fl ’ WWW/[WU l J _ R ‘_ — : _ THS_ VISCOSITY OF THE SEPARATE-CD [gOTQPES OF EWHLUM The“: €02 he Degree 0? M. S. 222 22222”; 2”? 2”” L riaVlH-fiz" U ’ Challes RE. Ra .ciaH 1962 ms 1111111111!flflflmilflflflflifl'HWIWTflflfilfiflfl 3 1293 01743 0251 ABSTRACT VISCOSITY OF THE SEPARATED ISOTOPES OF MOLTEN LITHIUM by Charles M. Randall In previous work at Michigan State University the viscosity of molten Li6 and Li7 was determined as a function of temperature. The dependence of viscosity on temperature followed an exponential law, in conformity with the finding for most liquids. The dependence on isotopic mass, however, was much stronger than predicted by simple theories. In the present work, a careful repetition of the experiment failed to give results significantly different from those obtained earlier. Hence the theoretical basis was re-examined. Fundamental theories of the liquid state are not sufficiently well established to permit mean- ingful predictions on the effect of isotopic mass, but some semiempirical laws are available that predict variation as the square root of the mass. Earlier dimensional-analysis arguments had indicated this same type of dependence. A complete treatment, however, shows that the square- root factor must be multiplied by a complementary function depending on the mass through quantum effects. Hence, a strong dependence on mass is not inconsistent with the results of dimensional analysis. In fact, in the only other unequivocal study on the effect of isotopic mass (with liquid hydrogen and deuterium) a similar anomaly is observed. It appears that further progress will depend on having many more measurements of the viscosity of liquids as a function not merely of temperature and pressure, but also of isotopic mass. VISCOSITY OF THE SEPARATED ISOTOPES OF LITHIUM BY Charles M. Randall A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Physics and Astronomy 1962 . \~‘ .3 I“ I; fii \s’ /‘; .I’ \~_: .pn ' . . p , ‘ . ’ ”v / l”~" ;. Y ' 1 L .1 .74‘ .I V. AC KNOWLEDGMENTS It is my pleasure to express my sincere appreciation to Doctor D. J. Montgomery who suggested the problem and who has guided the work and has been a source of encouragement as it has progressed. His wide knowledge, patience, and the unstinting contribution of his time are all gratefully acknowledged. 2 Thoughll have never personally meet Doctor N. T- Ban I would like to express my appreciation for the careful work he has done on the experimental method. . His carefully-kept records have made my work much easier. Mr. Frank Taylor has taken most of the data for Li6 and done many of the calculations for both isotopes. . His patient and careful work is certainly appreciated. I would like to express my thanks also to: Mr. Richard Haire of the Department of Chemistry for his assistance in the chemical analysis of some of the samples, and Mr. Jerry Tomecek who has helped with the data processing and drafting. Major financial support has come from the United States Atomic Energy Commission. More recent support has come fromthe National Science Foundation through a Cooperative Graduate Fellowship. To these sources of support I am grateful. The most important source of encouragement during this work has been my wife. In addition to the encouragement she has continually offered, she has taken time from her already crowded schedule to help with the material production of this thesis through typing of rough drafts and lettering the captions on the figures. For this help I am extremely grateful. ************ ii TABLE OF CONTENTS CHAPTER I. II. III. IV. V. INTRODUCTION . . .................. EXPERIMENTAL METHOD . . ........... Choice of Method . . . ............ Design of Apparatus ............ Observation of Oscillations . . . . ...... Measurement of Sphere Radius . . ...... .. Measurement of Moment of Inertia ...... Temperature Control and Measurement Calculation . . . . ............... PRWF’PP?’ SUMMARY OF PREVIOUS WORK ........... A. Theory . . ................... B. Results ................. 1) Parameters of the Apparatus. . . . 2) Results of Viscosity Coefficient . . . . . RE-EXAMINATION OF EXPERIMENT . ...... A. Analysis of Individual Measurements . . 1) Period . ....... . ...... . . 2) Logarithmic Decrement ........ 3) Radius . . . . ............ 4) Moment of Inertia ........... 5) Density .................. 6) Temperature ........... - . . B. Repetition of the Entire Experiment CONCLUSION . . . ..... ' ........... REFERENCES CITED . ................. APPENDICES......... .......... iii 16 16 17 18 21 21 21 22 22 27 27 27 27 28 37 37 37 39 49 57 59 TABLE II. III . IV. VI. VII. LIST OF TABLES Viscosity of Lithium-6 (Ban) Viscosity of Lithium-7 (Ban) . . . . . . . . . . . . . Decrement at Constant Temperature . . ..... Chemical Analysis of Lithium-7 After Measurements Determination of Moment of Inertia (Lithium-6) . . Viscosity of Lithium-6 (Randall) ..... Viscosity of Lithium-7 (Randall) . . ....... iv Page 23 23 29 38 42 43 LIST OF FIGURES FIGURE 1. General View of Apparatus .......... 2. Lithium Sample Holder . . . . . ....... 3. Photograph of Experimental Area . . . . . . ..... 4. Schematic View of Observation System 5. Counter Circuit . . . . . . ........ . ..... 6. Ektron Detector Amplifier Circuit . . ....... 7. Typical Record of Photocell Signals 8. Viscosity as a Function of Temperature (Ban). . . . 9. Loglo n as a function of l/T (Ban) . ...... 10. Height of Fluid as a FunCtion Of Cumulative Volume . . 11. 3V/Trh7‘ versus Height for Water . . ..... 12. 3V/1rhZ versus Height for Mercury. 13. Logarithmic Decrement Near the Melting Point (Li6) . 14., Logarithmic Decrement Near the Melting Point (Li?) . 15. Viscosity of Li6 as a Function of Temperature (Randall) l6. Viscosity of Li7 as a Function of Temperature(Randall) l7. Loglo n as a Function of l/T (Randall) 18. Viscosity as a Function of Temperature (Average Values). ........................ l9. Dimensional Analysis Model . . ......... 20. Temperature Calibration Curves . ......... Page 10 12 14 15 24 26 34 35 36 4O 41 45 46 47 48 51 62 LIST OF APPENDIC ES APPENDIX Page A. Determination of Moment of Inertia ...... . . 60 B. Temperature Calibration . ..... . ...... 61 C. Viscosity of Lithium-6 Data and Lithium-7 Data (Ban) ....................... . 63 D. Viscosity of Lithium-6 Data and Viscosity of Lithium-7 Data (Randall) ............. 65 E. Viscosity of Water . ................ 69 vi CHAPTER I INT R ODU CT ION When the existence of stable isotopes was established nearly a half century ago, it was immediately realized that isotopic mass could be utilized in studies of matter in the aggregate (1). Unfortunately the enrichments necessary to get meaningful results were not available until about 1945. Then high-current mass spectrometers left over from war-time atomic energy projects found utilization; the United States Atomic Energy Commission in particular instituting a program of supplying separated stable isotopes of most of the elements. Among the properties of liquids, viscosity appears to be a para- meter which would depend on the isotopic mass in some strong but non-trivial manner, for viscosity must be controlled both by the frequency of atomic vibration (depending on the square root of isotopic mass), and by the amplitude of atomic vibration (determined by quantum phenomena affected by isotopic mass). For simplicity of interpretation, a monatomic liquid would be best. For a large relative mass difference an element of low atomic number must be chosen. Helium (He3, He‘) forms a liquid with very special properties, and hydrogen (H1, H2) forms a molecular liquid; but both of these condense only at inconveniently low temperatures. Lithium (Li6,. Li?) forms a monatomic liquid at a convenient temperature (melting point,~ 180°C). In addition the separated isotopes of lithium are available in gram quantities at reasonable prices. . The next several elements in the periodic table are unsuitable for one reason or another. Because of its high specific heat, lithium has been suggested as a coolant for nuclear reactors both fission and fusion. Hence the viscosity of the separated isotopes has considerable technological interest. Values for the viscosity of natural lithium (7.4% Li", 92. 