NMR STUDIES OF SOLUTES IN NEMATIC LIQUID CRYSTALS: UNDERSTANDING THE NEMATOGENS By Leon Christiaan ter Beek Jr., Landbouwuniversiteit Wageningen, The Netherlands, 1988 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CHEMISTRY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January 1994 © Leon Christiaan ter Beek, 1994 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. 1 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. (Signature) Department of The University of British Columbia Vancouver, Canada Date DE-6 (2/88) Abstract NMR spectroscopy and dynamics of small solute molecules dissolved in several nematic liquid crystals provided insights toward a better understanding of nematogens. Expressions for all possible spin-relaxation rates for dideuterium are given. Exper imentally obtained Zeeman relaxation rates in several liquid crystals were fitted to the expression for the Zeeman relaxation rate, and correlation times were thus obtained. From our study of the dynamics of dideuterium in nematic liquid crystals it was also found that ortho and para dideuterium have different relaxation rates due to the inter ference of the dipolar and quadrupolar coupling constants and their spectral densities. The Zeeman relaxation rates of dideuterium show no liquid crystal solvent effect. Three different solute molecules with a single bond of rotation have been investigated in a special zero-electric field gradient nematic mixture: furfural, 2-chlorobenzaldehyde, and 2,2’-dithiophene. Dipolar coupling constants measured from NMR experiments are reported for these solutes and discussed in terms of the order parameters and conformer probabilities. In the course of the analysis a new geometry for 2,2’-dithiophene has been found. The spectrum of propane has been analyzed with the use of Multiple-Quantum NMR. Dipolar coupling constants of ethane, propane, benzene, 1 ,3,5-trifluorobenzene, hexaflu orobenzene, and 1 ,3,5-trichlorobenzene as an internal reference for liquid crystal orien tation, as solutes in three different nematic liquid crystals, were obtained from NMR spectra. The results are discussed in terms of orientation and solute—solvent interactions and especially the long-range molecular quadrupole moment—mean electric field gradi ent interaction. A method for obtaining molecular quadrupole moments of solutes in 11 nematic liquid crystals is discussed. The molecular quadrupole moment—mean electric field gradient interaction contributes to the orientation potential of these small alkanes and benzenes and will contribute to the orientation of alkyl chains and aromatic cores of nematogens as well. 111 Table of Contents Abstract ii List of Tables vi List of Figures viii Acknowledgement ix 1 Introduction and thesis outline 2 1.1 Nematic Liquid Crystals 1.2 Outline of Thesis — 1 an overview 1 4 Experimental 6 2.1 Liquid Crystals 6 2.2 Solutes and Sample Preparation 6 2.3 Instrumentation 7 3 Dideuterium as a solute in the nematic phase 3.1 Introduction 3.2 Basic Principles 9 9 12 3.2.1 Two Coupled Spin-i Particles as a Quantum Rotor 12 3.2.2 Equation of Motion 14 3.3 Relaxation Equations for two coupled spin-i particles 24 3.4 Illustrative Examples of the Solid- and Jeener-Echo 32 iv 4 5 3.4.1 Solid Echo 3.4.2 Jeener-Broekaert Echo . 32 3.5 Experimental Results for the Zeeman Spin-Lattice Relaxation Rate 3.6 Conclusions of this Chapter . . 34 37 Solutes with s-cis and s-trans isomers 39 4.1 Introduction 39 4.2 Theory 42 4.3 Results and Discussion 45 4.3.1 Geometry of Furfural 45 4.3.2 Geometry of 2-chlorobenzaldehyde 45 4.3.3 Geometry of 2,2’-dithiophene 49 4.4 Dealing with Flexible Molecules 56 4.5 Conclusions of this Chapter 61 Determination of Molecular Quadrupole Moments using Liquid Crys 62 tals 6 32 5.1 Introduction 62 5.2 Multiple-Quantum NMR 64 5.3 Molecular quadrupole moments 69 5.4 Experimental Results 71 5.5 Discussion 75 5.6 Conclusions of this chapter 79 80 Conclusions 83 Bibliography V List of Tables 3.1 A Spin-Basis for Two-Coupled Spin-i Nuclei 13 3.2 Density Matrix Elements for Two Coupled Spin-i Nuclei 16 3.3 Normalization Constants for the Elements in Table 3.2 17 3.4 Transition Frequencies for Two Coupled Spin-i Nuclei 22 3.5 Relaxation Rates Para under Dipolar and Quadrupolar Hamiltonian 3.6 Relaxation Rates Para under Spin-Rotation Hamiltonian 3.7 Relaxation Rates Ortho under Dipolar and Quadrupolar Hamiltonian 3.8 Relaxation Rates Ortho under Spin-Rotation Hamiltonian 31 3.9 Commutators of the MQ-Coherences with Secular Hamiltonian 33 25 . 27 28 . 3.10 Experimental Zeeman Relaxation Data of 2 D in Several Liquid Crystals . 35 4.11 EBBA-d2 Quadrupolar Splittings 44 4.12 Spectral Parameters of Furfural 46 4.13 Spectral Parameters 2-Chlorobenzaldehyde 47 4.14 Spectral Parameters 2,2’-Dithiophene 48 4.15 Products of Order Parameters and Weights, and Torsional Angles of 2,2’Dithiophene 52 4.16 Experimental Order and Geometric Parameters for 2,2’-Dithiophene . . 54 4.17 Experimental and Calculated Data for the Order Parameters of Furfural, 2-Chlorobenzaldehyde, and 2,2’-Dithiophene 57 4.18 Weight of Fufural, 2-Chlorobenzaldehyde, and 2,2’-Dithiophene 58 5.19 Spectral Parameters of Benzene, TFB, HFB, and TCB 72 vi 5.20 Spectral Parameters of Ethane 73 5.21 Spectral Parameters of Propane 74 5.22 Experimental Order Parameters of Propane, Ethane, Benzene, TFB, HFB, 75 and TCB 5.23 Calculated Order Parameters and Quadrupole Moments vii 78 List of Figures 1.1 Representation of Nematic Phase 3.2 H-NMR-spectrum of D 2 2 in PCH-7 10 3.3 Coupling Scheme of Certain Spin-Orders and the Lattice 26 3.4 Experimental Zeeman Relaxation Curves of D 2 in ZLI-1132 35 4.5 Structures and Molecular Fixed Axes for Furfural, 2-Chlorobenzaldehyde, 2 and 2,2’-Dithiophene 41 5.6 H-NMR Spectrum of Propane in ZLI-1132 1 66 5.7 Six and Seven-Quantum Spectra of Propane in ZLI-1132 67 viii Acknowledgement I wish to thank my supervisor Dr. E. Elliott Burneil for his guidance, encouragement, patience, and financial support, which has been a tremendous help for me over the years. The many hours of discussions we had in which he showed his enthusiasm, and positive attitude towards science, and topics which strayed rather far from science, kept me on the right track. I also wish to thank my colleagues and other people who passed the door to our lab: Dr. Dan S. Zimmerman, Kevin Y. Li, Chandrakumar, Dr. Z. Sun, and James Poison. As a chemist, it has been very rewarding for me to meet Dan Zimmerman and James Poison, who as physicists taught me not to rely to much on my intuition. I would like to thank Dr. Myer Bloom for teaching me the physics of NMR and for introducing me to the huge spectrum of culinary adventures in Vancouver. Most of my experiments wouldn’t have been possible without the excellert help from the people in the electronics shop, mechanical shop, and NMR lab. Their support has been great. I am also very grateful to my parents for their long-distance support. My deepest gratitude I must reserve for my wife Makimi, whose patience and understanding I am very thankful for. ix Chapter 1 Introduction and thesis outline More than a century after the first reported description of a nematic thermotropic liq uid crystalline phase by the Austrian botanist Reinitzer [1], many scientists are still investigating this interesting state of matter [2]. 1.1 Nematic Liquid Crystals — an overview There are several distinct kinds of liquid crystalline phases reported [2], and more phases are being discovered. The common characteristic of these phases is that they are stable in a temperature range which is between the temperature ranges where the isotropic liquid and the solid phase are stable. For this reason they are also referred to as mesophases, from the Greek word mesos meaning middle. In this thesis we will be concerned with only the nematic mesophase. It was Friedel [3] who first gave the name nematic from the Greek word nema, meaning thread, because of the thread-like discontinuities which can be observed under the polarizing microscope for this phase. The nematic phase has the lowest ordering of all the mesophases and, if present, precedes the transition to the isotropic liquid which occurs at the clearing point. The molecules making up the nematic phase are arranged in such a manner that there is no positional order of their centres of mass, like in the isotropic liquid, but there is a long-range orientational order. The molecules tend to orient on the average along a preferred direction within a large cluster of molecules, called the director . Within a sufficiently large magnetic field, however, all the local directors will be either parallel 1 Chapter 1. Introduction and thesis outline 2 Figure 1.1: Representation of Nematic Phase c phase. The molecules are An artist’s impression of a “snap shot” representation of the nemati r , but their centres aligned so that their long axes are on the average parallel to the directo of mass are distributed with equal probability. the molecular magnetic or perpendicular to the magnetic field depending on the sign of NMR experiments for susceptibility anisotropy of the molecules. This is important for our symmetric nematic it defines a single fixed direction in space for the uniaxial cylindrically at one instant in time. phase. Figure 1.1 gives an artist’s impression of the nematic phase ed by the The ordering of the molecules in the nematic phase is completely describ axis system with respect time or ensemble average of the orientation of a molecule-fixed tional ordering in to the director. However, the mechanisms responsible for this orienta ing insight into these liquid crystalline systems are not completely understood. Achiev of liquid crystal phases mechanisms may help us understand the fundamental principles better. investigation of these NMR is one of many techniques employed in the experimental averages over internal systems. The measurements usually yield parameters that are Chapter 1. Introduction and thesis outline 3 molecular motions. This averaging presents a problem in the interpretation of results for the liquid crystal molecules themselves because a general characteristic of molecules that form nematic phases is that they are elongated and non-rigid. The difficulty of dealing with such flexible molecules is that the orientation of each conformer is described by an independent order matrix [4]. However, no such problems exist in computer calculation experiments using models for the orientational potential. One of the most succesful descriptions of orientational order is the theory by Maier and Saupe [5]. The intermolecular potential is approximated by a single-molecule poten tial function and the use of a mean field description. They propose that the long-range anisotropic attractive interactions resulting from dispersion forces are responsible for ne matic formation. Other methods for describing the orientational order in liquid crystals are discussed in [6]. Since nematogens are difficult to deal with due to their size and flex ibility, small solute molecules have been used as probes dissolved in the liquid crystalline phase. A logical starting point for a systematic study to gain insight into the ordering mech anisms is to use small solute molecules dissolved in nematic liquid crystals. In recent years, a promising approach has been the use of hydrogen [7, 8, 9, 10, 11] and methane [11, 12, 13, 14] and their deuterated analogs as probe molecules for the orienting poten tial. Experiments with these molecules have demonstrated the presence of a mean electric field gradient (efg) in nematic liquid crystals [7, 9, 10]. This average efg of the liquid crystal can have a positive or negative sign depending on the particular liquid crystal. This efg is also a function of temperature and composition of the liquid crystal. In the case of molecular hydrogen the interaction between its molecular quadrupole moment and the efg of the solvent explains the sign and most of the magnitude of its orientation parameter in a series of liquid crystals. Many recent attempts at understanding the ordering of solute molecules have focussed Chapter 1. Introduction and thesis outline 4 around correlating the solute order parameters with some molecular property such as the polarizability [8, 15], moments of inertia [16, 17], and molecular shape. Several different modelling schemes for molecular shape have been used such as the chord model [18] and size and shape model [19, 20, 21]. Since liquid crystals with opposite average efg’s can be mixed together to obtain a zero-efg mixture where the average interaction with this zero-efg and the solute molecular quadrupole moment is also zero, the solute’s non-zero order tensor in this mixture must arise from an additional orienting mechanism. Also, as the size of the solute increases, the molecular quadrupole moment — efg mechanism no longer adequately describes the orientation in a non-zero efg liquid crystal. The general orientation of solutes in this special mixture can be described quite succesfully by modelling the short-range repulsive interactions between the solute molecule, modelled as a collection of Van der Waals’ spheres, and a mean restoring force of the nematic liquid crystal [19, 22, 23, 24, 25, 26, 27]. The interaction potential of this phenomenological mean field model is assumed to depend on the size and shape anisotropy of the solute molecule, and one or two force constants, which are a property of the liquid crystal environment only. The true nature of the orientational mechanisms in liquid crystals remains an intriguing problem. 1.2 Outline of Thesis The main objective of this thesis is to understand and learn how to deal with the molecules which make up the nematic phase. These molecules, the nematogens, are usually elon gated and flexible. These properties make it difficult to handle the statistics of the nematogens properly. We are using small solute molecules as probe molecules to learn more about the ne matogens themselves. The assumptions are that the solute molecules experience the same Chapter 1. Introduction and thesis outline 5 orienting potential as the nematogens and that solute—solute interactions are negligible. Therefore, low solute concentrations are used. A lot has been learned from the NMR spectroscopy of dideuterium in the nematic phase. In chapter 3, therefore, we expand our spin Hamiltonian to include a random part and try to learn more from the dynamics of this small solute molecule by means of the several relaxation rates. This will provide us with some details of the correlation times of dideuterium in the nematic phase. Many small rigid solutes have provided a lot of insight into the orienting potential in nematic liquid crystals. However, the nematogens themselves are usually flexible and for this reason we will discuss in chapter 4 solute molecules with a single bond of rotation as a starting point for understanding flexiblity. Dipolar coupling constants measured with NMR spectroscopy provide us with information, on the molecular level, about the different conformers present and allow us to test a model for the short-range anisotropic potential. Another aspect of nematogens is that they usually have an aromatic core and an alkyl chain. In this thesis we try to learn more about this aspect by using small alkanes and benzenes as salutes in the nematic phase. The molecular quadrupole moment, especially, which may contribute to the orienting potential, is of interest. We introduce a method of obtaining molecular quadrupole moments using liquid crystal solvents. Chapter 2 Experimental 2.1 Liquid Crystals The nematic liquid crystalline solvents used are • P CH- 7/Merck ZLI- 1115, trans-4-n-heptyl- (4-cyanophenyl)-cyclohexane • Merck ZLI-1 132, a eutectic mixture of alkylcyclohexylcyanobenzenes and alkyl cyclohexylcyanobiphenyls (see [28] for composition) • EBBA and EBBA-d2, N- (p-ethoxy-benzylidene)- {2,6-dideutero}-p’-n-butylaniline • A mixture of 55wt% Merck ZLI 1132 and 45wt% EBBA. These liquid crystals were used without further purification. The EBBA-d2 was synthesized at the Chemistry Department, Free University of Amsterdam. 2.2 Solutes and Sample Preparation The solutes furfural, 2-chlorobenzaldehyde, 2,2’-dithiophene, benzene, 1,3,5-trifluorobenzene, 1,3,5-trichlorobenzene, hexafiuorobenzene, ethane, and propane were commercially available and were used without further purification with the exception of furfural. Fur fural was used after double distillation. Dideuterium gas was prepared by electrolyzing 0. 