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An experimental study of static magnetic field effect on free diffusion of saccharides in aqueous solution Atwal, Virinder S. 1990

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AN EXPERIMENTAL STUDY OF STATIC MAGNETIC FIELD EFFECT ON FREE DIFFUSION OF SACCHARIDES IN AQUEOUS SOLUTION By VIRINDER S. ATWAL B. Eng. University of Bradford, England 1981 M. Eng. University of British Columbia, 1984 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES CHEMICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 1990 © VIRINDER S. ATWAL, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Chemical Engineering The University of British Columbia Vancouver, Canada Date June 20, 1990 DE-6 (2788) ABSTRACT The purpose of this work was to investigate experimentally the effects of an externally applied magnetic field on free diffusion of saccharides in aqueous solution. The diffusion coefficients of simple saccharides (deoxyribose, D(-)ribose, D(-f-)xylose, D-glucose, D-galactose, D(-)fructose, lactose, sucrose, maltose, raffinose) diffusing through a 0.4 micron pore diameter Nuclepore membrane were measured in applied magnetic field strengths ranging from zero to 1.1 T. The applied magnetic field strength was the only variable in these experiments. The initial saccharide concentration difference across the membrane was one percent by weight. The experiments were conducted at a constant temperature of 25 ±0.1° C. The diffusion coefficient was obtained by means of a modified Rayleigh interferometer-laser system. A Rayleigh interferometer measured refractive index profiles of dilute saccharide solutions contained in a diffusion cell. Refractive index profiles were converted to concentration profiles which were then used to calculate mass fluxes and the corresponding binary diffusion coefficients. A study of saccharide-water interactions indicates that these interactions are very complex in nature and that saccharide hydration depends not only on the number of equatorial hydroxyl (e-OH) groups in a saccharide molecule but also on their spatial orientation. The saccharide-water solutions exhibit properties that are considered to be the result of two factors (1) the elongated (non-spherical) shape of the oligosaccharides (2) effect of monosaccharides on the local water structure (i.e their ability to either enhance or destroy the local water structure). The observed magnetic field effect on diffusion coefficients of saccharides shows a strong dependency on these two factors. A decrease in binary diffusion coefficients ranging from two to eighteen percent has n been observed for applied magnetic fields up to 1.1 T. The diffusion coefficients evaluated at zero field strength (earth's magnetic field) agreed with literature values to within one percent. The noted decrease in diffusivity of monosaccharides ( ribose, xylose, galactose, glucose ) becomes larger with an increase in the number of equatorial hydroxyl (e-OH number) groups in the saccharide molecule. This is because an increase in e-OH number increases the microviscosity of the saccharide molecule ( structure making or stabilising effect ). Deoxyribose and fructose, on the other hand, are considered to be structure breakers. The observed decrease in diffusivity for these saccharides induced by the applied magnetic field seem to be the result of a general stabilizing effect of the applied field on the originally less stable saccharide-water solution. The effect of applied magnetic field on the binary diffusion coefficients of oligosaccharides (sucrose, lactose, maltose, raffinose) correlates with the e-OH numbers as well. In this case, however, the observed decrease in diffusivities is due directly to the orientation of these molecules by the externally applied magnetic field (Cotton-Mouton effect). The same membrane was used to study one complete saccharide system, six runs, (made possible by the changes introduced into the design of diffusion cell and diffusion cell holder in this work) so that variation between membranes would not be a factor. The new experimental procedure resulted in significant reduction in data scatter and highly improved measurement accuracy. Finally, it was shown that the membrane only presented an area reduction to diffusion i.e. the transport process through the membrane followed the assumption of free diffusion. in Table of Contents ABSTRACT ii List of Tables viList of Figures viii ACKNOWLEDGEMENT x1 INTRODUCTION 1 2 LITERATURE REVIEW 5 2.1 Biological Effects of Static Magnetic Fields 5 2.1.1 Natural and Man-made Magnetic fields 6 2.1.2 Mechanisms of Interaction 6 2.1.3 Summary of Experimental Data on the Biological Effects of Static Magnetic fields 9 2.2 Effect of Magnetic Field On Transport Properties of Gases . 10 2.3 Magnetic Field Effect on Liquid Transport Properties 12 2.4 Interferometry 16 2.5 Review of Saccharide-Water Interactions 19 3 THEORETICAL TREATMENT 33 3.1 A Theory of Molecular Diffusion3.2 Molecular Diffusion Through a Porous Membrane 35 iv 3.3 Effects of Magnetic Field on Diffusion 38 3.3.1 Basic Magnetic behaviour and Magnetochemistry 38 3.3.2 Orientation of Diamagnetic Molecules in a Magnetic Field .... 41 3.4 The Theory of Interferometry 43 3.4.1 Interference of Light Waves 5 3.4.2 Rayleigh Interferometer 50 3.4.3 Interpretation of Interference Fringes 53 3.4.4 Distortion of Wavefront by Refractive Index Gradient 57 4 EXPERIMENTAL EQUIPMENT AND PROCEDURE 64 4.1 Experimental Equipment 64.1.1 Laser and Collimating Lens Assembly 69 4.1.2 Masking Slit Assembly 64.1.3 Diffusion Cell 71 4.1.4 Plano-convex Lens and Cylinder Lens 74 4.1.5 Optical Component Mounts 77 4.1.6 Vibration Control 79 4.1.7 Temperature Control4.1.8 Electromagnet 83 4.1.9 Focal Plane Camera and Microscope 84.1.10 Membrane and Saccharide Solutions 6 4.2 Experimental Procedure 88 4.2.1 Diffusion Cell and Membrane Preparation 90 4.2.2 Start of an Experiment 92 4.2.3 Collection of Data Summary 94 v 5 DATA ANALYSIS 95 5.1 Interferometric Data5.2 Calculation of Refractive Index Profiles 97 5.3 Mass Fluxes and Diffusivity Calculations 101 6 RESULTS 117 CONCLUSIONS AND RECOMMENDATIONS 125 NOMENCLATURE 130 BIBLIOGRAPHY 4 APPENDICES 148 A Interference Fringe Pattern Data 14B Error Function Correlation Parameters (m, A and b) 205 C Raytracing Computer Program 22D Mass Flux and Diffusivity Calculation Computer Program 233 vi List of Tables 2.1 Magnetic field technologies 7 2.2 The values of diffusion coefficients and partial molar volumes at infinite dilution of saccharides at 25°C and e-OH numbers [112] 26 4.3 Plano-convex and cylinder lens parameters 77 4.4 Nuclepore membrane specification 88 5.5 Correlation parameters for equation 5.51 96 5.6 Correlation parameters for equation 5.525.7 Profile integral correlation results for maltose at -0.06 cm.(Run MAL16 Field = 1.1 T) 105 5.8 Molar fluxes for maltose and water at 2700 seconds for the bottom half of the cell. A negative flux is upwards (Run MAL16 Field = 1.1 T) .... 106 5.9 Binary diffusion coefficients for maltose calculated both with and without the bulk flow contribution to flux (Run MAL16 Field = 1.1 T) 107 5.10 Comparison of diffusion coefficients at 25°C from this work with literature at zero applied magnetic field strength( at same concentration) 109 6.11 Binary diffusion coefficients in applied magnetic field (T) 113 6.12 Linear regression parameters, equation 6.67 118 6.13 Reduced linear regression parameters for (D~) versus (H*), equation 6.68 120 vn List of Figures 2.1 Typical magnetic field effect on (a) - thermal conductivity and (b) - vis cosity of gases (OCS), x-axis (field strength / pressure) is the same for both graphs [15] 11 2.2 Lielmezs et al. results for the diffusion of electrolyte solutions at the ambient earth field (solid curve) and at the applied external transverse magnetic field (dashed curve) conditions [39] 15 2.3 The D-aldose family of saccharides [92] 21 2.4 The D-ketose family of saccharides and structure of deoxyribose [92] ... 22 2.5 Structure of maltose and sucrose [92] 23 2.6 Structure of lactose and raffinose [92] 4 2.7 Diffusion coefficient Do versus (^)1//3 for saccharides. 1,deoxyribose; 2, D(-)ribose; 3, D(+)xylose; 4, D(-)fructose; 5, D-glucose; 6, sucrose; 7, maltose; 8, raffinose [112] 27 2.8 Equatorial and axial positioning of functional groups in (a) cyclohexane (b) glucose [92] 9 2.9 The relation between diffusion coefficient D0 and mean number of e-OH groups in saccharide molecules. The numbers in the figure denote the same saccharide as found in Figure 2.7 [112] 31 3.10 Magnetic lines of force: (a) diamagnetic (b) paramagnetic [162] 40 Vlll 3.11 Molecular diffusion in a magnetic field, (a) Field applied transverse to diffusion ( vertical ) direction ( as in this work ), (b) Field applied parallel to diffusion direction 44 3.12 Propagation of light as an electromagnetic wave [164] 46 3.13 (a) Interference of light waves (b) Constructive interference (c) Destructive interference [163] 47 3.14 Principle of double beam interferometry 49 3.15 Young's double slit experiment [163] 51 3.16 Top view of Rayleigh interferometer-electromagnet system 52 3.17 Fringe pattern produced by a constant refractive index in the diffusion cell 54 3.18 Typical fringe pattern in the presence of a concentration gradient in the diffusion cell 56 3.19 Wavefront deflection through diffusion cell and refractive index gradient . 58 3.20 Ray tracing parameters 63 4.21 Howell's results for diffusion of sucrose showing large data scatter (a) 8.0 [i m pore diameter membrane (b) 0.8 \i m pore diameter membrane [63] 65 4.22 Experimental setup 66 4.23 Rayleigh interferometer and magnet assembly (I): LBE-Laser and beam expander, Si-Masking slits, DC-Diffusion cell, Ll-Plano- convex lens, L2-Cylindrical lens, EBC-Extension bellows and camera 67 4.24 General view of Rayleigh interferometer and magnet assembly (II) .... 68 4.25 Double masking slit assembly 70 4.26 Diffusion cell (showing proposed modification) 72 4.27 View of diffusion cell 3 4.28 Plano-convex lens and mount [63] 75 ix 4.29 Cylinder lens and mount [63] 76 4.30 Modified diffusion cell holder 8 4.31 Optical bench and mounting block [63] 80 4.32 Concrete and rubber sandwich block for vibration control 81 4.33 Temperature control system (showing electric resistance heater and electric fan) 82 4.34 Magnetic, field homogeneity [151] 84 4.35 Camera and extension bellows arrangement for taking fringe pattern pho tographs 85 4.36 Typical Nuclepore membrane showing round uniform pores [128] 87 5.37 Typical fringe profile (Run MAL16, field strength = 1.1 T) 98 5.38 Refractive index profile for maltose, field strength = 1.1 T 100 5.39 Mass flux and concentration profiles in diffusion cell 103 5.40 Molar flux, molar concentration and diffusivity versus cell position for the bottom half of the diffusion cell 108 6.41 Figure showing (using raffinose as an example) that the molar concen tration of any particular saccharide ( at the same time and at the same location in the diffusion cell ) remained constant for all the runs 114 6.42 Diffusion coefficients (D) versus applied magnetic field strength (H) . . . 115 6.43 Comparison of duplicate experiments for glucose. Curved lines are the 95% confidence limits for regression lines 117 6.44 Reduced diffusion coefficients in magnetic field (D*) versus reduced applied magnetic field (H*) 119 6.45 Reduced slope g (equation 6.68) versus e-OH number for oligosaccharides 121 6.46 Reduced slope g (equation 6.68) versus e-OH number for monosaccharides 122 x ACKNOWLEDGEMENT I would like to take this opportunity to thank my research supervisor Prof. Janis Lielmezs for his guidance, support and inspiration during this work. 1 am grateful for the financial support received from the Natural Sciences and Engineering Research Council of Canada. I would also like to thank Mrs. H. Aleman for many useful suggestions during the past four years. This thesis is dedicated to my grandfather whose wish to see this work finished could not be fulfilled because of his untimely death. Finally, this was all made possible by the continuous loving support and encourage ment from my wife, Susan, whose appreciation for the usefulness of home computer has increased dramatically since she helped me type this thesis. xi Chapter 1 INTRODUCTION Effects of magnetic field on various chemical and biological processes have intrigued sci entists for many years. Earliest research was conducted primarily on the living organisms [1]. Barnothy in 1948 found that young female mice, when placed in a magnetic field of 0.3 to 0.6 T, underwent a temporary retardation in growth. He also reports [2] that a magnetic field can have a retardation effect on the growth rate of cancer cells in mice. Many scientists [3-5] have reported an increase in the rate of healing of bone fractures in an applied magnetic field. A slight increase in the rate of healing has been observed when low frequency alternating magnetic fields are applied to the fracture. Several commercial enterprises [6] are offering magnetic field devices for therapeutic purposes. They claim that low frequency magnetic field will penetrate every single bod}r cell being exposed to the pulsating field. This in turn influences the ions within this cell to enhance ion ex change processes, thus improving the oxygen utilization of the cell. This is important for every healing and regeneration process. Barnothy [7] has compiled an extensive review (up to 1969) of other magnetic field effects on living systems. More recent results can be found in the report of the American Institute of Biological Sciences (1985) [8]. A new emphasis was added to magnetic field effect studies with the start of manned space flights in the 1960-1970's. Researchers wanted to know the effects of the absence of a magnetic field on the living organisms with respect to very long space flights [9]. Even today there is no consensus on a biochemical explanation for magnetic field effects on the living organism. Some researchers [10] propose that these effects can be 1 Chapter 1. INTRODUCTION 2 explained by a change in reaction rates within the cell while others [11] think that these changes are caused by alterations in the chemical bond formations ( structural changes ) due to the presence of a magnetic field. Liboff [12] has proposed that these effects are caused by the changes in diffusion rate of dissociated salts across the plasma membrane and nuclear membrane of the cell. Many researchers have reported a magnetic field effect on various time dependent physical and chemical processes. Svedberg [13] studied rates of diffusion of m-azonitrophenol in a mixed liquid crystal system of p-azoxyanisole and p-azoxyphenetole. He found that the magnetic field affected the rate of diffusion. The diffusion rate increased when the magnetic field was parallel to direction of flow, and decreased when the field was applied across the direction of flow. Magnetic field effects on transport properties of gases have also been reported [14-32]. Lielmezs et al [33-43] have studied magnetic field effects on liquid transport properties and diffusion rates of various aqueous salt solutions. Fahidy et al [44-53] have done valuable pioneering work in the relatively new field of magneto-electrolysis. In another recent study (1985) Gonet [54] failed to find the effect of magnetic field on dielectric constant, pH and surface tension of water as reported by many previous authors [55,56,57]. Yamagishi et al [58] in a 1986 paper discuss the behaviour of organic liquids in high magnetic fields. Kinouchi et al [59] in a 1988 theoretical study show that static magnetic fields affect the diffusion of biological particles in solutions through the Lorentz force and Maxwell stress. Their results show that the Lorentz force suppresses the diffusion of charged particles such as Na+, K+, Ca2+, Cl~, and plasma proteins. These results are discussed in more detail in chapter 2. The experimental evidence gathered from the literature review indicates that the ap plication of a magnetic field alters the dynamics of many physico-chemical and biological processes. While the kinetic theory has been used to explain the effects in gases, there is no satisfactory explanation for the observed effects in liquids. While there is a large Chapter 1. INTRODUCTION 3 body of published data ( sometimes contradictory ) on the magnetic field effects on living systems, very little work has been reported on magnetic field effects on physico-chemical processes. The observed magnetic field effects on living systems and various physico-chemical and biological processes are intriguing, but at this time no clear understanding of the molecular mechanisms responsible for these effects has been reached. Therefore, this work was undertaken to gather more experimental data to understand more fully and explain the different physical interactions influencing molecular motion in a magnetic field. The purpose of this work is to study the effects of an applied magnetic field on the diffusion rate of biologically significant organic molecules through a membrane. Simple saccharides in aqueous solution were selected for this investigation. The membrane used in this work consists of straight cylindrical pores (0.4 micron diameter) in a thin (10 mi cron) polycarbonate film. This combination of membrane and simple saccharides has a potential application to biological systems. There is ample diffusion data for comparison purposes available in literature for these saccharides in the absence of a magnetic field. Optical interferometry is one of the more accurate methods of studying liquid diffu sion. The Rayleigh interferometer was selected and in modified form adapted for this study for three primary reasons : 1) The concentration profile on each side of a membrane is obtained directly from a fringe photograph for a binary mixture, 2) the light source slit is oriented parallel to the concentration gradient and orthogonal to the membrane so that diffraction due to the membrane is minimized, and 3) the overall accuracy of the interferometer has been demonstrated [94,152,179]. A data reduction method devel oped by Bollenbeck [152] for use with membrane transport studies was used in this work to calculate concentration profiles, mass fluxes and the corresponding binary diffusion coefficients. The work described in this thesis consists of: Chapter 1. INTRODUCTION 4 1. A literature review of (a) previous studies made on magnetic field effects on transport properties of gases and liquids. (b) saccharide-water interactions and the use of interferometery as a tool to study mass transport in liquid systems. (c) the theory of molecular diffusion, diffusion through a Nuclepore membrane, theory of interferometry and magnetic field effects on molecular diffusion. 2. A detailed description of the experimental setup, experimental procedure and data collection. 3. (a) Description of a. data analysis method for interpreting interference fringes and reducing them to concentration profiles and mass fluxes (b) Discussion of a technique to correct for errors introduced by optical ray bending in a refractive index gradient. 4. A discussion of experimental results. 5. Conclusions and recommendations for further research. 6. Appendices containing (a) raw data (b) values of various correlation parameters (c) computer programs for raj'tracing and calculations of diffusion coefficient. Chapter 2 LITERATURE REVIEW 2.1 Biological Effects of Static Magnetic Fields Biological effects caused by magnetic fields form a wide subject matter that includes many different topics. In this work I have used a static magnetic field. Therefore, the review that follows leans strongly towards static magnetic field effects as opposed to those caused by time-varying magnetic fields. Even the more narrowed subject matter of static field effects covers many topics. These include the use of magnetic fields in spec troscopic studies of biological material, including NMR, electron paramagnetic resonance (EPR), and magnetic susceptibility and magnetization measurements. Static magnetic fields have also been used to orient cells or cell fragments in suspension. Applications of magnetic fields in physiology and clinical medicine include NMR imaging, magnetic tar geting and modulation of drug delivery; magnetic separation of biological materials, use of magnetic fields in surgical procedures, and noninvasive measurement of blood flow. A rapidly growing area spanning alternating current (AC) and direct current (DC) regimes is the measurement of magnetic fields generated by the human body, and the use of the knowledge of those measurements in medicine and physiology. A large study area involves mutagenic, mitogenic, metabolic, morphological, and developmental effects of exposure of organisms or biological materials to intense DC magnetic fields or to null field conditions. Another important research area includes studies of magnetic fields on behavioural patterns of living organisms. 5 Chapter 2. LITERATURE REVIEW 6 2.1.1 Natural and Man-made Magnetic fields The natural magnetic field consists of one component due to the earth acting as a per manent magnetic and several other small components, which differ in characteristics and are related to such influences as solar activity and atmospheric events [65,82,67,76]. The earth's magnetic field originates from electric current flow in the upper layer of the earth's core. There are significant local differences in the strength of this field. It varies from 33/xT to Q7fiT with an average of about 50/xT used by many workers [91]. The static and time-varying magnetic fields originating from man-made sources generally have higher field strengths than the naturally occurring fields. Apart from home appliances other man-made sources of magnetic field are to be found in research, industrial and medical procedures and in several other technologies related to energy production and transporta tion that are in the developmental stage [73,84,87]. A list of application of magnetic field technologies is given in Table 2.1 [88]. A detailed discussion about the technologies in table 2.1 and the magnetic field strengths encountered in those technologies can be found in the World Health Organisation publication [91]. 2.1.2 Mechanisms of Interaction A broad spectrum of interaction mechanisms can occur between magnetic fields and liv ing tissue. At level of macromolecular and larger structures, interaction of stationary magnetic field with biological systems can be characterised as electrodynamic or magne-tomechanical in nature. Electrodynamic effects originate through the action of magnetic fields with electrolyte flows, leading to the induction of electrical potentials and cur rents. Magnetomechanical phenomena include orientational effects of macro-molecular assemblies in homogeneous fields, and the movement of paramagnetic and ferromagnetic molecular species in strong gradient fields. At the atomic and subatomic levels, several Chapter 2. LITERATURE REVIEW 7 Table 2.1: Magnetic field technologies Energy technologies Thermonuclear fusion reactors, Magnetohydrodynamic systems, Superconducting magnet energy storage systems, Superconducting generators and transmission lines Research facilities Bubble chambers, Superconducting spectrometers, Particle accelerators, Isotope separation units Industry Aluminium production, Electrolytic process, Magnetoelectrolysis, Production of magnets and magnetic materials Transportation Magnetically levitated vehicles Medicine Magnetic resonance, Therapeutic applications Chapter 2. LITERATURE REVIEW 8 types of magnetic field interactions have been shown to occur in biological systems [72]. Two such interactions are the nuclear magnetic resonance in living tissues and the ef fects on electronic spin states and their relevance to certain classes of electron transfer reactions [72]. Recent reviews of the theoretical bases for magnetic field interactions include those of Bernhardt (1979,1986) [68], Schulten (1986) [83] , Pirusyan and Kuznetsov (1983) [81], Abashin and Yevtushenko (1984) [64], Swicord (1985) [85], Tenforde (1986) [88], Frankel (1986) [74]. A large number of diamagnetic biological macromolecules exhibit orientation in strong magnetic fields. In general, these macromolecules have a rod-like shape, and anisotropy in the magnetic susceptibility tensor (%) along the different axis of rotational symmetry. The saccharide-water combination used in this work is a diamagnetic system. The magnetic moment per unit volume (M) of these molecules in a field with intensity H is equal to %H. The theoretical calculation of the interaction energy per unit volume has been discussed by Tenforde [86] and Frankel [74]. The rod-like molecules Avill rotate to achieve a minimum energy in the applied magnetic field. For individual macromolecules, the magnetic interaction energy predicted theoretically will be small compared to the thermal energy kT, unless large field strengths are used. This fact has been demon strated for DNA solutions in which the extent of magneto-orientation has been studied from measurements of magnetically induced birefringence (the Cotton-Mouton effect). Measurements on Calf Thymus DNA [149] resulted in a degree of orientation of only 1% in an applied field of 13 T. Despite the weak interaction of individual macromolecules with intense magnetic fields, there are several examples of macromolecular assemblies that exhibit orientation in fields of 1.0 T or less. This phenomenon results from a sum mation of the individual molecules within the assembly, thereby giving rise to a large effective anisotropy and magnetic interaction energy for the entire molecular aggregate. Examples of biological systems that exhibit orientation in fields of 1.0 T or less are retinal Chapter 2. LITERATURE REVIEW 9 rod outer segments [71,77,90], muscle fibres [66] and "sickled" erythrocytes [79], A more detailed discussion can be found in Maret and Dransfield [149]. 2.1.3 Summary of Experimental Data on the Biological Effects of Static Magnetic fields Several comprehensive sources of experimental data on the biological effects of magnetic fields are available. Older results have been collected in two volumes edited by Barnothy [2] and the monograph by Kholodov [78]; more recent results can be found in the report of the American Institute of Biological Sciences (1985) [8]. Some recent reviews include those prepared by Bogolyubov (1981) [69] Galaktionova(1985) [75] and Tenforde et al. (1985) [89]. Valuable information and extensive references can be found in review papers by Budinger (1981) [70], Persson and Stahlberg (1984) [80] and Tenforde and Boudinger (1986) , [88]. All the above reviews are concerned with potential risks for human health from exposure to magnetic fields of a strength greater than of the geomagnetic field. Studies on the effects of static magnetic fields on enzyme reaction and cellular and tis sue functions have provided diverse and often contradictory findings. The occurrence of significant genetic or developmental alterations in cellular tissues and animal systems exposed to high intensity static magnetic fields appears unlikely from available evidence. Several reports have referred to changes in brain electrical activity and behaviour in an imals exposed to fields ranging from 0.1 to 9.0 T, but the data are inconsistent and at times contradictory. An inherent sensitivity to the weak geomagnetic field and correlated behavioural responses have been demonstrated for a number of different organisms and animal species. However, behavioural effects in higher organisms have not been estab lished at field strengths of less than 2.0 T. Although the data are inconsistent, effects on physiological regulation and circadian rhythms have been reported in animals, due to alterations in the local geomagnetic field. Negative findings in higher organisms have Chapter 2. LITERATURE REVIEW 10 been reported in studies involving field levels as high as 1.5 T. Reversible or transient effects have been reported in lower animals due to exposures to low-intensity static fields or due to alteration in the ambient geomagnetic field. However, no irreversible effects have been detected due to static magnetic field exposures of up to 2.0 T. The magnetic fields produced by transmission lines and computer monitors and their effects on living systems are at present areas of considerable controversy. 2.2 Effect of Magnetic Field On Transport Properties of Gases Senftleben [14] in 1930 first observed that the viscosity of oxygen changed upon the application of a magnetic field. Magnetic effects have since then been observed on the thermal conductivity and kinematic viscosity of a number of poly-atomic gases [14-32] including HC1, DC1, N20, C02, OCS, SF6, CH3F, CH3, CN, CRF3, Njf3. Figure 2.1 shows a typical effect of applied magnetic field on thermal conductivity and viscosity of gases. At high values of the magnetic field (H) the change in the transport coefficients reached saturation, in addition the effect showed a remarkable pressure (P) dependence in that (at constant temperature) saturation was approached as a function of the ratio H/P only. These effects have been explained by kinetic theory of gases. Molecular anisotropy will cause the molecular magnetic moment to precess in an applied field. The precession around the field direction is due to the interaction of the field with the rotational magnetic moment. The magnitude of the precession is dependent on the magnetic field strength, the molecular rotational magnetic moment, and angular momentum of the molecule. This precessional motion has the effect of changing the cross-sectional area of the molecule for collisions with other molecules. This change in cross-sectional area increases the collision probability and therefore the thermal conductivity and viscosity decrease. Chapter 2. LITERATURE REVIEW 11 Figure 2.1: Typical magnetic field effect on (a) - thermal conductivity and (b) - viscosity of gases (OCS), x-axis (field strength / pressure) is the same for both graphs [15] Chapter 2. LITERATURE REVIEW 12 The magnetic field effect on thermal conductivity and viscosity of gases is known as the Senftleben-Beenakker effect. In the case of thermal conductivity, this effect may be explained as follows. In a monatomic gas the heat flow cf is defined as [15]: in which f is the distribution function, m is the mass of particle, and v the particle veloc ity. The heat transport arises from the deviation of f from the Maxwellian distribution in which for the monatomic gases, the vector A can depend only on molecular velocity. For a poly-atomic gas the vector A depends both on the molecular velocity and the angular momentum of the molecule. When a magnetic field is applied to a gas, the angular momentum is polarised which results in an anisotropic distribution of energy. This in turn affects the thermal conductivity of the gas. The field behaviour of the viscosity of polyatomic molecules is completely analogous to the thermal conductivity, although somewhat more complex because of the higher tensorial rank of rj [15]. A detailed discussion of magnetic field effects on thermal conductivity and viscosity of gases can be found in the works of Thijsse et al. [15], Beenakker [23] and Beenakker and McCourt [32]. 