6% Li7) have been reported, but the values remain uncorroborated (2). The results, then, from any investigation on the viscosity of the isotopes of lithium would have technological as well as scientific interest. Such measurements on the separated isotopes of lithium were made at Michigan State University by Dr. N. T. Ban. (3). The behavior of each isotope singly was consistent with the usual laws found for the temperature dependence of viscosity, but the comparison between the isotopes showed that the dependence on isotopic mass was much stronger than had been expected. The theory of liquid metals is not highly developed, and hence it is uncertain just how reliable are the theoretical predictions. Furthermore, the experiment is difficult, in view of the small amounts of lithium available and its high chemical reactivity. The dependence on isotopic mass is a critical test of any fundamental theory of viscosity, and it is important to assess carefully the reliability of the experimental results on lithium, as well as the validity of the theoretical deductions. The purpose of the present work is to repeat some of the measurements made by Ban, searching for possible sources of error and evaluating‘thei’r” magnitude; and to re-examine some of the theoretical arguments put forth, to see if experimental results are truly discordant with theory. CHAPTER II EXPER IMENTAL MET HOD A. Choice of Method There are a number of experimental methods available to measure viscosity, each having its characteristic advantages and disadvantages. These methods are explained in detail in several books (4, 5). Common methods are: flow through a capillary, flow through an orfice, rise of a bubble in the fluid, descent of an object in the fluid, and drag between rotating concentric cylinders or cones. Because molten lithium is extremely reactive (6) and the separated isotopes are available only in limited quantities none of the conventional methods are useful. . In view of these considerations Ban adopted a method consisting of measuring the damping of the free oscillations of a torsion pendulum containing a spherical cavity filled with the liquid, as originally developed by Helmholtz and Piotrowski in !11816. ~-The damping is related to the viscosity by a mathematical formula allowing an absolute determination of the viscosity. 1 Although. original workers had difficulty with their mathematical analysis and experimental technique, the method has been improved through the years (7, 8) so that it will give reliable results, . In particular Andrade and Chiong (9) have developed a method of simplifying the calculations. . In the analysis, the fluid is assumed to move slowly in concentric spherical shells around a vertical axis, with no vertical component of motion. The major damping effect on the pendulum is assumed to be due to the viscous drag of the fluid at the liquid-sphere interface. . From the hydrodynamical equations of motion, the damping force is shown to be proportional to the time rate of change of the angular displacement 9 about a vertical axis. The proportionality constant L, is a complex number; however, it is still possible to describe the motion with the familiar second-order differential equation of the damped free harmonic oscillator: J . Ie+Le+Me=o (1) Here I is the moment of inertia of the oscillating system, and M6 is the restoring torque. When. this equation with the damping force as determined by the hydrodynamical equation is solved, the following expression for the coefficient of viscosity 1) is obtained in terms of measured quantities: aZszp 11 n: m [1-(1—u)T]T, where 2 us 3(2-§)15 ,_[%2 +1], (2) 21tapR gR-l (gR-l)z:(hR)z ’ .0 III a= 1- 6/417 + 52/3223, 1 g ‘= awn/T n)’. h = (up/Tn) (1+ 5/4n +5Z/32n2). The experimentally-measured quantities are: I = moment of inertia of oscillating system R = inside radius of sphere p = density of fluid 6 = logarithmic decrement of pendulum T = period of pendulum with damping present To = period of pendulum with damping absent The experimental set-up consists of a torsion pendulum oscillating in a vacuum, so that the major damping force is the viscous drag of the molten lithium. With the amounts of lithium available, this damping force is small, and consequently introduces special complications in its measurement. We now review the experimental apparatus in some detail, and see how each of the above parameters is measured. Later each type of measurement will be re-examined in an attempt to find possible sources of errors. B. Design of Apparatus The torsion pendulum is similar to the one described by Andrade and Chiong (9). The present apparatus was designed and constructed under the supervision of N. T. Ban. The details of the torsion pendulum are shown in Figure l. The head, A, which is the top support for the bifilar tungsten suspension, passes through the top of the vacuum chamber, B. 2 It is sealed in place by two O-rings, and can be rotated about a vertical axis to bring it to any desired location, or to start the pendulum oscillating. The top of the vacuum chamber can be leveled with three screws. The entire pendulum is enclosed in a vacuum chamber, H, connected to a conventional vacuum system consisting of a diffusion pump and a mechanical fore pump, along with the associated cold trap, valves, and gauges. Brass bellows have been installed part way down the vacuum enclosure to allow the lower part of the vacuum enclosure to be oriented for proper clearance of the pendulum inside. Near the bottom of the tubular part of the vacuum enclosure two windows are cemented 9Oo apart about a vertical axis. The bottom of the tungsten suspension is fastened to the oscillating system slightly above the level of these windows. At the level of the windows a mirror is fastened to the oscillating system so that the oscillations of the pendulum may be Scale: V4 D L I A ‘s 1’ a a R / zmfié . fir . KZLZAJCZ ,/ H H ' V . / . -—+ To diffusion pump 4 di ’ m: 1. J . J “‘1 5% I l/Jflé '0 j E 0 K N N ‘ N N K N N N N b cjgfie —-— l. 1 1 Water —r A .J m [c 1, :4 I L 1" \\\\1 “ES? '7] FIGURE I GENERAL VIEW OF APPARATUS observed by means of an optical lever. Below the mirror are two masses, 1, whose position may be adjusted to vary the period of the pendulum. A rubber vacuum hose is ”attached to a copper tube soldered into the vacuum enclosure, Q, and then clamped off with pinch clamps. 2 Thermocouples may be inserted in the vacuum chamber through this hose, which is then filled with vacuum grease and squeezed tight with two or three pinch clamps. Below the inertia arms, with their movable masses, the sphere is suspended from along stainless-steel tube, G, which has a number of holes drilled in it to increase the resistance of the thermal path from the sphere. . The chamber wall around the sphere is a hollow cylinder of copper attached by a short piece of stainless-steel tubing to the water jacket, U. A copper-constantan thermocouple is installed in the bottom of the copper block. This thermocouple is used to measure the temperature of the Sphere, and also to actuate the recorder-controller for the temperature of the furnace, . F. A detailed view of the sphere is shown in Figure 2. The sphere is made of ferritic stainless steel (Type 446), which is supposed to be especially resistant to corrosion by molten lithium (6). . Specific con- struction details are given in Ban's thesis. A photographof the experi- mental area is shown in Figure 3. C. Observation of Oscillations fl The measurementson the pendulum oscillations must yield the following data: the period T when theilitliium is liquid, and) thepieriomeo when the lithium is solid, and the logarithmic decrement 6 in both cases. The observation system is shown schematically in Figure 4, and is in principle identical to the system used by Ban. However, the auxiliary equipment has been almost completely rebuilt to increase its reliability and convenience. The Sanborn Millivac recorder used by Ban is no I INCH A__-‘ V 7’ / 45° BEVEL .039 x .039 IN. FOR GOLD O-RING / l2 / ’ / / , // /l / , , / 9 a E ; GOLD O°RING ,2 E ‘ZEV a; E? 27 j / . 2 / ‘ / ’ A / / HARD 4 SILVER SOLOER Tel? IN. THICK SLEEVE FIGURE 2 LI SAMPLE HOLDER MATERIAL‘ STAINLESS STEEL TYPE 446 OI .pcsohmoue ”Em: paw uopCoo on... 5 one .mBMHMQmm mcfipuoumu paw mzoo orfifimCmmopofiam 9.3... .oomausw mg new .Hmzouucoo 8.85360on 9.3 new mmsmw 8:309, 93 pmommm Em? 2.3 OH. .295 on”. 3 @0303 Mack/madam Hooum a 59G 3303.8». wfiwcmg HonEmao 835$, 05 5 30933 93 mmaodon u: and ”Em: 05 3 Emma Em: 9t muoofloum £033 .moudOm Haw: ofi mg $3 new 93 u< .uGoEomeHum 232852693 mo 33> HmpocmO I m whammh 10 .maoflmfidumo magnumno new 5333 m0 33> oSdEoAum I w oudwfih condom £34 ’ need .330 Em : uopuouom — .2... r .228 :3 AOL III .09me IHdUmuHo> fill vnmum . w III—HUM - ~300qu m .. 40 .M .39 In :93qu Esfispcunm do nouufiz. $5384 A I—_1 10 .mcomudzmomo M5>uumno new 533?... no 33> omumfiosum I v oudmmh ouusom £34 ’ I - need .308 Em H H : novucuum £35m _—I H3550 1— ,llll. 00G$m .4 ”“04 Tfi mofiuo> . 1F endow e . ~300qu m I 40 .h . L «on c933“ Esfisvnom - - no MOPS—z. _ “033954 A 11 longer available, and the apparatus has been modified touse a Brush two-channel recorder available‘from another project. . The oscilloscope and the recorder display the same information. . The oscilloscope allows adjustment of the equipment without wasting chart paper, and the recorder allows a permanent record to be made when data runs are underway. 