Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

The development and use of the Rayleigh interferometer to study molecular diffusion in an applied magnetic… Howell, Steven Kelley 1983

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1984_A1 H69.pdf [ 8.52MB ]
Metadata
JSON: 831-1.0058898.json
JSON-LD: 831-1.0058898-ld.json
RDF/XML (Pretty): 831-1.0058898-rdf.xml
RDF/JSON: 831-1.0058898-rdf.json
Turtle: 831-1.0058898-turtle.txt
N-Triples: 831-1.0058898-rdf-ntriples.txt
Original Record: 831-1.0058898-source.json
Full Text
831-1.0058898-fulltext.txt
Citation
831-1.0058898.ris

Full Text

THE DEVELOPMENT AND USE OF THE RAYLEIGH INTERFEROMETER TO STUDY MOLECULAR DIFFUSION IN AN APPLIED MAGNETIC FIELD by STEVEN KELLEY HOWELL B.Sc, Southern Methodist University, 1976 M.Sc, Southern Methodist University, 1977 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Chemical Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July, 1983 © Steven Kelley Howell, 1983 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requ i rements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Co lumb ia , I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copy ing o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g ran ted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s unders tood tha t copy ing o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l ga i n s h a l l no t be a l l owed w i thou t my w r i t t e n p e r m i s s i o n . Department o f Chemical Engineering The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver , Canada V6T 1Y3 D a t e 25,November, 1983 DE-6 (3/81) ABSTRACT The purpose of this work was to investigate the effects of an externally applied homogeneous steady magnetic fi e l d on liquid phase diffusion through a porous membrane. A physico-chemical model was developed to describe the effect of a magnetic f i e l d on magnetically anisotropic molecules diffusing through a porous membrane. An applied magnetic f i e l d is expected to cause a reduction in the diffusivity of an anisotropic molecular system. Orientation of the molecule in the magnetic f i e l d (Cotton-Mouton effect) w i l l change the effective cross-sectional area of the molecule, increasing the viscous drag between the molecule and membrane pore surface thereby reducing the diffusion coefficient. An optical interferometric technique was used to measure diffusion coefficients which offered advantage over other methods since concentration profiles could be locally observed adjacent to the membrane surface without disturbing the diffusive flows. A Rayleigh interferometer was designed and constructed to be placed between the pole pieces of a 30 cm electromagnet. The diffusion of an aqueous sucrose solution through General Electric Nucleopore membranes (pore diameters 0.8 \xm and 8.0 (j.m) was measured in applied f i e l d strengths from 0 to 12.5 kGauss. This combination of membrane and solution was selected for this i n i t i a l work because of potential applications to biological systems and to verify the validity of the measurement technique since widely accepted diffusion data for this system are - i i i -available in the literature for comparison. A computer program was developed to account for errors introduced by wavefront deflection in a refractive index gradient and to numerically calculate mass fluxes and di f f u s i v i t i e s from interference fringe data. Within the limits of experimental error a slight decrease (1% to 2%) in the diffusion coefficient of sucrose through the membrane has been observed for applied magnetic fields up to 12.5 KG. Free diffusion coefficients measured at no f i e l d conditions compared to accepted values measured at identical concentration and temperature to within ± 3%. While these results indicate some alignment of the sucrose-water clusters in an applied magnetic is taking place, further work is needed to improve the accuracy of the experimental technique and to study molecules possessing a higher degree of anisotropy than sucrose and water. Recommendations are made for modifications to the diffusion c e l l which should significantly reduce experimental errors. Magnetically anisotropic molecular systems are also recommended for study which should show a greater degree of magnetic orientation than sucrose and water. - iv -TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS ix CHAPTER 1 - INTRODUCTION 1 CHAPTER 2 - HISTORICAL BASIS AND PREVIOUS WORK 5 2.1 Magnetic Field Effects on Transport Properties of Gases 5 2.1.2 Magnetic Field Effects on Liquid Transport Properties 7 2.2 Historial Review of Interferometry 11 2.2.1 Early Work 11 2.2.2 Later Work 12 CHAPTER 3 - THEORETICAL CONSIDERATIONS 17 3.1 Theory of Molecular Diffusion in a Magnetic Field 17 3.1.1 Fick's Law of Binary Diffusion 17 3.1.2 Kinetic Interpretation of Diffusion 18 3.1.3 Molecular Diffusion Through a Membrane 20 3.2 Magnetic Field Effects on Diffusion 22 3.2.1 Basic Magnetochemistry 22 3.2.2 Molecular Orientation in a Magnetic Field and Diffusion 25 3.2.3 Aqueous Sucrose Solution Diffusion in a Magnetic Field 27 3.3 Theory of Interf erometry 32 3.3.1 Refractive Index 32 3.3.2 Wave Nature of Light and Interference 34 3.3.3 Double Beam Interf erometry 39 3.3.4 Interpretation of Interference Fringes 42 - v -CHAPTER 4 - EXPERIMENTAL APPARATUS AND PROCEDURE 50 4.1 Experimental Apparatus 50 4.1.1 Quartz Diffusion Cell 50 4.1.2 Piano-Convex Lens and Cylinder lens 54 4.1.3 Vertical S l i t Assembly 55 4.1.4 Laser and Collimating Lens System 56 4.1.5 Optical Bench and Component Mounts 57 4.1.6 Vibration Isolation 58 4.1.7 Temperature Control 63 4.1.8 Membrane and Sucrose Solution 66 4.1.9 Electromagnet 67 4.1.10 Camera and Measuring Microscope 69 4.2 Experimental Procedure 71 4.2.1 Aligning the Interferometer 72 4.2.2 Diffusion Cell and Membrane Preparation.... 74 4.2.3 Beginning an Experiment 77 CHAPTER 5 - FRINGE PATTERNS AND DATA ANALYSIS 79 5.1 Assumptions 79 5.1.1 One-dimensional Diffusion 80 5.1.2 Light Ray Deflection 80 5.1.3 Steady State Membrane Diffusion 82 5.1.4 Constant Diffusivity 82 5.2 Calculating Refractive Index Profile 82 5.2.1 Mass Flux and Diffusivity Calculations 87 5.2.2 Mass Fluxes 90 CHAPTER 6 - RESULTS 97 6.1 Raybending and Refractive Index Correlation 99 6.2 Diffusivities 99 CHAPTER 7 - CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH 108 7.1 Discussion • 108 7.2 Recommendations 110 7.3 Conclusions 113 NOMENCLATURE 115 REFERENCES 119 APPENDICES - A Interference Fringe Data 127 B Raytracing Refractive Index Profile Correlation Parameters 142 C Raytracing Computer Program 157 D Mass Flux and Diffusivity Calculation Computer Program....... 164 - v i -LIST OF TABLES Page Table I Membrane Parameters for G.E. Nucleopore Membranes 66 Table II Raybending Corrections of Data for a Typical Run 100 Table III Membrane and Free Diffusion Coefficients 101 Table IV Comparison of Aqueous Sucrose Solution Binary Diffusion Coefficients Measured by Interferometric Methods 104 Table V Results of Linear Regression Analysis for Sucrose Diffusion Coefficients in an Applied Magnetic Field 105 - v i i -LIST OF FIGURES Page Figure 1 Lielmezs' Results for the Diffusion of Aqueous Chloride Salts in an Applied Magnetic Field 10 Figure 2 Molecular Diffusion Through a Porous Membrane in a Magnetic Field 28 Figure 3 Structure of Sucrose 29 Figure 4 Young's Double S l i t Experiment 35 Figure 5 Propagation of Light as an Electromagnetic Wave Wave 36 Figure 6 Principles of Double Beam Interf erometry 38 Figure 7 Rayleigh Interferometer 43 Figure 8 Typical Rayleigh Interference Patterns 47 Figure 9 Experimental Setup 51 Figure 10 Quartz Diffusion Cell 52 Figure 11 Diffusion Cell Holder 59 Figure 12 Piano-Convex Lens and Mount 60 Figure 13 Cylinder Lens and Mount 61 Figure 14 Optical Bench and Mounting Block 62 Figure 15 Concrete and Rubber Vibration Isolation Platform 64 Figure 16 Magnetic Field Homogeneity 68 Figure 17 Optical Bench and Magnet 70 Figure 18 Wavefront Deflection Through Diffusion Cell and Refractive Index Gradient 81 Figure 19 Ray Tracing Parameters 89 Figure 20 Refractive Index Profiles for Typical Run 91 Figure 21 Mass Flux and Concentration Profiles in Diffusion Cell 92 - v i i i -Figure 22 Diffusion Coefficients vs. Field Strength for 8.0 um Pore Diameters 106 Figure 23 Diffusion Coefficients vs. Field Strength for 0.8 um Pore Diameters 107 - i x -ACKNOWLEDGMENTS I would l i k e to thank my research supervisor, Professor Janis Lielmezs, for his guidance, support, and i n s p i r a t i o n during this work. Acknowledgment i s made to the Natural Sciences and Engineering Research Council of Canada for t h e i r f i n a n c i a l support and to the Unive r s i t y of B r i t i s h Columbia for a Graduate Student Research Fellowship which made th i s work possible. A s p e c i a l thanks to Mrs. Hana Aleman for the many useful suggestions and ideas and timely assistance i n proofreading t h i s manuscript. I appreciate the assistance of Mr. Paddy J a r v i s of the UBC Chemical Engineering Workshop for the excellent f a b r i c a t i o n of the experimental equipment. None of th i s would have been possible without the loving support and encouragement from my wife, Debra. CHAPTER 1 INTRODUCTION In recent years a great deal of interest has been generated in magnetic f i e l d effects on various chemical and biological processes. With the advent of manned space flight in the 1960's, researchers became interested in the effects of the absence of a magnetic f i e l d on the 1 2 living organism with respect to extended outer space flight. Barnothy performed some of the earliest research in magnetic effects on the living organism. In 1948 he found that young female mice, when placed 3 in a (3 - 6) x 10 Oersted magnetic f i e l d , underwent a temporary retardation of growth. He also reports that a magnetic f i e l d can have a retardation effect on the growth rate of cancer cells in mice. Several researchers have also investigated the effect of an applied i+_7 magnetic f i e l d on the healing rate of bone fractures. A slight increase in the rate of healing has been observed when low frequency alternating magnetic fields are applied to the fracture. This method is currently being used on an experimental basis at the UBC Sports Medicine 8 Centre to influence the healing rate of athletic stress fractures. Q Barnothy has compiled an extensive survey of other interesting applied magnetic f i e l d effects to living systems including changing the germination rate of seeds, altering the navigational a b i l i t y of homing pigeons, and changing the pulse rates and metabolism of rats. No satisfactory agreement has been reached on a biochemical explanation for these effects. Bhatnager and Mathur 1 0 propose that these effects can be explained by a change in reaction rates within the c e l l , while Gross 1 1 - 2 -attributes these changes to alterations in the chemical bond formations 12 due to the presence of a magnetic f i e l d . Liboff proposes that the growth rate changes observed in cells in an applied magnetic f i e l d result from a change in the diffusion rate of dissociated salts across the plasma membrane and nuclear membrane of the c e l l . In addition to the magnetic effects on livi n g systems, several researchers have reported a magnetic f i e l d effect on various other physical and chemical dynamic processes. Lielmezs et a l . have studied magnetic effects on liquid transport properties and the diffusion rate of several aqueous salt s o l u t i o n s . 1 3 - 2 0 They also report a change in 2 1 the viscosity of calf thymus DNA during the thermal denaturation process which they relate to magnetic inhibition of the double he l i x - c o i l unwinding during denaturation. Changes in transport properties of gases in an applied magnetic f i e l d have also been 22—39 observed. The kinetic theory of gases has been extended to provide a theoretical basis for these observations. These results w i l l a l l be discussed in greater detail in Chapter 2. This experimental evidence indicates that a magnetic f i e l d applied to many dynamic chemical and biological processes changes the dynamics of that process. While the kinetic theory of gases has been successfully applied to predict these effects in gases, very l i t t l e theoretical basis exists for the prediction of magnetic effects in the liquid state. The effects of a magnetic f i e l d on the l i v i n g organism has intrigued researchers for many years, but at this time no clear understanding of the molecular mechanisms responsible for these effects - 3 -has been reached. Therefore, i t i s the purpose of this work to study the e f f e c t s of an applied magnetic f i e l d on the d i f f u s i o n rate of organic molecules through a membrane. A simple theory of molecular d i f f u s i o n i n a magnetic f i e l d through a porous membrane i s developed with a p p l i c a t i o n to molecules which possess a magnetic anisotropy. A system was selected for i n v e s t i g a t i o n which consists of a common organic molecule; sucrose, i n an aqueous solution d i f f u s i n g through a porous membrane. The membrane used consists of straight c y l i n d r i c a l pores i n a thi n (10 micron) polycarbonate f i l m . This combination of membrane and sol u t i o n has p o t e n t i a l a p p l i c a t i o n to b i o l o g i c a l systems, yet provides a simple system for analysis since the pores of the membrane act as c a p i l l a r i e s for the d i f f u s i n g molecules. In addition, data are r e a d i l y a v a i l a b l e i n the l i t e r a t u r e for t h i s system i n the absence of a magnetic i+0 f i e l d , providing a reference point for t h i s work. O p t i c a l interferometry was selected as the technique to measure these d i f f u s i o n c o e f f i c i e n t s . This method was chosen because i t provides a t o o l which gives a continuous concentration p r o f i l e of the d i f f u s i n g molecules at a l l locations i n the test c e l l . If the d i f f u s i n g medium i s transparent to the o p t i c a l wavelength used, then no energy i s absorbed by the d i f f u s i n g molecules and i t i s possible to measure the d i f f u s i o n rate without disturbing the d i f f u s i v e flows. Therefore, t h i s technique i s desirable for measuring l o c a l d i f f u s i o n c o e f f i c i e n t s through a membrane i n a magnetic f i e l d , since the only perturbations to the d i f f u s i o n process w i l l be due to the applied magnetic f i e l d and not from the measurement technique i t s e l f . - 4 -The work described i n this thesis consists of: I) previous work and l i t e r a t u r e survey — a survey of previous studies made on magnetic e f f e c t s observed on transport properties of gases and l i q u i d s , and a review of experimental techniques used to measure l i q u i d system d i f f u s i o n c o e f f i c i e n t s ; II) t h e o r e t i c a l considerations - development of a theory describing molecular d i f f u s i o n i n a magnetic f i e l d and a review of the theory of interferometry; III) experimental method - a detai l e d description of the experimental setup and method; IV) data analysis - a method of i n t e r p r e t i n g interference fringes and reducing them to concentration p r o f i l e s and mass fluxes, including a technique to allow for errors introduced by o p t i c a l ray bending i n a r e f r a c t i v e index gradient; V) r e s u l t s - a discussion of experimental data i n terms of the theory described i n Part II and VI) recommendations for areas of further research and conclusions. - 5 -CHAPTER 2 HISTORICAL BASIS AND PREVIOUS WORK 2.1. Magnetic Field Effects on Transport Properties of Gases The earliest work examining magnetic effects on the transport properties of gases and liquids was performed in 1930 by Senftleben when he discovered that an applied magnetic f i e l d changes the viscosity of 41 oxygen. He observed a decrease in viscosity up to a maximum of 0.4% as a function of H/P, where H was the applied magnetic f i e l d strength and P the pressure of O2 . Magnetic effects have since been observed on the thermal conductivity and kinematic viscosity of a member of polyatomic g a s e s 2 2 - 3 9 including HC1, DC1, N20, C02, OCS, SF 6, CH3F, CH3, CN, CHF3, CDF 3, NH3, ND3, NF3, PH3, P F 3 , AsH 3, 0 2, NO, N 2, CH4, CFL,, CO, nH2, HD, 39 oD2, ND2, pH2, and CDI+. Beenakker and McCourt give an excellent review of a l l work done prior to 1970, describing the various magnetic effects on the transport properties of gases. The kinetic theory of gases has been successfully applied to explain these results and elucidate several aspects of the interactions of polyatomic molecules. In the presence of a magnetic f i e l d , any molecular anisotropy w i l l precess around the f i e l d direction due to the interaction of the f i e l d with the rotational magnetic moment. This precession causes the axis of the molecule to spin about the magnetic f i e l d axis much as a rotating gyroscope "wobbles" about the gravitational vector. The frequency and magnitude of the precession is dependent upon the strength of the applied f i e l d , the molecular - 6 -r o t a t i o n a l magnetic moment, and angular momentum of the molecule. This precessional motion has the e f f e c t of changing the cross-sectional area of the molecule for c o l l i s i o n s with other molecules i n the gas phase. The net r e s u l t i s an o v e r a l l p o l a r i z a t i o n of the cross-sectional areas of the molecules with respect to the d i r e c t i o n of the applied f i e l d . This e f f e c t can best be v i s u a l i z e d by considering E i n s t e i n ' s model for the transport of heat i n a gas. For a monatomic gas the heat flow q i s defined as q = J f M 1/2 mV2 VdV* (2.1) M •* in which f i s the d i s t r i b u t i o n function, m the p a r t i c l e mass, and V the v e l o c i t y . A temperature gradient VT", produces a deviation of the d i s t r i b u t i o n function away from the i d e a l Maxwellian d i s t r i b u t i o n function f(®\ which r e s u l t s i n a transport of heat through the gas. In the f i r s t order, t h i s deviation i s proportional to the temperature gradient. f M = f ( 0 ) (1 + t V?) (2.2) For a polyatomic gas the vector X depends both upon the molecular v e l o c i t y and the angular momentum of the molecule. When a magnetic f i e l d i s applied to the gas the angular momentum i s polarized i n space. This p o l a r i z a t i o n causes an anisotropic d i s t r i b u t i o n of energy i n the gas which i s observable as an anisotropic thermal conductivity for the gas. Therefore, measurements of these anisotropies provide a test of - 7 -the k i n e t i c theory of gases for molecules with i n t e r n a l degrees of freedom. 2.1.2 Magnetic Field Effects on Liquid Transport Properties Unfortunately, experimental research i n magnetic f i e l d e f f e c t s on l i q u i d transport properties i s much more lim i t e d than for the gaseous state. While many aspects of the k i n e t i c theory of gases are f a i r l y well established and have been reasonably defined or derived such i s not the case f o r a k i n e t i c theory of l i q u i d s . In the l i q u i d state a molecule int e r a c t s simultaneously with several neighbors, whereas i n the gaseous state molecules generally react with only one other molecule at a time. Therefore a t h e o r e t i c a l basis for pr e d i c t i n g a magnetic e f f e c t on l i q u i d transport properties and d i f f u s i o n i s very l i m i t e d at t h i s time. B r e n n e r 4 2 3 derives a t h e o r e t i c a l expression for the e f f e c t s of an applied magnetic f i e l d on a d i l u t e suspension of spher i c a l p a r t i c l e s which possess a magnetic dipole. He discusses how an applied f i e l d hinders the free r o t a t i o n of the p a r t i c l e s and hence how the apparent v i s c o s i t y becomes anisotropic with respect to the d i r e c t i o n of applied f i e l d . 43 44 Lielmezs and Musbally, and Camp and Johnson have chosen to deal with magnetic e f f e c t s on l i q u i d d i f f u s i o n on a macroscopic l e v e l using the p r i n c i p l e s of i r r e v e r s i b l e thermodynamics. They define another d r i v i n g force term i n the thermodynamic f o r c e - f l u x r e l a t i o n s h i p s which i s the Lorentz force exerted on a d i f f u s i n g ion i n a magnetic f i e l d . - 8 -This force is defined as ? = Z (V* x h (2.3) K K K where Z^ is the electric charge on the diffusing ion, i t s average d r i f t velocity and B the magnetic induction. This force term is included with the other diffusive driving forces and applying the principles of irreversible thermodynamics they solve for the ratio of diffusion coefficients with and without an applied f i e l d as, D ° (v\ x i ) - (tf x h 1 + u, gradC ss s (2.4) where V*i and V*2 a r e t n e average dr i f t velocities for the two ions, LI s s is the partial derivative of chemical potential with respect to o^s concentration, -g^—, and gradC is the salt concentration gradient, s 45 O'Brien and Santhanan observed this Lorentz force in an aqueous solution of copper sulfate during electrodeposition in an applied magnetic f i e l d . They used a multiple beam interferometer to observe the concentration profile in the electrodeposition c e l l . When a magnetic f i e l d of 6.12 kG was applied to the c e l l the Lorentz force resulting from the motion of the ions in the applied magnetic f i e l d produced a convective driving force. This convection was observable as disturbances in the concentration gradient in the c e l l producing interference fringes. - 9 -Lielmezs and A l e m a n i D - z u have studied the d i f f u s i o n of various aqueous solutions of chloride s a l t s ( L i C l , NaCl, KC1, and CsCl) through a f r i t t e d glass diaphragm i n an applied magnetic f i e l d . Some of t h e i r r e s u l t s are depicted i n Figure 1. In an applied f i e l d of 5 kG the i n t e g r a l d i f f u s i o n c o e f f i c i e n t s for L i C l and CsCl show a s l i g h t decrease, while for the other s a l t s they show an increase, the most notable being KC1. The exact cause of these changes cannot be decided with any degree o f c e r t a i n t y . They note, however, that the KCI-H2O system, showing the largest magnetic e f f e c t , also shows the greatest s t r u c t u r a l disorder. These r e s u l t s are i n t r i g i n g and of a q u a l i t a t i v e nature, and at t h i s time no d e f i n i t e conclusions have been reached explaining them. Lielmezs et a l . 1 3 » 1 1 + have also observed a viscomagnetic e f f e c t f o r water and various aqueous solutions of paramagnetic s a l t s (Mn(N03), Cu(N03), Ni(N03) and Co(N03>). At a t r a n s v e r s a l l y applied f i e l d strength of 10 kG, they measured an increase i n the v i s c o s i t y of water between 0.1 and 0.2 %.' + 6' l t 7 They proposed that the magnetic f i e l d caused a s l i g h t change i n the angle of the hydrogen bonds i n water which i n turn a f f e c t s the t r a n s l a t i o n a l and r e o r i e n t a t i o n a l motion of the molecules and the v i s c o s i t y . They observed a s l i g h t decrease i n the v i s c o s i t y of the various water-paramagnetic s a l t s o lutions. They found that the v i s c o s i t y decreased at high s a l t concentrations, yet at low concentrations the observed v i s c o s i t y increases, approaching that of pure water i n an applied magnetic f i e l d , leading them to propose the existence of two competing microstructural i n t e r a c t i o n mechanisms; the d i p o l a r i n t e r a c t i o n s associated with pure diamagnetic water, and the - 10 -E o i Q 3 3 5 0 3 3 0 0 3 250 3 2 0 0 3130 3100 3O50 2-050 2 COO 1-950 1-900 I-B30 I 600 I 550 1-500 1^50 NoCt * 5 kG — No f i e l d __ i 0 2 0 4 0 - 6 0 6 l-O 12 K k6 1-820 c « / m o l I " 1 Arithmetic mean integral diffusion coefficient and concentration plot at 25°C temperature for HCI-, N a O -and K C 1 - H 2 0 solutions at the ambient earth field (solid curve) and at the applied external transversa magnetic field (dashed curve) condition. Figure 1 - Lielmezs' et a l . Results for the Diffusion of Aqueous Chloride Salts In an Applied Magnetic F ie ld , (used by permission). - l i -sp in-exchange mechanism characterizing the paramagnetic ion-water solution. A l l of these experiments were performed isothermally at room temperature for several different concentrations. 2.2 H i s t o r i c a l R e v i e w o f I n t e r f e r o m e t r y Optical interferometry is a technique which w i l l yield a continuous profile of the refractive index of the medium through which the light is being transmitted. In the case of liquid diffusion the refractive index of the diffusing liquid system can be observed continuously in space and time which can be related to concentration of the diffusing molecule or ion. With the advent of modern coherent laser light sources this technique has evolved to a powerful, sensitive tool for measuring small concentration changes in any diffusing molecular system. 