 773 0.639 0.018 1,4-dichlorobenzene 5.87 17.60 753 0.623 0.013 1,4-dibromobenzene 5.55 17.84 765 0.633 0.014 Benzene 17.51 715 0.591 0.005 1,3,5 -trifluorobenzene 17.81 743 0.614 0.009 1,3,5 -trichlorobenzene 17.76 693 0.573 -0.003 1,3,5 -tribr omobenzene 17.59 753 0.623 0.013 Results and Discussion / 108 Table 13 cont. Solute ka <E„2> - <E*> D H D d S(l.c.)e (dyne cm"1) (10l5v 2 cm" 2 ) (kHz) (Hz) SxX~Syy Hexafluorobenzene 16.95 733 0.606 0.015 2,4-hexadiyne 17.76 693 0.573 -0.003 2-butyne 17.59 753 0.622 0.013 Allene 17.69 735 0.608 0.008 Bromomethane 17.47 730 0.604 0.009 Iodomethane 17.70 780 0.645 0.019 Acetylene 17.54 752 0.622 0.014 Propyne 17.65 705 0.583 0.001 a) k is the force. constant which governs the short range interactions (see text). The values of k presented have been obtained by a least squares fit of the calculated values of S^, Syy, and Szz for the given solute to the experimental values. 2 „ 2 b) < E | | > — <Ej^  >is the anisotropy in the mean electric field squared due to the liquid crystal (see text). The values presented have been obtained by a least squares fit of the calculated values of Sxx, Syy, and Szz of the given solute to the experimental values. c) A P Q IS the quadrupolar splitting of the deuterons ortho to the nitrogen in the EBBA-d.2 contained in the 55wt per cent 1132/EBBA-d2 mixture (see figure 5). d) DJJTJ is the dipolar coupling between the deuterons and the adjacent protons of the EBBA-d2 contained in the 55wt per cent 1132/EBBA-d2 mixture (see figure 5). e) S(l.c.) refers to the order parameters of the aniline ring of the EBBA-d2 contained in the 55wt per cent 1132IEBBA-d2 (see figure 5). Results and Discussion / 109 Experimental versus calculated values of the asymmetry parameter: 17 = (SJJ — S22) I £>33 (see text) for solutes with C2v o r T>2h symmetry in 55wt per cent 1132/EBBA-d2. A)Short Range Interaction Model The asymmetry parameters have been calculated using a model for the short range interactions. The force constant describing the interaction has been adjusted separately for each solute (see tables 11 and 13). The labeling of the points refers to table 11. T = 301.4K, Fzz = 0.0 esu, correlation coefficient = 0.7620. The solid line of slope 1 has been drawn to aid the eye. Results and Discussion / 110 FIGURE 20B Asymmetry Parameters for C2v and ®2h Solutes Cd O 1.2 0 3 / 0.8 05 / 0.4 / 0 6 /CD2Q7 0.0 - 0 . 4 A i I I I I I I I I - 0 . 4 0.0 0.4 0.8 1.2 V (exp) Experimental versus calculated values of the asymmetry parameter: 17 = (SJJ — S22) I £>33 (see tiext) for solutes with C2v or D2h symmetry in 55wt per cent 1132/EBBA-d2. BJPolarizability Model The asymmetry parameters have been calculated using the interaction between the polarizability tensor of the solute and the mean electric field squared due to the liquid crystal. The mean electric field squared has been adjusted separately for each solute (see tables 11 and 13). The labeling of the points refers to table 11. T = 301.4K, F22, = 0.0 esu, correlation coefficient = —0.4795. The solid line of slope 1 has been drawn to aid the eye. Results and Discussion / 111 resonably good for the short-range interaction model. By comparison the asymmetry parameters calculated using the polarizability - electric field mechanism (figure 20b) do not agree as well with experiment. While this may be a result of inaccuracies in the polarizabilities we have used, it is unlikely that this could 2 2 account for all of the discrepancy. Because the values of k and (<E|) > - <E^ >) have been adjusted separately for each solute the scatter in figures 20a and 20b arises entirely because of inaccuracies in the form of the potential. These results suggest that the polarizability - electric field mechanism cannot account for all of the orientation of the solutes, although it cannot be ruled out entirely. This is also consistent with the results of a recent study by Emsley et al. [9] on anthracene dissolved in several liquid crystals. They found that dispersion forces, which depend on the solute polarizability, could not account for the solute order parameters. However, it should be pointed out that they did not take the electric field gradient - molecular quadrupole moment mechanism into account. c. Liquid Crystal Spectrum as an Internal Standard We now turn our attention to information which may be obtained from the deuteron spectrum of the liquid crystal. In the preceding discussion we have seen that slightly different values for the mean field are required to explain the experimental order parameters for each solute. One reason for this may be the variations in experimental conditions between samples. Because the deuteron spectrum is strongly dependent on the ordering of the liquid crystal molecules, which in turn has a strong dependence on temperature [57] and solute concentration, we can use the spectrum to monitor the experimental conditions. The deuteron spectrum of EBBA-d2 in the nematic phase consists of four lines Results and Discussion / 112 which arise from the quadrupolar couplings of the deuterons and the dipolar couplings between each deuteron and the adjacent ring protons [118]. The dipolar couplings between the deuterons and the other spins in the molecule are not resolved and contribute to the linewidth. In the 55wt per cent 1132/EBBA-d2 mixture containing no solute at 301.4K the value obtained for the quadrupolar splitting, A V Q , is 17.94 kHz and for the dipolar coupling, DJJJJ, is 774 Hz. The contribution of J^ C^—1 Hz) to the dipolar coupling has been ignored. We define the asymmetry parameter, rj^, for the electric field gradient tensor, q, along the CD bond as: <iaa - ^ c c 7?q = • (53) qbb where the a axis is in the plane of the aromatic ring and perpendicular to the CD bond, b is along the CD bond, and C is perpendicular to both a and b. A value of 0.04 has been obtained for rj from single crystal studies of benzene-dg [119] and we shall use this value for the aromatic deuterons in EBBA-d 2 . We will also assume a value of 185 kHz for the quadrupolar coupling constant, e 2qQD/h [118]. If we then take a value of 2.49X IO - 8 cm for the ortho HD distance [80] and 120.0° for the CCD angle the following order parameters for the anline ring (see figure 21) are obtained: S_ = —0.312, S„, = —0.322, and x x yy Szz= 0.634. FIGURE 21 Structure and Molecule Fixed Axes; EBBA-d2 Results and Discussion / 113 Values of Szz and - S for the EBBA-d2 aniline ring for samples containing various solutes at a variety of temperatures are given in tables 13, 14 and 15b. In all cases — is approximately zero so that the rigid aniline ring has almost axially symmetric ordering. The values of Sxx — Syy shown in table 14 are in good agreement with the value of —0.013 obtained by Diehl and Tracey for the same liquid crystal [118]. It is interesting to note that although the values of Sxx — Syy are small they are consistently negative for pure EBBA-d2 (table 14) and consistently positive for the 55wt per cent 1132/EBBA-d2 mixture (tables 13 and 15b). These order parameters may be used as an internal standard for comparing the experimental conditions between samples. However, because of the relatively large errors associated with D ^ we shall use the quadrupolar splitting A I > Q for such comparisons. For the purposes of making small corrections for the effects of finite dilution and temperature differences among the samples we shall assume (i) that F ^ for a given sample with a given liquid crystal quadrupolar splitting is the same as in the sample containing D2 with the same liquid crystal quadrupolar splitting, and (ii) that k for a sample with a given liquid crystal quadrupolar splitting is the same as for a reference sample with the same liquid crystal quadrupolar splitting. As our reference we will use