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Some dynamic properties of liquid helium Chopra, Kasturi Lal 1957

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S O M E D Y N A M I C P R O P E R T I E S O F L I Q U I D H E L I U M by K A S T U R I L A L C H O P R A B . S c . (Hons), De lh i .Univers i ty , 1952 M . S c . D e l h i Univers i ty , 1954 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F PHILOSOPHY i n the Department of PHYSICS We accept this thesis as conforming to the standard required from candidates for the degree of D O C T O R O F PHILOSOPHY T H E U N I V E R S I T Y O F BRITISH C O L U M B I A July , 1957 S O M E D Y N A M I C P R O P E R T I E S , O F L I Q U I D H E L I U M A B S T R A C T T h i s thesis describes three, independent investigations of the dynamic properties of the superfluid He II f i l m flow and the v iscous flow of l i q u i d he l ium i n acoustic streams generated by the gradient of the radiation pressure occuring due. to attenuation of sound propagating i n l i q u i d h e l i u m . The non-isothermal He II f i l m flow induced gravitationally i s studied for thecovered(Cu-Ni cover of the wire-f i l led- tube superleak intact) and the uncovered f i l m ( C u - N i cover peeled off) f lowing through a superleak when subjected to a thermal plus gravitational potential at a point midway between the two transfer po in ts . It i s found that the c r i t i c a l transfer rate of the covered f i l m i s not affected by the nature and magnitude of this potent ia l . For the case of uncovered f i l m warmed in the midd le , i n contrast to that of the covered f i l m , the transfer can be reversed by a suitable thermal potent ia l , the m a x i m u m rate being the same i n either d i rec t ion . In the case of cool ing by direct pumping over the uncovered f i l m , however, f i l m flow stops.either way; i n the l ight of other resul ts , we have given an explanation to this controversial observation in terms of thinning down of the f i l m to i ts non-superfluid l aye r s , a conclusion poss ib ly equivalent to a shift of X -point for thinner films . A c losed glass capsule , sealed off at room temperature w i th 750 p s ig o f He gas , sufficient to provide enough l i q u i d at he l ium temperatures, i s used to measure the gravitationally induced transfer rates of He II f i l m at temperatures between 0.3OK and the X - p o i n t . L i k e . Ambler and K u r t i , we f ind that the transfer rate r ises by 25% above the flat m i n i m u m near 1°K though, i n contrast to tjhe steady rise.observed by these authors, our results indicate s l ight flattening near 0 . 3 ° K . T h i s confirms that, l i k e thermal properties, the f i l m flow property of He II also undergoes a radical change below 1 ° K . A technique i s developed whereby fine part icles of a suitable mixture of s o l i d H2 p lus D2 are suspended i n l i q u i d h e l i u m . A number of experiments are suggested to study flow properties of l i q u i d He v i s u a l l y by us ing these part icles as indica tors . We have used these part icles as indicators i n acoustic streaming experiments designed to measure the ratio of the second to the first coefficient of v i s c o s i t y . These two coefficients occur in the expression for absorption coefficient of sound and are calculated by Khalatnikov for He II . The streaming i s observed to be turbulent to the lowest poss ible ultrasonic in tens i ty . Thus , such determination i s not feas ib le . The streaming i n He:II i s found to obey the c l a s s i c a l equation of turbulence. It i s independent of temperature and shows no anomaly at or above the A -point , poss ib ly due to complete absorption of sound i n the turbulent m e d i u m . Faculty of Graduate Studies PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of KASTURI LAL C H O P R A B.Sc. (Honours), Delhi University, 1952 M.Sc, Delhi University, 1954 FRIDAY, AUGUST 2nd, 1957, at 10:30 a.m. IN ROOM 300, PHYSICS BUILDING C O M M I T T E E I N C H A R G E D E A N G . M . S H R U M , Chairman J. B. BROWN A. HASLOW J. M. DANIELS T. E. HULL F. A. KAEMPFFER J. HALPERN W. OPECHOWSKI A. D. MOORE External Examiner: K. R. ATKINS 'University of Pennsylvania SOME DYNAMIC PROPERTIES OF LIQUID HELIUM A B S T R A C T This thesis describes three independent investigations of the dynamic prop-erties of the superfluid He II film flow and the viscous flow of liquid helium in acoustic streams generated by the gradient by the radiation pressure occurring due to attenuation of sound propagating in lio.uid helium. The non-isothermal He II film flow induced gravitationally is studied for the covered (Cu-Ni cover of the wire-filled-tube superleak intact) and the uncovered film (Cu-Ni cover peeled off) flowing through a superleak when subjected to a thermal plus gravitational potential at a point midway between the two transfer points. It is found that the critical transfer rate of the covered film is not affected by the nature and magnitude of this potential. For the case of uncovered film warmed in the middle, in contrast to that of the covered film, the transfer can be reversed by a suitable thermal potential, the maximum rate being the same in either direction. In the case of cooling by direct pumping over the uncovered film, however, film flow stops either way; in the light of other results, we have given an explanation to this controversial observation in terms of thinning down of the film to its non-superfluid layers, a conclusion possibly equivalent to a shift of lambda-point for thinner films. A closed glass capsule, sealed off at room temperature with 750 psig of He gas, sufficient to provide enough liquid at helium temperatures, is used to measure the gravitationally induced transfer rates of He II film at temper-atures between 0.317 K and the lambda-point. Like Ambler and Kurti, we find that the transfer rate rises by 25% above the flat minimum near 1 ° K though in contrast to the steady rise observed by these authors, our results indicate slight flattening near 0.3" K. This confirms that, like thermal properties, the film flow property of He II also undergoes a radical change below 1 0 K. A technique is developed whereby fine particles of a suitable mixture of solid E2 plus D2 are suspended in liquid helium. A number of experiments are suggested to study flow properties of liquid He visually by using these part-icles as indicators. We have used these particles as indicators in acoustic streaming experiments designed to measure the ratio of the second to the first coefficient of viscosity These two coefficients occur in the expression for absorption coefficient of sound and are calculated by Khalatnikov for He II. The streaming is observed to be turbulent to the lowest possible ultrasonic intensity. Thus, such determination is not feasible. The streaming in He II is found to obey the classical equation of turbulence. It is independent of temperature and shows no anomaly at or above the lambda-point, possibly due to complete absorption of sound in the tur-bulent medium. GRADUATE STUDIES Field ol Study: Physics Low Temperature Physics . . _ . . . J. M. Daniels Physics of the Solid State J . S. Blakemore and J. B. Brown Relativity F. A. Kaempffer Group Theory Methods in Quantum Mechanics W. Opechowski Topics in Theoretical Nuclear Physics J . B. Warren W. Opechowski G.M.Volkoff Other Studies: Advanced Differential Equations T. E . Hull Integral Equations T. E . Hull Magnetic Properties of Metals and Theory of Alloys . H. P. Meyer PUBLICATIONS 1. Helium Film Flow G. R. Hebert, K. L. Chopra, J . B. Brown, N. S. F. Conference on Low Temperature Physics and Chemistry, Louisiana State University, December, 1955 2. Helium II Film Filter K. L. Chopra, Rev. Sci. Instr. 28, 146, 1957 3. Helium II Film Flow Below 1° K G. R. Hebert, K. L. Chopra, J. B. Brown, Phys. Rev. 106, 391, 1957 4. Non-isothermal He II Film Flow K. L. Chopra, J . B. Brown, (Phys. Rev. August 1, 1957) 5. Suspension of Particles in Liquid Helium K. L. Chopra, J . B. Brown (Comm. International Conference on Low Temperature Physics, Madison, August, 1957) 6. Suspensions of Particles in Liquid Helium K.L.Chopra, J.B.Brown (to be published in Phys. Rev.) 7. Acoustic Streaming in Liquid Helium K.L.Chopra, J.B.Brown (to be published in Phys. Rev.) In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements fo r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available f o r reference and study. I further agree that permission f o r extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representative. I t i s under-stood that copying or publication of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Kasturi Lai Chopra Department of Physics  The University of B r i t i s h Columbia, Vancouver $, Canada. Date July , 1957  A C K N O W L E D G E M E N T S It is a pleasure to express my gratitude to m y supervisor, D r . J . B . .Brown, for his guidance and valuable help i n every phase of this work . I am grateful to M r . H . Zerbst for operating the liquefier to produce l i q u i d he l ium and h is continued technical assistance i n designing the cryostat . M r J . Lees performed the numerous intricate glass blowing operations required and I am indebted to h i m for contributing much to the project through h i s s k i l l and experience . in designing glass equipment. The members of the Physics Department Workshop Staff provided a great deal of helpful advice and assistance wi th a variety of problems and for t h i s , I w i s h to thank t hem. To the members of the Low Temperature Group, I am indebted for the many occasions on which they gave help wi th the experiments . F i n a l l y , I w i s h to acknowledge gratefully the. World Univers i ty Service of Canada for an Exchange Scholarship for two years and the Nat ional Research Counc i l of Canada for the award of a studentship. T A B L E O F C O N T E N T S P A G E I N T R O D U C T I O N 1 C H A P T E R I T H E P R O B L E M O F L I Q U I D H E L I U M 3 Some Experimental Aspects 3 Theoret ical Aspec t s 10 C H A P T E R II N O N - I S O T H E R M A L H E L I U M II F I L M F L O W 13 The Problem 13 Exper imental Arrangement 16 (a) General 16 (b) . Arrangement for Measuring F i l m F low Rates 18 Procedure 22 Resul ts 24 Di scuss ion 25 C H A P T E R LU H E L I U M II F I L M F L O W B E L O W 1°K 28 Introduction 28 Experimental Arrangement 33 (a) The Cryostat 33 (b) Sealed Glass Capsule 34 (c) Suspension and.Dumping Arrangement 36 (d) L igh t Source 39 (a) The Magnet 40 (f) The Suscept ibi l i ty Bridge 40 Procedure 43 Resul ts 45 D i scus s ion 46 C H A P T E R IV SUSPENSION O F P A R T I C L E S IN L I Q U I D H E L I U M Introduction 52 Poss ib i l i t i e s and Diff icul t ies .53 The Mixture 55 Experimental Arrangement 56 Methods of Obtaining Sma l l Particles 57 Nature of Particles 58 .Conclusions 59 V . P A G E C H A P T E R V A C O U S T I C S T R E A M I N G IN L I Q U I D H E L I U M 61 The Problem 61 Acous t i c Streaming 62 Theoret ical Explanat ion 63 Second Coefficient of V i s c o s i t y 67 Ultrasonic Attenuation-in L i q u i d He l i um 71 A n Exper imental Diff icul ty 74 Exper imental Arrangement 77 (a) Streaming Tube 77 (b) Ultrasonic Generator 80 (c) Crys ta l Holder 81 ,(d) Intensity Measurement Device 82 Method of Measurement 82 Resul ts 86 Di scuss ion and Conclusions 87 A P P E N D I X I A P P E N D I X II A P P E N D I X HI H E L I U M II F I L M F I L T E R C O N T I N U O U S M E A S U R E M E N T O F T E M P E R A T U R E B E L O W 1°K 94 S E A L I N G O F H I G H P R E S S U R E G L A S S BOMBS 99 102 -APPENDIX IV M E A S U R E M E N T OF U L T R A S O N I C I N T E N S I T Y IN LIQUID H E L I U M 105 R E F E R E N C E S 111 vi . L I S T O F I L L U S T R A T I O N S Fac ing Page 1 5 F i g . 1 Photograph of the Cryostat F i g . 2 The specif ic heat anomaly of l i qu id he l ium F i g . 3 The pressure attenuation coefficient of l i q u i d he l ium near A - p o i n t 5 F i g . 4 The ve loc i ty of second sound as a function of temperature 7 F i g . 5 F i l m Transfer rates over glass 7 F i g . 6 The vacuum sys tem 17 F i g . 7 The sys tem for measuring non-isothermal f i l m flow rates 18 F i g . 8 Block diagram of the thermometer 21 F i g . 9 Transfer rates of the covered f i l m 24 F i g . 10 Transfer rates of the uncovered f i l m (case of warming) 24 F i g . 11 Transfer rates of the uncovered f i l m (case of cooling) 25 F i g . 12 . A sealed glass capsule 34 F i g . 13 The b a l l i s t i c mutual inductance bridge 41 F i g . 14 T y p i c a l warm-up curves for the sealed capsule 45 F i g . 15 F i l m transfer rates below 1°K 46 F i g . 16 (a) Schemetic diagram of the system for m i x i n g H2 and D2 gases 56 (b) The heater arrangement for formation of part icles 56 F i g . 17 The different arrangements for acoustic streaming experiments 65 v i i . F a c i n g Page F i g . 18 Attenuation of first sound i n l iqu id he l ium 72 F i g . 19 (a) The streaming tube 78 (b) The-crystal holder 78 F i g . 20 Block diagram of 5 Mcps ultrasonic generator 80 F i g . 21 The ve loc i ty of acoustic streaming versus. temperature 86 F i g . 22 The ve loc i ty of acoustic streaming versus ultrasonic intensity 86 F i g . 23 Microphotograph of cross section of the wire -f i l led-tube 95 F i g . 24 Dependence of channel width on the.outer d iam eter of the wire - f i l l e d rtube 9 6 Figv 25 Apparatus for sea l ing high pressure glass bombs 99 F i g . 26 Photograph of the sealed glass bombs 101 L I S T O F T A B L E S P A G E T A B L E I Data on acoustic streaming ve loc i ty as a function of temperature 92 T A B L E II Data on acoustic streaming veloc i ty as a function of voltage across the quartz crys ta l 93 T A B L E III Data on wire-f i l led- tube fil ter 98 Fig. 1 Photograph of the Ciyostat facing page 1 I N T R O D U C T I O N T h i s thesis describes three indeperidant experimental investigations of some dynamic properties of l i q u i d he l ium below its X " point . The non-viscous flow of the creeping f i l m of l i q u i d he l ium and the viscous flow of the bulk l i q u i d he l ium II are the dynamic properties that are studied here . The experimental and theoretical background of the f i e ld of research is given in the first chapter on "The.Problem of L i q u i d H e l i u m " . The non-viscous or superfluid property of the creeping f i l m of hel ium II forms the subject of investigations described i n Chapters II and III. Exper iments , designed to study the transfer rates of He f i l m under non-isothermal conditions so as to be able to f ind an explanation of some controversial hypotheses, are described i n Chapter II . In Chapter III, the effect of change.of the behaviour of He II below 1°K on the transfer rates of the creeping f i l m are studied by v i sua l observations in a sealed glass capsu le . A technique i s developed to suspend i n l i q u i d he l ium s m a l l par t ic les of mixture of s o l i d H2 plus D2 of density approximately the same as that.of l i q u i d h e l i u m . T h i s . i s described i n Chapter I V . The suspension of part icles i n l i q u i d He i s used as an indicator i n the study of acoust ic streaming generated by radiation pressure gradient bccuring as a result of absorption of ultrasonics in the viscous l i q u i d . Acous t i c streaming experiments and their interpretation as,regards the v iscous forces involved i n the flow of l i q u i d h e l i u m , are described i n Chapter V . There are some s m a l l experimental projects w h i c h , though employed at one stage or the other during the.course of m a i n invest igat ions, are not direct ly related to the problem of l i q u i d he l ium discussed in the text . These projects contain useful data and other information and are described very briefly i n four appendices at the end of the text . These appendices have been referred to i n the main tex t . 3 . C H A P T E R I T H E P R O B L E M O F L I Q U I D H E L K J M It is quite essent ia l for the sake of completeness of the thesis as w e l l as understanding of the subject matter of the thesis without undue. outside references, that we should first describe the problem of l i qu id he l ium and the bas ic concepts underlying i t . Fo l lowing i s a brief account of only those salient features of the problem which are quite relevant to the text . . S O M E E X P E R I M E N T A L A S P E C T S Kamer l ingh Onnes succeeded i n l iquefying he l ium in 1908. After about thirty years , it became apparent that l i q u i d h e l i u m , i f cooled below 2 . 1 8 3 ° K , transforms into a substance which i s so very different from any other l i q u i d that i t has been sa id to constitute a new fourth state of matter unl ike any s o l i d , l i q u i d or gas . There i s no change i n density of l i q u i d he l ium at this temperature of 2 . 1 8 3 ° K , nor i s there any change i n the geometr ical arrangement of he l ium atoms in theTiquid at this temperature. The discontinuity of slope of density-temperature curve at th is temperature, c a l l ed the X - p o i n t , observed by K . Onnes and Bbks (1924) l e d Keesom and Wolfke (1928) to compare this discontinuity wi th 4. a phase transition and so they used the terminology He I and He II, He II being the low temperature f o r m . Whereas He I has properties approximating those- anticipated for a normal l i q u i d at very low temperatures, He II has remarkably unique properties. One could eas i ly v i sua l i s e that the quantum effects i n molecular interactions, so significant i n l ight atoms at low temperatures, can give r i se to interesting departures from c l a s s i c a l physics and ordinary properties though no str ikingly new phenomena are expected. The unusually large volume of l i q u i d He - 32 c m ^ per mole at 4 . 2 ° K as compared wi th 8 c m ^ for a c lose packing of He atoms (hard sphere) 2.7 x 10"^ c m i n diameter; low heat of vaporisation - 20 calories per mole as compared with 85 calculated fromTr'oiititon's law; unique s tabi l i ty of he l ium i n that there i s no condit ion of temperature and pressure at which three phases -s o l i d , l i q u i d and vapour can coexis t together i n equ i l ib r ium, are some of the properties which are a consequence of the quantum mechanica l effects i n molecular interact ions. But there are properties of l i q u i d He II that are not just unusual - they are unique. Some of them, relevant to the text , are: (1) Superfluidity ,(frictionless flow of the l i qu id at a c r i t i c a l ve loc i ty of about 20 to 30 c m per second through capi l lar ies of the order of 0.1 micron i n diameter) . (2) The creeping f i l m (frictionless flow of the l i q u i d in form of a f i l m of about 100 atom th ickness , against gravity over the surface of containers, at c r i t i c a l speeds of 20 to 30 c m per second independant of the driving pressure). 3 0 T ( K ) F ig - 2 Specific heat of liquid helium under its own vapour pressure-T E o 0 -9 0-6 0 1 3 0 • x T ( K ) 4-5 F i g - 3 Pressure attenuation coef f ic ientaVs temperature n e a r X - p o i n t fac ing poge 5 5. (3) Phenomenal rates of heat transfer (hundreds of t imes .that i n our best me ta l l i c conductors). (4) Second sound (temperature or thermal waves propagated through the l i q u i d He II undamped l i ke ordinary sound). Whi le the above mentioned properties appear right below the C\.-point, a number of s t r ik ing things happen to l i q u i d He right at the transi t ion point or the X -poin t . For example , the specif ic heat of l i q u i d He has infini tely large value at this point as shown in Figure 2 . T h i s is referred to as A - anomaly . Because of the s imi la r i ty of this curve wi th Greek letter A. , the transition in l i q u i d He at 2 . 1 8 3 ° K i s c a l l ed the X - t r a n s i t i o n . The pressure attenuation coefficient of first sound i n l i qu id he l ium undergoes an anomaly at the !X -point , as shown i n Figure 3 . The marked s imi l a r i t y between this anomaly and that of the specif ic heat at the ,X -point , i s quite interest ing. It is worth whi le to make some more comments on the superfluid property, of the bulk l i q u i d as w e l l as i ts f i l m . If the v i s cos i t y of He II i s measured by the damping,of an o sc i l l a t i ng d i sc immersed i n i t , i t i s found that the value derived i s of the same order of magnitude as that for He I but that the value decreases wi th f a l l i n temperature. On the other hand, i f the v i scos i ty i s measured by.observations on flow through fine capi l la r ies (PoiseuilleSmethod) or s l i t s , i t i s found that i ts apparent value decreases more and more as the width of the channel is reduced unt i l the apparent v i scos i ty of l i qu id He II becomes far l e ss than that of the vapour. 6 . The current explanation of this s t r ik ing phenomenon first given by T i s z a (1938, 1940) i s that He II may be considered to consis t of two superimposed "parts", one.of which has apparently no v i scos i ty and i s , therefore, c a l l ed the "superfluid", and theother a v i scos i ty of about the same magnitude as that for He I and therefore cal led , the "normal f l u i d " ; the two fluids are present i n a certain proportion depending only on the temperature. In the experiment wi th the osc i l l a t ing d i s c , the gradient of veloci ty of normal f lu id leads to viscous damping. The apparent v i s c o s i t y decreases wi th fa l l , i n temperature since the proportion of normal f luid.becomes smal ler the lower the temperature. On the other hand, i n extremely narrow capi l lar ies or s l i t s , only the superfluid i s capable of f low, for the normal f luid i s he ld stationary by the w a l l s . There i s , therefore, no gradient i n the ve loc i ty of normal f lu id and hence, no viscous effects. It should be mentioned that this result holds only up to a m a x i m u m veloc i ty ca l l ed the c r i t i c a l ve loc i ty for which superfluidity i s exhib i ted . Above this ve loc i t y , superfluid exhibits a compl ica ted behaviour characteristic of a d iss ipat ive flow and i s briefly d iscussed la ter . T i s z a ' s two f luid model of l i q u i d He II has stood the test of t i m e . It is. i m p l i c i t l y involved i n any theory of l iqu id H e . T h i s concept has proved to be of great importance i n not only designing various experiments to investigate the l i q u i d He problem, but also i n predicting and interpreting results of such experiments . One such prediction was that of second sound. Accord ing to T i s z a ' s m o d e l , l i qu id He II can have two modes of wave motion; to face page7 7 . one w i th normal and superfluid moving i n phase, and the other i n which they move out of phase . Whereas the first wave motion corresponds to density or pressure fluctuations (first sound), the second corresponds to entropy or temperature fluctuations (second sound). The variat ion of ve loc i ty of second sound wi th temperature i s shown i n Figure 4 . The curve has interesting features below 0 . 