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Gigahertz spectroscopy of hydrated polyunsaturated phospholipid membranes : an investigation of dielectric… Kurylowicz, Martin 2003

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G i g a h e r t z S p e c t r o s c o p y o f H y d r a t e d P o l y u n s a t u r a t e d P h o s p h o l i p i d M e m b r a n e s A n Invest igat ion of Die lec t r i c P rope r t i e s by Martin Kurylowicz B.Sc , The University of British Columbia, 2000 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T OF T H E R E Q U I R E M E N T S FOR T H E D E G R E E OF M A S T E R OF S C I E N C E in The Faculty of Graduate Studies (Department of Physics and Astronomy) We accept this thesis as conforming to the required standard EI~Z3 T H E U N I V E R S I T Y 0 F BRITISH C O L U M B I A September 9, 2003 © Martin Kurylowicz, 2003 . In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my depart-ment or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and Astronomy The University Of British Columbia 6224 Agricultural Rd., Vancouver, B C , V6T 1Z1, Canada Abstract ii Abstract A n experimental technique was developed to measure the complex dielectric constant for small volumes of lossy liquids between 45 MHz and 50 GHz, and from 5°C to 55°C. The study was motivated by a desire to investigate the role of water in bio-logical materials, and to explore the relevance of dielectric properties to fundamental questions in biophysics. Bulk samples of hydrated phospholipid membranes were used as a model system in which water plays a significant role, and polyunsaturated membranes were tested in comparison with saturated and mono-unsaturated model membranes. Membranes containing D H A (22:6) were of special interest since their biological role is suggestive of interesting electric or dielectric properties. The tech-nique proved to be effective in resolving phase transitions, the degree of hydration, the presence of macroscopic dielectric domains which are indicative of heterogenous structure, and diagnosing relaxation and conduction mechanisms. Contents i i i Contents Abstract i i Contents i i i List of Figures vi Acknowledgements vii 1 Introduction 1 1.1 Motivation 1 1.1.1 Fundamentals: Towards a Physics of the Living State 1 1.1.2 Water 3 1.1.3 Dielectric Properties of Water and Biopolymers 4 1.1.4 Microwave Spectroscopy 6 1.2 Phospholipid Bilayers: A Model Biophysical System 8 1.2.1 Dielectric Properties of Phospholipid Membranes 11 1.2.2 Polyunsaturated Phospholipids and D H A 13 2 Theory 15 2.1 Background 15 2.2 Basics 19 2.3 The relation of E to EL 20 2.3.1 The Claussius-Mossotti Relation 20 2.3.2 The Onsager Relation 23 2.4 Frequency Dependence 24 2.4.1 The Debye Equations 25 Contents iv 2.4.2 Cole Plots and Phenomenological Exponents 26 2.5 Temperature Dependence 28 2.5.1 Statics 28 2.5.2 Dynamics ; . . . . 30 2.6 Composite Spectra of Mixtures 31 3 Exper imen ta l Background 33 3.1 Dielectric Spectroscopy 33 3.2 Membrane Spectroscopy 35 4 Exper imen ta l Technique 37 4.1 One-Port Measurement of the Complex Dielectric Constant Using a Vector Network Analyzer 37 4.2 Matrix Formalism for Calibration and Data Analysis 38 ' 4.3 Calibration 42 4.4 Experimental Apparatus 42 4.4.1 Connector and Glass Bead Adapter 44 4.4.2 Sample Cell 45 4.4.3 Temperature Control 46 4.5 Sample Preparation 47 4.6 Temperature Sweep 48 5 Results and Discussion 50 5.1 Calibration 50 5.2 Standard Polar Liquids 51 5.2.1 Frequency Dependence at Fixed Temperature 51 5.2.2 Temperature Dependence 52 5.3 NaF Salt Solution: Ionic Conduction 54 5.4 Model Phospholipids: D M P C and D O P C 55 5.4.1 Phase Transition in D M P C .. 57 5.4.2 D O P C at Variable Hydration ' 58 Contents v 5.5 Sensitivity to Structure 59 5.6 Polyunsaturated Phospholipids 61 5.6.1 Comparisons with D M P C , D O P C and NaF.water 62 6 Conclusions 65 Bib l iography 67 List of Figures vi List of Figures 1.1 The relative permittivity of water between 50 MHz and 500 GHz . . . 5 1.2 The fluid mosaic model of membranes 8 1.3 Phospholipid Structure 10 1.4 A typical lyotropic phase diagram for phospholipid membranes. . . . 11 2.1 The internal fields in an isotropic material 21 2.2 Symmetric and Asymmetric Distributions of Relaxation Times. . . . 28 3.1 Previous dielectric measurement on D M P C 36 4.1 The experimental apparatus as a cascaded two-port network 39 4.2 Photos the measurement device 43 4.3 Schematic diagram of the measurement device 44 4.4 The Anritsu V102M connector 45 5.1 Dielectric spectra of standard polar liquids: ethanol, methanol and water 52 5.2 The temperature dependence of the ethanol e" spectrum 53 5.3 Dielectric spectra for NaF salt solutions 54 5.4 Dielectric spectra for D O P C and D M P C 56 5.5 Single-relaxation Debye fits to hydrated D O P C 57 5.6 The gel to fluid phase transition in D M P C 58 5.7 The hydration dependence of D O P C 59 5.8 Demonstration of sensitivity to structure in phospholipid samples. . . 60 5.9 Dielectric spectra for D A P C and D H P C 61 5.10 Comparison of phospholipids 62 Acknowledgements vii Acknowledgements First and foremost I would like to thank Walter Hardy and Doug Bonn, for bringing me into their lab 7 years ago and fostering my development as an experimentalist through many projects since then, and of course to Myer Bloom for inspiring my interest in biophysics and guiding the direction of research in this project. I am especially indebdted to Walter for the unique opportunity he offered to pursue my own scientific interests within a Master's thesis, and investing so much time away from his own research program to support my intellectual meanderings; the high T c lab is a very special place in today's scientific climate, and it has been a privilege to spend these years here. That said, a special thank you goes to Ruixing Liang and Pinder Dosanjh for their technical help in the lab, to Pinder especially for his moral support this year. I owe a giant debt of gratitude to David Broun for his experimental guidance throughout the entire project, in developing the experimental technique and in using the Vector Network Analyzer. I am indebted to Charles Huang for running many experiments on the V N A and implementing the mixing models in Chapter 2 (and to Doug for hiring him to help me out in my time of despair). Financial support for the purchase of polyunsaturated samples was generously pro-vided by the U B C Centre for Brain Research. MiliPore water and other experimental materials were provided by Dan Bizzotto's lab (Dept. of Chemistry), and I thank Dr. Bizzotto and Robin Stoodley for many fruitful discussions and the beginnings of some experimental collaboration. Darren Peets' help with all things computational has always been invaluable, as has the venerable wisdom of the older generation of High Tc students: Pat Turner, Richard Harris, Saeid Kamal and Ahmed Hosseini. I have Geoff Mullins, Dan Beaton and Jake Bobowski to thank for their comradery and for making Graduate school a memorable experience, and for keeping me from Acknowledgements viii working too hard. My thanks to Jen Babiak for her commitment to a more serious kind of comradery, along with Darren and Dan, during the long and emotionally exhausting days of the T A strike. I am grateful to C U P E 2278 for the most valuable lessons I take from graduate school - that the primary role of universities should be in defense of civil society - and to Martha Piper and her administration for highlighting the sad failure of our academic system in this regard. Last but not least, I have Lindsay and my family to thank for their support through everything, and for their remarkable endurance in tolerating me at my most stressed and self-absorbed times. To everyone, a heartfelt thank you, and Adios for now. Chapter 1. Introduction Chapter 1 Introduction 1.1 Motivation This experimental investigation is largely an exploratory exercise which arose from the opportunity to do interdisciplinary research in a laboratory which specializes in microwave spectroscopy. The following exposition outlines a rationale which argues that microwaves - especially in the higher frequencies bordering on mm-waves - might be a useful probe for investigating biophysical systems, and traces the train of thought which led to the choice of phospholipid membranes as an interesting model system for preliminary investigation. That said, it is also true that the project was partly a result of a serendipitous situation which brought together Dr. Myer Bloom's interest in polyunsaturated membranes with the experimental expertise of Dr. Walter Hardy and Dr. Doug Bonn's microwave spectroscopy lab. 1.1.1 Fundamentals: Towards a Physics of the Living State Biophysics is a young science, and results that would be considered of fundamental importance to our broader understanding of physical law are difficult to identity. This does not seem to be the case for some other subdisciplines of physics; e.g. condensed matter physics studies particular materials and extracts very general insights about physical law, which are then exported to other branches of physics. Why is the same not true of biophysics? It seems the majority of work in biophysics aims to contribute more to biology than to physics. That is, the field asks biological questions and answers them by applying the physicist's theoretical and experimental tools. It is not surprising that Chapter 1. Introduction 2 this approach rarely yields results of fundamental importance to either the physicist or the biologist. Consequently biophysics as practiced today is largely an applied science. A n appreciation of living systems as extraordinary physical states of matter might change this, and make biophysics a more fundamental discipline contributing to our understanding of physical law, and not only biological mechanism. From this per-spective living systems provide examples of matter and energy which support a state of ordered complexity far from thermal equilibrium; surely this demands unique phys-ical mechanisms that are worthy of study for their own sake. In addition to this, life is supported by intricate feedback loops and nested levels of complexity, which we might describe as highly nonlinear from the perspective of the differential equations which govern such processes. Hence from the physicist's perspective, biological phe-nomena may look like exemplary states of matter and energy in the nonequilibrium and nonlinear regime. That very little is understood about physics in this regime may motivate the investigation of biophysical systems as experimental models for the development of such physics. In the same sense that condensed matter physics makes use of High T c superconductors as model systems to explore the physics of highly correlated electrons, perhaps biological materials or even living organisms may be exploited to investigate the physics of the highly nonlinear and nonequilibrium regime, exploring phenomena of fundamental physical interest which are implied by the emergent order and organizational complexity so common to biological systems. Biophysics might be reinvented as a discipline more akin to condensed matter physics than biology; while a chemist may study a superconductor to understand its structure, and a biologist may study a living system to understand relations of structure and function, the physicist may study both as occasions of highly correlated or cooperative phenomena. Chapter 1. Introduction 3 1.1.2 Water Where are we to start looking for this new biophysics, a physics of the living state? If we are to begin by taking an approach towards biophysics that is akin to material science, a natural starting point is to consider the well known fact that most living systems are composed largely of water, and ask what is special about this medium, what physical characteristics make it stand out from all other materials as such a ubiquitous substrate for supporting the living state? What kinds of interactions or even excitations might be unique to this medium? If we are to develop a physics of the living state, understanding the physics of water and its interactions with biopolymers is likely to be an important part of the enterprise. It is probably no exaggeration to say that water is among the most thoroughly studied substances on earth. Yet a brief overview of the literature gives one a sense that there is still no single theoretical approach which is capable of accounting for many of water's extraordinary physical properties [1]. For example, the phase di-agram of water is extremely complicated, with static and dynamic anomalies still outstanding from the perspective of statistical mechanics [2]. Only recently has a theory been proposed [3] that captures many of its more elusive features (e.g. the deeply supercooled states) while maintaining a satisfying theoretical simplicity. Poole et al proposed an extension of the van der Waals equation by adding a term to the free energy to account for a dynamic lattice of fluctuating hydrogen bonds. This kind of lattice model may prove very fruitful in further developing a condensed matter approach to understanding water; what excitations might such a lattice support, and what physical picture might arise from perturbing such a lattice with active biopoly-mers? Such a dynamic picture of lattices is quite new, with unexplored consequences. New information-theoretic ideas are also being developed to understand dynamic net-works in water [4], and are also being applied to formulate novel descriptors for the physics of biopolymers [5], further demonstrating the potential of a condensed matter approach to biophysics. Chapter 1. Introduction 4 1.1.3 Dielectric Properties of Water and Biopolymers One of the outstanding characteristics of the materials which constitute living matter is that they are furnished with a great number of complex charge distributions on side chains and residues [6], so one may expect that the response of biological materials to electric fields might be an important property to understand. The macroscopic response of permanent or induced microscopic charge distributions in a material is the subject of dielectric theory. It seems appropriate that in seeking to wed bio-physics with condensed matter physics one should be brought to the consideration of dielectric properties. After all, a theory describing the dielectric response of materials constituted one of the first accounts of many-body interactions, and hence what we today call condensed matter physics; early ideas on how to calculate the