results for the solutes furan and thiophene. The experimental order parameters for these two solutes as a function of temperatures in the 55wt per cent 1132/EBBA-d2 mixtures are reported in references [24, 39] and are shown in table 15a. The liquid crystal splittings are given in table 15b. The value of k for each of the two solutes at each temperature has been obtained by fitting the calculated order parameters to the Results and Discussion / 114 T A B L E 14 EBBA-d2 AT 301.4K: Liquid Crystal Order Parameters", Deuteron Quadrupolar Splittings and Dipolar Couplings Sample A ^ Q** DHD*** Szz SxX Syy (kHz) (Hz) a) 1,3,5-tribromobenzene 21.30 805 0.666 -0.009 b) 1,3,5-trichlorobenzene 20.92 830 0.686 0.000 c) Benzene 21.29 823 0.680 -0.005 d) Hydrogen 21.18 753 0.623 -0.020 e) Propyne 21.28 785 0.649 -0.014 f) Acetylene 21.25 810 0.670 -0.008 g) 2-butyne 21.32 820 0.678 -0.006 h) 2,4-hexadiyne 21.46 835 0.690 -0.004 i) Allene 21.33 813 0.672 -0.008 j) 1,3,5-trifluorobenzene 21.22 823 0.680 -0.004 k) Hexafiuorobenzene 20.58 775 0.641 -0.010 1) Bromome thane 21.24 833 0.689 -0.002 m) Iodomethane 21.22 820 0.678 -0.005 n) Furan 21.48 814 0.673 -0.009 o) Thiophene 21.04 834 0.690 0.000 * The order parameters of EBBA-d2 refer to the aniline ring (see figure 5). * * A UQ is the quadrupolar splitting of the deuterons in EBBA-d2-*** ^KD I S the dipolar coupling between the deuterons and adjacent protons in EBBA-d2-experimental values. These results are shown in figure 22 as a function of A P Q . It is clear from the plot that the dependence of k on A P Q is roughly linear over the range that has been measured, and the slope is to a good approximation the the same for both solutes. It is also apparent that the small variation in the liquid crystal splittings shown in table 13 cannot account for the variation in k from one sample to another. These results suggests the solutes are governed by slightly different force constants, either because they sample different local environments, or because of inadequacies in our model, and not because of variations in the experimental conditions. Results and Discussion / 115 T A B L E 15A Furan and Thiophene in 55wt per cent 1132/EBBA-d2: Experimental Solute Order Parameters Solute Temp.a Sxx Syy (K) Furan 304 0.0907(1) -0.1365(1) 0.0457(2) 306 0.0884(1) -0.1335(2) 0.0452(3) 308 0.0863(1) -0.1311(4) 0.0448(5) 310 0.0841(1) -0.1285(2) 0.0443(3) 312 0.0820(1) -0.1258(1) 0.0437(2) 314 0.0798(1) -0.1227(1) 0.0430(2) 320 0.0735(1) -0.1152(4) 0.0418(5) 325 0.0669(1) -0.1055(6) 0.0389(7) Thiophene 304 0.0586(1) -0.1490(4) 0.0905(5) 306 0.0572(1) -0.1459(2) 0.0887(3) 308 0.0559(1) -0.1427(2) 0.0868(3) 310 0.0545(1) -0.1395(2) 0.0850(3) 312 0.0531(1) -0.1363(2) 0.0832(3) 314 0.0517(1) -0.1326(2) 0.0809(3) 320 0.0469(1) -0.1206(4) 0.0737(5) 325 0.0423(1) -0.1113(17) 0.0690(21) a) The temperature reported is the dial setting of the temperature control unit. A dial setting of 304K corresponds to an actual probe probe temperature of 301.4 ± 0.3K. The spectrum of D 2 in 55wt per cent 1132/EBBA-d2 has been measured as a function of temperature and the results of these measurements are given in table 16. In figure 23 the electric field gradient is plotted as a function of A P Q . It is clear, that to a good approximation, the electric field gradient also varies linearly with the liquid crystal splitting. It is interesting to note, however, that the best line through the points does not pass through the origin. Since both the electric field gradient and the liquid crystal splitting must go to zero at the nematic to isotropic phase transition temperature, the plot cannot be linear over the entire nematic range of the liquid crystal solution. It is also clear from the figure that Results and Discussion / 116 TABLE 15B Furan and Thiophene in 55wt per cent 1132/EBBA-d2: Liquid Crystal Order Parameters, Deuteron Quadrupolar Splittings and Dipolar Coupling Constants Sample Temp.* ** **Q D H D f szz SXx Syy-(K) (kHz) (Hz) Furan 304 17.93 778 0.643 0.016 306 17.58 732 0.605 0.008 308 17.33 715 0.591 0.007 310 17.02 698 0.577 0.006 312 16.74 722 0.597 0.014 314 16.46 698 0.577 0.011 320 15.49 663 0.548 0.012 325 14.44 628 0.519 0.013 Thiophene 304 17.46 750 0.620 0.014 306 17.12 715 0.591 0.009 308 16.81 732 0.605 0.016 310 16.57 715 0.591 0.014 312 16.25 698 0.577 0.013 314 15.83 663 0.548 0.008 320 14.68 610 0.504 0.007 325 13.57 575 0.475 0.009 * The temperature reported is the dial setting of the temperature control unit. A dial setting of 304K corresponds to an actual probe probe temperature of 301.4 ± 0.3K. ** &VQ is the quadrupolar splitting of the deuterons ortho to the nitrogen in the EBBA-a2 contained in the 55wt per cent 1132/EBBA-d2 mixture (see figure 5). t Dfjp is the dipolar coupling between the deuterons and the adjacent protons of the EBBA-d2 contained in the 55wt per cent 1132/EBBA-d2 mixture. $ S^, Syy and Szz are elements of the order parameter tensor of the aniline ring of the EBBA-d2 contained in the 55wt per cent 1132IEBBA-d2 mixture. Results and Discussion / 117 FIGURE 22 Force Constant in 55wt per cent 1132/EBBA-d2 1 s (dyne < 8 M constant 7 / 3 Force fi ty 1 1 1 1 1 13 15 17 A i / Q (kHz) 77ie /brce constant describing the short range interactions between solvent and solute versus the quadrupolar splitting of the deuterons of the EBBA-d2 contained in the 55wt per cent 1132IEBBA-d2- The variation in both quantities is obtained by varying the temperature. The value of the force constant is that obtained from a least squares fit of the calculated order parameters to the experimental values for furan (circles) and thiophene (triangles). The slope and the intercept of the best straight line through the points are; 0.6049 dyne cm~l kHz'l; —2.13 dyne cm-1. FZZ = °-° esu> T = 301.4K, correlation coefficient = 0.9817. Results and Discussion / 118 FIGURE 23 Electric Field Gradients in 55wt per cent 1132/EBBA-d2 14 15 16 17 18 AI/Q (kHz) The measured electric field gradient versus the quadrupolar splitting of the deuterons of the EBBA-d2 contained in the 55 wt per cent 1132/EBBA-d2. The electric field gradient is calculated [10, 31, 32] from the spectrum of D2 dissolved in 55wt per cent 1132/EBBA-d2. The variation in both quantities is obtained by varying the temperature. The slope and intercept of the best straight line through the points are 5.84X109 esu kHz-1 and -I.OIXIOU esu; correlation coefficient = 0.970. Results and Discussion / 119 T A B L E 16 D2 in 55wt per cent 1132/EBBA-d2 T * B * * D * * * F Z Z DHD* (K) (Hz) (Hz) (esuXlOll) (kHz) (Hz) 304 -125.26 5.08 0.023 17.71 760 306 -127.98 5.20 0.008 17.43 720 308 -130.25 5.29 0.012 17.24 715 310 -132.21 5.39 -0.019 16.98 740 312 -134.08 5.49 -0.057 16.80 760 314 -135.79 5.56 -0.058 16.57 720 320 -138.82 5.70 -0.083 15.68 695 325 -139.59 5.77 -0.144 14.90 663 * The temperature reported is the dial setting of the temperature control unit. A dial setting of 304K corresponds to an actual probe temperature of 301.4K ** B is the quadrupolar coupling of the deuterons in D2 *** D is the dipolar coupling between the deuterons in D2 tAvQ is the quadrupolar splitting of the deuterons ortho to the nitrogen in the EBBA-d2 contained in the 55wt per cent 1132/EBBA-d2 mixture (see figure 21). $ DHD is the dipolar coupling between the deuterons and the adjacent protons of the EBBA-d2 contained in the 55wt per cent 1132IEBBA-d2 mixture (see figure 21). over the range of liquid crystal splittings measured for the other samples at 301.4K (table 13) the electric field gradient experienced by D 2 is very small. Thus, if all solutes experience the same mean electric field gradient as D2, mechanisms involving this field gradient may be safely ignored for our samples. However, the possibility remains that each solute experiences a somewhat different distribution of local environments and hence a different mean electric field gradient. Results and Discussion / 120 d. Summary We will now briefly summarize the results for solutes in 55wt per cent 1132/EBBA before going on to discuss the results for other liquid crystals. First, we have shown that the order parameters of solutes dissolved in this mixture are quite strongly correlated with the size and shape of the solutes. Second, we have shown that a simple model for the short-range interactions gives somewhat better agreement with the experimental results than the interaction between the polarizability and a mean square electric field, due to the liquid crystal. Both of these mechanisms are, at least approximately, functions of the size and shape of the solutes. Finally, we have shown that the force constant necessary to describe the solute order parameters using the short-range interaction model is not exactly the same for all solutes, and that this is not a result of variations in the experimental conditions. 