2 The light source consists of a 10-watt concentrated arc lamp at the focal point of a converging lens. . The resulting parallel beam of light is passed through a slit, an image of which is focussed on the window of the photocell used to measure the logarithmic decrement. One detector is a cadmium-selenide photoresistor (ClairexCL-fi?) placed about halfway between the maximum excursions of the light beam. This cell is connected to the counter circuit shown in Figure 5. The circuit uses the 2D21 thyratron as a grid-controlled rectifier. Passage of the light beam over the face of the photocell causes its resistance to change. The resulting change in current; creates a positive voltage pulse which overcomes the negative bias on the grid of the 2D21 and causes the tube to conduct. The RC-network containing the relay coil, the counter coil, the resistor, R1, and the capacitor, C1, is designed to keep the tube in the conducting state long enough for the counter to be positively actuated and the relay firmly closed. The connector marked "to Scope Sweep" provides a pulse to trigger the sweep of the oscilloscope. The relay controls a second pen on the Brush recorder, and through appropriate latching relays and switches actuates a self- starting electric clock, which is started by one pulse and turned off by a succeeding one. In this way the periods T and To can be determined by allowing one pulse to start the clock and a later pulse to turn it off while the counter registers the number of intervening pulses representing oscillations of the pendulum. The logarithmic decrement is measured by a lead- sulphide photo- resistor (Kodak Ektron detector),having a sensitive area 0. 02 mm by z 1 .0.“ .> mflfi . OIIO/D mooBm omoom OH. Fla m... , x x oma L/\/\/\III._A [Es + $50.30 uuagoo m ondmmh wm 7: II omAI 29o tn .5 cm“ W. q mm he 0% 355.00 AlTéI/bpp: S I.\ _ H xa \ NM j zoo . moo _ 33 3 l—1 x 2 . - <52 13 2 mm. . It is placed in a circuit, shown in Figure 6, which contains a mercury battery serving as a current source for the detector, and a two-stage voltage amplifier with a long time constant. The output of this amplifier is fed to the vertical input of the oscilloscope and to the amplifier of the Brush recorder. The Ektron detector is mounted on a screw carriage to permit positioning of the detector to within 0. 01 mm over a range of about 5 cm. This screw carriage is in turn fastened to a sliding carriage having a vernier scale which likewise may be read to O. 01 mm. This sliding carriage has a travel of about 35 cm, and permits a coarse adjustment of the position over a wide range. The detector is placed near the point of maximum excursion of the oscillating light beam. When the light beam goes beyond the detector, the oscilloscope and the recorder reveal a pulse with two peaks (or even two separate pulses when the overshoot , is large). If the detector is kept at a constant position, the two peaks move closer and closer together as the amplitude decreases. Eventually there will be only one peak observed. (See Figure 7, a copy of an actual record.) This coalescence is taken to indicate the time when the amplitude of the oscillation corresponds to the position of the carriage. The carriage is then displaced a few tenths of a millimeter towards smaller amplitudes, and the process is repeated. From the number of swings between coalescence for a given displacement and from the initial amplitude, the logarithmic decrement may be computed.. In practice the process is repeated 30 to 40 times for each data run, and the results are averaged by a least-squares method to obtain a precise value. These calculations are programmed for MISTIC, the high-speed digital computer at Michigan State University. l4 , . ~12: :IIII «Soho noflfidE/w paw Hon—036n— Gouuvflm I o “:5th I l 20H Hmhm popcoaofl £0.3an I J.“ I y .I . Hi I I IL I... y n I . u . .- l I ”I I. -\e A . *1... II ”III a I I.’ | I a . on. . I -.\v .1- -.| PI 9.. ,I. .. Q I W - I I C ..: .I . e . I\. I. «I: .Ib. II . |\ ID... .1.- n‘s. .. L A V .wI. . . - I A V . .5 VII. 3.: I» Y 15 ._. , . a.» oucuunomdou \ . 323m :uogonm mo vacuum 053. A. '3... Hanna ~300qu cabana 303.3. I N. 0.3th gunman non—5.00 undom 03:0 . 16 D. Measurement of Sphere Radius The samples of Li6 and Li7 for the present experiments were those used by Ban. They were maintained at all times under seal in their spheres, and it was not possible to measure the radius of the sphere without destroying the sample. Hence his values of the radius have been taken for our calculations. Ban weighed the sphere both before and after filling it with water to the top of the neck, and thence computed the volume of the cavity. The volume of the cylindrical neck, as calcu- lated from its geometry, was subtracted from this total volume to give the volume of the spherical portion. E. Measurement of Moment of Inertia The geometry of the oscillating portion of the pendulum is so complicated that a precise calculation of the moment of inertia is not feasible. Therefore an experimental method has been adopted. The moment of inertia is changed by a known amount, and the effect of this change on the period is measured. The moment of inertia of the system can be expressed as, 1 =IO+I', (3) n n where In is the total moment of inertia for a particular position of the movable masses shown in Figure 1, 10 is the moment of inertia of the system not including the effects of the movable masses, and 1:1 is the moment of inertia of the two masses as given by the parallel-axis theroem, I; = Ilz + IZZ + Mldzm + Mzdzm, (4) where 112 and Izz are the moments of inertia of the two masses about an axis through their center of mass and parallel to the axis of rotation of 17 the entire system, and M1 and M; are the masses of the adjustable bodies at distances dm and dzn- The distances are set suchlthat dln = dZn = dn. If the damping is small, the period of the pendulum is given by 1 _ '2" Tn _ 21r(In/M) . Then In = T5, 34/411?- = 10 +1lz + Izz + (M1 + M2) (1;. A similar expression holds for Lm. The two equations are solved for 10 by eliminating M between them. (drnznTnz " dn2 Tmz) 2 z m'Tn I0 = "Ilz ' Izz + (M1 + M2) [ From equations 3 and 4 it is easy to get In in terms of the d's and T's: dmz - anz Tm; - Tn; In -'-'- (M; + M2) Tnz (5) The d's are fixed by means of a Special gauge block which slips over the end of the rods on which the masses slide, and moves the masses to a precise distance from the end of the rod. This rod is centered on the axis of rotation, so that use of the gauge block puts the masses at the same distance from the axis. F. Temperature Control and Measurement The furnace (F in Figure 1) has a 115-v, a-c, heating element wound on a ceramic frame just large enough to fit around the copper block which forms the bottom part of the vacuum enclosure and surrounds the sphere. The heating element is surrounded by a few inches of insulat- ing material. The entire furnace is housed in a steel shell resting on the floor. 18 The temperature is controlled with the aid of a copper-constantan thermocouple (T in Figure 1). To keep the measured potential within the range of the controller, a portion of the voltage is backed off with a potentiometer powered by a l-v, mercury battery. The difference between the thermocouple potential and the backing potential is applied to a Leeds and Northrup Type G Speedomax Recorder. The recorder is in turn connected to a Leeds and Northrup Series 60 Duration-Adjusting Type control unit which controls the current to the furnace by adjusting the fraction of the time the furnace is on. The Speedomax record shows that the temperature is maintained to well within 1; 0. 2. C-deg. The entire temperature-measuring system is calibrated in terms of potential difference measured with a Leeds and Northrup Type K-3 Potentiometer. At each run the temperature of the furnace as measured with the K-3 Potentiometer is recorded. To find the temperature of the sphere, we need a relation between it and the furnace temperature. This calibration is obtained by attaching thermocouples to the surface of the sphere and recording their potential as a function of the furnace temperature. The corresponding temperatures are obtained by use of National Bureau of Standards Tables for COpper-Constantan Thermo- couples (10). G. Calculation The fundamental equation (2) cannot be solved explicitly for the viscosity. The solution is obtained by iteration to the desired degree of accuracy. . In the present experiment the process is continued until successive approximations differ by less than 10”, a figure well beyond the limitations of the experiment. All the calculations and corrections are included in a single program for the calculation of viscosity on MISTIC, . The specific correction equations are now examined: 19 1.. Moment of inertia: Ban has calculated that any change due to temperature is not significant. 2. Radius of the Sphere: Since the equation for calculation of viscosity is very sensitive to small changes in the Sphere radius, thermal expansion of the sphere is taken into account to the first-order term: R = R20 [1+ d(t - 20)], where R is the radius at 200 C, o. is the coefficient of linear expansion (9. 9 x 10'6/C-deg. for type 446 stainless steel) and t is the temperature in degrees Centigrade. 