action of a dipole on itself through the collective influence of all its neighbors date back to eminent theorists in the 19th century such as Clausius, Lorenz and Lorentz [7], and continued to occupy some prominent condensed matter theorists early in this century [8]. The dielectric behavior of water is among its most extraordinary physical prop-erties. Combining one of the most electronegative elements with simple hydrogen into the characteristic V-shape of a water molecule gives rise to quite a large dipole moment on a very small molecule. This results in one of nature's largest naturally occurring static dielectric constants, which due to the size of the molecule maintains its mobility up to very high frequencies, sustaining a dielectric constant of ~ 80 up to 1 GHz. Figure 1.1 shows the relative permittivity of water in the frequency range relevant to it's dipole mobility [9]. This is a very rare combination of properties since a large dielectric constant usually requires a permanent dipole moment, whose size usually trades off against its mobility. Most dielectric materials with permanent dipole moments begin to roll off in the 10-100 MHz range [7]. The central dogma of modern biochemistry is that the structure of complex molecules determines their function. In large part the charge distributions found along most biopolymers play a prominent role in determining their secondary and Chapter 1. Introduction 5 P e r m i t t i v i t y o f W a t e r au • • • i 1—• • • • 70 : e 1 ^ \ 50 GHz 60 50 40 30 20 10 l . x l O 8 l . x l O 9 l . x l O 1 0 1 . x 1 0 1 1 L o g ( f ) Figure 1.1: The relative permittivity of water between 50 MHz and 500 GHz. The dotted line shows the frequency limit of the present experiment at 50 GHz. The curves are a fit to the Debye form (see Chapter 2) using measured parameters taken from [9]. tertiary structures by means of electrostatic forces, and therefore the charge distribu-tions along a single biopolymer strongly determine biological function [6]. However, electrical interactions between different molecules are very strongly screened by the permittivity of water, so long as they occur at timescales slower than the character-istic relaxation time of a water dipole. Hence water scales the strength of interaction for the many possible electrical interactions in biomolecular systems, and maintains very weak interactions for motions slower than ~ 1 ns. But what of charge interactions that occur on timescales faster than ~ 10 ps? The screening of electrical interactions would drop off as water dipoles cease to keep up with the A C field, and widely separated charge configurations might interact very strongly. Perhaps signatures of interesting electrodynamic or cooperative dielectric phenomena may become apparent at frequencies above ~ 20 GHz. This is a very speculative idea, but it serves to highlight the relevance of water's dielectric properties Chapter 1. Introduction 6 to understanding living systems at the molecular scale. While it is not likely that any permanent dipole moments on biopolymer residues would be more ..'mobile than water, other dipolar mechanisms are available, and in-deed common, in biomolecular systems. Life is largely built out of carbon polymers in an aqueous environment. Since these polymers are largely insoluble in water, most biopolymers are replete with residues which make the polymer soluble by dissociat-ing into a charged cloud around the groups in solution. Such polymers are known as polyelectrolytes, and D N A is the most common example of such a polymer. This ef-fect makes the polymer behave much like a material with permanent dipole moments, where the dielectric permittivity and loss are set by the mobilities and diffusive length scales of these charge clouds with respect to the polymer skeleton. Since these param-eters can vary a great deal for various polyelectrolytes, the frequencies across which they contribute to the dielectric response can span a very broad range. Investigations of the dielectric properties of polyelectrolytes have only recently been undertaken [10], but it is not unreasonable to suspect that this mechanism could exhibit fast dynamics on the timescale of water dipole mobilities, and possibly reveal interesting biophysics associated with these mechanisms' interactions with water. Polyelectrolytes are just one example of interesting dielectric phenomena in biopoly-mers, and there may be general reasons to believe there are more interesting mecha-nisms left to investigate. Consider that a typical phospholipid membrane maintains a potential of ~ 100 mV across a distance of ~ 10 nm, resulting in an electric field of 107 V / m [11]! Hence in a living cell the acyl chains and embedded regions of membrane proteins find themselves in a fairly extreme physical circumstances. One might expect that such large fields could drive interesting nonlinear behavior in the polarizability of these lipid chains, and in other molecules embedded in the membrane. 1.1.4 Microwave Spectroscopy One can study the relaxation of dipole moments in a material by measuring the macroscopic dielectric response, either in the time of frequency domain. The charac-Chapter 1. Introduction 7 teristic relaxation time of a permanent dipole can vary by many orders of magnitude depending on the size of the dipole moment as well as its mechanical and electri-cal environment, but experimentally it falls between 1 0 - 6 and 1 0 - 1 2 seconds. These timescales correspond to the MHz and GHz frequency regimes, hence microwave and mm-wave spectroscopy are appropriate techniques with which to investigate dielectric phenomena in the frequency domain. While a considerable amount of experimental work on dielectric spectroscopy of biological materials has been performed, most of the frequencies explored in broad-band experiments have fallen below 1 GHz, while work in the upper GHz regime has been limited to discrete frequencies (see Chapter 2). To undertake an exploratory ex-perimental investigation which touches upon the interactions of mobile water dipoles requires a broadband experiment, in order to catch any surprising features and avoid model-dependent fits to the data, one that reaches at least 20 GHz (the peak of di-electric loss in liquid water) and preferably goes all the way up to 500 GHz where water dipoles cease to keep up with A C fields. In this experiment, instrumentation was available to measure between 45 MHz and 50 GHz. This is high enough in fre-quency to access most of the action involving the dielectric mobility of water but low enough to catch the features of many slower dipole moments. Having considered directions from which to approach research which might address fundamental physics questions concerning the living state, we have arrived at the idea that water and biopolymer interactions are likely to be an appropriate starting place, and that interactions among their charge distributions, hence their dielectric behavior, might be a natural source of interest. Broadband microwave spectroscopy is an appropriate exploratory tool for such an investigation, and we have now to look for a model biophysical system to test for interesting behavior. r Chapter 1. Introduction S 1.2 Phospholipid Bilayers: A Model Biophysical System Phospholipid bilayers form the structural basis for all membranes in biological struc-tures, including the cell and nuclear membranes and those in organelles such as the mitochondria and the endoplasmic reticulum. As such, phospholipids are a very im-portant class of biopolymers whose function is to interact with water in a very specific way - to form selectively permeable barriers between different aqueous media, and to provide a substrate for the function of other active molecules. Figure 1.2: The fluid mosaic model of membranes. Protein and other biopolymers (rep-resented here by the ribbon structures) are embedded inside a fluid bilayer formed by amphiphilic molecules whose polar headgroups (green balls) inter-act with water dipoles (blue triangles) and whose oily chains shield each other from water due to the strong steric force which orders the membrane. The red triangles on the ribbons represent dipole moments or more complex charge distributions on the biopolymer, and the black lines indicate that these dipoles interact more strongly than ones in aqueous media due to the lower permittiv-ity of their environment. Note that any dipoles in the membrane must be on the interior regions of the molecule due to the hydrophobic environment. This image is adapted from a figure found at http://b2.cs.kent.edu/volkert/cells/ Chapter 1. Introduction 9 The amphophilic composition of a phospholipid molecule provides the main mech-anism for its function; a polar headgroup forms the hydrophilic part which orients preferentially towards water dipoles, and two acyl (oily) chains orient away from wa-ter towards other acyl chains whose headgroups shield the oily region from water. This forms a fluid bilayer structure, as shown in figure 1.2. There is a great deal of variability in the composition of naturally occurring phos-pholipids. There are many variations in both headgroup and acyl chain species, many headgroups can be found with any acyl chain, and the two chains on a single molecule are often of different species. There is a growing consensus that the phospholipid com-position of naturally occurring membranes is very specific to their function, and some estimates of the number of species present in a naturally occurring membrane number in the thousands [12]. Phosphocoline (PC), Phosphoserine (PS), Phosphoethelanomine (PE) are com-mon headgroups. A l l headgroups are charged or polar, although the mechanism may differ; P C and P E have permanent dipole moments due to a zwitterionic separation of negative charge at the phosphate group and a positive charge provided by variant of a nitrogen containing group, while PS carries both a dipole moment and an excess negative charge. The headgroup species determines the size and mobility of the dipole moment interacting with the aqueous environment. There is even more variation among the acyl chains. These hydrocarbon chains are typically between 14 and 22 carbons long, with many possible degrees of unsat-uration, that is, number of double bonds. One of the most striking things about the chains (which arises as a consequence of their biological synthesis) is that the double bonds are never conjugated, they are always separated by a fully saturated carbon. The conformational dynamics of the acyl chain species largely determine the mechanical properties of the membrane. For example, a single double bond causes a kink in the hydrocarbon chain allowing it to pack less efficiently and hence giving it more conformational space in the membrane. This decreases the density of the phospholipids and hence increases the fluidity of the membrane. Chain length plays a similar role [13]. Chapter 1. Introduction 10 Figure 1.3: Phospholipid Structure: Molecular structures of three acyl chains and three headgroups are shown, displaying various degrees of saturation, chainlength and polar structure. The acyl chain labelled 22:6 is DHA, which is discussed in section 1.2.2. This figure is reproduced from [14] Figure 1.3 shows typical molecular representation of various headgroup and acyl chain species. Common nomenclature uses 4 letters to denote a phospholipid, the first two indicating the first letter of each acyl chain (or using D as the first letter to denote 'di-'), and the last two indicating the headgroup. Thus, D O P C is di-oleoyl-phosphocholine. Acyl chains are often characterized by the number of carbons separated by a colon from the number of double bonds along the chain. Hence oleic acid is described as an 18:1 chain, and P O P C would be characterized as a 16:0/18:1-P C phospholipid, where the first P indicates palmitic acid. From the perspective of a condensed matter physicist, this structure forms a liquid crystal, which behaves like a fluid in two dimensions and like a solid normal to the membrane plane. Moreover, it is both a lyotropic and a thermotropic liquid crystal. The former term indicates that the system has many possible structural phases determined by the ratio of lipid to water molecules, while the latter indicates Chapter 1. Introduction 11 other phase transitions as a function of temperature. Figure 1.4 shows a typical lyotropic phase diagram for a phospholipid membrane, and illustrates a number of these phases. The main thermotropic transition is between the gel and liquid phases; at low temperatures the acyl chains are locked in a rigid gel of straight chains and at higher temperatures they move to a state where the chains have considerable conformational freedom, dramatically increasing the fluidity of the membrane. The liquid lamellar phase is most common in biological function, though there has been speculation that other lyotropic phases might play a biological role as well [15]. 100 j lom«llor a wafer Q9l a water • O 20 4 0 GO 80 Core** ration (%> ICO Figure 1.4: A typical phase diagram for the lyotropic states of a phospholipid membrane, taken from [16]. Various liquid crystal and solid phases occur at high am-phiphile concentration and have been omitted in this figure. 