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Chemical physics and the condensed phase : NMR studies in a liquid-crystal testing ground Weber, Adrian C. J. 2010

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Chemical Physics and the Condensed Phase: NMR Studies in a Liquid-Crystal Testing Ground by Adrian C. J. Weber  B.Sc., University of Manitoba, 2005  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Chemistry)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2010 c Adrian C. J. Weber 2010  Abstract Liquid crystals are an excellent media for the study of the condensed phase by NMR spectroscopy since the highly accurate proton dipolar couplings do not average to zero as they do in the isotropic condensed phase. Of course we can also take the opposite view and seek to understand the behavior of individual molecules and the effect of the condensed phase on them and so the impetus for studies of solutes in liquid crystals is two fold. By coupling theory to experiment via dipolar couplings one can gain insight into aspects of chemical physics and the condensed phase provided the spectra can be solved. As the number of spins of a molecule and its lack of symmetry increase so do the complexity of NMR spectra of solutes in orientationally ordered phases. Covariance Matrix Adaptation Evolutionary Strategies (CMA-ES) have proven to be remarkably useful towards the end of obtaining dipolar couplings from congested spectra. In essence this algorithm uses the principles of natural selection coupled with an aspect of cross-generational memory to find the set of spectral parameters at the global minima of an error surface which reproduce the experimental spectrum. It is not an overstatement to say this tool has significantly altered the allocation of efforts in the area of research presented here. In the research herein two approaches are employed which are complimentary. In the first chapters we use a diversity of solutes to test postulated interaction Hamiltonians intended to describe the intermolecular environment of nematic and smectic A phases. The putative Hamiltonians are fitted to solute order parameters obtained from dipolar couplings. Once an explicit form is obtained, reasonable speculation is made concerning what the Hamiltonian can tell us about the intermolecular environment of the condensed phases studied. In the latter chapters the complimentary view is taken. Specifically we attempt to understand how internal rotations of molecules are affected by the condensed phase enviii  Abstract ronment. To this end is considered the simplest example in n-butane. Again by obtaining dipolar couplings we can use a variety of theoretical tools in an attempt to exploit the full accuracy of these anisotropic spectral parameters and gain insight into the effect of a condensed phase on configurational statistics. These phenomena are also studied as a function of temperature.  iii  Table of Contents Abstract  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  List of Tables  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vii  Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  x  Dedication  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xi  Statement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xii  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Nematic Liquid Crystals . . . . . . . . . . . . . . . . . 1.1.3 Other Liquid-Crystal Phases . . . . . . . . . . . . . . . 1.2 Anisotropic Intermolecular Interactions . . . . . . . . . . . . . 1.2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Probing the Potential . . . . . . . . . . . . . . . . . . . 1.3 Model Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Maier Saupe . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Chord . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Size and Shape . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Kobayashi McMillan . . . . . . . . . . . . . . . . . . . 1.4 NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 General . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Dipolar Couplings and Orientational Order Parameters 1.4.4 Covariance Matrix Adaptation Evolutionary Strategies 1.5 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Tables and Figures . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . .  1 1 1 1 2 3 3 5 6 6 7 8 10 10 10 11 13 14 16 18  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  22  . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . .  iv  Table of Contents 2 Solute Order Parameters: Application of MSMS-KM Theory . . . . . . 2.1 Tables and Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  24 31  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  36  3 The 3.1 3.2 3.3 3.4 3.5 3.6 3.7  . . . . . . . .  38 39 41 42 45 47 51 52  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  62  4 The Butane Condensed Matter Conformational Problem . . . . . . . . 4.1 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Tables and Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  64 74 74  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  83  5 Condensed Phase Configurational Statistics and Temperature Effects 5.1 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Tables and Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  85 89 89  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  94  6 Conclusion  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  95  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  98  Smectic Effect Rationalized by Introduction . . . . . . . . . . . . Experimental . . . . . . . . . . . . The Nematic Potential: MSMS . . The Smectic Potential: MSMS-KM Results and Discussion . . . . . . Conclusion . . . . . . . . . . . . . Tables and Figures . . . . . . . . .  MSMS-KM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  Appendices A Experimental detail concerning chapters 2 to 5 . . . . . . . . . . . . . . .  99  B Experimental order parameters . . . . . . . . . . . . . . . . . . . . . . . . .  100  C Structural details of n-butane calculated with Gaussian 03  106  . . . . . . .  v  List of Tables 3.1  Maier-Saupe solute parameters . . . . . . . . . . . . . . . . . . . . . . . . .  4.1  Spectral parameters (Hz). The numbers in brackets are the errors in the last one or two digits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fitting parameters for simultaneous fits to all four sets of dipolar couplings. The CCH angle was varied in the top set of calculations and held constant in the bottom set. The numbers in brackets are the errors in the last one or two digits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculated and experimental dipolar couplings . . . . . . . . . . . . . . . . RIS Chord. The numbers in brackets are the errors in the last digit. . . . . Calculated conformer order parameters using the RIS approximation for the Cd and CI(2k) models. The calculated order parameters shown are from fits to n-butane dipolar couplings when dissolved in the MM and while varying the CCH bond angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4.2  4.3 4.4 4.5  5.1  5.2 5.3  Experimental dipolar couplings of n-butane in 1132 as a function of temperature. The numbers in brackets are the errors in the last one or two digits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CCd fitting parameters as a function of temperature. The numbers in brackets are the errors in the last one or two digits. . . . . . . . . . . . . . . . . CI(2k) fitting parameters as a function of temperature. The numbers in the brackets are the errors in the last one or two digits. . . . . . . . . . . . . .  B.1 B.2 B.3 B.4 B.5 B.6  Experimental Experimental Experimental Experimental Experimental Experimental  order order order order order order  parameters parameters parameters parameters parameters parameters  of of of of of of  sample sample sample sample sample sample  1 2 3 1 2 3  in in in in in in  C.1 C.2 C.3 C.4 C.5  Atom labels of n-butane nuclei and bonding Bond lengths and angles of trans n-butane . Dihedral angles of trans n-butane . . . . . . Bond lengths and angles of gauche n-butane Dihedral angles of gauche n-butane . . . . . .  8OCB 8OCB 8OCB 8CB . 8CB . 8CB . . . . . .  . . . . .  . . . . .  . . . . .  53 75  77 78 79  80  90 91 92  . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  100 101 102 103 104 105  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  106 107 108 109 110  vi  List of Figures 1.1 1.2 1.3 2.1  2.2  2.3  2.4 3.1  3.2  3.3  Example liquid crystal structure of EBBA . . . . . . . . . . . Liquid-crystal phases running left to right and top to bottom; matic, smectic A, smectic C, chiral nematic, crystalline . . . . The first four generations of an ES taken from [32]. . . . . . .  . . . . . . . . isotropic, ne. . . . . . . . . . . . . . . .  The upper plot is of the experimental 400MHz NMR spectrum while the lower is found using the CMA-ES. The peaks of the solutes (from left to right: tcb, clpro, fur and thio shown above in the molecule fixed coordinate system) are intersperced with one another. . . . . . . . . . . . . . . . . . . . The asymmetry in the order parameters (R) for fur, thio and clpro are plotted against their respective Sxx . Although vibrational corrections have been neglected the error in R is small. An arrow marks the phase transition and the lines are the best fit to the points in the nematic phase. . . . . . . . . . G8CB,ZZ (1) is plotted against G8CB,ZZ (2) where the black points signify the use of the smectic Hamiltonian and the open points are obtained with the nematic potential only. The ten points closest to the origin are from measurements in the nematic phase while the rest are in the smectic phase. Inset: RMS of fits to the potentials in both phases are plotted against G8CB,ZZ (2). The smectic order parameters τLs are plotted against temperature for each of the solutes tcb, fur, thio and clpro. . . . . . . . . . . . . . . . . . . . . . The structures of solutes in the molecule fixed (with the exception of hex and clpro for which the z-axis lies along the C-C and Cl-Cl directions) frame where the z axis protrudes from the plane of the page. . . . . . . . . . . . . The asymmetry in the order parameters R (Eq. 3.1) of each solute is plotted against its Sxx . Although vibrational corrections have been neglected the error in R is small. The open points correspond to measurements in the liquid crystal 8CB while the filled points are from 8OCB. Nematic points are to the left and smectic to the right. . . . . . . . . . . . . . . . . . . . . . . . GL,ZZ (1) is plotted against GL,ZZ (2) where the filled points signify the use of the smectic Hamiltonian and the open points are obtained with the nematic potential Eq. 3.7 only. The seven (8CB) or eight (8OCB) points closest to the origin are from measurements in the nematic phase while the rest are in the smectic phase. Inset: The RMS of fits to either potential in both phases are plotted against GL,ZZ (2). . . . . . . . . . . . . . . . . . . . . . . . . . .  19 20 21  32  33  34 35  54  55  56  vii  List of Figures 3.4  3.5 3.6  3.7  3.8  4.1  4.2  4.3 5.1  bs,MSMS is plotted against temperature for pdcb and dcnb in 8OCB and 8CB. Points in the smectic phase are to the left and those in the nematic to the right. The open circles were obtained using GL,ZZ (i) from the nematic potential Eq. 3.7 and the filled circles used GL,ZZ (i) from the smectic potential Eq. 3.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The κ′L (i) are plotted with error bars versus temperature with error bars for each liquid-crystal solvent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . mod (i) The modulation of each nematic mechanism from smectic layering HN tot in 8CB at 298.0K and 8OCB at 327.0K (see Eq. 3.11) and their sum HN for tcb (θs =π/2) and hex (θs =0) where the plane of the ring and symmetry axes are aligned along the director, respectively. The centre of the layer is at the origin. The dashed line is the total smectic Hamiltonian HA,Ls (Z) (Eq. 3.11) for each solute’s given orientation. The Hamiltonian prefactors used are G8CB,ZZ (1)=0.952, κ′8CB (1)=0.456, G8CB,ZZ (2)=0.369, κ′8CB (2)=-3.145, G8OCB,ZZ (1)=0.911, κ′8OCB (1)=0.160, G8OCB,ZZ (2)=0.326, κ′8OCB (2)=-1.303, τ8CB tcb =-0.182, τ8OCB tcb =-0.417, τ8CB hex =-0.147, and τ8OCB hex =-0.182. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . mod (i) The modulation of each nematic mechanism from smectic layering HN tot in 8CB at 298.0K and 8OCB at 327.0K for pdcb where and their sum HN each molecule-fixed axis is, in turn, oriented along the director. The centre of the layer is at the origin. . . . . . . . . . . . . . . . . . . . . . . . . . . . The smectic order parameters τLs are plotted against temperature for all solutes with error bars in each liquid crystal solvent. The odcb in 8OCB and phac in 8CB τLs values are fixed. . . . . . . . . . . . . . . . . . . . . . . . .  57 58  59  60  61  Calculated a) and experimental b) NMR spectrum of n-butane in 1132 at 298.5K. c) and d) show an expanded region of the calculated and experimental spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Energies, calculated from Gaussian 03 for n-butane as a function of dihedral angle φ. The points are for n-butane in the gas phase. The dashed line for the gauche conformer is shifted due to the isotropic part of the intermolecular potential in the condensed phase when the orientational order is described by the CI(2k) model. The solid line for the gauche conformer is shifted downward and obtained when describing the orientational order with the gas CCd model. The Etg calculated by Gaussian 03 is 651 cal mol−1 . . . . . . The probability P (φ) of finding n-butane at dihedral angle φ for the CCd model (solid line) and the CI(2k) model (dashed line) in the MM at 301.4K.  82  The Etg of n-butane as a function of temperature when the orientational potential is described by the CI (open points) and Cd (filled points) models. gas The dashed line represents the constant Etg calculated from Gaussian 03. .  93  76  81  viii  Acronyms CMA-ES = Covariance matrix adaptation evolutionary strategies Cd = Modified chord model C = Circumference model CI = Circumference integration model DFT = Density functional theory ES = Evolutionary strategy EF G = Average electric field gradient GA = Genetic algorithm I = Integration model KM = Kobayashi McMillan L = Liquid crystal ME = Maximum entropy MS = Maier Saupe NMR = Nuclear magnetic resonance RIS = Rotational isomeric state RMS = Root mean square s = Solute  ix  Acknowledgements I would like to thank Elliott, Ron, Kees and Leo for all the valuable advice over coffee.  I would also like to thank my Parents, Abbey and C´eline, for instilling in me the value of dedicating oneself to producing works of quality.  x  To Andrea, my wife  And what is good, Phaedrus, And what is not good– Need we ask anyone to tell us these things?  Robert Pirsig  Qu’on ne dise pas que je n’ai rien dit de nouveau: la disposition des mati`eres est nouvelle.  Blaise Pascal  xi  Statement of Co-Authorship Chapter 2 is a co-authored publication of A. C. J. Weber, X. Yang, R. Y. Dong, W. L. Meerts and E. E. Burnell. The research programs for this chapter were designed by A. C. J. Weber, R. Y. Dong and E. E. Burnell with technical assistance from W. L. Meerts. The research and data analysis were performed by A. C. J. Weber and X. Yang with the paper being written by A. C. J. Weber. Chapter 3 is a co-authored publication of A. C. J. Weber, X. Yang, R. Y. Dong and E. E. Burnell who were involved in similar capacities as in Chapter 2. Chapter 4 is a co-authored publication of A. C. J. Weber, C. A. de Lange, W. L. Meerts and E. E. Burnell. The research programs for this chapter were designed by A. C. J. Weber and E. E. Burnell with input from C. A. de Lange and technical assistance from W. L. Meerts. The research and data analysis was performed by A. C. J. Weber with guidance and suggestions from E. E. Burnell and the manuscript written by A. C. J. Weber. Chapter 5 is a work in progress by A. C. J. Weber and E. E. Burnell.  xii  Chapter 1  Introduction 1.1 1.1.1  Liquid Crystals General  In simpler times the condensed phase was understood as a dichotomy whereby there could be liquids which flow easily and crystals that do not. The liquid, with no regularity in any direction was the anti-thesis of the crystal which has regularity in all spatial directions. In 1888 Reinitzer [1] found that a turbid liquid was formed when solid cholesteryl benzoate was melted and became a clear isotropic liquid upon further heating. When the turbid liquid was characterized by Lehmann [2] it was found to be birefringent and therefore anisotropic. These phases which are anisotropic and still exhibit some degree of fluidity are described as ‘liquid-crystalline’. So it seems as though the condensed phase was actually a trinity of isotropic liquids, anisotropic liquid crystals and solids but this, as it turns out, is not quite so. Specifically, certain organic materials do not show a single anisotropic phase between liquid and solid, but instead a cascade of transitions involving new phases. These various phases are a rich source of fascinating physical behavior and have been well documented over the years [3, 4].  1.1.2  Nematic Liquid Crystals  Liquid crystal molecules forming nematic phases, or nematogens, are typically elongated with semi-rigid cores and flexible alkyl chains towards the end(s), an example of which is seen in Fig. 1.1. Within this phase are found ‘domains’, of the order a million molecules each, where there is long-range orientational ordering that can extend over distances of up to 1µm [3]. This orientational ordering describes the tendency of a nematogen’s long axis  1  1.1. Liquid Crystals to lie along the domains shared axis known as the director. The director is defined as the unit vector, n, parralel to the direction in which the aggregate mesogens long axis in a given domain is pointed toward. In the absence of external fields the orientation of the directors will vary throughout the sample. Thus, the turbid appearance comes from the scattering of light as it propagates through the phase, the dimensions of the domains being on the order of the wavelength of visible light. If a magnetic or electric field is applied all the nematic directors will align parallel or perpendicular to the field. The direction of the director alignment depends on the magnetic or dielectric susceptibility anisotropy of the domain. Since the interaction energy for the susceptibility anisotropy coupled with the field is very small compared to the thermal energy, the magnetic or electric field has a negligible effect on the relative orientational ordering of the individual molecules [5]. Over the entire collection of molecules within a domain however, there is sufficient energy to cause alignment of directors. Since a strong magnetic field will not significantly influence the relative orientational ordering of molecules within a phase we can attempt to use Nuclear Magnetic Resonance (NMR) parameters to learn about the intermolecular forces that create this ordering by exploiting the interplay between theory and experiment. The directors of the liquid crystals studied here all align with the magnetic field. Since there is no positional ordering of the component molecules, the nematic phase is cylindrically symmetric about the nematic director. The probabilities of a molecule aligning parallel or anti-parallel with the laboratory fixed Z direction are equal and so the nematic phase is apolar.  1.1.3  Other Liquid-Crystal Phases  Some liquid crystals, often after sufficient cooling of the nematic phase, form layers whose planes are normal to the director. In this smectic A phase then there is, in addition to orientational order, positional order in one of the spatial dimensions. In some cases the director of each layer will be tilted from the normal of the layers plane and is called the smectic C phase. Chiral phases (often used in displays) have the layers staked such that each director is rotated in the plane with respect to those above and below so as to trace 2  1.2. Anisotropic Intermolecular Interactions out a helix in the direction normal to the layers plane. In Fig. 1.2 can be found schematics for these phases. There are many more interesting phases one can consider [6] but here the focus will be on the relatively simple nematic and smectic A phases.  1.2 1.2.1  Anisotropic Intermolecular Interactions General  Anisotropic intermolecular interactions are responsible for the orientational ordering of liquid-crystal phases. The interaction can be characterized as anisotropic short-range repulsive or as anisotropic long-range. Short-range repulsive interactions depend on the details of the molecular structure such as size, shape and flexibility. Long-range interactions involve dipoles, quadrupoles, polarizabilities and other properties that describe the distribution of charges over a molecule and can be either attractive or repulsive. The average of any single-molecule property X(Ω) over the orientations of all molecules is defined by X =  dΩX(Ω)f (Ω)  (1.1)  L where f (Ω) can be expanded in terms of Wigner rotation matrices [7], Dm ′ m (Ω), of rank L ∞  L  f (Ω) = L=0 m,m′ =−L  2L + 1 L aLm′ m Dm ′ m (Ω). 8π 2  (1.2)  L∗ (Ω) and integrating over the angles, it follows that the Multiplying both sides by Dm ′m  expansion coefficients aLm′ m are L∗ aLm′ m = Dm ′ m (Ω) .  (1.3)  The averages are the microscopic orientational order parameters. The orientational ordering of an inflexible molecule is completely described by the orientational distribution function f (Ω) where Ω denotes the Eulerian angles that describe the orientations of the molecular fixed axes relative to the nematic director which is parrallel to the magnetic field direction in 3  1.2. Anisotropic Intermolecular Interactions the experiments reported here. Since these molecules are in a fluid phase f (Ω) is an average over all molecular reorientations and f (Ω)dΩ is the probability of finding the molecule in a small solid angle dΩ at the direction defined by Ω. The f (Ω) and ultimately the molecular orientational order parameters are supposed to originate from an orientational pseudo-potential U (Ω) which is characterized by the short-range repulsive and long-range interactions exp f (Ω) = exp  −U (Ω) kT −U (Ω) kT  .  (1.4)  dΩ  U (Ω) is the potential of mean torque experienced by a single molecule and is defined by eq. 1.4 [6]. The potential of mean torque is responsible for making molecules preferentially align parallel to each other and to the director. Because of the apolar nature of the nematic phase any measured properties are therefore invariant to rotations about the nematic director and all odd components of f (Ω) are necessarily zero. In principle, all components of the function can be assessed by X-ray diffraction techniques [8]. Up to the fourth rank component of the distribution function can be determined from neutron diffraction techniques [3] but this components effect is small compared to the second order Legendre polynomial. In practice though these techniques prove difficult to do on account of poor resolution and instrumental limitations [3]. The average second rank component, identified as the second rank orientational order parameter, of f (Ω)dΩ is accessible via the analysis of NMR spectra. 2 ) is then the leading The second rank orientational order parameter (analagous to D0m  term in the expansion of the anisotropic components of f (Ω) and can be written as  Sαβ =  3 2  cos(θαZ ) cos(θβZ ) − 12 δαβ exp( −UkT(Ω) )dΩ exp( −UkT(Ω) )dΩ  (1.5)  where α and β represent the molecule fixed axes and θαZ is the angle between these axes and the laboratory Z direction which in the studies presented here is coincident with the magnetic field and director direction.  4  1.2. Anisotropic Intermolecular Interactions  1.2.2  Probing the Potential  An important way to learn about U (Ω) is to compare real experimental Sαβ ’s with ones calculated from theory, models or simulations. Orientational order parameters of the constituent molecules of a liquid crystalline phase are difficult to study because these molecules are normally devoid of symmetry and exist in a number of symmetrically unrelated conformers. A proper description would require a multitude of orientational parameters as well as conformer probabilities and so to proceed one must assume some model for the pair potential in order to relate experimental measurements to single-molecule properties. Whatever molecular properties are involved in the potential it has become clear that size and shape anisotropy plays a dominant role for molecules of sufficient size [9]. However the ordering of very small and highly symmetric molecules such as D2 and CH4 show there are other effects to be considered. When NMR spectra where obtained of D2 , HD and DT it was suprising to find a positive order parameter in some nematic phases and a negative one in others. What was also intriguing was that the experimental spectra involving deuterons could not be reproduced with an intramolecular electric field gradient alone. Specifically there was an extra contribution to the quadrupolar splittings in these NMR spectra which suggested the notion of an external average electric field gradient ( EF G ) [10, 11]. With these observations in mind the quadrupolar coupling constant of the deuteron was written as  Bobs = −  3eQD ¯ (FZZ − eqS) 4h  (1.6)  where eQD is the deuteron nuclear quadrupole moment, F¯ZZ is the ZZ component of the EF G parallel to the director of the cylindrically symmetric nematic phase, S is the order parameter of the cylindrically symmetric solute and eq is the average electric field gradient due to the intramolecular charge distribution around the deuteron nucleus. With the S determined from the dipolar coupling the F¯ZZ could be adjusted for a given liquid crystal sample in fits to Bobs . The fitted F¯ZZ [12] could then be used to calculate orientational order parameters assuming a solute quadrupole to liquid-crystal EF G interaction Hamiltonian 5  1.3. Model Potentials which then reproduced the observed order of isotopomer orientational order [9]. Further confirming the idea of liquid crystal phases possessing an EF G was the discovery of ‘magic mixtures’ (MM) of the liquid crystals ZLI-1132 (see [13] for chemical composition) and p-ethoxybenzylidene-p ′ -n-butylaniline (EBBA) possessing F¯ZZ ’s of opposite sign that resulted in essentially zero EF G and orientational order of hydrogen and its isotopomers [14]. It is also found that descriptions of orientational ordering using molecular size and shape anisotropy work best in the MM [9]. Methane is not expected to orient on account of its small size and tetrahedral symmetry. Yet this molecule does show significant orientational order according to NMR spectra. This seemingly strange result can be rationalized by realizing that there is a coupling between vibrational and orientational molecular motions [15]. One can easily see that there can be a variety of significant contributions to molecular orientational order and there is still much debate as to the form that would apply equally well to all orientationally ordered molecules. For many intents and purposes one can postulate a potential that is not inconsistent with the various putative underlying mechanisms but is ambiguous as to which one.  1.3 1.3.1  Model Potentials Maier Saupe  One persistently popular way to render complicated problems, like liquid crystals, tractable is the ‘mean field’ approximation. In the theory of Maier and Saupe (MS) a single-molecule potential is sought assuming each molecule is moving in a field generated by its interactions with all the surrounding molecules and that this field is independent of the degrees of freedom of every molecule except the one being considered. The theory also assumes the liquid-crystal molecules can be well approximated as axially symmetric rods. The physical basis of the anisotropic mean field does not have to be specified and it is assumed that the pair potential between two rod molecules is of the form  UA,B = UA,B (rAB , θA , φA , θB , φB )  (1.7) 6  1.3. Model Potentials where θ and φ are the polar and azimuthal angles that the rods make with the vector rAB joining their centres. To obtain the single-molecule potential of a molecule interacting with its environment the intermolecular forces potential is averaged over all degrees of freedom of the other molecules as well as over the translational degrees of freedom of the given molecule. Short-range orientational effects are neglected. The most important feature of the MS mean field is its dependence on molecular orientational order leading to the longrange order characteristic of the nematic phase. Maier and Saupe calculated the mean field seen by one particle among many and showed [16, 17] that for a potential that varies with angle as P2 (cos θ) it can be written as  HN (θ) = −νS  3 1 cos2 (θ) − 2 2  (1.8)  where ν is a scale parameter that indicates the strength of the rod-rod interaction and θ is the angle between the ‘rod’ long axis and the director. This form (which is identified with the leading term in the potential of mean torque) of the interaction can represent short-range size and shape effects or longer-range interactions such as that involving the liquid-crystal EF G interacting with the solute quadrupole or the solute polarizability with the mean square electric field although this list is not exhaustive. Although the MaierSaupe theory does not specify the exact physical nature of the intermolecular potential, its relative success demonstrates that second-rank interactions dominate.  1.3.2  Chord  An extension of the potential of mean torque is to assign a ‘potential of mean torque’ to each bond or group of atoms in a molecule. This assumes a molecule is built up of rigid subunits each independently interacting with the liquid-crystal mean field to produce an orientational torque [18, 19]. A draw-back of this model is that molecules with substantially different shapes, and therefore different orientational ordering, can have the same potential of mean torque [20]. To better account for short-range interaction, Photinos, Samulski, and co-workers have  7  1.3. Model Potentials extended the idea of a potential of mean torque to a ‘chord’ model which attempts to account for the size and shape of the molecules [20–22]. This mean field model for molecular orientation in a uniaxial phase is specially constructed for molecules comprised of repeating identical units like alkanes [20, 21] and is derived from the leading terms in a rigorous expansion of the mean-field interaction w ˜0 P2 (si , si ) + w ˜1 P2 (si , si+1 )  Un (Ω) = −  (1.9)  i=1  The si is a unit vector describing the orientation of the ith C-C bond of the hydrocarbon chain and the sum is over all bonds in the chain. The factors P2 (si , si+m ) are given by P2 (si , si+m ) =  The fitting parameters w ˜i =  3 2 Swi ,  3 1 cos(θZi ) cos(θZi+m ) − si · si+m 2 2  (1.10)  where S is the liquid-crystal order parameter. The  first term in the chord model corresponds to the independent alignment of separate C-C bonds while the second term incorporates correlations between adjacent bond orientations and therefore distinguishes between conformations that may have equal numbers of trans and gauche bonds but significantly different shapes. In other words it accounts for shapedependent excluded-volume interactions.  1.3.3  Size and Shape  Simulations have shown that nematic phases can be realized based on size and shape anisotropy interactions alone [9, 23]. This suggests a large role for size and shape contributions to the potential of a liquid-crystal environment. Often times a molecule is approximated by a collection of van der Waals spheres that are centered on the nuclei. It is the anisotropy in the shape of the molecule interacting with the uniaxial nematic field that gives rise to the orientational dependence of the potential energy. In one size and shape model the liquid crystal is taken to be an elastic tube that must stretch to accommodate a solute. The Z or long axis of the tube is coincident with the nematic director. One can  8  1.3. Model Potentials imagine a restoring force resulting from this stretching which can be described with Hooke’s law F = −kdC(Ω)  (1.11)  where C(Ω) is the circumference about the tube perimeter for the solute oriented at the angle Ω. The energy associated with this distortion is 1 U (Ω) = kC 2 (Ω) 2  (1.12)  and is referred to as the Circumference (C) model. Another size and shape model was inspired by the idea that short-range interactions are those between the solute surface and the liquid-crystal mean field. In this model the circumference CZ (Ω) is calculated at points along the length of the molecule in the Z direction and integrated along Z such that 1 U (Ω) = − ks 2  Zmax  CZ (Z, ω)dZ  (1.13)  Zmin  This is known as the Integration (I) model. The I and C models give roughly equally good fits to the experimental order parameters calculated from NMR experiments [9]. However, for longer solutes the C model tends to overestimate, and the I model underestimate, the experimental order parameters. Given this observation it is reasonable to suspect that the combination of these two models would better describe experimental order parameters and so we write the CI model as 1 1 U (ω) = k(C(ω))2 − ks 2 2  Zmax  C(Z, ω)dZ  (1.14)  Zmin  This potential can be thought of as describing both an elastic distortion of the liquid crystal and an anisotropic interaction between a liquid-crystal mean field and the solute surface [24]. Modelling short-range forces this way can also successfully fit order parameters from Monte Carlo simulations [25].  9  1.4. NMR  1.3.4  Kobayashi McMillan  In the smectic-A phase the liquid crystal field is periodically modulated along the Zdirection. The Kobayashi-McMillan (KM) [6] model is mean field and includes both this periodic modulation as well as a term that accounts for the change in nematic order arising from the coupling of the nematic and smectic-A order parameters. The Hamiltonian is then  HA = −τ ′ cos(  2πZ ) + HN d  1 + κ′ cos(  2πZ ) d  (1.15)  where HN is the nematic Hamiltonian from eq. 1.8, κ′ is the magnitude of smectic-nematic coupling and τ ′ scales the pure periodic smectic effect. Z maps the direction normal to the layer planes and d is the repeat distance of the smectic-A translational periodicity. All of the above models are employed in the studies presented here.  1.4 1.4.1  NMR Introduction  The NMR spectra of solutes in isotropic liquids familiar to most chemists are usually interpreted in terms of scalar quantities such as the chemical shieldings (σ) and the indirect spin-spin couplings (J). Although in ‘normal’ NMR these properties appear as scalars, they are in fact tensorial properties. The Brownian movement of the molecule and the resulting isotropic tumbling leads to a situation where only the isotropic part of the tensor is expressed and so the Hamiltonian of NMR in isotropic liquids is  H=−  hBZ 2π  γi (1 − σiiso )Ii,Z + i  hJijiso Ii · Ij i<j  (1.16)  = HZiso + HJiso The magnetic field is along the laboratory-fixed Z-axis, γi is the nuclear gyromagnetic ratio and the ‘iso’ superscripts indicate the isotropic part of the relevant tensor. The summation  10  1.4. NMR indices i and j run over all nuclear spins in the molecule. Often times the first term expressing the larmour frequency is expressed relative to a reference compound (usually tetramethylsilane) such that δi = (  ωi − ω T M S ) ∗ 10−6 ωT M S  (1.17)  is expressed in units of ppm so that comparison amoung magnets can easily be made and is know has the chemical shift [26]. No such reference is used in this thesis.  1.4.2  General  In uniaxial anisotropic liquids the second-order tensorial character of several molecular properties plays a central role and so the general Hamiltonian must be written in terms of them. In the high field limit, where the chemical shielding and the direct and indirect interactions are small compared to the Zeeman interaction the NMR spectrum of orientationally ordered molecules is described by the spin Hamiltonian [27]  H = HZ + HJ + HD + HQ .  Since all the spins observed here are I =  1 2  (1.18)  we can neglect the Quadrupolar Hamiltonian, HQ  being concerned with nuclei of I > 12 . The classical interaction energy between a magnetic field B and a magnetic moment µ leads to the interaction energy  W = −µ · B  (1.19)  causing the compass needle to point north when we turn in the Earths magnetic field. The magnetic moment of a nucleus is related to the spin operators via  µ=γ  h I. 2π  (1.20)  11  1.4. NMR The splitting of otherwise degenerate spin states by a magnetic field along the Z direction can then we written in operator form as  HZ = −γ  h IZ BZ . 2π  (1.21)  Of course the field at the site of the nucleus is not exactly that of the external field BZ due to the electron cloud surrounding a given nucleus and so we must add the effects of local chemical environments HZ = −  hBZ 2π  γi (1Z − σi,Z )Ii .  (1.22)  i  The indirect spin-spin coupling Hamiltonian in Cartesian tensorial form is written as  HJ = h  Ii,α Jij,αβ Ij,β  Ii · Jij · Ij = h i<j  (1.23)  i<j α,β  where Jij is the second-rank tensor which describes the bilinear coupling between spin vector Ii and Ij , the indices i and j number the nuclear spins, and α and β label the laboratory-fixed axes X,Y and Z. Jij,αβ involves five contributions from the diamagnetic and paramagnetic nuclear spin-electron orbit operators, the Fermi contact and spin-dipole interactions and a cross term of the latter two which are all well described in [28]. However, for protons the anisotropy in the indirect spin-spin coupling is very small [28, 29] and so this term in the NMR Hamiltonian is approximately given by  HJ = h  Jij Ii · IJ  (1.24)  i<j  where Jij is the isotropic scalar coupling constant between spins i and j. If one places a magnet close enough to another it is found that rotating one will cause rotation in the other and so they are said to be coupled. Quantum spin magnets are similarly coupled and this interaction can be written as hDij (3IiZ IjZ − Ii · Ij ).  HD =  (1.25)  i<j  12  1.4. NMR Dij is the dipolar coupling constant between spins i and j and is expressed as  Dij = −  γi γj µ0 −3 3 1 rij ( cos2 θij,Z − ) 2 8π 2 2  (1.26)  where rij is the internuclear separation and θij,Z is the angle between the ij and magnetic field directions, and the angle brackets denote an average over solute internal and reorientational motions. Now that the Hamiltonian giving rise to the NMR spectra of orientationally ordered solutes has been established let’s turn to the matter of extracting order parameters before dealing with the analysis of these spectra in section 1.4.4.  1.4.3  Dipolar Couplings and Orientational Order Parameters  The second rank orientational order parameters Sαβ ’s can be determined from the analysis of NMR spectral data of orientationally ordered molecules. Specifically, the dipolar couplings between spins i and j contain information on the relative orientation of the internuclear vector between i and j. If there are no large amplitude internal motions, the Sαβ ’s can be calculated using the relationship  Dij = −  γi γj µ0 8π 2  Sαβ αβ  cos θij,α cos θij,β 3 rij  (1.27)  and the program SHAPE [30] which can be used do a least squares fit to dipolar couplings while assuming a strucure so that the order parameters can be varied. Here θij,α is the angle between the molecule-fixed α axis and the vector joining nuclei i and j which have internuclear separation rij . In isotropic systems Sαβ = 0 and therefore Dij = 0. In partially ordered fluids however the Sαβ are non zero and expressed as  Sαβ =  1 3 cos θα,Z cos θβ,Z − δαβ 2 2  (1.28)  where the angle brackets denote an average over molecular reorientational motions and θα,Z is the angle between the molecule-fixed α axis and the space-fixed Z axis, which for the liquid crystals considered here lies along both the director and magnetic field directions. 13  1.4. NMR If one is dealing with large amplitude motions, like when a molecule is undergoing rapid interconversion between symmetry unrelated conformers, eq. 1.27 needs to be replaced with a sum over the conformers multiplied by their populations as we shall see in chapters 4 and 5. Of course it’s fine and dandy to have equations but one can’t bring home the proverbial bacon unless one has a way to solve them.  1.4.4  Covariance Matrix Adaptation Evolutionary Strategies  For molecules of few spins and sufficient symmetry one can, with a reasonable initial guess of spectral parameters, use least-squares and line assignments to solve anisotropic NMR spectra. For molecules with relatively more spins and lower symmetry one can use selective deuteration and multi-pulse techniques to get better estimates for solving the single quantum spectra where sufficient lines can be identified for assignment. With sufficient spins however, one reaches a limit to these traditional techniques since the transitions overlap to the extent that one cannot assign a sufficient amount to be over-determined. One then needs a new strategy for these instances and, as it turns out, this new strategy is more expedient even when traditional methods could be employed. Covariance Matrix Adaptation Evolutionary Strategies (CMA-ES) combine the concepts of the Genetic Algorithm (GA) with a crossgenerational memory feature and turns out to be well suited to the analysis of complicated NMR spectra. With the GA, rather than least-squares and line assignments, the driving forces toward the solution (at the global minimum of the error surface) are selective pressure coupled with mutation and reproduction. First, the liquid crystal background must be removed from the experimental spectrum since the algorithm iterates on intensity. A qubic base spline is a simple way to accomplish this and when there are not sufficient identifyable zero’s, a spectral subtration of the liquid-crystal spectrum without the solute can be employed at an appropriate reduced temperature found by trial and error. In the GA parameters determining the spectrum (chemical shifts, scalar couplings and direct couplings) are thought of as genes and a vector of all the relevant genes is called a chromosome. The object is to find the chromosome corresponding to the global error minimum. To do this one must 14  1.4. NMR first decide what the upper and lower limits to each gene will be. The multiple dimensional space spanned by the upper and lower limits of all parameters becomes the search space which hopefully contains the point corresponding to the solution. These population size, number of generations, location of the experimental spectrum, spectral parameters and thier ranges and relationships are specified in an input file which the alrorithm will execute. A population of chromosomes are generated with initial parameter values chosen randomly within the specified limits. Selective pressure is then applied by first realizing that the lists of intensities and frequencies that represent the calculated and experimental spectra are simply vectors. The extent to which calculated and experimental vectors overlap can then be calculated with the fitness function  Ff g = cos(θ) =  (f · g) f  g  (1.29)  where f and g are the calculated and experimental spectra, θ the angle between them and f  is the magnitude of a given vector. Each calulated spectrum is generated by inputing  the genes of a chromosome into the subroutine LEQUOR [31]. LEQUOR, using a suitable set of basis functions calculates the matrix elements of the Hamiltonian in eq .1.18. These eigenvalues are the energy levels of the molecule. Based on the selection rules and symmetry of the molecule all possible transition frequencies and their corresponding intensities are calculated. Typically half the population, which best matches the experimental spectrum (Ff g closest to 1), will be used to create the next generation in a mutative step-wise fashion. These are the features that comprise the GA which is very successful but for more congested spectra, to get an initial convergence, one needed to broaden the experimental spectrum so that the mutation and reproduction forces could drive successive generations out of local error minima. But a broadened spectrum gives less certainty in the values obtained and so the applied broadening would have to be cut to zero in a couple of tries obtaining more refined spectral parameters each time. This can be avoided by use of evolutionary strategies (ES) like in the CMA-ES. The first four generations of an ES are shown in Fig. 1.3 where each quadrant is num-  15  1.5. Outline of Thesis bered by generation. In the first quadrant an initial population is generated, and the best offspring is used as the next parent. The offspring are spread over a larger area in the second generation due to the relatively large step made in the previous generation. The vector from the parent to the best offspring (dashed line) is combined with the (shortened) mutation vector of the last generation (dotted line) to generate the new parent (solid line). In the third generation, due to the correlation between the past two mutations, the search range has been extended again in the general direction of both mutations while it has been limited in the perpendicular direction. The best offspring is now a local minimum. The memory effect of the EA, which incorporates past mutation vectors into the calculation of the next parent, helps to overcome the local minimum and the next parent is still closer to the global minimum. In the fourth generation the barrier between the local and global minima has been overcome, and the optimization is progressing toward the global minimum. The CMAES was used to solve the spectrum of orientationally ordered n-pentane [32], with around 20000 overlapping transitions that would not have been possible by conventional means, and is used in spectral analysis throughout this thesis.  