2 99.8% liquid D 6 Chapter 2. Experimental 7 The solid solutes were dissolved in the liquid crystal in about 1—2 mol% concentration in 5 mm od standard NMR tubes. Each sample was thoroughly mixed on a vortex stirrer, capped and sealed with paraffin film. The gaseous solutes ethane and propane were condensed at liquid nitrogen temper ature into 5 mm od standard NMR tubes. A final pressure of about 2—3 atm at 304 K above the solvent was obtained. The dideuterium gas was condensed at liquid helium temperature into 9 mm od standard pyrex glass tubes, chosen so they fit snugly inside re high resolution 10 mm od NMR tubes. The tubes were flame sealed and a final pressu of dideutrium above the solvent of about 25 atm at 304 K was obtained. For the sam ples containing compounds which are gaseous at room temperature, the liquid crystal the solvents have been thoroughly degassed by several freeze-pump-thaw cycles before solutes were condensed. mid The NMR tubes with propane were equiped with a capillary tube, centred in the ents dle with the use of teflon spacers, and filled with acetone-d6. Long 2D-NMR experim n signal can thus be performed because the magnetic field remains locked on the deutero of d6-acetone. 2.3 Instrumentation H NMR Free Induction Decays were recorded on a Bruker WH-400 Fourier The ‘H and 2 data points transform NMR spectrometer and a Bruker AMX-500 spectrometer with 32 K gs in the . The temperatures in the probe were calibrated by measuring the splittin 2 in t picking spectra of ethylene-glycol and 4% methanol in methanol-d4. The Bruker peak most of routine for Lorentzian line shapes was used to measure the peak positions. For our spectra the line width is less than 5 Hz. and The multiple-quantum spectra were acquired with preparation times of 11 ms Chapter 2. Experimental a recycle delay of 4 s. 256 data points were acquired in t 2 and 1K increments in 8 ti. A digital phase shifter accomplished the 300 increments for the 6-quantum spectra and 25.75° increments for the 7-quantum spectra. All spectra were analyzed by performing a least squares fit to the spectral line po sitions using the computer program LEQUOR [29]. The experimental order parameters were calculated from the dipolar coupling constants obtained from LEQUOR using sev eral different assumed geometries of the molecule and the least squares computer program SHAPE [30]. No corrections were applied for vibrational motions. Chapter 3 Dideuterium as a solute in the nematic phase Most of this chapter is published in [31] or accepted for publication in [32]. 3.1 Introduction NMR Spectroscopy of dideuterium has provided a lot of insight into the orientational mechanisms of liquid crystaline systems [9, 11, 12]. It was found that the interaction between the molecular quadrupole moment of dideuterium and the mean electric-field gradient of the liquid crystal explains most of the orientation of this small solute. It seems a logical step to investigate the dynamics of dideuterium in order to obtain more information about the dynamics of small solutes in liquid crystals. Relaxation of nuclear spin systems is of considerable interest for studying the intermolecular interactions in gases and the dynamic behavior of molecules in general. The nuclear spins in most diatomic molecules are relaxed by the internal magnetic fields of the molecule which fluctuate as a result of molecular collisions. Since the internal interactions are known for two coupled spin-i particles, the nuclear spin relaxation rates can provide detailed information about the dynamics for this system. Both the ortho and para species of two coupled spin-i particles are observable with NMR and can be studied as one of the simplest systems. The 2 H-NMR spectrum of dideuterium, partially oriented in nematic liquid crystals, has been investigated by Burnell et al. [8, 9, 10]. It has been shown that the NMR signals from the ortho and para species are distinguishable and the transition frequencies can be 9 Chapter 3. Dideuterium as a solute in the nematic phase 10 Figure 3.2: H 2 -NMR-spectrum of D 2 in PCH-7 6D I * * 1000 500 0 -500 frequency/Hz -1000 A 400 MHz 2 H-NMR-spectrum of dideuterium partially oriented in the nematic liquid crystal PCH-7 at 304 K. The asterisks label the transitions from para-dideuterium. Note that B’ and D’ are of opposite sign, so the para splitting is 2(B’ 3D’). — calculated as a function of the coupling constants [8]. Although the solute dideuterium has a very small order parameter of about iO, justifying the isotropic average for our calculation, the couplings due to the quadrupolar and dipolar interactions are sufficient to give rise to observable splittings. The 1 2 1-NMR spectrum of dideuterium dissolved in an anisotropic liquid is dominated by these interactions as shown in Figure 3.2. Below liquid nitrogen temperatures Hardy [33, 34] observed that the spin-lattice re laxation rate of the ortho-species of dideuterium gas, D , is much longer than that of 2 Chapter 3. Dideuterium as a solute in the nematic phase 11 the para-species. This is mainly due to the fact that the averages over the even and odd rotational states lead to different spectral densities. Almost all of the on ho-species are in the J=0 rotational state at these low temperatures. This state is spherically symmetric and therefore not responsible for relaxation. The master equation for the evolution of the density operator under the effect of a random perturbation [35, 36, 37] from a Hamiltonian containing only the quadrupolar in teraction predicts that the Zeeman spin-lattice relaxation rate R 10 of dideuterium is equal for ortho and para configurations at high temperatures [38]. In this thesis we will show that including the dipolar term in the Hamiltonian predicts different relaxation rates for both species. This is due to the interference-terms between the dipolar and quadrupo lar couplings which are associated with the same correlation time 2 [39]. The ratio of the quadrupolar and dipolar interactions is about 8 for dideuterium, hence the dipolar coupling between the deuterons cannot be neglected as a mechanism for relaxation. We report 5 distinct relaxation rates for the para species, and 21 distinct relax ation rates for the ortho species, R,°°. For the ortho species there are 15 cross-relaxation rates, Rlm/km, between certain spin-orders which precess at a common frequency. We have evaluated the rates as a function of the second rank dipolar and quadrupolar coupling constants, the first rank spin-rotation coupling constant, and the spectral densities of the lattice. We will experimentally verify the expressions for the Zeeman spin-lattice relaxation rates . 10 The Zeeman spin-lattice relaxation rate is more commonly known as 1/Tie. R In this thesis we adopt a more general notation since we are dealing with many time dependences of the spin-orders represented by spherical tensors, Qim, most of which come from reference [40]. The remaining orders have been constructed such that they complete the orthogonal set and provide physical insight for our spin system. The choice of functions, e.g. Q2±m, are such that Qi±i evolves initially into Q2±1 under the secular Chapter 3. Dideuterium as a solute in the nematic phase 12 part of the Hamiltonian. Rim is the relaxation of the spin order represented by Qim. Cross-relaxation rates are symbolized by Rim/km where the coupling is between rank 1 and k. For the ortho configuration the presence of higher rank spin-orders renders the relax ation behavior more complicated than for the para. This is due to relaxation coupling be tween the various spin-orders given by the Rlm/km’S, hence giving rise to multi-exponential relaxation equations [41, 42]. In addition, the evolution of the spin-orders under a coher ent Hamiltonian involves commutators of the spin-orders with the Hamiltonian. Most spin-orders are not a constant of the motion and their relaxation equations will couple to the evolution equations of other spin-orders. Since the principle axes of the dipolar and the electric-field gradient tensors coincide , the spectral densities reported here involve only autocorrelation functions 2 N 4 2 and ‘ for D CI by Poupko 2 of the lattice. This is in contrast to the work on partially oriented CD ci al. [43] where cross-correlation terms between the motion of the two C-D directions also need to be included. They have derived some deuteron relaxation rates, but have included only the quadrupolar interaction. Most of our calculations have been aided with the Maple V Computer Algebra System [44], which provides symbolic computation for linear algebra. 3.2 3.2.1 Basic Principles Two Coupled Spin-i Particles as a Quantum Rotor For homonuclear diatomic molecules in which the nuclei obey Bose-Einstein statistics the product of the spin and vibration-rotation wave functions is symmetric with respect to permuting the individual nuclei. The states into which the molecule can be separated are ortho and para corresponding to the six symmetric spin states for total angular 13 Chapter 3. Dideuterium as a solute in the nematic phase Table 3.1: A Spin-Basis for Two-Coupled Spin-i Nuclei Para-states Ortho-states 12, 2>=ji; 1, 1> J {I1;i,0> +I1;0,1>} 1 I2,1>= 10, 0>=/ {ji; 1, —1> —Ii; 0,0> -Hi; —1, 1>} l2,0>/ {Ji;1,—i> +2J1;0,O> +Ji;—1,i>} i,i>=V1’ Ii, 0>= J1,—1> {I1;i,0> —li;0,l>} {i; 1, —1> —Ii; —1,1>) {i;0,—1> —Ji;—i,0>} I2,-1>’ {11;—1,0> +Ii;0,—1>} 12, —2>=I1; —1, —1> A basis set for two coupled spin-i particles by composition of angular momentum. There are six symmetric states and three anti-symmetric states. The total angular momentum wavefunction >, where m’ = 2 I IT, mT> is shown as a function of the single spin wavefunction Ii;mi, m . 2 1 + m m momentum IT = 0 and 2 and the three anti-symmetric spin states for IT = 1, respectively. A possible set of basis functions is given in Table 3.1. The para species can occupy the odd rotational levels (J=1,3,5,..j, and the ortho species the even ones (J=0,2,4,...). In the absence of magnetic field gradients the ortho and para states do not interconvert. These states are not averaged in our NMR experiments and give rise to separate spectra. For dideuterium the rotational states are well defined since the rotational levels J are well separated and thus not much affected by life-time shortening due to the many collisions in the dense medium. Therefore we can treat dideuterium as a well-defined quantum rotor and weigh each J-state by the appropriate Boltzmann factor. Since the lifetime of a J-state is short compared to the relaxation times of our system, the measured relaxation rate Rim can be expressed as a weighted average over the even or odd rotational states for ortho and para, respectively Rim = ‘Pj{Rim}j J (3.1) Chapter 3. Dideuterium as a solute in the nematic phase — 14 2J + 1 e(J+l)O/T E’(2J + l)e_J(1)6/T where S is the rotational constant for the molecule and T the absolute temperature. The primes on the summation symbols indicate that the summations are to be taken over even J (ortho) or odd J (para). Note that we have neglected orientational effects, i.e. we assume that all 3.2.2 mj states for a given J state are degenerate. Equation of Motion The coherent time independent Hamiltonian h7-€ 0 describing the spectrum of our spinsystem is = IWITz + UB(t) + W(t) + fl(t) (3.3) where the first term on the right hand side is the Zeeman Hamiltonian which is propor tional to the frequency offset w of the rotating frame, the second term is the secular part of the time averaged time dependent second rank dipolar and quadrupolar Hamil tonian, and 7-i (t) is the anisotropic part of the first rank spin-rotation Hamiltonian. 51 The scalar indirect coupling is ignored since it is only 6.554 Hz. We also neglect any contribution of the chemical shift anisotropy to relaxation. The relaxation Hamiltonian ti7t(t) is randomly fluctuating about its time averaged value (t) = flB(t) + 71’(t) + SR(t) — .(B(j) + W(t) + flsI(t). (3.4) For our system where the order parameter is of order iO, we shall use equations for isotropic motions. That is, flB(t) + D(t) + ‘Nst) will make negligible contribution to ?-(t) and can be ignored. The state of our coupled spin-system is described by the spin density operator a. This density operator can be expanded in the various nuclear spin-orders or coherences, Chapter 3. Dideuterium as a solute in the nematic phase , given in m Q’ 15 Table 3.2 with the normalization constants in Table 3.3. (3.5) The Qim form an orthonormal set: Tr ik 5 n, (QmQkfl) = m where the adjoint is Qm = (_l)mQi_m. The number of independent coherences is greatly reduced from 9x9—1 since there are no matrix elements connecting states with different permutation symmetry. It is therefore impossible to create phase coherence between the ortho and para states. There are thus 8 independent spin-orders for the para species and 35 independent spinorders for the ortho species. We will report the time dependences of these spin-orders under the random relaxation Hamiltonian in Tables 3.5—3.8 and under the coherent time independent Hamiltonian in Table 3.9 of this thesis. The evolution of the density operator under the full Hamiltonian of our spin-system and lattice is described by the quantum mechanical equation of motion. If the 8 and 35 elements of o, for para and ortho respectively, are arranged in a column vector , the equation of motion can be expressed in matrix form [45] = —iHoa(t) — R ((t) — (3.6) where H 0 is the matrix representation of the time independent Hamiltonian superoper ator, which describes the evolution of the density operator by connecting the elements which evolve into each other. The 8x8 evolution matrix and vector constructed from Table 3.9, is given below as an illustration: for the para species, Chapter 3. Dideuterium as a solute in the nematic phase 16 Table 3.2: Density Matrix Elements for Two Coupled Spin-i Nuclei = I 2 . 1 A/(2I — i) Qio = Qi±i = :FJVi±1IT± = [(3(I + 0 .A/ 2 1 Q2o 12z) 4) + D(21 22 1 12 — — (I1+I2— + 11—12+))] Q2±1 = ±+ 2 [i±]+ 1 aJr± I 2 ç([Ii + [12z,12±]+) + D(IiI Q2±2 = [--(1?± + ± 2 JV Q2’o = 2 3D)(21 2 1 .No(5B + 12 Q2’±l = 2 + 11±122) I 12 :F,±i(5B + 3D)(1 )] 2 4) + D(II — ±1 1 3D)1 ± ’ .A1 ( 2 5B ±+2 Q2’±2 Q 2”O I1±12z)] — (1I+I2_ + 11_12+)) — — (B + 9D)(3(I? 2 + 4) — 4) ±]± + [12212±1+) 1 12 (B + 9D)([1 (B + 9D)(I +4±) _i, Q2”lj— = A”o[Qn, Q2”—l]— 1 .)Vso[Q = a:A’ii±i[Qi_i, Q2”21— Q2”±2 = {[(1 + 211 Ar ± 2 1 Q3o = +(1—3IT{1T+1})IT] 2 .IV3o[54’ = 2 Jv±’[(5I Q3±2 = Jv3±2[1Tz,4’±]+ Q3±3 = Q4o = 2 .Mo(35I- — 4± — = Q Q2”—21— 11 [ 1 .A/”± , 2±),iTz}+ 1 i1± T 1 IT{ + 1} — (IT + 1)2 4 {30IT(IT + 1)— 25}4 + 3 — — iT(IT + 1)) 6 T(IT + 1) + i}ITz),IT±]+ 1 {3 = 21 2 IT(IT + 1) Ar4±3[ITz,IT±J+ r4 r Q4±2 = 2 A(4±2[(7IT ‘d4±4 = T± 1 1V4±4 — — 5),IT±J+ Coherences and populations which constitute spin-i particles. the elements of the density matrix for two coupled Chapter 3. Dideuterium as a solute in the nematic phase 17 Table 3.3: Normalization Constants for the Elements in Table 3.2 I 2 )j7 H— 2 00 2 H A’ H202 2 H 2±1 2 H 2±2 H; 2 H Ap—2 2’2 -,2 0 H— 2 2”±l 2 H 2”2 .N A’ H 3 2 33 H 2 H 1 H 42 H 3 H 4 H ortho 30 10 20 2 B 2 + 4BD + 18D 3 2 2 B BD + + 12D 9 2 B 2 + BD + 12D 9 2 2 + 504BD + 2268D 420B 2 2 + 336BD + 1512D 280B 2 2 + 336BD + 1512D 280B -- 10 1 5 96 360 1920 192 288 10080 2016 4032 288 576 para N.A. 2 4 2 3D) 2 3D) 2 3D) N.A. N.A. N.A. N.A. N.A. N.A. N.A. N.A. N.A. N.A. N.A. N.A. N.A. N.A. N.A. - - - Normalization constants for the elements of the density matrix (spin-orders) for two coupled spin-i particles of Table 3.2. N.A. indicates that these spin orders are not applicable to the para species. Chapter 3. Dideuterium as a solute in the nematic phase para — 18 00 0 0 0 0 0 0 00 0 0 0 0 0 0 (B—3D)S 0 0 0 0 0 0 0 0 0 0 —Aw 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 2w 0 00 0 0 0 0 0 —2w (B — 3D)S - —Aw (3D — B)S (3D — B)S Qio Q2o Qii para = Q21 Qi-’ Q2-1 Q22 Q2-2 R is the matrix representation of the relaxation superoperator, which drives the den sity operator to its equilibrium value 0 eq. The diagonal elements Rim of R describe the relaxation rate of Qim to its equilibrium value and the off-diagonal elements Rkm/lm de scribe the cross-relaxation of Qkm ortho species is illustrated below: to Qim. A part of the 35x35 relaxation matrix for the Chapter 3. Dideuterium as a solute in the nematic phase 00 R 0 19 0 0 , 2 Roof 0 00140 ft 0 0 0 0 1030 R 0 0 0 o o 0 20 R 20120 R 0 20140 R 0 0 2000 R 0 20120 R 20 R 0 2040 R 0 0 0 Ro,10 0 0 30 ft 0 0 0 40100 ft 0 40120 ft 4020 R 0 40 ft 0 0 o 0 0 0 0 0 11 R 0 = ... o 0 0 0 0 0 0 0 0 _ 4 R The basic principles of calculating the matrix elements of both H 0 and R are outlined in the following sections. Evolution under a coherent Hamiltonian From now on we will discuss the time dependence of the expectation values of the spinorders in the so called Heisenberg representation, > m <Qj (t) The evolution of a certain spin-order = Qim Tr (Qm0(t)) (3.7) under the secular part of the time indepen dent dipolar and quadrupolar Hamiltonian hfl , 0 = WITz + ç (3’ + I) —4) + D’ (211Z12Z — (I1÷I2- + 11_12+)) (3.8) is given by [35, 36, 37] ( t where B’ = m <Q > (t)) = —i < coherent BS and 3e q 2 Q 4h [Qim,o1> (3.9) Chapter 3. Dideuterium as a solute in the nematic phase 20 is the quadrupolar coupling constant of the spin-i nucleus with the electric-field gradient of the molecule. Note that the quadrupolar coupling constant corresponds to three times the quadrupolar interaction or one-half the quadrupolar splitting. The constant D’ = DS and D — = ph7 2 4r < r > 3 (3.11) is the dipolar coupling interaction between the nuclei, and S = (8)) 20 (Y = S —1> 2 <3cos (3.12) is the order parameter and 8 is the angle between the symmetry axis of our diatomic molecule and the magnetic field direction. The solution to Equation 9 is < m Q’ > (t) = (eHOt Q’ e_) m (3.13) A Taylor expansion of Equation 13 involves successive commutators with the Hamiltonian <Qim(i) > (Qirn Qjm]+ +ii[? , 0 [7to,[?o,Qzrn]]+ (3.14) 0 are The commutators of the elements of the density operator with the Hamiltonian ?