2.3 Magnetic Field Effect on Liquid Transport Properties While the kinetic theory of gases has been used to explain magnetic field effects on gases, a well developed kinetic theory for the liquid state does not exist at this time. In the liquid state a molecule interacts simultaneously with several neighbours whereas in the (2.1) function f(°\ This deviation - a first order proportional to temperature gradient- is expressed by [15]: Chapter 2. LITERATURE REVIEW 13 gaseous state molecules interact with only one other molecule at a time. There is a great deal of uncertainty and a theoretical basis for predicting a magnetic effect on liquid transport properties and diffusion is very limited at this time. Svedberg [13] studied rates of diffusion of m-nitrophenol in a mixed liquid crystal system of p-azoxyanisole and p-azoxyphenetole in the presence of a magnetic field. He found that magnetic field affected the rate of diffusion - the rate rose when the magnetic field was parallel to the direction of flow, and fell when the field was applied across the direction of flow. Camp and Johnson [62], Lielmezs and Musbally [33] have studied the magnetic effects on liquid diffusion of electrolytes on a macroscopic level using the principles of irreversible thermodynamics. Lielmezs and Musbally introduced the Lorentz force as the only force exerting influence on the diffusing particle in a magnetic field. This force is defined as [33]: magnetic induction. This force term is included with the other diffusive driving forces and applying the principles of irreversible thermodynamics Lielmezs and Musbally solved V\ and V2 are the average drift velocities for the two ions, LLSS is the partial derivative of chemical potential with respect to concentration, and grades is the electrolyte concen tration gradient. Lielmezs, Aleman and Fish [34] observed an increase in the viscosity of water (between 0.1 to 0.2 %) at a transversely applied magnetic field strength of 1.0 T. They proposed that the magnetic field caused a slight change in the angle of hydrogen bonds with water which in turn affects the translational and orientational motion of the molecules and therefore the viscosity. for the ratio of diffusion coefficient with and without magnetic field as [33]: £l = \i + (^i x B)-(V2 x B) D° (ist grade. (2.4) Chapter 2. LITERATURE REVIEW 14 Lielmezs and Aleman [33,34] also observed a small decrease in the viscosity of vari ous paramagnetic-water salt solutions. They found that the viscosity decreased at high salt concentrations, yet at low concentrations they observed a viscosity increase with an applied magnetic field. At low concentrations the observed viscosity increase ap proached that of pure water, leading them to propose the existence of two competing microstructural interaction mechanisms. The dipolar interactions associated with the pure diamagnetic water, and the spin exchange mechanism characterizing the paramag netic ion water solution. The effects of magnetic field on the diffusion of various chloride salts, (LiCl, NaCl, KC1 etc) through a fritted glass diaphragm has been observed by Lielmezs and Aleman [34-43]. Some of their results are depicted in Figure 2.2. They observed that for LiCl and CsCl the integral diffusion coefficient showed a decrease in an applied magnetic field of 0.5 T, while for other salts it showed an increase. They state that the exact cause of these changes cannot be decided with any degree of certainty. They note, however, that the KCl-water system, showing the largest magnetic field effect also shows the largest structural disorder. These results are intriguing and of qualitative nature, however at this time no definite conclusions have been reached explaining this. These results are important for understanding biological processes, as body fluids do contain salt-water solutions. Fahidy and his co-workers [44-53] have done pioneering research work in the relatively new field of magnetoelectrolysis. They show that mass transfer rates can be significantly altered in electrolytic processes when magnetic fields are externally imposed on the elec tric fields. Fahidy [44] gives an excellent review ( up to 1983 ) of the magnetic field effect on various electrochemical processes. Olivier in his Ph.D. thesis [60] also gives an appro priate earlier literature review of magnetic field effects on electrolyte properties. Brenner [61] presents a theoretical analysis of the effects of an external field on the rheological properties of a dilute suspension of spherical particles containing embedded magnetic Figure 2.2: Lielmezs et al results for the diffusion of electrolyte solutions at the ambient earth field (solid curve) and at the applied external transverse magnetic field (dashed curve) conditions [40] Chapter 2. LITERATURE REVIEW 16 dipoles. He discusses how an applied field hinders the free rotation of the particles. This gives rise to a system of body couples and, hence, to a state of antisymmetric stress. He shows that the apparent viscosity varies with the orientation of the viscometer relative to the direction of the external field. In other words the apparent viscosity becomes anisotropic with respect to the direction of applied field. In his preliminary work, Howell [63] observed the effect of transversely applied magnetic field on the diffusion of aqueous sucrose solution through a porous membrane ( Howell used only one saccharide (sucrose) for his work ). His results, although highly scattered, suggested a possible decrease of about 1 to 2% for the diffusion coefficient as the applied magnetic field increases to 1.25 T, Figure 4.21. These results indicate that some degree of alignment of the sucrose water clusters appears to be taking place in the magnetic field. In view of the literature surveyed it appears that there is a scarcity of data on magnetic field effects on diffusion of biological^ important systems and no satisfactory explanation of what a weak magnetic field can do to molecular level interactions seems to have been advanced. This work is undertaken to gather more experimental data to more fully understand and explain the different physical interactions influencing molecular motion in a magnetic fileld. 2.4 Interferometry Among the many physical processes, free diffusion is one where one wishes to measure the distribution of a solute in a solution without disturbing the solution in any way. One way of making continuous measurements while the experiment is in progress is to shine a beam of light through the solution and use some optical property of the solute (e.g. refractive index) to determine its distribution from the emergent beam. Optical interferometry is such a technique which will yield a continuous profile of the refractive Chapter 2. LITERATURE REVIEW 17 index of the medium through which light is being transmitted. There are many different types of optical interferometers, but the Gouy and a modified version of the Rayleigh interferometer have been the most widely used interferometers, for liquid diffusion studies. Philpot and Cook [93] in 1947 modified the Rayleigh interfer ometer with the introduction of a cylinder lens, which would focus a set of interference fringes representing the refractive index as a function of position in the test cell. They used this equipment to measure the diffusion of sodium thiosulphate in water. At the same time Longsworth [94] used a Gouy interferometer to measure diffusion coefficients of potasium chloride dissolved in water at 0.5°C. The Gouy interferometer does not use a cylinder lens and in addition uses a light source slit that is orthogonal (parallel in Rayleigh interferometer) to the refractive index gradient in a test ceil. This equipment produced interference fringes which were a function of refractive index gradient in the cell. Ogston [95] working independently, used the same method to measure diffusion co efficients of glycine, KC1 and sucrose dissolved in water. Gosting and Morris [105] in 1949 measured diffusion coefficients of an aqueous sucrose solution at 25°C and 1°C. Gosting and Akeley [96] in 1951 continued this work to measure diffusion coefficients for urea in water at 25°C. At the same time, Svensson [97] and Longsworth [98] were extending the original work done by Philpot and Cook on the Rayleigh interferometer to measure diffusion co efficients of aqueous sucrose solution at 25° C. Most of this work was done in the early nineteen forties and fifties and they used either sodium vapour or mercury vapour lamp as a light source. All of these researchers reported the accuracy of their measurement to be better that 0.2%. O'Brien [99] in 1964 and O'Brien et al. [100] in 1964 used a wedge interferometer, which was designed with the light beam passing through the diffusing medium many times, in contrast to a single pass as in the Gouy and Rayleigh interfer ometers. This technique is known as multiple beam interferometry. They applied this Chapter 2. LITERATURE REVIEW 18 method to measure the concentration gradients at electrode surfaces of Zn/ZnSC^/Zn and Cu/OuSC^/Cu electrochemical cells. Duda et al. [101] in 1969 used the same method to measure sucrose-water diffusion coefficients at 25°C. They estimate an accuracy of ± 3% with this technique. Rard and Miller [102,103] using a Rayleigh interferometer mea sured diffusion coefficients of various salt solutions at 25° C. They estimate an accuracy of better then 0.2% for these solutions which ranged in concentration from very dilute to highly concentrated. In 1982, Sorell and Myresson [104] used a Gouy interferome ter to measure the diffusion coefficient of an aqueous urea solution in a saturated and super-saturated solution. Renner and Lyons [106] in 1974 used a Gouy interferometer to measure the diffusivity of KCl-i^O solutions. They used an electronic photo-multiplier scanner to electronically measure the fringe spacing of the interference fringe pattern. The output of this device was directly fed to a digital computer which converted the fringe spacing to refractive index profiles, mass fluxes and diffusivities. This method minimized human errors in troduced when manually measuring fringes through a microscope. A similar technique was proposed by this author [107] to be used in this work. More recently double expo sure holography has been used to study mass transfer problems. Gabelman-Gray and Fenichel [108] used double exposure holography to measure the diffusion coefficient of a 10% sucrose solutions at 25°C. Their results agree to within 10% of literature values measured by other methods. O'Brien et al. [109] in 1982 used double holography to measure diffusion coefficients of respiratory gases in a perfluorocarbon liquid. Mass transport through membranes has been traditionally studied by placing a mem brane between two mechanically mixed fluid compartments and monitoring the compo sition of the solutions in each compartment. Interferometric techniques which directly measure the concentration profiles have been widely applied to the study of diffusion Chapter 2. LITERATURE REVIEW 19 in liquids. The opaque characteristics of membranes have inhibited the direct measure ment of concentration profiles in the membrane itself. However, Bollenbeck and Ramirez [110] in 1972 first used a modified Rayleigh interferometer to measure diffusion through membranes. They measured concentration profiles surrounding a membrane surface in an aqueous sucrose solution. They developed a technique to calculate mass flux at each membrane solution interface and therefore the diffusion coefficient through the membrane itself. Their diffusivity values agreed to within 1% of literature values for a 1% sucrose solution at 25°C. Their technique with slight a modification is used in this work and detailed discussions of this method are contained in appropriate sections of this thesis. Min et al. [Ill] used a wedge interferometer to study steady state diffusion of ethyl alcohol and water through a cellophane membrane at 25°C. The accuracy of their mea surements is ± 3%. The presence of a membrane in the diffusing medium presents several problems. The membrane, being opaque, does not allow the concentration profile to be measured through the membrane pores (recall that optical interferometers require trans parent medium). The wavefront deflection of light caused by a refractive index gradient also occurs near the membrane surface which produces a membrane shadow and a distor tion of the measured concentration profile. A technique is discussed in this work which is used to correct for deflection effects. 2.5 Review of Saccharide-Water Interactions Simple saccharides used in this work can be classified as follows. Deoxyribose, D(-)ribose, D(-(-)xylose, D(-)fructose, D-glucose and D-galactose are known as monosaccharides. Su crose, maltose and lactose contain two monosaccharides joined by a glycosidic linkage and are known as disaccharides. Raffinose consists of three monosaccharide units and is known as a trisaccharide. Saccharides that on hydrolysis yield 2 to 10 monosaccharides Chapter 2. LITERATURE REVIEW 20 are also known as oligosaccharides. Saccharides that give more that 10 monosaccharide units on hydrolysis are known as polysaccharides. Monosaccharides are further classified according to (1) the number of carbon atoms present in the molecule and (2) whether they contain an aldehyde or ketone group. All the monosaccharides studied in this work except fructose contain aldehyde group. Deox}rribose, D(-)ribose and D(+)xylose con tain four carbons and are known as aldotetroses. D-glucose and D-galactose have five carbons and are known as aldopentoses. Fructose is known as a ketopentose because it contains five carbons and a ketone group. Traditionally there are three ways used to show structures of sugars. (1) Fisher projection formula shows the straight chain forma tion of saccharide molecule, Figures 2.3 and 2.4 (2) H^'worth formula shows the cyclic nature of the saccharide molecule, Figure 2.6 (3) the chair form conformation usually adopted by the saccharides in solution, Figure 2.6. Figures 2.3 to 2.6 show all three types of structural representations for saccharides used in this study. In spite of the enormous progress in carbohydrate chemistry made by organic chemists, very little ef fort has been devoted to understanding the behaviour of carbohydrates in solution [118]. Saccharide-water interactions may be summarized in the following systematic way [118]: 1. Carbohydrates are polar molecules capable of hydrogen bonding with the solvent and with each other; therefore we would expect the mechanism of solvation to contribute significantly to an adequate description of the solution. 2. Water is a unique solvent because of its orientational asymmetry, i.e. its strong preferences for interactions in certain well denned orientations. 3. Saccharide stereochemistry also indicates preferred directions of hydrogen bonding. 4. It therefore seems reasonable that the resultant solute-solvent interactions should be influenced by the compatiblity and spacings with the saccharide stereochemistry. Chapter 2. LITERATURE REVIEW 21 CHO H OH CH<>H «K+)-CHXXTI Webvtk H-H-CHO -OH -OH CH3OH CK—»-£rythros< HO-H-CHO -H -OH CH;OM n-< — V-Thre<»sr H-H-H-CHO -OH -OH -OH CHiOH o-(-(-Ribos« HO-H-H-CHO -OH -OH CH:OH o^—)-Arab<oosc H-HO-H-CHO -OH -H -OH O-( + >-XY<<** HO-HO-H-CHO -H -H -OH CIU)H IM — M-vxosc H-H-H-H-CHO -OH -OH -OH -OH CH.OH o-<+»-A(los< CHO HO-H-H-H--OH -OH -OH CH,OH o-< + >-AI«roK H-HO-H-H-CHO -OH -H -OH -OH CH^OH CHO H0-HO-H-H--H -H -OH -OH CH:OH o-<+VMarm<»se H-H-HO-H-CHO -OH -OH -H -OH CH.OH o-(-K;«rfose HO-H-HO-H-CIK) -H -OH -H -OH t>-(-|-l<t<wc lI-HO-HO-H-CHO -OH -H —H -OH CI I-Oil HO-HO-HO-H-CHO -H -H -H -OH CIMHI «-< +HTiW.sc Figure 2.3: The D-aldose family of saccharides [92] Chapter 2. LITERATURE REVIEW 22 CHO > H—C—OH H—C—OH CH.OH 2-Dcoxy-i>-ribose CH,OH CH>OH I -M>ih vdroxy prop;* mine H-H-Cl 1.-4)11 O -OH -OH ("H;0!l o-< + )-Ril>u<<« OI-OII I •Oil CI I-Oil »-< - >-Er\1hralo5< CH.OH =o -OH -OH -OH HO-H-H-CH:OH =0 -H -OH CH;OI< o-<+)-Psicos« -OH CH;OH (M-)-Fr>*ctOK H-H0-H-aijon =o HO-H- OH CH;OH D-< +• (-Xylulose CH.-OH 0 -OH -H -OH HO-HO-H-CH-OII =0 -H -H -OH CH-OH o-i + hSorbose CH.OH D-<—)-Tagalos< The o-keioses (up to the ketohexoses). Figure 2.4: The D-ketose family of saccharides and structure of deoxyxibose [92] Chapter 2. LITERATURE REVIEW 23 conformational formula for maltose 4-<?-(a-D-glucopyranosyl)-i>-glucopyranose CH2OH OH a-glucopyranose CH2OH o HO HOCH, HO HO HO /}-fructofuranose HO CHjOH CH2OH OH sucrose /i-D-fructofuranosyl o-o-glucopyranoside Figure 2.5: Structure of maltose and sucrose [92] Chapter 2. LITERATURE REVIEW lactose 4-<7-(/!-D-galactopyranosyl)-D-gIucopyranose CrUOH Raffinosc Figure 2.6: Structure of lactose and raffrnose [92] Chapter 2. LITERATURE REVIEW 25 5. It follows from step 4 that conformational equilibria existing in solutions of car bohydrates contain solvent contributions i.e. solvation effects favour certain con formations over others (saccharides in solution usually exist as mixtures of various conformers). 6. Interaction between saccharide molecules in solution, or between saccharide residues within di-and oligosaccharides, are likely to be affected by such orientation-specific solvation. There are a number of theories of saccharide hydration. Differences in the observed hy dration properties of saccharides have been discussed in terms of a "specific hydration model" in which the compatibility between the spatial orientation of hydroxyl groups around the saccharide ring and the intermolecular order in the aqueous solvent is con sidered to be a significant influence on the extent of hydration [123]. Another viewpoint found in the literature of saccharide hydration is that of "water structure" [118]. This concept suggests that water has a fair degree of tetrahedral bonding, which structures it like ice. When ions are dissolved in water, they either destroy or enhance the ice like structure. If the ions destroy the structure, they are immediatly surrounded by water of a reduced apparent viscosity, equivalent to a higher local temperature. Such structure breaking ions would diffuse more rapidly than expected. In contrast, certain ions appar ently enhance water structure, these ions are immediately surrounded by water that is more organized than water in the bulk. Such more organized water has a higher viscosity, equivalent to lower temperature. Such "structure-making " ions diffuse more slowly than expected. Uedaira and Uedaira [112] report the partial molar volume and diffusion coefficients for ribose and deoxyribose in water at 25°C. They compared the data with the diffusion coefficients of various other saccharides extrapolated to infinite dilution and discuss the Chapter 2. LITERATURE REVIEW 26 Table 2.2: The values of diffusion coefficients and partial molar volumes at infinite dilu tion of saccharides at 25°C and e-OH numbers [112] Sugar Vo Do xlO6 e-OH cm3/mol cm2/sec numbers Deoxyribose 94.6 8.150 1.37 D(-)Ribose 95.2 7.795 2.14 D(+)Xylose 95.2 7.495 3.50 D(-)Fructose 110.4 7.002 3.01 D-Galactose 111.9 6.749 3.72 D-Glucose 112.2 6.750 4.51 Sucrose 211.6 5.230 6.21 Lactose 211.1 5.210 6.73 Maltose 211.3 5.201 7.22 Raffinose 303.2 4.359 8.82 saccharide hydration on the basis of the deviation from the Stokes-Einstein relation [112]. Table 2.2 shows the limiting (infinite dilution) values of the diffusion coefficients Do and the partial molar, volumes Vo f°r various saccharides at infinite dilution. According to the Stokes-Einstein law D0 is linear in 1/r (where r is the solute radius), provided its shape is spherical, for a given solvent and temperature. Hoiland and Holvik [137] suggested that the value of V0 is somewhat dependent on the degree of hydration of saccharides. Uedaira 1/3 and Uedaira therefore took V0 as a size of the saccharide molecule. A plot of D0 versus (i)1/3 is shown in Figure 2.7 where the straight line relation indicates Stokes-Einstein law. The deviation of saccharides 1,2,6,7,8 from Stokes-Einstein law arise from two effects :(1) the elongated shape of the molecule (2) a change in the local water structure [180]. By examining the translational frictional coefficients [140] and dielectric relaxation of aqueous sugar solutions [141], Uedaira and Uedaira did show that the deviation from Chapter 2. LITERATURE REVIEW 27 Figure 2.7: Diffusion coefficient D0 versus (^)^3 for saccharides. 1,deoxyribose; 2, D(-)ribose; 3, D(+)xylose; 4, D(-)fructose; 5, D-glucose; 6, sucrose; 7, maltose; 8, raffinose [112] Chapter 2. LITERATURE REVIEW 28 Stokes-Einstein law in Figure 2.7 for sucrose, maltose and rafHnose is due to the non-spherical shape of their molecules. Figure 2.7 shows that the behaviour of ribose, xylose and deoxyribose differs significantly from that of oligosaccharides. In spite of the similar ity of Vo (partial molar volume i.e size) for these saccharides (see Table 2.2), the values of D0 differ significantly from each other and increase in the order: Since the shape of these molecules is almost spherical, the deviation from Stokes-Einstein relation is based on the second effect mentioned above. The hydration of saccharides is affected by the mean number of equatorial hydroxyl (e-OH) groups [124], Uedaira and Uedaira [112] calculated the values of e-OH numbers for various saccharides from the data of Angyal and Picles [119], and Que and Gray [121] (Table 2.2). The number of e-OH groups decreases in the order Comparing equation 2.5 with 2.6 we see that the value of D0 for xylose, which has the largest number of e-OH groups in a molecule, is the smallest. The diffusion coefficients obtained experimentally support the idea that the microviscosity of a saccharide molecule increases with the number of e-OH groups in a molecule (Table 2.2). Figure 2.8 shows the equatorial and axial positioning of functional groups in organic compounds. The activity coefficient of ribose in water is nearly equal to 1, which suggests that water structure in the vicinity of ribose is close to that of bulk water. The activity coefficient of xylose (1.031 at 1M at 25°C) shows that xylose behaves as a weak structure maker [135]. Deoxyribose on the other hand is considered to destabilize water structure as a consequence of the competition of two interactions: one interaction between axial hydroxyl (a-OH) groups of the saccharide molecule and the water molecule and the deoxyribose > ribose > xylose (2.5) deoxyribose (1.37) < ribose (2.4) < xylose(Z.b) (2.6) Chapter 2. LITERATURE REVIEW 29 Figure 2.8: Equatorial and axial positioning of functional groups in (a) cyclohexane (b) glucose [92] Chapter 2. LITERATURE REVIEW 30 other between two water molecules [112]. Recently Stern and Huber [138] measured the enthalpies of transfer of ribose and deoxyribose from pure water to 3M aqueous solutions of ethanol and urea. According to them deoxyribose forms fewer hydrogen bonds with water than ribose. Uedaira and Uedaira [112] showed that the structure of water around a solute molecule can be established by considering deviations from Stokes-Einstein relation combined with the knowledge of the shape and conformation of the saccharide molecule. Figure 2.9 shows the relation between diffusion coefficient at infinite dilution (D0) and mean number of e-OH groups for various saccharides. The numbers in the figure denote the same saccharide as found in Figure 2.7. As seen, a good correlation exists between the two values, supporting experimentally the idea that was proposed by Franks [118] and Sugget et al. [124], i.e. that the saccharide molecule which has a larger number of e-OH groups in a molecule has a stronger stabilizing effect on the water structure. The discussion so far has established that the mean number of e-OH groups is a good parameter to describe the hydration of saccharide molecules. Several researchers [137,123] have indicated that not only is the number of e-OH groups important but also their relative positions in the sugar molecule. The partial molar volumes of saccharides published so far also indicate such orientation dependence. These differences are small, however, and Shahidi, Farrell and Edward [125] have demonstrated that the partial mo lar volumes of a whole range of carbohydrates can be calculated within 1-3% accuracy without considering any conformational differences. Such small differences in thermody namic properties may however, lie at the very root of differences in the shapes taken up by saccharides in solution and in determining the biological functions of such saccharides. To probe in more detail the manner in which hydration and conformation interact to produce the observed solution behaviour, it is necessary to investigate the interactions at the microscopic level. In this area the most detailed analysis of carbohydrate-water Chapter 2. LITERATURE REVIEW 31 Figure 2.9: The relation between diffusion coefficient D0 and mean number of e-OH groups in saccharide molecules. The numbers in the figure denote the same saccharide as found in Figure 2.7 [112] Chapter 2. LITERATURE REVIEW 32 interactions is that of Suggett and his colleagues [124,122,123] . This is based on mea surements of the dynamics of both molecular species in binary aqueous carbohydrate solutions. Using the complimentary techniques of dielectric and nuclear magnetic re laxation, Suggett has been able to interpret experimental data in terms of saccharide hydration and conformation. The important point that emerges from their investigation is that the orientation of e-OH groups has a small but significant effect on saccharide hydration. Uedaira and Uedaira [112] as mentioned previously calculated the size of the diffusing molecule using the partial molar volumes V0 (taking into account the small differences in Vo caused by saccharide hydration). Beck and Schultz [126] in their studies on hindrance of solute diffusion within membrane of known pore geometry also calculated the size of the diffusing molecule. They however recognized that there is some controversy as to the radii of these solutes that best characterize their apparent size in water. Their studies include such saccharides as glucose, sucrose and raffinose. They decided that the most accurate radii to use for this purpose are those calculated from the free diffusivity following the Stokes-Einstein equation i.e.: T>rp RSE = ^ A> NAV (2.7) where RSE is the Stokes-Einstein solute radius, R is the universal gas constant, T is the absolute temperature, r] is viscosity, D0 is diffusivity in free solution and NAV is Avogadro's number. A correction, derived by Gierer and Wirtz [127], to account for the fact that the solute molecules have a size comparable to that of a water molecule, was applied. This correction is: r F?„, i RSE (2-8) Rw , Rs 1-5 — h Rs Rs + 2Rw-where Rs is the equivalent solute radius and Rw is the radius of a water molecule. Numerical solution of equation 2.8 can be obtained by successive approximations. Chapter 3 THEORETICAL TREATMENT This chapter discusses the theoretical aspects as they apply to molecular diffusion, dif fusion in membranes, magnetic field effects on molecular diffusion and the use of inter ferometry in studying mass transport in liquids. 3.1 A Theory of Molecular Diffusion If pure water is layered upon an aqueous solution of sucrose in a test tube or in a diffusion cell, there occurs, simultaneous mass migration of sucrose molecules upward and water molecules downward. This process, whereby concentration differences in a solution spontaneously decrease until the solution finally becomes homogeneous, is called diffusion. Diffusion itself is a result of random Brownian motion of the molecules arising from the thermal energy of the molecule. Therefore, in the absence of any temperature and pressure gradients, the net molecular motion of the diffusing species will be from the region of higher concentration to lower concentration. Diffusion plays an important part in biological systems, where, a knowledge of the laws governing diffusion is basic to an understanding of transport through the walls of living cells and within the cell itself. For diffusion at constant temperature and pressure in binary systems showing no volume change on mixing, Fick's first law for one-dimensional transport of a solute is given by J = -D* (3.9) Fick's first law defines flux at steady state conditions. The above equation expresses the 33 Chapter 3. THEORETICAL TREATMENT 34 fact that at any time, t, and position, x, the flux, J, of solute is directly proportional to the solute concentration gradient dC / dx. D is the diffusion coefficient and its value is the same for solute and solvent in all two component systems if there is no volume change on mixing. For the one dimensional case the continuity of mass equation may be written as Ot OX Equation 3.10 states that the change in concentration of component i per unit time in a unit volume is equal to the differences of flows into and out of that volume. By combining Fick's first law (equation 3.9) with the continuity of mass equation (equation 3.10) we obtain for two-component systems the following differential equation: If the diffusion coefficient is independent of concentration and therefore of distance x, equation 3.11 reduces to Fick's second law The data analysis is based on a suitable solution of equation 3.12 such' as the error function solution [134] as discussed in chapter 5. Fick's first law (equation 3.9) can be extended to binary systems and written in terms of NA, the molar flux relative to stationary co-ordinates ( membrane in this work ) [129] NA = XA(NA + NB)- cDAB^ (3.13) where XA is the mole fraction of component A and c is the total molar concentration. The first term on the right hand side of this equation is the molar flux of component A resulting from bulk motion of the fluid. Bulk motion may be due to an imposed flow, density changes due to the diffusion process or both. In this work equation 3.13 is Chapter 3. THEORETICAL TREATMENT 35 used to calculate the binary diffusion coefficients of saccharides. Einstein [130] used the molecular kinetic theory of heat to develop a theory of Brownian motion and so provided a physical picture to describe the diffusion process in dilute solutions. Einstein proposed n kT , D = — (3.14) where D is the diffusion coefficient, k is Boltzmann's constant and f is a frictional co efficient of a solute molecule. Stokes [131] showed that the frictional coefficient f, for the special case of a spherical particle of radius r moving with uniform velocity in a continuum of fluid (with no slippage) of viscosity rj, is given by / = 67T7/r (3.15) Combining equation 3.14 with equation 3.15 gives • - kT This is the classical Stokes-Einstein equation for diffusivity. This equation is valid only in very dilute solutions of spherical particles which are large compared to the size of the solvent molecules and would characterize the free diffusion process in the frame work of this study. 3.2 Molecular Diffusion Through a Porous Membrane A complete description of the membrane used in this work is given in Chapter 4. The membrane made from a polycarbonate film is manufactured by Nuclepore Canada In corporated. The pores are essentially cylindrical and the pore size diameter used in this work is 0.4 micron. There are no significant physical or chemical interactions between the polycarbonate film and the saccharide solution [128]. The membranes used are 10 microns thick so that the applicability to thin membranes (e.g. biological membranes) is Chapter 3. THEORETICAL TREATMENT 36 also shown. Because the membrane is very thin it is assumed that the mass flux flowing into one membrane surface equals the mass flux out the other surface. The membrane used in this work represents only an area reduction to diffusion (discussed later in this section). The concentration at the membrane surfaces may be obtained from the profile correlations. The molar flux through the membrane may then be regarded as a simple diffusion phenomenon and written as [152]: AC NAm = XA(NAm +NBm)- AFDAB-~± (3.17) AX where Af is the effective free area for diffusion, DAB is the binary diffusivity, ACA is the concentration difference between the top and bottom surface of the membrane which has a thickness of Ax. The product of free area and binary diffusivity in equation 3.17 may be termed an apparent membrane diffusion coefficient [126]. Osmotic pressure, 7r, is expressed by Van't Hoff's law [132] for dilute solution as TT = RTAC (3.18) where R is the universal gas constant, T is absolute temperature and AC the solute concentration difference across the membrane which is permeable only to the solvent. Staverman [133] introduced concept of the reflection coefficient, cr, to account for the transient osmotic pressures that could arise when the solute did not pass easily through the membrane ( i.e. membrane permeability is different for solute and solvent ). Equation 3.18 may be combined with the reflection coefficient a , to give •K = aRTAC (3.19) Renkin [171] proposed an expression for a for cylindrical pores based entirely on geomet rical arguments. Renkin's equation for cr is [2(1 - /3)2 - (1 - /3)4][1 - 2.104/3 + 2.09/?3 - 0.95£5] [2(1 - 7)4][1 - 2.1047 + 2.0973 - 0.957E (3.20) Chapter 3. THEORETICAL TREATMENT 37 where P = j£ and 7 = ^ with R„, Rw and Rp being the radius of the solute molecule, solvent molecule and pore respectively. For raffinose, the largest saccharide used in this work Rs = 6.54 A[126]. The radius of water molecule Rw = 1.9 Aand pore radius Rp = 0.2 fim. Using these numbers gives the values of 0 and 7 (denned above) as /3 = 1.63 x 10~3 7 = 4.75 x 10"4 Introducing the obtained values of j3 and 7 in equation 3.20 yields the value of reflection coefficient a as a = 0.002452 When equation 3.19 is used with AC = 1% by weight we obtain an osmotic pressure 7T at 25°C as 0.091 centimeters of mercury. The osmotic pressure across the membrane serves only as a potential for water flow. An osmotic flow of water from the top side of diffusion compartment into the bottom side of the membrane must result in an upward displacement of the membrane. This movement can be easily observed with the focal plane microscope but no such upward displacement of membrane was noticed in this work. Faxen [172] used a frictional drag model to calculate the ratio of diffusion coefficient in a cylindrical pore to that in the bulk solution. His model was based on a sphere falling in a tube (filled with a viscous liquid) to predict this ratio. The Faxen equation is: Dm = Df(l - 2.104/3 + 2.09/33 - 0.95/?5) (3.21) where Dm and Df are the diffusion coefficients 01 solute in the membrane (pore) and in the bulk solution respectively. The numerical value of j3 in equation 3.21 is the same as for equation 3.20. The Faxen equation predicts that the diffusion coefficient observed Chapter 3. THEORETICAL TREATMENT 38 through the membrane (pore) is reduced due to the viscous drag between the diffusing molecule and pore wall surface. Iberal et al [153] support this observation by measuring the diffusion of differently sized molecules through a porous membrane with cylindrical pores. They observed a decrease in membrane permeability as the value of 8 increased. On the other hand Williamson et al [173] using membranes with different size pores observed a decrease in the diffusion rate of sucrose as the value of 8 increased, thereby confirming Faxen equation (equation 3.21). This observation becomes important when considering the diffusion of anisotropic molecules in a magnetic field. Using equation 3.21 and the value of 8 as calculated before, we obtain for raffinose, the largest saccharide used in this work Dm = 0.997D/ (3.22) This means that the 0.4 micron diameter pore size membrane used in this work should behave essentially as an area reduction to the saccharide-water diffusion process, i.e. the diffusion coefficient inside the pore is nearly the same as in the free solution. Any observed departure from this model would therefore be strictly due to geometric factors associated with solute, solvent and pore diameters. 