1.2.1 Dielectric Properties of Phospholipid Membranes The phospholipid membrane structure provides a substrate for a large number of intramembrane proteins and other cell infrastructure, whose orientations, conforma-tional freedom, mobility etc. are affected strongly by the material properties of the membrane, and hence by the acyl chain species which compose the membrane. For example, mechanical properties of the membrane such as fluidity largely determine Chapter 1. Introduction 12 the ease with which a protein can undergo a conformation change, which is often the primary mechanism of its function. The bending modulus and surface tension deter-mine how easily a protein can fit inside the membrane, and hence its solubility. Stearic properties also play a major role in determining the conformations of intramembrane proteins, as the hydrophobic portions will be embedded in the membrane, or line the extremal parts of the portion inside the membrane, while the hydrophilic parts will lie outside the membrane. The dramatic difference between the low frequency permittivity of water (~ 80) and oil (~ 2 — 4) is worth noting. In the space of ~ 10 nm the permittivity goes down and up again by about two orders of magnitude as one crosses the membrane, hence any biopolymer embedded in a membrane sits inside a very strong dielectric gradient. Considering the number and function of charged groups along proteins one might think that this could be significant to their function. The relevance of water permittivity to screening electrical interactions between and along biopolymers has already been noted. This property no longer holds in the low dielectric environment inside the membrane, and one might expect that charged groups along proteins inside the membrane (shielded by hydrophobic regions along the periphery) may interact much more strongly with each other, allowing more electrical communication between biopolymers inside the membrane than is possible outside of it in an aqueous environment. What about dynamic aspects? Conformational changes and molecular group mo-bilities span a similar range of timescales as is typical for dipole mobilities. Moreover, membrane fluctuations can also fall in this timescale, depending on the mechanical properties of the membrane, and give rise to a 'flexoelectric' effect whereby a collec-tive dipole moment is generated by curving a surface made up of permanent dipole moments [15]. These observations suggest the possibility that interesting phenomena related to the function of phospholipid membranes may arise from a coupling between the dynamic aspects of their mechanical and dielectric properties. These ideas about the relationship of dielectric properties of membranes to their biological function are speculative, but the fact remains that hydrated phospholipid Chapter 1. Introduction 13 bilayers form an interesting and rather unique dielectric structure that seems worthy of study given the likely relevance of dielectric behavior to biological function. But even if the connection to function does not prove fruitful, broadband investigation at these frequencies is useful to study structure alone; broadband lineshapes in this frequency regime have distinct characteristics indicative of conduction and relaxation processes, and can yield information which can be related to the structural proper-ties of the material. Given the dominant role that water dipoles play in membrane properties, access to frequencies near 10 1 1 Hz is important to understand the di-electric structure completely, and such a study fits nicely into the larger program of understanding the biophysics of water and its interactions with biopolymers. 1.2.2 Polyunsaturated Phospholipids and D H A There is an emerging consensus from a variety of disciplines which draws attention to one particular species of acyl chain, docosahexaenoic Acid (DHA, 22:6), shown in figure 1.3. This is the most polyunsaturated of all naturally occurring acyl chains (in mammalian cell membranes), and there is growing evidence that its presence is essential for human brain development [17]. D H A comprises up to 35 mol% of all acyl chains in grey matter and synaptically enhanced regions of the brain (in com-parison with 1-6 mol% for other acyl chains) [18], suggesting that this lipid forms a major constituent of membranes at the synaptic cleft whose polarization and depo-larization generate the firing events which are believed to encode information in the brain. Furthermore, the outer rod segment of the eye carries up to 50 mol% DHA, in membrane disks which embed large populations of rhodopsin [19]. This protein receives photons, and through a cascade of transformations generates a conformation change in rhodopsin that seems to be responsible for the neural firing event which encodes the light signal in the brain. Such a dominant presence of D H A and polyunsatruated lipids in general pose an interesting puzzle; organisms must expend a great deal of energy to maintain these lipids, which oxidize very easily and require large amounts of anti-oxidants Chapter 1. Introduction 14 (such as vitamin E) to sustain such large concentrations. Furthermore, mammals lack the enzymes required to synthesize D H A from scratch and must obtain essential fatty acid (EFA) precursors in their diet. The consequences of a shortage of D H A can be very serious, having a strong effect on neurodevelopment which results in impaired intellectual capacity and vision [20]. Hence this dietary dependence is rather significant, especially since plentiful sources or EFA's are not easily accessible in many natural habitats. This suggests that the polyunsaturates play a unique role which other, less biologically expensive and more stable phospholipids cannot replicate [14]. Given the prominence of D H A in membranes that are clearly involved in elec-trical signaling, it is natural to wonder if D H A plays any interesting electrical role when placed in a phospholipid membrane. A natural question to ask is whether it provides a unique dielectric environment for active biopolymers immersed in these membranes. Hence the specific question of the current investigation becomes: do polyunsaturated phospholipid bilayers, specifically those containing DHA, constitute a dielectric medium that significantly different than that of saturated or monounsat-urated bilayers? If so, one might expect that this difference plays a significant role in the function of such membranes, given the suggestive regions in which they are found. From a broader perspective, we are using phospholipids as an interesting start-ing point in the more general project of exploring electrical interactions between water and an important class of biopolymers, where the puzzles posed by polyunsaturated lipids naturally present themselves for investigation along the way. Chapter 2. Theory 15 Chapter 2 Theory 2.1 Background At frequencies below ~ 10 1 0 Hz, non-resonant microscopic mechanisms play a domi-nant role in the dielectric response of most materials. That is, atomic and electronic transitions are not excited, but slower mechanisms arising from molecular structure play a dominant role. Permanent dipole moments are among the simplest and most common of these structural features, and their dynamics dominate the dielectric re-sponse in this frequency regime. We limit our discussion of dielectric response to these and slower mechanisms. The study of low frequency dielectric response was one of the eariest examples of what we today call condensed matter physics, in that it established relationships be-tween microscopic material properties and macroscopic electrical response. Although much of the groundwork was laid down in the early part of the century [21], [8], the interpretation of these dielectric spectra continues to be an active and controversial field of study, as a brief survey of the literature will quickly reveal (for example see [22, 23, 24, 25, 26]). Indeed, even after decades of developments and improvements in experimental techniques (for reviews see [27, 28]) the interpretation of dielectric measurements can remain controversial for many years even within modern research programs [29]. One of the difficulties with the continued development of the field lies in the lack of sharp features exhibited by non-resonant dielectric spectra; relaxation phenomena generally show very broad features in the time and frequency domain. This poses a number of problems. There are several different physical mechanisms which give rise to relaxation-like behavior across a broad range of timescales, and it can be difficult Chapter 2. Theory 16 to interpret which mechanisms are responsible for which experimental features. For example, in reference [29], an artifact of interfacial polarization at the measuring electrodes could account for features which had been interpreted as colossal dielectric constants. On the other hand, the same interfacial mechanism can act on micro- or meso-scopic scales at domain boundaries within a material to give rise to real changes in material properties. This illustrates the difficulty in the theoretical interpretation of measurements even when experimental artifacts are controlled. A major dilemma arises from the fact that the very broad features characteristic in this field of study cannot easily discriminate between different microscopic models. There has been considerable progress recently on this front, with the development of sophisticated quantitative methods of analysis for ill-posed inverse problems [30]. This approach inverts the measured broadband spectra (in the time or frequency domain, both exhibiting little apparent structure) into a particular distribution of basis functions <p(i) that are meant to inform which microscopic mechanims that are at work. In this sense it is like Fourier analysis, but using relaxation processes (j)(t) = et/Ti as the basis set rather than oscillations elult. The former problem is more difficult than the latter, since more than one relaxation distribution may be capable of satisfying the desired function within experimental uncertainties, the solution may be unstable, or it may not exist at all. New computational techniques are proving useful in resolving the ill-posed na-ture of this problem, but the general approach falls in line with a long history of describing dielectric spectra as distributions of simple relaxations et^n (known as De-bye processes) [31, 32]. Different distributions of Debye processes may be used to fit the broad features common in dielectric spectra, but the question remains as to how well do these particular functions represent the microscopic processes at work in the material. In the 70's, Jonscher [33, 34] began another line of thought which suggested that non-Debyean macroscopic spectral shapes arose directly from non-Debyean micro-scopic processes in the material, i.e. that e^Ti was not an appropriate basis with which to fit many dielectric spectra . He examined a broad data set which encom-Chapter 2. Theory 17 passsed a very diverse array of materials and yet displayed almost identical charac-teristics in their dielectric spectra. He concluded that the microscopic mechanisms at work must be very different, and it is unreasonable to believe that individual Debye processes were all combining together in just the right distribution to yield the same macroscopic behaviour in all these cases. Thus, force-fitting the data to a basis set of Debye relaxations may have no relevance to the microscopic physics at work in the materials. Instead, he argued the spectra must arise from collective effects that are universal to a broad range of materials irrespective of their microscopic details. Specifically, Jonscher focussed on a particular dielectric response seen a many dif-ferent materials, exhibiting power law behaviour across many decades of frequency. To fit Debye relaxations to such a spectrum one must construct a very broad dis-tribution with relaxation times spanning many orders of magnitude, a contortion which Jonscher found implausible. Instead he argued that this behaviour arose from a 'universal' microscopic mechanism where the ratio of power lost to power stored was independent of frequency [34]. More recent developments continue this line of reasoning, bringing fresh ideas to bear on the problem. Jurlewicz et al [22, 35] propose a clustering model wherin Jonscher's 'universal' mechanism arises from superpositions of polarization responses across different length scales in a material. Many authors have noted the usefulness of fractals, either in the material's spatial structure (for example [24]) or in its time dynamics (for example [23]), in formulating a microscopic account of non-Debyan behavior [36]. These approaches make the argument that non-Debyean microscopic processes give rise to non-Debyean macroscopic response. The question which distinguishes the two approaches is a simple one; do non-Debyean macroscopic features in dielectric spectra arise from the average response of many dipoles Debye-relaxing at different rates, or from individual dipoles undergoing all undergoing the same, more complex process? Both kinds of processes may be present, but there is a physical distinction to be made nonetheless; the distinction is the same as that between an individual atom oscillating in a crystal lattice and the phonon which arises collectively in the material as a result of many such vibrations. Chapter 2. Theory 18 In this case there is a difference in the frequencies of the processes, and a dispersion relation clarifies and distinguishes the relationship between the two phenomena. It is not clear if the same kinds of relationships can distinguish the two competing pictures in dielectric relaxation. In simple homogeneous systems the question may only be one of perspective or mathematical convenience, but in complex heterogeneous systems with hierarchies of structure the question has very direct bearing on the microscopic physics at work. Take for example a multilamellar phase of hydrated phospholipids. Does the mono-layer of water closest to the headgroup dipoles interact more strongly than the next monolayer, resulting in a slower relaxation time for that monolayer? Does this lead to a continuous distribution of relaxation times in the water between bilayers, as claimed in [37]? Or is there a single non-Debyean relaxation process occurring throughout the water+headgroup system, determined by the complex reaction field associated with the system's structure? The answer depends on the degree and scale of structure in the local field inside a dielectric material. This perspective highlights another reason why the study of dielectric properties might have significant bearing on biophysics; the structure of local fields is an impor-tant aspect of the physical environment in which biopolymers perform their function. A complex biopolymer carries many charge distributions located on particular sites along its structure. The structure of the local field in which it finds itself would affect the forces experienced by different parts of the molecule, possibly altering its structure and thereby its function. The phospholipid membrane which houses many such macromolecules is a natural environment to investigate in this regard. Hence, questions concerning putative microscopic mechanisms for macrosopic di-electric response, still under dispute, are relevant to understanding the local struc-ture of electric fields in which biopolymers perform their function. Dielectric theory provides a window into the relationship between macroscopic and local microcopic electric fields in a material. Chapter 2. Theory 19 2.2 Basics The relative permittivity e is a convenient measurable property of a material, describ-ing by means of a simple ratio the displacement field D in the material in response to an applied field E, where e0 is the permittivity of free space, with the appropriate SI units F / m . It is convenient to separate the dielectric response into two components 6QE and P , such that P carries all the material properties D = e0eE = e0E + P. (2.2) Within the regime of linear response, we expect the macroscopic polarization Pmac to be proportional to the applied field E, so we define the electric susceptibility x Pmac = XE, (2.3) which by 2.2 defines the relationship e = x + 1-Now we define a microscopic polarizability Pmic = aTEL (2.4) where OLT is the total polarizability of a microscopic element of the material, and Er, is the local field at that element, ax is the sum of three contributions: orientational aQ, electronic ae and atomic aa polarizability. The orientational contribution is generally much larger than the latter two, which have much faster reaction times than a0 and hence make a constant contribution aae = ae + aa at the frequencies in which we are interested. To make the connection between the microscopic properties of the material and its macroscopic response to an applied field, we make a fundamental assumption about the linear scaling of the response: Chapter 2. Theory 20 Pmac — NPmic (2-5) where N is the number of polarizable elements per unit volume. By equations 2.3 and 2.4, this assumption results in the following relation: NaT E ( 2 6 ) eo(e - l ) EL We have written this relation in a form that highlights the fundamental problem of dielectric theory: to relate the microscopic properties of a material to its macroscopic response by understanding the relationship between the applied field E and the local field Ei inside the material. 