2.2.1 E a r l y Work Optical interferometry was f i r s t applied to liquid diffusion 48 measurements of Philpot and Cook in 1947. To do this they modified the Rayleigh interferometer 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 c e l l . Philpot and Cook used this original equipment to measure the diffusion of sodium thiosulfate in 4 9 water. Longsworth was working concurrently with Philpot and Cook and applied the Gouy method of interferometry to measure diffusion coefficients of KC1 dissolved in water at 0.5°C. This method is similar to the one used by Philpot and Cook, except he had no cylindrical lens to focus the interference fringes as a function of position in the test - 12 -c e l l . This equipment produced interference fringes which were a function of refractive index gradient in the c e l l . At the same time Longsworth was pioneering the applications of Gouy interferometry to 50 diffusion measurements, Ogston was working independently using the same method, applying i t to measure binary diffusion coefficients of glycine, KC1, sucrose, and lactoglobulin dissolved in water. In 1949 51 Gosting and Morris, continuing the development of the Gouy interferometric method published diffusion coefficients for an aqueous sucrose solution at 25°C and 1°C. In 1951, Gosting and Akeley 5 2 continued this work using the same method to evaluate diffusion coefficients for urea in water at 25°C. While these workers were developing and using the Gouy method of 5 3 54 interferometry, Svenson and Longsworth were extending the original work done by Philpot and Cook with the Rayleigh interferometer. They were both applying i t to the measurement of aqueous sucrose diffusion coefficients at 25°C. This early work was a l l performed before the advent of modern, laser light sources. These workers a l l used either sodium vapor or mer-cury vapor sources of light. The quality of interference fringes they reported is remarkable considering the poor coherence of those light sources as compared with the gas laser used today. A l l of these workers estimated the values of free diffusion coefficients they measured were accurate to 0.2% or less. 2.2.2 Later Work The advent of the continuous gas laser i n the mid nineteen - 13 -sixties provided an intense monochromatic coherent light source previously unobtainable with the conventional vapor lamps. A laser light source is essentially one frequency with the light waves in phase across the output beam cross section. This level of coherence produces interference fringes with a clarity and intensity unobtainable from a vapor light source. Principles of coherence w i l l be dealt with in more detail in Chapter 3. 55 56 In the early sixties O'Brien and O'Brien et a l pioneered the use of multiple beam interferometry for mass transport measurements. They used a wedge interferometer, which was designed with the light beam passing through the diffusing substance many times, in contrast to a single pass as in the Gouy or Rayleigh interferometric methods. This technique produced a set of interference fringes which were much more sensitive to small refractive index changes than the other single pass systems. They applied this method to measure the concentration gradients at the electrode surfaces of a Zn/ZnSO^/Zn and Cu/CuSOit/Cu electrochemical c e l l . 57 In 1969 Duda, Sigelko, and Vrentas developed a multiple pass wedge interferometer to measure sucrose-water diffusion coefficients at 25°C. They estimate an accuracy of ±3% with this technique. Multiple beam interferometry has since been applied to diffusion measurements for many different systems including the measurement of diffusion of 5 8 02,N2,H.2 and Ar into water. Rard and M i l l e r ^ 9 - 6 1 have used Rayleigh interferometry with a laser light source to measure the diffusion of BaCl2, KC1, CsCl, SrCl2, - 14 -NaCl, and CaCl2 in aqueous solutions at room temperature. They measured free diffusion coefficients of very dilute solutions (.01 moles/1) to high concentration solutions (5.5 moles/1). They estimate an accuracy of 0.1% to 0.2% for their results. 6 2 Sorell and Myerson used a Gouy interferometer with laser light source to measure the diffusivity of an aqueous urea solution in a saturated and supersatured solution. They estimate an error of less than 5% for their results. 6 3 Renner and Lyons developed a novel method of improving the data reduction techniques for interferometric measurements. They used an electronic photomultiplier scanner to electronically measure the fringe spacing of the interference fringes produced from a Gouy interferometer applied to the measurement of KCl-water diffusion coefficients. The output of this device was input directly to a di g i t a l computer which converted the fringe spacing to refractive index profiles and mass fluxes and d i f f u s i v i t i e s . This method minimized human errors introduced when manually measuring fringes through a microscope. More recently double exposure holography has found applications to mass transport problems. Holography has the primary advantage that abberrations in the optical components are "cancelled out" during the reconstruction process, enabling an experimenter to work with less expensive, more readily available lenses, mirrors, and optical components. .In this method a three dimensional hologram is made of the test c e l l during i t s i n i t i a l condition. Then, after the mass transport process begins another "double exposure" hologram is made of the object, - 15 -with virtually no movement allowed in either the object or photographic film. An interference fringe w i l l appear at every location on the reconstructed image of the test c e l l where an optical path difference of one wavelength occurred between the two exposures. The fringe results from an optical path length change where the light was transmitted through the object, thus enabling one to measure an index of refraction change at a particular location on the test c e l l . Gabelman-Gray and 6 4 Fenichel used double exposure holography to measure the diffusion coefficients of a 10% aqueous sucrose solution at 25°C. Their results are within 10% of other values measured by other methods. O'Brien, Langlais, and Seufort 6 5 used holography to measure the diffusion coefficients of respiratory gases into a synthetic blood substitute, a perfluorocarbon liquid. Interferometric techniques which directly measure concentration profiles have been widely applied to the study of free diffusion in liquids. The opaque characteristics of membranes have inhibited the direct measurement of concentration profiles in the membrane i t s e l f . However, Bollenbeck and Ramariz in 1972 4 0' 6 6 f i r s t applied interferometry to the measurement of diffusion through membranes. They used a Rayleigh interferometer to measure concentration profiles surrounding a membrane surface in an aqueous sucrose solution. They developed a technique to evaluate the mass flux at each membrane solution interface and therefore the diffusion coefficient through the membrane i t s e l f . An error of ±3% is estimated for their work. A modification of their technique is applied in this work to measure - 16 -membrane d i f f u s i o n c o e f f i c i e n t s i n an applied magnetic f i e l d . D e t a i l s concerning this technique are contained i n appropriate sections of th i s t h e s i s . 67 68 Min et a l and Forgacs et a l applied a wedge interferometer to the study of membrane d i f f u s i v i t i e s i n a cellophane and ion exchange membrane resp e c t i v e l y . Min et a l studied the steady state d i f f u s i o n of ethyl alcohol and water through a cellophane membrane at room temperature. They estimate an accuracy of at least ±3% for t h e i r r e s u l t s . Forgacs et a l studied the d i f f u s i o n of an aqueous KC1 solu t i o n through an ion-exchange membrane with an applied e l e c t r i c current following current r e v e r s a l . The applications of interferometry to d i f f u s i o n process measurements have been more widely applied to the measurement of free d i f f u s i o n c o e f f i c i e n t s rather than membrane d i f f u s i o n c o e f f i c i e n t s . The presence of a membrane i n the d i f f u s i n g medium presents several measurement problems not encountered with free d i f f u s i o n measurements. The membrane i s opaque to the transmitted l i g h t so i t i s not possible to act u a l l y observe the concentration p r o f i l e through the membrane. Simplifying assumptions must be made concerning the mass fluxes and concentration gradients through the membrane. In addition, wavefront d e f l e c t i o n of l i g h t transmitted through a r e f r a c t i v e index gradient occurs near the membrane surface which produces a d i s t o r t i o n of the measured concentration p r o f i l e and membrane shadow, so techniques must be developed to correct for d e f l e c t i o n e f f e c t s . - 17 -CHAPTER 3 THEORETICAL CONSIDERATIONS This chapter presents the t h e o r e t i c a l basis f o r molecular d i f f u s i o n through a membrane. Fick's law i s reviewed with a p p l i c a t i o n to a simple porous membrane. The e f f e c t s of a magnetic f i e l d on d i f f u s i o n i s discussed with a review of the basic theory of magnetism. A t h e o r e t i c a l model i s developed describing the e f f e c t of a magnetic f i e l d on the d i f f u s i o n rate of molecules through a membrane cons i s t i n g of c y l i n d r i c a l pores. In addition, the theory of o p t i c a l interferometry i s presented, with applications to d i f f u s i o n measurements. 3.1 Theory of Molecular Diffusion In a Magnetic Field 3.1.1 Fick's Law of Binary Diffusion D i f f u s i o n i s the transfer of a substance through a homogenous solution r e s u l t i n g from a difference i n concentrations at two d i f f e r e n t regions i n the mixture. D i f f u s i o n i s a r e s u l t of the random Brownian motion of the molecules a r i s i n g from the thermal energy of the molecule. In terms of t h i s assumption, the motion of the molecule may be considered random. Therefore the net molecular motion of the d i f f u s i n g species w i l l be from the d i r e c t i o n of higher concentration to lower concentration i n the absence of any temperature or pressure gradients. Fick's law of binary d i f f u s i v i t y expresses the bC molecular f l u x , J, as a l i n e a r function of concentration gradient -r— J = D ^ J U 5x (3.1) - 18 -where D is the constant of proportionality; the diffusion coefficient. Eq. 3.1 is Fick's f i r s t law and defines the flux at steady state conditions. This equation is valid for isothermal conditions at constant pressure and with no volume change on mixing. Expressing Fick's f i r s t law equation in terms of the conservation of mass (i.e. the change in concentration per unit time in a given volume is equal to the difference of the flows into and out of the volume) gives ac -5J Dae 2 ( 3 > 2 ) ot ox ox 2 which is Fick's second law. The work presented in this thesis w i l l only consider a concentration gradient in one dimension, so the partial derivatives in Eqs. 3.1 and 3.2 w i l l become ordinary derivatives. 3.1.2 Kinetic Interpretation of Diffusion Einstein used the molecular kinetic theory of heat to develop a theory of Brownian motion and provide a physical picture to describe the diffusion process in dilute solutions. According to the molecular kinetic theory, heat is simply a manifestation of the motion of the molecules in a system. The average _2 translational kinetic energy, 1/2 mv is proportional to the absolute temperature, T, of the system; 1/2 mv2 = 3/2 KT (3.3) - 19 -where K i s Boltzman's constant and v i s the mean square t r a n s i t v e l o c i t y i n any d i r e c t i o n . E i n s t e i n showed that the mean square Brownian _ 2 displacement along an axis, 6^ , for some time i n t e r v a l At i s 6 2 = 2KT At (3.4) x f where f i s a f r i c t i o n a l c o e f f i c i e n t of a solute molecule. The d i f f u s i o n c o e f f i c i e n t , D, i s related to the motion of the p a r t i c l e per unit time by 6 2 D = 2 A T ( 3 ' 5 ) Substituting Eq. 3.4 into 3.5 gives D = M (3.6) 70 which i s an E i n s t e i n d i f f u s i o n c o e f f i c i e n t . According to Stokes, the f r i c t i o n a l c o e f f i c i e n t f, of a r i g i d sphere of diameter d moving through a medium of v i s c o s i t y n i s f = 3imd (3.7) This r e l a t i o n s h i p i s v a l i d only i f the f l u i d medium i s continuous and i f 42 no s l i p occurs between i t and the sphere. Therefore, combining Eqs. 3.6 and 3.7 y i e l d s the c l a s s i c a l Stokes-Einstein equation f o r d i f f u s i v i t y ; - 20 -which is valid only in very dilute solutions of spherical molecules, which are large compared to the size of solvent molecules. Application of Fick's laws to mass transport through porous membranes w i l l be discussed in this section. The following assumptions w i l l be made: 1) Membrane pores are essentially cylindrical (details concerning Nucleopore membranes used in these experiments are given i n Chapter 4), 2) membrane is thin (10 Lim) and concentration difference through membrane is small (less than 1% by weight), therefore mass flux into one membrane surface equals mass flux out the other surface, and 3) membrane presents a simple cross sectional area reduction to diffusion, i.e., there are not chemical or physical interactions between membrane and solution; diffusion proceeds freely through membrane pores. If concentration difference is the only driving force through the membrane, then the mass flux through the membrane is where D^ , is the free diffusion coefficient, A^ is the pore cross-sectional area, AC is the concentration difference across the membrane which has a thickness of AX. This ignores the driving force due to osmotic pressure. In a system where the membrane permeability i s different for the solvent and solute molecules, an osmotic pressure, it, exists across the membrane as, 3.1.3 Molecular Diffusion Through a Membrane J = -] •D„ A £ F p Ax (3.9) % = oRTAC (3.10) - 21 -where a i s Staverman's reflection coefficient which accounts for the difference in permeability between the solvent and solute. R is the gas 7 2 constant and T is absolute temperature. Renkin proposes an expression for a for cylindrical pores which is based entirely on the geometry of the membrane, = i - U ( l - B ) 2 -(1-8)4][1-2.1048 + 2.09B3 - 0.95B5] ( 3 < n ) [2(l-y) 2 -(1-Y)4]U-2.104Y + 2.09y3 - 0.95Y5] where 6 = R /R and y = R /R with R , R , and R being the radius of r s p w p s w' p e the solute, solvent, and pore, respectively. For sucrose in water R = 5.3 A, 7 3 R = 1.9 A, 7 3 and R = 0.4 um and 4.0 urn. Therefore s w ' p 1 r — 3 a = 1.797 x 10 for 0.4 um pore radius a = 1.7886 x 10 - 1 + for 4.0 um pore radius Using the reflection coefficients calculated from Eq. 3.11 for a 1% by weight concentration difference at a temperature of 25°C, Eq. 3.10 predicts a maximum osmotic pressure difference of .10 centimeters of mercury for the .8 um diameter pore size and 0.01 centimeters of mercury for the 8.0 um diameter pore size. This osmotic pressure i s only a potential for water flow so there cannot be a water flux due to osmotic effects i f the volume is physically constrained. The membrane provides a physical barrier to volume change in the c e l l i f i t does not stretch during the diffusion process, so any water flow in the cel l w i l l only be due to diffusion and not osmotic effects. 74 Faxen derived an expression for the ratio of the diffusion coefficient in a cylindrical pore to the bulk diffusion coefficient. He - 22 -used a f r i c t i o n a l drag model based on a sphere f a l l i n g in a tube to predict this ratio. The Faxen equation is D = D (1 - 2.1046 + 2.0963 - 0.9585) (3.12) P * where is the pore diffusion coefficient, is the coefficient in bulk and 6 is the same as for Eq. 3.11. Therefore the diffusion coefficient observed through the membrane is reduced due to the viscous drag between the diffusing molecule and pore wall surface. Iberall i. o et al supports this observation by measuring the diffusion of different sized molecules through a porous membrane with cylindrical pores. The membrane permeability decreased as the value of 8 75 increased. Williamsen, et a l . , observed a decrease in the diffusion rate of sucrose through a porous membrane which confirmed the Faxen and Renkins equations. This observation becomes important when considering the diffusion of anisotropic molecules in a magnetic f i e l d discussed in Section 3.2. 3.2 Magnetic Field Effects on Diffusion 3.2.1 Basic Magnetochemlstry The most fundamental concept of magnetochemlstry is that of magnetic susceptibility, which is the response of a substance to a magnetic f i e l d . It i s commonly known that some substances are attracted to a magnetic f i e l d while others are repelled from i t . When a substance is placed in a f i e l d of H* Oersteds, the magnetic induction (expressed in Gauss) is given by the sum of the applied f i e l d (3) plus a contribution 4nM, where M is the intensity of magnetization or the magnetic moment - 23 -per unit volume i n the substance i t s e l f . The induction B* i s defined as the density of l i n e s of force per unit area A i n the substance. B* = 5 x 4 ^ (3.13) The magnetic s u s c e p t i b i l i t y , x> Is defined as the scalar r a t i o of the magnetization and applied f i e l d ; ft X-5 (3.14) H If a Cartesian coordinate system i s chosen within a specimen, the magnetic s u s c e p t i b i l i t y tensor i s diagonal, with xx» Xy» X z being referred to as the p r i n c i p a l s u s c e p t i b i l i t i e s . If a substance i s magnetically i s o t r o p i c , the magnetic induction i s independent of orientation i n the f i e l d and the magnetic s u s c e p t i b i l i t y tensor i s equal, X x = Xy = X z« If a substance i s anisotropic the magnetic s u s c e p t i b i l i t y depends upon the o r i e n t a t i o n of the molecule with respect to the d i r e c t i o n of the applied magnetic f i e l d . Anisotropic s u s c e p t i b i l i t y can only be observed i n a substance when a l l the molecules are oriented with respect to the magnetic f i e l d , as i n a single c r y s t a l . If a powder i s measured, the bulk magnetic s u s c e p t i b i l i t y Xfo* i s i s o t r o p i c and i s equal to the average of the three p r i n c i p a l magnetic s u s c e p t i b i l i t i e s . X + X + X 3 ~ ( 3 ' 1 5 ) Most substances can be c l a s s i f i e d as either dia or para -magnetic depending upon the sign of x« T n e sign of the magnetic - 24 -s u s c e p t i b i l i t y usually depends on whether the ground state electrons are paired or unpaired. The substance i s said to be diamagnetic i f the sign of x i s negative, i . e . , i t causes a reduction i n the l i n e s of force permeating the substance. This i s equivalent to the substance producing a magnetic f l u x In a d i r e c t i o n opposite to the applied f i e l d . In the presence of an inhomogeneous f i e l d , the molecule w i l l be repulsed from the region of higher f i e l d . Diamagnetism arises due to the motion of the electrons i n t h e i r atomic and molecular o r b i t s . An electron carry-ing a negative charge and moving i n a c i r c u l a r o r b i t i s the equivalent to a c i r c u l a r current. I f a magnetic f i e l d i s applied perpendicularly to the plane of the o r b i t , the revolving electron experiences a force 75 along the radius. Lenz's law, which predicts the d i r e c t i o n of motion of a current-carrying conductor placed i n a magnetic f i e l d ; predicts that the system as a whole w i l l be repelled away from the applied f i e l d . The degree of diamagnetism associated with an atom or molecules depends upon the size and shape of the o r b i t i n g electrons, the outermost electrons contributing the most to the diamagnetic s u s c e p t i b i l i t y . Paramagnetism, on the other hand, i s characterized by the magnetic induction being larger than the applied magnetic f i e l d . The sign of x i s therefore p o s i t i v e and a paramagnetic substance i s attr a c t e d to a region of higher magnetic f i e l d strength. Paramagnetism i s exhibited by substances which have unpaired electrons i n the ground state. I t i s generated by the tendency of magnetic angular momentum to orient i t s e l f i n a magnetic f i e l d . The magnetic angular momentum arises from the or i e n t a t i o n of the unpaired electrons with the magnetic f i e l d . Therefore, the magnitude of paramagnetic s u s c e p t i b i l i t y i s a function of - 25 -the numbers of unpaired electrons i n an atom or molecule. Most organic molecules are diamagnetic, including sucrose and water, the system studied i n t h i s work. 3.2.2 Molecular Orientation i n a Magnetic Field and Diffusion A magnetic f i e l d w i l l exert a force upon any molecule which possesses a magnetic anisotropy. This force w i l l tend to give a p r e f e r e n t i a l alignment of the molecular dipole i n the d i r e c t i o n of the applied magnetic f i e l d . The degree of alignment i s a function of the anisotropy of the molecule, the magnetic f i e l d strength, the i n t e r a c t i o n of the molecule with i t s neighbors, and the thermal k i n e t i c energy of the molecule. This section w i l l discuss how t h i s alignment a f f e c t s the d i f f u s i o n process. The o r i e n t a t i o n of diamagnetically anisotropic molecules i n a 77 fl 3 magnetic f i e l d i s termed the Cotton-Mouton e f f e c t . ~ An applied magnetic f i e l d produces a force on the molecule i f the molecular s u s c e p t i b i l i t y i s anisotropic. This force tends to orient the molecule p a r a l l e l to the directions of magnetic force which i s counteracted by the t r a n s l a t i o n a l and v i b r a t i o n a l motion of the molecule. The degree of o r i e n t a t i o n i s inversely related to the thermal energy of the system, since the random Brownian motion tends to d i s o r i e n t the molecules. The degree of o r i e n t a t i o n , 6 q, i s given by 6 o = (X11 " XP h 2 / K T ( 3* 1 6 ) with K being the Boltzmann constant, T the absolute temperature, H the - 26 -applied magnetic f i e l d strength, and » Xj_ being the diamagnetic susceptibilities parallel and perpendicular to a rotational symmetry axis, respectively. The degree of orientation has been measured by observing the magnetically induced birefringence An = n u - nj_ = CM X H 2 = 6 ( a n - aj)C (3.17) where n ^ > nj_ is the refractive index for light of wavelength X, when polarized parallel and perpendicular to H, respectively. <x^ , cxj_ are the molecular optical polarisabilities parallel and perpendicular to the molecular symmetry axis, C is the concentration and CM is the Cotton-Mouton constant for that molecule. This technique has been used to observe the orientation of some biological macromolecules such as 8 4 7 8 8 2 8 3 DNA , liquid crystals, ~ and micelles in a soap-water system. If a degree of orientation is achieved in a non-symmetrical molecule or molecular cluster, a change w i l l be observed on the diffusion rate of that molecule through a porous membrane. Equation 3.12 predicts a decrease in the diffusion coefficient of a molecule as 8, the ratio of molecule to pore diameter increases. If the axis of an anisotropic molecule is oriented orthogonal to the direction of the pore axis then the effective cross-sectional area of this molecule i s increased, increasing 8 and thereby reducing D^ , the membrane diffusion coefficient. Conversely, i f a molecule is oriented parallel to the pore - 27 -axis direction the effective cross-sectional area of the molecule is decreased, resulting in an increase in D^. Therefore, this effect can be used to observe and quantify the degree of anisotropic orientation of a solute in a magnetic f i e l d . This is depicted in Figure 2. 3.2.3 Aqueous Sucrose Solution Diffusion i n a Magnetic Field One objective of this work was to demonstrate the use of optical interferometry to measure diffusion coefficients In a magnetic f i e l d of organic molecules with applications to living systems. A sucrose-water solution was selected for these i n i t i a l experiments because i t is a simple organic solution for which diffusion measurements have been taken extensively using interferometry. Sucrose is a naturally occurring carbohydrate consisting of two monosaccharides bonded together to form 8 5 this common disaccharide, table sugar. A glycoside link joins one carbon of fructose with one carbon atom of glucose as shown in Figure 3. When in an aqueous solution i t forms hydration clusters as water molecules form hydrogen bonds with the H and OH groups on each ring. The geometry, bond angles and bond dimensions have been 9 0 9 1 determined from X-ray and neutron diffraction studies. The radius 73 of a sucrose molecule in water is 5.3 A. The magnetic properties of sucrose have not been measured to any great extent. The bulk magnetic susceptibility of sucrose has been calculated using Pascal's system, where the contributions to magnetic susceptibility from individual atoms and bonds is simply summed to yield DpiELD < D 0 D F I E L D > D 0 Fig. 2. Molecular Diffusion Through a Porous Membrane in a Magnetic Field: a. Field Applied Transverse to Pore Direction b. Field Applied Parallel to Pore Direction - 29 -a - D- g lucopyranosyl 0 CH 2 OH CH 2 OH / 3 - D - fructofuranoside Fig. 3. Structure of Sucrose, C^2r\220^ - 30 -the susceptibility for the moleculse as a whole. This value is X^  = -187.06 x IO - 6 emu/mole. Pascal's method has been applied to predict the magnetic susceptibility of complex organic molecules to better than 5% agreement with experimental values. 9 2 At this time no measurements are available for sucrose regarding magnetic susceptibility anisotropics, so one objective of this research is to evaluate any anisotropy by observing i f the degree of orientation in a magnetic f i e l d effects the membrane diffusion rate. Equation 3.16 predicts the degree of orientation of any molecular magnetic anisotropy in an applied magnetic f i e l d . While the bulk magnetic susceptibility for sucrose is readily available (xb = "187.06 emu/mole) data regarding the principal susceptibilities, x and x_j_ are not available. A highly anisotropic molecule such as benzene has 9 2 principal susceptibilities approximately two to one, i.e., the force exerted on the molecules In a homogenous f i e l d is twice as large in the parallel plane as the perpendicular plane. If the same degree of anisotropy were observed for sucrose, then x^ ~ "62 x 10 - 6 emu/mole and x_j_ ~ -124.7 x 10 - 6 emu/mole. Using these values Equ. 3.16 predicts that the degree of orientation for a single sucrose molecule in an applied magnetic field equal to 10 kG at 25°C would only be = 10 This is clearly beyond the limits of sensitivity for most experimental methods. However, 6 q 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 cluster is - 31 -proportional to N 6o = N ( X11 " U ) R 2 / K T ( 3 , 1 8 ) This situation is applicable in certain long macromolecules and almost f u l l magnetic alignment has been observed for various polymers, large 84 biological molecules such as nucleosides, chloroplasts, retinal rods, 77 82 and various liquid crystals. ~ This effect would also be observable in an aqueous sucrose solution i f long molecular clusters existed. Sucrose in water forms hydration clusters as water molecules bind to the 9 3 94 sucrose through hydrogen bonds. Shporer et a l . , Packer, and 95 , Resing discuss the 'ordering" of water near a solid surface in an applied magnetic f i e l d . Using NMR techniques, they believe partial ordering of water occurs in porous membranes placed in a magnetic f i e l d , observed by the splitting of the proton NMR spectrum. Therefore, i f an ordering of the sucrose hydration clusters occurs in a similar fashion i t should be observable as a decrease in the membrane diffusivity sucrose with a f i e l d applied orthogonal to the pores. The Faxen equation (Eq. 3.12) predicts that the membrane diffusivity w i l l decrease as 8, the ratio of molecular radius to pore radius, increases. This is due to increased drag between the diffusing molecule and pore surface. Any ordering of sucrose molecular clusters in the magnetic f i e l d would have the same effect as increasing the effective cross-sectional area of the molecule thereby increasing the drag between i t and the pore surface. Eqs. 3.12 and 3.18 can be combined to predict the change in the diffusion coefficient for molecular clusters aligned in a magnetic f i e l d . Defining 8f for a molecular cluster aligned in a magnetic f i e l d gives - 32 -Pf = ( r s + F 60>/ rp (3.18a) where r s and are the radius of the solute and pore respectively and F is an empirical geometrical factor relating the molecular geometry to the magnetic anisotropy. 