IIITERFERDMETRIC EXAMINATION .DF TEMPERATURE DISTRIBUTIONS IN LIQUIDS: REFRACT IVE INDEX AND THERMAL CONDUCTIVITY Thesis for the Degree of Ph. D. MICHIGAN. STATE UNIVERSITY JAMES DAVID DLSDN 1972 This is to certify that the thesis entitled INTERFEROMETRIC EXAMINATION 0F TEMPERATURE DISTRIBUTIONS IN LIQUIDS: REFRACTIVE INDEX AND THERMAL CONDUCTIVITY presented by JAMES DAVID OLSON has been accepted towards fulfillment of the requirements for PH.D. degree in CHEMISTRY Major professor (for F. H. Horne) Date August 24, 1972 0-7639 3mm av HMS & SUNS’ - 800K BINDER! INC. ABSTRACT INTERFEROMETRIC EXAMINATION OF TEMPERATURE DISTRIBUTIONS IN LIQUIDS: REFRACTIVE INDEX AND THERMAL CONDUCTIVITY BY James David Olson A powerful interferometric method of studying re— fractive index gradients is developed and is applied to the study of vertical temperature variations in pure liquids. New methods for the direct determination of (i) temperature dependence of refractive index and (ii) ther- mal conductivity are developed and carried out. An optical theory of the Bryngdahl wavefront shear- ing interferometer (J. Opt. Soc. Am., ii, 571 (1963)) is deduced in a general way without recourse to the mean value theorem or to phenomenological specification of the process producing the refractive index gradient. Full ac- count is taken of displacement of the light beam by dif- ferential refraction. A final working equation is pre— sented in terms of easily determined experimental quanti- ties. James David Olson The availability of the new optical theory makes possible the development of a new method of determining the temperature dependence of the refractive index of liquids. The basis of the method is the interferometric examination of a liquid contained in a glass rectangular parallelepiped, bounded above and below by metal thermo- stating walls, in which a temperature gradient is estab- lished and maintained. Experimental results are presented for water, carbon tetrachloride, cyclohexane and benzene. Steady state interference patterns produced by the nonelectrolytes do not have the appearance predicted by previous studies of the refractive index of these liquids: the interference fringes are parabolic rather than linear. Preliminary analysis of this anomalous behavior suggests that it is caused by slight nonlinearity in the steady state temperature distribution. Thermal conductivity of the liquid is studied by analysis of the time dependent fringe shift observed during the establishment of the temperature gradient in the liquid. This technique is named pure thermal conduction because, in principle, it is the simplest experimental arrangement: it does not require calorimetry, and the experiment is per— formed in a convection free apparatus. Experimental re- sults for water and carbon tetrachloride indicate that the new method, after technological refinement, will be of great usefulness. Iv ,V . V‘ ' ‘o ' ~T‘t. y ‘3. n . INTERFEROMETRIC EXAMINATION OF TEMPERATURE DISTRIBUTIONS IN LIQUIDS: REFRACTIVE INDEX AND THERMAL CONDUCTIVITY E 3; By g: I! :fl James David Olson F71 gf e $. 1. A THESIS Submitted to Michigan State University in.partial fulfillment of the requirements for the degree of DOCTOR or PHILOSOPHY , Department of Chemistry 1972 To my Parents I ACKNOWLEDGMENTS I would like to thank the Department of Chemistry for financial support, particularly for appointment as an Assistant Instructor for the academic year 1968—69. I also gratefully acknowledge financial support from the National Science Foundation. I especially appreciate the perceptive direction of Professor Frederick Horne during this research. His critical interest and encouragement were constantly avail- able. I express my thanks to Dr. Sara Ingle for many helpful discussions concerning thermal conductivity. Mr. William Waller provided valuable assistance with the com- putational aspects of this work. I wish also to thank Mr. Ping Lee who assisted me with some of the experiments. Finally, I am grateful to my wife Ellen for her encouragement and cheerfulness during the completion of this thesis. TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . LIST OF FIGURES. . . . . . . . . . . . . . Chapter I. II. III. IV. INTRODUCTION . . . . . . . . . . . . WAVEFRONT SHEARING INTERFEROMETRY. . A. Introduction. . . . . . . . . . . B. Optical Method. . . . . . . . . . C. Theory. . . D . Relationship between Cell Coordinates and Final Image Coordinates . . . E. Analysis of Optical Data. . . . . F. Applications. . . . . . . . . . . EXPERIMENTAL APPARATUS AND PROCEDURE Introduction. . . . . . . . . . . . Apparatus Improvements. . . . . . . Chemicals . . . . . . . . . . . . . Data Reduction. . . . . . . . . Experimental Procedure. . . . . . MUOW?’ TEMPERATURE DEPENDENCE OF REFRACTIVE A. Introduction. . . . . . . . . . . B Working Equations . . . . . . . . C. A priori Error Estimates. . . . D Experimental ResultsH for Water. . E. Results for CCl , C and C H F Discussion. . .4.. 6 H1?. 6 6 G. A Molecular Calculation Using (dn/dT) . H. Suggestions for Further Work. . . ANOMALOUS PARABOLIC FRINGE SHAPES. . A. Introduction. . . B. Analysis of the Parabolic Steady State Fringe Shapes . . . . . . C. Discussion. . . . . . . . . . . . iv Page viii Chapter Page VI. PURE THERMAL CONDUCTION. . . . . . . . . . . . 126 A. Introduction. . . . . . . . . . . . . . . 126 B. Experimental Methods. . . . . . . . . . . . 127 C. A New Transient Method. . . . . . . . 130 D. Theory of the Time Dependent Fringe Shift . 132 E. Experimental Results for Water and CCl4 . . 136 F. Discussion. . . . . . . . . . . . . . . . . 146 BIBLIOGMPHY I I I I I I I I I I I I I I I I I I I I 149 APPENDIX A I I I I I I I I I I I I I I I I I I I I I 155 APPENDIX B I I I I I I I I I I I I I I I I I I I I I 158 APPENDIX C I I I I I I I I I I I I I I I I I I I I I 163 LIST OF TABLES Table Page 4.1 Literature values for temperature dependence of refractive index at 25°C. . . . . . . . . . 64 4.2 Literature values for KT X 103, temperature dependence parameter for thermal conductivity at 25°C . . . . . . . . . 68 4.3 Typical values and experimental uncertainties of parameters in Eq. (4.11). . . 73 ’ 4.4 Summary of experimental results gor (dn/d'l‘)25° of water at A = 6328 A. . . . . . . 77 4.5 Summary of steady state fringe slope data for water. . . . . . . . . . 79 4.6 Summary of experimental results for (dn/dT)25°Oof carbon tetrachloride at X = 6328 A. . . . . . . . . . . . . . . . . . 83 4.7 Summary of experimental‘results for o (dn/dT)25° of cyclohexane at X = 6328 A. . . . 85 4.8 Summary of experimental results foro (dn/dT)25° of benzene at X == 6328 A . . . . . 86 4.9 Data for Béttcher plot of CC14 refractive index and specific volume temperature dependence. . . . . . . . . 90 4.10 Effective molecular radii and polarizability for CC14, C6H6 and C6H12. . . . 93 5.1 Results of curvefitting for alleged nonlinear temperature dependence of refractive index. . . . . . . . . . . . . . 102 5.2 Results of curvefitting for guassian nonlinear refractive index temperature dependence equation. . . . . . . . 106 Page Results of curvefitting for nonlinear temperature distribution equation. . . . . . . 118 Experimental and theoretical steady state temperature distributions for CC14 in 8 mm cell, AT = 2.80°C, KT = — 2.0 x 10'3 . 119 Summary of some physical properties of H20, CC14, C6le and C6H6 at 25°C. . . . . . . 125 Fringe shift data for experiment v-4 . . . . . 137 Summary of experimental results for thermal conductivity of water, Jm' sec"l K"l . . . . 139 Summary of experimental results for thermal conductivity of CC14, Jm' sec‘ K' . . . . . 145 LIST OF FIGURES Figure Page 2.1 Schematic diagram of the interferometer. . . . 9 2.2 Schematic diagram of the cell. . . . . . . . . 11 2.3 Interferogram from an optically homogeneous test object. . . . . . . . . . . . 15 2.4 Interferogram showing relocation of the cell center coordinate when a refractive index gradient is present in the cell. . . . . . . . 17 2.5 Light path through the cell when a refractive index gradient is present . . . . . 30 3.1 Schematic diagram of laser fringe counter. . . 46 3.2 Facsimile of computer output from program FRINGE. . . . . . . . . . . . . . 56 3.3 Strip chart record of phototransistor voltage output during fringe displacement of u-S. The arrow indicates position of t = O. . . . . . . . . . . . . . . 59 4.1 Interference fringe photographs: (a) isothermal slit photo, experiment v—l3; (b) isothermal cell profile for water, experiment v-12; (c) steady state cell profile for water, AT = 5.59K, experiment v—lz . . . . . . . . . . . . . . . . . . . . . 75 4.2 Battcher plot for cc14; nZ/[(n2—1)(2n2+1)V1 vs. (2n2—2)/(2n2+l). . . . . . . . . . . . . . 92 5.1 Interference fringe photographs: (a) steady state cell profile of CC14 in 8 mm cell, AT = 2.73K, solid line ( ) indicates midplane of cell, (b) steady state cell profile of C5H12 in 8 mm cell, AT = 2. 79K, (c) steady state cell profile of C6H6 in 8 mm cell, AT = 2. 79K. . . . . . . . . . . . 100 viii Figure Page 5.2 Temperature dependence of refractive index of C5H12. The solid line ( ) is Eq. (5.1) and the broken (— — —) is the linear behavior predicted by the literature (Table 4.1). . . . . . . . . . . . . . . . . . 105 5.3 Interference fringe photographs. This is the steady state cell profile of H20 in 30 mm cell, AT = 10.83K: (a) top (1/3) of cell, (b) middle (1/3) of cell, (c) bottom (1/3) of cell. . . . . . . . . . . . . . . . . . . . 108 5.4 Interference fringe photographs. This is the steady state cell profile of CCl in 30 mm cell, AT = 10.22K: (a) top (1/3) of cell, (b) middle (1/3) of cell, (c) bottom (1/3) of cell. . . . . . . . . . . . . . . . . . . . 110 5.5 Steady state cell profile of CCl4 and H20 in 30 mm cell. This composite drawing was made from Figures 5.3 and 5.4. The broken line (— - -) indicates the "break" in the CC14 fringes . . . . . . . . . . . . . . . 112 5.6 Interference fringe photographs: (a) isothermal slit photo for H20 in 5 mm cell, (b) steady state cell profile of H20 in 5 mm cell, AT = 10.09K, (c) fringe pattern produced by free convection in CCl4 heated from below . . . . . . . . . . . . 116 6.1 "Goodness of fit" plot for thermal con— ductivity fitting of data from experiment v. . 141 6.2 Plot of fringe shift at z = 0 as a function of time. The circles are the experimental points from v—4 and the solid line ( ) is the least squares curve calculated from KINET . . . . . . . . . . . . . . . . . . 144 ix CHAPTER,I INTRODUCTION Nonuniform concentration and temperature distribu- tions in liquids are two of the most difficult objects for the experimentalist to determine accurately. This type of measurement is required for determination of transport co— efficients related to nonequilibrium processes; diffusion coefficients, thermal diffusion factors and thermal con- ductivities are three examples. Examination of the history of these experiments indicates that accepted values of the coefficients have been constantly revised with the advent of more and more sensitive methods for their determination. In situ interferometry, which is the precision measurement of optical path length differences by observa— tion of alternating light and dark patterns produced by interfering light waves, has long been recognized as the most sensitive method of detecting mass and energy transfer in liquids (Tyrell, 1961; Wolter, 1956; Ingelstamm, 1957). The extensive applications of Longsworth (1940, 1945, 1946, 1950) are examples of the determination of diffusion co- efficients by an interferometric method. Gustafsson, et a1. (1965) and Anderson (1968) have determined thermal diffusion factors by an interferometric technique. An interferometric thermal conductivity method is described by Bryngdahl (1961). Any interference phenomenon is related to spatial variations of the refractive index of the material under study. Because of this, the refractive properties of the liquid may, in principle, be examined simultaneously with the transport processes occurring in the liquid. This has not been done previously because (i) the determination of the transport coefficients was of utmost interest and (ii) an interferometric technique had not been developed to facilitate this type of experiment. However, in addition to the purely experimental reasons for studying the proper— ties of the refractive index (Bauer, 1958; Coumou, 1964), this information can provide valuable fundamental knowledge about the molecular properties of the liquid (Waxler and Weir, 1963). Also, further refractive index studies are necessary to modify and improve the electromagnetic equa- tion of state (Amey, 1968) which relates the refractive index to density, temperature and some function of the molecular polarizability. The purpose of this thesis is to present an inter— ferometric method of studying optical inhomogeneities in liquids which can be used to determine simultaneously (i) the spatial temperature distribution in a liquid perturbed from equilibrium by a temperature gradient, (ii) the temperature dependence of the refractive index of the liquid and (iii) the thermal conductivity of the liquid. The basis of this method is the interferometric examination of a liquid in which a temperature gradient is established and maintained. We present experimental results for pure (one component) liquids although the technique could be extended to the study of mixtures. We begin in Chapter II with a general optical theory of the particular instrument, the Bryngdahl wavefront shear— ing interferometer. The defects of previous theoretical treatments are corrected and a general discussion of the instrument's application to the study of transport pro— cesses is presented. It is shown that the instrument pro- vides a sensitive yet easily executed method of measuring path length differences arising from mass and temperature distributions in liquids. Criteria for simultaneous study of transport processes and refractive index are discussed. Chapter III is a description of the experimental apparatus used in conjunction with the wavefront shearing interferometer to study the development and effects of temperature gradients in liquids. The sample vessel is a classical pure thermal diffusion cell: a rectangular glass parallelepiped bounded above and below by thermostated metal plates. By changing the temperature of the metal plates, a temperature gradient is established and main— tained in the liquid. Experimental determination of the temperature de- pendence of the refractive index is discussed in Chapter IV. The optical theory of Chapter II is used to calculate (dn/dT) from the interference fringe behavior observed when a liquid has a temperature gradient established in it. Ex- perimental results are presented for water, carbon tetra— chloride, cyclohexane and benzene. The anomalous appearance of some of the steady state fringe shapes noted during the (dn/dT) experiments is discussed in Chapter V. This phe- nomenon appears to be related to the details of the steady state temperature distribution. Finally, in Chapter VI we develop a new method for determining the thermal conductivity of the liquid. The basis of the method is the analysis of the time dependent behavior of the interference fringes. Experimental results are presented for water and carbon tetrachloride. CHAPTER II WAVEFRONT SHEARING INTERFEROMETRY A. Introduction Wavefront shearing interferometry, in which a dis- torted wavefront is compared with a sheared, or displaced, image of itself, is a highly sensitive method of determin— ing optical path length differences and, in principle, re— fractive index distributions. The most convenient instru- ment, a Cartesian coordinate wavefront shearing inter- ferometer, is the culmination of a study by Bryngdahl and co-workers (1957, 1960, 1961, 1963) of interferometers which employ Savart plate beam splitters. Advantages of Cartesian coordinate wavefront shearing interferometry in— clude: (i) ease of operation, (ii) elimination of the necessity for a reference cell, (iii) a sharp interference image which is a Cartesian representation of the test ob— ject and (iv) increased sensitivity in studies of gradients of refractive index. Unfortunately, the original optical theory (Bryng- dahl, 1963) includes two simplifications which prevent the complete determination of some types of refractive index distribution encountered in the laboratory: (1) While formally correct, the use of the mean value theorem for derivatives in order to relate the final interference image to the gradient of the refractive index normal to the optical axis in the test object introduces an unnecessary correction term in the final working equation. This correction vanishes only in the special case of a linear refractive index dis- tribution, when the gradient is the same everywhere in the object. (2) The optical path length expression in the original article (Bryngdahl, 1963) is valid only when the path of the light ray through the test object is rectilinear. Such a path is obtained only if the refractive index gradient in the object is vanishingly small. The curvilinear path traveled by a light ray when a refractive index gradient is present in any object is treated in general by Svennson (1954, 1956). The physical manifestation of this aberration is bending of the light beam away from the original optical axis. Other investigators (Gustafsson, et al., 1965; Wallin and Wallin, 1970; Mitchell and Tyrrell, 1972) have obtained optical theories for a similar interferometer in the course of investigations of thermal diffusion in dilute aqueous solutions. These treatments are consistent with ours, but they are not generally applicable since phenome- nological equations which describe thermal diffusion are introduced early. We present here a general optical theory for the Bryngdahl Cartesian coordinate wavefront shearing inter- ferometer. The theory can be used to determine the refrac- tive index distribution in an object, but it is independent of the phenomenological description of whatever produces the refractive index distribution. To obtain this improved theory we: (1) use an optical path length expression that includes corrections obtained by Svennson (1956) for a light ray traveling in a medium containing a refractive index gra- dient, (2) derive a method of relating the final inter- ference image to the refractive index distribution in the cell without recourse to the mean value theorem and (3) include explicitly the additional effects of the ray- bending caused by the refractive index gradient. B. Optical Method Figure 2.1 is a schematic diagram of the inter- ferometer arrangement called option II by Bryngdahl (1963). The only difference is the use of a laser as a light source because of its obvious advantages and ready availability. Detailed information about the modified Savart plates and other optical components is available in his article. We give here only an abbreviated description, with the help of the coordinate system* shown in Figure 2.2. *y represents the vertical direction in this chapter but 2 represents the vertical direction in subsequent chap- ters. This chapter conforms to optics literature, while subsequent usage conforms to thermal transport literature. .I ...u.I...L..F like. _ .. e .BIL. III I .1 :2 I . I . . . . . ,5 . . u u. {UCIIMUI L... (.02.. Ftuhrllrnlgr \au. Figure 2.l.--Schematic diagram of the interferometer. m0<2_ J. -______. ,1 .l 1' 1 [1.3.4 r .r . ..(pl,.. . i . , raca~ 10 Figure 2.2.-—Schematic diagram of the cell. 12 Collimated, monochromatic, coherent light which is polarized in the 6 direction (this bisects the x and y axes and is in- clined 45° from the plane containing the axes of the first Savart plate 01) passes through the test object. The test object may be any material in which an optical inhomogeneity exists or can be induced. In order to relate the general theory to the specific experiments discussed in later chap- ters, let the test object be a fluid-filled glass cell of length a and height h which is bounded above and below by opaque thermostating walls, as illustrated in Figure 2.2. Inside the cell, the wavefront is distorted by the refrac— tive index distribution in the fluid. The wavefront is next reduced by the simple telescope system, lenses L1 and L2 (see Figure 2.1), to a size compatible with the optical characteristics of the modified Savart plates 01 and 02. The modified Savart plates are identical, each consisting of a half-wave plate sandwiched between two birefringent crystal plates. The axes of these plates lie in the same plane but are perpendicular to each other. The first modi- fied Savart plate, 01, splits the wavefront into two iden- tical, vertically sheared wavefronts. The double image of the cell thus produced is focused by lens L3 into the second modified Savart plate 02. This one introduces a small shear angle between the converging wavefronts and thus produces a constant path length difference. The image now passes through a polarization analyser A which 13 makes an interference pattern visible in the final image plane. The final image is a sharply focused double image of the x—y profile of the cell with interference fringes in the region of overlap. The fringe pattern can be re- lated to the refractive index distribution in the cell by the optical theory of section C. Several types of fringe patterns can be produced. Figure 2.3 shows the fringe pattern produced by an optically homogeneous test object. When the test object is optically inhomogeneous, the fringe pattern of Figure 2.4 and a variety of others can occur. Moreover, experiments can be conducted both during transient periods, when the refractive index gradient is changing, and in steady states. In a transient period the fringes move horizontally across the final image plane, and the fringe shape can change; for example, the fringe shape might change from the straight vertical bands of Figure 2.3 to the tilted or curved bands of Figure 2.4. In the steady state, no fringe m0vement is observed; how- ever, the fringes may remain tilted and curved. The shape of the fringes in the steady state is directly related to the details of the refractive index distribution in the cell. As we show in section C, a linear refractive index distribution produces straight vertical fringes identical to the ones observed in the homogeneous case. Hence, the only experimental informa- tion available in this case is the number of fringes which 14 Figure 2.3.