2.3 The relation of E to EL The total dipole moment of a molecule at position XQ in an electric field is determined by the local field EL(x0) at that molecule. Thus the problem is to construct EL(X0) when that same field arises as a result of all the other dipoles at every x surrounding the molecule. The situation must be approached in a self consistent manner since EL(X) determines the dipole moments at every x, and arises partially from the action of the dipole at x0. The dielectric response of a material is intrinsically a collective effect. 2.3.1 The Claussius-Mossotti Relation Historically, the first approach to understanding the relationship between E and Ei (and fundamentally the only approach [7], at least until the 80's) was to represent the reaction of neighbouring dipoles on a given dipole by constructing a suitable " internal field". Consider an isotropic dielectric material, with a spherical cavity cut inside of it as a mathematical fiction. The dimensions of the hole are large compared to the average spacing between dipoles but small compared the the sample dimensions, as Chapter 2. Theory 21 E Figure 2.1: Internal fields in an isotropic material: An imaginary sphere is cut out from the material, centred on a point at which we wish to find Ej,. E is the external applied field, E\ is the field arising from all dipoles outside the sphere and E2 is the field due to the dipole configuration within the sphere. in figure 2.1. Then EL = E + Ex + E2 (2.7) where E\ is the field of all the dipoles outside the cavity and E2 is the field that would arise inside the cavity due to the specific configuration of dipole moments in it. It can be shown that E2 —• 0 at the centre of a simple cubic, F C C or B C C lattice of oriented dipoles. The same is true for a random assembly of dipoles non-oriented dipoles [38], and so for a homogenous polar liquid E2 is close to zero. E\ can be represented as the depolarizing field due to the geometry of the hole in the material, Ei = (2.8) A'eo where N = l / 3 is the depolarizing factor for a sphere, the shape most appropriate for an isotropic liquid. For anisotropic materials a different ellipsoidal shape would be chosen such that E2 —> 0 for the dipole configuration inside the chosen ellipsoid, and an appropriate value of N would be used for that ellipsoid. Hence from 2.7 and 2.8 Chapter 2. Theory 22 EL = E + ~ = 6-^E (2.9) 3e0 3 and by equation 2.6 we arrive at the Clausius-Mosotti equation: e - 1 N e + 2 3e0 On rearranging 2.10 we obtain cxT. (2.10) (2.H) 1 - iVa r / 3e which reveals the "dielectric catastrophe" inherent in this relation, since e —• oo when the material density N —> 3e/ctT- While in a. crystal lattice this effect may be indica-tive of ferroelectric behavior arising from the atomic and electronic polarizabilities, no such behavior is known to arise due to orientational polarization in liquids (where the dominant orientational contribution to ax might bring N into a range where this effect would appear to be commonplace). The formalism presented above bears a flaw in that the reaction field acts on CXT, which includes orientational polarization. This implies that not only does a dipole influence its own magnitude by acting through the reaction field it generates among its neighbours (which it does), but that the dipole may also affect its own orientation. This is impossible due to the vector addition of electric field lines. That said, the flaw in the Claussius-Mossotti relation arises only when the dynam-ics of permanent dipole moments are present. Thus the equation holds for non-polar liquids, and also at high frequencies where the field has exceeded the dipole mobility, and aa has ceased to play a role in the dielectric response. Thus we are naturally led to incorporate some frequency dependence in our expressions, and a more accurate statement of the Claussius-Mossotti relation is eoo + 2 3e0 where denotes the high frequency limit of e. e°° 1 = l L a . (2.12) Chapter 2. Theory 23 2.3.2 The Onsager Relation Onsager corrected the flaw in 2.10 by modifying the cavity field to encompass only a single dipole, and separating the contributions to EL as follows EL = G(a,E) + R(m(aT),EL) (2.13) where a = aa + ae does not include the orientational component of ar- The first term G is the internal cavity field due to the dipole's surroundings, which remains unaltered if the dipole in the cavity has its moment removed. The second term R is the reaction field caused by the dipole in the cavity. Due to the vector addition of fields this component will always be parallel to the dipole at its source, and hence this derivation avoids the 'catasrophe' noted above. The derivation of the Onsager relation is much more involved than the Claussius-Mossotti relation, and every standard text takes a different approach with various degrees of detail. I include an outline of the essential steps here; for a more detailed derivation see [39]. To find R and G one must obtain the potential 0(r, 9) by solving the boundary value problem of Laplace's equation inside and outside of a spherical region of volume V in a material of permittivity e. In the case of R a dipole field distribution is included inside the sphere, while this is excluded when calculating G. The fields are then determined by taking —d(f)/dr. The results are G = * T I ( 2 ' 1 4 ) R=V^km- (2'15) where m = [i + OLEL combines the permanent dipole moment p, with an induced dipole term arising from the local field, illustrating the action of the local field on itself as described above. We can use 2.12 to define a in terms of e^, separating it from the orientational term aa in equation 2.6 as well as in the treatment of R. It is easier to work with p instead of aQ in equation 2.15 to clarify the appropriate vector Chapter 2. Theory 24 relationship of the permanent moment to the field R, such that the moment is not capable of orienting itself. However, we can write the end result in terms of a0 and the instantaneous polarizability a expressed in terms of e^: V c coo )(2e + eoo) Ncx0 . . e( e o o + 2) 2 ~ 3e0 ' { ' ' This is known as the Onsager relation and is generally applicable to polar liquids. When e —> 1 equation 2.16 reduces to the Claussius-Mossotti relation. In the limit of a strongly polar liquid like water, where e = (eoo + 2 ) 2 ^ . (2.17) 2.4 Frequency Dependence The dielectric function has a frequency dependence which in the case of slow orienta-tional dynamics arises largely due to the phase lag between the material's response and the driving A C field. Hence it is most convenient to represent dielectric response as a complex quantity: e* = e' + ie". (2.18) Before moving on to the theoretical development of purely dielectric phenomena, it is worth noting that purely conductive components to the measured response of a material add on top of the dielectric response. A conductivity of a at the frequencies of interest here shows up has a characteristic l/u> dependence in the imaginary part of e: e* = e' + i{e" + a/u). (2.19) Chapter 2. Theory 25 2.4.1 The Debye Equations i The standard theory for the frequency dependence of the orientational component of dielectric response was formulated by Debye in 1929 [21]. Debye's equations are traditionally constructed by separating the polarizability into two components P = P\ + P<i such that there is an instantaeous component P\ = xiE from electronic and atomic polarizabilities, and a slow process Pi which lags behind P\. The key assumption is that P 2 approaches its final value XiE at a rate proportional to the difference (P2 — X2E), which yields the differential equation =-HX*E - P3). (2.20) at T Solving this equation for an alternating field E = E0eluJt, and taking P — Pi + P2 V 1 + IU)T J This corresponds to a complex permittivity of the form e = e00 + (2.22) 1 + ILOT which is the well known Debye equation. In light of the discussion at the start of this section concerning the relationship between microscopic and macroscopic mechanisms, a different derivation found in [23] may be illustrative. Consider a kernel function (f>(t), which represents the time evolution of an individual dipole in a dielectric material. The susceptibility to an alternating field can be defined as X(u) = / e^di-m- (2-23) Jo which we leave normalized to unity for now. This transformation is mathematically equivalent to a Laplace transform ~4>{t) = L{<Kt)} = f i - ^ ^ d t ) (2-24) Chapter 2. Theory 26 followed by the rotation p —> iui. The differentiation theorem states L{^-} = pfa) - 1 (2.25) which allows the complex susceptibility to be written in a more convenient form X H = [ 1 - P ^ ) U . (2.26) For the exponential relaxation process 4>(t) = eluJt, this evaluates in a straightforward way to yield X M = — ^ — • (2-27) 1 + IU)T If we now back up and normalize our definition of x to yield X2 (the equilibrium value for the slow processes we are representing), and add an instantaeous component Xi, we retrieve the form of x found in 2.21 X(") = X i + T T ^ — < 2 - 2 8 ) 1 + IU)T and hence the Debye equation 2.22. This derivation makes more apparent the connection between the microscopic process described by the kernel function and the macroscopic susceptibility, which will be useful in the discussion following the presentation of our results. 2.4.2 Cole Plots and Phenomenological Exponents The Debye equation can be separated into its real and an imaginary parts: £ ' ^ - T ^ 7 F + £ » ( 2 ' 2 9 ) = + ^ < 2 - 3 0 ) Chapter 2. Theory 27 Respectively, e' and e" describe the in-phase and out-of-phase components of the average dipole response in the material. As such the real part controls the magnitude of the reaction field in the material, while the imaginay part represents the loss. These quantities may be plotted separately against the log of frequency to yield the characteristic Debye shapes shown in 1.1, but it is often more useful to plot them against each other as a parametric plot of frequency, known as a Cole plot. Rearranging 2.29 and 2.30 to eliminate LOT gives (e' - ! i i ^ ) 2 + ^ 2 = ( ! i i ^ ) 2 ( 2 - 3 1 ) and we see that a Debye process plotted this way yields a semicircle whose centre and radius are related to the dielectric increment es — e^. Few dielectrics exhibit perfect Debye spectra and, historically, deviations from this circular shape have been characterized using one or two phenomenological exponents 0 and 7, e = £oo + jz —-r. r^T7- (2.32) (1 + (ILUTY) Setting (3 — 7 = 1 gives the original Debye equation, while modulating these expo-nents between 0 and 1 skews the shape of the Cole plot in characteristic ways that are recognizable in experimental data. Figure 2.2 shows the behavior of equation 2.32 for various values of (3 and 7. Plots adjusting (3 are known as Cole-Cole plots, while adjusting 7 gives Cole-Davidson plots. As discussed in the first section of this chapter, deviations from simple Debye behaviour have traditionally been analyzed in terms of distributions of simple relax-ation times. The simplest distribution that is often used is a gaussian centred on the dominant relaxation time, which has the effect of symmetrically broadening the drop in e' and the width of the peak in e". On a Cole-Cole plot this distribution looks similar to the effect of setting (3^1, and so the latter is often interpreted as the signature of a symmetric (though not necessarily Gaussian) distribution of relax-ation times. Similarly, spectra with 7 ^ 1 are interpreted as indicative of asymmetric Chapter 2. Theory 28 Symmetric Distributions Asymmetric Distributions 0.5 0.4 0.3 0.2 0.1 0 Y=l X 0=0.9 > v \ /3=0.7 \ A /3=0.5 ^ ^ ' ^ X . X M 0.2 0.4 0.6 0. e' Figure 2.2: Cole-Cole and Cole-Davidson Plots: These plots illustrate the behavior of equa-tion 2.32 for different values of the phenomenological parameters 8 and 7. De-creasing values of 8 (7) suggest broader symmetric (asymmetric) distributions of relaxation times. distributions. However, along the contrasting line of thought introduced by Jonscher, more recently there have been attempts to derive some of these exponents directly from the microscopic processes which reflect the physics governing individual dipoles. Metzler's treatment in [23] is a very good example, deriving the existance of the exponent /3 directly from a non-exponential kernel function 4>(t). 2.5 Temperature Dependence 2.5.1 Statics The orientational component of polarization is strongly influenced by temperature, since thermal agitation of molecules opposes their ability to line up with an applied field. The energy of a dipole moment p in a local field EL at an angle 9 is U = -p.