4. Ordering of solutes i n other liquid crystals Up to this point we have only considered the ordering of solutes in 55wt per cent 1132/EBBA where F z z is measured to be zero. We will now turn our attention to results in other liquid crystals. In these solvents F ^ is no longer zero so that if our model for the short range interactions is to find any general application, we must first be able to calculate the contribution to solute ordering from the electric field gradient - molecular quadrupole moment mechanism. Unfortunately, this requires a knowledge of the molecular quadrupole moments of the solutes which are not easily determined [56]. It is clear from the range of values which have been reported for solutes such as benzene [120-122] and acetylene [123-126], that the accuracy of the literature values is not always very Results and Discussion / 121 good. Values for the molecular quadrupole moments of 15 of the solutes we have studied are shown in table 17. We will use these values throughout to calculate the contribution to solute ordering from the electric field gradient - molecular quadrupole moment mechanism. We begin by considering the ordering of solutes in the two component liquid crystals, 1132 and EBBA, of the zero field gradient mixture. a. 1132 and EBBA Experimental order parameters for the solutes shown in table 17 have been measured in 1132 and EBBA-d2 and these results are shown in table 18. In order to calculate order parameters for these solutes we require both the force constant, k, and the electric field gradient, F z z , for the two solvents. We shall arbitrarily assume that the value of k is the same as found for the 55wt per cent 1132/EBBA-d2 mixture and we shall use the value of F z z obtained from experiments on D2 in 1132 and EBBA-d2 [31, 39]. If these assumptions are correct and if the exponential in expression (48) can be expanded in a Taylor series and truncated after the linear term then the difference between the order parameters in either 1132 or EBBA-d2 and the 55wt per cent 1132/EBBA-d2 mixture represents the contribution to ordering from the molecular quadrupole moment electric field gradient mechanism. Calculated versus experimental values for {S(EBBA-d2) - S(55wt per cent 1132)} and {S(1132) - S(55wt per cent 1132)} are shown in figures 24 and 25. Considering that there are no adjustable parameters involved and that differences in order parameters are being plotted, the agreement between the experimental and calculated values is really quite good. As can be seen, all three components of the order parameter tensor are Results and Discussion / 122 T A B L E 17 Molecular Quadrupole Moments Solute Molecular quadrupole momenta (10-26 e s u c m 2) Reference Benzene -7.80 ± 2.2 average of values [120] and [121] from 1,3,5-trifluorobenzene 0.94 [120] 1,3,5-trichlorobenzene -3.24 [128] 1,3,5-tribromobenzene -4.80 [128] Hexafluorobenzene 9.50 [122] 2,4-hexadiyne 2.59b 2 .15 c estimate [39] 2-butyne 2.38b 1.62C estimate [39] Allene 2.60b 1.56c 7.35 estimate [39] upper limit [127] Bromomethane 4.64 ± 1.1 average of values [129, 130] from Iodomethane 5.35 [130] Hydrogend H2 0.653 HD 0.648 D2 0.642 [55] Acetylene 5.5 ± 2.5 average of values [123, 124] from Propyne 4.82 [131] Fur an Qxx 5.9 Qyy -6.1 Qzz 0.2 [132] Thiophene Qxx 6.6 Qyy -8.3 Qzz 1-7 [132] a) For benzene, bromomethane, and acetylene the value presented is an average of the largest and smallest values reported in the literature. b) Estimated from solute order parameter in 1132 c) Estimated from solute order parameter in EBBA-d2 d) Average for ground vibrational state Results and Discussion / 123 T A B L E 18 Solute Order Parameters: 1132 and EBBA-d2 at 301.4K Solute 1132 EBBA-d2 a) 1,3,5-tribromobenzene -0.2883 -0.1848 b) 1,3,5-trichlorobenzene -0.2878 -0.2038 c) Benzene -0.2519 -0.1157 d) Hydrogen 0.0082 -0.0111 e) Propyne 0.2542 0.0553 f) Acetylene 0.1912 -0.0585 g) 2-butyne 0.2592 0.1524 h) 2,4-hexadiyne 0.4270 0.3144 i) Allene 0.1979 0.0925 j) 1,3,5-trifluorobenzene -0.2590 -0.1744 k) Hexafluorobenzene -0.2144 -0.3144 1) Bromomethane 0.0947 0.0368 m) Iodomethane 0.0915 0.0280 n) Furan Sxx 0.1727 0.0017 Syy -0.2063 -0.0565 Szz 0.0336 0.0548 o) Thiophene S>xx 0.1118 -0.0082 Syy -0.1990 -0.0737 Szz 0.0871 0.0819 a) For the solutes with C3V or higher symmetry Szz is given. predicted quite well for furan (labeled as circles) and thiophene (labeled as triangles) in both liquid crystals. All of the points (with the exception of the ones corresponding to 1,3,5-trifluorobenzene where Qmol is reported to be small and positive (table 17) he in the correct quadrant. The considerable scatter in the points is most likely a result of our assumption of the transferability of the force constant and electric field gradient. The experimental errors associated with molecular quadrupole moments, which are reflected in the error bars on the points, also contribute to the scatter. For three of the solutes, 2,4-hexadiyne, 2-butyne and allene, no values for the molecular quadrupole moments have been reported in the literature. However, from the experimental values of {S(1132) — Results and Discussion / 124 FIGURE 24 Solutes in EBBA-d2: Short Range Interaction Model - 0 .2 0.0 0.2 S (exp) Experimental versus calculated values of {S(EBBA-d2) — S(55 wt per cent 1132/EBBA-d2)} for furan (circles), thiophene (triangles), and solutes with C3V or higher symmetry. S(EBBA-d2) is the order parameter of the solute in EBBA-d2 and S(55 wt per cent 1132IEBBA-d2) is the order parameter of the solute in 55wt per cent 1132IEBBA-d2- For the solutes with C3V or higher symmetry only SZz I S plotted. The labeling of the points refers to table 10. The error bars are meant to represent the maximum possible error from Qmol I n the calculated values, and have been calculated as follows. Where more than one literature value for Qmol is available, the error bar represents the order parameters calculated using the limits of the literature values. Where only one literature value is available, the accuracy of Qmol is taken equal to that of a similar molecule. For example the benzene value has been used for all benzene derivatives. 55wt per cent 1132/EBBA-d2: T = 301.4K, k = 5.55 dyne cm-1, F z z = 0.0 esu. EBBA-d2 : T = 301 AK, k = 5.55 dyne cm-1, Fzz = —5.75X1QH esu. correlation coefficient = 0.858 Results and Discussion / 125 FIGURE 25 Solutes in 1132: Short Range Interaction Model -0 .2 0.0 0.2 S (exp) Experimental versus calculated values of (S(1132) — S(55wt per cent 1132/EBBA-d2)} for furan (circles), thiophene (triangles) and solutes with C3V or higher symmtery. S(1132) is the order parameter of a given solute in 1132 and S(55wt per cent 1132/EBBA-d2) is the order parameter of a given solute in 55wt per cent 1132/EBBA-d2- The calculated and experimental values have been obtained and are labeled in the same manner as for figure 24. 