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THESIS

(. Fir-ff
0k ..

This is to certify that the
dissertation entitled

THERMAL DESIGN STUDIES IN NIOBIUM AND
HELIUM FOR
SUPERCONDUCTING RADIO FREQUENCY
CAVITIES

presented by

Ahmad Aizaz

has been accepted towards fulfillment
of the requirements for the

Ph.D. degree in Mechanical Engineering

 

 

M

[' Major Pr 8 "s ignature
20 Nov. 2.006

 

 

Date

MSU is an Affirmative Action/Equal Opportunity Institution

 

LIBRARY
Michigan State
University

 

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE

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2/05 p:/ClRC/DateDue.indd-p.1

THERMAL DESIGN STUDIES IN NIOBIUM AND HELIUM FOR
SUPERCONDUCTIN G RADIO FREQUENCY CAVITIES

By

Ahmad Aizaz

A DISSERTATION

Submitted to
Michigan State University
In partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Mechanical Engineering

2006

ABSTRACT

THERMAL DESIGN STUDIES IN NIOBIUM AND HELIUM FOR
SUPERCONDUCTING RADIO FREQUENCY CAVITIES

By
Ahmad Aizaz
Liquid helium is now a common refrigerant for superconducting radio frequency (SRF)
technology. It helps in cooling the niobium (Nb) made SRF cavities below their
superconducting transition temperature (Tc). Testing of these cavities is routinely done in
saturated liquid helium-I (He-I) for low field measurements and in saturated liquid
helium-II (He-II) for high field measurements. The thermal design studies of these
cavities involve two thermal parameters, namely the temperature dependant thermal
conductivity of Nb at low temperatures and the Nb-He interface heat transfer coefficient.
During the fabrication process of the SRF cavities, Nb sheet material is plastically
deformed through a deep drawing process to obtain the desired shape. The effect of
plastic deformation on low temperature thermal conductivity as well as heat transfer
coefficients in the two states (He-I and He-II) has been studied. Strain induced due to the
plastic deformations reduces the thermal conductivity in its phonon transmission regime,
which may explain the performance limitations of the SRF cavities during their high field
operations. The effect of annealing the Nb samples at two different temperatures to
restore the phonon peak in the thermal conductivity curve has also been studied.
Measurements of heat transfer coefficient for nucleate pool boiling liquid helium are in
agreement with the theoretical predictions as well as with the existing experimental data.
These measurements reveal higher heat transfer for rough surfaces as compared with

smoother ones. Kapitza conductance measurements for Nb - He-II interface for rough

surfaces with surface index (SI) more than 3 as compared with the flat and smooth
surfaces have also been carried out before and afler the annealing. Here, SI is defined as
the ratio of exposed surface area to that of projected area. These measurements provide
helpful insight in understanding the problem of Kapitza conductance for different surface
topologies where the rough and annealed surface revealed increased Kapitza

conductance.

ACKNOWLEDGEMENTS

The contributions of many people, both at scientific level and personal level have been
fundamental to the completion of this dissertation.

First and the foremost, I would like to extend a huge thank you to my technical
advisor, Terry L. Grimm, for essentially teaching me to think like a scientist. Though
being fi'om outside of my parent department of mechanical engineering, he completely
understood my shortcomings and limitations. In this, he walked me through the difficult
process of objective learning with patience. His pushing and urging for more effort when
I needed to be pushed, and his general availability despite his hectic commitments
elsewhere in his office, have made him the ideal advisor. His guidance in the art of
presenting the research work like a salesman has simply been elegant.

Thanks also go to Neil T. Wright for acting as the Chair of my guidance committee.
His filII involvement in almost all the developmental stages of my research with valuable
suggestions and guidance on my write-up drafts remained crucial in completing my
thesis.

Special thanks to Robert McMasters for acting as Co-Chair on my guidance
committee despite his transfer to another university in Virginia. His polite and friendly
personality has been very instrumental, much beyond his professional obligations, in my
initial settling down in the new environment of MSU as an international student. I must
also acknowledge his support in getting me involved in this research project at the

cyclotron laboratory.

iv

I must thank Steve Bricker and Scott Hitchcock for extending all out helps in the
construction of my experimental test stand. Their vast experience in the laboratory helped
me a lot in diagnosing problems all along the way and the possible solutions to rectify
them. My special thanks to Steve for his untiring help during the long hours of calibration
runs before each and every test that I performed. I feel myself lucky to have such a highly
dedicated personality always near at hand whenever I needed it.

Many of the figures in this dissertation, as well as in various papers over the last few
years, are the work of John Bierwagen. He also was a great help whenever a quick
machine work was required, often on urgent basis. His dexterity both as a computer
graphic designer as well as a machinist has always been handy in a niche of a time.

Thanks are also due to Felix Marti, Walter Hartung, Chris Compton, John
Popielarski, Matt Johnson and Laura Saxton for their help and facilitating in many ways
to accomplish my research. Felix’s patience to listen intently to my often-simplistic
briefing during the gradutate conference meetings, helped me to assemble the ideas
coherently before presenting them. Solution to many of the mathematical riddles that
came across during my research work is especially due to Walter’s imaginative thinking.
Without even stepping into the clean room or getting closer to the dangerous chemicals of
BCP bath, my samples always got first class treatment in the hands of Chris and Laura
for which I am highly indebted. John’s help came in through multiple ways especially
whenever I needed to grab the instruments that were crucial to my test runs. Matt’s help
came in whenever computer simulations needed to be performed quickly and reliably.