3. Density: There are no values published for the density of separated isotopes of lithium as liquids. . It is reasonable to assume that the atomic volume of molten lithium does not change much with isotopic mass, and hence we may deduce the values of density required from the thermal expansion and the change of volume on melting for natural lithium, as given by Bernini and CantOni (11), L'osana (12.), and the hyalues of the bulk density of the solid separated isotopes, as measured by Snyder and Montgomery (13). The densities are then given by the following formulas: _ dLi 1L 7 pLi-é ‘ 1+ (174 x10'6) (t - 180.4) + (106 5210-9) (t - 180.4)Z dLi—7 p1.1-7 : 1+ (174 x10‘6) (t — 180.7) + (106 x 1047) (t -180.7)z . . . . -6 .7 Here dLi-6 and dLi-7 are the den31t1es of liquid L1 and L1 at the melting point: _ .460 [1 + (153.5 x10'6)x 20 + (92 x10'9)(20)21 Li-6 ‘ 1.0157 [1+ (153.5 x 1047) x180 + (92 x10”) (1230):] _ .537 [1 + (153.5 x10‘6)x 20 + (92 x10'9) (20)’-] Li-7 ‘ 1.0157[1+(153.5 x10'°)x 180 + (92 x10‘9) (180)Z_] 20 4.. Logarithmic Decrement: The residual damping occurring when the lithium is solid must be taken into account. Since this damping is small (of the order of one-twentieth of the viscous damping), the corresponding logarithmic decrement is simply subtracted from the total observed decrement. The data presented to MISTIC are in the form of a list of 18 para- meters appearing in the various equations. For each set of data there are in addition the three variables, furnace temperature (expressed in millivolts), logarithmic decrement (dimensionless), and period of the pendulum (seconds). The computer does all the intermediate calculations, and prints out the sphere temperature and the viscosity, as well as the results of intermediate calculations if desired. CHAPTER III SUMMARY OF PREVIOUS WORK A. Theory A critical review of the current theories of viscosity is given by Bondi (14). Ban reviews some of these theories, and shows from simple dimensional analysis that the expected relative change of the coefficient of viscosity 7), at a constant temperature T, for a change 1 in isotopic mass m should go as m7 At least two of the major types of theories reviewed by Bondi (rate-process theory, and molecular theory) predict this same dependence on mass. An exponential form of the temperature dependence of viscosity, suggested many years ago by Reynolds (15) and by Guzman (16), subsequently has been used by many other workers. For a great number of liquids it seems to fit the data quite adequately (17, 18).. The present experimental work is confined to investigating the dependence of viscosity on the two parameters, isotopic mass and temperature. B. Results Ban's experimental results are recalled in two sections. The first section contains values of parameters connected with the experi- mental apparatus and relevant to the experimental work described herein. These values are reproduced here for completeness. For the methods used to obtain the detailed data, the reader is referred to appendexes A and B, and to Ban's thesis. The second section consists of the results obtained by Ban for the viscosity of L16 and Li7 as a function of temperature. 21 22 1) Parameters of the Apparatus: a) Radius of the sphere: L16 sphere 1.2827 cm. _ L17 Sphere 1. 2842 cm. b) Moment of Inertia: Use of three positions of the inertia arms (d1, d2, d3) with observation of the corresponding periods, T10, T20, T30, permits three values of the moment of inertia to be calculated. The average value is taken for I, the moment of inertia. The experimental data needed are the masses, m1 and mg; the distances d1, d2, and d3, from the axis of rotation of the inertia weights; and the periods corresponding to the different positions. , The method of obtaining the masses and the distances is given in detail by Ban. The results are summarized here: m1 = 189.627 grams m2 = 189.629 grams m1+ m2 = 379. 256 grams df = 5.132 cm.2 d3 = 27.716 em.Z d; = 18.183 cm.2 The corresponding periods are given in Appendix A. The resulting values of I are: L16: I = 4443.86 gm-cmz L17: 1 = 4429.41 gm-cm?‘ c) The temperature-calibration curves are shown in Appendix B . 2) Results for Viscosity Coefficient: Ban's results for L16 and 1.17 are summarized in Tables I and II. The source data are found in Appendix C. Figure 8 shows the 23 Table I. Viscosity of Lithium 6 (Ban) 0 Temperature 0 Viscosity n Loglo 17 (mp) 103 C K mp T (0K) 188.3 461.5 4.39 0.642 2.17 198.0 471.2 4.28 0.631 2.12 217.4 490.6 4.00 0.602 2.04 236.2 509.4 3.85 0.585 1.96 236.8 510.0 3.83 0.583 1.96 266.6 539.8 3.69 0.567 1.85 266.8 540.0 3.67 0.565 1.85 Table II. Viscosity of Lithium-7 (Ban) w o Temperatureo Viscosity n Loglo 11 (mp) 103 C K mp T (UK) 191.2 464.4 5.61 0.749 2.15 200.8 474.0 5.45 0.736 2.11 221.2 494.4 5.13, 0.710 2.02 240.6 513.8 4.92 0.692 1.95- 271.8 545.0 4.56 0.659 1.84 24 .030.” .0de 03» .«o «000 0.31; 05 >3 m030> 31H 05 magamfidfi >o. 005030 300 .0 003000950» mo cofluna n on 953qu m0 m0m30mw p0u0umm0m 08. Ho P330032» 05 new 003.9, Pcdm .. w 0.3th 05 Eon.“ EA no.“ 039» 6000090 05 mm 0.9.30 6087.00 0gb AUOV u 0usudu0m50fi own ooN . ovN ONN OON emu _.a _ _ _ _ _ 1 fl‘ ‘1‘ N O m l4. AusoosyA ('dm) 25 experimental results for viscosity n, as a function of absolute temperature T. . It shows as well the dashed curve, 1 . (M )7 against T. Figure 9 shows 16g10 n "Li—6 Li-7/MLi-6 plotted against 1/r. The resulting straight line indicates that the exponential law is adequate to express the temperature dependence. The disagreement between the curve predicted for Li? and the dashed curve is unresolved by Ban. It is the purpose of the present work to repeat and refine the experiment, and to re-examine the theory in an effort to reconcile this disagreement. Z6 L17 Log10"7 - 0. 65 '- 0.55 1.80 2.00 2 20 103/1" (OK) Figure 9 Log10 Viscosity vs Reciprocal Temperature (Ban) CHAPTER IV RE-EXAMINATION OF THE EXPERIMENT The re-examination of the experiment has two phases: 1) a care- ful examination of each individual measurement, and 2) a careful repetition of the entire experiment. A. Examination of the Individual Measurements 1) Period: Time intervals were measured with an ordinary electric clock driven by a self-starting synchronous motor. On) several occasions the period was measured over long intervals of time (up to 24 hours). The period during these long intervals was constant. The only possible source of error in the period measurements was the clock itself, which is synchronous with the current furnished by the Michigan State University power plant. As a check, the clock was monitored with the time signals from the local telephone company. These signals are accurate within a least one or two seconds from day to day, as determined with signals broadcast from the Naval Observatory in Washington. The monitoring showed that the: clock was accurate within .1 part in 5, 000, an amount less than the error in reading the clock over the much shorter lengths of time measured in the actual runs. . Thus no significant error exists in the measurement of the period. 2) Logarithmic Decrement: The direct measurements involved in the calculation of this quantity are those of length of swing and of number of oscillations. The latter is obtained by counting and has no inherent error. The important requirement on the former is that the length scale be uniform and constant. . Probably the best indication of the accuracy of the measurement of logarithmic decrement is its constancy from 27 28 run to run, independent of initial amplitude. Table III shows the results of experiments with Li7, for ten runs over a period of 13 days at nominally the same temperature, Corresponding values of the viscosity coefficient have been calculated. The data have been analyzed statistically, and for what they are worth, 95% confidence limits have been established. The range of the data presented is 0. 1 millipoise, and even an error as large as this extreme is not sufficient to reconcile Ban's values for Li6 and Li7 to the square-root dependence. Consequently, inaccuracy in this determination must be rejected as an important source of error. 3) Radius of the Sphere: Ban has shown the high sensitivity of the viscosity to the radius of the sphere in the present method. Because of the high chemical reactivity of lithium, the anomalous results for Li6 and Li7 suggest the possibility of some sort of reaction between the lithium and the sphere, with consequent changes in the dimensions and the shape of the sphere, and in the density of the liquid metal. Moreover, a great increase in the viscosity of liquid metals upon the addition of small amounts of metallic impurities has been reported (19). Finally, the difference between Ban's values for lithium-7 and the present values suggest that some change may be taking place over a long period of time. The search for existence of corrosion was examined by two methods: 1) the lithium was analyzed for the constituents of the stainless steel, and 2) the radius of the sphere was remeasured. Since the analysis is destructive, only one sample, Li7, was tested. The stainless-steel Sphere containing the lithium was heated on a small electric hot plate under an argon atmosphere. * The molten lithium was transferred to a weighed nickle crucible by displacing it with argon forced into the sphere through a stainless-steel tube inserted into 3’ ‘(A controlled-atmosphere enclosure was kindly made available by Dr. H. A. Eick of the Department of Chemistry at Michigan State University. 