3.3 Theory of Interferometry Interferometry provides a tool which w i l l measure refractive index profiles in a transparent medium. Local refractive index changes in a diffusion process are due to variations in composition and concentration. Concentration profiles can be probed with light which permits a continuous observation of the profile with minimal disturbance to the diffusion process. 3.3.1 Refractive Index The refractive index of a solution depends upon i t s composition and is defined as the ratio of the phase velocity of light in a vacuum c to that of light in the medium, v (3.19) The Lorentz-Lorenz law .96-98 for a pure substance is RM M (n2 - 1) P (n 2 + 2) (3.20) which defines the molar refractivity RM as a function of molecular weight M, density p, and refractive index n. Equation 3.20 defines RM as a constant for any pure substance at isothermal conditions. For a - 33 -multicomponent, isothermal system with no chemical interactions, an average molar refractivity RM can be defined which is based upon the summation of the contribution from each individual component R: RM = E X. RM. (3.21) i where X^ is the mole fraction of component i . If we similarly define an average molecular weight M M = E X, M. (3.22) i 1 1 then Eq. 3.20 becomes for a mixture E X ± M± (n 2 - 1) RM. = E X. RM = 1 1 i 1 1 p ( n 2 + 2 ) <3'23> Equation 3.23 predicts that the refractive index of a mixture w i l l be a function of mole fraction and density for the system at a given wave-length of light. Since density is affected by temperature and pressure i t is necessary for the system to be maintained at isothermal and isobaric conditions to determine n from concentration. For most solutions at low concentrations, n is a linear function of concentration c n = ac + b n (3.24) where a and b n are empirical parameters. - 34 -Interferometry is a technique which measures local refractive index in a medium, therefore yielding local concentration of a diffusing molecule i f a and b are known for the system. 3.2.2 Wave Nature of Light and Interference Thomas Young f i r s t demonstrated that the light propagates as a wave, in 1802. Young performed an experiment securing light from two secondary sources through the use of two pinhole apertures as shown in Figure 4. A collimated light source illuminates a pinhole which diffracts a spherical wavefront to a second screen containing two additional pinholes. These two pinholes diffract two secondary phase-related spherical waves. When these two waves were observed on a screen Young saw a series of alternating bright and dark fringes. This phenomenon was accounted for by explaining the propagation of light as a wave phenomena. Maxwell's equations of electromagnetic theory predict the 99 propagation of visible light as two periodic transverse wave motions. The waves are oscillating magnetic f i e l d and oscillating electric f i e l d vectors at 90° to each other. The other direction of propagation is shown in Figure 5. Such behavior can be described as simple harmonic motion E(z,t) = Ag cos (ut + 0) (3.25) with Ag being the amplitude of the wave, t is time, u i s the angular speed and 0 is the phase angle constant at t = 0. Expressing Eq. 3.25 in complex notation gives - 35 -Source S1 Screen Fig. 4. Young's Double Slit Experiment - 36 -L Electric Field Fig. 5. Propagation of Light as an Electromagnetic Wave - 37 -E(Z,t) = (3.26) where Z = Ae . If two waves have the same amplitude, wavelength and general propagation direction and are superimposed upon each other, interference w i l l result. If one wave is 180° out of phase with the other so the positive peaks of one wave coincide with the negative peaks of the other, destructive interference w i l l occur and the amplitude of the two waves w i l l be zero. If a plane wavefront of light enters a medium with a locally variable refractive index, i t w i l l not remain plane, but the phase velocity of the front w i l l be reduced as the refractive index increases. The resulting local variation in phase is proportional to the change in refractive index An and distance travelled by the wave. A quantity termed optical path length, L, is defined as L = ns (3.27) A phase difference A0 for the wave travelling through a constant geometrical distance and refractive index difference An is A0 = ^ i 2 n (3.28) A. o where \ is the wavelength of the light In a vacuum. Figure 6 shows a plane wavefront of light passing through a A ^ Waves in Phase n No Refractive Index Gradient Fig 6. Principles of Double Beam Interferometry - 39 -medium of constant and variable refractive index f i e l d . The maximum change in optical path length between the edges of the refractive index gradient is one wavelength of light, X. If the two wavefronts are superimposed an Interference pattern would result with constructive interference occurring at the center and destructive interference occurring at the edges. From such an interference pattern i t is possible to determine the phase relationship between the two wavefronts. This is the basis for double beam interferometry; two light beams are generated which have a known phase relationship in both time and space and are superimposed such that both the interference phenomenon and object are imaged. Phase differences between the two beams are then visible as interference fringes which can be evaluated to determine the local refractive index differences between the object and reference. 3.2.3 Double Beam Interferometry This section w i l l consider a quantitative description of the interference phenomenon observable in double beam interferometry and the necessary conditions for the observation and interpretation of interference fringes. According to the definition of refractive index, Eq. 3.19, the ratio of distance z travelled by a wave of velocity v, i s - = — (3.29) v c From the definition of optical path length, L, Eq. 3.27, z/v can be expressed as - 40 -( 3 . 3 0 ) v c T h e e q u a t i o n o f m o t i o n f o r a wave d e s c r i b e s t h e e l e c t r i c f i e l d v e c t o r i n s p a c e a n d t i m e . A t some d i s t a n c e z f r o m t h e s o u r c e o f l i g h t , t h e h a r m o n i c w a v e c a n be d e s c r i b e d b y c o m b i n i n g E q s 3 . 2 5 , a n d 3 . 3 0 E ( z , t ) = A c o s [ u ( t - - ) + 0] c o r w r i t i n g t h i s e q u a t i o n i n c o m p l e x n o t a t i o n g i v e s E ( Z , t ) = R e [ Z e x p ( i w t ) ] ( 3 . 3 2 ) w h e r e Z i s t h e c o m p l e x a m p l i t u d e d e f i n e d a s Z = R e ( A e x p - i [ u L / c + 0] } ( 3 . 3 3 ) E q u a t i o n s 3 . 3 1 o r 3 . 3 2 s t a t e t h a t a t a n y g i v e n o p t i c a l p a t h l e n g t h away f r o m t h e s o u r c e , E i s a h a r m o n i c f u n c t i o n o f t i m e o n l y . T h e wave d e s c r i b e d b y E q . 3 . 3 2 i s a n i d e a l s i n g l e m o n o c h r o m a t i c w a v e . I n r e a l i t y s u c h a l i g h t s o u r c e d o e s n o t e x i s t . E v e n a l a s e r p r o d u c e s l i g h t w h i c h i s e m i t t e d b y a n u m b e r o f i n d i v i d u a l a t o m s 20 e x c e e d i n g 10 . E a c h a t o m i s m o v i n g w i t h some v e l o c i t y d ue t o t h e t h e r m a l e n e r g y o f t h e a t o m s . I n a d d i t i o n a l l a t o m s do n o t e m i t t h e i r r a d i a t i o n s i m u l t a n e o u s l y b u t i n b u r s t s o f some f i n i t e d u r a t i o n . T h e t h e r m a l m o t i o n o f t h e a t o m s p r o d u c e s a D o p p l e r s h i f t i n t h e f r e q u e n c y o f 9 9 l i g h t e m i t t e d b y t h a t a t o m . T h e n e t r e s u l t i s t h a t some p o i n t i n s p a c e i s i l l u m i n a t e d b y many w a v e t r a i n s o f d i f f e r e n t f r e q u e n c i e s d u r i n g t h e t i m e o f o b s e r v a t i o n . I n a d d i t i o n n o o p t i c a l d e t e c t o r ( p h o t o g r a p h i c - 41 -film, or the human eye) has a response time brief enough to observe individual waves of light. Rather the response of the detector is many orders of magnitude greater than the period of visible light ( 1 0 - 1 5 s). Therefore the detector measures a time-averaged intensity over many thousands of oscillations. Real light sources can be represented as a superposition of terms 9 9 like Eq. 3.32 using the Fourier integral i f the frequencies are strongly peaked about a certain frequency u. The light is considered to be quasimonochromatic and the light source can be expressed as E(t) = A(t) cos [wt + 0(t)] (3.34) where A(t) and 0(t) are slowly varying functions compared with cosut; they change only slightly during one period 1/to . Real quasimonochromatic light sources such as the laser have randomly varying functions A(t) and 0(t). The period of oscillation for visible light 1/w, is so short that two separate sources of quasimonochromatic _5 radiation would dr i f t out of phase with each other in 10 second or less, making i t very d i f f i c u l t i f not impossible to observe interference 99 between the two sources. For this reason virtually a l l laboratory interferometers employ a single source of light. The term double beam interferometry refers to a class of interferometers which use a single source of quasimonochromatic light which is "divided" into two separate beams which are focused by a suitable lens system to superimpose the two beams producing an interference phenomenon. The Rayleigh interferometer used in this work - 42 -employs a double s i l t to produce two coherent sources from a s i n g l e l a s e r l i g h t source. The Rayleigh i n t e r f e r o m e t e r i s depicted from above i n Figure 7. A c o l l i m a t i n g lens expands the l a s e r beam and produces a wavefront of l i g h t which i s c o l l i m a t e d and p a r a l l e l across the e n t i r e area of the beam. This beam i l l u m i n a t e s two v e r t i c a l s l i t s which d i v i d e the beam i n t o two separate l i g h t sources e x a c t l y In phase with each other.These two beams then pass through separate compartments of the d i f f u s i o n c e l l , each f i l l e d w i t h a medium of d i f f e r e n t r e f r a c t i v e index. The r e f r a c t i v e index i n one compartment of the d i f f u s i o n c e l l i s constant and known. The other compartment i s f i l l e d with a medium of unknown r e f r a c t i v e index and with a r e f r a c t i v e index gradient i n the v e r t i c a l plane. Lens L l i s a plano-convex lens which focuses the two beams on a f o c a l plane, FP. The beams are superimposed and since they are mutually coherent i n t e r f e r e n c e occurs at FP. Lens L2 i s a c y l i n d e r lens which only focuses i n one plane, i n t h i s case the v e r t i c a l plane. I t focuses an image of the d i f f u s i o n c e l l v e r t i c a l l y on FP, which gives an "image" of the i n t e r f e r e n c e p a t t e r n on FP. From t h i s i n t e r f e r e n c e p a t t e r n i t i s p o s s i b l e to evaluate the l o c a l r e f r a c t i v e index d i f f e r e n c e between the two compartments of the d i f f u s i o n c e l l . 3.2.4 I n t e r p r e t a t i o n of Interference Fringes The spacing of the centers of adjacent i n t e r f e r e n c e f r i n g e s i s determined by the wavelength of l i g h t used and the geometry of the o p t i c a l system. R e c a l l that the distance between each i n t e r f e r e n c e f r i n g e represents a phase d i f f e r e n c e between the two beams equivalent to one wavelength of l i g h t . In t h i s s e c t i o n simple geometrical p r i n c i p l e s Y - 44 -w i l l be used to derive a q u a n t i t a t i v e r e l a t i o n s h i p between f r i n g e spacing and r e f r a c t i v e index changes i n the d i f f u s i o n c e l l . R e f e r r i n g to F i g . 7 i t can be seen that the angle a between the two beams i s Y a = t a n - 1 ^ — (3.35) Z Z F P The enlargement of the f o c a l plane i n Figure 7 shows the f r i n g e spacing A which i s a f u n c t i o n of both a and X, the wavelength of l i g h t i n a i r . X can be r e l a t e d to X Q, the wavelength of l i g h t i n a vacuum, through the r e f r a c t i v e index r e l a t i o n s h i p . X X = — (3.36) n a where n a i s the r e f r a c t i v e index of a i r . The value B i n Figure 7 i s equal to \ X B = — — = 2 _ (3.37) cosa n cosa a By s i m i l a r t r i a n g l e s , A i s therefore B X X A = = 2 = ° (3 38) tan a n cosa tan a n s i n a * a a Therefore the f r i n g e spacing f o r a Rayleigh i n t e r f e r o m e t e r may be determined at a given wavelength by the spacing of the masking s l i t s and the f o c a l length of L^. - 45 -The phase difference between the two beams is equal to the difference in optical path lengths between the two beams. The optical path length for each half of the interferometer is a summation of optical path length contributions from the diffusion c e l l , quartz diffusion c e l l windows, transmission through the lenses and transmission of light through the air. The total optical path length for either the reference of diffusion side of the interferometer is then where n^ and z^ are the refractive index and geometrical path length for medium i which would be either a lens, air, sucrose solution, or a quartz optical f l a t . Eq. 3.39 is valid i f the refractive index i s constant through each distance, z^. A shift in the interference pattern equal to one fringe spacing, A, corresponds to a phase change between the reference and diffusion beams equal to one wavelength. This represents a difference in optical path length equal to \. Since the geometrical distance is fixed, then a change in optical path length must result from a change in refractive index. The phase difference between the two beams A0 corresponding to a fringe shift of A is In the Rayleigh interferometer the optical path length contribution from the lenses, air, and quartz optical components is identical between the (3.39) A0 2nz A(n, (3.40) - 46 -diffusion and reference sides. Therefore, any changes in optical path length must result from a change in refractive index in the diffusion compartment since the refractive index in the reference side is constant. The refractive index difference between the reference and diffusion sides of the diffusion c e l l producing one fringe shift is equal to an optical path length difference of \. A(n, - n.) - (3.41) 2 1 ZDC where z n (, is the geometrical distance through the diffusion c e l l . The cylinder lens, L2, produces a vertical image of the diffusion c e l l on the focal plane. This has the effect of creating an i n f i n i t e number of interferometers in the vertical plane (see Figure 7). Figure 8 shows a typical interference pattern produced by this interferometer. Starting at either end of the interference pattern the fringes are straight so a constant phase difference exists between the two sides of the diffusion c e l l . The refractive index is constant and known at each end of the diffusion c e l l , so the fringes are interpreted with respect to n Q, the known refractive index at each end of the c e l l . Moving towards the center 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 c e l l . At the location where the fringe pattern has shifted an amount equal to A, the phase change has been one wavelength of light between two consecutive fringes. From Eq. 3.41 the amount An can be calculated which is the difference in refractive index - 47 -Figure 8 - T y p i c a l Rayleigh Interference Patterns (a) No r e f r a c t i v e index gradient i n c e l l (b) D i f f u s i o n proceeding. - 48 -between the two vertical points. It i s therefore possible to evaluate the refractive index difference at a l l locations in the c e l l simply by counting the number of fringe shifts and measuring the vertical location, x^, at which a shift occurs. Therefore n(x), which is the refractive index at x. corresponding to the i th fringe shift i s : x^ is measured from the boundary condition where n(x Q) = n Q is known. This equation i s valid i f the refractive index in one of the diffusion c e l l compartments i s constant, which i s the case in these experiments since the reference compartment is f i l l e d with d i s t i l l e d water and is isothermal. Equation 3.42 defines a refractive index profile for the diffusion c e l l . Since the membrane provides a discontinuity in the profile, Eq. 3.42 is applied to each end of the c e l l where the refractive index is known. Counting fringes towards the membrane yields a continuous profile on either side of the membrane. Applying Eq. 3.24 to the refractive index profile determines directly the concentration profile in the c e l l X i z DC (3.42) C,(x) _ C i = o (3.43) i ZDC t where a is determined empirically from refractive index versus concentration data. The mass flux at any point i n the c e l l may be determined by the time rate of change of the i n t e g r a l of the concentration p r o f i l e 5t x J = / C(x,t)dx (3.44) x o Concentration p r o f i l e s are measured at several d i f f e r e n t times during the experiment and a numerical method i s used to evaluate the i n t e g r a l and time d e r i v a t i v e s i n Eq. 3.44. The concentration gradient at a given time and p o s i t i o n can be found by taking the de r i v a t i v e of the concentration p r o f i l e with respect to x. With the mass fl u x and concentration gradient known, the d i f f u s i o n c o e f f i c i e n t D can be evaluated from Fick's f i r s t law J dC(x,t) dx IF '* C ( X ' T ) D X D = o (3.45) dC(x,t) dx D e t a i l s concerning the numerical method used to evaluate the i n t e g r a l and derivative i n Eq. (3.45) are included i n Chapter 5. - 50 -CHAPTER 4 EXPERIMENTAL APPARATUS AND PROCEDURE 4.1 Experimental Apparatus An optical interferometer measures small changes in the refractive index of a given medium, by detecting the phase change between two beams of coherent light. One wavefront forms a reference beam having been transmitted through a medium of known refractive index, while the comparison wavefront is transmitted through a medium of unknown refractive index. Recall from Chapter 3 that a phase change —5 between the two beams equal to 1 wavelength of laser light (6.328 x 10" cm) w i l l distort the interference fringe pattern by one fringe spacing. Therefore, any phase change between the two beams greater than a fraction of the wavelength w i l l cause a distortion of the observed fringe pattern. This degree of sensitivity means that a l l optical components such as lenses and test cells must be designed and constructed to introduce a total wavefront aberration of less than a fraction of a wavelength into the transmitted laser beam. Details concerning these optical components and other components of the experimental setup are included in appropriate sections of this text. Figure 9 shows schematically the experimental setup. The following sections describe in detail the individual components of this setup. 4.1.1 Quartz Diffusion C e l l The fused s i l i c a diffusion c e l l was custom fabricated by Interoptics, LTD, of Ottawa, Ontario to state of the art optical tolerances. The c e l l is shown ln Figure 10. It is constructed of three V 4Y D CELL DC L1 RC / EM \ Fig. 9 Experimental Setup L2 A A A / \AAA D CELL Quartz Diffusion Cell DC Diffusion Compartment RC Reference Compartment L1 Plano-convex lens 12 Cylinder Lens FPC Focal plane camera LBE Laser and beam expander EM Electromagnet S Masking Slits - 52 -Membrane Front S ide Fig. 10. Quartz Diffusion Cell - 53 -quartz optical flats cemented together with a special optical epoxy. The c e l l was cut in half to allow a membrane to be inserted at the mid-plane. Two parallel slots were dr i l l e d out of the center optical fla t to provide the diffusion and reference compartments. The inside edges were chamfered so that any excess epoxy squeezed out during the assembly procedure would have a place to collect and not run out onto the interior windows, restricting the clear aperture. The flatness of a l l optical surfaces was maintained at X/20 (\ = 6328 A) and parallelism of a l l surfaces kept to at least one arc second. The thickness of the reference and diffusion compartments was chosen as 1 cm. This value was selected to produce an optimum number of interference fringes for a sucrose concentration difference of 1% by weight. A 1% by weight change of sucrose concentration at 25°C 40 corresponds to a refractive index change An, of approximately 0.0015. For a An = 0.0015, the number of fringe shifts is given by Eq. 3.42 as Anz D C using \ = 6.328 x 10 cm (He-Ne laser light) and z ^ = 1 cm ( c e l l thickness) yields a value of 24 fringes for SN. Therefore 24 is the number of fringes which must be measured manually with the microscope for a 1% by weight change in sucrose concentration. Therefore, a c e l l thickness of 1 cm w i l l be convenient for ease of fabrication and structural ri g i d i t y and also to produce an optimum number of interference fringes for the range of sucrose concentration studied. - 54 -4 . 1 . 2 P iano—Convex L e n s and C y l i n d e r L e n s The plano-convex lens and cylinder lens are high quality singlets, custom made by Interoptics, LTD. They are fabricated from SF4 A grade high index glass with a refractive index of 1.74999 at a wavelength of 6.328 x 10 - 5 cm. 1 0 0 These lenses were made to state of the art tolerances for minimal spherical aberration. The total wavefront distortion for these lenses is diffraction limited to X/4. The spherical lens has a focal length of 67.3 cm. The diameter of this lens is 10 cm and the thickness at the edge is 1 cm. The cylinder lens has a focal length of 15 cm. The length of the lens is 6 cm and the width Is 3 cm in the power plane. The edge thickness of this lens is 0.935 cm. The focal lengths of these lenses were selected to produce a clear, easily observable fringe pattern at the focal plane and also to meet the physical constraints introduced by the geometry of the magnet and laboratory space. The fringe spacing A is given by Eqs. 3.35 and 3.38 as a function of the plano-convex lens focal length f and s l i t width Y . s A = Y4- (4'2) s a where X is the wavelength of laser light and n is the refractive ct index of the air. Y G is equal to 1.0 cm which is determined by the spacing of the diffusion and reference compartments in the diffusion - 55 -c e l l . The focal length should be as large as practical to give an easily observable fringe spacing. However, simple physical limitations due to the geometry of the magnet and optical bench limit f to 1 meter or less. A value of 67.3 cm was selected because the tooling was readily available at Interoptics, LTD to manufacture this lens. This focal length produces a fringe spacing from Eq. 4.1 of A = 0.00426 cm, which was easily observable through the 30 power measuring microscope. The 15 cm focal length was selected for the cylinder lens to produce an image of the diffusion c e l l which would f i l l the frame of a 35 mm film. This lens produced an image of the test c e l l 0.9 cm long i n the focal plane of the camera. The vertical magnification of this configuration was determined experimentally to be 0.32. Figure 19 summarizes a l l optical component parameters and dimensions. 4 . 1 . 3 V e r t i c a l S l i t A s s e m b l y Interference fringes are observable as the result of the phase difference between two different superimposed wavefronts of light. A double masking s l i t assembly divides the spatially coherent laser beam into two rectangular vertical beams which illuminate the reference and diffusion compartments of the diffusion c e l l and are in phase with each other prior to being transmitted through the diffusion c e l l . When a s l i t i s illuminated with monochromatic and collimated light a diffraction phenomena occurs which is classified as Frauenhoffer 9 9 diffraction. This produces an "envelope" of coherent light, the width of which is determined by the width of the s l i t . The diffraction envelope width increases as the s l i t width decreases, while the - 56 -intensity of the envelope decreases with decreasing s l i t width. The selection of the optimum s l i t width is therefore a compromise between a width which gives adequate intensity of the diffraction envelope and yet is wide enough to easily observe the fringe pattern. The correct s l i t width was found by t r i a l and error until a well-illuminated, clear interference pattern was observed on the focal plane of the camera. The f i n a l s l i t width was chosen as 0.05 cm ± 0.0001 cm. The distance between the dual s l i t s is determined from the spacing of the diffusion and reference c e l l compartments and in this case is 1.0 ± 0.0001 cm. Ordinary razor blades provide an inexpensive source of straight edges for s l i t construction. Four razor blades were used to make the double s l i t assembly. They were cemented onto a special frame under the 30 power measuring microscope. The microscope was used to observe their position during construction and establish adequate spacing, parallelism, and uniform width. The s l i t geometry was measured and verified to the limits of accuracy for the measuring microscope, ± 0.0001 cm. This assembly produced clear, well defined interference fringes and the cost is a fraction of the cost of commercially available masking s l i t s . This is apparent from the photograph of a typical set of interference fringes shown in Figure 8. 