1.5  Outline of Thesis  A good way to do science is through the interplay between theory and experiment. One cannot have confidence in a theory until it has withstood many attempts at falsification. Conversely a plethora of numbers from measurements is useless without some framework to connect the dots. Testing a theory against experiment doesn’t only render a verdict on the theory. The inadequacy of theory to predict observables can suggest the nature of the missing terms. Experiment can point to the necessary theory and new theory demands new experiments. The primary research objective of this thesis is twofold as the title suggests. One is to better understand the intermolecular environment of partially ordered condensed phases that, in principle, gives rise to such phases. While the understanding of the intermolecular forces within a liquid-crystal phase is not complete it has become clear that size and shape  16  1.5. Outline of Thesis effects are dominant. The second is to understand how condensed phases effect the chemical physics of individual molecules. Specifically an understanding of large amplitude conformational motions in the simplest alkane possessing one will be sought not only because such motions play important roles in many areas of science but also because they play an important role in mesogen behavior since they are a component part of these molecules. Chapters two and three focus on the former objective while chapters four and five focus on the latter. Along the way the bar has been raised in terms of our facility and ability to solve complicated and multiple spin-system spectra with ES much of which is not presented here [29, 32, 33]. The use of this algorithm in the NMR spectroscopy of molecules in partially ordered phases afforded many unique opportunities some of which is presented in what follows. Chapters two and three employ the use of solutes dissolved in liquid-crystals at low concentrations in order to probe the intermolecular environment of the nematic and smectic A phase. By varying the temperature and obtaining NMR spectral data, solute order parameters could be calculated on either side of the nematic to smectic A phase transition for a given liquid-crystal. Then by combining a recent application of Maier-Saupe theory, which postulates two second rank terms with one representing size and shape effects and the other representing electronic influences on orientational ordering, for mesogens with the Kobayashi-McMillan theory for smectics we were able to make reasonable approximations and subsequently show that this combined Hamiltonian could fit the observables. When incorporating the Kobayashi-McMillan theory of smectics with the double Maier-Saupe theory of nematics we allowed each nematic mechanism to be modulated by the smectic layering independently which turned out to be important in terms of the interesting results we in turn obtained. The explicit form of the Hamiltonian then allowed for the calculation of solute smectic order parameters and speculation and insight into the nature of the intermolecular forces at play. In Chapters four and five the object of study is n-butane, which is the simplest alkane that undergoes a large amplitude conformational change between trans and gauche states, dissolved in a host of nematic liquid crystals. In order to make progress one needs to 17  1.6. Tables and Figures assume an orientational potential in order the separate out the conformational statistics and calculate the order parameters of each conformer. Two different mean-field models are employed to this end whose relative effectiveness can be compared. One must also account for the internal motion which can be done with Flory’s RIS approximation or with a continuous potential as a function of dihedral angle calculated with Gaussian 03. Having calculated a gas phase trans-gauche energy difference for the internal motion one is in a position to distinguish the condensed phase’s contribution to the total Etg whose remainder gas is the Etg calculated from Gaussian 03. Of interest is the effect of the condensed phase  on the rotational trans-gauch energy difference, Etg , and the commensurate Boltzmann statistics obtained by fitting to the highly accurate and conformationally sensitive dipolar couplings which turns out to have a model dependency. This ten-spin system gives rise to a very complex and featureless spectrum which is comprised of well-resolved lines. Since the solving of such spectra has become routine with ES the role of thermal energy in the effect of a condensed phase on configurational statistics is also investigated with the hope of sorting out the observed model dependency.  1.6  Tables and Figures  18  1.6. Tables and Figures  O  CH N  Figure 1.1: Example liquid crystal structure of EBBA  19  1.6. Tables and Figures  Figure 1.2: Liquid-crystal phases running left to right and top to bottom; isotropic, nematic, smectic A, smectic C, chiral nematic, crystalline 20  1.6. Tables and Figures  Figure 1.3: The first four generations of an ES taken from [32].  21  Bibliography [1] F. Reinitzer, Monatsh 9, 421 (1888). [2] O. Lehmann, Z. Phys. Chem. 4, (1889). [3] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Oxford, Clarendon, 1993). [4] S. Chandrasekhar, Liquid Crystals (Cambridge, Cambridge University Press, 2003). [5] P. G. de Gennes, C. R. Acad. Sci. (Paris) B 274, 142 (1972). [6] R. Y. Dong, Nuclear Magnetic Resonance of Liquid Crystals (New York, SpringerVerlag, 2nd edition, 1994). [7] R. N. Zare, Angular Momentum Understanding Spatial Aspects in Chemistry and Physics (Toronto, John Wiley Sons, 1988). [8] M. Kohli, K. Otness, R. Pynn and T. Riste, Z. Phys. 24B, 147 (1976). [9] E. E. Burnell and C. A. de Lange, Chem. Rev. (Washington, D.C.) 98, 2359 (1998). [10] G. N. Patey, E. E. Burnell, J. G. Snijders and C. A. de Lange, Chem. Phys. Lett. 99, 271 (1983). [11] J. G. Snijders, C. A. de Lange and E. E. Burnell, Israel J. Chem. 23, 269 (1983). [12] J. B. S. Barnhoorn and C. A. de Lange, Mol. Phys. 82, 651 (1994). [13] J. Lounila and J. Jokisaari, Prog. NMR spectrosc. 15, 249 (1982). [14] P. B. Barker, A. J. van der Est, E. E. Burnell, G. N. Patey, C. A. de Lange and J. G. Snijders, Chem. Phys. Lett. 107, 426 (1984). [15] E. E. Burnell and C. A. de Lange, Chem. Phys. Lett. 486, 21 (2010). [16] W. Maier and A. Saupe, Z. Naturforsch. A 14, 882 (1959). [17] W. Maier and A. Saupe, Z. Naturforsch. A 15, 287 (1960). [18] C. J. R. Counsell, J. W. Emsley, N. J. Heaton and G. R. Luckhurst, Mol. Phys. 54, 847 (1985). [19] C. J. R. Counsell, J. W. Emsley, G. R. Luckhurst and H. S. Sachdev, Mol. Phys. 63, 33 (1988). [20] D. J. Photinos, E. T. Samulski and H. Toriumi, J. Phys. Chem. 94, 4688 (1990). 22  Chapter 1. Bibliography [21] D. J. Photinos, E. T. Samulski and H. Toriumi, J. Phys. Chem. 94, 4694 (1990). [22] D. J. Photinos, E. T. Samulski and H. Toriumi, Mol. Cryst. Liq. Cryst. 204, 161 (1991). [23] M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford, Clarendon, 1987). [24] D. S. Zimmerman and E. E. Burnell, Mol. Phys. 78, 687 (1993). [25] R. T. Syvitski, J. M. Polson and E. E. Burnell, Int. J. Mod. Phys. 10, 403 (1999). [26] H. L. Malcolm, Spin Dynamics Basics of Nuclear Magnetic Resonance (Toronto, John Wiley Sons, 2001). [27] E. E. Burnell and C. A. de Lange (Eds.), NMR of Ordered Liquids (Dordrecht, The Netherlands, Kluwer Academic, 2003). [28] J. Vaara, J. Jokisaari, R. E. Wasylishen and D. L. Bryce, Progr. Nucl. Magn. Reson. 41, 233 (2002). [29] C. A. de Lange, W. L. Meerts, A. C. J. Weber and E. E. Burnell, J. Phys. Chem. A 114, 5878 (2010). [30] P. Diehl, P. M. Henrichs and W. Niederberger, Molec. Phys. 20, 139 (1971). [31] P. Diehl, H. Kellerhals and W. Niederberger, J. Magn. Reson. 4, 352 (1971). [32] W. L. Meerts, C. A. de Lange, A. C. J. Weber and E. E. Burnell, J. Chem. Phys. 130, 044504 (2009). [33] W. L. Meerts, C. A. de Lange, A. C. J. Weber and E. E. Burnell, Chem. Phys. Lett. 441, 342 (2007).  23  Chapter 2  Solute Order Parameters: Application of MSMS-KM Theory. ∗ We obtain dipolar couplings via a novel application of evolutionary algorithms solving for multiple spin-systems simultaneously and automatically from the NMR spectra of several solutes in several nematic and smectic liquid crystal solvents. The order parameters obtained from the dipolar couplings are used to test a novel Hamiltonian that includes two Maier-Saupe nematic terms plus Kobayashi-McMillan smectic A terms. It is shown that this Hamiltonian can rationalize the NMR experiments with physically reasonable smectic order parameters and Hamiltonian prefactors.  ∗  A version of this chapter has been published. A. C. J. Weber, X. Yang, R. Y. Dong, W. L. Meerts, and E. E. Burnell (2009) Solute Order Parameters in Liquid Crystals from NMR Spectra Solved with Evolutionary Algorithms: Application of Double Maier-Saupe Kobayashi-McMillan Theory. Chem. Phys. Lett. 476, 116  24  Chapter 2. Solute Order Parameters: Application of MSMS-KM Theory Two limits to the study of condensed matter by NMR are the ability to solve proton spectra and the availability of a Hamiltonian that describes the intermolecular interactions well. This is especially the case in the study of ordered fluids where the spectra of solute molecules are dominated by dipolar couplings [1]. The prospect of automated analysis of such spectra has been investigated with considerable success [2]. In later developments evolutionary algorithms (EAs) were found to yield solutions of moderately complicated molecules but often required operator intervention so as to avoid false minima [3, 4]. A later application of the genetic algorithm proved to be more robust, avoiding false minima without operator intervention [5]. More recently the use of covariance matrix adaptation evolutionary strategies (CMA-ES) [6] have made it possible to solve very complicated spectra (such as oriented pentane with many of its roughly 20000 transitions overlapping) that would not have been otherwise possible by conventional means [7]. An interesting property of liquid crystals is the orientational ordering of molecules that make up the nematic phase and their positional ordering in the smectic A phase [8]. Nematic liquid crystals have uniaxial orientational order along an average direction called the director. For many purposes these molecules can be approximated by axially symmetric rods that would then have a single nematic order parameter SL . If the nematic phase were to be perfectly ordered with all rods aligned along the director an SL value of 1 is obtained whereas if it were heated up to the isotropic phase the SL would become 0. The smectic A phase has positional as well as orientational order. More specifically liquid-crystal molecules will arrange, on average, into layers whose planes are perpendicular to the director. Kobayashi-McMillan theory provides a way to account for positional order with the use of additional order parameters [9, 10]. One of these will be τL which is zero if the centers of molecules are spread evenly across layers (as in a nematic) and 1 if they occupy a single position in the centre of the layer. But SL , the liquid crystal orientational order parameter, can change as the molecule goes deeper into the layer so there must be a nematic-smectic A coupling which will have an orientational-translational order parameter called κL . This theory has been applied in previous studies, one of which used DFT methods for the solute-solvent potential [11]. It has also been used in the context of solutes as probes 25  Chapter 2. Solute Order Parameters: Application of MSMS-KM Theory of the liquid crystal environment, but many assumptions were necessary. While the attempt was to test the Kobayashi-McMillan (KM) theory, the application was ironically hampered by an inadequate description of the nematic phase [12–14]. There is a wealth of literature dealing with the kinds of intermolecular forces that solutes experience when dissolved in liquid crystals [1, 15–20]. There is still much disagreement as to which effects dominate the nematic potential. However, there is compelling evidence to support a large role for size and shape effects [15]. More recently it has been shown that the use of two independent mean-field Maier-Saupe terms (MSMS) in the nematic potential can fit experimental order parameters to better than 5% [21]. In what follows we utilize advances provided by CMA-ES in spectral analysis along with a physically reasonable Hamiltonian that combines the MSMS and KM theory to rationalize NMR observables of solutes dissolved in smectic and nematic phases. The CMA-ES uses the principles of natural selection to evaluate potential solutions and the history of change in previous generations (a group of potential solutions) to guide a present generation towards a global minimum of an error surface. The solution at this minimum can be thought of as the NMR parameters (chemical shifts, scalar couplings and direct couplings) which best replicate the experimental NMR spectrum. First, an operator must choose reasonable upper and lower limits for each parameter. This will define the search or parameter space and one hopes that the correct solution is a point in this multidimensional space. A population of parameter sets are chosen randomly within the defined limits. Calculated spectra are generated from each member of the population and can be thought of as a vector f just as the list of intensities and frequencies of the experimental spectrum can be thought of as a vector g. Then by defining the Fitness function:  Ff g =  (f · g) f g  (2.1)  a number of the best parameter sets, with Ff g closest to 1, can be chosen as the next generation parents. This number is chosen by the operator and is typically 50% of the  26  Chapter 2. Solute Order Parameters: Application of MSMS-KM Theory population size. A population will be created from this parent in a mutative step-size fashion. The offspring of this next generation will be spread out over a larger region of the parameter space due to the movement from selection in the previous generation. A new ‘most fit’ solution will be selected according to Ff g but the next parents location in parameter space will be a weighted sum of the last two movements. This memory effect of the evolutionary algorithm, which uses past mutation vectors coupled with natural selection to calculate the parents of a next generation, helps to overcome local minima and move closer to the global error minimum until convergence is reached. The CMA-ES approach was quite successful (see fig. 2.1) in the current study of furan (fur), thiophene (thio), 2,2dichloropropane (clpro) and 1,3,5-trichlorobenzene (tcb) dissolved in the liquid crystal 4-noctyl-4′ -cyanobiphenyl (8CB) that forms both a nematic and a lower temperature smectic A phase. While the spectra are not as congested as the spectrum of oriented pentane [7], the parameter space is still comparably large as there are four independent parameter sets. Specifically, the pentane parameter space was defined by 11 dipolar couplings (Dij ), 3 chemical shifts (ωi ) and 9 indirect couplings (Jij ) while that of the combined solutes is composed of 11 Dij , 6 ωi and 3 scaling parameters that scale the relative intensities of solutes. The result is impressive since all solutes are solved for simultaneously and automatically. This achievement will be crucial for future more elaborate studies of solutes in the nematic and smectic phases of a given liquid crystal. All the solutes were dissolved in roughly equal amounts in 8CB for a total mole fraction of about 2% so that interactions amongst solutes can be ignored [22]. It is important that solutes be dissolved in the same sample tube (as was done here) so that they experience the same environment. Spectra were collected in 0.5 or 1K steps on both sides of the nematicsmectic A transition. The resulting dipolar couplings obtained from CMA-ES analysis were then used to calculate solute order parameters using a modified version of the computer program SHAPE [23] and structures from the literature [24–27]. One can see that the order parameters are affected by the smectic environment when plotting the solute order tensor asymmetry  27  Chapter 2. Solute Order Parameters: Application of MSMS-KM Theory  R = (Sxx − Syy )/Szz  (2.2)  versus solute Sxx and noting the change in slope at the phase transition as shown in fig. 3.2. The MSMS nematic potential for solutes with two or fewer independent order parameters can be written as:  HN,Ls (Ωs ) = −  3 4  2  GL,ZZ (i)βs,zz (i) × i=1  3 1 cos2 (θs ) − 2 2  +  bs (i) sin2 (θs ) cos(2φs ) 2  = H M S1 + H M S2 (2.3) where bs (i) =  βs,xx (i) − βs,yy (i) βs,zz (i)  (2.4)  Consistent with Maier-Saupe theory, the GL,ZZ (1) and GL,ZZ (2) are taken to be mean-field properties of the liquid crystal that interact with some solute properties which are denoted by the βs,γγ (1) and βs,γγ (2). The index γ runs over the molecule fixed x, y and z axes. Given the assumptions made in previous studies using Kobayashi-McMillan theory [12, 13] and the recent success of the MSMS potential in dealing with nematics, we propose the Hamiltonian in a smectic liquid crystal to be:  HA,Ls (Ωs , Z) = HM S1 1 + κ′L (1) cos ′ − τLs cos  2πZ d  2πZ d  + HM S2 1 + κ′L (2) cos  2πZ d  (2.5)  where κ′L (1) and κ′L (2) are the nematic-smectic coupling Hamiltonian prefactors, one to modulate each of the two nematic ordering mechanisms as we move across a layer, and it is ′ (the solute smectic reasonable to take them as a liquid-crystal property. There is one τLs  prefactor) for each solute in a given liquid-crystal and d is the width of the layer while Z  28  Chapter 2. Solute Order Parameters: Application of MSMS-KM Theory (Z=0 in the centre of a layer) maps the direction parallel to the director which lies along the magnetic field direction in the experiments reported here. Because the βs,γγ (i) are solute properties, and hence not functions of temperature or liquid-crystal solvent, their values are first obtained by fitting to order parameters of several molecules (including the ones of interest) in five different nematic liquid crystals with the MSMS potential. In particular we have used the liquid crystals: Merck ZLI-1132 (1132), a ‘magic mixture’ of 55wt% 1132 and 45wt% p-ethoxybenzylidene-p′ -n-butylaniline (EBBA), EBBA and both 8CB and 4-n-octyloxy-4′ -cyanobiphenyl (8OCB) at three different temperatures spanning the nematic phase. The analysis is carried out in the same manner as in [21] using the relationship:  Ss,γγ =  dΩs ( 32 cos2 (θs,γ ) − 21 )e dΩs e  −HN,Ls (Ωs ) kB T  (2.6)  −HN,Ls (Ωs ) kB T  With the βs,γγ (i)’s now fixed we proceed by fitting the experimental order parameters for the solutes of interest at each temperature in both the nematic and smectic phases of 8CB, using the nematic potential only by varying GL,ZZ (1) and GL,ZZ (2), and notice that the RMS of the fit increases significantly in the smectic phase (open circles in inset to fig. 2.3).We then attempt to fit the Ss,γγ in the smectic A phase to the combined MSMS′ ) with the KM Hamiltonian of Eq. (2.5) varying all free parameters (GL,ZZ (i),κ′L (i) and τLs  expression: Ss,γγ =  dΩs  d 3 2 0 ( 2 cos (θs,γ ) d dΩs 0 e  − 21 )e  −HA,Ls (Ωs ,Z) kB T  −HA,Ls (Ωs ,Z) kB T  dZ  (2.7)  dZ  to find that it does not converge and so are forced to make an assumption. In previous work, in order to be overdetermined