--Interferogram from an optically homogeneous test object. 15 QN UNUINIMIJ MINUINJUIH .11.... . . ... If: . . I. I I... .or.. o I o . . . . . . . £0,510, Dali‘s 1 t. 16 Figure 2.4.--Interferogram showing relocation of the cell center coordinate when a refractive index gradient is present in the cell. O>AI 973* I. II III III III III II L | '| 'l' 'l 'l' I- I1 18 move horizontally across the final image plane during the attainment of the steady state. Any curvature or tilting of the fringes in either a transient or a steady state is due to non—linearity of the refractive index distribution in the cell. To summarize, two types of experimental data are available when observing physical processes in the liquid-filled cell: (1) the total number of fringes that have passed a reference y—axis (this is related to the linear character of the refractive index gradient) and (2) the x—y shape of a particular interference fringe (this is related to the non-linear aspects of the gradient). C. Theory Light entering the cell is polarized in the E direc- tion (see Figure 2.1). Following Bryngdahl (1963), we describe the vertical component of the wave entering the cell by a transversal electric field strength vector, H, referred to the cell coordinate system shown in Figure 2.2, U = ( + j) (A//§)exp (i k z) (2.1) m m i ’b where A is the scalar amplitude, k = 2n/X, and i and j are ’\a the unit vectors in the x and y direction. If we denote the refractive index inside the cell by n(x,y,z), the optical path through the cell is a WIx,y) = Io{n(x,y.2)/cosm(x,y,z)]}dz (2.2) 19 where a(x,y,z) is the tangential angle of the ray with the optical axis. The amplitude vector of the wave leaving the cell is $3 = (% + a) (A/fi)exp{i k[w(x,y) + 20]} (2.3) where 20 is an arbitrary reference plane. Next the light passes through the image reducing telescope formed by lenses L1 and L2. We introduce a new path length function w(rx,ry) = W(x,y) (2.4) where r is the reduction factor of the lens system. The amplitude vector entering the first modified Savart plate 01 is then 2 = ( + j)(A/r/§)exp{i k[W(rx,ry) + 20]} (2.5) ’b i m In passing Q1, the component of H in the x direction becomes polarized in the y direction and is displaced down— ward by an amount (l/2)bl, _ 2_2 22 (1/2)bl - e(ne nO)/(ne+no) , (2.6) “fliers e is the thickness of each of the two birefringent Eilates and where ne and no are the principal refractive inmiices. For 11cm quartz plates in 6328 A light, (l/2)b1 = 53..5u. Likewise, the component in the y direction is dis— Placed upwards by the same amount and becomes polarized in 20 the x direction. The amplitude vector after passing Q1 is H = i(A/r/§)exp{i k[w(rx,ry + (1/2)bl) + z + (1/2)x]} 0 +3 (A/r/2)exp {i k[w(rx,ry — (l/2)bl) + 20- (1/2)xl} , (2.7) where the quantity x is introduced to take account of any path length difference due to the chance occurrence that the Savart plate may not be aligned so that the vector 6 is inclined 45° from the principal plane. The light now enters L3 which focuses it into 02 and magnifies it in order to produce the desired dimensions in the final image plane. The amplitude vector entering QZ is H = % (A/mr/§)exp{i k[V(mrx,mry + (1/2) mbl)+ 204-(1/2)x]} + j(A/mr/§)exp {i k[V(mrx,mry-(l/2) mbl)4-zo-(l/2)x]} (2.8) m where m is the magnification factor from L3 and V(mrx,mry) replaces w(rx,ry) as the optical path length function. On passing 02, the image is sheared in the x direc— tion, and a path length difference A is introduced which is related (Bryngdahl, 1961, 1963) to the incident and azimuthal angle of the light entering Q2, A = bl (mrx/l) cos i , (2.9) where 2 is the distance from the focal plane of L3 to the final image plane and i is the incident angle. The ampli- tude vector becomes I -21 1 U = i(A/mr/§)exp{i k[V(mrx-—(1/2)bl,mry-(l/2)mrb1) + Z0 + (l/2)A- (l/2)x']}4-g(A/mr/2)exp{i k[V(mrx + (l/2)b1,mry4-(l/2)mrbl)+-zO-(l/2)A4—(1/2)x']} , (2.10) where x' denctes any combined tilting of the Savart plates. C After passing the analyzer A, which is aligned in the n direction (see Figure 2.2), the resulting amplitude is U = U - (i + j)//2 W (\I (\J = (A/21r)exp{i k[V(mrx-(l/2)b1,mry -(l/2)mrbl) + 20 I (l/2)AF (l/2)x']} + (A/2mr)exp{i k[V(mrx+ (l/2)bl,mry + (1/2)mrb1) + 20 + (l/2)A — (1/2)x']} (2.11) Intensity at the final image is given by I = |Ul = U'U* = (1/2) (A/mr)2 (1 + cosw) , w = k[V(mrx-+(l/2)b1,mry-+(l/2)mbl) — V(mrx-(l/2)b1,mry -(l/2)mbl)- A-Px'] (2.12) The periodicity of the interference fringes is thus given by w = 2nn (n = 0,1,2,....). We now seek an equation to describe families of curves of the same intensity in the final image plane. Cos i may be expanded, Cos i = 1 - (1/2>0(i2) . 22 and Eq. (2.9) becomes A = b mrx/5?. — (blmrx/z) (1/2)o (12) . (2.13) 1 Substituting Eq. (2.13) into the second of Eqs. (2.12) and solving for mrx, we find mrx = (R/bl)[V(mrx-+(l/2)b1,mry-+(l/2)mb1) - V(mrx — (l/2)bl,mry-(l/2)mbl)] - (2/bl)[a5[n'(y)12n"(y)[n(yn‘2 , (2.17) where n'(y) = dn/dy and n"(y) = dZn/dyz. In using Eq. (2.17), we assume that the light ray entering the cell is parallel to the z-axis but that it then travels a curvilinear path through the cell. Equation (2.17) gives the path length in terms of the entrance (z=0) y coordinate of the light ray since this will be different from the exit (z=a) y coordinate of the same ray. Since W(y) = V(mry), Eqs. (2.15) and (2.17) yield x = (al/b1)[An(y) + (l/3)a2As'(y) + (1/15)a4As"(y)] - (i/b1)[(w/k) - x'] + (1/2)x 0(12) , (2.18) where X = mrx is the horizontal fringe shift in the co— ordinates of the final image plane, and An(y) = n(y+B) - n(y-B) As'(y) = [n'(y+s)121n(y+e)1‘1- [n'(y—e)121n(y-e)1‘1 As"(y) = [n"(y+B)l[n'(y+B)]2[n(y+B)]_2 — [n"(y—B)l[n'(y-B)]2[n(y-B)]_2 , (2.19) where B = (l/2)b1/r, the shearing distance in cell coordi— nates. 24 Equation (2.18) must be transformed to a more use- ful experimental form since A, the distance from the focal plane of L3 to the final image plane, is not easily measured accurately. Let 6 be the distance between points of equal intensity on adjacent fringes in the final image; for example, as illustrated in Figure 2.3. Passing from one fringe to another represents a change in w of 2n so that x + a = (aft/bl) [An(y) + (1/3)a2As' (y) + (1/15)a4As"(y)] - (1/b1){[(w—2n)/k1 — x'}+ (1/2)x 0(12) . (2.20) If we subtract Eq. (2.18) from Eq. (2.20) and recall that k = 2n/X, we find 6 = (AX/bl) or (l/bl) = (6/1), and the final working equation which relates the fringe image to the refractive index distribution in the cell is x = A[An(y) + (l/3)a2As' (y) + (1/15)a4As"(y)] + B where A = (ad/A) and B = - (6/X)[(w/k)- x']+-X 0(12) . (2.21) Equation (2.21) is comparable to Eq. (32) of Bryng- dahl (1963, see Appendix A) with the following particular advantages: (i) Our equation contains only one correction term, X 0(i2). If lens L3 is of suitable focal length, then i < 10—2, and the deviation from the Cartesian form of Eq. (2.21) will be considerably less than 0.1%. (ii) There are no mean value theorem corrections of the type found in Eqs. (30) and (31) of Bryngdahl (1963). -‘IIII"""""""""""""""""""'f""""'———————————————————---—-—— 25 (iii) Our equation retains explicitly the path length terms which arise from the curvilinear path followed by a light ray under the influence of a refractive index gradient. In some experimental instances, these additional terms might contribute negligibly to the total path length dif- ference. In such cases a simple working equation of the type X = A An(y) + B (2.22) may be used. Note also that if n is a linear function of y, then An is constant, n' is a constant and n" is zero. Therefore, As' = As" = 0, and the final image consists of straight vertical fringes. (iv) Although Bryngdahl's Eq. (32) looks like our Eq. (2.21), his equation contains a factor (Ay')—l = r/bl in the first term. This factor renders his result incorrect dimensionally and numerically (Bartelt, 1968). Experimental data for analysis by Eq. (2.21) are obtained from observation of both the fringe shift and the steady state fringe pattern in the final image plane. The fringe displacement X from an arbitrary vertical line selected before the refractive index gradient is developed in the cell is measured for each of a set of vertical co- ordinates in the final image plane. The fringe spacing 6 is also measured. The set of X's constitutes the data for the left-hand—side of Eq. (2221), and the fringe spacing 6 W——— 26 permits calculation of the apparatus constant, (ad/X). The set of vertical coordinates in the final image and the shear— ing distance are reduced to cell coordinates by division by the magnification factor, mr. (In the procedure described here, it is assumed that the vertical coordinate in the final image which corresponds to the center of the cell is chosen as Y e O. This point is discussed further in section D.) The experimental data for Eq. (2.21) are thus a set of ordered pairs of (i) the distance (in cm) of the fringe shift in the final image plane and (ii) the particular ver— tical coordinate in the cell (between —h/2 and h/2) which corresponds to the image plane vertical coordinate at which (i) is measured. Another experimental procedure is often more con- venient. Equation (2.21) can be cast into the form, (X/6) = (a/X)[An(y) + (l/3)a2As'(y) + (1/15)a4As"(y)] + (B/s) . (2.23) Note that the left-hand-side of Eq. (2.23) is dimensionless. This indicates that a number of fringes rather than a fringe shift distance is measured to provide data for the left-hand- side of Eq. (2.23). This is accomplished experimentally by placing an arbitrary vertical reference line on one of the fringe minima before the establishment of the refractive index gradient in the cell. As the fringes are shifted horizontally during the establishment of the refractive 27 index gradient, the experimenter counts (manually or by use of a photoelectric device) the number of fringe minima that pass the reference line. However, it in general happens that the total fringe shift is not an integral number of fringes; 143;, the fringe motion stops when no fringe mini— mum is colinear with the reference line. In this case a fractional fringe displacement is calculated by (a) measur- ing the distance between the reference line and the last fringe minimum to pass the reference and (b) dividing this distance by the fringe spacing 6. This function is added to the integral number of fringes. As before the data are a set of ordered pairs; (i) the dimensionless fringe shift measured for (ii) each chosen vertical cell coordinate. The latter are obtained by reducing the vertical coordi— nates of the final image by the magnification factor mr. D. Relationship between Cell Coordinates and Final Image Coordinates As mentioned above, the success of the use of Eq. (2.21) or (2.23) in experimental analysis of refractive index distributions depends on the ability to relate the coordinates of the final fringe image to the coordinates of the cell. This requires: (i) determination of the magnification factor and (ii) identification of the hori— zontal plane on the fringe pattern that corresponds to the center (y = 0) of the cell coordinate system. In the w——_—_— 28 — case of an optically homogeneous cell, whose fringe pattern is shown schematically in Figure 2.3, these relationships are straighforward: the center of the final image (where Y = Ymid) clearly corresponds to the center plane of the cell (where y = O), and the height H of either of the super— imposed final images is equal to the magnified height of the cell. The magnification factor, mr, is then simply the ratio of H to h, mr = H/h, and for any arbitrary set of Y coordinates, y = (Y — Ymid)/mr. (2.24) In most experimental arrangements, Ymid is chosen as Y = 0, and Eq. (2.24) becomes y = Y/mr. (2.25) In general, however, the refractive index distribution in the cell is nonlinear, and the emergent ray is bent. The physical manifestation of the ray-bending effect of a refrac- tive index gradient is shown in Figure 2.5. Ray-bending affects the relationship of the fringe coordinate system to the cell coordinate system in two ways: (a) The amount of (bending changes with height in the cell, so that there is a net focusing of the light beam leaving the cell and a subsequent change in the magnification factor; (b) Part of the light beam may be cut off by the top or bottom of the cell, and consequently the center of the final fringe image 29 Figure 2.5.--Light path through the cell when a refractive index gradient is present. 30 I I 1 I I I I Ou> I4 [—7 >. <—-—--- 31 no longer corresponds to the center plane of the cell. The relationship between the cell coordinate y and the final image coordinate Y is then 3! = (Y - Yo)/(mr)* (2.26) where (mr)* is the magnification factor in the general case and where Yo corresponds to y = 0. Both Y0 and (mr)* must be determined in order to analyze data. If it were possible to determine directly the amount of bending and the amount of light beam cutoff, then one could calculate (mr)* directly by equating it to H*/h*, the ratio of apparent cell height in the final image plane to the effective optical cell height (see Figures 2.4 and 2.5). Instead, it is ordinarily easier to measure (mr)* directly by temporarily inserting a slit of accurately known aperture width c just after the cell and measuring the width C* of the image of c in the final image plane. Then (mr)* = C*/c. Note that measurement of B(or 8*), the shearing distance in cell coorindates, yields r (or r*) according to r = ((1/2)b1)B-1, and m (or m*) is then obtainable from mr [or (mr)*]. The quantity 28 is obtained by dividing the distance 2D (see Figure 2.3) by the magnification factor mr, and 28* is obtained by dividing 2D* by (mr)*. Once the correct magnification factor is obtained, the position Yo can be determined from (see Figure 2.4) Y0 = Ymid i O , (2.27) 32 where (see Figures 2.4 and 2.5) G = (mr)*e , 6 = h - h* . (2.28) Alternatively, since (mr)* = H*/h*, G is obtained from more easily measured quantities, O = (mr)*h - H* . (2.29) The choice of signs in Eq. (2.27) is necessary because although 6 and G are positive, the light beam may bend either up (+) or down (-), depending on the refractive index distribution in the cell. Substitution of Eqs. (2.27) and (2.29) into Eq. (2.26) yields the final, general formula which relates Y to y for an arbitrary Y coordinate system, -_- ._ 'k .. * 9: Y [(Y Ymid)/(mr) ] i [h H /(mr) ] (2.30) Note that Eq. (2.30) reduces to Eq. (2.24) only when both (mr)* = mr and h* = h. The procedure can be simplified if a horizontal slit is placed before the cell and the aperture arranged so that the light beam entering the cell is symmetric about the y = 0 plane. In this case none of the light beam will be cut off, and the center of the fringe image will cor- respond to the center of the cell at the entrance coordinate. In practice, it is difficult to align the slit symmetrically about the y = 0 plane. Also, the amount of cell observable is decreased by this type of masking. 33 Wallin anui Wallin (1970) suggest changing the angle of incidence of the light entering the cell to counteract the bending effect. This has the advantage of producing an exit beam that has the same exit coordinate as entrance coordinate but has the disadvantage of requiring a path length expression different from Eq. (2.17) in order to in- clude terms which contain the entrance angle. A suitable expression could no doubt be derived, starting with the work of Svennson (1956), but its use would require accurate measurement of the angles of exit and accurate control of the angles of entrance. The techniques so far suggested for taking account of demagnification and coordinate re- location caused by ray-bending are probably not the only ones which might be useful; the particular physical arrange- ment of the interferometer and the properties of the system being studied may suggest other appropriate means to deal with these effects. E. Analysis of Optical Data After the correct final image coordinate relation— ship and the fringe displacement have been obtained, it is possible to calculate the refractive index distribution in the cell. Let the refractive index distribution in the cell be represented by the following equation: n(y) = no + f(y) . (2.31) 34 The leading term in Eq. (2.31), no, which is the absolute refractive index at the center of the cell, cannot be de- termined from analysis of the final image because it appears in the difference expressions, n(y) = f(y +8) - f(y -B) , (2.32) which appear in Eq. (2.21). Rather, the specific values of the parameters in f(y), which describes the functional varia— tion of the refractive index about no, are determined. The general form of f(y) may be suggested by the shape of the fringes in the final image plane. The method of multiple linear regression (Anderson, 1968; Efroymson, 1960; see Appendix C) for choosing a set of variables is valuable in this regard. To consider a specific example, the results might be expressed by a simple Taylor's series about y = 0, n(y) = nO-I-(dn/dy)y=0 y + .... +(i!)