3.3 Effects of Magnetic Field on Diffusion 3.3.1 Basic Magnetic behaviour and Magnetochemistry If a substance is placed in a magnetic field of strength H, the substance becomes mag netized. The magnetic induction ( = magnetic flux density ) B is given by the sum of two parts: (1) the applied magnetic field strength H, and (2) the induced magnetization ( = magnetic moment per unit volume ) I. Quantitatively, the magnetic induction B is defined as the density of lines of force per unit area in the substance and is given by [162] B = H + 4tt/ (3.23) Chapter 3. THEORETICAL TREATMENT 39 where B , H and / are expressed in the unit Gauss, G ( 1 T = 10 kG). Dividing equation 3.23 by H gives the magnetic permeability \i per unit volume of substance : H = 5 = 1 + 4ttX (3.24) H where % = / / H is termed the susceptibility per unit volume and is a dimensionless quan tity. Magnetic susceptibility, x, the most fundamental concept in magnetochemistry, in general can be defined as the extent to which a substance is susceptible to magnetiza tion induced by an applied magnetic field strength H. In isotropic media B, H and I have the same value in all spatial directions and thus x becomes both a scalar and di mensionless. For anisotropic substances, the magnetic susceptibility % depends upon the orientation of the molecule with respect to the direction of the applied magnetic field. The anisotropic magnetic susceptibility can only be observed in a substance if all the molecules are oriented with respect to the direction of magnetic field, for instance as in a single crystal. Most substances may be classified as dia, para or ferromagnetic. In diamagnetic substances the permeability (equation 3.24), fi is less than 1 (/i < 1) and the intensity of induced magnetization I and hence the susceptibilities x are an negative. A diamagnetic substance causes a reduction in the lines of force as shown in Figure 3.10 (a). This is equivalent to the substance producing a magnetic flux in a direction opposite to the applied field. Thus if a substance is placed in an inhomogeneous field the substance will move to a region of the lowest field, and the net effect of applied magnetic field manifests itself as one of repulsion. Hence the susceptibilities are shown with a negative sign and the per kilogram susceptibility for diamagnetic substances is usually very small (of the order of -0.126 X 10-6 in S.I. units). Diamagnetism arises due to the motion of the electrons in their atomic and molecular orbits. According to the classical theory, an electron carrying a negative charge and moving in a circular orbit is equivalent to a Chapter 3. THEORETICAL TREATMENT •3»- H (b) a Diamagnetic body in a magnetic field showing its permeability (/x) to the field to be less than 1. ie \i < 1 b Paramagnetic body in a field with /*> 1. Figure 3.10: Magnetic lines of force: (a) diamagnetic (b) paramagnetic [162] Chapter 3. THEORETICAL TREATMENT 41 circular current. If a magnetic field is applied perpendicular to the plane of the electron orbit, the moving electron will experience a force along the radius, the direction of which depends on that of the magnetic field and the moving electron. The well known Lenz's law, which predicts the direction of motion of a current carrying conductor placed in a magnetic field, when applied to this situation shows that the system as a whole will be repelled away from the applied field. In paramagnetic materials the permeability fi is greater than one (fi > 1) and produces an increase in the density of fines of force as shown in Figure 3.10 (b). Thus the intensity of magnetization I and susceptibilities are all positive. This implies that the susbtance is producing a flux in the same direction as the applied field, and the substance when placed in an inhomogeneous field will tend to move to regions of the highest field, showing an attraction between the two. The susceptibilities are shown with a positive sign. The per kilogram susceptibility \ 1S numerically much greater than in the diamagnetic case and ranges between 1.26 and 12.6 X 10~6 in S.I. units. Paramagnetism is exhibited by substances which have unpaired electrons in the ground state. It is generated by the tendency of magnetic angular momentum to orient itself in a magnetic field. The magnetic angular momentum arises from the orientation of the unpaired electrons with the magnetic field. Most organic compounds are diamagnetic, including the saccharide-water system studied in this work. 3.3.2 Orientation of Diamagnetic Molecules in a Magnetic Field The orientation of diamagnetically anisotropic molecules in a magnetic field may be described by the Cotton-Mouton effect [149]. An applied magnetic field produces a net force on the molecule if the molecular susceptibility is anisotropic. This net force tends to orient the molecule parallel to the direction of magnetic force. The degree of orientation Chapter 3. THEORETICAL TREATMENT 42 is a function of the anisotropy of the molecule, the magnetic field strenth, the interaction of the molecule with its neighbours and the thermal kinetic energy of the molecule. The degree of orientation S0 is given by [149] *0 = (X||-Xx)^ (3-25) where k is the Boltzmann constant, T the absolute temperature, H the applied magnetic field strength, and x\\, Xi are the diamagnetic susceptibilities parallel and perpendicular to a rotational symmetry axis, respectively. The degree of orientation of an individual molecule is very small. Measurements on calf thymus DNA has resulted in a degree of orientation of only 1% in an applied field strength of 13 T [149]. However, 80 can be increased dramatically when a great number N of such molecules are fixed together parallel to one another, since the effective diamagnetic anisotropy of such a molecular aggregate or cluster is proportional to N [149] 6o = N(Xll-x±)1j^ (3.26) The degree of orientation has been measured by observing the magnetically induced birefringence [149] An = n(| - nx = CMXH2 = S(a[{ - ax)c (3.27) where n\\, n± is the refractive index for fight of wavelength A, when polarized parallel and perpendicular to H, respectively. Q||, a± are the molecular optical polarisabilities parallel and perpendicular to the molecular symmetry axis, c is the concentration and CM is the Cotton-Mouton constant. This technique has been used to observe the orientation of some biological macro-molecules such as DNA [149], liquid crystals and micelles in soap-water system [148]. This situation is applicable in certain long macro-molecules and almost full alignment has been observed for polymers, large biological molecules such as nucleotides, chloroplasts, retinal rods [149] and various liquid crystals . This effect should Chapter 3. THEORETICAL TREATMENT 43 also be observable in aqueous saccharide solutions if elongated molecular aggregates or clusters existed. The magnetic properties of saccharides have not been measured to any great extent. The molar magnetic susceptibilities have been reported for several saccharides [186]. Pas cal's method can be used to calculate bulk magnetic susceptibilities for organic molecules, where the susceptibilities from individual atoms and bonds are summed up to yield the susceptibility for the molecule as a whole [162]. The susceptibility values predicted by this method agree to within 5% with experimental values [162]. At this time no measurements are available for saccharides regarding magnetic susceptibility anisotropics. Faxen's equation (equation 3.21) predicts that the membrane diffusivity will decrease as j3, the ratio of molecular radius to pore radius, increases. Any ordering of saccharide clusters in transversally applied magnetic field would have the same effect as increasing the effective cross-sectional area of the molecule, thereby increasing (3. Therefore, the membrane diffusivity should decrease with a field applied orthogonal to the pores. This is shown in Figure 3.11. 3.4 The Theory of Interferometry There are three optical properties of a solvent which may be modified by the presence of a solute. These are the absorption of light, refractive index and rotation of the plane of polarized light. Interferometry provides a tool which will measure refractive index profiles in a transparent medium. Refractive index changes in a diffusion process are due to concentration changes, in the absence of any temperature gradients. The Refractive index n of a medium is defined as the ratio of the velocity of light wave in a vacuum, c, to that in a medium, v, and is always greater than unity. n = - (3.28) Chapter 3. THEORETICAL TREATMENT 44 Dreld < DC Diffusion Reid DF,eld>D, Diffusion -1 Reid Figure 3.11: Molecular diffusion in magnetic field, (a) Field applied transverse to diffusion ( vertical ) direction ( as in this work ), (b) Field applied parallel to diffusion direction Chapter 3. THEORETICAL TREATMENT 45 For most solutions at low concentration, including the saccharide-water system studied in this work, n is a linear function of concentration C [152]: n = acC + 8c (3.29) where etc and 8c are empirical parameters. Values of etc and 8c at 25°C for all the saccharides used in this work are listed in Table 5.5. Equation 3.29 can be used to convert measured refractive index profiles into concentration profiles if otc and 8c are known for the system. 3.4.1 Interference of Light Waves According to the electromagnetic theory, light is the result of simultaneous propagation of an electric and a magnetic field, at right angles to each other, Figure 3.12. Suppose two sources of fight A and B have exactly the same frequency and amplitude of vibration and their vibrations are always in phase with each other, Figure 3.13 (a). Their combined effect at a point such as X, is obtained by adding algebraically the displacements at the point due to the sources individually; this is known as the principle of superposition. Figure 3.13 (b) illustrates the constructive interference at X or Q due to A and B, which have the same amplitude and frequency. A bright band is obtained at X. Generally, a bright band is obtained at any point Y if the path difference, BY - AY is given by BY - AY = mX (3.30) where A is the wave length of the souces A, B, and m is an integer. On the other hand Figure 3.13 (c) illustrates the case for destructive interference at P whose distance at P is half a wavelength longer than its distance from A ie AP - BP = A/2. The vibrations at P due to B will then by 180° out of phase with the vibrations there due to A. The resultant effect at P is then zero as the displacements at any instant are equal and opposite to Chapter 3. THEORETICAL TREATMENT 46 E z E Figure 3.12: Propagation of light as an electromagnetic wave [164] Chapter 3. THEORETICAL TREATMENT 47 time time time Vibrations at X—constructive interference. (b) time time (iii) resultant ___ Vibrations at P—destructive interference. (c) Figure 3.13: (a) Interference of light waves (b) Constructive interference (c) Destructive interference [163] Chapter 3. THEORETICAL TREATMENT 48 each other. A dark band is therefore obtained at P. If the path difference, AP - BP were 3 A/2, 5 A/2, instead of A/2, a dark band would again be seen at P at the vibrations there due to A and B would be 180° out of phase. In general, if the path-difference is zero or a whole number of wavelengths, a bright band is obtained; if it is an odd number of half-wavelengths, a dark band is obtained. As discussed above for the phenomena of interference to be observed, the two beams of light must be coherent, which means that the two beams must have the same wavelength and be always in phase with each other or have a constant phase difference. If a plane wavefront of light passes through a medium with a locally changing refractive index, it will not remain plane, but the phase velocity of the wavefront will be reduced as the refractive index increases, Figure 3.14. The resulting local variation in phase is proportional to the change in refractive index and geometrical distance travelled by the wave (product of refractive index and geometrical distance). The phase of a wave arriving at a point is affected by the medium through which it travels. Suppose light travels a distance I in a medium of refractive index difference, A n. The phase difference, A 6 is then [63] A , 27rAnZ Acp = — (3.31) where A0 is the wavelength of light in vacuum. In equation 3.31 the quantity nl, the product of the refractive index and path length, is called the optical path in the medium. As discussed previously in this chapter, the constructive interference of two light wave occurs if there optical path difference is mA (see equation 3.30). Figure 3.14 shows a plane wavefront of light passing through a medium of constant and variable refractive index field. One of the first demonstrations of the interference of light waves was given by Young in 1881 [163]. He placed a source, S, of monochoromatic light in front of a narrow slit C, Chapter 3. THEORETICAL TREATMENT 50 and arranged two very narrow slits A, B close to each other, in front of C, Figure 3.15. Young observed bright and dark bands on either side of 0 on a screen T, where 0 is on the perpendicular bisector of AB, Figure 3.15. Young's observations can be explained by considering the light from S illuminating the two slits A, B. Since the light diverging from A has exactly the same frequency as, and is always in phase with, the light diverging from B, A and B act as two coherent sources. Interference thus takes place in the shaded area, where the light beams overlap. Figure 3.15. On either side of O dark or bright bands are obtained depending on optical path differences between the interfering light waves. Young's two slit experiment is an example of interference by division of wavefront, also known as double beam interferometry (Figure 3.14). Most interferometers including the Rayleigh interferometer used in this work use a single source monochromatic light which is divided into two separate beams which are focused by a suitable lens system to superimpose the two beams producing an interference pattern. 3.4.2 Rayleigh Interferometer A top view of the Rayleigh interferometer is shown in Figure 3.16. A collimating lens expands the laser beam and produces a wave front of light which is collimated and parallel across the entire area of the beam. This beam illuminates two vertical slits, S, which divide the beam into two separate light sources exactly in phase with each other. These two beams then pass through the two separate compartments of the diffusion cell, each filled with a medium of different refractive index. The reference side compartment (RC) is filled with a medium (doubly distilled water) of constant and known refractive index. The diffusion side compartment (DC) is filled with a medium of unknown refractive index and with a refractive index gradient in the vertical plane. The plano-convex lens, LI, focuses the two beams on a focal plane, FP. The beams are superimposed and since they are mutually coherent, interference occurs at FP. Chapter 3. THEORETICAL TREATMENT 53 Lens L2 is a cylindrical lens, which only focuses in the vertical plane. It focuses an image of the diffusion cell vertically on FP, which then gives an image of the interference pattern on FP. From this interference pattern it is possible to calculate refractive index profiles and hence concentration profiles of the fluid mixture in the diffusion cell. When these profiles are suitably recorded as a function of time, it is possible to obtain mass fluxes, concentration gradients and diffusivities for a binary system, as shown in Chapter 5. 3.4.3 Interpretation of Interference Fringes Equation 3.31 states that the phase difference for a wave is proportional to the refractive index difference and the distance travelled by the wave. If the refractive index is constant in the diffusion cell then the fringe pattern obtained would consist of straight vertical fringes, Figure 3.17. This is because of the constant phase difference between the two sides of the diffusion cell. The distance between the centers of two fringes (either bright or dark) is determined by the wavelength of light used and the geometry of the optical system. This distance is given by [63] (3.32) where f is focal length of the plano-convex lens LI, A is the wavelength of the He-neon laser light, Ys is the distance between the two vertical slits and na is the refractive index of the air. The parameters used in this work are f = 67.3 cm, A = 0.0000623 cm (He-neon laser), Ys = 1.0 cm and na (for air) = 1.0. These values when used in equation 3.32 give A = 0.0042 centimeters. If rii is the refractive index in the reference side compartment (RC) (i.e. nx is constant ) and n2 is the refractive index in diffusion side compartment (DC) ( where n2 is a function of vertical position x ) then the phase change Ar/> at any vertical position between the Chapter 3. THEORETICAL TREATMENT 54 Figure 3.17: Fringe pattern produced by a constant refractive index in the diffusion cell Chapter 3. THEORETICAL TREATMENT 55 waves passing through the two sides of the diffusion cell is given by [152] 2-KI Acp = — A[n2(x) - na] (3.33) Equation 3.33 states that the phase change Acp at the focal plane is a function of vertical position x. The introduction of a cylinder lens L2 to the system allows the observation of phase change on the focal plane as a function of x. When moving along a vertical line in the fringe pattern there will be a horizontal shift of one fringe whenever A[n2{x)-m] = - (3.34) where I is the geometrical distance in the diffusion compartment. Equation 3.34 states that a shift in the interference pattern equal to one fringe spacing corresponds to a phase change or optical path difference between the reference and diffusion compartments equal to one wave length. The fringe pattern now resembles Figure 3.18. Starting at either end of the interference pattern the fringes are straight indicating that a constant phase difference exists between the two sides of the diffusion cell. The refractive index is constant and known at each end of the diffusion cell, so the fringes are interpreted with respect to ra0, the known refractive index at each end of the cell. Moving towards the centre of the interference pattern the fringes bend, resulting from a phase change between the two light rays passing through a given vertical position in the cell. At the location where the fringe pattern has shifted an amount equal to A, the phase change is now one wavelength of light between any two consecutive fringes. From equation 3.34 the amount A n can be calculated which is the difference in refractive index between the two vertical points. It is therefore possible to evaluate the refractive index difference at all locations in the cell simply by counting the number of fringe shifts Nj and measuring the vertical location, Xj, at which a shift occurs. Therefore n(x), which is the refractive index at Figure 3.18: Typical fringe pattern in the presence of a concentration gradient in the diffusion cell Chapter 3. THEORETICAL TREATMENT 57 location X{ corresponding to the i th fringe shift is rii(x) = n0 + Ni X 7 (3.35) Location X{ is measured from the boundary condition where n(a;o) = n0 is known see Figures 3.18 and 5.39. Equation 3.35 is valid if the refractive index in one of the diffusion cell compartments is constant, which is the case in these experiments since the reference compartment is filled with distilled water and experiments have been performed at con stant temperature. Equation 3.35 defines a refractive index profile for the diffusion cell. Since the membrane provides a discontinuity in the refractive index profile, equation 3.35 is applied to each end of the cell where the refractive index is known. The counting of fringes towards membrane produces a continuous profile on either side of the membrane. Applying equation 3.35 to the refractive index profile (equation 3.29) determines the concentration profile in the cell where cxc is an empirical constant determined from refractive index versus concentration data, see Chapter 5. Mass fluxes and diffusion coefficients can be then calculated from the obtained concentration profiles (equation 3.36), as discussed in chapter 5. 3.4.4 Distortion of Wavefront by Refractive Index Gradient If a collimated wavefront is transmitted through a medium in which there is refractive index gradient with components normal to the direction of propagation, all points along the wavefront will not proceed with the same velocity. The wavefront will be distorted and the outward normal vectors will have components in the direction of increasing refractive index, Figure 3.19. Wavefront deflection by the refractive index gradient has been discussed by a number of workers [175,176,152]: In this case equation describing a d{x) = C0 + Ni X (3.36) OLCl Chapter 3: THEORETICAL TREATMENT 58 Figure 3.19: Wavefront deflection through diffusion cell and refractive index gradient Chapter 3. THEORETICAL TREATMENT 59 ray passing through a refractive index gradient is [175,152]: dx dz (3.37) where s is the arc of the ray, no is the initial refractive index, n is the refractive index, x is the vertical direction and z is taken along the optical axis, Figure 3.19. Equation 3.37 shows that the ray bends in the direction of increasing refractive index. The downward deflection of the light ray introduces errors into the data analysis because the vertical position of a ray on the focal plane does not correspond to the actual vertical position of the ray entering the diffusion cell. Bollenbeck [152] using Rayleigh interferometer calculated for sucrose-water solution an error of 0.8 percent in refractive index gradients at the focal plane in the case of an uncorrected wavefront deflection. This is significant since Rayleigh interferometer is capable of measurement accuracy to within 0.1% [105]. Therefore, wavefront deflection effect correction must be made before mass fluxes and diffusivities can be calculated. When traversing the fringe pattern in a vertical direction, (Figure 3.18) the location where each fringe crosses the vertical axis of traverse is recorded with the corresponding fringe number. The final result is a set of fringe numbers and displacements counting from the ends (top and bottom) of the cell towards the membrane. If bending (wavefront deflection) is not considered each fringe shift represents a change in optical path length difference (An/) between the diffusion and reference sides of the cell of one wavelength where A is the wavelength of the light and I is the geometric path length through the solution. When bending is considered, the optical path length L through the solution in the diffusion cell is defined by the arc, s, of the ray and the refractive index along the An/ = A (3.38) Chapter 3. THEORETICAL TREATMENT 60 ray so that solution n(x)ds (3.39) Furthermore, the optical path length L through the last optical flat, lenses, and air are not the same as for a horizontal ray, Figure 3.16. As a result the final position of the ray on the focal plane may not correspond to the position of initial entry into the cell. In addition since the refractive index gradient is the greatest near the center of the cell and zero at the ends of the cell, the bending effect becomes function of position in the cell. The fringe pattern readings may be corrected for bending effects if the original entry positions and optical paths of the rays can be determined. In this way the refractive index of the solution at the initial glass-solution interface can be found for each point along the glass-solution interface, Figure 3.19. A simple iterative scheme was used to derive the correct refractive index profile. An approximate refractive index profile was first determined from the observed fringe pattern data assuming no ray bending. Snell's law 1 was then used to trace the ray through the optical system to the film plane. The calculated location of this light ray is compared with the actual location of the same raj' observed on the film and the refractive index profile then was modified. The entrj' position in the diffusion cell was modified by an amount proportional to the error between the same ray and corresponding position on the focal plane observed for this ray. The procedure is repeated until the "corrected" refractive index profile produces a fringe pattern corresponding to the one observed. The steps involved in this ray tracing scheme are summarized below [63]. 1. Assuming no ray bending, calculate the approximate refractive index profile from 1Snell's law states nxsinQi = ri2 sin 0:2, where nx and n.2 are the refractive indices for two different media and ai, a2 are the angles of light ray being transmitted through these respective media, Figure 3.19. Chapter 3. THEORETICAL TREATMENT 61 fringe data using the following equation n(xi) = n0 + An(xi) (3.40) 2. Using the profile determined from step 1, find a suitable correlation for n(x) (details about this correlation function are given in the following section). n(x) = f(x) (3.41) 3. From step 2, evaluate dn/dx. 4. Assuming that the arc of ray passing through the diffusing fluid can be represented by a straight line (verified by Bollenbeck for dilute solutions), the refractive index along the path may be obtained from [152] dn n(x) = n(x0) + dx (Xi - x0) (3.42) 5. The average refractive index n along the line is then n = n(x0) dn dx (x1 - x0) (3.43) "0 6. The path of a ray through refractive index gradient is given by a modification of equation 3.37 as dx = — f n(x) Jo dn ds ds n(x) Jo dx 7. Integrating equation 3.44 with a constant refractive index gradient gives (3.44) dx dn dx (3.45) ds ti(xi) where Xo,i is the original entry of a ray in the diffusion cell corresponding to the ith fringe shift. Chapter 3. THEORETICAL TREATMENT 62 8. Substituting n(x) from equation 3.42 into equation 3.45, integrating, and then using equation 3.43 gives [152] dn dx which is the vertical deflection through the cell thickness. LU (3-46) 0,i 9. Substituting the definition of optical path length for L0,; ( i.e. Lo,{ = n s = n (zj — z0) ) into equation 3.46 gives [152] (ajj - s0) = x=r dn dx (zi - zo)2 (3.47) where (z\ — z0) is the horizontal geometrical distance through the cell, see figure 3.19. We now have the current estimates for (xi — x0), dx/dz and dn/dx. According to the straight line approximation the path length through the cell can be found by the Pythagorean's Theorem, see Figure 3.19. s] = (asi - x0f + (zx - z0)2 (3.48) Knowing the path length, equation 3.48, average refractive index, equation 3.43 and angle of deflection a.\ in the concentration gradient (Figure 3.19), it is possible to trace the ray through the remaining optical system using Snell's law. Figure 3.20 lists all parameters used in the ray tracing program for the various lenses and optical components. The ray tracing scheme then gives the position of this ray on the focal plane. It is compared with the position of the actual fringe pattern corresponding to the same ray observed on the film. Based on the difference between the two values, the x0 position of the original ray entering the cell is updated giving a new refractive index profile n(x). Steps 1 to 9 are then repeated again until the calculated refractive index profile produces a theoretical calculated fringe pattern which corresponds to the observed (experimental) one. This process takes four to five iterations to converge to a tolerance of 1 x 10~5 [63]. The raytracing computer program is listed in Appendix C. Chapter 3. THEORETICAL TREATMENT 63 Radius of Curvature x(cm) Axial Distance JL'(cm) 0.0 1.0 2.0 3.0 Refractive index JL nquartz = 1.45709 Diffusion Cell 99.29 58.2 59.2 L1 Plano-convex Lens 22.4 110. 111. L2 n lens = 1.7499 Cylinder Lens 126.6 FP Focal Plane "air - 1.000276 Figure 3.20: Ray tracing parameters Chapter 4 EXPERIMENTAL EQUIPMENT AND PROCEDURE 4.1 Experimental Equipment The general apparatus set-up for this work was adapted from the work of Howell [63]. His results ( Howell used only one saccharide (sucrose) for his work) however did show a large scatter, Figure 4.21. To reduce this observed data scatter his method of measure ment was modified by removing the difficulties associated with mounting the membrane in the diffusion cell. This is further explained in the section regarding diffusion cell and membrane preparation. The design of diffusion cell and diffusion cell holder was modified accordingly to allow the same membrane to be used over again for each saccharide system (note Howell used a new membrane for each run). Six runs (using the same membrane) at different applied magnetic field strength (H = 0.0tol.lT) were made for each sac charide system. This resulted in a significant reduction in data scatter and considerably improved the measurement accuracy. An additional suggestion for improvement in diffu sion cell design is included in the recommendations section in Chapter 7. The schematic experimental setup is shown in Figure 4.22. Photographs of the optical bench assembly and magnet are shown in Figures 4.23 and 4.24. The following sections describe in detail the individual components of the system. 64 Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE 65 c o o -O -O, x 1CH crrv.s -D„ x ".f>: cm-is a c ) .1 2 3 A .5 .6 .7 -8 .9 1-0 1.1 1.2 Applied RekJ Strength (T) 6.0 O 5.0 o o O-0F xlO* erne's c tr. ^-OuXlO-6 cm!;s 3.0 Dill b 2.0 1.0 - <-c i < i i ii i i i i II 0 .1 J .3 .5 .6 .7 .8 .9 1.0 1.1 1-2 Applied Reid Strength (T) Figure 4.21: Howell's results for diffusion of sucrose showing large data scatter (a) 8.0 /x m pore diameter membrane (b) 0.8 /z m pore diameter membrane [63] Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE 66 o CO I J LU cjl-O .tr LU CO c CD o C o co 3 £ £ CO £0 b o x o § CO C <u c o M •— o c — fli S O CiCC Q. E" CO o <D —I C t_ <D £ TO O u_ 0) c 03 Q. X CD S 1 <2 -o <2 co C fc CD ra O E *-- i: Ii£ w o co <2 tu ra ra r— _l UJ ^ LU O O UQC Q a. co ^ co o CO c CD CD CL X LU pi to It C e •n ex 05 to Figure 4.23: Rayleigh interferometer and magnet assembly (I): LBE-Laser and beam expander, Si-Masking slits, DC-Diffusion cell, Ll-Plano- convex lens, L2-Cylindrical lens, EBC-Extension bellows and camera Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE 68 Figure 4.24: General view of Rayleigh interferometer and magnet assembly (II) Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE 69 4.1.1 Laser and Collimating Lens Assembly A Spectra-Physics model 124B Helium-neon laser of an output of 15 mW with a wave length ( visible red light ) of 0.6328 micron together with Spectra-Physics model 332 and 336 spatial filter and collimating lens provided a coherent, collimated and spatially filtered beam for the experiment. The beam is first focused and passed through a pinhole aperture. This has the effect of removing any spatial noise in the beam and providing a smooth Gaussian intensity profile across the output beam. Spatial noise is produced from diffraction effects as the beam encounters small irregularities on the inside bore of the laser tube. This spatially filtered beam is passed through the beam expander and collimator which produces a beam 5.0 cm in diameter, which is amply sufficient to illuminate the 3.5 centimeters long masking slits. Total wavefront deformation for this assembly is less than A/10. 4.1.2 Masking Slit Assembly As discussed in Chapter 3, interference fringes are observeable as the result of the phase difference ( or optical path difference ) between two different superimposed wavefronts of light. A double masking slit assembly, S (Figures 4.22 and 4.25) divides the spatially coherent laser beam into two rectangular vertical beams which illuminate the reference and diffusion compartments of the diffusion cell. These two beams are exactly in phase with each other prior to being transmitted through the diffusion cell. Ordinary razor blades were used to provide a double slit assembly. Four razor blades were mounted on to a rectangular cardboard frame. This frame was then mounted on laser end of the styrofoam enclosure, Figure 4.25. Equation 3.32, shows that the horizontal spacing between two fringes ( either dark or Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE Figure 4.25: Double masking slit assembly Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE 71 bright ) near the center of pattern is A = A/ (4.49) Y. n, 'a Fringe spacing A thus becomes a function of the masking slit separation, Ys once the wave length A and focal length f are fixed for the system. The masking slit separation is determined by the design of the diffusion cell. The reference and diffusion side com partments of the diffusion cell are 1.0 cm apart. A masking slit separation of 1.0 cm produced a horizontal fringe spacing which agreed quite well with the value predicted by equation 4.49. The two values are The optimum slit width is found by experimental trial and error until a clear well illu minated fringe pattern was observed on the focal plane of the camera. The slit width used in the experiments is 0.05 centimeters. This assembly produced a clear well defined interference pattern as shown in Figure 3.18. 