Thank you very much all of you for your time and help.

In the same vein, 1 would like to thank all my student colleagues, both graduate and
undergraduate, in the SRF group who have worked on my experimental setup or shared
with me their emotions and feelings through thick and thin, especially Susan Musser,
Dave and Mandy Meidlinger, Hairong J iang and Tim Kole.

The collaborative work with Pierre Bauer, Claire Antoine and Chris Boffo, from
Fermi National Accelerator Laboratory, who helped me with providing flat plate Nb
samples, was pivotal to my research validation. This dissertation would be incomplete
without their contributions. Titanification processing on the Nb samples done at Cornell
was only possible through the generous help extended by Rongli Geng. This help is well
appreciated.

Finally, I’d like to thank my best friend and wife Asma Aizaz, for all the late night
dinners brought to the lab when I had to run experiments, and the help in encouraging me
to finish the thesis well in time. Without your tremendous patience, support and love, this
work would not have been completed.

Last of all, I would like to thank both my parents, Malik Sher Afgan and Mrs.
Munawar Malik, for moral and spiritual support through their well-needed prayers. This

work would not have been completed without their prayers.

vi

TABLE OF CONTENTS

1 Introduction .............................................................................................. 1
1.1 SRF Cavities ...................................................................................... 1
1.2 Advantages of SRF Cavities ............................................................. 3
1.3 Limitations of SRF Cavities Through Thermal Breakdown ............. 6

1.3.1 Thermal Breakdown Due to Defects ........................................................... 6
1.3.2 Global Thermal Instability (GT1) ................................................................ 7
1.3.3 Thermal Breakdown Due to Field Emission ............................................... 8
1.4 Thermal Magnetic Interactions in SRF Cavities ............................... 8
1.4.1 Effect of Thermal Conductivity on SRF Cavities ..................................... 11
1.4.2 Effect of Heat Transfer Through Liquid Helium on SRF Cavities ........... 14

2 Thermal Conductivity at Low Temperatures ......................................... 15
2.1 Phonon Conduction ......................................................................... 15
2.2 Electronic Conduction ..................................................................... 16
2.3 Molecular Conduction ..................................................................... 17
2.4 Thermal Conduction in Niobium .................................................... 18
2.5 Phonon Scattering Mechanisms ...................................................... 19

2.5.1 Phonon-Electron Scattering ...................................................................... 20
2.5.2 Boundary (Grain or Specimen) Scattering ................................................ 20
2.5.3 Impurity Scattering ................................................................................... 20
2.5.4 Dislocation Scattering ............................................................................... 20
3 Liquid Helium and Its Pool Boiling Theory .......................................... 21

vii

3.1 Pool Boiling Heat Transfer in He-I ................................................. 21

3.1.1 Regimes of Pool Boiling Heat Transfer in He-I ....................................... 23
3.1.2 Factors Affecting Pool Boiling Heat Transfer in He—I .............................. 24
3.1.3 Nucleate Boiling Heat Transfer in He-I .................................................... 26
3.2 Pool Boiling Heat Transfer in He-II ................................................ 29
3.3 Kapitza Conductance ....................................................................... 31
3.3.1 Acoustic Mismatch Model (AMM) .......................................................... 34
3.3.2 Diffuse Mismatch Model (DMM) ............................................................ 35
3.3.3 Phonon Radiation Limit (PRL) ................................................................. 36
3.4 Factors Affecting Kapitza Resistance ............................................. 38
3.4.1 Effect of Boundary Layer ......................................................................... 40
3.4.2 Effect of Pressure ...................................................................................... 41
3.4.3 Effect of Attenuation ................................................................................. 42
3.4.4 Effect of Conduction Electrons ................................................................. 43
3.4.5 Effect of Localized States at the Interface ................................................ 44
3.4.6 Effect of Surface Roughness ..................................................................... 44
3.4.7 Effect of Titanium Getter Material ........................................................... 47
3.4.8 Effect of Deposited Thin Films on Kapitza Conductance ........................ 48
Experimental Apparatus ......................................................................... 50
4.1 Cryogenic Dewar ............................................................................. 50
4.2 New lid Insert Assembly ................................................................. 52
4.3 Dewar Lid Top Plate ....................................................................... 54
4.4 Thermal Radiation Shields .............................................................. 54
4.5 Sensors and Wiring on Liquid Helium Side ................................... 56