_l(dln/dyl)y=oyl + .... - (2.33) In this case, Eq. (2.33) is truncated after a sufficient number of terms and substituted into Eq. (2.23) which is written in cell coordinates for this purpose. This yields an expression containing (X/é) as a function of y with the Spatial derivatives as parameters, (X/G) = X(y,dn/dy, dzn/dyz, ..... din/dyi) . (2.34) 35 The optical experiment yields a set of [(X,6),y] data from the number of fringes displaced and the fringe shape. As discussed following Eq. (2.22), the fringe data and the shear distance, D, can be reduced to cell coordinates by use of the magnification factor. The spatial derivatives [or similar parameters in other choices of f(y)] are ob- tained by least squares adjustment of Eq. (2.34) to the eXperimental data. This can be done by a number of pa- rameter estimation techniques (see Appendix C) which are facilitated by the use of a digital computer. It should be noted that in this analysis, there are no assumptions made about the process producing the refractive index dis- tribution. F. Applications The interferometer may be used to study physical processes such as mass and heat transport in the cell. Sup- pose we wish to relate the optical data to some property P which varies as a function of cell height, P = P(y), and we also know that the refractive index is a function of P, n = n(P). The refractive index can then be written as a function of cell height, n = n[P(y)] . (2.35) Equation (2.35) in effect specifies a particular f(y) for use in Eq. (2.32). For example, in the case of initially 36 sharp boundary-layered isothermal diffusion experiments, a gaussian form of the refractive index gradient would be applicable. However, we must know explicitly either P(y) or n(P) to determine the other: that is, it is not possible to separate the property spatial distribution from the re- fractive index prOperty dependence in one experiment. Hence, either of two mutually exclusive experimental ob- jectives may be achieved: (i) Determination of the de- pendence of n on some property P whose y dependence is known: (ii) Determination of the y dependence of some property P whose relationship to n is known. Mass and heat transport experiments would fall into the latter cate- gory because the temperature and concentration dependence of the refractive index can be determined experimentally. This method of analysis can be extended to account for more than one property which affects the refractive index and varies in the y direction, as, for example, in a pure thermal diffusion experiment. Equation (2.35) would then be written, n = n[Pl(Y)I P2(Y)r 00-: Pi(YH . (2.36) In this case, all but one of the property spatial distribu— tions and property refractive index dependences must be known completely to determine the process. Application of the method to systems in which a very small An exists (for example, flow profiles in gases) 37 would require the shear distance D or the geometrical light path a to be increased in order to preserve high sensitivity. However, as D is increased, the region of overlap between the sheared wavefronts decreases, and the added sensitivity would eventually be countered by the decrease in the area of the test object studied. A compromise between these two effects must be found according to the needs of the particu- lar experiment. Wallin and Wallin (1970) discuss a method of increasing sensitivity by rotation of the second Savart plate. The ultimate sensitivity is available in holographic analogs (Bryngdahl, 1969; Becsey, et al., 1970) of these methods; phase multiplication during reconstruction of the image increases sensitivity to any desired level. The specific application of the interferometer to the study of the establishment and decay of a temperature gradient in a pure liquid will be discussed in Chapters IV, V and VI. CHAPTER III EXPERIMENTAL APPARATUS AND PROCEDURE A. Introduction The method of obtaining experimental fringe shift and fringe shape data is discussed in this chapter. These data can be analyzed by means of the optical theory de— veloped in Chapter II to determine the temperature depen— dence of refractive index and thermal conductivity of the liquid. The experimental apparatus used to study the effects of temperature nonuniformity in pure liquids is essentially the thermal diffusion apparatus built previously in this laboratory (Anderson, 1968). The liquid is contained in a rectangular glass sample cell bounded on the top and bottom by metal thermostating plates whose temperatures are con— trolled by circulating water reservoirs. The source of circulating water to each plate reservoir is controlled by a valve system so that the cell may be changed from an isothermal configuration (both plates thermostated by water from the same mean temperature bath, TM) to a temperature difference configuration (water from a warmer bath, TH’ circulating in the upper reservoir and water from a cooler 38 39 bath, TC, circulating in the lower reservoir). This arrange- ment allows a temperature difference to be applied vertically across the cell, a temperature gradient to be developed in the liquid as a function of time, and finally, a steady state temperature distribution to be maintained in the liquid. Observation of both the establishment and decay of a tem- perature gradient is possible. Plate temperatures are measured with copper con- stantan thermocouples and a microvolt potentiometer. The continuous change of the plate thermocouple voltages is monitored by two strip chart recorders, each modified to record in 200 uv full scale steps. The circulating baths' temperatures are controlled by continuous heating and cool- ing regulated by a proportional temperature controller. The plates' temperatures are maintained constant within 0.003K fluctuations. The cell is aligned in an optical train in which the vertical cell profile is illuminated by a laser and the image is optically analyzed by a Bryngdahl wavefront shearing interferometer. In some experiments, the inter- ference fringes are recorded with a Polaroid MP-3 camera in the final image plane (see Figure 2.1). In other ex- periments, the Polaroid unit is removed from the final image plane and the horizontal movement of the fringes is monitored with a phototransistor. 40 A complete description of the apparatus design and objectives and details of the original apparatus construc- tion are available elsewhere (Anderson, 1968, pps. 76-116). Anderson required maintenance of a carefully controlled temperature difference for ten to sixteen hours in order to obtain a thermal diffusion steady state. The present experiments, however, require only 15 to 45 minutes of con- trolled temperature difference. The effects of problems such as ambient temperature drift and temperature controller artifacts are therefore much less important for the present experiments. Short term temperature control is usually easy to obtain. B. Apparatus Improvements Several parts of the apparatus have been improved in the course of this research. However, the temperature monitoring equipment, the cell filling technique, the cell insulating technique, and the use of the MP-3 Polaroid camera are the same as described by Anderson (1968). Im- provements are: 1. Temperature Control. The first step in improvement of temperature control was the installation of prOportional temperature controllers to replace the mercury thermoregu- lators supplied by the bath manufacturer (Tamson). Model 2156 Versa-Therm controllers were purchased from Cole-Parmer, Chicago, Ill., and Model 408 "Banjo" probes were purchased 41 from YSI Co., Yellow Springs, Ohio. These eliminated the "temperature ripple" in the water baths by supplying a con- stant amount of power to the heating element in the bath to balance the constant cooling rate provided by a coil cooled with water from a refrigerated temperature bath. Any fluc— tuations in the bath temperature are balanced by prOpor- tionally greater or lesser amounts of power to the heating elements. Fluctuations were further reduced by heat ex- changers (Anderson, l968, p. 92). The plate temperatures are routinely maintained to within £0.003K of the desired temperature. Optimum use of proportional temperature controllers requires a constant cooling rate. The Lab-line Tempmobile used by Anderson as a source of cooling water for the three baths was used with one modification. Flow meters and stop- cocks were installed in order to monitor empirically and to reproduce cooling rates. Typically, cooling is maintained by a flow of about 250 ml/min of 13°C water. Slightly dif- ferent rates are used to satisfy the individual requirements of the T T and TC baths. M' H' The two position valve (Anderson, 1968, p. 73) used to switch the bath water inlets to the plate reservoirs from the isothermal configuration to the temperature dif- ference configuration was retained unchanged. However, the pinch clamps used to control access to the water bath by- passes were replaced by stopcocks for more convenient switchover. 42 2. Optical Components. The original optical train was replaced by equipment purchased from The Ealing Corporation, Cambridge, Massachusetts. The goals in this replacement were (a) reproducibility of the position of optical com- ponents and (b) capability of vertical and horizontal ad- justment of lenses and modified Savart plates. A two meter optical bench, equipped with a mm scale, replaced the two unconnected optical benches used previously. The optical carriers are equipped with a 0.1mm vernier which allows accurate reproduction of component position on the optical bench. Two carriers which can be adjusted ver- tically and horizontally are used to support the laser light source and align it along the optical axis. L1 is mounted in a lens holder also supported by a vertically and horizontally adjustable optical carrier. L2 is mounted in a focusing microscope objective holder which was modi- fied by the Chemistry Department machine shop to serve as a lens holder. 01 and 02 are mounted in rotatable compo- nent holders which can be tilted 45° from the vertical axis as described in the optical theory (see Chapter II, section B). L3 is also mounted in a focusing microsc0pe objective holder. A precision polarization analyzer completes the Optical components of the interferometer. In order to pro- vide the magnification of the final image necessary to take Polaroid photographs, a fourth lens is used after the analyzer. 43 The light source is a He-Ne gas laser, Spectra- Physics Model 120, equipped with a Model 332-337 spatial filter and beam expander. The beam expander eliminated the first two lenses of the original optical setup (Ander- son, l968, p. 104) and produces an optically pure, coherent, monochromatic light beam which is sufficiently intense to observe and record the interference fringes. The power output of the laser is about eight milliwatts. The laser beam has a diameter of 5 cm as it leaves the beam expander. It is focused on the midplane of the sample cell and masked to the region of interest by a 1 cm X 3 cm slit. This pre- vents spurious reflections of laser light from the metal plates and reservoirs. Biconvex positive lenses obtained from Special Optics, Cedar Grove, New Jersey, are used for L2 and L3. L1 and the modified Savart plates were the same as used previously (Anderson, 1968, pp. 104-5). A 3x microscope objective is used to focus the final image on the Polaroid camera back. A 0.01mm x-y vernier telescope was purchased from The Ealing Corporation to replace the 0.1mm vernier micro- scope stage used previously to record the horizontal dis- placement of fringes (Anderson, 1968, p. 113). The frosted glass back of the Polaroid camera target does not produce a suitable image for monitoring fringe shifts with the x—y telescope. A target of translucent bond (typing) paper is 44 placed about one meter behind the camera. During an experi— ment the camera back is removed and the cell profile image focused on the paper. The x-y telesc0pe is placed about 30cm behind the paper and used to record fringe motion. The previous changes are improvements upon the original apparatus design. Two additions were made that have no analog in the original optical equipment: (1) a rotatable parallel plate has been placed in the optical train between the sample cell and L1; (2) a phototransistor is used to monitor fringe shifts during an eXperiment. The parallel plate makes it possible to adjust the cell profile vertically in the final image plane without adjusting the relative position of the lenses in the inter— ferometer as Anderson had to do. Also, the cell image can be relocated on the center of L1 if the "bending" effect of the refractive index gradient moves the light beam out of the optical axis. Finally, it is possible to align the laser and sample cell on a different optical axis from the optical axis of the interferometer. The parallel plate is used to raise or lower the cell image into the interferometer Optical axis. A Photo-Darlington Ll4B (Allied Electronics, Chicago, Ill.) phototransistor is used in conjunction with a mechani- cal fringe counter and signal amplifier, built by Mr. Ronald Haas of the Chemistry Department electronics shOp, to monitor the horizontal travel of the interference fringes. Figure 3.1 45 Figure 3.l.--Schematic diagram of laser fringe counter. 46 mmbzaoo. Nazi“. mmm<4 5&3 «use... 3248mm cub—5:00 m0“. cur—Prom”. 05¢... x3... an... 13d JG 04 4.3 mm: W 252. 238. A( >2. yaks-:00 x I DID A. I , #1.! «((IITL 03363 yuan» 1....ng >Emzum =5... l_ W. . _ «no. v.— .m 3.0 We.“ 00» 80.. on: one no» (5083» c twat 80 r t ‘0 o . x“ .ooo x” 032“ m .............. . 3°— 33 x. a s _ own on»: ,0. 0 k . ....... .0.'..n_. ...... 47 is a schematic diagram of the fringe counter. The transistor is mounted in the vernier microscope stage (previously used to measure fringe travel) and horizontally centered on the fringe chart area by locating the position of minimum voltage output. The transistor is vertically centered on the line corresponding to the midplane in the sample cell by means of a wire which is mounted in the middle of the back of the phototransistor and which extends horizontally 1 cm in each direction. Two forms of data are available from the amplifier: (1) the digital output from a mechanical counter which re- cords the total number of integral fringes that pass the transistor (a square wave pulsed by a Schmitt trigger ac- tivates the counter) and (2) the amplified output voltage of the transistor which is displayed on a strip chart re- corder. The voltage output goes from a minimum (0.0 fringe displaced) to a maximum (0.5 fringe displaced) and back to a minimum (1.0 fringe displaced). Of course, the fringe displacement is seldom exactly an integral number of fringes. The fractional fringe is measured directly by the x—y vernier telesc0pe as described in section C. 3. Data from Fringe Photographs. The vernier micro- scope stage was used by Anderson (1968, p. 113) to obtain spatial data from the Polaroid photographs of the inter— ference fringes. Determination of 2B, the shearing dis- tance in cell coordinates, is also calculated from this 48 type of measurement. A more sensitive method of measuring distances on the photos was suggested by Dr. Marc DeBacker. The photo of interest is mounted on an x-y recorder equipped with crosshairs instead of a pen. A stable and sensitive (£0.00001v) variable voltage supply is used to buck the recorder and move the crosshairs from position to position on the photo. At each point, the x and y input voltages are read to 0.lmv on a Heath UDI digital voltmeter and are then used as the coordinates of that point. This method provides fast, efficient analysis of fringe distances and shapes to three significant figures. C. Chemicals Water used in the refractive index and thermal con- ductivity experiments was obtained from the MSU Brillouin Lightscattering group. It was triply—distilled deionized conductivity water. The carbon tetrachloride, cyclohexane and benzene used were purchased from the J. T. Baker Chemical Company. These were spectrophotometric grade reagents and were used without further purification. D. Data Reduction Raw data obtained from a refractive index tempera- ture derivative experiment includes the following: 1. Cell height and cell length. 2. Shear distance. 49 3. Temperatures of the upper and lower plates in the o M. 4. The temperature difference at the steady state. isothermal configuration, T 5. Total number of fringes displaced during the estab- lishment of the temperature gradient. 6. X-Y fringe shape in the steady state. Thermal conductivity experiments require all of these data, except 6. In this case the following are also required: 7. Time dependence of the change of the plate tempera- tures from the isothermal configuration to the steady state, iiiir strip chart recording of thermocouple voltage. 8. Time dependence of the interference fringe shift, iLgL, strip chart recording of the phototransistor voltage. Note that both experiments can be done simultaneously; the transient data lead to determination of the thermal con- ductivity and the steady state data lead to determination of the refractive index temperature dependence. This section will be a discussion of how each datum is obtained experimentally and how it is reduced to a form that can be used to determine the temperature dependence of refractive index or the thermal conductivity. 1. Cell height and cell length. The cell height was measured to £0.0005cm by repeated micrometer readings. The cell length was measured to i0.001cm with a vernier 50 calipers. The results for the two cells are: 0.8103cm Cell 1 Length 6.828cm Height Cell 2 Length 6.805cm Height 3.0124cm 2. Shear distance. The shearing distance, 28, is determined by measurements made on the photograph of a slit of known aperture, h, as described in Chapter II, section D. The slit photos resemble Figure 2.3. Two example photos are Figures 4.1(a) and 5.6(a). The dis- tances between the four parallel lines are measured in a vertical plane chosen at random on the photograph. The vertical coordinate of each boundary is obtained by mea- surement with the x-y recorder as described in section B, 3. These coordinates are called Y(l), Y(2), Y(3), Y(4). The differences lY(l) - Y(3)| and lY(2) - Y(4)| are the magnified slit heights and the distances IY(2) - Y(1)I and IY(4) - Y(3)| are the magnified shear distances, 2D. The shear distance in cell coordinates is obtained from the following equations (see Chapter II, section D): mr IY(1) — Y(3)I/h , IY(4) - Y(2)I/h , (3.1) 28 [Y(Z) — Y(l)|/mr , IY(3) - Y(4)|/mr . (3.2) Note that each measurement of four coordinates yields two determinations of the shear. 51 Numerical Example: Isothermal Slit Photo Slit = 0.4580cm Y(l) = 17.60 Y(2) = 41.10 Y(3) = 140.1 Y(4) = 163.5 mr = |14o.1 - l7.6l/0.4580cm = 267.5cm-1 mr = [163.5 - 4l.lI/0.4580cm = 267.3cm’l Absolute Shear, 28 = 41.1 - 17.6I/267.4cm-1==0.0879cm = [163.5-l40.lI/267.4cm-l==0.0875cm A computer program, SHEAR, was written to calculate the average shear from data collected from experimental photographs according to Eqs. (3.1) and (3.2). As many determinations of 28 as desired can be carried out. 3. Isothermal mean temperature, T3. (0) means measurements conducted in the isothermal configu- (The superscript ration to distinguish from the temperature symbols referring to the steady state; T and T . M' TU' L ) thermocouples connected to the upper and lower plates are The voltages of the measured with a Leeds and Northrup K-3 potentiometer im— mediately before the application of the temperature dif— ference. The voltages are converted to temperatures from a calibration chart obtained previously (Anderson, 1968, o = To = To. At times, the temperature M U L readings are different for the two plates. These differ- p. 103). Usually, T ences are less than 0.006K. T; was calculated to be (l/2)(T3 + T2) where T3 and T: are the temperatures of the upper and lower plates in the isothermal configuration. 4. Temperature difference at the steady state, AT. When the steady state is attained (the fringes have stopped 52 moving), TU and TL are again measured by recording the thermocouple voltages. Their difference, AT = TU - TL' used directly in the calculations, and therefore any errors is such as thermocouple reference bath drift or potentiometer null offset is present in both the TU and TL readings and is eliminated in the difference. T the mean temperature M’ in the steady state, is equal to (l/2)(TU + TL). 5. Total number of fringes displaced during the estab- blishment of the temperature gradient. A strip chart re- cording of the phototransistor output yields the total number of integral fringes that pass an arbitrary refer- ence point. The fraction of fringe displaced after the last integral fringe is measured with the x-y vernier tele- scope. The telesc0pe crosshairs are centered on a dark area of a fringe before the start of an experiment. The horizontal coordinate of this point is called X0. It is noted in which horizontal direction the fringes are travel- ing during the experiment. After the steady state is reached, the horizontal coorindate x of the last fringe L to pass the crosshairs is measured. The position XN of the fringe that would next pass the crosshairs is also measured. The fractional fringe displacement is calcu- lated from this equation: Fringe Fraction, X/G = [x0 - xLl/lxL - le (3.3) 53 NumeriCal Example: Experiment 0-15 X0 = 13.794 XL = 14.535 XN = 13.431 Fringe fraction = I13.794 -l4.535|/|l4.535 —l3.43l| = 0.671 Strip chart indicated 6 integral fringes plus 0.5 to 1.0 fraction fringe Total fringe displacement = 6.671 6. X—Y fringe shape. The steady state fringe shape data are collected from Polaroid photographs using the x-y recorder. These fringe shapes are often parabolas similar to Figure 2.4. Two types of reduced data are calculated from the photographs: (1) the total number of fringes dis- placed during the establishment of the temperature differ- ence as a function of vertical coordinate and (ii) devia- tions from the midplane fringe shift as a function of vertical coordinate. Data of type (i) are obtained by adding the horizontal fringe shift measured previously in part 5 to the data of type (ii). Both of these types of data are tabulated in ordered pairs of dimensionless fringe shifts, X/G, for each vertical photo position converted to cell coordinates (see Chapter II, section C). Photos of the cell profile and a slit of known aperture are taken in the isothermal configuration and in the temperature difference steady state. The magnifications factors, (mr) and (mr)*, are calculated from the slit photos. 54 Often it is discovered that the cell height calculated from the isothermal cell profile photograph is not equal the actual cell height. This is due to masking of the cell- plate boundary by sealant that leaks out of the boundary upon tightening of the plates. The amount of sealant cut off, assumed to be the same at the top and bottom cell boundaries, is calculated from the following equation: Ahcutoff = h — H/mr (3.4) where h is the measured cell height, H is the magnified apparent cell height, and (mr) is the isothermal magnifica- tion factor calculated from the isothermal slit photograph. Equation 2.30, Y = [(Y - Ymid)/(mr)*] i [h - H*/(mr)*] , (2.30) for conversion of photograph vertical coordinates to cell coordinates was modified to account for sealant cut-off: y = [(Y - Ymid)/(mr)*] — {[h - H*/(mr)*] - Ah f/2.o}. (3.5) cutof This allows for the downward bending of the light beam that is required to lower the beam past the upper sealant boundary. A computer program, FRINGE, was written which takes raw data from the x-y recorder and calculates the fringe shape data. A facsimile of a partial output from this program for a set of test data is shown in Figure 3.2. 