-t given in Table 3.9. After an ‘Tx = (IT++IT_) pulse to the equilibrium magnetization Qio, the coherence Qi±i is created. From Table 3.9 and Equation 14 it can be seen that after a time t long compared to the evolution period many single quantum coherences will be created for the ortho species. To solve the time evolution for the single quantum coherences during the time t one has to solve 6 coupled differential equations for the ortho and the 2 for the para species, which amounts to diagonalizing that part of the matrix H which is associated with the single quantum coherences. The Fourier-Transform of the solution to these Chapter 3. Dideuterium as a solute in the nematic phase 21 equations gives the experimentally observed single-quantum spectrum as in Figure 3.2. For the para species, for example, the solution for Q (t) 1 1 is (t) 1 Q 1= ({Qii(O) — (O)}ei 2 Q 1 B’_3D’)i i(O)}ei_+3D’)t). 2 + {Q(O) + Q (3.15) Therefore, there are two transitions of equal intensity from the para species. The single quantum coherence part of the evolution matrix for the ortho species can also be diagonalized to give the spectrum. The intensity of four of the six lines from the ortho species are a function of B’ and D’ [8]. In a rather cumbersome way, all the multiple quantum spectra for our two coupled spin-i particles can be calculated from Table 3.9. The transition frequencies reported by Poupko et al. [43] contain algebraic errors. We report the transition frequencies as a function of w, B’, D’, and J in Table 3.4. As expected, for D’ = 0 the time evolution for the ortho and para coherences from Table 3.9 are the same. Evolution under a random Hamiltonian In order to calculate the relaxation expressions in the Redfield approximation we write our total time dependent intra-molecular interaction Hamiltonian in spherical tensor form. The coupling to the lattice is through the time dependent first rank spin-rotation and second rank quadrupolar and dipolar Hamiltonians: hflSR, B, and hflD respectively [35] (_1 )m2Ym(9(t), (t)) (3.16) ml where 12 is B, D, or We,.. Wsr is the spin-rotation constant for the interaction of the spin = with the effective magnetic field due to the molecular rotation. }‘m(9, ) is a spherical harmonic of the angles 8 and q5 associated with the vector joining the two nuclei and Chapter 3. Dideuterium as a solute in the nematic phase 22 Table 3.4: Transition Frequencies for Two Coupled Spin-i Nuclei rn-Quantum Transition O-QT [ para/ortho para Transition Frequency 0 0 0 ortho i-QT para ortho 2-QT para ortho 3-QT ortho 4-QT ortho 0 0 0 0 0 0 -R B’ + 3D’ &-‘+B’—3D’ B’ 3D’ &,—J—R ‘— J+ R R 1 iw+J+ &+ J— R i+B’+3D’ 2& R 1 2&’—B’—3D’—J— 2&—B’—3D’—J+R 2&.’ J+ 2 2&+B’+3D’+ R 1 2+B’+3D’+J— R 3w — B’ — 3D’ 3iw + B’ + 3D’ 4, — — — Transition frequencies for para- and ortho-species in a partially oriented solvent. Note that the total number of transition frequencies equals the number of basis operators Qim for m 0 plus 2 + i2D’ 2 — 8B’D’ + 9J2 — 4B’J 12D’J. the identity. R = %/4B’ — Chapter 3. Dideuterium as a solute in the nematic phase 23 the term 7 is the m-th spherical component of the normalized irreducible tensor of rank 1 associated with either the quadrupolar, 7, dipolar, ‘1, or spin-rotation SR 7 Hamiltonian. Jio = = = 1Ji(211z12z ‘-rB — — — 4) (3.19) (3.20) (I1+I2— + 11—12+)) (3.21) (3.22) F(1Iz12± + 11±12z) 1112 ‘T — 22 = T ‘) (II± + 11±Ilz + 12z12± + I2±I2) = = (3.18) ±) 2 çIT±=Rl/(Il±+I 2 = \,/i(3(Iiz + T Tf 3.17 ‘Tz’lz+’2z 12 (3.24) (I1±I2±) Since the ortho and para states don’t mix we can treat them as two distinct particles and write the ‘Tim as a direct sum of the q1O and the 7 m and evaluate each of them separately. The transitions between the IT = 0 and IT = 2 states do, however, play an important role for the ortho species as will be shown in section 3. This mixed state forms the ortho-configuration. The master equation for relaxation of a spin-order Qim under the random fluctuating Hamiltonian < B(t) Qirn>) random + ?“(t) + = (S1(t) 7 is given by Tr ([[Qi 1+ 2 m DT — DT + u](c, 7 — oeq)) n——2 Tr ([[Qi , WsrT], Wsr7t](U m — — aeg)) J (w). 1 (3.25) n-1 The conditions for the validity of this master equation have been discussed by Abragam [35]. The coupling constants are in angular frequency units. Chapter 3. Dideuterium as a solute in the nematic phase 24 Equation 25 is derived using a semi-classical treatment where the coupling with the lattice is represented by the autocorrelation functions Jim. In our case, the nuclear spin relaxation is mainly caused by molecular collisions which reorient the molecule or cause the rotational angular momentum S to change direction, hence producing transitions between the rotational mj states. Because we are dealing with a quantum rotor the Jim are a function of the expectation vaiues Kim (1(i)) inside the J-manifold of the random lattice operators Yim(8(t), 4(t)) [35]. Because dideuterium has a very small orientational order, to an excellent approximation the Kim’S have zero average = 0 (i.e. we assume an isotropic average for our molecule) and are normalized such that KkmK?m = J(J+ 1) for k = 1 = 1 and (2J+3) for k = 1 = 2 and zero otherwise [35]. Needler et al. [46] suggested that the correlation functions Kim(O)Km(t) decay exponentially with the correlation time Ti associated with the l-th rank interaction as lKim l2eimwjte where the term with the wj represents the free precession of !between collisions. The spectral density Jim() is then given by = where ii m (w) is j = (3.26) the reduced spectral density. In the short correlation time limit the reduced spectral density im is simply the correlation time 3.3 IKi,nI j 2 im() TI. Relaxation Equations for two coupled spin-i particles For the para configuration, the equations of motion under a randomly fluctuating Hamil tonian fl(t) are effectively the same as for a single spin-i particle and are given by ( <Q:ra>) = random —R° < > — < ç,ara > (327) The five relaxation rates, Rra, as calculated with Equation 25 are given in Tables 3.5 and 3.6 and are the diagonal elements of the relaxation matrix Note that the effective Chapter 3. Dideuterium as a solute in the nernatic phase 25 Table 3.5: Relaxation Rates Para under Dipolar and Quadrupolar Hamiltonian R(B, D) = (B — Ra(B, D) = — R7’(B,D) = — R7a(B, D) = (B R(B,D) = — — 21 { 2 3D) J + 4J } 22 2 {J 3D) 20 + + 3D) 2 J 2 1 20 { 2 3D) J+2 J 1+ 2 {J 3D) 21 + 2J } 22 The five relaxation rates for the para-state under the dipolar and quadrupolar Hamiltonian. These expressions are known as the Zeeman spin-lattice, Zeeman spin-spin, second rank spinlattice, second rank spin-spin, and the double quantum relaxation rates for a single spin-i particle. The effective coupling constant is B 3D. — coupling constant in the rate equations under the second rank interactions is B — 3D. This is in agreement with expectation based on the splitting between the para-lines in the spectrum of dideuterium (Figure 3.2). For the ortho configuration, the equation of motion couples spin-states of rank 1 with k. In general, the recovery will be multi-exponential. ( = <Q°>) _R0 (< <Qr h t o Qrho> random Rrm(< — Q0 > t <Q’ >eq) (3.28) For example, in the case of Zeeman relaxation, the Zeeman order, Qio, is coupled by relaxation to the octupolar order, Q3o [31]. This coupling is schematically illustrated in Figure 3.3. A consequence of this relaxation coupling is that certain coherences can be populated through the random fields. This route of populating coherences is in addition to those that occur under the evolution of a coherent Hamiltonian and pulses. Using Equation 25, the results obtained for the time dependence of the expectation value of all of the spin states for the ortho-species are given in Tables 3.7 and 3.8. The Chapter 3. Dideuterium as a solute in the nematic phase 26 Figure 3.3: Coupling Scheme of Certain Spin-Orders and the Lattice k para orthö 10/30 .oñhà 10 k00 10 k” 10 k0hhl0 30 LATTICE: The coupling of the Zeeman order, octupolar order, and lattice among each other through the quadrupolar, dipolar, and spin-rotation interactions (Tables 3.5—3.8). The lattice represents the rotational motions of the molecules. There is no coupling between the ortho and para states. This figure also illustrates that certain spin-orders can be populated by means of relaxation. The relation between the rate constants k and relaxation rates R is as follows: Rra = ROTtI kth0 R0rtl 1_ k0h0 ROrtiw R0h0 z0 1 k01t ’ 1 k°’ 10 10 30 30 30/10’ ‘ 10/30’ a d 10/30 30/10 30/10 10/30 — — — — — — — — — — — Chapter 3. Dideuterium as a solute in the nematic phase 27 Table 3.6: Relaxation Rates Para under Spin-Rotation Hamiltonian DparafTxy \ 1”gr) .L(1o — rflxT2 7 — nparoTzT T172 r ‘,rlJ1O+J11 ‘8r) — DparafTxr \ .‘v8r) — — VVVsrIhl Dpara,Txr \ .“sr) — — TX7 f 7 2 I 7 Y.,1or.w11 ‘11 2O ‘21 2 £TXT 7 W r 3 ) = 2Wr{2Ji 11 J 0+} The five relaxation rates for the para-state under the first rank spin-rotation Hamiltonian. cross-relaxation rate, is the cross-relaxation between the spin order Qim and Qkm. Note that the equality Rim/km = Rkm/im is satisfied. This is because of the invariance of the trace and the Hermitian property of the Hamiltonian: Tr ([[Qirn, ], ?]Qkm) = Ti’ ([[Qkm, H], ‘H]Qim). In the short correlation time limit, Ti << 1/(w 0 j), all cross— terms except the R2m/2’m’S vanish. If there were no mixing between the J2, 0> and 10,0> states the effective interaction for the on ho-state would be B + 3D, but due to mixing the effective interaction is more complicated. As expected, the rates for ontho become equal to those for para when the spins are not coupled (i.e. D = 0) and the equations become identical to those for a single spin-i particle. The only difference will be the different averaging over the J states [38]. When B taken, + = 0 and the appropriate average over the ortho and para states is the spin-lattice relaxation rate is as given by Abragam for two coupled spins [35]. For experimental verification of the Zeeman relaxation for the ortho and para-species see section 5. 28 Chapter 3. Dideuterium as a solute in the nematic phase Table 3.7: Relaxation Rates Ortho under Dipolar and Quadrupolar Hamiltonian Rg0(B, D) (2B jho(B D) (B2 + R”°(B, D) çD2){J + o (B2 + BD + 2 — 21 + 2J } 22 J + 2J 20 { 2 3D) LD2){J + 4J } 22 BD + 21 2 } 2 +J +189D 3 18OBD +4 D 2 + 138B 12B 3 + 4 5B D 21+ +18BD+81D 15B 2 9D 16B 24BD+ ( 4D + ) 2 22 +6BD+27D 5B 2 BD B 9D +138B 18 180 12 D + D 2 + 3 +4 4 5B R0(B, D) 20+ 162D 36BD+B 30 + 2 4 3 + 405 D D + 756 BD 2 D + 522 B 3 4 + 12 B 5B } 22 21 + 2J {J 2 + 108BD+486D 2 90B 9D B 24BD+ 2 2 2D (16 + ) pho(B D) 20 J + +6BD+27D 5B 2 = 4 D + 756 BD + 405 D 2 D + 522 B 3 4 + 12 B 5B 21+ 243D B 54BD+ 45 + 2 4 3 + 324D D + 216 BD 2 D+ 180 B 3 4 + 24B lOB 243D 54BD+B 45 + 2 4 3 + 783 D D 252 BD 2 D 78 B 3 4 + 100 B 35 B 21 J + 6D2)J + B2+BD+ ( o R0(B,D) = 2 B +567D 126BD 105 + 2 4 3 + 2322 D D —468 BD 2 D 42 B 3 4 + 280 B 100 B 126BD-f-567D 105 B + 2 2 BD 783D —252 + 3 D—78B 4 3 100B D 4+2 35B R°(B, D) D B 252BD+1134 210 + 2 4 3 + 8343 D D + 108 BD 2 D + 1062 B 3 4 + 900 B 335 B 21 J + D B +3402 756BD 630 + 2 4 3 + 5427 D D 1188 BD 2 D 162 B 3 4 + 660 B 235 B 2 315 B +378BD+ 1701D 2 D l161D —234B + 3 4 —21B D 3 140B D +2 4 50B 20+ R’°(B,D) = 126BD-f567D +2 2 105B 4 3 + 5427 D D 1188 BD 2 D 162 B 3 4 + 660 B 235 B D B 378BD+1701 315 + 2 4 3 + 1863 D D + 378 BD 3 D + 180 B 2 4 + 477 B 70 B 2 2 + 378BD + 1701 D 315 B 21 + 9J } 22 21 + 2J } + D2{11J 22 21 + 8J ) + BD{7J 22 R°(B,D) = B2{3J R’°(B, D) — — — R0(B,D) = 20 !2j D — — — — o+ 2 i +J 2 } + BD{J 22 +J o+ 2 {J + ThJ22} 2+ ) 2 + J Chapter 3. Dideuterium as a solute in the nematic phase R°(B, D) 0 B 1 2{4J +j +VJ22) o+ 2 + BD{4J 29 ) 2 2+ +J D {3J 20 + J i+ } 2 22 J 7 R0(B, D) o 2 !B2{J i + 3J 2 +!J ) + BD{3J 22 20 + 7.121 + 22 2J + ) D {J 20 + J 21 + 2.122) R’°(B, D) (B2 R”(B, D) 20 + B2{J — BD + 20 ){2J + } 2 D 21 + (B2 + 22 J )J 2 2D + 20 + D2{J 20 + BD{J — ) 2 2+ +J + R°(B, D) 20 + B2{2J R0(B, D) 20 + J B2{J 21 + ) 22 + jBD{—12J2o + 31.121 + 16J J } 22 + — 21 + 8J BD{J }+D 22 2 {J 20 + J 22 J i+ } 2 21 + 34J } 22 +D2{cJ2O + 94J RthQ(B, D) 20 + B2{’J jJ2l + J22) o+ 2 + BD{—J J21 + J22} + D {J o + --J 2 i + 8.122) 2 R0(B, D) 20 + .B2{J R0(B, D) ‘B2{J R0(B, D) 21 + 2J B2{J } + BD{J 22 } + 21 21 {J + .122) 2 2D 22 + J + } 21 + 2.122) + BD{3J 20 +J — D2{J + J J 2 ) + 20 2 1 + 2J i + 2J 2 } 22 2/V’5B2+6BD+27D2(2B_3D) Rg,’O(B, D) 315 — Rg ( 0 ° 4 B, D) D2{9J + 21 20 + } BD{J 22 + 20 J 8J + 11J } 22 2) 4v’T(2B—3D 945 } 22 —2J 21 +J 20 {J 21 —4 20 {3J } 22 +J J Rh O 3 (B, D) 16 (BD+ D 2 ){J 21 Rh ( 3 B, D) ç(BD + 20 ){J 2 D R,’ ( 0 B, D) 2 TiD (B + 3 D)(2 B —3 D)(4 B +3 D) } 22 {2J21 + 5J 105 B 2 + 126 BD + 567 D 2 Rh O 4 O(B, D) 4v’TD(B+3D)(4B+3D) 63v’ + 2 6BD+2 5B 7D — } 2 J 2 } 2 J 2 21J221 Chapter 3. Dideuterium as a solute in the nematic phase r.’orho 1210140 (B, D) = 30 2//5B2+6BD+27D2(2B_3D) 20 + J 315 ) 2 2 ..ji (2 B 3 D) (35 B 2 + 162 BD + 279 D — — — +6BD+27D 2 945/5B 4v(2B_3D)(10B2+72BD+99D2) .122 R/I(B, D) B, D) R/° ( 1 = — — RI(B, D) — — B, D) R’ ( 2 R,(B,D) = — = = .-(B + 3D)(2B = porho 3212112 (B, D) — — +6BD+27D 21v’5B 2 ) 2 2 + 22BD+39D v’(2B— 3D) (5B .120+ +6BD+27D 21/5B 2 5B 14BD+33D v’(2B—3D)( + ) 2 21 + J +6BD+27D 21/5B 2 2/D(2B-3D)(4B+3D) +6BD+27D 21’’5B 2 /E(2B+6D)(2B—3D){J J} 27 — B,D) RI ( 1 +6BD+27D 2 945/5B 2viD(2B—3D)(B+3D)(4B+3D) 21 + 4.122) {J20 + 2J 2 2 + 126 BD + 567 D 105 B /iD(B÷3D)(4B+3D) } 22 —3J 21 +2J 20 {J +6BD+27D 21/5B 2 ) 2 2 + 6 BD + 9 D (20 B 30 D) (B D (4 B 6 D)(4 B + 3D) 20 + J B 6BD+27D 5 1 63v + 2 +6BD+27D 21v’5B 2 +42BD+54D (4B—6D)(5B ) 2 22 J +6BD+27D 63v’5B 2 vTD(B+3D)(2B-3D)(4B+3D) 20 + 8J {5j 21 + .122) 2 105 B 2 + 126 BD + 567 D (4B+3D){ } 2.flD(B+3D) J 2 J 4 — 20 3D){—3J — .121+ } 22 21 + 4J J All independent relaxation rates for the ortho-state under the dipolar and quadrupolar Hamil tonian. The cross-relaxation rates, Rim/km = Rkm/lm form the off-diagonal elements of the relaxation matrix for the ortho species. Chapter 3. Dideuterium as a solute in the nematic phase 31 Table 3.8: Relaxation Rates Ortho under Spin-Rotation Hamiltonian RW(Wsr) = 0 Rh0(Wsr) = 11 2W,.J ) 8 R0(W = W r 2 8 {Jio+Jii} = 6WrJii = = W:r{Jio+5J } 11 10 + } 2W{2J 11 J 11 12W.J R°(Wa ) 7 Rth0(Wsr) R’°(Wsr) R,°(Wsr) Rth0(W ) 87 = = } 11 Wr{JiO + 11J o+2J 1 4W.{J } 11 (Wsr) R0(Wsr) = = 3Wr{3Jio+Jii} Rth0(Wsr) = R’°(W r 3 ) Rth0(W,r) = = R(Wsr) porho (Wsr) 1i2lO = R0(Wsr) = = Rth0(Wsr) porho 4I33 porho “2’2 (Wsr) = 20W?rJii Wr{Jiø+19Jii} 4W:r{Jio+4Jii} Wr{9Jio+11Jij} 4W?,.{4J } 1 +Ji 10 r.?orho = “2O ( 4 1 ’ar) R?0(Wsr) = Rh10(Wsr) R°(Wsr) — — 22 (r) The relaxation rates for the ortho-state under the first rank spin-rotation Hamiltonian. Note that the relaxation matrix for the spin-rotation interaction is diagonal; first rank interactions do not connect different coherences. 32 Chapter 3. Dideuterium as a solute in the nematic phase Illustrative Examples of the Solid- and Jeener-Echo 3.4 To illustrate the complexity of a strongly coupled spin-system we will qualitatively follow, using Table 3.9, what happens when we subject the orE ho configuration to the solid echo () () t2 pulse sequence, (j) — 1 t — ii — — ( (, — — — — echo [47], and the Jeener-Broekaert echo pulse sequence, echo [48]. The first pulse, I = (IT+ + IT—), converts , Qi± evolves into 1 the Zeeman order Qio into transverse magnetization Qi±i. During t 1 short compared to the all other single quantum coherences Qi±i. However, for a time t evolution time of the interaction, where we can neglect the higher order terms in the Taylor expansion (Equation 14), or for small enough values of the dipolar constant, D’, it is possible to create mainly Q2±1 coherences. Solid Echo 3.4.1 The second pulse will transform Q2±l into into Q and Q21 which after a time 7 = 1 will refocus i thus forming the solid echo with an amplitude reduced by the contribution from Q3±l. The formation of Q3±l, which is proportional to D’, and which can create the other single quantum coherences, causes in essence the dipolar dephasing of the solid echo for the orE ho species. 3.4.2 Jeener-Broekaert Echo In general, the second pulse in the Jeener-Broekaert pulse sequence populates all the elements of the density matrix. However, for short t, we obtain Q20 and Q2*2 only. Suppose we want to follow the relaxation of Q20 2 it seems . When varying t 2 during t 0 However, even in the short correlation . 