viii

4.6 Housing for Sample Holder Assemblies ......................................... 59

4.7 Sample Holder Assembly ................................................................ 59
4.8 Sample Heater ................................................................................. 62
4.9 Temperature Sensors and Wiring in the Vacuum ........................... 62
4.10 Calibration of Carbon Sensors ........................................................ 65
4.11 Flat Plate Sample ............................................................................. 66
4.12 Cold Shock and Leak Test .............................................................. 68
4.13 Steady State Measurement Technique ............................................ 69
5 Results and Analysis .............................................................................. 71
5.1 Cylindrical Sample Preparation ...................................................... 71
5.2 Flat plate Nb Sample Preparation .................................................... 74
5.3 Thermal Conductivity Measurements ............................................. 75
5.3.1 Repeatability Analysis .............................................................................. 75
5.3.2 Validation Comparison with Literature Data ............................................ 79
5.3.3 A New Trend in the Data .......................................................................... 80
5.4 Nucleate Boiling Heat Transfer Measurements .............................. 87
5.4.1 Surface Roughness Measurements ........................................................... 87
5.4.2 Comparison of Machine Cut and BCP etched Surfaces ........................... 91
5.4.3 Effect of Mechanical Polishing ................................................................. 92
5.4.4 Effect of Surface Index (SI) ...................................................................... 93
5.4.5 Overall Effect of Surface Roughness ........................................................ 95
5.5 Kapitza Conductance Measurements .............................................. 96
5.5.1 Comparison with Literature Data .............................................................. 96
5.5.2 Repeatability ............................................................................................. 98

ix

5.5.3 Effect of Surface Index ............................................................................. 99

5.5.4 Effect of Low Temperature Annealing ..................................................... 99
5.5.5 Effect of Moderate Temperature Annealing ........................................... 101

6 Discussion ............................................................................................ 105
6.1 Thermal Conductivity Above 3 K ................................................. 105
6.2 Thermal Conductivity Below 3K .................................................. 108
6.3 Nucleate Boiling Heat Transfer ..................................................... 111
6.4 Kapitza Conductance ..................................................................... 1 13

7 Conclusions .......................................................................................... 115
7.1 Effect of Phonon Peak in Nb Thermal Conductivity .................... 115
7.2 Effect of Surface Roughness on Nucleate Boiling Heat Transfer
Coefficient ............................................................................................... 116
7.3 Effect of Kapitza Conductance ..................................................... 1 16
7.4 Looking Ahead: Large Phonon Peak and Large Kapitza
Conductance ............................................................................................ 1 17

8 Appendixes ........................................................................................... 118
8.1 Appendix A Pool Boiling Heat Transfer ..................................... 119
8.2 Appendix B Estimation of Measurement Errors .......................... 123
8.2.1 Thermal Conductivity Measurement Uncertainty .................................. 123
8.2.2 Interface Heat Transfer Measurement Uncertainty ................................. 124

8.3 Appendix C Parallel Heat Flow Losses ....................................... 126

8.4 Appendix D Estimation of Heat Leak into Stainless Steel Tube .. 128
8.5 Appendix E Sensors Sensitivity Comparison .............................. 130
8.6 Appendix F Specification Sheet for Nb Samples ......................... 131

8.7 Appendix G Surface Roughness Parameters ................................. 132

8.7.1 Introduction ............................................................................................. 132
8.7.2 Ra, The Roughness Average ................................................................... 132
8.7.3 Rq, The Root Mean Square (rrns) Roughness ........................................ 134
8.7.4 Rp, Rv, Rt, Rpm, Rvm, Rz ..................................................................... 136
8.7.5 Application .............................................................................................. 137
References ............................................................................................ 138

xi

LIST OF FIGURES

FIGURE 1.1: A TYPICAL SRF ELLIPTICAL CAVITY, SHOWING PARTICLE BEAM AND
FUNDAMENTAL OR LOWEST RF FREQUENCY MODE (TM 010) OF THE SRF
CAVITY. ........................................................................................................................ 2

FIGURE 1.2: HEAT TRANSFER FROM RF SURFACE OF SRF CAVITY TO THE
SURROUNDING LIQUID HELIUM AS DEPICTED THROUGH ELECTRICAL ANALOGY. .......... 8

FIGURE 1.3: THERMAL-MAGNETIC SIMULATIONS WITH CONSTANT THERMAL
PROPERTIES, K AND H, FOR THE TWO CASES ARE SHOWN. ABOUT FIVE TIMES
INCREASE IN K AND THREE TIMES INCREASE IN H RESULTED IN ALMOST 50%
IMPROVEMENT IN APPLIED MAGNETIC FIELDS. PARAMETERS USED FOR THESE
SIMULATIONS ARE RRES = 5 NOHMS, E = 3 MM, T3 = 2 K, F = 1.3 GHz, RR =
230. RISE IN SURFACE RESISTANCE (RIGHT AXIs) DUE To RF SURFACE
TEMPERATURE INCREASE IS ALSO PLOTTED. ............................................................... l 1

FIGURE 1.4: TEMPERATURE DEPENDANT THERMAL CONDUCTIVITY CLEARLY SHOWS
THE FORMATION OF A PHONON PEAK AT ~ 2 K. ADAPTED FROM PADAMSEE ET

AL [91. ......................................................................................................................... 12
FIGURE 2.1: THERMAL CONDUCTIVITY BY PHONON MECHANISM “41 .................................. 16
FIGURE. 2.2. THERMAL CONDUCTIVITY BY ELECTRON MECHANISM “41 .......................... 17