55wt per cent 1132/EBBA-d2: T = 301.4K, k = 5.55 dyne cm-1, F z z = 0.0 esu. 1132: T=301.4K, k = 5.55 dyne cm-1, F z z = 6.07X10H esu, correlation coefficient = 0.833. Results and Discussion / 126 S(55wt per cent 1132)} and {S(EBBA-d2) - S(55wt per cent 1132)} estimates of the molecular quadrupole moment of each solute can be made and are presented in table 17. There is little variation between the two values obtained for each solute and the signs of the molecular quadrupole moments are consistent with the molecular structure. The value obtained for allene is also consistent in magnitude and sign with the upper limit reported in reference [127]. It is also pleasing to note that the negative order parameters observed for the symmetry axes of acetylene and hydrogen in EBBA-d 2 are correctly predicted without invoking special mechanisms for these two solutes. These results demonstrate very clearly the importance of the electric field gradient - molecular quadrupole moment mechanism for the ordering of solutes in these two liquid crystals. b. 5CB-a,B-d4 As a final test of our model for the short-range interactions we shall consider the order of solutes dissolved in the nematic liquid crystal 4-(a, j3-d4-n-pentyl)-4-cyanobiphenyl (5CB-a,B-d^). Values of in this liquid crystal at various temperatures (from reference [40]) are presented in table 19 with the measured dipolar and quadrupolar coupling constants of D 2 and the quadrupolar splittings of the a- and ^-deuterons of the 5CB-a,B-d^. The temperature T r is defined as T/T^_j where TJJ_J is the nematic to isotropic phase transition temperature of the solution. The values of &re about 30 per cent as large as observed in EBBA and 1132 [31, 39] and are the same sign as in 1132. Measured order parameters for 9 solutes with C g y or higher symmetry dissolved in 5CB-a,B-d^ (also from reference [40]) are shown in table 20 along with the quadrupolar splitting of the liquid crystal deuterons. These experimental order Results and Discussion / 127 T A B L E 19 D 2 in 5CB T T r * Bt D* F Z Z At>a (K) (Hz) (Hz) (lOH esu) (kHz) (kHz) 275 0.8929 901.41±0.18 -38.92+0.13 3.60±0.13 67.21 50.78 285 0.9253 709.19±0.33 - 3 0 . 4 3 ± 0 . 1 5 2.52±0.24 62.65 45.70 295 0.9578 524.85±0.23 -22.8010.10 2.3010.16 56.25 39.33 300 0.9740 424.93±0.18 -18.64+0.08 2.1510.13 51.09 34.91 305 0.9903 299.37±0.20 -13.00+0.08 1.3010.13 42.39 28.05 307 0.9968 219.45±0.20 - 9 . 7 0 ± 0 . 0 9 1.2310.14 34.23 22.27 * Tr = TIT^j where 7JJI is the nematic isotropic phase transition temperature. t B is the quadrupolar coupling of the deuterons. $ D is the dipolar coupling between the two deuterons T A B L E 20 Order Parameters of Solutes in 5CB at 294K* Solute S(exp) S(calc)** S(calc)*** Aiv/J (kHz) (kHz) a) l,3,5tribromo- -0.2160 -0.2124 -0.2170 50.05 34.21 benzene b) l,3,5trichloro- -0.2139 -0.1920 -0.2116 53.61 37.21 benzene c) benzene -0.1352 -0.1353 -0.1158 44.03 29.50 d) hydrogen 0.0033 0.0175 0.0206 56.25 39.33 e) propyne 0.1668 0.1475 0.1618 53.17 36.62 f) acetylene 0.1686 0.1429 0.1712 58.11 41.18 g) 2-butyne 0.1624 0.1774 0.1738 48.04 32.61 h) 2,4-hexadiyne 0.3545 0.3684 0.4445 55.68 39.03 i) allene 0.1160 0.1105 0.1209 52.20 36.43 * All samples were measured at T=294K except benzene which was measured at T= -296K. ** k = 4.22 dyne cm-1 , Fzz = 2.30X1011 esu for all solutes. *** k and Fzz for each solute scaled using Aj>a (see text). k 0 = —1.26 dyne cm-1 and kf for the D2 sample at 394K = 5.15 dyne cm'l. Results and Discussion / 128 parameters may then be predicted using our model for the short range interactions and taking into account the interaction between the molecular quadrupole moments of the solutes and the electric field gradients given in table 19. For 2,4-hexadiyne, 2-butyne and allene, averages of the estimates of the molecular quadrupole moments obtained from the solute order parameters in 1132 and EBBA-d2 have been used. The value of k has again been obtained by a least squares fit and is found to be 4.26 dyne cm -*. The results of the fit are shown in table 20 and are plotted in figure 26. Excellent agreement between the calculated and experimental values is obtained. It is interesting to note that the values of the molecular quadrupole moments for 2,4-hexadiyne, 2-butyne and allene needed to explain the order parameters of these solutes in EBBA-d2 and 1132, also predict their orientation in 5CB-a,/~-d4. In calculating the solute order parameters we have assumed that the samples are infinitely dilute and at precisely the same temperature. The experiments, however, do not satisfy these criteria. Differences in temperature and concentration among the samples are apparent in the variation of liquid crystal deuteron quadrupolar splittings (A»»a and Ap^ in table 20). We will now use these splittings to correct the values of k and F Z 2 for these differences. For all spectra of the samples reported here, the value of Ai> ^  obtained for a given value of A»>a is not affected by the presence of the solute. This implies that the solutes do not significantly affect conformational averaging or the value of 7j (expression 52) of the solvent molecules and means that either Ai>a or Ap^ may be used to make corrections for the effects of temperature and finite dilution. Results and Discussion / 130 A plot of the calculated field gradient (table 19) versus the measured quadrupolar splitting of the a-deuterons of the 5CB-a,/3-d4 (figure 27) shows a roughly linear correlation. The best straight line through the points has a slope of 6.62X10^ esu kHz"* and an intercept of —1.28X10** esu. (A similar curve for the /3-deuterons has a slope of 7.82X10^ esu kHz"* and an intercept of — 0.70X10** esu.) There is considerable scatter in the results, primarily because the dipolar and quadrupolar splittings of D 2 cannot be measured accurately enough to give accurate values of F z z > This graph, however, can be used as a calibration curve for F z z . We obtain a similar curve for the force constant by assuming that the value of k corresponding to a given liquid crystal splitting, Ai>a, is the same for all solutes. The dependence of k on Ai>a for some reference solute may then be used as a calibration curve for k for all solutes. We have shown in figure 22 that the force constant obtained for furan and thiophene in 55wt per cent 1132/EBBA-d2 depends roughly linearly on the liquid crystal deuteron quadrupolar splittings over a small temperature range. Thus, it seems reasonable to expect that a similar behaviour will be found for 5CB-a,/3-d4. The force constant kj for a given sample is then: ki= ko+ (3k /3Ai» a ) , (54) where &vla is the quadrupolar splitting due to the a- deuterons in the 5CB-a,^-d^ for sample i. The values of kQ and bk/dAva will be obtained from the dependence of k on A v a for a reference sample. For this purpose we will use 1,3,5-trichlorobenzene. A plot of k versus A»>a for this sample is shown in figure 28. From this plot we obtain a value of 0.1141 dyne cm' * kHz"* for Results and Discussion / 129 FIGURE 26 Solutes in 5CB-a,/3-d4: Short Range Interaction Model -0 .2 0.0 0.2 0.4 S (exp) Experimental versus calculated order parameters in 5CB-a,B~d^. The points are labeled in the same manner as in table 20. The force constant describing the short range interactions has been obtained from a least squares fit to the solid line of slope 1. T = 294K; k = 4.22 dyne cm-1; F z z = 2.305X10H esu; correlation coefficient = 0.997 Results and Discussion / 131 FIGURE 27 Electric Field Gradients in 5CB-a,^-d4 9 on fc 2 30 40 50 60 A v a (kHz) 70 The measured electric field gradient versus the quadrupolar splitting of the a-deuterons of the 5CB-a, {}-d4. The electric