7. Time dependence of plate temperature change. The following empirical equations (Anderson, 1968, p. 97) are used to describe the warming of the upper plate and the 55 Figure 3.2.--Facsimile of computer output from program FRINGE. 56 SAMPLE DATA FOR FRINGE SHAPE DATA CONVERSION FRINGE NUMBER 1 OF 2 FRINGES SHEARCCM) CELL HGHTICM) CELL LNTH(CM) KTII/OEG C) DEL T(DEG C) .10 .5 7.0 .0020 3.000 AMOUNT OF CELL CUTOFF DUE TO SEALANT = .05033 CM MI H1 M2 H2 .100000E+04 .449968E403 .100000E004 .399971E003 APPARENT CELL HTIISOTHERMAL) APPARENT CELL HT(DELTA T1 DEL(Z=0) .449967 CM .399971 CM .075013 CM DISTANCE BETWEEN FRINGES 3 .149989E003 (PHOTO UNITS) STD DEV = .101685E-11 FRINGE SHIFT WITHOUT lST DERIVATIVE ZTCM) X(FRINGES) -607246 .44524 -.05222 .31351 0.03198 .18177 -&01174 .05004 602849 -.08009 .07373 -.20983 .11849 -.07289 014824 .06284 617788 .26524 621716 .66844 FRINGE SHIFT WITH IST DERIVATIVE SHIFT ADDED ZTCM) XIFRINGES) *o07246 40.26825 '605222 40.13652 -.03l98 40.00478 -401174 39.87305 .02849 39.74291 .07373 39.61318 .11849 39.75011 .14824 39.88585 617738 40.08825 .19752 40.28985 .21716 40.49145 57 cooling of the lower plate during the change from the isothermal configuration to the temperature difference configuration: = o _ o _ -t/Y Tupper TM + (TU TM) (1 e U) , (3.6) T = T3, t = 0 T = TU, t = w _ o _ o _ _ -t/YL Tlower TM (TM TL) (1 e . ) (3.7) T = T3, t = 0 T = TL, t = m In general, YU did not equal YL' The raw data are obtained by reading the thermo- couple voltage data in arbitrary units from the strip chart. A computer program, GAMMA, converts the strip chart coor- dinates to time and temperature and punches a data deck in a format appropriate for use in the generalized curvefitting program KINET (Nicely and Dye, 1971). KINET is used to ob- tain an ordinary least squares estimate of y (all data point weights equal 1.0). Appendix C contains an outline of the procedure used to obtain parameter estimates. 8. Time dependence of the interference fringe shift. A typical strip chart recording (reduced in scale) of the phototransistor output voltage is shown in Figure 3.3. Note that the maxima are not equal. This prevents 58 Figure 3.3.--Strip chart record of phototransistor voltage output during fringe displacement of u—S.‘. The arrow indicates position of t = 0. 59 60 interpolation between half integer fringes. The horizontal scale reading is recorded for each of the maxima and minima. A computer program, TAU, is used to convert the arbitrary time points to real time by multiplying by the sec/unit factor obtained from the chart speed and subtracting the scale reading of zero time marked by the downward blip on the recording. Estimates of the experimental variance of the time and fringe positions are also made. These data are punched out in a KINET format. Typical fringe shift vs. time data are tabulated in Table 6.1. E. Experimental Procedure A description of how an experiment was carried out is best given in a step by step outline similar to that of Anderson (1968, pp. 128-137). 1. Adjust temperature baths to give the desired plate temperatures in the isothermal configuration and in the temperature difference configuration. 2. Fill and insulate the cell in the isothermal con- figuration using the technique described by Anderson (1968, pp. 132-3). 3. Adjust T; if necessary. 4. Take slit photo for shear measurement. Take cell profile photo if desired. 0 and To U L Start strip charts for thermocouples and phototransistor 5. Measure T and record on the strip charts. readings. 61 6. Adjust vernier telescope crosshairs to center of dark fringe. Adjust phototransistor to same position on an adjacent fringe. 7. Start application of temperature difference by switching water bath lever. Simultaneously, close shorting switch on phototransistor recorder momentarily to mark time zero. 8. When fringes have stOpped moving, stop strip chart recorder. Measure TU and TL and record on strip charts. Measure fractional fringe with the vernier telescope. Take slit and cell profile photos if desired. 9-a. If data are to be recorded during decay of the temperature gradient, repeat steps 5, 6, 7 and 8, start- ing the experiment by switching back to the isothermal configuration. 9-b. If data during the decay of the temperature gra- dient are not desired, switch valve and wait for the fringes to stop moving. Repeat experiment from step 3. Experiments were identified by the following code. Each filling of the cell was given a greek letter. Each repeated experiment on the same cell filling was assigned a consecutive number starting at one for each new filling. If the data were obtained during the establishment of a temperature gradient, the symbol E was added to the desigf nation. If the data were obtained from the decay of the temperature gradient, the symbol D was added to the desig- nation. 62 After many runs on a particular sample of liquid, the cell was disassembled, cleaned and stored. The raw data were reduced and were ready for analysis as described in Chapters IV, V and VI. CHAPTER IV TEMPERATURE DEPENDENCE OF REFRACTIVE INDEX A. Introduction This chapter contains working equations for the determination of the temperature dependence of the refrac- tive index of liquids and experimental results for H O, 2 CCl4, C6H12 and C6H6' The method uses the optical theory developed in Chapter II to calculate the temperature de- rivative behavior directly from the eXperimental fringe shift data observed when a temperature gradient of known magnitude is established in a liquid. There has been little previous work on (dn/dT) for liquids. Bauer, Fajans and Lewin (1958) state that (dn/dT) is always negative for liquids. They note that the value of the derivative is unusually small, -l.0 x 10-4K-1, for water and is relatively large, -8 x 10-4K-l, for liquids with high absolute indices such as carbon disulfide. They also point out that the value, -4.75 x 10-4K-1, commonly used to adjust data for organic liquids, is not adequate in many situations. Other references to (dn/dT) are found in light scattering work, for example, Coumou, et a1. (1964). Previous experimental work is summarized in Table 4.1. 63 64 Table 4.1—-Literature values for temperature dependence of refractive index at 25°C. 0 6328A Liquid n25° -(dn/dT)X104 -(d2n/dT2)X106 (d3n/dT3)X1010 H20 1.33162d 1.04a 3.0a 100.a 0.98b 0.93c 0.16C 2200.c 1.05d 1.6d 1.00e 2.6e , cc14 1.45586d 5.75f 0.0f 0.0f 5.82g 0.0g 0.0g 6.08d 0.0d 330.0d 5.96h 5.5i (6.48)j (0.0004)j (-194.0)j (5.95)k c6312 1.42205f 5.47f 0.0f 0.0f 5.349 0.0g 0.0g 5.44h . (5.72)j (0.509)j (-0.66)3 (5.44)k C6H6 1.49511d 6.35f 0.0f 0.0f 6.30g 0.09 0.09 . (7.13)j (0.017)j (753.0)J (6.45)k o All refractive index data are adjusted to 6328A.‘ I I indi- cates values calculated from theory. References. aTilton and Taylor, 1938. bWashburn (ICT), 1933. cBryngdahl, 1961. 65 References (continued). dWaxler, et al., 1964. eAndréasson, et al., 1971. fTimmermans, 1952; 1959 (see text). gCoumou, et al., 1964. hAnderson, 1968. 1Bauer, et al., 1958. JCalculated using Lorentz-Lorenz equation and density data of Wood, et al.; 1945, 1952. kCalculated from density data of Wood, et al., using modified L-L equation of Looyenga, 1965. Table 4.1 lists previous literature values for the temperature dependence of the refractive index in terms of Taylor's series derivatives evaluated at T = 25°C, M n(T) = n + 2 (1/11) (din/dTi) (T - T )1 T=TM i=1 M . (4.1) All of these values were obtained by curvefitting absolute refractive index data to various truncated forms of Eq. (4.1). Corrections to 6328 X.were made when the data were obtained at other wavelengths. The many values from Tim— mermans (1950, 1959) were combined by Anderson (1968), who used a multiple regression least squares routine MULTREG. 66 This routine was also used in this work to obtain derivative values from the data of Waxler, et a1. (1964). The values in brackets were calculated from the temperature dependence of the density by use of the Lorentz-Lorenz (L-L) equation, 2 2 (n -1)/(n + 2) = (4n/3M)NApa (4.2) where p is density, a is molecular electronic polarizability, M is the molecular weight and N is Avagadro's number, or A from a modified form of the L-L equation (Looyenga, 1965) 2 in which (n - l)/(n2 + 2) is replaced by (nz/3 - l). B. Working Equations The previous work tabulated in Table 4.1 suggests that for practical purposes Eq. (4.1) may be truncated after four terms, n(T) = nO + n'(T - TM) + (n'I/2)(T - TM)2 + (n"'/6)(T - TM)3 (4.3) where no = n(T)T=TM , n' = (dn/dT)T=TM , n" = (dzn/dT2)T=TM and n"' = (d3n/dT3)T=TM . This is a specific refractive index property dependence equation as discussed in general in the optical theory, Eq. (2.35). 67 The temperature distribution in the liquid after the steady state is established has been determined from the transport equations of nonequilibrium thermodynamics by Horne and Anderson (1970)*: 2 T(z) = TM + (AT/h)z + (l/2)(AT/h)2KT[(h/2)2 - z 1 (4.4) where — h/2 i z i h/2 , AT = TH - TC = TU - TL , TM = (TH + TC)/2 , KT = (1/K)(3K/3T)T=TM , Thermal Conductivity. K This is the specific property spatial distribution as dis- cussed in the optical theory, Eq. (2.35). Eq. (4.4) was obtained by a perturbation method (Horne and Anderson, 1970). In using Eq. (4.4), terms of order K2 may not be retained without simultaneous retention T of higher order terms such as KTT==(l/K)(32K/8T2). If we keep this restriction in mind, it is possible to obtain an ‘expression for the refractive index as a function of the ‘Vertical coordinate; Eq. (4.4) is solved for (T — TM) and substituted into Eq. (4.3) to produce the following expres- sion for n(z): *z is the vertical coordinate of the cell and y is the optical axis in this chapter. h is the cell height' commonly labeled a. 68 n(z) = [no + n'(AT)2KT/8] + [n'AT/hj. + inf) (AT)3KT/8h]z + [n"(AT)2/h2 - n'(AT)2KT/2h2 +~ n"'(AT)4KT/16h2]zz + [n"'(AT)3/6h3 - n"(AT)3KT/2h3]z3 + [n"'(AT)4KT/4h4]z4 . (4.5) Terms in KT are likely to be very small because [K = 10"3 (see Table 4.2). T | Table 4.2--Literature values for K x 103, temperature dependence parameter fog thermal conductivity at 25°C. H20 CC14 C6H12 C6H6 2.68a -2.1ob -o.71e e1.7se -l.26d References. aMcLaughlin, 1964. bTree and Leidenfrost, 1969. cTouloukian, et al., 1970. dChalloner, et al., 1958. eBriggs, 1957. fWeast (Handbook of Chemistry and Physics), 1959. We now refer to the optical theory to predict the interference fringe behavior which results from the estab- lishment of the refractive index distribution Eq. (4.5) in 69 the liquid. In order to obtain working equations of manage- able size for this discussion, Eq. (2.22) in the form XI (X/d) = (a/A)An(z) + B , (4.6) An(z) n(z + 8) - n(z - 8) , a = cell length, will be used as the optical equation. However, the actual numerical analysis and the curvefitting performed on experi— mental data utilized Eq. (2.23) which contains the path length corrections, AS'(z) and AS"(z), due to the presence of a refractive index gradient. These terms contribute less than 1% to the total optical path length for our ex— perimental arrangement. If the total fringe shift from a reference line is measured during the establishment of the refractive index gradient, 112;! X'(T=w) — X'(T=0), the constant B vanishes. Eq. (4.6) is then effectively, X' = (a/A)[n(z + 8) - n(z - 8)] . (4.7) Upon substitution of Eq. (4.5), Eq. (4.7) becomes: x' = (a/A){[28n'AT/h + 83n"'(AT)3/3h3 + 8n"(AT)3KT/4h — 83n"(AT)3KT/h3] + [4Bn"(AT)2/h2 - 28n'(AT)2KT/h2 3 + 8n"'(AT)4KT/4h2 + 23 n"'(AT)4KT/h4]z + [Bn"'(AT)3/h3 - 38n"(AT)3KT/h3]22 + [28n"'(AT)4KT/h4]z3} . (4.8) 70 The expression for the total horizontal fringe shift measured at z=0 is, x' = (a/A)28[n'AT/h + 82n"'(AT)3/6h3 + n"(AT)3KT/8h — 82n"(AT)3KT/2h3] (4.9) and 2 X'[A/a(28)] = (n'AT/h)[l + e n"'(AT)2/6n'h2 + n"(AT)2KT/8n' - 82n"(AT)2KT/2n'h2]. (4.10) Tables 4.1 and 4.2 list typical values for the parameters of Eq. (4.10). If ln"| i 10“4 and In"'l i 2.4 X 10-6, only the leading term of the R.H.S. of Eq. (4.10) is retained to 0.1% accuracy. Eq. (4.10) can then be solved for the first derivative of the refractive in- dex with respect to temperature: (dn/dT)T=TM = x'Ah/(AT)a(2B) . (4.11) This is the final working equation for the determination of (dn/dT). All of the quantities on the R.H.S. of Eq. (4.11) can be obtained by the experimental technique de- scribed in Chapter III. Tilton and Taylor (1938) found that the refractive index temperature dependence for water exhibited second derivative behavior on the order of 3% of the first de- rivative (see Table 4.1). This would cause a parabolic refractive index distribution in the liquid that would be 71 manifested by straight fringes that are tilted from the vertical. The lepe, m, of these tilted fringes at the center of the final image can be measured from the steady state photographs. Eq. (4.8) can be differentiated with respect to z to obtain a formula for (dzn/de): (3X'/32’z=o = m = (4a8/A)[(AT)2/2h2](n" - n'KT). (4.12) Eq. (4.12) can be rearranged to obtain (dzn/dT2)T=T = mxhz/(28)a(AT)? + n‘K M T . (4.13) Eqs. (4.11) and (4.13) are calibration—free expres- sions for determining the temperature derivative behavior of the refractive index of liquids with a wavefront shear- ing interferometer. C. A priori Error Estimates We now estimate the expected uncertainty in (dn/dT) and (dzn/de) based on the precision with which the experi- mental quantities in Eqs. (4.11) and (4.13) can be deter— mined. Let a derived quantity U be a continuous differ- entiable function of measurable quantities X1, X2, X3, ... defined by, U = U(Xl’ X X3, ...) . (4.14) 2' 72 The estimated variance (the square of the standard deviation) of U is obtained by the familiar prOpagation of error formula (Parratt, 1961; Wentworth, 1965): 2 _ 2 2 2 CU — (BU/3X1) o + (BU/3X2) ox 1 2 h.) XN + (BU/3X3) a: + ... (4.15) 3 .The a priori estimated variance in (dn/dT) is given by the following expression derived from Eqs. (4.11) and (4.15): [o:./(n')2] = (ox./X')2+I(ox/M2 + (oh/h)2 2 2 + (ca/a) + (028/28) . (4.16) Typical values and experimental uncertainties for the quantities in Eq. (4.16) are listed in Table 4.3. These are the experimental standard deviations (rms error) for repeated observations of these quantities. Note that the relative uncertainty in 28, the shearing distance in cell coordinates, is larger than any of the other relative un- certainties. When these numerical estimates are substi- tuted into Eq. (4.16), we obtain an estimate for o , nl on, = 1.14 x 10-2In'l . (4.17) The determination of (dn/dT) with an experimental uncertainty of about 1% is thus to be anticipated from this analysis. Note again that this uncertainty is due essentially entirely to the uncertainty in determining 28. 73 Table 4.3--Typical values and experimental uncertainties of parameters in Eq. (4.11). Quantity Value (Xi) Uncertainty (OX1) I(d§i/X§)I x' 5.0 (fringes) 10‘2 (fringe), 4 x 10'6 A 6.328 x 10'5em 10'9em 3 x 10‘9 0.81cm 0.001cm 1.5 x 10"6 AT 5.0°K 0.003°K 1.6 x 10"7 28 0.09cm 0.001cm 1.2 x 10"4 a 6.82cm 0.005cm 6 x 10"7 - The derived value of (dzn/de) is directly prOpor- tional to m, the lepe of the steady state fringes. The error in determining the slope would be expected to be larger than the uncertainty in the other quantities in Eq. (4.13). The following equation gives the uncertainty in the second derivative assuming that the error in the slope is the only important source of uncertainty: 2| I l l 2 _ 2 on /(n ) - (om/m) . (4.18) The slope was determined experimentally from a series of (AX'/Az) measurements on the "best line" drawn by hand through a fringe. This measurement was repeated for several fringes in a family of steady state fringes (see Figure 4.1 (c)). Typical values were m = 1.0 fringe cm-1 and cm = 0.07. Using these data, .. ‘2 II o — 7 x 10 |n | (4.19) 74 Figure 4.l--Interference fringe photographs: (a) isothermal slit photo, experiment v—13; (b) isothermal cell profile for water, experiment v-12; (c) steady state cell profile for water, AT = 5.597K, ex- periment v-l2. 76 Thus, we anticipate a 7% uncertainty in the determination of the second derivative (dzn/de). D. Experimental Results for Water Experiments were conducted upon water to determine (dn/dT)25° at 6328 A. The qualitative behavior of the interference fringe pattern during the establishment of the temperature gradient in the water filled cell was as follows: The fringes were straight and vertical as shown in the photo Figure 4.1 (b) taken in the isothermal con- figuration. At the onset of the temperature difference, the fringes became extremely parabolic and the family of parabolas was shifted horizontally across the final image plane. Finally, the parabolas decayed away to the straight tilted fringes of Figure 4.1 (c) when the steady state temperature distribution was established. This is the behavior we would predict from the literature values of K and (dzn/de). The parabolic T appearance of the fringes observed during the establish- ment of the temperature gradient is caused by the tran- sient nonlinear temperature distribution in the liquid. When the temperature distribution reaches its essentially linear steady state, the parabolas become straight fringes which are slightly tilted due to the influence of the second order terms, Eq. (4.13). Table 4.4 lists the experimental results for 30 determinations of (dn/dT)25° for water. As described in 77 Table 4.4--Summary of experimental results for (dn/dT) A 0 25° of water at A = 6328 A. h = 0.8103cm a = 6.828cm 28 = 0.0905cm ID TH Tc TM AT x' - (dn/dT)x104 v-lE 27.903 22.183 25.043! 5.720. 6.873 0.997 v-2E 27.855 22.210 25.033 5.645 6.749 0.992 v-3E 27.857 22.240 25.049 5.617 6.747 0.997 v-4E 27.837 22.190 25.013 5.647 6.672 0.981 0-53 27.793 22.260 25.027 5.533 6.579 0.987 v-6E 27.820 22.253 25.037 5.567 6.618 0.987 v-7E 27.813 22.283 25.048 5.530 6.596 0.990 v-8E 27.883 22.260 25.072 5.623 6.656 0.983 v-9E 27.863 22.350 25.107 5.513 6.579 0.991 0-108 27.865 22.303 25.084 5.562 6.669 0.995 0-118 27.843 22.350 25.097 5.493 6.537 0.988 0-128 27.830 22.130 24.980 5.700 6.690 0.974 0-138 27.805 22.307 25.056 5.498 6.602 0.997 v-14E 27.833 22.303 25.068 5.530 6.622 0.994 V-lSE 27.843 22.280 25.062 5.563 6.671 0.995 (dn/dT)E = -0.990 x10"4 8 = 6.4x10"7 0.69% 10 TH Tc TM AT x' - (dn/dT)x104 v-lD 27.890 22.183 25.036 5.707 6.838 0.994 v-2D 27.860 22.210 25.035 5.650 6.720 0.987 0-30 27.867 22.250 25.059 5.617 6.632 0.980 0-40 27.805 22.215 25.010 5.590 6.625 0.983 0-50 27.788 2 22.266 25.027 5.521 6.531 0.982 v-6D 27.823 I 22.262 25.042 5.565 6.597 0.984 0-70 27.796 1 22.273 25.034 5.523 6.518 0.979 v-8D 27.865 I 22.293 25.079 5.573 6.557 0.977 0-90 27.855 ; 22.330 25.093 5.525 6.573 0.987 0-100 27.865 22.317 25.091 5.547 6.669 0.998 0-110 27.840 22.350 25.095 5.490 6.549 0.990 0—120 27.827 22.270 25.049 5.557 6.704 1.00 0-130 27.796 . 