7 ° K though not shown in Figure 4 . S i m i l a r l y , the assumption that the superfluid i s non v iscous as w e l l as does not carry entropy, gives a clear picture of fountain effect and i t s reverse, e . g . thermomechanical effect. Incidentally, the existence of these effects c lear ly demonstrates that i t i s imposs ib le to treat the hydrodynamical and thermal properties of l i q u i d He independently. A n interesting case involv ing flow of superfluid alone is the R o l l i n creeping f i l m . T h i s f i l m of about 100 atoms thickness covers any s o l i d surface dipping i n He II and i f f i l m can flow from one container to another without being evaporated, i t starts transferring the l i q u i d from the container wi th He II at a higher l eve l to the other wi th He II at a lower l e v e l l i k e a syphon. L i k e the superflow of bulk l i q u i d He II through narrow channels , f i l m flow i s independent of pressure head and has a c r i t i c a l veloci ty, of the order of 20 to 30 c m per second. .Although rate of transfer i s independent of pressure head, yet a hydrostatic or a thermomechanical pressure head is absolutely necessary to init iate the f low. The rate of flow of He II f i l m i s strongly temperature dependent and is shown in Figure 5 . The rate of transfer i s the same whether flow i s induced gravitationally or thermal ly . Fairbank and Lane (1949) have shown that 8. He.II f i l m can evaporate at a temperature very near to that of the parent l i q u i d . Thus , the f i l m f low, whether induced thermally or gravi tat ional ly, is i so thermal . The.resemblance.between the f i l m transfer rate curve and that of second sound should be noted. There i s no satisfactory theory to expla in a l l the properties of the creeping f i l m . However, i t appears that the V a n der Waals type of forces between the w a l l and he l ium atoms i n the f i l m and the. zero-point energy of He atoms i n the f i l m play an important role in the behaviour of the f i l m . F i l m flow as w e l l as He II flow through narrow channel , above the c r i t i c a l ve loc i t y , i s i n a way s imi l a r to the flow of He II i n wide channels characterised by prac t ica l ly zero c r i t i c a l ve loc i t y . Both .types of flow exhibit non-linear fr ict ional forces . These forces are receiving a lot of attention at present from both theoretical as w e l l as experimental investigators . On the bas is of London'sand Landau's theories (described later) , the superfluid i s incapable of rotational m o t i o n . If we accept Reynold ' s cri terian for turbulent mot ion , we would expect the flow of superfluid wi th any, finite ve loc i ty to be turbulent. Although on one hand there is enough experimental evidence to support the hypothesis that the flow of superfluid i s rotational only above a certain c r i t i c a l ve loc i ty , yet on the other, the.experiments oh rotation of He II c lear ly prove that wi th any observable ve loc i ty of rotation, both normal and superfluid rotate indicat ing the existence of fr ict ional force in the flow of the superfluid. 9 . T h i s brings us to two unanswered questions: 1. What are. the cr i ter ia of setting up of diss ipat ive forces i n a superfluid flow? 2 . What i s the mechanism by which superfluid i s dragged by normal fluid? It i s not poss ible to comment much on the first question although we could say that the weight of evidence does indicate that Reynold 's cri terian for set t ing up .of turbulence i s not applicable, to the case of l i q u i d H e . A s to the second question, there have been a few empi r ica l approaches to explain the observed behaviour •though: Gorter and M e l l i n k ' s (1949) seems to expla in a good number of observations. .Accord ing to these authors, there exis ts a mutual fr ict ion force proportional to the cube of the relative ve loc i ty between the normal and the superfluid. Although results of wide channels can be approximately explained by this type of mutual friction force, yet the same results fit very n ice ly the c l a s s i c a l equation for turbulence (Daunt and Smith 1954). Furthermore, experiments of H o l l i s - H a l l e t t (1955) on large amplitudes of o sc i l l a t i ng d i s c , definitely prove that by i t se l f the mutual fr ict ion could bring superfluid i n motion and i n mot ion , the latter is able to exert a tangential retarding force on a moving s o l i d boundary. F i n a l l y , i t i s observed that the superproperties of He II undergo a radica l change at or below 0 , 6 ° K . The. temperature dependence of the specif ic heat, thermal conduct ivi ty , entropy, second sound, relative density,of normal f l u i d , e t c . undergo a marked change i n the v i c in i t y of 0 . 6 ° K . T h i s behaviour suggests a Debye- l ike structure of l i q u i d He II i n this region of temperature. T H E O R E T I C A L A S P E C T S We sha l l now briefly describe two .theories of l i q u i d He II which interpret T i s z a ' s vague picture of two penetrating f luids-i n a definite manner. These are macroscopic quantum mechanica l theories due to London (1938) and Landau (1941, 1944). In addit ion, there are a number of other theories and the reader is referred to the review article by Dingle (1952). London's Theory; London (1938) pointed out the s imi l a r i ty between the transformations of He I into He II and. the degeneration which can occur at sufficiently low temperature in a perfect Bose-Eins te in g a s . In the perfect Bose-Eins te in gas , the degeneration or condensation consis ts i n a finite fraction of a l l the part icles dropping into the lowest energy state where they are characterised by a uniform distribution i n configuration space but by,only a single momentum; i . e . they are ordered or "condensed" in momentum space . Th i s condensation corresponds to a second order phase change (there i s a discontinuity in the slope of the specif ic heat). Below the degeneracy .temperature, given by the.temperature of the peak appearing i n the specific heat curve, the part icles drop into the lowest energy l eve l i n increasing numbers and the entropy of the gas drops rapidly towards zero . Since he l ium atoms, obey Bose-Eins te in s t a t i s t i c s , London sought to e x p l a i n the properties of He II by this gas m o d e l . The,atoms i n the lowest energy state correspond to "superfluid" postulated by T i s z a . London's ideas have been investigated a great deal by a number of other invest igators . Landau ' s Theory: Landau (1941, 1944) rejected the idea that the type of s ta t i s t ics obeyed by He atoms has anything to do wi th the superfluid properties of He II . Accord ing to Landau , at absolute zero the l i qu id would be i n i ts ground state, which state was assumed to be free of vort ic i ty i . e . , in this state, only longitudinal phonons could be exc i t ed . It i s natural to suppose that vorticity would increase the entropy of the system and hence.excitation of vort ici ty which corresponds to f in i te , quantised energy increment, would represent departure from the zero temperature state. Landau pointed out that the exci ta t ion of one or more units of sound-wave energy or phonons could also give r ise to departures from ground state . The unit of vortex motion was ca l l ed a "roton" i n analogy to phonon. States near the ground state^therefore, were characterised by the numbers and energies of the phonons and rotons superimposed on the ground state . A s long as this superimposit ion i s va l id , , the specif ic heat and entropy of the sys tem are ident ical wi th the specif ic heat of the excitat ions which quantities may be calculated when their energy spectra are known. On.this m o d e l , the ground state and the exci ta t ions , respect ively , p lay the role of the superfluid and the normal f l u i d . We. should note that London's theory of Bose-Eins te in l i q u i d has .toad considerable success i n expla ining the properties of l i q u i d He i n the neighbourhood of T ^ . It i s interesting to remark here that although London's theory predicts a A - anomaly in specif ic heat, yet neither London 's theory nor any other theory put forward so far has been able to expla in successful ly the experimental ly observed shape,of this A - anomaly . The entirely different behaviour of l i q u i d He^ obeying F e r m i - D i r a c s ta t is t ics as compared wi th that of Bose-Eins te in l i q u i d H e 4 indicates the important role of s ta t is t ics i n the unique, behaviour of H e 4 I I . On the other hand, the model proposed by Landau , which is v a l i d at lower temperatures, i s unique i n i t se l f in predicting a radica l ly different phys i ca l behaviour of He II at or below 0 . 6 ° K . Accord ing ly , the properties of He II at or below 0 . 6 ° K should be characterist ic of Debye- l ike structure of phonon gas of He II . Temperley (1952) outlined a theory attempting to reconcile the two models of London and Landau by assuming that while the former i s v a l i d above 1 . 5 ° K , the latter holds below 0 . 6 ° K . For detai led account of both experimental and theoretical aspects.of the l i q u i d He problem, the reader is referred to the recent review art icles by A t k i n s (1952), Daunt (1952), Dingle (1952), Jackson (1952), Daunt and Smith (1954) and several other art icles i n Volumes I and II of "Progess i n L o w Temperature P h y s i c s " edited by Gorter (1955 ,1957) . C H A P T E R II N O N - I S O T H E R M A L H E L I U M II F I L M F L O W We know that l i qu id he l ium II can transfer i t se l f from one container into another by means of a superfluid f i l m of about 100 atoms th ickness . T h i s phenomenon, ca l l ed " F i l m F l o w " was discovered by R o l l i n and Simon in 1923 and since then has been investigated a great deal by a number of research workers . L i spite of the fact that there exis ts a lot of data on the superfluid property of he l ium f i l m , there are a number of controversial i s sues , investigation of some of which i s responsible for the project described i n this chapter. The Problem F i l m flow has generally been studied under isothermal condi t ions, flow of f i l m being induced by gravity or thermal potential or a combination thereof, and further the moving f i l m being under two different condi t ions , namely , covered and uncovered. The f i lm, f lowing through a w i r e - f i l l e d rtube superleak w i th an outer meta l cover intact , i s an example of covered f i l m . Here the f i l m i s i n thermal equi l ibr ium wi th the immobi le vapours trapped i n the superleak. On the other hand, the f i l m flowing through a wire f i l l e d tube superleak wi th i ts meta l cover peeled off, or the f i l m along a s o l i d surface dipped i n l i q u i d he l ium bath, are examples of the uncovered f i l m . In this case , the f i l m is in equi l ibr ium wi th vapours of the l i q u i d i n the container. The case of f i l m flow under non-isothermal conditions rather than isothermal has some interesting features which we w i l l d iscuss now. It should be mentioned that experiments s imi l a r to that reported by Daunt and Mendelssohn (1939, 1950) i n which the uncovered f i l m flowing out of a glass beaker immersed i n a bath of l i qu id he l ium II i s warmed at a point in between the two transfer points , cannot be c l a s s i f i ed under non-isothermal condi t ions . T h i s is. due to the fact that,with a very sl ight temperature gradient, the f i l m evaporates (Fairbank and.Lane 1949). Furthermore, the experiments in which, f i l m flow i s induced thermally a s , for example , i n the.case.of Daunt and Mendelssohn (1939) and Kap i t za (1941), the flow i s under isothermal conditions and temperature gradient i s . i n the form of a sharp temperature jump extending over a distance of the order of the thickness of the boundry layer (Kapi tza 1941). Non-isothermal conditions can be achieved, very Conveniently, by letting,the f i l m , f lowing between the two end points (reservoirs), flow through a temperature w e l l such that the f i l m does not evaporate. The covered f i l m cannot evaporate-as long as i ts temperature i s below & -point whi le the uncovered one can be prevented from doing so by pass ing through the temperature w e l l f i l l e d wi th l i q u i d h e l i u m . T h i s forms the bas is of the arrangement used i n the present inves t igat ions . Brown and Mendelssohn (1948) made an interesting observation i n studying non-isothermal flow of the uncovered he l ium f i l m . In this experiment, the f i l m connecting two reservoirs at the same temperature but different gravitational heights, was exposed at i ts midpoint to coo l ing by pumping . It was found that the flow i n either direction was stopped when a s m a l l temperature reduction appeared at the midpoin t . Two explanations of the effect seemed p o s s i b l e . One was that the f i l m was unable to flow through a temp-erature w e l l , i . e . , over a thermomechanical potential barrier which consequently would be in marked contrast to the behaviour of a syphon flow characterised by a flow independent of any gravitational potential present during the course of the f low. Another explanation was suggested later by observations of Bowers, Brewer and Mendelssohn (1951) and L o n g and Meyer (1952, 1953) of a possible shift i n X- tempera ture wi th a reduction in saturation of the f i l m . There has been disagreement between different workers, however, as to the interpretation of experiments on superfluidity of unsaturated f i l m . The unsaturated f i l m i s one which is in equi l ibr ium wi th i ts vapour at l e s s than the vapour pressure. For detailed account of the divergent results on unsaturated f i l m , the reader i s referred to a review article by L o n g and Meyer (1953). It i s uncertain as to whether the superfluid flow i n the unsaturated f i l m begins at the normal \ -temperature and was inhibited i n particular experiments by thermomechanical effects produced by pumping over the f i l m , or the flow begins at an "onset" temperature lower than the ^ - p o i n t by an amount depending on the saturation of the f i l m . Long and Meye r (1953) concluded that a l l . the experiments which had shown a_sharp onset temperature had i n common the feature that, " a l l superfluid moving i n the process must evaporate.completely to keep the process go ing" . Thus , the problem consis ts i n designing experiments to clarify the above mentioned resu l t s . A t the same t i m e , opportunity i s taken to test further a suggestion .made by Waring (1955) that, under certain condi t ions , the . thermally induced transfer rates may be quite different from the gravity induced ones . In.fact, on basis of a single observation of gravity induced flow in his sys t em, he has found the thermally induced flow rate to be about two t imes that induced by gravi ty . It i s mteresting to remark here that.in Waring's arrangement for measuring thermally induced f i l m flow rates below 1 ° K , the f i l m flow i s essent ia l ly non- i so thermal . T h i s point i s d iscussed later i n the next chapter. Exper imenta l Arrangement (a) General: The cryostat and the auxiliary, equipment used for low temperature experiments are of sausual des ign . The auxi l iary equipment, p r imar i ly cons is t ing of vacuum sys t em, i s schemat ica l ly drawn i n To He storage vacuum gouge To cryostat -($)-o'ut-<T-4> Hg Oil manometers F ig . 6 The Vacuum System to face page 17 Figure 6 . The vacuum sys tem consis ts of a single stage mercury diffusion pump P2 preceded by a rotary pump P^ and followed by a l i qu id air t rap. F is a f lask acting as a ba l las t and.also, because of i ts .connect ions, . as a trap for o i l or mercury rushing i n or out of the equipment in case of acc ident . and rrij are mercury and o i l manometers respec t ive ly . The density of this o i l (Apiezon B) i s 15.85 t imes smal ler than that of mercury: A . s m a l l o i l manometer ni2 i s connected to the inner brass chamber (to be described later) to measure the vapour pressure of l i qu id he l ium i n this chamber. F 2 , F 3 are two 5 l i t re reservoirs for storing clean he l ium gas . G , D are vacuum gauge and discharge tube respec t ive ly . The discharge tube i s located such that i t can be.connected independently to different parts of vacuum system and hence { i s useful i n hunting leaks i n different parts . There are outlets provided for pumping the jackets of transfer tube (used to transfer l i q u i d hel ium) and the l i q u i d he l ium dewar. The experiments are done inside the l i q u i d he l ium dewar which f i ts into a l i q u i d air dewar. Both dewars are made of pyrex glass and are s i lvered such that two thin ver t ica l windows are. left to enable v i s u a l observat ions. The he l ium dewar (inner dewar) i s c losed at the top by means of a brass cap . A rubber gasket seals the dewar vacuum t ight . The.cap has german s i lver tubing outlets for Kinney pump, transfer tube, vapour pressure l ine and he l ium gas return l i n e . Both the vacuum sys tem as w e l l as cap. assembly are bui l t on a r ig id dexion stand as shown i n photograph of Figure 1. Fig. 7 System for measuring Non-Isothermal Film Flow Rates facing page 18 C M - H i C O V E R R E M O V E D WIRE F I L L E D T U B E -PUMPING TUBE - B E A K E R (b) Cu G L A S S S E A L - ' B A T H Cu B O T T O M r Cu-Ni C O V E R _ - INTACT C -/ The l i q u i d he l ium i s .obtained from a C o l l i n s ' l iquefier commerc ia l ly available from A . D . L i t t l e Company, and i s stored i n a 25 l i tre l i q u i d he l ium can (SuperairCo) which serves as transport v e s s e l . L i q u i d he l ium is transferred from this can to the cryostat v ia . a transfer tube.by means of overpressure i n the c a n . The transfer tube i s . a U-shaped double wa l l ed meta l tube wi th an arrangement to.evacuate i ts annular space . The horizontal part i s made of copper tubing whi le the two l i m b s , one leading.to the cryostat and the other to the can , are made of german s i lver tubing. The sys tem i s quite efficient i n transferring l i q u i d h e l i u m . In fact , for a w e l l pre-cooled dewar, there i s . a loss of about 0 .3 , l i t re of l i q u i d he l ium by evaporation i n a transfer of about 2 l i t r e . To pump over the l i q u i d he l ium bath, a large capacity K inney mechanical pump has been ins ta l led in the laboratory and is l i nked to the apparatus by a 4 inch p i p e . The rate of pumping is. controlled by means of a s m a l l and.a b i g valve i n para l le l to each other. The pressure over the he l ium bath can be varied continuously and read by mercury manometer for higher pressure.and o i l manometer for lower pressure (below about 40 m m . of H g . ) . . A t max imum pumping speed, a temperature of 1 .3°K can be obtained. Th i s i s helped by keeping the cap seal ing the he l ium dewar f i l l e d wi th l i q u i d a i r . (b) Arrangement for Measur ing F i l m F l o w Rates: The arrangement used to measure f i l m flow rates i s shown in Figure 7 . It consis ts of a U-shaped w i r e - f i l l e d tube ( w . f . t . ) f i l ter or superleak which a l lows.only f i l m to flow from one container (ca l led bath) into the other (cal led beaker) or v ice ve r sa . The wire-f i l led- tube (w.f . t . ) fi l ter i s described i n deta i l i n Appendix I . It essent ia l ly consis ts of a copper-nickel tube i n which there are 1200 constantan.wires (0.002" diameter) . It i s drawn down through holes of success ive ly decreasing diameters i n a s teel die plate unt i l c lose packing of wires starts resul t ing i n deformation of individual wires into hexagonal cross sec t ions . The filter used i n the present experiments has a mean channel width between the sides of the neighbouring hexagons of about 2.14 m i c r o n . ,For f i l m f low, the superleak has.an effective perimeter of about 19 c m . The ends of the w . f . t . superleak are bared and brushes are formed from the wires (as shown i n Figure 7) for good thermal contact wi th the l i q u i d he l ium bath . i The w . f . t . passes through the bottom of a brass chamber c and i s soldered in posi t ion as shown i n the figure. The brass chamber i s a 3 c m . long cylinder of 3 c m . diameter . A t one side of the U i s soldered a kovar sea l wh ich i s the top of a glass tube wi th a copper-seal bottom to ensure good thermal contact wi th the bath. Th i s glass tube, made.of pyrex g l a s s , is.about 5 c m . long and has inner diameter of 6 m m . and thickness of 1/2 m m . T h i s tube i s referred to as the beaker. The term'bath' refers to the l i q u i d he l ium i n the dewar. It shoukLbe mentioned that the terminology,of bath and beaker i s conventional i n f i l m flow measurements . A 1/2 m m . hole i n the kovar seal at the top of the beaker helps equal i s ing vapour pressures in the bath and the beaker. , J$ i s a capil lary,tube having a kovar sea l at the top which i s soldered into the bottom of the brass.chamber. Th i s i s used as an indicator of l e v e l of l i q u i d he l ium condensed inside the chamber. Whereas Figure 7 (a) refers to the case of covered f i l m , Figure 7 (b) refers to the arrangement for studying uncovered f i l m . In the latter case , the C u - N i tube i s peeled off at the bend of U for about a centimeter inside the brass chamber. Figure 7 (c) shows a double w a l l pyrex glass jacket surrounding the brass.chamber a s sembly . The annular space i s f i l l e d wi th l i q u i d he l ium by means of two t iny holes at the top and this acts: as a radiation s h i e l d . It was found that, as such , i t d id not prove to be . an efficient radiation sh ie ld unless i t was covered wi th a luminium f o i l or was s i lve red . A kovar sea l at the top of the jacket i s soldered to the 3 m m . german s i lver tubing leading to the brass chamber. The german s i lver tube i s connected to a 1/4" brass tube outside the cryostat v i a a radiation s h i e l d . T h i s radiation sh ie ld consis ted of a tube wi th two approximately right angle bends. The cryostat i s sealed from outside by means of 0-ring sea ls around the above mentioned brass tube. S l id ing 0 - r ing seals enable the brass chamber assembly to be moved up and down by about 20 c m . The sys tem i s connected to he l ium gas f lasks v i a a needle valve which i s used to control the rate of condensation of the gas in the chamber. The vapour pressure inside the chamber i s measured by an o i l manometer. F i g - 8 B l o c k Dioqrom of the Therm ometer fac ing poge2i; Furthermore, connection i s provided for pumping over the l i q u i d he l ium in the chamber independently of the.main bath. Inside the brass chamber, are f ixed two carbon resistors :(each 56 ohm Allan-Bradley type) . One i s used as. a heater and the other as .a thermometer. The four leads of heater and thermometer are of #40 Constantan (for low heat inf lux) . These are taken out of the cryostat through the. annular space: between moving brass tubes and are sealed at the room temperature end by means of seal ing w a x . The temperature of the chamber is also measured by vapour pressure measurement. The temperature of the bath and beaker are measured only by vapour pressure measurements wi th mercury and o i l manometers . The arrangement employed to measure the resistance of the carbon thermo-meter i s shown schemat ica l ly i n Figure 8 . It i s a part of the.thermo-regulators described by Boyle and Brown (1954). It i s essent ia l ly an alternating current Wheatstone bridge whose output i s ampl i f ied i n two stages and displayed on an-osc i l loscope . Whereas i n the thermoregulator the output i s fed back to an auxi l iary heater, here.the output i s balanced to n i l . The source i s a 400 c p s . generator. The two ratio arms are from 1000 ohm wire wound res is tors , whi le the variable i s a decade box . For balancing out the.reactive part, a variable condenser i s necessary. Th i s -arrangement measures temperature wi th an accuracy p f the order of 1 mi l l i deg ree . The measurement of the. rate of change of the l e v e l of meniscus of l i q u i d he l ium i n the beaker i s done v i sua l ly by means of cathetometer and stop wa tch . The cathetometer can read the l eve l of meniscus wi th an accuracy of 0.01 m m . The meniscus was i l luminated by ,a 2 watt neon bulb and la te r on by another source cons is t ing o f a 6 watt tungsten f i lament bulb supplied wi th variable power, the l ight being fil tered through a cuprous chloride c e l l .