field gradient is measured from the spectrum of D2 dissolved in the 5CB-a, $-d4. The variation in both quantities is obtained by varying the temperature. The slope and intercept of the best straight line through the points are 6.64X10$ esu kHz'l and —1.28X10U esu; correlation coefficient = 0.941. Results and Discussion / 132 FIGURE 2 8 Force constant in 5CB-a,^-d4 7 i 30 40 50 60 70 A v a (kHz) The force constant describing the short range interactions between solvent and solute versus the quadrupolar splitting of the a-deuterons of the 5CB-a,0-d4. The value of the force constant is that needed to explain the orientation of l,3,5-trichiorobenzene;dkJdhVa = 0.1141 dyne cm-lkHz-1; k 0 = —1.26 dyne cm-1; correlation coefficient — 0.992. Results and Discussion / 133 9k/9FAi>a and —1.26 dyne cm - 1 for kQ (a similar curve using the /3-deuterons gives 9 k/9Ai>0 = 0.1329 dyne cm - 1 kHz - 1 and kQ=-0.19 dyne cm - 1). As is evident from figure 28, our assumption that the variation of k with A f a is linear is valid over the range of the experimental results. The values of k- and for each sample may now be obtained and order parameters for the other solutes may be calculated. The values obtained from this calculation are shown in table 20 and are plotted against the experimental order parameters in figure 29. Again excellent agreement with experiment is obtained. However, it is important to point out that unlike the order parameters presented in figure 26 only the value for 1,3,5-trichlorobenzene has been obtained from a fit to the experimental value. The excellent agreement for the other solutes indicates very clearly that in this liquid crystal the force constant and electric field gradient may be transferred from one solute to another provided that the variation in experimental conditions is taken into account. It is also interesting to note that the values of the molecular quadrupole moments of 2,4-hexadiyne, 2-butyne and allene, obtained from the order parameters of the solvents in EBBA-d2 and 1132, predict their ordering in SCB-a^-d^ very well. The order parameters calculated using the B -deuteron splitting to scale F ^ and k differ from those reported in table 20 by at most 3.1 per cent. The values obtained for kt- for the D 2 solution at 295K using the a and B deuterons are 5.15 and 5.02 dyne cm - 1 respectively. Hence the calculated order parameters do not depend on which quadrupolar splitting is used to calculate F ^ and k. To further test the above scaling method we compare the experimental and Results and Discussion / 134 F I G U R E 29 Solutes in 5CB-a,/3-d4: Short Range Interaction Model - 0 . 2 0.0 0.2 0.4 S (exp) Calculated versus experimental order parameters for solutes dissolved in 5CB-a, &-d4. The calculated order parameters have been obtained in the same manner as for figure 26 except that values of F Z z and k have been obtained from figures 28 and 29 using the quadrupolar splittings of the a-deuterons of the 5CB-a,(}-d4 (see text). The labeling of the points refers to table 20. correlation coefficient = 0.994. Results and Discussion / 135 FIGURE 30 2,4-hexadiyne in 5CB-a,/J-d4 0.6 0.4 o 00 0.2 0.0 0.0 0.2 0.4 0.6 S ( e x p ) Calculated versus experimental order parameters for 2,4- hexadiyne in 5CB-a, B-d.4. The calculated order parameters have been obtained in the same manner as for figure 26 except that values of F z z and k have been obtained from figures 28 and 29 using the quadrupolar splittings of the a-deuterons of the 5CB-a, B-d4 (see text), correlation coefficient = 0.991. circles = 1 mole per cent, squares = 9 mole per cent. Results and Discussion / 136 calculated order parameters obtained for two 2,4-hexadiyne samples at various temperatures [133] with those calculated using the values of kj from figure 27 and F z z from figure 28. The results of this calculation are presented in figure 30. As can be seen fairly good agreement is obtained between the calculated and experimental results. However, the points lie on a straight line whose slope is slightly different from one. This could be a result of several factors. First, our assumption that all solutes experience exactly the same environment is not entirely valid as we have seen in 55wt per cent 1132/EBBA-d2. Second, the calculated values depend on the value of the molecular quadrupole moment which has been estimated from the order parameters of the solute in 1132 and EBBA. Given the number of assumptions we have made in estimating this value, the agreement in figure 30 is surprisingly good. Finally, the linear correlations we have assumed from figures 28 and 29 may not be valid over the range of values of A<va nd LvQ for these samples. Nonetheless, for the purposes of making relatively small correction to k and F z z our method would appear to be good. In addition, as can be seen from figure 30 the method predicts order parameters for both dilute and concentrated samples. IV. CONCLUSIONS The main point of this thesis has been an attempt to describe the ordering of small solutes in nematic liquid crystals. As we have seen our approach of beginning with very simple solutes and studying their spectra in some detail has been quite successful. The discovery, from the spectrum of deuterium [10, 15, 31, 32, 36], of the presence of an external electric field has lead to a better understanding of the ordering of molecular hydrogen and other solutes. It is now clear that the interaction between the molecular quadrupole moment and this field gradient accounts for a considerable portion of the observed ordering. This interaction does not account for all of the subtle effects observed for hydrogen where the field gradient is small. It is apparent that other mechanisms are involved in the ordering of this solute. For highly symmetric solutes such as methane the coupling of the internal motions and the reorientations of the solute also plays a significant role in determining the observed dipolar couplings. These effects are also seen for acetylene, although they account for a much smaller portion of the observed couplings. It is also clear that the liquid crystal dependence of the ratios of the dipolar couplings in acetylene results from non-rigid contributions to the couplings and not because of changes in the average structure as had previously been assumed [21]. We have also seen that the size and shape of a solute governs its ordering in 55wt per cent 1132/EBBA where the external field gradient is zero. This suggests that the short range interactions are the dominant factor in determining solute ordering and a very simple model for these interactions successfully predicts the order parameters of a wide range of solutes in this mixture. However, because solute properties such as the polarizability are also functions of the size and shape it is difficult to 137 Conclusions / 138 distinguish clearly between mechanisms involving such properties and the size and shape. The short range interaction model is somewhat more successful in predicting the anisotropy in the order matrix of solutes with C2 V and &2h symmetry. This result gives support to the idea that it is indeed the size and shape of a solute which determines its ordering. Finally we have shown that in liquid crystals where the field gradient is not zero that the combination of the short range interaction model and the electric field gradient quadrupole moment mechanism predicts the order parameters quite well. From this work, then, a very clear picture has been gained of the general form of the interactions involved in the orientation of solutes in nematic liquid crystals. This should provide a basis for a description of the system on a molecular level. The size and shape model may also provide a method of explaining the ordering of the liquid crystal molecules