FIGURE 2.3. MEASURED THERMAL CONDUCTIVITY FOR A NB SAMPLE As COMPARED
WITH A THEORETICAL PREDICTION. STEEP DECREASE IN ELECTRONIC THERMAL
CONDUCTIVITY IS CLEARLY SEEN BELOW TC. THE CURVE ON THE RIGHT SHOWS
THE RISE IN THE THEORETICAL PREDICTION OF PHONON THERMAL
CONDUCTIVITY WHEN SCATTERING OF PHONONS FROM ELECTRONS Is
SUBSTANTIALLY REDUCED IN THE SUPERCONDUCTING STATE AS THE
TEMPERATURE Is DECREASED. THIS IS THE CAUSE OF PHONON PEAK VISIBLE IN
THE CURVE ON THE LEFT [9]. ....................................................................................... 19

FIGURE 3.1 TYPICAL HEAT TRANSFER RELATIONSHIP FOR POOL BOILING HE-I “71 ............. 24

FIGURE 3.2: NUCLEATE BOILING HEAT TRANSFER To HE I. VARIATION IN THE DATA
Is SEEN THROUGH A DIFFERENCE IN SURFACE ROUGHNESS WHERE INCREASING
SURFACE ROUGHNESS LOWERS THE SURFACE SUPER HEAT. PLOTTED
EMPIRICAL FIT 13 BY SCHMIDT AND ADAPTED FROM VAN SCIVER “71 ......................... 28

FIGURE 3.3: PHASE DIAGRAM OF LIQUID HELIUM .............................................................. 29

xii

FIGURE 3.4: INCIDENT AND TRANSMITTED PHONON ANGLES, RELATED ACCORDING
To SNELL’S LAW. IF THE VELOCITIES AND ANGLES OF THE INCIDENT AND
TRANSMITTED PHONONS ARE Cm, 0m, Cm, AND BM, RESPECTIVELY, THEN
SIN BIN/C,N = SIN OWN/Cram (ADAPTED FROM SWARTz ET AL. “81) ............................ 34

FIGURE 3.5: EXPERIMENTAL DATA OF VARIOUS RESEARCHERS SHOWN AS A,B,C,D
AND, E (ADAPTED FROM SWARTZ ET AL. “81) FOR THE MEASUREMENT OF
KAPITZA BOUNDARY RESISTANCE (RBD=RK) BETWEEN COPPER AND HE-II,
MULTIPLIED BY T3 To REMOVE THE STRONG TEMPERATURE DEPENDENCE.
NOTE THE DROP IN RBDT3 WITH TEMPERATURE ABOVE 0.1 K. THE UPPER SOLID
LINE IS THE PREDICTION OF AMM AND THE LOWER ONE Is THE PREDICTION OF
DMM, ALMOST TWO ORDERS OF MAGNITUDE DIFFERENT. THE KINK IN THE
CURVE B IS THE RESULT OF IRREPRODUCIBILITY OF KAPITZA RESISTANCE EVEN
ON THE SAME SAMPLE FROM RUN TO RUN, AND DOES NOT REFLECT ANY
PHYSICAL PHENOMENON. ........................................................................................... 38

FIGURE 3.6: PLOT OF THE REDUCED THERMAL BOUNDARY RESISTANCE P AS A
FUNCTION OF T00. THE STARS INDICATE MEASUREMENTS BETWEEN THE SOLID
AND 3HE. THE DATA IS BY SEVERAL RESEARCHERS AND COMPILED BY SWARTz
[18]
ET AL . ................................................................................................................... 40

FIGURE 3.7: MEASURED KAPITZA RESISTANCE BETWEEN ELECTRO-POLISHED
COPPER AND 3HE, MULTIPLIED BY T3 TO REMOVE STRONG TEMPERATURE
DEPENDENCE. THE PRESSURE DEPENDENCE Is SEEN WELL BELOW 0.5K, BUT
NEAR 0.5K AND ABOVE, THE KAPITZA RESISTANCE IS NEARLY PRESSURE
INDEPENDENT. ADAPTED FROM SWARTz ET AL “8] ..................................................... 42

FIGURE 3.8: A KAPITZA RESISTANCE OF THE FOUR SAMPLES, ALONG WITH THEIR
SURFACE TREATMENT AND CORRESPONDING ROUGHNESS DATA ARE SHOWN IN
THE CHART. THE GRAPH SHOWS THE RELATIVE EFFECT OF SURFACE
ROUGHNESS ON RK WITH TEMPERATURE VARIATION. CE STANDS FOR
CHEMICAL ETCH, EP MEANS ELECTROPOLISHING AND, A STANDS FOR
ANNEALING. (ADAPTED FROM AMRIT ET. AL. [3”) ..................................................... 45

FIGURE 3.9: PICTORIAL DISPLAY OF SURFACE CHARACTERISTICS WITH ASSOCIATED
DIFFERENT LEVELS OF LENGTH SCALES. ..................................................................... 47

FIGURE 3.10: EFFECT OF TI ATOMS PRESENT IN NB NEAR THE INTERFACE ON RK
(ADAPTED FROM J. AMRIT ET AL [30]) ......................................................................... 48

FIGURE 4.1: SCHEMATIC DIAGRAM OF THE CRYOSTAT DEWAR WITH EXPERIMENTAL
SETUP HOOKED UP ON A LID ASSEMBLY INSIDE THE DEWAR. ...................................... 51