(which has a cut off frequency at about 8 0 0 0 A 0 ) and then focussed by a l e n s . Procedure The procedure followed in.transferring i l i qu id he l ium i s a conventional one. The pre-cool ing of the inner he l ium dewar i s done by f i l l i n g the outer dewar wi th l i q u i d ni trogen. The pre -cool ing can be made quite fast by r ins ing me inner dewar wi th l i q u i d nitrogen. s Th i s procedure has one disadvantage due to the poss ib i l i t y of clouding up of the dewar and hence making the v i s i b i l i t y quite poor. The rate of cool ing i s controlled by the pressure of exchange gas i n the annular space of the dewar. Usua l ly pre-cool ing i s done at a pressure of about 1 m m . of H g . and then later on , the dewar i s pumped hard. The pressure read on manometer gives a fair indicat ion of the cool ing p rocess . The he l ium dewar and i ts contents are first evacuated and then flushed wi th he l ium gas . In f i l m flow measurements, the effect of impuri t ies i s known to be very serious.. Hence, i n the present case , extraordinary precautions were taken to remove any water vapours or other foreign gases which also could.block the w . f . t . superleak after so l id i f i ca t ion . The he l ium dewar w a s , thus, kept under vacuum for as long as.24 hours and flushed wi th c lean he l ium gas for a few t i m e s . After the he l ium dewar i s at about l i q u i d nitrogen temperature, l i q u i d he l ium i s transferred into the dewar. The l i q u i d i s then pumped by means of the Kinney pump to obtain lower temperatures. The pumped gas i s returned to the so -ca l l ed dirty he l ium storage from which i t i s cleaned for using again i n Miquef iCat ion . The gravity induced flow rates are measured by observation of the rate of change.of the meniscus l e v e l i n the beaker when the beaker i s ra ised up or lowered down as compared wi th the l eve l of the l i qu id i n to the bath . Empty ing (flow in»the beaker) and f i l l i n g (flow out of the beaker) rates induced gravitationally are measured i n two different c a s e s . F i r s t l y , the bath beaker and the chamber are kept at the same temperature. T h i s i s .the case of isothermal f low. Secondly, the chamber i s at a temperature different from that of bath and beaker which are at the same temperature. In this case of non-isothermal flow, the chamber i s warmed by means of a heater in i t and is cooled by pumping over the l i q u i d he l ium i n the chamber. Furthermore, above mentioned measurements are made under-two different condit ions; (1) W h e n C u - N i cover of the w . f . t . superleak i s intact (covered'fi lm) (2) When C u - N i cover of the w . f . t . superleak i s peeled off to about 1 c m . length inside.the chamber (uncovered f i l m ) . - 4 0 T J T = 1-96 K B a t h 1 1 1 -30 - 2 0 -10 AT (milfi d e g r e e ) O Cr O XT 8 o a» E a rM0 0 - 6 0 - 0 6 - 1 - 2 H - 8 Fi9-9 (a) Transfer rates of the Covered Film V r s Temperature Difference between Bath and the Chamber (Cool in g Case) facing page 2.^ _ 1*8 6 rsl O X T>|T> J*- 0-6 - 0 - 6 -1-2 <5=-10 •Cr TT • a o T = 1-96 K Bath 1 1 20 30 4 0 A T (millidegree) O -XT •O Fig- 9(b) Transfer Rotes of the Covered Film V s A T ( Warming Case) facing page£4 3-0 u 0> E 2 0 \ L\ o <\) O X \ b - 1 0 * \ \ \ \ \ \ 1 T = 1 4 7 4 K Bath 1 1 4 6 , 8 A T (millidegree) - 2 0 \ - 3 - 0 & - o - o F ig lO Transfer Rates of the Uncovered Film V's a T (Warming Case) facing p a g e ^ Resul t s : Resul ts for different cases are shown graphically in Figures 9, 10 and 1 1 . The emptying (-dh)and f i l l i ng (-}- dh ) rates for aT ~dt the beaker are plotted against temperature difference . A T between that of chamber and ba th . A T i s posi t ive or negative as the chamber i s warmer or colder as compared wi th the. bath. Isothermal F l o w : The flow rates for this case ( A T sr .o) for both covered as w e l l as uncovered f i l m are the same as u s u a l . These results are not shown. Non-isothermal F l o w : (1) Covered F i l m : The transfer rates are shown i n Figure 9 (a) and,(b) and are found to be independent of heating or cool ing of the chamber. It should be mentioned that whereas the max imum amount of cool ing achieved i s l i m i t e d to about 40 mil l idegrees because.of the pumping speed and imperfect thermal i so la t ion of the chamber from l i q u i d he l ium bath, there.is no l i m i t to heat ing. In fact , excess ive heating ( A T about 100 mi l l idegrees) leads to compl icated effects l ike decrease of flow rate and even reversal of the f low. Th i s can be attributed to evaporation and possible, differential thermomechanical pressures i n the two l i m b s of U . (2) Uncovered F i l m : T h i s case i s quite interesting and is shown in Figures 10 and 1 1 . In case of heating (Figure 10), the beaker empties at the same rate whi le i t f i l l s at a rate which goes on decreasing unt i l T = I 9 6 0 K Bath - 8 - Q - - O - ^ 0 — ^ J -- 6 - 4 A T ( m i f l i d e g r e e ) - 2 - j ) l - 8 / I \ I U Q> 1-2 e o X 0 - 6 "° - 0 - 6 \ 4 r l - 8 Fig 110 Transfer Rates of the Uncovered Film V s Temperature Difference between Bath 8» Chamber (Cooling Case) facing page2.5 at A T = 2 mi l l idegrees , it stops f i l l i n g . For A T 2 mi l l idegrees , it starts emptying at the same max imum rate as i n the. case of isothermal f low. The case of coo l ing (Figure 11) i s more s t r ik ing . Here the flow stops either way wi th cool ing of the order of 1 mi l l idegree The cool ing i s absolutely necessary for the flow to s top. With further coo l ing , the beaker neither empties nor f i l l s . D i s c u s s i o n T h e results indicate that the covered f i l m under non-isothermal conditions flows at the same c r i t i c a l rate .as both covered as w e l l as uncovered f i l m flow under isothermal conditions; flow i n both cases being induced by gravi ty . The case of flow of uncovered f i l m under non-isothermal conditions i s , however, compl ica ted by additional effects. The results are e a s i l y explainable i n case, of warming up of the uncovered f i l m i f we superimpose the thermomechanical pressure on the gravitational pressure keeping i n m i n d , however, that the flow rate cannot exceed the c r i t i c a l va lue . Since there exis ts a thermomechanical pressure from the bath as w e l l as the beaker to the warmer chamber, both bath and beaker empty into the chamber. T h i s w i l l not effect theempty ing rate of the beaker w h i c h , although i t has an additional pressure head, cannot empty at a faster rate. A s the f i l l i ng of the beaker by gravity i s opposed by the rmal emptying, the f i l l i n g rate decreases unti l the latter overcomes the former, result ing in emptying . The emptying rate approaches the c r i t i c a l rate after pass ing through the subcr i t ica l rate of f l o w . We can draw two conclusions from the above d i s cus s ion . F i r s t l y , we have a further confirmation of the fact that the f i l m transfer rates are the same whether produced by thermal or gravitational potentials either singly, or superimposed. T h i s conclus ion supports that of a number of earl ier experiments i n contradiction to the poss ib i l i t y suggested by War ing . Secondly, results show that the covered f i l m can flow through a thermomechanical potential barrier without any change in flow rate . T h i s property i s analogous to the flow of a l i q u i d through a syphon. However, we know that whereas f i l m flow is independent of the pressure head as w e l l as length of the.channel, the flow through syphon does depend on both parameters as given by the fo l lowing relation for rate of volume flow dv = . A p d 3 dt where A p / 4 . is the pressure gradient over the l e n g t hZo f syphon, d i s the diameter of the syphon and i s the v i s cos i t y of the l i q u i d . Nevertheless i t i s poss ib le to mainta in the analogy between syphon flow and he l ium f i l m flow because of the existence of c r i t i c a l rate of transfer along wi th van i sh -ingly s m a l l v i s c o s i t y of l i q u i d he l ium f i l m which characterise the f i l m flow as a c r i t i c a l ve loc i ty l i m i t e d syphon f low. Now we sha l l d iscuss the case.of cool ing by pumping. The results are s i m i l a r to those.observed by Brown and L o n g and M e y e r . In conjunction wi th other results as d iscussed above, a possible explanation can be suggested. The fact that when covered, the f i l m can flow through a temperature w e l l , makes . less l i k e l y a sp l i t t ing of the f i l m by thermo-mechanica l effects alone i n the case, of cool ing the uncovered f i l m by direct pumping . On the.other hand, the latter si tuation is not one which fits Long;and Meyer ' s c l ass i f i ca t ion of. experiments showing a sharp onset temperature, that " a l l the superfluid moving in the process must evaporate to keep the process go ing" . Rather i t would seem that the . i m m o b i l i t y of the f i l m under pumping i s due to a thinning down of the f i l m to its non-superfluid layers or a shift of the A - p o i n t , poss ibly two aspects of the same effect. In this connection, Atk ins (1954, 1955) has recently made some interesting remarks . In h is theory, of formation of h e l i u m f i l m , Atkins-has establ ished that the mean density of the he l ium f i l m i s s l ight ly greater than that of the bulk l i q u i d . Th i s increase in density becomes quite large for th in f i l m s . Th i s effect could be very important in the case-of unsaturated f i l m s . According to A t k i n s , increase in density may also expla in the low temperatures for the onset of superfluidity, observed in thin f i l m s . However, this.suggestion is merely based on qualitative arguments. C H A P T E R III H E L I U M II F I L M F L O W B E L O W 1°K  Introduction It i s w e l l establ ished that the "super" properties of l i q u i d he l i um II e . g . , superfluidity, superthermal conduct ivi ty , second sound e t c . , are strongly temperature dependent. They have been extensively studied above 1 ° K . How they,are modif ied by progressive lowering of the temperature, i s . a subject of great interest to both the theoretical and the experimental p h y s i c i s t . It should be mentioned that the dis t inct ipn between l i q u i d he l i um above and below 1°K was or ig inal ly made only for purely technical reasons . Whereas temperatures above 1°K can be.attained by ordinary pumping on the l i q u i d , temperatures below 1°K can be attained readily only be.a recourse to the magnetic coo l ing techniques. However, now the phys i ca l behaviour of the l i q u i d , wh ich undergoes radica l changes at or below 0 . 6 ° K , just if ies this d i s t inc t ion . The transfer rates of l i q u i d he l ium f i l m above 1°K have been studied by a number of investigators reference to whom i s already given i n the previous chapter. A typ ica l transfer rate, versus temperature curve has already been shown i n Figure 4 . It i s interesting to note that although various investigators find essen t ia l ly s imi l a r temperature dependence, yet the absolute values of the transfer rates measured vary by a factor of as much as three. The major part of such variations can be accounted for by impuri t ies or irregularities and evaporation effects. However, these s t i l l remains an unexplained.though complete ly random variation to which attention has been drawn by M c C r u m and E i sens t e in (1955). In general, the transfer rate curve can be fitted in the equation where R is the transfer rate. in c m 3 / c m . s e c . , R 0 i s a constant, T i s temperature and T ^ i s temperature at A - p o i n t . A s we know that the transfer of He II i s due to the creeping f i l m which consis ts of superfluid of density J * s , has a thickness d c m and moves wi th a c r i t i c a l ve loc i ty of y c c m / s e c . , therefore, assuming the density of He II transferred as ^ ( f => ^ s f pn where p n i s density of normal f lu id) , the observed rate of transfer can a lso be written as R - £ s v c d (2) f L e t us interpret equation 2 . E a r l y investigators (Bijl „ deBoer and M i c h e l s , 1941) observed from data on the f i l m that the product v c d i s approximately,constant and in fact i t i s ~> ^4»v w h i c h , inc identa l ly , i s an expression of Heisenberg's uncertainty re la t ion . Th i s hypothesis seems to have been taken for granted as variations of R wi th temperature could s i m p l y be explained due to variat ion of relative c o n c e n t r a t i o n / p of superfluid. T h i s i s borne out by experimental .results of Andron ikashv i l l i (1948) which can be expressed in the form - , 5.6 ff° - f - - (I) T h i s , being true from 1 .3°K to Arpo in t , explains equation 1. A t present, refined data on . f i lm thickness clearly indicates that d i s not constant and i t probably varies wi th temperature. The variat ion of v c d , i f any, could be significant below 1°K w h e n l i q u i d he l ium i s completely superfluid (99.99% at 1°K) and so variation of R i s so le ly due to the variat ion of v c d . We sha l l now review c r i t i c a l l y some previous investigations wh ich negate the hypothesis that v c d i s constant . The problem of attaining low temperatures by magnetic cool ing i s no longer ser ious . However, maintaining of low temperatures for a reasonably long t ime taken to measure the f i l m transfer rates i s not so s imple because of warming-up of l i q u i d due to heat in f lux . Since vapour pressure i s extremely s m a l l at low temperatures (1.6 x 10 " 'mm. of H g . at 0 . 5 ° K ) , evaporation and re condensation contribute l i t t l e to the heat in f lux . The m a i n source of heat influx wi th an open ve s se l i s the f i l m flow i t s e l f . The f i l m can creep up, evaporate,somewhere i n the upper parts of the cryostat and recondense i n the low temperature ve s se l del iver ing a disastrous amount of condensation heat . Incidental ly, one can see why the knowledge of f i l m transfer rates below 1°K would be of prac t ica l importance i n designing an open v e s s e l cryostat us ing l i q u i d heliumfas a coolant below 1 ° K . Although an open ves se l has the disadvantage of heat inf lux , yet it i s convenient because he l ium can be eas i ly condensed. A compromise can be found by us ing a narrow cap i l l a ry . Th i s method has been applied both by Ambler and K u r t i (1952) and Waring (1955) i n their study of f i l m flow below 1 ° K . In contrast to these techniques, a c losed capsule technique has the advantage of very s m a l l warming-up rate . Th i s latter technique,. to be described later , i s used i n the present inves t iga t ion . Amble r and Kur t i (1952) were the first to measure f i l m flow rates below 1°K by us ing an open bath and beaker arrangement f i l l e d wi th l i q u i d he l i um and cooled by adiabetic demagnetisation of manganous ammonium sulphate. They could observe only emptying rates of the beaker and their results are shown, i n Figure 15 . Th i s indicates a steady increase of flow rate below 0 . 8 ° K . A t 0 . 3 ° K , the flow rate is about 30% higher as compared wi th the m i n i m u m .value just above 1°K , although different runs show fluctuations as big.as 15%. The method used is subject to some sources of error. A s already pointed out, the heat influx is the major source of error i n temperature measurement. Since only f i l l i ng rates are observed, therefore, any anomolous evaporation could not be detected by a comparison wi th f i l l i n g ra tes . Furthermore, impuri ty effects are quite obvious in their measurements as the magnitudes of transfer rates above 1°K are a lmost twice that of Daunt and.Mendelssohn va lue . In addi t ion, further experiments , part icularly wi th different geometry, are quite desi rable . Lesensky and Boorse (1952) attempted further measurements below 1°K using copper surfaces and a capaci t ive method of measuring leve ls wh ich obviated the need for v i s u a l observations. Their results seemed to confirm a r ise i n flow rate below 1°K but difficulties, i n obtaining ;a suff iciently long warm-up t ime prevented their going below about 0 . 7 5 ° K . EXiring the course of the present work, a prel iminary report of which was given i n Conference i n Low Temperature Physics and Chemis t ry , L o u i s i a n a State Univers i ty (December 1955), Waring (1955) published his results on f i l m flow below 1°K which are also shown i n Figure 15 . There i s a marked m i n i m u m i n flow rate curve near 1°K and then i t r i ses by 10% from this m i n i m u m va lue . The curve i s flat below 0 . 5 ° K and has indicat ion of being quite independent of temperature below 0 . 5 ° K . The absolute value of R above 1°K i s about three t imes that of Daunt and Mendelssohn va lue . Measurements of flow rate are made by co l lec t ing the he l ium gas obtained as a result of evaporation of the f i l m flow which i s induced thermal ly . On the bas is of a soli tary observation of f i l m flow rate induced gravi ta-t iona l ly i n the same apparatus, Waring suggested that the flow rates induced thermal ly are higher than those induced by gravi ty . Th i s suggestion is contrary to the observations of a number of investigators and has already been d iscussed i n the previous chapter. A n explanation to Waring's result has been afforded by us (Hebert, Chopra and Brown, 1957) and is given i n the d i scuss ion at the end of this chapter. The above review c lear ly suggests the desirabi l i ty of further investigations of f i l m flow rates below 1°K wi th improved experimental technique to overcome sources of error present i n experiments of previous workers . Moreover, rel iable data on f i l m flow i n this region of completely different though highly s impl i f i ed phys ica l behaviour of l i q u i d He II would be of great importance to theoretical phys ic i s t s i n formulating a consistent theory, of f i l m f l o w . Exper imenta l Arrangement (a) The Cryostat The cryostat and i ts auxi l iary equipment i s s im i l a r to one already shown i n Figure 6 . But because of i ts use in magnetic coo l ing , i t has some additional features. The vacuum system i s very efficient and consis ts of a two stage rotary pump preceding a two stage o i l diffusion pump designed to give high speed a t low pressures. The speed i s important i n decreasing t ime required i n adiabatic demagnetisation whi le the ult imate vacuum reached i s in t imately related to warm-up t i m e . The nitrogen as w e l l as the he l ium dewar have a t a i l in which the paramagnetic sal t i s brought for magnet isa t ion. The t a i l i s a common feature of magnetic cool ing cryostats . T o get an idea of the max imum s ize of paramagnetic sample that can be used, fo l lowing are the outer diameters of the t a i l s : He dewar 1.50" Nitrogen dewar 2 .00" Inside the he l ium dewar i s another pyrexglass jacket wi th a . ta i l of outer diameter of 12 m m . T h i s jacket i s also s i lvered and has two s l i t s for v i s u a l observations. The sealed glass capsule containing paramagnetic sal t i s suspended i n this j acke t . This jacket acts as a thermal swi tch and this act ion i s due to the exchange gas present in i t . The efficiency of the thermal switch i s controlled by the amount of exchange gas .which i s given to i t from a s m a l l capi l lary tube f i l l e d wi th clean he l ium gas . The pressure of exchange gas i s measured by a suitable gauge which in our case , due to troubles i n the Ph i l l i p s gauge, was M c L e o d gauge. Although the latter i s quite inconvenient to use , yet because of i ts absolute measurement, i t proved to be of great value i n the success of Our experiments , (b) Sealed Glass Capsule Hebert (1956) successful ly sealed glass capsules or "bombs" . A t y p i c a l bomb i s shown schematical ly" in Figure 12 . (a l soshown i n photograph in Figure 26) . It contains paramagnetic sa l t for magnetic coo l ing , bath and beaker arrangement used i n a conventional way to measure flow rate, and high pressure he l ium gas . A t l i q u i d he l ium temperatures, he l ium gas condenses to give enough l i q u i d for studying transfer rate of the f i l m . The c losed capsule technique was or iginal ly devised by K u r t i , Rol l i n and Simon (1936) in an experiment on therma 1 relaxation t i m e . La t e r , a number of other investigators used this technique. Whereas these investigators used meta l capsules , Hebert succeeded i n sea l ingpyrex glass capsules making v i s u a l observations p o s s i b l e . Hebert used these capsules to measure f i l m transfer rates above 1°K but due to the diff icul t ies .of quick warm-up , he was unable to extend these observations below 1°K which were later on undertaken by the author and are reported here. The sealed capsule underwent a few developments. Two of the models are shown i n photograph in Figure 26 . Mode l (h) i s used i n the present invest igat ion. It consis ts of an ordinary pyrex glass tubing of 8 m m . outer diameter, 1/2 m m . thickness and about 5 c m . l o n g . T h i s tube acts as the bath whi le inside i t i s the beaker in the form of a capi l lary bore 0.059 c m . The lower part of this capi l la ry tube, ca l l ed the shoulder, fits the bath s i n g l y . To avoid strains in the glass beaker, the insert i s not fused to the w a l l . A f i l m of glycerine i n the gap between the two reduces the amount of he l ium required to be sealed into the capsu le . The upper part of the capi l lary has i ts w a l l s ground to the m i n i m u m possible thickness without harming the bore of the cap i l l a ry . The empty space, above the shoulder so formed, is f i l l e d wi th paramagnetic sal t - manganous ammonium sulphate. T h i s sa l t , a lso used by Amble r and K u r t i , has some advantages. It i s possible to pump over i t at s a l t - i ce temperature without decomposing i t . Pumping i s necessary to remove foreign gases which would give sea led- in impur i t i e s . The sal t has high specif ic heat down to about 0 . 