4.1.4 Laser and Collimating Lens System A Spectra-Physics model 124B helium-neon laser provides a coherent light source for the experiment. It is rated at 15 mW optical output at a wavelength of 6328 A . It operates in the TEM Q O mode.1"1 A Spectra-Physics model 332 and 336 spatial f i l t e r and - 57 -collimating lens assembly are used to provide a collimated, spatially 102 filtered beam. The beam is f i r s t focused and passed through a pinhole aperture. This has the effect of removing any spatial noise in the beam and providing a smooth Gaussian intensity profile across the output beam. Spatial noise is produced from diffraction effects as the beam encounters small irregularities on the inside bore of the laser tube. To eliminate the scattered light from these diffraction effects the beam is focused and passed through an aperture 1.5 times the minimum focused spot diameter. This eliminates any noise and transmits about 99% of the Gaussian beam power. This spatially filtered beam is then passed through the beam expander lens and collimator which produces a beam 5.0 cm in diameter. Total wavefront deformation with this assembly is less than \/10. 4.1.5 Optical Bench and Component Mounts Since the objective of this work is to measure the effects of a homogeneous applied magnetic f i e l d on diffusion, i t is imperative that no ferromagnetic material be placed in the f i e l d which would destroy the homogeneity. This criterion eliminated most commercially available optical benches and component mounts which usually contain some quantity of steel. Therefore, the optical bench and component mounts were custom made by the UBC chemical engineering workshop using only non-ferromagnetic materials; aluminum and nylon. The optical bench consisted of a two meter long aluminum beam placed between the poles of the magnet. It rests upon two concrete and rubber vibration isolation platforms placed at each end of the magnet - 58 -which center the bench between the poles of the magnet, 70 cm above the floor. The beam is 10 cm wide and 5 cm thick. A strip of aluminum 4 cm wide by 0.5 cm thick is mounted on the optical bench along the entire length. The special mounting blocks slide along this strip and can be clamped in place by number 8 machine screws. The aluminum diffusion c e l l holder is shown to scale in Figure 11. The c e l l f i t s in rectangular groove cut in a circular plate. Four vertical retaining posts keep i t vertical and two nylon shim screws are used to align the two halves of the diffusion c e l l into the same plane. The c e l l is held in place by elastic retaining strips. The plano-convex lens and mount are shown to scale in Figure 12. A threaded retaining ring holds i t in the circular holder between Teflon washers. The cylinder lens and mount are shown to scale in Figure 13. The lens slides In between two grooved retaining posts and is clamped in place by two nylon retaining screws. A l l component mounts are placed in the clamping blocks by adjustable aluminum rods so the height can be adjusted. The axial position of the components can then be changed by moving the clamping block along the bench. A clamping block is depicted to scale in Figure 14. 4.1.6 Vibration I s o l a t i o n The configuration of the Rayleigh interferometer makes i t relatively insensitive to vibrations. Vibrations could introduce errors into the measurements only i f the optical path length between the two - 59 -Top V iew ( « * — 5 c m — • ] Side V iew Nylon © Retaining (D Screws 1 1 CD Fig. 11. Diffusion Cell Holder - 60 -0 pzzzzzzzzzfy I L I L- J I J o3 i , — t T i i Side V iew 0 _L Front V iew i _ Fig. 13. Cyl inder Lens and Mount Fig. 14. Optical Bench,and Mount ing Block - 63 -interfering beams was changed. Because the diffusion c e l l is small (distance between reference and diffusion compartments is 1 cm) and rigidly constructed, any relative displacement between the two compartments is minimized, thereby reducing optical path length differences from vibrations. The optical bench rests on two vibration isolated bases located on either side of the magnet. The base is constructed of concrete patio blocks sandwiched between high density foam rubber. Each base weighs approximately 500 kilograms. Figure 15 shows one of the vibration isolation platforms. The massive layered blocks and rubber combination acts as a spring mass system which provides sufficient dampening action to attenuate any vibration which might be transmitted through the floor. The fringes were observed through the microscope to verify that any building vibrations would not cause any movement. When a building compressor in an adjacent room was switched on, no perceivable movement in the fringe pattern could be observed through the 30 power microscope. 4.1.7 Temperature Control Maintaining a constant temperature in the diffusion c e l l Is essential in order to eliminate an important independent variable in the diffusion process. The Einstein-Stokes equation (Eq. 3.8) for a liquid diffusion coefficient predicts that the diffusion coefficient is directly proportional to the absolute temperature of the system. Therefore any change in temperature w i l l have a direct effect on the diffusion coefficient. Temperature fluctuations in the air adjacent to the diffusion - 64 -Figure 15 - Concrete and Rubber Vibration Isolation Platform. - 65 -c e l l are controlled to ± 0.1°C. This i s achieved by a dual temperature control system. The ambient air temperature in the room was controlled to ± 1°C by a Koldwave model K16DF water-cooled air conditioner, rated at 4835 W. This was also used to remove the heat produced by the magnet and other electronic equipment. The optical bench is enclosed in a 1 cm thick styrofoam box to provide thermal insulation. A 25 watt fan, the type used to cool electronic equipment, is mounted along the enclosure 30 cm from the diffusion c e l l . An electric resistance heater is mounted inside the enclosure across the output of the fan. It was constructed by wrapping nichrome wire around a 1.5 cm diameter glass tube. It i s rated at 300 W at 115 VAC input. The air temperature adjacent to the diffusion c e l l (4 cm away) i s measured by an Omega model 100 R 30 platinum RTD, thermometer. The electric resistance of this thermometer is proportional to temperature. An Omega model 4201 proportional and on-off controller measures the resistance and therefore temperature of the thermometer. The sensitivity of the equipment is ± 0.05°C. This controller has an adjustable bandwidth between 0 and 5% of f u l l scale. An error signal i s generated by the controller which is proportional to the difference between the measured temperature and the setpoint temperature. This error signal in turn is used to drive a variable transformer which powers the electric heater. The bandwidth adjustment and voltage output of the variable transformer were adjusted simultaneously to provide a constant cycle time for the heater. This proved to be between 15 and 20 - 66 -VAC with a cycle time of 10 to 15 seconds. The temperature fluctuations measured by the platinum thermometer and temperature controller combination were always within 0.1°C of the 25°C setpoint temperature. Turning on the magnet produced no change in the temperature indicated by the controller. 4.1.8 Membrane and Sucrose Solution The membranes selected for this work are General Electric 103 Nucleopore membranes. They are made from a polycarbonate film which is bombarded by high speed sub-atomic particles from a nuclear reactor. The particles pass straight through the film, leaving tracks of molecular damage which can be etched preferentially in a chemical bath to round, linear pores of uniform diameter. The maximum pore diameter certified by the manufacturer varies no more than + 0% to -10% for each membrane. The surface is smooth and f l a t , maximum peak to valley distance on the membrane surfaces is less than 0.1 um. The membrane dimensions used in this experiment are shown in Table I. Table I Membrane Parameters for GE Nucleopore Membranes Pore Size, um Pore Density pores/cm Thickness, um Porosity cm /cm 8.0 0.8 1 x 105 3 x 107 50 10 .050 .151 The solution was made from AMCHEM reagent grade sucrose dissolved and diluted volumetrically with doubly d i s t i l l e d water at room temperature. - 67 -4 . 1 . 9 E l e c t r o m a g n e t A Varian Associates 30 cm Model V-7300 electromagnet was used to apply the magnetic f i e l d to the diffusion c e l l . The magnet was fit t e d with an 18 centimeter diameter pole piece with a 10 cm gap width. A Varian Associates Model V-7800 DC power supply and V-7872 heat exchanger were used to cool this magnet which could generate fields as high as 12.5 kG. Homogeneity of the f i e l d has been measured to better than 7 x 10"H kG over the diameter of the pole piece at an applied f i e l d strength 21 of 9.0 kG The f i e l d strength homogeneity measurements are shown in Figure 16. The f i e l d strength was measured by a Hall-effect crystal probe mounted on one pole piece. The crystal is excited by an electronic oscillator and the output voltage from the crystal is proportional to the magnetic f i e l d . This signal is then used to control the current produced by the power supply. This configuration enables the magnetic f i e l d to be controlled to within 1PPM for a ± 10% change in line voltage or load resistance. 1 0 4 The power supply produces a continuously variable DC output between 5 amperes and 114 amperes. The magnet is cooled by a two loop water to water heat exchanger. D i s t i l l e d water is continuously recirculated through the magnet and one side of the heat exchanger. Filtered city water flows through the other side, with waste heat being drained to the city sewer lines. A temperature sensor at the reservoir inlet controls a proportional flow control value in the raw water circuit, thus providing automatic temperature control of the magnet assembly water inlet temperature, maintained at less than 50°C. - 68 -k6 r> . " • ' I ' • ' — ' — i — ' — i — i — i — r — i — i — i — i — i — . — i — i — i distance from the center ot magnet in cm Figure 16 - Magnetic Field Homogeneity . (used by permission). - 69 -An experiment was conducted to establish that the magnetic f i e l d did not affect the optical characteristics of the diffusion c e l l , laser, and optical components. The fringe pattern was examined through the 30 power measuring microscope when the magnet was brought up to f u l l power (12.5 kG) and then switched off during a preliminary experiment with d i s t i l l e d water and 1% sucrose solution in the c e l l . No change in the fringe pattern could be seen through the microscope. Therefore, i t was assumed that the magnetic f i e l d had no observable effects on the optical properties of this system. 4.1.10 Camera and Measuring Microscope The interference fringes are photographed with a Contax model 139 Quartz 35mm single lens reflex camera. Kodak Panatonic-X film was used with Edwal FG-7 developer to provide a fine grain and high resolution medium for recording the fringe patterns. The Model 139 camera has an electronic automatic exposure control. However, the best exposure was found by t r i a l and error to be 1/500 second, which was sufficiently short to give sharp photographs. The unfiltered laser beam was i n i t i a l l y too intense to yield satisfactory exposure, even at the fastest shutter speed available. A Kodak Wratten No. 59 f i l t e r was used to attenuate the beam. This provided over 95% attenuation in the red-orange region of the spectrum which proved satisfactory for the 25 ASA film speed. Since the f i l t e r was placed less than 6 cm from the focal plane there was no noticeable distortion of the interference fringes. A Contax Model S infrared controller set was used to provide Figure 17 - O p t i c a l Bench and Magnet LBE) - Laser and Beam Expander, Sl)-Masking S l i t s , DC)-Diffusion C e l l , LI)-Piano-Convex Lens, L2) Cylinder Lens, EB) Extension Bellows. - 71 -remote control of the camera, thereby avoiding any vibration caused from a manual shutter release. The camera was mounted on a Yashica Model F adjustable extension bellows providing a light-tight interface with the experimental enclosure and horizontal adjustment for proper focusing. The extension bellows was mounted on the optical bench using one of the aluminum mounting blocks. Fringes were measured with a Gaertner model M-1160 measuring microscope. This consisted of a 32 power microscope mounted on a precision vernier stage capable of measuring ± 0.0001 cm distances. A 90° spider silk cross hair was used with the moveable stage to accurately measure fringe spacing. The precision lead screw has been calibrated to a standard scale, the accuracy of which is certified by the National Bureau of Standards. A photograph of the optical bench assembly and magnet is shown in Figure 17. 4.2 Experimental Procedure The complete procedure used to set up the interferometer and obtain interference fringe data is described below. Proper alignment of the interferometer involved four main steps: 1) Collimation and spatial f i l t e r i n g of the laser beam through focus and position adjustments on the collimating lens spatial f i l t e r assembly; 2) Proper height adjustment for a l l optical components; 3) Focusing the plano-convex lens L l ; 4) and proper focus for the cylinder lens in order to focus on a plane in the center of the diffusion c e l l . When the interferometer - 72 -was satisfactorily aligned the membrane was mounted in the c e l l and an experiment began. Appropriate details are described in the following sections. Figure 9 shows the respective components described in the alignment procedure. 4.2.1 Aligning the Interferometer (1) The spatial f i l t e r and collimating lens are adjusted to provide maximum beam Intensity and good collimation. The X and Y motion adjustment screws on the pinhole aperture are f i r s t adjusted to center the aperture on the focused beam. The aperture is then adjusted axially to bring It into the focal plane of the beam, yielding maximum beam intensity with the most uniformly illuminated output. The collimating lens is then screwed on the spatial f i l t e r . To collimate the beam, i t is directed to a screen placed across the room, approximately five meters away from the laser. The beam diameter is measured upon exit from the laser and the lens adjusted to produce an equivalent spot size on the screen. The beam is now considered collimated and f i l t e r e d . (2) The laser is placed on the optical bench. It is directed down the centerline of the bench. A calibrated ruler is used to check the height and verify that the beam is parallel with the optical bench centerline. Shims are placed under the laser head assembly to adjust the height and vertical angle of propagation of the beam. (3) The c e l l holder and lenses are next placed on the optical bench. The clamping screws in the mounting block are kept loose and the lenses are placed in approximate position. The c e l l holder is centered between the pole pieces of the magnet. The height of each component is - 73 -adjusted to bring i t into l i n e with the beam center. (4) Next the s l i t assembly i s mounted over the opening to the insul a t e d enclosure. A square i s used to orient the s l i t s orthogonal to the o p t i c a l bench axis. The p o s i t i o n i s adjusted to center the s l i t s i n the 50 mm diameter laser beam. (5) The b i f o c a l lens system i s then brought into focus. This i s accomplished by placing a small screen i n the f o c a l plane of the camera body. A fine wire grid (1 mm spacing) i s placed i n the d i f f u s i o n c e l l holder at a l o c a t i o n corresponding to the correct f o c a l plane f o r the d i f f u s i o n c e l l . The cylinder lens, L2, i s then removed from the mounting block. The plano-convex lens, L l , i s moved a x i a l l y along the o p t i c a l bench u n t i l a spot was focused on the screen of the f o c a l plane camera. The clamping screws are then tightened. The cylinder lens i s placed back i n the mounting block and i t s a x i a l p o s i t i o n on the o p t i c a l bench adjusted u n t i l an image of the wire mesh i s focused on the camera screen. The measuring microscope i s then used to examine the image of the wire mesh and interference pattern to v e r i f y sharp focus and cl e a r f r i n g e s . (7) F i n a l l y the lenses are checked to v e r i f y that the surfaces are orthogonal to the propagation d i r e c t i o n of the laser beam. They are rotated i n the mounting blocks u n t i l the r e f l e c t i o n from the masking s l i t s i s r e f l e c t e d back into the masking s l i t s . The heights of the lenses are measured again before the clamping screws are tightened to v e r i f y no change during the alignment. The alignment of the interferometer i s now complete. The wire mesh i s removed from the - 74 -diffusion c e l l holder and the fringe pattern examined with the measuring microscope. The fringes should be straight since the refractive index is constant in the c e l l at this time. 4.2.2 Diffusion C e l l and Membrane Preparation (1) To remove dissolved gasses from the d i s t i l l e d water i t is boiled for approximately 30 minutes. A 5 cc hypodermic syringe is f i l l e d with hot d i s t i l l e d water, before the water cools and air is re-dissolved in i t . The remaining water is then cooled in an air tight sealed flask with no air space. 200 ml of this cooled water is then used to make a 1% by weight sucrose solution. This solution is then evacuated by a mechanical vacuum pump for approximately one hour to remove any gases which may have dissolved in the water during the mixing process. There may be a slight concentration change due to the evaporation of water during the gas removal. Therefore, the refractive index of this solution was measured with a Bausch and Lomb model 3L refractometer at 25°C to four decimal places before each experimental run. A 5 cc syringe is then f i l l e d with sucrose solution. The syringes f i l l e d with water and solution are placed in the insulated enclosure which has been set to the desired temperature. A minimum of four hours are allowed for the system to reach thermal equilibrium before an experiment is started. (2) Next the diffusion c e l l and membrane are prepared. The membrane is boiled in d i s t i l l e d water for approximately thirty minutes before mounting to remove any unrelaxed stresses and entrapped air in the pores. This makes i t easier to mount the membrane without any warps - 75 -o r w r i n k l e s . A new membrane i s used f o r each r u n . (3) The d i f f u s i o n c e l l i s c l e a n e d w i t h c o t t o n Q - t i p s and s o a p . I t i s r i n s e d w e l l w i t h d i s t i l l e d w a t e r . (4) A v e r y t h i n coat of s i l i c o n vacuum g rease i s a p p l i e d to b o t h i n t e r f a c e s o f the d i f f u s i o n c e l l . The g r ea se p r o v i d e s a w a t e r - t i g h t s e a l between the membrane and g l a s s and i s not s o f t e n e d by w a t e r . I f t he coa t i s too t h i c k excess g r ea se w i l l e x t r u d e , d i s t o r t i n g the membrane shadow. (5) The lower h a l f o f the c e l l i s p l a c e d on a t a b l e top and the r e f e r e n c e and d i f f u s i o n s i d e compartments f i l l e d w i t h water and s u c r o s e s o l u t i o n , r e s p e c t i v e l y , f rom the hypode rmic s y r i n g e s . The compartments a re f i l l e d u n t i l the men iscus i s j u s t o ve r the top s u r f a c e and then the membrane i s removed f rom the b o i l i n g water and p l a c e d on the g r e a s e d g l a s s s u r f a c e . The membrane i s smoothed f l a t w i t h a c o t t o n Q - t i p , e x p e l l i n g any e x t r a water o r s o l u t i o n . G r e a t c a r e must be t aken to a s s u r e t h e r e a re no warps o r w r i n k l e s i n the membrane and i t makes comp le t e c o n t a c t w i t h the g r e a s e d s u r f a c e w i t h o u t any g r e a s e e x t r u d i n g onto the d i f f u s i o n a r e a o f the membrane. ( 6 ) The lower h a l f o f the d i f f u s i o n c e l l i s p l a c e d i n the base o f the c e l l mount ing p l a t e . The top h a l f i s then p l a c e d on the membrane, u s i n g c a r e t o a s s u r e no w r i n k l e s a r e f o r m e d . The r e f l e c t i o n of the l a b o r a t o r y f l u o r e s c e n t l i g h t tubes f rom the membrane s u r f a c e w i l l be u n d i s t o r t e d i f the membrane i s smooth and no warps o r w r i n k l e s e x i s t . The hypodermic s y r i n g e i s then used to f i l l the r e f e r e n c e s i d e compartment w i t h d i s t i l l e d water i f t h e s e t e s t s a r e s a t i s f i e d . The top - 76 -half of the d i f f u s i o n c e l l side i s f i l l e d with 1% sucrose s o l u t i o n with the hypodermic syringe from the insulated enclosure. Rubber stoppers are placed i n the openings to the d i f f u s i o n c e l l . When the experiment i s started, the sucrose so l u t i o n i n the upper half of the d i f f u s i o n side compartment i s removed and replaced with d i s t i l l e d water. The water, being of lower density than the sucrose s o l u t i o n , was placed In the top half of the c e l l to eliminate bulk flow e f f e c t s which would r e s u l t from the more dense l i q u i d being above the lower density one. (7) The d i f f u s i o n c e l l holder i s assembled and placed i n the mounting blocks on the o p t i c a l bench. The c e l l alignment i s adjusted by turning on the las e r and observing the r e f l e c t i o n of the las e r beam from the d i f f u s i o n c e l l surfaces back into the masking s l i t s . The two nylon shimming screws are adjusted to bring the two h a l f s of the c e l l s into the same plane, making both surfaces orthogonal to the laser beam. When the r e f l e c t i o n s of the masking s l i t form v e r t i c a l l i n e s , both surfaces are f l a t and aligned i n the same plane. The c e l l holder i s then rotated i n the mounting block u n t i l the s l i t r e f l e c t i o n i s directed exactly back into the masking s l i t s . The height i s checked again to v e r i f y the c e l l i s centered i n the las e r beam. When a l l of these steps are completed the c e l l i s then considered aligned and the experiment i s ready to begin. The cover of the enclosure i s replaced and at least two add i t i o n a l hours are allowed for the temperature to e q u i l i b r a t e . The o p t i c a l components are now i n alignment and the experiment i s ready to begin. - 77 -4.2.3 Beginning an Experiment I n i t i a l l y the diffusion c e l l is completely f i l l e d with sucrose solution in the diffusion side compartment and the reference side is f i l l e d with d i s t i l l e d water until temperature equilibrium i s reached. An experiment is initiated by removing the sucrose solution from the top half of the diffusion side compartment and replacing i t with water. (1) The sucrose solution in the top half of the diffusion c e l l side is carefully removed with a hypodermic syringe. It is essential that the membrane surface not be touched with the needle or else warping w i l l occur and the run must be aborted. As much solution as possible i s removed without actually touching the membrane. Only a slight meniscus in the corners of the c e l l remains with careful solution removal. It is estimated that the amount of solution remaining is less than one drop or 0.03 ml out of a total volume of the upper half of the compartment of 1.3 ml. The magnet is then turned on and water is inserted into the empty top half of the diffusion c e l l to replace the sucrose using another hypodermic syringe. A l l syringes have been kept inside the enclosure during the time the temperature i s equilibrating so they are a l l at a constant temperature of 25°C. The stop watch is started after the water i s injected. The rubber stopper is then replaced in the opening of the diffusion c e l l . (2) The fringe pattern is observed through the measuring microscope, sighting the cross hairs onto a fringe near the membrane shadow. When that fringe has shifted by a distance approximately equal to two fringes, the camera is attached to the bellows and a photograph - 78 -taken using the remote control. The f i r s t photograph is not taken until 15 minutes have elapsed to allow any convection effects from the i n i t i a l c e l l f i l l i n g to dissipate. The rate of change of the fringe pattern is an exponential function, so i n i t i a l l y the time intervals are quite close together (approximately 15 minutes), while at the end of the experiment the change is much more gradual so photographs are only taken every hour. A minimum of five photographs are taken for each experiment. If the fringes at either end of the c e l l have moved, the run is terminated since the boundary conditions have changed. The run was terminated after 3 hours. - 79 -CHAPTER 5 FRINGE PATTERNS AND DATA ANALYSIS A t y p i c a l interference fringe p r o f i l e i s shown i n Figure 8. The pattern on the l e f t i s observed with a uniform sucrose concentration through the c e l l , and therefore a uniform r e f r a c t i v e index i n the c e l l . The phase difference between the reference and d i f f u s i o n side beams i s constant through the ent i r e length of the d i f f u s i o n c e l l , therefore the interference fringes are straight l i n e s . However, i n the presence of a r e f r a c t i v e index gradient i n one side of the d i f f u s i o n c e l l , the phase difference i s no longer equal between the two sides, but becomes a function of v e r t i c a l p o s i t i o n i n the c e l l . This causes a bending of the fringe pattern as shown i n the right h a l f of Figure 8. The degree of bending i n the pattern i s d i r e c t l y proportional to the r e f r a c t i v e index at that l o c a t i o n i n the d i f f u s i o n c e l l . The equations used to determine r e f r a c t i v e index, concentration, mass fluxes, and d i f f u s i v i t i e s from fringe data are described i n d e t a i l i n th i s section. 