-EL = -p.EL cos 9 (2.33) The Boltzman distribution allows us to solve for the thermal average of cos 6 Chapter 2. Theory 29 fexV(-U/kBT) cos 0<m J exp (—U/kBT)d\l Carrying out the integration over the solid angle O yields the Langevin function L(y) (cos 6) = L(y) = cothiy) - - (2.35) y where y = exp (—pEL/kBT). Under normal experimental conditions EL < 105V/rn, and taking fx = 1.8 debye for water as a typical dipole moment we find /J,EL — 6 • 10~ 2 5 J. At room temperature kBT = 4 • 10~ 2 1 J so we are in the limit where ! / < l . Expanding coth (y) in this limit we find L ( ? / ) = y + 3 - 4 5 + - - - - y * 3 ' ( 2 ' 3 6 ) hence ( C O S ^ ^ 3 0 - ( 2 - 3 7 ) The average moment per dipole in the direction of EL is thus M = ^ ( c o s 0 ) = g | , (2-38) which yields the orientational polarizability: a0 = , (2-39) ZkBT y ' Decomposing ar into its permanent and induced components also separates the temperature dependent part of ar- Thus we can now add temperature dependence to the results developed in section 2.3. The Claussius-Mossotti relation now reads £ _ 1 = ^ ( ^ ^ ) (2-40) e + 2 3e0"3kBT and the Onsager relation gives (e s-eoo)(2e, + 6^) = Np2 es(eoo + 2) 2 9e0kBT' Chapter 2. Theory 30 2.5.2 Dynamics The Debye-like spectra presented in section 1.4 have another temperature dependent feature in addition to es. Standard rate theory can be used to interpret the relaxation time r as the inverse of an 'attempt rate' for a dipole to hop an energy barrier and take on a different orientation. Thus the relaxation time should follow an Arrhenius law r = - ^ - e x p ( ^ ) (2.42) kBT *KRT' v ' where F is the free energy of activation, h is Planck's constant and R is the gas constant. The free energy is a function of enthalpy H and entropy S, so using AF = AH — TAS and taking the natural log of both sides we obtain - " IT + W ( 2 4 3 ) and differentiating both sides with respect to 1/T yields 9HrT) _ AH d{l/T) R ' y ' ' Hence a plot of ln{rT) against 1/T should give a straight line with slope AH/R. Moreover, Debye [21] showed that the dielectric relaxation time r can be calculated using the relation T = T e,+ 2 Coo + 2 r* is the intrinsic (mechanical) relaxation time given by (2.45) r« - ^ ? (2.46) kBT where n is the viscosity surrounding a sphere of radius a which encompasses the relaxing dipolar molecule. This relation arises from Stokes law for rotary diffusion of spheres in a viscous liquid. Chapter 2. Theory 31 2.6 Composite Spectra of Mixtures The presentation of dielectric theory so far has been structured around a theme which highlights the relationship between the microscopic and macroscopic properties of materials. The reason for this becomes clear when one considers that biological materials are composite, heterogeneous structures, and that the dielectric spectrum of such materials takes very different forms depending on the scale of the structure, that is, the sizes of separate material domains. Consider a mixture of two dipolar substances. If the mixture is homogeneous then one might expect the dielectric properties of the two substances to average together at the microscopic scale; an average polarizability aAB, dipole moment piAB and mobility rAB would give rise to a single macroscopic relaxation mechanism. On the other hand if there were large domains of A and B then the resulting spectrum might be a weighted superposition of the macroscopic spectra of each substance. To test and illustrate this hypothesis two models were implemented to compare with the data presented in chapter 5. Given a molar ratio NA : NB for two substances x — A or B, let N'=N*TN-B' ( 2 ' 4 7 ) In the heterogenous mixing model, the total dielectric response is simply the weighted sum of the macroscopic (Debye) response of A and B: ^ ) = " ' ( T ^ + £~ ) + reB(T^ + £~ ) < 2 ' 4 8 ) where ex, ej^ , and rx are taken from the literature and inserted into the Debye equation 2.22. The homogeneous-mixing model takes the weighted average of microscopic quanti-ties to calculate macroscopic quantities before inserting them into the Debye equation. We use literature values of ax and the high frequency Claussius-Mossotti relation 2.12 to calculate e £ B : Chapter 2. Theory 32 ^ - « W + B V ) . (2.49) This quantity, along with literature values of px are then used to calculate an average eAB using the temperature dependent Onsager equation 2.41: (eAB - e^){2eAB + e^B) (NA + NB){nAu^A + nBfxB)2 eAB(eAB + 2)2 9e0kBT r A B is caluculated using equations 2.45 and 2.46: (2.50) ^_AB 4n(nAaA + nBaBf{nAriA + nBr)B) eAB + 2 kBT + ( 2 - 5 1 ) where ax is the radius of the dipole, taken from literature values of molecular polar-izability (which has dimensions of volume). rf is extracted from literature values of rx, using equations 2.45 and 2.46. Finaly, the total dielectric response is given by AB _ eAB e A » = V AB+tiB)- (2-52) 1 + lLOTAa The results of these models for A=ethanol and B=water are presented in chapter 5 for various NA : NB, and compared with measured spectra for an ethanol-water mixtures and phospholipid-water mixtures. Chapter 3. Experimental Background 33 Chapter 3 Experimental Background 3.1 Dielectric Spectroscopy The dielectric response of a material can be measured in either the time or frequency domain, and by means of reflection or transmission techniques. A broad overview of experimental techniques used to measure dielectric properties between 1 MHz and 1500 GHz can be found in [27]. The review gives a sense of the breadth of the field as it stood almost 20 years ago, covering developments up to 1986 including time- and frequency-domain methods; reflection, transmission and resonant methnods; guided and free-space methods; discrete-frequency and broadband methods. Since this time the field has no doubt broadened even further, with increased attention on higher frequency measurements in biological materials being driven by the development of wireless technology [40], and more effort devoted to broadband measurements due to the increased availability of Vector Network Analyzers (VNAs). For a more recent discussion of the field and its relevence to applied biophysics, see [41]. The experimental technique presented in Chapter 4 combines aspects of three experiments found in the literature. We use a single V N A port (as in [42]) to measure the reflection coefficient of a lumped capacitive load (as in [43]), using a transfer matrix formalism to calibrate the measurement plane (as in [44]). Measuring permittivity from reflections off of a lumped capacitor has been quite thoroughly explored by Stuchly et al [28, 43, 45]. This is a discrete frequency slotted-line technique which uses a calculated bare capacitance to compare the reflection coefficient of an empty capacitor to one filled with a lossy liquid. The technique is limited to ~ 10 GHz, and avoids the need to calibrate by reflecting directly off a modified commercial connector (GR900-BT), which is used as the sample holder. Chapter 3. Experimental Background 34 Thus the technique is stongly constrained by the geometry of available connectors. The open-probe technique uses a truncated transmission line to contact an arbi-trarily shaped and sized sample, and has recieved considerable attention since it lends itself to probing biological materials in situ. Marsland and Evans [42] have developed an open coaxial probe technique which makes use of a V N A and approximate admit-tance models, relying on three calibrations which measure the reflection coefficient of known admittances. There are two main problems with the open probe technique; it is limited in the frequencies it can attain due to radiative losses into the sample, and is dependent on approximate formulas solved by numerical techniques (with limita-tions that may not be obvious to the experimentalist). The lack of control over the geometry of the measured sample can also lead to experimental artifacts arising from resonances or transmission losses in the sample. Pelster [44], has developed the most comprehensive experimental technique to date, making full use of V N A technology to make temperature dependent, broad-band measurements of dielectric response up to 2 GHz. The technique is very similar to the one described in this work, in that it uses a cascading matrix formalism to achieve a full calibration using measurements on known standards, but using a full 2-port transmission measurement rather than a single port reflection. As such it requires a more extensive calibration procedure, and the sample cell has more param-eters to control than a simple capacitive load (the length of the cell as well as bare capacitive and inductive contributions from both sides of the cell). However, being a transmission measurement the technique is not so limited by loss of sensitivity at lower frequencies, and more by an upper cutoff frequency. Only recently have broadband measurements pushed into the millimeter wave regime [46] to study biological and organic liquids. The technique used by Duhamel et al is very similar to that of Pelster, but using waveguide instead of coaxial line to make measurements at higher frequencies, and applying a different calibration technique than used in [44]. The experimental technique described in Chapter 4 combines the simplicity of a single reflection measurement with the matrix formalism for calibration which cir-Chapter 3. Experimental Background 35 cumvents the need for commercial calibration loads at the measurement plane. It also achieves higher frequencies than have been reported without the use of waveg-uide. The sample chamber has a well defined geometry that is convenient for sample preparation and control of hydration, and sample volumes are ~ Q.5pL, an advantage when dealing with biological materials. 3.2 Membrane Spectroscopy The broadband dielectric response of model phosphilipids has been measured between 1 MHz and 1 GHz in two experiments, both using the V N A s . Kloesgen et al [37] used the technique described in [42] to measure reflections from an open coaxial probe inserted into a large volume of vapour-hydrated phospholipid surrounded by excess water. The experiment requires a relatively large volume of lipid (~ 1 mL), and does not control sample hydration in a very reliable way. The size of the sample is also not controlled and approaches dimensions where artifacts may arise due to resonances and reflections; despite claims in the study that an oriented sample is achieved, this is difficult to imagine given the size of the sample and the deposition and hydration technique that is used. The study also claims to extract the peak relaxation frequency of a distribution representing bound water, based on the broadening of the dielectric spectra visible up to 1 GHz. Haibel et al [47] used Pelster's transmission technique [44] to study D M P C , DSPC and a 50/50 mix of the two phospholipids. Figure 3.1 compares the data for D M P C from [37] and [47]. The dominant peak in e" and drop in e' is attributed to the relaxation of the P C headgroup. Both studies state that their samples have 50% (by weight) water content, although the study reported in [47] claims to be measuring multilamellar vesicles while in [37] the researchers claim their macroscopic sample has planar ordering. While it is difficult to make a detailed comparison due to the differences in presentation, it seems clear that the quality of the data from Haibel et al is superior to that of Kloesgen et al, and also that the broadening upon which the latter study bases its conclusions is not present in the former. This suggests that Chapter 3. Experimental Background 36 sample preparation is not adequately controled in one of the studies, particularly with respect to hydration, and that measurements at higher frequency are necessary to characterize this aspect of the sample's dielectric behavior. 50 40 30 20 10 0 •10 [ e & (b) . . . . i . , 10 100 f [MHz] 1000 I0; 10* Frequency [Hz| Figure 3.1: Previous dielectric measurements on DMPC: On the left are shown measure-ments at 35.5°C from [37] using the open probe technique described in [42], for both e' and e". On the right are measurements of e" from [47] at various temperatures using the transmission technique described in [44]. Chapter 4. Experimental Technique 37 Chapter 4 Experimental Technique 4.1 One-Port Measurement of the Complex Dielectric Constant Using a Vector Network Analyzer We use a single port measurement to obtain the complex reflection coefficient of a reflection load containing a dielectric sample over a broad range of frequencies. That is, we measure the phase and amplitude of the outgoing wave relative to the incident wave at each frequency. The reflection load is a cylindrical capacitor terminated with a dielectric slab followed by a metallic geometry designed to prevent radiation. The reflection coefficient is measured using an Agilent 85IOC Vector Network Analyzer (VNA), with a frequency range from 45 MHz to 50 GHz. This is an instrument capable of measuring the amplitude and phase of all reflected and transmitted waves in a two port device, at a given frequency. In this work, we use only one port and do not make use of the transmission measurement capability. The reflection coefficient Sn of a load Z on a line of characteristic impedance ZQ is given by The V N A measures reflections from an electrical plane defined by a calibration procedure which requires three standard loads. A variety of calibration sets are avail-able for any given instrument, which are a set of precision electrical loads (typically a short, an open and a 50 Q broadband load) which mate to standard connector types. Measurements with the V N A face a very general problem when calibration sets are Chapter 4. Experimental Technique 38 not available at the desired measurement plane. One can imagine many kinds of experiments where the design of the sample cell is constrained in such a way that it would not be possible to mate the sample to a standard calibration set. Generic ex-amples of such considerations include the size of sample that is available, its electrical characteristics in a given frequency regime, and constraints due to the requirements of in-situ sample preparation. This experiment offers a case in point, where the calibration plane is separated from the measurement plane by a connector which takes a 2.4 mm connection to a 1.2 mm glass bead coax line, for which commercial calibration sets are not available. The desire to work at high frequencies on small, lossy samples of high e' requires a load cell with low capacitance in order that reflections on a 50 Q transmission line can be resolved. This in turn requires small sample cell dimensions. Furthermore, the desire to prepare the sample in the measurement cell, and the need to use chloroform in the sample preparation, also severely constrains the allowed connector materials. Effectively this dictates that the dielectric in the coaxial line at the measurement plane be made of glass, and metallic surfaces be limited to gold due to the chemical sensitivity of biological samples. In such experimental circumstances a technique had to be developed for calibrating out the effects of the hardware intervening between the calibration and measurement planes. This technique is developed below. 