2 that one has a handle on the relaxation R time limit, when the relaxation coupling to Q40 vanishes, Q20 is still coupled to Q2’o. Although Q20 is a constant of the motion, Q2’o is not and is coupled to Q2”o, Qoo, and Chapter 3. Dideuterium as a solute in the nematic phase 33 Table 3.9: Commutators of the MQ-Coherences with Secular Hamiltonian para: ortho: — 3D’)Q 1 j 2 — 3D’)Q a 3 [?Io,Qi±i] = ±wQi*i ± (B’ ±i ± (B’ 2 ±wQ [7o,Q2±2] = j 2 ±2wQ J ± 1 [flo,Q = ±I 2 ±wQj ±/5B’2+6BDl+27DI2Q [1io,Q2±1] = 27D’ Qi±i ± 21 ± /5B12 + 6B’D’ + 2 ±wQ [1€o,Q3±1] = i ± ±wQ ± 3 2v’5D’(4B’ + 3D’) Q3±1 2 2 + 6B’D’ + 27D’ 5’./5B’ 2v’D’(4B’+3D’) Q2*1 ± 2 2 +6B’D’ + 27D’ 5/5B’ 2../i(B’+3D’)(2B’—3D’) (B’+D’) Q4±1 ± ’±i 2 Q 2 21”+6B’D’+27D’ ‘ 21 JTh5 (B’ + 3D’) Q (2B’ ± [No, Q4±1] = j ± 4 ±AwQ [No, Q2’±ll = j 1 s 2 ±twQ 2v’i(B’+3D’)(2B’—3D’) Q3th1 2 21 + 6B’D’ + 27D’ [No, 1 ”± 2 Q = ij ± 1 ’ 2 ±wQ I (2B’ [No, Q ’o] 2 = ?_/5B12 , 0 , Q 2 + 6B’D’ + 27D’ ] 20 [No,Q = _! (2B’_SD’)Qoa + 15 ,Qoo] 0 [N = ! (2B’_3D’) Q2”o Q 40 , 0 [N ] = ] ± 2 [No,Q = 2v’D’(4B’ + 3D’) ± ± v,5B,2+6BlD,+27D,23*2 2 ±2wQ [No, Q3±21 = 2 ± ±2wQ j 3 21 3D’) Q ji 4 — ,j 1 i 2 3D’) Q j 1 i Q 2 —/5B’ + 6B’D’ + 27D’ /5B’2 + 6B’D’ + 27D’ ’±i Q 2 2T ___/5B’2 +6B’D’+27D’ Qvo 2 — 2V’O 40 (2B’ _.3D’)Q 15 35 (2B’ — 3D’) Q2”o 2v’D’(4B’+ 3D’) Q2±2 ± 2 2 + 6B’D’ + 27D’ /5B’ 5/(B’+3D’)(2B’—3D’) Q2’*2 ± 2 2 + 6B’D’ + 27D’ 21 /5B’ ’± 2 [o,Q ] = ±o ±2wQ i 2 ”± 2 [o,Q ] = ±2wQo’s±2 ] 2 j 4 [No,Q = 2 ± j 4 ±2wQ [No,Q] [No,Q4j3] = ] ± 4 [No,Q = = 4/T —n— (B’ + 3D’) Q4±2 2/i 5v(B’+3D’)(2B’—3D’) _j._\/5B’2 si± Q 27D’ + 6B’D’ + 2 Q3±2 2 ./5B’ 2 27D’ 6B’D’ 21 + + 2T _/sB’2 +6B’D’ + 27D’ 1 _j i Q 2 (B’ 3D’) Q42 .-— 4JT (B’ + 3D’) Q32 21 (2B’ “ — 3D’) — 2 Q2”± 3 ± 4 ±3wQ ± (B’ + 3D’)Q ± 3 ± 3 3 ±(B’+3D’)Q ± 4 ±3wQ quantum Commutators of the zero-, Qio, one-, Qii, two-, Q12, three-, Q13, and fourcoherences under the secular part of the Hamiltonian ?to. The scalar spin-spin coupling is ignored. Qio, Q2o, and Q30 are constants of the motion under flo. Chapter 3. Dideuterium as a solute in the nematic phase Q 4O Therefore, one has to solve a set of 5 coupled differential equations to follow the time dependence for 3.5 34 Q20. Hence, the relaxation of Q2c will not be single-exponential. Experimental Results for the Zeeman Spin-Lattice Relaxation Rate Only the measurement of the Zeeman spin-lattice relaxation rate R 10 is straightforward in high field. At the most there are two coupled differential equations due to the relaxation coupling of the Zeeman order and the octupolar order. In addition, both orders are constants of the motion under flo. The experimental results were obtained by a least squares fit of the peak intensities from the dideuterium spectrum for each of the art ho and parcz species to three parameters , C 1 C , and C 2 3 ) 1 S(t which in the case of = Ci — C e 2 _t13 (3.29) the spin-inversion recovery experiment with perfect 1800 pulses gives the equilibrium magnetization M 0 time R = = 1 C = C / 2 2, and the Zeeman spin-lattice relaxation . The relaxation times obtained for each of the four peaks in one half 3 C of the dideuterium spectrum are given in Table 3.10. The experimental points and the fitted curve for the 1132 sample are shown in Figure 3.4. It is interesting to note that similar results were obtained for methane where the 2 H Zeeman relaxation rate was found to be nearly independent of solvent [50]. Using the short correlation time limit, the assumption normally used for dideuterium in the gas phase that relaxation does not depend on transitions between J levels (Equa tion 26), and neglecting the J dependence of Wsr, B, and D, we end up with the following equations for the relaxation of Qio of a dideuterium molecule in a given J-state [31]: (R°)j = J 2 WJ(J+ 1)r + (B2 + BD + çD2)( 3) (3.30) 35 Chapter 3. Dideuterium as a solute in the nematic phase D in Several Liquid Crystals Table 3.10: Experimental Zeeman Relaxation Data of 2 [ (Rti0)_lt Liquid Crystal PCH-7 ZLI1132* 55% 1132/EBBA 12.5±3.4 10.6±0.6 12.7±2.0 12.4±1.5 10.4±1.3 12.3±1.5 (R,’°)’ (s) 6.2±1.5 6.6±0.6 6.3±0.9 (s) 12.1±1.0 11.7±0.6 12.3±1.9 The Zeeman relaxation times, RjJ’, for artho and para dideuterium in several nematic solvents at 304 K. I These values are measured from the three different ortho-peaks of one half of the spectrum (see Fig. 3.2). * Data from [49]. 2 in ZLI-1132 Figure 3.4: Experimental Zeeman Relaxation Curves of D I I I I I I I I 1 0.5 - C -0.5 - -1 I o 1111111 20 I I I 60 40 delay tIme (a) 80 I I 100 Experimental relaxation data for ortho • and para o dideuterium in 1132. The solid lines are calculated from the fits to Equation 3.29. Chapter 3. Dideuterium as a solute in the nematic phase (R)j = WJ(J + 1)Ti + — 2 3D) (2J —l)(2J+ 3) 36 T2 (3.31) To compare the relaxation equations with experiment we use Equations 1, 30, and 31, with the values 8=43.826 K, W , B = 1060.25x10 1 ,. = 55.090x10 3 3 s 3 s , and 1 D = —43.007x10 3 s for dideuterium from reference [51, 52]. We also use the relationship between r 1 and r 2 for weak collisions [53] 1 r 2 T — 3(4J + 2 4J—7) (2J—1)(2J+3) 332) to calculate the ratio R0/R independent of the absolute value of the correlation times. 0th0 1 R — EoddJ{RcO(Wsr) + R7(B, D)}j Eevenj{Rroth(wsr) + Rjh0(B, D)}j Using Equation 33, we find a ratio of 1.98 at 304 K in good agreement with the experi mental data in Table 3.10. /r ratio of 3 is a good approximation for Equation 32 1 r For high temperatures, a 2 2 of 4.3x10’ 3 s which gives the best fit [54]. With this assumption we find a value for r 2 we obtain a spin-lattice to our experimental results. From the one parameter fit to r relaxation time RQ 1 of 6.3 s for para and 12.5 s for ortho dideuterium at 304 K. These values agree very well with the experimental values given in Table 3.10. With this high temperature approximation we obtain the same ratio of 1.98 for RTa/RthO as from Equation 32. It is interesting to note that our theory predicts the experimental ratios of ortho and para relaxation rates observed in the low temperature studies on dideuterium by Hardy [33, 34], where the relaxation rate of para is many times larger than ortho. For low temperatures, the smaller J values are important for the relaxation and Equation 32 will be used. At T = 60K we predict a ratio Rr/RthO of 24, at T = 50K a ratio of 64, and Chapter 3. Dideuterium as a solute in the nematic phase at T = 37 30K a ratio of 2070. These ratios have been calculated without any adjustable parameters and they agree within the error of the experimental values. Note that all the above is applicable for dinitrogen, ‘ , as well. However, the ratio 2 N 4 of the dipolar to the quadrupolar coupling constant is much lower than for dideuterium. Ortho and para dinitrogen will therefore have nearly identical relaxation rates [51]. 3.6 Conclusions of this Chapter We have calculated all 5 independent relaxation rates for the para and all 21 relaxation and 15 cross-relaxation rates for the ortho species of two coupled spin-i particles under the quadrupolar, dipolar, and spin-rotation interactions. Interference effects between the second rank dipolar and quadrupolar interactions give different expressions for the relaxation of ortho and para species of a homonuclear diatomic molecule with spin-i nuclei. Also the different J dependencies for ortho and para species contribute to the different measured relaxation rates. In the short correlation time limit most cross-terms vanish for the ortho species. Experimental results on the spin-lattice relaxation time 10 of dideuterium are in excellent agreement with our calculated expressions for the R ’ time dependence of the Zeeman order, showing a ratio of about 2 between the para and ortho spin-lattice relaxation rates at 304 K. Good agreement has been obtained with low temperature studies as well. The time dependence of the density operator under the secular part of the dipolar and quadrupolar Hamiltonian and Zeeman Hamiltonian is given to illustrate the complexity of the evolution of a strongly coupled spin-system. In this chapter we have seen that the dynamics of a small solute in a liquid crystal quickly becomes very complicated even though we have treated our diatomic as a rigid quantum rotor. One of the interesting results of this study of dideuterium in the nematic phase is that the Zeeman relaxation is independent of liquid crystal solvent. In the next Chapter 3. Dideuterium as a solute in the nematic phase 38 chapter we will look at solutes with a single bond of rotation. These solutes are much larger and can be treated classically with respect to their orientation in the liquid crystal. Chapter 4 Solutes with s-cis and s-trans isomers Most of this chapter has been published in [55, 56] 4.1 Introduction The mechanisms responsible for orientational ordering in liquid crystalline systems are not completely understood. Achieving insight into these mechanisms may help us un derstand the fundamental principles of liquid crystal phases better. A host of techniques has been employed in the experimental investigation of these systems. The measure ments, using those techniques, usually yield parameters that are averages over internal molecular motions. This averaging presents a problem in the interpretation of results for the liquid crystal molecules themselves because a general characteristic of molecules that form nematic phases is that they are elongated and non rigid. The difficulty of dealing with such flexible molecules is that the orientation of each conformer is described by an independent order matrix [4]. Using small solutes with one single bond of rotation would seem to be the simplest starting point for understanding flexible molecules in nematic phases. NMR is an excellent tool for investigating experimentally orientational ordering in liquid crystalline systems. However, the determination of order matrices of flexible 39 Chapter 4. Solutes with s-cis and s-trans isomers 40 molecules is not straightforward since only products of order parameters S and proba bilities p’ can be determined from the anisotropic coupling constants of the NMR spec trum. These products cannot be separated without knowing in advance either the con former probabilities or the relationship among the order parameters of the conformers. One approach that has been used to separate the order parameters and probabilities is to equate the order parameters of the conformers [57]. This is at best a rough guess for cases where the different conformers are not related by symmetry. A better approach would be to use a physically reasonable model to estimate the relative magnitudes of the order parameters in the different conformers. This is the approach we shall use in this chapter. In our research group the orientational order of small solute molecules with high symmetry has been studied in a special nematic mixture in which the mechanism involv ing the molecular quadrupole moment and the mean electric field gradient [7, 9, 11, 12] of the nematic environment is negligible for dideuterium. It has been demonstrated [27, 10, 22, 23, 24, 58, 59, 60] that in this mixture the solute orientation can be described by a short range anisotropic potential acting on the solute molecule, which is modelled as a collection of Van der Waals spheres. The calculated order parameters, obtained from this anisotropic potential acting on the solute, compare very well with the experimen tal ones determined from the NMR spectra of these molecules as solutes in the special nematic mixture. In this chapter we describe our approach for dealing with the small flexible molecules furfural, 2-chlorobenzaldehyde and 2,2’-dithiophene (Figure 4.5) in the zero mean electric field gradient nematic mixture 55% 1132/45% EBBA (w/w). Our set of three solutes with internal rotation require more orientational parameters than do the molecules previously studied in this mixture [19, 24, 25, 26]. Thus they provide a useful test of models of the anisotropic potential in our special mixture. We use ‘H-NMR and published geometries to 41 Chapter 4. So)utes with s-cis and s-trans isomers Figure 4.5: Structures and Molecular Fixed Axes for Furfural, 2-Chlorobenzaldehyde, and 2,2’-Dithiophene 4 H 4 H H c#o 113 H14SC 2 H S— trans s— ci.s x z 5 —H 2 H S— trans 4 H H s— cis 4 113 H ).H 5 J 2 H H?’’H6 3 H H 6 HIS’S’ s—trans S —cis The solutes furfural (a), 2-chlorobenzaldehyde (b) and 2,2’-dithiophene (c) in the s-trans con formation and the s-cis conformation. The choice of molecular fixed Cartesian axes and the labeling of the proton positions are shown. For 2,2’-dithiophene the x-aids is chosen to be the 2 axis of symmetry for both conformations. C Chapter 4. Solutes with s-cis and s-trans isomers 42 obtain information about the orientation of these molecules. For 2,2’-dithiophene we have shown that the dipolar coupling constants can be fitted with physically reasonable order parameters to a mixture of planar s-trans and non-planar s-cis conformers. In order to separate the order matrices and conformer probabilities we use our model for anisotropic short range interactions to estimate relative order parameters for each conformer. Theory 4.2 NMR is an excellent tool for investigating molecules dissolved in the liquid crystaline phase [57, 61, 62]. In this anisotropic phase the NMR spectrum is dominated by the intramolecular dipolar coupling constants between the spins i and j 11 = — 8r2r gyromagnetic ratio of a proton , — 1 > (4.34) z is the angle between the magnetic field direction 8 Z and the axis connecting nuclei i and j. <3cos 2 9 ijZ is the magnetic permeability in vacuum, 27rh is Planck’s constant, 7H is the where and () The term < 3cos z O 2 — j, 13 is the distance between the nuclei i and r 1 > is the order parameter along the ij-direction with respect to the magnetic field direction Z. The angle brackets denote an ensemble average over all angles 0. The molecules that constitute the nematic phase contain many spins with complex dipole-dipole interactions which give rise to very broad unresolved lines. However, the fast molecular translation and tumbling of molecules motionally average intermolecular dipolar couplings and give rise to well resolved solute lines in the NMR spectrum. In the case of molecules which occur in two rigid conformations, provided that the frequency of interchange between the two states is much faster than the largest difference of dipolar coupling constants between the conformers, the measured dipolar coupling Chapter 4. Solutes with s-cis arid s-trans isomers 43 constants are values averaged over the two conformations <D:, >... PCSDCiS + ptTsDtfl3. (435) If the s-cis state is non-planar it will be degenerate, thus the fractional occupancy for this state is p = ‘() + (—) and for the s-trans state is pt1 = 1 — The orientational order of each rigid conformer n can be described by its own order matrix, given by [63] S= <3cos9zcosSz (4.36) — where a, /3 are the molecule fixed x, y and z axes, 8 is the angle between the molecule fixed a, /3 axis and the magnetic field direction Z, and 6 is the Kronecker delta. The order parameters S of the internuclear vectors r, between spins i and j are related to the order parameters of the molecule in conformation n by the direction cosines of the angles and j q j 5 3 and the molecule fixed axes a, 3. The equation which between r relates the dipolar coupling constants to the product of the fractional occupancies and the corresponding order parameters is given by 3 <D [S >= — piScOs7IScOs$ + SpcoscosçS]. (4.37) 22 Only in cases with symmetry relationships between the conformers can the fractional occupancies be separated from the order parameters. For our solutes this is not the case. In our research group we have tested several models for the short range anisotropic potential. The one that provides the best fit of results for 46 different solutes in a zero average electric field gradient nematic phase is given by [19] 1 2 U(IZ)SR = kC(fl) — 1 k 3 tZmax J Zm,n Cz(IZ)dZ (4.38) 3 are constants of the nematic environment. The solute is represented as a where k and k collection of Van der Waals spheres. C(f) is the projected circumference of the solute at Chapter 4. Solutes with s-cis and s-trans isomers 44 Table 4.11: EBBA-d2 Quadrupolar Splittings v(EBBA-d2)/kHz with solute furfural 2-chlorobenzaldehyde 2,2’-dithiophene 300 K 17.77 304 K 17.91 17.38 17.36 314 K 324 K 14.24 16.04 Quadrupolar splittings of the EBBA-d2 ring deuterons as a function of temperature for the three different samples. orientation f onto a plane perpendicular to the director, and Cz(f) is the circumference of a slice that is perpendicular to the director and that cuts the solute at position Z along the director. The parameters k = 2.04 mN m and k 3 = 48.0 mN m 1 were determined from a simultaneous fit to a collection of 46 solutes at 301.4 K [19]. In this chapter we shall use these k values for our three solutes. The order parameters can now be calculated by numerical integration of the orientation of the molecule over all solid angles and are given by — f(3cos z 8 cosOz 2f e_U()/kBTdf — 4 39 ) . where kB is the Bolzmann constant and T is the absolute temperature. We have also investigated the temperature dependence of the order parameters. We have to assume some temperature dependence of the anisotropic potential. We shall assume, as would be the case for the Maier-Saupe mean field theory [5], that the force constants k and k 3 are proportional to the orientational order of the liquid crystal. The quadrupolar splitting of the EBBA-d2 ring deuterons will be used as a measure of the or der of the liquid crystal. The quadrupolar splittings for several temperatures are given in Table 4.11 and are used to scale the force constants in the potential to the constant deter mined at 301.4 K for a sample containing dideuterium as a solute: v(EBBA-d2)=17.71 kllz [19]. The H 2 -NMR spectra of the EBBA-d2 were measured under identical con ditions without removing the sample from the probe and the quadrupolar splittings Chapter 4. Solutes with s-cis and s-trans isomers 45 obtained are reported in Table 4.11. 