so as to solve for smectic Hamiltonian prefactors, we linearly extrapolated the interaction energy from the nematic into the smectic phase once for each solute (where some solutes showed some curvature) and used the extrapolation to estimate the smectic effect. This crude assumption has been circumvented by the MSMS theory. Here we make the single assumption that GL,ZZ (1) is linearly related to GL,ZZ (2). Hence, we extrapolate once for the liquid crystal by drawing a line through the nematic 29  Chapter 2. Solute Order Parameters: Application of MSMS-KM Theory points of GL,ZZ (1) versus GL,ZZ (2) and forcing this relationship into the smectic phase as seen in fig. 2.3. Under these circumstances we now vary GL,ZZ (2) (from which GL,ZZ (1) is ′ for each solute to find that we calculated), κ′L (1) and κ′L (2) for the liquid crystal and a τLs  obtain excellent fits (see inset of fig. 2.3). With the smectic Hamiltonian in hand we go on to calculate the solute smectic order parameters τLs using:  τLs =  dΩs  d 2πZ 0 cos( d )e  dΩs  d 0 e  −HA,Ls (Ωs ,Z) kB T  −HA,Ls (Ωs ,Z) kB T  dZ  (2.8)  dZ  the results of which are shown in fig. 3.8. The overall trend is satisfying in that the magnitude of τLs (with the exception of fur) becomes smaller with increasing temperature as would be expected when approaching the nematic phase. A negative value of τLs implies a preference for the interlayer region while a positive value indicates a preference to the centre of a layer. The closer the τLs of either sign is to zero the less pronounced is its partitioning, with τLs = 0 meaning positional isotropy. It is interesting to note that clpro has the largest negative τLs which means it displays the most partitioning to the interlayer region, as expected because the size and shape of clpro is fairly well approximated by a sphere. Fur and thio show less partitioning, but exhibit opposite trends with changing temperature. In particular fur tends to partition more to the interlayer region upon increasing temperature. Turning now to the liquid crystal prefactors we note that the GL,ZZ (2) calculated with the smectic potential of Eq. (2.5) is smaller than that calculated with only the MSMS nematic potential of Eq. (3.7) for fits in the smectic phase as seen in fig. 2.3. This shows that the nematic potential was accounting for the extra order of the smectic phase which was later picked up in a more appropriate way in the smectic potential. We obtain a κ′L (1) value of 0.38 which is constant and in good agreement with our previous studies. However, previously we had only one coupling prefactor which modulated all nematic mechanisms that were brought together as one term. With these restrictions now relaxed it is interesting to find a constant κ′L (2) value of -2.5, greater than 1 in absolute value. In other words, the second MS mechanism in Eq. (2.5) undergoes a sign change as a function of Z due to the 30  2.1. Tables and Figures nematic-smectic coupling in 8CB. This seems to imply that there are planes along Z where the second MS mechanism imposes no order and that there would be isotropy here were it not for the dominance of the first MS mechanism. In summary, we have shown that an MSMS-KM potential can be used to obtain valuable information on solutes in the smectic phase in addition to some clues regarding the phase itself. These matters clearly merit further, more elaborate investigation which are much more manageable now that we can quickly solve spectra of multiple solutes automatically and simultaneously in liquid crystals using Evolutionary Algorithms.  2.1  Tables and Figures  31  2.1. Tables and Figures  Figure 2.1: The upper plot is of the experimental 400MHz NMR spectrum while the lower is found using the CMA-ES. The peaks of the solutes (from left to right: tcb, clpro, fur and thio shown above in the molecule fixed coordinate system) are intersperced with one another.  32  2.1. Tables and Figures  Figure 2.2: The asymmetry in the order parameters (R) for fur, thio and clpro are plotted against their respective Sxx . Although vibrational corrections have been neglected the error in R is small. An arrow marks the phase transition and the lines are the best fit to the points in the nematic phase.  33  2.1. Tables and Figures  Figure 2.3: G8CB,ZZ (1) is plotted against G8CB,ZZ (2) where the black points signify the use of the smectic Hamiltonian and the open points are obtained with the nematic potential only. The ten points closest to the origin are from measurements in the nematic phase while the rest are in the smectic phase. Inset: RMS of fits to the potentials in both phases are plotted against G8CB,ZZ (2).  34  2.1. Tables and Figures  Figure 2.4: The smectic order parameters τLs are plotted against temperature for each of the solutes tcb, fur, thio and clpro.  35  Bibliography [1] E. E. Burnell and C. A. de Lange (Eds.), NMR of Ordered Liquids (Dordrecht, The Netherlands, Kluwer Academic, 2003). [2] F. Castiglione, G. Celebre, G. De Luca and M. Longeri, J. Magn. Reson. 142, 216 (2000). [3] H. Takeuchi, K. Inoue, Y. Ando and S. Konaka, Chem. Lett. 11, 1300 (2000). [4] K. Inoue, H. Takeuchi and S. Konaka, J. Phys. Chem. A 105, 6711 (2001). [5] W. L. Meerts, C. A. de Lange, A. C. J. Weber and E. E. Burnell, Chem. Phys. Lett. 441, 342 (2007). [6] N. Hansen and A. Ostermeier, Evolut. Comput. 9(2), 159 (2001). [7] W. L. Meerts, C. A. de Lange, A. C. J. Weber and E. E. Burnell, J. Chem. Phys. 130, 044504 (2009). [8] R. Y. Dong, Nuclear Magnetic Resonance of Liquid Crystals (New York, SpringerVerlag, 2nd edition, 1994). [9] K. K. Kobayashi, Mol. Cryst. Liq. Cryst. 13, 137 (1971). [10] W. L. McMillan, Phys. Rev. A 4, 1238 (1971). [11] G. Celebre, G. Cinacchi and G. De Luca, J. Chem. Phys. 129, 094509 (2008). [12] A. Yethiraj, Z. Sun, R. Y. Dong and E. E. Burnell, Chem. Phys. Lett. 398, 517 (2004), A typographical sign error in the definition of P2,asymm following Eq. (4) is corrected in Eq. (5) of the present paper; b in Fig. 2 should be replaced by −b. [13] A. Yethiraj, A. C. J. Weber, R. Y. Dong and E. E. Burnell, J. Phys. Chem. 111, 1632 (2007). [14] A. Yethiraj, E. E. Burnell and R. Y. Dong, Chem. Phys. Lett. 441, 245 (2007) [15] E. E. Burnell and C. A. de Lange, Chemical Reviews 98, 2359 (1998), and references therein. [16] A. di Matteo, A. Ferrarini and G. J. Moro, J. Phys. Chem. B 104, 7764 (2000). [17] A. Ferrarini and G. J. Moro, J. Chem. Phys. 114, 596 (2001). [18] A. di Matteo and A. Ferrarini, J. Phys. Chem. B 105, 2837 (2001).  36  Chapter 2. Bibliography [19] G. Celebre and G. De Luca, Chem. Phys. Lett. 368, 359 (2003). [20] G. Celebre, J. Phys. Chem. B 111, 2565 (2007). [21] E. E. Burnell, L. C. ter Beek and Z. Sun, J. Chem. Phys. 128, 164901 (2008). [22] G. S. Bates, E. E. Burnell, G. L. Hoatson, P. Palffy-Muhoray and A. Weaver, Chem. Phys. Lett. 134, 161 (1986). [23] P. Diehl, P. M. Henrichs and W. Niederberger, Molec. Phys. 20, 139 (1971). [24] B. Bak, D. Christensen, W. B. Dixon, L. Hansen-Nygaard, J. Rastrup-Andersen and M. Schottlander, J. Mol. Spectrosc. 9, 124 (1962). [25] B. Bak, D. Christensen, L. Hansen-Nygaard and J. Rastrup-Andersen, J. Mol. Spectrosc. 7, 58 (1961). [26] A. Almenningen, I. Hargittai, J. Brunvoll, A. Domenicano and S. Samdal, J. Mol. Struct. 116, 199 (1984). [27] M. Hirota, T. Iijima and M. Kimura, Bull. Chem. Soc. Jpn. 51, 1589 (1978).  37  Chapter 3  The Smectic Effect Rationalized by MSMS-KM Theory. ∗ From the dipolar couplings obtained by NMR spectroscopy we have calculated the order parameters of a wide variety of solutes in the nematic and smectic A phases of the liquid crystals 8CB and 8OCB. These measurements are then rationalized with the previously tested two Maier-Saupe Kobayashi-McMillan Hamiltonian from which smectic order parameters are calculated.  ∗  A version of this chapter has been published. A. C. J. Weber, X. Yang, R. Y. Dong and E. E. Burnell (2010) The Smectic Effect on Solute Order Parameters Rationalized by Double Maier-Saupe KobayashiMcMillan Theory. J. Chem. Phys. 132, 034503  38  3.1. Introduction  3.1  Introduction  Over the years there has been significant effort put forth towards understanding the properties of liquid crystal phases [1, 2] and the behaviour of molecules dissolved therein [3]. Of particular interest is the smectic A phase where the liquid crystal molecules arrange, on average, into layers [4]. That is the smectic A phase has, in addition to the orientational order observed in the nematic phase where the long axis of liquid crystal molecules point on average along a common axis called the director, positional ordering of layers whose normal lies parallel to this director. The extent of orientational order can be quantified with the so called orientational order parameters. To understand this parametrization it is useful to consider the idealization of a liquid crystal molecule as an axially symmetric rod as in the theory of Maier and Saupe (MS) [5, 6]. In this reasonable approximation the liquid crystal order parameter SL would be 1 if all molecules were aligned perfectly along the director, -0.5 if all were perpendicular and 0 if heated to the isotropic phase where all molecules tumble randomly. This measurable property is defined as the average of the second Legendre polynomial. Much effort has been devoted to elucidating the exact nature of the nematic ordering mechanisms of solutes [3, 7–25] but a clear consensus has yet to be reached. However, it does seem clear that there is a large role for size and shape effects especially in the so called ‘magic mixture’s (MM) where molecular dideuterium experiences an average electric field gradient of zero at 301.4K [20]. To quantify the extent of smectic ordering one needs to invoke additional order parameters. The most pragmatic such formulation to this end is that of Kobayashi and McMillan (KM) [26, 27]. The first of these parameters is the ‘pure smectic’ liquid-crystal L order parameter τL which is 0 when a molecule is spread evenly across layers, 1 if it occupies the centre of the layers only and -1 (for the case of a solute) if completely restricted to the interlayer region. Also, the orientational ordering potential exerted on liquid crystal molecules can change across a layer, and the second parameter in KM theory accounts for this change via a nematic-smectic coupling which gives κL an orientational-translational order parameter.  39  3.1. Introduction A variety of techniques have been employed to measure the positional order of smectics ranging from X ray [28] to static field gradient NMR diffusometry [29–31]. There have been previous attempts at using the KM theory with solutes, one of which used Density Functional Theory methods to describe the solute-solvent potential [32]. Other studies of smectics focused on the NMR of solutes as probes of the intermolecular environment [33–35]. In these solute studies it was necessary to extrapolate the nematic interaction energy asymmetry for each solute in each liquid crystal into the smectic phase so that smectic information could be obtained. A problem was the observation that the points used for extrapolation showed curvature for some of the solutes. In addition the interaction energy asymmetry changed with temperature, suggesting that there is more than a single orientational mechanism: however, it was assumed that the various interactions could be brought together into a single MS term. In later investigations of nematic phases it was shown that using two MS mechanisms (termed double Maier-Saupe theory, MSMS) to describe the nematic environment could predict solute order parameters to better than 5% [36]. As it turns out much of the difficulty in the previous smectic work on solutes was due to an inadequate description of the nematic phase rather than any shortcomings of the KM theory [37]. By bringing together the MSMS theory of nematics and the KM theory of smectics it became possible to rationalize NMR measurements of orientational order parameters using a single assumption, that the two MS mechanisms change the same way with respect to each other in both the smectic and nematics phases. This assumption gave physically reasonable smectic order parameters. This preliminary success clearly warranted further testing of the MSMS-KM potential. To this end in what follows we show the details of such experiments. Specifically we have greatly expanded our array of probe molecules from four to twelve solutes which comprise a diversity of size, shape and electronic configurations. In addition we have carried out these studies in a second liquid crystal which should be a check against obtaining fortuitous results. With these efforts it is our hope to more rigorously and comprehensively test the value and validity of the MSMS-KM theory in describing the intermolecular environment of the smectic phase.  40  3.2. Experimental  3.2  Experimental  The solute molecules (see figure fig. 3.1 for structures and coordinate system) ortho-dichlorobenzene (odcb), meta-dichlorobenzene (mdcb), para-dichlorobenzene (pdcb), 1,3,5-trichlorobenzene (tcb), furan (fur), thiophene (thi), 2,2-dichloropropane (clpro), phenylacetylene (phac), para-bromobenzonitrile (pbbn), 2,4-hexadiyne (hex), ortho-dicyanobenzene (dcnb) and fluorobenzene (fb) were co-dissolved in five different liquid crystals (or mixtures thereof), as groups of four or five for a total mole fraction of about 4% with tcb in each group to serve as an orientational reference. The liquid crystals used were p-ethoxybenzylidene-p′ -nbutylaniline (EBBA), Merck ZLI-1132 (1132), a ‘magic mixture’ (MM) of 55wt% 1132 and 45wt% EBBA, 4-n-octyl-4′ -cyanobiphenyl (8CB) and 4-n-octyloxy-4′ -cyanobiphenyl (8OCB). Each sample was mixed thoroughly in the isotropic phase and then placed into an NMR tube. Specifically phac, dcnb, pbbn, hex and tcb were in one tube while fur, thi, clpro and tcb were placed in a second tube in each liquid crystal. The solutes odcb, mdcb, pdcb and tcb were co-dissolved in a third tube in 1132, EBBA and MM and the data for these solutes in 8CB and 8OCB were taken from [34]. The order parameters of fb in 8CB were taken from reference [33] and in 1132, EBBA and MM from reference [36]. Capillary tubes filled with deuterated acetone and centred by Teflon spacers were placed axially symmetric in each NMR tube to provide a lock signal. After mixing thoroughly in the isotropic phase and sealing the tube, samples were placed into a Bruker Avance 400 MHz NMR spectrometer in the nematic phase in which the director is oriented by the magnetic field along the field direction. With the temperature controlled by the Bruker air-flow system spectra were acquired in 0.5 or 1 K steps in the nematic and smectic A phases of 8CB and 8OCB and at one temperature in the nematic phases of EBBA, 1132 and MM. The spectral parameters (dipolar couplings (Dij ) and chemicals shifts (wi )) of each group of molecules in each liquid crystal were then obtained simultaneously and automatically with the use of covariance matrix adaptation evolutionary strategies (CMA-ES) [37–40]. Having obtained the very accurate dipolar couplings we can go on to calculate the solute order parameters, with the help of the program SHAPE [41]. The values of the order parameters  41  3.3. The Nematic Potential: MSMS obtained can be found in the supplementary material [42]. Since not all the solutes were co-dissolved in the same sample tube of a given liquid crystal we must correct the order parameters so they correspond to a given liquid crystal environment. That is to say we need to apply our potential (nematic or smectic) to a single liquid crystal environment rather than over the small ranges given by different samples at the same temperature but with slightly different concentrations. To this end we plot (not shown) each solute’s order parameters against Szz of tcb (the orientational reference) from a given tube and interpolate them to the Szz of tcb in the arbitrarily chosen reference tube (the one with fur, thi, clpro and tcb). To best visualize the effect of the smectic A phase on the order parameters we plot the solute order tensor asymmetry  R = (Sxx − Syy )/Szz  (3.1)  versus solute Sxx and observe the change in slope at the phase transition, as shown in fig. 3.2 for all solutes that have two independent orientational order parameters (except fb).  3.3  The Nematic Potential: MSMS  Before pursuing any smectic information one must first adequately describe the nematic potential which is responsible for the orientational order. The classic theory of mesogen ordering is that of Maier and Saupe [5, 6] who developed a mean-field theory for axiallysymmetric liquid-crystal rods in which the rods sit in a potential that varies as the second Legendre polynomial HN,L (θL ) = −νSL  3 1 cos2 (θL ) − 2 2  (3.2)  where θL is the angle between the ‘rod’ long axis and the director and ν is a scale parameter that indicates the strength of the rod-rod interaction. The liquid-crystal rod’s order  42  3.3. The Nematic Potential: MSMS parameter can then be calculated using  SL =  sin(θL )dθL ( 23 cos2 (θL ) − 12 )e sin(θL )dθL e  −HN,L (θL ) kB T  −HN,L (θL ) kB T  (3.3)  In the case of solutes of sufficiently low concentration so that solute-solute interactions can be ignored [43], we can write a similar potential (assuming the solute ‘feels’ the same environment as the liquid-crystal molecules) for an axially symmetric solute s in liquid crystal L as 1 3 cos2 (θs ) − 2 2  HN,Ls (θs ) = −νLs SL  (3.4)  where νLs is the solvent-solute interaction parameter and θs is the angle between the solute symmetry axis and the liquid-crystal director. Now assuming that all solutes feel the same mean field the potential for a solute s of arbitrary symmetry in a liquid crystal of arbitrary symmetry can be written as [9]  HN,Ls (Ωs ) = −  1 2  GL,αǫ βs,αǫ α  (3.5)  ǫ  where GL is a property of the liquid crystal and represents the anisotropic part of the liquid-crystal mean-field interaction for the experimental conditions and is traceless; βs is the solute property that interacts with the liquid-crystal mean field and Ωs represents the orientation of the solute with respect to the phase. For the axially-symmetric liquid-crystal mean fields we consider here Eq. 3.5 can be written 3 HN,Ls (Ωs ) = − GL,ZZ 4  cos(θs,γ ) cos(θs,δ )βs,γδ γ  (3.6)  δ  where Z is the mean-field symmetry axis and θs,γ is the angle between the solute γ axis and Z . However, there is ample evidence in support of two or more mechanisms being responsible for nematic ordering [20] and, given the recent success of using two MS terms (the MSMS theory) [36], we write the nematic Hamiltonian for solutes (like the ones considered here)  43  3.3. The Nematic Potential: MSMS of sufficient symmetry so as to have only one or two independent order parameters as:  HN,Ls (Ωs ) = −  3 4  2  GL,ZZ (i)βs,zz (i) × i=1  1 3 cos2 (θs ) − 2 2  +  bs (i) sin2 (θs ) cos(2φs ) 2  = H M S1 + H M S2 (3.7) where bs (i) =  βs,xx (i) − βs,yy (i) βs,zz (i)  (3.8)  Examples of liquid-crystal/solute interactions GL,ZZ (i)βs,zz (i) could be short-range size and shape effects or longer-range interactions such as that involving the liquid-crystal average electric field gradient interacting with the solute quadrupole or the solute polarizability with the mean square electric field: this list is not exhaustive. A problem arises in trying to use Eq. 3.7 since the GL,ZZ (i) and βs,zz (i) always appear as products and they occur with the same angular dependency. It turns out that there are four degrees of freedom and hence four values must be fixed in order to use the set of equations in a least-squares fit to the experimental order parameters. A convenient place to start is with the MM, where the short-range size and shape mechanism is considered to be the only significant interaction [20]. Thus we set GMM,ZZ (1)=1 and GMM,ZZ (2)=0 therefore implying that the first nematic mechanism is affiliated with size and shape effects. In order to provide a scale for the second interaction we can arbitrarily set G1132,ZZ (2)=1. At this point we have no obvious choice for the final degree of freedom and having no need, for our intents and purposes, to specify the nature of the second mechanism we set GEBBA,ZZ (1)=G1132,ZZ (1) (as was done in [36]). In effect we are making the reasonable assumption that size and shape play an equal role in the two liquid crystals EBBA and 1132. As the βs,γγ (i) are solute properties, and consequently not functions of liquid-crystal solvent or temperature, their values are obtained by fitting to the order parameters (of the solutes at a single nematic temperature in 1132, MM and EBBA and at three temperatures spanning the nematic ranges of 8CB and 8OCB) with the MSMS potential (Eq. 3.7) and the relationship  44  3.4. The Smectic Potential: MSMS-KM  Ss,γγ =  dΩs ( 32 cos2 (θs,γ ) − 21 )e dΩs e  −HN,Ls (Ωs ) kB T  (3.9)  −HN,Ls (Ωs ) kB T  With the solute properties (βs,γγ (i)) fixed in this way (see Table 1) one can proceed to fit the experimental order parameters of all solutes at each temperature in both the nematic and smectic phases of 8CB and 8OCB using the nematic potential Eq. 3.7 only by varying GL,ZZ (1) and GL,ZZ (2) and obtain the values depicted by the open circles in fig. 3.3. We find that the RMS of the fit increases significantly in the smectic phase (open circles in inset to fig. 3.3) in 8CB as well as 8OCB. This shows that the MSMS theory works for the entire nematic phase and also, by virtue of the less good fit in the smectic phase, that we need additional terms in the potential to more accurately describe this latter phase. In order to understand the effect of two nematic mechanisms on the effective bs parameter used in the earlier studies [33–35], we note that the nematic potential in Eq. 3.7 includes the term [GL,ZZ (1)(βs,xx (1)−βs,yy (1))+GL,ZZ (2)(βs,xx (2)−βs,yy (2))]. In the spirit of Eq. 3.7, one can express the asymmetry in the βs,γγ parameters to give bs,MSMS as  bs,MSMS =  GL,ZZ (1) GL,ZZ (2) (βs,xx (1) − βs,yy (1)) GL,ZZ (1) GL,ZZ (2) βs,zz (1)  + (βs,xx (2) − βs,yy (2)) + βs,zz (2)  .  (3.10)  In Fig. 3.4 we show bs,MSMS as a function of temperature (using open symbols and GL,ZZ (i) from the nematic potential Eq. 3.7 only) for the two solutes pdcb and dcnb in both the nematic and smectic A phases.  3.4  The Smectic Potential: MSMS-KM  Given the recent preliminary success of the MSMS-KM theory on a limited range of solutes, we write our smectic Hamiltonian as before [37]  45  3.4. The Smectic Potential: MSMS-KM  HA,Ls (Ωs , Z) = HM S1 1 + κ′L (1) cos  2πZ d  + HM S2 1 + κ′L (2) cos ′ −τLs cos  mod mod ′ = HN (1) + HN (2) − τLs cos  2πZ d  2πZ d (3.11)  2πZ d  where κ′L (1) and κ′L (2) (taken to be purely liquid-crystal properties that can depend on temperature) modulate independently each of the two nematic ordering mechanisms as we move across a layer. The κ′L (i) are called the nematic-smectic coupling Hamiltonian prefactors. Because different solutes may partition differently within the smectic layer, one ′ is ascribed to each solute in each liquid crystal at each temperature and is thus termed τLs  the solute smectic prefactor. The width of a layer is d while Z (Z=0 in the centre of a layer) maps the direction parallel to the director which for a smectic A phase lies along the layer normal. We can now attempt to fit this Hamiltonian to our observables in the smectic A phase with the relationship  Ss,γγ =  dΩs  d 3 0 (2  cos2 (θs,γ ) − 21 )e  d dΩs 0 e  −HA,Ls (Ωs ,Z) kB T  −HA,Ls (Ωs ,Z) kB T  dZ  (3.12)  dZ  When for a given liquid crystal at a given temperature in the smectic A phase we ′ ) we do not obtain vary all free parameters (GL,ZZ (1), GL,ZZ (2), κ′L (1), κ′L (2) and τLs  convergence and thus are forced to make an assumption. In order to proceed we assume that the amount each nematic mechanism contributes to orientational ordering changes the same in the smectic phase as in the nematic. This amounts to saying that the ratio of the liquid crystal fields of these two mechanisms is a function of temperature only. To effect this assumption we simply linearly extrapolate the GL,ZZ (2) of GL,ZZ (1) function into the smectic phase (see line in fig. 3.3) and therefore no longer have to independently vary GL,ZZ (1) in our fits to the MSMS-KM Hamiltonian. Now by varying (at each temperature in the smectic phase of a given liquid crystal) this reduced set of parameters (GL,ZZ (2), ′ ) in a least squares fit to orientational order parameters S κ′L (1), κ′L (2) and τLs s,γγ we  46  3.5. Results and Discussion obtain convergence. The resulting GL,ZZ (i) are displayed in fig. 3.3 (filled circles), the ′ of odcb βs,MSMS in Fig. 3.4 (filled circles) and the κ′L (i) in fig. 3.5. Unfortunately the τLs  in 8OCB and of phac in 8CB tend to infinity. These large values only give a negligible improvement in the RMS relative to physically reasonable values of these prefactors, which in turn corresponds to the physically unreasonable τLs (see Eq. 3.13) of 1.0. This situation is likely not helped by the fact that the smectic effect on order parameters is not a large one, especially compared to the nematic ordering mechanisms, and consequently our global ′ of these error minimum is shallow. In light of these considerations we arbitrarily fix the τLs  two solutes in their respective liquid crystals to 3.0 (which corresponds to a τLs of 0.8). ′ ) and temperature can be found in the The Hamiltonian prefactors (GL,ZZ (i), κ′L (i) and τLs  supplementary material [42]. Once equipped with an explicit form of the smectic Hamiltonian one can calculate the smectic order parameters (τLs ) using  τLs =  dΩs  d 2πZ 0 cos( d )e d dΩs 0 e  3.5  −HA,Ls (Ωs ,Z) kB T  −HA,Ls (Ωs ,Z) kB T  dZ  (3.13)  dZ  Results and Discussion  While the focus of this study has been on obtaining smectic information it has also served as a further test of the MSMS theory. It is therefore good to see that all the βs,γγ (i) of tcb, fb and pdcb agree within 2.2 times the sum of the errors in each βs,γγ (except for βtcb,zz (1) which differed by 3.9 times) of those found previously [36] where it was shown that the MSMS Hamiltonian could reproduce nematic order parameters to within 5 % at a single temperature in various liquid-crystals. Here we find that the MSMS theory holds up across the entire nematic temperature range of 8CB and 8OCB with roughly equally good fits throughout, as shown by the inset of fig. 3.3. Order parameters are reproduced for all solutes in both liquid crystals to within 5 % (except for mdcb in 8CB and clpro which has order parameters of the order of the RMS). As noted above the RMS’s of fits using the nematic Hamiltonian only in the smectic A phase of 8OCB and 8CB increase relative to  47  3.5. Results and Discussion those in the nematic phase. It is perhaps debatable weather or not the KM formulation is the best way to describe the smectic environment but what is clear is that extra terms are needed as we observe a twofold increase in the RMS for 8OCB and a threefold increase for 8CB when applying only the MSMS potential across both phases. In Fig. 3.3 and 3.4 the open points give values of GL,ZZ (i) and bs,MSMS as a function of temperature for fits with the smectic Hamiltonian Eq. 3.11. As can be seen the effect of adding the smectic potential is to shift the smectic points closer to the nematic ones. This shift is consistent with expectations of a more or less continuous change of the nematic potential as one crosses over to the smectic A phase; i.e. a jump would not be consistent with the second-order nature of this phase transition as seen for the open circles (nematic potential only) in Fig. 3.4. It is interesting to see that in our efforts to find a better formulation to get smectic information, some of the most interesting smectic phenomena found here came about from having to accommodate two explicit nematic mechanisms. Specifically, we could think of no reason why both nematic mechanisms would be modulated in the same way as a result of the formation of layers and so postulated two independent κ′L (i). Once this is done we find the κ′L (i) to be constant (within error) and have average values of κ′8CB (1)=0.45, κ′8OCB (1)=0.16, κ′8CB (2)=-3.15 and κ′8OCB (2)=-1.37 (see fig. 3.5). Since the κ′L (i) multiply mod (i) terms of Eq. 3.11 we find H a cosine in the HN M S1 modified by a peak to peak factor of  0.90 and 0.32 for 8CB and 8OCB moving across a layer. HM S2 turned out to be modulated considerably more by maximum amplitudes of 6.30 and 2.74 for 8CB and 8OCB. It is fascinating to find the absolute values of κ′L (2) to be greater than 1 as this implies a sign mod (2) moving across a layer and is perhaps a clue to the nature of this second change in HN  orientational mechanism. In order to gain more insight into the modulation of the nematic potential by smectic layering we have plotted the two HNmod (i ) (the first and second terms in Eq. 3.11) and their sum (HNtot ) for solutes tcb, hex (fig. 3.6) and pdcb (fig. 3.7). Since each nematic potential is obviously a function of solute orientation we have plotted them for a fixed orientation. For tcb and hex we have chosen the orientation each molecule is most likely to 48  3.5. Results and Discussion adopt (from intuitive shape arguments) so as to be most physically relevant. Specifically the plane of tcb (θs =π/2 in Eq. 3.7) and the long axis of hex (θs =0 in Eq. 3.7) are made to lie along the director. The HNmod (1) of tcb is observed to be at a minimum in the centre of the layer (Z=0) which implies a stronger size and shape orientational influence here. This of course makes good sense as the mesogen cores are densely packed here and hence enhance short range orientational interactions; the interlayer region is dominated by the liquid-crystal flexible chains which provide a weaker short-range size and shape orientational influence than do the cores, and hence the potential is maximum. Furthermore the minima of HNmod (1 ) is even deeper for hex, as expected, since it is elongated (not unlike a liquidcrystal molecule) and is therefore not likely to adopt an orientation from a size and shape interaction such that its long axis is perpendicular to the director while positioned in the middle of a smectic layer. We also find the expected result that HNtot is dominated by the size and shape potential (HNmod (1 )). It is observed that HNmod (2 ) shows the opposite behaviour of HNmod (1 ) and so HNtot varies less than do its components as a result. If we consider the solute property tensor elements βs,γγ (2) of the second nematic mechanism to be associated with its quadrupole then the behaviour of HNmod (2 ) can be seen as consistent with a quadrupole-quadrupole interaction with the liquid crystal. Specifically, when we recall that like quadrupoles prefer to align perpendicular and unlike quadrupoles parallel, it then appears sensible that HNmod (2 ) would peak in the centre of the layer where the negative (symmetry axis) quadrupole of tcb is aligned with the negative quadrupoles of the liquid-crystal benzene cores. A similar consistency is found in the HNmod (2 ) of hex which is also maximum at the centre of the layer where its positive (symmetry axis) quadrupole is perpendicular to the negative quadrupole of the benzenes in the mesogen cores. For pdcb with its lower symmetry we have plotted in Fig. 3.7 the HNmod (i ) and HNtot with each molecule-fixed symmetry axis (θs =0 in Eq. 3.7) aligned along the director in turn where the z direction is perpendicular to the ring, the x axis coincident with the Cl-Cl direction and the y axis normal to the other two. With the z axis aligned the plane of the ring is perpendicular to the director and so, as expected, the HNmod (1 ) (and therefore HNtot ) is minimum in between the layers where the molecules relative orientation can best 49  3.5. Results and Discussion be accommodated. However, this is not a favourable situation for pdcb as HNtot is positive for all Z. It is easy to see this graph is the opposite of that for tcb, where the ring plane was parallel to the director, as one would expect. Conversely, as in the previous two examples, when pdcb is in its preferred orientation with its x axis (the long Cl-Cl direction) aligned with the director we find that HNmod (1 ) and HNtot are minimal in the centre of the layer and also minimum compared to the other orientations plotted. The HNmod (2 ) for the y direction of pdcb show similar behaviour to those plotted of tcb and hex, being negative in the interlayer region before turning positive around the layers centre. Similarly, the HNmod (2 ) for the z direction of pdcb is consistent with the tcb plot (when we recall that the ring plane for pdcb is perpendicular to the director, which is opposite to that for tcb). We notice that HNmod (2 ) for the x direction (and HNmod (1 ) for the y direction) of pdcb is almost zero for all Z which is commensurate with the fact that βs,xx (2) (and βs,yy (1)) is very small. The dashed lines of fig. 3.6 are plots of the smectic Hamiltonian Eq. 3.11 for the solutes tcb and hex in the orientations indicated above. Their behaviour shows the energy (including orientational and positional) of the solute as a function of Z. Since a more negative energy results in a higher probability, and vice versa, the effect is that tcb (with its plane aligned along the director) prefers the interlayer region. This behaviour is then translated into the τLs order parameters via Eq. 3.13 where all orientational angles are considered and a negative value results. A similar result obtains for hex at 327.0K in 8OCB. However, a different behaviour is observed in 8CB. For hex (with its symmetry axis aligned along the ′ term is not large enough to overcome the H tot term and so at director) in 8CB the τLs N  this orientation the centre of the layer is slightly more preferred. For other hex orientations (away from the long axis being aligned with the director) the sinusoidal modulation ′ term is unchanged and H of HNtot is reduced while the τLs A,Ls becomes maximum at Z=0  and minimum at Z = ±d/2. Thus when all angles with respect to the director are included with their appropriate weighting, there is a greater tendency for hex to partition toward the interlayer region as indicated by the negative τLs order parameter for this solute in 8CB at 298.0 K. Fig. 3.8 shows as a function of temperature the τLs order parameters obtained from 50  3.6. Conclusion Eq. 3.13 after fitting solute orientational order parameters in the smectic A phase. It is found that they are constant (within error) or decrease with increasing temperature as one would expect. The τLs of a given solute show roughly the same behaviour in both liquid crystals which is sensible as the liquid crystals are not drastically different in structure. In a previous study where a single MS mechanism was used [34] the sign of τL  odcb  was found  to depend on which assumptions were made, whereas here we find that it is positive. We also find that the relative magnitudes of τL magnitude of τL  mdcb  is greater than τL  mdcb  pdcb )  and τL  pdcb  are the reverse (the absolute  of what was found previously when a less  realistic treatment of the nematic potential was employed. It should be noted that the τLs ′ (and therefore τ ) not being well determined have fairly large errors as a result of τLs Ls  because the smectic effect on orientational order parameters is small. In two cases a limit ′ had to be imposed. on τLs  3.6  Conclusion  We have tested the KM theory in a way that can be described as comprehensive with a broad range of probe molecules. It can be seen that the double Maier-Saupe potential works as a function of temperature and that the double Maier-Saupe Kobayashi-McMillan theory can rationalize smectic phase observables in a simple, practical and physically reasonable way. This is accomplished by allowing two explicit mean-field nematic mechanisms to account for the orientational ordering so that when we extrapolate their behaviour into the smectic phase the smectic order is separated and exposed. This was done using the single assumption that the ratio of the two nematic orienting mechanisms had the same interdependence in the smectic phase that is displayed in the nematic. With the smectic order exposed in such a way it became possible to account for it with the smectic order parameters and nematic-smectic coupling Hamiltonian prefactors. It is interesting to find, according to the theory presented here, that the second orientational mechanism is substantially modulated by the smectic layering and undergoes periodic sign changes in the Z direction. When the modulation of the two nematic mechanisms are considered, they seem to be consistent with  51  3.7. Tables and Figures one being a size and shape effect and the second involving the solute quadrupole.  3.7  Tables and Figures  52  3.7. Tables and Figures  solute tensor component tcbzz pdcbyy pdcbzz mdcbyy mdcbzz odcbyy odcbzz f uryy f urzz thiyy thizz hexzz phacyy phaczz dcnbyy dcnbzz pbbnyy pbbnzz clproyy clprozz f byy f bzz  solute property (βs,γγ (i)) M S1 M S2 -0.990(13) -0.207(17) -0.024(9) 0.229(13) -0.951(11) -0.236(16) 0.669(6) 0.079(8) -0.933(10) -0.266(14) 0.333(7) 0.156(9) -0.927(10) -0.285(13) 0.331(6) 0.227(8) -0.529(8) -0.246(10) 0.246(6) 0.185(8) -0.570(8) -0.262(10) 1.066(6) 0.210(8) -0.033(8) 0.023(11) -0.949(12) -0.359(14) 0.393(7) 0.187(11) -1.126(12) -0.215(17) -0.204(10) 0.190(16) -1.019(12) -0.235(19) 0.028(6) 0.056(8) -0.035(6) -0.002(8) 0.259(8) 0.207(9) -0.751(11) -0.272(12)  Table 3.1: Maier-Saupe solute parameters  53  3.7. Tables and Figures  y  Cl  Cl Cl  x  Cl  Cl  1,3,5-trichlorobenzene (tcb)  Cl  ortho-dichlorobenzene (odcb)  Cl  meta-dichlorobenzene (mdcb) N  Cl  Cl  Br  N N  para-dichlorobenzene (pdcb)  1-bromo-4-cyanobenzene (pbbn)  1,2-dicyanobenzene (dcnb) Cl  F Cl  phenylacetylene (phac)  fluorobenzene (fb)  O  S  thiophene (thi)  2,2-dichloropropane (clpro)  furan (fur)  2,4-hexadiyne (hex)  Figure 3.1: The structures of solutes in the molecule fixed (with the exception of hex and clpro for which the z-axis lies along the C-C and Cl-Cl directions) frame where the z axis protrudes from the plane of the page.  54  3.7. Tables and Figures  Figure 3.2: The asymmetry in the order parameters R (Eq. 3.1) of each solute is plotted against its Sxx . Although vibrational corrections have been neglected the error in R is small. The open points correspond to measurements in the liquid crystal 8CB while the filled points are from 8OCB. Nematic points are to the left and smectic to the right.  55  3.7. Tables and Figures  0.006 RMS 0.004 0.002 0.2  0.3  0.4  0.5  0.2  0.3  0.4  0.5  0.015 RMS 0.01 0.005  Figure 3.3: GL,ZZ (1) is plotted against GL,ZZ (2) where the filled points signify the use of the smectic Hamiltonian and the open points are obtained with the nematic potential Eq. 3.7 only. The seven (8CB) or eight (8OCB) points closest to the origin are from measurements in the nematic phase while the rest are in the smectic phase. Inset: The RMS of fits to either potential in both phases are plotted against GL,ZZ (2).  56  3.7. Tables and Figures  Figure 3.4: bs,MSMS is plotted against temperature for pdcb and dcnb in 8OCB and 8CB. Points in the smectic phase are to the left and those in the nematic to the right. The open circles were obtained using GL,ZZ (i) from the nematic potential Eq. 3.7 and the filled circles used GL,ZZ (i) from the smectic potential Eq. 3.11.  57  3.7. Tables and Figures  Figure 3.5: The κ′L (i) are plotted with error bars versus temperature with error bars for each liquid-crystal solvent.  58  3.7. Tables and Figures  mod (i) (see Figure 3.6: The modulation of each nematic mechanism from smectic layering HN tot Eq. 3.11) and their sum HN in 8CB at 298.0K and 8OCB at 327.0K for tcb (θs =π/2) and hex (θs =0) where the plane of the ring and symmetry axes are aligned along the director, respectively. The centre of the layer is at the origin. The dashed line is the total smectic Hamiltonian HA,Ls (Z) (Eq. 3.11) for each solute’s given orientation. The Hamiltonian prefactors used are G8CB,ZZ (1)=0.952, κ′8CB (1)=0.456, G8CB,ZZ (2)=0.369, κ′8CB (2)=-3.145, G8OCB,ZZ (1)=0.911, κ′8OCB (1)=0.160, G8OCB,ZZ (2)=0.326, κ′8OCB (2)=-1.303, τ8CB tcb =0.182, τ8OCB tcb =-0.417, τ8CB hex =-0.147, and τ8OCB hex =-0.182.  59  3.7. 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Lett. 134, 161 (1986).  63  Chapter 4  The Butane Condensed Matter Conformational Problem. ∗ From the dipolar couplings of orientationally ordered n-butane obtained by NMR spectroscopy we have calculated conformer probabilities using the modified Chord (Cd) and Size-and-Shape (CI) models to estimate the conformational dependence of the order matrix. All calculation methods make use of Gaussian structures for the gauche and trans conformers. Calculations were performed for both the Rotational Isomeric State (RIS) approximation, as well as a continuous gas-phase potential for the dihedral angle rotation. Conformational probability distribution functions for butane as a solute in the ordered liquid-crystal solvent are obtained.  ∗ A version of this chapter has been submitted for publication. A. C. J. Weber, C. A. de Lange, W. L. Meerts and E. E. Burnell (2010) The Butane Condensed Matter Conformational Problem.  