4.2 Matrix Formalism for Calibration and Data Analysis Figure 4.1 shows a single port reflection measurement where some admittance Y is separated from the calibration plane by a device whose characteristics are unknown. The situation reflects the main experimental difficulty in this study: the electrical properties of our sample control the admittance of the sample cell (a capacitor), and this information must be extracted from the unknown effects of a connector on the Chapter 4. Experimental Technique 39 V N A Connector Calibration Plane Shunt Y Measurement Plane Figure 4.1: A n experimental dilemma: the V N A calibration plane is separated from the desired measurement plane by an uncharacterized two-port device. other side of a calibration plane. Following [48], the problem can be understood using a matrix formalism to represent n-port systems. A voltage wave travelling down a transmission line at a given frequency can be represented by a complex number indicating its amplitude and phase at any point along the line. For an n-port device, the incident waves an at each port n can be related to the reflected waves bn via the scattering matrix S, which characterizes the device completely. That is, b = S-a where a and b are n-element vectors representing the normalized incoming and outgoing waves at each of the n ports . For a 2-port network: ' Sn S12 a1 _b2 S21 S22 a.2 (4.2) In the case of a one-port measurement, S\2 = 0 and S n = bl/al is the reflection coefficient. This is the quantity measured directly by the V N A in the present exper-iment. Si 1 is a complex number at any given frequency, representing the amplitude and phase change of the reflected wave at the calibration plane. There are a number matrix formalisms equivalent to the scattering matrix formu-lation. Among them are the transmission matrix T which takes {ai,6i} to {a2,b2}, and the transfer matrix A which takes {Vi, Ii} to {V2, —12}- These latter formalisms Chapter 4. Experimental Technique 40 are particularly useful because they transform all the information at a given electrical plane to that at another plane further along the transmission line, and hence cascade across a chain of devices. While T is formulated in the same normalized basis as S, A is formulated in a basis that corresponds to measurable physical quantities. The bases are related by (4.3) (Vn — ZonIn)/V Zon where Zon is the characteristic impedance at port n. Hence our measured quantity Znl 0^ 1 S n = h = Yi ai Vi + ZQII (4.4) where ZQ = 50 fl for the V N A transmission line. The task is to find Vi and I\ at the calibration plane due to a reflection from our connector and sample cell combination, and extract the electrical information of the latter from the former. The total transfer matrix A * at the calibration plane is the product of the connector's transfer matrix A c and that of a shunt admittance, A y : A * = A c A y = A* = a b 1 0 c d Y 1 i + Yb b + Yd d (4.5) (4.6) The condition that the transfer matrix represent a terminating element in a cascade forces l2 — 0. We can now solve for Vi and l\ from the definition of A and eqn. 4.6: Vi v2 = A * h - h I2=0 (a + Yb)V2 (c + Yd)V2 (4.7) and finally, from eqn. 4.4 we find a relationship between our measured Sn and our unknown Y : 5 n = (g + Yb) -Z0(c + Yd) (a + Yb) + Z0{c + Yd)" (4.8) Chapter 4. Experimental Technique 41 Since we are only interested in one element of the scattering matrix, we do not need to know the absolute values of {a, b, c, d}: ratios of these parameters are adequate to characterize the connector such that we can extract Y from S\\. Setting A=a/d, B=b/d and C=c/d we eliminate d and obtain our main result: (A + YB)-Z0(C + Y) b n ~ (A + YB) + Z0(C + YY { ' The problem of calibrating out the effect of the connector reduces to finding three independent relationships, by means of calibration data sets, in order to solve for {A, B, C} as functions of frequency. One particularly convenient calibration is a short where Y —> oo. Hence a short circuit calibration completely determines B: D + Zo 1 — on A and C must be found from two calibrations on known Y . Consider a simple shunt admittance consisting of a terminal capacitance. Then Y = icoC*, where C* is a complex capacitance which depends on the complex per-mittivity €* = €' + ie" of the material inside the capacitor. It is essential to work in a capacitor geometry where the field distribution is not strongly perturbed by insertion of different materials. That is, there must be a constant bare capacitance Co such that Y = iuC0e*. (4.11) For a cylindrical capacitor, Co = ^p^- (4.12) log(r 0/ri) where and rQ are the inner and outer conductor radii, h is the length of the cylinder, and e0 is the permittivity of free space. Using 4.11 and 4.12, equation 4.9 can now be used twice to calibrate for A and C from measurements on two substances of known Chapter 4. Experimental Technique 42 Once {A, B,C} are known from the calibration sets, eqn. 4.9 can be rearranged in combination with with eqn. 4.11 to yield an unknown e* from a single measurement of Sn-= - 1 A(SU ~ 1) + ZpCjSu + 1) f 4 1 3 v iuC0 B(Sn - 1) + Z0(Sn + 1) ' 1 ' ' 4.3 Calibration Before calibrating at the measurement plane with the procedure outlined above, the V N A was calibrated at the calibration plane (see figure 4.1) using an Agilent 85056A 2.4 mm calibration set, terminating with standard short, open and 50 Vt loads. Three separate calibrations were saved, each spanning one decade in frequency, in order to take more data points than the maximum 801 allowed by the instrument. The first spanned 45 to 500 MHz, the second from 500 MHz to 5 GHz, and the third from 5 to 50 GHz. Each calibration setting ramped through 801 equally spaced points averaging 10 times per frequency step. The connector parameters {A, B, C} were determined by measurements of Sn on a short and two substances of known e*. The short was made by pressing an indium cap across the end of the cylindrical capacitor. The calibration substances, ethanol and water, were chosen for having spectra with features comparable to that of a typical phospholipid spectrum. The former has a dispersion peak in a range similar to that of the phosphocoline headgroup, while the latter matches the features in the phospholipid signal due to hydration. 4.4 Experimental Apparatus Figure 4.2 shows a photographs of each half of the measurement device. Figure 4.3 shows a schematic diagram of the measurement device. There are three main components: the connector, the sample cell and the sample cap. The main design criteria are discussed below. Chapter 4. Experimental Technique 43 Figure 4.2: Photos of the measurement cell. The top panel shows the cylindrical sample cell. The bottom panel shows the cap to the sample cell, including a quartz glass window on R T V glue, alignment pins, an o-ring and vent holes. See figure 4.3 for details. Chapter 4. Experimental Technique 44 1 mm u o, a. o U 8" & Anritsu VI00 Connector bead (glass and gold plated coax) Solder RTV elastic silicone glue Anritsu VI02M Connector Vent hole for o-ring seal Screws and alignment pins Sample Cell (cylindrical capacitor) O-ring Quartz cylinder Microscope for viewing sample Figure 4.3: The major components of the measurement device are shown roughly to scale. There are three main components: the connector, the sample cell, and the sample cap. 4.4.1 Connector and Glass Bead Adapter The Anritsu V102M male Sparkplug Launcher brings a 2.4 mm connection down to mate to an Anritsu V100 glass bead coax. This combination is designed to have a characteristic impedance of 50 Q, from DC to 60 GHz, from -55°C to 125°C with 15 dB return loss. Figure 4.4 shows the design of the connector, featuring three sections of transmission line (air, teflon, air dielectric) which scale the connection down to the glass bead. The bead has a glass dielectric, and gold plated kovar conductors. The glass dielectric and gold conducting surfaces provide desirable surfaces for preparation of biological samples. The bead's small dimensions also provide a base upon which to design a cell capable of measuring small samples, and with a capacitance small Chapter 4. Experimental Technique 45 Figure 4.4: The Anritsu V102M connector: Three sections of transmission line (air, teflon, air dielectric) bring a 2.4 mm connection to a fourth section pro-vided by the Anritsu V100 bead, whose glass dielectric provides an ap-propriate sample surface of small dimension. enough to work at high frequencies and large permittivity. 4.4.2 Sample Cell The sample cell is a cylindrical capacitor formed by the center pin of the VI00 bead and hole in the copper base whose diameter matches that of the bead's outer con-ductor. The hole was gold plated using an electrode-less plating solution. The bead was soldered into the copper base while pressed into position by an aluminum plug of dimensions similar to the V102M connector, and alignment of the center pin was confirmed by viewing the other side under a microscope. The inner and outer diameters of the cylindrical cavity were determined using a scaled grating under a microscope, and the depth measured using the focal dial, yielding h = 0.42 mm, = 0.12 mm and r0 = 0.68 mm. Using equation 4.12 this gives a bare capacitance Co = 0.014 pF. At 45 MHz this yields | Z |= \ / U C Q = 253fcf2 and using equation 4.1 | Sn \— 0.9995 which lies at the limit of the V N A resolution, Chapter 4. Experimental Technique 46 while at 50 GHz | Sn |= 0.6395. Inserting any dielectric into the capacitor will bring Z closer to Z0 and bring the signal farther from 1 at the low frequency limit, while at 50 GHz with the largest permittivity tested (e'water(bOGHz) = 28.8), | Sn |= 0.7273. The capacitor is capped with a quartz cylinder, to seal the sample into the ca-pacitor while providing a viewing window to the sample. The cap's diameter is 1.5 mm, just large enough to cover the capacitor but small enough that any resonance modes are above 50 GHz. To avoid picking up any low frequency tail to a resonant mode in the cap, quartz was used for it's relatively low dielectric constant and loss. This cylinder is held in place 0.05 mm above the copper mating surface by R T V Sili-cone glue, a rubbery adhesive that allows the window to make first contact with the sample, and which takes up any strains caused by thermal expansion of the copper. The cap is backed by a hole 3 mm long and 1.4 mm in diameter. This is large enough to view the sample on the other side of the quartz cap, but small enough that it allows no propagation of frequencies below 50 GHz, providing a geometric constraint to radiation losses in the measurement cell. A n 1/4 inch o-ring surrounds the window, with a 1 mm di. vent hole between it and the window. This hole prevents pressure being built up in the sample chamber when the o-ring is squeezed to make a seal. A #30 teflon tube is epoxied into the outside of the vent hole, which can be sealed by inserting a greased copper wire (0.3 mm di.). 4.4.3 Temperature Control Temperature is controlled by a water bath regulated between 5°C and 60°C, cir-culating through a brass cylinder which fits over the sample cell. Silicone grease provides the thermal contact between the walls of the cylinder and measurement cell. A thermocouple is connected to the measurement cell close to the sample, to read the sample temperature (which varies slightly from the bath temperature). The brass cylinder and sample cell within it was encased in styrofoam, and the temperature was regulated to within 0.2°C. Chapter 4. Experimental Technique 47 4.5 Sample Preparation Pure phospholipids were purchased from Avanti Polar Lipids at 5 mg/mL in chloro-form, and stored at -20°C in 5mg ampules. Since polyunsaturated samples oxidize easily, samples were prepared in a glove bag under argon flow, having evacuated and flushed the bag five times before starting sample preparation. The technique de-scribed below was developed to prepare the sample in as short a time as possible while controlling the lipidrwater ratio. The numbers below refer to a 50:50 preparation. Since the density of hydrated lipid is very close to that of water p = 1 g/mL [16], and the volume of the cavity is 0.6 pg, the hydrated phospholipid mass capacity of the cavity was M = 0.6 mg. Hence the required mass of dry lipid was simply n M where n is the mass ratio of lipid to water. For a 50:50 lipid to water mixture, 0.3 mg of lipid had to be deposited into the sample cell. Concentration of the lipid solution was necessary in order that the required mass of lipid could be deposited in the sample cavity using a reasonable number of drops (15 - 20), whose volume was smaller than that of the cavity (0.60 pL). Upon opening an ampule, 50 fj,L was transferred into a 1.5 mm di. capillary test tube, leaving 0.5 cm headspace. A teflon tube connected to argon flow was inserted into the top of the capillary and the chloroform was evaporated from the sample within 20 minutes, leaving dry lipid on the walls of the capillary. 