4.3 Results and Discussion The analysis of the ‘H-NMR spectra was done by performing a least squares fit to the spectral lines with the aid of the computer program LEQUOR [29]. Chemical shift differences and dipolar coupling constants are given in Tables 4.12—4.14. Spin-spin cou plings were taken from high resolution spectra reported in the literature: furfural [64], 2-chlorobenzaldehyde [65], and 2,2’-dithiophene [66]. Without assumptions, it is not possible to separate the conformer probabilities from the order parameters in equation 4.37. However, experimental products of order pa rameters and probabilities of the conformers can be calculated from the dipolar coupling constants and the geometry of the molecule by the computer program SHAPE [30]. There are minor differences in the geometries published in the literature. 4.3.1 Geometry of Furfural For furfural, some authors assumed that the C-C-CHO angle for the different conforma tions is the same [67, 68], while Mönnig et al. [69] indicated from microwave data that the C-C-CHO angle for the s-trans conformer is 133.900, whereas for the s-cis conformer it is taken to be 124.66°. Our dipolar couplings are not consistent with the same angle for both conformations. However, we obtain very good fits to the dipolar couplings using the MSnnig et a!. microwave geometry from [69]. 4.3.2 Geometry of 2-chlorobenzaldehyde The geometry of 2-chlorobenzaldehyde was taken from electron diffraction measurements [70]. The RMS deviations of the fits using this geometry with the phenyl ring distorted Chapter 4. Solutes with s-cis and s-trans isomers 46 Table 4.12: Spectral Parameters of Furfural furfural T=304 K exp. frequency (Hz) 1 v V2 1F3 4 V. D 1 2 13 D 14 D 23 D 24 D .1334 RMS T=324 K — 1 v 1/2 3 v 4 V D 1 2 13 D 14 D 23 D .1324 .034 RMS 348.08± 0.29 1180.07±0.32 246.64 ± 0.38 0.00 ± 0.34 -156.45± 0.11 -47.86± 0.24 45.26± 0.15 -731.98± 0.26 -159.97± 0.32 -948.04± 0.27 0.30 365.17 ± 0.27 1195.64± 0.59 244.35 ± 0.72 0.00 ± 0.64 -126.55± 0.47 -41.12± 0.26 16.05± 0.32 -581.65± 0.76 -126.64± 0.97 -758.37± 0.33 0.56 AD:j II D 1 1 III -19.72 20.79 88.12 -71.28 -27.37 -100.10 58.20 1.55 3.70 -0.32 1.56 -9.12 1.97 2.76 -10.63 23.78 70.78 -171.50 -32.97 -91.87 67.53 2.69 0.51 -0.15 1.15 -4.77 -0.35 1.25 Experimental parameters of the 400 MHz spectrum of furfural in a 55% 1132/EBBA (w/w) nematic phase. The J.j couplings used are J 12 = J 13 = 0.8, J 14 = 1.7, J 24 = 0.3 and J 34 = 3.7 Hz. All other J ‘s are fixed to zero [64]. The resonance frequencies v have arbitrary zero. = D (exp) .D,(ca1c) from the calculation reported in Table 4.17 and 4.18 for Methods 3 II and III (see text). — Chapter 4. Solutes with s-cis and s-trans isomers 47 Table 4.13: Spectral Parameters 2-Chlorobenza.ldehyde 2-chlorobenzaldehyde T=304 K V2 3 v 114 115 D 1 2 13 D 14 D 15 D .023 24 D .025 .034 35 D .045 RMS T=314 K ii 112 113 114 115 D 1 2 D3 14 D .023 1324 25 D .034 D 3 5 45 D RMS exp. frequency (Hz) 0.00± 0.29 183.62± 0.32 62.88± 0.38 274.40± 0.34 1288.98± 0.14 -746.56± 0.39 -107.64± 0.39 -92.42± 0.20 -162.17± 0.18 -711.05± 0.58 -210.40± 0.95 -91.02± 0.21 -1255.26± 0.56 -100.58± 0.24 -322.59± 0.26 0.32 0.00± 0.20 166.43± 0.25 55.43± 0.39 266.77± 0.29 1283.44± 0.19 -693.89± 0.55 -103.98± 0.74 -86.37± 0.19 -147.20± 0.15 -665.51± 0.92 -189.88± 1.15 -82.63± 0.14 -1138.59± 0.91 -92.33± 0.17 -302.14± 0.20 0.09 II III 154.90 42.58 5.87 -30.56 75.20 -34.72 -16.85 -221.48 -9.17 -16.71 54.69 0.08 -0.95 -2.88 -5.20 5.21 -4.41 -1.89 7.36 0.27 -0.98 3.34 143.25 35.86 4.65 -26.96 62.67 -28.18 -14.55 -187.82 -7.24 -7.35 29.52 0.28 -2.40 -1.87 -3.50 10.51 0.74 -0.94 18.27 0.89 0.75 2.62 Experimental parameters of the 400 MHz spectrum of 2-chloro-benzaldehyde in a 55% 1132/EBBA (w/w) nematic phase. The J,, couplings used are J 12 = 8.141, J 13 = 1.055, = 7.433, 2 J 4 = 1.893, 34 = 7.758, J 23 4 = 0.473, J Ji 15 = 0.035, J J = 0.050, J 35 = 0.784, and ’s are fixed to zero [65]. 3 45 = 0.078 Hz. All other J J Chapter 4. Solutes with s-cis and s-trans isomers 48 Table 4.14: Spectral Parameters 2,2’-Dithiophene 2,2’-dithiophene T=300 K V1 = 2 = 1)5 1)3 = 1)4 12 = D D 56 D 1 3=D 46 14 = D D 36 15 = D D 26 16 D 23 = D D 45 24 = D D 35 25 D 34 D RMS T=304 K -‘1 = Y2 = 1)5 113 = 1)4 12 = D D 56 D 1 3=D 46 14 = D D 36 15 = D D € 2 16 D 23 = D D 45 1324 = .1335 25 D 34 D RMS exp. frequency (Hz) 0.00± 0.04 229.79± 0.04 113.38± 0.04 614.68± 0.04 -148.09± 0.05 -216.24± 0.02 -84.15± 0.02 -72.92± 0.02 -2189.46± 0.02 -184.19± 0.02 -82.31± 0.04 -741.47± 0.04 0.13 0.00± 0.02 217.84± 0.04 107.32± 0.04 583.00± 0.04 -143.43± 0.06 -208 .43± 0.03 -81.14± 0.03 -70.31± 0.02 -2109.82± 0.02 -178.34± 0.02 -79.58± 0.05 -717.29± 0.05 0.16 1 II AD , III 1 AD 246.29 1.24 -6.06 -7.55 -10.28 -265.42 -38.01 -12.99 -184.47 151.42 0.06 -0.99 -0.59 -0.07 1.51 0.01 0.32 -0.32 -0.70 0.59 228.96 6.84 -26.91 -10.04 -6.43 -232.33 -18.78 -9.89 -66.80 130.44 0.07 -1.07 -0.39 0.07 1.42 0.01 0.15 -0.39 -0.41 0.53 Experimental parameters of the 400 MHz spectrum of 2,2’-dithiophene in a 55% 1132/EBBA (w/w) nematic phase. The J,, couplings used are J 12 = 5.0, J 13 = 1.3, and J 23 = 3.8 Hz. All other J ’s are fixed to zero [66]. 3 Chapter 4. Solutes with s-cis and s-trans isomers 49 along the C-CHO axis are not as good as the fits obtained when a regular hexagon is used for the ring and all other parameters are fixed to the electron diffraction values. For our analysis we shall, therefore, use a regular hexagonal ring and other geometric parameters from reference [70]. 4.3.3 Geometry of 2,2’-dithiophene The geometry we are using in our work of 2,2’-dithiophene is constructed mainly from microwave data [71] and the bridging parameters from MNDO calculations [72]. We elaborate our discussion on the geometry of 2,2’-dithiophene since none of the published geometries fit our experimental dipolar coupling constants well. Many authors claimed that 2,2’-dithiophene exists in only one conformation and quoted values for the dihedral angle 4 between the two thiophene rings the planar s-cis and 950 = ( = 00 for 1800 for the planar s-trans conformation) which vary between and 1800. The X-ray diffraction studies by Visser et al. [73] indicated that, in the solid phase, all molecules are in the planar s-trans conformation. Almenningen et al. [74] obtained an angle 4 of 146°, using Electron-diffraction in the gas phase, but also suggested that the molecule may actually rotate freely between = 95° and = 146°. Aroney et al. [75] found an angle of (157 ±10)° from the apparent polarizations and molar Kerr constants of 2,2’-dithiophene in carbon tetrachioride. From ESR studies Cavalieri d’Oro et al. [76] found a superposition of two closely related radical anions of 2,2’-dithiophene in a 4:1 ratio at 193 K. The authors suggested that these are the s-trans and s-cis conformers. There are two published NMR studies on the structure of 2,2’-dithiophene dissolved in nematic solvents. Bucci et al. [66] reported that 2,2’-dithiophene dissolved in Merck Phase IV at room temperature exist in two planar conformations: the occurence of the s-trans state is estimated to be (70 ±4)% and the rest of the molecules are in Chapter 4. Solutes with s-cis and s-trans isomers 50 the s-cis state (in another analysis in which they assume an intramolecular potential which is a function of they estimated p’ of 0.64). The analysis was performed using the structure of 2,2’-dithiophene obtained from X-ray spectroscopy [73] and assuming that the motional constants C3z2_r2 and are equal for both the s-cis and s-trans conformers. These results were calculated from NMR spectra obtained at 110 and 220 MHz. The spectral line width was 4 Hz for both spectra, the RMS error in the spectral fit was more than 5 Hz, and the dipolar coupling constants were fitted with an RMS error of 3 to 4 Hz. In another experiment, Khetrapal et at. [77] obtained the spectrum of 2,2’-dithiophene in 80% N- (4-ethoxybenzylidene)--4’-n-butylaniline/20% O-carbobutoxy-4-oxybenzoic acid ethoxy phenyl ester at 301 K. Their analysis also suggested that 2,2’-dithiophene exists as a mixture of planar s-cis and s-trans conformations. In order to force the order pa rameters associated with the directions perpendicular to the rings to have the same sign in both conformations they assumed that the ratio equals unity (see Figure 4.5 for the definition of the molecular axis). Their spectral fit yielded an RMS error of 1.2 Hz and the dipolar couplings were fitted with an RMS error of 0.2 Hz by iterating upon the four proton distance ratios of the molecule (assuming 1 r 2 = 2.64 A). In this section of the thesis we report measurements of 2,2’-dithiophene dissolved in another nematic liquid crystal. We shall show that the dipolar coupling constants can be fitted when rigid planar s-trans and non-planar s-cis conformers are considered. We shall perform this analysis for our new set of dipolar coupling constants and with both published independent sets of dipolar coupling constants [66, 77]. In all cases the RMS error is small and a mixture of planar s-trans and non-planar s-cis conformers is supported. By assuming that both S’/S and S/SZ equals unity, we shall estimate the s-trans state probability. It is not possible to use Equations 4.37 and the experimental D:, in Table 4.14 to Chapter 4. Solutes with s-cis and s-trans isomers 51 determine the structure, order matrices and fractional occupancies of the conformers of 2,2’-dithiophene without making some assumptions. First, there are too many unknowns: three order parameters for the planar s-trans and two for the planar s-cis conformation, the fractional occupancy of the trans conformer pt8, and the coordinates for the pro ton positions in both conformations. In this study we shall fix some of the structural parameters to values determined in several published studies using different techniques. Secondly, the products of fractional occupancies and S are not separable. To obtain distinct values for these parameters we must either assign a value to pt73 or fix the ratio of at least one of the to S. There are several published structures of 2,2’-dithiophene. We have found that the best least squares fits to the experimental dipolar couplings are obtained using the struc ture of the thiophene ring measured by microwave spectroscopy [71] in combination with the ring linking parameters, 1.455 A for the C-C bridge length and 119.4° for the S-C-C angle between the thiophene rings, taken from an MNDO-SCF-MO calculation [72]. In all cases the molecule fixed x-axis is chosen to be the C 2 axis of symmetry. Many of the previous structural studies of 2,2’-dithiophene report the presence of a single conformer. However, we are unable to account for the experimental dipolar couplings with a single conformer. For example: a two parameter fit for a planar s-cis molecule gives an RMS error of 303 Hz; a three parameter fit for a planar s-trans molecule gives a 105 Hz RMS error; and a four parameter fit for a non-planar structure gives a 40 Hz RMS error and a twist of 100 degrees from the planar s-cis structure. In addition the values otained in all these fits for the order parameters are outside the allowable range (— S 1). It is not possible to obtain acceptable fits by adjusting the geometric parameters within reasonable limits. Thus we come to the same conclusion reached by Bucci et at. [66] and Khetrapal et al. [77] that 2,2’-dithiophene dissolved in a nematic liquid crystal does not exist as a single conformer. 52 Chapter 4. Solutes with s-cis and s-trans isomers Table 4.15: Products of Order Parameters and Weights, and Torsional Angles of 2 ,2’-Dithiophene strans TI/K Sir t’ yij 0.2907 300 0.2804 304 0.2078 300 0.2014 304 XX I’ -0.4320 -0.4202 -0.1022 -0.1064 tjz S 0.0758 0.0729 0.1601 0.1532 I’ -0.0133 -0.0122 -0.0109 -0.0100 $ 4 c rans 4 i 00t 0t 180 O° 0j 180 24.40 180.0° 180.0° zz 1-’ 0.1623 0.1588 -0.1780 -0.1647 24.4° RMS 0.85 0.84 0.48 0.48 Products of order parameters and fractional occupancies, and geometrical parameters for 2,2’dithiophene from two different independent sets of spectra. The geometry is taken from Mi crowave data [71], with bridging bond and angle from MNDO data [72]. S is fixed to zero (see text). tThe parameter has been fixed during the iteration. Previous studies using liquid crystal solvents [66, 77] assumed that both s-cis and s-trans conformers are planar. We also obtain a good five parameter fit with small RMS error using planar geometries (Table 4.15). The fit produces the product of conformer probabilities times order parameters. However, the numbers obtained from the fit are unacceptable. First, there is no way to choose conformational probabilities where all the order parameters S are greater than the minimum allowed value case in reference [77] the positive value of SS -. Second, as is the is counter to normal observation where it is found that in planar molecules the order parameter associated with the direction perpendicular to the plane of the molecule is negative. A negative value for S is also predicted by a model calculation where the short range potential is based on the size and shape of the solute molecule [55]. Therefore planar s- cis and s-trans conformers in the nematic phase can be ruled out. Since many of the previous studies of 2,2’-dithiophene predict non-planar structures, it seems reasonable to attempt a fit of the liquid crystal solution results with such structures. Thus we performed fits where both the s-cis and the s-trans conformers are allowed to deviate from planarity by rotation about the C-C bridging bond. These fits require three Chapter 4. Solutes with s-cis and s-trans isomers extra parameters: the dihedral angles 4 and 53 and the off diagonal order parameter A difficulty with this eight parameter fit is that the angle to the value of We anticipate that the value of cis 4 is strongly correlated should lie between that for the planar s-cis structure (zero) and that for the planar s-trans conformer (S”). Since the dihedral angle decreases by less than two degrees when we let S change from zero to the value for in separate seven parameter fits, and since the dihedral angles obtained from the fits correspond to structures that are close to that of a planar s-cis structure, we expect S to be close to zero. Also from model calculations [56] S is calculated to be close to zero. Thus we fix the value of S to zero. The seven remaining cis, trans (which contribute to the six inter-ring dipolar couplings) and free parameters, 4 the five products of order parameters times conformer probabilities (which contribute to all nine dipolar couplings), are fitted by least squares to the nine dipolar couplings and the results for 300 and 304 K are shown in Table 4.15. The dihedral angle for the s-cis conformer is found to be 240 and for the s-trans conformer 1800. A parameter of great interest is the conformer probability — unfortunately equations 4.37 for the dipolar couplings do not allow separation of these probabilities from the values of the order parameters. To obtain either the order parameters or the conformer probabilities from the fitted values reported in Table 4.15 some method of assigning at least one of the values must be used. In previous studies, Bucci et al. [66] assumed ,2_ are the same in both conformers. They justified this 2 that values of 2_r2 32 and C C assumption by studies of the cis and trans isomers of thienothiophenes. In our laboratory we have been investigating the mechanisms of orientational order of solutes dissolved in the special mixture of liquid crystals used in this study. It has been found that in this mixture the average electric field gradient is zero [9, 11, 12] and that there is no contribution to the orientational order resulting from the interaction between the solute molecular quadrupole tensor and this gradient. In addition we have been quite Chapter 4. Solutes with s-cis and s-trans isomers 54 Table 4.16: Experimental Order and Geometric Parameters for 2,2’-Dithiophene TI‘K 300 304 RT’ 301 2 S’”0.3676 0.3545 0.2196 0.1512 St18 -0.2742 -0.2660 -0.1524 -0.1084 -0.0184 -0.0169 0.0095 0.0075 rans or 0.61 0.61 0.59 24.1° 24.2° 23.7° 24.0° RMS 0.50 0.49 0.37 0.28 Order and geometrical parameters for 2,2’-dithiophene from four independent sets of dipolar coupling constants. The fractional occupancy for the s-trans state pft is obtained by equating the two largest order parameters. S is fixed to zero and is fixed to 1800 (see text). ‘Coupling constants used from [66]. 