2 ° K which makes it quite suitable for use i n magnetic cool ing down to this temperature. Furthermore, i ts properties as a magnetic refrigerant and thermometer are w e l l known. T h e m a x i m u m amount of sal t that could be used in the biggest possible bomb was about 0 .5 g m . whereas weight of the capsule , cooled by the thermal contact of about 0.05 g m . of l i q u i d he l ium inside i t , was 120 g m . The glass capsule has its neck drawn out to about 2 m m . . diameter and 1 c m . length . It i s p laced i n a brass case to which clean he l ium gas can be admitted to a pressure of 750 p s i g . A conica l tungsten heater surrounding the tip of capsule i s heated hot enough to seal off the end. Detai ls of the seal ing operation are described i n Appendix II . Amount of he l ium gas sealed when condensed produces a fraction of c . c . of l iqu id h e l i u m . It should be noted that both bath and beaker are under the same adiabatic conditions which i s an additional advantage in this technique. (c) Suspension and Dumping Arrangement The l i qu id formed by condensation of he l ium gas i n the capsule f i l l s both the bath and the beaker (the capi l la ry tube.)!. For measuring the f i l l i n g rate of the beaker by gravitationally induced f i l m f low, the capi l la ry has to be empt ied . Th i s i s done by dumping the capsule in the inner jacket and.then reversing i t aga in . In addition to this operation, we have to have an arrangement to move the capsule down into the t a i l for magnetic cool ing process as w e l l as measurement of temperature of the capsule and then up into the bulge of the inner vacuum jacket for f i l m flow measurements . The stringent requirements of suspension and dumping sys tem can be rea l i sed from the fol lowing conditions which have to be sat isf ied: F i r s t l y , the capsule of outer diameter of 8 m m . has to move in and out of a t a i l of 12 m m . inner diameter without touching i t . These are max imum possible dimensions and are l i m i t e d by the s ize of dewars ava i l ab le . Secondly, the capsule of total length of 53 m m . has to be dumped inside the inner jacket of inner diameter of 60 m m . (again l i m i t e d by the s i ze of the He dewar) without touching the wa l l s of the dewar. Touching w i l l give r ise to anomdlous warming up of the capsule . Th i rd ly , the whole process of moving up and down for f i l m transfer as w e l l as temperature measurements has to be done almost b l indly so that after every operation, the capsule occupies the same pos i t ion . Furthermore, the operations have to be gentle enough to prevent vibrations (which could also result i n quite warming up of capsule) and at the same t i m e , the operations have to be performed quickly so as to be able to make measurements before the capsule warms up appreciably. A n attempt was made to overcome most of these d i f f icu l t i es . Before we describe the f inal arrangement, i t i s interesting to describe the different developments . In a l l cases to be described the suspension system i s essent ia l ly the s a m e . The capsule i s suspended by means of thin nylon threads which are known to have very low thermal conduct iv i ty . On both top as w e l l as the bottom of the bomb are cemented teflon rings by means of household cement (Duco). Two threads are attached to the bottom ring and one to the top r i n g . The threads at the bottom pass through two d iamet r ica l ly opposite hooks i n the bulged part of the inner j acke t . The thread at the top was kept free i n a l l other arrangements except the f inal one i n wh ich i t i s a lso passed through one of a number of holes i n the third hook. T h i s hook consis ts of a long horizontal glass strip s i tuated symmet r i ca l ly w .r . t . the above two hooks and has a number of holes d r i l l ed i n i t wh ich are.very near to the centre of the inner jacke t . It i s poss ible to select one of these holes so that when the capsule i s dumped to be hor izonta l , it does not touch the w a l l s of the j acke t . It should be mentioned that a sp l i t brass r ing (spl i t to avoid eddy,currents) is cemented at the top of the capsu le . T h i s brass r ing i s heavy enough to make C . G . of the.capsule as near to the centre as p o s s i b l e . T h e two threads.at the bottom are t ied together to a single one. Thus , i t i s clear that the relative movement of this s ingle one and the thread at the top can be used to perform reversing operation whi le the two threads moving together enable the capsule to move up and down. The arrangement for the relative and combined movement of these threads, ca l l ed here the dumping arrangement, underwent a series of developments. F i r s t two and, later on , three independently rotating ground glass winches were used to move the individual threads. Then two concentric ground glass joints were used so that the two threads could be moved together as w e l l as independently. T h i s was replaced by the concentric meta l winch sys tem wi th s l ight modifications and proved to be more satisfactory than any other. However, the m a i n sources of troubles i n a l l these cases were: (1) Jamming and breaking of threads during movements . (2) Long t i m e taken for movements resul t ing i n warm-up capsule before any measurements could be taken. (3) Due to s l ipp ing and loose windings of threads, winding followed by unwinding d id not result i n the same posi t ion of the bomb. (4) Large vibrations of the capsule result ing i n touching, of the capsule to the w a l l s . These troubles were overcome by the use of concentric s l i d i n g O-r ing seal arrangement. The threads are t ied to the hooks at the ends of non-magnetic thin m o l y b d i u m . These wires pass through two separate s m a l l holes i n a radiation sh ie ld and are then soldered to two concentric brass rods . TheTatter can be moved together as w e l l as independently through s l id ing O- r ing s e a l s . The radiation sh ie ld consists of a brass tube about 5 c m . long , wi th i ts ends soldered by half d i scs wh ich have s m a l l holes for molybdium wires to pass through, (d) L igh t Source Ambler and Kur t i had reported anomcftous warm -up of paramagnetic sa l t by absorption of l ight by the s a l t . A s a 'precaution, a spec ia l source of l ight was designed which was also found quite convenient for v i s u a l observat ions, by means of a cathetometer. The. l ight source consis ted of a 12 watt tungsten fi lament e lect r ic bulb wi th i ts input controlled by a va r i ac . The l ight was fi l tered through a cuprous chloride c e l l of 4 c m . length, 3 c m . diameter and normal concentration. Th i s fil ter has a cut o f fa t about 8000A° and so completely : e l iminates infra-red . Th i s l ight i l luminates a spec ia l ly designed s l i t whose length as w e l l as width can be va r i ed . The s l i t image i s then focussed by a converging lens mounted i n an adjustable tube.. The focussed image can be al lowed to f a l l only on the cap i l l a ry but not on the s a l t . Although a movable a lumin ium shutter around the window of the jacket was also provided, yet we d id not find i t too convenient to u se . Instead, switching the l ight source off and on seemed to be quite a convenient operation. (e) , The Magnet For magnetisation of paramagnetic sa l t , an electromagnet , wi th 4" pole face.diameter, iron core, adjustable gap, water cooled type, i s ava i l ab le . The current i s controlled by a pair of water cooled rheostats. It i s mounted on a carriage ro l l i ng on ra i l s so that i t could be eas i ly moved to and fro from the cryostat . The current i s s tab i l ized manua l ly . The f ie ld cal ibrat ion against current, effected by proton resonance method, is already ava i l ab le . The max imum f ie ld available is about 6 .5 k i lo -gauss at a current of 45 amperes for 1" gap though only a f ie ld of 5 k i lo -gauss at a current of 35 amperes and a gap of 3 .15" i s used i n the present inves t iga t ions . (f) The Suscept ibi l i ty Bridge The temperature of the capsule i s measured by us ing paramagnetic sal t as a thermometer. The salt used follows Curie ' s law quite accurately down to 0 . 1 ° K . Cookt(1955) gives X - — - 4.375-T " T where 5C is the suscept ib i l i ty per mole and T is the absolute temperature. Measurement of suscept ibi l i ty i s effected by a m u t u a l inductance bridge. The latter essent ia l ly consis ts of two mutual inductance c o i l s . One surrounds the sample whereas the second, c a l l e d the compensator, i s farther away from the sample and has a variable part. The secondaries in the two co i l s are i n series opposition and i t i s possible to adjust them so that the two mutual inductances effectively cancel each other when there is no sample i n the test c o i l . A s a resul t , as current is reversed or continually 2 - 2 4 V I l lF-A Ammeter with 0 0 5 0 t o 5 0 0 ompereshunts C External compensator • G Tinsley galvanometer with telescope 8> scale P Paramagnet ic sample S Secondary surrounding sample in the helium bath Secondary compensating part ial ly S| T Revers ing sw i t ch 5 6 • • -VVVWvW 4 6 * —'WJWWWV— 175 • — V W W V W 7IO f ~ ->VWWVW Li quid Helium Bath The D ' C M u t u o l inductance Br idge f a c i ng page changed in the pr imar ies , the net e . m . f . in the secondary c i rcui t is n i l . However, as a paramagnetic sample i s moved inside the test c o i l , the two mutual inductances cannot buck each other and one get a net e . m . f . output. Th i s output i s direct ly proportional to suscep t ib i l i ty and hence, inversely proportional to the temperature of the sample . There are two forms of mutual inductance br idges . One based on reversal of current i n the primary and measuring the e . m . f . produced by a h ick i n b a l l i s t i c galvanometer, i s the d . c . (bal l is t ic) bridge . The other, based on continuous change of current, i s a . c . br idge. The advantages of a . c . bridge are obvious . It measures temperature continu-ously and, furthermore, i t i s much easier to amplify i t for graphic detect ion. A s i n this case , i t i s quite desirable to measure temperature continuously part icularly during the measurements of f i l m transfer rates , so a . c . bridge was t r ied . M spite of the fact that the a . c . bridge had some obvious disadvantages over the d . c . bridge, the method was tr ied but i t d id not prove to be promis ing . Hence, i t was discarded and the d . c . bridge was employed . The a . c . bridge and i ts meri ts and demerits relative to the d . c . bridge are d i scussed i n Appendix III. The c i rcui t of the d . c . or ba l l i s t i c mutual inductance bridge i s shown i n Figure . 13 . The mutual inductance c o i l , around the paramagnetic sample , i s wound on a bakelite former. It consis ts of two secondaries in series opposition separated by a gap of 1" and a primary extending over a total length of 5" . One secondary i s 1" i n length whi le the other i s 1 .25". The sample i s p laced inside the shorter secondary at the centre of the former. The longer secondary, near the end of the former serves to compensate largely the effect of the other. The c o i l t ightly fits the t a i l of the inner jacket and thus is completely immersed i n l i q u i d h e l i u m . Another compensator, at room temperature, can be varied simultaneously from 0 to 4 m i l l i h e n r i e s . Th i s helps balancing the mutual inductances f inely when the capsule is .out of the c o i l . The calibrat ion of the bridge i s . done by reversing 250 mi l l iampere i n the primary c o i l by means of a spec ia l reversing sw i t ch . The current produced i n the secondary c i rcui t by induced e . m . f . i s measured by a c r i t i c a l l y damped galvanometer. The cal ibrat ing current is reduced to 100 mi l l i ampere for measurements below 1 ° K . Th i s reduces heat of magnetisation of the sample though,due to large deflexions, sens i t iv i ty i s not much affected. Reference to the diagram of Figure 13 can be supplemented by the fo l lowing data on the suscept ib i l i ty c o i l . S i has 3200 turns whi le S 2 has 3620 turns of #40 B and S, s ingle s i l k covered copper wire which at room temperature has a . d . c . resistance of 1600 ohms . The primary has 576 turns of #36 B and S, double s i l k covered copper wire wi th a r e s i s t anceo f 74 ohms at room temperature. At l i qu id he l ium temperature, the secondary has a resistance of 21 ohms whi le the primary has a resistance of 4 .1 ohms . Procedure The pre-cool ing of the he l ium dewax to l i q u i d nitrogen temperature i s done gradually by control l ing the amount of exchange gas present i n the he l ium dewar as Wel l as the.inner j acke t . Th i s is dictacted by the fact that sudden cool ing cracks the glycerine sea l and effects seriously the v i s i b i l i t y of the cap i l l a ry . With slow cool ing , glycerine can sol idify into transparent c rys t a l s . After transferring l i q u i d h e l i u m by the process already described i n the previous chapter, the bath i s cooled by pumping over it by the .Kinney mechanica l pump. A s the. temperature is lowered, ca l ib ra -t ion of the suscept ib i l i ty bridge i s effected . The balance point (with sample out of the co i l ) i s checked off and o n . E a c h observation i s taken.after a few reversals. of current. For the sake of checking data, a few measurements of the transfer rates are taken above 1°K as the l i q u i d i s being cooled . The rate i s measured by measuring the rate at which the l eve l of meniscus of l i qu id he l i um II r ises i n the capi l lary tube. The rise of l e v e l i s measured wi th a cathetometer having a b u i l t - i n graduated s c a l e . T ime , measured by a spl i t second hand watch , i s recorded as the l eve l moves from one b ig d iv i s ion of this scale to the other. Th i s enables a plot of the meniscus l eve l against t ime to be made for each observation. T h i s helps to check the nature of the flow as being pure superflow and, furthermore, the mean value of the transfer rate can be ca lcu la ted . For measurements below l ° K , t h e capsule i s reversed to be i n dumped posi t ion and lowered into the t a i l . The beaker empties i n the dumped pos i t i on . The magnet i s brought i n pos i t i on . A s m a l l amount of exchange gas i s introduced i n the inner jacke t . The amount of exchange gas to be used depends upon two factors . F i r s t l y , i t has to be sufficient to provide a good thermal contact to the bomb in the shortest t ime p o s s i b l e . Secondly, i t should be possible to pump i t out in the shortest t ime p o s s i b l e . The two conditions being contrary to each other, one has to find, an optimum amount of exchange gas to satisfy the above requirements. It was found by t r i a l and error that optimum amount was about 1 m i c r o n . With the exchange gas introduced, the magnet i s switched on and pumping of exchange gas stopped for five minutes so as to have isothermal magnet isa t ion. The exchange gas carries away heat of magnet i -sa t ion . After this the pump i s started again and i t takes about 25 minutes more to evacuate the exchange gas to a pressure of about 10"^ c m . of H g . Then , the magnet i s switched off resul t ing i n adiabatic demagnetisation of the paramagnetic sal t and hence i ts coo l i ng . T h i s process , essent ia l ly a routine one, cools the capsule down to 0 . 2 ° K which then takes about 15 minutes to reach the l i q u i d he l ium bath temperature. It should be mentione d that we tr ied to cut down the pumping time to 15 minutes so as to avoid excess ive heating of the c o i l s of the electromagnet and also save t ime for more experiments . However, i t was not useful as warm-up of the capsule was quite fast though the amount of cooling.achieved was the same . _^ , I— ~ b - - . . -<>-•.. N x a> Q E o c o 4 h to 8 h - N ^ - JO | 6 l _ . ^ ^ - - K CD "A S 0 2 4 6 8 10 12 14 16 . 18 T ime (m inu tes ) Fig - 14 Some Typ ica l Warm-up Curves for the Sealed Glass Capsule facing page^-i A s soon as the salt i s demagnetised, i ts temperature i s measured by the mutual inductance bridge. The capsule i s then ra i sed , reversed to be . in observation posi t ion and rate of f i l l i n g of capi l lary measured. After th i s , it i s reversed again in dumping posi t ion for emptying which normal ly should take about a minute or so because of a drop forming on the end of the beaker. During this process of emptying, the capsule i s lowered again into the t a i l for temperature measurement. It i s l i f ted again for flow rate measurement. Thus , this cyc le of measurement of temperature and flow rates alternately i s repeated unt i l the capsule warms up to its i n i t i a l temperature. h i connection wi th warm -up, we have some more comments . H u l l e t . a l . (1951) have mentioned that vibrations can warm the sal t very q u i c k l y . In our case , vibrations seemed to be beyond control , espec ia l ly quite violent vibrations occurring in the process of dumping which can be e a s i l y seen wi th the naked eye . Some typ ica l warm-up curves are given in Figure 14. These curves do not show any outstanding warm-up due to vibrations .or any other cause . However, in almost a l l of our runs, we found that just after the first reversal of the bomb, for flow rate measurement, the temperature jumped to about 0 . 4 ° K . Th i s could be due to the l a g in thermal equi l ibr ium of sal t and the other parts of the bomb. Resul ts The transfer rates ( c m ^ / c m . s e c . ) observed are plotted against temperature (°K) i n Figure 15 . The data represented are from a number of runs on two different days but no normal i s ing factor (See Amble r and K u r t i , 1952) was found to be required. For the sake of compar ison, flow rates below 1°K as obtained by other investigators, are shown on the same graph. .Above 1 ° K , the curve, through the present result seems to be ident ica l wi th that of Mendelssohn and White (1950). Below 1 ° K , the general shape, of the curve i s l i k e that of Amble r and K u r t i . Below approximately 0 . 3 ° K , however, our curve shows a poss ible indicat ion of a s l ight flattening i n contrast to the complete flattening of Waring's case and no flattening at a l l in that of Amble r and K u r t i . In fact , i n the case of the latter authors, a steady increase i s indica ted . D i scuss ion L i k e Ambler and Kurii, we have also found that f i l m transfer rate increases below l ° K . a n d i s about 25% higher at 0 . 3 ° K as compared wi th the m i n i m u m value at about 1 ° K . A s far as.absolute values are concerned, a.more l ikely, explanation of the higher rates observed by both Ambler and K u r t i , and Waring as compared wi th Our resul ts , i s poss ib le i n terms of some surface effects, poss ib ly impuri t ies or irregularities over the short sect ion of l i m i t i n g cross sec t ion . E v e n i n the case.of the sealed capsule used here, i t was noted that a first bomb, on which meas -urements had been made above 1°K before i t was accidental ly exploded at room temperature, gave results approximately 100%higher, presumably due to sea led- in impuri t ies (Hebert 1956). We feel that the discrepancy between Waring's results on one hand and those of Amble r and Kur t i and the present work on the other, might be explained at least insofar as the fractional increase i n flow rate at lowest temperature, i s concerned, as a consequence of Waring's method of measuring flow rate induced thermal ly . While i t i s known that above 1 ° K , f i l m can evaporate at a temperature very close to that of bulk l i q u i d (Fairbank and L a n e , 1949), i t may not be the case i n the region below 0 . 6 ° K where vapour pressure fa l ls to very s m a l l v a l u e s . F r o m the k ine t ic theory,of gases , we-know that the weight "w" i n g m s . evaporated per cm2 per second from a substance of molecular weight M , and at a temperature T (°K) and pressure p in dyne per c m ^ , i s given by (Loeb , 1934) If the f i l m which i s carrying a bulk l i qu id of the order of 8 x 10" c . c . per second per c m . wid th , at a temperature very c lose to absolute zero, has to evaporate wi th in a c m . then one.can calculate from the.above equation the pressure necessary to do s o . By substituting magnitudes we find that pressure required i s about 7 x 10"^ c m . of H g . w h i c h , as vapour pressure, corresponds to a temperature l ess than 0 . 5 ° K . Thus we see that a rough ca lcula t ion of m a x i m u m evaporation rate based on k ine t i c theory indicates that the l i q u i d , carried by the f i l m at the .observed transfer rate, could be evaporated wi th in a few centimeters of the surface only when raised to about 0 . 5 ° K . T h u s , i t appears that in Waring's case , flow rates are. l i m i t e d to a value of transfer rate at the temperature of evaporation which i s about 0 . 