5 .1 Assumptions The following assumptions and s i m p l i f i c a t i o n s are made i n t h i s a n a l ysis: 1) d i f f u s i o n occurs only i n one-dimension, 2) l i g h t rays are deflected as they are transmitted through a r e f r a c t i v e index gradient, but t r a v e l through the c e l l i n a straight l i n e , 3) d i f f u s i o n through the membrane i s steady state at any given time, i . e . , the mass fluxes are equal on both surfaces of the membrane, 4) the d i f f u s i o n c o e f f i c i e n t i s independent of concentration at any given time i n the d i f f u s i o n process, 5) only binary d i f f u s i o n i s considered, i . e . sucrose and water, impurities being ignored. - 80 -5 . 1 . 1 One—d imens i ona l D i f f u s i o n The assumption of one-dimensional diffusion ignores any boundary layer effects occurring at the walls of the diffusion c e l l . The plane of focus is located near the center of the diffusion c e l l (approximately 5 mm from the wall). I n i t i a l l y the concentration gradient is only one-dimensional in the vertical direction. Viscous drag between the diffusing molecules and wall would cause a three dimensional concentration profile, however in the midplane of the c e l l the profile should be flat since boundary layers are no thicker than several molecular diameters (10.6 A for sucrose). Therefore, diffusion near the c e l l midplane w i l l remain essentially one-dimensional. 5 . 1 . 2 L i g h t R a y D e f l e c t i o n The path of a light ray through a refractive index gradient in the diffusion c e l l is shown in Figure 18. The path, L, is given by L = J* n(x)ds (5.1) and the slope of the deflection is given b y , 1 0 5 - 1 0 6 ds n(x) 1 dx o This predicts that s w i l l be an arc bending towards the direction of increasing refractive index. In these experiments with a dilute sucrose solution of 1% by weight the amount of deflection is small, so i t w i l l be assumed the ray is bent, but travels through the c e l l in a straight line as shown in Figure 18. Wavefront deflection is discussed in greater detail in Section 5.2. - 81 -Fig 18. Wavefront Deflection Th rough Diffusion Cell and Refractive Index Gradient. 5.1.3 Steady State Membrane Diffusion The Nucleopore membranes used i n th i s experiment are thi n (10 urn). Also, they are saturated with sucrose solution at the st a r t of a run since the c e l l i s f i l l e d with s o l u t i o n at least two hours p r i o r to the s t a r t of an experiment. I f there i s no swelling or shrinkage of the membrane ( e a s i l y observable during the duration of the run through the microscope by a change i n the membrane shadow), then conservation of mass requires that the mass flux be equal at both membrane surfaces, i . e . there are no sources or sinks of sucrose within the membrane i t s e l f . 5.1.4 Constant Diffusivity This assumption constitutes i d e a l F i c k i a n d i f f u s i o n . It i s v a l i d i f the maximum concentration change i n the d i f f u s i o n c e l l i s small, since the d i f f u s i o n c o e f f i c i e n t decreases with increasing concentration. The constant d i f f u s i v i t y condition i s assumed f or d i f f u s i o n through the membrane at any given time i n t e r v a l during the run. The maximum difference i n concentration during these experiments i s 1% by weight between the two halves of the c e l l at zero time. The concentration change across the membrane i s even smaller as the run proceeds since d i f f u s i o n i s not steady state and the concentration difference through the membrane decreases with time. This assumption has been v e r i f i e d experimentally by Bollenbeck for a 1% sucrose , 40 change. 5.2 Calculating Refractive Index Profile The obtained fringe pattern i s traversed v e r t i c a l l y with the measuring microscope. Figure 8 shows a t y p i c a l interference pattern - 8 3 -d u r i n g a n e x p e r i m e n t . T h e r e f e r e n c e p o i n t f o r t h e m e a s u r e m e n t s s t a r t a t e a c h e n d o f t h e d i f f u s i o n c e l l w h e r e t h e r e f r a c t i v e i n d e x i s c o n s t a n t a n d t h e f r i n g e s s t r a i g h t . T h e l o c a t i o n w h e r e e a c h f r i n g e c r o s s e s t h e v e r t i c a l a x i s o f t r a v e r s e i s r e c o r d e d w i t h t h e c o r r e s p o n d i n g f r i n g e n u m b e r . T h i s i s r e p e a t e d w i t h t h e o t h e r h a l f o f t h e c e l l . T h e f i n a l r e s u l t i s a s e t o f f r i n g e n u m b e r s a n d d i s p l a c e m e n t s c o u n t i n g f r o m t h e e n d s o f t h e c e l l t o w a r d s t h e m e m b r a n e . E a c h f r i n g e s h i f t r e p r e s e n t s a p h a s e d i f f e r e n c e l i g h t . T h i s p h a s e d i f f e r e n c e A0 i s e q u a l t o t h e o p t i c a l p a t h l e n g t h d i f f e r e n c e b e t w e e n t h e t w o b e a m s A0 = X = A n s (5.3) S i n c e s i s t h e g e o m e t r i c a l d i s t a n c e t h e b e a m t r a v e l s a n d i s k n o w n f r o m t h e g e o m e t r y o f t h e i n t e r f e r o m e t e r , A n c a n b e c a l c u l a t e d f r o m E q u . 5.3. T h e r e f o r e , k n o w i n g t h e v e r t i c a l l o c a t i o n s i n t h e c e l l w h i c h c o r r e s p o n d t o a p h a s e s h i f t o f X, f r o m t h e i n t e r f e r e n c e f r i n g e s , i t i s p o s s i b l e t o c a l c u l a t e a s e t o f p o i n t s g i v i n g n a s a f u n c t i o n o f x ^ , t h e v e r t i c a l p o s i t i o n i n t h e c e l l c o r r e s p o n d i n g t o a f r i n g e s h i f t . n ( x ± ) = n r o + A n ( x ± ) (5.4) n r o i s t h e k n o w n r e f r a c t i v e i n d e x a t e a c h e n d o f t h e c e l l e q u a l t o t h e i n i t i a l r e f r a c t i v e i n d e x a t t i m e = 0 . T h i s s i m p l e a n a l y s i s i g n o r e s t h e d e f l e c t i o n o f a l i g h t r a y t h r o u g h a r e f r a c t i v e i n d e x g r a d i e n t n o r m a l t o t h e d i r e c t i o n o f p r o p a g a t i o n , a s s u m i n g t h e l i g h t r a y t r a v e l s i n a s t r a i g h t h o r i z o n t a l - 84 -line through a l l locations in the diffusion c e l l . The physical reason for this deflection lies in the dependence of propagation velocity on refractive index. The different elements of a wavefront propagate through the medium at different times, causing the wavefront to t i l t in the direction of increasing refractive index (see Figure 18). The downward deflection of the light ray introduces errors into the analysis because the vertical position of a ray on the focal plane do not correspond to the actual vertical position of the ray entering the diffusion c e l l . If the original entry position of the ray and the optical path of the ray through the c e l l can be determined, then a corrected refractive index profile may also be derived for the one-dimensional diffusion process. A simple iterative scheme was used to derive the correct refractive index profile. The approximate refractive index profile was f i r s t determined from the observed fringe pattern data and Eq. 5.4. Snell's law* 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 ray observed on the film and the refractive index profile is modified. The entry position in the diffusion c e l l is modified by an amount proportional to the error between the same ray and corresponding position on the focal plane observed for this ray. This procedure is repeated until the "corrected" *Snell's law states nisinai = n 2 s i n a 2 , where ni and n2 are the refractive indices for two different mediums and a i , 0:2 are the angles of a light ray being transmitted through these respective media. refractive index profile produces a fringe pattern corresponding to the one observed. The calculations are summarized below: (1) Calculate refractive index profile from fringe data using Eq. 5.4 assuming no ray bending. (2) Using profile determined from step 1, find a suitable correlation for n(x) (details concerning correlation are included in following section). (4) If the arc of the ray passing through the diffusing fluid i s represented by a straight line passing through a locally constant refractive index gradient then the refractive index along the path is given by n(x) - f(x) (5.5) (3) From Eq. 5.5 evaluate dn/dx. (5.6) (5) The average refractive index, n, along the line is then n = n(x ) + 1/2 o (5.7) (6) The path of the ray is given by a modification of Eq. 5.2 as - s e -cy) Integrating Eq. 5.8 with a constant refractive index gives 4^- = — 7 — r [ 4^  ] s (5.9) ds n(x.) dx x . i o,i (8) Substituting n(x) from Eq. 5.5 and integrating gives 1 2(n .) 3 L d x -| 0 , i 0 1 which is the vertical deflection through the c e l l thickness. (9) Substituting the definition of optical path length for LQJ» into Eq. 5.10 gives L Q 1 = n s (5.11) therefore 1 |~dn"1 (z. - z n ) 2 11 v I N . , , 1 ° <*, - * , > - - M £ " l - V ( 5 . 1 2 ) where (z^ - ZQ) is the horizontal distance through the c e l l . We now have current estimates for (x^ - X Q ) , dx/ds, and dn/dx. According to the straight line approximation the path length through the c e l l can be found by Pythagorean's Thorem s ± 2 = (xj^ - x Q ) 2 + (z x - z Q ) 2 (5.13) - 87 -Knowing the arc length, average refractive index, and angle of deflection in the concentration gradient i t is possible to trace the ray through the remaining optical system using Snell's law. Figure 19 l i s t s a l l parameters used in the raytracing program for the various lenses and optical components. The results of the raytracing program then give 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 upon the deviation between the two values, the X q position of the original ray entering the c e l l is updated giving a new refractive index profile n(x). Steps 1 to 11 are then repeated until the calculated refractive index profile produces a hypothetical fringe pattern which corresponds to the observed one. This process takes four to five iterations to converge to a tolerance of 1 x 10 percent. The maximum deflections, refractive index gradients and percent errors for a typical run are given in Table II. A complete l i s t i n g of the ray tracing computer prgram is given in Appendix C. 5.2.1 Mass Flux and Diffusivity Calculations The fringe profiles were checked and corrected for deflection errors before molar fluxes and d i f f u s i v i t i e s were calculated. This required a suitable correlation function for the refractive index as a function of position, as was discussed in the previous section. The profiles shown in Figure 20 indicate that the refractive index function has a sigmoid shape. The solution to Fick's second law with constant diffusivity (Eq. 3.2) predicts a sigmoid concentration - 88 -profile. If boundary conditions are such that the concentration profile i s a step change from zero to C Q at x equal to zero and time equal to zero and constant diffusivity is assumed, then the solution to Eq. 3.2 107 is the complemented error function c(x,t) = 1/2 C erfc (5.14) 0 2/Dt where x is the diffusion coordinate, t is time, and D the binary diffusion coefficient. Since the membrane creates a discontinuity in concentration through the length of the c e l l , the data must be fitted to the correlation function separately on each side of the membrane. Recall from Section 3.3.1 that the refractive index is a linear function of concentration for the dilute sucrose solution used in this work. Eq. 5.14 implies that the refractive index could be fitted to an error function on each side of the membrane as n - m erf (Ax) + b (5.15) where m, A, and b are constants, derived from the best f i t of the refractive index data to Eq. 5.15. The parameters in Eq. 5.15 were found by using a linear 108 least-squares curve f i t t i n g routine to find the values m and b which gave the best f i t for the experimental data, m and b were correlated for the linear function: n = ncc^£n + b (5.16) where x.. = erf(Ax). A value of A was f i r s t assumed, then m and b - 89 Radius of Curvature x ( c m ) Axia l Distance _Z (cm) 0.0 1.0 2.0 3.0 Refractive index JJL "quar tz = 1.45709 Diffusion Cell 99.29 58.2 59.2 L1 Plano-convex Lens oo 22.4 "110. 111. L2 Cylinder Lens n lens = 1.7499 126.6 FP Focal Plane n air = 1.000276 Fig. 19 Ray Tracing Parameters - 90 -found by a least squares method. The RMS error for the linearized function, Eq. 5.16, was calculated and a parameter search performed on A to give the minimum RMS error. _ 3 The RMS error for this function was less than 1 x 10 percent for most runs. Figure 20 shows a sample experimental run with the solid lines being the error function correlation for the experimental points. The f i t of the experimental points to Eq. 5.15 error function correlation shows excellent agreement. This correlation function i s then used for the wavefront bending calculations described in Section 5.1. 5 . 2 . 2 Mass Fluxes To calculate the binary diffusivity i t i s necessary to know both the concentration gradient and mass flux at the membrane surface. The refractive index function w i l l give the refractive index at the membrane surfaces and in turn the concentration from the relation c = na +6 (5.17) c c where acand 8 c are determined experimentally. In this work the values used are a = 0.204014 c 8 = -.0271606 r c which give c in units of gm-moles/ml. These parameters were measured by Bollenbeck 4 0 for a sucrose-distilled water system at 25°C and for a wavelength of 6330 A using a Pulfrich refTactometer and monochromatic light source. He used a linear regression analysis for data yielding - 9 1 -Figure 20 - Refractive Index Prof i les for Typical Run. Fig. 21. Mass Flux and Concentration Profiles in Diffusion Cell - 93 -these values of a and 8 with a c o r r e l a t i o n c o e f f i c i e n t of 0.9995. c r c This equation i s applicable from 0% to 5% by weight sucrose concentration. Consider the mass flux of sucrose at the membrane surface i n each half of the d i f f u s i o n c e l l . Performing an unsteady state mass balance on t h i s volume of c e l l between X q and x i n the v e r t i c a l d i r e c t i o n , along a uniform cross s e c t i o n a l area gives where J i s the mass fl u x at the membrane surface, x , and C(x) i s the xo o continuous concentration p r o f i l e determined from Eqs. 5.15 and 5.17. The boundary conditions are taken at the surface of the membrane and the end of the d i f f u s i o n c e l l . This assumes no sources or sinks between x and x. Finding the mass fl u x involves f i r s t i n t e g r a t i n g concentration p r o f i l e s i n the c e l l with respect to distance for several d i f f e r e n t times, then evaluating the concentration i n t e g r a l s with respect to time and d i f f e r e n t i a t i n g . This analysis i s only v a l i d when the concentration at the ends of the c e l l i s constant, i . e . no mass f l u x at either end of the c e l l . This condition i s met during experiments when the experiment i s terminated before the interference pattern changes at either end of the c e l l . J xo = ^ / C(x)dx (5.18) xo o Combining Eqs. 5.17 and 5.18 gives for the mass f l u x J -5 J X n(x)dx + 8 c ( x 1 - (5.19) xo xo - 9 4 -Using the error function c o r r e l a t i o n , Eq. 5 . 1 5 , y i e l d s for the concentration i n t e g r a l x. x, / C(x)dx = a mf erf(Ax)dx + a b(x. - x n ) + 6 (x. - x ) ( 5 . 2 0 ) c c i u c i o X I xo xo 40 Integrating the error function i n Eq. 5 . 2 0 gives J X C(x)dx = <x {f [Axerf(Ax) + 7- e " ( A x ) 2 ] X l xo C A V % X 0 ( 5 . 2 1 ) + b(x - x 0)} + S c ( x - x Q) When a suitable number of concentration i n t e g r a l s at various times have been evaluated using Eq. 5 . 2 1 then the mass flux can be determined from Eq. 5 . 1 9 . The time d e r i v a t i v e i n Eq. 5 . 1 9 may be evaluated by f i n i t e difference methods, graphical methods, or from a suitable c o r r e l a t i o n . A c o r r e l a t i o n method i s preferred, since any i r r e g u l a r i t i e s i n the data are smoothed. Based upon the error function 40 s o l u t i o n to the d i f f u s i o n equation, Bollenbeck proposes a fu n c t i o n a l form of the concentration i n t e g r a l as a function of time 2 xo SL di J C(x)dx - a x + a 2 e r f (—nj) + a^t 1 / 2 exp ( 5 . 2 2 ) x t The four constants i n Eq. 5 . 2 2 can be determined from f i v e d i f f e r e n t concentration p r o f i l e i n t e g r a l s at d i f f e r e n t times. A UBC computer centre non-linear least-squares curve f i t t i n g r o u t i n e 1 0 8 was used to - 95 -find the best f i t for the constants in Eq. 5.22. This correlation function predicted the concentration profile integrals versus time with an RMS percent error better than 0.1 percent. Eq. 5.22 can then be differentiated with respect to time to yield the mass flux in Eq. 5.19. The molar flux of sucrose at the membrane surface then becomes1+0 The membrane diffusivity can now be calculated from Fick's f i r s t law since the molar flux and concentration gradient across the membrane are known. Equation 5.23 is only valid when the flux at the end of the cell is zero. This condition is met at the c e l l bottom as long as the concentration gradient is zero. Since the membrane is only 10 microns thick i t can be assumed that a steady state condition exists through the membrane, i.e., the mass flux i s equal on the top and bottom surfaces. The mass flux predicted by Eq. 5.23 is also the mass flux through the membrane, so Fick's f i r s t law can be applied to the membrane using this flux and knowing the sucrose concentration at each membrane surface 2 xo " 2 t i / 2 "372 " 7- -T72 - a4 , a4 a3 1 a2 a3 (5.23) J xo = D, ) A c M "ox (5.24) where Ac is the concentration driving force or the concentration difference across the membrane, 6x is the membrane thickness, and D M is - 96 -the membrane diffusion coefficient. Ac can be evaluated from the interference fringe results. Fick's law for binary diffusion can be written as J = X ( J + J ) - CD T (5.25) s s w s F dx where J is the molar flux of sucrose, J is the molar flux of water, D„ s ' w ' F is the free diffusion coefficient, and X is the mole fraction of s sucrose. For a binary system the concentration of sucrose c g and concentration of water can be related, 4 0 assuming no chemical reactions between the water and sucrose c w = Y l c g + E l (5.26) the values of y^ and are determined from the density as a function of 40 composition by Bollenbeck as yl = 11.68744 (5.27) e x = 0.0553512 (5.28) If the concentration of water and sucrose are linearly related then the flux of sucrose can be related to the flux of water as J = - y , J (5.29) s '1 w and Eq. 5.25 can be used to calculate Dp, at the free diffusion coefficient determined at the location x, in the c e l l . - 97 -CHAPTER 6 RESULTS A total of twenty-three experiments were conducted for two different membrane pore sizes and conditions ranging from no magnetic f i e l d to an applied f i e l d strength of 12.5 kG. The only variables in these experiments were applied f i e l d strength and pore size. In a l l experiments the i n i t i a l sucrose concentration difference across the membrane was 1% by weight of sucrose. The reference side of the diffusion c e l l was always f i l l e d with d i s t i l l e d water. The temperature of the enclosure was kept constant at 25°C ± .05 for a l l runs. Table III l i s t s a l l experimental parameters and variables. The raw data consisted of a set of fringe displacements taken at different times for each run. The fringe locations were measured with respect to a datum taken at 1.0 cm from the membrane at each end of the c e l l . The concentration of sucrose was constant at this location for a l l runs (0% in top half of c e l l and 1% by weight in lower half). Each distance measured by the microscope corresponded to a refractive index change equivalent to one fringe shift or one wavelength of laser light. These data were stored on permanent f i l e s for each run and were then used as input for the various data analysis computer programs. The f i r s t program RAYTRACE took the raw data and converted i t into refractive index profiles for each time interval using the equations derived in Chapter 5. These profiles were used to calculate the degree of raybending due to the refractive index gradient. A corrected refractive index profile was determined to account for the ray - 98 -bending. RAYTRACE then evaluated the constants A, b, and m for Eq. 5.15 to correlate the corrected refractive index gradient with the error function correlation. These parameters were stored on a permanent f i l e for use by the mass flux and diffusivity calculation program. RAYTRACE is listed in Appendix C. The program DIFFCALC used the correlation parameters A, b, and m to integrate the concentration profile in the c e l l . This integral was evaluated at each time interval and a partial derivative with respect to time was evaluated to determine the mass flux at the membrane surface for a given time, (see Section 5.2 for equations and derivations). The mass flux was evaluated for the lower half of the diffusion c e l l only and the flux through the membrane was assumed equal at both membrane surfaces. Ray deflection effects cause the rays entering the c e l l to be deflected downward towards the direction of increasing refractive index. This produces a thickening of the membrane shadow in the focal plane. Due to deflected light rays striking the upper surface of the membrane (see Figure 16) there is a loss of information near the upper surface of the membrane. Therefore, only mass fluxes in the lower half of the c e l l are used for diffusivity calculations. The concentration gradient producing the driving force for this mass flux is determined from the concentration difference at each membrane surface using the refractive index correlation evaluated by RAYTRACE. The diffusivity through the membrane is then calculated from Eq. 5.24 and free diffusion coefficient determined from Eq. 5.25. DIFFCALC is listed in Appendix D. - 99 -6.1 Raybendlng and Refractive Index Correlation Figure 20 shows a typical refractive index profile for run M. The solid lines are the profiles corrected for raybending effects and correlated to the error function, Eq. 5.15. The points are the refractive index values calculated directly from the raw data, not accounting for wavefront deflection. As would be expected, the deflection is most noticeable near the membrane surface where the refractive index gradient is greatest. Near the end of the experiment, when diffusion has considerably reduced the magnitude of the refractive index gradient, wavefront deflection is much less, which is apparent from Figure 20 since the corrected profile and experimental profile are identical. Table II l i s t s the values of the corrected refractive index profile with the values of the uncorrected profile for each point in the run. Each x coordinate listed corresponds to one shift in the fringe pattern. Again i t is apparent that the largest differences occur near the membrane, early in the experiment where refractive index gradients are largest. The correlation parameters A, b and m for the refractive index profile correlation are listed in Appendix B for a l l runs. 6.2 Dlff u s i v l t l e s The diffusion coefficient through the membrane and free diffusion coefficient evaluated in the lower half of the diffusion c e l l for a l l runs are listed in Table III. These diffusion coefficients are calculated from the mass flux and concentration difference at the Table II Ray Bending Corrections of Data for Run L at 900 Seconds, Lower Half of C e l l n ^observed X 1 1 o , c e l l Pass 1 ax Pass 2 AX Pass 3 AX Pass 4 AX Pass AX 5 Pass AX 6 x i i o . c e l l 1.332146 -.0005 -.01485 .004567 -.000799 .000121 -1.8 X l O " 5 3 X l O " 6 > 1 X IO" 6 .010019 1.33210 -.0030 -.008910 .004533 -.000707 .000106 -1.6 X l O " 5 2 X l O " 6 > 1 X IO" 6 .002727 1.332273 -.0054 -.016038 .004448 -.000612 .000093 -1.4 X l O " 5 2 X l O " 6 > 1 X IO" 6 -.004404 1.332336 -.0089 -.026433 .004241 -.000476 .000080 -1.1 X l O " 5 2 X l O " 6 > 1 X i o - 6 -.015045 1.332400 -.0130 -.038610 .003892 -.000338 .000069 -9.0 X l O " 6 1 X l O " 6 > 1 X IO" 6 -.027874 1.332463 -.0173 -.051381 .003435 -.000233 .000061 -7.0 X l O " 6 1 X l O " 6 > 1 X IO" 6 -.041707 1.332526 -.0220 -.065340 .002878 -.000164 .000052 -5.0 X l O " 6 1 X l O " 6 > 1 X IO" 6 -.057137 1.332589 -.0285 -.084644 .002108 -.000124 .000037 -4.0 X l O " 6 1 X i o - 6 > 1 X IO" 6 -.078652 1.332653 -.0368 -.109295 .001274 -.000099 .000021 -3.0 X l O " 6 >1 X l O " 6 > 1 X IO' 6 -.105749 1.332716 -.333 -.989003 -.000001 >- lxlO"" 6 > l x l 0 " 6 >-1.0 X l O " 6 >1 X IO" 6 > 1 X i o - 6 -.989006 undeflected corrected l i g h t rays for def l e c t i o n = fringe location observed on f i l m (cm) " o r i g i n a l entry position l n c e l l for ray corresponding to X observed (cm) «• deviation between observed fringe and calculated fringe location (cm) • re f r a c t i v e Index l n c e l l at position X Q , c e l l - 101 -Table III Membrane and Free Diffusion Coefficients Pore diameter, d = 8.0 um Free area, = 0.050 cm2/cm2 Run Field Strength kG 2 °M (cm /s x 10 ) 2 ° F (cm Is x 10 ) D 2 a M ,cm . 