2 constants Coupling used from [77]. tThe parameter has been fixed during the iteration. successful in modeling the remaining orientational forces in this mixture in terms of a model for the short range repulsive forces that treats the solute as a collection of Van der Waals spheres and the nematic phase as an axial symmetric repulsive force acting on the solute molecule. Calculations using this model [55] predict that the ratios deviate from unity by at most 2% for the values of 4 reported in this and paragraph. In principle the model calculations can be used in conjunction with either of the order parameter ratios and the results of Table 4.15. Examination of Table 4.15 shows that such an analysis would yield quite different values for the conformer probabilities, depending on which ratio is used. However, we find that the order parameters are correlated. If, as suggested by our model calculations [55], we fix both the ratios to 1, and 4 and sisans to 180°, the value reported in Table 4.15, the quality of the least squares fits to the dipolar couplings (Table 4.16) are hardly affected, and fits of equal quality to those in Table 4.15 are obtained. As the model calculations predict equality for these two order parameter ratios, we take the results of Table 4.16 as having physical significance. Assuming that the accuracy of the model in predicting order parameters is about 10%, Chapter 4. Solutes with s-cis and s-trans isomers we estimate the error of the s-trans probability to be about ±0.03. The dihedral angle for the s-cis conformer remains at 240 and the fractional occupancy for the s-trans state is 0.61. The above results are based on a specific assumed structure for 2,2’-dithiophene. If we use other published structures for 2,2’-dithiophene, such as those from MNDO [72] or Electron Diffraction [74] studies, we obtain almost the same order parameters, probabilities and dihedral angle that are reported in Table 4.16. In these cases the RMS error is much larger due to small differences in the geometries compared to the microwave geometry and MNDO linking parameters. It is gratifying that our results are independent of the assumed geometry. In order to substantiate further our conclusions we repeated our analysis on the two totally independent sets of published dipolar coupling constants from Veracini et al. [66] and Khetrapal et al. [77]. The fits are also reported in Table 4.16, and give the same angle 4’ and conformer probability as obtained from our experiments. The RMS error from the couplings of Khetrapal et al. increases slightly from 0.2 to 0.28 Hz, but we are adjusting one less parameter than was done in that study. The RMS error from Veracini et al. reduced from 4 to 0.4 Hz. The fact that our results and the published data using quite different liquid crystal solvents all give the same result that p’ is 0.61, and for the s-cis state is about 240, gives us confidence in our analysis. Recently, an investigation by Berardi et al. [78) has confirmed our work on the conformations of 2,2’-dithiophene by re-analysing our accurately determined dipolar coupling constants with the maximum entropy internal order method (MEIO). Chapter 4. Solutes with s-cis and s-trans isomers 4.4 56 Dealing with Flexible Molecules Using the best geometries, the six dipolar couplings of furfural, the ten dipolar couplings of 2-chlorobenzaldehyde and the nine dipolar couplings of 2,2’-dithiophene are sufficient to determine the products of order parameters and the probability of each conformation. No corrections were applied for vibrational motions in all three cases and any anisotropy in the indirect couplings is neglected. The experimental products of order parameter and probability Sp’(exp) obtained from the fits are presented in the first data column of Table 4.17. Since the 3 D: ’ s are averages over at least two conformations (eqn. 4.35), and since the products of order parameters and statistical weight cannot be separated, we must make assumptions in order to obtain the order parameters and statistical weights separately for each conformation. To do this we use the model potential for short range anisotropic interactions (eqn 4.38), using the best fit parameters k = 2.04 mN m 1 and k = 48.0 mN m’ [19], to predict with equation 4.39 the order parameters for each conforma tion, S(calc) (fourth data column Table 4.17). These calculated order parameters can now be used in several different ways in order to extract conformer probabilities from our experimental results. Here we shall present three different ways of performing the analysis. Method I: One way of separating the order parameters from the conformer probabili ties is to fix relative values of any one of the order parameters using the values calculated with the model potential (data column 4 of Table 4.17): Sp(exp) S;n3prs(exp) — — S(calc) p S7z8(calc) ptrans (4 40) Conformer probabilities are then calculated from equation 4.40 and the relation p = 1 where pcis is the total population of the possible cis conformations [55]. The — Chapter 4. Solutes with s-cis and s-trans isomers 57 Table 4.17: Experimental and Calculated Data for the Order Parameters of Furfural, 2-Chlorobenzaldehyde, and 2,2’-Dithiophene c43;n furfural T304K furfural T=324K 2-chlorobenzaldehyde T=304K 2-chlorobenzaldehyde T=314K 2,2’-dithiophene T=300K 2,2’-dithiophene T=304K xx;cis zz;cis xy;cis xx;trans zz;trans xy;trans RMS/Hz xx;cis zz;cis xy;cis xx;trais zz;trans xy;trans RMS/Hz xx;cis zz;cis xy;cis xx;trans zz;trans xy;trans RMS/Hz xx;cis zz;cis xy;cis xx;trans zz;trans xy;trans RMS/Hz yy;cis zz;cis yz;cis xx;trans yy;traxis yz;trans RMS/Hz yy;cis zz;cis yz;cis xx;trans yy;trans yz;trans RMS/Hz p (exp) 0 S 0.0728(-) -0.0657(.) .0.0681(-) 0.0916(.) .0.1211(-) .e.0133(.) 0 0.0575(-) .0.0582(-) -0.0528(-) 0.0740(-) -0.0946(-) -0.0117(-) 0 0.021(5) -0.07(2) -0.008(6) 0.134(5) -0.15(2) 0.011(6) 2.01 0.020(3) -0.06(2) -0.006(5) 0.122(4) -0.15(2) 0.008(5) 1.61 0.15(2) -0.15(8) -0.001 (2) -0.13(7) 0.21(2) -0.011(1) 0.49 0.15(2) -0.14(8) .0.001(2) -0.13(8) 0.21(2) .0.010(2) 0.49 p (exp) 0 pfl(jJfl 0.132(5) .0.119(5) -0.124(5) 0.20(1) .0.27(1) 0.030(1) 0.109(3) .0.111(3) -0.101(2) 0.156(4) .0.199(5) -0.0246(6) 0.22(5) -0.7(2) -0.08(6) 0.149(6) -0.17(2) 0.012(7) 0.19(3) -0.6(2) -0.06(5) 0.137(5) .0.17(2) 0.009(6) 0.39(5) -0.4(2) .0.002(5) -0.2(1) 0.35(3) -0.018(2) 0.38(5) -0.4(2) -0.003(5) -0.2(1) 0.34(3) -0.016(2) (,eip) 0 S 0.139(2) -0.184(2) -0.107(6) 0.194(2) -0.185(2) -0.043(2) 2.76 0.1118(5) -0.1521(8) -0.089(2) 0.1532(6) -0.1528(8) .0.036(1) 1.25 0.163(1) -0.224(1) 0.0108(4) 0.156(1) -0.226(1) 0.0012(4) 3.34 0.149(1) -0.1960(4) 0.006(5) 0.143(1) -0.2107(4) 0.0007(6) 2.62 0.3687(2) -0.2691(3) -0.0042(2) .0.2772(3) 0.3669(2) -0.0179(7) 0.59 0.3554(2) -0.2609(3) -0.0039(1) -0.2690(3) 0.3539(2) -0.0164(6) 0.54 S(calc) 0.1273 -0.1742 -0.0754 0.1775 -0.1751 -0.0302 0.0973 -0.1368 -0.0541 0.1333 -0.1375 -0.0217 0.1338 -0.2239 0.0544 0.1281 -0.2253 0.0062 0.1226 -0.2069 0.0477 0.1178 -0.2083 0.0054 0.3227 -0.2589 -0.0050 -0.2666 0.3211 -0.0212 0.3121 -0.2528 -0.0048 -0.2605 0.3107 -0.0203 Experimental and calculated products of order parameter and probability (columns 1: raw data with no assumptions and 2: calculated model order parameters times the probability from Method II) and order parameters for each conformation (column 3: order parameters from Method II and 4: model order parameters). The geometry for furfural is taken from [68], and the geometry for 2-chlorobenzaldehyde comes from [70], except that all aromatic ring angles are set at 1200. The dithiophene structure is reported in [55]. Chapter 4. Salutes with s-cis and s-trans isomers 58 Table 4.18: Weight of Fufural, 2-Chlorobenzaldehyde, and 2,2’-Dithiophene rons t p (furfural 304K) RMS/Hz prans (furfural 324K) RMS/Hz pirans (2-chlorobenzaldehyde 304K) RMS/Hz pirans (2-chlorobenzaldehyde 314K) RMS/Hz pirQTIs(2,2dithiophene 300K) RMS/Hz rans t p (2,2’-dithiophene 304K) RMS/Hz Method I 0.48±0.16 0.49±0. 13 0.88±0.21 0.89±0.20 0.59±0.15 0.57±0.09 Method II 0.39±0.08 58.20 0.51±0.16 67.53 0.96±0.05 54.69 0.94±0.03 29.52 0.76±0.11 151.42 0.60±0.14 130.44 Method III 0.45±0.02 2.76 0.475±0.012 1.25 0.901 ±0.006 3.34 0.893±0.007 2.62 0.6103±0.0009 0.59 0.6094±0.0008 0.54 Probabilities with their standard errors for the trans conformation of each solute as calculated by Method I, II, and III (see text). average value of ptrans obtained from the three order parameters reported for each ex periment in Table 4.17 are given in Table 4.18. The errors are standard deviations of the three ratios used from Table 4.17 , and are rather large. This is partly due to the large errors in data column 1 of Table 4.17, which arise from correlations among the experimental probabilities times order parameter values. The choice of geometry con tributes to the quality of the fit as well. As an example, if we use the geometry of 2-chlorobenzaldehyde from ref. 24, Method I gives conformer probabilities for s-trans of 0.80±0.38 and 0.78±0.37 for 304 and 314 K respectively; the errors are much larger than those obtained for 2-chlorobenzaldehyde with the hexagonal ring (Table 4.18, Method I), and the averages are 10% smaller. Unfortunately, Method I does not give accurate values of pirans, and it is desirable to search for a better method of analysis. Method II: In this method all order parameters are fixed to values calculated from 2 the model and only the probability pi3 is adjusted in a fit to the dipolar couplings D Chapter 4. Solutes with s-cis and s-trans isomers 59 from Table 4.14. The probabilities obtained for the s-trans conformation in this way are given in Table 4.18, Method IL The errors associated with these one parameter fits are the square root of the variances from the minimization and are large, but the calculated probabilities agree (within two standard errors) with those obtained from Method III below. The and the weighted RMS errors from this fit are given in Table 4.14. 4 S 1 , 4 is small, because (exp) D 14 for furfural. However, 1 The largest deviation is in zD is small. One reason for this is demonstrated by the model calculations which give 4 that are small and of opposite sign for cis and trans conformers. D contributions to 1 Method II is a stringent test of the model. The agreement between experimental and calculated dipolar couplings are rather good. This is excellent considering the order matrix for each conformer is calculated using parameters from a fit to 46 different solutes. One objective of this chapter is to obtain estimates of ptT that rely as little as possible on any model calculation. The problem with method I is it assumes that the ratios of order parameters are those given by the fit of products of probability and order parameter to the dipolar couplings. However, because the dipolar coupling equations for s-cis and s-trans conformers are not always sufficiently different, these products are not well determined; the large errors in the products lead to large errors in the ratios, and hence to large errors in The difficulty ‘with method II is that it assumes the model for the short-range potential is perfect. However, in the excellent fits obtained for 46 solutes [19], differences between calculated and experimental order parameters of about pu73 of 10% are common. Such differences lead to the large uncertainly found for the method IL Method III: In order to separate probability from order parameter in eqn. 4.37, it is sufficient to use the model calculation simply to relate how any one element of the order matrix changes with conformational change. As there is no a priori way of choosing which order parameter ratio should be fixed, we fix the ratio of all order parameters to Chapter 4. Solutes with s-cis and s-trans isomers the 60 values given by the model calculation (Table 4.17, data column 4), and we iterate on the three S 3 and the pt8 in a fit to the experimental dipola.r couplings from Table 4.14. The results of these fits are presented in Tables 4.14, 4.18, and as S in the third data column of Table 4.17. The errors for these four parameter fits are very gratifying. Thus, the model potential (eqn. 4.38) seems to predict relative order parameter changes with conformer changes very well. The probabilities reported in Table 4.18 agree with those obtained using Methods I and II, but the uncertainty is much less, making the probabilities from Method III more stable. Thus, the probabilities do not depend to any great extent on the assumptions made. In addition, the most predominant conformer is found to be the same as in other studies reported in the literature on furfural [67, 68] 2-chlorobenzaldehyde [79] and 2,2’-dithiophene [66, 76, 77]. At this point it is interesting to examine the experimental products of order parameter and conformer probability (Table 4.17, data column 1) obtained directly from the dipolar couplings by dividing them by the probability calculated from Method Ill. The results from these divisions are shown in the second data column of Table 4.17. The errors are calculated from both the errors in Sp’(exp) and pZ from Method III. The order parameters compare reasonably well with those obtained from Method III (Table 4.17, data column 3), although the error is larger. As in the case of Method I, the large errors arise from correlations in the Sp1’L (exp) between the two conformers. Further, for 2-chlorobenzaldehyde S is less than the lowest possible value — . Unfortunately, for flexible molecules the products of order parameter and conformer probability cannot always be determined very reliably and, as is shown in this chapter, analysis based on some extra assumption is required. Chapter 4. Solutes with s-cis and s-trans isomers 4.5 61 Conclusions of this Chapter In this chapter we have investigated the orientational order of three small, flexible molecules as solutes in a special mixture of nematic liquid crystals for which the av erage electric field gradient is zero. The order parameter matrix for each conformer of these three flexible molecules has been predicted with the aid of a model for the short range anisotropic potential in nematic liquid crystals. The model is seen to provide a good description of the orientational order of each conformer. In each case, the ratios of predicted order parameters between conformers have been used to obtain the probability of the s-trans conformer. The numbers obtained agree with previous values determined in other phases. Using a model to predict changes in order parameter associated with con formational change is a reasonable way to approach the problem of dealing with flexible, partially oriented molecules. In the course of the analysis we have found a new conforma tion of 2,2’-dithiophene in the nematic phase. The s-trans conformer of 2,2’-dithiophene is planar and the s-cis conformer is non-planer with a dihedral angle of ±24° between the thiophene rings. This chapter examines the flexibilty aspect of nematogens. Another aspect of nematogens is that they usually have an aromatic core with an alkyl chain. The contribution of the molecular quadrupole moment of small alkanes and benzenes to the orientation will be discussed in the next chapter. Chapter 5 Determination of Molecular Quadrupole Moments using Liquid Crystals 5.1 Introduction In chapters 3 and 4 of this thesis NMR has proven to be a very useful tool for the study of the dynamics, structure, and orientation of molecules dissolved in liquid crystalline solvents. A proposed orienting potential can be tested once the molecular order param eters are known. In this chapter we will investigate the possiblity for obtaining numbers for the molecular quadrupole moment of small alkanes and several benzenes from the order parameters using a potential which will include a short-range and a long-range in teraction part. The long-range potential describes the interaction between the molecular quadrupole moment of a solute and the mean electric field gradient (efg) of the liq uid crystal environment. Buckingham [80] has pointed out that one of the major uses of molecular quadrupole moment data is the prediction of long-range contributions to inter molecular potentials. Unfortunately, accurate experimental data for quadrupole moments are rather scarce and high quality theoretical results