5 ° K . Consequently, by this method, one would observe a f la t tening of flow rate curve below 0 . 5 ° K . w - 43.74 x 1 0 " ° p g m . A s regards the difference between absolute values of f low rate i n Waring*s case for gravitat ionally and thermally induced flow rates, we have already mentioned that weight of evidence is against Waring's suggestion that thermally induced flow rates are different from those induced by gravi ty . There i s no doubt that our measurements of temperature are subject to c r i t i c i s m . T h e temperature of the capsule i s measured before and after the. f low rate measurements and then a mean is taken. The situation may look s t i l l worse i n view of the fact that there i s additional non-l inear warm-up of the capsule as i t i s moved up and down for temperature and flow rate measurements . The uncertainty in tempera-ture is max imum at lowest temperature when capsule warms up quite fast . However, i t i s gratifying to note from the warm-up curve that max imum error is about 20% at the lowest temperature whi le it i s much less at higher temperatures. T h i s error could be reduced a lot by us ing a larger bomb mounted i n an e l l i p t i c a l t a i l dewar when i t could be tipped wi th a m i n i m u m of disturbance. We can conclude that the hypothesis; that the product v c d occurring in equation 2 i s constant, i s definitely negated below 1°K because variat ion of R in this region can only be attributed to variation of this product. We have already mentioned that the weight of evidence (Jackson and co-workers 1951, 1953, 1954)suggests that the f i l m thickness d varies wi th temperature and i t i s poss ible that below 1 ° K , the variat ion of v c d i s the pr incipal source of variation of R . A variat ion of thickness d seems probable i n the l ight of data on measurement of f i l m thickness (Burge and Jackson, 1951). Measurements of f i l m thickness below 1°K would prove very useful to clar i fy this quest ion. The change i n transfer rate dependance on temperature below 1°K is not unexpected i n view of the strange behaviour of l i q u i d he l ium as regards other properties i n this temperature range. For example , specif ic heat of He II which follows a T ^ variat ion above I ° K has a change in i ts temperature dependance to T^ at about 0 . 6 ° K (T^ variat ion is typ ica l of Debye sol id); thermal conductivity changes i ts temperature dependance from T D to law at 0 . 6 ° K ; second sound*shows a marked r ise at about T = 0 . 6 ° K and so i s true for other properties l ike superfluid density and entropy. The only significance of this new addition of the case of f i l m flow to above mentioned properties i s that whereas a l l other properties are thermal , that of f i l m flow is a mechanical one and hence, we have here a .case. in which mechanica l property of He II changes i ts behaviour below 1 ° K . Furthermore, whereas the above mentioned thermal properties have been explained on the bas is of Landau's theory i n which the exci ta t ion in He II below 1°K merely consists of a phonon gas that can be treated as a Debye s o l i d , the changed hehaviour of the mechanical propertyof f i l m flow s t i l l l acks a theoretical explanat ion. A n interesting point to note i s the s imi l a r i ty between the second sound curve and the f i l m transfer rate curve which has already been pointed out in Chapter I (see Figure 4 , 5 ) . We can see that the s imi l a r i ty persis ts to the lowest temperature used for f i l m flow measurements. 50. However, we should notice that though the shapes of the two.curves are s i m i l a r , yet fractional increase below 1°K i s quite.different i n two cases: By quasi-thermodynamic argument, T i s z a suggested that f - (if which relat ion i s a fair approximation to the experimental results i n temperature range 1 .6°K to T^ . U s i n g this re la t ion, we get rate, of transfer of the f i l m as R = R j i - Ck) (5) T i s z a arrived at an expression for second sound as when A i s a constant. A s s u m i n g that the behaviour of He II below 1°K i s best explained by Landau's phonon-gas m o d e l , the density of phonon gas or normal f luid i s given by 6 = 3 (7, where u i i s first sound ve loc i ty , a i s a constant and i s equal to 5 4- - \ V Iff ^ On this b a s i s , equation (2) can be re-written as K 3 ) Equation for second sound i n this region given by Landau i s It i s interesting to compare equation (4) and (5)j,and (7) and (8). We know that equation (8) i s a good approximation to experimental results (neglecting other effects due to mean free path e tc . ) though at the same t i m e , we cannot say anything about equation (7) unless variat inn of v c d are known. Further measurements on the f i l m flow rate below 0 . 2 ° K should prove interesting i n d i sc los ing how far the s i m i l a r i t y between two curves pe r s i s t s . C H A P T E R IV  SUSPENSION O F P A R T I C L E S IN L I Q U I D H E L I U M Introduction It would c lear ly be very useful i n the study of the properties of l i q u i d he l ium to be able to suspend in the l i q u i d fine part icles which would render v i s i b l e the motion of the f l u i d . Studies of the onset of turbulence s imi l a r to the c l a s s i c a l ones of Reynolds should be interesting i n the case of l i q u i d he l ium at low temperatures, where on one hand, the v i s c o s i t y becomes vanishingly s m a l l whi le on the other, Landau 's theory suggests that me. M o t i o n l e s s or superfluid component of He II should be capable only of irrotational motion and hence cannot be dragged along by the viscous.or normal component unless some type of mutual fr ict ion mechanism i s postulated. It would be interesting a lso to examine a rotating ves se l of l i q u i d he l ium for the vortex structure suggested by Onsager (1949) and Feynman (1953, 1954). Studies of ultrasonic streaming (also ca l l ed acoustic streaming or quartz wind) such as undertaken by Liebermann (1947) and others, i n their study of the second coefficient of v i s cos i t y of water and a number, of other l iquids showing absorption of sound much i n excess of the theoretical va lue , might also be undertaken to examine further the anomalous absorption of sound at and below the \ - p o i n t i n l i qu id h e l i u m . Analogous to first sound streaming caused by gradient of first sound radiation pressure, one could observe streaming due to attenuation of second sound i n l i qu id He II by means of the v i s ib l e suspension of part ic les and hence measure second sound radiation pressure. L a s t l y , it might be poss ible wi th such a suspension of part icles to study. Brownian movement in a Bose-Eins te in l i q u i d . Poss ib i l i t i e s and Diff icul t ies Of course, the extremely s m a l l v i scos i ty of l i q u i d he l ium (1.3 x 10"^ poise at 1 .5°K) renders diff icult the suspension of even very s m a l l part icles for any length of t i m e . For example , i f we. assume.Stoke's law for a body fa l l ing i n a f lu id of v i s c o s i t y ^ , then terminal ve loc i ty v i s given by v s ; | £ 1 (p.-6)g where a i s radius of the body, $is i t s densi ty, C i s density of the l i q u i d and g i s acceleration of gravi ty . For a terminal ve loc i ty of about 0.1 m m / s e c o n d , - 6*= 1 g m / c c . and ^ = 1 . 3 x 10"^ p o i s e ; we get a = 2 .4 x 10"^ c m . which i s not very prac t icable . F rom the above re la t ion, one can see that the only practicable way to suspend s m a l l v i s i b l e part icles is to make the density of the mater ia l of part icles approach that of l i q u i d h e l i u m . Calculat ions for s m a l l thin wa l l ed hollow spheres f i l l e d wi th l i q u i d he l ium so as to approximate the l i q u i d in their average density, do not give promis ing results for practicable d imens ions . For example , a hollow sphere of radius R , thickness t , and density 3 g m / c c . (glass) has an average density P z 4 R 2 t (g - C ) _ 3 t ( f - € T ) 4/3 7T R 3 -R Us ing R = 0.5 c m . , t = 0.01 c m . , we get £ =r 0.18 g m / c c . which i s very nearly equal to the density of l i q u i d h e l i u m . However, ca lcula t ion of terminal veloci ty y ie lds v r 7 . 7 x 1 0 ^ c m / s e c . Thus even for t = 10~ 4 c m . , v = 77 c m / s e c . These fantastic magnitudes render this method imprac t icable . It is quite clear that i t i s essent ia l to try to obtain part icles of a s o l i d whose density i s as near as possible that of l i q u i d h e l i u m . We know that so l i d hydrogen is lighter than l i qu id he l ium whi le a l l other so l id i f i ed gases , for example , O2, N2> M, N e , e t c . are heavier than l i q u i d h e l i u m . It i s poss ib l e , at l eas t , theoret ical ly , to have a mixture of H2 and some other heavier gas to form a s o l i d of the same density as that of l i q u i d h e l i u m . However, i n pract ice , the m i x i n g gases have to have their solidif icat ion :temperatures very close to each other so that in the process of so l id i f ica t ion of mixture , the components sol id i fy a lmost at the same t i m e , resul t ing in a mixture of right proportion. Th i s condition i s w e l l sat isf ied by D2 and H 2 mix tu re . The bo i l ing points of H2 and D ^ a t atmosphere pressure are 20 .4°K and 2 3 . 6 ° K respec t ive ly . Incidental ly, this mixture i s very economica l as compared wi th the mixture of H2 and some other rare gas , say neon (boi l ing point at 1 atmospheric pressure = 2 7 . 2 ° K ) . The Mixture •v. T o calculate the m i x i n g ra t io , we proceed as fol lows: Densi ty of H 2 ' • at 4 . 2 ° K = 0.088 g m / c c . Density of D 2 at 4 . 2 ° K = 0.205 g m / c c . To get a resultant density of 0.145 g m / c c . , we have to satisfy 0.088 H 2 -+ 0.205 (I - H 2 ) = 0.145 where H 2 . = volume i n c c . of hydrogen i n a mixture of 1 c c . Solving we get the amount of H 2 , D 2 . i n 1 c c . of mixture as 0.514 c c . and 0.486 c c . re spec t ive ly . Since density of l i qu id H 2 = 0.0708 g m / c c . at normal boiling-point ( N . B . P ) and that of l i q u i d D 2 = 0.168 g m / c c . at N . B . P . , and also 1 l i t re of l i q u i d H 2 = 780 l i t r e s o f H 2 gas at N . T . P . and 1 l i t re of l i q u i d D 2 - 935 l i t res of D 2 at N . T . P . , therefore, amount of H 2 gas required to produce 0.514 c c . of so l i d H 2 i s given by 7 8 0 x 0.088 x 0.514 = 5 0 0 c c . at N . T . P . 1 x 0.0708 S i m i l a r l y , to produce 0.486 c c . . of s o l i d D 2 , amount of D2 required is 935 x 0 . 2 0 5 x 0.486 = 555 c c . at N . T . P . 1 x 0.168 Thus , the m i x i n g ratio o - H 2 / D 2 i s 1/1.11 It should be mentioned that the amount of mixture required to get a number of fine par t ic les , i s .quite s m a l l . To produce spherical part icles of radius r and hence volume K = 4/3 7T r 3 , we need to m i x 500 K c c . of H 2 wi th 555 K c c . of D 2 at N . T . P . For r = 0.1 m m . (100 micron) , T 4 - e To , pump-4—(£-T 3 ^ To ->xryostat — o 1 0 0 c c 5 0 0 c c " T, r v Hg mono meter Fig 16 (a) Sche.metic diagram of the mixing s y s t e m To pump -4— Dewar cap Glass tube Porcelaintube Constonton he ater_ Capil lary tube Liquid helium Fig 16 (b ) The heater arrangement for form ation of Particles facing page 56 total quantity of mixture required i s = 0 . 0 0 4 4 . c . c . Thus the amount of mixture (to be ca l l ed a "shot") required to get a good number of part icles is quite s m a l l . Exper imental Arrangement A s imple arrangement for m i x i n g H2 and D2 i s shown schemat ica l ly i n Figure 16 (a) . Th i s system i s evacuated to remove . a l l foreign gases . The m i x i n g flask (500 c e . ) i s f i l l e d wi th deuterium and then w i th gydrogen through stopcock T 2 . Hydrogen i s taken from a commerc ia l ly available cylinder of pure, e lec t ro ly t ica l ly produced hydrogen. Deuter ium i s obtained from e lec t ro lys is of heavy water . A part of the mixture can be al lowed to f i l l a s m a l l glass, tube between stopcocks T3 and T 4 and a l so , i f necessary, a s m a l l 100 c c . f l a sk . The amount of mixture in this glass tube determines the. "shot" to be used for formation of pa r t i c l e s . The s m a l l flask helps to vary the amount of the shot. The mixture i s l ed into the cryostat through a narrow channel occupied by a long sp i ra l type of heater of constantan. In our c a se , the heater is 40 c m . long and i s wound on a porcelain tube.of diameter 4 m m . which i s enclosed in a glass tube of 6 m m . inner diameter . Th i s i s shown in Figure 16 (b). The heater tube i s connected to the apparatus i n wh ich part icles have to be suspended. By means.of an 0-r ing s l i d i n g seal arrangement outside the cryostat , it i s possible to move-the heater tube up and.down wi th respect to he l ium bath l e v e l . A s explained later , the function of the heater depends upon i ts locat ion w i th respect to the bath l e v e l . The usual procedure followed to suspend particles i s as fol lows: The m i x i n g sys tem i s evacuated and f i l l e d wi th 5 .5 c m . of D2 gas and then wi th 5 . 0 c m . . of H2 gas at room temperature. The mixture i s usual ly prepared a few hours before the experiment and so i t i s thoroughly m i x e d at the time of i ts u s e . When l i q u i d he l ium i n the cryostat has been pumped to the lowest possible pressure of about 2 m m . of H g . , then a shot o f mixture i s f i l l e d i n the glass tube between T 3 and T 4 . The heater i s switched on from a variac wi th power input of about 1 to 4 wa t t s . Simultaneously, T4 i s opened to let the shot go into the cryostat . Methods ;of Obtaining Sma l l Particles Without the heater, the mixture enters the cryostat i n the form of a thick coagulated c l o u d . However, wi th the heater, i t breaks up into p i e c e s . .One can eas i ly get part icles of various s izes by varying the amount of heat supplied to the heater w h i c h , being dependent on the locat ion of heater wi th respect to the l i q u i d he l ium bath, has to be determined by t r i a l and error. With this .method, one.can e a s i l y get approximately spherical part icles of about 100 micron d4;ajmeter or even l e s s . The second method i s analogous to sputtering. It consis ts in first condensing the mixture on the heater wire and then evaporating i t by a sudden heat pulse in the heater. Th i s method also results i n quite fine and independent pa r t i c l e s . The third method essent ia l ly consis ts in vigorous mechanical s t irr ing up of the coagulated cloud by means of a strong ultrasonic beam which has an intensity l eve l at the l i m i t of cavitat ion in the l i q u i d . T h i s method was found as a result of the use of the suspension as indicator in acoustic streaming experiments,to be described i n the next chapter. Ul t ra sound i s generated by an x-cu t 5 M c p s . quartz c r y s t a l . Nature of Part icles When particles, are formed, i t i s quite easy to see that they have a good range of densi ty ,dis t r ibut ion. A s a resul t , some are heavy and f a l l down; some are l ight and float on the surface and only a s m a l l part is of the right density and remains suspended i n the l i q u i d . Th i s merely indicates that part icles are-aggregates of so l i d H 2 and D 2 molecules and the ratio of the two types of molecules in any particle is probably determined by chance rather than any chemica l aff ini ty . The s m a l l part icles that remain suspended i n the l i q u i d , start moving s lowly towards the glass w a l l s of the dewar and have a tendency to s t ick to the w a l l s . Moreover, quite often, the part icles combine together to form bigger and bigger pa r t i c l e s . Th i s process taken a few minutes after which no more pa r t i c l es are left suspended. Th i s has proved to be a great nuisance. A s i t was suspected to be due to a charge effect, a part of the cryostat was s i lvered and connected to ground so as to be able to discharge the par t i c les . However, no change i n behaviour was observed. La te r , part icles were irradiated by a 1/2 cur ie , C o ^ radioactive source p laced at a distance of about 10 c m . outside the two dewars. There was no observable change i n behaviour of pa r t i c les . The irradiation resul t ing i n charging of the part icles by knocking out the electrons, should produce enhanced charge effect and hence increased probability, of s t i ck ing of part icles wi th each other. It appears from these observations that the above mentioned behaviour i s probably not due to change effect. It should be noted that above the ^X.-point, the part icles do not s t ick w i th each other and are quite free, to move about. Th i s could be attributed to st irr ing of the l i q u i d due to escape of bubbles of he l ium gas wh ich disappear below the X -point . Conclusions We have used this type of suspension in v i s u a l study of acoustic streaming i n l i q u i d h e l i u m . Th i s inves t iga t ion i s described i n the next.chapter. A n attempt was .a lso made, to carry out an experiment s imi l a r to that of Reynolds for observing the onset of turbulance i n the flow of l i q u i d he l ium II . Due to the difficulty of-s t icking of part icles to the g lass w a l l s , i t was imposs ib le to have the relat ively large par t ic les move along wi th the l i q u i d f lowing through a narrow capi l la ry tube. Although the problem of s t i ck ing of part ides i s not quite serious in case of a large diameter flow tube, yet the flow of l i q u i d being too fast, it does not permit any observations . Some striking.observations of the internal convection currents as a result of heat pulse i n l i q u i d he l ium II were also made wi th the suspension of pa r t i c l e s . On the basis of the two f lu id mode l , when heat i s supplied to a source of l i q u i d h e l i u m , superfluid rushes towards it whi le normal f lu id rushes.outwards. Th i s type of convection is beautifully i l lustrated by the movement of part icles away from the source. F i n a l l y , we should add that the. difficulty, of s t i ck ing of par t ic les i s quite ser ious . A number of experiments cannot be performed due-to this d i f f icu l ty . We hope that this problem w i l l be solved by further research. C H A P T E R V A C O U S T I C S T R E A M I N G IN L I Q U I D H E L I U M  The Problem Eckar t (1948) suggested from his.theoretical.treatment of acoustic s treaming that the streaming ve loc i ty depends direct ly upon the ratio of second to first coefficient of v i s c o s i t y . Th i s suggestion was successful ly followed by Liebermann (1949) and others to determine the so c a l l e d second coefficient of v i s cos i t y of a number of l i q u i d s . However, we now know that Eckar t ' s treatment was incomplete and that the ve loc i ty of streaming merely depends upon the ratio of sound absorption coefficient to the shear v i s c o s i t y . The absorption coefficient, i n turn, i s proportional to a generalised coefficient of v i s c o s i t y wh ich is the.sum of the f i r s t , second and some other hypothetical v i s cos i t y coefficient equivalent to any other relaxation process responsible for additional absorption of sound i n the l i q u i d . The present investigations were undertaken to study acoustic streaming i n l i q u i d he l ium i n an attempt to determine the ratio of absorption coefficient of sound to shear viscosity., of l i q u i d He and hence ratio of the second to first (shear) coefficient of v i s cos i t y where these coefficients of v i s c o s i t y occur i n the general expression of absorption coefficient and are calculated by Khalatnikov i n his theory of absorption of first sound i n l i q u i d h e l i u m . Although there: exis ts a considerable literature on this interesting subject of acoustic streaming and second coefficient of v i s c o s i t y (Royal Society Symposium on Second.Coefficient of V i s c o s i t y , 1954), yet there.being no up-to-date review on this subject, a brief account of the general features of this subject is given i n the fol lowing sec t ions . Furthermore, this account is quite essent ia l for an interpretation of the results of acoust ic s t reaming. Acous t i c Streaming It i s a we l l -known fact that i n a sound f ie ld generated by an essent ia l ly s inusoidal sound source, the part icles of the medium do not have a s imple s inusoidal mo t ion . In addition to the "sound f i e l d " which corresponds to a to and fro motion of each f lu id element of the m e d i u m , one often finds a pattern of steady vortices or time-independent c i rcula t ion in the body of the f lu id through which sound wave propagates. (The term "sound" covers both audible and ultrasonic compressional waves) . Faraday (1831) was the first to make such an observation wi th vibrat ing p la tes . A century la ter , Andrade (1931) made smoke photographs of the curious arrangement of vortices which occur in a Kundt 's tube wi th and without the presence of s m a l l obs tac les . In l i q u i d s , a very familar k ind of streaming i s the so - ca l l ed "quartz w i n d " which may be generated by any source which projects a high intensity beam of sound into the body of f luid (Meissner 1926). Though typ ica l ly associated wi th quartz. osci l la tors in l i qu ids , the effect (also referred to as "sonic w i n d " or acoustic streaming) to be henceforward c a l l e d "acoustic s t reaming", has also been noted i n a i r . It i s necessary to dis t inguish between the types of flow which depend in turn upon whether the soundwave interacts wi th the medium alone or wi th the medium and i ts s o l i d boundries i n combination (Westervelt 1953). .We.shal l here be concerned wi th the former case on ly . Theoret ical Explanat ion Although the first theoretical explanation of such phenomena was described by Rayle igh (1884), yet the first serious theoretical analysis of streaming phenomena was undertaken by Eckart (1448) who developed equations of second-order acoustic phenomena i n a systematic manner, Accord ing to Ecka r t , i f we neglect v iscous forces, the second order effects, for example , inertia of acoustic energy, radiation pressure and variable compress ib i l i ty of the m e d i u m , only result i n production of overtones and do not cause an acceleration of the f l u i d . (In such cases , e las t ic rather than v iscous forces are restoring forces) . However, when v i scous forces are ; introduced into the ca lcu la t ions , Eckar t finds another second order effect, namely steady vor texmotion. Since forces res i s t ing steady motion depend only on the shear v i scos i ty of the medium whi le the generating