FIGURE 4.2: NEW LID ASSEMBLY CARRYING TWO SAMPLE HOLDER ASSEMBLIES. .............. 53

FIGURE 4.3: DEWAR LID TOP PLATE WITH VARIOUS PORTS AND RECEPTACLES .................. 55

xiii

FIGURE 4.4: TOP PLATE OF LID ASSEMBLY HAS VARIOUS PORTS AND FEED-THROUGH
ASSEMBLIES. MULTI-CONNECTER ELECTRICAL FEED-THROUGH IS A CUSTOM
MADE ASSEMBLY DESIGNED AND ASSEMBLED AT NSCL [39] ...................................... 57

FIGURE 4.5: RIBBON CABLES FROM THE MULTI-CONNECTOR FEED-THROUGH ON THE
TOP PLATE ARE PASSING THROUGH THE RADIATION SHIELDS. A PLATINUM
SENSOR IS SHOWN ON THE LOWER SHIELD. ................................................................. 57

FIGURE 4.6: HELIUM LEVEL AND TEMPERATURE SENSORS ARE PLACED IN LIQUID
HELIUM. ..................................................................................................................... 58

FIGURE 4.7: CUT AWAY VIEW OF THE HOUSINGS FOR SAMPLE HOLDER ASSEMBLIES
MOUNTED ON THE SIX-WAY CROSS CONFLATE FLANGE. ............................................. 60

FIGURE 4.8: THE TWO HOUSING (A) FOR THE Two SAMPLE HOLDER ASSEMBLIES (B)
ARE CONNECTED TO A COMMON VACUUM LINE. THIS SAMPLE HOLDER
ASSEMBLY Is USED To MEASURE THERMAL CONDUCTIVITY As WELL AS THE
INTERFACE HEAT TRANSFER COEFFICIENT OF THE SAMPLE SIMULTANEOUSLY IN
ONE EXPERIMENTAL SETTING. .................................................................................... 61

FIGURE 4.9: UNASSEMBLED COMPONENTS OF CYLINDRICAL SAMPLE HOLDER
ASSEMBLY .................................................................................................................. 63

FIGURE 4.10: ASSEMBLED NB SAMPLES IN A SAMPLE HOLDER ASSEMBLY. THREE
CARBON SENSORS WITH MANGANIN WIRES ARE ALSO VISIBLE. SENSOR C1 13
CLOSER To THE HEATER END OF THE NB SAMPLE AND SENSOR C3 Is CLOSER TO
THE HE INTERFACE. .................................................................................................... 64

FIGURE 4.11: RECTANGULAR FLAT PLATE NB SAMPLE (11 X 1.4 X 0.3 CM)
ATTACHED TO A 2 %” CONFLAT FLANGE WITH THE HELP OF A BRASS SCREW.
THREE COLLINEAR CARBON SENSORS ARE PLACED To MEASURE THE THERMAL
CONDUCTIVITY OF THE SAMPLE. (DRAWING NOT TO THE SCALE) ............................... 67

FIGURE 4.12: FLAT PLATE NB SAMPLE Is ATTACHED To A CONFLAT FLANGE. THREE
COLLINEAR CARBON SENSORS (C1, C2 AND C3) ARE PLACED TO MEASURE THE
THERMAL CONDUCTIVITY OF THE SAMPLE. A THIN FOIL HEATER ON THE FREE
END OF THE SAMPLE IS ALSO SHOWN. ......................................................................... 67

FIGURE 4.13. LID ASSEMBLY Is ON A TEST STAND AND HOOKED UP WITH A LEAK
DETECTOR. THE HOUSING ASSEMBLY Is IMMERSED IN A POOL OF LIQUID
NITROGEN FOR COLD SHOCK ....................................................................................... 68

FIGURE 5.1: Two CYLINDRICAL SAMPLES Sl (LEFT VIEW) AND S2 (RIGHT VIEw)
ARE SHOWN. S2 SAMPLE HAS A ROUGH SURFACE (~ 1-2 MM) AS COMPARED
WITH SMOOTH SURFACE OF SAMPLE Sl. ..................................................................... 73

xiv

FIGURE 5.2: THERMAL CONDUCTIVITY OF SAMPLE 81 AT THREE DIFFERENT BATH
TEMPERATURES, As SHOWN IN THE LEGEND, MEASURED IN NOV 2005. ERROR
BARS ARE ON THE GRAPH WITH BATH TEMPERATURE OF 2.0 K ................................... 76

FIGURE 5.3: THE PHONON PEAK IS SHOWN NEAR AVERAGE TEMPERATURE OF 2 K IN
THERMAL CONDUCTIVITY MEASUREMENTS OF THE TWO SAMPLES, 81 AND 82,
IN NOVEMBER 2005 TESTS. (TB=2 K). ....................................................................... 78

FIGURE 5.4: VARIATION OF THERMAL CONDUCTIVITY WITH TEMPERATURE FOR
SAMPLE 82 Is SHOWN ALONG WITH DATA [4°]. (LITERATURE DATA ADAPTED
FROM PADAMSEE “21). ................................................................................................ 79

FIGURE 5.5: PHONON PEAK Is CLEARLY ABSENT IN THE THERMAL CONDUCTIVITY
MEASUREMENTS OF SAMPLE 81 IN THE TESTS CARRIED OUT AFTER 3% STRAINS
INDUCED IN THE SAMPLE (JUNE 2006 TEST) AS COMPARED WITH ITS TESTS
PERFORMED IN MAY 2006 .......................................................................................... 80