22.323 25.059 5.473 6.601 1.00 0-140 27.835 22.295 25.065 5.540 6.646 0.996 0-150 27.843 22.283 25.063 5.560 6.662 0.994 -4 . -7 (dn/dT)D = -0.989><10 0 = 8.l><10 = 0.82% 78 Chapter III, each experiment had data collected during two different periods: (1) the fringe shift measured during the establishment of temperature gradient, labeled E and (2) the fringe shift measured during the decay of that same temperature gradient, labeled D. The experimental estimate of the standard deviation (scatter) is calculated from 3' {[l/(j — 1)] 2 (8' - ni) i=1 A O' 2 1/2 n' } , . (4.20) I) (1/j) Z n; . i=1 fit There appears to be no statistical difference be— tween the E and D experiments. A t-test on the averages from the two sets of experiments indicated that the null hypotheses, fig = 36 , level. Moreover, it was possible to verify the randomness could not be rejected at the 99% of errors by use of the run test (Wilson, 1952) on the sign of the quantity (0.989-ni x 104) for the sequence of experi- mental trials. This test indicated that the measurements were independent with random errors. The final result for 30 determinations on water is 63283 4 1 (dn/dT)25° = - 0.989 x 10 K , (4.21) 8 . = 7.35 x 10‘7 . n The experimental standard deviation is about 0.75%, which is comparable to but lower than the a priori estimate of 1% obtained in section C. 79 The steady state fringe pattern was photographed during four experiments, and the slope was obtained as dis- cussed at the end of section C. The data and the values of the second derivatives calculated from Eq. (4.13) are shown in Table 4.5. The average of these results is O (dZn/dT2)g§§8A = — 2.34 x 10"6K"2 , (4.22) A _ "7 II on,, _ 1.1 x 10 = 4.6% |n | The experimental standard deviation of 4.6% is lower than the a priori 7% estimate. Table 4.5--Summary of steady state fringe slope data for water. ID TM AT —m -(d2n/dT2)X106 0-12 24.980 5.597 0.906 2.21 0-13 25.056 5.543 0.972 2.39 0-14 25.068 5.563 0.936 2.30 0-15 25.062 5.616 1.03 2.46 The value obtained for the first derivative, - 0.989 x 10-4K-1, may be compared to the previous values listed in Table 4.1. Our value is about 5% lower than Tilton and Taylor (1938) and Waxler, et a1. (1964) and about 6% higher than the number obtained by Bryngdahl (1961). The closest agreement is with the ICT value of - 0.98 x 10"4K-l 4 l and with Andréasson, et a1. (1971), - 1.00 x 10‘ K’ . Tilton and Taylor's work on the absolute index of water has long 80 been considered definitive. However, their (dn/dT) values are obtained from an empirical equation used to interpolate the absolute index results. The uncertainties in their least squares parameters when applied to calculations of the derivative values are unknown and may be the source of the discrepancy. In any case our value is preferable be- cause it is measured directly with an interferometric tech- nique (112;! a differencing technique) over a small tem- perature range. Bryngdahl's value was obtained in con- junction with a hot wire thermal conductivity experiment. His table of eXperimental results suggests that the scatter in his determination of (dn/dT) may be as great as 3%. Moreover, (dn/dT) is obtained as a derived quantity. The ICT value was obtained from a compilation of the data available in 1933. The work of Tilton and Taylor had not been completed at this time and is probably more accurate than the data used in the ICT determination. Andréasson, et a1. (1971) used an interferometric technique which they claimed is comparable in precision to the method of Tilton and Taylor. The work of Andréasson, et a1. is the most re- cent precision study on the retractive index of water. The value determined for (dzn/de) is the same order of magnitude but slightly smaller than that obtained from Tilton and Taylor's data. This may again be due to the artifacts of their curvefitting process. Our value agrees well with Andréasson, et a1. However, the effect 81 of an anomalous temperature distribution (see Chapter V) would cause the fringes to be tilted with a slope different from that predicted by Eq. (4.13). Our results may be in error if this effect is present for temperature distribu- tions in water. E. Results for CCl C6H12’ and C6H6 4’ Experiments were conducted on three sensibly spheri- cal nonelectrolytes; carbon tetrachloride, cyclohexane and benzene. Anderson (1968, p. 196) first observed anomalous parabolic steady state fringe patterns for the first two of these liquids: The theory of the interferometer predicts that a uni- form nonzero refractive index gradient should produce interference fringes identical to those for a zero gradient but shifted horizontally by some fixed dis- tance. What we in fact observed for both CCl4 and C6H12' when a temperature difference was imposed ver- tically, and a steady state temperature distribution develOped, were curved interference fringes of a generally parabolic shape. This anomalous parabolic behavior, observed also in the present work, will be discussed in detail in Chapter V. Even though the fringes are parabolic in the steady state it is still possible to observe the gross horizontal shift of the family of fringes. Determinations of (dn/dT) could be made by measuring the midplane (z = 0) fringe shift and calculating the derivative from Eq. (4.11). Changes in (AT/h) were restricted to values of l K cm-1 or 82 less to prevent the light beam from being too far bent away from the optical axis. The most extensive study was made on CCl4, with the experimental results shown in Table 4.6. CCl4 was studied under three different sets of experimental conditions: (i) observation of the fringe shift in the 8 mm cell on changing AT from 2.7K to 3.5K, 28 = 0.0844cm; (ii) obser- vation of the fringe shift in the 8 mm cell on changing AT from 0.0K to 0.95K, 28 = 0.0685cm and (iii) observation of the fringe shift in the 30 mm cell on changing AT from 0.0K to 5.6K, 28 = 0.0857cm. The missing numbers in Table 4.6 correSpond to determinations that were not completed or were rejected because of experimental difficulties. Usually these were related to maintaining and measuring the temperature difference, AT. Also, some of the numbers were rejected because the upper and lower plates were not' isothermal at the start of an experiment. (dn/dT)25° dif- fered about 1% for the three sets of experiments. The grand average for 36 determinations and the calculated standard deviation are, CCl __4. 63280 4 1 (dn/dT)25° A = - 5.70 x 10' K , (4.23) 8 = 8.2 x 10‘6 = 1.4% In'l . n. 83 Table 4.6--Summary of experimental results for (dn/dT) of carbon tetrachloride at A 6328 3. 25° cc14(i) a = 6.828cm h =-0;81026cm 28 e 0.0844cm 10 AT(1) AT(2) (AT) TM x' -(dn/dT)x104 v-2E 2.730 3.484 0.754 24.90 5.046 5.95 v-3E 2.730 3.491 0.761 24.93 4.823 5.64 0-40 3.490 2.716 0.774 25.09 5.000 5.75 v-SE 2.716 3.501 0.786 24.90 5.000 5.66 :v-GD 2.838 3.501 0.663 25.04 4.238 5.69 0-20 3.431 2.765 0.666 24.98 4.282 5.72 v-BE 2.765 3.398 0.633 24.90 4.101 5.77 v-SE 2.738 3.485 0.747 24.84 4.959 5.90 v-6D 3.485 2.739 0.746 24.94 4.790 5.71 v-7E 2.739 3.416 0.677 24.93 4.255 5.58 0-100 3.470 2.757 0.713 24.99 4.536 5.66 . . n .. —4 A —5 (dn/dT)i.= ~5.73x 10 o = l.l><10 = 1.9% CC14‘11) a = 6.828cm h = 0.81026cm 28==0.06854cm 10 TH Tc TM AT x' -(dn/dT)xu) v-2E w25.513. 24.568 25.040 0.945 4.814 5.58 v-4E 225.508 24.573 25.040 0.935 4.948 5.80 v-SE -25.495 24.542 25.019 0.953 4.879 5.61 v-6E .25.393 24.518 24.955 0.875 4.714 5.90 v-7E '25.498 24.505 25.001 0.993 5.137 5.67 v-8E 25.533 24.573 25.053 0.960 5.023 5.73 v-9E 25.493 24.598 25.045 0.895 4.708 5.76 v-lOE 25.520 24.520 25.028 1.000 5.165 5.66 v-llE (25.490 24.541 25.010 0.949 4.905 5.67 v-lZE 25.503 24.560 25.031 0.943 4.895 5.69 0-133 25.478 24.583 25.030 0.895 4.639 5.67 v-14E 25.465 24.585 25.025 0.880 4.601 5.73 v-lSE 25.495 24.582 25.038 0.913 4.721 5.67 - 5 70x10‘4 “ - ‘6 - - - . o — 8.3><10 — 1.5% (dn/dT)ii 84 Table 4.6--(continued). CC14(111’ = 6.805cm h = 3.0124cm 28 = 0.08565cm ID TH Tc TM AT x' -(dn/dT)x104 g-3E 27.787 22.210 ,24.999 5.577 9.682 5.67 6-58 27.750 22.240 24.995 5.510 9.509 5.64 E-GE 27.796 22.193 24.995 5.603 9.738 5.68 6-78 27.863 22.258 25.061 5.605 9.727 5.68 E-BE 27.835 22.270 25.053 5.565 9.613 5.65 6-98 27.950 22.285 25.118 5.660 9.786 5.65 6-108 27.848 22.363 25.106 5.485 9.515 5.67 E-llE 27.857 22.343 25.100 5.514 9.642 5.72 E-lZE 27.855 22.257 25.056 5.598 9.703 5.67 6-138 27.813 22.270 25.042 5.543 9.539 5.63 6-14E 27.817 22.255 25.036 5.562 9.758 5.74 E-lSE 27.884 22.268 25.076 5.616 9.681 5.64 _ '4 A ‘5 _ (an/6T)iii - -5.67 x10 0 = 3.3><10 — 0.58% Comparison of this result with the values listed in Table 4.1 shows that it agrees well with the literature values, - 5.82 x 10m4K”l (Coumou, et al., 1964) and - 5.75 x 10.4K-l (Timmermans; 1950, 1959). The value obtained by Anderson (1968), - 5.96 x 10-4K-1, is the result of a small number of experiments and depends on a calibration experiment. Waxler's value (1964) was obtained by curve- fitting data from three widely spaced temperatures, 25°C, 35°C and 55°C. The results of 10 determinations of (dn/dT)25° for cyclohexane are shown in Table 4.7. The average value and calculated standard deviation are, Table 4.7--Summary of experimental results for (dn/dT)95° of cydlohexane at A = O 6328 A. C6312 a 6.8280m n = 0.81026cm 28 0.08698cm ID TH Tc TM AT x' -(dn/dT)x104 K-lE 25.415 24.623 25.02 0.792 5.000 5.45 K-lD -- -- 24.96 0.792 50009 5.46 2-28 25.405 24.628 25.01 0.777 5.000 5.56 K-2D -- -- 24.98 0.777 4.919 5.47 K-3E 25.430 24.613 25.02 0.817 5.000 5.28 K-3D -- -- 25.01 0.817 5.000 5.28 K-4E 25.430 24.610 25.02 0.820 5.025 5.29 2-58 25.417 24.622 25.02 0.795 4.965 5.39 r-sn -- -- 25.01 0.795' 4.965 5.39 C6H12 63280 4 1 (dn/dT)25° A = - 5.38 x 10‘ K‘ , (4.24) 8n. = 1.0 x 10"5 = 1.9% In'l This result agrees well with the previous values, - 5.34 x 10"“:("1 (Coumou, et al., 1964), - 5.47 x 10'4K'l (Timmer- mans; 1950, 1959), and - 5.44 x 10’4K‘l (Anderson, 1968). The results of eight determinations of (dn/dT)25° for benzene are shown in Table 4.8. The average value and calculated standard deviation are, C6H6 0 6328A 4K-1 ' 25. (4.25) (dn/dT) - 6.21 x 10‘ an: 4.9 x 10"6 = 0-79% In'l . 86 Table 4.8-—Summary of experimental results for (dn/dT)25° of benzene at A = 6328 A. C H a = 6.828cm h = 0.81026cm 28 = 0.08698cm ID TH Tc TM AT x' -(dn/dT)x104 A-l 25.437 24.612 25.02 0.825 5.900 6.17 A-l -- -— 25.04 0.825 5.849 6.12 A-2 25.432 24.615 25.02 0.817 5.933 6.27 A-Z —- -- 25.05 0.817 5.928 6.26 1-3 25.437 24.6112 25.03 0.825 5.964 6.24 A-3 —- -- 25.03 0.825 5.937 6.21 A-4 25.426 24.607 25.02 0.819 5.882 6.20 A—4 -- -— 25.03 0.819 5.883 6.20 This value is only slightly lower_in magnitude than the previous results, - 6.30 x 10-4K—l (Coumou, et al., 1965) and - 6.35 x 10"":('1 (Timmermans; 1950, 1959). F. Discussion These results for H O, CC14, C6H12 and C6H6 indicate 2 that this technique is an accurate and precise method of determining (dn/dT). It is superior to previous methods, where results were obtained by curvefitting of absolute index data over a large range of temperatures, typically 10°C to 50°C. Our measurements were made over a much smaller temperature range (24°C to 26°C) and, most impor- tant, (dn/dT) was determined directly. The results indicate that values of (dn/dT) cal- culated from the temperature dependence of density using the Lorentz-Lorenz equation, Eq. (4.2), are too high by 87 4, 14%; C6H12' 6%; C6H6’ 15%. This suggests that further work is necessary, the following percentage amounts: CCl perhaps in the direction suggested by Looyenga (1965), whose equation yields results in better agreement with. our values. G. A Molecular Calculation Using (dn/dT) The temperature dependence of the refractive index may be interpreted molecularly for nonpolar sensibly spheri- cal molecules (this restriction excludes water from this discussion) by electromagnetic equations of state (Amey, 1968) similar to the Lorentz—Lorenz equation, Eq. (4.2). Battcher (1952) has derived a two parameter version of this equation by starting from a more accurate approxi— mation of the local field acting on a molecule in the liquid. His expression is n2/[(n2 -1) (2n2 + 1W] = 3 (M/anA) [6‘ — r"‘3(2n2 - 2)/(2n2+1)] (4.26) where r is the effective molecular radius. The model used by Bottcher assumes that the fluid is composed of sensibly spherical molecules whose polarizability and radius are independent of temperature and pressure. Langer and Montalvo (1968) and Waxler and Weir (1963) have used Eq. (4.26) to calculate r and a from 88 their experimental measurements of the pressure dependence of the refractive index. They obtain the pressure depen- dence of the specific volume from one of two sources in the literature; (i) direct measurements of the density as a function of pressure or (ii) eXperimentally determined constants for the Tait equation for liquids. They plot 2 + 1)?) vs.(2n?-2)/(2n2+1) and calculate 112/[(192 - 1) (2n 0 and r from, respectively, the slope and the intercept of the straight line ordinarily obtained. We may use our experimental results for (dn/dT) and the literature values for the temperature dependence of specific volume in a similar analysis. The specific volume of CCl4 and C6H12 are given by Wood and Gray (1952) and the specific volume of C6H is given by Wood and Austin (1945) 6 (T in these equations is in °C): CCl _£ — . 3 -1 —4 V(1n cm gm ) = 0.612334-7.294 X 10 T + 7.22x10'7T2 + 5.52x 10‘9T3 . (4.27) C6H12 — . 3 —1 -3 V(1n cm gm ) = 1.2548 + 1.4362 x 10 T + 2.529><10"°T2 + 5.37x 10‘9T3 . (4.28) C6H6 — . 3 —1 —3 V(1n cm gm ) = 1.1106 + 1.3105 X 10 T + 1.477x10"°T2 + 7.65x 10‘9T3 . (4.29) 89 The refractive 1nd1ces of CC14, C6H12 and C6H6 as a function of temperature are cc14 63283 4 n(T) = 1.45586 - 5.70 x 10‘ (T - 25) (4.30) C6H12 63283 4 n(T) = 1.42205 — 5.38 x 10 (T - 25) (4.31) C6H6 63280 4 n(T) A = 1.49511 - 6.21 x 10‘ (T - 25). (4.32) 63283 The values of n are interpolated from the absOlute 25° index data of Waxler, et a1. (1964) and Timmermans (1950, 1959). The refractive index and the specific volume are obtained from Eqs. (4.27 - 4.32) for a range of temperatures 2-2)/(2n2-+1) and n2/[(n2 - 1)(2n2+-1)VJ and the quantities (2n are calculated. These numbers are fit to Eq. (4.26) by or- dinary linear least squares to obtain the slepe and inter- cept from which r and a are calculated. Table 4.9 shows values for a BBttcher plot from 20°C to 30°C calculated from Eqs. (4.27) and (4.30) for CCl Figure 4.2 is a 4. graph of these numbers and is typical of the plots obtained for the liquids studied. 90 Table 4.9--Data for Béttcher plot of CCl4 refractive index and specific volume temperature dependence. T°(C) n(63283) V (2n2-2)/(2n2+l) nz/(n2-1)(2n2+1)V 20.00 1.45871 .62725 .4291 .57229 21.00 1.45814 .62801 .4288 .57235 22.00 1.45757 .62878 .4284 .57242 23.00 1.45700 .62955 .4281 .57248 24.00 1.45643 .63032 .4277 .57254 25.00 1.45586 .63110 .4273 .57259 26.00 1.45529 .6318? .4270 .57265 27.00 1.45472 .63265 .4266 .57271 28.00 1.45415 .63344 .4262 .57277 29.00 1.45358 .63422 .4259 .57283 30.00 1.45301 .63501 .4255 .57289 0 6328A It was found empirically that the value of n25° used in the calculation does not greatly affect the values of r or 0 obtained. However, r and a are very sensitive to the value of (dn/dT). Therefore, small errors intro- duced by interpolation to 63283 would not affect the final estimates of r and a. Table 4.10 summarizes the polarizabilities and effective molecular radii calculated from our data. Also listed are the previous values obtained from the analysis of pressure dependence data and the values calculated by Waxler and Weir (1963) directly from molar volume data. Our values for the effective molecular radius are about 16% higher than those obtained by Waxler and Weir while the polarizabilities agree within 6%. However, our values for r agree well with the radii calculated by 91 Figure 4.2-—Battcher plot for CCl pZ/[(n2-l)(2n2+l)V] 4; vs. (2n2-2)/(2n2+1). 92 QNV. 2+~cNV\ANu~:NV hmv. mmc. l nubm. wmhm mmkm. wmhm. hmhm. mmbm. mth. A (u +2112) (I 'zu1/z“ 93 Table 4.10--Effective molecular radii and polarizability for CC14, C6H6 and C6Hl2’ . . 0 . ° 3 Source Liquid r(A) n(A) ‘ a CCl4 3.46 10.55 C6H6 3.25 10.30 C6H12 3.24 10.65 b CCl4 2.93 9.88 C6H6 2.93 9.86 c CCl4 3.33 C6H6 3.40 aThis work. bCalculated from pressure dependence of n, Waxler and Weir, 1963. cCalculated from molar volume, op. cit. Waxler and Weir directly from the molar volume data. It appears that the pressure dependence and temperature de- pendence experiments are not equivalent. The precise physical meaning of the "effective" radius is not clear. It may correspond to a correlation distance between molecules or to the "cavity" occupied by a simple molecule in the.liquid. Since the refractive index is manifested molecularly by the displacement of the molecule's electrons, the effective radius should be re- lated to the scattering cross—section of the individual 94 molecules. Clearly, a more complete theoretical treatment is necessary here; perhaps these results can suggest direc- tions for future work. H. Suggestions for Further Work Although accurate and reproducible results were obtained with the apparatus described in the preceding pages, the basic experimental techniques can be improved in four ways. (1) Improved Temperature Control. Anderson (1968) sug- gests methods of maintaining and changing the temperature of the metal plates that may be more efficient than cir— culating water baths. The possibility of electrical heat- ing and constant temperature cooling coils being balanced by a pr0portiona1 controller should also be considered. (2) Determination of Shear. As pointed out in section C, the error in determining the absolute shearing distance 28 is the largest contributor to the experimental uncer- tainty in (dn/dT). A slit of precise and uniform aperture would improve the determination of 28 as described in Chapter III. Wallin and Wallin (1970) have described a method for determining the shear by passing the laser beam backwards through 01, L1 and L2 and measuring the beam separation directly. They claim that use of this technique lowered the uncertainty in the absolute shear by an order of magnitude. 95 (3) Cell Sealant. The liquid of interest is sealed in the sample cell with a viscous stopcock grease (Dow "FS" Fluorosilicone) applied between the metal plates and the tOp of the glass cell walls. Use of a teflon gasket on top of the glass walls would eliminate the problem of sealant extrusion from the metal-glass boundary when pres« sure is applied to the metal plates. This would improve the optical quality of the fringe image (compare Figure 4.1(a) and Figure 4.1(b)) and would also eliminate danger of contamination of the sample liquid with the sealant. (4) Digital Data Acquisition. A significant modification of the apparatus would be the installation of a small digital computer (for example, PDP 8E) to record the data during a run. This would eliminate the tedious and time consuming process of obtaining the time dependent temperature change and fringe shift data from a strip chart recorder output. This changeover would be relatively easy because the data are available as an electrical outputs and these could be easily interfaced to the analog input of a computer. It is likely that accuracy would also be im— proved. Many further eXperiments can be performed in the determination of (dn/dT) for liquids. The most obvious is the determination of (dn/dT) at several different mean temperatures. This would make possible the calculation of the 2nd and 3rd derivatives from the change (if any) 96 in the first derivative with temperature. Tabulation of data from a series of chemically related compounds would indicate trends that may suggest further improvements in the electromagnetic equation of state. (dn/dT) values are useful to workers performing light-scattering and depolar- ization-experiments. Finally, this technique could possibly be used to determine (dn/dT) for mixtures. However, the fringe shift would have to be measured before the onset of thermal dif— fusion because the formation of the mass fraction gradient also causes the fringes to move horizontally (Horne and Anderson, 1971). A small temperature gradient in a large cell would minimize the effects of thermal diffusion. CHAPTER V ANOMALOUS PARABOLIC FRINGE SHAPES A. Introduction The steady state fringe shapes observed for carbon tetrachloride, cyclohexane and benzene in the 8mm cell were anomalously parabolic (see Chapter IV, section E). Examina- tion of the literature (Table 4.1) indicates that for CC14, C6H12 and C6H6’ unlike water which has a significant (d2n/dT2), the first derivative is the only nonzero term in the power series expansion of refractive index, Eq. (4.3). Second and third derivatives of the order of mag- 8 to 10-10 would not cause detectable fringe nitude of 10- shifts. A priori, we would therefore expect a refractive index derivative experiment on, for example, CCl4 to pro- ceed as follows: straight vertical fringes in the iso- thermal configuration would become parabolic due to the initial nonlinear temperature distribution at the onset of the temperature difference. These parabolas would be shifted horizontally across the final image plane and would decay away to vertical or very slightly tilted straight fringes. What in fact was observed experimentally was behavior identical to that described by Anderson (1968): 97 98 the steady state fringes remain parabolic in shape with the ends of the parabolas extending in the direction of travel of the horizontal fringe shift. Typical steady state photographs for the three liquids are shown in Fig- ure 5.1, (a), (b), (c). These parabolas do not move hori- zontally with time after reaching the steady state (5-8 min.), nor do they change shape even after 36 hours of a maintained temperature gradient. B. Analysis of the Parabolic Steady State Fringe Shapes There are several possible explanations for the anomalous parabolic fringe shapes for CC14, C6H12 and C6H6' These include (1) nonlinear temperature dependence of the refractive index, (2) nonlinear (sigmoidal) steady state temperature distribution, and (3) 2nd order optical effects. (1) Nonlinear temperature dependence of the refractive index. This is the interpretation we put forth at an early stage of this research (Olson, et al., 1970). The notion was first described by Bartelt (1968) and Anderson (1968, pp. 196-200): Analysis of the refractive index data from the literature for the two pure compounds showed that only a linear dependence on temperature was statistically significant. Use of those data and the working equa- tions for the interferometer required that the tempera- ture distribution inside the liquid be sigmoidal in shape in order to explain the shape of the interference fringes. 