are limited to small molecules. Most nematogens have an aromatic core and a flexible alkyl chain. Small alkanes and several benzenes have been used as parts of a nematogen to investigate the contribution of the molecular quadrupole moment to the orientation. The spectral analysis becomes more and more complex when there is little or no sym metry of the molecule and/or the number of spins become larger than 8. The ‘H-NMR spectrum of benzene, l,3,5-trifluorobenzene (TFB), hexafiuorobenzene (HFB) and the 62 Chapter 5. Determination of Molecular Quadrupole Moments using Liquid Crystals 63 internal standard 1,3,5-trichlorobenzene (TCB) are trivial to solve. Even the spectrum of ethane is easy to solve since there are only two independent coupling constants and the orientation is completely described by one order parameter. For propane, however, it becomes a very cumbersome task even though there are only four different coupling constants and two order parameters to describe the orientation. The single quantum spectrum of propane dissolved in several nematic liquid crystals contains hardly any structure, and therefore the determination of the molecular orientation from this spec trum seems complex. Usually, the approach has been to get a good initial estimate for the dipolar coupling constants from an estimated structure and trial order parameters and see if the simulated spectra resemble the experimental spectra. But since for propane there is hardly any structure and the appearance of the spectrum is very sensitive to small changes in the order parameters and dipolar couplings, other methods of analysis may seem useful. For the spectral analysis of propane we have relied on the technique of Multiple-Quantum NMR (MQ-NMR). A recent analyses on a complex 1-Q spectrum of biphenelyne has been solved using MQ-NMR [81]. The six and seven quantum spectrum of propane has been used to obtain very good starting values for the four dipolar coupling constants in the final analysis. Once the spectra of our molecules have been solved, the dipolar coupling constants provide details of the structure and orientation of these molecules in the three liquid crys tal solvents used: EBBA, 1132, and a 55 wt% 1132/EBBA mixture. From dideuterium studies these liquid crystals are known to have different electric field gradients. The three aromatic molecules, benzene, TFB, and HFB, have been the subject of research with respect to their quadrupole moments by many scientists [82, 83]. Benzene and HFB have quite similar molecular properties, excepting their quadrupole moments which have nearly the same magnitude but differ in sign [83]. The negative sign of the zz-component of the molecular quadrupole moment O of benzene, where z is the molecular symmetry Chapter 5. Determination of Molecular Quadrupole Moments using Liquid Crystals 64 axis, suggests that the electronic charge is concentrated along the z-axis and depleted along the axes of the plane of the molecule. In contrast, the positive value of O of HFB suggests the opposite distribution of the electronic charges to that for benzene. A larger transfer of charge away from the center of HFB is the origin of the different sign of its quadrupole moment. We analyzed our solute molecules and the solvents in terms of two contibutions to the orientational potential. The short-range size and shape potential described in chapter 4 and the long range molecular quadrupole moment—efg interaction potential. This analysis may provide us, in general, with insight into obtaining molecular quadrupole moments of solutes in liquid crystals. In addition, the ethane and propane results allow us to estimate the contribution of the quadrupole moment of alkyl chains to the whole liquid crystal molecule in particular. 5.2 Multiple-Quantum NMR What can be measured in an NMR experiment is the expectation value of I <1+ > (t) = Tr{Icr(t)}, = I + iI_ (5.41) where o is the spin-density operator describing the state of the spin-system. The other so-called forbidden transitions can be measured by indirect detection schemes as a special form of 2D-spectroscopy. The technique of Multiple-Quantum NMR (MQ-NMR) has the ability of simplifying complex spectra [84, 85} by revealing some of the forbidden transitions. Figure 5.6 shows a single quantum spectrum of propane in a nematic liquid crystal. It has virtually no structure and analysis seems almost impossible on a PhD time scale. As can be seen in Figure 5.7 the MQ-NMR spectra of propane, and in particular the 6- and 7-quantum spectra, contain far fewer lines than the conventional 1-quantum spectrum as can be Chapter 5. Determination of Molecular Quadrupole Moments using Liquid Crystals 65 seen in Figure 5.7, and hence the assignment of lines in the fitting process will be greatly simplified. In a system of N coupled spin- nuclei, the N—i and N —2 quantum spectra contain a sufficient number of lines to obtain all the chemical shifts and dipolar coupling constants. k-quantum coherences are defined as coherences between states where im = ±k. For a system of N spin-i nuclei the maximum coherence achievable is of order k = ±N. As an example, the following operators represent +1 and +3-quantum coherences. 1+ k = +1 (5.42) 121+ k = +1 (5.43) 1—1+IzI+ k=+1 (544) 1+1+1+ k=+3 (5.45) (5.46) 1+121+1+ IILI1 k = (5.47) +3 The order of quantum coherence produced by a product of operators is equal to the num ber of raising operators minus the number of lowering operators. Most MQ-coherences are not directly detectable with conventional coils. Even for m = ±1 this may be true, since although 1+ contributes to the signal, the other 1-Q coherences do not because their product with 1+ have vanishing traces. In this thesis, however, when we mention i-Q spectra we mean the conventional spectra where we observe I and not all l-Q coher ences in the strict sence of the word. Other coherences have to be measured indirectly by means of 2D-techniques. A pulse sequence commonly used to generate and detect MQ-coherences is given by — r — — — () — acquire). (t 2 (5.48) After the first pulse, various l-Q coherences will evolve during the preparation time r. Chapter 5. Determination of Molecular Quadrupole Moments using Liquid Crystals 66 Figure 5.6: ‘H-NMR Spectrum of Propane in ZLI-1132 —4000 —2000 0 2000 Frequency (Hz) 4000 The 500 MHz ‘H-NMR spectrum of propane in ZLI-1132 at 304 K: the calculated spectrum (top) and experimental spectrum (bottom). Chapter 5. Determination of Molecular Quadrupole Moments using Liquid Crystals 67 Figure 5.7: Six and Seven-Quantum Spectra of Propane in ZLI-1132 7Q 6Q —4000 0 —2000 Frequency (Hz) 2000 The 7-Q (top) and 6-Q (bottom) ‘H-NMR spectra of partially oriented propane in ZLI-1132. A preparation time r of 11 ms was used. Chapter 5. Determination of Molecular Quadrupole Moments using Liquid Crystals 68 This time is fixed during the experiment and modulates the intensities of the peaks in the MQ-spectrum as a function of the dipolar couplings. The second pulse converts . 1 these coherences to all possible orders. The various MQ-coherences will evolve during t The third pulse is the detection pulse which converts a part of the MQ-coherences back into 1-Q coherences which then evolve into 1 to be detected in the FID. A 2D-Fourier transform (magnitude) of the resulting interferogram over the variable times t 2 and i provide information about the evolution of the various MQ-coherences during t. Since 1 frequency axis yields the the information in F 2 is redundant, a projection onto the F final MQ-spectrum. The orders will be separated by virtue of their dependence of the 1 and hence the accuracy of the spectral . Digital resolution in F 2 transmitter offset in F 1 increments. When q analysis is limited by the number of t = a large spectral width 1 is required to completely separate all MQ-orders. The necessary spectral width in F 1 can be narrowed by exciting or detecting [86, 87, 88] only the MQ-orders of interest. F Selective detection of MQ-coherences of order k can be achieved using the sequence in Equation 5.48 by coadding 2k FID’s at each value of t and cycling the phase of the first two pulses, , incremented 2k times by = [89]. Addition and subtraction of 1 except the orders ±nk, the resultant FID’s result in cancellation of all MQ-orders in F where n=l,2,3... 1 since both The above phase cycling is performed with no quadrature detection in F the +k and —k orders will appear in the spectrum. To further narrow down the spectral 1 [90] of order k can be width in F , quadrature detection of the MQ-spectrum in F 1 achieved by coadding 4k FID’s. The phase q is incremented by receiver phase 412 incremented in steps of . = with the Chapter 5. Determination of Molecular Quadrupole Moments using Liquid Crystals 69 53 Molecular quadrupole moments The molecular quadrupole moment — electric field gradient interaction is for dihydrogen and its deuterated analogs the most important mechanism for orientation. In this chapter we investigate the importance of this mechanism for larger molecules. It is assumed that these molecules can still be regarded as point quadrupoles with respect to the liquid crystal molecules creating the mean electric field gradient (efg) in the nematic phase. It is also worthwhile investigating the possiblity of obtaining molecular quadrupole moments using the strong efg present in liquid crystals. A full charge distribution provides all information for an interaction potential, but an expansion in electric multipole moments will characterize the charge distribution and can be used to describe the interaction energy of molecules whose separation is large compared to their dimensions. The monopole or charge q is given by (5.49) q= where e, is the i-th element of charge. The electric dipole moment j3 is the first moment of charge and is given in tensor notation by = 2 is the charge at where c = x, y, z and e moment i er (5.50) relative to an origin. The molecular quadrupole e is the second moment of the charge and is defined in tensor notation as 9a = ej(3rjarjfl — r6aø). (5.51) The quadrupole moment is a tensor of second rank and is symmetrical, i.e. = Oj3a, (5.52) Gaa = 0. (5.53) and traceless, i.e. Chapter 5. Determination of Molecular Quadrupole Moments using Liquid Crystals 70 The quadrupole moment gives an indication of how much the charge distribution deviates from spherical symmetry. The value of the quadrupole moment is independent of the choice of the origin only if the net charge, q, and the dipole moment of the molecule, new origin at zi is zero. Relative to the away from the previous origin, the new quadrupole moment is = — pf3Ic, +p 6+ — — , . 3 6 2 A ) (5.54) It is common to define the origin as the center of mass, where the first moment of mass is zero. The Hamiltonian describing the interaction between an arbitrary distribution of charges in a potential 4 U produced by external charges is given by [91] = 0 + (Vq)orja + (VaVi)oricrri + e{ 1= e + where Vc. = . .. (5.55) (5.56) } U/Oc and the subscript o indicates a value at the origin and where a repeated Greek index implies summation. With the definitions of the multipoles for q, p, and q 2 € and Laplace’s equation V U = = 0 qç 0 for the potential 4 of the external charges — where the field at the origin is Fa — = — ... (5.57) (Va)o, the field gradient at the origin is Fcr = —(VaVq)a, etc. The interaction energy is independent of the origin, but the contributions to the multipoles may vary. The solutes used in this chapter have either no or a negligible net dipole moment. Propane has a small dipole moment of 0.083 D [92]. In this chapter we are trying a new method of obtaining molecular quadrupole mo ments, using the large electric field gradient component, Fz in the liquid crystal solvent. Chapter 5. Determination of Molecular Quadrupole Moments using Liquid Crystals 71 The second rank interaction of the molecular quadrupole moment and the Fz is an ori enting mechanism for small solutes in liquid crystals [8, 9, 10, 11, 12, 13, 14] and as such should in principle be obtained by means of measuring the order parameter and knowing the other orienting mechanisms. In chapter 4 we have used for the short range potential Equation 4.38 1 U(IZ)SR = 2 kC(l) 1 — 5 k tZmax J Zm*n Cz(fZ)dZ (5.58) and in this chapter we will add the long-range contribution due to the non-zero Fz for the liquid crystals 1132 and EBBA. ULR = 5.4 EFaOa (5.59) Experimental Results The spectra of benzene, 1,3,5-trifiuorobenzene, hexafiuorobenzene, ethane, and the in ternal standard 1,3,5-trichlorobenzene in the three different liquid crystals 1132, 55wt% 1132/EBBA (MM), and EBBA have been solved. Chemical shift differences and dipolar coupling constants of the fits are given in Table 5.19 and 5.20. The single quantum spectrum of propane is very difficult to solve due to the many lines which do not form distinguishable groups to help the assignment (Figure 5.6). MQ NMR techniques have been employed to help the spectral analysis for propane. For a particular sample tube, the single and MQ-NMR spectra were obtained without removing the tube from the probe to ensure identical conditions. The FID’s were recorded at 304 K. 1 from Field et al. [90] we were able Using the quadrature detection technique in F to obtain good quality 6 and 7 quantum spectra of propane dissolved in the 55wt% 1132/EBBA nematic mixture, see Figure 5.7. A mixing time r of 11 ms has been used. Chapter 5. Determination of Molecular Quadrupole Moments using Liquid Crystals 72 Table 5.19: Spectral Parameters of Benzene, TFB, HFB, and TCB Parameter BENZENE 1132 MM EBBA Dortho Dmeta Dpara -994.55±0.03 192.97±0.03 125.89±0.04 0.34 -673.42±0.03 430.40±0.04 84.89±0.04 0.24 -448.95±0.03 86.06±0.04 55.74±0.07 0.27 -878.05±0.04 -138.99±0.12 -110.98±0.06 -205.24±0.03 0.24 -668.11±0.08 -106.80±0.10 -84.47±0.15 -155.75±0.34 0.35 -587.48±0.08 -93.34±0.30 -74.50±0.14 -137.21±0.06 0.20 -627.85±0.07 -117.56±0.09 -83.52±0.17 0.37 -222.34±0.04 -604.18±0.06 -113.00±0.10 -80.60±0.11 0.42 -171.02±0.01 -867.12±0.07 -162.18±0.10 -115.53±0.12 0.40 -154.49±0.17 RMS 1 ,3,5-TRIFLUOROBENZENE DHFO DFF DHFP DHH RMS HEXAFLUOROBENZENE Dortho Dmeta Dpara RMS D(TCB) Experimental parameters of the 200 MHz spectrum of benzene, 1,3 ,5-trifluorobenzene, hexafluo robenzene, and TCB in three different nematic liquid crystal solvents. The resonance frequency v has arbitrary zero. The dipolar coupling constant for TCB in each sample tube is given as a internal reference. Note that au aromatic solutes in one liquid crystal solvent are in the same sample tube. Chapter 5. Determination of Molecular Quadrupole Moments using Liquid Crystals 73 Table 5.20: Spectral Parameters of Ethane Parameter 1132 Frequency/Hz MM Frequency/Hz EBBA Frequency/Hz 0.00±0.05 680.28±0.02 270.48±0.02 0.19 222.15±0.02 0.00±0.04 545.51±0.02 -217.30±0.01 0.16 184.34±0.02 0.00±0.02 586.40±0.01 234.18±0.01 0.07 160.00±0.02 ETHANE ii Dmi,mi 2 Dmi,m RMS D(TCB) Experimental parameters of the 500 MHz spectrum of ethane in three different nematic liquid crystal solvents. ml and m2 represent the protons on methyl group 1 and 2. The Jml,m2 has been fitted to 8.00 Hz. The resonance frequency v has arbitrary zero. The dipolar coupling constant for TCB in each sample tube is given as an internal reference. Using the propane geometry from microwave experiments [92] and the short range po tential, Equation 5.59 with k parameters from chapter 4, we obtained a good starting point for the 6 and 7 quantum spectra. In the six quantum spectrum there are two lines whose positions only depend on the chemical shift. One of them is determined by 6 VmehyZ and the other one by 4 Vmell + 2 ’1 mehylene. From our initial fit for the dipolar coupling constants we could estimate the chemical shift differerence between the methyl protons and the methylene protons from our experimental spectrum by associating the two lines from the calculated spectrum. The chemical shift difference was measured to be about 0.6 ppm. This value is larger than the 0.44 ppm measured in the isotropic phase [93]. The 7-quantum spectra aided in the analysis of the 6-quantum spectra, since in principle all the information about the dipolar couplings are in the 6-quantum spectra. Difficulties arise when lines are missing in the experimental 6-quantum spectra due to our choice for the delay r. The 6-quantum spectra are fit in 1132, 55wt% 1132/EBBA, and EBBA with an RMS of 1.64, 1.42, and 0.85 respectively. The qualtity of the fit gives an excellent prediction for the l-Q spectra; there is virtually no difference between the Chapter 5. Determination of Molecular Quadrupole Moments using Liquid Crystals 74 Table 5.21: Spectral Parameters of Propane Parameter PROPANE Urnl 2 Vm 2 Jmi,m Dmi,mi Dmi,m 2 3 Dmi,m ,m Dm 2 RMS D(TCB) 1132 Frequency/Hz MM Frequency/Hz EBBA Frequency/Hz 0.00±0.10 336.61±0.15 7.37±0.12 690.76±0.05 457.68±0.07 327.92±0.04 1620.02±0.14 0.12 218.52±0.02 0.00±0.09 299.60±0.13 7.36±0.12 508.70±0.05 419.75±0.06 237.02±0.04 1228.03±0.12 0.06 171.65±0.02 0.00±0.10 309.00±0.18 7.36±0.15 578.50±0.05 -144.53±0.07 258.28±0.04 1476.24±0.15 0.10 161.00±0.02 Experimental parameters of the 500 MHz 1-Q spectrum of propane in three different nematic liquid crystal solvents. ml and m3 represent the protons on the two methyl groups and m2 the protons on the methylene group. The Jml,m3 has been fixed to zero. The resonance frequencies v have arbitrary zero. The dipolar coupling constant for TCB in each sample tube is given as a internal reference. calculated spectrum using dipolar couplings from the fit to the 6-Q spectrum and the experimental l-Q spectrum. The results of the final fits are shown in Table 5.21 and the calculated spectrum from the results of the fit of propane in 1132 is shown in Figure 5.6 (top). The internal standard l,3,5-trichlorbenzene (TCB) is used to relate the different sam ple tubes. Dipolar coupling constants for TCB are given in the tables for the experimental parameters for each sample tube. Molecular order parameters can be calculated from the dipolar couplings and the geometry of the molecule. The geometries are taken from the literature. Benzene [94], TFB [94], TCB [95], HFB [95], ethane [94], and propane [92] all gave reasonable fits to the dipolar couplings. The aromatics and ethane have C 3 and higher symmetry and it can be shown [63] that for such molecules there is only one independent order parameter, Chapter 5. Determination of Molecular Quadrupole Moments using Liquid Crystals 75 Table 5.22: Experimental Order Parameters of Propane, Ethane, Benzene, TFB, HFB, and TCB Parameter PROPANE S S TCB S ETHANE S TCB S BENZENE S TFB S HFB S TCB S 1132 Frequency/Hz MM Frequency/Hz EBBA Frequency/Hz -0.02620 0.09000 -0.02890 0.08191 -0.04310 0.11299 -0.29504 -0.23175 -0.21737 0.06401 0.05134 0.05521 -0.29993 -0.24888 -0.21602 -0.25244 -0.17091 -0.11392 -0.27132 -0.20650 -0.18214 -0.23517 -0.22693 -0.32529 -0.30019 -0.23089 -0.20858 Experimental order parameters for several solutes in three different liquid crystals. For each sample tube the order parameter of TCB is given. , symmetry and there 2 S, where z is the symmetry axis. Propane has an effective C will be two independent order parameters, S, and S, to describe the orientation for propane, where x is the C 2 symmetry axis and y is along the long axis of the molecule. The methyl groups were allowed to rotate with a 12.5 kJ/mol potential in step of 100. The order parameters are given in Table 5.22. 