2 Dp cm L 0.0 0.229 5.50 0.042 N 1.0 0.243 6.07 0.040 K 2.5 0.270 5.56 0.049 R 4.0 0.191 6.20 0.031 M 10.0 0.230 5.35 0.043 T 10.5 0.242 5.41 0.045 P 11.0 0.234 5.91 0.040 H 12.0 0.192 5.83 0.033 Pore diameter, = 0.8 um Free area, A = 0.151 cm /cm A 2 0.0 0.889 5.22 0.170 E 2 2.5 0.436 5.17 0.084 C 2 5.0 0.897 5.04 0.178 D 2 10.0 0.414 4.98 0.083 B 2 12.5 1.246 5.25 0.237 aThe ratio %/D F is an indicator of experimental error, since i t should be constant for a given membrane. - 102 -membrane surface. The concentration was extrapolated to x = 0 from the concentration c o r r e l a t i o n . The membrane c o e f f i c i e n t given was time averaged for each run to smooth errors r e s u l t i n g from fringe pattern reading near the membrane surface. The free d i f f u s i o n c o e f f i c i e n t was taken at T = 3600s which was the t h i r d set of fringes out of a t o t a l of f i v e d i f f e r e n t times to a time of 3 hours. Faxen's equation p r e d i c t i n g the r a t i o of the d i f f u s i o n c o e f f i c i e n t i n a c y l i n d r i c a l pore to the bulk d i f f u s i o n c o e f f i c i e n t (Eq. 3 .12 ) was used to calculate t h i s r a t i o f o r the membrane-solution systems used i n this work. For the 0 . 8 \m pore diameter the r a t i o Dp/Dp equals 0 .9972 and for the 8 .0 \im pore diameter i t i s 0 . 9 9 9 7 2 . Therefore, the membrane can be considered as a simple cro s s - s e c t i o n a l area reduction to d i f f u s i o n , pore-solute interactions are n e g l i g i b l e . The membrane d i f f u s i o n c o e f f i c i e n t s measured i n th i s work divided by the e f f e c t i v e free area should y i e l d the bulk d i f f u s i o n c o e f f i c i e n t . The e f f e c t i v e free area can be calculated from the membrane manufacturer's s p e c i f i c a t i o n s for pore size and surface porosity. For the 8 .0 |j,m diameter pore size the free area f o r d i f f u s i o n i s 0 . 0 5 0 cm /cm . For 2 2 the . 8 \m diameter pore size the free area i s 0 .151 cm /cm . Table I I I l i s t s the e f f e c t i v e pore area calculated by d i v i d i n g the experimental membrane d i f f u s i v i t y by the d i f f u s i o n a l free area. Table IV l i s t s the values of sucrose-water binary d i f f u s i o n c o e f f i c i e n t s measured by other workers compared with the values determined i n th i s work for the same temperature and concentration. Figures 22 and 23 show the d i f f u s i o n c o e f f i c i e n t s as a function of applied f i e l d strength for the 8 .0 vim and 0 . 8 \m. pore diameter - 103 -membranes. A linear regression analysis was performed on each data set. The straight line approximation is the best f i t for the data predicted by the linear regression analysis. Table V l i s t s the slope, y-intercept, and correlation coefficient for each data set. A correlation coefficient equal to zero indicates no correlation between the data and straight line and a value of 1.0 indicates a l l points are equal to the correlated values. - 104 -T a b l e I V C o m p a r i s o n o f A q u e o u s S u c r o s e S o l u t i o n B i n a r y D i f f u s i o n C o e f f i c i e n t s M e a s u r e d b y I n t e r f e r o m e t r i c M e t h o d s ( c m 2 / s x 1 0 6 ) R e s e a r c h e r s Commen t s E s t i m a t e d A c c u r a c y (%) 5 . 2 1 5 G o s t i n g e t a l . 51 G o u y i n t e r f . , f r e e d i f f . , v a p o r l i g h t s o u r c e 0 . 2% 5 . 2 1 ' 5 . 2 6 2 5 . 5 6 D u d a e t a l . 57 40 B o l l e n b e c k G a b l e m a n - G r a y 64 Wedge i n t e r f . , f r e e d i f f . R a y l e i g h i n t e r f . , memb. d i f f . H o l o g r a p h y , f r e e d i f f . 3% 3% 2% 5 . 3 6 ' T h i s w o r k R a y l e i g h i n t e r f . , memb. d i f f . 3% v - A l l v a l u e s m e a s u r e d a t 2 5 ° C a n d 0 .5% w e i g h t f r a c t i o n s u c r o s e . V a l u e e x t r a p o l a t e d t o 0 . 5% w e i g h t f r a c t i o n s u c r o s e . A v e r a g e v a l u e m e a s u r e d f o r n o f i e l d c o n d i t i o n s . " V a l u e s m e a s u r e d a g r e e w i t h a c c e p t e d v a l u e s m e a s u r e d a t s i m i l a r c o n d i t i o n s t o w i t h i n ± 3%. - 105 -Table V Results of Linear Regression Analysis for Sucrose Diffusion Coefficients in Applied Magnetic Field. Correlation Data Set Slope Y-intercept Coefficient 8.0 u m D F pore, -0.0122 5.808 X IO"6 0.191 8.0 L i m DM pore, -0.0016 0.239 X IO"7 0.296 0.8 L i m DF pore, -0.0038 5.155 X IO"6 0.169 0.8 L i m DM pore, 0.0186 0.665 X IO"6 0.274 0.8 Lim Dp* pore, -0.025 5.212 X IO"6 0.959 0.8 L i m DM* pore, -0.0332 0.804 X IO - 6 0.525 *Data set evaluated with point at 12.5 kG removed (see text for just i f i c a t i o n ) . - 106 -6.0 L. ° O O 5.0 -4.0 3.0 2.0 1.0 A o ° o o o D F x 1 0 6 cm2/s c A D M x 1 0 7 cm2/s o 3 I I 1 I I 1_ I I L_ 1_ L 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 1.2 App l i ed Field Strength kG x .1 Fig. 22. Diffusion Coeff ients vs Field Strength for 8.0 ixrr\ Pore Diameter - 107 6.0 5.0 "O " o •0. 4.0 3.0 0 o O c o (0 3 - a O D F x 10 6 c m 2 / s A D M x 1 0 6 cm2/s 2.0 1.0 A A JL _L 0 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 1.2 Applied Field Strength kG x .1 Fig. 23. Diffusion Coefficients vs Field Strength for 0.8/im Pore Diameter - 108 -CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH 7.1 Discussion A Rayleigh interferometer was designed and constructed to measure molecular diffusion coefficients for an aqueous sucrose solution through a porous membrane in an applied magnetic f i e l d . The interferometer was constructed to satisfy the design constraints requiring to f i t betweeen the 10 cm gap of an electromagnet without disturbing the homogeneity of the magnetic f i e l d . The interferometer produced an image of the refractive index profile in the diffusion test c e l l which was evaluated as a function of time to yield concentration profiles, mass fluxes and dif f u s i v i t i e s for the diffusing molecules at a constant temperature and different applied magnetic f i e l d strengths. A computer program was written to reduce the interference fringes to refractive index and concentration profiles, mass fluxes, and diffusion coefficients. Optical errors introduced by wavefront deflection through a refractive index gradient were considered and a computer program was written to correct the refractive index profiles for this effect. This equipment was used to study the diffusion of an aqueous sucrose solution through a General Electric Nucleopore membrane in magnetic f i e l d strengths from 0 to 12.5 kG. A 1% by weight solution of sucrose was allowed to diffuse through membranes with pore diameters 0.8 um and 8.0 um at a temperature of 25 ± .05°C. Any molecular magnetic anisotropics w i l l cause a partial or - 109 -complete alignment of the molecule in a magnetic f i e l d . This would be observable as a change in the molecular diffusion coefficient through the porous membrane. The results of the linear regression analysis indicate a slight decrease (1 to 2%) for three sets of diffusion coefficients as the applied magnetic f i e l d increases to 12.5 kG. While the limited number of data points and relatively large amount of data scatter prevent a definite correlation from being made between the diffusivity and f i e l d strength, the results clearly indicate a trend. The correlation coefficient from 0.19 to 0.9 indicates that this trend is more than just random experimental error. The fact that a new membrane was used for each run may account for some of the data scatter. Therefore, some degree of alignment of the sucrose-water clusters appears to be taking place in the magnetic f i e l d . The membrane diffu s i v i t i e s measured for the 0.8 um pore size do not show a decrease in the magnetic f i e l d as the other three sets of data do. If the value measured at 12.5 kG is weighted less than the other four points, the linear regression analysis would show a slight decrease in the diffusion coefficients in an applied magnetic f i e l d . The values of free diffusion area, Ap, tabulated in Table III would justify a lower level of confidence in this point. The calculated value of Ap, the ratio of membrane to free d i f f u s i v i t i e s , should be a constant for each membrane. The value calculated for this point is much higher than the ones corresponding to the other experimental points which would indicate some unknown experimental error in the diffusivity measured for 12.5 kG f i e l d strength. The correlation coefficients evaluated for Dp and D^  with this point removed indicate a better f i t than with the point included (see Table V). - 110 -An aqueous sucrose solution was selected for this i n i t i a l work because accurate, widely accepted diffusion data over a concentration range are available for comparison. The bulk binary diffusion coefficient measured in these experiments for no f i e l d conditions at 25°C agree with the values measured at the same temperature and 51 concentration by Gosting and Morris within 3%. 7.2 Recommendations Based upon the results obtained in these experiments, two further areas of research and development can be recommended; 1) improving the accuracy and sensitivity of the interferometer and, 2) examining diffusion of molecules exhibiting a higher degree of anisotropy than sucrose. Temperature variations were not fel t to be a major source of experimental error. The maximum temperature change of the air surrounding the diffusion c e l l was 0.1°C during a run. The Stokes-Einstein equation (Eq. 3.8) predicts the diffusion coefficient to be a linear function of absolute temperature, the Boltzman constant being the constant of proportionality. For a maximum temperature change of 0.1°C at 25°C, the diffusion coefficient would change by .03%, clearly less than the data scatter in this work. From the experience gained in this work, i t is f e l t that a major source of error lies in accurately measuring the fringe patterns near the membrane surface. The refractive index gradient is greatest immediately adjacent to the membrane surface, so therefore the wavefront deflection is also greatest at that location. Wavefront deflection is - I l l -observable as a thickening of the membrane shadow and a downward displacement of that shadow on the f i l m f o c a l plane. The membrane surface shadow provides a f i d u c i a r y point to reference the interference fringe locations i n each half of the c e l l . The edges of the membrane shadow tend to be blurred i n the image of the interference pattern on the f i l m when viewed through the measuring microscope. I t i s f e l t that the membrane shadow i s not the best f i d u c i a r y point for referencing interference fringe l o c a t i o n s . Therefore, i t i s recommended that the d i f f u s i o n c e l l be modified to produce a more d e f i n i t e f i d u c i a r y mark on the interference pattern. This could be accomplished by constructing a wire scale which could be placed inside the d i f f u s i o n compartment. The wire g r i d would produce a shadow i n the f o c a l plane providing reference locations on the fringe pattern thereby eliminating the need to c l e a r l y define the membrane shadow boundaries. This may introduce some problems i n the analysis i f the one-dimensional nature of the d i f f u s i o n i s interrupted by the wire g r i d . In addition to improving the accuracy of the equipment and experimental procedure e f f o r t s could be made to reduce the time required to measure fringes and input f r i n g e data to the computer. Measuring fringes through the microscope and recording several hundred data points i s a very tedious and time consuming process. The t o t a l elapsed time required to set up an experiment and obtain d i f f u s i v i t i e s i s approximately one week. Any magnetic f i e l d e f f e c t on d i f f u s i o n i s small, so i t i s e s s e n t i a l that a large number of experiments be performed to give s t a t i s t i c a l s i g n i f i c a n c e to any observed change i n - 112 -d i f f u s i o n rate. Therefore, a more e f f i c i e n t data evaluation system i s desirable to minimize the time required to conduct an experiment. Renner and Lyons report on a computer recorded automated interferometric system. They u t i l i z e a photoraultiplier tube correlated with a motor driven measuring microscope to automatically measure the interference fringes and provide input to a numerical method computer program. Adams 1 0 9 et. a l and W a t k i n s 1 1 0 et. a l discuss s i m i l a r methods using e l e c t r o n i c l i g h t sensing elements to provide an automated data a c q u i s i t i o n system. An e l e c t r o n i c data a c q u i s i t i o n system would considerably reduce the time required to measure and evaluate data enabling one to conduct a greater number of experiments for a given amount of time. The next l o g i c a l extension of t h i s work i s to study d i f f u s i o n of strongly anisotropic molecules i n a magnetic f i e l d . P a r t i a l or complete alignment i n a magnetic f i e l d has been observed for several macromolecules and molecular clusters by measuring the magnetically induced birefringence (Cotton-Mouton e f f e c t ; discussed i n Chapter 3). Any molecular system e x h i b i t i n g a Cotton-Mouton e f f e c t should also show a change i n d i f f u s i o n rate i n a porous membrane. A Cotton-Mouton e f f e c t has been observed for polypeptides (Poly (Try-Glu)), nucleic acid fragments, 8 4 r o d l i k e v i r u s e s , 8 4 DNA, 8 4 l i q u i d c r y s t a l s , 7 8 - 8 2 c h l o r o p l a s t s , 8 4 r e t i n a l r o d s , 8 4 and m i c e l l a r aqueous soap s o l u t i o n s . 8 3 Interferometric measurement of macromolecular d i f f u s i o n presents several problems which must be addressed. The size of these molecules range from a few Angstroms to several thousand Angstroms. In t h i s s i z e - 113 -range o p t i c a l dispersion of the helium-neon l a s e r l i g h t (wavelength equals 6382 A) may be s i g n i f i c a n t enough to destroy the s p a t i a l coherence of the transmitted l a s e r beam. In addition, these systems are strongly absorbing i n the v i s i b l e l i g h t spectrum being used. This absorption may introduce heating and convection e f f e c t s into the d i f f u s i n g system. H a l l * * * et. a l discuss these e f f e c t s i n o p t i c a l absorption by hemoglobin using laser l i g h t . Macromolecular d i f f u s i o n c o e f f i c i e n t s tend to be highly concentration dependent, so the si m p l i f y i n g assumptions of constant d i f f u s i v i t y would not be applicable. The method developed i n th i s work has been applied to binary d i f f u s i o n . However, most proteins and organic macromolecules require a buffer s o l u t i o n to prevent denaturation, so d i f f u s i o n i s no longer binary, but multicomponent. This introduces another independent variable into the analysis since the d i f f u s i o n rate of the buffer and i t s coupling to the macromolecule must be evaluated. 7.3 Conclusions A physical model i s developed describing the e f f e c t of an applied magnetic f i e l d on the d i f f u s i o n rate of anisotropic molecules through a porous membrane. The model predicts a decrease i n the d i f f u s i o n rate of a molecular system d i f f u s i n g through a porous membrane i n a magnetic f i e l d applied transverse to the pore d i r e c t i o n . Rayleigh interferometry has been applied to measure the l o c a l d i f f u s i o n c o e f f i c i e n t i n a s p e c i a l l y designed d i f f u s i o n c e l l for an aqueous sucrose system d i f f u s i n g through a GE Nucleopore membrane i n an applied magnetic f i e l d . A s l i g h t decrease (1 to 2%) i n the measured d i f f u s i o n - 114 -coefficient was observed in applied f i e l d strengths up to 12.5 kG as was predicted by the model developed in this work. Therefore, optical interferometry has been demonstrated to be a useful technique to elucidate certain aspects of molecular-magnetic interactions and their effects on molecular transport properties. Free diffusion coefficients for a dilute aqueous sucrose solution compare with accepted value to within ± 3%. However, the di f f i c u l t y i n clearly defining the membrane surface boundary in the image of the diffusion c e l l limits the experimental accuracy to a magnitude comparable to expected change in sucrose diffusion rate in a magnetic f i e l d . It is therefore, recommended that modifications be made to the diffusion c e l l to eliminate the need to use the membrane shadow as a fiduciary point for the data analysis and improve the accuracy of the interferometer. To further verify the proposed model developed in this work i t is also recommended that further work be done, using molecules with a higher degree of anisotroy than sucrose, which would be expected to exhibit a larger magnetic f i e l d effect on their diffusion rate. - 115 -NOMENCLATURE a, , Constants in Eq. 5.22 a Constant in Eq 3.24 Ap Cross-sectional area for free diffusion A Constant in Eq. 5.15 Ag Amplitude of light wave B* Magnetic induction vector b Constant in Eq. 3.24 n M b Constant in Eq. 5.15 c Velocity of light in a vacuum C Concentration C„ Cotton-Mouton constant n D Diffusion coefficient d Diameter E Harmonic motion of light wave F Geometrical factor in Eq. 3.18a M f Maxwellian distribution function in Eq. 2.1 f Stoke's f r i c t i o n a l coefficient H Magnetic f i e l d strength J Molar flux K Boltzman's constant L Optical path length defined in Eq. 3.27 M Molecular weight M* Intensity of induced magnetization vector - 116 -mv m n q r R RM s t T v V x Y s z Z Zk Greek Symbols a a c P 6 x Momentum Constant ln Eq. 5.15 Refractive index Heat flow Radius Gas constant Molar refractivity Geometrical path length of a light ray Time Temperature Velocity vector Mean square transit velocity for molecular motion Velocity of light through a given medium Distance Masking s l i t spacing Distance Complex amplitude of a light wave Electric charge on diffusing ion Molecular optical polarisabilities Constant in Eq. 5.17 Ratio of solute to pore radii in Eqs. 3.11, 3.12, and 3.18a Constant in Eq. 5.17 Mean square Brownian displacement - 117 -6q Degree of orientation for a molecule aligned in magnetic f i e l d 6x Membrane thickness A Difference Constant in Eq. 5.26 Y Ratio of solvent to pore radii in Eq. 3.11 Y Constant in Eq. 5.26 1 X Wavelength of light n Viscosity u Chemical potential u Osmotic pressure p Density a Staverman's reflection coefficient <j> Phase angle of light wave at time = 0 X Magnetic susceptibility tensor a) Angular speed of light wave Subscripts a Air b Bulk DC Diffusion c e l l f Magnetic f i e l d F Free diffusion FP Focal plane of interferometer l i n Linearized M Membrane P Pore - 118 -s w II 4-X y z Superscripts H o Solute Water Denotes value p a r a l l e l to applied magnetic f i e l d Denotes value perpendicular to applied magnetic f i e l d Value along p r i n c i p a l x-axis Value along p r i n c i p a l y-axis Value along p r i n c i p a l z-axis With applied magnetic f i e l d Without applied magnetic f i e l d - 119 -R E F E R E N C E S 1. Busby, D.E., "Biomagnetics - Considerations Relevent to Manned Space Flight", NASA CR-889 (Sept. 1967). 2. Barnothy, J.M., and Barnothy, M.F., and Boszormenyi-Nagey, J., "Influence of a Magnetic Field Upon the Leucocyte of a Mouse", Nature, 177, 577 (1956). 3. Barnothy, M.F., and Barnothy, J.M., "Biological Effect of a Magnetic Field and the Radiation Syndrome", Nature, 181, 1785 (1958). 4. Lechner, F., "Influencing of Bone Formation Through Electromagnetic Potentials", Langenbecks Arch. Chir. 631 (1974). 5. Ketchen, E.E., Porter, W.E., Bolton, N.E., "Biological Effects of Magnetic Fields on Man", Am. Indst. Hygiene Assoc. J., 39_, 1 (1978). 6. Lechner, F., Kraus, W., "Influencing of Disturbed Fracture Healing Through Electromagnetic Fields", Hefte Zur Unfallheilkunde, 114, 325 (1973). 7. Freedman, A.P. "Medical Significance of the Magnetic Activities of the Human Body", Biomagnetism, Walter de Gruyter and Co., Berlin (1981). 8. Personal communication with C. Smith, U.B.C. Sports Medicine, May (1982). 9. Barnothy, M.F., Biological Effects of Magnetic Fields, Plenum Press, N.Y., Vol. 1 (1964), VII (1969). 10. Bhatnagar, S.S., Mathur, K.N., Physical Principles and  Applications of Magnetochemistry, Macmillan and Co., London (1935). 11. Gross, L., Biological Effects of Magnetic Fields, Ph.D. Thesis, New York University (1963). 12. Liboff, R.L., "A Biomagnetic Hypothesis", Biophysical J., _5, 845 (1965). 13. Lielmezs, J., Aleman, H., "Weak Transverse Magnetic Field Effect on the Viscosity of Mn(M>3)2 - H2O Solution at Several Temperatures", Thermochim. Acta, 20, 219 (1977). 14. Lielmezs, J., Aleman, H., "Weak Transverse Magnetic Field Effect on the Viscosity of Ni(N03)2 - H2O Solution", Thermochim. Acta, 21_, 233 (1977). 15. Lielmezs, J., Aleman, H. "External Transverse Magnetic Field Effect on Electrolyte Diffusion in HCI-H2O Solution", Electrochim. Acta, 21, 273 (1976). - 120 -16. Lielmezs, J., Aleman, H., "Weak Transverse Magnetic Field Effect on the Viscosity of KCI-H2O Solution", Thermochim. Acta, 18, 315 (1977). — 17. Lielmezs, J. , Aleman, H., Musbally, G.M., "External Transverse Magnetic Field Effect on Electrolyte Diffusion in KC1-H20 Solution", Z. Phys. Chem. Neue Folge, 90, 8 (1974). 18. Lielmezs, J., Aleman, H., "External Transverse Magnetic Field Effect on Electrolyte Diffusion in NaCl-H^O Solution", Thermochim. Acta, 9, 247 (1974). 19. Lielmezs, J., Aleman, H., "External Transverse Magnetic Field Effect on Electrolyte Diffusion in LiCl-H 20 Solutions", Bioelectrochem. and Bioenerg., _5» 2 8 5 (1978). 20. Lielmezs, J., Aleman, H., "External Transverse Magnetic Field Effect on Electrolyte Diffusion in CuCl-H.20 Solution", Thermochim. Acta, _15_, 63 (1976). 21. Lielmezs, J. Aleman, H., "External Magnetic Field Effect on Viscometer Efflux Times of Calf Thymus DNA in 0.0172 M KC1 Solution During the Thermal Denaturation Process", Thermochim. Acta, 43, 9 (1981). 22. Thysse, B.J., Van Ditghuyzen, P.G., Hermans, L.J.F., Knaap, H.F.P., "The Influence of a Magnetic Field on the Transport Properties of Symmetric Top Molecules", Proc. 7th Symp. Thermophys. Prop. 638 (1978). 23. Beenakker, J.J.M., Hermans, L.J.F., Hulsman, H., Knaap, H.F.P., Korving, J. "The Influence of an External Field as a Tool in the Study of Transport Properties of Rotating Molecules", Proc. 5th Symp. Thermophys. Prop. 168 (1970). 24. McCourt, F.R., Knaap, H.F.P., Moraal, H., "The Senfteben-Beenakker Effects for a Gas of Rough Sperical Molecules", Physica, 43_, 485 (1969). 25. Tip, A., "On the Senfteben-Beenakker Effect in Mixture", Physica, 37, 411 (1967). 26. Beenakker, J.J.M., Hulsman, H., Knaap, H.F.P., Korving, J., and Scoles, G., "The Influence of a Magnetic Field on the Viscosity and Other Transport Properties of Gaseous Diatomic Molecules", Adv. in Therophys. Prop, at Extreme Temp, and Press., ASME, 216 (1965). 27. Mehaffey, J.R. , Garisto, F. , Kapral, R. , Desai, R.C, "Magnetic Field Dependence of Thermal Correlations in Molecular Gases", J. Chem. Phys., 58, No. 10, 4084 (1973). - 121 -28. Moraal, H., McCourt, F.R., "The Seriftleben Effects for a Gas of 2E Molecules", Physica, 46, 367 (1970). 29. Hermans, L.J.F., Schutte, A., Knaap, H.F.P., Beenakker, J.J.M., "Transverse Heat Transport in Polyatomic Gases Under the Influence of a Magnetic Field", Physica, 46, 491 (1970). 30. Beenakker, J.J.M., "The Influences of Electric and Magnetic Fields on the Transport Properties of Polyatomic Dilute Gases", Festkorperproblems, 8^, 276 (1968). 31. Udayan De, Bhanja, S., Barua, A.K., "The Effect of Magnetic Field on the Heat Conductivity of O2--N2 and O2-H2 Gas Mixtures", J. Chem. Phys., 68, n7, 3226 (1978). 32. Hulsman, H., Van Kuik, F.G., Walstra, K.W., Knaap, H.F.P., Beenakker, J.J.M., "The Viscosity of Polyatomic Gases in a Magnetic Field", Physica, 57_, 501 (1972). 33. Korving, J., "Viscosity of NH3 in High Magnetic Fields", Physica, 46, 619 (1970). 34. Eggermont, G.E.J., Oudeman, P., Hermans, L.J.F., "Experiments on the Influence of a Magnetic Field on the Thermal Diffusion in Polyatomic Gases", Physics Ltrs., 50A, N3, 173 (1974). 35. Vugts, H.F., Tip, A., Los, J., "The Senftleben Effect on Diffusion", Physica, 38, 579 (1968). 36. Burgmans, A.L.J., van Ditzhuyzen, P.G., Knaap, H.F.P., "The Viscomagnetic Effect in Mixtures", Z. Naturflorsch., 28a, 849 (1973). 37. Burgmans, A.L.J., van Ditzhuyzen, P.G., Knaap, H.F.P., Beenakker, J.J.M., "The Temperature Dependence of the Viscomagnetic Effect", Z. Naturforsch., 28a, 835 (1973). 38. Vevai, J.E., E l l i o t , D.G., Honeywell, W.I., "Senfleben-Beenakker Thermal Conductivity Effect Apparatus", Rev. Sci. Inst. 49_, N8, 1116 (1978). 39. Beenakker, J.J.M., McCourt, F.R., "Magnetic and Electric Effects on Transport Properties", Am. Rev. Phys. Chem., 2l_, 47 (1970). 40. Bollenbeck, P., A Modified Rayleigh Interferometer for Membrane  Transport Studies", Ph.D. Dissertation, University of Colorado (1972). 41. Senfteben, H., "Magnetische Beeinflussing des Warmeleitvertische Paramagnetischer Gase", Physik Z., 31, 961 (1930). - 122 -42. Iberall, A., Schindler, A. Physics of Membrane Transport, General Technical Services, Inc., Upper Darby, Pa. (1973). 42a. Brenner, H., "Rheology of a Dilute Suspension Dipolar Spherical Particles in an External Field", J. Colloid and Interface Sci., 32, Nl,141 (1970). 43. Lielmezs, J., Musbally, G.M., "Effect of External Magnetic Field on Diffusion of Electrolytes in Solution", Electrochim. Acta, 17, 1609 (1972). 44. Camp, F.W., Johnson, E.F., "Magnetic Effects in Certain Systems of Chemical Engineering Interest", I & EC Fundamentals, ^, N2, 145 (1965). 45. 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). 46. Lielmezs, J., Aleman, H., Fish, L.W., "Weak Transverse Magnetic Field Effect on the Viscosity of Water", Z. fur Physik., Chem. Neue Folge, 99, 117 (1976). 47. Lielmezs, J., Aleman, H., "Weak Transverse Magnetic Field Effect on the Viscosity of Water at Several Temperatures", Thermochim. Acta, 21, 225 (1977). 48. Philpot, J. St. L., Cooke, C.H., "A Self-Plotting Interferometric Technique for the Ultracentrifuge", Research, 1_, 234 (1947). 49. Longsworth, L.G., "Experimental Tests of an Interference Method for the Study of Diffusion", J. Am. Chem. Soc, 69, 2510 (1947). 50. Ogston, A.G., "The Gouy Diffusiometer, Further Calibration", Proc. Roy. Soc. (London), A196, 272 (1949). 51. Gosting, L.J., Moris, M.S., "Diffusion Studies on Dilute Aqueous Sucrose Solutions at 1 and 25° with the Guoy Interference Method", Am. Chem. Soc J. , 71_, 1521 (1949). 52. Gosting, L.E., Akeley, D.F., "A Study of the Diffusion of Urea in Water at 25° with the Gouy Interference Method", Am. Chem. Soc J. , 74, 1113 (1952). 