64  Chapter 4. The Butane Condensed Matter Conformational Problem Comprehension of the configurational statistics of chain molecules is essential to an understanding of the properties of these ubiquitous compounds. For example, the facility with which cyclic structures are formed in a chemical reaction from acyclic chains is related to the statistical distribution of the two ends of the chain relative to one another and so depends on their configurational characteristics. Constitutive properties of a chain molecule, which are dependent up on its configuration, include the mean-square dipole moments, the optical anisotropy and the spectral dichroism of the molecule. The behavior of flexible hydrocarbons in condensed phases is particularly important in the field of liquid crystals as they are an essential component of mesogens that form partially ordered phases [1]. Butane (fig. 4.1) is the simplest flexible alkane with only one conformational degree of freedom and is well suited to the study of the effect of the condensed phase on the conformational behavior of non-rigid molecules. Flory suggested that the intramolecular potential which gives rise to hydrocarbon conformers should closely resemble that in the gas phase with the configurational space populated according to the Boltzmann distribution over conformations as a function of intramolecular energy with intermolecular effects being ignored [2]. This view was later challenged by Chandler et al . whose rigorous theory of hydrocarbons predicted an increase in the gauche population resulting from short-range packing in the liquid phase [3, 4]. Experiments have given credence to the latter view with gas-phase studies typically reporting the trans-gauche energy, Etg , to be 788-884 cal/mol [5–9] while studies of n-butane, both as a liquid and dissolved in other isotropic liquid solvents, consistently report a lower range of 502-681 cal/mol [10–13]. Proton dipolar couplings from the NMR spectra of molecules in partially ordered phases are an excellent means to obtain conformational information as they are very accurate and highly sensitive to the relative distances between hydrogens on a given solute molecule [1]. Obtaining these values for spin systems with more than 8 spins is difficult because of the multitude of lines that are characteristic of such spectra, and the normal approach is to use selective deuteration and/or multiple quantum techniques to solve the congested spectra. Recent applications of the Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) 65  Chapter 4. The Butane Condensed Matter Conformational Problem have reduced this task to a much more routine exercise [14]. In order to say something useful on the subject of condensed-phase effects on conformational statistics it would be good to extract a probability distribution function to compare with that of the gas phase. A problem in this respect arises when extracting conformational information from dipolar couplings since the probability of a conformer, pn , appears n , multiplied by its order matrix, Sαβ  Dij =  2 3  n n n Sαβ Dij,αβ  pn n  (4.1)  αβ  and there is no straightforward way to determine each separately [15]. One way around this is to use mean-field models (that describe the orientational potential which gives rise n ) in order to estimate how the order parameters change with conformation. One such to Sαβ  orientational model describes the interaction between solute and liquid crystal as arising from the size and shape anisotropy of the solute. A particularly successful variant of this treatment is the Circumference Integral (CI) model [16]. Another is specially tailored to chain molecules, called the modified Chord (Cd) model, and treats the C–C bond orientation relative to the director and correlations between orientations of neighboring C–C bonds as key factors in molecular ordering [17]. Another important aspect of this problem is how we think of the different accessible conformations that can be populated and what their relative importance and properties are. Since there is no axis system for butane dissolved in a liquid-crystal solvent that would allow the separation of internal and reorientational motion [15], one is forced to make assumptions n . One way of addressing these issues is by using the threein order to determine pn and Sαβ  state rotational isomeric state (RIS) model [2] to describe the accessible conformational states of each C–C bond of the molecule. Another way is to allow all dihedral angles to be populated assuming a continuous rotational potential and commensurate Boltzmann statistics. One can also allow each member of this continuum to have its own order matrix, or alternatively one can restrict all conformers to have the order matrix of the trans or gauche conformers according to the well in the potential in which they are found. 66  Chapter 4. The Butane Condensed Matter Conformational Problem Various orientational potentials were used in conjunction with the RIS approximation in fits to the dipolar couplings of n-butane dissolved in a so-called ‘magic mixture’ (MM) liquid crystal and it was found that the Chord model gave the best fits although the full accuracy of the experimental dipolar couplings was not exploited [18]. Since solving these spectra has become routine with our sophisticated Evolutionary Algorithm analysis [19], we now revisit this general problem with more experiments and improved methods of accounting for the potentials used previously [18] with the hope of better understanding the contributions of the gas and condensed phases as well as orientational ordering on the conformational problem. n-Butane (see Fig. 4.1 for structure) and 1,3,5-trichlorobenzene (tcb) were co-dissolved in the four different liquid crystals p-ethoxybenzylidene-p ′ -n-butylaniline (EBBA), Merck ZLI-1132 (1132), 4-n-pentyl-4′ -cyanobiphenyl (5CB) and a MM of 55wt% 1132 and 45wt% EBBA (data taken from [18]). Since n-butane is a gas at room temperature and ambient pressure it was allowed to flow into a vacuum and then condensed into the 5 mm o. d. standard NMR tube which was submerged in liquid nitrogen. Enough solute was added to obtain roughly 5.0 mole % and 0.50 mole % for n-butane and tcb. After each sample was sealed and mixed thoroughly in the isotropic phase it was placed into a Bruker Avance 400 MHz NMR spectrometer magnet. With the temperature controlled by the Bruker air-flow system proton NMR spectra were acquired at 300.5 K in EBBA, 298.5 K in 1132 and 298.5 K in 5CB. Each liquid crystal is in the nematic phase at these respective temperatures and its director will be aligned with the magnetic field direction. The spectral parameters (dipolar couplings (Dij ), indirect couplings (Jij ) and chemical shifts differences (δ1 − δ4 )) of n-butane in each liquid crystal were then obtained automatically with the use of CMA-ES [14, 19–22]. Since the algorithm iterates on both frequencies and intensities, the liquid-crystal background is removed prior to the fitting with a cubicbase spline. An excellent fit to the experimental spectrum is obtained as shown for 1132 in Fig. 4.1. The NMR parameter values are shown in Table 4.1. The dipolar couplings and chemical shift differences are significantly different between liquid-crystal solvents, whereas the indirect couplings are constant within experimental error as expected. 67  Chapter 4. The Butane Condensed Matter Conformational Problem All computational chemical calculations were carried out using Gaussian 03 [23] and the structural parameters obtained can be found in the appendix. M¨ oller-Plesset 2ndorder perturbation theory [24] was employed using Dunning’s cc-pvdz basis set [25]. The designated minima were confirmed to be minima by using analytical 1st and 2nd energy derivatives as is routine with MP2. The dipolar couplings between protons i and j in an orientationally ordered molecule is given by Dij = −  γ 2 µ0 8π 2  3 2  Z) − cos2 (θij  1 2  3 rij  (4.2)  Z is the angle between the internuclear vector where rij is the internuclear distance and θij  and the static magnetic field. A pragmatic way of analyzing dipolar couplings of partially oriented flexible molecules is to assume that the molecule exists in several discrete conforn . This assumption is necessary since mations each having its own Saupe order matrix Sij  the trans and gauche conformers are not related by symmetry [15]. An important model used for approximating the conformations of hydrocarbon chains is Flory’s RIS model [2] in which each C–C bond is assumed to exist in three different orientations with respect to an adjacent C–C bond, trans and ±gauche, with dihedral angles of 0o and ±φg corresponding to the angles at the minima of the rotational potential. The order matrix of the n th conformer in eq. 4.1 can be written as n Sαβ =  3 1 n n cos(θα,Z ) cos(θβ,Z ) − δαβ 2 2  (4.3)  n where θα,Z is the angle between the α-molecular axis of the n th conformer and the nematic  director which, in these experiments, is aligned with the magnetic field along the Z axis. n The Dij,αβ tensor elements are then defined by  n Dij,αβ =−  µ0 γ 2 3 8π 2 rij  3 1 cos(θαij,n ) cos(θβij,n ) − δαβ 2 2  (4.4)  where θαij,n is the angle between the ij and the molecule-fixed α directions. Assuming a  68  Chapter 4. The Butane Condensed Matter Conformational Problem mean-field ordering potential one can write the order matrices as follows n Sαβ  =  n ) cos(θ n ) − 1 δ ) exp(−U aniso (Ω)/kT )dΩ ( 23 cos(θα,Z n β,Z 2 αβ  exp(−Unaniso (Ω)/kT )dΩ  (4.5)  where Unaniso is the anisotropic nematic ordering potential of the n th conformer. The conformer probability is a function of both the isotropic (Uniso ) and anisotropic (Unaniso ) parts of the intermolecular potential and can be written pn = where Gn =  Gn exp(−Uniso /kT ) exp(−Unaniso (Ω)/kT )dΩ n iso aniso (Ω)/kT )dΩ n G exp(−Un /kT ) exp(−Un  (4.6)  n I n I n is a rotational kinetic energy factor which is dependent on the Ixx yy zz  principal values of the moment of inertia tensor for each conformer. The isotropic part of iso , and the intermolecular potential, Uniso , is composed of an intramolecular component, Uint,n iso . The trans-gauche energy difference, E , is an intermolecular part, Uext,n tg  iso iso int ext Etg = Ugauche − Utrans = Etg + Etg  (4.7)  int ≡ E gas is taken as the energy difference as calculated by Gaussian 03. As where Etg tg  discussed above, since the conformer probabilities and molecular order parameters appear as products, they cannot be determined independently and so one must use a model for Unaniso in order to proceed. In the current study we shall focus on two different models, namely the Cd and CI models. When these two models are employed without the RIS approximation we will employ a gas-phase rotational potential which is a function of dihedral angle and calculated with Gaussian shown as the points in fig. 4.2. The size-and-shape potential 1 1 UnCI (Ω) = k(Cn (Ω))2 − ks 2 2  Zmax,n  Cn (Z, Ω)dZ  (4.8)  Zmin,n  involves two terms, the first of which involves a Hooke’s law restoring force [26] where Cn (Ω) is the minimum circumference traced out by the projection of the solute onto a  69  Chapter 4. The Butane Condensed Matter Conformational Problem plane perpendicular to the nematic director; the second term represents the anisotropic surface potential where the area of the infinitesimally thin ribbon Cn (Z, Ω)dZ is summed over its projection onto the plane parallel to the nematic director [27]. Note that UnCI (Ω) is comprised of both isotropic and anisotropic components; therefore, in the calculations we use UnCI,aniso (Ω) = UnCI (Ω) − UnCI  (4.9)  where UnCI is the isotropic average over all angles. The Cd model is especially tailored for the orientational order of molecules comprised of repeating identical units in a uniaxial phase [17, 28], and is derived from a rigorous expansion of the mean-field interaction potential in which only the leading terms are retained w ˜0 P2 (si , si ) + w ˜1 P2 (si , si+1 )  UnCd,aniso (Ω) = −  (4.10)  i=1  where w ˜i =  3 2 Swi ,  S being the liquid-crystal order parameter. The si is a unit vector  describing the orientation of the i th C–C bond of the hydrocarbon chain and the sum is over all bonds in the chain. The factors P2 (si , si+m ) are given by P2 (si , si+m ) =  3 1 cos(θZi ) cos(θZi+m ) − si · si+m 2 2  (4.11)  where θZi is the angle between the i th bond and the nematic director. The parameters, w ˜m , are proportional to the liquid-crystal order parameter. The first term in eq. 4.10 corresponds to the independent alignment of separate C–C bonds. The second term incorporates correlations between adjacent bond orientations, and therefore distinguishes between conformations that may have equal numbers of trans and gauche bonds, but significantly different shapes and so accounts for shape-dependent excluded-volume interactions. It should be realized that values given for w ˜0 and w ˜1 in the literature do not always conform to the definition of Eq. 4.10. In Table 4.2 the results of six different calculations are shown. In addition to the RIS approximation we employ a modified version, RIS(± 20), which includes the original  70  Chapter 4. The Butane Condensed Matter Conformational Problem dihedral angle calculated by Gaussian 03 for both trans and gauche conformers (weighted 50 %) and the original dihedral angle ± 20 degrees (each weighted 25 %). These RIS calculations were performed with the Cd model for orientational order. The Cd model was also applied with a continuum of conformers. In this case two different variants were calculated. The continuous chord model, CCd, allows each conformer to have its own order matrix while the discontinuous chord model, DCd, restricts all conformers to have the order matrix of the trans or gauche conformer (calculated at the well minimum) according to in which potential well a given conformer is found. The CI model was employed in a manner similar to the CCd calculation where the two parameters in the orientational potential were varied independently (CI(2k)) but was also performed restricting the ratio of these two parameters (CI(k)) to the value found in a study of 46 solutes in the MM [27]. All of these calculations are divided into those where we varied the methyl CCH bond angle and those where we did not. Detailed studies of methyl groups in methyl fluoride, methyl iodide and ethane have shown that vibration-reorientation interaction effects are transferable among methyl groups in different molecules and lead to an apparent increase in HCH angle when neglected [29–32]. The variation of the CCH angle is intended to account for, in a crude way, the vibration-reorientation effects on the methyl protons. As found for an earlier study of pentane [19] this crude vibration-reorientation correction obtained by adjusting the CCH angle gives a significant improvement for all fits. We show in Table 4.3 the experimental and recalculated dipolar couplings for the CCd and CI(2k) model calculations. The worst relative disagreement is found for the intermethylene couplings D46 and D47 . These couplings are particularly sensitive to conformational aspects of the problem. These differences could form the starting point for the testing of new ideas for the anisotropic intermolecular potential. The fits can also be carried out on a single liquid crystal at a time (i .e. a separate Etg for each). It is found that the fitting parameters do not change substantially and a lower RMS (around 2-3 Hz) is obtained. However, even if one does not vary the CCH methyl angle we have a slightly better fit than in [18] as can be seen in Table 4.4 where we compare the chord model with the RIS assumption for the previous and present studies. 71  Chapter 4. The Butane Condensed Matter Conformational Problem This repeated calculation is performed with the single parameter chord model (w ˜0 = w ˜1 ) and the RIS(±20) approximation. The difference in the parameters obtained is likely due to the use of a rough geometry (φg = 116.0◦ ) in the old calculations and high-quality Gaussian geometries (φg = 116.6756◦ ) for the present study (see appendix). For the CCd and DCd calculations we used a continuous gas-phase potential barrier ext in these calculated with Gaussian at 5 degree intervals over the entire rotation. The Etg  instances is the shift due to the condensed phase (see Fig. 4.2) of all points in a gauche well relative to those in the trans. We also apply the same idea with the CI(2k) and CI(k) calculations. It is interesting to note that the shift from the calculated Gaussian value (651 cal mol−1 ) is upward in the case of the CI models when the CCH angle is varied and roughly the same when the CCH angle is constant. On the other hand the shift is markedly downward in the case of the CCd and DCd models. There seems to be no significant benefit to assuming a continuous order matrix over a discontinuous one as the goodness of fits in the CCd and DCd calculations are roughly equal. The RIS(±20) approximation seems to give the best fit but only slightly better than the other Cd model calculations. There were only two distinct probability distributions as a function of dihedral angle that emerged from the different calculations, and these are presented in Fig. 4.3. The Cd model probability distribution favors the gauche conformer more than does the CI models regardless of which liquid crystal is used for the condensedphase environment, how we treat the available conformers, what rotational potential is used or whether or not we vary the CCH angle to correct for vibrational-reorientational effects. We find the effect of the condensed phase on the configurational statistics obtained are determined not by the particular liquid crystal or assumptions with regards to the available butane conformations and order matrices. Instead, the effect appears dependent only on the model chosen to describe the orientational potential. In order to better understand this observation the calculated order parameters of the trans and gauche conformers are shown in Table 4.5. The order parameters are calculated from dipolar couplings taken from orientationally ordered butane in the MM at 301.4K where comparisons with the CI model are most appropriate. The angle required to diagonalize the order matrices are the 72  Chapter 4. The Butane Condensed Matter Conformational Problem same for a given conformer in the two calculations indicating both models identify the same principal ordering axes. While the assymetry of the order matrix, η n , are fairly similar for both models concerning the trans conformer, they differ significantly for the gauche. To assess the extent of orientational ordering of a conformer we define its ‘average orientational ordering’ to be S2  n  =  2 + S2 + S2 Sxx yy zz  It can be seen that the mean-square orientational ordering,  (4.12) n  S 2 , of the trans conformer  calculated by the Cd model is significantly larger than that calculated by the CI model whereas the  S2  g  for the gauche conformer is lower when calculated by the Cd model.  This explains why the Cd model favors the gauche conformation since the order matrix multiplies the conformer probability function in eq. 4.1 and both calculations are fitting to the same dipolar couplings. In summary, we find that the effects of a liquid-crystal condensed phase on the configurational statistics of butane are largely dependent on the model chosen to describe the ext (which from Eq. 4.7 is orientational potential. In particular, the value obtained for Etg int , where E int =651 cal mol−1 from Gaussian 03) is the main difference between Etg − Etg tg ext is of order -200 cal mol−1 , while for the CI model models. Thus for the chord models, Etg  it is of order 0. This number represents the different influences of the isotropic part of the intermolecular potential on the conformers of butane in the condensed phase. The fact that this number is model dependent indicates the need for further investigation to determine which (if either) model is in any way related to the actual intermolecular potential of butane in these condensed phases. The precise description of the ordering potential is of course a fundamental and recurring question in chemistry and physics. Perhaps changing the Boltzmann factor via the temperature could help sort things out. Clearly further investigation is warranted and is in progress [33].  73  4.1. Acknowledgement  4.1  Acknowledgement  We thank Steve Hepperle for assistance with computational chemistry considerations. We acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada.  4.2  Tables and Figures  74  4.2. Tables and Figures  Table 4.1: Spectral parameters (Hz). The numbers in brackets are the errors in the last one or two digits. Parameter EBBA (300.5K) 1132 (298.5K) 5CB (298.5K) MMa (301.4 K) D12 924.55(7) 1142.45(4) 662.79(4) 817.63(3) D14 -243.66(6) -272.42(4) -157.56(3) -199.57(5) D16 -407.90(5) -553.43(3) -324.40(3) -388.84(2) D18 -205.34(5) -277.85(3) -162.92(3) -196.14(2) D45 1812.29(12) 2196.44(8) 1313.25(7) 1601.09(4) D46 80.57(19) 94.37(12) 50.44(11) 65.61(8) D47 34.30(19) 63.55(12) 24.26(11) 33.98(8) δ1 − δ4  -258.04(18)  -289.87(11)  -228.14(10)  -309.40(7)  J45 J14 J16 J46 J47  -13.5b 7.29(12) -0.23(10) 8.85(34) 5.84(34)  -13.5(4) 7.40(8) -0.22(6) 8.73(23) 6.02(23)  -12.3(5) 7.47(6) -0.28(6) 8.74(19) 6.26(20)  -14.9(1.5) 7.37(5) -0.19(2) 8.83(2) 6.04(2)  Dtcb  156.50(25)  225.46(47)  147.03(11)  173.03(2)  a  taken from reference [18] which used a 500 MHz NMR spectrometer. b Was fixed to the average of the other three spectra since an error could not be calculated.  75  4.2. Tables and Figures  Figure 4.1: Calculated a) and experimental b) NMR spectrum of n-butane in 1132 at 298.5K. c) and d) show an expanded region of the calculated and experimental spectra.  76  4.2. Tables and Figures  Table 4.2: Fitting parameters for simultaneous fits to all four sets of dipolar couplings. The CCH angle was varied in the top set of calculations and held constant in the bottom set. The numbers in brackets are the errors in the last one or two digits. Model Etg (cal mol1 ) w ˜0ebba /k ebba w ˜01132 /k 1132 w ˜0M M /k M M w ˜05CB /k 5CB w ˜1ebba /ksebba w ˜11132 /ks1132 w ˜1M M /ksM M w ˜15CB /ks5CB CCH angle increase (deg) RMS (Hz) Etg (cal mol1 ) w ˜0ebba /k ebba w ˜01132 /k 1132 w ˜0M M /k M M w ˜05CB /k 5CB w ˜1ebba /ksebba w ˜11132 /ks1132 w ˜1M M /ksM M w ˜15CB /ks5CB RMS (Hz)  CCd 442(3) 197.8(1.0) 212.3(8) 161.7(5) 126.6(6) 117.7(1.6) 193.8(1.1) 135.0(8) 119.5(9) 1.05(5) 3.5 417(5) 196.1(1.8) 208.7(1.3) 158.5(9) 125.6(1.0) 123.6(2.8) 202.8(1.7) 143.3(1.3) 123.5(1.6) 6.1  DCd 441(3) 197.4(1.0) 211.7(8) 161.5(5) 126.4(6) 118.4(1.6) 194.6(1.1) 135.6(9) 119.9(1.0) 1.00(6) 3.6 417(4) 196.0(1.7) 208.5(1.3) 158.5(9) 125.7(1.0) 123.8(2.7) 203.0(1.6) 143.4(1.2) 123.5(1.6) 5.9  RIS 476(3) 191.6(1.3) 205.0(1.0) 156.2(7) 122.5(7) 121.4(2.1) 198.3(1.4) 138.6(1.1) 122.0(1.2) 0.90(5) 4.6 456(4) 190.2(1.8) 202.0(1.3) 153.5(9) 121.7(1.0) 126.4(2.9) 206.1(1.7) 145.8(1.3) 125.4(1.7) 6.3  RIS(±10) 481(3) 195.9(1.0) 210.3(7) 159.9(5) 125.3(6) 119.4(1.5) 195.4(1.0) 136.6(8) 120.6(9) 1.00(5) 3.3 458(4) 194.2(1.7) 207.0(1.2) 156.8(8) 124.4(1.0) 64.2(2.6) 47.8(1.6) 35.9(1.2) 25.1(1.5) 5.8  CI(2k) 676(8) 1.0(5) 3.8(2) 2.7(2) 2.5(2) 67.4(6.0) 49.5(3.0) 38.3(1.9) 27.1(2.7) 1.43(11) 7.8 618(8) 1.4(1) 4.0(5) 2.9(2) 2.7(2) 85.2(2) 86.4(6.0) 54.8(3.1) 2.4(1.9) 10.9  CI(k) 697(8) 2.195(10) 2.696(9) 1.976(6) 1.605(6)  1.52(13) 9.0 641(5) 2.228(12) 2.734(9) 2.004(6) 1.628(7)  12.2  The w ˜nL and k L /ksL are in units of cal mol−1 and N m−1 .  77  4.2. Tables and Figures  Table 4.3: Calculated and experimental dipolar couplings Liquid crystal EBBA  1132  MM  5CB  Calculation experimental CI CCd experimental CI CCd experimental CI CCd experimental CI CCd  D12 924.55 902.91 932.86 1142.45 1146.87 1140.52 817.63 817.00 819.14 662.79 667.96 664.49  D14 -243.66 -227.79 -246.63 -272.42 -286.76 -273.50 -199.57 -204.28 -201.06 -157.56 -167.04 -158.27  D16 -407.90 -424.16 -409.48 -553.43 -542.73 -552.83 -388.84 -386.46 -387.60 -324.40 -316.02 -322.97  D18 -205.34 -222.40 -209.16 -277.85 -278.57 -279.28 -196.14 -201.15 -199.50 -162.92 -164.03 -167.13  D45 1812.29 1817.74 1791.64 2196.44 2197.47 2199.75 1601.09 1601.03 1599.02 1313.25 1308.46 1309.05  D46 80.57 61.23 76.44 94.37 94.01 88.40 65.61 61.59 60.15 50.44 50.95 45.97  D47 34.30 35.05 64.99 63.55 82.91 76.18 33.98 45.59 46.68 24.26 37.57 33.30  78  4.2. Tables and Figures  Table 4.4: RIS Chord. The numbers in brackets are the errors in the last digit. Parameter Olda New −1 Etg (cal mol ) 518(9) 468(9) w ˜0 (cal mol−1 ) 143.2(3) 151.6(3) RMS (Hz) 6.7 6.5 a  taken from [18]. The parameters w ˜0 and w ˜1 reported here correspond to the original definitions in [28]. Hence, the w ˜0 of [18] are multiplied by the factor 2/3 to be consistent with [28].  