10 fJ,L of the stock solution was then used to collect the dry lipid, leaving the desired mass of lipid (0.3 mg) in the capillary. The concentration of this solution could not be controlled, since chloroform evaporates quickly in the dry argon environment. A 0.5 pL syringe was used to take up solution from the capillary test tube and deposit 0.5 /JL drops in the cavity. A microscope was positioned above the glove bag to facilitate the deposition and view the lipid drying in the cavity. Enough time was allowed between drops for most of the chloroform to evaporate (30 seconds to a few minutes, depending on how much lipid was already in the cavity). When all the solution was gone from the capillary, another 2 pL of pure chloroform was used to gather any excess lipid from the capillary walls, and deposited into the cavity as Chapter 4. Experimental Technique 48 above. To assure all the lipid had been taken up from the bottom of the capillary, pure chloroform was taken into the 0.5 fiL syringe, expelled into the bottom of the capillary and immediately taken back into the syringe, and finally deposited into the cavity. The deposition procedure took about 20 minutes. After all the lipid had been deposited, the sample cell was held directly under argon flow for 30 minutes to assure that all the chloroform had evaporated. This was more effective for evaporating chloroform than holding the sample under reduced pressure. Deionized water of 18.2 MQcm resistivity was used to hydrate the sample. The water was taken from a MiliPore water purifier, and purged of oxygen by bubbling with argon for 30 minutes before inserting into the glove bag. Since argon solubility in water goes down with temperature, the water sample was bubbled at 90°C to avoid supersaturating it with argon, which could lead to bubble formation in samples at lower temperatures. Once dry, the lipid sample was covered with 0.1 /uL of water and allowed to sit for a few minutes to allow any holes in the dry lipid to form bubbles and work themselves out of the sample. The 0.5 pL syringe was used to deposit water into the cavity, under a microscope. Then the remainder of the cavity was filled with water, checking that the approximate volume of water deposited was 0.3 fiL. The cavity was slightly overfilled to assure that no gas was trapped when the quartz cylinder capped the cavity. The cap was lowered onto the o-ring using the alignment pins and then screwed to a tight seal as quickly as possible (within 30 seconds of depositing the water) to assure the overfilling meniscus did not evaporate away. The vent hole was then sealed and the entire device was removed from the glove box and immediately attached to the V N A . 4.6 Temperature Sweep Immediately after removing the sample from the glove box, spectra were taken on the V N A every 10 minutes to observe sample homogenization. Generally, spectra ceased Chapter 4. Experimental Technique 49 to change after 1 hour, at which point we consider the sample to have reached equi-librium. The entire sample preparation procedure to this point took approximately 2.5 hours. Once the sample reached equilibrium it was cooled to 5 °C and allowed to equi-librate for a few minutes before spectra were measured with the V N A . Temperature was changed in 10°C steps up to 65°C, each temperature step taking approximately 15 minutes, with 5 minutes for measurements and equilibration at each tempera-ture. Hence a complete temperature sweep took about 2 hours, which should be long enough to assure that the sample has come to equilibrium at each temperature. It was an empirical question as to whether the total sample preparation and mea-surement time is longer than the sample lifetime, given the degree to which the glove bag environment is free from oxygen. The literature concerning measurements on DHA-containing membranes is still fairly young, with some reports claiming mean-ingful results having used a glove bag for sample preparation and much longer mea-surement times [49], and some experiments taking much more stringent precautions which included a glove box with an oxygen scrubbing mechanism [50]. Chapter 5. Results and Discussion 50 Chapter 5 Results and Discussion The results presented below are meant to demonstrate the general capabilities of the experimental technique described in chapter 4, presenting comparisons and trends rather than an exhaustive quantitative analysis. This is largely due to the time constraints on the project, since the majority of the effort went into developing the experimental technique. The data obtained is quite rich however, with full frequency and temperature dependence for both the real and imaginary parts of the dielectric function; there is much more room for a quantitative analysis by means of curve-fitting and parameter extraction, which must be left for the future. 5.1 Calibration To calibrate out the effects of the connector, measurements were made on polar and nonpolar liquids which have been well characterized in the literature. As dis-cussed in Chapter 3, broadband measurements are scarce in our frequency range, but monochromatic measurements at a number of different frequencies are often used to fit spectra to the Debye form 2.22, extracting the parameters es, and r . With the exeption of the methanol spectrum in section 5.2, the same calibration data set was used for all of the data presented below: a short, methanol and heptane. The methanol spectrum presented below used ethanol for calibration instead of methanol, for obvious reasons. Parameters for ethanol and methanol were taken from [51]. For the nonpolar liquid n-heptane frequency independent parameters were taken from [52]. The data was taken over the course of many weeks, the V N A being recalibrated to the calibration plane in figure 4.1 on a number of occasions. It is significant that a Chapter 5. Results and Discussion 51 single calibration set was used to obtain reasonable results out of all the data presented here; this indicates that the calibration procedure described in section 4.2 behaves as it should, characterizing only the Anritsu connector, an unchanging element for all the measurements taken. 5.2 Standard Polar Liquids 5.2.1 Frequency Dependence at Fixed Temperature To demonstrate the validity of results obtained with our technique, figure 5.1 shows the measured dielectric spectra of ethanol, methanol and water. The ethanol and methanol spectra are smooth and fit the reference spectra very well. The water spectrum suffers fairly serious deviations above ~ 10 GHz, although the general shape of the curve follows that expected for water. This manner of deviation at high frequency appears in all the data where the measured |e| is above ~ 20, but does not appear for low values of |e|. For example, zooming in on the high frequency tails of the alcohol spectra does not reveal significant deviations of the same kind seen for water. It is likely that the calibration technique breaks down in the high |e| regime due to the lack of calibration substances with comparable |e| at high frequency, and an amplification of subtraction errors results. The value of |e| in the hydrated heterogenous systems presented below are much smaller than for pure water. The general features apparent in these spectra are typical of materials with per-manent dipole moments and form the basis for the discussion below. At the low frequency end e'(u) saturates to a value close to es, arising from the total polariza-tion of the molecule including orientational, atomic and electronic polarizabilities. The drop in e'(u>) towards its value at high frequencies reflects the orientational component dropping out as the mobility of the permanent dipole ceases to keep up with the applied A C fields. Thus the dielectric increment es — is a measure of the magnitude and number of permanent dipoles in the system. The broad peak in e"(u>) shows the same process in the out-of-phase component of the permittivity; it Chapter 5. Results and Discussion 52 Ethanol and Methanol 8 8.5 9 9.5 10 10.5 log f Water 8 8.5 9 9.5 10 10.5 log f Figure 5.1: The left hand plot shows e'(to) and e"(u) for ethanol (green, lower pair of curves) and methanol (purple, upper pair) at 25°C. The right hand plot shows the measured permittivity of water (blue, with deviations from the Debye shape above 10 GHz). The smooth black curves compare the fitted Debye spectra based on measured parameters cited in [51]. is zero at low frequencies when the dipoles are in phase because they keep up with the applied field, zero again at high frequencies with the dipoles cease to move at all, and maximum in the intervening region where on average they are 90° out of phase. The frequency at which e"(u) peaks corresponds to the characteristic relaxation time of the permanent dipoles in the system. Fitting a single Debye process to the data presented in figure 5.1 yields Tethanoi = 15bAps,Tmethanol = 5Q.5ps,Twater = lO.Sps, though it is difficult to fit the last curve to e" due to the high frequency deviations, and Twater is extracted by matching e'Debye to the data at (es — too)/2. 5.2.2 Temperature Dependence Figure 5.2 shows the temperature dependence of the e" spectrum for ethanol. As expected, the relaxation frequency 1/r moves up with temperature, indicating in-creased dipole mobility due to the decreased density (and viscosity) of the liquid (see equations 2.45 and 2.46. The decreasing height of the loss peak e"max = (es — £00)72 is indicative of the decrease in es with temperature, since doesn't change much Chapter 5. Results and Discussion 53 with temperature. This trend reflects the effect of thermal fluctuations in broaden-ing the distribution of dipole alignments around the applied field axis, reducing the magnitude of the total dipole moment in the direction of the field (see equation 2.37). The plot on the right of figure 5.2 shows the temperature dependence of es, ex-tracted from the low frequency end (45 MHz)of the e' spectra (es(45 MHz) is taken as the average of the first 50 points in the e'(T) spectra corresponding to the e"(T) spectra shown in 5.2). Shown for comparison are two sets of values taken from [53] for the temperature dependence of the low frequency es. While the values between 15°C and 35°C compare favorably, the slope of the trend does not, and there is significant deviation at the higher temperatures. The latter effect may be due to evaporation of the sample from the cavity or formation of gas bubbles, since these temperatures approach the boiling point of ethanol. The deviation of the slope is of more concern, though the discrepancy may arise due to the relatively high frequency at which we extract es, fairly close to the roll-over, while the literature data are taken further away at lower frequencies. e''(w,T) for Ethanol 6 g of Ethanol 8 8.5 9 9.5 10 10.5 0 10 20 30 40 50 Log(f) Temperature (°C) Figure 5.2: The plot on the left shows the temperature dependence of the e"(u>) spectrum for ethanol. The smooth black curve shows the fitted Debye spectrum at 25°C from [51] for reference. The plot on the right shows e s(T) compared with two sets of values taken from [53]. Chapter 5. Results and Discussion 54 5.3 NaF Salt Solution: Ionic Conduction Salt solutions are often used as calibration standards in dielectric spectroscopy, since the conductivity o in an ionic solution is easily characterized, and the l/u dependence (equation 2.19) cancels the loss of sensitivity due to the frequency dependence of admittance for a capacitive load Y = IUICQC* . This frequency dependence is most easily recognized as a straight line of slope -1 on a log-log plot, which we will use to view e" when a conduction signature is apparent. 7 7-5 8 8.5 9 9.5 10 10.5 7 7.5 8 8.5 9 9.5 10 10 5 Log(f) Log(f) Figure 5.3: The concentration dependence (top) and temperature dependence (bot-tom) of a NaF solution between 45 MHz and 50 GHz. The low frequency tail of e" shows a 1/UJ frequency dependence, as shown on the bottom right plot where o/u) is plotted for comparison. Here a = 5 • 10 1 1. NaF is often used as a standard salt in electrochemistry, in preference to NaCl which is known to cause problems with electrodes [54]. Figure 5.3 shows the concen-Chapter 5. Results and Discussion 55 tration dependence of this conductivity tail, as well as the temperature dependence of a 0.01 mol% solution. In e" only the scale of the conductivity o changes with temper-ature and concentration, not the frequency dependence (ie slope of the conductivity tail). The relaxation of the water dipoles is apparent at the higher frequencies, and has a temperature dependence similar to that of pure water. The temperature de-pendencies of e'(u)) in these spectra are not strongly affected by the salt content, and show a temperature dependence characteristic of pure water. The salt content changes the low frequency es, but does not appreciably alter the frequency depen-dence of the dielectric response. These plots will be useful for comparison with the spectra of polyunsaturated phospholipids presented below. 5.4 Model Phospholipids: D M P C and DOPC As outlined in Chapters 1 and 3, the fully saturated and mono-unsaturated phos-pholipids have been thoroughly studied over the course of about 20 years. A great deal is known about their physical properties, and the development of membrane bio-physics has largely relied on these lipids as model systems. Only recently has serious investigation begun into the more exotic polyunsaturated phospholipid species. Figure 5.4 shows the temperature dependence of e'(u>) and e"(u>) for hydrated D O P C (18:1) and D M P C (14:0), both at 1:1 molar ratio of lipid to water. Both plots show a distinct relaxation near 10 GHz (at 5°C) which is attributable to the water in the system. A lower frequency relaxation is also apparent near 50 MHz in D O P C above ~ 35°C, while in D M P C the loss peak only moves into our frequency range at ~ 55°C. The process is also apparent in e' at the low frequency end, though only at the highest temperatures can we seen the leveling towards es. This relaxation process is attributable to the P C headgroup, whose dynamics are strongly affected by the properties of the acyl chains attached to it. Figure 5.5 shows the D O P C e"(u) spectrum with two Debye processes fit to each peak and their sum. The figure shows that the high frequency peak is that expected for bulk water at 45°C, and the low frequency peak arises from a largely independent relaxation process. Chapter 5. Results and Discussion 56 DOPC 18 :1 (50 mol% water) DOPC 18 :1 (50 mol% water) 8 8.5 9 9.5 10 10.5 8 8.5 9 9.5 10 10.5 Log(f) Log(f) Figure 5.4: Temperature dependence of the real (left) and imaginary (right) permittivity of DMPC (top) and DOPC (bottom). A T = 10°C between adjacent curves. The difference between the data and a simple sum of these two relaxations should give some measure of the interaction between the populations of dipoles, although a quantitative analysis is left for future studies. These kinds of fits to the low frequency peak in the D M P C and D O P C data yield TDMPC = 1-8 ns and TDOPC = 0.94 ns at 45°C. Hence, the D M P C spectra show a much less mobile headgroup, since the chains are fully saturated and hence pack more densely than those in DOPC, which have a single kink due to the steric hindrance of a single double bond half way down the chain. This kink does not allow the oleic acyl chains to pack together easily, resulting in more dilute packing of the P C headgroups and thus the increased D O P C headgroup mobility evident in the data. The properties of the D M P C spectra in figure 5.4 also exhibit a discontinuity between 15°C and 25°C, with the spectra at lower temperatures not following the Chapter 5. Results and Discussion 57 8 8 . 