5.5 Discussion Burnell et al. have determined the efg-component, Fz, where Z is the direction parallel to the director, in several liquid crystals. They found that 1132 has a negative Fz and Chapter 5. Determination of Molecular Quadrupole Moments using Liquid Crystals 76 EBBA has a positive Fzz of almost equal magnitude. A 55wt% 1132/EBBA mixture was measured to have a zero mean Fz at 301.4 K as experienced by dideuterium. A striking feature of the order parameters from Table 5.22 of benzene, 1,3,5-trifiuorobenzene, and hexafluorobenzene as a function of liquid crystal solvent is that the order parameter of benzene is decreasing from a liquid crystal with negative Fzz to a liquid crystal with a positive Fez. HFB is observed to obey the opposite trend, while TFB behaves somewhere between benzene and HFB. This is striking since this trend follows the sign and magnitude of the quadrupole moments [83, 96, 97] of these molecules. Benzene, TFB, and HFB have a quadrupole moment € of -8.69, 0.94, and 9.50 x 10—26 esu cm . 2 It therefore seems promising to obtain values for quadrupole moments from an analysis of the order parameters and a model for the orienting potential. Using the short-range potential from chapter 4 U(1)SR = 1 kC(l) 2 — 1 k 8 fZmax J Z,, Cz(fl)dZ (5.60) and the long-range molecular quadrupole moment—electric field gradient potenial ULR = (5.61) it should, in principle, be possible to fit the k’s for each liquid crystal and the product to the order parameters. Different conditions like solute concentration and probe temperature effect the order of the liquid crystalline matrix. Ideally, all solutes should be put in one sample tube containing one particular liquid crystal. Then only three different sample tubes with either 1132, 55wt% 1132/EBBA, or EBBA have to be made up. Unfortunately, it will be almost impossible to analyse the NMR spectrum, since all solute spectra will be superimposed. In order to relate the order parameters of solutes in the same liquid crystal but from different sample tubes, we have used TCB as an internal reference in each sample tube. Chapter 5. Determination of Molecular Quadrupole Moments using Liquid Crystals 77 The short-range potential for each liquid crystal is scaled using the order parameters of TCB with respect to the samples containing propane in the respective liquid crystals (cos8) potential. The scaling factors are shown as CLC in Table 5.23. 2 using a P To make up for imperfections in the geometry and the short-range model potential, and to confine the error in the geometry to the specific molecule, we choose to scale for each solute the constant k in the short-range potential. We used the order parameters of the solutes in the 55wt% 1132/EBBA mixture and fixed the Fz to zero. A numerical , 3 fit was performed to obtain a perfect fit of the order parameters of propane to k and k 3 constant. The scaling and k for all other solutes were scaled keeping the ratio of k/k factors CSOL are shown in Table 5.23. We have tried to fit the 6 k’s, two Fzz’s (keeping one fixed), and the 7 quadrupole mo ments to the 21 order parameters from results in all three liquid crystals. Unfortunately, there are strong covariances in the variables and the fit gives unphysical results, such as negative k’s. Fixing more variables doesn’t result in a more realistic fit. The equations 3 equal are too correlated, since there is a sum of three terms in the potential. Setting k to zero makes the determinant of the problem bigger. The resulting short-range potential is the one proposed by Van der Est et al. [22]. Fixing the Fzz for 1132 and EBBA to 6.15 and -6.39 x 1O1 esu from [9], and fixing the quadrupole moments of benzene and HFB, the remaining molecular quadrupole moments were fitted to the order parameters and the results are shown in Table 5.23. From Table 5.23 it can be seen that the value of the quadrupole moment of TFB is fitted between that of benzene and HFB. This result is in agreement with the literature [83, 96, 97]. Also the value of the quadrupole moment for TCB is close to the value reported by Ritchie [97]. He obtained a. value of -3.24 for of TCB. Ethane is calculated to have a rather small value for the quadrupole moment. Although the absolute value calculated is much smaller than j-1.OOj as measured by Buckingham [98], the sign of the Chapter 5. Determination of Molecular Quadru pole Moments using Liquid Crystals 78 Table 5.23: Calculated Order Parameters and Quadrupole Moments Parameter 1132 c 1 S 13 PROPANE xx yy CLC MM EBBA CLC -0.03625 0.09516 if -0.05531 0.13250 if CLC 0.07853 1.0312 0.06378 1.1126 0.07011 0.9913 CLC -0.16123 0.9947 -0.27957 1.0329 -0.20145 0.9947 -0.08421 0.9446 CLC -0.18759 0.9446 CLC -3.96 0.9304 -0.18681 1.0329 -0.21338 0.99476 -0.31642 if -0.23762 if -0.29769 0.9446 TCB zz -8.69f 1.0529 HFB zz -0.07 1.1557 -0.27098 1.0329 TFB zz 0.65 -1.00 1.0021 BENZENE zz O if -0.03861 0.11038 if ETHANE zz CSOL S’ 9.50f 0.9763 -0.22516 if -5.62 Calculated order parameters for several solutes in three different liquid crystals. The experimen tal order parameters are given in Table 5.22. The calculated order parameters were obtained to a fit to the short-range and long-range orienting potential. The fitted k are 8.768 for 1132, 6.657 for the mixture, and 8.007 mN m 1 for EBBA. The Fz for the mixture was fitted to ’ esu. 1 esu = 2.998x10 1 O.061x10 6 Vm . t indicates that this value has been fixed during the 2 iteration. Chapter 5. Determination of Molecular Quadrupole Moments using Liquid Crystals 79 quadrupole moment °Z2 of ethane is negative. The component of the quadrupole tensor along the long axis of propane, is also calculated to be negative as is the case for ethane. The negative sign may not be too surprising considering the fact that methyl groups tend to be electron withdrawing, making the negative charges on the end of our alkanes contribute more to the quadrupole moments. 5.6 Conclusions of this chapter The work in this chapter has been a first attempt to extract quadrupole moments of small solutes dissolved in nematic liquid crystals with high electric field gradients. High correlations between parameters in our equations tend to overshadow a clear-cut method of obtaining reliable numbers for quadrupole moments. The data obtained in this chap ter suggest a contribution to the orientation of the molecular quadrupole moment of a nematogen consisting of an aromatic core with a flexible alkyl chain. Chapter 6 Conclusions To investigate the dynamics of a small solute molecule, the time dependence of the density operator for two coupled spin-i nuclei, such as dideuterium D 2 and dinitrogen 14 , under 2 N the second rank intra-molecular dipolar and the first rank spin-rotation Hamiltonians has been derived. It is shown that the relaxation rates for the ortho and para species are not identical. We report the 5 independent rates for the para and the 21 independent rates for the orho species. Due to cross-relaxation between spin-orders which precess at a common frequency the relaxation for the ortho configuration is not always single exponential. There are 15 cross-relaxation terms between certain spin-orders for the ortho species in the slow motion regime. The complexity of our coupled spin-system is discussed for the solid and the Jeener-Broekaert echo sequences. The Zeeman spin-lattice relaxation rates, , 10 for ortho- and para-dideuterium have R been measured experimentally in partially oriented nematic solvents at 304 K using HNMR. The solvents used are: PCH-7, 1132 and a 55 wt% 1i32/EBBA mixture. 2 These oriented solvents make it possible to separate the resonances of the ortho and para configurations thereby facilitating the R 10 determination directly without multi10 of para- is about twice that of ortho-dideuterium in all exponential analysis. The R 10 based on quadrupolar, dipolar, and experiments. Using our calculated expressions for R spin-rotation interactions, and performing the proper averaging over the rotational states, we obtain excellent agreement with our experimental results. We obtain a value for the 3 s for dideuterium at room temperature. Good agreement correlation time of 4.3xi0’ 80 Chapter 6. Conclusions 81 has been obtained with published low temperature studies as well. Nematogens are usually flexible and can therefore occur in several conformations. Larger solute molecules with a single bond of rotation have been investigated in a zeroelectric field gradient nematic mixture in order to understand the interpretation of mo tionally averaged spectra better with the goal of ultimately studying the nematogens themselves. The anisotropic couplings obtained from the ‘H-NMR spectra of partially oriented molecules with internal rotation are difficult to analyze because the order param eters for each conformer cannot be deduced without assumptions. One purpose of this thesis is to show how a model for the anisotropic short range potential can be useful in the interpretation of the high resolution ‘H-NMR spectra of the flexible molecules furfural, 2-chlorobenzaldehyde, and 2,2’-dithiophene dissolved in a 55% Merck ZLI 1132 and 45% N-(p-ethoxybenzylidene)-p’-n-butylaniline (w/w) mixture. In this nematic mixture there is no contribution to the anisotropic potential from the interaction between the solute molecular quadrupole moment and the average electric field gradient of the liquid crystal solvent; the dominant anisotropic intermolecular potential in this mixture is taken to result solely from short range interactions. We find good agreement between experiment and the order parameters calculated by the model of the short range interactions. The ratios of order parameters for the s-cis and s-trans conformers predicted by the model are used to calculate the statistical weights of the s-trans conformation for the solutes: at 304 K, ptraflS(furfural) pirans 0.45 ± 0.02, ph18(2chlorobenzaldehyde) 0.901 ± 0.006, and (2 ,2’-dithiophene) = 0.6094 ± 0.0008. For the analysis of 2,2’-dithiophene we were not able to obtain a good fit using pub lished geometries. We report a determination of the structure of 2,2’-dithiophene dis solved in different nematic liquid crystal solvents. The ‘H-NMR spectra of this molecule dissolved in a 55% Merck ZLI 1132 and 45% N-(4-ethoxybenzylidene)-4’-n-butylaniline Chapter 6. Conclusions 82 (w/w) mixture are recorded at 300 and 304 K and are analyzed. The dipolar coupling con stants obtained are used to show that the solute molecule exists in both the s-cis and the s-trans conformation. The s-trans conformer is planar and the absolute value of the dihe dral angle between the two thiophene ring planes for the s-cis conformer is found to be (24 ±1)°. Using the same method of analysis we reinterpreted two additional independent sets of published dipolar coupling constants from 1 H-NMR spectra of 2,2’-dithiophene dissolved in Merck Phase IV at room temperature and in 80% N-(4-ethoxybenzylidene)4’-n-butylaniline/20% O-carbobutoxy-4-oxybenzoic acid ethoxy phenyl ester at 301 K These additional studies yield the same results for the structure of 2,2’-dithiophene. Another aspect of liquid crystal molecules is that they usually have an aromatic core with an alkyl chain. Once we gained enough confidence in our short-range model po tential, the contribution of the quadrupole moment of the alkyl chain and the aromatic core to the nematogen was investigated using ethane, propane, and several substituted benzenes in three different nematic liquid crystals. 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NMR studies of solutes in nematic liquid crystals: understanding the nematogens Ter Beek, Leon Christiaan 1994
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Title | NMR studies of solutes in nematic liquid crystals: understanding the nematogens |
Creator |
Ter Beek, Leon Christiaan |
Date Issued | 1994 |
Description | NMR spectroscopy and dynamics of small solute molecules dissolved in several nematic liquid crystals provided insights toward a better understanding of nematogens. Expressions for all possible spin-relaxation rates for dideuterium are given. Exper- imentally obtained Zeeman relaxation rates in several liquid crystals were fitted to the expression for the Zeeman relaxation rate, and correlation times were thus obtained. From our study of the dynamics of dideuterium in nematic liquid crystals it was also found that ortho and para dideuterium have different relaxation rates due to the inter- ference of the dipolar and quadrupolar coupling constants and their spectral densities. The Zeeman relaxation rates of dideuterium show no liquid crystal solvent effect. Three different solute molecules with a single bond of rotation have been investigated in a special zero-electric field gradient nematic mixture: furfural, 2-chlorobenzaldehyde, and 2,2’-dithiophene. Dipolar coupling constants measured from NMR experiments are reported for these solutes and discussed in terms of the order parameters and conformer probabilities. In the course of the analysis a new geometry for 2,2’-dithiophene has been found. The spectrum of propane has been analyzed with the use of Multiple-Quantum NMR. Dipolar coupling constants of ethane, propane, benzene, 1 ,3,5-trifluorobenzene, hexaflu- orobenzene, and 1 ,3,5-trichlorobenzene as an internal reference for liquid crystal orient- tation, as solutes in three different nematic liquid crystals, were obtained from NMR spectra. The results are discussed in terms of orientation and solute—solvent interactions and especially the long-range molecular quadrupole moment—mean electric field gradi- ent interaction. A method for obtaining molecular quadrupole moments of solutes in nematic liquid crystals is discussed. The molecular quadrupole moment—mean electric field gradient interaction contributes to the orientation potential of these small alkanes and benzenes and will contribute to the orientation of alkyl chains and aromatic cores of nematogens as well. |
Extent | 2250202 bytes |
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Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-06-05 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0059644 |
URI | http://hdl.handle.net/2429/8845 |
Degree |
Doctor of Philosophy - PhD |
Program |
Chemistry |
Affiliation |
Science, Faculty of Chemistry, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1994-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
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