99 Figure 5.1--Interference fringe photographs: (a) steady state cell profile of CCl4 in 8mm cell, AT = 2.73K, solid line ( ) indicates midplane of cell, (b) steady state cell profile of Cklein.8mm cell, AT = 2.79K, (c) steady state cell profile of C6H6 in 8mm cell, AT = 2.79K. 101 Such a temperature distribution would, in turn, re- quire either anomalous variations in the thermal con- ductivity of the liquid or some inexplicable apparatus effect. Believing that the thermal conductivity is a well-behaved function of the temperature, and that our apparatus caused no strange effects (since the same temperature difference, applied to water, gave the expected straight fringes), we turned our attention to the validity of the reported values Of the tempera- ture dependence Of the refractive index for CCl4 and C6H12. Since only second order thermal conductivity tem- perature dependence affects the value Of (dT/dz)0, and since the effect is less than 0.l°, we have, at the center of the cell, (dT/dz) = AT/a...., therefore second and higher temperatuge derivatives are Obtain- able from fringe shape analysis. If the temperature distribution is essentially linear, 1121' described by Eq. (4.4),.then parabolic fringes would be explained by a cubic form of the refrac- tive index temperature dependence formula, Eq. (4.3). KINET was used to curvefit the steady state parabolic fringe shape data (see Appendix C) using Eqs. (2.23), (4.3) and (4.4). Literature values of KT (Table 4.1) and our ex- perimental values Of (dn/dT) (Eqs. (4.23), (4.24) and (4.25)) were used so that (dZn/dT) and (d2n/dT3) were the only adjustable parameters. The average results and un- certainties calculated with KINET are shown in Table 5.1. Note that (AT/h) for this series Of experiments was = 3.5Kcm-l. This caused the light beam from the cell to be bent out Of the interferometer Optical axis and re- quired that the rotatable parallel plate and L1 be adjusted 102 Table 5.l--Results of curvefitting for alleged nonlinear temperature dependence of refractive index. Liquid (62n/6T2)x105 8x105 -(d3n/dT3)x105 8x105 cc14 1.77 0.13 7.10 0.30 C6H12 1.94 0.30 5.84 0.30 c686 1.51 0.24 2.78 0.25 to return the light beam to the Optical axis. This type Of adjustment was not necessary during the (dn/dT) experi- ments where (AT/h) = 1.0Kcm'l. Superficially, it appears that the nonlinear n(T) explanation is entirely satisfactory since the data were easily fit tO this model. However, there are two objections to this interpretation: (1) The refractive index equation Obtained would not reproduce the literature refractive index behavior (the values Obtained for (dzn/de) and (d3n/dT3) are about 103 larger than indicated by previous experimental or theoreti- cal studies (Table 4.1)). Upon substitution Of the numeri- cal values from Table 4.7 into Eq. (4.2), we Obtain for C6H12' n(T) - 5.38 X 10-4(T - 25) + 9.70><10-6(T--25)2 = n25° 6 3 - 9.73 x 10‘ (T - 25) (5.1) 103 Figure (5.2) is a plot both of Eq. (5.1) and the absolute refractive index from the literature. Note the gross dif- ferences at 25°C and 30°C. This type Of behavior would be easily detectable with a conventional critical angle re- fractometer since variations would occur in the 3rd decimal place. However, Coumou, et a1. (1964) measured the absolute index at 20°, 25°, 30° and 40° and noted only linear be- havior. In order to eliminate this problem, a gaussian form Of the refractive index temperature equation was postulated, n(T) + (dn/dT)25°(T - 25) = n25° 2 ’°(T ‘ 25) [¢(T - 25) 2 +6 +8(T-25)3] . (5.2) This equation was fit to the steady state fringe data and the results are shown in Table 5.2. Note that the expo- nential parameter 8 is very poorly determined as indicated by the very large estimated standard deviation. The 2nd and 3rd derivatives are of the same order of magnitude as before. Eq. (5.2) produces the correct absolute index be- havior at all temperatures. However, this type of equation suggests that [(dzn/dT2)l and I(d3n/dT3)I are maximal at 25° and decay away to zero everywhere else. This sort of singular behavior is not physically reasonable nor theoreti- cally satisfying. 104 Figure 5.2--Temperature dependence of refractive index of C6H12' The solid line ( ) is Eq. (5.1) and the broken line (- - -) is the linear behavior predicted by the literature (Table 4.1). 105 85 ~525an o- 6.3 OflN OéN me¢4 m_¢; ON¢; (1) U 1NN¢4 ¢N¢4 106 Table 5.2--Results of curvefitting for guassian nonlinear refractive index temperature dependence equation. Liquid 8x103 8x103 8x106 8x106 8x105 8x105 cc14 -7.9 38.0 8.73 0.31 —1.13 0.050 C6H12 24.4 57.0 10.4 0.50 -0.998 0.057 c6116 -223.0 130.0 6.68 0.50 -0.461 0.028 lOSX(d2n/dT2) = 20 105x(d3n/dT3) = 60 cc14 1.74 -6.78 C6H12 2.08 —5.99 C6H6 1.34 -2.77 (ii) Experiments on CCl4 in the 30mm cell gave results for the higher derivatives which were quite different from those Obtained in the 8mm cell. Steady state fringe data for water and CCl4 were Obtained using the 30mm cell with (AT/h) = 3.3K cm"l (AT 2 10K, h x 3cm). This is comparable to (AT/h) = 3.5K cm-1 for the 8mm cell experiments (AT 2 2.80K, h = 0.8cm). The mean temperature was also 25°C in the large cell. The entire cell profile could not be placed on one photograph. Consequently, three photos Of the cell profile were taken for water (Figure 5.3) and CCl4 (Figure 5.4). A composite drawing of the complete cell profile made from these photographs is shown in Figure (5.5). These drawings have been marked to indicate the region of an 8mm cell. 107 Figure 5.3--Interference fringe photographs. This is the steady state cell profile of H20 in 30mm cell, AT = 10.83K: (a) tOp (1/3) Of cell, (b) middle (1/3) of cell, (c) bottom (1/3) of cell. 109 Figure 5.4--Interference fringe photographs. This is the steady state cell profile Of CC14 in 30mm cell, AT = 10.22K: (a) tOp (1/3) Of cell, (b) middle (1/3) Of cell, (c) bottom (1/3) of cell. 111 Figure 5.5--Steady state cell profile of CC14 and H O in 30mm cell. This composite drawing was made from Figures 5.3 and 5.4. The broken line (- - -) indicates the "break" in the CCl4 fringes. 112 '280’4 1 )- 230.4 )- z = 0.0 - z--0.4 -Z=0.0 Zia-0.4 113 The general appearance of the steady state fringes for water is the same as in the 8mm cell: straight fringes that are slightly tilted from the vertical. -However, the CCl4 fringe shape is vastly different in the two different cells. In the center region Of Figure (5.5) which corre- sponds to the central 8mm, the fringes are nearly straight. If (dzn/de) and (d3n/dT3)_were actually the values in. Table 5.1, the fringes in the center Of Figure (5.5) should be similar to Figure (5.1(a)) since the mean temperature, (AT/h) and the shear are almost identical. The data from Figure (5.5) were fit to the cubic refractive index temperature equation using KINET. The following results were Obtained: CCl4 30mm cell (dzn/dT2)25° 5 -4.4 x 10‘7 = 0 (5.3) 8 = 10 x 10'7 (d3n/dT3)25° = -0.75 x 10‘5 5 8 = 0.077 x 10‘ The value of the 3rd derivative (responsible for para- bolicity) has decreased by an order of magnitude. An in- tensive physical property Of the liquid should be inde- pendent Of the height of the cell in which the liquid is contained. 114 Another experimental test was carried out. Since the (dn/dT) Of water is about (1/6) the value Of (dn/dT) for CC14, the refractive index gradient is smaller in water than in CCl4 for identical (AT/h) conditions; (3n/3z)2=0 = (AT/h)z=o(dn/dT)T=25.. (5.4) For (AT/h) = 3.0K cm-l , H20 ~ -4 -1 (an/az)z=o — 3.0 x 10 cm , (5.5) CCl4 ~ —4 -l (an/82)z=o - 17 x 10 cm . (5.6) We decided to subject water to a large temperature difference to see if parabolic behavior could be produced if the re- fractive index gradient was similar in magnitude to that in the nonelectrolyte experiments. Figure (5.6(b)) shows photo- graphs Of steady state for water with (AT/h) = 20K cm-l. The fringes continue to tilt from the vertical in response tO the large (AT/h) as predicted by Eq. (4.13). Parabolic steady state fringes could not be produced for water. (2) Nonlinear temperature distribution. If the refrac- tive index is well represented by the formula, n(T) = 11250 + (dn/dT) (T " 25) p (5.7) 115 Figure 5.6--Interference fringe photographs: (a) Isothermal slit photo for H20 in 5mm cell, (b) steady state cell profile Of H20 in 5mm cell, AT a 10.09K, (c) fringe pattern produced by free convection in CCl4 heated from below. 117 then a sigmoidal spatial temperature distribution in the steady state would produce parabolic fringes. We know that the temperature distribution should be sigmoidal dur- ing the establishment of the temperature difference (Ingle, 1971, Figure l, p. 97), but this nonlinearity should decay away so that the steady state T(z) is Eq. (4.3). Even for water we observed parabolas at the beginning of the eXperi- ment that decayed away to straight fringes in the steady state. We write Eq. (5.8) as an empirical model for an essentially linear steady state temperature distribution with parameters S1 and $2 to introduce nonlinearity, T(z) = (AT/h)z + 81(22 - (h/2)2] 2 2 + $2 {z[z - (h/2) ]} + TM - (5.8) 82 causes the distribution to become sigmoidal while S1 causes the distribution to be skewed (unsymmetric) about 2 = 0. Eqs. (5.8) and (5.7) were used with the complete Optical equation, Eq. (2.23), to fit the steady state parabolic fringe data with KINET. Table 5.3 shows re- sults for representative CC14, C6Hl2 and CGH6 data. The experimental values of S1 and 82 were used to calculate the steady state temperature distributions from Eq. (5.8) and Eq. (4.4). The results are tabulated in Table 5.4. 118 Table 5.3--Results Of curvefitting for nonlinear tempera— ture distribution equation. Liquid 81 x 10 I O ><10 82 x 10 0x 10 CC14 -14.4 0.64 7.54 0.21 C6H12 -7.30 0.38 7.92 0.15 C6H6 -7.06 0.42 3.17 0.16 Note that the maximum difference in the two tempera— ture distributions is 0.028K. It is apparent that this is not a drastic deviation. The s-shaped temperature distribu- tion shown would not be inconsistent with the theoretical analysis of the steady state temperature distribution be— cause the deviation is less than the claimed accuracy Of the temperature equation (Horne and Anderson, 1970). The steady state fringe shape data for CCl4 in the 30mm cell were not fit to the empirical temperature equa- tion because of the sharp breaks in the fringe shape indi- cated in Figure 5.5. The break is more apparent when look— ing at the actual photographs, Figure 5.4. There appear to be three distinct regions: (a) sharply tilted fringes near the tOp of the cell, (b) a center region Of almost straight fringes, and (c) sharply tilted fringes near the bottom of the cell. Curvefitting difficulties were en- countered because the Optical equation delta functions, An(z), AS'(z) and AS"(z), are not continuous across the 119 Table 5.4--Experimental and theoretical steady state tempera- ture distributions for CC14 in 8mm cell, AT = 2.80°C, KT = -2.0 x 10-3. z(cm) T°C (Theory) T°C (Experiment) .405 26.4000 26.4000 .385 26.3298 26.3263 .365 26.2596 26.2533 .344 26.1895 26.1809 .324 26.1193 26.1091 .304 26.0491 26.0380 .284 25.9790 25.9673 .263 25.9089 25.8972 .243 25.8387 25.8274 .223 25.7686 25.7582 .203 25.6985 25.6892 .182 25.6284 25.6207 .162 25.5584 25.5524 .142 25.4883 25.4843 .122 25.4182 25.4165 .101 25.3482 25.3489 .081 25.2781 25.2814 .061 25.2081 25.2140 .041 25.1381 25.1466 .020 25.0680 25.0793 -.000 24.9980 25.0120 -.020 24.9280 24.9446 -.041 24.8581 24.8771 -.061 24.7881 24.8094 -.081 24.7181 24.7416 -.101 24.6482 24.6736 -.122 24.5782 24.6053 -.142 24.5083 24.5367 -.162 24.4384 24.4678 -.182 24.3684 24.3985 -.203 24.2985 24.3287 -.223 24.2286 24.2586 -.243 24.1587 24.1879 -.263 24.0889 24.1167 -.284 24.0190 24.0449 -.304 23.9491 23.9725 -.324 I 23.8793 23.8995 -.344 23.8095 23.8257 -.365 23.7396 23.7513 -.385 23.6698 23.6760 -.405 23.6000 23.6000 120 three regions. The physical significance of these three regions will be discussed in section C. (3) 2nd Order Optical Effect. There is a possibility that the parabolicity could in some way be the result of the optical components of the interferometer acting on the wavefront after it leaves the cell. The wavefront, having passed through a refractive index gradient, is distorted. This possibility is remote in that explicit account Of the gradient is made in the theory derived in Chapter II. Empirical tests were performed which included tilt- ing the Savart plates at different angles, adjusting the polarization analyzer, and slightly changing the angle of incidence of the light beam on L1. None of these affected the parabolicity in any unusual way that would suggest that critical adjustment Of a particular component is necessary. Note that benzene has the largest value of (dn/dT) for the liquids studied. Therefore the benzene filled cell contained the largest refractive index gradient for the experiments on CC14, C6Hl2 and C6H6 when (AT/h) = 3.5K cm_1. However, Figure 4.1 shows that the degree of parabolicity Of the benzene fringes is the smallest of the three. This suggests that the size Of the gradient and the resulting nonlinear Optical effects are not respon-. sible for the parabolicity. 121 C. Discussion Rejecting the possibility Of an Optical effect, there are two alternatives for explanations of the parabolic steady state fringe shape. We pointed out earlier in the Optical theory that to analyze the fringe shape with respect to a prOperty varying in the vertical coordinate in the liquid, either the refractive index prOperty dependence, in this case n(T), or the property spatial distribution, T(z), must be specified beforehand. If we claim to know T(z) exactly as Eq. (4.4), then the data indicate nonlinear refractive index behavior that is drastically different from anything ever Observed or predicted before. However, if we accept n(T) as a linear function, then the data in- dicate that T(z) is slightly different (a maximum Of=:0.2%) from the theoretically predicted temperature distribution. We therefore conclude that the temperature distribution in the steady state is slightly sigmoidal and that this per- turbation is the cause of the parabolic fringes observed for CCl , and C 4' C6H12 6H6“ It is important to note that the effect is small in magnitude compared to the gross linear character of the tem- perature distribution as described by Eq. (4.4). Because of this, thermal diffusion experiments that use Eq. (4.4) to describe the steady state temperature distribution (Anderson, 1968; Horne and Anderson, 1971) are not in any 122 way affected by the discovery of a sigmoidal temperature perturbation. The mass distribution response tO this small a temperature variation would not be detectable. In like manner, the development Of the thermal conductivity experi- ment discussed in Chapter VI is not affected by this dis— covery. The experimental evidence at this time suggests that the physical cause Of the sigmoidal perturbation is a wall effect related to the liquid-metal boundaries at the tOp and bottom of the sample cell. Radiation heat flux, horizontal heat flux through the glass walls, the heterogeneous mechanism Of "wetting" of the metal surfaces, adsorption Of the liquid on the metal and dissolution Of the sealant at the liquid—metal interface are possible causes Of a wall effect. The theory of metal—liquid inter— actions at a boundary has not been developed so that a quantitative theoretical analysis is not possible. In addition, the vertical heat flux through the liquid does not vanish at the boundaries. Anomalous behavior of the heat flux would, of course, be manifested in a perturba- tion of the temperature distribution. The suggestion of a wall effect is supported by the peculiar appearance Of the CCl4 steady state fringes in the 30mm cell. We have already noted the sharp break in the fringes near the tOp and bottom boundaries of the cell profile. The region of extreme curvature extends 123 about 4.6mm from the top boundary and about 3.9mm from the bottom boundary (taking into account the cutoff due to the light beam bending). If these two regions were joined to— gether to form an 8.5mm fringe pattern, the resulting pa- rabolas would be similar to those actually Observed in 8mm cell. This suggests that the boundary influence on the temperature distribution extends about 4-5mm into the liquid from each Of the metal boundaries. The suggestion that the metal-liquid boundary could influence the temperature distribution in this type Of ap- paratus was also made by Longsworth (1957). He noted peculiar behavior Of the temperature gradient established in water contained in a sandwich cell whose upper and lower boundaries were, at different times, silver, stain- less steel Or Koroseal (a gasket material). By use Of Rayleigh interferometry, he determined that the gradient Of refractive index was markedly larger near the upper and lower boundaries of the cell. This behavior could not be explained by the known behavior of refractive in- dex or thermal conductivity. Longsworth referred the origin of the wall effect notion to Bates (1933) who made the following statement concerning the temperature dis- tribution in water and red oil contained in a similar sandwich type apparatus with a vertical heat flux: "A surface effect is present at the interface Of a stationary liquid and a solid during the transmission Of heat normal to the face, and should properly be considered." 124 In our investigation, water, so Often the exception in physical properties Of liquids, exhibits "normal" steady state fringe behavior. Some physical properties Of water, CC14, C6H12’ and C6H6 are compared in Table 5.5. Signifi- cantly, the thermal conductivity of water exhibits unusual behavior in that it increases with temperature and has noticeable nonlinear temperature dependence (McLaughlin, 1964). This indicates that the sigmoidal perturbation is not due to anomalous behavior in the thermal conductivity. Moreover, the fact that water is probably structured due to hydrogen bonding would tend to reduce the influence of a wall effect. Further experiments should be performed for a series Of sample cells whose heights are intermediate be- tween 8mm and 30mm. Different materials should be tried as the upper and lower thermostating boundaries. These experiments would indicate if the distance from the upper and lower walls to the "break" in the fringes (see Figures 5.4 and 5.5) is a constant. Observation Of the steady state fringe pattern for other liquids, especially hydro— gen bonded liquids, would indicate trends, which would be useful in beginning a theoretical treatment of the phe- nomenon. Finally, experiments should be conducted at dif- ferent mean temperatures. 