53. Svensson, H., "On the Use of Rayleigh-Philpot-Cook Intereference Fringes for the Measurement of Diffusion Coefficients", Acta Chemica Scand., _5> 7 2 (1951). 54. Longsworth, L.G., "Tests of Flowing Junction Diffusion Cells with Interference Methods", Rev. Sci. Inst., 21, N6, 524 (1950). 55. O'Brien, R.N., "Interferometer for Electrochemical Investigations", Rev. Sci. Inst., 35, N7, 803 (1964). - 123 -56. O'Brien, R.N., Yakymyshyn, W.F., Leja, J., "Interferometric Study of Zb/ZnSOit/Zn System", J. Electrochem. Soc, 110, 820 (1963). 57. Duda, J.L., Sigelko, W.L., Ventras, J.S., "Binary Diffusion Studies with a Wedge Interferometer", J. Phys. Chem., 73_, Nl, 141 (1969). 58. O'Brien, R.N., Hyslop, W.F., "A Laser Interferometric Study of the Diffusion of 0 2, N 2, H 2 and Ar into Water", Can. J. Chem., 55, 1415 (1977). 59. Rard, J.A., Miller, D.G., "The Mutual Diffusion Coefficients of NaCl-H20 and CaCl-H20 at 25°C from Rayleigh Interferometry", J. Solution Chem., 8_, N10, 701 (1979). 60. Rard, J.A., Miller, D.G., "Mutual Diffusion Coefficients of SrCl 2-H 20 and CsCl-H20 at 25°C from Rayleigh Interferometry", J. Chem. Soc, Faraday Trans., 1, 78, 887 (1982). 61. Rard, J.A., Miller, D.G., "Mutual Diffusion Coefficients of BaCl 2 - H20 and KC1-H20 at 25°C from Rayleigh Interferometry", J. Chem. Eng. Data, 25, 211 (1980). 62. Sorell, L.S., Myerson, A.S., "Diffusivity of Urea in Saturated and Supersaturated Solutions", AIChE J., 28, N5, 772 (1982). 63. Renner, T.A., Lyons, P.A, "Computer-Recorded Gouy Interferometric Diffusion and the Onsager-Gosting Theory", J. Phys. Chem., 78, 2050 (1974). 64. Gableman-Gray, L., Fenichel, H., "Holographic Interferometric Study of Liquid Diffusion", Applied Optics, j_8, N3, 343 (1979). 65. O'Brien, R.N., Langlais, A.J., Seufert, W.D., "Diffusion Coefficients of Respiratory Gases in a Perfluorocarbon Liquid", Science, 217, 153 (1982). 66. Bollenbeck, P.H., Ramirez, W.F., "Use of a Rayleigh Interferometer for Membrane Transport Studies", I&EC Fund., L3, N4, 385 (1974). 67. Min, S., Duda, J.L., Nother, R.H., Vrentas, J.S., "An Interferometric Technique for the Study of Steady State Membrane Transport", AIChE J., 22, Nl, 175 (1976). 68. Forgacs, C , Liebovitz, J. , O'Brien, R.N. , Spiegler, K.S., "Interferometric Study of Concentration Profiles in Solutions Near Membrane Interfaces", Electrochim, Acta, ^ 0_, 555 (1975). 69. Einstein, A., Investigations on the Theory of Brownian Movement, Dover, New York (1956). 70. Stokes, G.G., Trans. Cambridge Phil. Soc, 9_, Part II, 8 (1851). - 124 -71. Staverman, A. J. , "Non-Equilibrium Thermodynamics of Membrane Processes", Rev. Trav. Chim., 70, 344 (1951). 72. Renkin, E.M., "Filtration, Diffusion and Molecular Sieving Through Porous Cellulose Membranes", J. Gen. Physiol., 38, 225 (1954). 73. Dawson, H., Textbook of General Physiology, 4th Ed., J & A Churchill, London (1970), p. 414. 74. Faxen, H., "Movement of a Rigid Sphere Along the Axis of a Pipe F i l l e d with a Viscous Fluid", Ark, Mat., Astron. och Fysik, _17_, 27 (1922). 75. Williamsen, B.G., Geankopolis, C.J., "Diffusion of Sucrose in Protein Solutions with Pore Restriction Effects Present", J. Chem. Eng. Data, 26, 368 (1981). 76. Bueche, F., Introduction to Physics for Scientists and Engineers,  3rd Ed., McGraw-Hill, New York (1980). 77. Schmalz, T.G., Norris, CL., Flygare, W.H., "Localized Magnetic Susceptibility Anisotropics", J. Am. Chem. Soc, 95_, N 2 4 (1973). 78. Fujuwara, F.Y., Reeves, L.W., "Liquid Crystal/Glass Interface Effects on the Orientation of Lyotropic Liquid Crystals in Magnetic Fields", Can. J. Chem., 5_6, 2178 (1978). 79. Ratna, B.R., "Magnetic an Electric Birefringence in the Isotropic Phase of a Nematic of Large Positive Dielectric Anisotropy", Mol. Cryst. Liq. Cryst., 58, 205 (1980). 80. Buckingham, A.D., "Experiments with Oriented Molecules", Chem. in Britian, 54, (Feb, 1965). 81. Lizuka, E., Kondo, Y., "Magnetic-Field Orientation of the Liquid Crystals of Polyribonucleotide Complexes", Mol. Cryst. Liq. Cryst., 57_, 285 (1979). 82. Muta, K., Takezoe, H., Fukados, A., Kuze, E., "Mixing Effects on the Cotton-Mouton Constant of 30CB and 50CB in Isotropic Phase", Jap. J. Appl. Phys., 20, N3 (1981). 83. Porte, G., Poggi, Y., "Behavior of Micellar Aqueous Solutions in High Magnetic Field", Phys. Rev. Ltrs., 41, N21 (1978). 84. Maret, G., Dransfeld, K., "Macromolecules and Membranes in High Magnetic Fields", Physica, 86-88B, 1077 (1977). 85. Fessenden, R.J., Fessenden, J.S., Organic Chemistry, 2nd Ed., Prindle, Weber & Schmidt Publishers, Boston (1982). 86. Harvey, J.M., Symons, M.G.R., "The Hydration of Monosaccharides -an NMR Study", J. Solution Chem., J 7 , N8, 57 (1978). - 125 -87. 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). 88. Mathlouthi, M., "X-Ray Diffraction Study of the Molecular Association in Aqueous Solutions of D-Fructose, D-Glucose, and Sucrose", Carbohydrate Res., 91_, 113 (1981). 89. Jeffry, G.A., "Intramolecular Hydrogen-Bonding in Carbohydrate Crystalate Structures", Carbohydrate Res., 28_, 233 (1973). 90. Hanson, J.C, Sieker, L.C, Jensen, L.H., "Sucrose: X-Ray Refinement and Comparison with Neutron Refinement", Acta Cryst., B29, 797 (1973). 91. Brown, G.M., Levy, H.A., "Further Refinement of the Structure of Sucrose Based on Neutron-Diffraction Data", Acta Cryst, B29, 790 (1973). 92. Mulay, L.N., Boudreaux, E.A., Theory and Applications of Molecular Diamagnetism, John Wiley & Sons, New York (1976). 93. Shporer, M., Vega, A.J., Frommer, M.A., "Diamagnetic Susceptibility Effects in NMR Measurements of the Properties of Water in Polymeric Membranes", J. Polymer Sci., _12_, 645 (1974). 94. Packer, K.J., "Nuclear Spin Relaxation Studies of Molecules Adsorbed on Surfaces", Progr. Nucl. Magnetic Resonance Spectroscopy,_3, 87 (1968). 95. Resing, H.A., "Nuclear Magnetic Resonance Relaxation of Molecules Adsorbed on Surfaces", Adv. Mol. Relaxtion Processes, 1_, 109 (1967-1968). 96. Beysens, D., Calmettes, P., "Temperature Dependence of the Refractive Indices of Liquids: Deviation from the Lorentz-Lorenz Formula", J. Chem. Phys., N2 (1977). 97. Reisler, E., Eisenberg, H., Minton, A.P., "Temperature and Density Dependence of the Refractive Index of Pure Liquids", J. Chem. Soc, Faraday Trans., II, 68_, 1001 (1972). 98. Lin, CT. , Marques, A.D.S. , Pessine, F.B.T. , Guimaraes, W.V.N. , "The Shape of the Refractive Index Versus Composition Curves for Hydrogen-Bonded Liquid Mixtures", J. Molecular Structure, 7_3_, 159 (1981). 99. Klein, M.V., Optics, John Wiley & Sons, New York, 1970. 100. Driscoll, W.G., and Vaughn, W., Handbook of Optics-Optical Society  of America, McGraw-Hill Co., New York (1978). - 126 -101. Model 124B Nelium-Neon Laser, Spectra Physics Mountain View Calif., G/124B (1979). 102. Models 332, 336, and 338 Telescopes and Spatial Filters -Spectra-Physics, Mountain View, Calif., B/332-338 (1977). 103. Rickles, R.N., Membranes; Technology and Economics, Noyes Development Corp. (1967). 104. V-7560 F i e l d i a l Regulator Mark I, Varian Associates Publication, 87-170-406 Rev., A1171 (1971). 105. Beach, K.W., Muller, R.H., Tobias, C.W., "Light Deflection Effects in the Interferometry of One-Dimensional Refractive-Index Fields", J. of Optical Soc, of Am., 63, N5, 559 (1973). 106. Clifton, M., Sanchez, V., "Optical Errors Encountered in Using Holographic Interferometry to Observe Liquid Boundary Layers in Electrochemical Cells", Electrochim. Acta, 24, 445 (1979). 107. Carrier, G., Pearson, CE., Partial Differential Equations, Academic Press, New York (1976). 108. UBC Curve, Curve Fitting Routines, University of British Columbia Computing Centre, 1981. 109. Adams, W.A., Davis, A.R., Seabrook, G., Ferguson, W.R., "A Digitized Laser Interferometer for High Pressure Refractive Index Studies of Liquids", Can. J. of Spectroscopy, 2l_, N2, 40 (1976). 110. Watkins, L.S., Tvarusko, A., "Lloyd Mirror Laser Interferometer for Diffusion Layer Studies", Rev. Sci. Inst., 41^ , N12, 1860 (1970). 111. Hall, R.S., Young, S.O. , Johnson, CS., "Photon Correlation Spectroscopy in Strongly Absorbing and Concentrated Samples with Applications to Unliganded Hemoglobin", J. Phys. Chem. JJ4_, N7, 756 (1980). - 127- -APPENDIX A Interference Fringe Data r 128 -This appendix contains a l l experimental data points. The points are tabulated for each run i n columns corresponding to each time, given i n seconds. Each point i s the l o c a t i o n i n centimeters where an interference fringe "bends" by an amount equal to one fringe spacing. Fringes are measured from each end of the d i f f u s i o n c e l l where they are s t r a i g h t , i . e . , no r e f r a c t i v e index gradient. Negative values denote fringes measured i n lower half of c e l l . The membrane surface i n each half of c e l l i s located at 0.0 cm. - 129 -R U N L , F I E L D = 0 . 0 , PORE D=8.0 M I C R O N * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 900.0 1800.0 3600.0 5400.0 7200.0 ***************************************************** 0 . 3330 0.3300 0 . 3300 0.3300 0 . 3330 0 . 0231 0.0386 0 .0575 0.0837 0 . 1 050 0 . 01 67 0.0278 0 . 0450 0.0643 0 . 0823 0 .0110 0.0205 0 . 0344 0.0502 0 . 0635 0 . 0065 0.0147 0 .0260 0.0413 0 .0507 0 . 0033 0.0090 • 0 .0183 0.0303 0 .0398 0 . 0002 0.0045 0 .0126 0.0225 0 . 0305 - . 0005 0.0007 0 .0070 0.0151 0 .0215 - . 0030 -.0026 0 .0010 0.0081 0 .0129 - . 0054 -.0064 - .0009 0.0015 0 . 0050 - . 0089 -.0108 - . 0059 -.0020 - . 0023 - .0130 -.0150 - .0118 -.0087 - . 0095 - .0173 -.0199 - .0169 -.0157 - .0177 - . 0220 -.0257 - . 0225 -.0224 - . 0257 - . 0285 -.0320 - .0300 -.0297 - .0349 - . 0368 -.0400 - . 0376 -.0388 - . 0438 - . 3330 -.0525 - . 0466 -.0470 - . 0542 -.3330 - . 0588 - . 0 5 8 5 - . 067 1 - .0768 -.0730 - .0837 - .3330 - . 0 9 3 5 - . 1 086 -.3330 - . 3330 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * - 1 3 0 -RUN N, FIELD= l.OkG, PORE D=8.0 MICRON *************************************** 900.0 1800.0 3600.0 5400.0 9000.0 ************************************************** 0 . 3330 0 .3300 0 . 3300 0.3300 0.3330 0. 0279 0.0450 0 .0650 0.0931 0. 1245 0.0193 0.0335 0 . 0492 0.0700 0.0926 0.0 136 0.0252 0 . 0383 0. 0549 0.0737 0.0096 0.0184 0 . 0300 0. 0429 0.0602 0.0053 0.0128 0 . 0228 0.0336 0.0471 0.0013 0.0084 0 . 01 52 0.0249 0.0360 -.0028 0.0036 0 . 0095 0.0172 0.0259 -.0061 0 . 0001 0 . 0036 0.0100 0.0171 -.0090 -.0032 - .0002 0.0031 0.0081 -.0127 -.0071 - .0055 -.0071 0.0001 -.0166 -.0108 - .0120 -.0125 -.0058 -.0204 -.0155 - .0170 -.0198 -.0134 -.0263 -.0200 - .0230 -.0276 -.0220 -.0350 - . 0257 - . 0302 -.0360 -.0310 -.3330 -.0317 - .0378 -.0457 -.0414 -.0406 - .0470 -.0561 -.0520 -.0520 - .0590 -.0692 -.0645 -.3330 - .0765 -.0914 -.0793 - . 3330 -.3330 -.0970 -.1255 -.3330 ***************************************************** - 131 -RUN K, FIELD=2.5 kG , PORE D=8.0 MICRON * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 900.0 1800.0 3600.0 5400.0 9000.0 ***************************************************** 0.3330 0.3300 0.3300 0 . 3300 0 . 3300 0.0256 0.0437 0.0670 0 .0828 0 . 1 094 0.0177 0.031 1 0.0499 0 .0628 0 .0842 0.0118 0.0225 0.0376 0 .0495 0 .0671 0.0079 0.0164 0.0287 0 .0380 0 .0521 0.0036 0.0116 0.0205 0 .0287 0 .0395 0.0010 0.0066 0.0138 0 .0200 0 .0300 -.0013 0.0016 0 . 0079 0 .01 24 0 .0194 -.0045 -.0035 0.0022 0 . 0056 0 .0106 -.0074 -.0083 - .0046 - .0053 0 .0025 -.0107 -.0124 - .0097 - .0118 - .0006 - .0139 -.0161 -.0152 - .0189 - . 0090 -.0176 -.0218 - .0209 - .0258 - .0172 -.0225 -.0272 -.0271 - .0338 - .0264 -.0280 -.0327 - .0347 - .0427 - .0358 -.0370 -.0404 - . 0431 - . 0529 - .0462 -.3330 -.0505 - .0525 - .0645 - .0584 -.3330 - .0650 - .0793 - .0708 - .0868 - .1031 - .0857 - .3330 - . 3330 - . 1 063 -.1378 -.3330 ***************************************************** - 132 -RUN R, FIELD = 4.0 kG, PORE D= 8.0 MICRON ************************************************n.*i;i. 900.0 1800.0 3600.0 5400.0 9000.0 *************************************** 0 .3330 0.3300 0 .3300 0.3300 0.3330 0 . 0278 0.0411 0 . 0645 0. 0930 0.1182 0 . 01 88 0.0300 0 . 0484 0.0653 0.0884 0 .0131 0.0217 0 .0378 0.0508 0.0704 0 .0081 0.0155 0 . 0289 0.0393 0.0547 0 . 0050 0.0110 0 . 021 1 0.0302 0.0429 0 .0014 0.0062 .0 . 01 50 0.0221 0.0274 — . 001 6 0.0015 0 .0084 0.0143 0.0219 - . 004 1 -.0001 0 .0016 0.0074 0.0133 — .0076 -.0044 - . 0033 0.0001 0.0051 — . 0 1 1 1 -.0083 - . 0087 - -.0003 -.0009 — .0151 -.0126 - .0146 -.0066 -.0093 — .0190 -.0178 - .0201 -.0131 -.0170 - . 0251 - .0227 - . 0272 -.0201 -.0265 — .0337 -.0292 - . 0348 -.0273 -.0373 - . 3330 -.0371 - .0433 -.0353 -.0484 -.0494 - .0553 -.0455 -.0587 -.3330 - .0728 -.0571 -.0745 . 3330 -.071 1 -.0928 - .3330 -.0918 -.1186 -.3330 ***************************************************** - 133 -RUN M, FIELD=10.0kQPORE D=8.0 MICRON *********************************** 900.0 1800.0 3600.0 5400.0 9000.0 ***************************************************** 0.3330 0.3300 0.3300 0.3300 0.3330 0.0298 0.0476 0.0656 0.0870 0.1083 0.0192 0.0336 0.0489 0.0651 0.0829 0.0134 0.0246 0.0368 0.0493 0.0650 0.0082 0.0180 0.0280 0.0380 0.0499 0.0045 0.0124 0.0194 0.0280 0.0378 0.0009 0.0072 0.0129 0.0195 0.0272 -.0003 0.0024 0.0062 0.0114 0.0170 -.0031 -.0027 -.0003 0.0048 0.0083 -.0061 -.0070 -.0061 -.0018 -.0037 -.0087 -.0117 -.0121 -.0086 -.0115 -.0126 -.0162 -.0182 -.0160 -.0210 -.0167 -.0218 -.0245 -.0238 -.0300 -.0219 -.0278 -.0322 -.0315 -.0410 -.0291 -.0349 -.0408 -.0410 -.0520 -.0378 -.0425 -.0509 -.0509 -.0651 -.3330 -.0564 -.0636 -.0630 -.0809 -.3330 -.0856 -.0805 -.1004 -.3330 -.1043 -.3330 -.1301 -.3330 *********************************************** * * * * * * * * * * * * if * * * * * * * * * * I I I I I I I I I O O O O O O O C O O O O O O O O O O O O O O O O C O W W W N - - - ^ O O O O O O - * ^ N J C O w a n o r o o a w o - j w ^ o >t> -j ro CD 03 w O V D t O O O x l C D * > O U ) i I i W > l i C D ( ^ W O O I I I I I o o o o o o o o CO O o o O o LO cn CO ro N J Co a i —' N J CTl o O cn CO 00 00 o o o o o o o — - * o o o o o O O O O O CO O — 1 — 1 NJ CO CO W C O ^ C O O l O V D - J W - ^ -O VO CO it* o> o I I I I I I I I I I O O O O O O O O O C O O O O O O O O O O O O O O O O O O O C O tAJCOCMJl^WIO-'-'OOOO-'-'WW^CMO WN)--l£)OWCriU5*«)WOff\N)CO(nuiOOCCIO 000\OW«)-'IO-'OCMTiCfMtiCr\iC>CriN)ITiO I I I I I I I I I I I I O O O O O O O O O C O O O O O O O O O O O O O O O O O O O O C O w u 3 - J ( j i ^ * > w w - " - ' O o o - - - - w ( j j ^ o i a i w W v i ^ o y j o w u i v i - ' t n o i i i - ' V D x i m c D ^ s i o o ^ a n D c o m w - ' i o c r i C D - ' C D L n M a i C D - v j f o u i o I I I I I I I O O O O O O O O O O C 0 - - O O O O O O O O O O O O O O O O O O - - C 0 UWU)Nl (TIUlpt>WN)-'OOO-'-'Wp(i(JlUlC0-'U W s l V O i D W W W W i C ' U l v I - ' t i J O V D V D O - ' -O >C> CO o * > w « } * > s j u i u ) w * > c o w c n w w o - * > w - J M O * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * co O CO O O o CO o> o o cn o o CO o * * * * * * * * * * * * * * * * * * * * * * * * * * * a Tl CT a cn a o Kl a oo o » o z CO - 135 -RUN P, F I ELD=l l . kG ,PORE D=8.0 MICRON * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 900.0 1800.0 3600.0 5400.0 9000.0 ***************************************************** 0 . 3330 0.3300 0 . 3300 0.3300 0.3330 0.0329 0.0536 0 . 0778 0.0997 0.1317 0.0237 0.0374 0 .0562 0.0748 0.0983 0.0167 0.0282 0 .0443 0.0587 0.0779 0.0120 0.0208 0 .0346 0.0468 0.0634 0.0077 0.0147 0 .0264 0.0362 0.0501 0.0043 0.0106 0 .0188 0.0284 0.0394 0.001 1 0.0061 0 .01 29 0.0202 0.0293 -.0003 0.0019 0 .0067 0.0132 0.0194 - .0039 -.0034 0 .0016 0.0064 0.0105 -.0071 -.0074 - .0017 -.0035 0.0021 -.0102 - .0115 - .0069 -.0101 -.0003 -.0134 * - .0167 - .01 37 -.0170 -.0077 - .0177 - .0229 - .01 95 -.0245 -.0170 - .0240 - .0287 - .0260 -.0330 -.0266 -.0321 -.0372 - .0343 -.0421 -.0368 - . 3330 -.0492 - .0429 - .0529 -.0467 - .3330 .0551 .0731 -.0677 -.0887 - .0586 -.0732 .3330 -.3330 -.0938 -.1187 - .3330 ***************************************************** -136 -RUN H, FIELD = 12.0 kG, PORE D = 8.0 MICRON ************************************* 900.0 1800.0 3600.0 7200.0 10800.0 ***************************************************** 0.3300 0. 3300 0.3300 0.3300 0. 3300 0.0259 0. 0494 0.0702 0.1037 0 . 1 1 56 0.0165 0. 0332 0.0509 0.0779 0. 0857 0.0105 0. 0249 0.0384 0.0587 0. 0668 0.0062 0. 0176 0.0286 0.0460 0. 0522 0.0020 0. 0112 0.0202 0.0346 0. 0384 -.0040 0. 0061 0.0131 0.0246 0. 0276 -.0076 0. 001 3 0.0064 0.0143 0. 01 56 -.0103 0034 0.0011 0.0060 0. 0052 -.0133 -. 0086 -.0067 -.0043 0064 -.0164 0 124 -.0110 -.0107 • 01 52 -.0203 • 0166 -.0172 -.0186 - # 02 45 -.0247 0209 -.0226 -.0267 - _ 0352 -.0302 0268 -.0296 -.0353 0455 -.0403 0325 -.0369 -.0445 0584 -.3330 0400 -.0452 -.0554 07 1 6 0498 -.0539 -.0663 0883 3330 -.0677 -.0789 1 092 -.3330 -.0962 -.3330 1332 3330 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * - 137 -RUN A2, FIELD=0.0 , PORE D=0.8 MICRON * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 900.0 1800.0 3600.0 7200.0 10800.0 ***************************************************** 0.3300 0.3300 0 .3300 0.3300 0.3330 0. 0206 0.0335 •0 .0519 0.0889 0.1149 0.0136 0.0225 0 .0371 0.0648 0.0853 0.0076 0.0146 0 .0272 0.0484 0.0663 0.0031 0.0093 0 .0184 0.0362 0.0513 0.0001 0.0051 0 .0114 0.0248 0.0368 - . 0039 0.0001 0 . 0051 0.0160 0.0250 - .0070 - .0059 - .0081 0.0075 0.0138 - .0096 -.0094 - .01 34 0.0003 0.0045 - .0128 - .0132 - .0189 -.0093 -.0094 - .0142 - .0173 - . 0242 -.0172 -.0184 -.0171 - .0216 - .0304 - .0235 -.0270 - .0155 - .0269 - .0365 - .0315 -.0363 - .0190 - .0323 - .0444 -.0394 - .0459 - . 0234 - .0385 - .0513 - .0480 -.0576 - .0272 - .0458 - .0616 -.0573 -.0700 - .0327 - .0567 - .0750 - .0679 -.0823 - . 0402 - .0768 - .0906 -.0811 -.0972 - .0562 - .3330 - .3330 -.0961 -.1175 - .3330 -.1214 - . 1437 -.3330 -.3330 ***************************************************** - 138 -RUN E2, FIELD= 2..5kO, P O R E D=0.8 MICRON *************************************** 915.0 1840.0 3600.0 7200.0 10800.0 ***************************************************** 0.3300 0.3300 0.3300 0.3300 0.3330 0.0264 0.0412 0.0616 0. 1033 0.1394 0.0155 0.0297 0.0466 0.0764 0.1007 0.0100 0.0206 0 .0344 0.0574 0.0778 0.0050 0.0143 0.0249 0.0442 0.0611 0.0008 0.0085 0.0170 0.0318 0.0466 -.0057 0.0023 0.0091 0.0216 0.0332 -.0093 -.0031 0 . 0027 0.0119 0.0219 -.0125 -.0072 -.0093 0.0027 0.0104 -.0158 -.0124 -.0154 -.0059 -.0123 -.0200 -.0169 -.0222 -.0151 -.0226 -.0244 -.0219 -.0288 -.0242 -.0333 -.0288 -.0268 -.0371 -.0343 -.0455 -. 0344 -.0335 -.0444 -.0444 -.0564 - . 0440 -.0405 -.0537 -.0550 -.0709 -.3330 -.0493 -.0674 -.0673 -.0860 -.0636 -.0862 -.0819 -.1293 -. 3330 -.3330 -.1017 -. 1336 -.3330 -.1694 -.3330 ***************************************************** - 139 -RUN C2, FIELD=5.0kG,PORE D=0.8 MICRON ***************************************** 900.0 1800.0 3600.0 7200.0 10800.0 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 0.3300 0.3300 0 . 3300 0.3300 0 .3330 0.0273 0.0447 0 . 0649 0.0921 0 . 1 1 62 0.0199 0.0318 0 .0509 0.0730 0 . 0889 0.0141 0.0240 0 . 0387 0.0572 0 .0704 0.0093 0.0178 0 .0289 0.0464 0 . 0562 0.0053 0.0124 0 . 0224 0.0367 0 . 0430 0.0023 0.0078 0 . 01 58 0.0270 0 . 0323 -. 0041 0.0032 0 . 0097 0.0171 0 .021 5 -.0066 -.0042 0 . 0044 0.0094 0 .0121 -.0096 -.0080 - . 0022 0.0026 0 .0031 -.0123 -.0119 - . 0089 -.0014 - .0084 -.0152 -.0156 - .0152 -.0099 - .01 68 -.0185 -.0202 - .0191 -.0183 - . 027 1 -.0219 -.0246 - .0253 -.0254 - . 0364 -.0253 -.0294 - .0317 -.0338 - . 0463 -.0302 -.0351 - . 0378 -.0436 - .0563 -.0358 -.04 1 2 - . 0448 -.0526 - . 0688 -.044 1 -.0490 - . 0544 -.0628 - .0821 -.3330 -.0613 - . 0648 -.0754 - .0976 -.3330 - . 0794 -.0926 - .1184 - . 3330 -.1105 - . 1 466 -.3330 - .3330 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * - 140 -RUN D2, FIELD=10.kG,PORE D=0.8 MICRON * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 900.0 1800.0 3600.0 7200.0 10800.0 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 0.3300 0.3300 0.3300 0.3300 0.3330 0.0222 0.0380 0.0558 0.0959 0.1218 0.014 1 0.0260 0.0395 0.0703 0.0915 0.0081 0.0175 0.0295 0 . 0532 0.0704 0.0031 0.0110 0.0203 0.0405 0.0547 -.0063 0.0057 0.0122 0.0290 0.0398 -.0093 0.0012 0.0053 0.0194 0.0279 -.0122 - .0044 - .0060 0.0105 0.0167 -.0151 - .0076 -.0128 0.001 1 0.0006 -.0183 - .0110 - .0179 -.0102 - .0116 -.0213 - .0159 -.0234 -.0181 -.021 1 - .0246 - .0197 -.0293 - .0256 - .0305 -.0291 - .0245 -.0358 -.0333 - .0405 - . 0340 - .0290 -.0425 -.0414 - .0503 - .0418 - .0340 -.0512 -.0507 - .0630 - .3330 - .0399 -.0617 -.0612 - .0759 - .0477 - .0780 -.0744 -.0922 - .0594 - .3330 -.0904 - .1128 - .3330 -.1171 -.3330 -.1424 - .3330 ***************************************************** - 141 -RUN B2, FIELD = l2.5kG,PORE D=0.8 MICRON * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 900.0 1800.0 3600.0 7200.0 10800.0 ***************************************************** 0.3300 0.3300 0 .3300 0.3300 0 . 3330 0.0323 0.0480 0 .0695 0. 1028 0 . 1 209 0.0222 0.0343 0 .0534 0.0787 0 .0949 0.0158 0.0257 0 .041 5 0.0616 0 . 0754 0.0113 0.0188 0 .0322 0.0490 0 .0609 0.0072 0.0134 0 .0245 0.0387 0 . 0482 0.0035 0.0083 0 .01 73 0.0285 0 . 0365 0.0004 0.0038 0 .0119 0.0199 0 . 0264 - .0056 - .0010 0 .0060 0.0119 0 .0149 - .0086 -.0053 0 .0004 0.0038 0 . 0062 - .0114 - .0100 - .0003 -.0014 - .0060 -.0143 -.0141 - .0056 -.0098 - . 01 64 - .0175 -.0183 - .01 03 -.0183 - .0258 -.0212 - .0227 - .01 60 -.0259 - . 0357 - .0245 -.0273 - .0218 -.0340 - .0456 - .0289 -.0326 - .0274 -.0421 - .0569 - .0349 -.0392 .0339 -.0519 - .0676 - .0434 - .0466 - .0409 -.0626 - .0830 - . 3330 -.0602 - ,0487 -.0750 - .0994 - .333C - .0582 -.0928 - . 1 209 - .0698 - . 1149 - . 1 578 - .091 9 -.3330 - . 3330 ***************************************************** - 142 -APPENDIX B Raytracing Refractive Index P r o f i l e C o r r e l a t i o n Parameters - 143 -This appendix tabulates the c o e f f i c i e n t s A, b, and m derived from the raytracing computer program for the r e f r a c t i v e index p r o f i l e i n each hal f of the d i f f u s i o n c e l l n = m erf(Ax) + b where x i s the l o c a t i o n i n centimeters, n i s r e f r a c t i v e index, and A, b and m are determined from the best f i t of the r e f r a c t i v e index p r o f i l e corrected for wavefront d e f l e c t i o n . The values tabulated for each run are l i s t e d In columns under the respective times they are taken, i n seconds. The f i r s t three rows are the parameters tabulated for the upper h a l f of the d i f f u s i o n c e l l , the l a s t three rows corresponding to the lower ha l f of the d i f f u s i o n c e l l . RUN L, F I ELD=0.0 . PORE S IZE=8.0 MICRON * * * * * * * * * * + * + * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * + * * * + + + • * * + + * • * • TIME 9 0 0 . 0 1800.0 3600.0 5 4 0 0 . 0 7200 .O A * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * M-UP -0 .000446 -0.000493 -0.000553 -0.000603 -0.000617 A-UP 10.557301 7.465138 5.278650 4 .310000 3.732570 B-UP 1.331726 1.331776 1.331834 1.331903 1.331930 • i t * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * M-D0WN -0.000484 -0.000545 -0.000598 -0.000626 -0.00063 1 A-00WN 10.557301 7.465138 5.278650 4 .310000 3.732570 B-DOWN 1.332232 1.332170 1.332117 1.332089 1.332086 RUN N. FIELD= 1 KG. PORE SIZE =8.0 MICRON * + ***+***•# + + ****# + ***+*•**•• + + + * + • * • + + + + + • + + + • • + + + * * + + + * • • * * + + •+ •••* TIME 900.0 1800.0 3600.0 5400.0 9000.0 + * * * + * * * * + * * * + + * + • * + + + * + + + + •* + * + + + + + + + + + + + + + + + • + + * + + + + + •••••* + +••* + + +* + + *• M-UP -0.000474 -0.000550 -0.000573 -0.000606 -0.000634 A-UP 10.557301 7.465138 5.278650 4.310000 3.3385 11 B-UP 1.331768 1.331846 1.331869 1.331922 1.331950 M-DOWN A-DOWN B-DOWN -0.000469 -0.000556 -0.000599 -0.000601 -0.000656 10.557301 7.465138 5.278650 4.310000 3.338511 1.332249 1.332161 1.332116 1.332116 1.332058 - 146 -* •f m — r- CN — r-• o • 01 — a + CD — m m co l£J in O +- o + O co — O CO CN + o O n n + o r> ro o • O n n * o n ro +-o * • • O ro — • o r> +- * • * t # 1 * * * * * * * • * + * * * O O * r> O CO o O r- « I D O •rr * m O co * I D o O + * o • O O - * O o CN + * o O — ro O ro + + rr O n n * O ro ro * in * • O rr *- O <s -r- * t I + * * • + * * * + + * * 2 * * O * * r- O r- * o O ^ ~ cz * o ^ lO ^ in in in I D co * I D I D o * b * O co - * O CO CN £ * o * O t— r> * o r-- ro * u> « O cs n # o CN ro O ro * •r O in — *• o in # CO * * i 1 *-ii * * * UJ * * * rsi ¥ * * >—< *• « * * in * * * * f - CO CN * r> CO LU O * O ro o * r~ n rr * * * in — co * in * o * O * O in — * O in CN * CL * o O co * o I D ro * * 00 * O ^  n * o ro + - * * CJ * O r- - * o »- * * i * 1 * + <* * in + « # * « * * * CM * « * * II * + * * D * * (£> CN * »- -^ rr _j * o * in O ^ * o o ••-Li-) * * 'j n r~ * in r> CN t * o * O r- - O r~ CN * o * O m ro O m ro * * 0) * O tn co * O in ro * - * * * * * * o o - « O O •»- * * * 1 * 2 * * * * Z) * * * Q: z 2 2 UJ a a a 3 s. 3 Z) 3 o O O i i i D o o £ < CO 1 i 1 s < CO RUN R. FIELD=4.0 KG, PORE 0=8.0 MICRON * * * * * * * * * * * * * * * * * * • * * * * * * • * * * * * * * + + # + + *< TIME 900.0 1800.0 3600.0 .*•*•** M-UP A-UP B-UP -0.000466 10.557301 1.331758 -0.000508 7.465138 1 .331797 -0.000555 5.278650 1.331852 M-DOWN A-DOWN B-DOWN -0.000440 10.557301 1.332278 -0.000510 7.465138 1.332208 -0.000567 5.278650 1.332151 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * **+******+*++•*++****+*•*• 5400.0 9000.0 k + + +• + f -* -0.000581 -0.000599 4.3 10000 3.3385 11 1.331890 1.331909 • • * * * * • * • * + * + * * * * * * * • * • « * -0.000600 -0.0006 16 4.3 10000 3.338511 1.332116 1.332100 4>-• * * * < * • * * + * • * * * * * * • • • * + I RUN M, F I E L D = 1 0 K G , PORE S I Z E = 8 . 0 M I C R O N * * * * * * * * * * ( * * * * * * * * * + + * + * * + * + * * + * * * • • < T I M E 9 0 0 . 0 1 8 0 0 . 0 3 G 0 0 . 0 A * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * M-UP A-UP B-UP - 0 . 0 0 0 4 4 9 1 0 . 5 5 7 3 0 1 1 . 3 3 1 7 4 9 - 0 . 0 0 0 5 0 9 7 . 4 6 5 1 3 8 1 . 3 3 1 8 1 5 - 0 . 0 0 0 5 2 0 5 . 2 7 8 6 5 0 1 . 3 3 1 8 2 3 fc*************************************** M-DOWN A-DOWN B-DOWN - 0 . 0 0 0 4 8 2 1 0 . 5 5 7 3 0 1 1 . 3 3 2 2 3 2 - 0 . 0 0 0 5 6 3 7 . 4 6 5 1 3 8 1 . 3 3 2 1 5 0 - 0 . 0 0 0 6 0 9 5 . 2 7 8 6 5 0 1 . 3 3 2 1 0 1 5 4 0 0 . 0 9 0 0 0 . 0 * * * * * * * + * + * + •*+•+* + * * * • + 0 . 0 0 0 5 5 1 - 0 . 0 0 0 5 6 6 4 . 3 1 0 0 0 0 3 . 3 3 8 5 1 1 1 . 3 3 1 8 6 1 1 . 3 3 1 8 7 0 0 . 0 0 0 6 3 4 4 . 