4.1.3 Diffusion Cell The fused silica diffusion cell was custom fabricated by Interoptics, Ltd, of Ottawa, Ontario. The diffusion cell is shown in Figures 4.26 and 4.27. The cell dimensions are 3x3x6 centimeter. The flatness of all optical surfaces was maintained at A/20 and parallelism of all surfaces kept to at least one arc second. The cell was cut in half to form an interface for a membrane. Two parallel slots (size 0.5 x 1.0 x 2.0 centimeters) were drilled out of the center optical flat to provide the diffusion and reference compartments. Ports were drilled in the end of the fluid compartments for transferring fluids in and out of the cell. The light path through the fluid compartments is 1 cm. Equation 3.35 can A (measured) = 0.0042 centimeters A (calculated)= 0.0043 centimeters Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE 72 o J°L Top [<— 3 cm—•] Membrane Side Proposed Modification Figure 4.26: Diffusion cell (showing proposed modification) Figure 4.27: View of diffusion cell Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE 74 be used to calculate the total number of fringe shifts, Y^, N as An/ (4.50) A A 1% by weight change of deoxyribose concentration at 25°C corresponds to a refractive index change, An, of approximately 0.0014 (calculated from the refractive index profile). Using A=6.328 x 10-5 cm , / = 1 cm yields from equation 4.50 a value of 22 fringes (11 on either side of the membrane). To further improve the cell design a port should be drilled on the bottom side of the reference compartment. This is shown by the dashed line on the front view of the diffusion cell in Figure 4.26. This would further facilitate the mounting of the membrane before distilled water is introduced into the reference compartment. This procedure will reduce the risk of air bubbles trapped near the bottom side of membrane surface. 4.1.4 Plano-convex Lens and Cylinder Lens Figures 4.28 and 4.29 show the Plano-convex lens and cylindrical lens assemblies respec tively. These two lenses are fabricated from (the same material) SF4 A grade high index glass with a refractive index of 1.749999 at a wave length of 6.328 x 10~5 cm. These lenses were made by Interoptics Ltd of Ottawa, to the state of the art tolerances for minimal spherical aberration. The total wavefront distortion for these lenses is limited to A/4. The lens specifications are shown in Table 4.3. The focal lengths of these lenses were selected such as to produce a clear, easily observable fringe pattern at the focal plane FPC, see Figure 4.22 and also to meet the physical design constraints introduced by the geometry of the magnet and laboratory space. The chosen focal length for the spheri cal lens produced a fringe spacing of 0.0042 cm ( see equation 4.49 ), which was easily observable through the 30 power measuring microscope. The cylindrical lens produces a vertical image of the diffusion cell at the focal plane Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE 75 Figure 4.28: Plano-convex lens and mount [63] Chapter A. EXPERIMENTAL EQUIPMENT AND PROCEDURE 76 Top View f-<—• 5 cm H 0 <3z 1 J 1 Side View i « —fT I I Front View i Figure 4.29: Cylinder lens and mount [63] Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE 77 Table 4.3: Plano-convex and cylinder lens parameters Plano-convex Lens Cylinder Lens Focal length (cm) 67.3 15.0 Diameter (cm) 10.0 Length (cm) 6.0 Width (cm) 3.0 Edge thickness (cm) 1.0 0.935 that would fit the frame of a 35 mm film. The 3.5 cm long masking slits produced a 0.9 cm long image at the focal plane of the camera. The vertical magnification of the system is then 0.2571. 4.1.5 Optical Component Mounts The plano-convex lens and cylinder lens mounts are shown in Figures 4.28 and 4.29 re spectively. To preserve the magnetic field homogeneity, it is imperative that optical bench and all the component mounts are made from non- ferromagnetic materials. Therefore, the optical bench and component mounts were custom made by the UBC Chemical En gineering workshop from aluminum and nylon. The optical bench that consists of two meter long aluminum beam rests upon the concrete and rubber vibration isolation plat forms placed at each end of the magnet which center the bench between the poles of the magnet, 70 cm above the the floor, Figure 4.24. The aluminum beam is 10 cm wide and 5 cm thick. A strip of aluminum 4 cm wide by 0.5 cm thick is mounted on the optical bench along the entire length. The special mounting blocks slide along this strip and can be clamped in place by number 8 machine nylon screws. The modified aluminum diffusion cell holder is shown in Figures 4.30 and 4.27. The cell fits in rectangular groove cut in a circular plate. The two nylon thin screws are used Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE Top View 6 CD Nylon Retaining Screws -r—i 1—r < • • • New CD I I Modification Figure 4.30: Modified diffusion cell holder Chapter 4, EXPERIMENTAL EQUIPMENT AND PROCEDURE 79 to align the two halves of the diffusion cell into the same plane. The vertical height of the component mounts can be changed by the adjustable alu minum rods. The axial position of the components can then be changed by moving the clamping block along the bench. A clamping block is shown to scale in Figure 4.31. 4.1.6 Vibration Control The optical bench rests on two vibration isolated bases each weighing, approximately 250 kilograms. The base is constructed of concrete patio blocks sandwiched between high densit}- foam rubber. The massive layered blocks and rubber combination acts to dampen any vibrations which might be transmitted through the floor. Figure 4.32 shows one of the vibration isolation platforms. Vibration isolation is necessary to prevent any optical path length changes between the two interfering light beams during the measurement. 4.1.7 Temperature Control The temperature in the interferometer encloser is maintained at 25 ± 0.1°C. This is achieved by a dual temperature control system. The ambient air temperature in the room was controlled to ±1°C by.a Koldwave model K16DF water cooled air condition unit. The optical bench is enclosed in a 1 cm thick styrofoam box to provide thermal insulation. A 25 watt fan is mounted along the encloser. An electric resistance heater is mounted inside the encloser across the output of the fan, Figure 4.33. It was constructed by wrapping nichrome wire around a 1.5 cm diameter glass tube. It is rated at 300 watt, 115 VAC input. The air temperature adjacent to the diffusion cell (5 cm away) is measured by an Omega model 100R30 platinum RTD thermometer, Figure 4.33. An Omega model 4201 proportional on-off controller measures the resistance and therefore temperature of the thermometer. The sensitivity of the controller is ±0.05°C. An error signal is generated by the controller which is proportioned to the difference between Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE 80 Top View Figure 4.31: Optical bench and mounting block [63] Figure 4.32: Concrete and rubber sandwich block for vibration control Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE 82 Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE 83 the measured temperature and the set-point temperature. This error signal in turn is used to drive a variable transformer which powers the electric heater. The band width adjustment and voltage output of the variable transformer were adjusted simultaneously to provide constant cycle time for the heater. This proved to be between 15 and 20 volts with a cycle time of about 10 seconds. The temperature fluctuations measured (in this work) by the platinum thermometer and temperature controller combination were always within ±0.1°C of the 25°C set point temperature. 4.1.8 Electromagnet A Varian Associates 30 cm model V-7300 electromagnet was used to apply the magnetic field to the diffusion cell, Figure 4.23. The magnet was fitted with an 18 centimeter diameter pole piece possessing a 10 cm gap width. A model V-7800 DC power supply and V-7872 heat exchanger were used to cool this magnet generating fields as high as 1.25 T. Homogeneity of the field has been measured to be better than 7 x 10~5 T over the diameter (10 cm) of the pole piece (6.0 cm long diffusion cell fits in the region where the field strength is homogeneous) at an applied field strength of 0.9 T [151], Figure 4.34. The power supply produces a continuously variable DC output between 5 amperes and 114 amperes. The magnet is cooled by a two loop water to water heat exchanger. 4.1.9 Focal Plane Camera and Microscope The camera for obtaining fringe patterns consists of a Contax model 139 Quartz 35 mm single lens SLR reflex camera and a Yashica model F adjustable extension bellows focusing attachment, Figure 4.35. The extension bellows provide a light-tight interface with the experimental enclosure and horizontal adjustment for proper focusing. Kodak Panatomic-X film was used with Edwal FG-7 and Rodinal developer to provide fine grain and high resolution fringe patterns. The best exposure was found by trial and error to Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE k6 «L Figure 4.34: Magnetic field homogeneity [151] Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE Figure 4.35 tograpns Camera and extension bellows arrangement for taking fringe pattern ph Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE 86 be 1/500 second, which was sufficiently short to give sharp photographs. The laser beam was initially too intense to yield satisfactory exposure, even at the fastest shutter speed available. A Kodak Wratten No. 59 filter was used to attenuate the laser beam. The filter was placed less than 6 cm from the focal plane and there was no noticable distortion of the interference fringes. A Contax model S infra-red controller set was used to provide remote control of the camera, thereby avoiding vibrations coming from a manual shutter release. Fringes obtained were measured with a Gaertner Model M-1160 measuring microscope. This consisted of a 32 power microscope mounted on a, precision Vernier stage capable of measuring ± 0.0001 cm lengths. A 90° spider silk cross-hair was used with the movable stage to accurately measure fringe spacing. 4.1.10 Membrane and Saccharide Solutions The Nuclepore membranes used in this work are manufactured by Nuclepore Canada Incorporated [128]. These membranes contain straight through cylindrical pores. They are made from polycarbonate film which is bombarded by high speed sub-atomic particles from a nuclear reactor. The particles pass straight through the filter leaving tracks of molecular damage which can be etched preferentially in a chemical bath to round linear pores of uniform diameter, Figure 4.36. Table 4.4 shows dimensions of the membrane used in this work. The manufacturer's specifications on pore size variation are -f 0 to - 20 percent and for pore density ± 15 percent. All the saccharides (deoxyribose, D(-)ribose, D(+)xylose, D-glucose, D-galactose, D(-)fructose, sucrose, maltose, lactose, rafhnose) used were of ACS analytical grade with a better than 99% purity. D-galactose was provided by the Aldrich Chemical Inc. and the rest of the saccharides were provided by the BDH Chemicals Limited. All the saccharides were used without any further purification. A 1% by weight solution was made with doubly distilled water at room Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE 87 Figure 4.36: Nuclepore membrane showing round uniform pores [128] Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE Table 4.4: Nuclepore membrane specification 88 Pore size /xm Pore density pores/cm2 Thickness 0.4 1 x 108 10.0 temperature. This is further explained in the diffusion cell and membrane preparation subsection. 4.2 Experimental Procedure This section describes all procedures used to set up the interferometer and obtain in terference fringe data. Before an experiment can begin, the interferometer should be properly aligned to obtain a clear, well focused interference pattern. This alignment procedure involves 1) collimation and spatial filtering of the laser beam, 2) proper height adjustment for all optical components, 3) focusing the plano-convex lens LI and cylinder lens L2 to obtain an image of the diffusion cell on the focal plane of the camera. The following steps give in detail the proper alignment procedure. 1. Attach the spatial filter and collimating lens assembly to the laser. The X and Y motion adjustment screws on the laser are first adjusted to center the aperture on the focused beam. The aperture is then adjusted axially to provide the maximum ^ beam intensity with the most uniformly illuminated output. To collimate the beam, it is directed to a white paper screen placed across the room. The collimating lens is adjusted until the diameter of the beam a few inches in front of the lens and the diameter of the beam on the screen become equivalent. A caliper is used to measure the beam diameters. When the diameters have been found to be equivalent Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE 89 the beam is considered to be filtered and collimated. 2. The laser is placed on the optical bench. The laser is directed down the centerline of the bench. The height of each component is adjusted to bring it into line with the beam center. A calibrated ruler is used to measure the height and verify that the beam is parallel with the optical bench centerline. Shims are placed under the laser head assembly to adjust the height and vertical angle of propagation of the beam. 3. The cell holder and lenses are next placed on the optical bench. The clamping screws in the mounting blocks are kept loose and the lenses are placed in their approximate positions. The cell holder is centered between the pole pieces of the magnet. 4. Next the vertical masking slit assembly is mounted over the opening to the insulated enclosure, making sure the slits are orthogonal to the optical bench axis. The position is adjusted to center the slits in the 5.0 cm diameter laser beam. 5. Next a fine wire grid (1 mm spacing) is placed at center of the diffusion cell holder, cylindrical lens being temporarily removed. A small white paper screen is placed at the focal plane of the camera. The plano-convex lens is moved axially along the optical bench until a spot was focused on the paper screen. The cylinder lens is placed back on the optical bench and its axial position adjusted until an image of the wire mesh is focused on the camera screen. The clamping screws are then tightened. 6. Finally the lenses are checked to verify that their surfaces are horizontal to the propagation direction of the laser beam. They are rotated in the mounted blocks until the reflection from the masking slits is reflected back into the masking slits. Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE 90 The alignment of the interferometer is now complete. The wire mesh is now removed from the diffusion cell holder. If the obtained fringes are now examined with a microscope placed at the focal plane, they should be straight since the refractive index difference between the reference and diffusion campartments of the diffusion cell is constant at this time. 4.2.1 Diffusion Cell and Membrane Preparation 1. Distilled water to be used for both the reference and the sample solution is boiled for approximately 45 minutes to remove any dissolved gases. For filling the reference side compartment of the diffusion cell a 10 cc hypodermic syringe is filled with hot distilled water, before the water cools and air is re-dissolved in it. The remaining water is then cooled in an air tight sealed flask. 100 ml of this cooled water is then used to make a 1% by weight saccharide solution. A 10 cc syringe is then filled with this saccharide solution. Both syringes, one filled with distilled water, and the other with saccharide solution, are then placed in the insulated enclosure (apparatus) which has been pre-set to keep at 25°C. At least 4 hours waiting period is allowed for the system to come to thermal equilibrium before starting the experiment. 2. Next the membrane is boiled in distilled water for approximately forty five minutes before mounting to remove any unrelaxed stresses and any entrapped air in the pores. This also makes it easier to mount the membrane without any warps and wrinkles. 3. Before placing solutions in it, the diffusion cell is cleaned with Q-tips and soap. It is rinsed well with distilled water and left to dry. After the diffusion cell has dried, a very thin coat of silicon grease (Dow Corning high vacuum silicon grease) is applied to both interfaces of the diffusion cell. The grease provides a water tight Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE 91 seal between the membrane and glass and is not softened by water. Extreme care is required here if the coat is too thin or not uniformly applied over the entire interface, leakage ( distilled water or saccharide solution ) may occur, or if it is too thick, excess grease will extrude into diffusion cell compartments, distorting the membrane shadow. 4. The lower half of the diffusion cell is placed on a clean and dry table top. Following the new experimental procedure, at this time only the reference side compartment is filled with distilled water, using the already filled hypodermic syringe. It should be noted that the compartment is filled until the meniscus is just over the top surface. The membrane is removed from the boiling water and placed on the greased glass surface of the lower half of the diffusion cell. The membrane is smoothed flat with a cotton Q-tip, expelling any extra water. The proposed modification to the design of the diffusion cell (Figure 4.26) would further facilitate the mounting of the membrane in the diffusion cell 1. 5. The top half of the diffusion cell is then placed over the membrane making sure that no wrinkles have been formed. If the membrane is free from wrinkles and/or warps, the reflection of the laboratory fluorescent light tubes from the membrane surface will be undistorted. If these tests are satisfied then the reference side of the diffusion cell is filled with distilled water while the diffusion side is filled with 1% saccharide solution. Rubber stoppers are placed in openings (ports) to the diffusion cell. When the experiment is started, the saccharide solution from the upper half 1The present procedure of mounting the membrane while the bottom half of the reference side com partment is already filled with water often leaves air bubbles trapped underneath the membrane. Hence, the membrane must be remounted to make sure that the membrane is free from any entrapped air bubbles. In the proposed modification to the diffusion cell (Figure 4.26) insertion of distilled water into the reference side compartment of the cell can be performed after mounting the membrane. This makes it much easier to mount the membrane and eliminates the possibility of any air bubbles left entrapped underneath the membrane Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE 92 of the diffusion side compartment is removed ( using a hypodermic syringe ) and replaced with distilled water. The water is placed in the upper half of the diffusion cell to eliminate any bulk flow effects, which would result from the more dense liquid being above the less dense one. 6. The diffusion cell is placed in the diffusion cell holder and this assembly is then placed in a mounting block on the optical bench. The mounting block for the dif fusion cell holder has been already centered between the pole pieces of the magnet. The cell alignment is adjusted by observing the reflection of the laser beam from the diffusion cell surface back into the masking slits. The two nylon shimming screws are adjusted to bring the two halves of the cell into the same plane, thus making both surfaces orthogonal to the laser beam. When the reflection of the masking slits form vertical lines, both surfaces (top and bottom halves of the diffusion cell) are flat and aligned in the same plane. The cell holder is then rotated in the mounting block until the slit reflection is directed exactly back into the masking slits. The cell and the optical equipment are now in alignment and the experiment is ready to be started. 4.2.2 Start of an Experiment Before the first experimental run the diffusion cell which is kept in the enclosure (appara tus) at 25°C is completely filled with saccharide solution in the diffusion side compartment and distilled water in the reference side compartment. The experiment is started by re moving the saccharide solution from the top half of the diffusion side compartment and replacing it with distilled water. The saccharide solution in the top- half of the diffusion compartment is carefully and slowly removed with a hypodermic syringe. It is estimated that the amount of solution remaining is less than one drop or 0.03 ml out of a total Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE 93 volume of the upper half of the compartment of 1.3 ml. Extreme care is needed to make sure that the membrane surface is not touched with the syringe needle or else warping may occur and the run must be aborted. When this is done the magnet is then turned on and water is inserted into the empty half using another hypodermic syringe. The stop watch is started after the water is injected. The camera is attached to the bellows and photographs are taken of the fringe pattern for five time intervals. The first photograph is not taken until fifteen minutes have elapsed to allow any convective currents from the initial cell filling to dissipate. The rate of change in fringe pattern is an exponential function, (Figure 3.18) (in the beginning, diffusion is fast because of large concentration gradient) so initially the time intervals are quite close together ( fifteen minutes ), while at the end of the experiment the change is much more gradual so the last photograph is taken after a half hour interval [152]. The run was terminated after two hours, well before the diffusion would have changed the fringes (i.e. boundary conditions) at either end of the cell. If a horizontal fringe shift is observed at either end of the pattern at any time during a run, the boundary conditions have changed and no more fringe photographs need be taken since the refractive index profile can not be determined when the boundary conditions are unknown. It is convenient to check the fringe pattern (boundary conditions) with a microscope mounted at the focal plane (by temporarily removing the camera) before a photograph is taken. In this work, six runs, each at a different magnetic field strength (ranging from zero to 1.1 T) were made for each saccharide system. The same membrane was used for all six runs. The hypodermic syringes which are filled with saccharide solution and distilled water are kept in an enclosure (apparatus) with the temperature set at 25°C. For the subsequent runs (run two to six) the solution from the diffusion side compartment of the diffusion cell (both from the top and bottom side compartment) is carefully removed with a hypodermic syringe. A fresh saccharide solution is inserted into the bottom half Chapter 4. EXPERIMENTAL EQUIPMENT AND PROCEDURE 94 of the diffusion side compartment and distilled water is inserted into the top half of the compartment as explained before. Now, the next experimental run ( at a different chosen magnetic field strength ) is ready to begin. 4.2.3 Collection of Data — Summary A total of five fringe pattern photographs (at known time intervals) is taken for each experimental run. The film is developed as explained previously. The film is washed, dried, cut into individual frames, and mounted between glass microscope slides. Fringe patterns were measured using a Gaertner model M-1160 measuring microscope. The measuring microscope is mounted on its own stand when measuring the fringe patterns. The stand has a plane mirror for illuminating the microscope stage from below. A mounted fringe photograph is clipped on to the microscope stage and oriented so that the horizontal hair is centered on and parallel to a straight fringe at one end of the pattern. Traverse the pattern with the microscope and record the location of the center of each fringe and the corresponding fringe number. It is important to turn the micrometer screw of the microscope in only one direction while making a set of readings or backlash in the micrometer mechanism may introduce error. The entire procedure is repeated for the other side of the pattern. Appendix A contains the raw data obtained for all the runs. The interpretation of raw data to obtain refractive index profiles, concentration profiles, mass fluxes and the corresponding binary diffusion coefficients is discussed in Chapter 5. Chapter 5 DATA ANALYSIS 5.1 Interferometric Data The use of interferometric data to calculate mass fluxes, concentration gradients and diffusion coefficients requires that the refractive index and density as a function of com position be known for all saccharides. For dilute saccharide solutions the relationship between refractive index and concentration may be expressed by the following linear equation n = acCA+3C (5.51) where n is the refractive index at the given temperature ( 25°C ) and CA is the saccharide concentration expressed as gram-moles per cubic centimeter. The values of density as a function of composition were taken from the literature, Table 5.6. The general linear equation describing water concentration CB (gmoles per cubic centimeter) as a function of saccharide concentration is [152]: CB = liCA + £l (5.52) Tables 5.5 and 5.6 give the correlation parameters (ctcPcyi, ei) for equations 5.51 and 5.52 with a correlation coefficient of 0.999. The refractive index versus composition data for deoxyribose, D(-)ribose, D(+)xylose and raffinose were obtained experimentally (to four decimal places) at 25°C by the author using a Bausch and Lomb reffactometer. The diffusion experiments of this study were conducted at 25 ± 0.1°C according to the procedure described in Chapter 4. The fringe 95 Chapter 5. DATA ANALYSIS 96 Table 5.5: Correlation parameters for equation 5.51 Sugar reference Deoxy-ribose 17.845674 1.332870 this work D(-)ribose 19.974436 1.332670 this work D(+)xylose 19.980289 1.332619 this work D-glucose 28.041606 1.333254 [182] D-galactose 28.607077 1.332865 [182] D(-)fructose 31.559187 1.328253 [183] Maltose 53.866882 1.333256 [182] Sucrose 49.016210 1.331313 [152] Lactose 55.314708 1.333000 [182] Raffinose 76.463182 1.331918 this work Table 5.6: Correlation parameters for equation 5.52 Sugar 7i ei reference Deoxy-ribose -5.254400 0.0553448 [185] D-ribose -5.400445 0.0553472 [137] D(-j-)xylose -4.942885 0.0553175 [137] D-glucose -6.829681 0.0554623 [182] D-galactose -6.586536 0.0553999 [182] D-fructose -7.985082 0.0567105 [184] Maltose -12.663880 0.0554429 [182] Sucrose -11.687440 0.0553512 [182] Lactose -12.083347 0.0554775 [182] Raffinose -27.814300 0.0553440 [137] Chapter 5. DATA ANALYSIS 97 position data taken from the fringe pattern photographs are given in Appendix A. Note that the distances are measured at the focal plane i.e. the vertical magnification of the optical system is not 3'et taken into account. Dividing each distance by the vertical mag nification (0.2571) of the optical system used in this work (Rayleigh interferometer) gives the fringe location relative to the cell. The fringe numbers are for accounting purposes only. Figure 5.37 shows a typical plot of fringe numbers as a function of position at the focal plane. Figure 5.37 indicates that the fringe profiles exhibit asymptotic behaviour at the extreme ends of the fringe pattern. This is due to the fact that these extreme concentrations remained constant throughout a run, because the run is terminated well before the diffusion process could change the concentration at the extreme ends of the diffusion cell. 5.2 Calculation of Refractive Index Profiles As described in Chapter 3, wavefront deflection by the refractive index gradient affects the fringe pattern in that the refractive index difference between any two fringes is no longer constant but becomes a function of the refractive index gradient in the cell (equation 3.37). Chapter 3 discusses the possible magnitude of this effect and presents an iterative scheme for correcting the fringe data. The fringe profiles ( Chapter 3 ) must be corrected for deflection errors before molar fluxes and diffusivities can be calculated. This requires finding a suitable correlation for refractive index as a function of position as discussed in Chapter 3, equation 3.41. Examination of Figure 5.37 indicates that the fringe profiles have a definitive sigmoid shape. A sigmoid type correlation is desirable in the extremes of the profiles since the sigmoid function possesses the necessary asympotic properties. Such a correlation must be fitted to the data on one side of the membrane at a time because of the existing discontinuity in concentration across the membrane. A sigmoid Distance From Membrane, cm. Figure 5.37: Typical fringe profile (Run MAL16, field strength = 1.1 T) to 00 Chapter 5. DATA ANALYSIS 99 concentration profile is given by the error function solution to the ideal diffusion equation ( equation 3.12) as follows [134] c(x,t) = ^C0erfc^= (5.53) where c is the concentraton scaled from zero to one , x is diffusion coordinate, t the time, D the binary diffusion coefficient, and erfc is the complimented error function. Equation 5.53 implies the existance of a sigmoid function in refractive index of the form [152] n = m erf (Ax) + b (5.54) where m, A and b are empirical, saccharide dependent constants derived from the best fit of the refractive index data to equation 5.54. As discussed previously in this chapter, equation 5.54 must be fitted to the data separately on each side of the membrane. The parameters in equation 5.54 were found by using a linear least-squares curve fitting routine [177] to find the values m and b which gave the best fit for the experimental data. Parameters m and b were correlated linearly as: n — m xun 4- b (5.55) where xun — erf(Ax). A value of A was first assumed, then m and b were found by a least squares method. The RMS error for the linearized function, equation 5.55, was cal culated and a parameter search performed on A to give the minimum RMS error. Figure 5.38 shows a sample experimental run for maltose with the solid lines being the error function correlation ( equation 5.55 ) for the experimental points. The fit of the experi mental points to the error function correlation equation shows excellent agreement. This correlation function is then used in the wavefront bending calculations (equation 3.41) described in chapter 3. Appendix B gives complete listing of the correlation parameters m, A and b found in this manner. Chapter 5. DATA ANALYSIS 100 Figure 5.38: Refractive index profile for maltose, field strength = 1.1 T Chapter 5. DATA ANALYSIS 101 5.3 Mass Fluxes and Diffusivity Calculations The membrane yields a discontinuity in the concentration profile, therefore, methods that require continuous profiles over the entire distance between the boundaries can not be used to calculate diffusivities in this work. Also, most of these methods assume that the diffusion coefficient and or density are constant over the profiles. To circumvent these difficulties Bollenbeck [152] proposed a method that solves the continuity of mass equation numerically to obtain mass fluxes. The method as developed by Bollenbeck for membrane transport studies is limited to binary mixtures and is independent of diffusivity and density changes. It needs a flux boundary condition on each side of the membrane in addition to the concentration profiles from the fringe patterns. The mass fluxes and diffusion coefficients may be calculated at any point in the diffusion cell [152]. The details of the proposed method are given below. Consider the mass flux of component A , NA , in the x direction, and along a uniform cross-sectional area. The unsteady state mass balance between any x0 and x position is NA L -NA \X= Q-tJ2°A (5-56) where Yl &A is the total amount of component A in the volume between x0 and x. For a continuous concentration profile, and when there are no sources or sinks of A between x0 and x , equation 5.56 becomes NA |X = NA \Xo ~ f CAdx (5.57) Ot Jxo Equation 5.57 can be used to calculate the flux at any position x in the cell, if the concentration profiles are known for several different times and a reference flux is known at XQ. The reference flux at x0 is known at the bottom limit of the cell where it (flux) is zero as long as the concentration gradient is zero. The data is first integrated and then differentiated. If diffusivities are to be determined the concentration gradient must Chapter 5. DATA ANALYSIS 102 be found by yet another differentiation of the data. Figure 5.39 shows mass flux and concentration profiles in diffusion cell. The flux at the top end of the cell should also be zero, but there will in general be a nonzero flux at the top of the cell even in the absence of a concentration gradient because of density changes caused by the diffusion process in the middle portion of the cell. This means that the flux at the bottom surface of the membrane can be easily found but the flux at the top surface may only be found by integrating equation 5.57 across the membrane or by independentaly measuring the mass velocity at the upper limit of the cell [152]. The membrane used in this work is very thin (10 microns) so that the flux should be essentially the same at both surfaces. Therefore, it is sufficient to find the flux only on one side of the membrane. Equation 5.51 indicates that for dilute saccharide solutions there is linear relationship between molar concentration and refractive index. Integrating equation 5.51 between x0 and x and rewriting in terms of molar concentration gives [X [X / CAdx = a-i j ndx + (3i(x — x0) (5.58) J X\) Jxu where ai = 1 / OLC and /3i = 1 / /3c- Using the error function correlation, equaton 5.54, equation 5.58 becomes rx rx / CAdx = aam / erf(Ax)dx + a^b^x — x0) + 0i(x — x0) (5.59) JxO JxO The integration of the error function in equation 5.59 gives [152] / CAdx = ax 7^\Axerf(Ax) + ~^e~{Ax)2)l0 + b(x - x0) Jxn A v"' + /31(:c-a!0) (5.60) Equation 5.60 is used to evaluate the integral in equation 5.57. The bottom integration limit, x0, is taken at the extreme of the fringe pattern ( -1.8 cm ) where the fringes are straight, and the mass flux is zero. Once a set of integrals is obtained at different times, Chapter 5. DATA ANALYSIS 103 Figure 5.39: Mass flux and concentration profiles in diffusion cell Chapter 5. DATA ANALYSIS 104 the flux then may be calculated from equation 5.57 as: d N Ax = -jr fX CAdx (5.61) Of Jx<\ Equation 5.52 states that for dilute solutions the molar concentration of water and sac charide show linear dependency . Then the water flux NBx becomes NBx = ~ I' CBxdx = -lrNAx (5.62) ot Jxo The time derivative in equation 5.61 may be calculated by finite differences, a suitable correlation or graphically. For this work the correlation approach is preferred, since any irregularities in the data would be smoothed. Figure 5.38 shows that the error function relationship equation was successful in representing the concentration profiles individually at particular times. Integration of equation 5.53 with respect to x gives [152] Co Co F x CoVDt -| Based on equation 5.63 Bollenbeck proposed a functional form of the concentration in tegral as a function of time given as [cdx=^x-^xerf^-^^e-lM +k (5.63) J 2 2 JWt JTT K J I' Jxo p -P2 CAdx = —ay — 6i erf j2 — ca r1/2e T (5.64) The four constants a,\ , by, P and Cy in equation 5.64 were determined from concentration profile integrals ( equation 5.60 ) at five different times. A UBC computer center non linear least-squares curve fitting routine [177] was used to find the best fit for the constants in equation 5.64. As shown in Table 5.7 the profile integral data were represented quite well by equation 5.64 with deviations between the observed ( equation 5.60) and predicted values ( equation 5.64) being less than 0.1%. The molar flux of saccharide at any point in the diffusion cell is found by differentiating equation 5.64 with respect to time and substituting the result in equation 5.61 to give [152]. N Ax cy ^ P2 by P 2^/2 ^3/2 ^£3/2 e-p2" (5.65) Chapter 5. DATA ANALYSIS 105 Table 5.7: Profile integral correlation results for maltose at -0.06 cm.