125 Table 5.5—-Summary Of some physical properties of H20, 0 CC14, C6H12 and C6H6 at 25 C. Liquid K KT 0 Cp n 7 TB H o 60.8C 2.68C 0.997a 4.18a 0.894a 72.0a 100.0a 2 b b e b h a a CCl 10.3 -1.84 1.584 0.85 0.904 26.1 76.7 4 d d e g h a a C6H12 12.4 -0.71 0.774 1.84 0.900 26.5 80.1 C6H6 16.0a -1.7sb 0.874f 1.72a 0.608a 28.2a 80.7a K = Thermal conductivity x 102, J m—1 sec"1 K_1. _ 3 KT _ (1/r)(ar48T)25o x 10 . p = Density, 9 cm'3.' C = Specific heat capactiy at constant pressure, P J g-1 K-1 n = Viscosity X 102, g cm.1 sec-l. y = Surface tension, dynes cm‘l. T = Boiling point at 1atm.,°C. 4‘ References. ,3- aWeast (Handbook Of Chemistry and Physics, 44th edition), 1963. b Touloukian, 1970. CMcLaughlin, 1964. dBriggs, 1957. eWood and Gray, 1952. fWood and Austin, 1945. gTimmermans, 1950. hHammond and Stokes, 1955. CHAPTER VI PURE THERMAL CONDUCTION A. Introduction The Observation and theory of heat flow in matter is the Oldest and most fundamental problem in the study Of transport phenomena. It was the first transport process to be precisely described by a linear law when Fourier (1822) stated his famous relation for heat flow in one dimension, q = —K(8T/8X) . (5,1) If a system is perturbed by a temperature gradient, Eq. (6.1) indicates that a heat flux, q, will occur (112;! heat will flow) toward the colder part Of the object. This heat flux will tend to restore the system tO equilib- rium. The dimensions Of the heat flux are (energy) (time).l (distance)-2, and the proportionality factor K, the thermal conductivity, has dimensions of (energy) (time).l (distance)—l (temperature)-l. In a recent review, McLaughlin (1964) considers the importance Of the study of thermal conductivity Of liquids: 126 127 The subject Of the thermal conductivity of liquids and dense gases is important for two main reasons: firstly, because its study can help in giving a better understand- ing of the basic molecular processes involved in trans- port phenomena in general, and secondly, because of its significance in technological applications. B. Experimental Methods The experimental methods of determining thermal conductivity are divided into two types: (1) steady state methods and (2) transient methods. A recent general dis- cussion can be found in Tyrrell (1961). (l) Steady state methods. A liquid is contained in a vessel with geometrically well defined boundaries, for example, between two concentric cylinders. If a constant heating source is maintained in one of the boundaries, a steady state heat flux will be present in the liquid. In this case Eq. (6.1) can be approximated by Q/A = K(AT/d) . (6.2) where Q is the energy produced in the heating surface per unit time, A is the area of the heating surface, d is the separation Of the boundaries of the liquid and AT is the steady temperature difference between the liquid boundaries. This treatment assumes that K is independent of temperature over this AT. The thermal conductivity experiment is be- gun by applying a constant heat flux through an initially isothermal liquid (this is usually accomplished by means 128 Of electrical heating with constant current passing through a resistor Of known value). The steady state temperature difference AT is measured when no further temperature varia- tion is detected in the liquid. This number and the geo- metric constants can be used to calculate K from Eq. (6.2). Energy may be transported through the liquid by two mechanisms in addition to pure conduction described by Eqs. (6.1) and (6.2). These are: (i) convection, which is the bulk transfer Of mass from the hot area Of the liquid to the cold area, and (ii) radiation Of energy from one boundary of the liquid to the other. Care must be taken to eliminate these mechanisms when designing a thermal conductivity apparatus. Heating from the tOp of a horizontal layer downward is commonly done tO minimize convection, while maintaining a small temperature differ— ence tends to minimize the effect of radiation. The main sources of error in steady state methods are (1) measurement of Q (calorimetry), (ii) elimination Of "heat leaks" so that the entire heat flux generated at the boundary of the liquid is transported through the liquid, (iii) minimization Of the effects of convection and radiation energy transfer and (iv) the careful machin- ing and measurement Of the cell geometry necessary to Ob- tain the surface area, A. A recent careful application of this method including a systematic discussion of the experimental errors has been made by Tree and Leidenfrost (1969). 129 (2) Transient methods. The only previous time dependent experimental method Of determining thermal conductivity is the "hot wire" experiment. In this method the develOpment of the temperature Of an electrically heated wire immersed in the liquid of interest is Observed. In principle the temperature rise Of the wire will depend on the rate Of the conduction Of heat away from the wire in the surrounding liquid and therefore on the thermal conductivity Of the liquid. Extensive theoretical treatments have produced several equations describing the temperature Of the wire as a function of time (McLaughlin, 1964). A commonly used equation is, T2 - T1 = (Q/24TK)1n(t2/tl) , (6.3) where T is the temperature of the wire at time t2, T1 is 2 the temperature Of the wire at time '81 and 8 is the length Of the wire. The thermal conductivity is calculated from the lepe of a plot of the temperature of the wire against log time. However, after a time convection begins in the liquid surrounding the wire. Consequently all measurements must be made before the onset of convection, which is in— dicated by deviations from Eq. (6.3). Additional experi- mental errors include those discussed above for steady state methods. A recent exhaustive theoretical treatment of this method and a carefully designed computerized eXperi- ment on toluene are described by McLaughlin and Pittman (1971). 130 C. A New Transient Method Tyrrell (1961, p. 299) noted that in principle the eXperimental apparatus used for steady state measurements could be used in transient methods: The above methods are all steady state methods, but cells Of similar geometrical form (parallel plates) could in principle be used tO determine the heat con- ductivity from the rate Of establishment of the thermal gradient in the liquid layer. It is possible to en- visage several methods by which this might be done, for example by adapting optical methods used for dif— fusion measurements. ..., but the practical difficul- ties cannot be negligible because no accurate measure- ments have been carried out using techniques Of this kind. The fringe shift produced by a wavefront shearing interferometer during the establishment Of a temperature gradient in a liquid as described in Chapters III and IV should be a time dependent measurement Of the rate of es— tablishment of the thermal gradient. In order to design a quantitative transient experiment to determine the thermal conductivity, three basic problems must be solved: (1) Description Of the boundary conditions. A system- atic and reproducible method of applying and maintaining a temperature difference across the boundaries Of the liquid is necessary. The thermostated metal plates are the boundaries of the liquid contained in the sample cell. The time dependence of the temperature Of the upper and lower plates is described by the following equations, TU(t) = T; , TL(t) = TM , t i ts ; TU(t) - T; + (T; - T3) (1 - e‘t/YU) , (6.4) TL(t) = T; - (T; - TE) (1 - e‘t/YL) , (6.5) t 3 t = 0 , where tS is the time when the plates are switched from the isothermal configuration to the temperature difference con- figuration, T; is the temperature of the plates in the iso- thermal configuration, T: and T: are the steady state values (t = 00) of the plate temperatures and YU and YL are adjustable relaxation parameters determined by curvefitting of eXperimental data (Chapter II). (2) A method of monitoring temperature changes. A sensitive method of monitoring changes Of temperature in the vertical coordinate Of the liquid is necessary. The fringe shift produced by a wavefront shearing inter- ferometer can be used in conjunction with the already determined temperature dependence of refractive index to measure the temperature gradient in the sample cell at the midplane (z = 0). (3) Theoretical description Of the temperature in the liquid. An equation for the spatial and temporal develop- ment of the temperature of the liquid is necessary to analyze the temperature changes. Ingle (1971) has Ob- tained a T(z,t) function that can be combined with our 132 Optical theory to analyze the horizontal fringe shifts in terms Of the thermal conductivity. The method outlined above is called pure thermal conduction because it is the simplest experimental arrange- ment. It requires no calorimetry, nO complex geometrical measurements and, moreover, it is performed in an apparatus demonstrated to be free of convection (Anderson, 1968). Convection in the cell would be manifested by abnormal fringe behavior similar to that shown in Figure 5.2 (c). D. Theory Of the Time Dependent Fringe Shift Ingle (1971, pp. 77-101) developed an equation to describe the temperature Of a liquid as a function Of time and position when the liquid is contained in a rectangular parallelepiped cell such as described in Chapters III and IV and is subject to boundary conditions Eqs. (6.4) and (6.5). Her analysis is based on the perturbation scheme described by Horne and Anderson (1970) for the temperature distribution in a binary liquid undergoing thermal diffu- sion. This approach takes into account the variability with temperature of the transport coefficients. Ingle assumed that (i) the temperature gradient is applied only in the vertical direction with no horizontal effects and (ii) the lateral cell walls are adiabatic so that there is no horizontal heat flux through the lateral 133 walls. Hence, heat and matter flow only in the vertical direction, and there is no steady state convection when the liquid is heated from above (for liquids which expand upon heating). She began with the appropriate transport equa- tions, simplified where possible, and then applied the per- turbation scheme to the set Of partial differential equa- tions describing the transfer of mass and energy in the cell. The temperature equation Obtained is of the form, T(z,t) = TM + To(z,t) + Tl(z,t) + T2(z,t) ... (6.6) where T0 is the result for constant coefficients, T1 is the result when first derivatives of coefficients are retained, T2 is the result when second derivatives and products of first derivatives are retained, and so forth to any order. Each Of the Tn, n = 0,1,2,..., was Obtained by Fourier transforms. Numerical analysis indicated (Ingle, 1971, pp. 98- 101) that T alone is a very good representation of the 0 temperature distribution because each succeeding term in 3 to 10-4 smaller in magnitude the solution Eq. (6.6) is 10— than its predecessor. In addition, T1 was found tO be symmetric about z = 0, so that its derivative is zero at z =’0 for all t and would add nothing to a temperature dif- ference measured about z = 0. Her Eq. (4.20) will be used as T(z,t) for the analysis Of the time dependent fringe shift: 134 T(z,t)=:'I‘M + [(z/h + (1/2)](TU — TM)[l-exp(-t/YU)] + [(z/h) + (1/2)](TM - TL)[1 eXP(-t/YL)] + 21 (-l)n(nn'l{[4n2(YU/T)-11_1(TU-TM) n: x [exp(-t/YU) - eXp(-4n2t/T)] 1 + [4n2(yL/r)-1]' (TM-TL)[exp(-t/YL)-exp(-4n2t/T)]} x sin(2nwz/h) m + ) (2/w)(-1)A(21+1)‘1([(21+1)2(yU/r)-11‘1(TU-T M) x {exp[-(2£+1)2t/T] — exp(-t/YU) - [(22+1)2(YL/r)—11‘1(TM - TL) x {exp[—(22+l)2t/T]-exp(-t/YL)})cos[(22+1)n2(z/h)]. (6.7) The thermal relaxation time T is _ 2 — 2 T — h (pCp)0/K K0 , (6.8) where C? is the specific heat capacity, p is the density and K is the thermal conductivity. The subscript 0 indi- cates the value Of these quantities at T Note that the M' "thermal diffusivity," a = pCé/K, is the quantity determined by the experiment. If values for p and C? are previously known, K can be determined. It was shown in Chapter IV that for experimental purposes, the refractive index is adequately represented by the equation, 135 n(T) = nT=T + né=T (T - T ) . (6.9) M M M If Eq. (6.7) is solved for [T(z,t) - TM] and substituted into Eq. (6.9), the result is a time dependent refractive index spatial distribution in the liquid, _ I ... n(z,t) - nz=0 + nT=TM[T(z,t) T M] . (6.10) We now refer to the Optical theory to predict the inter- ference fringe behavior resulting from the establishment of the time dependent refractive index distribution, Eq. (6.10), in the liquid. The Optical equation, Eq. (4.7), is X' = (X/G) = (a/A)[n(z+8) - n(z-8)] . (6.11) If we consider the case of horizontal fringe shifts, X6 measured as a function of time for the midplane (z = 0) Of the cell, Eq. (6.11) may be written, X5(t) = (a/A){AT=TM[T(B' t) — T(—B, t)]} . (6.12) where Eq. (6.10) has been solved for z = 0 and substituted into the difference expression Of Eq. (6.11). Eq. (6.12) may be used to analyze the time dependent fringe shift data Obtained experimentally as described in Chapter III. The explicit Optical equation, Obtained by substituting Eq. (6.7) into Eq. (6.12), is a very long and complicated expression. A Fortran subroutine, TEMP, was written to evaluate Eq. (6.7) for any given values Of 8, t, 136 YU' YL' TU, TL, TM and T. The infinite series were auto- matically truncated in each calculation using the criteria discussed by Ingle (1971, p. 94). This subroutine was sub- sequently used with a special subroutine EQN for the general curvefitting program KINET to determine thermal conductivity from curvefitting of fringe shift vs. time data. These sub- routines are listed in Appendix B. E. Experimental Results for Water and CCl4 Fringe shift vs. time data measured at (z = 0) were available from refractive index temperature dependence ex- periments vand u (see Tables 4.4 and 4.6) and these data were analyzed as an experimental test of Eq. (6.12). A typical set of data are listed in Table 6.1. The variance Of the time and fringe shift values for each data point were estimated from the reading error on the strip chart recording. The fringe maxima and minima were not as clearly defined in the final stages of the experiment as in the early stages, so the variances are larger for longer times. This is because the fringes were "slowing down" and consequently the recorder pen stayed at a maxi- mum value for a longer period Of time (see Figure 3.3)- One major experimental difficulty was encountered. During some of the runs the photo transistor was not exactly in the center Of a fringe minimum at-the start of Table 6.l--Fringe shift data for experiment 0-4. 137 Data Point Time(sec) 0: X'(fringe number) Oi, l. 0.0 0.360 0.00 0.010 2. 19.1 0.014 0.50 0.010 3. 23.4 0.014 1.00 0.012 4. 27.8 0.014 1.50 0.010 5. 31.6 0.014 2.00 0.010 6. 36.7 0.014 2.50 0.010 7. 41.3 0.014 3.00 0.010 8. 47.2 0.014 3.50 0.010 9. 53.2 0.014 4.00 0.010 10. 62.3 0.032 4.50 0.010 11. 72.4 0.130 -5.00 0.014 12. 89.2 0.360 5.50 0.023 13. 114.1 0.706 6.00 0.026 14. 194.8 5.760 6.50 0.032 15. 402.4 0.032 6.67a 0.040 aThis number was Obtained from the steady state measurement Of the fractional fringe shift, see Table 4.4. an experiment. This was due to movement Of the fringe pat- tern, variations in the phototransistor voltage amplifier characteristics and changes in the laser intensity. Con- sequently, these runs exhibited gross errors in the cal- culated thermal conductivity. They are labeled with an (*) in the tables of experimental results and are not in— cluded in the calculation of the final average or the ex- jperimental standard deviation. 138 Experimental results for 15 trials on water from the data of experiment 0 are listed in Table 6.2. Note that the and YL values are relatively constant. Three YU sets Of results are listed corresponding tO the number of data points included in the curvefitting. This number of data points corresponds to a starting time, t for start' the first data point. The values of K are different for the three sets. The early time data should not be included in the curvefitting Of the fringe shift data for two a priori reasons: (1) the perturbation scheme's higher order con- tributions would be largest at the start Of an experiment, and (2) the initial temperature change Of the plates is probably not completely exponential as described by Eqs. (6.4) and (6.5). Because Of this we would not expect Eq. (6.12) to fit the very short time data points. There was experimental evidence for choosing the data set with N = 11, tstart = Blsec: Figure (6.1) is a plot of the average standard deviation of the thermal con- ductivity calculated by KINET for each set of data. This number indicates the "goodness Of fit" Of the data with Eq. (6.12). It is a minimum for the data with N = 11; 112;! for data beginning with points at t = Blsec. Ap- parently there are two Opposing influences on the good- ness Of fit: (1) the inclusion of early time data points is not theoretically allowed and (2) fitting only the long 139 .Ummmm I uumum 0 .ma 2 .oonem u shapes .oa n 26 .oonam u humans .Ha n 26 u mucwom want no Hwnfiszn .mmma .AGOHUHGO nuvv .mUHmNnm flaw Nuumflfimsu MO xoonflammv Ummmzm 6H.a me.oo no.a mo.oo o~.a ~a.eo I- I- .ma.5 oa.a ma.oo oo.H em.oo ma.o am.am m.om m.am «Hue ea.a mm.ao oo.a om.mo 6H.H Hm.eo e.aa ~.am .ma-» ee.o mm.mo me.o am.~o ea.a ma.oo H.me ~.am «a-» ea.o oa.mo om.o Ho.vm aa.o mo.mo o.ea e.am .HH-» o~.a em.om mo.a me.om ea.o oo.om 5.5. e.mm .oaa mm.o am.mm oe.o ma.mm am.o ma.mo H.~a o.am a-» mm.o mo.mo ae.o mm.mo am.o mm.ao m.aa o.am m-> ao.a oo.oo aa.o am.oo oa.o ma.am a.am m.am hue Ha.o m~.om oo.o aa.oo om.o me.am e.ae 6.0m 6-5 em.o eo.~o me.o mm.ao om.o ma.om 6.56 m.om m-» ae.o em.ao 56.0 ma.oo am.o em.oo m.aa m.om a-» om.a mm.mm mo.a ee.mm ma.o aa.em m.Ha m.om «m-» oo.a mo.mo mm.o ao.mo Ho.a mm.~o I- I: .~.9 NH.H om.am ma.o oa.am ma.o am.om «.ma m.om a-» Noaxsm moaxommu moaxum oaxomMs moaxsm moaxomMs 16065:» 160655» oH .mmo mm ans 5.6 o oaea.e up n.80 a eeoeaa.o u . he H .HIM Hlumm :8 h .umum3 mo >DH>HuOsocoo Hmfiumsu How muHSmmH Hmucmfifiummxm mo MHmEESmII~.m magma 140 Figure 6.l--"GOOdness Of fit" plot for thermal conductivity fitting Of data from experiment 0. 141 8.3 Eon 28 E... .6 .6: 06¢ 0.0m 0.0N 0.0. omd V A 9 J D D 00.. a 80v N O .V o _ ._ 142 time "flat” end Of the curve also produces a poor fit. Further evaluation of this process is necessary before definitive experiments can be performed. Figure 6.2 is a plot of the data from experiment v—4 and the calculated curve from KINET. The average for nine acceptable determinations Of the thermal conductivity of water at 25°C is, H20 r . = 61.1 x 10‘23 m‘l sec"l K"l , ‘ (6.13) 25 7 8K = 1.26 x 10‘2==2.06% K . This number compares well with the value, 60.74 x 10"2 J mml sec-1 K-l, recommended by McLaughlin (1964) from a least square fit Of the results Of several workers. Our value is quite close to that determined by Challoner, 2J“m-lAsec719KIl.. However, the et a1. (1958), 61.2 x 10‘ experimental precision Of our method reflected in the 2% scatter is large compared to the 0.5% to 1% precision claimed for the steady state and hot wire experiments. Experimental results for 13 trials on CCl4 from the data of experiment 0 are listed in Table 6.3. Only data collected after t = 3lsec were used in the curve- fitting.' Note that values of YU and YL contain more scatter compared tO the water experiment. This is probably due tO the difficulty involved in maintaining the smaller AT required in the case of CCl4 to prevent excessive ray bending. 143 Figure 6.2--Plot of fringe shift at,z =‘0 as a function of time. The circles are the experimental points from v-4 and the solid line ( ) is the least squares curve calculated from KINET. 144 g 8mm. m2: 089. comm coon comm coo... com. 080. ooh - - _ q . . d O.N 145 Table 63.--Summary of experimental results for thermal conductivity Of C014, J m-1 sec-l K-l. a -3 -l -l p = 1.584 g-cm Eb = 0.854 J g K 25° P250 ID YU(sec) YL(sec) K25°Xl0. onlo u-2 42.3 30.6 ‘1.156‘ 0.023 '3! 1.1-4 33.4 30.5 1.110 0.014 1 g u-S 39.4 30.4 . 1.168 0.024 u—7 39.7 29.2 1.213 0.018 ;"*E (1-8 36.0 31.0 1.165 0.013 J u-9 38.3 31.4 1.176 0.015 u-lo 37.4 28.6 1.168 0.014 u—ll 36.2 27.9 1.139 0.008 u—l2 39.1 29.1 1.175 0.014 p—l3 31.2 30.1 1.067 0.007 u-l4 34.4 28.4 1.041 0.016 u-lS 36.1 31.4 1.135 0.012 aWood and Gray, 1952. 7‘? Touloukian, 1970.’ The average for 12 acceptable determinations of i g the thermal conductivity Of CCl4 at 25°C is, H CCl4 225, = 1.143 x 10'13 m‘l sec‘l K"l , (6.14) 0.049 x 10"1 = 4.3% K . A O' K 146 This result can be compared to the recommended 1J m-1 sec.l K-l, Obtained by Touloukian value, 1.04 x 10‘ (1970) from a weighted least squares fit of the "best" data. Our value is about 10% higher than this accepted value. The thermal conductivity of CCl4 has been a source Of controversy for many years. Compilations by Tyrrell (1961) and Tree and Leidenfrost (1969) show that values 1 l -1 -1 as high as 1.5 - 1.6 X 10- J m- sec K have been de- termined over the years. Woolf and Sibbitt (1954) Obtained 1J m-1 sec.1 K"1 a value Of 1.14 X 10_ in a series Of experi- ments on different liquids during which they Obtained cor- rect values for the thermal conductivity of water. In view Of these divergent results, CCl4 was a poor choice as a standard to test a new method. Toluene would have been a much better choice (McLaughlin; 1964, 1971). Again the precision Of our determinations is worse than the normal scatter associated with the classical methods. F. Discussion The experiments discussed in section E indicate that the method Of pure thermal conduction is a viable technique and warrants further investigation and develop- ment. Because of the small number Of experiments and in view Of the experimental scatter, the specific results for water and carbon tetrachloride must be considered as 147 preliminary and the determination of definitive values postponed until the technology Of the experiment is Opti- mized. The National Bureau of Standards recently (NSRDS News, March 1972) listed criteria for use in the critical evaluation of thermal conductivity experiments: 1. A direct experimental assessment of radiative losses. 2. Experimental proof of the absence of convection. 3. A discussion of parasitic conduction and of the efforts made to estimate its magnitude and correct for it. 4. A discussion of the temperature-gradient measure- ment including specification of the size of the temperature difference and a discussion of the relation of the measured temperature difference and the gradient in the fluid. 5. A discussion Of the method Of measuring heat flow and its accuracy. 6. Experimental confirmation that the measured thermal conductivity is independent Of the magnitude Of the temperature gradient (Fourier's law). 7. The determination of the geometrical constants Of the system. 8. The geometry of the temperature field. 9. Accommodation coefficients. 10. If the experimental method is a relative method, the calibration and proof Of validity Of the method. 11. The purity and composition Of the sample. 