29 Table III. Decrement at Constant Temperature Lithium-7 (Randall) Temperature T Logarithmic Viscosity 7) Date Furnace Sphere Decrement 6 (mp) (mV) (0C) 6/20/61 9.9715 187.1 578.484 5.88 6/23/61 9.9794 187.2 577.040 5.84 6/21/61 9.9805 187.2 580.143 5.94 6/19/61 9.9867 187.3 577.179 5.83 6/22/61 9.9869 187.3 576.269 5.80 6/15/61 9.9929 187.5 577.338 5.83 6/25/61 10.0021 187.6 577.354 5.84 6/26/61 10.0029 187.6 576.892 5.83 6/13/61 10.0038 187.7 577.338 5.84 6/16/61 10.0087 187.7 579.200 5.91 Statistical Analysis of Logarithmic Decrement Mean 577.7 x10'6 Standard Deviation 1. 18 x 10-6 95%-Confidence Interval (577.7 1 2. 32) x 10-6 Statistical Analysis of Viscosity Mean 5. 85 mp Standard Deviation 0. 014 mp . 95%-Confidence Interval (5. 85 _+_ 0. 014 ) mp 30 the sphere through the neck. When the lithium had cooled, the crucible was reweighed in the controlled-atmosphere enclosure. The crucible containing the lithium was placed in a desiccator which was then removed from the argon atmosphere. There was some question concerning the weighings made in the dry box, because the balance could not be closed and the pans may have been disturbed by atmospheric currents set up by the hot plate. In view of this uncertainty, the crucible with the lithium was later quickly transferred from the desiccator to a single-pan balance and reweighed. The results differed from those made in the controlled atmosphere by only slightly more than 0. 1 percent. The crucible and the lithium were then placed in a 600 ml. beaker. Water was added slowly until all the lithium had reacted to form the hydroxide. The resulting liquid was filtered through a sintered-glass funnel and made up to standard volume. The funnel was then washed with hot nitric and hydrochloric acids, and the resulting solution was analyzed to detect any materials that might have been not in actual solution in the lithium but rather in the form of inclusions. The small amount of lithium remaining in the sphere was also reacted with water. After the resulting lithium hydroxide was removed, the sphere was washed with acid (dilute HCl and HNO3). The four solutions thus obtained were analyzed separately by colorimetric methods for the major constituents of the type-446 stainless steel, namely, iron, chromium, and nickel. The amounts in the individual analyses are added to get the total quantity of each element present. . Chromium was determined by oxidizing the chromium to valence Six and forming a soluble red-violet product with di-phenylcarbazide in acid >1: solution. The iron was determined by the formation of a red complex >’ FThe analytical procedures were selected and carefully carried out by Mr. Richard Haire of the Department of Chemistry at Michigan State University. 31' between O-phenanthroline and the ferrous ion, in a controlled-pH solution. Nickel was determined by forming nickel (II) dimethylglyoxime and extracting it into chloroform (20). The results of the analysis are presented in Table IV. The amounts listed represent upper limits only, Since the contamination was so slight that the color changes were difficult to detect. The first section of the table includes the results from all four solutions. Since there was possibly attack of the stainless steel in the acid washing, there is some question whether the analysis of this solution should be included in the total metals found. Therefore the data excluding the contributions from the final washings of the sphere are presented in the second part of the table. The original semi-quantitative spectrographic analysis supplied by Oak Ridge is not available. Analyses of the L16 sample and of an earlier Li7 sample, however, are available. They indicate the possibility of an even greater amount of contamination than that found in the present analysis. These negative results lead us to believe there has been. no significant contamination of the lithium by the stainless steel. .As an additional check the radius of the sphere was remeasured, by a slightly different technique from that of Ban. , The Sphere is filled with fluid in small precisely measured amounts, and the corresponding change in the height of the fluid, h, is noted- If 3V/1Th2, where V is the cummulative volume of the fluid in a vessel, is plotted versus the height, h, the result for any ellipsoid of revolution about a vertical axis is a straight line. If the ellipsoid is a sphere, the slope of the line is minus one. This procedure then tells something about the shape of the cavity as well as its dimensions, which can be found from the intercepts of the straight line. For a true sphere either intercept is equal to three times the radius . 32 Table IV. . Chemical Analysis of Lithium-7 After Measurements Sample size 4.80 grams (I Material Maximum Amount Percent Present grams Including Acid wash of Sphere Iron 1 x10'3 0.02 Nickel 6 x 10"4 0.01+ Chromium 3.4 x 10-4 0.007 Excluding Acid wash of sphere Iron 6 x10"4 0.01 Nickel 5 x10"4 0.01 Chromium 3 x 10'4 0.006 33 For measuring, the sphere was attached to a milling—machine bed to obtain calibrated vertical motion as well as convenient motion in a horizontal plane. A vertical rod was passed into the sphere, clearing the neck and serving as an electrode to contact the surface of the liquid. The height of the liquid was determined by raising the sphere until the electrode touched the surface of the liquid, and triggered a thyratron actuating a relay. The liquid was added from a S-ml. precision bore burette. The results of a typical measurement for volume, V, as a function of height, h, are shown in Figure 10. Figure 11 gives the same data in different form with 3V/1ThZ plotted against h. . The filling liquid was water, with 0. 1% wetting agent (an aerosol) added. The best-fitting line (shown dashed) is slightly less steep than a line of slope minus-one drawn through the points (Shown dot-dash). Another line (shown solid)! is obtained upon assuming the container to be a true sphere of the radius measured by Ban (1. 28 cm). .Examination suggests a radius slightly larger (1. 30 cm). Repetition of the measurements, however, with mercury instead of water (Figure 12) givesara‘di‘us Slightly smaller (1. 25 cm). It seems more reasonable then to explain the divergence between measurements through interfacial tension of the liquid to the walls, rather than by a long-term increase in the size of the chamber. Interfacial tension gives rise to a positive contact angle in the case of water, and a negative contact angle in the case of mercury. The bracketing of Ban's value by our values leads us to believe that there has been no significant change in the radius of the sphere during the experiments. In summary, there seems to be no reason to suspect occurrence either of long-term change in the radius of the sphere, or of contamination of its contents, for the following reasons: Height h (cm.) .34, ~‘% ~ 1.1-1 1 '1 1 1 1 . 3 5 7 9 Volume V (cc. ) Figure 10 - Height h of fluid (water) as a function of the cumulative volume V of fluid added for the Li7 sample container. - 3,5 ‘ /, / 2.. E _ 3 N ,1: (1. h. \ > .. m I- .4 \ \\ \ \ . x}. 2 Height h (cm.) 4 Figure 11 - The data of figure 10, plotted with 3V/11'h2 against height h of fluid (water). The dashed line is a straight line through the experimental points. through the points. container of the volume measured by Ban. The dot-dash line is a line of Slope minus one The solid line is the curve expected for a sperical 3V/1r hz (cm. ) 39 \\~\ \ \ )— \\ ‘ \ \ 1.. \ \ \\\‘ .— \§\ - \\ r- o o \\ " o \3\ _ \ _ \§\ _ \ \ .. \ .\\\ .. \\\ )— \ \ _ ~ \ ' \, \\ ' \. \ .. \ \ 1 1 - 1 L 1 1‘ 1' 1 \ 1 z 3 Height h (cm. ) Figure 12 - The results of fillin the Li7 sample container with mercury plotted with 3V/1rh against height h. The dashed line is a straight line through the experimental points. The dot-dash line is a line of slope minus one through the points. The solid line is the curve eiqiected for a spherical container of the volume measured by Ban.' 37 a) Measurement of volume versus height gives no indication of an increase in the radius. b) Analysis of the lithium shows that the constituents of the stainless steel are not present in the lithium to any greater extent than they were in the original sample. c) The Li6 sample, which has been in its container for several months longer than the Li7 sample, Shows no significant long- period change in viscosity. 4) Moment of inertia: The calculation of the viscosity coefficient is not critically dependent on I, the moment of inertia of the oscillating system. The moment of inertia for the Li6 Sphere was redetermined, however, as a check on the reproducibility of this measurement. The method of calculation is the same as that adopted by Ban. The results are summarized in Table 5. When the average value for I from Table V is compared with Ban's value, there is less than 0. 3% difference. . This difference would have only a very slight effect on the viscosity coefficient. 