forces depend also on the bulk v i s c o s i t y , the steady mot ion , therefore, involves the ratio of the bulk to. shear coefficients of v i s c o s i t y . This.affords an independent method of measuring this r a t io . T h i s suggestion has been followed by several recent investigators . (Libermann, 1949; K a r i m , 1953; L a m b and Piercy, 1$54) who have successful ly determined the ratio defined i n Eckar t ' s theory. However, Eckar t ' s theory has been c r i t i c i zed by a number of authors (Fox and Herzfe ld , 1950; Markham, 1952; Nyborg , 1953; Westervelt , 1953; Doak, 1954; L a m b and Piercy, 1954; Richardson, 1954) who concluded from their theoretical analysis. that the veloci ty , of s treaming i s determined by the coefficient of sound absorption, no matter from what source the absorption may a r i se . In fact , L a m b and Piercy have gone a l i t t l e further i n their statement that the sound attennation rather than the second coefficient of v i s c o s i t y i s direct ly responsible for the s t reaming. The .conc lus ion , that the ve loc i ty of streaming i s dependent upon the total absorption rather than any select ion mechanism of absorption, seems to be w e l l establ ished now. Eckar t ' s conclusion that the streaming veloc i ty depends upon the second coefficient of v i s c o s i t y appears to have arisen because of the fact that he introduced v iscous d iss ipa t ion as the only absorption mechanism in h is i n i t i a l equations. L a m b and Percy have given a very s imple treatment of the streaming problem. It. i s equivalent to that of Eckart except that the dr iving force i s assumed to be the gradient of radiation pressure. The propagation of a plane progressive wave through an attenuating f luid medium results i n a time-independent gradient of radiation pressure which produces acceleration of the f luid provided there exis t s a return path for the f lu id . "Th i s is the phys ica l basis of s t reaming. "It i s to be noted that choice of such a dr iving force i s dictated by J conservation law... Fo r , any mechanism contributing:tio the absorption of acoust ic energy would make: a corresponding contribution to the absorption of acoustic momentum. Thus the dependence of streaming Crystal ! w H 1 Crystal Crystal Fig-1 7 ! (a) Libermann's Arrangement (b) Lamb 8 Piercy8 Arrangement (c) Present Arrangement P : ^ i' :  r " "" The Diftrent Arrangements for Acoustic Streaming Experiments  facing pagesS veloc i ty on the total absorption of sound rather than on any selected mechanism may be considered.to be a consequence of the conservation of momen tum." (Quoted from article by L a m b and Piercy, 1954). T o find an expression for the ve loc i ty of s treaming, le t us consider phys ica l systems: (to be ca l led ' s t reaming tube" adopted by (a) Liebermann, (b) L a m b and Piercy , (c) i n present invest igat ions, as shown i n Figure 17 . In the case of Liebermann 's arrangement, one could write down the Navier -Stokes ' equation i n cy l ind r i ca l polar coordinates for the steady flow i n the ax ia l z-di rect ion assuming, of course, that the driving force is due to gradient of radiation pressure or energy density of sound . T h e solution of the equation i s ident ica l to that of Eckar t i f the f inal expression for ve loc i ty of streaming i s expressed i n terms of attenuation coefficient rather than bulk v i s c o s i t y . However, one can get approximately the same-result by application of Poisseui l le 's l aw to the circulatory flow of l i q u i d which is assume d to be a stream l ine f low. A l l the three arrangements shown i n Figure 17 are essent ia l ly s imi l a r to each other i n that the l i qu id moves in the central tube f i l l e d wi th the sound beam under the influence of the gradient of radiation pressure and returns along the return path provided. T h e ratio of ve loc i t i es of direct and return flows i s governed by geometry. If we make the following assumptions: (1) the flow i s stream l ine (2) sound energy is complete ly absorbed at the end of streaming tube and none is reflected back to the central tube (3) the end effects due to corners of side tube axe negl igible (4) the beam i s uniform and f i l l s the central tube completely then, the average ve loc i ty v Q of l i q u i d i n the central tube of radius "r" and length "1" i s given by Po i s seu i l l e ' s relation as v 0 (1) I 4 » i £ where AP/fc is . the pressure gradient along the tube and ^ i s the coefficient of shear v i scos i ty of the l i q u i d . Since the pressure gradient arises from decrease of radiation pressure due to the attenuation of sound energy,over a path length t> in the ma in tube, therefore, p = E G (1 - e " 2 ^ ) (2) where E Q i s acoustic energy density (or radiation pressure) at the source, (or at the entrance of side tube i n case of L a m b and Piercy ' s arrangement Where v Q i s the ve loc i ty of f lu id in the side or return tube of length t). The relat ion (2) i s a consequence of the fact that energy density E of a wave propagating i n the Z-d i rec t ion i s given by = E (r) e " 2 * z Incidental ly, this equation also defines the sound absorption coefficient Substituting (2) i n (1), we get v 0 = E 0 r 2 (1 - e"2**8) ( 3 ) 4 ' > " If the exponent i s sufficiently s m a l l , equation (3) can be approximated to v Q = E o r 2 oC (4) 2 n It i s interesting to compare (4) wi th expression of v 0 found by Eckart on the bas is of a selected mechanism of absorption. In the s imple arrangement of L iebermann, Eckar t ' s theory gives V o = « i ^ L ( 2 t l ) ( 5 ) P C \ z. where G = ( -^ 2. ~ ' ) 12 ( r ) ' r ' r o 3 X 1 2 r a c u u s 'of-outer To 0 (return) tube and the m a i n tube respectively; , ^ are coefficients of first and second v i s c o s i t y . Since 1/ = Eo is ' the acoustic energy density or radiation pressure at the source, and further (2 + ZL (total absorption coefficient) , we c a n , therefore, rewrite equation (5) as * 2 (2 + n ) > v 0 = E 0 r 2 ^ G (6) For the given geometry in Liebermann's case , equations (6), (4) are approximately equa l . Second Coefficient of V i s c o s i t y We have seen that the streaming veloc i ty is proportional to the ratio of total absorption coefficient to. shear v i s c o s i t y . The next question that we have to d iscuss i s : what factors contribute to the total absorption coefficient? We are part icularly interested in the contribution, i f any, of the second coefficient of v i s cos i t y to absorption coeff icient . The passage of sound through any medium is always accompanied by absorption of some of the energy w h i c h , unt i l recent ly, has been attributed to three causes: 1) v i s cos i t y 2) heat conduction 3) heat radia t ion. Out of these three, v i s cos i t y is the predominant factor. Stokes (1845) deduced a formula for the coefficient of absorption of acoustic energy in v iscous fluids due to the effect of v i s c o s i t y , namely , where eC i s absorption coefficient already defined, W- 27^ t imes the frequency, f , ^ is density of the m e d i u m , c i s the velocity, of propagation of the sound waves and i s the coefficient of shear v i s c o s i t y . It i s w e l l known that the observed sound absorption, wi th few except ions, i s always in excess of the values calculated from Stokers re la t ion . T h i s discrepancy between theory and experiment has been investigated by different workers though no satisfactory explanation has yet emerged. We know that energy diss ipat ion i s a manifestation of a mechanica l relaxat ion, that i s , difference i n phase between the stress and strain i n case of a per iodical ly varying deformation. In a l l transport phenomena, for example , heat f low, mass flow or momentum f low, i f sufficient t ime i s not al lowed for these phenomena to reach a stationary state, a relaxation occurs (Kneser , 1954). B a s i c a l l y , any approach to expla in the above mentioned:discrepancy, can be correlated wi th a transport phenomenon. Th i s i s beautifully i l lustrated by" the Relaxat ion Theory developed by Herzfeld and R i c e (1928) and the Molecular Structure Theory proposed by H a l l (1947, 1948). In the former, a slow rate of exchange of energy between the translational movements of the molecule and its internal degrees.of freedom, and i n the latter, a t i m e - l a g between the re-arrangement of the mo lecu le s , lead to relaxation and hence absorption of energy. However, i t i s to be noted that both theories involve bas i ca l ly the effect of v i scous forces wh ich are, of course, different from shearing viscous forces. One type of viscous force, wh ich had hitherto been neglected i n hydrodynamical theory, can be of great importance i n generation of acoust ica l waves i n l iquids and gases which produces divergent motion as dis t inct from divergenceless flow of an incompressible l i q u i d often treated i n most of the problems in hydrodynamics . Th i s i s a v iscous force that arises when a volume of f luid is compressed or di lated without change i n shape, and i ts magnitude depends oh the rate of compression or d i la ta t ion . This , is measured by the so-ca l led .second coefficient of v i s c o s i t y or coefficient of di la ta t ional v i s c o s i t y , denoted by ^ . In hydrodynamics, i t has been customary i n deal ing wi th actual l iqu ids to ass ign for the value of di latat ional v i s cos i t y that, calculated for an ideal gas; the d i la t ional viscosi ty, of an ideal gas may be shown to be equal to - 2 \ /3 (where ^ is coefficient of shear viscps i t y ) . Th i s proof i s attributed to Stokes (1851) though as to its jus t i f ica t ion , Stokes h imse l f was aware of the deficiencies of h is arguments: " I have always felt that the correctness of the value \ /3 for the las t term of this equation (the term which i s the sum of shear and di latat ional v i scos i ty ) does not rest on as f i rm a bas is as the correctness of the equation of motion of an incompressible f lu id for which this las t term does not come at a l l . " In a divergenceless mot ion , stress i s independent of \ whi le i n a divergent motion associated wi th acoust ical waves , i t i s not true and so both d i l a t a -t ional as w e l l as shear v iscos i ty , can be expected to exert an important influence on the propagation of these waves . The effect of the two v i scos i t i e s i s conveniently studied by beginning w i th the usual second-order equation of hydrodynamics f Br + v f v v - u ( 9 ) where u is the part icle ve loc i t y , P i s density and p i s the pressure. This equation may be considerably s imp l i f i ed for the study of acoustic waves: for plane waves i n the x -d i rec t ion , equation (9), f j*+ - (2n + <) do) D t 2lx ^ 2 Equation (10), together wi th the equation of continui ty, gives the usual plane wave solut ion U = U 0 exp - i (Wt+kx) (11) i n which k i s complex i f luTis r e a l . The imaginary part of k i s the negative of absorption coefficient Ji . T o a sufficiently good approximation, absorption coefficient i s given by toP-(2\f\ ) /2fc$ (12) where I»r = 27f t imes frequency f and c i s the ve loc i ty o f sound. In the ideal gas approximation (sometimes ca l l ed Stokes' approximation) X ~- - 2 1 / 3 (13) so that equation (13) y ie lds oC - 2 n W 2 (14) 3 ^ c 3 which is the same as equation (7) and i s ca l l ed the c l a s s i c a l absorption coeff ic ient . Because of Stokes' assumption (equation 13), another coefficient k ca l l ed the coefficient of bulk v i s cos i t y is .defined by k = Xf -2JI (15) 3 so that Stokes' assumption s i m p l y means that k » o . It i s very importantito note that a conventional viscometer does not measure the di latat ional v i s c o s i t y nor does there exis t a di latat ional v i scometer . The only known phenomena in which i t contributes i s the sound absorption i n the m e d i u m . Unfortunately, i n this phenomenon too, there is no way to discr iminate between the effect of true dilatat ional v i s c o s i t y and any other mechanism of sound absorption which can also be. described by an equivalent coefficient of v i s c o s i t y . However, one could overcome this difficulty by g iv ing a very general.definition of second coefficient of v i s c o s i t y so that this coefficient plus shear coefficient of v i s cos i t y determine absorption coefficient comple te ly . Th i s concept would , no doubt, be complete ly pheriomenological and far from being fundamental. Experiments of Liebermann (1949) have left no doubt that the observed absorption, i n excess , of the theoretical va lue , can be explained by invoking such a generalised coefficient of v i s c o s i t y . The results show that this coefficient of v i s cos i t y i s not only posit ive and much greater than ^ i n contradiction to what is expressed i n equation (13) (Stokes' assumption) but a l so , i s quite independent of ^ • Th i s leaves no doubt as to the failure of Stokes* assumption i n high absorption coefficient l i q u i d s . Ultrasonic Attenuation i n L i q u i d He l ium We sha l l now attempt to describe the problem of anomalous attenuation of sound in l i qu id he l ium below i ts A - p o i n t and then, d iscuss i t i n the l ight of the concepts aiready.described. The first measurements on the attenuation were done by , Pe l l am and Squire (1947). Above 3 . 0 ° K , in the He I region, their measurements of absorption coefficient are approximately m accord w i t h the theoretical va lues . Below 3 . 0 ° K , just above the ^ -po in t , Pe l l am and F ig • 18 Attenuation of First Sound in LiquidHelium facing page7Z. Squire found a strong increase of the attenuation and at the A - p o i n t , a very sharp increase . Immediately below the A - p o i n t , a strong decrease i s observed and below 2 . 0 ° K , a new increase appears. Atk ins and Chase (1950) carried out measurements down to 1 .3°K using the same experimental technique as Pe l l am and Squire. The results are shown i n Figure 18 . The theoretical value of attenuation coefficient oC above 1 .6°K was calcula ted by Pe l l am and Squire by assess ing separately the contribution to absorption due- both to v iscous losses (using viscos i ty of normal f lu id a s .v i scos i ty coefficient) and to the thermal conduction by s t r ic t ly c l a s s i c a l methods. It i s seen that down to 3 ° K , the agreement between theory and experiment i s good due probably, as pointed out by Pe l l am and Squire, to the monatomic character of he l ium w h i c h , therefore, disposes of relaxation phenomena connected wi th inner degrees of freedom i n m o l e c u l e s o r wi th a s soc ia t ion . On the other hand, the marked increase i n the experimental value of <?Cbelow 3°K is outstanding, apparently leading to infinity at T ^ . Explanat ion of the anomalous absorption has been given by a number of invest igators . Noteworthy i s that of Pippard (1957) who has considered in deta i l the question of l o c a l fluctuational transitions from he l ium I to he l ium II and.vice ve r sa . (This was first pointed out by Pellam-and Squire). T h i s transition is characterised by a thermal relaxation process wh ich would provide anomalous absorption of pressure waves . T h i s theory can expla in attenuation near the A - p o i n t . However, Khala tnikov (1950) has given a very satisfactory theory of attenuation of sound i n he l ium II . The broken curve i n Figure 18 below 1 .6°K gives Khala tn ikov 's . ca lcula t ion of oC which he predicted before any measurements were made . One can see how w e l l h i s prediction was verif ied by experiments . Khala tn ikov 's theory i s essent ia l ly an extension of the Landau - Khalatnikov theory (1949) of v i s c o s i t y . Basis of the theory i s . that in a pressure wave the l o c a l non-equi l ibr ium phonon and roton densit ies can revert back to the equi l ibr ium values only in finite t i m e s . Th i s provides a relaxation mechanism which causes attenuation of sound. Two most impor tancco l l i s ion processes by which equ i l i b r ium is established are: phonon-phonon co l l i s ions and phonon-roton c o l l i s i o n s . Khalatnikov has calculated the parameters i n the expression for absorption coefficient in the fo l lowing fami l ia r form: Khala tnikov c a l l s k the coefficient of bulk v i scos i ty resul t ing from relaxation effects of the type d iscussed already. Furthermore, k in Khala tn ikov ' s theory can be calculated except for two parameters, the values of which were derived by Khalatnikov from data of Pe l l am and Squire on absorption coeff icient . Without going into the detailed expression of & and i ts interpretation, we should merely,note that Stokes' assumption which apparently is v a l i d for monatomic l i q u i d he l ium above the A - p o i n t , i s no longer v a l i d at or below the >i-point. In fact , l i k e the findings of Liebermann for other anomalous cases of c l a s s i c a l l i q u i d s , the experimental (16) results for l i qu id he l ium II indicate that there is a contribution from generalised second coefficient of v i s cos i t y which is about ten;times that due to sheaf v i s c o s i t y . In this connection, i t should be mentioned that this conclus ion i s based on experimental results obtained from the we l l -known pulse technique' (Pel lam and Gait) for measuring sound absorption coefficient but not on acoustic streaming experiments l i ke that of L iebermann . A s a matter of fact , the present investigations are primari ly,designed. to measure absorption coefficient i n He I I b y acoustic s t reaming measurements . A n important advantage, of the method of acoust ic s treaming over the conventional methods for measuring absorption coefficient i s that one can see v i s u a l l y i f other effects l ike, cavi tat ion and complex vortex structure, which can e a s i l y lead to anomolous absorption of sound energy, are also present or not . Th i s advantage i s part icularly important i n the case of l i q u i d h e l i u m having vanishingly s m a l l v i s cos i t y and so eas i ly susceptible to turbulent f l o w . . T h i s point sha l l be quite clear from the result wh ich reports failure of the experiment because of the existence of turbulence. A n Exper imenta l Dif f icu l ty A s seen from equation (4), to determine oCand hence T/.^ we have to measure v 0 and E 0 in the streaming experiment. Liebermann had. an ingenious idea for determining v 0 and E 0 together by studying the radiation force and the s treaming force on a: torsion vane suspended i n the medium i n wh ich streaming i s produced. The radiation force on a d i sc of diameter d and reflection coefficient of sound as R due to an energy density of sound E x at the posi t ion of the d i s c , i s given by F R = JL d 2 E X R ( 1 7 ) 4 The streaming force on the same d i sc due. to f lu id of density f and moving wi th a ve loc i ty ,v 0 c m / s e c . , i s given by F s :- kfv02 ( 1 8 ) when k i s constant of the geometry. The bas is of Liebermann 's idea i s that whereas radiation force is instantaneous, s treaming force lags behind due to bu i ld up of s t reaming. T h u s , due to separation of two effects i n ^ t i m e , one can separate the two forces and hence, determine both v Q and E x . T h i s idea sounds l og i ca l as i s also clear from the fol lowing equation of motion of f lu id (in an arrangement l i ke that of Liebermann) i n transient state: TTr2 E Q 2 cdt ~- * i 2 f t t o + IT t2 l ^ L v dt r ^ ( 1 9 ) where i i s the length, r i s radius of the inner tube. The dr iving force 7fr2 E 0 ( 1 - e " 2 o C f ) — 7fr2 E Q 2 oCi (if exponent i s smal l ) i s equated to the accelerat ion force (7f t 2 t ) and the viscous force (shear dt v i scos i ty ) 1( x2 v . Here v i s the instantaneous ve loc i ty of r 1 steady s t reaming. Equation ( 1 9 ) can be s imp l i f i ed to dv + 1 1 v = 2 * E n (20) dt f The transient solut ion of this,equation is whi le the steady state solution i s and i s the same as equation (4) -Therefore, from equation (21), we get the relaxation t ime ^- as given by Z - f_Z_ -- vpfr o v ^ f c (22) 8 ^ 2 o C E 0 2 ^ 1 We see that the relaxation t ime taken for streaming to set up depends only upon ^ and \ i f shear v i s cos i t y i s the only type of retarding force. On the bas is of this s imple ana lys i s , for a tube of radius r = 0.75 c m . , we get t for water.at 2 0 ° C = 1.x (0 .75 ) 2 = 7 seconds 10 ~ 2 and for l i q u i d he l ium at about 1 . 