themselves. V. APPENDIX I Summary of Results for H 2 , HD and D 2 in Several Liquid Crystals Temperature Solute B (Hz) D (Hz) S(exp) EBBA 298.0 H 2 3059.3010.50 -1.063X10-2 298.0 HD -1817.20±0.50 506.5010.50 -1.144X10-2 298.0 D 2 -1943.60±0.50 83.1010.20 -1.220X10-2 320.0 H 2 2632.0010.30 -9.151X10-3 320.0 HD -1556.70 + 0.40 434.9010.40 -0.983X10-2 320.0 D 2 -1654.10±0.40 71.2110.10 -1.046X10-2 1132 298.0 H 2 -2340.9010.30 8.134X10-3 298.0 HD 1351.80±0.60 -382.5010.60 0.864X10*2 298.0 D 2 1461.4910.30 -63.2710.30 0.929X10-2 320.0 H 2 -1796.1010.30 6.244X10-3 320.0 HD 1029.0010.10 -292.8011.60 0.662X10-2 320.0 D 2 1106.8011.40 -48.0010.80 0.705X10*2 Phase V 298.0 H 2 2670.5010.30 -9.279X10-3 298.0 D 2 -1695.3010.90 72.5010.30 -1.064X10-2 298.0 HD -1588.1010.90 443.2010.90 -1.001X10-2 310.0 D 2 -706.9810.05 30.6310.02 -0.450X10-2 1167 310.0 H 2 1141.2010.30 -3.967X10-3 310.0 HD -655.8610.06 185.6810.06 -0.420X10-2 310.0 D 2 -706.9810.05 30.6310.02 -0.450X10-2 330.0 D 2 -572.7010.60 24.9810.20 -0.367X10-2 330.0 HD -534.3010.30 152.3810.30 -0.345X10-2 23.9wt per cent 1132/EBBA 310.0 D 2 -788.2610.43 33.6710.14 -0.494X10-2 49.3wt per cent 1132/EBBA 310.0 D 2 -267.6310.55 11.1510.18 -0.164X10-2 61.3wt per cent 1132/EBBA 303.0 D 2 -13.7510.11 0.4510.04 -0.007X10-2 66.1wt per cent 1132/EBBA 310.0 D 2 116.0610.42 -5.3810.14 0.079X10-2 76.0wt per cent 1132/EBBA 310.0 D 2 373.0710.20 -16.4610.60 0.242X10-2 SCB-a^-d, 310.0 D 2 524.8510.23 -22.8010.10 0.335X10-2 139 Appendix I / 140 Summary of Hydrogen Results cont. Temperature Solute B (Hz) D (Hz) S(exp) 50wt per cent 1132/EBBA (sample 1) 293.7 H 2 448.7310.47 -1.559X L0-3 295.4 H 2 455.4810.47 -1.582X 10-3 298.9 H 2 457.8410.47 -1.591X 10-3 300.6 H 2 459.3610.47 -1.596X 10-3 302.4 H 2 459.7210.47 -1.598X LO-3 304.1 H 2 460.5410.47 -1.600X 10-3 305.9 H 2 459.9910.47 -1.599X 10-3 307.6 H 2 458.8210.47 -1.595X 10-3 309.4 H 2 456.9810.47 -1.588X 10-3 311.1 H 2 454.5510.47 -1.580X 10-3 312.9 H 2 451.4610.47 -1.569X 10-3 314.6 H 2 447.6710.47 -1.556X 10-3 316.4 H 2 443.4010.47 -1.541X 10-3 318.2 H 2 438.5110.47 -1.524X 10-3 290.1 HD -251.93±0.07 65.3310.14 -1.476X 10-3 293.7 HD -254.20±0.07 65.8710.14 -1.488X 10-3 295.4 HD -255.85±0.07 66.4210.14 -1.501 X 10-3 297.2 HD -257.96±0.07 66.9610.14 -1.513X 10-3 298.9 HD -259.21±0.07 67.3910.14 -1.523 X 10-3 301.6 HD -260.27±0.21 67.7110.14 -1.530X 10-3 303.3 HD -260.08±0.21 68.0410.14 -1.537X 10-3 305.0 HD -260.07±0.21 67.8110.14 -1.532X 10-3 306.7 HD -259.9810.21 67.7110.14 -1.530X 10-3 290.1 D 2 -251.9310.07 10.5010.02 -1.541X 10-3 293.7 D 2 -254.2010.07 10.6110.02 -1.557X 10-3 295.4 D 2 -255.8510.07 10.6610.02 -1.565X 10-3 297.2 D 2 -257.9610.07 10.7810.02 -1.582X 10-3 298.9 D 2 -259.2110.07 10.8110.02 -1.587X 10-3 301.6 D 2 -260.2710.21 10.9510.07 -1.607 X 10-3 303.3 D 2 -260.0810.21 10.8610.07 -1.594X 10-3 305.0 D 2 -260.0710.21 10.8910.07 -1.599X 10-3 306.7 D 2 -259.9810.21 10.8910.07 -1.599X 10-3 Appendix I / 141 Summary of Hydrogen Results cont. Temperature Solute B (Hz) D (Hz) S(exp) 50wt per cent 1132/EBBA (sample 2) 293.6 H 2 448.7310.47 -1.559X 10-3 295.4 H 2 455.4810.47 -1.582X 10-3 298.9 H 2 457.8410.47 -1.591X 10-3 300.6 H 2 459.3610.47 -1.596X 10-3 302.4 H 2 459.7210.47 -1.598X 10-3 304.1 H 2 460.5410.47 -1.600X 10-3 305.9 H 2 459.9910.47 -1.599X 10-3 307.6 H 2 458.8210.47 -1.595X 10-3 309.4 H 2 456.9810.47 -1.588X 10-3 311.1 H 2 454.5510.47 -1.580X 10-3 312.9 H 2 451.4610.47 -1.569X 10-3 314.6 H 2 447.6710.47 -1.556X 10-3 316.4 H 2 443.4010.47 -1.541X 10-3 318.2 H 2 438.5110.47 -1.524X L0-3 293.7 HD -282.9110.35 77.7012.00 -1.755X 10-3 295.4 HD -284.9410.49 78.8210.71 -1.781X 10-3 298.9 HD -287.6910.30 78.0610.71 -1.764X 10-3 300.7 HD -287.4410.35 78.7710.71 -1.780X 10-3 302.4 HD -288.1410.35 78.6210.71 -1.776X 10-3 304.2 HD 79.6210.71 -1.799X 10-3 305.9 HD 78.3910.71 -1.771X 10-3 307.7 HD 78.0410.71 -1.764X 10-3 309.4 HD 77.4710.71 -1.751X 10-3 311.2 HD 77.1910.71 -1.745X 10-3 312.9 HD 76.4210.71 -1.727X 10-3 314.7 HD 76.0410.71 -1.719X 10-3 316.4 HD 74.9510.71 -1.694X 10-3 318.2 HD 74.1310.71 -1.676X 10-3 293.7 D 2 -294.1715.00 11.5311.65 -1.692 X 10-3 295.4 D 2 -298.3810.69 12.2010.23 -1.791X 10-3 298.9 D 2 -301.3910.35 12.6010.20 -1.850X 10-3 300.7 D 2 -301.0410.35 12.5810.20 -1.847X 10-3 302.4 D 2 -302.3010.35 13.0910.12 -1.922X 10-3 Appendix I / 1 4 2 Summary of Hydrogen Results cont. Temperature Solute B (Hz) D (Hz) S(exp) 55wt per cent 1132 / E B B A 300.6 H 2 200.3710.09 - 6 . 9 6 3 X ] LO-4 302 . 4 H 2 203.6910.09 -7.078X] LO-4 304 . 1 H 2 207.0610.09 - 7 . 1 9 6 X ] LO-4 305.9 H 2 209.6410.09 - 7 . 2 8 6 X ] LO-4 307 . 6 H 2 212.1710.09 -7.374X] LO-4 309 . 4 H 2 214.5010.23 -7.455X] LO-4 311.1 H 2 216.4310.23 - 7 . 5 2 3 X ] LO-4 312 . 9 H 2 217.8310.23 - 7 . 5 7 2 X ] LO-4 314 . 6 H 2 218.3510.23 - 7 . 5 9 0 X ] LO-4 300.7 HD -135 . 3 9 + 0.14 35.7810:14 - 8 . 0 8 4 X ] LO-4 302 . 4 HD - 1 3 7 . 2 2 ± 0 . 1 4 36.2810.14 - 8 . 1 9 8 X ] LO-4 304 . 2 HD -138 . 6 8 + 0.14 36.7310.14 - 8 . 3 0 0 X ] LO-4 305 . 9 HD -139 . 9 4 + 0.14 37.1510.14 - 8 . 3 9 5 X ] LO-4 307 . 7 HD - 1 4 1 . 0 0 ± 0.14 37.5010.14 - 8 . 4 7 5 X ] LO-4 309 . 4 HD - 1 4 1 . 9 2 ± 0 . 1 4 37.7910.14 - 8 . 5 4 1 X ] LO-4 311.2 HD - 1 4 3 . 5 3 ± 0 . 1 4 38.0210.14 - 8 . 5 9 3 X ] LO-4 312 . 9 HD - 1 4 2 . 6 6 ± 0 . 2 6 38.2010.26 - 8.634 X ] LO-4 314 . 7 HD -142 . 6 4 + 0.14 3 8.1510.14 - 8 . 6 2 3 X ] LO-4 300.7 D 2 - 1 3 8 . 0 2 ± 0 . 2 9 5.5910.14 - 8 . 2 0 6 X ] LO-4 302 . 4 D 2 - 1 4 0 . 4 0 ± 0 . 4 3 5.7710.13 - 8 . 4 7 0 X ] LO-4 304 . 2 D 2 - 1 4 2 . 1 0 ± 0 . 3 0 5.8810.10 - 8 . 6 3 2 X ] LO-4 3 0 5 . 9 D 2 - 1 4 3 . 6 2 ± 0 . 3 5 5.9910.12 - 8 . 7 9 4 X ] LO-4 307 . 7 D 2 - 1 4 5 . 1 0 ± 0 . 1 4 6.1210.05 - 8 . 9 8 6 X ] LO-4 309 . 4 D 2 - 1 4 6 . 1 2 ± 0 . 1 4 6.1510.05 - 9 . 0 3 0 X ] LO-4 311.2 D 2 -146 . 8 3 + 0.14 6.1610 . 05 - 9 . 0 4 5 X ] LO-4 312 . 9 D 2 -146 . 8 8 1 0.14 6.1310 . 05 - 9 . 0 0 2 X ] LO-4 314 . 7 D 2 - 1 4 6 . 8 3 ± 0 . 1 4 6.0810.05 - 8 . 9 2 9 X ] LO-4 Appendix I / 143 Summary of Hydrogen Results cont. Temperature Solute B (Hz) D (Hz) S(exp) 60wt per cent 1132/EBBA 291.3 H 2 -42.6110.23 1.480X10-4 293.2 H 2 -34.2510.12 1.190X10-4 294.2 H 2 -29.7510.12 1.034X10-4 295.2 H 2 -27.4210.24 0.953X10-4 296.2 H 2 -24.0010.19 0.834X10-4 297.1 H 2 -19.3910.47 0.674X10-4 298.1 H 2 -16.0110.21 0.556X10-4 299.1 H 2 -13.5910.14 0.472X10-4 300.0 H 2 -7.3410.19 0.255X10-4 301.0 H 2 -2.7610.38 0.096X10-4 302.0 H 2 0.6010.23 -0.021X10-4 303.0 H 2 3.9610.14 -0.138X10-4 304.9 H 2 10.2510.16 -0.356X10-4 291.3 HD 4.4210.21 -3.8210.28 8.629X10-5 293.3 HD 1.5510.14 -2.7910.14 6.303X10-5 294.3 HD -3.4010.11 -1.4410.11 3.253X10-5 295.2 HD -3.9410.11 -1.3010.11 2.937X10-5 296.2 HD -5.5810.11 -0.9310.11 2.101X10-5 297.2 HD -8.1510.11 -0.0410.11 0.090X10-5 298.1 HD -9.8610.14 0.4610.14 -1.039X10-5 299.1 HD -11.5110.14 0.9110.14 -2.056X10-5 300.1 HD -14.2810.14 1.6210.14 -3.660X10-5 301.1 HD -16.1310.18 2.1010.18 -4.745X10-5 302.0 HD -18.2110.11 2.8010.11 -6.327X10-5 303.0 HD -20.6310.11 3.4910.11 -7.886X10-5 293.3 D 2 9.4610.11 -0.7210.04 1.057X10-4 302.0 D 2 -12.1010.11 0.2110.04 -0.308X10-4 303.0 D 2 -13.7510.11 0.4510.04 -0.661X10-4 Appendix I / 144 Summary of Hydrogen Results cont. Temperature Solute B (Hz) D (Hz) S(exp) 70wt per cent 1132/EBBA 293.7 HD 363.4810.07 -52.0710.14 1.176X10-3 295.4 HD 352.0510.07 -49.3710.14 1.115X10-3 297.2 HD 339.8610.07 -46.5010.14 1.051X10-3 305.9 HD 280.8210.07 -31.2610.14 0.706X10-3 307.7 HD 270.1210.07 -28.8910.14 0.653X10-3 309.4 HD 258.3610.07 -26.1410.14 0.591X10-3 311.2 HD 246.6710.07 -23.4410.14 0.530X10-3 312.9 HD 235.8510.07 -19.7510.14 0.446X10-3 293.7 D 2 363.4810.07 -16.0010.02 2.348X10-3 295.4 D 2 352.0510.07 -15.4810.02 2.272X10-3 297.2 D 2 339.8610.07 -14.9810.02 2.199X10-3 305.9 D 2 280.8210.07 -12.4010.02 1.821X10-3 307.7 D 2 270.1210.07 -11.9710.02 1.757X10-3 309.4 D 2 258.3610.07 -11.4510.02 1.681X10-3 311.2 D 2 246.6710.07 -10.9610.02 1.609X10-3 312.9 D 2 235.8510.07 -10.4410.02 1.533X10-3 The results for 1132, EBBA, Phase V and 1167 are from reference [10]. See also tables 2, 3 and 9. VI. APPENDIX II A. CALCULATION OF MAXIMUM AND MINIMUM CIRCUMFERENCES In this appendix we will describe the calculation of the minimum and maximum circumferences of the projection of a solute onto the XY plane. These circumferences are used in section D of chapter III in the model for the short range interactions. 