@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix skos: . vivo:departmentOrSchool "Applied Science, Faculty of"@en, "Chemical and Biological Engineering, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Atwal, Virinder S."@en ; dcterms:issued "2011-01-10T23:01:58Z"@en, "1990"@en ; vivo:relatedDegree "Doctor of Philosophy - PhD"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description """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(+)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 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."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/30560?expand=metadata"@en ; skos:note "AN EXPERIMENTAL STUDY OF STATIC MAGNETIC FIELD EFFECT ON FREE DIFFUSION OF SACCHARIDES IN AQUEOUS SOLUTION By VIRINDER S. A T W A L B. Eng. University of Bradford, England 1981 M . Eng. University of British Columbia, 1984 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF D O C T O R OF PH ILOSOPHY in T H E FACULTY OF G R A D U A T E STUDIES CHEMICAL ENGINEERING We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA May 1990 © VIRINDER S. A T W A L , 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 C h e m i c a l E n g i n e e r i n g The University of British Columbia Vancouver, Canada D a t e J une 2 0 , 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 vii List of Figures viii ACKNOWLEDGEMENT xi 1 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 Diffusion 33 3.2 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 45 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 4.1.1 Laser and Collimating Lens Assembly 69 4.1.2 Masking Slit Assembly 69 4.1.3 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 Control 79 4.1.8 Electromagnet 83 4.1.9 Focal Plane Camera and Microscope 83 4.1.10 Membrane and Saccharide Solutions 86 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 Data 95 5.2 Calculation of Refractive Index Profiles 97 5.3 Mass Fluxes and Diffusivity Calculations 101 6 RESULTS 111 7 CONCLUSIONS AND RECOMMENDATIONS 125 NOMENCLATURE 130 BIBLIOGRAPHY 134 APPENDICES 148 A Interference Fringe Pattern Data 148 B Error Function Correlation Parameters (m, A and b) 205 C Raytracing Computer Program 225 D 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 5.52 96 5.7 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] 24 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] 29 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 V l l l 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 73 4.28 Plano-convex lens and mount [63] 75 ix 4.29 Cylinder lens and mount [63] 76 4.30 Modified diffusion cell holder 78 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 N M R , electron paramagnetic resonance ( E P R ) , 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 N M R 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 ( A C ) 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 D C 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 N a t u r a l a n d M a n - m a d e M a g n e t i c 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 M e c h a n i s m s of I n t e r a c t i o n 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 D N A 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 D N A [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]. A l l 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, SF 6 , CH3F, CH3, C N , CRF3, Njf 3 . 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 L iC l 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. Al l 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 K C l - 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. Al l 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 C H O H O H CH<>H « K + ) - C H X X T I Webvtk H -H -C H O - O H - O H C H 3 O H CK—»-£rythros< H O -H -C H O -H -OH C H ; O M n-< — V-Thre<»sr H -H -H -C H O - O H - O H - O H C H i O H o-(-(-Ribos« H O -H -H -C H O - O H - O H C H : O H o^—)-Arab-XY<<** H O -H O -H -C H O - H - H - O H C I U ) H IM — M-vxosc H -H -H -H -C H O - O H - O H - O H - O H C H . O H o-<+»-A(los< C H O H O -H -H -H -- O H - O H - O H C H , O H o-< + >-AI«roK H -H O -H -H -C H O - O H - H - O H - O H C H ^ O H C H O H0-H O -H -H -- H - H - O H - O H C H : O H o-<+VMarm<»se H -H -H O -H -CHO - O H - O H - H - O H C H . O H o-(-K;«rfose H O -H -H O -H -CIK) - H - O H - H - O H t>-(-|-l H—C—OH H—C—OH CH.OH 2 - D c o x y - i > - r i b o s e C H , O H CH>OH I -M>ih vdroxy prop;* mine H -H -Cl 1.-4)11 O - O H - O H (\"H ; 0 ! l o-< + )-Ril>u<<« OI-OII I •Oi l CI I-Oil »-< - >-Er\\1hralo5< C H . O H =o -OH - O H - O H H O -H -H -C H : O H =0 - H -OH CH ;OI< o-<+)-Psicos« - O H C H ; O H (M-)-Fr>*ctOK H -H0-H -aijon =o H O -H - O H C H ; O H D-< +• (-Xylulose CH.-OH 0 - O H - H - O H H O -H O -H -CH-OII =0 - H - H - O H C H - O H o-i + hSorbose C H . O H 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--glucopyranose C H 2 O H O H a-glucopyranose C H 2 O H o H O H O C H , H O H O H O /}-fructofuranose H O C H j O H C H 2 O H O H sucrose /i-D-fructofuranosyl o-o-glucopyranoside Figure 2.5: Structure of maltose and sucrose [92] Chapter 2. LITERATURE REVIEW l a c t o s e 4-<7-(/!