5) Density: The values of this parameter were deduced from the work of others (11, 12, 13), and were not checked experimentally. The atomic volumes for Li6 and L17 differ by only a few hundred parts per million at room temperature, and surely by considerably less at temperatures above the melting point. The atomic masses are very closely in the ratio of mass numbers. The density for natural lithium is well established, and even significant errors in its value would not affect the ratio of the densities of the separated isotopes. 6) Temperature: The temperature is measured by indirect means, and there is a possibility of appreciable error in this measurement. But it is extremely unlikely that the error could amount to the approxi- mately 60 C-deg. necessary to reconcile the viscosity measurements with the simple theory. 38 Table V. Determination of Moment of Inertia Lithium-6 (Randall) Period H S 11 8.8098 sec. T20 = 12.8078 58C. 15.0720 sec. 1-1 w o I Distance Used With d1 and d2 with d1 and d3. With <12 and d3 Average value Results (Distance)z di- = 5.13204 cmz d3 = 27.71601 cm2 d§ = 18.18255 cmz Moment of Inertia I = 4,444. 58 gm-cmz I = 4,444.84 gm-cmZ I = 4,446.17 gm-cmZ I = 4,445.20 gm-cmz 39 Fortunately the calibration point at melting can be ascertained directly, since the melting points of the separated isotopes are known (21). The damping should increase sharply when the lithium melts. The abrupt rise is Shown for Li6 in Figure 13 and- for Li? in. Figure 14. . In each case the abscissas are taken as furnace temperature (in milli— volts) and sphere temperature (in degrees Centigrade) as given by the calibration relation. The melting point is shown by a vertical line. During the calibration experiments the temperature of the sphere was continuously monitored. Equilibrium was attained about 16 hours after the furnace was first turned on. For a 1-mv equivalent change in temperature, about-.12h0urs were. required for the Sphere to come to equilibrium. . Summary: A careful examination of the individual measurements reveals no source of error of the magnitude necessary to reconcile the experimental results with the predictions of simple theory. B. Repetition of the Entire Experiment The results for Li6 and Li7 are summarized in Tables VI and VII, respectively. The data from which the values were derived are tabulated in Appendix D. . Here each entry for the logarithmic decrement represents one run in which the logarithmic decrement was obtained at about every half millimeter over a decrease in amplitude of about 20 millimeters at an initial swing of 20 cm. Thus, thirty to forty points were available from which the average logarithmic decrement could be obtained. The runs were usually spaced about a day apart to make sure that sufficient time had elapsed for temperature equilibrium to be achieved. To verify attainment of equilibrium, two consecutive runs were made at each temperature. The consistency of suchdata attests to the constancy of the temperature. 40 I 1. 1' 3O - m E. S x - l *E 0 E 0 H U 0 Q .E.’ E b :5: 10 H 1- ‘“ I ADD 0 D .4 co‘ " U‘ I _———_ lmelting point Furnace I Temperature (mv) 1 1 1 ~ I 1 1 9.3 9.5 178.95 182.62 Sphere Temperature (0C) Figure 13 - Logarithmic decrementaas a function of temperature for Li . The melting is evidenced as an abrupt increase in the decrement when the melting temperature (180.49C) is reached. The upper abscissa scale is the furnace temperature in-millivolts; the lower scale is the sphere temperature as given by the calibration relation. 41 I .50 - O O 0 ._ O 30 - l .. l 3 x l a - I 0 E l 0 H 0 0 ° l .3 E l .c‘. t: . '14 10 1'" I «a DD 1:- .3 . r I .. I . l 5 " l _. o I I melting ‘point 3 7' Furnace Temperature (mv) 1 1 l ‘ l 1 9. 6 9. 7 180. 16 182. 03 Sphere temperature (0C) _ Figure 14. - The melting of Li7 is observed as an abrupt increase in the logarithmic decrement when the melting temperature (180.70C) is reached. he upper absCiSsa scale is the furnace temperature in millivolts; the. lOWer scale is the sphere temperature as given by the calibration relation. 42 Table VI. Viscosity of Lithium-6 (Randall) 0 Temperature To Viscosityr) Loglo r) _1_0_:5__ C K mp T( K) 182.4 455.6 4.20 0.623 2.195 188.2 461.4 4.14 0.617 2.167 198.2 471.4 4.14 0.617 2.121 203.7 476.9 4.04 0.606 2.097 221.3 494.5 3.93 0.594 2.022 243.3 516.5 3.77 0.576 1.936 265.4 538.2 3.63 0.560 1.858 267.1 540.3 3.60 0.556 1.851 287.4 560.6 3.48 0.642 1.784 43? Table VII. Viscosity of Lithium-7 (Randall) .____ 43-“. 0 Temperature T o Viscosity n Loglo r) 19;— C Kl rnp T( K) 184.8 458.0 5.88 0.769 2.183 185.6 458.8 5.97 0.776 2.180 186.7 459.9 5.86 0.768 2.174 187.5 460.7 5.89 0.770 2.171 199.6 472.8 5.71 0.757 2.115 215.3 488.5 5.41 0.733 2.047 238.8 512.0 5.12 0.709 1.953 240.2 513.4 5.15 0.712 1.948 264.2 537.4 4.84 0.685 1.861 265.7 538.9 4.79 0.680 1.856 286.3 559.5 4.61 0.664 1.787 44 The results presented in Tables VI and VII along with Ban's results are plotted in Figures 15 and 16. The agreement is good, especially in the case of Li6. The average between Ban's work and the present work is also shown (dot-dash). . The fit of the present data to the exponential law can be seen in Figure 17, where the logarithm of the viscosity is plotted against reciprocal temperatures for both isotopes. Figure 18 shows the average curves for lithium-6 and lithium-7 along with the curve for the former multiplied by the square root of the mass ratio. The results are not significantly different from those given by Ban (Figure 8). Our examination of the individual measurements and the corrobora- tion of Ban's findings, upon our repetition of the experiment lead us to believe that the experiment has been conducted with proper care. 45. .n0>u:0 025 05 000209 0w0u0>0 :0 mm 0?;0 605.36 03H. .0303 «:0m0um 05 5 600 0.003 mafimm 5 650m mm J 000530950» mo £03003 0 mm «A .«0 8. 55003.» 05H. 1 mm 0.5th o 00 J .0u30u0m509 owN chm. o¢~ oNN oo~ o3 4 fl _ fl _ _ m a . _ a I. o .m 16.6 1N.m :m6smm 0 00m Q .l ‘h AnsoasiA (dw) 46 .m0>.30 030 05 .«o 0mmu0>m c0 mw 0>u=0 @0336 03A. 5 undo“ mm J 0u3000m50u 05 mo coflugm .0 mm .0203 0:0m0um 05 cm cam x003 0,:mm Fwd mo 613300me 08H - A: 0usmmh 00 J .003000QE0H own cow. 3% CNN oom _ J _ _ J a Suva—mm o cam d 1 _ d. _ N.m ed (dm) °lL AusoosgA 140310” 47 .75!- (Randall) .65w- I l l l l 1. 80 2.00 2.20 103/T Figure 17 - Results of the present work in the form of the logarithim of the viscosity as a function of the reciprocal absolute temperature. 48 .333 .0de 030 m0 goon 0.2250. 0:“ >0, 0051; odd 0A0 msgfiaflfide >0, 605.3070 0006 24 0%" 50.3 EA 00m 030> 0000033 050 m“ 0>n50 00:93.0 0AM. .0Hsumn0mrc0u m0 £030qu .0 mm o,“ pad 3 0005mm m0 m0>udo 0m000>0 09H u m: 005mg 00 J .0udumn0m50fi owm SN 3%. SN com 2: _ _ _ r _ _ _ q _ . _ _ II. 0 om : T3 ‘ III s: s: I [0.0 1~.m .06 '11. AiisoostA (dw) CHAPTER V CONCLUSION A careful re-examination of the experimental method has failed to reveal any serious sources of error. Repetition of the experiment gives essentially the same results as those found by Ban, leaving us with values of the viscosity that do not agree with a simple dependence on the square root of the mass. There are two explanations: First, the principle of the experimental method is not correct, but with the present form of the apparatus such an explanation does not seemlikely. .Andrade and Dobbs (2) with this type of apparatus have obtained values for the viscosity of natural lithium (92. 6% Li7, 6.4% Lib) which are not in severe disagreement with the present results obtained with L17. These workers have measured the viscosities of a number of other materials and obtained results which agree with published values (22). Ban has measured the viscosity of water with the present apparatus, obtaining values only a few percent lower than the accepted values (Appendix E). I . The second explanation is that the theory is not sufficiently. developed to predict the results of the substitution in isotopic mass. The liquid state is less susceptible to theoretical attack than either the crystalline or gaseous state. .In crystals the interactions are persistent, but the order is high. A In gases the disorder is high, but the interactions are infrequent. In liquids there is the unhappy combination of high dis— order with constant interaction between any particle and several neighbors. Ban has reviewed some of the theories of the liquid state. None of them can be applied directly to predict the effect of isotopicrmrass; on 49 50 viscosity at moderate temperatures. Consequently we resort to general arguments such as those of dimensional analysis. The results of a careful analysis of this type on the problem at hand do not seem to have been published. Consider a fixed number of spherically- symmetric particles each of mass, m, and in a container of volume v, at a temperature T, and under an external pressure p. Let the interaction of the particles be described by a potential functioncb (r) representing perhaps a weak attraction at large distances and a strong repulsion at short distances. (See Figure 19.) We characterize the potential by two parameters, the first a spatial coordinate representing a range of some kind, the second an energetic coordinate representing a depth of some kind. . For con- venience, we take. as the former s, the distance of the potential minimum from the center of the particle, and we take as the latter b, the second derivitive of the potential curve evaluated at the minimum, recognizing that for curves of a fixed form the depth of the well and the second derivative of the function at the bottom are proportional. The para- meter b has the interpretation, of course, of the spring constant for an oscillator in the harmonic approximation. We now apply the methods of dimensional analysis to this model (2.3), which in the present form could apply to any state of matter. We shall consider as the parameters likely to appear in the equations of motion for the system the set m, b, and s, characterizing the individual particles; the set p, v, and T, characterizing the conditions of the system; and’h (Planck's constant) which will enter when any quantum effects are present. The fundamental variables could have been taken as mass, length, time, and temperature; however, such a procedure merely introduces another constant, k (Boltzman's constant), and complicates the algebra without yielding additional information.’ We believe it is better just to take kT together in the form of a thermal energy. 51 d! r) b :- 6(a) Figure 19 - Diagram to, illustrate parameters relevent to dimensional analysis. 52 Moreover, we introduce 41 by combining it with an angular frequency (b/m)%-, to get expressly a term in the form of a quantum of energy'fi. times (b/m)%-. Because there is an equation of state only two of three quan- tities, p, v, and T are independent,~'and need be included in the analysis. The seven variables to be related are: m = [mass]l, b = [mass]l times [time YE p = [mass]l [length]’l[time]'z (alternatively, v = [length]3), kT, “h (b/mffr: [mass]l [length]‘2 [time]'2, s = [length]l and n = [mass]l [length]1 [time]'1, where we have expressed each variable in terms of three fundamental units. According to the principles of dimensional analysis there exists a functional relation among 7 - 3 = 4 dimensionless products of these variables. There is some arbitrariness in the manner of expression of these variables, but for the sake of definiteness we write the relation in this form: n s2 «Vbh’n kT b I _ F —— , T . ——t'z— . — ‘0 r—-—'ka bs sp Since we are concerned with an analysis of viscosity it is more revealing to rewrite the equation in the form _ 'Jmkll‘. ::< b A kT ”h'Vb7m T? T n Asp' '. ET— . RT (6) .4 Alternatively it may be written: _ «I ka >:<>=< v kT 16-1/me ' .922 n '5’ bsz ' KT (7) ‘ >:£ *3]; I O 0 0 Here n and n are d1mens1on1ess functions whose detailed form can be determined only by constructing and solving the actual equations of motion for the liquid. I We can conclude immediately from equation (6) that the viscosity of separated isot0pes of the same element at the same temperature and l at atmospheric pressure must be proportional to m7, apart”, from quantum effects. With reSpect to these, we can say only that at such high 53 temperatures that quantization energies become negligible with respect to thermal energies, the dependence of n on m must follow the square- root law. But we have no sure way of knowing how high these tempera- tures are. . Regrettably, a technique of comparing between isotopes at "corresponding temperatures, " that is, temperatures for which 41(b/m)%-/kT has the same value for eachisotope, fails because of the presence of the other agument in 17* that contains T, namely, kT/bsz° We must therefore go beyond dimensional analysis to gain further information or at least estimates of the importance of quantum effects. As we have stated the full theoretical treatments are too compli- cated to permit estimating the effect of isotopic mass. Therefore we turn to the semiempirical formulas. The traditional relation, suggested in 1886 by Reynolds (15) and given explicit form by Guzman in 1913 (16), for the viscosity n, as a function of temperature T and at constant pressure p is: B/RT = Ae (Reynolds; p constant) Here A and B are supposed to be constants for a given material, but in fact are not. We have seen a typical example of the applicability of this law in the data presented earlier in this thesis. The lack of constancy is annoying but what is more serious“, is that the effect of isotopic mass cannot be disentangled from the quantities A and B. The effect of temperature is two-fold--the direct effect of the larger thermal energy for the individual molecules, and the indirect effect of increasing the free volume available to them. Yet this latter effect is masked in the exponential form of the equation. . Hence we seek other laws that display the dependence separately. The boldest approach is that of Batschinski (24), who argued in 1913 that the viscosity of a given substance‘is affected only by the increase in 54 free volume, regardless of the particular combination of temperature and pressure to attain it. He proposed the very simple relation n = C/(v - b) (Batschinski) where C and b are supposed to be constant, the latter being very nearly the same as the covolume occurring in the van der Waals equation for the substance. Regrettably the relation is not accurate. To take into account both the increase in thermal energy and the increase in free volume, van der Waals Jr. in 1918 (25), proposed the relation 1 _ CTT B/T - —— e (van der Waals Jr.) v(v-b) n a three-constant formula that has not drawn much attention, presumably because of its complexity without a sound theoretical foundation. MacLeod in 1923 (26) suggested a generalization of Batschinski's law in modifying the form of the function containing v, but still free from the explicit dependence on T: T) = C/(V-b)q (MacLeod I) where q is a third constant. Of course, this formula can no more account for the observed variation of n with T at constant v than can the original formula. MacLeod later (1936) (27) did introduce a term containing T explicitly: B Ce /Tv 7’) = fl (MacLeod II) Then van Wijk and Seeder (28) in 1937 compounded the complexity by taking B in van der Waals equation not as a constant but as a function of T and v. Their theoretical foundation for the expression, however, 55 enabled them to show that sometimes B should be proportional to T and independent of v, whence the equation reduced to a two-constant formula. . Indeed, Brinkman (29) with this formula in 1940 was able to explain the relation between the viscoSities of liquid hydrogen and liquid deuterium as observed experimentally. Here was a clear-cut case where both the masses and the moments of inertia of the mole- cules were in the same proportion, 2:1. The square root of this ratio is 1.4, yet the observed ratio of the viscosities was about 2. 9, and nearly independent of temperature. By taking the simplified relation, 1 n = CT? /v(v-b) p—s with C wmy, and b estimated from the observed molar volumes for liquid hydrogen and the volumes for solid hydrogen and extrapolated to 00K, Brinkman was able to fit the data. This suggests the same analysis be tried with the present data. The equation is adequate to describe the temperature dependence. The best values for both isotopes are given below: C b »Li6 2. 02 mp/(cc)z(Ko) 12. 93 cc/mol Li7 2. 28 mp/(cc)z(KO) 13. 04 cc/mol The difference in the values for b is 0.11 cc/moLa little less than 1 part per hundred. This difference is somewhat larger than we might expect in view of the likely very small difference in atomic volume, but not significantly so. The ratio of C for the two isotopes turns out to be about 1. l3, much larger than the ratio of the square root of the masses, 1.08. There seem to be no other cases in which the viscosity of two substances which have had both their moment of inertia and mass changed by the same ratio. Data for deuterated methane and 56 water are available.(30), but it has been pointed out (31) that in these substances the moment of inertia has been changed as well as the mass. Hence any agreement here with a square root dependence would likely be fortuitous. We may state then the following: 1) The dimensional'analysis argument advanced by Rowlinson, Pople, and others that the viscosity of isotopes of the same element at a given temperature is proportional to mi- is incomplete. The viscosity is rather proportional to (kaYL', multiplied by a complementary function of‘h(b/m)%-/kT. In fact, in the two unequivocal instances of the examination of an isotopic substitution (that is, where the mass and the moment of inertia are changed in the same ratio), the complementary function varies more strongly with isotopic mass than does the factor In}: (H1, H2: mass square root ratio, 1.4, viscosity ratio, 2.9, therefore the complementary function ratio 2. 1; Li", Li7: mass square root ration 1. 8, viscosity ratio, 1. 3, therefore the complementary function ratio 1. 24:) 2. Present semi-empirical theories when simple enough to predict the effect of isotopic mass give the incorrect result. 3. Before the theories can proceed farther, many more measure— ments will be needed on the combined effects of temperature and pressure on viscosity, as emphasized by the experiencedtheorist, Moelwyn-Hughes (32). We submit that measurements on the effect of isotopic mass on viscosity will be at least as important, in that this effect can usually be predicted more surely than the effect of temperature or pressure, and hence may afford a critical test. y—a *7 O 10. *11. >#12. 