(Run MAL16 Field = 1.1 T) /-0.06cm p P2 CAdx = -ax - herf—j- - c1t1/2e~~ -1.8cm tl''! CA= Maltose concentration , Gm-moles/cc oi = 5.030 x 10"5 6x = -2.796 x 10-6 C! = -1.155 x 10~8 P = -4.973 Time Profile Integral G m-moles/cm2 x 105 percent Seconds Observed Predicted difference 900 5.0471 5.0479 -0.016 1800 5.0199 5.0186 0.026 2700 5.0059 5.0008 0.102 3600 4.9787 4.9874 -0.176 5400 4.9699 4.9670 0.060 Chapter 5. DATA ANALYSIS 106 Table 5.8: Molar fluxes for maltose and water at 2700 seconds for the bottom half of the cell. A negative flux is upwards (Run MAL16 Field = 1.1 T) cell position cm. flux g-moles/cm2-sec x 1010 maltose water 0 (membrane) - 1.994 25.252 -0.0200 - 1.936 24.517 -0.0400 - 1.826 23.124 -0.0600 - 1.675 21.212 -0.0800 - 1.494 18.920 -0.1000 - 1.297 16.425 -0.1200 - 1.095 13.867 -0.1400 - 0.895 11.334 The molar flux of water can be calculated from equation 5.62. Table 5.8 gives a sample of the calculated molar fluxes of maltose and water in the diffusion cell. As expected, the fluxes decrease uniformly with distance from the membrane. The reliability of the calculated molar fluxes can be checked by using the fluxes and profile gradients (at zero field strength) to calculate binary diffusion coefficients and then checking the coefficients against literature values. Fick's law for binary diffusion may be extended and defined as [129] NA=XA(NA + NB)-cDAB^ (5.66) where XA is the mole fraction of component A and c is the total molar concentration. The first term on the right-hand side of equation 5.66 defines the molar flux of component A resulting from bulk motion of the fluid. The binary diffusion coefficients are calculated both with and without the bulk motion contributions as follows. Equation 5.64 is fitted to the concentration profile integral equation 5.60 to determine the values of constants aii cij and P at any value of x. The membrane is located at x = 0.0 cm. The upper integration limit x is varied from x = 0.0 to - 0.16 cm.; the bottom limit is kept Chapter 5. DATA ANALYSIS 107 Table 5.9: Binary diffusion coefficients for maltose calculated both with and without the bulk flow contribution to flux (Run MAL16 Field = 1.1 T) Maltose cone. Gm-moles/cc x 105 Binary Diffusivity,cm2/sec x 106 percent difference no bulk bulk 1.6977 4.7701 4.7872 -0.357 1.8625 4.6977 4.7162 -0.392 2.0235 4.6331 4.6529 -0.425 2.1752 4.5758 4.5969 -0.459 2.3140 4.5274 4.5496 -0.488 2.4372 4.4897 4.5128 -0.512 2.5435 4.4634 4.4874 -0.534 2.6324 4.4284 4.4530 -0.552 constant at x = - 1.8 cm., where molar flux is zero. The constants determined under these conditions from equation 5.64 are then put in equation 5.65 to calculate molar flux at the corresponding value of x. To calculate the binary diffusion coefficient from equation 5.66, we need to know the concentration gradient. The concentration gradient at any value of x is calculated from equations 5.51 and 5.54. Now we know both the molar flux and the concentration gradient at a particular value of x, and so from equation 5.66 we can calculate the corresponding binary diffusion coefficient. Table 5.9 gives the diffusion coefficients of maltose calculated from the flux and pro file data. The diffusion coefficients for saccharide-water system should decrease with increasing saccharide concentration. This trend is present in all the saccharides used in this work. Table 5.9 shows that the bulk flow contribution also increases with increasing maltose concentration due to the increasing mole fraction of maltose. Figure 5.40 shows a plot of changes in molar flux, molar concentration and diffusivity in the bottom half of the diffusion cell. The data used in Figure 5.40 was taken from Tables 5.8 and 5.9. Diffusion coefficients calculated at zero applied magnetic field strength ( i.e. earth field 6.0 5.5 5.0 H 4.5 4.0 3.5 3.0 2.5 2.0* < 1.5-1.0 .5 0.0 ^Diffusion Coefficient (cm^sec) x 106 0.00 —i— .02 Molar Concentration (gm-mol|crr?)x105 secJ x JO70 > .04 ~06 ,08. .10 —•— .12 —i— .14 .16 s I CO Cell Position (cm) Figure 5.40: Molar flux, molar concentration and diffusivity versus cell position for the bottom half of the diffusion cell. o Chapter 5. DATA ANALYSIS 109 Table 5.10: Comparison of diffusion coefficients at 25°C from this work with literature at zero applied magnetic field strength( at same concentration). Sugar Sugar Cone. Gm-moles/cc x 105 Binary Diffusivity, cm2/sec x 106 percent difference this work literature Deoxy-ribose 3.657 8.147 8.110 [112] 0.14 D(-)Ribose 3.155 7.753 7.778 [112] 0.36 D( + )Xylose 3.199 7.443 7.455 [115] 0.16 D-Glucose 2.175 6.776 6.731 [113] 0.66 D(-)Fructose 3.320 7.044 7.002 [140] 0.60 Maltose 1.677 5.122 5.158 [115] 0.70 Sucrose 1.413 5.272 5.224 [105] 0.53 Lactose 1.654 5.254 5.238 [139] 0.36 Raffinose 1.065 4.337 4.359 [114] 0.51 ) are compared with literature values in Table 5.10. The diffusion coefficients have been compared at the same concentration. The agreement with literature values, Table 5.10 is within one percent. This is not surprising since the interferometer is capable of measur ing diffusion coefficients to an accuracy of 0.1 to 0.2%. Bollenbeck [152] and Howell [63] evaluated binary diffusion coeffficients for sucrose using the same data analysis technique. Their results agreed with literature values to within 1%. Bollenbeck [152] developed his data analysis technique to show that a Rayleigh Inter ferometer can be used for membrane transport studies. In the application of his method he showed that the porosity (effective free area for diffusion) of an unknown membrane could be calculated using this particular technique. In Chapter 3 it was shown that the Nuclepore membrane as used in this work would represent only an area reduction to diffusion. The flux through the membrane has already been found, equation 5.65. The concentration at the membrane surface is available from the concentration profile correlations, at x=0 cm. The flux through the membrane may then be regarded as a Chapter 5. DATA ANALYSIS 110 simple diffusion process and written as shown in equation 3.17 [152]. The edges of the membrane shadow tend to be blurred in the image of the interference pattern on the film when viewed through the measuring microscope. Therefore, there is great deal of uncertainty in deciding the proper value for the membrane shadow thickness, Ax for use in equation 3.17. Aa; should be constant for all runs. Therefore, it is felt that until such time as the diffusion cell is modified to produce a more definite fiduciary mark on the interference pattern, use of equation 3.17 would not give reliable surface porosity, Af values, for a membrane. For the reason mentioned above it was decided not to calculate the effective free areas (surface porosity), Aj, values in this work. Chapter 6 RESULTS A total of eighty runs were made for the ten saccharide-water aqueous solutions. The saccharides chosen for the measurements were mono, di and tri-saccharides of biological interest. The experiments were conducted in random fashion for conditions ranging from zero applied magnetic field to an applied field strength of 1.1 T. The applied magnetic field strength was the only variable in these experiments. In all experiments the initial saccharide concentration difference across the membrane was 1% by weight and the refer ence side of the diffusion cell was always filled with distilled water. The experiments were conducted at a constant temperature of 25°±0.1C. The raw data (given in Appendix A) consisted of a set of measured fringe displacement values taken at different times for each run. Each distance measured by the microscope corresponded to a refractive index change equivalent to one fringe shift or one wavelength of laser light. The fringe displacements at the focal plane were converted into cell positions by using the vertical magnification of the optical system (Rayleigh interferometer). The concentrations at extreme ends of the cell remained constant yielding a fringe pattern that consisted of straight parallel fringes near the ends of the diffusion cell. The fringe locations were measured with respect to a reference location taken at a distance of 1.3 cm from the membrane at each end of the cell. At this position the fringe pattern as observed through the microscope consisted of a straight parallel fringes, Figure 3.18. A raytracing computer program converted raw data into refractive index profiles for each time, using equations developed in Chapter 3. As already mentioned in Chapter 111 Chapter 6. RESULTS 112 3, refractive index gradients in the diffusion cell cause the rays to bend in the direction of increasing refractive index gradients. Raytrace computer program using an iterative scheme (Chapter 3) calculates the corrected refractive index profiles. Raytrace then eval uates the constants A, b and m for equation 5.54 to correlate the corrected refractive index profile with the error function correlation, equation 5.54. Figure 5.38 shows a re fractive index profile for maltose. The solid lines are the profiles corrected for raybending effects and correlated to the error function equation (equation 5.54). The points are the refractive index values calculated directly from the raw data not accounting for raybend ing effects. Figure 5.38 shows an excellent fit and as expected the deflection error is most noticable near the membrane surface where the refractive index gradient is greatest. The correlation parameters A, b and m for each run for all saccharide solutions are given in Appendix B. The correlation parameters A, b and m were used by the computer program Diffcalc to integrate the concentration profile (equation 5.60) in the cell. This integral was evaluated at each time interval and a partial derivative with respect to time was evaluated (equation 5.65) to determine the mass flux at the membrane surface for a given time, (see also Chapter 5 for these equations). As explained in Chapter 5 the mass flux was calculated for the lower half of the cell only. The membrane is only 10 microns thick so the flux at the top surface should be essentially the same as on the bottom surface. It was shown in Chapter 3 that the membrane should present only an area reduction to diffusion. This means that the diffusion inside membrane pores could be considered to be the same as free diffusion in the solution. The concentration gradient producing the driving force for this mass flux is calculated from the refractive index correlation evaluated by Raytrace. The corresponding binary diffusion coefficient is then obtained by using equation 5.66. The diffusion coefficients for all the runs and for all saccharide solutions are listed in Table 6.11. Table 6.11 indicates that for oligosaccharides ( sucrose, maltose, lactose, raffinose Chapter 6. RESULTS Table 6.11: Binary diffusion coefficients in applied magnetic field (T) 113 Saccharide Binary Diffusion Coefficients, cm2/sec x 106 0.0 T 0.3 T 0.5 T 0.7 T 0.9 T 1.1 T Deoxy-ribose 8.147 7.923 7.741 7.489 7.525 7.310 D(-)Ribose 7.753 7.615 7.564 7.568 7.500 D(+)Xylose 7.444 7.101 7.035 6.983 6.782 6.781 D-Glucose 6.775 6.252 6.167 5.813 5.747 5.407 D-Galactose 6.754 6.628 6.398 5.932 5.625 D(-)Fructose 7.044 6.402 5.736 5.323 4.910 Sucrose 5.272 5.263 5.242 5.146 5.080 5.080 Maltose 5.122 4.930 4.873 4.789 4.787 Lactose 5.252 5.222 5.114 5.058 4.973 Raffinose 4.337 4.173 4.191 4.032 3.850 4.020 ) and monosaccharides ( ribose, xylose ) the effect of applied magnetic field on diffusion coefficients is tapering off (compare diffusion coefficient values for these sugars at 0.9 and 1.1 T field strength ). The values of diffusion coefficients listed in Table 6.11 are calculated at the membrane surface (i.e. at x = 0 cm.). The concentration at this location (and at the same time) remained constant for all the runs ( magnetic field strength increasing from zero to 1.1 T ) made for a particular saccharide. For illustration, Figure 6.41 shows that the molar concentration of raffinose remained constant for all the runs made at different applied magnetic field strength. Table 5.10 lists the molar concentrations for all the saccharides at x = 0 cm. ( membrane surface ) in the diffusion cell. Computer programs Raytrace and Diffcalc are given in Appendices C and D respectively. The validity of the calculated mass fluxes was verified by comparing binary diffusion coefficients (calculated at zero magnetic field strength) with the accepted values found in the literature, Table 5.10. The agreement was within 1%. Figure 6.42 shows the binary diffusion coefficient as a function of applied magnetic field stength. A linear regression O 1— X £ o w 4) o E i E CD o c o O w_ TO O 2 •8 to 0.000 "V 0.0 Applied Magnetic Field Strength ( T ) Figure 6.41: Figure showing (using raffinose as an example) that the molar concentration of any particular saccharide ( at the same time and at the same location in the diffusion cell ) remained constant for all the runs. o Deoxyribose » Lactose — + Ribose •= Raffinose I Xylose < Maltose • Sucrose o Glucose Fructose Applied Magnetic Field Strength ( T ) p;-,.r, 6.42: Diffusion coefficients (D) versus applied magnetic field strength (H) Chapter 6. RESULTS 116 analysis was performed on each data set. Figure 6.42 clearly shows that the applied magnetic field has a statistically significant effect on diffusion process. This indicates that within the given range of experimental conditions, a linear relationship (equation 6.67) may exist between the measured diffusion coefficient D and the applied magnetic field strength H. D =• a + g H (6.67) Table 6.12 gives the obtained straight line linear regression parameters for equation 6.67. The somewhat lower value of correlation coefficient for raffinose is due to the unusually higher value of diffusion coefficient at the applied field strength of 1.1 T ( Table 6.11 ). If this point is not included in data analysis the new correlation coefficient ( - 0.95283 ) compares favorably with the correlation coefficients calculated for other saccharides ( Table 6.12 ). A hypothesis test (H0 : 8=0) was run for all saccharides to check if the regression slopes were significantly different from zero. The resulting t-statistics (Table 6.12) for the regression slopes are highly significant at the 95% confidence limits and at the appropriate degrees of freedom. The reproducibility of experimental results at randomly selected magnetic field strengths was checked by performing duplicate runs for most of the saccharides. Glucose, which shows one of the largest effects of magnetic field on diffusion coefficient, is used as an example to show the excellent reproducibility of experimental results in this work. In addition to the initial six runs made for glucose at different magnetic field strengths ( 0.0 to 1.1 T ), four duplicate runs were made at magnetic field strengths of 0.0 T, 0.5 T, 0.7 T and 1.1 T. These two experiments each consisting of six runs and four runs respectively were analysed by linear least squares regression. To illustrate, Figure 6.43 shows the regression lines and the 95% confidence limits for the duplicate experiments. The slopes of the regression lines are very similar thus indicating that the same effect was 8.0 7.5 H Applied Magnetic Field Strength ( T ) Figure 6.43: Comparison of duplicate experiments for glucose. Curved lines are the 95% confidence limits for regression lines. Chapter 6. RESULTS 118 Table 6.12: Linear regression parameters, equation 6.67 Sugar Slope xlCT7 Intercept xlO 6 Correlation t-statistics g a Coefficient of slope Deoxy-ribose -7.533 8.129 - 0.97979 9.80 D(-)ribose -2.198 7.741 - 0.97814 8.15 D(+)xylose -5.873 7.364 - 0.96126 6.98 D-glucose -11.700 6.702 - 0.98590 11.78 D-galactose -11.162 6.848 - 0.99905 9.60 D(-)Fructose -19.020 7.024 - 0.99905 39.67 Sucrose -2.099 5.303 - 0.93935 5.50 Maltose -2.934 5.076 - 0.95180 5.38 Lactose -2.396 5.262 - 0.99791 7.86 Raffinose -3.649 4.313 - 0.86661 3.47 observed in duplicate experiments. Figure 6.43 clearly shows that the applied magnetic field has a statistically significant effect on diffusion process. The introduction of reduced parameters D" (D* = DH/DHO), the reduced binary saccharide diffusion coefficient and H' (H~ = H/i/0), the reduced applied magnetic field strength such that DJJ0 is the binary diffusion coefficient measured in the earth magnetic field (H0 = 70 /iT [187]) and DJJ is the binary diffusion coefficient observed at the given applied magnetic field yields correspondingly, equation 6.68 and allows us to examine these parameters (D', H') with respect to a fixed coordinate. D" = a + gH~ (6.68) Figure 6.44 shows a plot of reduced parameters (D" versus H"). A linear least squares regression analysis was performed on each data set. Figure 6.44 gives us a picture of the relative effect of magnetic field on binary diffusion of saccharides. Figure 6.44 shows that magnetic field reduces the diffusion coefficients of all the saccharides studied in this work and that some of them are affected more than others. Table 6.13 gives the linear least I Q CD O 3 •o DC Saccharides <r Raffinose 2001 4001 "i 1 1 r 6001 8001 10001 12001 d • Lactose o Maltose * Sucrose Galactose I Fructose < Glucose • Xylose o Deoxyribose Ribose 8 •S 14001 16001 18001 Reduced H* ( H / H0 ) Figure 6.44: Reduced diffusion coefficients in magnetic field (D~) versus reduced applied magnetic field (H~) Chapter 6. RESULTS 120 Table 6.13: Reduced linear regression parameters for (D~) versus (H"), equation 6.68 Sugar Slope xlO-6 Intercept Correlation 9 a Coefficient Deoxy-ribose -6.472 0.998 -0.97979 D(-)ribose -1.980 0.998 -0.97814 D(+)xylose -5.523 0.989 -0.96126 D-glucose -12.089 0.990 -0.98590 D-galactose -10.928 1.015 -0.98412 D(-)fructose -18.901 0.997 -0.99905 Sucrose -2.789 1.006 -0.93935 Maltose -4.010 0.991 -0.95180 Lactose -3.203 1.002 -0.95633 Raffinose -5.890 0.994 -0.86613 squares parameters for the reduced parameters (i.e. D' versus H~) for equation 6.68. It was shown in Chapter 3 that saccharide hydration is affected by the mean number of e-OH groups existing in various conformers of saccharide usually found in saccharide solutions. The slopes of linear regression lines (Figure 6.44 and Table 6.13) are plotted against the mean number of e-OH groups for oligosaccharides and monosaccharides in Figures 6.45 and 6.46 respectively. Examination of Figure 6.45 shows that the magnetic field influence on free diffusion of oligosaccharides ( sucrose, lactose, maltose, raffinose) increases with the increase in the e-OH number. It was discussed in Chapter 3 that oligosaccharide molecules are non-spherical (elongated) in shape. It appears from Figure 6.45 that the magnetic field effect becomes larger with an increased size and the associated elongation of the oligosac charides. The transversly applied magnetic field (as in this case) then orients these hydrated (elongated) saccharide-water clusters parallel to the direction of applied field (Cotton-Mouton effect). As shown in Figure 3.11 this increases the cross-sectional area Sucrose • Laclose • OLIGOSACCHARIDES Maltose a Raflinose • 5.0 —(  5.5 6.0 6.5 7.0 —,— 7.5 8.5 9.0 —i— 9.5 10.0 8 •8 Number of e-OH groups Figure 6.45: Reduced slope g (equation 6.68) versus e-OH number for oligosaccharides to I,, X ID) •o <u o 3 TJ O Q. 0--2--4 • -6--8« -10--12« -14-1 -16' -18--20' 0.0 .5 Ribose D Deoxyribose • MONOSACCHARIDES —i— 1.0 —i— 1.5 2.0 Xylose • Galactose • Glucose • —i— 2.5 Fructose o 3.0 —i— 3.5 —i— 4.0 s Co ! 4.5 5.0 5.5 Number of e-OH groups Figure 6.46: Reduced slope g (equation 6.68) versus e-OH number for monosaccharides Chapter 6. RESULTS 123 of the diffusing molecular cluster thus resulting in decrease in diffusion coefficient. This is because according to the Stokes-Einstein law (equation 3.16), diffusion coefficient is inversely proportional to the size of the diffusing molecule. While there appears to be a good correlation between magnetic field effect and e-OH number of all oligosaccharides, this is not the case for all monosaccharides ( Figure 6.46 ). The shape of monosaccharides is nearly spherical. The orienting effect of magnetic field would not be significant in this case. The difference in diffusion coefficients for these saccharides is attributed to the effect these saccharides may have on the local water structure. Deoxyribose, which has only 1.37 e-OH groups destabilizes the water structure [112] as a consequence of the competition of two interactions: one interaction between a-OH (axial) groups of the saccharide molecule and the water molecule and the other between two water molecules. The destabilizing effect by a-OH groups depends on their numbers [112]. The observed large decrease in the free diffusion coefficient value of deoxyribose induced by the applied magnetic field would seem to be the result of a general stabilizing effect of the applied field on the originally less stable configuration of the deoxyribose-water structure. Several authors have discussed the structure-making or structure-breaking properties of mono-saccharides [112,124,138]. The diffusion coefficients obtained experimentally support the idea that the microviscosity of a saccharide molecule increases with the increase in number of e-OH groups in a molecule. This means that the ability of aldoses ( ribose, xylose, galactose, glucose) as structure-makers increases with the increase in the number of e-OH groups in saccharide molecules. This is apparent from Figure 6.46, which shows that the magnetic field effect on diffusion coefficients of these saccharides also increases with the increase in number of e-OH groups. This is because the micro-viscosity of saccharide molecules, as discussed above, increases with an increase in the number of e-OH groups. Chapter 6. RESULTS 124 Fructose, which shows the largest effect on free diffusion in an applied magnetic field, is structurally different from all the saccharides used in this work. Fructose has a ketone group as compared to an aldehyde group for all the other saccharides considered, Figures 2.3 and 2.4. Whether the large magnetic field effect shown by fructose is due to the ketone group being more susceptible to the applied magnetic field or to it's structure making/breaking ability is not clear at this time. Also, as the e-OH number for fructose is much lower than for other pentoses (Table 2.2), this implies that fructose would probably form fewer hydrogen bonds with water. In this case the observed large decrease in the free diffusion coefficient value of fructose (Figure 6.46) induced by the applied magnetic field, would seem to be the result of a general stabilizing effect of the applied magnetic field on the originally less stable configuration of the fructose-water structure. Suggett and co-workers [123] have shown that not only the number of e-OH groups but also their spatial orientation has a small but significant effect on saccharide hydra tion. Whether the applied magnetic field affects the orientation of e-OH groups and hence the pattern of saccharide hydration (stabilization of water structure in solution) as to change (reduce) the values of the diffusion coefficients especially for structure "breakers" is a question that should be addressed in further study. Anisotropy in magnetic field susceptibility for saccharides is not known at this time. Equation 3.26 indicates that the degree of orientation of molecules, Cotton-Mouton effect (orienting effect) is directly proportional to anisotropy in magnetic susceptibility. Therefore, the exact relationship expressing the effect of applied magnetic field on binary diffusion coefficients of saccha rides should take into account any anisotropy in magnetic susceptibility of saccharides. Chapter 7 CONCLUSIONS AND RECOMMENDATIONS A modified Rayleigh interferometer laser system was used to study the effect of an exter nally applied magnetic field on free diffusion of saccharides in aqueous solution through a porous membrane. Diffusion cell and diffusion cell holder were modified in such a way that the new design permitted the use of the same membrane to study one complete saccharide system ( six runs, magnetic field strength changing from zero to 1.1 T in a random fashion). This new experimental procedure eliminated any run to have varia tions caused by a new membrane and resulted in a significant reduction in data scatter and improved measurement accuracy. Binary diffusion coefficients calculated at no field conditions agreed with literature values to within one percent. The difference between bulk and no bulk flow diffusion coefficients was less than 0.6 percent. The results of the linear regression analysis indicate a decrease (two to eighteen per cent) of diffusion coefficients as the applied magnetic field increases to 1.1 T. A study of saccharide-water interactions indicates that saccharide solutions are very complex in nature and that saccharide hydration depends not only on the number of equatorial hy-droxyl (e-OH) groups in a saccharide molecule but also on their (i.e. e-OH numbers) spatial orientation. The saccharide-water solution complexes may be characterised by two factors (1) the elongated (non-spherical) shape of the oligosaccharides (2) the effect of monosaccharides on the local water structure (i.e. they are either structure makers or structure breakers). The effect of applied magnetic field on the binary diffusion coeffi cients of oligosaccharides (sucrose, lactose, maltose, raffinose) shows a strong correlation 125 Chapter 7. CONCLUSIONS AND RECOMMENDATIONS 126 with e-OH numbers , Figure 6.45 and the associated molecular size of the oligosaccha rides. The observed decrease in diffusivities of oligosaccharides is due to the orientation of these molecules in the magnetic field ( Cotton-Mouton effect). Orientation of the molecule by the applied magnetic field (if the magnetic susceptibility is anisotropic) will increase the cross-sectional area of the diffusing molecule as in this case, Figure 3.11 (a), thus reducing the diffusion coefficient. At this time no measurements are available for saccharides regarding magnetic sus ceptibility anisotropics. Results obtained in this work show that oligosaccharides must exhibit anisotropic behaviour. On the other hand, the shape of monosaccharides is es sentially spherical, therefore the orientation of these molecules in applied magnetic field is not expected to have any significant effect on diffusion of these saccharides. The de crease in diffusivity of monosaccharides ( D(-)ribose, D(+)xylose, D-galactose, D-glucose ) becomes larger with an increase in e-OH number. This is because an increase in e-OH number increases the microviscosity of the saccharide molecule ( structure making or stabilizing effect ). Deoxyribose and D(-)fructose on the other hand are considered to be structure breakers. The observed decrease in diffusivity for these saccharides induced by the applied magnetic field seem to be the result of a general stabilizing effect of the applied field on the originally less stable saccharide-water solution. This compares well with the findings for the diffusion of KC1-water solutions [43]. The analysis in chapter 3 showed that the system was free of osmotic and hindered diffusion effects so that the membrane presented only an area reduction to diffusion. The membrane used in this work was 0.4 micron diameter pore size polycarbonate film manufactured by Nuclepore Canada Inc. Free area calculations for the membrane were not performed in this work. The limiting factor was the membrane shadow thickness. The membrane shadow thickness should ideally be constant for all runs but significant difference was observed for each saccharide system due to the inconsistencies involved Chapter 7. CONCLUSIONS AND RECOMMENDATIONS 127 while mounting the membrane. This is made worse by the fact that the edges of the membrane shadow tend to be blurred in the image of the interference pattern on the film when viewed through the measuring microscope. The manufacturer's specifications on pore size variation are + 0 to - 20 percent and for pore density ± 15 percent. So there is a possibility of a quite large variation in surface porosity. The minimum theoretical surface porosity for a 0.4 micron diameter pore size membrane is therefore 0.06836 cm2/cm2 (i.e. 6.84 percent) and the maximum possible is 0.1445 cm2jam2 (i.e. 14.45 percent) based on the manufacturer's specifications. Generally, the results of this work show that a modified Rayleigh interferometer laser system, when properly focused and corrected for wave-front deflection by the refractive index gradient, is capable of giving highly accurate diffusion coefficients in applied mag netic fields. The results for membrane parameters (surface porosity) also gave consistent values. But it was felt that membrane shadow thickness was a limiting factor for free area calculations. The apparent accuracy of the interferometer used in this work is ± 1% based on the diffusivity results. Based on the experience gained in this work it is recommended that the diffusion cell be further modified to facilitate mounting of the membrane on the diffusion cell inter face. This should be done by drilling a port at the bottom side of the reference side compartment (see Figure 4.26). After this modification the fluids can be introduced into the diffusion cell after the membrane has been successfully mounted on the cell interface. This would reduce the possibility of any air bubbles being left entrapped at the bottom surface of the membrane. In addition to improving equipment accuracy and experimen tal procedure, efforts could be made to reduce the time taken to measure fringes and input data to the computer. Measuring fringes through the microscope and recording several thousand data points is a tedious and time consuming process. Therefore a more efficient data evaluation system is desirable to minimize the time required to conduct Chapter 7. CONCLUSIONS AND RECOMMENDATIONS 128 an experiment. Based upon the discussions with the technical staff at Optikon Corpo ration Ltd., an automated fringe measurement system was proposed by the author in a previous research proposal [107]. The cost of the total system which includes data acquisition and analysis came to about $20000. Rennor and Lyons [106] also report on a computer recorded automated interferometric system. They utilize a photomultiplier tube to automatically measure the interference fringes and provide input directly to a digital computer. Adams et al. [178] and Watkins et al. [179] discuss similar methods using electronic light sensing elements to provide an automated data acquisition system. Further research and