5 9 9 . 5 10 1 0 . 5 L o g ( f ) Figure 5.5: A 50/50 DOPC-water mixture at 45°C, with two Debye curves and their sum shown for illustrative purposes. The water curve is 0.2 e^ a 4 e r (w,45°C) using parameters extrapolated from [9]. The PC plot is fit to the low frequency peak in order to extract the relaxation time of the PC headgroup. same trends as the higher temperature D M P C spectra or those apparent in all the D O P C spectra. This is due to a well known phase transition from the gel to the fluid state. 5.4.1 Phase Transition in D M P C D M P C is known to have a thermotropic phase transition from the gel state to the fluid state at 24°C [16]. The response of the low frequency features is clearly discontinuous with temperature, while the trends in the water portion of the spectrum are smooth as in other measurments on similar systems. Figure 5.6 shows more detail of the phase transition's signature. On the left is e"(u), with more temperatures taken near the transition and only the lower frequency end of the spectra shown. A line is drawn at 200 MHz on this plot, which is used somewhat arbitrarily to plot the cross section of e"(200MHz) as a function of temperature, displayed to the right of the same figure. Chapter 5. Results and Discussion 58 DMPC: Gel- F l u i d Phase T r a n s i t i o n Gel- F l u i d T r a n s i t i o n i n DMPC : 5 5 ° C ^ 2 5 ° c \ : 2 2 ° C ^ ; 2 0 ° C t Z ? 5°C <*B 200 MHz ^ 24°C 1.2 8.4 8.6 8.8 Log(f) 9.2 9.4 10 20 30 40 Temperature (°C) 50 Figure 5.6: The gel to fluid phase transition in D M P C This response at a single frequency is used only as an indicator of the behaviour of the system to show a discontinuity at 24 °C, where the phase transition is expected to be. (Ideally, one would like to fit r or es to each curve and plot r(T) or es(T) to demonstrate the phase transition. This is complicated by the fact that below the transition the dielectric response is much broader than a single debye response, and fitting becomes a more involved and theory-laden.) 5.4.2 D O P C at Variable Hydration A significant amount of experimental effort went into controling the amount of water present in the samples. The degree to which this undertaking was successful is shown in figure 5.7. e"(u>) of D O P C is shown for two mole ratios of D O P C to water (1:3 and 3:1) in addition to the ratio shown in the corresponding plot of figure 5.4 (1:1). The relative heights of the loss peaks roughly track the amount of water in the system. The ratio of the peaks does not correspond quantitatively to the mole ratio, although it would not be surprising if the relationship between these quantities was not linear. Chapter 5. Results and Discussion 59 DOPC 18 :1 (25 mol% water) — , — , n DOPC 18 :1 (75 mol% water) 8 8.5 9 9.5 10 10.5 8 8.5 9 9.5 10 10.5 Log(f) Log(f) Figure 5.7: The hydration dependence of DOPC. A T = 10°C between adjacent curves. 5.5 Sensitivity to Structure The hydration-dependent results on DOPC can be used to show that our experimen-tal technique is capable of resolving heterogenous structure in hydrated samples of phospholipid membrane. The top panel of figure 5.8 contrasts the results of the two mixing models described in section 2.6, for an ethanol-water mixture at the various molar ratios. These plots show the difference between loss spectra of homogenous mixtures given by the microscopic averaging model in equation 2.52, and heteroge-nous mixtures given by the macroscopic averaging model in equation 2.48. The middle panel in figure 5.8 shows measurements on two different kinds of water mixtures, for comparison with these models. On the right are loss spectra for various molar ratios of a water-ethanol mixture, and on the left are the same ratios for a DOPC-water mixture. The difference between these two plots corresponds qualitatively to the difference between the microcopic and macroscopic averaging models. To make the comparison to D O P C data more direct, included at the bottom of the figure are re-sults from the macroscopic averaging model for a mixture of water with a PC-like dipole, whose dielectric parameters were chosen roughly to match the observed prop-erties of the headgroup response in D O P C (es = 30, r = 7.5ns). This comparison also draws attention to one of the interesting features in the measurements; the response of the P C headgroup 'saturates' at a 1:1 molar ratio, suggesting an interesting avenue Chapter 5. Results and Discussion 60 ro-model: Ethanol-Water Mixture at 25°C Micro-model: Ethanol-Water Mixture at 25°C 8 8.5 9 9.5 10 10.5 9 9.5 10 10.5 Log(f) Log(f) Macro-model: PC-Water M i x t u r e a t 40°C i .el I 8 8.5 9 9.5 10 10.5 Log(f) Figure 5.8: Demonstration of sensitivity to structure in phospholipid samples. The top graphs present macroscopic (left) and microscopic (right) averaging models for a mixture of ethanol and water. The middle panel prsents data on a DOPC-water mixture (left) and an ethanol-water mixture. The bottom graph presents the results of the macroscopic averaging model for a mixture of a PC-like dipole with water. Chapter 5. Results and Discussion 61 of research (is the headgroup response suppressed at higher hydration levels, or fully expressed at some critical hydration, and what is the role of the acyl chains in this?). These results demonstrate the ability of the technique to gain information about dielectric domains in a system. This is clearly an advantage in measurements of phospholipid bilayers, demonstrating the organization of the sample into aqeous and non-aqueous regions large enough to exhibit their own bulk dielectric response. Fur-thermore, we see here the potential for this technique to begin addressing questions about the role of water in modulating the dielectric response of other dipolar species, a significant result in light of the motivation behind this investigation. 5.6 Polyunsaturated Phospholipids Log(f) Log(f) Fi gure 5.9: Temperature dependence of the real (left) and imaginary (right) permittivity of DAPC (top) and DHPC (bottom). AT = 10°C between adjacent curves. Chapter 5. Results and Discussion 62 Figure 5.9 shows e' and Log(e") for D A P C (20:4) and D H P C (22:6). The log-log plot is used for e" to capture the power law frequency dependence of the low frequency end of the spectra, while the semi-log plot of e' is shown for comparison with figure 5.4. While e'{u) is not significantly different from that of D M P C and DOPC, e"(u) is very different and worthy of closer examination. 5.6.1 Comparisons with D M P C , D O P C and NaF:water Comparison of PhLipids and NaF, 45°C 7.5 8 8.5 9 9.5 10 10.5 7.5 8 8.5 9 9.5 10 10.5 Log(f) Log(f) Figure 5.10: Comparison of phospholipid species of various degrees of unsaturation with 0.01 mol% NaF solution. Figure 5.10 shows e'(u) and e"(cu) at 45°C for all the phospholipid species measured as well as NaF solution. The e'(ui) spectra are quite similar for all the phospholipids tested, with the exception of D A P C which attains a higher low-frequency es than the other lipids. This suggests that if the low frequency features are due to the P C head-group, somehow the orientational polarizability of the headgroup in D A P C is larger than in the other lipids. This is difficult to account for, though one might imagine that a different degree of ordering among the membranes within the macroscopic sample - determined by the acyl chain packing and water content - could align more of the P C headgroups in the direction of the probing fields, leading to a higher es. On the other hand, the low frequency features of e'(u) may not be related to the P C headgroup; more data at lower frequencies would be required to establish whether the low frequency tails are due to a relaxation process or have a different form which may Chapter 5. Results and Discussion 63 arise from the conduction process evident in e"(u). Without the low frequency loss peaks in e"(u) apparent in D M P C and D O P C , which may be present but masked by the conduction process in the polyunsaturated, it is difficult to draw any conclusions about the relaxation time of the P C headgroups for D H P C and D A P C unless data is taken at lower frequencies. The similarity of the e"(cu) spectra for the polyunsaturates to that of a simple salt solution is discouraging. One obvious conclusion is that the ~ \/u> dependence at low frequencies is indicative of ionic conduction in the sample. While this might be construed as an exciting result in the context of a search for interesting electrical properties, it is more likely that this is an artifact of sample oxidation rather than an intrinsic property of the polyunsaturated membranes; the oxidation of double bonds is likely have an ionic by-product which is released into solution in the aqueous portion of the sample. If we attribute the conduction effects seen in the data as resulting from oxidation of the sample, the measurements suggest that a glove bag is not ah adequately oxygen-free environment for the preparation of polyunsaturated phospholipid samples. A less likely but more interesting interpretation is possible, however. Since the log-log slope of the polyunsaturates' low frequency behaviour is slightly lower than that of the salt solution, this suggests a power law 1/u/* with a < 1. In figure 5.10 &DHPC ~ 0.75 and OLDAPC ~ 0.8. This kind of power law corresponds to Jonscher's 'universal' dielectric response and is commonly seen in ionic solids where charge hop-ping between loosely bound sites is the conduction mechanism. Perhaps the low frequency tail is indicative of an interesting charge transport mechanism in polyun-saturated membranes, presumably fostered by the non-conjugated double bonds along the acyl chains. This would be an extraordinary claim, requiring extraordinary evidence which this study cannot support at this stage. The e'(u) spectra suggest that there is a heterogenous mixture of two macroscopic dielectric domains, and the deviation in a may be due to the superposition of an ionic conduction domain in the aqueous portion with a Debye-like domain in the lipid portion of the sample. The e'(u>) spectra also Chapter 5. Results and Discussion 64 resemble that of D M P C and D O P C closely enough that they suggest two independent Debye processes rather than a power law in e'(u>) which is common in the 'universal' response. Again, more data at frequencies below 45 MHz is required to resolve these issues. Chapter 6. Conclusions 65 Chapter 6 Conclusions We have demonstrated an experimental technique that is capable of investigating interesting features of biological materials which stem from their dielectric behavior. The results presented in Chapter 5 show that reliable dielectric spectra can be ob-tained for small liquid samples of ~ 0.5 pL between 45 M H z and ~10 GHz. The accuracy progressively deteriorates in the region above 10 GHz, but useful spectra are available up to 50 GHz. The spectra can yield values for the dielectric increment es — Coo which is a measure of the orientational component of polarizability arising from permanent dipole moments in the material, as well as r , the relaxation time of these dipole moments. The temperature dependence of the spectra are also reliable between 5°C and 35°C, and follow appropriate trends up to 55°C. Used on the standard phospholipids D O P C and D M P C , the technique is capable of resolving the gel to liquid phase transition, the degree of hydration, and the pres-ence of macroscopic dielectric domains which are indicative of heterogenous structure. The sample preparation technique was capable of controling water content in a fairly reliable way, which is crucial for obtaining consistent results in experimental inves-tigations of phospholipid membranes. There is enough information in the measured spectra to extract the relaxation time of headgroup dipoles, which reflects struc-tural properties of the membrane related to acyl chain packing, and to infer that the headgroup dipoles are interacting with the water dipoles. For the polyunsaturated membranes D A P C and D H P C , the e' spectra were similar to D M P C and D O P C , with similar signatures of two relaxation processes attributable to the headgroup and to water. These features suggest the presence of macroscopic dielectric domains, in-dicating that the samples retain their lyotropic structure. However, e" indicated a dominant ionic conduction process which is most likely due to sample oxidation, al-Chapter 6. Conclusions 66 though it may be attributed to a more interesting charge hopping mechanism in the membrane. With regard to the larger project described in Chapter 1, the investigation at its present stage did not yield any surprising features pertaining to the physics of water and its interactions with biopolymers. In its present form, the technique is limited in the reliability of data above ~ 10 GHz, frequencies where the shape of the dielectric spectra begins to yield information about the behavior of water in the system. We believe there is information in the data regarding the interaction of dipoles in the spectral shape at frequencies between dominant loss peaks, however this is left for future analysis and investigation. There were also no conclusive results indicating a dramatic difference in the dielectric properties of polyunsaturated phospholipids compared to D M P C and D O P C . In this case, we believe that the quality of the sample preparation environment has to be imporved before definitive statements can be made. The conclusions of this investigation are strongly limited by the frequency range that is attainable at both the high and low ends of the spectra. If a V N A was available to push further in either direction, the lower end would still be noise limited and a sample cell with larger capacitance would be necessary to resolve the reflection coefficients. At the high frequency end, the technique is limited in its sensitivity by factors that are not understood at present, but may arise from the calibration standards that were used. In the case of the polyunsaturated phospholipids, a more oxygen-free sample preparation environment is probably needed before unambiguous results can be ob-tained. Also, it would be desirable to work on more ordered samples rather than the bulk multilamellar state. This would require the development of a technique capable of measuring much smaller samples, ideally on the order of a few bilayers. 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