5 ° K , X- - 700 seconds. Th i s seems to just ify Liebermann 's idea though, unfortunately, things in practice behave quite differently. Darna and L i d e (1954) made measurements of the time for streaming to start by observing thermal s t r ia set up in front of a sound source and reached the conclusion that streaming i s very suddenly establ ished on turning on the source, i n about 1/50 second,in fact . Our observations of streaming i n water, methyl a l coho l , benzene and l i q u i d he l ium also indicate that streaming i s established almost suddenly. . It thus appears that the s imple analysis given above i s quite inadequate. Perhaps the complex vort ices at the boundry, as observed by Liebermann i n c l a s s i c a l l iquids and turbulent flow i n l i q u i d he l ium observed by u s , have something to do wi th the setting up o f s t reaming. It i s thus clear that we have to design an experimental arrangement to measure v Q and E D independently. It should be pointed out (see Appendix TV) that by studying variat ion of force on an immersed vane as a function of energy density E 0 , i t i s possible to separate the two effects of radiation and streaming because of the fact that radiation force i s proportional to E 0 wh i le streaming force i s proportional to E 0 . However, as we sha l l see later , i t i s a quite futile suggestion for the peculiar case of l i q u i d h e l i u m . Experimental Arrangement (a) Streaming Tube: Streaming tubes of the type used by, Liebermann and L a m b and Piercy (shown i n Figure 17) were first tr ied for some organic . l iquids . Also .one s imi la r to that of Liebermann was used for l i qu id h e l i u m . We found that a strong counterflow of l i q u i d along ,the wa l l s of the ma in tube i n the case of L a m b and Piercy's-arrangement, as compared wi th a s m a l l return flow i n the side.or return tube, complicates streaming measurements . On the other hand, i n the case of Liebermann's arrangement, strong vortex motions at the boundries of the sound beam, affect quite seriously the streamline motion of the.fluid i n the main beam. T h i s effect i s c l e a r l y shown i n Liebermann 's photograph of streaming and cou ld , i n h i s case as w e l l as i n our case , be main ly due to inhomogenity of the beam in tens i ty . Because of these troubles, the streaming tube, detai ls of which are shown i n Figure 19 (A) , was designed. In this arrangement, the main stream i s separated from the symmetr ica l counter s t ream. Moreover , due to smal ler resistance to the flow of counter s t ream, there is no reverse flow i n the m a i n tube. Thus this; arrangement has features i n common with both Liebermann 's and Lamb, and Piercy 's arrangements. (A) \ Crystal N Holder v — - f -Giass rod Return tube Main tube Cotton (B) Kovar Seal Brass split- spring Glass tube-^ ,p rystal Brass ring (A) The Streaming Tube (B) The Crys ta l Ho lde r However, unlike the arrangement of these authors, this streaming tube i s designed to be used ver t ica l ly rather than horizontally because of the severe l imi ta t ions of the s izes of the he l ium dewars ava i l ab le . Although a miniature s ize horizontal tube similar , to that of Figure 19 (a) was designed and used, yet i t proved to be a failure and the reasons for failure sha l l be d i scussed la ter . T y p i c a l dimensions of the s treaming tube, used ver t i ca l ly are: Diameter of inner tube = 1.5 c m . Diameter of outer tube = 4 .5 c m . Mean length of inner tube = 10 c m . Length of the return l ine entrance and exit holes » 2 c m . each Another streaming tube w i th s im i l a r dimensions but wi th diameter of the inner tube of 0.75 c m . was also used to check dependence of streaming ve loc i ty on geometry. T o get an idea of the s ize of the largest possible streaming tube for horizontal set t ing, we have the dimensions of the tube used as fol lows: Diameter of outer tube = 1.9 c m . Diameter of inner tube = 6 m m . Mean length of inner tube = 2 .5 c m . Length of entrance and exi t holes - 6 m m . each The dotted l ines shown i n the figure are about 2 c m . long glass.rods (2 m m . diameter) used to hold the central tube in pos i t ion . Since i t was found that streaming is quite irregular in the immediate v i c in i t y of the transducer, the m a i n tube i s , therefore, extended by about 4 c m . so that the transducer i s 4 c m . away from the entrance of the circulatory f low. On the other end of the inner tube i s put a tapered plug to absorb sound completely whi le being impermeable to f l u i d . The plug consis ts of a 4 c m . long glass tube of inner diameter about 3 m m . having a glass d isc at the top wi th a hole at centre. Around the glass tube i s packed cotton in a conica l form. The s m a l l tube i s used as entrance for part icles of s o l i d H2 plus D 2 mixture to be used as indicators i n observation of s treaming. The streaming tube has 1" kovar seal at top which is soldered to a german s i lver tube about 60 c m . long which goes out of the cryostat through l i qu id he l ium dewar cap . Inside the german s i lver tube i s a long spira l type of constantan heater about 40 c m . long , wound on porcelain tube and enclosed i n a glass tube which extends right to the end of the p l u g . A s already mentioned, due to l imi t a t i on of the s ize of the dewards ava i lab le , the streaming tube is used v e r t i c a l l y . The conventional horizontal set t ing has- the advantage of avoiding convection currents as w e l l as that of select ing out l ight part icles of suspension so that gravity correction is e l iminated . In c l a s s i c a l l i q u i d s , convection currents would complicate matters considerably. In a ver t ica l streaming tube, the convection currents can be avoided by p lac ing the heating source (which i n this case is the dielectr ic heating of the crystal) at the top of the streaming tube. In our Amplifier Matching circuit Crystal 5 M c p s Ultrasonic Generator facing p o q e So case , this arrangement, too, would make.the cryostat and particularly the arrangement for getting par t ic les :of suspension inside.the cryostat, quite compl ica ted . The high thermal conductivity of . l iquid hel ium below the X-point, however, i s of great help here and the convection currents are not a problem.in this ca se . T h i s , as shown later , was confirmed by substituting the crystal by a heater. But the difficulty of applying gravity correction i n this case as d iscussed later , i s . a serious disadvantage.of the arrangement. The horizontal arrangement, already described, was also tried but i t proved to be a failure because of the fo l lowing reasons: 1) Completely irregular s treaming, due to the crys ta l being very close to the s m a l l streaming tube predominate over the. regular s t reaming. 2) St icking of part icles (due to reasons so far unknown to us), i s a serious problem in narrow tubes. For streaming to be. v i s i b l e , acoustic energy required i s quite: large which produces turbulent mot ion . The motion of part icles i s too compl icated to admit a possible interpretation. A s we sha l l see. later that although turbulent streaming i s unavoidable even in the case.of a ver t ica l arrangement, yet there exis ts a regular pattern of streaming which enables some quantitative observations to be made and interpretted. Hence, this type.of arrangement is used i n a l l experiments. (b) Ultrasonic Generator The circui t of the r . f . generator to drive 5 M c p s . quartz crys ta l is shown i n Figure 20 . In block diagram, i t consis ts of three components, namely , osc i l la tor , amplif ier and matching network. The osci l la tor is a combination of electron coupled (E . C O . ) and resonant output Colpi t ts type osc i l l a to r . Th i s c i rcui t has not only the advantage of a stable and high output, but also i t could be used as a frequency mul t ip l ie r so that the crys ta l could be driven at i ts harmonics . The r . f . amplif ier i s of usual des ign . The necessi ty . of matching network i s quite obvious . The radiation resisteSnce R L offered to crys ta l being of the.order of 1 0 to 5 0 k i l o ohm, the .matching c i rcu i t has to be such that the output stage of the amplifier should see .resistance. R S of about 5 0 0 ohm - to 5 k i l o o h m . Several networks.; for example , p i network , L network, tapped tank c i rcui t (Smith and Stumpf, 1 9 4 6 ) were t r ied but only the tapped tank c i rcui t worked sa t i s fac to r i ly . For this c i rcu i t , i f Lj_ = L,2 and coupling constant k is 1 . In the present c i rcu i t , (c) Crys ta l Holdex The crys ta l holder, to hold x - c u t , 5 M c p s . quartz c rys ta l of 3 / 4 " diameter, i s schemat ica l ly shown i n Figure 19"(b)-. The crys ta l in this arrangement is . loaded wi th l i qu id on both s i d e s . On the back, i t i s pressed by a double brass sp l i t - sp r ing attached to a circular r ing which s i ts on the c ry s t a l . A brass disc was also used for a r ig id backing but the r ing arrangement was found to be more eff icient . The front side of Rs R L L I s= L 2 and both are movable powdered iron-core type . the s i lvered crystal (radiating side) i s pressed against the holder case which i s connected to ground. A connection to the other side is.taken out of the press cap by way of a kovar sea l and forms the active end. The brass holder is put at the end of the.extension of the streaming tube by means of a rubber cement or de-Khot insky cement or araldite cement , (d) Intensity Measurement Device It was planned to measure the intensity of ultrasonics by measuring radiation pressure. However, due to certain di f f icul t ies , it was not possible to measure . i t . The arrangements used , diff icult ies met wi th and Mother possible methods of measuring intensity, of ultrasonics i n the peculiar case of l i qu id he l i um, are d iscussed separately in Appendix I V . Although i t i s quite desirable to measure intensity absolutely, yet in view of the semi-quantitative nature of the data obtained, i t was not worth designing an elaborate equipment to measure intensity absolutely when the possible conclusions from the data could just as w e l l be drawn by knowing relative magnitude of.intensity from r . f . voltage across the c rys t a l . In this connection, i t should be noted that the matching network used enables voltage/and. current to be i n phase so that voltage across the crystal i s direct ly a measure of power input to the c r y s t a l . Method of Measurement After fol lowing the usual procedure (described i n Chapter I) for transferring l i qu id he l ium and then cool ing i t down to theTowest possible temperature (at a pressure.of 3 c m . of o i l ) , a shot of ( H 2 t D 2 ) mixture (as discussed in Chapter IV) is given to suspend s m a l l part icles i n l i q u i d hel ium in the streaming tube. When the ultrasonic generator is switched on , these part icles start moving wi th the streaming f l u i d . They move up in-the maintube along the sound beam and return along the outer tube, thus, completing c i rcu la t ion . To verify whether or not the observed streaming, wh ich i s exact ly s imi l a r to thexonvect ion currents in f lu ids , was a true sonic effect, the.following experiment was performed: The crystal was replaced by a "disc type of heater madeout of a d isc of bakelite on the surface of which was wound constantan wire (#42) to serve as a heater. The heater had a r e s i s t anceo f about 50 ohms . It was possible to supply both d . c . as w e l l as a . c . of frequency varying from 60 cps . to 10 K c p s . to the heater. The power input could b e v a r i e d from 0 to 1/10 watt . It was found that this heating neither generated any observable streaming nor would i t affect the motion of freely fa l l ing part icles i n the m e d i u m . The ve loc i ty of streaming was measured by following an individual particle-and measuring the time taken by i t to traverse a known dis tance. The distance i s known by means :of graduations .on the streaming tube. A t one stage, i t was decided to measure the ve loc i ty of the part icles photographically. For this purpose, an osci l lograph camera, Cossor Mode l 1428, wi th a continuous f i l m drive (speed variable from 0.05 to 25 inches of f i l m per second) was used . The f i l m used was Kodak. Linograph Panchromatic (relative speed of 640) wi th l ight source of stroboscope xenon l a m p . However, tediOusness of this method as compared wi th the quite convenient method of v i sua l ly following the part icle over the longest possible-dis tances, l ed us to adopt the latter method for measuring veloci ty of s t reaming. Moreover , the poss ib i l i ty of fol lowing a particle moving in a straight l ine over a long distance makes this method much more-accurate than the photographic one . The ve loc i ty of moving part icles measured as above would normal ly need two important but difficult corrections to be made so.as to be able to determine the velocity, of s treaming of the. l i q u i d . The first one is the gravity,correction. The ve loc i ty of par t ic les , which usual ly have a density s l ight ly different from that of the m e d i u m , is either increased by, or decreased by the terminal ve loc i ty of free (gravity) f a l l of the part icles depending on whether the part icles are fa l l ing down i n the return tube or r i s ing up against gravity in the ma in tube. The gravity,correction could be eas i ly made i f we could measure the. velocity, of the.same particle while moving upwards as w e l l as downwards. However, the difficulty of the l i m i t e d f ie ld of view wh ich , in.no case , ensures the. poss ib i l i ty of fol lowing the same par t ic le . in its circulatory mot ion , coupled wi th the fact thatthere is a poss ib i l i t y of s t ick ing of the particle. to the glass w a l l or the p lug temporarily espec ia l ly when going from inner to outer tube, complicates the problem. The only resort, under these condit ions, is . to take an average over a number of observations on part icles which appear to be approximately of the same s i z e . Th i s average is corrected for gravity as fol lows: If and Vp2 are the magnitudes of the ve loc i t ies of part icles in the inner and outer tube of radi i T, 7 Yz respectively; v ^ and Vj_^ are ve loc i t ies .of l i qu id i n the inner and outer tube and Vg is the terminal veloci ty of the freely fa l l ing par t ic le , then from the-equation of continuity, we get v L l r x 2 = v L 2 ( T 2 2 - r 2 ) (23) Substituting numerical values, of Ti ,12. > we get v L i = 6 ' 4 v L 2 < 2 4 ) A l s o V P 1 = v L i " v g <25> V P 2 " V L 2 t v g .(26) . F rom these equations, we get V L 2 = V P 1 + V P 2 7.4 (27) v g = 5.4 v L 2 - ( V p l - V p 2 ) ( 2 8 ) 2 The second correction i s . l i kewise a pract ica l difficulty i n a stream l ine f low. There being veloc i ty gradient from tube boundry.to i ts a x i s , one has to take account of i t . The observations show quite a different feature. Although there are variations in the ve loc i ty of par t ic les , yet these, variations are i n no way of the type expected from stream line f low. We sha l l see later that this i s due to the fact that the flow is .turbulent. Effort i s made.to take as many observations.as possible so as.to be able to take the average. A s mentioned earl ier , intensity is measured relat ively by measuring the r . f . voltage across the.crystal by means of a V T V M . F i g - 2 I The Velocity of Acoustic Streaming in Liquid Helium as a function of Temperature ( Ul t rasonic Intensity being constant at E = 30Volts across Crystal 0 facing page 86 function of Ultrasonic Intensity(©c^) facing page 86 Resul ts The observations are divided into two parts. The first part studies temperature dependence of mean ve loc i ty of streaming ( V L J . ) i n the main?) tube as the intensity i s kept constant. The second part studies depence of V L I on the intensity (square of the voltage) at a constant temperature. A s the streaming data are.interesting, these are given in Tables I, II . Furthermore, the data i n Table I and Table II are graphical ly represented i n Figures 21 , 22 respect ive ly . To determine the dependence of vj^^ on the diameter of the.inner tube, another streaming tube, wi th diameter half that of the previous one, was used in s imi la r experiments. The results of dependence of YI_,J on temperature and intensity are. the same as those for the first tube. In addit ion, the results es tabl ish that - v L ° ^ d X / 2 (yt<"V{) (29) where d i s . the diameter of the inner tube. Some more qualitative observations of interest are as follows: (1) The quantitative data .mentioned above establishes.definitely that the streaming mot ion observed i s turbulent rather than stream l i n e . However, i t is.quite hard to decide from v i sua l observations in the. case of He II whether or not flow is turbulent because of the fact that the particles seem to move along approximately straight l i n e s . In contrast to i t , the case.of l i qu id He I is quite different. Here wi th the same intensity of sound, part icles (which are quite free to move without s t i ck ing to wa l l s ) have a large random motion superimposed on.the streaming mot ion . Th i s .large random motion is typ ica l of turbulence in c l a s s i c a l l i q u i d s . This remarkably different behaviour of part icles i n turbulent He I and He II i s note worthy. (2) Due to the behaviour of He I as mentioned above, quantitative measurements of velocity, of streaming i n He I are not very s ignif icant . Anyhow, these crude measurements at and.above the ^ - p o i n t show np sign of abrupt change i n the.velocity of streaming as might be expected from anomalous attenuation at the JVpoint . In fact, the ve loc i ty i s s l ight ly higher at. and above X -point as compared wi th that below the A -point . (3) When l iqu id he l ium I i s pumped on the top, a s m a l l ultrasonic energy creates bubbles in the l i qu id which make streaming quite compl ica ted . Th i s energy is much less than that required to produce cavitat ion in l i q u i d he l ium devoid.of any bubbles. When pumping i s stopped, higher intensity is required to produce bubbles. Th i s .could be due to (Blake, 1949) the presence of nuc le i such as s m a l l gas bubbles. It i s significant to note that formation of bubbles is quite pronounced at the X-point. D i scuss ion and Conclusions We have succeeded in observing v i sua l ly acoustic streaming i n l iqu id he l ium both above and below its X - p o i n t . Furthermore, we have quantitative results to prove that in a state of turbulence, as i t i s in our present invest igations, flow of l i q u i d He II obeys the c l a s s i c a l formula for turbulence. T h i s w i l l be d iscussed la ter . We have fa i led to measure absorption coefficient and second coefficient of v i scos i ty of He II because of the complicated interpretations of the observed turbulent f low. Nevertheless , we have -demonstrated the feas ib i l i ty of such an inves t igat ion. However, any further investigation has to consider the fol lowing factors: A s s u m i n g the va l id i ty of Reynolds ' cri terian (which in case .of He II i s doubtful and is also d iscussed later) , for stream l ine f low, R e = has.to be l ess than 2000. Because of the vanishingly s m a l l value of , this.requirement can be met only by having very s m a l l values of d and v 0 . It is possible to have v Q s m a l l by decreasing intensity or s t i l l better, by studying acoustic streaming at low frequencies. One can eas i ly work at frequencies as low as 100 K c p s . and s t i l l get observable s t reaming. Th i s case , part icularly true of H e , has a significant advantage i n that i t could provide more information on absorption at low frequencies. The possibi l i ty , of decreasing d i s not p romis ing at present in v iew of the affinity of part icles. to the wa l l s of the streaming tube which makes them s t ick and not take part i n the circulatory mot ion . The ser ious-ness.of this problem of s t i ck ing has already been emphas ized . Another d i f f icul ty , a purely technical one, arises from the fact that due to l imi ta t ions of s ize .of dewars avai lable i the streaming tube has to be used v e r t i c a l l y . The streaming tube has not only to be of large diameter so as to m i n i m i s e the poss ib i l i ty , of s t ick ing of par t ic les , but a lso quite long so as to m i n i m i s e the effect of l o c a l irregularities near the cyrs ta l on streaming i n the streaming tube. The dimensions of the streaming tube used i n the present experiments are j u s t reasonable to meet these requirements. These.requirements thus make the technical problem also quite d i f f icul t . However, wi th the arrangement used, i t seems to be.almost imposs ib le to keep Reynolds ' number l ess than 2000 because no streaming is observed (movement of part icles) for low enough in tens i t i es . For example no s t r eamingcou ld be observed for voltages l e s s than 20 vol ts across the c r y s t a l . A t about 10 v o l t s , one can just see some part icles suspended in the l i q u i d wi th streaming force balancing gravity force . A n attempt was made to. reduce the diameter of inner tube to half, but due to s t i ck ing of pa r t i c les , a lo t of difficulty was experienced i n making observations on s t reaming. We can sum up by saying that the diff icult ies of s t ick ing and that offered by gravity in a ver t ica l set-up have to be overcome for further experiments . The absence of any discontinuity or an anomalous change i n ve loc i ty at the ,A~pbint, tempts one to offer an explanation of observed anomalous attenuation at the \ -point i n terms ;of attenuation by gas bubbles eas i ly formed i n the l i q u i d by a very s m a l l ultrasonic in tensi ty . It i s not possible to support this hypothesis.because of a more consistent alternative .explanation of our observed resu l t s . The observed temperature dependent s treaming ve loc i ty , corresponding to complete absorption of sound at a l l temperatures, i s poss ib ly due to the existence of turbulence Which gives r ise to complete absorption. Since ve loc i ty i s found to be independent of temperature and so the concentration of superfluid and furthermore, the flow as a whole- obeys the c l a s s i c a l equation of turbulence, i t i s , therefore, obvious that both the normal and superfluid are indist inguishably moving together i n the s t reaming. Th i s observation can be reconci led to Landau 's assumption Cur l Vs = o only i f we assume that the observed ve loc i ty of streaming i s above the c r i t i c a l ve loc i ty l i m i t e d up to wh ich Landau's assumption is v a l i d . A n interesting thing to be noted is that were i t possible to have streamline acoustic s t reaming, another poss ib i l i t y would then be offered for making a c ruc ia l observation as to at what c r i t i c a l ve loc i ty the.superfluid i s dragged by normal f l u i d . The c l a s s i c a l experiments of A l l e n and Misener (1939) and others on studies of flow of He II through wide channels revealed neither c r i t i ca l , ve loc i ty nor complete superfluidity (characteristic, of narrow channels) . Although Landau's assumption Cur l vs = 0 prohibits turbulence i n He II, yet A l l e n and Misener ' s and A t k i n s ' data at 1 .2°K fits i n a s e m i - e m p i r i c a l formula (Daunt and Smi th , 1954) v = b d ' 4 (grad p ) / 2 - (30) where a - 0.07;. b = 30 . The first term corresponds to Mot t ' s expression for c r i t i c a l ve loc i ty and the second i s approximately the c l a s s i c a l expression for turbulent f l ow. In our case , the first term i s negligible while the second term represents streaming data very n i c e l y , i . e . '/2. I/* v 0 - b-d (gradp) (31) A n important thing to be noted i s that whereas equation (29) represents flow for Reynolds ' number R > 750, equation (30) represents flow for R = 30 ,000 . Reynolds ' cri terion demands turbulence for any finite movement of superfluid - a fact significant in experiments on rotation of l i q u i d he l ium but certainly contradicted by a number of other invest igat ions, part icular ly those of H o l l i s - H a l l e t (1955rl956) on the damping of an osc i l l a t i ng d i s c . H o l l i s - H a l l e t has calculated R for which large damping of d i sc set in as varying from 20 at 1 .2°K to 6000 at 2 . 