3 1 0 0 0 0 1 . 3 3 2 0 7 6 - 0 . 0 0 0 G 4 4 3 . 3 3 8 5 1 1 1 . 3 3 2 0 6 7 RUN T, FIELD=10.5 KG, PORE SIZE=8.0 MICRON * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * + * * * * * * * * * * * * + * * * * * * * * * * * + • * TIME 960.0 1800.0 3600.0 5400.0 9 120.0 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * • * * * * * * * * + + * + + + * • * * * * * M-UP -0.000442 -0.000481 -0.000527 -0.00055 1 -0.000589 A-UP 10.222062 7.465138 5.278650 4.310000 3.316475 B-UP 1.331734 1.331762 1.331827 1.331858 1.331893 *****++*********+***********+******•*+**•*••+*+•*********•*+•++++++-M-OOWN -0.000496 -0.000573 -0.000635 -0.000659 -0.000674 A-DOWN 10.222062 7.465138 5.278650 4.310000 3.316475 B-DOWN 1.3322 19 1.332 143 1.332076 1.332056 1.332040 - 150 -• * + * + o * * * ID * o O * o * O * o *• o * o * * * o • * 1 • * * * * +• * ft • ft * ft CN * o ft co • *S> * 6 * O • o ft o + ft o in * + ft o • ft 1 • ft * + + * * * * 0) 2 * O * 0) D + * in cc O ft o (J + O * o * CD * o * CO • ft 6 O + * 1 • * CO * * II * LiJ + * M * t—i * ft r-(/) + O * (D * ft LT) UJ * O ft o a * o * o o * CO * o CL * * * * o • * * 1 * ft * ft « * * ft II * * Q # CO _i * O * LU « LO *-* * 6 * o u . * o * O * 0) * O - * * a * # O * ft i z * * # LU a 2 t~< i »- 2 * o ID f- * *- — o * in cn * 10 in * CO — * o CO CN # ro n * o ro ro * D O # o ro ro « * * ro -- * o ro * i * * * • * * * • * « O iD + o O CO + O in * r- O * O 0) + m O * O - + O O fN * — ro • O ro « CO ro * O ro ro * * * *j — * O * i * * * ft * * O in * r~ O O in *- * in * IP 00 * in IS> * co * O CO CM * r- ro * o t - ro * CN ro * o CN ro * * + in — * 6 in «- « # 1 * * * * * • * * * * co O * r- CO O ro co * 0) ro CN * * - CO * *T ~— <N * in — * o in CN * ir> ro * o in ro • ro o ro * « r» -~ # 6 r-* 1 * *  « * * * * « — r- p- -r— ro * O O CD * ro co * ro CN * • O r- CN * in ro * O in ro • in ro * o in ro # 6 - * 6 O * -^ « 1 * * * * * z Z a a s o r ) o o O i i o a a < CO 1 i i < CO - 151 -* * ¥ + +• r- O ¥ CO o o • * o + rr ro rr ¥ VP ro in *- * LD IP CO ¥ IP IP O + • o + O r- ¥ O CN ¥ o O rj ro ¥ O ro ¥ ¥ CO * O O ro ¥ o O CO ¥ ¥ o • ¥ ¥ — * C co ¥ o ro — ¥ * i ¥ 1 ¥ + • ¥ ¥ * ¥ ¥ * * ¥ ¥ + ¥ ¥ + ¥ ¥ • CN O ip ¥ rr O Ln ¥ o * LT) f - <s> ¥ rr r- r- * + in in CO ¥ tP in O ¥ ¥ o • O CN ¥ O cv CN ¥ o * O co ro * o cn CO ¥ * CN * O r- CO * o r- CO ¥ * r- • * ¥ * * O ro ¥ o co »- ¥ * i ¥ 1 ¥ + ¥ ¥ • ¥ ¥ +• * ¥ ¥ + ¥ * ¥ • * r- O in o O o ¥ 2 O • CN in ro ¥ o in CN * D * * in ip CO ¥ IP ip * Ct O * O co — ¥ o CO CN * U * O * O r~-ro ¥ o r- CO ¥ t—t IP + O CN ro ¥ o CO £ * co * ¥ * + O in * o in ¥ o + * i 1 ¥ * ¥ ¥ CO * * * ¥ tl * * * ¥ UJ ¥ * ¥ rsj * ¥ ¥ »—i *- ¥ in co * 03 co in ¥ (/> * O ¥ co ro 00 # r- ro rr ¥ + ¥ rj — * in » - ¥ UJ * O ¥ O tn ••- * O in CN ¥ ct • O ¥ O <£) CO * O CO ¥ o « CO ¥ O rr ro ¥ O CO ¥ a * ••— ¥ ¥ ¥ * ¥ O f - ¥ 6 r- ¥ • # ¥ i * ¥ * • ¥ * * * ¥ ¥ * * ¥ CN ¥ * ¥ * * * ¥ tl * ¥ r- -r- rr * 00 »- CP ¥ Q * O ¥ co O CO * in o in ¥ - J ¥ * ro ro IP * in CO »— ¥ LU d * O r- * O r- CN ¥ •—* ¥ o * O in ro * O IT) CO ¥ ¥ 0 ) * O in ro * O in CO ¥ * * * ¥ • * # 6 6 * 6 O ¥ X * * i * i ¥ # * ¥ 2 * ¥ * ¥ 3 Ct z 2 2 UJ CL CL CL 3 j * £ 3 3 3 o O O i t i o O Q £ < CO 1 i t s. < CO RUN A2, FIELD=0.0. PORE SIZE=0.8 MICRON ******•***+*****•*******************•*+***************+*+*******+*•* TIME 900.0 1800.0 3600.0 7 200.0 10800 0 M-UP -0.000352 -0.0004 11 -0.000439 -0.0005 17 -0.000537 A-UP 10.557301 7.465138 5.278650 3.732570 3.047630 B-UP 1.331638 1.331691 1.331724 1.331807 1.331837 *********************************+******+*************************++ M-DOWN -0.000769 -0.000731 -0.000778 -0 000775 -0.000764 I A-OOWN IO 557301 7.465138 5.278650 3.732570 3.047630 B-DOWN 1.331936 1.331966 1.331925 1.331935 1.331946 £ N> • A * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * . RUN E2, FIELD=2.5 KG. PORE 5IZE=0.8 MICRON ft**********************************************^ TIME 915.0 1840.0 3600.0 7 200.0 10800.0 r****** + + + *-% + *****«** + * + * + **** + ** + ** + + M-UP A-UP B-UP -0.000363 10.470408 1.331661 -0.000439 7.383551 1.331739 -0.000486 5.278650 1 .331787 -0.000527 3.732570 1 . 33 1842 -0.000561 3.047630 1 . 331886 M-DOWN A-DOWN B-DOWN -0.000602 10.470408 1.332105 -0.000645 7.383551 1.332062 -0.000659 5 . 278650 1.332049 -0.000686 3.732570 1.332016 -0.000639 3.047630 1.332068 CO RUN C2. FIELD=5KG, PORE SIZE=0.8 MICRON ********************************************************** TIME 900.0 1800.0 3600.0 7200.0 10800.0 ************************************************* ******************* M-UP A-UP B-UP -0.000488 10.557301 1.331779 -0.000529 7.465138 1.331825 -0.000576 5.278650 1.331872 -0.000609 3.732570 1.331903 -0.000602 3.047630 1 .33 1896 **************** ** ******************************** r + **+****H M-DOWN A-DOWN B-DOWN -0.000671 10.557301 1.332036 -0.000707 7.465138 1.332000 -0.000717 5 . 278650 1.331997 -0.0007 13 3.732570 1.332000 -0.000756 3.047630 1.331953 • • • A * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * RUN D2, FIELD =10KG, PORE SIZE=0.8 MICRON ( + * • + + + * • * * * * * - + + + +••*-•-TIME 900.0 1800.0 3600.0 7 200.0 10800.0 * * * * * * * * * f * * * * * * * * * * * * * * * . * * * * * * * * • * * * * * * * * + • * * • • * + + • + *< M-UP A-UP B-UP -0.000317 10.557301 1.331616 -0.000418 7.4G5138 1.331709 -0.0004 4 1 5.278650 1.331734 -0.000525 3.732570 1.331828 -0.000526 3.047630 1 .33 184 1 **********************************•+*#* > + * + * * + * + + * * + * i k + + •* + + + + M-DOWN A-DOWN B-DOWN -0.000666 10.557301 1.332044 -0.000704 7.465138 1.332006 -0.000684 5.278650 1.332031 -O.000708 3.732570 1.332006 -O.000709 3.047630 1.332003 RUN B2, FIEID=12.5 KG, PORE SIZE=8.0 MICRON TIME 900.0 1800.0 3600.0 7200.0 10800.0 ********************************************** M-UP -0.000528 -0.000530 -0.000600 -0.000610 -0.00062 1 A-UP 10.557301 7.465138 5.278650 3.732570 3.047630 B-UP 1.331829 1.331836 1.331903 1.331919 1.331924 A * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * - * M-DOWN -0.000648 -0.000673 -0.000735 -0.0007 11 -0.000744 I A-DOWN 10.557301 7.465138 5.278650 3.732570 3.047630 ^ B-DOWN 1.332060 1.332037 1.331971 1.332001 1.331963 -, 157 -APPENDIX C Raytracing Computer Program - 158 -IMPLICIT REAL*8(A-H,L,0-Z) C***********RAYTRACE TRACES RAYS THROUGH OPTICAL SYSTEM ACCOUNTING FOR C » * * * * * * * * * * R A Y BENDING FROM REF INDEX GRADIENT AND FITTING REF INDEX C***********PROFILE TO AN ERROR FUNCTION CORRELATION******************** REAL*8 M DIMENSION XO(40),F(40),IV(100),ND0BS(4O),PAR(3),XFP(40) , 1LD0BS(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, 120,2 1 ,22,23,Z5,27,N0URT2,NAIR,NGLASS REA L *8 NAIR,NGLASS,NWATER,NOURTZ,LAMBDA,LDCALC,LDOBS 1 ,NT0P,NB0TTM,ND0BS.NXO,NX1 C*********DEFINE PARAMETERS************************* PI=3. 14 127 XMAG=.337 D E L X M =.005 XMAGO=XMAG-.001 LAMBDA =.6328E-4 NA I R = 1 .000276 NWA T E R =1 .3340 NOURTZ=1.45709 NT I ME T =0 NGLASS=1.7499 DATA M.A.B/-7.D-4.7.DO,1.3300/ C * * * * * * » * * * * * * * * p r j ^ Q jfg DATA FOR RUN * * * * * * * * * * * * * * * * * * * * * * * * * * * READ( 10,629)NTI ME 1 ' CONTINUE ITER=0 IF L AG =1 c . * * , . * * * . . * S E T COUNTER FOR NUMBER OF TIME STEPS DURING EXPERIMENT-**** NT I ME T = NTI ME T+1 NE LMT =0 ISHIFT=0 IF(NTIMET.GT.NT IME)STOP C***********READ TITLE AND REFERENCE REFRACTIVE INDICES************ READ(10 j 610)NT 1,NT2,NT3,NT4,NT5,NT6,NT7,NT8,NT9,NT 10,NT 1 1 ,NT12 R E AD( 10,629)NPTS,NWATER.NTOP.NBOTTM,TIME IF(NTIMET.NE.1)G0 TO 589 WRITE(9,610)NT1,NT2,NT3,NT4,NT5,NT6,NT7.NT8.NT9,NT 10,NT 11 . NT 1 2 WRITE(9,629)NTIME 589 NE LMNT =NPT S DO 599 1=1,NPTS READ( 10,629)NFRNG(I ) ,X0( I ) 599 CONTINUE 50 FORMAT ( 'LREF LDOBS(I) NDOBS(I) X0( I ) X7(I) LL1 LL2 ' ) 5 CONTINUE NELUP =0 XMAGO=XMAG DO 1000 1=1,NELMNT 610 FORMAT(20A4) C** * *»**CALCULAT I0N5 FOR TOP HALF OF CELL ARE PERFORMED FIRST ASSUMING C , , * * * . * N Q BENDING IN REFRACTIVE INDEX GRADIENT********************** 629 FORMAT(I5.4G10.5) NDOBS(1)=NTOP XOC E L L(I)=X0(I)/XMAG IF(XOCELL(I).GT.O.)NELUP=NELUP+1 L12 = NQURTZ*(Z2-Z1 ) CALL RTRACE(XOCELL(I),0.D0,REF0PL,X7I) X7(I)=X7I LREF(I)=NWATER*(Z1-Z0)+REF0PL+L12 - 159 -C*******SUM ALL UNDEFLECTED PATH LENGTHS*************** IF(I.EQ.1)LREFO=LREF(I) C*******USING OPTICAL PATH LENGTHS FIND INITIAL REF INDEX PROFILE**** C***********ASSUMING NO RAY BENDING THROUGH REF INDEX GRADIENT******* NDOBS(I)=NTOP + (LAMBDA/(21-Z0))*(NFRNG(I )-1 ) NDOBS(1)=NTOP 60 FORMAT(.1X,7F10.7) 1000 CONTINUE. XMAG=X7(1)/XOCELL(1) IF(DABS(XMAG-XMAGO ) .GT. .00001)G0 TO 5 C ********INITIALIZE BOTTOM HALF OF C E L L * * * * * * * * * * * * * * * * * * * * * * * * ISTART=NELUP+1 DO 75 I=ISTART,NELMNT NDOBS(I)=NBOTTM-(LAMBDA/(Z1-Z0))*(NFRNG(NE LMNT)-NF RNG(I )) 75 NDOBS(NE LMNT)=NBOTTM PRINT 610.NT 1,NT2,NT3,NT4.NT5,NT6,NT7,NT8,NT9,NT 10.NT 11.NT 12 PRINT 55 DO 76 1=1,NELMNT 76 PRINT 56, NDOBS(I ) .NFRNG(I),XOCELL(I),X7(I).XO(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******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 * * * * * 10 IF(IFLA^.LT.0)NELMAX = NE LMNT ICONV= 1 C * * * * » * * F I T REF INDEX PROFILE TO ERROR FUNCTION************** ITER = ITER+ 1 IF(ITER . LT . 10. )G0 TO 57 PRINT 570 570 FORMAT(//,1X,'PROGRAM FAILED TO CONVERGE IN 10 PASSES',//) GO TO 1100 C************START WITH TOP HALF OF CELL. THEN DO BOTTOM HALF***** C * * * » * * * - - * * * S E T FLAGS FOR TOP OR BOTTOM** * * * * * * * * * * * * * * * * * * * * * * * * 57 NF =NE LMAX IF(IFLAG.LT.O)NE LMAX = NELMNT-NE LUP IF(I FLAG.LT.0)ISHIFT = NELUP DO 300 IC=1.NELMAX Y(IC)=NDOBS(IC+1 SHI FT) 300 XF(IC)=XOCELL(IC+ISHIFT) C*****CALL LINEARIZED ERROR FUNCTION FITTING ROUTINE CALL ERRFIT(PAR,NELMAX,TIME) 3000 FORMAT('RETURN CODE =',I5) M = PAR( 1 ) A = PAR(2) B = PAR(3) ISTART=ISHIFT+1 IFINAL = ISHIFT+NELUP IF(IFLAG.LT.O)IFINAL=NELMNT DO 500 I=ISTART,IFINAL c * * » » » * , * » * * * . P E R p 0 p M REFRACTIVE INDEX GRADIENT CALCS AND RAYBENDING** * IF(ITER.EO.1)NDOLD=NDOBS(I) DNDOBS = 2.*M*A*OEXP(-(A*A*XOCELL(I)*XOCELL(I)))/DSORT(PI ) 9999 DX1XO = DNDOBS*(Z1-ZO)*(Z1-ZO)/(2.*ND0BS(I)) X1=X0CELL(I)+ 0X1X0 NXO = NDOBS(I) - .5*DND0BS*(DX1XO) - 160 -S(I) = (DX1XO*DX1XO + (Z1-20)*(Z1-ZO))**.5 SINALP = DNDOBS * S(I)/NDOBS(I) L01=NDOBS(I)*S(I) NX 1 = NDOBS(I) + DNDOBS*DX1XO L12 = NQURTZ*(Z2-Z1)/DCOS(DARSIN((NOURTZ/NX1)*SINALP)) BETA 1 = DARSIN(SINALP*NQURTZ/NX1) X2=X1+(Z2-Z1)*DTAN(BETA1) C***********PERFORM RAYTRACING THROUGH REST OF SYSTEM****************** CALL RTRACE(X2.BETA1.OPTPL,XFINAL) LDCALC(I)=OPTPL +L12+L01 XFP(I )=XFINAL C********CHECK ERRORS BETWEEN BENT RAY POSITION ON FOCAL PLANE AND C********ACTUAL FRINGE AND CORRECT REFRACE INDEX PROFILE THEN c «* * * .» . * * START ING ITERATIONS ALL OVER AGAIN UNTIL CONVERGENCE IS OK**** DELXFP=XO(I)-XFP(I) XOCELL(I)=XOCE LL(I)+DELXFP/XMAG IFIDABS(DELXFP).GT. 1.E-6)IC0NV = -1 500 CONTINUE I F ( I CONV.GT.0)G0 TO 1100 GO TO 10 1100 CONTINUE IF(IFLAG.GT.0)PRINT 505 1 I F(I FLAG.LT.0)PRINT 5052 IF( IFLAG.GT.0) C******WRITE CORRELATION PARAMETERS ON UNIT 9 IF CONVERGENCE HAS BEEN ACHIEVED** 1WRITE(9,504 5)TIME,M.A.B.X0CELL(IFINAL) IF(I FLAG.LT.0)WRITE(9,504 5)M.A.B.XOCELL(I FINAL) 5045 FORMAT(1X.5E15.8) PRINT 5050. M,A.B PRINT 5060 00 1105 I=ISTART,IFINAL DELXFP = X0(I )-XFP(I ) 1105 PRINT 5000,LDCALC(I),LDOBS(I).S( I ) .DELXFP.NDOBS(I).X0CELL( I ) . 1XFP(I) IF(IFLAG.LT.O)G0 TO 1200 IFLAG=-1 ITER=0 GO TO 10 1200 GO TO 1 5060 F0RMAT(//,4X, 'LDCALC' . 1X, ' ' ,5X, 'S ' . 5X, 'DELXFP '.5X. 1 'NDOBS ' ,5X, 'XOCELL ' ,5X, 'XFP ' , / / ) 5000 FORMAT(1X,8(1X.F10.6)) 5051 FORMAT(//,1X,'TOP HALF OF C E L L ' . / / ) 5052 FORMAT(//,1X,'BOTTOM HALF OF C E L L ' , / / ) 5050 FORMAT(//,'M = ' , F 9 . 6 , ' A = ' . F 9 . 6 , ' B = ' .F9.6) END BLOCK DATA C********INITIALIZE COMMON BLOCKS WITH EQUIPMENT GEOMETRICAL PARAMETERS***** IMPLICIT REAL*8(A-H.N.0-Z ) COMMON /SET1/Y(40),XF(40)/SET2/RL1.TL1.XL1.RL2,TL2.XL2, 1ZO.Z1,Z2,Z3,Z5,Z7,N0URTZ.NAIR,NGLASS DATA RL1,TL1 .XL1/99.29, 1 .0.5.0/ DATA RL2,TL2,XL2/22.4,1.0,1.5/ DATA Z0.Z1,Z2,Z3.Z5,Z7/0. .1 . .2. ,58.2, 1 10. , 126 .6/ END SUBROUTINE ERRFIT(PAR.NEL.T) 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 IMPLICIT REAL*8(A-H,0-Z) DIMENSION XLIN(40),YFIT(40).WT(40) .E1(2),E2(2),P1(2),P2(2) - 161 -1,PAR(3) COMMON /SET 1 / Y(40),XF(40) / 1 CELL/IF LAG EXTERNAL AUX C*****SUBROUTTNE ERRFIT FITS REF INDEX PROFILE TO AN ERROR FUNCTION C * * » * * B Y FIRST LINEARIZING ERR FCT THEN FITTING WITH LINEAR CURVE FIT C*****INITIALIZE PARAMETERS IFLAG= 1 . IF(XF(2).LT.0.D0)IFLAG=-1 DO 1 1=1.NEL WT(I ) = 1.DO C*****IP T H E DEFLECTED X VALUE NEAR MEMBRANE IS IN A LOCATION THROUGH C*****THE MEMBRANE THIS POINT IS IGNORED FOR CORRELATION CALCULATIONS** I IF(XF(3).LT.O.DO.AND.XF(1).GT.0.DO)WT(1)=1.D-6 IF(XF(2).LT.O.DO)WT(NEL)=10.DO IF(XF(2).GT.0.DO)WT(1)=1.DO C******THIS WEIGHTS THE ENDPOINTS 10 TIMES OTHER POINTS*"**** IC0N=1 ITER=0 D=5.D-6 CUT =1,D-9/DS0RT(D*T) TAU = (1.+S0RT(5. ) ) /2.0 ALOW=O.DO AHIGH = 3./DSORT(D* T ) AT2 = (AHIGH-A LOW)/TAU +ALOW AT 1 = (AHIGH-AT2 +ALOW) EPS=1.D-10 P1(1)=0.D0 P1(2)=0.D0 P2( 1 ) =P 1 ( 1) P2(2)=P1(2) DO 100 I=1,NEL 100 XLIN(I)=DERF(AT1*XF(I)) C******DL0F IS UBC CURVE FITTING LINEAR LEAST-SQUARES CURVE FITTING ROUTINE** CALL DLQF(XLIN,Y,YFIT,WT,E1,E2.P1,1.DO.NEL.2.-5,ND,EPS.AUX) CALL ERRCAL(Y.YFIT.ERR0R1,NEL) DO 101 1=1.NEL 101 XLIN(I)=DERF(AT2*XF(I) ) CALL DLQF(XLIN,Y,YFIT.WT,E1.E2.P2,O.DO,NEL.2,-9.ND.EPS,AUX) CALL ERRCAL(Y,YFIT,ERR0R2,NEL) 10 IF(ERR0R1 .LE.ERR0R2)G0 TO 18 II AHIGH = AT2 IF(AHIGH-ALOW.LE.CUT)GO TO 21 AL 1 =AT1-ALOW IF(AHIGH-AT1.LT.AT1-AL0W)G0 TO 15 12 AT2 = AT 1 AT 1=AHIGH-(AT 1-ALOW) ERR0R2 = ERR0R 1 P2( 1 )=P1 ( 1 ) P2(2)=P1(2) ITER=ITER+ 1 112 DO 102 1=1,NEL 102 XLINfI)=DERF(AT1 *XF(I ) ) CALL DLQF(XL IN.Y,YFIT,WT,E1.E2,P1,1.D0.NEL,2.-5.ND.EPS.AUX) CALL ERRCAL(Y,YFIT,ERR0R1 .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.E1 ,E2.P2, 1 .DO,NEL,2.-5,ND,EPS,AUX) CALL ERRCAL(Y,YFIT,ERR0R2,NEL) - 162 -GO TO 10 ALOW=AT 1 IF(AT2-AL0W.LE.CUT)G0 TO 21 IFUT2-AL0W.LT.AHIGH-AT2) GO TO 20 AT 1= AT2 AT2 = AL0W+(AHIGH-AT 1 ) ERROR 1=ERR0R2 P1(1 )=P2( 1 ) P1(2)=P2(2) GO TO 115 AT 1=AHIGH-(AT2-AL0W) GO TO 112 CONTINUE FORMAT( 1X,6( 1X.F 15 . 6) ) IF(ITER.GT.20)ICON=-1 IF;ICON.LT.0)GO TO 200 ERR0R=ERR0R1 IF(ERROR 1.GT.ERR0R2)ERROR = ERR0R2 PAR(2 ) =AT 1 IF(ERROR 1 .GT.ERR0R2 )PAR(2 ) =AT2 PAR(3 ) = P1 (2 ) IF(ERROR 1 .GT.ERR0R2 )PAR(3 ) =P2(2 ) PARI 1 )=P1 ( 1 ) IF(ERROR 1 .GT.ERR0R2 )PAR( 1 ) = P2( 1 ) RETURN PRINT 5000 F0RMAT( 1X,//, 'ERRFIT FAILED TO CONVERGE IN 20 PASSES'.//) RETURN END SUBROUTINE ERRCAL(Y.YFIT,ERROR,N) ***SUBROUTINE ERRCAL CALCULATES RMS ERROR IN FITTED ERROR FUNCTION*" IMPLICIT REAL*8(A-H.0-Z) DIMENSION Y(40).YFIT(40) ERROR=0.0 DO 1 I = 1 , N ERROR=ERROR+(Y(I)-YFIT(I))*(Y(I)-YFIT(I)) RETURN END FUNCTION AUX(P,D,XLIN,L) **FUNCTIN AUX CALCULATES PARTIAL DERIVATIVES FOR LINEAR CURVE FITTING **ROUTINE USED TO FIT ERROR FUNCTI O N * * * * * * * * * * * * * * * * * * * * * * * * IMPLICIT REAL*8(A-H,0-Z ) COMMON /ICELL/IFLAG DIMENSION P(1),D(1) D ( 1 ) = X LIN D(2)=1.DO AUX=P(1)*XLIN+P(2) RETURN END SUBROUTINE LNSTRC(XL I .TLI .RLI.XIN.ALPHA 1 .XOUT.BETA.OPTPL.ZL) . IMPLICIT REAL*8(A-H,0-Z) RE A L * 8 NAIR,NGLASS ••SUBROUTINE TO CALCULATE OPTICAL PATH LENGTH THROUGH PLANO-CONVEX * * * **LENS USING GEOMETRICAL OPTICAL RAY TRACING************************ **TL1=LENS THICKNESS AT THINNEST POSITI O N * * * * * * * * * * * * * * * * * * * * * * **RLI=RADIUS OF CURVATURE OF LENS* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * **ZL=Z DISTANCE OF RAY THROUGH LENS* * * * * * * * * * * * * * * * * * * * * * * * * * * **XLI=OUTSIDE RADIUS OF LENS* * * * * * * * * * * * * * * * * * * * * * * **XINT=INITIAL X COORDINATE OF RAY ENTERING LENS***** **XOUT=FINAL X COORDINATE OF RAY LEAVING LENS****** - 163 -C******ALPHA1=ANGLE OF ENTERING RAY NORMAL TO FLAT LENS SURFACE**** C******ALPHA2 = ANGLE OF RAY THROUGH LENS MAT SRIAL * * * * * * * * * * * * C******BETA=EXIT ANGLE OF RAY LEAVING LENS*************** C******THETA=ANGLE RAY CUTS WITH TANGENT TO CURVED SURFACE INTERNAL TO LENS** N = 0 NAIR=1.00027G NGLASS=1.57 IF(XIN.LT.0.)XLI=-XLI THE TAI=DARSIN(XLI/RLI ) 2L=TLI ZLO=ZL 100 N=N+1 ALPHA2 = DARSIN((NAIR/NGLASS)*DSIN(ALPHA 1)) XOUT=XIN + 2L*DTAN(ALPHA2) THETA = DARSIN(XOUT/RLI ) ZL=RLI*(DCOS(THETA)-DCOS(THETAI)) + TLI OPTPL=(NGLASS/DC0S(ALPHA2))*2L IF(N.GT.2O)G0 TO 150 IF(DABSC2L-ZL0 ) . LT , 1 .E-8 )G0 TO 150 55 F0RMAT(1X.'N XOUT ALPHA2 THETA 2L ZLO OPTPL ') 50 FORMAT(1X,12,6F10.6) 2L0=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 = ' ,F10.6, 'BETA = ' .F10.6) RETURN END SUBROUTINE RTRACE(XIN.ALPHIN.OPTPL.XFINAL) c * * . . * » * « S U B R O U T I N E RTRACE TRACES A RAY THROUGH OPTICAL SYSTEM FROM C * * * * * * » * D I F F U S I O N CELL EXIT TO FOCAL PLANE* * * * * * * * * * * * * * * * * * * * * * * IMPLICIT REAL*8(A-H,L.0-2) C0MM0N/SET2/RL1,TL1,XL1,RL2,TL2.XL2,Z0.21.22.23.25,27, 1NQURTZ,NAIR.NGLASS REAL-8 NAIR.NGLASS,NOURTZ X2 = XIN + (Z2-Z1)*DTAN(ALPHIN) A L PHA 3 = DARSIN((NAIR/NOURTZ)*DSIN(BETA 1 )) L23 = NAIR*(Z3-Z2)/DC0S(ALPHA3) X3 = X2 + (23-22)*DTAN(ALPHA3) C*****LENS 1 RAY TRACING************************ CALL LNSTRC(XL'1 . TL1 ,RL1 ,X3,ALPHAS,X4.BETA4 , LL 1 , 2L 1 ) C* * * * * LENS 1 TO LENS 2************************** 24=Z3+ZL1 X5 = X4-r (Z5-24)*DTAN(BETA4) ALPHA5 = BETA4 L45 = NAIR*(Z5-Z4)/DC0S(ALPHA5) C*****LENS 2 RAY TRACING************************ CALL LNSTRC(XL2.TL2.RL2,X5.ALPHA5.X6,BETA6,LL2,ZL2) C*****LENS 2 TO FOCAL PLANE RAY TRACING********* Z6=Z5+ZL2 XFINAL = X6 + (27-26)*DTAN(BETA6) L6FP = NAIR*(27-26)/DC0S(BETA6) OPTPL = L23+LL1+L45+LL2+L6FP RETURN END - .164 -APPENDIX D Mass Flux and D i f f u s i v i t y C a l c u l a t i o n Computer Program - 165 -I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) C * * * * * D I F F C A L C C A L C U L A T E S M A S S F L U X E S . C O N C E N T R A T I O N P R O F I L E S A N D C * * * * * C O N C E N T R A T I O N I N T E G R A L S F R O M R E F R A C T I V E I N D E X C O R R E L A T I O N S C * * * * * E V A L U A T E D F R O M P R O G R A M R A Y T R A C E A N D F I T T O E R R O R F U N C T I O N C * * * * * C 0 R R E L A T I 0 N * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * R E A L * 8 N A X M L , N A X M E M , N A X M U , N B X M L , N B X M U D I M E N S I O N C X D I F F ( 1 0 ) , C X M L ( 1 0 ) , T I M E ( 1 0 ) , A U ( 1 0 ) , B U ( 1 0 ) , P M U ( 1 0 ) , 1 A L ( 1 0 ) , C X ( 1 0 ) , C X M L C ( 1 0 ) , B L ( 1 0 ) , P M L ( 1 0 ) . P A R L ( 4 ) 1 , C L S G A R ( 1 0 ) , C W A T E R ( 1 0 ) , X S U G A R ( 1 0 ) C O M M O N X ( 1 0 ) , Y ( 1 0 ) C * * * * * * * * R E A D D A T A F I L E F O R P R O F I L E C O R R E L A T I O N P A R A M E T E R S * * * * * * * C * * * * * * R E A D Tlfl £************************ **•**»*****»#*. *****»*********»* + * R E A D ( 1 0 . 5 0 0 ) N T 1 , N T 2 , N T 3 , N T 4 , N T 5 . N T 6 . N T 7 . N T 8 . N T 9 , N T 1 0 C * * * * * * R E A D N U M B E R O F T I M E S D U R I N G R U N * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * R E A D ( 1 0 , 5 0 1 ) N T I M E 5 0 0 F O R M A T ( 2 0 A 4 ) 5 0 1 F O R M A T ( 1 5 . 3 E 1 5 . 8 ) C * * * * * * * * * * S E T P A R A M E T E R V A L U E S F O R C A L C U L A T I O N S * * * * * * * * * * * * * * A L P H A = 1 . D 0 / 4 9 . 0 1 6 2 1 D O X 6 0 T T M = - 1 . 8 D 0 B E T A = - 1 . 3 3 1 3 1 3 D 0 / 4 9 . 0 1 6 2 1 D 0 D X M = . 0 0 5 0 G A M M A = - 1 1 . 6 8 7 4 4 2 3 D 0 X M L = 0 . D 0 E P S I = 0 . 0 5 5 3 5 1 2 0 1 5 D O X T O P = X M L P I = 3 . 1 4 1 5 9 2 7 D O C * * * * * * * * * * D E L T X I S X I N C R E M E N T S F O R W H I C H C A L C U L A T I O N S A R E P E R F O R M E D I N C » . . . * , , , ; , X P O S I T I O N I N D I F F U S I O N C E L L , S T A R T I N G A T M E M B R A N E S U R F A C E C , » * A N D W O R K I N G D O W N T H R E E V A L U E S T O - 0 . 1 0 C M . * * * * * * * * * * * * * * * * * * * * * N I N T = 3 . 0 D E L T X = ( - . 1 0 - X T 0 P ) / N I N T P R I N T 5 5 5 5 5 5 F 0 R M A T ( 1 X , / / , ' D I F F C A L C W I T H C O N C E N T R A T I O N A T T I M E = 0 ' , / / ) P R I N T 5 0 0 , N T 1 , N T 2 , N T 3 . N T 4 , N T 5 , N T 6 , N T 7 , N T 8 , N T 9 , N T 1 0 D O 1 0 0 1 = 1 , N T I M E C * * * * * * » R E A D V A L U E S M , A , B F O R R E F R A C T I V E I N D E X P R O F I L E C O R R E L A T I O N P A R A M E T E R S C * * * * * * . . A T E A C H T I M E V A L U E * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * R E A D ( 1 0 , 5 0 2 ) T I M E ( I ) , P M U ( I ) , A U ( I ) . B U ( I ) . X U F R E A D ( 1 0 , 5 0 2 ) P M L ( I ) , A L ( I ) , B L ( I ) . X L F 1 0 0 X ( 1 + 1 ) = T I M E ( I ) X ( 1 ) = 1 . 0 0 C * . * * * * * * - P E R F O R M P R O F I L E S I N T E G R A L S F O R E A C H T I M E F R O M C O N S T A N T S * * * * D O 1 0 5 I N T = 1 , N I N T D O 1 0 6 I = 1 , N T I M E C » * . . * » F U N C T I O N C I N T G L C A L C U L A T E S C O N C E N T R A T I O N I N T E G R A L . I N C E L L * * * * * * * * * * * * * C X M L ( I ) = C I N T G L ( P M L ( I ) . A L ( I ) , B L ( I ) , X T O P ) -1 C I N T G L ( P M L ( I ) , A L ( I ) , B L ( I ) , X B O T T M ) C * * » * * « » E V A L U A T E C O N C E N T R A T I O N A T B O T H S U R F A C E S O F M E M B R A N E F O R M E M B R A N E C * * * * * * * D I F F U S I V I T Y C A L C U L A T I O N S * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C X ( I + 1 ) = C X M L ( I ) C X ( 1 ) = ( X T O P - X B O T T M ) * ( A L P H A * 1 . 3 3 2 7 1 6 + B E T A ) 1 0 6 C O N T I N U E C * * * * * * * * * F I N D C O N S T A N T S F O R F I T T I N G C O N C E N T R A T I O N P R O F I L E V S T I M E * * * * C A L L E O U F I T ( C X . P A R L . N T I M E ) P R I N T 5 0 3 . X T O P C * * * * * * * * * F J N D F L U X E S * * * * * * * * * * * * * * * * * * * * P R I N T 5 0 0 0 P R I N T 5 0 2 , P A R L ( 1 ) , P A R L ( 2 ) , P A R L ( 3 ) , P A R L ( 4 ) P R I N T 5 0 0 2 - 166 -DO 101 1 = 1,NT I ME 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***** NAXML = DCDT(PARL,TIME(I)) C*******NAX IS SUCROSE MOLAR FLUX, NBX IS WATER MOLAR FLUX* * * * * * * * * * * * * * * * NAXMEM=NAXML NBXML = GAMMA*NAXML CXMLC ( I ) =PARL ( 1 ) + PARL(2)*DERF(PARLU)/TIME( I ) * * . 5 ) 1 + PARL(3)*TIME(I)**.5*DEXP(-PARL(4)*PARL(4)/TI ME(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*********FIND SUCROSE CONCENTRATION AT MEMBRANE SURFACES FOR C*********DIFFUSIVTIY CALCULATIONS*************** 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.DO*XTOP)) DMEMB=NAXMEM/DCDMEM CDELXA=(i.0-CLSGAR(I)*(1.0+GAMMA)/((1.O+GAMMA) 1*CLSGAR(I) + EPSI) )*DSUCDY C*******CALCULATE FREE DIFFUSION COEFFICIENTS AND MEMBRANE DIFFUSION COEF F .* * DNOB LK = - NAXML/CDE LXA DBULK=(-NAXML + XSUGAR(I)*(NAXML + NBXML ) )/CDELXA CXDIFF(I ) = ( (CXMLCI )-CXMLC(I ) )/CXML(I ) )* 100.O PRINT 5001 ,TI ME(I ) ,CLSGAR(I ),NAXML.CXML(I ) ,CXMLC(I ) , 1DMEMB.DNOBLK,DBULK . 101 CONTINUE 502 FORMAT(1X,8E15.8) 5001 FORMAT( 1X,F8.2,8( 1X.E15.8) ) 5002 FORMAT(//,1X,'TIME',8X,'SUC CONC',9X.'SUC FLUX',9X.'OBS CINTGL' 1,4X.'CALC CONCINTGL',7X,'DMEMB',10X,'DNOBULK',10X,'DBULK',//) 5000 FORMAT(//,4X,'APARAMETER',4X,'BPARAMETER',4X,'CPARAMETER', 14X.'DPARAMETER',//) 503 F0RMAT(1X.//, 'INTEGRATION LIMITS -1.80 CM TO ' . F10 .7 . / / ) XTOP =XTOP +DELTX 105 CONTINUE STOP END FUNCTION CINTGL(M.A,B,X) C*********FUNCTION CINTGL CALCULATES CONCENTRATION INTEGRALS FROM C*********ERROR FUNCTION PARAMETERS DETERMINED FROM RAYTRACE**** IMPLICIT REAL*8(A-H,0-Z) REAL"8 M ALPHA =1.DO/49.01621D0 BETA=-1.331313D0/49.0162100 CINTGL = ALPHA*((M/A)*(A*X*DERF(A*X) + 1.564 18958 * DEXP(-A*A * X * X)) + B*X) + BETA * X RETURN END FUNCTION DCDT(P.T) C********FUNCTION DCDT FINDS PARTIAL DERIVATIVE WITH RESPECT TO TIME C********FOR PROFILE CONCENTRATIN I NTEGRALS * * * * * * * * * * * * * * * * * * * * * * * * IMPLICIT REAL*8(A-H,0-Z) DIMENSION P(4) DCDT = (P (3 ) / (2 . *T** .5 ) + P ( 3 ) *P(4)*P(4)/(T* * 1 . 5) 1 -P(2 ) * . 5 6 4 1 8 9 5 8 * P ( 4 ) / ( T » * 1 . 5 ) ) * D E X P ( - P ( 4 ) * P ( 4)/T) RETURN - 167 -E N D S U B R O U T I N E E O U F I T ( C . P A R , N T I M E ) C * * * * * * * * * E Q U F I T F I T S C O N C E N T R A T I O N P R O F I L E S A T T I M E I N T E R V A L S T O E R R O R F U N C T . * * £ * * * * . * * * * * P A R ( \ ) = A * * * * * * * * * * * * * * ******* * p ^ p ( 2 ) = B * * * * * * * * * * * * * * £ * * * * * » . * * * P A R (3)=c************** C * * * * * * * * * P A R ( 4 ) = D * * * * * * * * * * * * * * I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) -C O M M O N X ( 1 0 ) , Y ( 1 0 ) D I M E N S I O N I V ( 6 4 ) , C ( 1 0 ) , V ( 4 0 0 ) , P A R ( 4 ) E X T E R N A L C A L C R , C A L C U C A L L D F A L T ( I V . V ) I V ( 2 3 ) = 1 N = N T I M E + 1 M = 4 V ( 2 9 ) = 1 . D - 2 0 V ( 4 0 ) = 1 . D - 2 0 V ( 4 2 ) = 1 . D - 2 0 P A R ( 1 ) = 6 . 7 D - 5 P A R ( 2 ) = 1 . 9 0 - 6 P A R C 3 ) = - 4 . 1 D - 8 P A R ( 4 ) = 2 5 . 6 D 0 D O 1 0 0 I = 1 . N 1 0 0 Y ( I ) = C ( I ) C * * * * * * N L 2 S 0 L I S U B C C U R V E F I T T I N G N O N - L I N E A R L E A S T S Q U A R E S F I T T I N G R O U T I N E * * * * C A L L N L 2 S 0 L ( N , M , P A R , C A L C R . C A L C J . I V , V . I P A R M . R P A R M , F P A R M ) R E T U R N E N D S U B R O U T I N E C A L C R ( N , M , P A R . N F , R , I P A R M . R P A R M , F P A R M ) C * . , * * . * * * » C A L C R C A L C U L A T E S E R R O R F U N C T I O N E Q U . A N D R E S I D U A L S * * * * * * * * * * I M P L I C I T R E A L * 6 ( A - H , 0 - Z ) C O M M O N " X ( 1 0 ) . Y ( 1 0 ) D I M E N S I O N P A R ( 4 ) , R ( N ) D O 1 0 0 I = 1 , N F X = P A R ( 1 ) + P A R ( 2 ) * D E R F ( P A R ( 4 ) / ( X ( I ) * * . 5 ) ) 1 + P A R ( 3 ) * ( X ( I ) » * . 5 ) * D E X P ( - P A R ( 4 ) * P A R ( 4 ) / X ( I ) ) 1 0 0 R ( I ) = F X - Y ( I ) R E T U R N E N D S U B R O U T I N E C A L C J ( N , M , P A R , N F , D . I P A R M , R P A R M . F P A R M ) c * * , , » * . » * C A L C j C A L C U L A T E S P A R T I A L D E R 1 V I T I V E S O F E R R O R F U N C T . C O R R E L A T I O N * * I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) C O M M O N X ( 1 0 ) . Y ( 1 0 ) D I M E N S I O N P A R ( 4 ) , D ( N . 4 ) P I = 3 . 1 4 1 S 9 2 7 D 0 D O 1 0 0 1 = 1 , N D ( I . 1 ) = 1 . D O D ( I . 2 ) = D E R F ( P A R ( 4 ) / ( X ( I ) * * . 5 ) ) D ( I , 3 ) = ( X ( I ) * * . 5 ) * D E X P ( - P A R ( 4 ) * P A R ( 4 ) / X ( I ) ) D U . 4 ) = D E X P ( - P A R ( 4 ) * P A R ( 4 ) / X ( I ) ) * 2 . / ( X ( I ) * * . 5 ) 1 * ( P A R ( 2 ) / P I * * . 5 - P A R ( 4 ) * P A R ( 3 ) ) 1 0 0 C O N T I N U E R E T U R N E N D 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0058898/manifest

Comment

Related Items