1. Projection of solute In order to project the molecule onto the XY plane we define the transformation from space fixed coordinates to molecule fixed coordinates in terms of two rotations of the axis system. The first rotation is in a clockwise direction about the space fixed Y axis through an angle 8. This is then followed by a clockwise rotation through an angle <f> about the resulting z axis. In terms of the angles 8 and 4> the coordinates of the projection of the i t n nucleus of the solute onto the XY plane are given by: Xj- = *j- cosfl cos<f> — yj cos0 sin$ + z± sin0 (55) Yj = x- s'm<j> + yj- cos0 . (56) The projection of the molecule in the XY plane is thus an array of overlapping circles centred at the points (Xj,Yj). The projected circle associated with the fa nucleus is given by: (X - Xj) 2 + (Y - Yj) 2 = r f 2 . (57) 145 Appendix II / 146 The maximum and minimum values of X and Y on the projected circle associated with the j nucleus are given by: Xmaxj = X ; + rj (58) Xminj = Xj - r £ (59) Ymaxj = Yj + r f (60) Yminj = Y ; - r ; (61) If Xmaxy and Ymax^ are the largest of the various Xmaxj and Ymaxj, and Xmin7 and Ymin^, are the smallest of the various Xmin- and Ymin; then the l m i i • points (Xmaxy,Yy), (Xmin^Yj), (X^Ymax^), and (X m ,Ymin m ) lie on the outer surface of the array of projected circles. 2. Maximum circumference The maximum circumference may then be calculated from the points satisfying eqn.(57) by starting at (X-,YmaxO and moving around circle j in small steps until the intersection with an adjacent circle, k, is found. Then starting at the intersection point we move around circle k in the same manner until the next intersection point is found and so on until the point (Xy,YmaxO is found again. The sum of the distances between adjacent points will be equal to the circumference in the limit of a small enough step. 3. Minimum circumference For the minimum circumference we begin by drawing line segments between all pairs of nuclei such that each line segment is tangential to the two circles involved and does not intersect the internuclear vector at a point which lies between the two nuclei. We then define the angle 6 between the Y axis and a Appendix II / 147 given line segment. Starting with the nucleus m which has the smallest Ymin we take the nuclei with Xmax- >^  Xmax m and calculate 6 for the line segments between each of these nuclei and nucleus m. We label the nucleus whose line segment has the largest 6 as m'. This line segment is then taken as a section of the projection of the outer surface of the molecule. Its contribution to the circumference is the distance between the intersection points of the line segment with the two circles plus the arc length of circle m between (X m , Yminm) and the intersection point of the line segment with circle m. The procedure is then repeated starting at nucleus m' and we continue in this manner until the nucleus with the maximum Xmax is found. The contribution to the circumference between the other three pairs of outer points may be calculated in the same manner to give the total circumference. VII. APPENDIX III A. SOURCE CODE FOR T H E CALCULATION OF EXPRESSION (51) PROGRAM SIZE(INPUT, OUTPUT); ^ ************************************************** This program calculates the order parameters a set of molecules based on their size and shape. The order parameter i s the ca l ss ica l average weighted using the potential calculated by the function POTEN which uses the minimum circumference of the projection of the molecule in the XY plane. * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ^ CONST MAXATOM = 20; MAXMOL =20; K = 1.38044E-16; PI = 3.141592645; TYPE ATOMRECORD = RECORD P, Q, R, RADIUS, X, Y: REAL; RIJ: ARRAY [ l . . MAXATOM] OF REAL; END; MOLRECQRD = RECORD NAME: PACKED ARRAY [ l . . 30] OF CHAR; NATOM, XMAXNUC, YMAXNUC, XMINNUC, YMINNUC: INTEGER; FKON, EFG, QXX, QYY, QZZ: REAL; DIFF, SEXP, SXX, SYY, SZZ: ARRAY [ l . . 3] OF REAL; ATOM: ARRAY [ l . . MAXATOM] OF ATOMRECORD; END; MOLARRAY = ARRAY [ l . . MAXMOL] OF MOLRECORD; VAR MOLECULE: MOLARRAY; MOLPT, NMOL: INTEGER; FCON, FZZ, T: REAL; LXTAL: PACKED ARRAY [1 . . 30] OF CHAR; OUTCODE: (REED, RITE); CIRCFILE: FILE OF REAL; FUNCTION DASIN(VAR X:REAL):REAL; FORTRAN; FUNCTION DATAN2(VAR X, Y:REAL):REAL; FORTRAN; PROCEDURE READINPUT; ! 148 Appendix III / 149 VAR I, MOLPT, ATOMPT: INTEGER; CODE: CHAR; BEGIN PAGE(OUTPUT); WRITELNC INPUT ' ) ; WRITELN (' + ' ) ; WRITELN; READLN(CODE); CASE CODE OF 'R' : OUTCODE := REED; •W : OUTCODE := RITE; END(*CASE*); FOR I := 1 TO 30 DO LXTAL [ I ] := ' ; I := 0; WHILE NOT EOLN(INPUT) DO BEGIN I := I + 1; READ(LXTAL[I]); END (*WHILE*); READLN(NMOL, T); WRITELN(' 'LIQUID CRYSTAL: ' , LXTAL); WRITELNC ' , 'TEMPERATURE : ' , T: 7: 1); WRITELN; WRITELN(' MOLECULE','COORDINATES':26,'RADIUS':19, 'QUADRUPOLE MOMENT *: 26); WRITELN('(XE-8 CM)':34,'(XE-8 CM)':22, '(XE-26 ESU CM2)':23); WRITELN('X' : 23, 'Y' : 8, ' Z ' : 8, 'XX': 30, 'YY': 6, 'ZZ':8) ; WRITELN(' + ' , ' ' :42, ' ' :42); FOR MOLPT := 1 TO NMOL DO" WITH MOLECULE[MOLPT] DO BEGIN FOR I := 1 TO 30 DO NAME[I] := ' ' ; I : = 0 • WHILE NOT EOLN(INPUT) DO BEGIN I := I + 1; READ(NAME[I]) END (*WHILE*); READLN(FKON, EFG); WRITELNC ' , 'FIELD GRADIENT:', EFG: 12); WRITELNC ' , "FORCE CONSTANT:', FKON: 12); READ(SEXP[1J,SEXP[2],SEXP[3], NATOM, QXX, QYY, QZZ); WRITELN; WRITELNC ' , NAME, QXX: 37: 2, QYY: 6: 2, QZZ: 6: 2); QXX := QXX * 1.0E-26; QYY := QYY * 1.0E-26; QZZ := QZZ * 1.0E-26; FOR ATOMPT := 1 TO NATOM DO WITH ATOM[ATOMPT] DO Appendix III / 150 BEGIN READLN(P, Q, R, RADIUS); WRITELNC ' , P: 24: 4, Q: 8: 4, R: 8: 4, RADIUS:12: 2); END (*WITH*); END (*WITH*); END (*READINPUT*); PROCEDURE PROJECTNUCLEKTHETA, PHI: REAL); ^************************************************** This procedure projects the coordinates of the solute onto the XY plane as a function of i t s orientation which are given by the angles 8 and t/>. ******************************************** VAR ATOMPT: INTEGER; BEGIN WITH MOLECULE[MOLPT] DO BEGIN FOR ATOMPT := 1 TO NATOM DO WITH ATOM[ATOMPT ] DO BEGIN Y := P * SIN(PHI) + Q * COS(PHI); X := P * COS(THETA) * COS(PHI) -Q * COS(THETA) * SIN(PHI) + R * SIN(THETA); END (*WITH*); END (*WITH*); END (*PROJECTNUCLEI*) ; PROCEDURE OUTERNUCLEI; ^ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * This procedure finds the points with the maximum and mimimum X and Y values for the projection of a given molecule i n the XY plane ******************************************* ^ VAR ATOMPT: INTEGER; YMAX, YMIN, XMAX, XMIN: REAL; BEGIN WITH MOLECULE[MOLPT] DO BEGIN YMAX := 0.0; YMIN := 0.0; XMAX := 0.0; XMIN := 0.0; FOR ATOMPT := 1 TO NATOM DO WITH ATOM[ATOMPT] DO Appendix III / 151 BEGIN IF (Y + RADIUS - YMAX) > 1.0E-5 THEN BEGIN YMAX := Y + RADIUS; YMAXNUC := ATOMPT; END (*IF*); IF (Y - RADIUS - YMIN) < - l.OE-5 THEN BEGIN YMIN := Y - RADIUS; YMINNUC := ATOMPT END (*IF*); IF (X + RADIUS - XMAX) > l.OE-5 THEN BEGIN XMAX := X + RADIUS; XMAXNUC := ATOMPT; END (*IF*); IF (X - RADIUS - XMIN) < - l.OE-5 THEN BEGIN XMIN := X - RADIUS; XMINNUC := ATOMPT END (*IF*); END (*WITH*); END (*WITH*); END (*OUTERNUCLEI*); PROCEDURE ROTATEAXES; ^***************************************************** This procedure rotates the space