FIGURE 5.6. THE PHONON PEAK DID NOT REAPPEAR AFTER IT WAS LOST DURING THE
COMPRESSION PROCESS IN PREPARATION OF (MAY TEST) SAMPLE 82. LOW
TEMPERATURE HEAT TREATMENT (JUNE TEST) HAS SMALL EFFECT ON THE
THERMAL CONDUCTIVITY OF THE NIOBIUM. ............................................................... 81

FIGURE 5.7 (A) AND (B): THE LOW TEMPERATURE HEAT TREATMENT (JULY 2006)
HAS SLIGHTLY INCREASED THE THERMAL CONDUCTIVITY OF FLAT SAMPLES IN
PHONON CONDUCTION REGIME, FROM THEIR PREVIOUS TEST IN MARCH 2006
ON THE As RECEIVED SAMPLES ................................................................................... 83

FIGURE 5.8: MODERATE TEMPERATURE ANNEALING DURING TITANIFICATION
PROCESS RESULTED IN THE REFORMATION OF THE PHONON PEAK (OCT 06)
AFTER IT WAS LOST DUE TO INDUCED STRAINS (JUNE 06) IN THE As RECEIVED
NB SAMPLE 81 (MAY 06) ........................................................................................... 85

FIGURE 5.9: MODERATE TEMPERATURE ANNEALING DURING TITANIFICATION
RESULTED IN THE REFORMATION OF PHONON PEAK (OCT 06) AFTER IT WAS
LOST DUE TO INADVERTENT STRAINS INDUCED (MAY 06) IN THE As RECEIVED
NB SAMPLE 82 (Nov 05) ........................................................................................... 86

FIGURE 5.10: HISTOGRAM AS WELL AS DIFFERENT SURFACE ROUGHNESS
PARAMETERS IS SHOWN FOR SAMPLE S1. ..................................... 89

FIGURE 5.11: HISTOGRAM As WELL AS DIFFERENT SURFACE ROUGHNESS
PARAMETERS IS SHOWN FOR SAMPLE S2. NOTE THAT THE VALID DATA IS ONLY
31.2 % IN THIS MEASUREMENT. ................................................................................. 90

FIGURE 5.12: NUCLEATE BOILING HEAT TRANSFER MEASUREMENTS FOR THE Two
CYLINDRICAL SAMPLES MADE IN NOVEMBER 2005 TESTS AT BATH
TEMPERATURE OF 4.2 K. SCHMIDT CORRELATION IS THE SAME AS THE ONE IN
FIGURE 3.3. ................................................................................................................ 91

XV

FIGURE 5.13: EFFECT OF MECHANICAL POLISHING ON SAMPLE Sl HAS RESULTED IN
AN INCREASE OF SUPERHEAT FOR THE SAME APPLIED HEAT FLUX. ............................. 92

FIGURE 5.14: DECREASE IN SUPERHEAT DUE TO INCREASE IN SI OF SAMPLE 82 IS
SEEN IN MAY 06 MEASUREMENTS As COMPARED WITH NOV 05. ................................ 94

FIGURE 5.15: THE EFFECT OF INCREASING SURFACE ROUGHNESS IS TO REDUCE THE
SUPERHEAT FOR A GIVEN HEAT FLUX. ........................................................................ 95

FIGURE 5.16: KAPITZA CONDUCTANCE MEASUREMENTS FOR THE TWO CYLINDRICAL
SAMPLES, 81 AND 82, CARRIED OUT IN NOVEMBER 2005, ARE COMPARED
WITH THE ONE GIVEN BY S. BOUSSON ET AL. [38]. ....................................................... 97

FIGURE 5.17: KAPITZA CONDUCTANCE MEASUREMENTS OF THE SAMPLE S1 BEFORE
AND AFTER THE PLASTIC DEFORMATIONS REVEAL NO SIGNIFICANT
DIFFERENCE. .............................................................................................................. 98

FIGURE 5.18: THE KAPITZA CONDUCTANCE OF SAMPLE 82 SURFACE WITH SI > 3
(MAY 06) AND THE EFFECT OF LOW TEMPERATURE ANNEALING WITH SI > 3
(JUNE 06) IS COMPARED WITH THE MACHINE CUT SURFACE (NOV 05).
LITERATURE DATA IS ADAPTED FROM BOUSSON ET AL. [38] ...................................... 100

FIGURE 5.19: KAPITZA CONDUCTANCE FOR SAMPLE Sl COMPARISON BEFORE (MAY
AND JUNE 06) AND AFTER TITANIFICATION (OCT 06). .............................................. 101

FIGURE 5.20: KAPITZA CONDUCTANCE INCREASE HISTORY FOR SAMPLE 82
THROUGH ALL THE PROCESSES ESPECIALLY AFTER TITANIFICATION (OCT 06). ........ 102

FIGURE 5.21: COMPARISON OF KAPITZA CONDUCTANCE VALUES OF THE TWO
SAMPLES, 81 AND S2, AFTER TITANIFICATION (OCT 06). ......................................... 104