development work could be done with membrane pore sizes that offer sufficient resistance to flow (i.e. hindered diffusion in relation to molecular diame ters). An experimental run was made in this work with a 0.015 micron pore size diameter membrane. This membrane offered a great deal of resistance to sucrose diffusion with the result that it was possible to measure only a total of four fringes after the experiment was allowed to run for 10 hours as compared with 21 fringes for 0.4 micron pore size membrane for an experimental time of two hours. So a membrane pore size should be chosen that would give enough fringes in about 3 hours for calculation purposes. Also ex periments could be conducted at different temperatures and saccharide concentrations. It is known that saccharide-water clusters (hydration) may be temperature sensitive [122]. To be able to observe this effect in relation to applied magnetic field influence on the diffusion process, the temperature would have to much higher or lower than 25°C. This may require modification to the current temperature control system. The next logical extension to this work would be to choose different biologically important systems showing strong anisotropic behaviour. Partial or complete alignment in a magnetic field has been observed for several macromolecules and molecular clusters by measuring the magnetically induced birefringence (Cotton-Mouton effect, Chapter 3). Any molecular system exhibiting a Cotton-Mouton effect should also show a change in Chapter 7. CONCLUSIONS AND RECOMMENDATIONS 129 diffusion rate in a porous membrane, when subjected to an applied magnetic field. A Cotton-Mouton effect has been observed for polypeptides, nucleic acid fragments [149], rodlike viruses [149], DNA [149], liquid crystals, chloroplasts [149], retinal rods [149], and micellar aqueous soap solutions. Most macromolecules of biological importance require a buffer solution. The method used in this work has been developed for binary systems so a different data analysis procedure will have to be used to evaluate diffusion coefficients of multicomponent systems. The results of this work point the way for further research, i.e. by combining the traditional methods of studying saccharide hydration (NMR, nuclear magnetic relax ation and dielectric relaxation) with the applied magnetic field. The combined study may provide some fundamental information about the observed diffusion behaviour of saccharide-water solutions in applied magnetic field. This information together with the knowledge of the anisotropics in magnetic susceptibilities of saccharides may be suffi cient to formulate a detailed mathematical model to describe the diffusional behaviour of saccharide-water solutions in applied magnetic field. NOMENCLATURE Af Effective free area for diffusion, equations 3.17 A Constant in equation 5.54 A Velocity vector, equation 2.2 a, b, c, P, a Constants in equations 5.64, 6.67 and 6.68 B Magnetic induction vector, equation 2.3 b Constant in equation 5.54 c Concentration equation 3.27, total molar concentration equations 3.13 and 5.66, velocity of light in vacuum, equation 3.28 gradcs Electrolyte concentration gradient, equation 2.4 C Concentration CM Cotton-Mouton constant, equation 3.27 D, D' Diffusion coefficient, reduced diffusion coefficient, equation 6.68 DF, DM Binary and membrane (pore) diffusion coefficients f° Maxwellian distribution function in equation 2.2 f Distribution function, equation 2.1; Stokes friction coefficient, equation 3.14; focal length, equation 3.32 g, g Constants in equations 6.67 and 6.68 G Gauss, Figure 2.1 H, H' Magnetic field strength, equations 3.23 and 6.68 H0 Null hypothesis / Induced magnetization, equation 3.23 130 NOMENCLATURE 131 J Molar flux in equation 3.9 k Boltzmann's constant, equation 3.25 I Geometrical distance in the diffusion compartment, equation 3.34 mv2 Momentum, equation 2.1 m constant in equation 5.54, membrane in equation 3.17 n refractive index, equation 3.28 N Any number, equation 3.26 NA, NB Molar flux of species A (saccharide) and B (water) relative to stationary coordinates, equations 3.13 and 5.66 NAV Avogadro's number, equation 3.16 q heat flow in equation 2.1 r radius, equation 3.15 R Universal gas constant Rs, Rw Radius of solute and water, equation 2.8 RSE Stokes-Einstein radius, equation 3.16 S Masking slits, Figure 4.23 s arc of a ray of light, equation 3.39 t time T Tesla, Absolute temperature ^2) velocity vector in equation 2.4 Vk Average drift velocity of an ion, equation 2.3 v Velocity of light in a medium, equation 3.28 NOMENCLATURE v Particle velocity, equation 2.1 x Vertical distance XA Mole fraction of component A, equations 3.13 and 5.66 Ys Masking slit separation, equation 3.32 z Distance along optical axis Zf. Electric charge on diffusing ion, equation 2.3 Greek Symbols ct|], ct± Molecular optical polarisabilities, equation 3.27 etc Constant in equation 5.51 8 Ratio of solute to pore radii, equation 3.20 3c constant in equation 5.51 8m Membrane thickness, Figure 5.39 80 Degree of orientation of a molecule abgned in magnetic field, equation 3.25 A Difference, distance A X Membrane shaddow thickness, equations 3.13 and 5.66 e-y Constant in equation 5.52 7 Ratio of solvent to pore radii, equation 3.20 7X Constant in equation 5.52 A Wavelength of light, thermal conductivity (Figure 2.1) rj Viscosity, equations 3.16 and 3.15, Figure 2.1 fi Magnetic permeability, equation 3.24 fiSi Chemical potential, equation 2.4 7r Osmotic pressure, equation 3.19 NOMENCLATURE 133 p Density a Staverman's constant, equation 3.19 Acp Phase difference of light wave,equation 3.31 x± Parallel and perpendicular magnetic susceptibilities, equation 3.25 Subscripts B Bottom, Figure 5.39 T Top, Figure 5.39 lin Linearized function, equation 5.55 0 Without applied magnetic field, equation 6.68 Superscripts H With applied magnetic field, equation 2.4 0 Without applied magnetic field, equation 2.4 BIBLIOGRAPHY Barnothy, J.M., and Barnothy, M.F., and Boszormenyi-Nagey, J., Influence of a Magnetic Field upon the Leucocyte of a Mouse, Nature, 177, 577 (1956). 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Adams, W.A., Davis, A.R., Seabrook, G., Ferguson, W.R.m A Digitized Laser Interferometer for High Pressure Refractive Index Studies of Liquids, Can. J. of Spectroscopy, 21, N2, 40 (1976). Watkins, L.S., Tvarusko, A., Lloyd Mirror Laser Interferometer for Diffusion Layer Studies, Rev. Sci. Inst, 41, N12, 1860 (1970). BIBLIOGRAPHY 147 [180] Edward. J.T. Molecular Volumes and Stokes- Einstein Equation, J. of Chem. Ed ucation, Vol. 47, No4 (1970). [181] James, D.W. and Frost, R.L., Structure of Aqueous Solutions. Structure Making and Strurcture Breaking in Solutions of Sucrose and Urea, J. Phys. Chem., Vol. 78, No 17, (1974). [182] International Critical Tables of Numerical Data, Vol 2, McGraw-Hill Book Com pany., New York (1927). [183] Emmerich, E.,Prowe, B., Rosenbruch, K.J.,Die Abhangigkeit der Brechzahl Vom Massengehalt Wassriger Losungen Von Glucose, Fructose Invertzuker, Zuckerind ,109, Nr6 , (1984). [184] Emmerich, A, Emmerich, L., Die Dickie Wassriger Losungen Von Glucose, Fruc tose and Invertzucker Sowie ihre Messung, Zuckerind 111 Nr5 (1986). [185] Personal Communication (Letter) With Dr. H. Uedaira of Hokkaido University. Sapporo, Japan , April (1989). [186] Landolt-Bornstein Numerical Data and Functional Relationships in Science and Technology, Group II, Vol 16 (1986). [187] Williamsom, S.J., Romani, G., Kaufman,L., Modena, I. (Editors), Biomagnetism, Plenum Press, New York (1983). Appendix A Interference Fringe Pattern Data This Appendix contains all experimental data points. Fringe positions are measured at the focal plane for five times. Each point is the location in centimeters where an interference fringe bends by an amount equal to one fringe spacing. Fringes are measured from each end of the diffusion cell where they are straight, i.e., show no refractive index gradient. Negative values denote fringes measured in lower half of the cell. The membrane surface in each half of the cell is located at 0.0 cm. 148 Appendix A. Interference Fringe Pattern Data 149 RUN DE0X1 FIELD = 0.0 T fringe location relative to the membrane, cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0458 0.0682 0.0815 0.1003 0.1179 3 0.0339 0.0529 0.0646 0.0763 0.0888 4 0.0265 0.0407 0.0502 0.0590 0.0710 5 0.0201 0.0314 0.0386 0.0467 0.0565 6 0.0144 0.0234 0.0287 0.0352 0.0425 7 0.0095 0.0169 0.0198 0.0252 0.0288 8 0.0044 0.0104 0.0126 0.0157 0.0177 9 0.0010 0.0039 0.0050 0.0066 0.0076 membrane 10 -0.0054 -0.0049 -0.0047 -0.0050 -0.0053 11 -0.0092 -0.0103 -0.0086 -0.0103 -0.0118 12 -0.0145 -0.0163 -0.0174 -0.0180 -0.0196 13 -0.0185 -0.0210 -0.0221 -0.0265 -0.0287 14 -0.0232 -0.0276 -0.0298 -0.0351 -0.0395 15 -0.0292 -0.0347 -0.0390 -0.0451 -0.0514 16 -0.0356 -0.0421 -0.0476 -0.0553 -0.0653 17 -0.0436 -0.0519 -0.0581 -0.0671 -0.0801 18 -0.0550 -0.0625 -0.0707 -0.0819 -0.1000 19 -0.3330 -0.0795 -0.0909 -0.1050 -0.1220 20 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 150 RUN DEOX3 FIELD = 0.3 T fringe location relative to the membrane, cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0490 0.0747 0.0900 0.1062 0.1255 3 0.0372 0.0552 0.0664 0.0773 0.0973 4 0.0281 0.0420 0.0513 0.0591 0.0768 5 0.0212 0.0321 0.0392 0.0460 0.0607 6 0.0155 0.0248 0.0301 0.0364 0.0475 7 0.0100 0.0181 0.0221 0.0273 0.0347 8 0.0052 0.0113 0.0142 0.0180 0.0236 9 0.0012 0.0050 0.0072 0.0095 0.0121 membrane 10 -0.0054 -0.0037 -0.0038 -0.0046 -0.0047 11 -0.0092 -0.0057 -0.0076 -0.0075 -0.0092 12 -0.0141 -0.0107 -0.0125 -0.0143 -0.0178 13 -0.0179 -0.0167 -0.0193 -0.0199 -0.0264 14 -0.0231 -0.0223 -0.0268 -0.0276 -0.0362 15 -0.0290 -0.0288 -0.0355 -0.0386 -0.0468 16 -0.0363 -0.0372 -0.0455 -0.0511 -0.0593 17 -0.0450 -0.0452 -0.0544 -0.0652 -0.0747 18 -0.0559 -0.0548 -0.0658 -0.0804 -0.0960 19 -0.3330 -0.0683 -0.0830 -0.1002 -0.1230 20 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 151 RUN DE0X5 FIELD = 0.5 T fringe location relative to the membrane, cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0480 0.0667 0.0829 0.0973 0.1193 3 0.0365 0.0492 0.0637 0.0736 0.0950 4 0.0280 0.0378 0.0530 0.0564 0.0757 5 0.0218 0.0283 0.0418 0.0422 0.0592 6 0.0155 0.0213 0.0324 0.0318 0.0427 7 0.0104 0.0145 0.0234 0.0229 0.0294 8 0.0062 0.0085 0.0149 0.0149 0.0190 9 0.0017 0.0026 0.0069 0.0065 0.0093 membrane 10 -0.0054 -0.0043 -0.0044 -0.0040 -0.0048 11 -0.0092 -0.0080 -0.0091 -0.0060 -0.0092 12 -0.0139 -0.0134 -0.0153 -0.0120 -0.0166 13 -0.0179 -0.0174 -0.0209 -0.0186 -0.0244 14 -0.0230 -0.0240 -0.0279 -0.0278 -0.0354 15 -0.0287 -0.0313 -0.0367 -0.0392 -0.0476 16 -0.0350 -0.0395 -0.0462 -0.0497 -0.0628 17 -0.0426 -0.0478 -0.0563 -0.0630 -0.0786 18 -0.0523 -0.0595 -0.0732 -0.0767 -0.0956 19 -0.3330 -0.0731 -0.0948 -0.0927 -0.1143 20 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 152 RUN DE0X4 FIELD = 0.7 T fringe location relative to the membrane, cm. fringe no 900 sec 1800 sec. 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0503 0.0692 0.0824 0.1051 0.1224 3 0.0380 0.0522 0.0656 0.0787 0.0900 4 0.0301 0.0396 0.0521 0.0590 0.0687 5 0.0227 0.0301 0.0396 0.0459 0.0545 6 0.0172 0.0217 0.0292 0.0348 0.0425 7 0.0114 0.0150 0.0202 0.0257 0.0304 8 0.0061 0.0080 0.0115 0.0183 0.0207 9 0.0017 0.0012 0.0036 0.0091 0.0088 membrane 10 -0.0058 -0.0042 -0.0042 -0.0044 -0.0055 11 -0.0101 -0.0066 -0.0081 -0.0091 -0.0104 12 -0.0149 -0.0117 -0.0163 -0.0167 -0.0181 13 -0.0196 -0.0169 -0.0224 -0.0231 -0.0276 14 -0.0249 -0.0227 -0.0295 -0.0338 -0.0383 15 -0.0313 -0.0293 -0.0382 -0.0443 -0.0484 16 -0.0384 -0.0364 -0.0454 -0.0569 -0.0607 17 -0.0464 -0.0451 -0.0565 -0.0689 -0.0749 18 -0.0587 -0.0575 -0.0718 -0.0827 -0.0950 19 -0.3330 -0.0772 -0.0963 -0.1015 -0.1254 20 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 153 RUN DE0X6 FIELD = 0.9 T fringe location relative to the membrane, cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0441 0.0689 0.0835 0.0958 0.1153 3 0.0324 0.0514 0.0633 0.0745 0.0899 4 0.0255 0.0397 0.0497 0.0592 0.0688 5 0.0183 0.0302 0.0383 0.0455 0.0525 6 0.0127 0.0232 0.0285 0.0347 0.0386 7 0.0083 0.0160 0.0198 0.0239 0.0278 8 0.0034 0.0095 0.0117 0.0160 0.0181 9 -0.0058 0.0029 0.0040 0.0056 0.0072 membrane 10 -0.0100 -0.0039 -0.0044 -0.0053 -0.0051 11 -0.0143 -0.0064 -0.0073 -0.0074 -0.0092 12 -0.0180 -0.0124 -0.0136 -0.0132 -0.0160 13 -0.0231 -0.0165 -0.0197 -0.0192 -0.0249 14 -0.0287 -0.0231 -0.0277 -0.0287 -0.0350 15 -0.0354 -0.0303 -0.0365 -0.0378 -0.0453 16 -0.0439 -0.0374 -0.0456 -0.0483 -0.0591 17 -0.0564 -0.0459 -0.0560 -0.0599 -0.0764 18 -0.3330 -0.0561 -0.0697 -0.0746 -0.0959 19 -0.3330 -0.0722 -0.0885 -0.0944 -0.1169 20 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 154 RUN DE0X2 FIELD = 1.1 T fringe location relative to the membrane, cm. fringe no 907 sec. 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0470 0.0688 0.0832 0.0921 0.1148 3 0.0345 0.0527 0.0635 0.0709 0.0878 4 0.0269 0.0417 0.0507 0.0561 0.0668 5 0.0201 0.0314 0.0396 0.0438 0.0520 6 0.0147 0.0247 0.0301 0.0331 0.0407 7 0.0097 0.0178 0.0210 0.0243 0.0285 8 0.0058 0.0106 0.0135 0.0141 0.0173 9 0.0017 0.0042 0.0055 0.0055 0.0065 membrane 10 -0.0054 -0.0048 -0.0058 -0.0050 -0.0063 11 -0.0092 -0.0075 -0.0099 -0.0093 -0.0112 12 -0.0139 -0.0116 -0.0150 -0.0164 -0.0170 13 -0.0187 -0.0178 -0.0205 -0.0227 -0.0242 14 -0.0233 -0.0239 -0.0288 -0.0318 -0.0355 15 -0.0290 -0.0306 -0.0373 -0.0414 -0.0465 16 -0.0355 -0.0389 -0.0459 -0.0521 -0.0596 17 -0.0433 -0.0469 -0.0557 -0.0639 -0.0741 18 -0.0548 -0.0569 -0.0710 -0.0799 -0.0934 19 -0.3330 -0.0722 -0.0884 -0.1010 -0.1161 20 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 155 RUN RIB1 FIELD = 0.0 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0446 0.0653 0.0853 0.0991 0.1159 3 0.0332 0.0491 0.0657 0.0762 0.0906 4 0.0255 0.0386 0.0528 0.0602 0.0715 5 0.0155 0.0305 0.0402 0.0482 0.0570 6 0.0143 0.0231 0.0316 0.0376 0.0444 7 0.0097 0.0163 0.0234 0.0276 0.0334 8 0.0055 0.0100 0.0170 0.0185 0.0224 9 0.0010 0.0039 0.0096 0.0106 0.0128 10 0.0017 0.0018 0.0023 0.0027 membrane 11 -0.0074 -0.0043 -0.0052 -0.0050 -0.0048 12 -0.0106 -0.0089 -0.0112 -0.0133 -0.0098 13 -0.0154 -0.0145 -0.0173 -0.0198 -0.0180 14 -0.0191 -0.0194 -0.0246 -0.0283 -0.0292 15 -0.0240 -0.0256 -0.0321 -0.0370 -0.0396 16 -0.0288 -0.0328 -0.0408 -0.0457 -0.0507 17 -0.0348 -0.0390 -0.0495 -0.0565 -0.0634 18 -0.0428 -0.0477 -0.0590 -0.0684 -0.0786 19 -0.0540 -0.0579 -0.0721 -0.0834 -0.0933 20 -0.3330 -0.0733 -0.0913 -0.1059 -0.1212 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 156 RUN RIB7 FIELD = 0.5 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0450 0.0649 0.0867 0.0950 0.1203 3 0.0343 0.0492 0.0657 0.0733 0.0948 4 0.0261 0.0393 0.0517 0.0571 0.0740 5 0.0194 0.0303 0.0413 0.0460 0.0598 6 0.0140 0.0225 0.0319 0.0341 0.0463 7 0.0086 0.0153 0.0232 0.0250 0.0344 8 0.0045 0.0092 0.0160 0.0160 0.0238 9 0.0005 0.0037 0.0079 0.0077 0.0125 10 0.0017 0.0014 0.0005 0.0023 membrane 11 -0.0064 -0.0030 -0.0033 -0.0020 -0.0020 12 -0.0096 -0.0071 -0.0094 -0.0079 -0.0086 13 -0.0142 -0.0139 -0.0165 -0.0149 -0.0173 14 -0.0176 -0.0190 -0.0225 -0.0222 -0.0268 15 -0.0230 -0.0250 -0.0305 -0.0321 -0.0380 16 -0.0282 -0.0324 -0.0385 -0.0413 -0.0506 17 -0.0347 -0.0399 -0.0474 -0.0522 -0.0631 18 -0.0426 -0.0489 -0.0591 -0.0639 -0.0776 19 -0-0533 -0.0586 -0.0729 -0.0793 -0.0952 20 -0.3330 -0.0749 -0.0915 -0.1000 -0.1210 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 157 RUN RIB6 FIELD = 0.7 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0449 0.0647 0.0871 0.0968 0.1197 3 0.0337 0.0500 0.0667 0.0742 0.0935 4 0.0260 0.0394 0.0527 0.0590 0.0736 5 0.0196 0.0302 0.0411 0.0463 0.0587 6 0.0141 0.0227 0.0325 0.0351 0.0450 7 0.0094 0.0165 0.0237 0.0252 0.0329 8 0.0051 0.0099 0.0155 0.0163 0.0224 9 0.0009 0.0036 0.0082 0.0077 0.0117 10 0.0017 0.0022 0.0004 0.0011 membrane 11 -0.0076 -0.0043 -0.0052 -0.0050 -0.0048 12 -0.0109 -0.0089 -0.0112 -0.0133 -0.0098 13 -0.0157 -0.0145 -0.0175 -0.0168 -0.0205 14 -0.0191 -0.0204 -0.0232 -0.0245 -0.0301 15 -0.0238 -0.0270 -0.0316 -0.0327 -0.0412 16 -0.0292 -0.0325 -0.0394 -0.0430 -0.0530 17 -0.0354 -0.0405 -0.0481 -0.0535 -0.0651 18 -0.0433 -0.0485 -0.0593 -0.0648 -0.0804 19 -0.0532 -0.0595 -0.0730 -0.0793 -0.0994 20 -0.3330 -0.0745 -0.0905 -0.1003 -0.1250 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 158 RUN RIB8 FIELD = 0.9 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0490 0.0704 0.0794 0.0988 0.1090 3 0.0373 0.0529 0.0622 0.0774 0.0860 4 0.0292 0.0415 0.0501 0.0611 0.0674 5 0.0194 0.0314 0.0393 0.0487 0.0533 6 0.0162 0.0248 0.0300 0.0389 0.0409 7 0.0104 0.0171 0.0217 0.0284 0.0289 8 0.0064 0.0109 0.0137 0.0182 0.0180 9 0.0015 0.0054 0.0071 0.0094 0.0075 10 0.0017 0.0006 0.0021 0.0010 membrane 11 -0.0050 -0.0030 -0.0035 -0.0027 -0.0067 12 -0.0093 -0.0072 -0.0095 -0.0091 -0.0133 13 -0.0144 -0.0128 -0.0160 -0.0172 -0.0210 14 -0.0182 -0.0177 -0.0215 -0.0245 -0.0310 15 -0.0238 -0.0238 -0.0291 -0.0332 -0.0411 16 -0.0296 -0.0304 -0.0377 -0.0430 -0.0531 17 -0.0364 -0.0379 -0.0460 -0.0538 -0.0660 18 -0.0445 -0.0467 -0.0566 -0.0649 -0.0816 19 -0.0558 -0.0573 -0.0701 -0.0804 -0.0994 20 -0.3330 -0.0715 -0.0886 -0.1020 -0.1257 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 159 RUN RIB2 FIELD = 1.1 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0444 0.0689 0.0837 0.0944 0.1155 3 0.0333 0.0519 0.0654 0.0733 0.0919 4 0.0262 0.0407 0.0519 0.0577 0.0726 5 0.0204 0.0322 0.0409 0.0464 0.0578 6 0.0143 0.0246 0.0318 0.0354 0.0447 7 0.0098 0.0180 0.0238 0.0266 0.0328 8 0.0057 0.0111 0.0159 0.0162 0.0219 9 0.0008 0.0056 0.0084 0.0088 0.0115 10 0.0017 0.0011 0.0006 0.0018 membrane 11 -0.0073 -0.0033 -0.0031 -0.0034 -0.0025 12 -0.0109 -0.0076 -0.0080 -0.0103 -0.0094 13 -0.0152 -0.0139 -0.0163 -0.0170 -0.0183 14 -0.0190 -0.0183 -0.0214 -0.0249 -0.0276 15 -0.0240 -0.0247 -0.0292 -0.0335 -0.0382 16 -0.0293 -0.0314 -0.0379 -0.0436 -0.0493 17 -0.0359 -0.0384 -0.0466 -0.0529 -0.0633 18 -0.0435 -0.0475 -0.0570 -0.0657 -0.0775 19 -0.0536 -0.0580 -0.0698 -0.0809 -0.0945 20 -0.3330 -0.0739 -0.0875 -0.1001 -0.1192 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 160 RUN XYL1 FIELD = 0.0 T fringe location relative to the membrane, cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0461 0.0670 0.0851 0.0970 0.1235 3 0.0347 0.0510 0.0642 0.0760 0.0950 4 0.0268 0.0402 0.0512 0.0592 0.0769 5 0.0207 0.0326 0.0412 0.0465 0.0613 6 • 0.0156 0.0247 0.0319 0.0367 0.0476 7 0.0108 0.0180 0.0238 0.0276 0.0360 8 0.0067 0.0125 0.0161 0.0190 0.0264 9 0.0023 0.0065 0.0098 0.0107 0.0164 10 0.0017 0.0028 0.0038 0.0078 membrane 11 -0.0071 -0.0030 -0.0046 -0.0079 -0.0048 12 -0.0113 -0.0076 -0.0115 -0.0151 -0.0123 13 -0.0150 -0.0125 -0.0167 -0.0210 -0.0203 14 -0.0185 -0.0183 -0.0236 -0.0289 -0.0303 15 -0.0235 -0.0244 -0.0310 -0.0376 -0.0414 16 -0.0289 -0.0308 -0.0391 -0.0471 -0.0511 17 -0.0352 -0.0382 -0.0480 -0.0576 -0.0604 18 -0.0432 -0.0471 -0.0589 -0.0699 -0.0790 19 -0.0544 -0.0577 -0.0709 -0.0853 -0.0970 20 -0.3330 -0.0721 -0.0903 -0.1085 -0.1231 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 161 RUN XYL4 FIELD = 0.3 T fringe location relative to the membrane, cm. fringe no 905 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 o 0.0483 0.0695 0.0887 0.0985 0.1155 3 0.0362 0.0525 0.0684 0.0767 0.0917 4 0.0271 0.0422 0.0545 0.0615 0.0741 5 0.0207 0.0338 0.0434 0.0484 0.0579 6 0.0156 0.0259 0.0347 0.0386 0.0449 7 0.0111 0.0187 0.0261 0.0289 0.0345 8 0.0067 0.0128 0.0176 0.0193 0.0231 9 0.0024 0.0072 0.0108 0.0125 0.0141 10 0.0018 0.0041 0.0039 0.0039 membrane 11 -0.0055 -0.0047 -0.0047 -0.0051 -0.0042 12 -0.0084 -0.0083 -0.0103 -0.0099 -0.0116 13 -0.0140 -0.0146 -0.0170 -0.0176 -0.0188 14 -0.0173 -0.0195 -0.0236 -0.0250 -0.0294 15 -0.0220 -0.0255 -0.0310 -0.0339 -0.0392 16 -0.0275 -0.0324 -0.0389 -0.0434 -0.0506 17 -0.0334 -0.0398 -0.0479 -0.0537 -0.0628 18 -0.0410 -0.0491 -0.0583 -0.0656 -0.0777 19 -0.0520 -0.0600 -0.0717 -0.0802 -0.0957 20 -0.3330 -0.0745 -0.0904 -0.1016 -0.1215 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 162 RUN XYL2 FIELD = 0.5 T fringe location relative to the membrane, cm. fringe no 900 sec 1837 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0487 0.0698 0.0882 0.0933 0.1142 3 0.0369 0.0526 0.0677 0.0759 0.0903 4 0.0286 0.0409 0.0535 0.0579 0.0719 5 0.0221 0.0329 0.0424 0.0465 0.0568 6 0.0168 0.0247 0.0339 0.0358 0.0442 7 0.0114 0.0176 0.0252 0.0261 0.0335 8 0.0070 0.0114 0.0180 0.0170 0.0216 9 0.0031 0.0063 0.0108 0.0095 0.0112 10 0.0005 0.0047 0.0016 0.0022 membrane 11 -0.0057 -0.0036 -0.0062 -0.0053 -0.0042 12 -0.0087 -0.0070 -0.0119 -0.0117 -0.0111 13 -0.0141 -0.0143 -0.0178 -0.0186 -0.0188 14 -0.0174 -0.0181 -0.0244 -0.0263 -0.0285 15 -0.0220 -0.0244 -0.0322 -0.0342 -0.0388 16 -0.0279 -0.0315 -0.0407 -0.0444 -0.0489 17 -0.0336 -0.0394 -0.0498 -0.0545 -0.0632 18 -0.0412 -0.0489 -0.0595 -0.0679 -0.0773 19 -0.0524 -0.0590 -0.0744 -0.0818 -0.0950 20 -0.3330 -0.0737 -0.0926 -0.1033 -0.1202 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data RUN XYL7 FIELD = 0.7 T fringe location relative to the membrane, cm. fringe no 900 sec 1803 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0487 0.0677 0.0847 0.0951 0.1164 3 0.0369 0.0505 0.0651 0.0743 0.0910 4 0.0287 0.0405 0.0508 0.0587 0.0739 5 0.0220 0.0325 0.0408 0.0465 0.0587 6 0.0170 0.0238 0.0314 0.0359 0.0460 7 0.0117 0.0173 0.0223 0.0268 0.0345 8 0.0073 0.0113 0.0153 0.0182 0.0224 9 0.0031 0.0070 0.0092 0.0106 0.0141 10 0.0041 0.0046 0.0053 0.0067 membrane 11 -0.0069 -0.0065 -0.0079 -0.0069 -0.0081 12 -0.0108 -0.0115 -0.0143 -0.0145 -0.0163 13 -0.0147 -0.0164 -0.0194 -0.0202 -0.0228 14 -0.0184 -0.0218 -0.0260 -0.0281 -0.0330 15 -0.0238 -0.0274 -0.0342 -0.0365 -0.0443 16 -0.0292 -0.0346 -0.0422 -0.0460 -0.0548 17 -0.0355 -0.0417 -0.0505 -0.0559 -0.0671 18 -0.0441 -0.0508 -0.0613 -0.0681 -0.0821 19 -0.0549 -0.0614 -0.0742 -0.0820 -0.1014 20 -0.3330 -0.0781 -0.0941 -0.1043 -0.1266 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 164 RUN XYL3 FIELD = 0.9 T fringe location relative to the membrane, cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0486 0.0698 0.0885 0.0982 0.1167 3 0.0369 0.0528 0.0668 0.0756 0.0920 4 0.0288 0.0415 0.0532 0.0590 0.0762 5 0.0220 0.0329 0.0420 0.0465 0.0596 6 0.0173 0.0249 0.0330 0.0373 0.0458 7 0.0120 0.0179 0.0240 0.0269 0.0348 8 0.0077 0.0123 0.0168 0.0178 0.0234 9 0.0032 0.0064 0.0103 0.0106 0.0143 10 0.0014 0.0032 0.0021 0.0049 membrane 11 -0.0071. -0.0066 -0.0035 -0.0035 -0.0036 12 -0.0113 -0.0111 -0.0090 -0.0092 -0.0128 13 -0.0151 -0.0167 -0.0160 -0.0167 -0.0202 14 -0.0191 -0.0221 -0.0224 -0.0242 -0.0309 15 -0.0239 -0.0283 -0.0307 -0.0330 -0.0411 16 -0.0295 -0.0353 -0.0386 -0.0426 -0.0532 17 -0.0358 -0.0424 -0.0474 -0.0528 -0.0653 18 -0.0433 -0.0513 -0.0578 -0.0649 -0.0806 19 -0.0557 -0.0621 -0.0706 -0.0792 -0.0978 20 -0.3330 -0.0793 -0.0900 -0.1003 -0.1246 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 165 RUN XYL5 FIELD = 1.1 T fringe location relative to the membrane, cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0479 0.0680 0.0905 0.0992 0.1133 3 0.0358 0.0511 0.0693 0.0784 0.0891 4 0.0284 0.0405 0.0542 0.0612 0.0713 5 0.0213 0.0324 0.0429 0.0491 0.0563 6 0.0165 0.0245 0.0349 0.0387 0.0444 7 0.0115 0.0181 0.0258 0.0297 0.0339 8 0.0069 0.0125 0.0183 0.0198 0.0225 9 0.0028 0.0062 0.0109 0.0126 0.0136 10 0.0014 0.0038 0.0042 0.0032 membrane 11 -0.0071 -0.0057 -0.0047 -0.0055 -0.0057 12 -0.0109 -0.0102 -0.0089 -0.0112 -0.0121 13 -0.0149 -0.0154 -0.0161 -0.0183 -0.0198 14 -0.0186 -0.0200 -0.0228 -0.0254 -0.0290 15 -0.0239 -0.0260 -0.0294 -0.0344 -0.0401 16 -0.0294 -0.0328 -0.0383 -0.0440 -0.0514 17 -0.0356 -0.0400 -0.0465 -0.0553 -0.0634 18 -0.0442 -0.0490 -0.0575 -0.0656 -0.0793 19 -0.0550 -0.0604 -0.0709 -0.0816 -0.0954 20 -0.3330 -0.0757 -0.0895 -0.1029 -0.1208 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 166 RUN FRU4 FIELD = 0.0 T fringe location relative to the membrane, cm. fringe no 902 sec 1800 sec 2710 sec 3620 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0452 0.0647 0.0824 0.0988 0.1161 3 0.0335 0.0507 0.0647 0.0755 0.0905 4 0.0255 0.0395 0.0507 0.0602 0.0731 5 0.0196 0.0311 0.0393 0.0480 0.0591 6 0.0140 0.0231 0.0302 0.0374 0.0460 7 0.0098 0.0170 0.0226 0.0279 0.0350 8 0.0060 0.0114 0.0154 0.0199 0.0250 9 0.0016 0.0057 0.0089 0.0123 0.0153 10 0.0017 0.0016 0.0048 0.0056 membrane 11 -0.0068 -0.0041 -0.0044 -0.0082 -0.0101 12 -0.0109 -0.0091 -0.0098 -0.0148 -0.0178 13 -0.0155 -0.0150 -0.0179 -0.0209 -0.0269 14 -0.0188 -0.0197 -0.0243 -0.0295 -0.0363 15 -0.0228 -0.0258 -0.0322 -0.0375 -0.0464 16 -0.0285 -0.0328 -0.0397 -0.0458 -0.0581 17 -0.0346 -0.0401 -0.0475 -0.0570 -0.0702 18 -0.0430 -0.0476 -0.0588 -0.0683 -0.0848 19 -0.0535 -0.0587 -0.0721 -0.0828 -0.1016 20 -0.3330 -0.0728 -0.0882 -0.1039 -0.1291 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 167 RUN FRU6 FIELD = 0.3 T fringe location relative to the membrane, cm. fringe no 901 sec 1815 sec 2700 sec 3602 sec 5405 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0463 0.0657 0.0842 0.1016 0.1188 3 0.0348 0.0508 0.0649 0.0790 0.0930 4 0.0273 0.0400 0.0514 0.0627 0.0746 5 0.0211 0.0307 0.0402 0.0498 0.0603 6 0.0157 0.0245 0.0322 0.0389 0.0470 7 0.0116 0.0175 0.0240 0.0303 0.0363 8 0.0075 0.0115 0.0174 0.0222 0.0251 9 0.0031 0.0057 0.0104 0.0150 0.0163 10 0.0009 0.0035 0.0073 0.0067 membrane 11 -0.0040 -0.0048 -0.0042 -0.0099 -0.0075 12 -0.0084 -0.0106 -0.0100 -0.0164 -0.0170 13 -0.0144 -0.0161 -0.0179 -0.0224 -0.0257 14 -0.0193 -0.0208 -0.0244 -0.0304 -0.0348 15 -0.0232 -0.0273 -0.0310 -0.0388 -0.0466 16 -0.0281 -0.0333 -0.0390 -0.0481 -0.0561 17 -0.0345 -0.0404 -0.0486 -0.0578 -0.0685 18 -0.0426 -0.0491 -0.0588 -0.0697 -0.0832 19 -0.0541 -0.0591 -0.0709 -0.0852 -0.0989 20 -0.3330 -0.0749 -0.0889 -0.1060 -0.1267 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 168 RUN FRU7 FIELD = 0.7 T fringe location relative to the membrane, cm. fringe no 902 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0468 0.0669 0.0822 0.0968 0.1253 3 0.0344 0.0511 0.0634 0.0747 0.0945 4 0.0269 0.0406 0.0515 0.0601 0.0747 5 0.0209 0.0327 0.0407 0.0486 0.0614 6 0.0156 0.0250 0.0321 0.0383 0.0489 7 0.0112 0.0184 0.0236 0.0288 0.0381 8 0.0068 0.0131 0.0162 0.0203 0.0273 9 0.0025 0.0070 0.0095 0.0127 0.0173 10 0.0018 0.0030 0.0044 0.0091 membrane 11 -0.0062 -0.0044 -0.0048 -0.0049 -0.0049 12 -0.0103 -0.0089 -0.0113 -0.0111 -0.0155 13 -0.0146 -0.0152 -0.0179 -0.0189 -0.0230 14 -0.0188 -0.0194 -0.0253 -0.0271 -0.0334 15 -0.0235 -0.0254 -0.0323 -0.0357 -0.0439 16 -0.0290 -0.0318 -0.0404 -0.0458 -0.0536 17 -0.0350 -0.0391 -0.0491 -0.0552 -0.0647 18 -0.0433 -0.0476 -0.0598 -0.0659 -0.0788 19 -0.0548 -0.0585 -0.0722 -0.0812 -0.0967 20 -0.3330 -0.0732 -0.0917 -0.1002 -0.1243 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 169 RUN FRU5 FIELD = 0.9 T fringe location relative to the membrane, cm. fringe no 900 sec 1875 sec 2745 sec 3603 sec 5466 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0480 0.0728 0.0851 0.1040 0.1222 3 0.0363 0.0566 0.0648 0.0803 0.0949 4 0.0284 0.0443 0.0515 0.0635 0.0755 5 0.0222 0.0347 0.0418 0.0513 0.0625 6 0.0172 0.0275 0.0328 0.0404 0.0498 7 0.0125 0.0205 0.0250 0.0309 0.0375 8 0.0079 0.0139 0.0177 0.0223 0.0277 9 0.0039 0.0084 0.0104 0.0147 0.0174 10 0.0037 0.0042 0.0072 0.0085 membrane 11 -0.0045 -0.0043 -0.0053 -0.0053 -0.0099 12 -0.0102 -0.0089 -0.0122 -0.0106 -0.0173 13 -0.0160 -0.0159 -0.0181 -0.0160 -0.0255 14 -0.0201 -0.0212 -0.0250 -0.0229 -0.0350 15 -0.0238 -0.0277 -0.0333 -0.0309 -0.0440 16 -0.0299 -0.0341 -0.0408 -0.0381 -0.0558 17 -0.0364 -0.0411 -0.0497 -0.0473 -0.0664 18 -0.0453 -0.0503 -0.0596 -0.0578 -0.0825 19 -0.0579 -0.0614 -0.0731 -0.0696 -0.1001 20 -0.3330 -0.0768 -0.0929 -0.0855 -0.1289 21 -0.3330 -0.3330 -0.1060 -0.3330 22 -0.3330 Appendix A. Interference Fringe Pattern Data RUN FRU3 FIELD = 1.1 T fringe location relative to the membrane, cm. fringe no 1056 sec 1860 sec 2705 sec 3680 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0512 0.0690 0.0908 0.0980 0.1157 3 0.0382 0.0531 0.0681 0.0751 0.0910 4 0.0301 0.0412 0.0544 0.0607 0.0724 5 0.0229 0.0326 0.0435 0.0479 0.0585 6 0.0180 0.0242 0.0346 0.0373 0.0451 7 0.0126 0.0185 0.0268 0.0289 0.0346 8 0.0079 0.0129 0.0192 0.0205 0.0244 9 0.0033 0.0073 0.0118 0.0117 0.0138 10 0.0010 0.0052 0.0038 0.0055 membrane 11 -0.0065 -0.0054 -0.0058 -0.0048 -0.0072 12 -0.0126 -0.0109 -0.0119 -0.0119 -0.0167 13 -0.0179 -0.0169 -0.0191 -0.0203 -0.0261 14 -0.0218 -0.0223 -0.0261 -0.0293 -0.0345 15 -0.0269 -0.0276 -0.0332 -0.0372 -0.0450 16 -0.0324 -0.0346 -0.0409 -0.0467 -0.0563 17 -0.0394 -0.0428 -0.0496 -0.0564 -0.0682 18 -0.0481 -0.0512 -0.0606 -0.0675 -0.0834 19 -0.0606 -0.0618 -0.0726 -0.0833 -0.0989 20 -0.3330 -0.0778 -0.0917 -0.1031 -0.1235 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 171 RUN GLU7 FIELD = 0.0 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0440 0.0634 0.0784 0.0874 0.1255 3 0.0325 0.0491 0.0606 0.0695 0.0977 4 0.0252 0.0387 0.0481 0.0554 0.0786 5 0.0195 0.0311 0.0382 0.0437 0.0640 6 0.0144 0.0234 0.0296 0.0342 0.0517 7 0.0095 0.0175 0.0226 0.0256 0.0407 8 0.0057 0.0106 0.0154 0.0172 0.0299 9 0.0016 0.0055 0.0084 0.0099 0.0214 10 0.0016 0.0020 0.0039 0.0113 membrane 11 -0.0066 -0.0049 -0.0068 -0.0092 0.0036 12 -0.0109 -0.0092 -0.0119 -0.0152 -0.0060 13 -0.0148 -0.0146 -0.0179 -0.0204 -0.0128 14 -0.0183 -0.0202 -0.0237 -0.0285 -0.0207 15 -0.0224 -0.0252 -0.0309 -0.0370 -0.0295 16 -0.0276 -0.0320 -0.0387 -0.0452 -0.0408 17 -0.0329 -0.0387 -0.0473 -0.0559 -0.0499 18 -0.0410 -0.0471 -0.0570 -0.0670 -0.0629 19 -0.0514 -0.0575 -0.0702 -0.0809 -0.0757 20 -0.3330 -0.0702 -0.0872 -0.1020 -0.0932 21 -0.3330 -0.3330 -0.3330 -0.1136 22 -0.3330 Appendix A. Interference Fringe Pattern Data 172 RUN GLU8 FIELD = 0.3 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0480 0.0702 0.0800 0.0921 0.1153 3 0.0359 0.0536 0.0617 0.0731 0.0887 4 0.0274 0.0416 0.0502 0.0566 0.0700 5 0.0213 0.0332 0.0406 0.0454 0.0552 6 0.0153 0.0263 0.0307 0.0361 0.0439 7 0.0106 0.0193 0.0237 0.0269 0.0342 8 0.0059 0.0126 0.0159 0.0188 0.0222 9 0.0018 0.0071 0.0088 0.0105 0.0130 10 0.0016 0.0024 0.0019 0.0035 membrane 11 -0.0051 -0.0057 -0.0055 -0.0072 -0.0078 12 -0.0117 -0.0093 -0.0106 -0.0116 -0.0146 13 -0.0153 -0.0141 -0.0173 -0.0183 -0.0223 14 -0.0196 -0.0194 -0.0229 -0.0261 -0.0316 15 -0.0246 -0.0258 -0.0300 -0.0343 -0.0428 16 -0.0298 -0.0323 -0.0381 -0.0431 -0.0540 17 -0.0361 -0.0384 -0.0474 -0.0524 -0.0664 18 -0.0441 -0.0474 -0.0569 -0.0637 -0.0790 19 -0.0565 -0.0582 -0.0684 -0.0773 -0.0975 20 -0.3330 -0.0718 -0.0871 -0.0974 -0.1259 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 173 RUN GLU3 FIELD = 0.5 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0439 0.0618 0.0790 0.0895 0.1112 3 0.0329 0.0471 0.0608 0.0692 0.0854 4 0.0249 0.0363 0.0477 0.0541 0.0680 5 0.0181 0.0279 0.0367 0.0426 0.0532 6 0.0129 0.0205 0.0280 0.0318 0.0410 7 0.0081 0.0138 0.0197 0.0223 0.0303 8 0.0044 0.0074 0.0125 0.0134 0.0194 9 0.0019 0.0021 0.0049 0.0063 0.0088 membrane 10 -0.0046 -0.0040 -0.0054 -0.0092 -0.0078 11 -0.0094 -0.0080 -0.0108 -0.0171 -0.0165 12 -0.0141 -0.0142 -0.0175 -0.0229 -0.0256 13 -0.0176 -0.0195 -0.0247 -0.0318 -0.0344 14 -0.0216 -0.0255 -0.0323 -0.0402 -0.0447 15 -0.0269 -0.0323 -0.0400 -0.0502 -0.0560 16 -0.0323 -0.0401 -0.0496 -0.0598 -0.0687 17 -0.0391 -0.0493 -0.0603 -0.0719 -0.0829 18 -0.0482 -0.0591 -0.0728 -0.0870 -0.1003 19 -0.0629 -0.0746 -0.0898 -0.1089 -0.1215 20 -0.3330 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 174 RUN GLU4 FIELD = 0.7 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0431 0.0657 0.0773 0.0925 0.1109 3 0.0317 0.0492 0.0585 0.0720 0.0846 4 0.0237 0.0380 0.0455 0.0553 0.0670 5 0.0170 0.0287 0.0347 0.0422 0.0512 6 0.0111 0.0217 0.0257 0.0327 0.0383 7 0.0059 0.0142 0.0176 0.0222 0.0276 8 0.0028 0.0076 0.0097 0.0133 0.0166 9 0.0024 0.0031 0.0042 0.0053 membrane 10 -0.0040 -0.0087 -0.0107 -0.0061 -0.0035 11 -0.0072 -0.0141 -0.0173 -0.0117 -0.0089 12 -0.0106 -0.0194 -0.0239 -0.0193 -0.0184 13 -0.0153 -0.0252 -0.0310 -0.0274 -0.0274 14 -0.0192 -0.0320 -0.0393 -0.0359 -0.0376 15 -0.0237 -0.0389 -0.0478 -0.0448 -0.0497 16 -0.0289 -0.0469 -0.0579 -0.0557 -0.0613 17 -0.0352 -0.0576 -0.0694 -0.0676 -0.0760 18 -0.0428 -0.0725 -0.0874 -0.0821 -0.0945 19 -0.0542 -0.3330 -0.3330 -0.1051 -0.1168 20 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 175 RUN GLU5 FIELD = 0.9 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0430 0.0659 0.0793 0.0955 0.1207 3 0.0315 0.0490 0.0581 0.0746 0.0942 4 0.0234 0.0386 0.0471 0.0586 0.0743 5 0.0170 0.0296 0.0360 0.0456 0.0594 6 0.0115 0.0226 0.0271 0.0349 0.0477 7 0.0069 0.0161 0.0193 0.0254 0.0362 8 0.0030 0.0100 0.0120 0.0173 0.0254 9 0.0040 0.0045 0.0095 0.0164 membrane 10 -0.0079 -0.0061 -0.0067 0.0032 0.0060 11 -0.0129 -0.0104 -0.0121 -0.0074 -0.0067 12 -0.0165 -0.0164 -0.0192 -0.0150 -0.0134 13 -0.0199 -0.0217 -0.0258 -0.0238 -0.0220 14 -0.0246 -0.0287 -0.0334 -0.0317 -0.0351 15 -0.0301 -0.0347 -0.0412 -0.0413 -0.0425 16 -0.0361 -0.0420 -0.0509 -0.0512 -0.0536 17 -0.0437 -0.0514 -0.0606 -0.0627 -0.0660 18 -0.0563 -0.0616 -0.0727 -0.0753 -0.0803 19 -0.3330 -0.0770 -0.0916 -0.0948 -0.0990 20 -0.3330 -0.3330 -0.3330 -0.1238 21 -0.3330 Appendix A. Interference Fringe Pattern Data 176 RUN GLU6 FIELD = 1.1 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0435 0.0674 0.0832 0.0912 0.1142 3 0.0329 0.0510 0.0644 0.0724 0.0886 4 0.0258 0.0392 0.0507 0.0572 0.0717 5 0.0200 0.0315 0.0408 0.0467 0.0570 6 0.0149 0.0249 0.0321 0.0366 0.0451 7 0.0100 0.0175 0.0237 0.0279 0.0336 8 0.0061 0.0119 0.0172 0.0192 0.0241 9 0.0025 0.0062 0.0101 0.0112 0.0141 10 0.0025 0.0036 0.0045 0.0050 membrane 11 -0.0076 -0.0052 -0.0068 -0.0045 -0.0058 12 -0.0112 -0.0115 -0.0116 -0.0095 -0.0133 13 -0.0155 -0.0178 -0.0180 -0.0164 -0.0207 14 -0.0194 -0.0247 -0.0249 -0.0239 -0.0305 15 -0.0235 -0.0298 -0.0319 -0.0327 -0.0400 16 -0.0292 -0.0371 -0.0390 -0.0415 -0.0621 17 -0.0347 -0.0453 -0.0479 -0.0507 -0.0747 18 -0.0406 -0.0547 -0.0573 -0.0613 -0.0904 19 -0.0528 -0.0684 -0.0700 -0.0760 -0.1170 20 -0.3330 -0.3330 -0.0868 -0.0948 -0.3330 21 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 177 RUN GLA1 FIELD = 0.0 T fringe location relative to the membrane, cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0415 0.0638 0.0786 0.0956 0.1086 3 0.0313 0.0491 0.0614 0.0722 0.0857 4 0.0247 0.0384 0.0483 0.0588 0.0684 5 0.0194 0.0307 0.0391 0.0476 0.0542 6 0.0148 0.0240 0.0298 0.0373 0.0414 7 0.0105 0.0172 0.0230 0.0283 0.0302 8 0.0068 0.0116 0.0158 0.0209 0.0206 9 0.0032 0.0063 0.0093 0.0135 0.0132 10 0.0016 0.0030 0.0063 0.0055 membrane 11 -0.0044 -0.0050 -0.0051 -0.0041 -0.0045 12 -0.0075 -0.0076 -0.0084 -0.0083 -0.0085 13 -0.0117 -0.0120 -0.0140 -0.0139 -0.0149 14 -0.0149 -0.0168 -0.0189 -0.0190 -0.0207 15 -0.0183 -0.0216 -0.0255 -0.0274 -0.0292 16 -0.0226 -0.0269 -0.0319 -0.0348 -0.0394 17 -0.0272 -0.0337 -0.0390 -0.0428 -0.0481 18 -0.0328 -0.0398 -0.0468 -0.0522 -0.0599 19 -0.0394 -0.0472 -0.0554 -0.0637 -0.0768 20 -0.0488 -0.0568 -0.0680 -0.0771 -0.0929 21 -0.3330 -0.0700 -0.0852 -0.0971 -0.1134 22 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 178 RUN GLA5 FIELD = 0.3 T fringe location relative to the membrane, cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0495 0.0653 0.0792 0.0952 0.1113 3 0.0376 0.0508 0.0630 0.0747 0.0856 4 0.0298 0.0406 0.0504 0.0600 0.0692 5 0.0246 0.0326 0.0414 0.0484 0.0558 6 0.0194 0.0257 0.0326 0.0400 0.0455 7 0.0146 0.0201 0.0261 0.0316 0.0355 8 0.0112 0.0143 0.0189 0.0234 0.0255 9 0.0072 0.0087 0.0124 0.0159 0.0164 10 0.0035 0.0035 0.0063 0.0078 0.0063 membrane 11 -0.0034 -0.0054 -0.0040 -0.0040 -0.0043 12 -0.0065 -0.0090 -0.0075 -0.0071 -0.0075 13 -0.0105 -0.0145 -0.0117 -0.0119 -0.0117 14 -0.0144 -0.0184 -0.0167 -0.0181 -0.0186 15 -0.0177 -0.0244 -0.0225 -0.0241 -0.0281 16 -0.0223 -0.0307 -0.0305 -0.0331 -0.0375 17 -0.0267 -0.0374 -0.0380 -0.0413 -0.0465 18 -0.0331 -0.0446 -0.0463 -0.0501 -0.0582 19 -0.0401 -0.0540 -0.0554 -0.0608 -0.0614 20 -0.0501 -0.0689 -0.0673 -0.0734 -0.0866 21 -0.3330 -0.3330 -0.0858 -0.0915 -0.1092 22 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 179 RUN GLA3 FIELD = 0.5 T fringe location relative to the membrane, cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0442 0.0673 0.0777 0.0906 0.1073 3 0.0346 0.0503 0.0612 0.0718 0.0846 4 0.0273 0.0411 0.0490 0.0577 0.0690 5 0.0216 0.0332 0.0386 0.0467 0.0560 6 0.0166 0.0260 0.0311 0.0369 0.0449 7 0.0126 0.0199 0.0237 0.0277 0.0338 8 0.0083 0.0144 0.0164 0.0211 0.0238 9 0.0052 0.0089 0.0097 0.0131 0.0143 10 0.0022 0.0039 0.0038 0.0060 0.0049 membrane 11 -0.0049 -0.0039 -0.0043 -0.0028 -0.0044 12 -0.0078 -0.0068 -0.0079 -0.0066 -0.0082 13 -0.0113 -0.0116 -0.0139 -0.0116 -0.0160 14 -0.0147 -0.0170 -0.0182 -0.0179 -0.0216 15 -0.0183 -0.0212 -0.0250 -0.0243 -0.0302 16 -0.0227 -0.0267 -0.0321 -0.0322 -0.0390 17 -0.0268 -0.0332 -0.0388 -0.0403 -0.0492 18 -0.0327 -0.0398 -0.0473 -0.0499 -0.0605 19 -0.0399 -0.0472 -0.0570 -0.0602 -0.0750 20 -0.0500 -0.0565 -0.0697 -0.0735 -0.0925 21 -0.3330 -0.0715 -0.0873 -0.0923 -0.1138 22 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. interference Fringe Pattern Data 180 RUN GLA6 FIELD = 0.7 T fringe location relative to the membrane, cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0450 0.0698 0.0809 0.0919 0.1113 3 0.0350 0.0544 0.0623 0.0727 0.0884 4 0.0280 0.0438 0.0519 0.0595 0.0724 5 0.0224 0.0353 0.0417 0.0474 0.0593 6 0.0173 0.0290 0.0326 0.0385 0.0462 7 0.0134 0.0224 0.0259 0.0296 0.0354 8 0.0089 0.0170 0.0186 0.0225 0.0255 9 0.0053 0.0109 0.0123 0.0148 0.0174 10 0.0016 0.0060 0.0061 0.0075 0.0085 membrane 11 -0.0059 0.0016 -0.0038 -0.0043 -0.0041 12 -0.0083 -0.0041 -0.0065 -0.0073 -0.0072 13 -0.0122 -0.0088 -0.0120 -0.0124 -0.0142 14 -0.0158 -0.0146 -0.0176 -0.0185 -0.0206 15 -0.0197 -0.0184 -0.0240 -0.0246 -0.0305 16 -0.0238 -0.0241 -0.0308 -0.0332 -0.0397 17 -0.0289 -0.0305 -0.0390 -0.0410 -0.0502 18 -0.0352 -0.0374 -0.0464 -0.0512 -0.0622 19 -0.0420 -0.0445 -0.0559 -0.0622 -0.0749 20 -0.0527 -0.0543 -0.0686 -0.0745 -0.0908 21 -0.3330 -0.0672 -0.0853 -0.0931 -0.1117 22 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 181 RUN GLA4 FIELD = 0.9 T fringe location relative to the membrane, cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 0.0456 0.0635 0.0829 0.0889 0.1064 3 0.0354 0.0488 0.0654 0.0691 0.0841 4 0.0280 0.0392 0.0527 0.0558 0.0677 5 0.0224 0.0319 0.0427 0.0449 0.0527 6 0.0180 0.0249 0.0349 0.0366 0.0414 7 0.0139 0.0194 0.0263 0.0287 0.0313 8 0.0098 0.0139 0.0194 0.0201 0.0223 9 0.0062 0.0079 0.0132 0.0129 0.0140 10 0.0028 0.0026 0.0065 0.0058 0.0065 membrane 11 -0.0045 -0.0042 -0.0039 -0.0038 -0.0045 12 -0.0074 -0.0065 -0.0075 -0.0070 -0.0086 13 -0.0112 -0.0094 -0.0133 -0.0125 -0.0145 14 -0.0144 -0.0152 -0.0177 -0.0185 -0.0209 15 -0.0173 -0.0194 -0.0245 -0.0250 -0.0292 16 -0.0232 -0.0249 -0.0313 -0.0328 -0.0387 17 -0.0280 -0.0317 -0.0392 -0.0408 -0.0478 18 -0.0336 -0.0379 -0.0473 -0.0506 -0.0598 19 -0.0411 -0.0455 -0.0579 -0.0611 -0.0736 20 -0.0515 -0.0553 -0.0694 -0.0756 -0.0911 21 -0.3330 -0.0700 -0.0871 -0.0941 -0.1131 22 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data RUN GLA2 FIELD = 1.1 T fringe location relative to the membrane, cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0459 0.0675 0.0783 0.0946 0.1147 3 0.0356 0.0515 0.0612 0.0734 0.0883 4 0.0290 0.0416 0.0499 0.0601 0.0695 5 0.0232 0.0340 0.0399 0.0478 0.0575 6 0.0181 0.0269 0.0319 0.0381 0.0463 7 0.0138 0.0211 0.0253 0.0301 0.0365 8 0.0100 0.0150 0.0176 0.0215 0.0265 9 0.0062 0.0095 0.0109 0.0130 0.0175 10 0.0025 0.0049 0.0046 0.0065 0.0076 membrane 11 -0.0060 -0.0035 -0.0040 -0.0034 -0.0045 12 -0.0093 -0.0073 -0.0075 -0.0065 -0.0086 13 -0.0132 -0.0109 -0.0118 -0.0119 -0.0145 14 -0.0163 -0.0162 -0.0173 -0.0178 -0.0209 15 -0.0198 -0.0210 -0.0238 -0.0248 -0.0292 16 -0.0244 -0.0262 -0.0302 -0.0333 -0.0387 17 -0.0290 -0.0327 -0.0383 -0.0409 -0.0478 18 -0.0350 -0.0398 -0.0459 -0.0497 -0.0598 19 -0.0423 -0.0472 -0.0556 -0.0592 -0.0736 20 -0.0526 -0.0571 -0.0672 -0.0730 -0.0911 21 -0.3330 -0.0716 -0.0834 -0.0920 -0.1131 22 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 183 RUN MAL12 FIELD = 0.0 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0387 0.0586 0.0773 0.0839 0.1017 3 0.0287 0.0462 0.0574 0.0650 0.0796 4 0.0221 0.0350 0.0463 0.0524 0.0636 5 0.0166 0.0282 0.0372 0.0415 0.0505 6 0.0109 0.0223 0.0288 0.0329 0.0409 7 0.0064 0.0152 0.0222 0.0252 0.0309 8 0.0034 0.0091 0.0148 0.0182 0.0220 9 0.0017 0.0047 0.0086 0.0088 0.0145 10 0.0041 0.0037 0.0065 membrane 11 -0.0062 -0.0055 -0.0053 - 0.0049 -0.0070 12 -0.0097 -0.0091 -0.0107 - 0.0104 -0.0131 13 -0.0123 -0.0127 -0.0145 - 0.0154 -0.0206 14 -0.0153 -0.0175 -0.0194 - 0.0219 -0.0277 15 -0.0193 -0.0220 -0.0258 - 0.0291 -0.0376 16 -0.0236 -0.0280 -0.0325 - 0.0371 -0.0458 17 -0.0289 -0.0345 -0.0393 -0.0461 -0.0560 18 -0.0359 -0.0411 -0.0482 -0.0549 -0.0678 19 -0.0444 -.0.0504 -0.0577 -0.0670 -0.0837 20 -0.3330 -0.0618 -0.0744 -0.0849 -0.1049 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 184 RUN MALI7 FIELD = 0.3 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0391 0.0573 0.0726 0.0837 0.1019 3 0.0298 0.0450 0.0556 0.0663 0.0808 4 0.0221 0.0341 0.0447 0.0523 0.0650 5 0.0173 0.0268 0.0356 0.0423 0.0526 6 0.0126 0.0208 0.0277 0.0331 0.0420 7 0.0085 0.0149 0.0211 0.0253 0.0318 8 0.0051 0.0092 0.0136 0.0185 0.0235 9 0.0019 0.0046 0.0079 0.0102 0.0137 10 0.0034 0.0045 0.0052 membrane 11 -0.0062 -0.0065 -0.0051 -0.0059 -0.0049 12 -0.0082 -0.0085 -0.0097 -0.0105 -0.0105 13 -0.0107 -0.0115 -0.0133 -0.0147 -0.0163 14 -0.0136 -0.0156 -0.0187 -0.0199 -0.0252 15 -0.0174 -0.0204 -0.0255 -0.0271 -0.0336 16 -0.0222 -0.0263 -0.0320 -0.0351 -0.0431 17 -0.0276 -0.0327 -0.0383 -0.0434 -0.0546 18 -0.0344 -0.0408 -0.0478 -0.0538 -0.0669 19 -0.0436 -0.0499 -0.0585 -0.0658 -0.0829 20 -0.3330 -0.0655 -0.0738 -0.0828 -0.1037 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 185 RUN MALI4 FIELD = 0.7 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0381 0.0546 0.0747 0.0747 0.1013 3 0.0276 0.0418 0.0591 0.0591 0.0776 4 0.0223 0.0335 0.0472 0.0472 0.0629 5 0.0168 0.0257 0.0374 0.0374 0.0502 6 0.0125 0.0198 0.0287 0.0287 0.0398 7 0.0085 0.0144 0.0219 0.0219 0.0298 8 0.0049 0.0086 0.0133 0.0133 0.0220 9 0.0024 0.0036 0.0058 0.0058 0.0117 10 0.0039 membrane 11 -0.0063 -0.0046 -0.0052 -0.0052 -0.0043 12 -0.0092 -0.0080 -0.0106 -0.0106 -0.0092 13 -0.0115 -0.0114 -0.0159 -0.0159 -0.0160 14 -0.0145 -0.0155 -0.0215 -0.0215 -0.0250 15 -0.0186 -0.0204 -0.0286 -0.0286 -0.0339 16 -0.0224 -0.0260 -0.0362 -0.0362 -0.0435 17 -0.0271 -0.0323 -0.0453 -0.0453 -0.0535 18 -0.0341 -0.0392 -0.0545 -0.0545 -0.0640 19 -0.0435 -0.0482 -0.0672 -0.0672 -0.0760 20 -0.3330 -0.0614 -0.0852 -0.0852 -0.0989 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 186 RUN MALI8 FIELD = 0.9 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0392 0.0576 0.0730 0.0840 0.0947 3 0.0285 0.0453 0.0550 0.0660 0.0745 4 0.0223 0.0341 0.0450 0.0522 0.0603 5 0.0169 0.0270 0.0354 0.0420 0.0478 6 0.0125 0.0210 0.0280 0.0327 0.0374 7 0.0084 0.0150 0.0210 0.0252 0.0280 8 0.0050 0.0093 0.0138 0.0184 0.0187 9 0.0020 0.0043 0.0080 0.0100 0.0090 10 0.0039 0.0040 0.0021 membrane 11 -0.0060 -0.0062 -0.0054 -0.0060 -0.0057 12 -0.0080 -0.0088 -0.0100 -0.0100 -0.0104 13 -0.0110 -0.0118 -0.0135 -0.0140 -0.0166 14 -0.0138 -0.0160 -0.0188 -0.0210 -0.0254 15 -0.0176 -0.0210 -0.0258 -0.0267 -0.0350 16 -0.0225 -0.0264 -0.0326 -0.0350 -0.0442 17 -0.0277 -0.0328 -0.0387 -0.0440 -0.0544 18 -0.0343 -0.0410 -0.0480 -0.0535 -0.0663 19 -0.0435 -0.0500 -0.0580 -0.0650 -0.0810 20 -0.3330 -0.0650 -0.0740 -0.0838 -0.1028 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 187 RUN MALI6 FIELD = 1.1 T fringe location relative to the membrane , cm. fringe no 908 sec 1800 sec 2700 sec 3603 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0380 0.0563 0.0745 0.0829 0.0990 3 0.0287 0.0425 0.0562 0.0639 0.0774 4 0.0225 0.0337 0.0440 0.0512 0.0627 5 0.0167 0.0262 0.0351 0.0405 0.0499 6 0.0119 0.0203 0.0272 0.0313 0.0391 7 0.0073 0.0149 0.0212 0.0238 0.0293 8 0.0042 0.0087 0.0144 0.0147 0.0218 9 0.0025 0.0042 0.0075 0.0076 0.0104 10 0.0023 0.0024 0.0031 membrane 11 -0.0052 -0.0051 -0.0058 -0.0064 -0.0043 12 -0.0082 -0.0072 -0.0091 -0.0102 -0.0092 13 -0.0109 -0.0116 -0.0132 -0.0162 -0.0162 14 -0.0135 -0.0161 -0.0185 -0.0230 -0.0256 15 -0.0173 -0.0208 -0.0247 -0.0305 -0.0340 16 -0.0221 -0.0257 -0.0313 -0.0379 -0.0400 17 -0.0273 -0.0326 -0.0392 -0.0471 -0.0539 18 -0.0344 -0.0398 -0.0477 -0.0567 -0.0644 19 -0.0434 -0.0489 -0.0583 -0.0695 -0.0763 20 -0.3330 -0.0621 -0.0750 -0.0894 -0.0992 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 188 RUN LAC13 FIELD = 0.0 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0378 0.0548 0.0710 0.0847 0.0954 3 0.0294 0.0426 0.0557 0.0666 0.0762 4 0.0231 0.0341 0.0447 0.0537 0.0615 5 0.0182 0.0271 0.0366 0.0442 0.0507 6 0.0143 0.0218 0.0288 0.0354 0.0403 7 0.0111 0.0164 0.0226 0.0274 0.0315 8 0.0078 0.0190 0.0172 0.0216 0.0240 9 0.0045 0.0077 0.0113 0.0152 0.0159 10 0.0018 0.0035 0.0064 0.0090 0.0088 11 0.0017 0.0029 0.0027 membrane 12 -0.0037 -0.0031 -0.0034 -0.0043 -0.0044 13 -0.0064 -0.0060 -0.0068 -0.0090 -0.0077 14 -0.0095 -0.0097 -0.0122 -0.0153 -0.0146 15 -0.0130 -0.0139 -0.0170 -0.0195 -0.0203 16 -0.0156 -0.0175 -0.0220 -0.0258 -0.0278 17 -0.0191 -0.0225 -0.0281 -0.0332 -0.0367 18 -0.0238 -0.0274 -0.0341 -0.0403 -0.0448 19 -0.0286 -0.0333 -0.0406 -0.0480 -0.0546 20 -0.0345 -0.0403 -0.0495 -0.0580 -0.0667 21 -0.0431 -0.0480 -0.0596 -0.0712 -0.0790 22 -0.3330 -0.0593 -0.0761 -0.0898 -0.1018 23 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 189 RUN LAC12 FIELD = 0.3 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0385 0.0565 0.0684 0.0792 0.0962 3 0.0290 0.0438 0.0540 0.0626 0.0759 4 0.0235 0.0358 0.0430 0.0507 0.0624 5 0.0180 0.0285 0.0352 0.0410 0.0510 6 0.0143 0.0223 0.0281 0.0331 0.0404 7 0.0109 0.0173 0.0210 0.0254 0.0318 8 0.0074 0.0119 0.0152 0.0184 0.0232 9 0.0042 0.0078 0.0099 0.0125 0.0156 10 0.0011 0.0040 0.0046 0.0064 0.0084 11 0.0006 0.0012 0.0017 membrane 12 -0.0041 -0.0033 -0.0036 -0.0048 -0.0038 13 -0.0088 -0.0059 -0.0074 -0.0075 -0.0095 14 -0.0101 -0.0096 -0.0139 -0.0134 -0.0161 15 -0.0135 -0.0146 -0.0176 -0.0180 -0.0222 16 -0.0162 -0.0178 -0.0229 -0.0243 -0.0304 17 -0.0197 -0.0229 -0.0288 -0.0316 -0.0388 18 -0.0242 -0.0275 -0.0356 -0.0385 -0.0479 19 -0.0294 -0.0337 -0.0424 -0.0463 -0.0569 20 -0.0361 -0.0404 -0.0502 -0.0548 -0.0690 21 -0.0449 -0.0494 -0.0607 -0.0665 -0.0835 22 -0.3330 -0.0604 -0.0768 -0.0853 -0.1043 23 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 190 RUN LAC14 FIELD = 0.5 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0384 0.0553 0.0702 0.0814 0.0979 3 0.0290 0.0432 0.0546 0.0623 0.0769 4 0.0225 0.0350 0.0443 0.0506 0.0623 5 0.0184 0.0282 0.0355 0.0411 0.0498 6 0.0142 0.0224 0.0292 0.0327 0.0394 7 0.0106 0.0168 0.0219 0.0264 0.0310 8 0.0070 0.0118 0.0167 0.0188 0.0230 9 0.0037 0.0076 0.0110 0.0126 0.0156 10 0.0007 0.0035 0.0059 0.0064 0.0077 11 0.0007 0.0008 0.0011 membrane 12 -0.0033 -0.0028 -0.0031 -0.0019 -0.0035 13 -0.0060 -0.0054 -0.0069 -0.0056 -0.0083 14 -0.0094 -0.0103 -0.0121 -0.0116 -0.0155 15 -0.0128 -0.0149 -0.0167 -0.0170 -0.0209 16 -0.0159 -0.0184 -0.0221 -0.0228 -0.0291 17 -0.0191 -0.0228 -0.0282 -0.0290 -0.0374 18 -0.0236 -0.0274 -0.0339 -0.0365 -0.0459 19 -0.0283 -0.0337 -0.0407 -0.0444 -0.0567 20 -0.0340 -0.0411 -0.0483 -0.0532 -0.0676 21 -0.0431 -0.0498 -0.0578 -0.0649 -0.0831 22 -0.3330 -0.0620 -0.0739 -0.0811 -0.1051 23 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 191 RUN LAC15 FIELD = 0.9 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0383 0.0555 0.0718 0.0834 0.1037 3 0.0293 0.0439 0.0564 0.0660 0.0806 4 0.0230 0.0348 0.0449 0.0534 0.0674 5 0.0182 0.0284 0.0362 0.0435 0.0544 6 0.0141 0.0221 0.0293 0.0348 0.0439 7 0.0108 0.0169 0.0227 0.0278 0.0352 8 0.0074 0.0114 0.0166 0.0209 0.0265 9 0.0041 0.0076 0.0108 0.0142 0.0187 10 0.0014 0.0035 0.0057 0.0076 0.0107 11 0.0017 0.0021 0.0040 membrane 12 -0.004-0 -0.0032 -0.0042 -0.0037 -0.0033 13 -0.0063 -0.0057 -0.0102 -0.0065 -0.0077 14 -0.0092 -0.0099 -0.0162 -0.0124 -0.0148 15 -0.0125 -0.0141 -0.0211 -0.0175 -0.0209 16 -0.0154 -0.0176 -0.0271 -0.0240 -0.0282 17 -0.0195 -0.0226 -0.0334 -0.0304 -0.0366 18 -0.0233 -0.0278 -0.0398 -0.0374 -0.0442 19 -0.0283 -0.0334 -0.0477 -0.0454 -0.0546 20 -0.0347 -0.0405 -0.0577 -0.0538 -0.0661 21 -0.0430 -0.0493 -0.0735 -0.0673 -0.0810 22 -0.3330 -0.0605 -0.3330 -0.0833 -0.1006 23 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 192 RUN LACll FIELD = 1.1 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0388 0.0564 0.0701 0.0793 0.0935 3 0.0308 0.0443 0.0550 0.0627 0.0752 4 0.0243 0.0352 0.0446 0.0513 0.0609 5 0.0192 0.0287 0.0359 0.0417 0.0496 6 0.0151 0.0225 0.0286 0.0334 0.0389 7 0.0118 0.0175 0.0230 0.0264 0.0315 8 0.0085 0.0127 0.0166 0.0196 0.0229 9 0.0053 0.0082 0.0108 0.0133 0.0147 10 0.0022 0.0041 0.0061 0.0074 0.0075 11 0.0013 0.0022 0.0016 membrane 12 -0.0038 -0.0034 -0.0037 -0.0037 -0.0038 13 -0.0069 -0.0064 -0.0076 -0.0080 -0.0082 14 -0.0103 -0.0101 -0.0128 -0.0136 -0.0147 15 -0.0134 -0.0143 -0.0169 -0.0192 -0.0208 16 -0.0159 -0.0181 -0.0221 -0.0249 -0.0293 17 -0.0195 -0.0229 -0.0285 -0.0313 -0.0368 18 -0.0242 -0.0284 -0.0349 -0.0495 -0.0451 19 -0.0288 -0.0342 -0.0416 -0.0470 -0.0552 20 -0.0350 -0.0414 -0.0501 -0.0561 -0.0659 21 -0.0450 -0.0501 -0.0599 -0.0684 -0.0800 22 -0.3330 -0.0604 -0.0750 -0.0850 -0.0985 23 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 193 RUN SUC3T FIELD = 0.0 T fringe location relative to the membrane, cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0461 0.0554 0.0788 0.0918 0.1100 3 0.0346 0.0425 0.0609 0.0701 0.0878 4 0.0279 0.0340 0.0501 0.0560 0.0688 5 0.0213 0.0274 0.0406 0.0448 0.0554 6 0.0172 0.0214 0.0322 0.0362 0.0444 7 0.0127 0.0158 0.0240 0.0277 0.0350 8 0.0087 0.0114 0.0175 0.0199 0.0263 9 0.0057 0.0066 0.0111 0.0138 0.0174 10 0.0017 0.0020 0.0047 0.0063 0.0090 11 0.0017 membrane 12 -0.0063 0.0079 -0.0080 -0.0052 -0.0097 13 -0.0101 0.0119 -0.0142 -0.0112 -0.0174 14 -0.0129 0.0152 -0.0192 -0.0163 -0.0254 15 -0.0174 0.0188 -0.0271 -0.0217 -0.0342 16 -0.0217 0.0247 -0.0337 -0.0295 -0.0440 17 -0.0272 0.0308 -0.0422 -0.0376 -0.0543 18 -0.0346 0.0380 -0.0520 -0.0472 -0.0667 19 -0.0452 0.0462 -0.0646 -0.0580 -0.0811 20 -0.3330 0.0617 -0.0847 -0.0709 -0.1033 21 0.3330 -0.3330 -0.0904 -0.3330 22 -0.3330 Appendix A. Interference Fringe Pattern Data 194 RUN SI FIELD = 0.3 T fringe location relative to the membrane, cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0464 0.0593 0.0882 0.0992 0.1145 3 0.0343 0.0464 0.0675 0.0789 0.0923 4 0.0264 0.0367 0.0542 0.0633 0.0740 5 0.0210 0.0300 0.0438 0.0509 0.0589 6 0.0155 0.0236 0.0353 0.0399 0.0456 7 0.0112 0.0175 0.0270 0.0299 0.0353 8 0.0082 0.0124 0.0193 0.0211 0.0242 9 0.0045 0.0068 0.0123 0.0130 0.0155 10 0.0022 0.0040 0.0055 0.0050 0.0060 membrane 11 -0.0056 -0.0060 -0.0053 -0.0054 -0.0051 12 -0.0095 -0.0100 -0.0098 -0.0104 -0.0104 13 -0.0120 -0.0120 -0.0131 -0.0142 -0.0151 14 -0.0145 -0.0160 -0.0169 -0.0210 -0.0236 15 -0.0187 -0.0211 -0.0243 -0.0296 -0.0334 16 -0.0229 -0.0266 -0.0319 -0.0374 -0.0428 17 -0.0282 -0.0328 -0.0401 -0.0478 -0.0535 18 -0.0343 -0.0400 -0.0482 -0.0587 -0.0662 19 -0.0421 -0.0482 -0.0580 -0.0708 -0.0799 20 -0.3330 -0.0615 -0.0708 -0.0861 -0.0969 21 -0.3330 -0.0893 -0.1064 -0.1211 22 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 195 RUN SUC2 FIELD = 0.5 T fringe location relative to the membrane, cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0440 0.0547 0.0699 0.0788 0.1007 3 0.0331 0.0430 0.0535 0.0605 0.0779 4 0.0265 0.0334 0.0427 0.0482 0.0633 5 0.0211 0.0263 0.0318 0.0380 0.0500 6 0.0155 0.0199 0.0251 0.0282 0.0398 7 0.0113 0.0149 0.0186 0.0213 0.0295 8 0.0070 0.0094 0.0116 0.0137 0.0208 9 0.0029 0.0035 0.0050 0.0056 0.0129 10 0.0030 membrane 11 -0.0035 -0.0041 -0.0049 -0.0038 -0.0050 12 -0.0070 -0.0070 -0.0092 -0.0092 -0.0127 13 -0.0105 -0.0115 -0.0142 -0.0153 -0.0195 14 -0.0142 -0.0160 -0.0213 -0.0219 -0.0281 15 -0.0189 -0.0213 -0.0272 -0.0296 -0.0370 16 -0.0233 -0.0268 -0.0338 -0.0371 -0.0462 17 -0.0291 -0.0332 -0.0442 -0.0457 -0.0565 18 -0.0360 -0.0410 -0.0519 -0.0560 -0.0685 19 -0.0467 -0.0506 -0.0626 -0.0683 -0.0845 20 -0.3330 -0.0638 -0.0790 -0.0848 -0.1048 21 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 196 RUN SUC5 FIELD = 0.7 T fringe location relative to the membrane, cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0366 0.0632 0.0870 0.1033 0.1339 3 0.0282 0.0472 0.0661 0.0791 0.1035 4 0.0208 03626 0.0527 0.0635 0.0801 5 0.0158 0.0275 0.0417 0.0497 0.0647 6 0.0109 0.0209 0.0323 0.0384 0.0499 7 0.0063 0.0147 0.0234 0.0287 0.0367 8 0.0022 0.0090 0.0155 0.0192 0.0256 9 0.0033 0.0073 0.0100 0.0140 10 0.0020 0.0024 0.0031 membrane 11 -0.0026 -0.0042 -0.0043 -0.0053 -0.0078 12 -0.0066 -0.0078 -0.0113 -0.0151 -0.0179 13 -0.0104 -0.0126 -0.0189 -0.0231 -0.0277 14 -0.0148 -0.0186 -0.0253 -0.0315 -0.0380 15 -0.0194 -0.0240 -0.0328 -0.0405 -0.0494 16 -0.0247 -0.0302 -0.0411 -0.0501 -0.0622 17 -0.0314 -0.0375 -0.0501 -0.0603 -0.0745 18 -0.0387 -0.0457 -0.0602 -0.0725 -0.0880 19 -0.0500 -0.0556 -0.0733 -0.0868 -0.1057 20 -0.0685 -0.0687 -0.0876 -0.1034 -0.1264 21 -0.3330 -0.3330 -0.3330 -0.3330 -0.1560 22 -0.3330 Appendix A. Interference Fringe Pattern Data 197 RUN S3 FIELD = 0.9 T fringe location relative to the membrane, cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0457 0.0541 0.0804 0.0946 0.1128 3 0.0362 0.0402 0.0622 0.0768 0.0882 4 0.0280 0.0319 0.0481 0.0622 0.0703 5 0.0214 0.0240 0.0379 0.0464 0.0546 6 0.0169 0.0178 0.0289 0.0362 0.0411 7 0.0124 0.0122 0.0206 0.0264 0.0293 8 0.0078 0.0065 0.0129 0.0173 0.0180 9 0.0035 0.0016 0.0058 0.0100 0.0061 10 0.0012 membrane 11 -0.0044 -0.0042 -0.0067 -0.0048 -0.0075 12 -0.0081 -0.0087 -0.0142 -0.0122 -0.0177 13 -0.0114 -0.0135 -0.0212 -0.0204 -0.0281 14 -0.0157 -0.0190 -0.0288 -0.0297 -0.0397 15 -0.0198 -0.0247 -0.0375 -0.0369 -0.0511 16 -0.0247 -0.0313 -0.0473 -0.0470 -0.0638 17 -0.0306 -0.0375 -0.0582 -0.0583 -0.0781 18 -0.0372 -0.0462 -0.0717 -0.0725 -0.0916 19 -0.0477 -0.0559 -0.0907 -0.0869 -0.1113 -0.3330 -0.1076 -0.3330 -0.1560 20 -0.3330 -0.3330 -0.3330 -0.1070 -0.1386 21 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 198 RUN S4 FIELD = 1.1 T fringe location relative to the membrane, cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0403 0.0579 0.0829 0.1079 0.1297 3 0.0299 0.0435 0.0647 0.0839 0.1012 4 0.0220 0.0342 0.0513 0.0667 0.0823 5 0.0164 0.0259 0.0407 0.0546 0.0649 6 0.0118 0.0200 0.0323 0.0413 0.0510 7 0.0079 0.0144 0.0234 0.0326 0.0390 8 0.0040 0.0090 0.0165 0.0231 0.0267 9 0.0042 0.0090 0.0144 0.0164 10 0.0019 0.0055 0.0049 membrane 11 -0.0046 -0.0061 -0.0061 -0.0068 -0.0019 12 -0.0081 -0.0112 -0.0124 -0.0160 -0.0121 13 -0.0118 -0.0154 -0.0195 -0.0241 -0.0209 14 -0.0161 -0.0208 -0.0270 -0.0326 -0.0318 15 -0.0211 -0.0267 -0.0349 -0.0422 -0.0428 16 -0.0261 -0.0337 -0.0432 -0.0523 -0.0566 17 -0.0332 -0.0406 -0.0542 -0.0669 -0.0717 18 -0.0436 -0.0512 -0.0655 -0.0820 -0.0857 19 -0.3330 -0.0671 -0.0823 -0.0994 -0.1049 20 -0.0477 -0.3330 -0.3330 -0.3330 -0.1303 21 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 199 RUN RAF1 FIELD = 0.0 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0325 0.0524 0.0640 0.0751 0.0908 3 0.0240 0.0400 0.0540 0.0581 0.0712 4 0.0184 0.0316 0.0393 0.0471 0.0569 5 0.0137 0.0249 0.0321 0.0380 0.0465 6 0.0097 0.0190 0.0255 0.0308 0.0373 7 0.0067 0.0139 0.0192 0.0233 0.0281 8 0.0035 0.0094 0.0136 0.0168 0.0205 9 0.0011 0.0054 0.0081 0.0103 0.0125 10 0.0017 0.0034 0.0049 0.0059 membrane 11 -0.0056 -0.0058 -0.0051 -0.0054 -0.0044 12 -0.0084 -0.0106 -0.0087 -0.0097 -0.0099 13 -0.0113 -0.0153 -0.0134 -0.0141 -0.0164 14 -0.0144 -0.0187 -0.0170 -0.0193 -0.0219 15 -0.0167 -0.0234 -0.0224 -0.0254 -0.0289 16 -0.0200 -0.0275 -0.0270 -0.0317 -0.0368 17 -0.0234 -0.0332 -0.0330 -0.0381 -0.0457 18 -0.0279 -0.0404 -0.0403 -0.0463 -0.0548 19 -0.0334 -0.0491 -0.0483 -0.0552 -0.0661 20 -0.0425 -0.0597 -0.0580 -0.0665 -0.0809 21 -0.3330 -0.3330 -0.0712 -0.0814 -0.0993 22 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 200 RUN RAF3 FIELD = 0.3 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0344 0.0524 0.0646 0.0758 0.0919 .3 0.0263 0.0406 0.0502 0.0587 0.0724 4 0.0206 0.0324 0.0407 0.0477 0.0590 5 0.0159 0.0259 0.0321 0.0388 0.0475 6 0.0118 0.0196 0.0260 0.0305 0.0382 7 0.0083 0.0145 0.0199 0.0235 0.0300 8 0.0050 0.0103 0.0142 0.0170 0.0216 9 0.0024 0.0064 0.0089 0.0111 0.0142 10 0.0017 0.0037 0.0053 0.0070 membrane 11 -0.0069 -0.0051 -0.0058 -0.0051 -0.0052 12 -0.0078 -0.0074 -0.0086 -0.0088 -0.0098 13 -0.0112 -0.0111 -0.0139 -0.0144 -0.0164 14 -0.0139 -0.0145 -0.0175 -0.0185 -0.0216 15 -0.0164 -0.0184 -0.0219 -0.0241 -0.0286 16 -0.0199 -0.0227 -0.0278 -0.0301 -0.0368 17 -0.0235 -0.0273 -0.0333 -0.0374 -0.0457 18 -0.0285 -0.0331 -0.0407 -0.0457 -0.0543 19 -0.0337 -0.0398 -0.0484 -0.0544 -0.0650 20 -0.0434 -0.0483 -0.0575 -0.0650 -0.0791 21 -0.3330 -0.0595 -0.0733 -0.0825 -0.0988 22 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 201 RUN RAF7 FIELD = 0.5 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0347 0.0525 0.0649 0.0756 0.0935 3 0.0262 0.0499 0.0502 0.0593 0.0726 4 0.0202 0.0316 0.0389 0.0476 0.0594 5 0.0156 0.0244 0.0320 0.0385 0.0481 6 0.0111 0.0191 0.0250 0.0311 0.0382 7 0.0080 0.0141 0.0191 0.0241 0.0299 8 0.0046 0.0097 0.0135 0.0168 0.0215 9 0.0018 0.0056 0.0078 0.0110 0.0146 10 0.0017 0.0030 0.0056 0.0075 membrane 11 -0.0051 -0.0049 -0.0059 -0.0057 -0.0057 12 -0.0083 -0.0074 -0.0082 -0.0087 -0.0103 13 -0.0120 -0.0110 -0.0132 -0.0142 -0.0170 14 -0.0145 -0.0153 -0.0175 -0.0185 -0.0229 15 -0.0169 -0.0184 -0.0223 -0.0249 -0.0294 16 -0.0204 -0.0222 -0.0278 -0.0310 -0.0375 17 -0.0240 -0.0273 -0.0342 -0.0380 -0.0464 18 -0.0286 -0.0327 -0.0408 -0.0456 -0.0555 19 -0.0353 -0.0398 -0.0488 -0.0543 -0.0671 20 -0.0441 -0.0480 -0.0586 -0.0655 -0.0803 21 -0.3330 -0.0598 -0.0726 -0.0815 -0.0996 22 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 202 RUN RAF6 FIELD = 0.7 T fringe location relative to the membrane , crn. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0348 0.0522 0.0641 0.0765 0.0907 3 0.0261 0.0410 0.0497 0.0593 0.0709 4 0.0204 0.0320 0.0396 0.0485 0.0582 5 0.0149 0.0263 0.0313 0.0388 0.0464 6 0.0114 0.0203 0.0250 0.0307 0.0372 7 0.0079 0.0151 0.0185 0.0242 0.0291 8 0.0045 0.0111 0.0133 0.0173 0.0208 9 0.0016 0.0065 0.0078 0.0112 0.0139 10 0.0024 0.0031 0.0058 0.0067 membrane 11 -0.0061 -0.0056 -0.0048 -0.0046 -0.0040 12 -0.0085 -0.0083 -0.0089 -0.0095 -0.0089 13 -0.0114 -0.0118 -0.0127 -0.0154 -0.0153 14 -0.0143 -0.0159 -0.0172 -0.0196 -0.0211 15 -0.0168 -0.0190 -0.0221 -0.0250 -0.0285 16 -0.0201 -0.0231 -0.0270 -0.0313 -0.0359 17 -0.0238 -0.0281 -0.0334 -0.0384 -0.0454 18 -0.0284 -0.0341 -0.0403 -0.0462 -0.0537 19 -0.0344 -0.0408 -0.0482 -0.0551 -0.0653 20 -0.0431 -0.0490 -0.0583 -0.0658 -0.0786 21 -0.3330 -0.0600 -0.0710 -0.0830 -0.0988 22 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 203 RUN RAF4 FIELD = 0.9 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0344 0.0529 0.0652 0.0758 0.0912 3 0.0266 0.0406 0.0517 0.0594 0.0719 4 0.0198 0.0327 0.0409 0.0481 0.0575 5 0.0153 0.0258 0.0332 0.0383 0.0470 6 0.0115 0.0197 0.0264 0.0308 0.0377 7 0.0076 0.0151 0.0202 0.0242 0.0283 8 0.0046 0.0104 0.0149 0.0171 0.0206 9 0.0017 0.0061 0.0097 0.0111 0.0135 10 0.0018 0.0041 0.0053 0.0068 membrane 11 -0.0058 -0.0048 -0.0056 -0.0052 -0.0055 12 -0.0087 -0.0085 -0.0088 -0.0091 -0.0102 13 -0.0115 -0.0116 -0.0130 -0.0142 -0.0161 14 -0.0146 -0.0152 -0.0174 -0.0182 -0.0217 15 -0.0168 -0.0193 -0.0219 -0.0244 -0.0291 16 -0.0203 -0.0237 -0.0271 -0.0312 -0.0367 17 -0.0238 -0.0287 -0.0331 -0.0377 -0.0448 18 -0.0284 -0.0345 -0.0400 -0.0454 -0.0542 19 -0.0344 -0.0415 -0.0483 -0.0546 -0.0650 20 -0.0437 -0.0494 -0.0568 -0.0658 -0.0790 21 -0.3330 -0.0616 -0.0724 -0.0829 -0.0988 22 -0.3330 -0.3330 -0.3330 -0.3330 Appendix A. Interference Fringe Pattern Data 204 RUN RAF2 FIELD = 1.1 T fringe location relative to the membrane , cm. fringe no 900 sec 1800 sec 2700 sec 3600 sec 5400 sec 1 0.3330 0.3330 0.3330 0.3330 0.3330 2 0.0345 0.0527 0.0634 0.0759 0.0933 3 0.0260 0.0406 0.0488 0.0590 0.0722 4 0.0197 0.0323 0.0388 0.0473 0.0586 5 0.0156 0.0249 0.0308 0.0386 0.0473 6 0.0109 0.0195 0.0241 0.0304 0.0381 7 0.0074 0.0147 0.0183 0.0232 0.0295 8 0.0047 0.0103 0.0129 0.0163 0.0216 9 0.0015 0.0059 0.0075 0.0108 0.0140 10 0.0019 0.0025 0.0052 0.0073 membrane 11 -0.0050 -0.0048 -0.0058 -0.0055 -0.0052 12 -0.0086 -0.0081 -0.0085 -0.0095 -0.0107 13 -0.0117 -0.0116 -0.0130 -0.0147 -0.0170 14 -0.0146 -0.0152 -0.0169 -0.0194 -0.0223 15 -0.0172 -0.0189 -0.0218 -0.0248 -0.0300 16 -0.0204 -0.0233 -0.0267 -0.0311 -0.0378 17 -0.0244 -0.0282 -0.0327 -0.0388 -0.0466 18 -0.0293 -0.0336 -0.0404 -0.0467 -0.0559 19 -0.0351 -0.0410 -0.0483 -0.0549 -0.0665 20 -0.0444 -0.0493 -0.0574 -0.0667 -0.0805 21 -0.3330 -0.0600 -0.0719 -0.0823 -0.0995 22 -0.3330 -0.3330 -0.3330 -0.3330 Appendix B Error Function Correlation Parameters (m, A and b) 205 Appendix B. Error Function Correlation Parameters (m, A and b) 206 RUN DEOX1 FIELD = 0.0 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000596 -0.000584 -0.000563 -0.000564 -0.000548 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333511 1.333508 1.333487 1.333493 1.333475 bottom m -0.000608 -0.000682 -0.000667 -0.000675 -0.000668 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333578 1.333502 1.333514 1.333514 1.333524 RUN DEOX3 FIELD = 0.3 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000595 -0.000599 -0.000591 -0.000601 -0.000595 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333520 1.333528 1.333517 1.333528 1.333529 bottom m -0.000604 -0.000622 -0.000638 -0.000632 -0.000649 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333582 1.333572 1.333557 1.333562 1.333548 RUN DEOX5 FIELD = 0.5 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000621 -0.000561 -0.000598 -0.000562 -0.000553 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333543 1.333476 1.333527 1.333482 1.333485 bottom m -0.000605 -0.000647 -0.000655 -0.000620 -0.000642 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333583 1.333543 1.333535 1.333482 1.333553 Appendix B. Error Function Correlation Parameters (m, A and b) 207 RUN DEOX4 FIELD = 0.7 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000613 -0.000535 -0.000541 -0.000593 -0.000571 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333545 1.333462 1.333473 1.333519 1.333493 bottom m -0.000618 -0.000632 -0.000659 -0.000655 -0.000660 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333563 1.333562 1.333531 1.333533 1.333536 RUN DEOX6 FIELD = 0.9 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000587 -0.000568 -0.000551 -0.000559 -0.000549 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333493 1.333491 1.333476 1.333486 1.333469 bottom m -0.000615 -0.000632 -0.000642 -0.000635 -0.000642 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333573 1.333560 1.333550 1.333564 1.333555 RUN DEOX2 FIELD = 1.1 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000624 -0.000591 -0.000575 -0.000560 -0.000549 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333537 1.333517 1.333500 1.333480 1.333467 bottom m -0.000607 -0.000642 -0.000667 -0.000658 -0.000654 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333580 1.333549 1.333527 1.333536 1.333544 Appendix B. Error Function Correlation Parameters (m, A and b) 208 RUN RIB1 FIELD = 0.0 T time 900 sec 1800 sec 2700 sec 3602 sec 5400 sec top m -0.000617 -0.000596 -0.000600 -0.000585 -0.000581 A 10.557301 7.465139 6.095260 5.277185 4.310000 b 1.333320 1.333311 1.333329 1.333317 1.333306 bottom m -0.000643 -0.000663 -0.000687 -0.000695 -0.000659 A 10.557301 7.465139 6.095260 5.277185 4.310000 b 1.333346 1.333326 1.333301 1.333293 1.333334 RUN RIB7 FIELD = 0.5 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000585 -0.000587 -0.000584 -0.000562 -0.000572 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333291 1.333303 1.333316 1.333288 1.333331 bottom m -0.000617 -0.000664 -0.000661 -0.000652 -0.000654 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333373 1.333326 1.333327 1.333341 1.333339 RUN RIB6 FIELD = 0.7 T time 900 sec 1800 sec 2700 sec 3600 sec 5445 sec top m -0.000606 -0.000596 -0.000589 -0.000559 -0.000562 A 10.557301 7.465139 6.095260 5.278650 4.292153 b 1.333317 1.333308 1.333321 1.333288 1.333295 bottom m -0.000648 -0.000665 -0.000684 -0.000678 -0.000665 A 10.557301 7.465139 6.095260 5.278650 4.292153 b 1.333341 1.333322 1.333305 1.333316 1.333326 Appendix B. Error Function Correlation Parameters (m, A and b) 209 RUN RIB8 FIELD = 0.9 T time 1015 sec 1800 sec i 2700 sec 3613 sec 5400 sec top m -0.000603 -0.000597 -0.000577 -0.000575 -0.000553 A 9.944253 7.465139 6.095260 5.269145 4.310000 b 1.333317 1.333323 1.333300 1.333311 1.333271 bottom m -0.000606 -0.000652 -0.000660 -0.000676 -0.000686 A 9.944253 7.465139 6.095260 5.269145 4.310000 b 1.333384 1.333341 1.333332 1.333317 1.333306 RUN RIB2 FIELD = 1.1 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000614 -0.000604 -0.000587 -0.000568 -0.000568 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333326 1.333328 1.333318 1.333292 1.333297 bottom m -0.000642 -0.000665 -0.000673 -0.000662 -0.000648 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333346 1.333327 1.333320 1.333329 1.333346 RUN XYL1 FIELD = 0.0 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000657 -0.000620 -0.000610 -0.000601 -0.000619 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333318 1.333289 1.333285 1.333276 1.333230 bottom m -0.000641 -0.000657 -0.000681 -0.000721 -0.000675 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333299 1.333285 1.333256 1.333217 1.333270 Appendix B. Error Function Correlation Parameters (m, A and b) 210 RUN XYL4 FIELD = 0.3 T time 905 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000651 -0.000619 -0.000620 -0.000606 -0.000590 A 10.528096 7.465139 6.095260 5.278650 4.310000 b 1.333315 1.333296 1.333306 1.333286 1.333267 bottom m -0.000601 -0.000661 -0.000671 -0.000671 -0.000664 A 10.528096 7.465139 6.095260 5.278650 4.310000 b 1.333340 1.333296 1.333262 1.333272 1.333280 RUN XYL2 FIELD = 0.5 T time 900 sec 1837 sec 2700 sec 3600 sec 5400 sec top m -0.000665 -0.000595 -0.000630 -0.000577 -0.000571 A 10.557301 7.389577 6.095260 5.278650 4.310000 b 1.333335 1.333269 1.333310 1.333253 1.333247 bottom m -0.000605 -0.000659 -0.000696 -0.000682 -0.000661 A 10.557301 7.389577 6.095260 5.278650 4.310000 b 1.333336 1.333282 1.333242 1.333258 1.333283 RUN XYL7 FIELD = 0.7 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000671 -0.000640 -0.000616 -0.000610 -0.000603 A 10.557301 7.458926 6.095260 5.278650 4.310000 b 1.333341 1.333304 1.333288 1.333280 1.333270 bottom m -0.000633 -0.000703 -0.000727 -0.000711 -0.000711 A 10.557301 7.458926 6.095260 5.278650 4.310000 b 1.333305 1.333234 1.333210 1.333229 1.333230 Appendix B. Error Function Correlation Parameters (m, A and b) 211 RUN XYL3 FIELD = 0.9 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000681 -0.000608 -0.000611 -0.000585 -0.000593 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333350 1.333282 1.333291 1.333263 1.333272 bottom m -0.000643 -0.000703 -0.000660 -0.000656 -0.000671 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333295 1.333231 1.333279 1.333285 1.333327 RUN XYL5 FIELD = 1.1 