12. Specification of the state variables, including the temperature, at the position in the cell at which the thermal conductivity is measured. Although this list was intended for the steady state experi- ments discussed in section B, we note: (a) numbers 2, 3, 4, 6, 7, 8, 11 and 12 are automatically or easily satisfied by our method, (b) numbers 5, 9 and 10 do not apply and (c) criterion number 1 requires further investigation for our method. 148 Experimental improvements would include all Of the modifications discussed at the end of Chapter IV, especially digitalized data collection. In addition, there are some suggestions in the literature (Tyrrell, 1961; McLaughlin and Pittman, 1971) that a smaller cell height would be advantageous. A cell height in the 4-5mm range would tend to eliminate any possibility of a horizontal heat flux. Bartelt (1968) has considered this problem in some detail for a pure thermal diffusion experiment. Finally, a critical review Of the development Of Eq. (6.7) should be made in order explicitly to consider the problem of radiation energy transfer and metal—liquid boundary effects. BIBLIOGRAPHY BIBLIOGRAPHY Amey, R. L., "The Electromagnetic Equation Of State Data," Pure Dense Fluids, H. L. Frisch, Ed. (Academic Press, New York, 1968). .' -‘J . Em; Anderson, T. A., Ph.D. Thesis, Michigan State University, 1968. Andréassor, S. P., S. E. Gustafsson, and N. O. Halling, J. Opt. Soc. Am., 61, 595 (1971). Bartelt, J. L., Ph.D. Thesis, Michigan State University, £;"' 1968. Bates, O. K., Ind. Eng. Chem., 25, 431 (1933). Bauer, N., K. Fajans, and S. Z. Lewin, Technique Of Organic Chemistry, Vol. I., A. Weissberger, Ed. (WiIey (Interscience), New York, 1958). Beck, J. V., Parameter Estimation in Engineering and Science, Preliminary Edition (Mich. State Univ., E. Lansing, Mich., 1972). Becsey, J. G., Gene E. Maddox, Nathaniel R. Jackson, and' J. A. Bierlein, J. Phys. Chem., 24, 1401 (1970). Bevington, P. R., Data Reduction and Error Analysis for the Physical Sciences TMcGraw Hill Book Co., New York, 1969). BOttcher, C. K., Theory of Electric Polarization (Elsovier Publ. Co., Amsterdam, 1952). Briggs, D. K. H., Ind. Eng. Chem., 49, 418 (1957). Bryngdahl, 0., Acta Chem. Scand., 11, 1017 (1957). and S. Ljunggren, J. Phys. Chem., 64, 1264 (1960). , Arkiv. Fysik, 31, 289 (1961). , J. Opt. Soc. Am., 53, 571 (1963). , J. Opt. Soc. Am., 59, 142 (1969). 150 151 Challoner, A. R., H. A. Grundy, and R. W. Powell, Proc. Roy. Soc. (London), A245, 259 (1958). Coumou, D. J., E. L. Mackor, and J. Hijmans, Trans. Faraday Soc., _2, 1539 (1964). Deming, W. E., Statistical Adjustment of Data (John Wiley and Sons, Inc., New York, 1943). Efroymson, M. A., Mathematical Methods forgQigitalComputers, Vol. 1, A. Ralston and H. S. Wilf, Eds. (John Wiley and Sons, New York, 1960). Fourier, J. B. L., Théorie Analytique de la Chaleur (Paris, 1822). Gustafsson, S. E., J. G. Becsey, and J. A. Bierlein, J. Phys. Chem., 42, 1016 (1965). Hamilton, W. D., Statistics in Physical Science: Estimation, Hypothesis Testing, and Least Squares (Ronald Press Co., New York, 1964). Hammond, B. R.. and R. H. Stokes, Trans. Faraday Soc., _1_, 1641 (1955). Horne, F. H. and T. G. Anderson, J. Chem. Phys., _5__3_, 2332 (1970). , J. Chem. Phys., §§, 2831 (1971). Ingelstam, E., J. Opt. Soc. Am., 41, 536 (1957). Ingle, S. E., Ph.D. Thesis, Michigan State University, 1971. Langer, D. W. and R. A. Montalvo, J. Chem. Phys., 42, 2836 (1968). Longsworth, L. G., Ann. N. Y. Acad. Sci., 44, 211 (1945). , Ind. Eng. Chem. (Ana1.),‘l§, 219 (1946). , J. Am. Chem. Soc., 42, 2510 (1947). , J. Am. Chem. Soc., 14, 4155 (1952). , J. Phys. Chem., 44, 1557 (1957). Looyenga, H., Molecular Phys., 3, 501 (1965). 152 McLaughlin, E., Chem. Rev., 44, 389 (1964). and J. F. T. Pittman, Phil. Trans. R. Soc. (Lon- don), 270A, 557 (1971). Mitchell, M. and. H. J. V. Tyrrell, J. Chem. Soc., Faraday Trans. II, 44, 385 (1972). Nicely, V. A. and J. L. Dye, J. Chem. Ed., 48, 443 (1971). Olson, J. D., T. G. Anderson, J. L. Bartelt, and F. H. Horne, American Chemical Society National Meeting (September, 1970). Parratt, L. G., Probability and Experimental Errors in Science (John WiIey and Sons, New York, 1961). Svennson, H., Opt. Acta, 4, 25 (1954). and R. Forsberg, Opt. Acta, 4, 90 (1954). , Opt. Acta, 4, 164 (1956). Tilton, L. W. and J. K. Taylor, J. Research NBS, _2_9_, 419 (1938). Timmermans, J., Physico—Chemical Constants of Pure Organic Compounds (Elsevier, New YOrE) 1950). , Physic97Chemical Constants Of Binary Systems, Vol. I (Interscience, New York, 1959). Touloukian, V. 8., Ed. Thermophysical Properties Of Matter (Plenum Press, New York, 197’). Tree, D. R., and W. Leidenfrost, "Thermal Conductivity Measurements of Liquid Toluene," Thermal Conduc- tivit , Proceedings Of the Eighth Conf., Ho and Taonr, Eds. (Plenum Press, New York, 1969). Tyrrell, H. J. V., Diffusion and Heat Flow in Liquids (Butterworths, London, 1961). Wallin, L. E., J. Chem. Phys., 23' 552 (1970). and K. Wallin, Opt. Acta, 41, 381 (1970). Washburn, E. W., Ed., International Critical Tables Of Numerical Data (McGraw-Hill, New York, 1933). 153 Waxler, R. M., and C. E. Wier, J. Research NBS, 67A, 163 (1963). , C. E. Weir, and H. W.-Schamp, Jr., J. Research NBS, 68A, 489 (1964). Weast, R. C., Ed. Handbook Of Chemistry and Physics (Chemical Rubber Pub. Co., Cleveland,gl959). , Ed. Handbook Of Chemistry and Physics (Chemical Rubber Pub. Co., Cleveland, 1963). Wentworth, W. E., J. Chem. Ed., 43, 96 (1965). Wilson, E. B., An Introduction to Scientific Research (McGraw Hill, Inc., New York, 1952). Wolberg, J. R., Prediction Analysis (D. Van Nostrand Co., Inc., Princeton, N. J., 1967). Wolter, H., Handbuch der Physik, Vol. XXIV, S. Flugge, Ed. (Springer Vorlag, Berlin, 1956). Wood, S. E. and A. E. Austin, J. Am. Chem. Soc., _61, 480 (1945). and..J..A. Gray, III, J. Am. Chem. Soc., 24, 3729 (1952). Woolf, J. R. and 7% In Sibbitt, Ind. Eng. Chem., 44, 1947 (1954). APPENDICES APPENDIX A ORIGINAL OPTICAL THEORY OF BRYNGDAHL In the original derivation Of the Optical theory for the wavefront shearing interferometer, Bryngdahl (1963) used the mean value theorem to replace the difference function AV Of Eq. (2.15). We give here a brief summary of his treatment for comparison to the results of Chapter II. The equation number from his original article is given after the Appendix equation number. Bryngdahl uses the mean value theorem: If a i X i b , (f(b) - f(a))/(b-a) = Af(x)/Ax = f'(a + eAx) for some 0 , 0 i 8 i l , to write Eq. (2.14) as, mrx = 8{V;[mrx + 01(b1/2), mry + 82(mb1/2)]} + m£{V§[mrx + 01(b1/2), mry + 02(mb1/2)]} - (8/b1)[(¢/k) — x‘] + mrx 0(12), (A.l)(27) where -l < 01 < 1 and -l < 02 < 1 Next, Bryngdahl limits consideration to refractive index variations in the y coordinate and expands v; by use Of the mean value theorem to give, 155 156 mrx = m2V'(y) + 82(m22b1/2)V"[mry + 8203(mb2/2)] - (8/bl)[(¢/k) — x'] + mrx0(i2) , (A.2)(29) where -l < 83 < 1 . Since, V(mry) = W(ry) = W(y). V'(mry) = (l/mr)W'(y), V"(mry) = (1/m2r2)W"(y). Eq. (A.2) may be written, mrx = (2/r)W'(y) + (821b1/2r2)w"[y + 0203(b1/2r)] — (l/bl)[(¢/k) - x') + xo(12) . (A.3)(3l) At this point, Bryngdahl substitutes the path length expression, Eq. (2.16), into Eq. (A.3) to Obtain his final working equation: W(y) an(y) . X mrx = A(An/Ay) + B (A.4)(32) where A = (dz/bl) , Ay = (bl/r) and B = - (i/bl)[(¢/k) - x']. Equation (A.4) may be rewritten, x = (al/bi)rAn + B . (A.5) Equation (A.5) is dimensionally incorrect because the quantity (ax/bi) should have the units Of distance. 157 The first quantity in Eq. (A.3), (i/r), does not contain bl so that the subsequent definition Of A in Eq. (A.4) should be, A = (afi/r) . (A.6) APPENDIX B SUBROUTINES EQN AND TEMP These subroutines were used to evaluate Eq. (6.7) during least squares fitting of experimental data from thermal conductivity experiments. KINET was used to per- form the parameter estimation. A program listing and in- structions for use Of KINET can be Obtained from the authors (Nicely and Dye, 1971). SUBROUTINE EQN is the part of KINET that can be modified to the particular user's application. EQN evalu- ates the difference between the experimental fringe shift and the value calculated by use of SUBROUTINE TEMP for a Specific choice Of the thermal conductivity. An iterative procedure minimizes the sum Of the squares Of these dif— ferences. The following pages are a listing Of these sub— routines exactly as they were used in Chapter VI. 158 159 SUBROUTINE EON COMMON KOUNTOITAPEQJTAPE!IWT9LAP9XINCR9NOPT9NOVAR9NOUNK9X9U9ITMAX9 lWTXcTESToIoAVoRESIDvIAR9EPS¢ITYPQXXQRXTYPQDXIIQFOPOFOQFUvPszOTOQE ZIGVAL9XST9TQDTQLQMQJJJOYODYQVECTvNCSTQCONST DIMENSION X(49100)9 U(20)9 WTX(49100)9 XXI4)! FOP(100)9 FO(100)9 F IU(100)9 P(20921)9 VECT(20921)9 ZL(100)9 TO(20)9 EIGVAL(20)9 XST(10 20)9 Y(10)9 DY(10)9 CONST(16) GO TO (293949591), ITYP CONTINUE ITAPE=60 JTAPE=61 WRITE (JTAPE96) FORMAT(* THERMAL CONDUCTIVITY FITTING*) NOUNK 3 I NOVAR=2 RETURN CONTINUE RHO 3 0.9970445 CPBAR = 4.1796 PI 3 3.14159 0F 3 1.0 GAMMAU = CONST(7) SGAMMAL = CONSTIB) DELTU 3 CONST(9)-CONST(11) SDELTL = CONST(11)-CONST(10) CLLT = CONST(1) CLHT = CONST(2) SHEAR = CONST(3) DNNT = CONST(4) 5 = S/CLHT TAU 3 ((CLHT*“Z)PRHO*CPBAR)/((P1992)“U(I)) TIME 3 XX(I) CALL TEMP (TAUvTIME9 SyGAMMAUoGAMMAL9DELTU¢DELTL9TZ) TPLUS 8 T2 CALL TEMP (TAU,TIME9-SoGAMMAUoGAMMALvDELTUoDELTLoTZ) TMINUS = TZ FRIN = (DFPCLLT/(6.328E-05))PDNNT“(TPLUS-TMINUS) FRIN 3 ABS(FRIN) RESID 3 FRIN-XX(2) RETURN CONTINUE RETURN CONTINUE RETURN CONTINUE RETURN END SUBROUTINE TEMP (TAU,TIMEoZOAoGAMMAUvGAMMALvDELTUoDELTL9T2) DIMENSION TERM(150)9TEST(150) IF(TIME.EQ.0.0) GO TO 500 DENOM = ABS(4.0*GAMMAU/TAU - 1.0) IF (DENOM.LT.I.0E-06) PRINT 20 DENUM 8 ABSI4.0*GAMMAL/TAU - 1.0) 200 1005 22 160 IF I DENUM.LT.I.0E'06) PRINT 20 DENOM 3 ABSIGAMMAU/TAU ' 1.0) IF (DENOMoLT.I.OE-O6) PRINT 40 DENUM 3 ABS(GAMMAL/TAU - 1.0) IF (DENUM.LT01.0E-06) PRINT 50 TT 3 TIME/TAU PI 3 3.14159 T2 8 000 ITMAX 3 0 DO 200 I319150 TESTII) 3 0.0 TERMII) 3 0.0 TA 3(DELTU”(1.0 " EXP(-TIME/GAMMAU)) *DELTL*(I.0 - EXP(-TIME/ I GAMMAL)))*ZOA TB 3 0.5*DELTU”(1.0'EXPI'TIME/GAMMAU))-0.5*DELTL§(I.O'EXPI-TIME l/GAMHAL)’ T2 3 TA ’ TB NT 3 0 NT 3 NT * I A 3 FLOATINT) AA 3 2003A AA? 3 AA332 DUMSIN 3 1.0/(A‘PI)3(DELTU/(AA2*GAMMAU/TAU-I.0)3(EXPI-TINE/GAMNAU I) ' EXPI-AAZPTTTI 0 DELTL/(AAZ‘GAMMAL/TAU - I.0)*(EXP('TIME/OAMM BALI ' EXP(-AA2“TT))T'SINIAA*PI§ZOA) L 3 NT/Z RND 3 L RNDUM 3 A/2.0 IF (RND.LT.RNDUM) DUMSIN 3 -DUMSIN NMIN 3 NT -1 B 3 FLOATINMIN) BB 3 20038 3' 1.0 982 3 38332 DUMCOS 3 2.0/(PIPER)”(DELTU/(BBZPGAMMAU/TAU - I.0)*(EXP(-HBZ§TT) I-EXP('TINE/GAMMAU)T'DELTL/(BBZ*GAMMAL/TAU - 1.0)PIEXP(’B.2*TT) a-EXPI'TIME/GAMHAL)TTPCOSIBBRPIPZDA) L 3 NT/z RND 3 L RNDUM 3 B/2.0 IF IRND.LT.RNDUN) DUMCOS 3 ODUMCDS TERMINT) 3 DUNSIN 0 DUMCOS IFINT.GT.3) DO TO 1005 T2 3 T2 * TERMINT) IF(NT.E..7) GO TO 3 GO TO 2 CALL TESTI(TERMQTESTOTAQITMAXQNT) NT 3 NT ’ I NTD 3 NT 0 I IFIITMAX.E..I) GO TO I A 3 FLOATINT) AA 3 2003A AA? 3 AA”? DUMSIN 3 I.O/IARPII.(DELTU/(AAZ*GAMMAU/TAU-I.0)*(EXP(-TIME/GAHMAU I) ’ EXP(-AA2*TT)) 0 DELTL/(AAZ’GAMMAL/TAU - I.0)*(EXP(-TIME/GAMM 500 10 20 30 40 50 161 ZAL) - EXP(-AA2*TT))T'SINIAA3P1320A) L 3 NT/Z RND 3 L RNDUM 3 A/2.0 IF (RND.LT.RNDUM) DUMSIN 3 -DUMSIN NMIN 3 NT -1 B 3 FLOAT(NMIN) DB 3 2.038 0 1.0 982 3 33332 DUMCOS 3 2.0/(PI*BB)'(DELTU/(882*GAMMAU/TAU - l.O)§(EXP(-DRZ3TT) l-EXP('TIME/GAMMAU))‘DELTL/(33296AMMAL/TAU - l.O)*(EXP(-BMZ*TTT 2-EXP(-TIME/GAMMAL))IPCOSIBB3PIRZOA) L 3 NT/Z ' RND 3 L RNDUM 3 3/2.0 IF (RND.LT.RNDUM) DUMCOS 3 -DUMCOS TERM(NT) 3 DUMSIN 6 DUMCOS A 3 FLOAT(NTD) AA 3 2003A AA? 3 AA“? DUMSIN 3 I.O/(AfiPI)3(DELTU/(AA2*GAMMAU/TAU-l.O)*(EXP(-TIME/GAMMAU I) - EXPI-AAZ’TT)) + DELTL/(AA236AMMAL/TAU - l.0)*(EXP(-TIME/OAMM ZAL) ' EXPI-AAE’TT))1*SIN(AA3PI*ZOA) L 3 NT/Z RND 3 L RNDUM 3 A/2.0 IF (RND.LT.RNDUM) DUMSIN 3 -DUMSIN NMIN 3 NTD - l I 3 FLOAT(NMIN) BB 3 2.038 * 1.0 932 3 88*”? DUMCOS 3 2.0/(PI‘HB)3(DELTUI(BBZ‘GAMMAU/TAU - 1.0)3(EXP(-IBZ*TT) l-EXP(-TIME/GAMMAU))-DELTL/(BBZ*GAMMAL/TAU - l.0)”(EXP(-DDZ*TT) 2-EXP(-TIME/GAMMAL))1'COSCBB’PI“ZOA) L 3 NT/Z RND 3 L RNDUM 3 8,200 IF (RND.LT.RNDUM) DUMCOS 3 -DUMCOS TERM(NTD) 3 DUMSIN ‘ DUMCOS T2 8 T2 0 TERM¢NT) DTO 3 T2 0 TERM(NTD) CALL TESTZ(TERM9TEST9TZ!DT09ITMAXQNTQNTD) IF(NT.GT.100) GO TO 6 GO TO 22 T2 3 0.0 GO TO I PRINT 10 FORMATIR THE SERIES HAS FAILED TO CONVERGE AFTER 100 TERMS”) FORMATI“ 4.06AMMAU 3 TAU”) FORMATI’ 4.00AMMAL 3 TAU*) FORMATI“ GAMMAU 3 TAU') FORMAT(* GAMMAL 3 TAU“) GO TO 1 CONTINUE 7 101 100 162 RETURN END SUBROUTINE TESTl(TERMoTESTvTAoITMAX9NT) DIMENSION TEST(150)9TERM(150)oSUM(5) NT 3 3 DO I I 3103 S 3 0.0 DO 2 J319I S 3 TERM(J) * S SUMIIT3S TEST(1) 3(TERM(1)/(TA+ SUM(1)))*100.0 TEST(2)3((TERM(I)+TERM(2))/(SUM(2)+TA))*100.0 TEST(3)3((TERM(3)*TERM(2))/(SUM(3)*TA))*100.0 DO 7 13193 TEST(I) 3 ABS(TEST(I)) IF(TEST(1).LT.0.01) GO TO 4 IF(TEST(2).LT.0.1) GO TO 5 IF(TEST(3).LT.0.I) GO TO 6 GO TO 100 CONTINUE GO TO 101 CONTINUE GO TO 101 CONTINUE ITMAX 3 l CONTINUE RETURN END SUBROUTINE TEST2(TERM.TEST9T290T0vITMAXyNToNTD) DIMENSION TERM(ISO)9TEST(ISO) TST 3 0.0 TST 3 ((TERM(NT) ‘ TERM(NT-l))/TZ)*100.0 TEST(NT) = ABS(TST) TEST(NTD) = ABS(((TERM(NT)*TERM(NTO))IDTO)*100.0) TST 3 ABS(TST) IF(TST.LT.O.1) GO TO 1 GO TO 2 CONTINUE ITMAX 3 l RETURN END APPENDIX C PARAMETER ESTIMATION A. Introduction The theory of parameter estimation from experimental data has been considered by many workers (Bartelt, 1968; Wentworth, 1965; Hamilton, 1964; Beck, 1972; Wolberg, 1967). The maximum likelihood estimation method described by Went- worth has been successfully translated into a versatile computer program KINET (Nicely and Dye, 1971). During the course of our research it was often necessary to apply these various curvefitting methods to various experimental techniques (for example, the thermal conductivity eXperiment of Chapter VI). This Appendix gives an outline of the procedure used to obtain parameter estimates. The outline was written for use with KINET but the goals and pitfalls of the procedure apply equally to other least squares methods. The purpoSe of this outline is (i) to suggest a systematic way of approaching the prac- tical application of parameter estimation methods and (ii) to describe some of the problems commonly encountered. 163 164 B. Procedure for obtaining parameter estimates I. Construct a Test Problem A. Select an equatiOn to fit the data.. This can be a theoretical equation of definite form or can be a phe- nomenological approximation like a Taylor's series expan- sion. MULTREG (Anderson, Appendix H, 1968) is useful for choosing among a set of linearly related parameters if only an empirical equation is required. B. Take the selected equation and generate (by hand or computer) a set of test data using an eXperimentally reason- able range of variables and values of the parameters. Put the equation into subroutine EQN of KINET and read in the test data. Use estimates of the parameters determined as you would from an experiment. KINET should give back exact replicas of the parameters with very small standard devia- tions. If this does not happen, there are several possi- bilities: (l) Subroutine EQN may not be doing what you think it is. Computers do exactly what they are told. (2) The equation may be "too non-linear" for the Taylor series in method one of KINET to be valid; that is, the initial guesses of the parameters may be outside the radius of convergence for the truncated series. Try minimization by method two in KINET. (3) Do not attempt to obtain too much information from the data. An equation having a 165 large number of parameters for a small number of data points will possibly produce meaningless (or nonunique) results. C. Add randomly generated errors to the test data to see if KINET can still calculate the true input parameters. Increase the variance until the program no longer gives the correct results. This will give an indication of how well determined the experimental data must be and will help in design of the experiment. II. Transform the Experimental Data A. Reduce the data to a form containing the experimental observations and their variance. Decide on values of the variance in each experimental variable or, if that is not possible, estimate the relative variance of the variables and use arbitrary variances reflecting this ratio. Con- sider alternate estimates of the variance, especially under varying experimental conditions. Avoid combining a number of experimental variables into a derived quantity and using it as the experimental observation because this tends to produce correlations among the data points which may bias the parameter estimates. However, repeated ex- periments of the same type may be combined into a smaller set of data using the "spread" of the data points as a guide for estimating the variance. Finally, obtain esti— mates for the initial guesses of the parameters by a linear approximation of the equation or by other means such as graphing or the values of previous workers. 166 B. Punch data in the prescribed format and read into KINET using the form of subroutine EQN tested in part I. III. Analyze KINET Output A. Examine the parameter estimates. Are they mathe- matically and chemically reasonable? Is there any other external information that can be used to check the reason- ableness of the estimates? A common example of this type of check occurs in chemical kinetics. Rate constants must be independent of the initial concentrations of the reacting species. If data from different initial concentration con- ditions give widely varying values of the rate constant, suspicion should be cast on the rate law even though the equation appears to "fit" the data in each individual case. Compare the parameter estimates to literature values or theoretically derived values. B. Calculate the relative error of each parameter, relative error = |(8U )/Ui| , (C.l) i . ' where 8U is the estimated standard deviation of parameter 1 Ui' Relative errors greater (especially much greater) than one indicate an equation of the wrong form or a parameter which cannot be determined from the experimental data. C. Examine the variance-covariance matrix and multiple correlation coefficients. The multiple correlation coef- ficient is a measure of the total linear dependence of 167 the parameter on all the other parameters. There will usually be some dependence among the parameters. However, as a qualitative guide, multiple correlation close to one may indicate a parameter or parameters that cannot be de- termined uniquely from the data. In this case a different equation (or data) should be considered. Bevington (1969) and Hamilton (1964) discuss tests for the statistical sig- nificance of the correlation coefficients. D. Perform "Goodness of Fit" Tests. It is often neces- sary to know how well a particular equation "fits" the data. This can provide experimental evidence that one theory is "more likely" than an alternate. Several tests of this type are available. In addition, the quality of the pa- rameters as discussed in sections A and B would also in- dicate a "goodness of fit" for the data. Three methods are generally used to test "good- ness of fit": (1) The randomness of the residuals. (2) The fraction of data points lying outside of a statistical confidence limit interval constructed from the parameters and their estimated errors. (3) The x2 test. (1) The sign of the residuals should be distributed randomly among the data points. If not, this indicates a poor choice for the fitting equation. See Wilson (1952). (2) A confidence interval can be constructed about the derived line using the estimated variances and co- variances for the parameters, the prOpagation of error 168 formula, and an appropriate "t" value. An equation for the construction of .an interval can be found in Wolberg (1967, p. 64). If a large percentage of the data points lie outside the constructed interval, the validity of the equation may be suspect. Individual data points may also be examined in order to investigate the source of their error. Obvious sources of error such as miscopied data or gross error in measurement should be considered before casting doubt on the equation. (3) Under certain conditions, the x2 test can be ap— plied to the quantity, S/(N - m). S = the minimum sum of the weighted residuals (N - m) = the number of degrees of freedom; N = experimental observations m = parameters This test applies rigorously only when the true variances of the observations are known. The method of application is discussed in Wolberg (1967, p. 67). A x2 table lists the probability that S > x2 for various values of (N — m). Two different equations could be tested with the x2 dis- tribution and the ratio, (SI/$2), may constitute an F test for statistically significant difference between the two theories. At the completion of these tests, the investigator may have evidence that the "goodness of fit" is not satis- factory. As Wentworth (1965) states, "he may (1) re-examine 169 his experimental procedure and techniques for bias errors not properly taken into account, and/or (2) reinvestigate the problem from a theoretical standpoint to find a more representative function."