13. 14. REFER ENC ES CIT ED . F. W. Aston, Mass Spectra and Isot0pes, London: Edward Arnold, 1941. . E. N. da C. Andrade and E. R. Dobbs, Proc. Roy Soc., (London) A211, 12-30 (1952). . N. T. Ban,. "The Viscosity of Molten Lithium-6 and Lithium-7, " Unpublished Ph. D. dissertation, Michigan State University, 1960. . A. Merrington, Viscometry, London: E. Arnold, 1949. G. Barr, A Monograph of Viscometry, London, Oxford University Press, 1931. . R. M. Lyon, Ed. . Liquid Metals Handbook, Washington, D- C. , U. S. Govt. Print. Office, 1954. J. E. Verschaffelt, Commun. Phys. Lab. Univ. Leyden. vNo. 148b, 17 (1915). R. Ladenburg, Ann. Physik, 21, 157, (1908). E. N. da C. Andrade and Y. S- Chiong, Proc. Phys. Soc. (London) fig, 247, (1936). Shenker, e_t a_._l. ,. Reference Tables for Thermocouples, (National Bureau of Standards circular 561, April 27, 1955), pp. 34-36. A. Bernini and C.,Cantoni, Nuovo Cimento, 3, 241, (19-14). L- Losana, Gazz. Chim. Ital. é_5_, 851 (1935). D. D. Snyder and D. J. Montgomery, J. Chem. Phys. _2_'_7_, 1033, .(1957). A. Bondi, "Theories of Viscosity, " Rheology. Theory and Applications, vol. 1, ed. Frederick R- Eirich, (New York: Academic Press, 1956), 321-356. J 'l~ Indicates references which were not seen in the original. 57 *15. *16. 17. 18. 19. 20. 21. 22. 23. 26. *27. 28. 29. 30. 31. 32. 58 O. Reynolds, Phil. Trans. 177, 157 (1886). J. de Guzman, Anales de la’Sociedad Espanola de Fisica y Quim'ica, 11, 353, (1913),_1_2, 432, (1914). M. P. Venkatarama Iyer, Indian Journal of Physics, _5_, 371-383, (1930). E. N. da C. Andrade, Phil. Mag. _L7_, 497, and _ll, 698 (1934). T. P. Yao, V. Kondic, J. Inst. Metals. _8_l_, 17 (1952). E. B. Sandell, Colorimetric Determination of Traces of Metals, 2nd Ed. Interscience Publ. New York, 1950, 257-271, 362-388, 469-478. J. A. Crawford, and D. J. Montgomery, Bull. Am. Phys- Soc. .11, _2_, 299, (1957). M. F. Culpin, Proc. Phys. Soc. B70, 1069-1086, (1957). P. W. Bridgman, Dimensional Analysis, Yale University Press, 1931. A. J. Batschinski, z. Phys. Chem. 8_4, 643 (1913). . J. D. van der Walls Jr., Proc. Ac. Amsterdam, H, 743, 1283, (1918/19). D. B. MacLeod, Trans. Faraday SOC. _1_9_ 6, 1923,-£1_ 151, 1925. D. B. MacLeod, Trans. Faraday Soc. 32, 875, (1936). W..R. van Wijk and W. A. Seeder, Physica, 4, 1073. H. C. Brinkman, Physica, 1, 447, (1940). 'J. s. Rowlinson, Physica, 1_9_, 303 (1953). J. A. Pople, Physica, 13, 668 (1953). E. A- Moelwyn-Hughes, States of Matter, New York: Interscience Publishers, 1961. ' 3| m |< O I I Indicates references which were not seen in the original. APPENDICES 59 APPENDIX A DETERMINATION OF MOMENT OF INERTIA (Ban) Lithium - 6 .Sphe r e Period (Distance)z Tm: 8.795 sec. ai- = 5.13204 cmz T20 = 15.048 sec. 83 = 27.71601cmz T30 = 12.787 sec. 8; = 18. 18255 cmz Results Data Used Moment of Inertia With 8, and dz I = 4,443.79 gm-cmz With 8, and d3 ’1 = 4,443.75 gm-cmz With 8, and d3 I = 4,444.01 gm-cmz Average value I = 4,443.86 gm—cmz Lithium - 7 Sphe r e Period (Distance)2 T10 - 8.821 sec. di‘ = 5.13204 cmz T20 = 15.109 sec. d§ = 27.71601cm2 T30 = 12.839 sec. 8; = 18. 18255 cm?- Results Data Used Moment of Inertia With 8, and dz I = 4,428.82 gm—cmz With 8, and d3 I = 4,424.99 gm-cmz With 82 and d3 I = 4,434.43 gm-cmz Average value I = 4,429.41 gm-cm2 60 APPENDDC B TEMPERATURE- CALIBRATION 61 Sphere temperature (mv) .12 b 62 10... 1 1.'1 1 1 1 8 10 12 14 Furnace Temperature. (mv) Figure 20 - Calibration relation between the sphere temperature (ordinate) and furnace temperature (abscissa). APPENDDCC VISCOSITY OF LITHIUM 6 DATA (Ban) Parameters: Moment of Inertia 4443. 86 gm--cmz . Period (Lithium Solid) 8.795 sec. Residual Logarithmic Decrement 40.8819 x 10“6 Radius of Sphere (at 20° C) 1. 2827 cm ,Density (At 20° C) 0.460 gm/cm3 Sphere T%mperature ‘Logarithmic Decrement Period Viscosity C x 106 sec. mp 188.3 482.776 8.794 4.3239 188.3 481.167 8.793 4.2768 188. 3 487. 582 8.794 4.4649 188.3 488.628 8.794 4.4962 188.3 485.038 8.794 4.3897 198.0 476.442 8.794 4. 1533 198.0 483.129 8.793 4.3425 198.0 483.182 8.793 4.3440 198.0 481.242 8.793 4.2787 217.4 471.045 8.795 4.0231 217.4 468.990 8.794 3.9670 217.4 470.401 8.794 4.0049 217.4 470.146 8.794 3.9980 236.2 463.459 8.794 3.8374 236. 2 465.757 8.794 3.8975 236.2 462.393 8.794 3.8098 236.2 462.969 8.794 3.8480 236.8 462.466 8.798 3.8144 236.8 463.749 8.798 3.8477 236.8 463.108 8.798 3.8311 266. 5 456. 381 8.795 3.6832 266.5 457.315 8.795 3.7067 266.5 456.848 8.795 3.6949 266.8 455.686 8.795 3.6665 266.8 456. 209 8.795 3.6797 266.8 455.948 8.795 3.6731 continued 63 64 APPENDIX C -» Continued VISCOSITY OF LITHIUM 7 DATA (Ban) Parameters: Moment of Inertia 4438. 8928 gm-cmz Period (Lithium Solid) 8. 7492 sec. Residual L0garithmic Decrement 41.8851 x 10'6 Radius of sphere (at 20° C) 1.2841 cm Density (at 20° C) 0. 537 gm/cm3 Sphere Temperature Logarithmic Decrement . Period Viscosity 0C x 106 sec. mp 191.2 579.735 8.819 5.664 191.2 580.046 8.819 5.614 200.8 575.222 8.819 5.474 200.8 573.335 8.819 5.416 200.8 574.591 8.819 5.474 200.8 573.830 8.820 5.433 221.2 565.118 8.820 5.194 221. 2 563.419 8.820 5.144 221.2 563.923 8.820 5.159 240.6 554.083 8.820 4.899 240.6 555.613 8.820 4.942 240.6 554.270 8.820 4.904 240.6 555.061 8.819 4.926 271.8 540.354 8.820 4.560 271.8 539.344 8.820 4.534 271.8 541.171 8.820 4.582 271.8 540.581 8.820 4.566 APPENDIX D VISCOSITY OFFLITHIUM 6 DATA (Randall) Parameters: Moment of Inertia: 4445. 1964 gm-—cmz Period (Lithium Solid) 8. 8098 sec. Residual Logarithmic Decarement 41.0963 x 10'6 Radius of sphere (at 20 C ) 1. 2828-cm. Density (on melting) .440803 gm/cm3 Temperature Sghere Furnace Logarithmic Decrement Period Viscosity C mv x 106 sec. mp 180.8 9.4031 480.026 8.809 4.2251 181.8 9.4534 482.173 8.809 4.2869 182.4 9.4900 476.081 8.809 4.1165 182.7 9.5056 481.684 8.809 4.2738 183.9 9.5710 480.842 8.809 4.2509 184.6 9.6092 482.404 8.809 4.2961 185.5 9.6555 481.106 8.810 4.2608 187.9 9.7924 474.529 8.810 4.0795 188.2 9.8044 474.506 8.808 4.0773 188.3 9.8116 479.574 8.808 4.2181 188.4 9.8179 478.740 8.808 4.1947 191.3 9.9790 476.036 8.811 4.1248 191.8 10.0034 474.086 8.810 4.0707 192.1 10.0196 478.400 8.811 4.1912 192.1 10.0215 478.496 8.810 4. 1930 203.3 10.6442 473.113 8.808 4.0524 203.5 10.6560 471.997 8.809 4.0232 203.9 10.6780 472.530 8.811 4.0397 204.1 10.6894 473.254 8.811 4.0596 continued 65 66 APPENDIX D - Continued Temperature Sphere Furnace Logarithmic Decrement Period Viscosity OC mv x106 sec. mp, 221.2 11.6539 468.286 8.813 3.9424 221.2 11.6560 467.822 8.812 3.9293 221.4 11.6656 465.500 8.808 3.8651 221.6 11.6739 471.707 8.808 4.0303 221.8 11.6884 466.334 8.810 3.8889 243.1 12.9136 459.979 8.811 3.7434 243.3 12.9274 461.477 8.811 3.7818 243.5 12.9356 461.757 8.808 3.7867 265.3 14.2182 455.317 8.812 3.6453 265.5 14.2321 453.765 8.811 3.6063 267.1 14.3211 453.793 8.811 3.6082 267.1 14.3236 453.414 8.811 3.5989 287.2 15.5251 448.673 8.812 3.5005 287.7 15.5549 447.032 8.811 3.4609 67 APPENDIX D - Continued VISCOSITY OF LITHIUM 7 DATA (Randall) Parameters: Moment of Inertia: 4549. 6853 gm-cm2 Period (Lithium Solid) 8. 9398 sec. Residual Logarithmic Decorement 40.8761 x 10"6 Radius of Sphere (at 20 C ) 1. 2841 cm Density of Lithium (on melting) . 51458 gm/cm3 Temperature 5%here Furnace Logarithmic Decrement Period Viscosity C mv x 106 sec. mp 181.1 9.6477 449.360 8.921 2.7461 181. 3 9.6590 581.420 8.920 6.0255 184.8 9.8516 577.031 8.922 5.8815 185.6 9.8947 579.558 8.924 5.9725 186.7 9.9506 576.508 8.918 5.8606 187.1 9.9715 578.484 8.924 5.9373 187.2 9.9794 577.040 8.930 5.8962 187.2 9.9805 580.143 8.924 5.9951 187.3 9.9849 573.763 8.941 5.8000 187.3 9.9867 578.318 8.923 5.9306 187.3 9.9869 576.269 8.919 5.8548 187.5 9.9929 577.338 8.916 5.8873 187.5 9.9955 575.873 8.939 5.8692 187.6 10.0021 577. 354 8.923 5.8978 187.6 10.0029 576.892 8.926 5.8862 187.7 10.0038 577.338 8.926 5.9015 187.7 10.0087 579. 200 8.929 5.9702 187.8 10.0108 574.815 8.938 5.8322 187.8 10.0141 575.243 8.941 5.8510 198. 3 10.5852 568.943 8.937 5.6489 198.4 10.5895 550.634 8.938 5.0788 continued 68 APPENDIX D - Continued Temperature S%here Furnace Logarithmic Decrement Period Viscosity C mv x106 sec. mp 199.5 10.6495 571.214 8.920 5.7030 199.6 10.6531 571.266 8.937 5.7275 199.6 10.6550 570.513 8.940 5.7065 215.3 11.5209 559.806 8.936 5.3761 215.3 11.5234 562.031 8.939 5.4501 216.6 11.5926 559.190 8.938 5.3609 217.3 11.6320 551.695 8.940 5.1345 238.7 12.8391 550.837 8.938 5.1323 238.8 12.8434 549.985 8.938 5.1068 240.2 12.9264 550.997 8.940 5.1412 240.3 12.9321 551.557 8.941 5.1594 264.1 14.3000 540.297 8.939 4.8548 264. 3 14. 3134 539.236 8.936 4.8217 265.7 14.3936 538.487 8.935 4.8010 265.7 14.3955 537.335 8.939 4.7727 286.1 15.5935 530.774 8.936 4.6109 286.4 15.6117 531.051 8.936 4.6188 69 APPENDIX E VISCOSITY OF WATER Toemp. n (expt.) r) (std.) per cent C . mp . mp deviation 16.6 10.48 10.94 -4.2 17.1 10.42 10.80 -3.5 18.6 9.87 10.40 -5.1 PHYSICS-MATH L151 . f .4,“ .--'_ HICHIGQN STQTE UNIV. LIBRQR IllllN111"llllllllllllllillllfllllllllllllllllllllHlll“HI 293017430251