fixed axis system through 90 degrees about Z so that the procedure GETCIRCUM need only calculate the section of the circumference from YMIN to XMAX after each of four 90 degree rotations. ******************************************************\ VAR ATOMPT, SAVINT: INTEGER; XOLD: REAL; BEGIN WITH MOLECULE[MOLPT] DO BEGIN FOR ATOMPT := 1 TO NATOM DO WITH ATOM [ATOMPT] DO BEGIN XOLD := X; X := Y; Y := - XOLD; END (*WITH*); SAVINT := XMAXNUC; XMAXNUC := YMAXNUC; YMAXNUC := XMINNUC; XMINNUC := YMINNUC; YMINNUC := SAVINT; END (*WITH*); END (*ROTATEAXES*); Appendix III / 152 PROCEDURE GETRIJ; (*************************************************** This procedure calculates the distances between pairs of nuclei for the projection of the solute in the XY plane. These distances are used in calculating the circumference. ******************************************** VAR XI, YI: REAL; I, J : INTEGER; BEGIN WITH MOLECULE[MOLPT] DO BEGIN FOR I := 1 TO NATOM DO BEGIN WITH AT0M[I] DO BEGIN XI := X; YI := Y END (*WITH*); FOR J := 1 TO NATOM DO WITH ATOM[j] DO RIJ[I] := SQRT((XI - X) * (XI - X) + (YI - Y) * (YI - Y)); END (*FOR*); END (*WITH*); END (*GETRIJ*); PROCEDURE GETCIRCUM(VAR SEG: REAL); ^ *************************************************** This procedure calculates the section of the circumference between YMIN and XMAX ****************************************************\ VAR ATOMPT, If J : INTEGER; XI, YI, RI, XMAXJ, XMAX, XMAXI, ALPHAJ, ALPHA, GAMMA, DELTAJ, DELTA, COSGAM, SINGAM, BETA: REAL; BEGIN CASE OUTCODE OF Appendix III / 153 REED: READ(CIRCFILE,SEG); RITE: BEGIN WITH MOLECULE[MOLPT] DO BEGIN J := 0; SEG := 0.0; BETA := 0.0; I := YMINNUC; WITH ATOM[XMAXNUC] DO XMAX := X + RADIUS; WHILE I O XMAXNUC DO BEGIN DELTA := 0.0; WITH AT0M[I] DO BEGIN XI := X; YI := Y; RI := RADIUS; XMAXI := X + RADIUS; END (*WITH*); IF ABS (XMAX - XMAXI) < 1E-5 THEN BEGIN WITH ATOM[XMAXNUC] DO IF RIJ[I] O 0.0 THEN BEGIN ALPHA := (RADIUS - RI) / RIJ [ l ] ; ALPHA := DASIN(ALPHA); SINGAM := (X - XI) / R l J f l l ; COSGAM := (Y - YI) / RIJ[IJ; GAMMA := DATAN2(SINGAM,COSGAM); DELTA := ALPHAJ + GAMMA; J := XMAXNUC; END (*IF*) ELSE J := XMAXNUC; END (*IF*) ELSE BEGIN FOR ATOMPT := 1 TO NATOM DO WITH ATOM[ATOMPT] DO BEGIN XMAXJ := X + RADIUS; IF ((XMAXJ - XMAXI) > - 1.0E-5) AND (ATOMPTOI) AND (RIJ[I] o 0.0) THEN BEGIN ALPHAJ :=(RADIUS-RI)/RIJ[I]; ALPHAJ := DASIN(ALPHAJ); SINGAM := (X - XI) / RIJ [ l ] ; COSGAM := (Y - YI) / RIJ [ l ] ; GAMMA:= DATAN2(SINGAM,COSGAM); DELTAJ := ALPHAJ + GAMMA; IF (DELTAJ - DELTA) > -1.0E-5 THEN BEGIN DELTA := DELTAJ; ALPHA := ALPHAJ; Appendix III / 154 J := ATOMPT; END (*IF*); END (*IF*); END (*WITH*); END (*ELSE*); WITH AT0M[J] DO BEGIN SEG := SEG + RI * ((PI / 2.0 - DELTA) - BETA); BETA := PI / 2.0 - DELTA; SEG := SEG + RIJ[l] * COS(ALPHA); END (*WITH*); I 5= J * END (*WHILE*); SEG := SEG + ATOM[ I ] .RADIUS * (PI / 2.0 - BETA); WRITE(CIRCFILE,SEG); END (*WITH*); END(*RITE*); END (*CASE*); END (*GETCIRCUM*); FUNCTION POTEN(THETA, PHI: REAL): REAL; ^******************************************** This function calculates the potential energy of the molecule as a function of the angles 8 and 4>. The potentila has the form U = ~i K C 2 Where C i s the circumference of the projection of the molecule in XY plane *******************************************\ VAR C, SEG, PXX, PYY, PZZ: REAL; J : INTEGER; BEGIN (*P0TEN*) WITH MOLECULE[MOLPT] DO BEGIN C : = 0.0" IF OUTCODE = RITE THEN BEGIN PROJECTNUCLEI(THETA, PHI) ; GETRIJ; OUTERNUCLEI; END(*IF*); GETCIRCUM(SEG); C := C + SEG; FOR J := 1 TO 3 DO BEGIN IF OUTCODE = RITE THEN Appendix III / 155 ROTATEAXES; GETCIRCUM(SEG); C := C + SEG; END (*FOR*); PXX := (3.0 * SIN(THETA) * SIN(THETA) * COS(PHI) * COS(PHI) - 1.0) / 2.0; PYY : = (3.0 * SIN(THETA) * SIN (THETA) * SIN(PHI) * SIN(PHI) - 1.0) / 2.0; PZZ := (3.0 * COS (THETA) * COS (THETA) - 1.0) / 2.0; U := (FCON * C * C - FZZ * (QXX * PXX + QYY * PYY + PZZ * QZZ)) / (2.0 * K * T); U := EXP(- U) * SIN(THETA); END (*WITH*); POTEN := U; END (*POTEN*); PROCEDURE INTEGRAL(FUNCTION F(X, Y: REAL): REAL; L, U, LI , U l : REAL); /************************************************** This procedure calculates the order parameters of the molecule by integrating the function F(X,Y) from X=L to X=U and Y=L1 to Y=U1. INTEGRAL i s called with: F(X,Y) = POTEN(THETA,PHI) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * VAR F l , F2, F3, F4, INC,INC1, THETA, THETA1, SUM, PHI1, PHI: REAL; I, J i NLOOPS1, NLOOPS: INTEGER; BEGIN WITH MOLECULE [MOLPT] DO BEGIN FOR I := 1 TO 3 DO BEGIN SZZ[I] := 0.0; SYY[I] := 0.0; SXX[I] := 0.0; END (*FOR*); SUM := 0.0; F2 := 0.0; NLOOPS1 :=36; NLOOPS := 36; INC := (U - L) / NLOOPS; INC1 := (Ul-Ll) /NLOOPS1; Appendix III / 156 THETA := L; PHI := LI; F4 := F(THETA, PHI); FOR I := 1 TO NLOOPS1 DO BEGIN F3 := F4; PHI1 := PHI; PHI := I * INC1 + LI ; F4 := F(THETA, PHI); F2 := F2 + (F3 + F4) * INC1 / SXX[2] := SXX[2] + (COS(PHI) * * F4 + COS(PHIl) * * F3) * INC1 / 2.0; SYY[2] := SYY[2] + (SIN(PHI) * SIN(PHI) * F4 + SIN(PHIl) * 2.0; COS(PHI) COS(PHIl) SIN(PHIl) * END (*FOR*); SZZ[2] := F2 * (1 / 2) * (3 * COS(THETA) F3) * INC1 / 2.0; [2] := SYY[2] * (3 / 2) [2] := SXX[2J * (3 / 2) = 1 TO NLOOPS DO SYY SXX FOR I BEGIN * COS(THETA) * SIN(THETA) * SIN(THETA) - i ) ; * SIN(THETA); * SIN(THETA); • F2; 1' := SZZ 2' 1' := SYY '2' ' l ' := SXX '2' SZZ SYY SXX THETA1 := THETA; THETA := INC * I + L; F2 := 0.0; SZZ[2] := 0.0; SYY[2l := 0.0; SXX[2] := 0.0; PHI := LI; F4 := F( THETA, PHI); FOR J := 1 TO NLOOPS 1 DO BEGIN F3 := F4; PHI1 := PHI; PHI := J * INC1+ LI ; F4 := F(THETA, PHI); F2 := F2 + (F3 + F4) * INC1 SXX[2] := SXX[2] + (COS(PHI) * F4 + COS(PHIl) * COS(PHIl) * F3) SYY[2] := SYY[2] + (SIN(PHI) * F4 + SIN(PHIl) * SIN(PHIl) * F3) END (*FOR*); SZZ[2] := F2 * (1 / 2) * (3 * COS (THETA) COS(THETA) - 1); / 2.0; COS(PHI) * INC1 / 2.0; * SIN(PHI) * INC1 / 2.0; SYY SXX SUM : = [21 := SYY[2] [2] := SXX[2J SXX SYY SZZ SYY SXX SUM + (FI := SXX[3] := SYY" := SZZ END (*FOR*); SXX[3] := SXX[3] / ] (3 / 2) * SIN(THETA) (3 / 2) * SIN(THETA) F2) * INC / 2, (SXX[1' (SYY[1" (SZZ[1 * SIN(THETA); * SIN(THETA); ] + SXX 2' ) * INC / 2.0; + SYY '2' ) * INC / 2.0; J + SZZ '2' ) * INC / 2.0; SUM - 1 / 2; Appendix III / 157 SYY[3] := SYY[3] / SUM - 1 / 2 ; SZZ[3] := SZZ[3] / SUM; WRITELN; WRITELN('SXX = ' : 20, SXX[3l); WRITELN('SYY = ' : 20, SYY[3J); WRITELNCSZZ = ' : 20, SZZ[3]); END (*WITH*); END (*INTEGRAL*); PROCEDURE PRINTOUT; ^ ************************************* This procedure writes the output **************************************^ VAR If MOLPT: INTEGER; BEGIN PAGE(OUTPUT); WRITELN(' OUTPUT ' ) ; WRITELN (' + ' ) ; WRITELN; WRITELN(' ' , 'LIQUID CRYSTAL: ' , LXTAL); WRITELN(' ' , "FIELD GRADIENT:', FZZ: 12); WRITELNC ' , 'FORCE CONSTANT:', FCON: 12); WRITELNC *, 'TEMPERATURE : ' , T: 7: 1); WRITELN; WRITELN(' ' , 'MOLECULE': 8, 'S(CALC)': 33, 'S(EXP)': 8, 'PERCENT ERROR': 17); WRITELN; FOR MOLPT := 1 TO NMOL DO WITH MOLECULE[MOLPT] DO BEGIN WRITEC NAME); WRITE('SXX',SXX[3]:8:4,SEXP[l]:8:4,DIFF[l]:14:3); WRITELN; WRITELN('SYY*:34,SYY[3]:8:4,SEXP[2]:8:4,DIFF[2]:14:3); WRITELN('SZZ':34,SZZ[3]:8:4,SEXP[3]:8:4,DIFF[3]:14:3); WRITELN; END (*WITH*); END (*PRINTOUT*); (* This i s the main body of the program*) BEGIN (*SIZE*) RESET(INPUT,'UNIT=SCARDS'); REWRITE(OUTPUT,'UNIT=SPRINT,NOCC*); READINPUT; CASE OUTCODE OF REED: RESET(CIRCFILE,'UNIT=1,BLOCK'); RITE: REWRITE(CIRCFILE,'UNIT=1,BLOCK'); END(*CASE*); PAGE(OUTPUT); FOR MOLPT := 1 TO NMOL DO WITH MOLECULE [MOLPT] DO Appendix III / 158 BEGIN FZZ := EFG; FCON := FKON; WRITEC ',NAME); WRITELN; INTEGRAL(POTEN, 0.0, [ 1 ] 11} DIFF DIFF DIFF END (*WITH*); PRINTOUT; CLOSE(CIRCFILE); END (*SIZE*). = ABS((SXX = ABS((SYY = ABS((SZZ PI 3 ] -I]: / 2.0 SEXP - SEXP SEXP ,0,PI); ?.) / / / SEXP SEXP SEXP 100.0; 100.0; 100.0; VIII. 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