FIGURE 6.1: A LINEAR CORRELATION (LEAST SQUARE FIT) IS SHOWN FOR THE
VARIATION OF RRR VALUES WITH THOSE OF THERMAL CONDUCTIVITY AT 4.2
K. ............................................................................................................................. 107

FIGURE 1.3: (REDRAWN FOR ILLUSTRATION PURPOSES) THERMAL-MAGNETIC
SIMULATIONS WITH CONSTANT THERMAL PROPERTIES, K AND H, FOR THE TWO
CASES ARE SHOWN. ABOUT FIVE TIMES INCREASE IN K AND THREE TIMES
INCREASE IN H RESULTED IN ALMOST 50% IMPROVEMENT IN APPLIED
MAGNETIC FIELDS. ................................................................................................... 109

FIGURE 6.2: THERMAL-MAGNETIC SIMULATIONS FOR A DEFECT FREE 300 MHz
CAVITY AT 4.2 K. VARIATION IN SURFACE RESISTANCE Rs IS ALSO PLOTTED
(RIGHT AXIS) AGAINST THE SURFACE TEMPERATURE Ts. ......................................... 1 12

FIGURE A. 1: TYPICAL BOILING CURVE, SHOWING QUALITATIVELY THE DEPENDENCE
OF THE WALL HEAT FLUX, Q, ON THE WALL SUPERHEAT, AT, DEFINED As THE
DIFFERENCE BETWEEN THE WALL TEMPERATURE, Tw, AND THE SATURATION

xvi

TEMPERATURE, Ts”, OF THE LIQUID. SCHEMATIC DRAWINGS SHOW THE
BOILING PROCESSES IN REGIONS I—V. THESE REGIONS AND THE TRANSITION

POINTS A—E ARE DISCUSSED IN THE TEXT. (ADAPTED FROM BURMEISTER [44]) ........ 121
FIGURE C.1: THERMAL CIRCUIT SHOWING PARALLEL HEAT LEAK .................................... 127
FIGURE D. 1: ESTIMATION OF HEAT LEAK INTO THE STAINLESS STEEL TUBE .................... 128

FIGURE G.1: PROFILES ARE SHOWN ABOVE FOR SIMPLICITY FOR THE
UNDERSTANDING OF BASIC DEFINITION OF RA. WHEN EVALUATING THE 3D
PARAMETERS THE VARIOUS SURFACE FUNCTIONS ARE UNDERSTOOD To APPLY
To THE COMPLETE 3D DATASET. ADAPTED FROM COHEN “81 .................................. 133

FIGURE G.2: PROFILES FOR RQ ARE SHOWN ABOVE FOR SIMPLICITY. WHEN
EVALUATING THE 3D PARAMETERS THE VARIOUS SURFACE FUNCTIONS ARE
UNDERSTOOD To APPLY To THE COMPLETE 3D DATASET. ADAPTED FROM
COHEN [48] ................................................................................................................ 135

FIGURE 6.3: PEAK AND VALLEY STRUCTURE OF A TYPICAL SURFACE IS SHOWN.
ADAPTED FROM COHEN “81 ...................................................................................... 136

xvii

LIST OF TABLES

TABLE 5.1: SURFACE AND BULK PREPARATION OF THE TWO CYLINDRICAL NIOBIUM
SAMPLES .......................................................................................................... 72

TABLE 5.2: BULK PREPARATION OF TWO FLAT PLATE NIOBIUM SAMPLES .......................... 74

TABLE 6.1: RRR VALUES OF NB SAMPLES AGAINST THE MEASURED VALUES OF
THERMAL CONDUCTIVITY AT 4.2 K. ............................................................... 106

TABLE C1: PERCENTAGE OF HEAT LOSS CALCULATION BASED ON THERMAL CIRCUIT
GIVEN IN FIGURE C.1 ...................................................................................... 127

TABLE E1: COMPARISON OF SENSITIVITY OF CARBON SENSOR WITH GERMANIUM
SENSOR [39‘ ..................................................................................................... 130

xviii

1 Introduction

Advances in particle physics are closely linked to advances in particle accelerators.
Among many factors affecting the performance of the accelerator, the availability of
higher energy particle beams is one of the most important.

A key component of the modern particle accelerator is the device that imparts energy
to the charged particles. This is the superconducting electromagnetic cavity resonating at
a microwave radio frequency (SRF). The resonating frequency is typically between 50
MHz and 3000 MHz. Among many practical limitations of achieving high performance
in SRF cavities, this study deals with their thermal limitations. The work described in this
thesis is an attempt to improve the performance of SRF cavities by studying its thermal
design parameter.

1.] SRF Cavities

An SRF cavity is the device through which power is coupled into the particle beam of
an accelerator. AS mentioned earlier, SRF accelerating cavities are microwave resonators,
which generally derive from a “pillbox” shape (right circular cylindrical), with
connecting tubes to allow particle beam to pass through for acceleration. Figure 1.1
shows a typical cylindrically symmetric cavity. The electric field is roughly parallel to the
beam axis, and decays to zero radially upon approaching to the cavity walls. Boundary
conditions dictate that the electric field must be normal to the metal surface with peak
electric field occurring near the iris; a region where the beam tube joins the cavity. The
magnetic field is in the azimuthal direction, with the peak magnetic field located near the

cavity equator but decays to zero on the cavity axis.