79  4.2. Tables and Figures  Table 4.5: Calculated conformer order parameters using the RIS approximation for the Cd and CI(2k) models. The calculated order parameters shown are from fits to n-butane dipolar couplings when dissolved in the MM and while varying the CCH bond angle. Model Cd CI S2 t SZZ ηt θt S2 g SZZ ηg θg  t  g  0.130 0.184 0.097 132.3 0.071 0.093 -0.802 112.9  0.118 0.164 0.144 133.5 0.077 0.108 -0.428 113.4  n is the principal value of the θ is the angle needed to diagonalize the order matrix; Szz S −S diagonalized order matrix; the asymmetry of the diagonalized order matrix is η n = xxSzz yy .  80  4.2. Tables and Figures  6  4  2  0 0  100  200  300  Figure 4.2: Energies, calculated from Gaussian 03 for n-butane as a function of dihedral angle φ. The points are for n-butane in the gas phase. The dashed line for the gauche conformer is shifted due to the isotropic part of the intermolecular potential in the condensed phase when the orientational order is described by the CI(2k) model. The solid line for the gauche conformer is shifted downward and obtained when describing the orientational order gas with the CCd model. The Etg calculated by Gaussian 03 is 651 cal mol−1 .  81  4.2. Tables and Figures  0.2  0.15  0.1  0.05  0 0  100  200  300  Figure 4.3: The probability P (φ) of finding n-butane at dihedral angle φ for the CCd model (solid line) and the CI(2k) model (dashed line) in the MM at 301.4K.  82  Bibliography [1] E. E. Burnell and C. A. de Lange (Eds.), NMR of Ordered Liquids (Dordrecht, The Netherlands, Kluwer Academic, 2003). [2] P. G. Flory, Statistical Mechanics of Chain Molecules ( New York, Wiley-Interscience, 1969). [3] L. R. Pratt and D. Chandler, J. Chem. Phys. 68, 4202 (1978). [4] L. R. Pratt and D. Chandler, J. Chem. Phys. 68, 4213 (1978). [5] A. L. Verma, W. F. Murphy and H. J. Bernstein, J. Chem. Phys. 60, 1540 (1974). [6] J. R. Durig and D. A. C. Compton, J. Phys. Chem. 83, 265 (1979). [7] D. A. C. Compton, S. Montero and W. F. Murphy, J. Phys. Chem. 84, 3587 (1980). [8] R. K. Heenan and L. S. Bartell, J. Chem. Phys. 78, 1270 (1983). [9] K. B. Wiberg and M. A. Murcko, J. Am. Chem. Soc. 110, 8029 (1988). [10] P. B. Woller and E. W. Garbisch Jr, J. Amer. Chem. Soc. 94, 5310 (1971). [11] L. Colombo and G. Zerbi, J. Chem. Phys. 73, 2013 (1980). [12] S. Kint, J. R. Scherer and R. G. Snyder, J. Chem. Phys. 73, 1599 (1980). [13] D. A. Cates and A. MacPhail, J. Phys. Chem. 95, 2209 (1991). [14] E. E. Burnell, C. A. de Lange and W. L. Meerts, Nuclear Magnetic Resonance Spectroscopy of Liquid Crystals, edited by R. Y. Dong. (Singapore, World Scientific Publishing Co, 2009). [15] E. E. Burnell and C. A. de Lange, Chem. Phys. Lett. 76, 268 (1980). [16] E. E. Burnell and C. A. de Lange, Chem. Rev. (Washington, D.C.), 98, 2359 (1998). [17] D. J. Photinos, E. T. Samulski and H. Toriumi, J. Phys. Chem. 94, 4694 (1990). [18] J. M. Polson and E. E. Burnell, J. Chem. Phys. 103(16), 6891 (1995). [19] W. L. Meerts, C. A. de Lange, A. C. J. Weber and E. E. Burnell, J. Chem. Phys 130, 044504 (2009). [20] A. C. J. Weber, X. Yang, R. Y. Dong, W. L. Meerts and E. E. Burnell, Chem. Phys. Lett. 476, 238 (2009). 83  Chapter 4. Bibliography [21] W. L. Meerts, M. Schmitt and G. Groenenboom, Can. J. Chem. 82, 804 (2004). [22] W. L. Meerts and M. Schmitt, Int. Rev. Phys. Chem. 25, 353 (2006). [23] H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J. E. Knox, H. P. Hratchian, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, P. Y. Ayala, K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg, V. G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. V. Ortiz, Q. Cui, A. G. Baboul, S. Clifford, J. Cioslowski, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, C. Gonzalez and J. A. Pople, Gaussian, Inc 03, revision E.01, Wallingford, CT 2007. [24] C. M¨ oller and M. S. Plesset, Phys. Rev. 46, 61 (1934). [25] T. H. Dunning, Jr. J. Chem. Phys. 90, 1007 (1989). [26] A. J. van der Est, M. K. Kok and E. E. Burnell, Mol. Phys. 60, 397 (1987). [27] D. S. Zimmerman and E. E. Burnell, Mol. Phys. 78, 687 (1993). [28] D. J. Photinos, E. T. Samulski and H. Toriumi, J. Phys. Chem. 94, 4688 (1990). [29] J. B. S. Barnhoorn and C. A. de Lange, Mol. Phys. 88, 1 (1996). [30] E. E. Burnell, C. A. de Lange, J. B. S. Barnhoorn, I. Aben and P. F. Levelt, J. Phys. Chem. A 109, 11027 (2005). [31] E. E. Burnell and C. A. de Lange, Solid State Nucl. Magn. Reson. 28, 73 (2005). [32] C. A. de Lange, W. L. Meerts, A. C. J. Weber and E. E. Burnell, J. Phys. Chem. A 114, 5878 (2010). [33] A. C. J. Weber and E. E. Burnell, to be submitted for publication and is displayed in its current form in chapter 5 of this thesis (2010).  84  Chapter 5  Condensed Phase Configurational Statistics and Temperature Effects. ∗ Experimental dipolar couplings are obtained for n-butane in the liquid crystal 1132 over an 80 degree temperature range with a Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES). These observables are rationalized using a size and shape potential for orientational order, CI(2k), and also with a modified chord model, CCd, built especially for alkyl chains. The trans-gauche energy difference, Etg , is found to be temperature dependent and this observation is discussed.  ∗ A version of this chapter will be submitted for publication. A. C. J. Weber and E. E. Burnell (2010) Configurational Statistics of n-Butane in the Condensed Phase and the Effects of Temperature.  85  Chapter 5. Condensed Phase Configurational Statistics and Temperature Effects In the previous chapter [1] it was found that the effect of a liquid-crystal condensed phase on the configurational statistics of n-butane was largely dependent on the model chosen to describe the orientational potential. Specifically the CI(2k) model favoured the trans conformer whereas the Cd model favoured the gauche. The question arises; which orientational potential, if any, has got it right? One way around this could be a multitechnique maximum entropy (ME) approach [2] which requires the RIS approximation and a large amount of independent experimental results, but avoides the use of any mean-field orientational models. The method was initially tested on simulated data of substituted alkanes and it is found that while combining dielectric with 1 H NMR data improved ME results, this synergy is small if 2 H NMR are merged with 1 H NMR results [2]. However, that maximizing entropy defines the conformational-orientational solution via multiple data sets is not certain and much of the stated impetus for such methods was the difficulty of solving complicated 1 H NMR of numerous spectra which has been removed by the use of the CMA-ES. Later, ME methods in conjunction with a moment of inertia model for orientational order were used to fit proton dipolar couplings of the alkyl chains n-hexane through to n-decane in a nematic liquid crystal solvent and qualitative agreement with mean-field models was obtained [3]. Another approach combined an additive potential (AP) for orientational order with ME methods and successfully fit the dipolar couplings calculated from molecular dynamics simulations of the liquid-crystal 5CB in a low orientational order regime without assuming an intramolecular potential [4]. This hybrid additive potential maximum entropy approach was then applied to the experimental dipolar couplings of diphenylmethane dissolved in a nematic phase [5]. A conformational distribution function was obtained and it was found to be in agreement with the mean-field AP interpretation whereas the ME approach failed to extract any conformational information [5]. Considering the equation for conformation probability pn =  Gn exp(−Uniso /kT ) exp(−Unaniso (Ω)/kT )dΩ n iso aniso (Ω)/kT )dΩ n G exp(−Un /kT ) exp(−Un  (5.1)  it becomes clear, in the context of the previous chapter [1], that the only remaining variable  86  Chapter 5. Condensed Phase Configurational Statistics and Temperature Effects we can control in order to sort things out is the temperature and therefore the thermal energy kT . This is because Unaniso (Ω) is modeled with the CCd or CI(2k) orientational potential iso (which has been calculated with Gaussian 03) and U iso and Uniso is composed of Uint,n ext,n  which is the very effect we wish to determine. Now that obtaining spectral parameters for orientationally ordered n-butane is no longer a prohibitive task, we can obtain spectra over a large nematic temperature range with the hope that sufficiently changing the Boltzmann factor will reveal which orientational model is accounting for things properly and yield the correct Boltzmann statistics. n-Butane and 1,3,5-trichlorobenzene (tcb) were co-dissolved in the liquid-crystal solvent 1132 to concentrations of 5.0 mole % and 0.50 mole %. Since n-butane is a gas at room temperature and ambient pressure it was allowed to flow into a vacuum and then condensed into the 5 mm o. d. standard NMR tube which was submerged in liquid Nitrogen. After the sample was sealed and mixed thoroughly in the isotropic phase it was placed into a Bruker Avance 400 MHz NMR spectrometer magnet. With the temperature controlled by the Bruker air-flow system proton NMR spectra were acquired every 5 degrees from 253.5 K to 333.5 K. 1132 is in the nematic phase in this temperature range and its director will be aligned with the magnetic field direction. As seen in the previous chapter the CMA-ES can successfully fit the experimental spectra of orientationally ordered n-butane and the dipolar couplings obtained are presented in Table. 5.1 for an 80 degree temperature range. The seven highly accurate independent dipolar couplings are used to obtain four fitting parameters for each of two models namely two model prefactors for orientional ordering (w ˜n for CCd and k and ks for CI(2k)) with the ext and the CCH angle increase which accounts for vibrationalremaining two being the Etg  reorientational effects [6–9]. These fitting parameters are shown in Table. 5.2 and Table. 5.3 for the CCd and CI(2k) orientational models. The CCH angle increase is constant within error for the CCd model but changes slightly for the CI(2k) model. In Figure. 5.1 is presented the trans-gauche energy difference as a function of temperature. Etg is the sum gas ext Etg = Etg + Etg  (5.2) 87  Chapter 5. Condensed Phase Configurational Statistics and Temperature Effects and the dashed line in Figure. 5.1 represents the first term in this sum. Both models predict Etg values below the range of 788-884 cal/mol obtained from experiments in the gas phase [5–9] as expected. However, it is interesting to note that the CI(2k) values are within experimental Etg values in liquids while the CCd values are for the most part below this range of 502-681 cal/mol [10–13]. It is also interesting to note that while the CCd gas Etg values stay well below our Etg value calculated with Gaussian 03, the CI(2k) Etg  values are only slightly lower than this value and actually become slightly greater at lower temperature although the difference is just beyond three standard deviations and may not be significant. In fact both models show an increase (more significantly the CCd) in the sum Etg as temperature decreases which implies a greater preference for the trans conformer than given by the Boltzmann factor involving a constant Etg . One would expect the trans conformer population to increase as the thermal energy needed to populate the gauche conformation is decreased but it is perhaps not obvious why the Etg value would change so much thereby adding to the effect, especially by the amount displayed by the CCd model. It is customary to think of Etg as temperature independent [19, 20] but if it really does need to be written as gas ext Etg (T ) = Etg + Etg (T )  (5.3)  for a liquid crystal according to ones results than it seems appropriate to come up with some kind of hypothesis to account for the observation. A nematic phase will flow more readily when it is at a higher temperature than when the temperature is lower. When the temperature, and hence the thermal energy, kT , is higher attractive intermolecular interactions are broken up more frequently and so liquid crystal molecular motions are less constrained than when the thermal energy is reduced until eventually the nematic-crystal transition is reached when the attractive intermolecular interactions overcome the thermal energy. This is of course why the higher temperature nematic will be less dense than the lower temperature one. Of course one of the first things students learn in chemistry is that ‘like dissolves like’. So if liquid crystal rods are taking on less perpendicular-like angles as they pack closer at lower temperatures they’ll better dissolve the more rod-like trans  88  5.1. Acknowledgement n-butane conformer than the gauche based on space filling ‘entropic’ arguments. This may or may not be the case but it is interesting to find two substantially different mean-field models yield the same trend. It would be interesting to see what result obtains when one measures these things as a function of temperature in both the nematic and smectic A phases and such experiments are in progress.  5.1  Acknowledgement  Tables 5.2 and 5.3 are dedicated to Professor Emeritus Ronald Y. Dong.  5.2  Tables and Figures  89  5.2. Tables and Figures  Table 5.1: Experimental dipolar couplings of n-butane in 1132 as a function of temperature. The numbers in brackets are the errors in the last one or two digits. T (K) 253.5 258.5 263.5 268.5 273.5 278.5 283.5 288.5 293.5 298.5 303.5 308.5 313.5 318.5 323.5 328.5 333.5  D12 1736.06(5) 1667.50(5) 1601.40(6) 1539.36(5) 1482.70(5) 1416.19(5) 1352.90(4) 1290.98(4) 1230.41(4) 1142.45(4) 1078.84(4) 1013.60(4) 946.88(4) 879.72(4) 801.35(4) 714.14(4) 600.30(4)  D14 -436.90(4) -416.47(4) -397.31(4) -379.56(4) -363.57(4) -345.09(3) -327.79(3) -311.15(3) -295.11(3) -272.42(4) -256.13(3) -239.64(3) -223.00(3) -206.44(3) -187.41(3) -166.49(3) -139.47(3)  D16 -804.76(4) -777.60(4) -751.25(4) -725.89(3) -702.44(3) -674.36(3) -647.17(3) -620.16(3) -593.38(3) -553.43(3) -524.35(3) -494.20(3) -463.00(3) -431.36(3) -394.00(3) -352.01(3) -296.69(3)  D18 -369.21(4) -360.55(4) -352.35(4) -344.00(4) -335.91(4) -325.92(3) -315.80(3) -305.47(3) -294.87(3) -277.85(3) -265.46(3) -252.22(3) -238.12(3) -223.42(3) -205.49(3) -184.86(3) -156.75(3)  D45 3090.73(9) 2995.74(9) 2904.19(10) 2815.63(9) 2733.08(9) 2633.70(8) 2537.03(8) 2440.35(7) 2343.68(7) 2196.44(8) 2089.43(7) 1977.38(8) 1860.22(7) 1740.12(7) 1596.29(7) 1432.65(7) 1213.52(7)  D46 194.40(15) 180.89(15) 167.87(17) 156.99(14) 146.85(14) 135.22(13) 124.85(13) 115.16(11) 105.96(12) 94.37(12) 85.93(11) 77.80(11) 70.10(11) 62.91(11) 55.18(11) 47.04(11) 37.67(11)  D47 209.62(15) 188.01(15) 169.23(18) 151.26(14) 136.22(14) 119.56(13) 104.57(13) 90.95(11) 78.51(12) 63.55(12) 52.90(11) 43.11(11) 34.38(11) 26.31(11) 18.58(11) 12.06(11) 6.64(11)  90  5.2. Tables and Figures  Table 5.2: CCd fitting parameters as a function of temperature. The numbers in brackets are the errors in the last one or two digits. T (K) w ˜0 (cal mol−1 ) w ˜1 (cal mol−1 ) CCH angle increase (deg) RMS (Hz) 253.5 246.4(1.6) 211.1(1.6) 0.70(9) 6.5 258.5 243.8(1.3) 211.8(1.4) 0.71(9) 5.5 263.5 241.9(1.2) 211.9(1.2) 0.78(8) 4.6 268.5 239.1(9) 212.0(1.0) 0.77(7) 3.7 273.5 236.9(8) 211.8(9) 0.79(6) 3.1 278.5 233.4(7) 210.0(7) 0.85(5) 2.6 283.5 229.3(6) 208.1(6) 0.85(4) 2.2 288.5 225.0(6) 205.6(6) 0.86(4) 2.1 293.5 220.4(5) 202.6(6) 0.88(4) 1.8 298.5 210.5(6) 195.4(7) 0.89(5) 2.0 303.5 204.4(6) 190.3(7) 0.92(5) 2.1 308.5 197.2(7) 184.5(7) 0.94(6) 2.2 313.5 189.3(7) 177.7(8) 0.96(6) 2.3 318.5 180.5(7) 170.1(8) 0.97(7) 2.5 323.5 168.7(7) 159.8(9) 0.99(8) 2.6 328.5 154.3(7) 146.9(9) 1.01(9) 2.5 333.5 133.1(7) 127.9(8) 1.03(10) 2.3  91  5.2. Tables and Figures  Table 5.3: CI(2k) fitting parameters as a function of temperature. The numbers in the brackets are the errors in the last one or two digits. T (K) k (N m−1 ) ks (N m−1 ) CCH angle increase (deg) RMS (Hz) 253.5 5.3(2) 45.7(2.3) 1.14(9) 6.0 258.5 5.3(2) 45.9(2.5) 1.16(10) 6.8 263.5 5.2(2) 46.1(2.9) 1.20(12) 7.4 268.5 5.3(2) 44.1(3.1) 1.37(12) 7.5 273.5 5.2(2) 44.1(2.7) 1.40(12) 7.8 278.5 5.0(2) 45.8(3.0) 1.27(14) 8.3 283.5 4.8(2) 45.5(3.1) 1.30(14) 8.2 288.5 4.7(4) 45.1(4.7) 1.32(20) 8.2 293.5 4.6(3) 44.3(3.4) 1.34(16) 8.1 298.5 4.4(3) 43.1(3.8) 1.55(19) 7.6 303.5 4.3(3) 40.7(3.3) 1.37(17) 7.7 308.5 4.2(3) 38.9(3.4) 1.38(18) 7.5 313.5 4.1(3) 37.2(3.1) 1.39(18) 7.1 318.5 3.9(3) 35.2(3.1) 1.39(19) 6.7 323.5 3.7(2) 32.2(2.9) 1.40(19) 6.3 328.5 3.5(2) 28.6(2.6) 1.41(19) 5.7 333.5 3.1(2) 23.2(2.2) 1.41(19) 4.8  92  5.2. Tables and Figures  700  600  500  400 260  280  300  320  Figure 5.1: The Etg of n-butane as a function of temperature when the orientational potential is described by the CI (open points) and Cd (filled points) models. The dashed line gas represents the constant Etg calculated from Gaussian 03.  93  Bibliography [1] A. C. J. Weber, C. A. de Lange, W. L. Meerts and E. E. Burnell, has been submitted for publication in chem. phys. lett. (2010). [2] R. Berardi, F. Spinozzi and C. Zannoni, J. Chem. Phys. 109, 3742 (1998). [3] Y. Sasanuma, Polym. J. 32, 883 (2000). [4] B. Stevensson, D. Sandstr¨om and A. Maliniak, J. Chem. Phys. 119, 2738 (2003). [5] J. Thaning, B. Stevensson and A. Maliniak, J. Chem. Phys. 123, 044507 (2005). [6] J. B. S. Barnhoorn and C. A. de Lange, Mol. Phys. 88, 1 (1996). [7] E. E. Burnell, C. A. de Lange, J. B. S. Barnhoorn, I. Aben and P. F. Levelt, J. Phys. Chem. A 109, 11027 (2005). [8] E. E. Burnell and C. A. de Lange, Solid State Nucl. Magn. Reson. 28, 73 (2005). [9] C. A. de Lange, W. L. Meerts, A. C. J. Weber and E. E. Burnell, J. Phys. Chem. A 114, 5878 (2010). [10] P. B. Woller and E. W. Garbisch Jr, J. Amer. Chem. Soc. 94, 5310 (1971). [11] A. L. Verma, W. F. Murphy and H. J. Bernstein, J. Chem. Phys. 60, 1540 (1974). [12] J. R. Durig and D. A. C. Compton, J. Phys. Chem. 83, 265 (1979). [13] D. A. C. Compton, S. Montero and W. F. Murphy, J. Phys. Chem. 84, 3587 (1980). [14] R. K. Heenan and L. S. Bartell, J. Chem. Phys. 78, 1270 (1983). [15] K. B. Wiberg and M. A. Murcko, J. Am. Chem. Soc. 110, 8029 (1988). [16] L. Colombo and G. Zerbi, J. Chem. Phys. 73, 2013 (1980). [17] S. Kint, J. R. Scherer and R. G. Snyder, J. Chem. Phys. 73, 1599 (1980). [18] D. A. Cates and A. MacPhail, J. Phys. Chem. 95, 2209 (1991). [19] P. G. Flory, Statistical Mechanics of Chain Molecules ( New York, Wiley-Interscience, 1969). [20] J. W. Emsley, Nuclear Magnetic Resonance of Liquid Crystals ( Boston, D. Reidel Publishing Company, 1983).  