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000663 -0.000614 -0.000620 -0.000609 -0.000590 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333331 1.333285 1.333305 1.333290 1.333261 bottom m -0.000637 -0.000682 -0.000666 -0.000679 -0.000673 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333301 1.333257 1.333275 1.333262 1.333270 RUN FRU4 FIELD = 0.0 T time 902 sec 1800 sec 2710 sec 3620 sec 5400 sec top m -0.000635 -0.000613 -0.000596 -0.000615 -0.000610 A 10.545590 7.465139 6.084004 5.264048 4.310000 b 1.328924 1.328912 1.328903 1.328925 1.328918 bottom m -0.000638 -0.000665 -0.000678 -0.000723 -0.000742 A 10.545590 7.465139 6.084004 5.264048 4.310000 b 1.329343 1.329315 1.329301 1.329257 1.329238 Appendix B. Error Function Correlation Parameters (m, A and b) 212 RUN FRU6 FIELD = 0.3 T time 901 sec 1815 sec 2700 sec 3602 sec 5405 sec top m -0.000689 -0.000605 -0.000625 -0.000648 -0.000615 A 10.551440 7.434227 6.095260 5.277185 4.308006 b 1.328982 1.328907 1.328932 1.328961 1.328927 bottom m -0.000643 -0.000684 -0.000677 -0.000747 -0.000727 A 10.551440 7.434227 6.095260 5.277181 4.308000 b 1.329335 1.329295 1.329303 1.329233 1.329258 RUN FRU7 FIELD = 0.7 T time 902 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000665 -0.000627 -0.000613 -0.000618 -0.000636 A 10.545590 7.465139 6.095260 5.278650 4.310000 b 1.328959 1.328932 1.328921 1.328928 1.328948 bottom m -0.000624 -0.000665 -0.000688 -0.000682 -0.000698 A 10.545590 7.465139 6.095260 5.278650 4.310000 b 1.329355 1.329316 1.329290 1.329298 1.329285 RUN FRU5 FIELD = 0.9 T time 900 sec 1875 sec 2745 sec 3603 sec 5466 sec top m -0.000710 -0.000642 -0.000630 -0.000642 -0.000632 A 10.557301 7.314312 6.095093 5.276452 4.283900 b 1.329009 1.328855 1.328938 1.328960 1.328945 bottom m -0.000668 -0.000671 -0.000696 -0.000735 -0.000732 A 10.557301 7.314312 6.045093 5.276452 4.283900 b 1.329308 1.329307 1.329284 1.329244 1.329252 Appendix B. Error Function Correlation Parameters (m, A and b) 213 RUN FRU3 FIELD = 1.1 T time 1056 sec 1860 sec 2705 sec 3600 sec 5400 sec top m -0.000678 -0.000609 -0.000616 -0.000602 -0.000635 A 9.746354 7.343746 6.089624 5.278650 4.310000 b 1.328981 1.328918 1.328930 1.328916 1.328948 bottom m -0.000710 -0.000685 -0.000698 -0.000686 -0.000683 A 9.7463541 7.343746 6.089624 5.278650 4.310000 b 1.329269 1.329294 1.329284 1.328916 1.329304 RUN GLU7 FIELD = 0.0 T time 900 sec 1810 sec 2700 sec 3600 sec 5400 sec top m -0.000636 -0.000613 -0.000608 -0.000609 -0.000645 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333924 1.333910 1.333907 1.333904 1.333967 bottom m -0.000635 -0.000671 -0.000700 -0.000728 -0.000682 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.334156 1.334118 1.334090 1.334061 1.334111 RUN GLU8 FIELD = 0.3 T time 1023 sec 1860 sec 2700 sec 3600 sec 5400 sec top m -0.000611 -0.000615 -0.000608 -0.000595 -0.000590 A 9.902306 7.343746 6.095260 5.278650 4.310000 b 1.333904 1.333925 1.333914 1.333900 1.333895 bottom m -0.000683 -0.000672 -0.000595 -0.000693 -0.000701 A 9.902306 7.343746 6.095260 5.278650 4.310000 b 1.334106 1.334117 1.333900 1.334099 1.334088 Appendix B. Error Function Correlation Parameters (m, A and b) RUN GLU3 FIELD = 0.5 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000612 -0.000556 -0.000572 -0.000564 -0.000575 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333898 1.333848 1.333872 1.333861 1.333874 bottom m -0.000724 -0.000654 -0.000687 -0.000750 -0.000721 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.334053 1.334130 1.334098 1.334033 1.334067 RUN GLU4 FIELD = 0.7 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000547 -0.000552 -0.000543 -0.000543 -0.000541 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333835 1.333853 1.333839 1.333848 1.333841 bottom m -0.000681 -0.000682 -0.000709 -0.000691 -0.000650 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333835 1.334108 1.334081 1.334096 1.334142 RUN GLU5 FIELD = 0.9 T time 905 sec 1800 sec 2700 sec 3606 sec 5400 sec top m -0.000566 -0.000593 -0.000569 -0.000588 -0.000612 A. 10.528096 7.465139 6.095260 5.274257 4.310000 b 1.333850 1.333891 1.333865 1.333895 1.333925 bottom m -0.000671 -0.000694 -0.000709 -0.000656 -0.000696 A 10.528096 7.465139 6.095260 5.274257 4.310000 b 1.334114 1.334087 1.334075 1.334136 1.334092 Appendix B. Error Function Correlation Parameters (m, A and b) 215 RUN GLU6 FIELD = 1.1 T time 900 sec 1825 sec 2700 sec 3600 sec 5400 sec top m -0.000661 -0.000625 -0.000624 -0.000618 -0.000604 A 10.557301 7.413831 6.095260 5.278650 4.310000 b 1.333948 1.333925 1.333931 1.333921 1.333909 bottom m -0.000652 -0.000635 -0.000703 -0.000663 -0.000615 A 10.557301 7.413831 6.095260 5.278650 4.310000 b 1.334136 1.334156 1.334086 1.334030 1.334179 RUN GLA1 FIELD = 0.0 T time 900 sec 1800 sec 2704 sec 3600 sec 5400 sec top m -0.000705 -0.000624 -0.000621 -0.000643 -0.000601 A 10.557301 7.465139 6.090750 5.278650 4.310000 b 1.333582 1.333521 1.333520 1.333547 1.333497 bottom m -0.000651 -0.000711 -0.000722 -0.000707 -0.000697 A 10.557301 7.465139 6.090750 5.278650 4.310000 b 1.333791 1.333725 1.333718 1.333734 1.333747 RUN GLA5 FIELD = 0.3 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000816 -0.000670 -0.000677 -0.000675 -0.000632 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333719 1.333571 1.333578 1.333581 1.333531 bottom m -0.000628 -0.000669 -0.000695 -0.000692 -0.000685 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333812 1.333771 1.333745 1.333753 1.332766 Appendix B. Error Function Correlation Parameters (m, A and b) RUN GLA3 FIELD = 0.5 T time 900 sec 1800 sec 2700 sec 3600 sec 5404 sec top m -0.000754 -0.000674 -0.000633 -0.000644 -0.000613 A 10.557301 7.465139 6.095260 5.278650 4.308406 b 1.333642 1.333576 1.333532 1.333544 1.333512 bottom m -0.000649 -0.000710 -0.000712 -0.000684 -0.000702 A 10.557301 7.465139 6.095260 5.278650 4.308406 b 1.333792 1.333727 1.333726 1.333760 1.333742 RUN GLA4 FIELD = 0.9 T time 900 sec 1800 sec 2700 sec 3600 sec 5405 sec top m -0.000794 -0.000659 -0.000675 -0.000649 -0.000619 A 10.557301 7.465139 6.095260 5.278650 4.308006 b 1.333685 1.333757 1.333584 1.333543 1.333509 bottom m -0.000639 -0.000683 -0.000704 -0.000693 -0.000698 A 10.557301 7.465139 6.095260 5.278650 4.308006 b 1.333800 1.333757 1.333733 1.333751 1.333748 RUN GLA2 FIELD = 1.1 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000780 -0.000693 -0.000651 -0.000643 -0.000642 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333677 1.333596 1.333551 1.333550 1.333544 bottom m -0.000669 -0.000704 -0.000699 -0.000689 -0.000698 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333766 1.333733 1.333741 1.333756 1.333748 Appendix B. Error Function Correlation Parameters (m, A and b) 217 RUN SUC1 FIELD = 0.0 T time 900 sec 1620 sec 3010 sec 3686 sec 5160 sec top m -0.000749 -0.000666 -0.000648 -0.000651 -0.000640 A 10.557301 7.868947 5.772858 5.216707 4.409093 b 1.332099 1.332200 1.331995 1.332001 1.332000 bottom m -0.000549 -0.000635 -0.000646 -0.000663 -0.000657 A 10.557301 7.868947 5.772858 5.216707 4.409093 b 1.332157 1.332072 1.332058 1.332042 1.332049 RUN SUC3 FIELD = 0.3 T time 903 sec 1800 sec 3600 sec 5400 sec 9000 sec top m -0.000574 -0.000583 -0.000586 -0.000586 -0.000582 A 10.539749 7.465139 5.278654 4.310000 3.338511 b 1.331900 1.331978 1.331945 1.331929 1.331925 bottom m -0.000599 -0.000708 -0.000748 -0.000771 -0.000776 A 10.539749 7.465139 5.278654 4.310000 3.338511 b 1.332114 1.331970 1.331937 1.331916 1.331916 RUN SUC2 FIELD = 0.5 T time 900 sec 1800 sec 3600 sec 5400 sec 7200 sec top m -0.000735 -0.000671 -0.000644 -0.000618 -0.000614 A 10.557301 7.465139 5.278650 4.310000 3.732569 b 1.332080 1.332012 1.331992 1.331956 1.331953 bottom m -0.000593 -0.000657 -0.000706 -0.000707 -0.000703 A 10.557301 7.465139 5.278650 4.310000 3.732569 b 1.332115 1.332052 1.332000 1.332000 1.332000 Appendix B. Error Function Correlation Parameters (m, A and b) 218 RUN SUC5 FIELD = 0.7 T time 960 sec 1820 sec 2700 sec 3600 sec 5400 sec top m -0.000677 -0.000612 -0.000591 -0.000584 -0.000607 A 10.222062 7.424008 6.095260 5.278650 4.310000 b 1.332020 1.331942 1.331926 1.331916 1.331947 bottom m -0.000566 -0.000710 -0.000660 -0.000652 -0.000673 A 10.222062 7.424008 6.095260 5.278650 4.310000 b 1.332140 1.332072 1.332041 1.332054 1.332033 RUN SUC3 FIELD = 0.9 T time 900 sec 1800 sec 3600 sec 5400 sec 9000 sec top m -0.000704 -0.000568 -0.000670 -0.000583 -0.000554 A 10.557301 7.465139 6.095260 4.310000 3.338512 b 1.332060 1.331895 1.332593 1.331918 1.331874 bottom m -0.000591 -0.000657 -0.000713 -0.000674 -0.000702 A 10.557301 7.465139 6.095260 4.310000 3.338512 b 1.332110 1.332736 1.332037 1.332028 1.332000 RUN SUC4 FIELD = 1.1 T time 913 sec 1800 sec 3600 sec 5400 sec 9000 sec top m -0.000613 -0.000608 -0.000603 -0.000618 -0.000601 A 10.481870 7.465139 5.278650 4.310000 3.338512 b 1.331942 1.331941 1.331946 1.331968 1.331941 bottom m -0.000518 -0.000623 -0.000717 -0.000631 -0.000645 A 10.481870 7.465139 6.095260 4.310000 3.338512 b 1.332189 1.332080 1.334027 1.332076 1.332061 Appendix B, Error Function Correlation Parameters (m, A and b) 219 RUN MAL12 FIELD = 0.0 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000296 -0.000485 -0.000527 -0.000552 -0.000546 A 10.557301 7.465138 5.278650 3.732570 3.047630 b 1.221684 1.331797 1.331835 1.331826 1.331847 bottom m -0.000558 -0.000573 -0.000600 -0.000644 -0.000663 A 10.557301 7.465138 5.278650 3.732570 3.047630 b 1.332156 1.332145 1.332120 1.332120 1.332075 RUN MAL17 FIELD = 0.3 T time 900 sec 1833 sec 2710 sec 3603 sec 5556 sec top m -0.000654 -0.000646 -0.000626 -0.000630 -0.000624 A 10.557301 7.397635 6.084000 5.273526 4.249061 b 1.333323 1.333891 1.333901 1.333915 1.333908 bottom m -0.000557 -0.000646 -0.000654 -0.000654 -0.000649 A 10.557301 7.397635 6.084000 5.273526 4.249062 b 1.334245 1.334175 1.334168 1.334170 1.334174 RUN MAL14 FIELD = 0.7 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000672 -0.000608 -0.000618 -0.000594 -0.000610 A 10.557301 7.465138 6.095260 5.278650 4.310000 b 1.333933 1.333880 1.333901 1.333863 1.333893 bottom m -0.000593 -0.000637 -0.000639 -0.000661 -0.000648 A 10.557301 7.465138 6.095260 5.278650 4.310000 b 1.334230 1.334185 1.334185 1.334161 1.334177 Appendix B. Error Function Correlation Parameters (m, A and b) 220 RUN MAL18 FIELD = 0.9 T time 900 sec 1800 sec 2734 sec 3631 sec 5500 sec top m -0.000658 -0.000612 -0.000631 -0.000625 -0.000594 A 10.545590 7.465139 6.057241 5.256069 4.270638 b 1.333923 1.333892 1.333912 1.333910 1.333867 bottom m -0.000578 -0.000650 -0.000658 -0.000653 -0.000655 A 10.545590 7.465139 6.057240 5.256069 4.270638 b 1.334244 1.334170 1.334165 1.334172 1.334168 RUN MAL16 FIELD = 1.1 T time 908 sec 1800 sec 2700 sec 3603 sec 5400 sec top m -0.000644 -0.000614 -0.000617 -0.000596 -0.000602 A 10.510690 7.465139 6.095260 5.276452 4.310000 b 1.333909 1.333888 1.333902 1.333881 1.333884 bottom m -6.000572 -0.000638 -0.000652 -0.000671 -0.000647 A 10.510690 7.465139 6.095260 5.276452 4.310000 b 1.334251 1.334183 1.334170 1.334149 1.334177 RUN LAC13 FIELD = 0.0 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000782 -0.000710 -0.000687 -0.000689 -0.000669 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333783 1.333722 1.333707 1.333716 1.333682 bottom m -0.000608 -0.000671 -0.000694 -0.000716 -0.000696 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.334000 1.333938 1.333914 1.333891 1.333916 Appendix B. Error Function Correlation Parameters (m, A and b) 221 RUN LAC12 FIELD = 0.3 T time 900 sec 1800 sec 2700 sec 3600 sec 5448 sec top m -0.000755 -0.000691 -0.000666 -0.000665 -0.000660 A 10.557301 7.465139 6.095260 5.278650 4.290971 b 1.333761 1.333707 1.333682 1.333683 1.333675 bottom m -0.000633 -0.000674 -0.000706 -0.000702 -0.000710 A 10.557301 7.465139 6.095260 5.278650 4.290971 b 1.333975 1.333934 1.333900 1.333908 1.333899 RUN LAC14 FIELD = 0.5 T time 900 sec 1800 sec 2700 sec 3603 sec 5400 sec top m -0.000737 -0.000687 -0.000678 -0.000664 -0.000652 A 10.557301 7.465139 6.095260 5.276452 4.310000 b 1.333743 1.333701 1.333697 1.333626 1.333668 bottom m -0.000605 -0.000676 -0.000697 -0.000674 -0.000699 A 10.557301 7.465139 6.095260 5.276452 4.310000 b 1.334004 1.333931 1.333913 1.333938 1.333910 RUN LAC15 FIELD = 0.9 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000760 -0.000683 -0.000678 -0.000609 -0.000676 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.333764 1.333698 1.333701 1.333703 1.333706 bottom m -0.000604 -0.000671 -0.000671 -0.000690 -0.000694 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.334005 1.333937 1.333938 1.333921 1.333917 Appendix B. Error Function Correlation Parameters (m, A and b) 222 RUN LAC11 FIELD = 1.1 T time 900 sec 1810 sec 2700 sec 3600 sec 5400 sec top m -0.000800 -0.000700 -0.000682 -0.000681 -0.000658 A 10.557301 7.444488 6.095260 5.278650 4.310000 b 1.333808 1.333715 1.333701 1.333698 1.333669 bottom m -0.000620 -0.000679 -0.000701 -0.000703 -0.000699 A 10.557301 7.444488 6.095260 5.278650 4.310000 b 1.333988 1.333920 1.333907 1.333902 1.333913 RUN RAF1 FIELD = 0.0 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000648 -0.000650 -0.000647 -0.000654 -0.000641 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.332548 1.332568 1.332571 1.332577 1.332563 bottom m -0.000644 -0.000684 -0.000712 -0.000718 -0.000710 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.332807 1.332758 1.332734 1.332727 1.332736 RUN RAF3 FIELD = 0.3 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000664 -0.000660 -0.000643 -0.000654 -0.000655 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.332574 1.332580 1.332560 1.332578 1.332778 bottom m -0.000654 -0.000703 -0.000711 -0.000719 -0.000720 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.332789 1.332741 1.332735 1.332727 1.332726 Appendix B. Error Function Correlation Parameters (m, A and b) 223 RUN RAF7 FIELD = 0.5 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000671 -0.000630 -0.000647 -0.000661 -0.000657 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.332582 1.332562 1.332568 1.332585 1.332581 bottom m -0.000652 -0.000696 -0.000715 -0.000713 -0.000720 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.332792 1.332750 1.332729 1.332733 1.332726 RUN RAF6 FIELD = 0.7 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000665 -0.000675 -0.000647 -0.000663 -0.000653 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.332577 1.332595 1.332567 1.332588 1.332573 bottom m -0.000648 -0.000710 -0.000710 -0.000718 -0.000702 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.332797 1.332734 1.332736 1.332727 1.332745 RUN RAF4 FIELD = 0.9 T time 900 sec 1800 sec 2700 sec 3600 sec 5400 sec top m -0.000668 -0.000660 -0.000670 -0.000660 -0.000649 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.332578 1.332582 1.332593 1.332584 1.332570 bottom m -0.000648 -0.000706 -0.000713 -0.000711 -0.000713 A 10.557301 7.465139 6.095260 5.278650 4.310000 b 1.332796 1.332736 1.332733 1.332735 1.332734 Appendix B. Error Function Correlation Parameters (m, A and b) 224 RUN RAF2 FIELD - 1.1 T time 902 sec 1800 sec 2700 sec 3600 sec 5400 sec top m. -0.000693 -0.000660 -0.000662 -0.000659 -0.000656 A 10.545590 7.465139 6.095260 5.278650 4.310000 b 1.332602 1.332581 1.332584 1.332583 1.332579 bottom m -0.000645 -0.000696 -0.000717 -0.000709 -0.000712 A 10.545590 7.465139 6.095260 5.278650 4.310000 b 1.332801 1.332750 1.334027 1.332737 1.332735 Appendix C Raytracing Computer Program 225 Appendix C. Raytracing Computer Program IMPLICIT REAL*8(A-H,L,0-Z) C C *********************************************************** C * RAYTRACE TRACES RAYS THROUGH OPTICAL SYSTEM ACCOUNTING FOR ' C * RAY BENDING FROM REF INDEX GRADIENT AND FITTING REF INDEX ' C * PROFILE TO AN ERROR FUNCTION CORRELATION C REAL*8 M DIMENSION X0(40),F<40),IV(100),NDOBS(40),PAR(3),XFP(40), 1LDOBS(40),LREF(40),X7(40) ,NFRNG(40) ,V(400),S(40),LDCALC(40) 2,XSHIFT(40),X0CELL(40) COMMON /SET1/Y(40) ,XF(40) /SET2/RL1,TL1 ,XL1,RL2,TL2,XL2, 1Z0 Z2,Z3,Z5,Z7, NQURTZ, NAIR, NGLASS REAL*8 NAIR,NGLASS, NWATER, NQURTZ, LAMBDA, LDCALC, LDOBS 1 ^TOP/NBOTTM/NDOBS^CNXl C Q******** *DEFINE PARAMETERS* ******** * *************** C PI=3.14x27 XMAG=.2571 DELXM=.005 XMAGO=XMAG--001 LAMBDA=.6328E-4 NAIR=1.000276 NWATER=1.3340 NQURTZ=1.45709 NTIMET=0 NGLASS=1.7499 DATA M,A,B/-7.D-4,7.D0,1.33D0/ C f_;***************j^E;^j3 j^j DATA FOR RUN *************************** C READ(10,629)NTIME 1 CONTINUE ITER=0 IFLAG=1 C C*»*********SET COUNTER FOR NUMBER OF TIME STEPS DURING EXPERIMENT* NTIMET=NTIMET+1 NELMT=0 ; ISHIFT=0 IF(NTIMET.GT.NTIME)STOP C £***********$££[) TITLE AND REFERENCE REFRACTIVE INDICES* *********** C READ(10,610)NT1,NT2,NT3,NT4,NT5,NT6,NT7,NT8,NT9,NT10,NT11,NT12 READ(10,629)NPTS,NWATER,NTOP,NBOTTM,TIME IF(NTIMET.NE.1)GO TO 589 WRITE(9,610)NT1,NT2,NT3,NT4,NT5,NT6,NT7,NT8,NT9,NT10,NT11,NT12 WRITE(9,629)NTIME 589 NELMNT=NPTS DO 599 I=1,NPTS READ(10,629)NFRNG(I),X0(I) 599 CONTINUE 50 FORMAT('LREF LD03S(I) NDOBS(I) X0(I) X7(I) LLI LL2' ) 5 CONTINUE NELUP =0 XMAGO=XMAG DO 1000 1=1,NELMNT 610 FORMAT(20A4) Appendix C. Ra.ytia.cing Computer Program 227 c C******"CALCULATIONS FOR TOP HALF OF CELL ARE PERFORMED FIRST ASSUMING C"******NO BENDING IN REFRACTIVE INDEX GRADIENT**""****************** C 629 FORMAT(I5,4G10.5) NDOBS(l)=NTOP XOCELL(I)=X0(I)/XMAG IF(XOCELL(I) .GT.0.)NELUP=NELUP+1 L12=NQURTZ*(Z2-Z1) CALL RTRACE(XOCELL(I),0.DO,REFOPL,X7I) X7(I)=X7I LREF(I)=NWATER*(Z1-Z0)+REFOPL+L12 C C*******SUM ALL UNDEFLECTED PATH LENGTHS*********""**"* C IF(I.EQ.l)LREFO=LREF(I) C C**»****USING OPTICAL PATH LENGTHS FIND INITIAL REF INDEX PROFILE**** C*******ASSUMING NO RAY BENDING THROUGH REF INDEX GRADIENT******* C NDOBS(I)=NTOP + (LAMBDA/(Z1-Z0))*(NFRNG(I)-1) NDOBS(l)=NTOP 60 FORMAT(1X,7F10.7) 1000 CONTINUE XMAG=X7(1)/XOCELL(1) IF(DABS(XMAG-XMAGO).GT..00001)GO TO 5 C C *******INITIALIZE BOTTOMO HALF OF CELL************************ C ISTART=NELUP+1 DO 75 I=ISTART/NELMNT •NDOBS(I)=NBOTTM-(LAMBDA/(Z1-Z0))*(NFRNG(NELMNT)-NFRNG(I)) 75 NDOBS(NELMNT)=NBOTTM PRINT 610/NTi/NT2,NT3/NT4,NT5/NT6,NT7/NT8,NT9/NTI0,NTll,NT12 PRINT 55 DO 76 1=1,NELMNT 76 PRINT 56, NDOBS(I),NFRNG(I),X0CELL(I),X7(I),X0(I) 55 FORMAT(' NINITIAL FRINGE NO. XCELL INITIAL XFOCAL PL. INIT',//) 56 FORMAT(F10.6,I8,3(3X,F10.6) ) RMS=0.0 • NELMAX=NELUP PAR(1)=M PAR(2)=A PAR (3) =B C C******IFLAG IS A FLAG WHICH SIGNALS WHETHER CALCULATIONS ARE BEING PERFORMED C******ON UPPER OR LOWER HALF OF CELL, IFLAG=1 FOR UPPER, -1 FOR LOWER***** C 10 IF(IFLAG.LT.0)NELMAX=NELMNT ICONV=l C*******FIT REF INDEX PROFILE TO ERROR FUNCTION************** ITER=ITER+1 IF(ITER.LT.10.)GO TO 57 PRINT 570 570 FORMAT(//,IX,"PROGRAM FAILED TO CONVERGE IN 1C PASSES',//) GO TO 1100 C C***"********START WITH TOP HALF OF CELL, THEN DO BOTTOM HALF***** C***«*******«SET FLAGS FOR TOP OR BOTTOM************************** C Appendix C. Raytracing Computer Program 228 57 NF=NELMAX IF(IFLAG.LT.0)NELHAX=NELMNT-NELUP IF(IFLAG.LT.O)ISHIFT=NELUP DO 300 IC=1,NELMAX Y(IC)=NDOBS(IC+ISHTFT) 300 XF(IC)=XOCELL(IC+ISHIFT) C C*****CALL LINEARIZED ERROR FUNCTION FITTING ROUTINE CALL ERRFIT(PAR,NELMAX,TIME) C 3000 FORMAT('RETURN CODE =',15) M = PAR(l) A = PAR(2) B = PAR(3) ISTART=ISHIFT+1 IFINAL=ISHIFT+NELUP IF(IFLAG.LT.O)IFINAL=NELMNT DO 500 I=ISTART,IFINAL C C****«*.******PERF0RM REFRACTIVE INDEX GRADIENT CALCS AND RAYBENDING*** C IF(ITER.EQ.1)NDOLD=NDOBS(I) DNDOBS = 2 . *M*A*DEXP(-(A*A*X0CELL(I)*X0CELL(I)))/DSQRT(PI) 9999 DX1X0 = DNDOBS*(21-Z0) *(Z1-Z0) /(2.*NDOBS(I) ) X1=X0CELL(I)+ DX1X0 NX0 = NDOBS(I) - .5*DND03S*(DX1X0) S(I) = (DX1X0*DX1X0 + (Z1-Z0)"(Z1-Z0))**.5 SINALP = DNDOBS * S(I)/NDOBS(I) L01=NDOBS(I) *S(I) NX1 = NDOBS(I) + DND03S*DX1XO L12 = NQURTZ*(Z2-Z1) /DCOS(DARSIN((NQURTZ/NX1)"SINALP)) BETA1 = DARSIN(SINALP*NQURTZ/NX1) X2=X1+(Z2-Z1)*DTAN(BETA1) C C***********PERFORM RAYTRACING THROUGH REST OF SYSTEM****************** C CALL RTRACE(X2,BETA1,OPTPL,XFINAL) LDCALC(I)=OPTPL +112+L01 XFP(I)=XFINAL C ' C********CHECK ERRORS BETWEEN BENT RAY POSITION ON FOCAL PLANE AND C********ACTUAL FRINGE AND CORRECT REFRACE INDEX PROFILE THEN C********STARTING ITERATIONS ALL OVER AGAIN UNTIL CONVERGENCE IS OK**** C DELXFP=X0(I)-XFP(I) XOCELL(I)=X0CELL(I)+DELXFP/XMAG IF(DABS(DELXFP).GT.l.E-6)ICONV=-l 500 CONTINUE IF(ICONV.GT.0)GO TO 1100 GO TO 10 1100 CONTINUE IF(IFLAG.GT.0)PRIKT 5051 IF(IFLAG.LT.O)PRINT 5052 IF(IFLAG.GT.O) C C******WRITE CORRELATION PARAMETERS ON UNIT 9 IF CONVERGENCE HAS BEEN ACHIEVED"* C 1WRITE(9,5045)TIME,M, A, B,X0CELL(IFINAL) IF(IFLAG.LT.O)WRITE(9, 5045) M,A,3,X0CELL(IFINAL) 5045 F0RMAT(1X,5E15.8) Appendix C. Ra.ytra.cing Computer Program 229 PRINT 5050, M,A,B PRINT 5060 DO 1105 I=ISTART,IFINAL DELXFP=X0(I)-XFP(I) 1105 PRINT 5000,LDCALC(I),LD0BS(I) ,S(I),DELXFP,NDOBS(I),X0CELL(I), lXFP(I) IF(IFLAG.LT.O)GO TO 1200 IFLAG=-1 ITER=0 GO TO 10 1200 GO TO 1 5060 FORMAT(//,4X,'LDCALC' , IX, ' ',5X,'S',5X,'DELXFP ',5X, 1 'NDOBS',5X,'X0CELL',5X,'XFP', //) 5000 FORMAT(1X,8(1X,F10.6) ) 5051 FORMAT(//,IX,'TOP HALF OF CELL',//) 5052 FORMAT(//,IX,'BOTTOM HALF OF CELL',//) 5050 FORMAT(//,'M = ',F9.6,'A = ',F9.6,'B = ',F9.6) END BLOCK DATA C C********INITIALIZE COMMON BLOCKS WITH EQUIPMENT GEOMETRICAL PARAMETERS* * * * * C IMPLICIT REAL*8(A-H,N,0-Z) COMMON /SET1/Y<40) ,XF(40) /SET2/RL1,TL1,XL1,RL2,TL2,XL2, 1Z0,Zl,Z2,Z3,Z5,Z7,NQURTZ,NAIR,NGLASS DATA RL1,TL1,XL1/99.29,1.0,5.0/ DATA RL2,TL2,XL2/22.4,1.0,1.5/ DATA Z0,Zl,Z2,Z3,Z5,Z7/0. ,1.,2.,58.2,110.,126.6/ END SUBROUTINE ERRFIT(PAR,NEL,T) C C****« "SUBROUTINE ERRFIT FITS REF INDEX PROFILE TO ERROR FUNCTIN CORRELATION C********N = M * ERF(A*X) + B TO FIND PARAMETERS M,A,AND B FOR BEST FIT C IMPLICIT REAL*8(A-H,0-Z) DIMENSION XLIN(40),YFIT(40),WT(40),E1(2),E2(2),P1(2),P2(2) 1,PAR(3) COMMON /SET1/ Y(40),XF(40) /ICELL/IFLAG EXTERNAL AUX C . C*****SUBROUTINE ERRFIT FITS REF INDEX PROFILE TO AN ERROR FUNCTION C*****BY FIRST LINEARIZING ERR FCT THEN FITTING WITH LINEAR CURVE FIT C*****INITIALIZE PARAMETERS C IFLAG=1 IF(XF(2)-LT.0.D0)IFLAG=-1 DO 1 1=1,NEL WT(I)=1.D0 C C«««««IF THE DEFLECTED X VALUE NEAR MEMBRANE IS IN A LOCATION THROUGH C*****THE MEMBRANE THIS POIKT IS IGNORED FOR CORRELATION CALCULATIONS** C 1 IF(XF(3) .LT.O.DO.AND.XF(l) -GT.0.DO)WT(1)=1.D-6 IF(XF(2) .LT.O.DO)WT(NEL) =10.DO IF(XF(2).GT.0.D0)WT(1)=1.DO C C******THIS WEIGHTS THE ENDPOINTS 10 TIKES OTHER POINTS***"" C ICON=l ITER=0 Appendix C. Raytracing Computer Program 230 D=5.D-6 CUT =l.D-9/DSQRT(D*T) TAU = (l.+SQRT<5.))/2.0 ALOW=O.DO AHIGH = 3./DSQRT(D*T) AT2=(AHIGH-ALOW)/TAU +ALOW AT1 =(AHIGH-AT2+ALOW) EPS = 1.D-10 PI(1)=O.D0 PI(2)=O.D0 P2(l) =P1(1) P2(2)=P1(2) DO 100 1=1,NEL 100 XLIN(I)=DERF(AT1*XF(I)) C C******DLQF IS UBC CURVE FITTING LINEAR LEAST-SQUARES CURVE FITTING ROUTINE** C CALL DLQF(XLIN,Y,YFIT,WT,E1,E2,P1,1.D0,NEL,2,-5,ND,EPS,AUX) CALL ERRCAL(Y/YFIT,ERROR1,NEL) DO 101 1=1,NEL 101 XLIN(I)=DERF(AT2*XF(I)) CALL DLQF(XLIN,Y,YFIT/WT/E1/E2/P2,0.D0/NEL,2,-9,ND,EPS,AUX) CALL ERRCAL(Y,YFIT,ERROR2,NEL) 10 IF(ERROR1-LE.ERROR2)GO TO 18 11 AHIGH=AT2 IF(AHIGH-ALOW.LE.CUT)GO TO 21 AL1=ATI-ALOW IF(AHIGH-AT1.LT.AT1-ALOW)GO TO 15 12 AT2=AT1 ATI=AHIGH-(AT1-ALOW) ERROR2 = ERROR1 P2(1)=P1(1) P2(2)=P1(2) ITER=ITER+1 112 DO 102 1=1,NEL 102 XLIN(I)=DERF(AT1*XF(I) ) CALL DLQF(XLIN,Y,YFIT/WT,El,E2,Pl,l.D0,NEL/2,-5,ND,EPS,AUX) CALL ERRCAL(Y,YFIT,ERROR1,NEL) GO TO 10 15 AT2=ALOW+(AHIGH-AT1) 115 DO 16 1=1,NEL 16 XLIN(I)=DERF(AT2*XF(I) ) CALL DLQF(XLIN,Y,YFIT,WT,El,E2,P2,1.DO,NEL,2,-5,ND,EPS,AUX) CALL ERRCAL(Y,YFIT,ERROR2,NEL) GO TO 10 18 ALOW=ATl IF(AT2-ALOW.LE.CUT)GO TO 21 IF(AT2-ALOW.LT.AHIGH-AT2) GO TO 20 19 ATI=AT2 AT2=ALOW+(AHIGH-AT1) ERRORl=ERROR2 PI (1) =P2(1) PI(2)=P2 (2) GO TO 115 20 AT1=AHIGH-(AT2-ALOW) GO TO 112 21 CONTINUE 5010 FORMAT(IX,6(IX,F15.6)) IF(ITER.GT.20)ICON=-l IF(ICON.LT.0)GO TO 200 Appendix C. Raytracing Computer Program 231 ERROR-ERROR1 IF(ERROR1•GT.ERROR2)ERROR=ERROR2 PAR(2)=AT1 IF(ERROR1.GT.ERROR2)PAR(2)=AT2 PAR(3)=P1(2) IF(ERR0R1.GT.ERROR2)PAR(3) =P2(2) PAR(l)=P1(1) IF(ERROR1.GT.ERROR2)PAR(l)=P2(1) RETURN 200 PRINT 5000 5000 FORMAT(IX,//,'ERRFIT FAILED TO CONVERGE IN 20 PASSES',//) RETURN END SUBROUTINE ERRCAL(Y,YFIT,ERROR,N) C C*******SUBROUTINE ERRCAL CALCULATES RMS ERROR IN FITTED ERROR/FUNCTION*** C IMPLICIT REAL*8(A-H,0-Z) DIMENSION Y(40),YFIT(40) ERROR=0.0 DO 1 1=1,N 1 ERROR=ERROR+(Y(I)-YFIT(I))*(Y(I)-YFIT(I)) RETURN END FUNCTION AUX(P,D,XLIN,L) C C******FUNCTIN AUX-CALCULATES PARTIAL DERIVATIVES FOR LINEAR CURVE FITTING C******ROUTINE USED TO FIT ERROR FUNCTION"**********"*****"***** C IMPLICIT REAL*8(A-H,0-Z) COMMON /ICELL/IFLAG DIMENSION P(l),D(1) D(l)=XLIN D(2)=1.D0 AUX=P(1)*XLIN+P(2) RETURN END SUBROUTINE LNSTRC(XLI,TLI,RLI,XIN,ALPHA1,XOUT,BETA,OPTPL,ZD IMPLICIT REAL*8(A-H,0-Z) REAL*8 NAIR,NGLASS C C******SUBROUTINE TO CALCULATE OPTICAL PATH LENGTH THROUGH PLANO-CONVEX*** C******LENS USING GEOMETRICAL OPTICAL RAY TRACING************************ C******TL1=LENS THICKNESS AT THINNEST POSITION********************** C******RLI=RADIUS OF CURVATURE OF LENS****************************** C******ZL=Z DISTANCE OF RAY THROUGH LENS*************************** C******XLI=OUTSIDE RADIUS OF LENS*********************** C******XINT=INITIAL X COORDINATE OF RAY ENTERING LENS***** C******XOUT=FINAL X COORDINATE OF RAY LEAVING LENS****** C******ALPHA1=ANGLE OF ENTERING RAY NORMAL TO FLAT LENS SURFACE**** C******ALPHA2=ANGLE OF RAY THROUGH'LENS MATERIAL************ C******BETA=EXIT ANGLE OF RAY LEAVING LENS*************** C******THETA=ANGLE RAY CUTS WITH TANGENT TO CURVED SURFACE INTERNAL TO LENS"* C N=0 NAIR=1.000276 NGLASS=1.57 IF(XIN.LT.0-)XLI=-XLI THETAI=DARSIN(XLI/RLI) ZL=TLI Appendix C. Raytracing Computer Program 232 ZLO=ZL 100 N=N+1 ALPHA2 = DARSIN((NAIR/NGLASS)*DSIN(ALPHA1)) XOUT=XIN + ZL*DTAN(ALPHA2) THETA=DARSIN(XOUT/RLI) ZL=RLI*(DCOS(THETA) -DCOS{THETAI)) + TLI OPTPL=(NGLASS/DCOS(ALPHA2))*ZL IF(N.GT.20)GO TO 150 IF(DABS(ZL-ZLO).LT.l.E-8)GO TO 150 55 FORMAT(IX,'N XOUT ALPHA2 THETA ZL ZLO OPTPL *) 50 FORMAT(1X,I2,6F10.6) ZLO=ZL GO TO 100 150 IF(N.GT.20)WRITE(6,40) 40 FORMAT('ERROR IN LNSTRC-DID NOT CONVERGE IN 20 PASSES*) ALPHAP=ALPHA2-THETA BETAP=DARSIN((NGLASS/NAIR)*DSIN(ALPHAP)) BETA = BETAP+THETA 3000 FORMAT('BETAP = ',FI0.6,'BETA = *,F10.6) RETURN END SUBROUTINE RTRACE(XIN,ALPKIN,OPTPL,XFINAL) C C********SUBROUTINE RTRACE TRACES A RAY THROUGH OPTICAL SYSTEM FROM C********DIFFUSION CELL EXIT TO FOCAL PLANE****************"*"*** C IMPLICIT REAL*8(A-H,L,0-Z) COMMON/SET2/RL1,TLI,XL1,RL2,TL2,XL2,Z0,Zl,Z2,Z3,Z5,Z7, 1NQURTZ,NAIR,NGLASS REAL*8 NAIR,NGLASS,NQURTZ X2 = XIN + (Z2-Z1) *DTAN(ALPHIN) ALPHA3 = DARSIN((NAIR/NQURTZ)*DSIN(BETA1)) L23 = NAIR*(Z3-Z2)/DCOS(ALPHA3) X3 = X2 + (Z3-Z2)*DTAN(ALPHA3) C C*****LENS 1 RAY TRACING************************ C CALL LNSTRC(XL1,TLI,RL1,X3,ALPHAS,X4,BETA4,LL1,ZL1) C Q** * * *LENS 1 TO LENS 2*******^****************** C Z4=Z3+ZL1 X5 = X4+ (Z5-Z4)*DTAN(BETA4) ALPHA5 = BETA4 L45 = NAIR*(Z5-Z4) /DCOS(ALPHA5) C Q** * **LENS 2 RAY TRACING* * ********************** c CALL LNSTRC(XL2,TL2,RL2,X5,ALPHAS,X6,BETA6,LL2,ZL2) C C*****LENS 2 TO FOCAL PLANE RAY TRACING********* C Z6=Z5+ZL2 XFINAL = X6 + (Z7-Z6)*DTAN(BETA6) L6FP = NAIR*(Z7-Z6)/DCOS(BETA6) OPTPL = L23+LL1+L45+LL2+L6FF RETURN END Appendix D Mass Flux and Diffusivity Calculation Computer Program 233 Appendix D. Mass Flux and Diffusivity Calculation Computer Program 234 IMPLICIT REAL*8(A-K,0-Z) C C * DIFFCALC CALCULATES MASS FLUXES, CONCENTRATION PROFILES AND * C * CONCENTRATION INTEGRALS FROM REFRACTIVE INDEX CORRELATIONS * C * EVALUATED FROM PROGRAM RAYTRACE AND FIT TO ERROR FUNCTION * C * CORRELATION * Q It*******************************-******-*-*******************-******** C REAL*8 NAXML,NAXMEM,NAXMU,NBXML,NBXMU DIMENSION CXDIFF(IO),CXML(10),TIME(10),AU(10),BU(10),PMU(10), 1AL(10),CX(10),CXMLC(10),BL(10),PML(10),PARL(4) 1,CLSGAR(10),CWATER(10) ,XSUGAR(10) COMMON X(10),Y(10) C C********READ DATA FILE FOR PROFILE CORRELATION PARAMETERS******* C** * * **READ TITLE** ****************************************** * ************ READ(10,500)NT1,NT2,NT3,NT4,NT5,NT6,NT7,NT8,NT9,NT10 C C******READ NUMBER OF TIMES DURING RUN************************************ READ(10,501)NTIME 500 FORMAT (20A4) 501 FORMAT(15) C C**«****«**SET PARAMETER VALUES FOR CALCULATIONS************** C ALPHA=1.DO/49.01621D0 XBOTTM=-1.8D0 BETA=-1.331313D0/49.01621D0 DXM=.0050 GAKMA=-11.6874423D0 XML=0.D0 EPSI=0.055351201D0 XTOP=XML PI=3.1415927D0 C C***.******DELTX IS X INCREMENTS FOR WHICH CALCULATIONS ARE PERFORMED IN C**********X POSITION IN DIFFUSION CELL, STARTING AT MEMBRANE SURFACE C******»***AND WORKING DOWN EIGHT VALUES TO -0.16 CM.********************* C NINT=8.0 DELTX=(-.10-XTOP)/5.0 PRINT 555 555 FORMAT(IX,//,'DIFFCALC WITH CONCENTRATION AT TIME=0',//) PRINT 500, NT1,NT2,NT3,NT4,NT5,NT6,NT7,NT8,NT9,NT10 DO 100 I=1,NTIME C C*******READ VALUES m,A,b FOR REFRACTIVE INDEX PROFILE CORRELATION PARAMETERS C*******^T EACH TIME VALUE*************************************************** C READ(10,502)TIME(I),PMU(I) ,AU(I),BU(I),XUF READ(10,502)PML(I),AL(I) ,BL(I),XLF 100 X(I+1)=TIME(I) X(l)=1.DO C C*********PERFORM PROFILES INTEGRALS FOR EACH TIME FROM CONSTANTS**** C DO 105 INT=1,NINT DO 106 I =1,NTIME C Appendix D. Mass Flux and Diffusivity Calculation Computer Program C*****~FUNCTI0N CINTGL CALCULATES CONCENTRATION INTEGRAL IN CELL*************** C CXML(I) = CINTGL(PMLd) ,AL<I) ,BL(I) ,XTOP) -1CINTGL(PML(I),AL(I),BL(I),XBOTTM) C C*******EVALUATE CONCENTRATION AT BOTH SURFACES OF MEMBRANE FOR MEMBRANE Q*******DIFFUSIVITY CALCULATIONS* ** * ****************************************** C CX(I+1)=CXML(I) CX(1> =(XTOP-XBOTTM)*(ALPHA*1.332716+BETA) 106 CONTINUE C*********FIND CONSTANTS FOR FITTING CONCENTRATION PROFILE VS TIME**** CALL EQUFIT(CX,PARL,NTIME) PRINT 503,XTOP C Q****** ** *FIND FLUXES* ************* ****** C PRINT 5000 PRINT 502, PARL (1) , PARL (2) , PARL (3) , PARL (4) PRINT 5002 DO 101 1=1,NTIME C C*******FUNCTION DCDT SOLVES FOR PARTIAL DERIVATIVE OF CONCENTRATION INTEGRAL C*******WRT TO TIME TO DETERMINE MASS FLUX IN CELL AT ANY VALUE X, TIME***** C NAXML = DCDT(PARL,TIME(I)) C C*******NAX IS SUCROSE MOLAR FLUX, NBX IS WATER MOLAR FLUX**************** C NAXMEM=NAXML NBXML = GAMMA*NAXML CXMLC(I)=PARL(1)+PARL(2)*DERF(PARL(4)/TIME(I)**.5) 1+PARL(3)*TIME(I)**.5*DEXP(-PARL(4)*PARL(4)/TIME(I)) CLSGAR(I) = ALPHA*(PML(I)*DERF(AL(I)*XTOP)+BL(I))+BETA CWATER(I) = GAMMA*CLSGAR(I) + EPSI XSUGAR(I)=CLSGAR(I)/(CWATER(I)+CLSGAR(I) ) DNDY=PML(I)*AL(I)*DEXP(-AL(I)*AL(I)*XTOP*XTOP)/PI**.5 DSUCDY=ALPHA*DNDY C C*********FIND SUCROSE CONCENTRATION AT MEMBRANE SURFACES FOR C*********DIFFUSIVTIY CALCULATIONS*************** C CLMEMB=ALPHA*(PML(I)*DERF(AL(I)*XTOP)+BL(I))+BETA CUMEMB=ALPHA*(PMU(I)*DERF(-AU(I)*XTOP) +BU(I))+BETA DCDMEM=(CUMEMB-CLMEMB)/(DXM+DABS(2-D0*XTOP)) DMEMB=NAXMEM/DCDMEM CDELXA=(1.0-CLSGAR(I)*(1.0+GAMMA)/((1.0+GAMMA) 1*CLSGAR(I)+EPSI))*DSUCDY C C*******CALCULATE FREE DIFFUSION COEFFICIENTS AND MEMBRANE DIFFUSION COEFF.** C DNOBLK = -NAXML/CDELXA D3ULK=(-NAXML+XSUGAR(I)*(NAXML+N3XKL) ) /CDELXA CXDIFF(I)=((CXML(I)-CXMLC(I))/CXML(I))*100.0 FAREA = DMEMB/DBULK PRINT 5001,TIME(I),CLSGAR(I),NAXML,CXML(I),CXMLC(I), 1FAREA,DNOBLK,DBULK 101 CONTINUE 502 F0RMAT(1X,9E15.8) 5001 F0RMAT(1X,F8.2,8(1X,E15.8)) Appendix D. Mass Flux and Diffusivity Calculation Computer Program 236 5002 FORMAT(//,IX,'TIME',8X,'SUC CONC',9X,'SUC FLUX',9X,'OBS CINTGL' 1,4X,'CALC CONCINTGL',7X,'FAREA',10X,*DNOBULK',10X,'DBULK',//) 5000 FORMAT(//,4X, 'APARAMETER' ,4X, 'BPARAMETER' , 4X, 'CPARAMETER' , 14X,'DPARAMETER',//) 503 FORMAT(IX,//,'INTEGRATION LIMITS -1.80 CM TO ',F10.7,//) XTOP =XTOP +DELTX 105 CONTINUE STOP END FUNCTION CINTGL(M,A,B,X) C C*********FUNCTION CINTGL CALCULATES CONCENTRATION INTEGRALS FROM C*********ERROR FUNCTION PARAMETERS DETERMINED FROM RAYTRACE * * * * C IMPLICIT REAL*8(A-H,0-Z) REAL*8 M ALPHA =1.DO/49.01621D0 BETA=-1.331313D0/49.01621D0 CINTGL = ALPHA*((M/A)*(A*X*DERF(A*X) + 1.56418958 * DEXP(-A*A*X*X)) + B*X) * BETA*X RETURN END FUNCTION DCDT(P,T) C C********FUNCTION DCDT FINDS PARTIAL DERIVATIVE WITH RESPECT TO TIME C********FOR PROFILE CONCENTRATIN INTEGRALS ************************ C IMPLICIT REAL*8(A-H,0-Z) DIMENSION P(4) DCDT = (P(3)/(2.*T**.5) + P(3) *P(4>-?(4)/(T**1.5) 1 -P(2)*.56418958*P(4)/(T**1.5) ) *DEX?(-P(4)*P(4)/T) RETURN END SUBROUTINE EQUFIT(C,PAR,NTIME) C C*********EQUFIT FITS CONCENTRATION PROFILES AT TIME INTERVALS TO ERROR FUNCT.* C Q*********p£RQ)_^************** C*********PAR(2) =£** *********** * Q*********p^J^ =£************** Q*********p^R(4)=D************** C IMPLICIT REAL*8(A-H,0-Z) COMMON X(10),Y(10) DIMENSION IV(64),C(10),V(400) ,PAR(4) EXTERNAL CALCR,CALCJ CALL DFALT(IV,V) IV(23)=1 N=NTIME+1 M = 4 V(29)=1-D-20 V(40)=l.D-20 V(42)=l.D-20 PAR(l)=6.7D-5 PAR(2)=1-9D-6 PAR(3)=-4.1D-8 PAR(4)=25.6D0 DO 100 1 = 1,N 100 Y(I)=C(I) Appendix D. Mass Flux and Diffusivity Calculation Computer Program 237 c C******NL2S0L IS UBC CURVE FITTING NON-LINEAR LEAST SQUARES FITTING ROUTINE**** C CALL NL2SOL< N , M,PAR,CALCR,CALCJ, IV , V , IPARM,RPARM,FPARM) RETURN END SUBROUTINE CALCR(N,M,PAR,NF,R,IPARM,RPARM,FPARM) C C*********CALCR CALCULATES ERROR FUNCTION EQU. AND RESIDUALS***""**" C IMPLICIT REAL*8(A-H,0-2) COMMON X(10),Y(10) DIMENSION PAR(4),R(N) DO 100 1 = 1,N FX=PAR(1)+PAR(2)*DERF(PAR(4)/(X(I) **.5) ) 1 + PAR(3)*(X(I)**-5)*DEXP(-PAR(4)*PAR(4) /X(I)) 100 R(I)=FX-Y(I) RETURN END SUBROUTINE CALCJ(N,M,PAR,NF,D,IPARM,RPARM,FPARM) C C*********CALCJ CALCULATES PARTIAL DERIVITIVES OF ERROR FUNCT. CORRELATION** C IMPLICIT REAL*8(A-H,0-Z) COMMON X(10),Y(10) DIMENSION PAR(4),D(N,4) PI=3.1415927D0 DO 100 1=1,N D(I,1)=1.D0 D(I,2) =DERF(PAR(4)/(X(I) **.5) ) D(I,3) = (X(I)**.5)*DEXP(-PAR(4)*PAR(4) /X(I)) D(I,4) =DEXP(-PAR(4)*PAR(4)/X(I))*2./(X(I)**.5) 1*(PAR(2)/PI**.5 - PAR(4)*PAR(3)) 100 CONTINUE RETURN END 

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