-D-galactopyranosyl)-D-gIucopyranose C r U O H 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 x l O 6 e-OH cm 3 /mol cm 2/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 D 0 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> N A V (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 = X A ( N A + N B ) - c D A B ^ (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 • - k T 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 + N B m ) - A F D A B - ~ ± (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 = R T A C (3.18) where R is the universal gas constant, T is absolute temperature and A C 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 - 0.95 7 E (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 A C = 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 D m = 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 + 4 t t / (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 + 4 t t X (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 a r e a n 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 1 0 - 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 \\ 1 S 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 | | - X x ) ^ (3-25) where k is the Boltzmann constant, T the absolute temperature, H the applied magnetic field strength, and x\\\\, Xi a r e 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 ( X l l - x ± ) 1 j ^ (3.26) The degree of orientation has been measured by observing the magnetically induced birefringence [149] An = n ( | - n x = 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 C M is the Cotton-Mouton constant. This technique has been used to observe the orientation of some biological macro-molecules such as D N A [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 T h e T h e o r y o f I n t e r f e r o m e t r y 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 D r e l d < D C Dif fus ion Reid D F , e l d > 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, B Y - 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 t i m e t i m e t i m e Vibrations at X—constructive interference. (b) t i m e t i m e (i i i) r e s u l t a n t _ _ _ 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 A 0 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 A B , 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, L I , 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 L I , 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 n 2 is the refractive index in diffusion side compartment (DC) ( where n 2 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) - n a] (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 A n / = 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 nxs inQi = ri2 sin 0:2, where n x and n.2 are the refractive indices for two different media and a i , 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) = n 0 + 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 L 0 , ; ( i.e. Lo,{ = n s = n (zj — z 0) ) into equation 3.46 gives [152] (ajj - s0) = x = r dn dx (zi - zo)2 (3.47) where (z\\ — z 0) 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 + (z x - z 0 ) 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 \"a i r - 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 . 0 t o l . l T ) 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 - 0 F x lO* erne's c tr. ^ - O u X l O - 6 cm!;s 3.0 Dill b 2.0 1.0 - <-c i < i i i i i i i i I I 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 L U cjl-O . t r L U C O c CD o C o co 3 £ £ CO £ 0 b o x o § CO C .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. C h a p t e r 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, cm 2/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 x l C T 7 Intercept x lO 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. 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Physics of Membrane Transport, General Technical Ser-vices, Inc., Upper Darby, Pa. (1973). O'Brien, R.N. , Santhanam, K.S.V. , Magnetic Field Effects on the Diffusion Layer at Vertical Electrodes During Electrodeposition, J. Electrochem. Soc, 129, N6 1266 (1982). Lielmezs, J . , Aleman, H., Weak Transverse Magnetic field Effect on the Viscosity of Water at Several Temperatures, Thermochim. Acta, 21, 225 (1977). Forgacs, C , Liebovitz, J. , O'Brien, R.N. , Spiegler, K.S., fnterferometric Study of Concentration Profiles in Solutions Near Membrane Interfaces, Electrochim, Acta, 20, 555 (1975). James, D.W., Frost, R.L., Structure of Aqueous Solutions, Structure Making and Structure Breaking of Sucrose and Urea, J. Phys. Chem., 78, N17, 1754 (1974). Mathlouthi, M . , X-Ray Diffraction Study of Molecular Association in Aqueous So-lutions of D-Fructose, D-Glucose, and Sucrose, Carbohydrate Res., 91, 113 (1981). 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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 R U N D E 0 X 1 F I E L D = 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 R U N D E O X 3 F I E L D = 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 R U N D E 0 X 4 F I E L D = 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 R U N D E 0 X 6 F I E L D = 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 R U N D E 0 X 2 F I E L D = 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 R U N R IB1 F I E L D = 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 R U N R I B7 F I E L D = 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 R U N R IB6 F I E L D = 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 R U N R IB8 F I E L D = 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 R U N R IB2 F I E L D = 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 