94  Chapter 6  Conclusion Spectral parameters are obtained simultaneously and automatically by CMA-ES analysis for judiciously chosen molecules, orientationally ordered in liquid-crystal phases, in order to better understand nematic and smectic A phases and the properties of molecules dissolved therein [1, 2]. To learn about the potential giving rise to orientational ordering in the nematic and smectic A phases of the liquid crystals 8CB and 8OCB a sizable collection of solutes with a variety of shapes, sizes and electronic configurations were used as probes. The very accurate dipolar couplings of these probe molecules are used to obtain orientational order parameters. These order parameters are related to the ordering potential via statistical mechanics so one is in a position to postulate some general form of the potential and then obtain an explicit form by least-squares fitting to the measured observables. Because of the previous success of a double Maier-Saupe (MSMS) potential in reproducing orientational order parameters to within 5 % for solutes dissolved in several nematic liquid-crystal solvents [3], it was again postulated for the nematic phases of 8CB and 8OCB which also form smectic A phases at lower temperature. Because the smectic effect on orientational order parameters is relatively small, it was necessary to extrapolate the change in this nematic potential from the nematic phase into the smectic. The orientational ordering that the MSMS potential was unable to account for in the smectic A phase is then allowed to be absorbed by Kobayashi-McMillan terms which involved two nematic-smectic coupling terms (one for each MS mechanism) and a purely smectic term. This combined MSMS-KM Hamiltonian can then be seen to rationalize NMR experiments in the smectic A phase reasonably well and allowed for the calculation of solute smectic order parameters. When one obtains a Hamiltonian by fitting to observables the best thing to do is to ask it some questions. It is interesting to find that the second orientational mechanism is substantially modulated by the smectic layering and undergoes periodic sign changes in 95  Chapter 6. Conclusion the direction normal to the layers. Moreover, when the modulation of the two nematic mechanisms are considered they are observed to be consistent with one being due to size and shape effects and the second involving the solute quadrupole. Having spent some time asking what solutes can tell us about a condensed phase environment one is compelled to ask what is the effect of a condensed phase environment on a solute. The effect of a liquid-crystal phase on molecular structure is negligible [4]. However, in molecules undergoing large amplitude vibrations the effect of a condensed phase is not as clear. To better understand these phenomena dipolar couplings of n-butane were measured in four different nematic phases [5]. The first problem that arises is that the conformer probability function is multiplied by the order matrix in the equation for the observed dipolar couplings and so to proceed one needs to assume a model for the orientational ordering potential. To this end we employed one model based on size-and-shape and another invoking orientational ordering torques associated with each bond and lines joining mid-points of bonds known as the CI and CCd models. A potential for the internal motion was calculated with Gaussian 03 and the gauche wells in the potential were allowed to be ext which represents the shifted with respect to the trans well by the fitting parameter Etg  contribution of the condensed phase to the trans-gauche energy difference. It was then fascinating to find that the configurational statistics obtained were not dependent on which liquid crystal is used for the condensed-phase environment, how we treat the available conformers and order matrices or whether or not any vibrational-reorientational effects are accounted for. Instead the crux of the matter is dependent only on which orientational ordering potential is assumed. It seems the extent to which a model calls for more or less ordering in a given conformer is compensated by a decrease or increase in its population in order to fit the dipolar couplings. To better understand this observation, dipolar couplings of n-butane were obtained in the liquid crystal 1132 over an 80 degree temperature range. The trans-gauche energy difference is observed to be a decreasing function of temperature and this is tentatively explained with simple entropic space filling arguments. The Etg ’s calculated with the CI model are found to fall within the range found from condensed phase experiments while those calculated with the Cd model for the most 96  Chapter 6. Conclusion part did not. In summary, it has been demonstrated that a combined MSMS-KM Hamiltonian can be used to rationalize NMR observables in smectic phases reasonably well and yields valuable information regarding the nature of the phase and the orientational ordering mechanisms therein. It has also been shown that the MSMS potential is capable of describing nematic orientational ordering as a function of temperature. The effect of the condensed phase on the configurational statistics of n-butane is observed to depend on the choice of orientational ordering model and the trans-gauche energy difference is observed to be a decreasing function of temperature but much more so according to the Cd model compared to the CI model where the temperature variation in the Etg may not be significant. Solutes were used to probe the intermolecular environments of simple liquid-crystal phases and the effects of the nematic phase on molecular conformational statistics was investigated. In future research one could consider what effect a layered liquid-crystal phase would have on the configurational statistics of n-butane. It would also be interesting to see if theoretical considerations could make any sense of the n-butane trans-gauche energy difference being temperature dependent in the condensed phase. An interesting challenge would be to see what the limits of the CMA-ES in NMR spectroscopy are. It is likely that the single quantum spectrum of n-hexane can be solved with CMA-ES but at what point (length of alkyl chain) does the spectrum become too featureless for ES to decipher a global minimum on the error surface? Although some progress towards refining our picture of the intermolecular environment of some simple liquid-crystal phases was made, the problem of the exact nature of the orientational potential reared its head when the question of the effect of the condensed phase on individual molecules was asked. This problem is as stubborn as it is old but future experiments should shed light on this problem incrementally as was found here.  97  Bibliography [1] A. C. J. Weber, X. Yang, R. Y. Dong, W. L. Meerts and E. E. Burnell, Chem. Phys. Lett. 476, 116 (2009). [2] A. C. J. Weber, X. Yang, R. Y. Dong and E. E. Burnell, J. Chem. Phys. 132, 034503 (2010). [3] E. E. Burnell, L. C. ter Beek and Z. Sun, J. Chem. Phys. 128, 164901 (2008). [4] C. A. de Lange, W. L. Meerts, A. C. J. Weber and E. E. Burnell, J. Phys. Chem. A 114, 5878 (2010). [5] A. C. J. Weber, C. A. de Lange, W. L. Meerts and E. E. Burnell, has been accepted for publication in Chem. Phys. Lett. (2010).  98  Appendix A  Experimental detail concerning chapters 2 to 5 The temperature was calibrated with a sample of 4% methanol in methanol-D4 . Specifically the chemical shift difference, ∆δ, between the OH and CH3 protons is used in conjunction with the empirical relationship T = (4.109 − ∆δ)/0.008708  (A.1)  to correct the temperature reported by the thermocouple which is accurate to ± 0.5 K. Sample spinning was employed to sharpen line widths and to mitigate concentration gradients. The following acquisition parameters were used: acquisition time=1.0 second number of scans=80 number of points=32768 line broadening=0.1 Hz pulse width=9.0 µ seconds  99  Appendix B  Experimental order parameters  T/K 320.96 321.96 322.96 323.96 324.96 325.96 326.96 327.96 328.96 329.96 330.96 331.96 332.96 333.96 334.96 335.96 336.46 336.96 337.46 337.96 338.46  Table B.1: tcbSzz -0.23310 -0.23094 -0.22967 -0.22738 -0.22627 -0.22392 -0.22141 -0.21907 -0.21685 -0.21494 -0.21007 -0.20827 -0.20140 -0.19454 -0.18882 -0.18331 -0.18026 -0.17790 -0.17428 -0.17143 -0.16863  Experimental order furSzz furSxx -0.15538 0.04177 -0.15387 0.04163 -0.15224 0.04142 -0.15050 0.04126 -0.14872 0.04103 -0.14693 0.04081 -0.14503 0.04050 -0.14295 0.04022 -0.14079 0.03989 -0.13868 0.03964 -0.13540 0.03903 -0.13339 0.03867 -0.12878 0.03776 -0.12277 0.03646 -0.11855 0.03556 -0.11473 0.03470 -0.11274 0.03426 -0.11061 0.03374 -0.10858 0.03330 -0.10644 0.03278 -0.10445 0.03229  parameters of sample 1 in 8OCB thiSzz thiSxx clproSzz clproSxx -0.16634 0.09077 -0.00400 -0.00919 -0.16466 0.08995 -0.00414 -0.00891 -0.16293 0.08910 -0.00429 -0.00860 -0.16107 0.08822 -0.00448 -0.00826 -0.15920 0.08728 -0.00462 -0.00796 -0.15726 0.08634 -0.00481 -0.00760 -0.15528 0.08535 -0.00501 -0.00723 -0.15304 0.08425 -0.00523 -0.00682 -0.15079 0.08310 -0.00548 -0.00636 -0.14845 0.08202 -0.00569 -0.00600 -0.14491 0.08027 -0.00598 -0.00541 -0.14278 0.07918 -0.00621 -0.00499 -0.13781 0.07667 -0.00661 -0.00424 -0.13134 0.07331 -0.00695 -0.00340 -0.12696 0.07105 -0.00699 -0.00305 -0.12270 0.06881 -0.00718 -0.00243 -0.12056 0.06768 -0.00713 -0.00233 -0.11829 0.06644 -0.00710 -0.00216 -0.11611 0.06528 -0.00705 -0.00206 -0.11380 0.06403 -0.00696 -0.00197 -0.11160 0.06286 -0.00693 -0.00180  100  Appendix B. Experimental order parameters  Table B.2: Experimental order parameters of sample 2 in 8OCB T/K 316.50 317.50 318.50 319.50 320.50 321.50 322.50 323.50 324.50 325.00 325.50 326.00 326.50 327.00 327.50 328.00 328.50 329.00 329.50 330.00 330.50 331.00 331.50 332.00 332.50 333.00 333.50 334.00 334.50 335.50 336.50 337.50 338.50 339.50 340.50 341.50  tcbSzz -0.24532 -0.24377 -0.24204 -0.24039 -0.23881 -0.23714 -0.23497 -0.23356 -0.23159 -0.23079 -0.22971 -0.22870 -0.22777 -0.22681 -0.22562 -0.22484 -0.22345 -0.22227 -0.22123 -0.21984 -0.21825 -0.21679 -0.21551 -0.21365 -0.21188 -0.20973 -0.20587 -0.20286 -0.20058 -0.19539 -0.19082 -0.18605 -0.18124 -0.17565 -0.16996 -0.16339  pbbnSyy -0.13049 -0.12906 -0.12776 -0.12627 -0.12481 -0.12336 -0.12203 -0.12048 -0.11897 -0.11828 -0.11749 -0.11648 -0.11577 -0.11493 -0.11413 -0.11335 -0.11249 -0.11157 -0.11064 -0.10967 -0.10866 -0.10784 -0.10650 -0.10537 -0.10407 -0.10240 -0.10004 -0.09842 -0.09682 -0.09309 -0.09032 -0.08738 -0.08407 -0.08104 -0.07746 -0.07368  pbbnSxx 0.40875 0.40560 0.40252 0.39917 0.39593 0.39246 0.38904 0.38559 0.38188 0.37995 0.37801 0.37585 0.37408 0.37193 0.36998 0.36778 0.36554 0.36324 0.36066 0.35823 0.35549 0.35252 0.34965 0.34640 0.34264 0.33847 0.33104 0.32596 0.32151 0.31178 0.30355 0.29492 0.28623 0.27682 0.26683 0.25557  dcnbSyy 0.06686 0.06733 0.00000 0.06837 0.06882 0.06923 0.06961 0.07002 0.07043 0.07065 0.07075 0.07093 0.07106 0.07122 0.07139 0.07152 0.07163 0.07174 0.07185 0.07198 0.07203 0.07210 0.07214 0.07216 0.07217 0.07210 0.07164 0.07134 0.07108 0.07015 0.06927 0.06818 0.06700 0.06552 0.06380 0.06190  dcnbSxx 0.21123 0.20907 0.22763 0.20457 0.20230 0.19999 0.19767 0.19532 0.19280 0.19150 0.19029 0.18898 0.18768 0.18631 0.18495 0.18354 0.18204 0.18058 0.17903 0.17733 0.17566 0.17384 0.17194 0.16989 0.16755 0.16485 0.16031 0.15737 0.15485 0.14960 0.14532 0.14109 0.13670 0.13217 0.12745 0.12216  hexSzz 0.39608 0.39258 0.38904 0.38541 0.38183 0.37817 0.37442 0.37065 0.36672 0.36473 0.36275 0.36073 0.35862 0.35651 0.35435 0.35216 0.34987 0.34755 0.34513 0.34257 0.33995 0.33713 0.33417 0.33096 0.32741 0.32319 0.31617 0.31152 0.30730 0.29734 0.28922 0.28086 0.27212 0.26280 0.25285 0.24182  phacSxx 0.36803 0.36484 0.36145 0.35756 0.35356 0.35006 0.34629 0.34153 0.33787 0.33557 0.33380 0.33098 0.32939 0.32721 0.32503 0.32279 0.32040 0.31803 0.31483 0.31234 0.30975 0.30706 0.30405 0.30088 0.29798 0.29324 0.28653 0.28176 0.27744 0.26794 0.26001 0.25239 0.24462 0.23673 0.22818 0.21747  phacSyy -0.10015 -0.09847 -0.09693 -0.09514 -0.09335 -0.09194 -0.08995 -0.08824 -0.08645 -0.08552 -0.08470 -0.08380 -0.08302 -0.08213 -0.08115 -0.08021 -0.07923 -0.07835 -0.07721 -0.07627 -0.07518 -0.07429 -0.07327 -0.07211 -0.07086 -0.06938 -0.06733 -0.06586 -0.06437 -0.06124 -0.05864 -0.05629 -0.05377 -0.05150 -0.04883 -0.04580  101  Appendix B. Experimental order parameters  T/K 316.30 316.80 317.30 317.80 318.30 318.80 319.30 319.80 320.30 320.80 321.30 321.80 322.30 322.80 323.30 323.80 324.30 324.80 325.30 325.80 326.30 326.80 327.30 327.80 328.30 328.80 329.30 329.80 330.30 330.80 331.30 331.80 332.30 332.80 333.30 333.80 334.30 334.80 335.80 336.80 337.80 338.80  Table B.3: Experimental order tcbSzz odcbSyy odcbSxx -0.24221 0.05199 0.18858 -0.24130 0.05219 0.18742 -0.24070 0.05232 0.18665 -0.23979 0.05250 0.18548 -0.23896 0.05267 0.18440 -0.23831 0.05279 0.18360 -0.23733 0.05299 0.18237 -0.23647 0.05313 0.18127 -0.23579 0.05325 0.18043 -0.23478 0.05342 0.17917 -0.23407 0.05353 0.17827 -0.23314 0.05368 0.17714 -0.23218 0.05382 0.17595 -0.23148 0.05392 0.17509 -0.23038 0.05408 0.17376 -0.22952 0.05418 0.17273 -0.22863 0.05429 0.17163 -0.22760 0.05438 0.17040 -0.22683 0.05449 0.16947 -0.22565 0.05461 0.16806 -0.22476 0.05469 0.16700 -0.22366 0.05478 0.16571 -0.22249 0.05487 0.16436 -0.22165 0.05492 0.16339 -0.22026 0.05500 0.16179 -0.21917 0.05505 0.16054 -0.21796 0.05509 0.15918 -0.21657 0.05514 0.15762 -0.21544 0.05514 0.15635 -0.21371 0.05516 0.15449 -0.21218 0.05510 0.15287 -0.21044 0.05509 0.15100 -0.20860 0.05504 0.14903 -0.20679 0.05495 0.14714 -0.20383 0.05477 0.14414 -0.19878 0.05416 0.13941 -0.19482 0.05360 0.13577 -0.19151 0.05311 0.13277 -0.18664 0.05234 0.12852 -0.18008 0.05117 0.12299 -0.17370 0.04992 0.11785 -0.16731 0.04858 0.11281  parameters of sample mdcbSxx mdcbSyy 0.17175 0.06509 0.17133 0.06459 0.17106 0.06427 0.17064 0.06377 0.17025 0.06330 0.16995 0.06298 0.16950 0.06245 0.16910 0.06197 0.16878 0.06161 0.16832 0.06105 0.16797 0.06068 0.16755 0.06017 0.16709 0.05965 0.16676 0.05928 0.16625 0.05869 0.16584 0.05822 0.16541 0.05773 0.16493 0.05716 0.16455 0.05676 0.16398 0.05614 0.16354 0.05564 0.16301 0.05505 0.16244 0.05444 0.16203 0.05399 0.16134 0.05326 0.16079 0.05269 0.16018 0.05207 0.15946 0.05137 0.15889 0.05073 0.15799 0.04991 0.15719 0.04914 0.15627 0.04828 0.15529 0.04737 0.15428 0.04658 0.15270 0.04510 0.14966 0.04334 0.14732 0.04138 0.14505 0.04039 0.14178 0.03877 0.13716 0.03675 0.13248 0.03524 0.12774 0.03366  3 in 8OCB pdcbSyy -0.04732 -0.04710 -0.04696 -0.04674 -0.04655 -0.04640 -0.04619 -0.04599 -0.04584 -0.04562 -0.04547 -0.04527 -0.04508 -0.04493 -0.04471 -0.04453 -0.04436 -0.04416 -0.04401 -0.04379 -0.04363 -0.04343 -0.04322 -0.04308 -0.04284 -0.04265 -0.04246 -0.04223 -0.04205 -0.04178 -0.04156 -0.04130 -0.04105 -0.04079 -0.04041 -0.03992 -0.03923 -0.03853 -0.03754 -0.03604 -0.03458 -0.03311  pdcbSxx 0.29201 0.29090 0.29016 0.28904 0.28800 0.28722 0.28603 0.28496 0.28414 0.28291 0.28202 0.28091 0.27973 0.27888 0.27755 0.27652 0.27542 0.27417 0.27323 0.27180 0.27071 0.26939 0.26799 0.26697 0.26530 0.26399 0.26254 0.26087 0.25950 0.25748 0.25564 0.25359 0.25142 0.24927 0.24584 0.24010 0.23527 0.23118 0.22516 0.21692 0.20894 0.20094  102  Appendix B. Experimental order parameters  Table B.4: Experimental order parameters of sample 1 in 8CB T/K 292.50 293.50 294.50 295.00 295.50 296.00 296.50 297.00 297.50 298.00 298.50 299.00 299.50 300.00 300.50 301.00 301.50 302.00 302.50 303.00 303.50 304.00 304.50 305.00 305.50 306.00 307.00  tcbSzz -0.24646 -0.24400 -0.24143 -0.24049 -0.23906 -0.23771 -0.23605 -0.23434 -0.23275 -0.23090 -0.23065 -0.22839 -0.22618 -0.22373 -0.22034 -0.21636 -0.20752 -0.20186 -0.19735 -0.19296 -0.18887 -0.18419 -0.17584 -0.17110 -0.16538 -0.15845 -0.14168  hexSzz 0.39328 0.38889 0.38412 0.38161 0.37906 0.37640 0.37367 0.37055 0.36720 0.36357 0.36283 0.35918 0.35517 0.35051 0.34504 0.33782 0.32273 0.31318 0.30547 0.29801 0.29056 0.28300 0.26879 0.26034 0.25076 0.23978 0.21271  dcnbSyy 0.04108 0.04292 0.04481 0.04563 0.04662 0.04750 0.04854 0.04942 0.05040 0.05126 0.05159 0.05245 0.05348 0.05446 0.05543 0.05645 0.05739 0.05765 0.05760 0.05753 0.05720 0.05669 0.05542 0.05466 0.05348 0.05203 0.04759  dcnbSxx 0.24437 0.23997 0.23526 0.23284 0.23034 0.22780 0.22507 0.22215 0.21898 0.21567 0.21497 0.21164 0.20796 0.20379 0.19901 0.19284 0.18077 0.17381 0.16860 0.16367 0.15908 0.15457 0.14640 0.14154 0.13628 0.13028 0.11578  phacSxx 0.38601 0.38072 0.37452 0.37138 0.36833 0.36542 0.36218 0.35869 0.35495 0.35077 0.34971 0.34580 0.34095 0.33625 0.32949 0.32232 0.30653 0.29666 0.28883 0.28126 0.27324 0.26630 0.25276 0.24408 0.23451 0.22470 0.19975  phacyy -0.11293 -0.11017 -0.10755 -0.10622 -0.10498 -0.10355 -0.10209 -0.10055 -0.09897 -0.09722 -0.09676 -0.09513 -0.09330 -0.09136 -0.08890 -0.08616 -0.08087 -0.07743 -0.07476 -0.07216 -0.06949 -0.06717 -0.06267 -0.05997 -0.05704 -0.05399 -0.04678  pbbnSyy -0.13854 -0.13688 -0.13416 -0.13282 -0.13156 -0.12980 -0.12858 -0.12709 -0.12554 -0.12426 -0.12367 -0.12195 -0.12035 -0.11809 -0.11597 -0.11289 -0.10705 -0.10312 -0.10010 -0.09723 -0.09440 -0.09156 -0.08611 -0.08287 -0.07946 -0.07557 -0.06603  pbbnxx 0.42071 0.41583 0.41061 0.40794 0.40525 0.40265 0.40033 0.39729 0.39364 0.38935 0.38885 0.38477 0.38029 0.37510 0.36895 0.36083 0.34385 0.33337 0.32491 0.31680 0.30888 0.30081 0.28589 0.27704 0.26699 0.25545 0.22700  103  Appendix B. Experimental order parameters  Table B.5: Experimental order parameters of sample 2 in 8CB T/K 292.46 293.46 294.46 295.46 296.36 297.36 297.96 298.36 298.86 299.46 299.96 300.46 300.96 301.46 301.96 302.46 302.96 303.46 303.96 304.46 304.96 305.46 305.96  tcbSzz -0.242809 -0.240622 -0.238475 -0.236072 -0.233412 -0.230483 -0.228390 -0.227188 -0.225069 -0.222395 -0.221234 -0.218615 -0.214726 -0.206045 -0.198619 -0.194056 -0.189762 -0.187130 -0.180446 -0.175505 -0.170361 -0.168592 -0.163111  furSxx 0.040413 0.040232 0.039970 0.039782 0.039562 0.039149 0.038899 0.038820 0.038628 0.038204 0.038009 0.037671 0.037156 0.035768 0.034688 0.034019 0.033362 0.032928 0.031941 0.031134 0.030306 0.030048 0.029126  furSzz -0.168961 -0.166991 -0.164801 -0.162576 -0.160169 -0.157458 -0.155544 -0.154397 -0.152525 -0.150218 -0.149196 -0.146898 -0.143695 -0.136690 -0.131003 -0.127492 -0.124196 -0.122040 -0.117219 -0.113527 -0.109745 -0.108485 -0.104522  thiSxx 0.094450 0.093478 0.092409 0.091289 0.090073 0.088714 0.087765 0.087171 0.086250 0.085076 0.084553 0.083375 0.081723 0.078032 0.075015 0.073100 0.071307 0.070188 0.067482 0.065407 0.063333 0.062613 0.060348  thiSzz -0.178213 -0.176151 -0.173872 -0.171548 -0.168957 -0.166118 -0.164135 -0.162888 -0.160904 -0.158498 -0.157404 -0.154975 -0.151560 -0.144096 -0.138105 -0.134312 -0.130802 -0.128571 -0.123400 -0.119460 -0.115530 -0.114185 -0.109913  clproSxx -0.014538 -0.014021 -0.013419 -0.012839 -0.012177 -0.011552 -0.011073 -0.010766 -0.010299 -0.009752 -0.009525 -0.009021 -0.008314 -0.006930 -0.006011 -0.005499 -0.005091 -0.004877 -0.004361 -0.004022 -0.003719 -0.003637 -0.003359  clproSzz -0.001969 -0.002272 -0.002642 -0.002986 -0.003387 -0.003749 -0.004040 -0.004227 -0.004494 -0.004800 -0.004941 -0.005228 -0.005626 -0.006330 -0.006699 -0.006860 -0.006951 -0.006972 -0.006997 -0.006971 -0.006903 -0.006862 -0.006747  104  Appendix B. Experimental order parameters  T/K 290.96 291.96 292.96 293.46 293.96 294.46 294.96 295.46 295.96 296.46 296.96 297.46 297.96 298.46 298.96 299.46 299.96 300.46 300.96 301.46 301.96 302.46 302.96  Table B.6: Experimental order parameters of sample 3 in 8CB tcbSzz odcbSxx odcbSyy mdcbSxx mdcbSyy pdcbSxx -0.24382 0.19987 0.04473 0.07593 0.16428 0.28912 -0.24226 0.19666 0.04551 0.07450 0.16367 0.28666 -0.23925 0.19349 0.04615 0.07268 0.16314 0.28412 -0.23828 0.19185 0.04647 0.07180 0.16283 0.28280 -0.23717 0.19020 0.04690 0.07090 0.16252 0.28150 -0.23586 0.18838 0.04719 0.06992 0.16215 0.27999 -0.23464 0.18658 0.04755 0.06901 0.16173 0.27851 -0.23322 0.18472 0.04791 0.06795 0.16134 0.27693 -0.23173 0.18280 0.04820 0.06716 0.16080 0.27529 -0.23026 0.18075 0.04857 0.06592 0.16035 0.27356 -0.22876 0.17874 0.04887 0.06495 0.15977 0.27174 -0.22701 0.17651 0.04917 0.06360 0.15922 0.26979 -0.22511 0.17413 0.04945 0.06251 0.15844 0.26758 -0.22011 0.16832 0.04974 0.05949 0.15625 0.26184 -0.21680 0.16455 0.04996 0.05754 0.15484 0.25807 -0.21262 0.15988 0.05012 0.05567 0.15278 0.25326 -0.20620 0.15293 0.05000 0.05192 0.14955 0.24570 -0.19314 0.14032 0.04890 0.04627 0.14208 0.23046 -0.18641 0.13419 0.04814 0.04379 0.13793 0.22246 -0.18071 0.12911 0.04731 0.04195 0.13410 0.21545 -0.17542 0.12456 0.04648 0.04026 0.13048 0.20894 -0.16926 0.11947 0.04540 0.03847 0.12617 0.20139 -0.16260 0.11396 0.04418 0.03682 0.12128 0.19311  pdcbSyy -0.04249 -0.04230 -0.04193 -0.04178 -0.04167 -0.04142 -0.04129 -0.04110 -0.04093 -0.04074 -0.04056 -0.04026 -0.04009 -0.03936 -0.03904 -0.03853 -0.03770 -0.03582 -0.03466 -0.03340 -0.03232 -0.03104 -0.02960  105  Appendix C  Structural details of n-butane calculated with Gaussian 03 Table C.1: Atom labels of n-butane nuclei and bonding Atom Atom type Atom attached to 1 C 2 C 1 3 H 1 4 H 1 5 H 1 6 C 2 7 H 2 8 H 2 9 C 6 10 H 6 11 H 6 12 H 9 13 H 9 14 H 9  106  Appendix C. Structural details of n-butane calculated with Gaussian 03  Table C.2: Bond lengths Parameter label R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 A19 A20 A21 A22 A23 A24  and angles of trans n-butane Nuclei Value R(1,2) 1.5302 R(1,3) 1.1026 R(1,4) 1.1037 R(1,5) 1.1037 R(2,6) 1.5311 R(2,7) 1.1066 R(2,8) 1.1066 R(6,9) 1.5302 R(6,10) 1.1066 R(6,11) 1.1066 R(9,12) 1.1026 R(9,13) 1.1037 R(9,14) 1.1037 A(2,1,3) 111.6531 A(2,1,4) 110.8641 A(2,1,5) 110.8641 A(3,1,4) 107.8379 A(3,1,5) 107.8379 A(4,1,5) 107.6183 A(1,2,6) 112.8465 A(1,2,7) 109.6932 A(1,2,8) 109.6932 A(6,2,7) 109.0822 A(6,2,8) 109.0823 A(7,2,8) 106.2241 A(2,6,9) 112.8465 A(2,6,10) 109.0822 A(2,6,11) 109.0822 A(9,6,10) 109.6932 A(9,6,11) 109.6932 A(10,6,11) 106.2241 A(6,9,12) 111.6531 A(6,9,13) 110.8641 A(6,9,14) 110.8641 A(12,9,13) 107.8379 A(12,9,14) 107.8379 A(13,9,14) 107.6183  107  Appendix C. Structural details of n-butane calculated with Gaussian 03  Table C.3: Dihedral angles of trans n-butane Parameter label Nuclei Value D1 D(3,1,2,6) 180.0 D2 D(3,1,2,7) -58.157 D3 D(3,1,2,8) 58.1571 D4 D(4,1,2,6) -59.7333 D5 D(4,1,2,7) 62.1096 D6 D(4,1,2,8) 178.4238 D7 D(5,1,2,6) 59.7334 D8 D(5,1,2,7) -178.4237 D9 D(5,1,2,8) -62.1096 D10 D(1,2,6,9) 180.0 D11 D(1,2,6,10) -57.8132 D12 D(1,2,6,11) 57.8132 D13 D(7,2,6,9) 57.8132 D14 D(7,2,6,10) 180.0 D15 D(7,2,6,11) -64.3736 D16 D(8,2,6,9) -57.8132 D17 D(8,2,6,10) 64.3736 D18 D(8,2,6,11) -180.0 D19 D(2,6,9,12) -179.9998 D20 D(2,6,9,13) -59.7331 D21 D(2,6,9,14) 59.7335 D22 D(10,6,9,12) 58.1573 D23 D(10,6,9,13) 178.424 D24 D(10,6,9,14) -62.1094 D25 D(11,6,9,12) -58.1569 D26 D(11,6,9,13) 62.1098 D27 D(11,6,9,14) -178.4236  108  Appendix C. Structural details of n-butane calculated with Gaussian 03  Table C.4: Bond lengths Parameter label R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 A19 A20 A21 A22 A23 A24  and angles of Nuclei R(1,2) R(1,3) R(1,4) R(1,5) R(2,6) R(2,7) R(2,8) R(6,9) R(6,10) R(6,11) R(9,12) R(9,13) R(9,14) A(2,1,3) A(2,1,4) A(2,1,5) A(3,1,4) A(3,1,5) A(4,1,5) A(1,2,6) A(1,2,7) A(1,2,8) A(6,2,7) A(6,2,8) A(7,2,8) A(2,6,9) A(2,6,10) A(2,6,11) A(9,6,10) A(9,6,11) A(10,6,11) A(6,9,12) A(6,9,13) A(6,9,14) A(12,9,13) A(12,9,14) A(13,9,14)  gauche n-butane Value 1.5317 1.1027 1.1024 1.1044 1.5348 1.1069 1.1054 1.5317 1.1054 1.1069 1.1027 1.1024 1.1044 111.1545 111.7955 110.7421 107.4638 107.7576 107.746 113.7255 109.5375 109.0882 109.0601 108.8857 106.2746 113.7255 108.8857 109.0601 109.0882 109.5375 106.2746 111.1545 111.7955 110.7421 107.4638 107.7576 107.746  109  Appendix C. Structural details of n-butane calculated with Gaussian 03  Table C.5: Dihedral angles of gauche n-butane Parameter label Nuclei Value D1 D(3,1,2,6) 175.9641 D2 D(3,1,2,7) -61.7287 D3 D(3,1,2,8) 54.2134 D4 D(4,1,2,6) -63.9499 D5 D(4,1,2,7) 58.3574 D6 D(4,1,2,8) 174.2994 D7 D(5,1,2,6) 56.2125 D8 D(5,1,2,7) 178.5198 D9 D(5,1,2,8) -65.5381 D10 D(1,2,6,9) 63.3244 D11 D(1,2,6,10) -174.8126 D12 D(1,2,6,11) -59.2459 D13 D(7,2,6,9) -59.2459 D14 D(7,2,6,10) 62.6171 D15 D(7,2,6,11) 178.1839 D16 D(8,2,6,9) -174.8126 D17 D(8,2,6,10) -52.9496 D18 D(8,2,6,11) 62.6171 D19 D(2,6,9,12) 175.9644 D20 D(2,6,9,13) -63.9496 D21 D(2,6,9,14) 56.2128 D22 D(10,6,9,12) 54.2137 D23 D(10,6,9,13) 174.2997 D24 D(10,6,9,14) -65.5379 D25 D(11,6,9,12) -61.7284 D26 D(11,6,9,13) 58.3577 D27 D(11,6,9,14) 178.5201  110  

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