1 7 ° K . Quoting h i m from his comments on the role of R i n deciding the on-set of turbulence .as mot ion of l i q u i d he l ium I i s not turbulent for R = 50,000, i t i s most unl ikely to be turbulent for lower numbers encountered in l i qu id he l ium II. " These conf l ic t ing conclusions seem to reveal that it i s poss ib le that Reynolds ' cri terian i s not applicable to the case of ^iquid h e l i u m . T A B L E I R . F . Voltage r 30 v r . m . s . T M e a n V p l Mean V p i V L I V L 2 V g ( O K ) c m / s e c c m / s e c c m / s e c c m / s e c c m / s e c 1.394 1.65 0 .77 2.09 0.33 0.44 .1.446 1.89 0.65 2 .2 0.34 0.30 1.496 1.40 0.77 1.81 0.30 0.47 1.651 1.76 0.77 2.18 0.34 0.43 1.748 1.62 .0.68 1.98 0.31 0.37 1.758 1.65 0.82 2.03 0.33 0.49 1.797 1.54 0.70 1.94 0.30 0.40 1.824 1.54 0.66 1.91 0 .30 0.36 1.850 1.97 0.68 2 .3 0.36 0.33 1.956 1.71 0.74 2 .12 0.33 0.41 .2.069 1.70 0.78 2 .15 0.34 0.45 2.160 1.51 0.77 1.98 0.31 0.47 2.164 1.53 0.85 2.06 0.32 0.53 2.182 1.86 0.87 2.36 0.37 0.50 2.203 2 .4 .1.54 3.41 0.54 1.00 2.450 2 .6 1.76 3.80 0.59 1.16 3.371 1.6 0.88 2.15 0.34 0.54 3.527 1.55 0.82 2 .05 0.32 0.50 T A B L E II Mean Temperature = 1.407OK Voltage Mean V p i Mean Vp2 V n V L 2 V g (volts) c m / s e c c m / s e c c m / s e c c m / s e c c m / s e c 10 No Observable Streaming 20 1.26 0.64 1.64 0.26 0.39 25 1.48 0.60 1.80 0.28 0.32 30 1.64 0.72 2 .05 0.32 ,0.42 35 1.97 0.61 2.24 ( 0.35 0.33 40 2.01 0.79 2.43 0.39 0.44 50 2.17 0.74 2.52 0.39 0.40 60 2.37 0.89 2.83 0.44 0.44 A P P E N D I X I H E L I U M II F I L M F I L T E R During the preparation of wire f i l l e d tubes for a l i qu id he l i um II.film'.filter or superleak for the problem described in.Chapter II, some observations of prac t ica l interest i n their construction and use have been made . The procedure of preparing such fil ters has been described in brief by A l l e n and Misener (1939) and Brown and Mendelssohn(1947). In our case , i t consis ts of drawing down a copper-nickel tube packed wi th 1200 wires of Eureka (0.0002" diameter) through steel, die-plate holes of success ive ly decreasing diameters . A typ ica l table of data representing four different filters i s given at the end . The channel wid th is determined experimentally from the rate of flow of he l ium gas at 90°K under a pressure head of one atmosphere. Rate of flow is given by Po i seu i l l e ' s relat ion x for a rectangular channel of length ' h ' and width ' a ' . Q = (PI - P2) a 3 h 12 <\t where the different letters have usual s igni f icance . (Al len and Misener ,1939) x Under these condit ions, "Knudsen" gas flow shal l , contribute to flow channel widths of the order of 0.08 m i c r o n . However, there i s no treatment at the present t i m e , for the case of flow between ordinary viscous . f low and "Knudsen" gas flow which should occur in this case of rectangular channels of width much smal ler than length. Mlcropiotograph of cross section of a Wire-Filled-Tube (16«nifi cation X 890) facing page 95 A microphotograph x of a c r o s s sect ion of such a filter showing regular hexagonal cross-sect ion of individual wires is also given in Figure 23 . From this microphotograph, length ' h ' and width ' a ' can be es t imated . However, given the;diameter of the wire and change in i ts length, length ' h ' of hexagon i s known wi th in the accuracy of measure-ments . From this parameter and a lso flow rate measurements, ' a ' Can be ca lcu la ted . A l s o , i t should be mentioned, that from the symmetry of regular hexagons, i f we associate hal f of each channel to each of the two contributing hexagons, there w i l l be '3n ' channels for ' n ' hexagons :or 'n ' w i r e s . T h i s , of course, is true when ' n ' i s very large, but even for s m a l l value of 'n*, the correction introduced by wires along the boundary is wi th in the accuracy of observations. . A logari thmic plot of channel width versus outer diameter of the wire- f i l led- tube (Figure 24) gives a .straight l i n e . Th i s holds when close packing of wires i n tube.hegins. The existence of such a s imple relation i s of pract ical importance i n determining the required outer diameter (and hence dieplate hole number wh ich is related logar i thmical ly to the outer diameter) to give the required channel width without the. r isk of b lock ing the fi l ter or even taking the trouble to make flow rate measurements at every stage of drawing. Another consideration: the b l o c k i n g o f such filters by heat treatment, pointed out by A l l e n and Misener (1939) had also to be investigated, x I am indebted to D r . J . G . Parr of M i n i n g and Metal lurgy Departmeht, U .B , C . and D r . F . D . Manchester , for this microphotograph. i 1 1 1 r L o g D i a m e t e r Fig 2 4 Dependence ot Channel Width on the Outer Diameter of the WireFilledTube ~ ~ ~ ~ facing page 86 for in our case , the filter had to be bent in;the form of ' U ' . We annealed the fil ter i n an atmosphere of he l ium gas at a pressure of about 20 c m . of mercury at 450°C and then could bend i t very e a s i l y . It was found that annealing decreased the flow rate of gas by a few percent which amounts .to about 1 percent decrease of channel w id th . A l s o i t was found that even l o c a l heating at points of soldering d id not produce any appreciable change i n flow rate . But an interesting observation of b locking of these fil ters after a few runs i n l i q u i d he l ium II could not be ignored. On chopping about a hal f centimeter from each end, the fi l ter was found normal aga in . Th i s may be due to impur i t i e s . The measurements of flow rate of l i q u i d he l ium II through superleaks of different channel width have been found interest ing. T y p i c a l curves of emptying rates at the same temperature (1 .284°K). , for two superleaks cut from the same wire f i l l e d tube but drawn through different ho les , c lear ly show that whi le the 2.21 micron superleak shows a pure f i l m flow which is independent of the pressure head, the 4.77 micron superleak shows a peculiar f low rate dependence on the pressure head. It should be stressed that impuri t ies have no role to play i n this, case as they cannot be effective at one stage whi le ineffective at the next . The flow rate observed i n the latter case can be fitted i n a rapidly converging 'power series of the pressure head. The relation so obtained i s much more compl icated than the s imple 1/3-law reported by. other authors l i k e Swinn and Rorschach (1955) who have analysed their data on the bas is of the relation dh , - a h 1 ? 3 dt where dh i s proportional to rate of f low, h i s the pressure head dt and a i s a constant. It i s to be noted that this relation does not have a constant t e rm. Thus , as the pressure head approaches zero, dh , dt that i s , the v e l o c i t y , approaches zero . T A B L E 3 Data for Wire F i l l e d Tube F i l t e r  Before Drawing Length of wires Outer diameter ( O . D . ) of wire f i l l e d tube ( W . F . T . ) Inner diameter of W . F . T . Number of wires Size of wires 36 c m . 0.364 c m . 0.272 c m . 1200 0 .002" Data After Drawing Hole No . 31 O . D . of W . F . T . ( cm. ) 0.364 Length of Wires ( cm. ) 36 Channel Width (micron) 39 0 .295 . 38 5.15 Close Packing Starts 40 0.280 .40 4.77 41 0.268 42 .3.20 42 0.256 44 2.24 43 0.244 48 1.42 44 0.233 53 0.95 45 0.223 58 0.67 Fig-25 Apparatus For Sealing GlassBomb f a c i n g porge 93 { ^ TUNGSTEN T HEATER -GLASS «BOMB GLEAN HELIUM A P P E N D I X II S E A L I N G O F HIGH P R E S S U R E G L A S S BOMBS The technique of sea l ing glass capsules or 'bombs' containing high pressure h e l i u m gas described below is. due to Hebert (1956) of our Low Temperature,Section. These capsules consis t of an annealed Pyrex glass tube of 8 m m . o . d . , 1.7 m m . w a l l thickness and about 5 c m . l o n g . The original tubing has a bursting strength of the. order of 4000 p s i g . although the pressure, at which they are sea led , ranges from 700 to 1200 p s i g . The sea l ing apparatus used i s schemat ica l ly shown i n Figure 25 . It i s an a l l -brass made.enclosure cons is t ing of two pa r t s . The glass capsule si ts i n the lower part, c a l l e d the 'holder ' . The upper part i s screwed on to the lower one. The. upper part contains a con ica l tungsten heater made from 0.(6)2" tungsten w i r e . It surrounds the drawn out part of the neck of the capsule (as shown i n the figure) and its posi t ion can be f inely adjusted. The two ends of the heater are s i lver-soldered to 1/8" brass rods . One of the rods i s connected to the ma in body of the enclosure by means.of a set-screw whi le the other i s insulated from the body of the enclosure by means of fibre washers which also serve as seals for high pressure. One one side of the upper part i s a glass post held i n posi t ion by the seal ing-nut . One can see the sea l ing operation through this pos t . The lower part i s soldered to a copper tubing w h i c h , i n turn, i s connected to a high pressure he l ium gas cy l inder . The capsule i s very carefully f i l l e d wi th manganous ammonium sulphate around the capi l la ry and, then, i s p laced in posi t ion i n the sea l ing chamber. Every effort i s made to keep the.capsule free of impuri t ies part icularly those.due to absorbed and adsorbed gases which are pumped by means of a vacuum pump. The capsule i s kept at sa l t - i ce temperature during pumping so as to avoid decomposit ion of the paramagnetic sa l t . After evacuating, the capsule i s flushed wi th clean he l ium gas and then connected to the high pressure source. The brass container has to be tight at this high pressure. For sea l ing , the heater is switched on and current i s increased s t ead i ly . To avoid decomposit ion due to heat from heater at the top of capsu le , the lower part of the sys tem (up to the neck of the capsule) is dipped i n i c e - c o l d water . In a typ ica l operation of heating, the current i s steadily raised to 22 amperes at 20 volts across the heater i n about 5 minu tes . The voltage i s controlled by an auto-transformer and the current is .regulated by a movable carbon-plate rheostat. The glass i s red hot at a heating power of about 450 wat t s . After reaching 22 amperes, the current i s s lowly decreased. The he l ium gas pressure i s maintained throughout the heating as w e l l as cool ing operation, so that pressure inside as w e l l as outside the bomb i s the same . The cool ing i s a slow process so that the g lass . i s w e l l annealed. The current i s decreased from 22.amperes to zero in about 5 minu tes . (A) (B) Fig. 2 6 Photograph of the Sealed Glass Bombs facing pqge 101 The chances of sea l ing such a bomb at a pressure of about 1200 p s i g . are only 10 to 20 percent. However, la ter , we found i t quite easy to sea l at pressures of about 800 p s i g . The capsules used by Hebert at high pressures had a l i m i t e d l i fe i n m o s t of the.cases whereas those used by us at lower pressures seemed to have quite.a long l i f e . The only way to know whether the capsule i s successful ly sealed or not, i s to examine i t carefully under a microscope for poss ible streaks. leading outwards through the neck . These streaks are.the source of leak of the high pressure gas and are formed in.the cool ing operation. The use of polar ised l ight to detect these streaks and other s imi l a r strains could prove usefu l . It should be mentioned that the weighing of the bomb i s not a rel iable method at a l l for knowing whether i t i s sealed or not . It i s due to the very s m a l l increase i n weight , a s .we l l as uncertainty.of the weight of the base bomb when i t undergoes seal ing operation i n which the s t i ck ing of mol ten glass to the heater i s quite p o s s i b l e . A photograph of two sealed bombs i s shown i n Figure 26 . The model (A) has been used throughout the experiments described i n Chapter III. A P P E N D I X JJJ C O N T I N U O U S M E A S U R E M E N T OF T E M P E R A T U R E S B E L O W 1°K The temperatures below 1°K are generally measured by us ing a paramagnetic salt as the thermometer. By measuring the suscept ibi l i ty 0C)<>f the sa l t , one determines the temperature of the salt from the Cur ie -Weis s law wh ich i n our case.of manganous ammonium sulphate i s s imply X = S- = 4.375 T T The suscept ib i l i ty (X) i s measured by a mutual inductance bridge. The principle of this bridge and description of i ts d . c . of ba l l i s t i c version has already been d iscussed i n Chapter III. A s i t i s desirable , i n measurement of f i l m transfer rates below l 'OK, to measure the temperature s imul taneously , continuously and, i f poss ib le , automat ical ly , the choice of a . c . mutual inductance bridge i s quite obvious . T h i s bridge essent ia l ly consis ts of an R - C osci l la tor wi th a frequency range.from 40 to 200 cps; a phase shift detector and an ampl i f i e r . A mutual inductance c o i l surrounds the sample of paramagnetic s a l t . T h i s mutual inductance is bucked by the mutual inductance of a s i m i l a r , separate and var iab le .co i l (cal led the "compensator) so that without the sample inside the first c o i l , the net e .m . f . i nduced in the secondary c i rcui t by alternating current i n the primary is zero. With the sal t i n , a net e . m , f . i s produced, ampl i f ied and the d . c . output i s fed to self-recording Ester l ine - Angus d . c . m i l l i a m e t e r . The .deflexion of this d . c . meter i s proportional to the suscep t ib i l i ty . 103. In contrast to the d . c . bridge, i n a . c . bridge, the mutual inductance c o i l has„ to have a large secondary impedance (unless inconvenient matching transformer i s used) so.as to be able to connect i t direct ly to the X-mete r and hence this c o i l i s so large that i t cannot be put right inside he l ium dewar or even l i q u i d nitrogen dewar. Th i s situation i s further compl ica ted by the fact that we l i k e to measure temperature at the same t ime as we make f i l m flow measurements. For this purpose, instead of one c o i l , two ident ical mutual inductance c o i l s forming a Helmhol tz system are used so that the capsule hangs i n the uniform f i e ld between the two co i l s and also i s v i s i b l e for observations. The use of these two b ig co i l s c a l l s for a third separate but s im i l a r c o i l sufficiently b ig to be able to buck the mutual inductances of the first two c o i l s . To get an idea of the huge dimensions invo lved , the fol lowing is the data on these c o i l s : He lmhol tz C o i l s : Primary 750 turns (#36.Cu) d . c . resistence = 28 ohm Secondary 30,000 turns (#38 Cu) d . c . resistence = 16,000 o h m . The primary i s wound on the secondary. Inner diameter of one of the. co i l s is 10 c m . wh i l e outer is about 13 c m . Compensator C o i l s : Secondary 30,063 turns (#38 Cu) Primary (#36 Cu) has separate taps for 750, 750, 500, J250, 250, 250 turns These c o i l s are on luci te formers and are put around the l iqu id nitrogen dewar and are at room temperature as compared wi th the d . c . mutual inductance c o i l which i s inside the l i q u i d he l ium dewar. Although due to the large diameter of these c o i l s , sens i t iv i ty of a . c . bridge i s much less .as compared wi th that of d . c . bridge, yet the sys tem is sensit ive enough to detect the paramagnetic sal t used i n the experiment. However, an unpredictable drift i n the output due to temperature fluctuations i n the b i g suscept ib i l i ty co i l s and i n addition s m a l l s ignal to thermal noise ra t io , forced us to abandon this bridge i n favour of the more inconvenient, though quite re l i ab le , d . c . bridge. A P P E N D I X IV M E A S U R E M E N T O F U L T R A S O N I C I N T E N S I T Y IN L I Q U I D H E L I U M We have already mentioned i n Chapter V that different methods used to measure absolute value of the ultrasonic intensity proved to be a fa i lure . Here we sha l l d iscuss the various methods that were or can be used to make such a measurement i n l i q u i d he l ium and the diff icul t ies to be encountered i n these methods. 1. Radiat ion Pressure Method Two arrangements were used to measure intensity by measuring the radiation pressure of u l t rasonics . The first arrangement consis ted of an a lumin ium disc suspended from a spring which is connected to a l ight iron core of a Linear Variable Differential Transformer X . ( L V D T ) . The radiation pressure on the d isc makes the spring contract w h i c h , i n turn, causes the iron core to move from i ts nu l l pos i t ion , result ing in net output e . m . f . from the two symmetr ica l secondaries of L V D T , which are connected i n series opposi t ion . The output i s ampl i f ied and can be measured. The springs were made from 0 .003" tungsten wire and had a sensitivity, of the order of 1.5 m m . per m g . The quartz springs available to us were much less sensit ive than.the tungsten springs and i n spite of some disadvantages, the latter were used . Unfortunately, no movement was detected . The reason l i e s i n the fact Ava i l ab le from Schaevitz Engineering and described i n their bul le t in A A - 1 A . that we cannot put i n enough energy without producing cav i ta t ion . For example , at about 1 . 5 ° K , l i qu id he l ium w i l l cavitate at a pressure of 6 x 103 d y n e / c m . ^ .Us ing this .value for acoustic pressure P , we.can calculate the radiation pressure P^ from IL = P £ _ I / C (1) 2fc2 where I i s the acoustic intensi ty , fc i s the acoustic impedance of the medium and c i s the velocity, of sound in the m e d i u m . The radiation force F R on a d i sc of radius r i s given by F R = 2 R if r 2 P r (2) where R i s the reflexion coefficient and is unity for thick A l d i s c . For a d i sc of 10 m m . diameter, the force on i t due to radiation pressure i s about 350 m i l l i g r a m s . Although there i s a poss ib i l i ty of detecting movement of the spring which would be about 0 . 4 m m . , yet due to the unknown decrease of the sens i t iv i ty of the spring at he l ium temperatures and the fr ict ional forces due to possible touching of the spring to the w a l l s , no movement was detected v i sua l l y through a microscope as w e l l as e l ec t r i ca l ly by the arrangement described already. The second arrangement, much more sensit ive than the first one, employed a pendulum, 53 c m . long , made from 0 .001" thick nylon thread and having an a luminium disc of 10 m m . diameter (1/16" thick) at the end of the thread. If m i s the weight of the bob (neglecting weight of the thread), t i s length of pendulum and x i s displacement of the pendulum under the steady force of radiation pressure of p of sound, then, approximately, (3) Us ing m = 27.4 m g . , t <= 53 c m . , T = 5 m m . , we get P r = 0.32 x d y n e / c m . ^ wh ich gives x «• 3 c m . for a radiation force of the order of 1 m i l l i g r a m . In this arrangement, the s treaming tube was used horizontally and the pendulum was suspended i n the path of the b e a m . However, the presence of streaming of ve loc i ty of a few c m : per second over shadowed the effect due to radiation pressure. The streaming force F s i s given by that F s i s much larger, than E R . Th i s case for l i q u i d he l ium i s quite the converse.of that for c l a s s i c a l l iquids where F R i s larger than F s . It appears that the only hope, of measuring F R i s in being able to separate the two effects due to F R and F s . However, we sha l l see now that the poss ib i l i ty of doing so is s m a l l . by assuming that whi le the radiation force i s instantaneous, the streaming force takes some t ime . i n bui ld ing up . However, this c l a i m i s challenged by a number of authors to whom reference has been given i n Chapter V . In l i qu id h e l i u m , too, the two effects appear to be instantaneous so that separation i s not pos s ib l e . Another way to separate the two effects, which we have used successful ly i n the case of water, consis ts i n studying dependence of pendulum (4) For ve loc i ty .of s treaming v about 2 . c m / s e c . , we can see Liebermann (1949) c l a i m e d to have separated the two effects . deflexion on voltage E across the c r y s t a l . Since, veloci ty of streaming i s proportional to the intensi ty of sound which i t se l f is proportional to the square of voltage (E ) , therefore , F s o C E 4 (5) and F R oC E 2 (6) The total Ref lexion, which is proportional to the sum of F s and F R i s given by x r A E 2 + B E 4 (7) where A and B are constants . T h u s , a plot of X / E 2 against E 2 can be used to determine.A and B and, hence,;the"contributions.of radiation and streaming forces . But the; case.of l i q u i d he l ium i s not so s i m p l e . Streaming i s observed to be turbulent i n the arrangement used and so the above equations do not h o l d . It should be mentioned that one might think that i t i s not necessary to separate the two effects experimental ly because, of the existence of the theorem of conservation of momentum according to which the sum of radiation and streaming term i s .approximately constant. However, i t i s clear that to separate the two effects.by ca lcu la t ion , one has to know the absorption coefficient - the quantity we are looking forward to calculate by measuring in tens i ty . 2 . Ray le igh D i s c Method Th i s absolute method i s not practicable i n the case of high frequency for the thickness of d isc has to be much smal ler than the wavelength of sound used ( K i n g , 1935) wh ich i n this case, i s 0.05 m m . 3 . Sel f -Reciproci ty Method In this method (Cartenson, 1947), the transducer i s caused to emi t short pulses which are reflected from a perfect reflecting surface and, in turn, received by the same e lement . T h u s , one can calibrate the transducer. However, due to lack of electronic equipment and further,it being not very essen t ia l to have-absolute measurement of intensi ty , i t was not thought worthwhile to pursue this lengthy project . 4 . Q - Measurement Method In p r inc ip le , one could measure. Q of the crys ta l i n the medium and then knowing the power input, calculate.the power output. It i s not so s imple i n practice and one can only expect to get an order of magnitude, rather than the absolute value of intens i t y . The major sources of error are: (a) The unknown amount of power los t ind ie lec t r ic heating of the c rys ta l i n l i qu id h e l i u m . (b) Measured Q , being the Q of the quartz crys ta l i n the holder and the matching c i rcu i t , ca lcu la t ion of Q of c rys ta l alone i s approximate. In our case , matching c i rcui t has .a Q smal ler than that of the c ry s t a l , wh ich complicates matters a l o t . 5 . Comparison Method By measuring the. radiation pressure at a f ixed ultrasonic . intensity i n l iqu ids of different acoustic impedance (Pc), one can determine the dependence of radiation pressure on the acoustic impedance and then extrapolate i t for the case of l i q u i d h e l i u m . 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