R U N X Y L 1 F I E L D = 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 R U N X Y L 4 F I E L D = 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 R U N X Y L 2 F I E L D = 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 R U N X Y L 7 F I E L D = 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 R U N X Y L 3 F I E L D = 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 R U N X Y L 5 F I E L D = 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 R U N F R U 4 F I E L D = 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 R U N F R U 6 F I E L D = 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 R U N F R U 7 F I E L D = 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 R U N F R U 5 F I E L D = 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 R U N F R U 3 F I E L D = 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 R U N G L U 7 F I E L D = 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 R U N G L U 8 F I E L D = 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 R U N G L U 3 F I E L D = 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 R U N G L U 4 F I E L D = 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 R U N G L U 5 F I E L D = 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 R U N G L U 6 F I E L D = 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 R U N G L A 1 F I E L D = 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 R U N G L A 5 F I E L D = 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 R U N G L A 3 F I E L D = 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 R U N G L A 6 F I E L D = 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 R U N G L A 4 F I E L D = 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 R U N G L A 2 F I E L D = 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 R U N M A L 1 2 F I E L D = 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 R U N M A L I 7 F I E L D = 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 M A L I 4 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 R U N M A L I 8 F I E L D = 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 R U N M A L I 6 F I E L D = 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 R U N L A C 1 3 F I E L D = 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 R U N L A C 1 2 F I E L D = 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 R U N L A C 1 4 F I E L D = 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 R U N L A C 1 5 F I E L D = 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 R U N L A C l l F I E L D = 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 R U N S U C 3 T F I E L D = 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 R U N SUC2 F I E L D = 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 R U N SUC5 F I E L D = 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 R U N S3 F I E L D = 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 R U N S4 F I E L D = 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 R U N R A F 1 F I E L D = 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 R U N R A F 3 F I E L D = 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 R U N R A F 7 F I E L D = 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 R U N R A F 6 F I E L D = 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 R U N R A F 4 F I E L D = 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 R U N R A F 2 F I E L D = 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 F IELD - 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*»********* S E T 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 3 0 0 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 * * * * « * . * * * * * * P E R F 0 R M 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 2 2 9 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 « « « « « I F 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 F I T TO ERROR FUNCTION * C * CORRELATION * Q I t * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * - * * * * * * - * - * * * * * * * * * * * * * * * * * * * - * * * * * * * * 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 F I L E FOR PROFILE CORRELATION PARAMETERS******* C** * * **READ TIT L E * * ****************************************** * ************ 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 * * « * * * * « * * S E T 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 * * * . * * * * * * D E L T X I S 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) , A U ( I ) , B U ( I ) , X U F READ(10,502)PML(I),AL(I) , B L ( I ) , X L F 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 =(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 £ R Q ) _ ^ * * * * * * * * * * * * * * 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 "@en ; edm:hasType "Thesis/Dissertation"@en ; edm:isShownAt "10.14288/1.0059032"@en ; dcterms:language "eng"@en ; ns0:degreeDiscipline "Chemical and Biological Engineering"@en ; edm:provider "Vancouver : University of British Columbia Library"@en ; dcterms:publisher "University of British Columbia"@en ; dcterms:rights "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en ; ns0:scholarLevel "Graduate"@en ; dcterms:title "An experimental study of static magnetic field effect on free diffusion of saccharides in aqueous solution"@en ; dcterms:type "Text"@en ; ns0:identifierURI "http://hdl.handle.net/2429/30560"@en .