Cavity Equotor‘ Magnetic Field

Portmfle $“J/L mea\

 

 

 

D 2;"—

b Axis

Beom

   
 

COVitY INS Electric Field

Figure 1.1: A typical SRF elliptical cavity, showing particle beam and fundamental
or lowest RF frequency mode (TM 010) of the SRF cavity.

An important figure of merit in evaluating various SRF cavity structures is its quality
factor (Q0), which is defined as the product of RF angular frequency (0)) and the ratio of
the stored energy in the electromagnetic fields (U) to the dissipated power (Paiss), and can
be written as

Qo=wU/Pdiss (1-1)

It is known that surface resistance causes power dissipation by the surface currents,
which arise in order to support the magnetic fields at the RF surface. The dissipated
power can be written as the ratio of the square of accelerating voltage (V acc) and the shunt
impedance (R,), that iS,.

misc/“92m. (l-la)
and Q0 is also frequently written as

Qo=G/Rs (1-1b)

where, G is the geometric constant depending upon the shape of the cavity and R5 is the
RF surface resistance.

Therefore, the dissipated power in the cavity can be related with Rs, as:

Ribs =TE—“L—=—R—“—c—Rs (l-lc)
(“QiQo (“QJG

Here (R. / Q.) is another figure of merit ofien quoted for determining the level of mode

V2 V2

excitation by charges passing through the cavity. Note that it is independent of RF
surface resistance as well as of cavity size. Often one quotes the average accelerating
electric field (Em) that the electron sees during transit. This is given by

Eacc = Vacc/d (1-1d)
where, d is the active length of an accelerating structure and is usually an iris to iris
distance of a typical elliptical cavity such as shown in figure 1.1.

Wall losses usually occur in the form of heat dissipation through the wall material of
the cavity into the liquid helium surrounding the cavity. As the power increases, a stage
comes when the losses become too great to allow the liquid helium to remove the heat
{Tom the cavity. At this point, the temperature of the material starts rising higher than the
material’s superconducting critical temperature (Tc) above which the material becomes a
normal conductor, bringing about a sudden degradation of the performance of the SRF
cavity. This mechanism of performance degradation of SRF cavities is usually known as
thermal breakdown or “quench” and is described more in the subsequent paragraphs.

1.2 Advantages of SRF Cavities

It is well known that many materials, known as superconductors, lose all DC

electrical resistance when the temperature drops below the Tc. However, unlike DC

resistance, RF resistance is zero only at T=0 K (absolute zero). At temperatures above
absolute zero, but below the critical temperature, the surface resistance is greatly reduced,
yet non-zero. This phenomenon is usually explained through the “London two-fluid
model”. According to this model, for temperatures less than Tc/2, the superconducting

surface resistance can be well represented as:

f Ao()
R— =A-T—eXP(- K——0T—)+R . (1-2)

The first term on the right hand side of equation 1-2 is the BCS surface resistance; named
after the three scientists who first presented the BCS theory i.e. Bardeen, Cooper, and
Schrieffer. For surface temperatures (T,) less than TJZ, the binding energy, A(0), is
nearly unchanged from its value at absolute zero. The coefficient A is a function of
material parameters such as the superconducting coherence length, the penetration depth,
the electron mean free path, and the Fermi velocity. Here, k3 is the Stefan-Boltzmann
constant and the value of A (1.333x10“1 for RRR ~230) can be evaluated computationally
via programs by Turneasurell lor Halbritter [21.Material purity is usually described via
the Residual Resistivity Ratio (RRR), which is the ratio of electrical resistivity at room
temperature to the normal conducting resistivity at 4.2 K. The BCS surface resistance in a
typical f = 3 GHz cavity varies from 3 uOhms at T=4.2 K to less than 1 nOhm at T=1.4
K. It is important to note that cavities operating at 4.2 K in He-I, i.e. when the
temperature is not significantly less than TJ2, the use of relation 1-2 may not produce
accurate results predicted by BCS theory.

The second term on the right hand side of equation 1-2 is the residual resistance R0,

which is temperature independent. Mechanisms for R0 are not well understood, though

several possibilities have been proposed and investigated [31’ [4]. Residual resistance values
are generally found to be between 5 and 100 nOhm, although values as low as 1 nOhm
have been measured [5 1.

The chief advantage in the use of SRF cavities is the reduced wall loss power
dissipation. The wall loss power dissipation is proportional to the surface resistance,
which is reduced by a factor of 106 in superconducting cavities as compared to copper
cavities. The total power usage does not reflect all of this gain, however, due to the need
to refiigerate the cavities to liquid helium temperatures. Even including refrigerator
power, using typical refrigerator efficiencies, the net power usage drops by a factor of
several hundreds to a thousand in superconducting cavities. In continuous operation
(CW) this means greatly reduced power and higher accelerating gradients. In pulsed
operation, SRF cavities offer long pulse lengths and high duty cycles compared to normal
conducting cavities.

Niobium is currently the material of choice for superconducting cavities. The main
reason for this choice is that niobium has the highest critical temperature of all pure
metals (T6 = 9.25 K), and in addition, is relatively simple to fabricate.

Experimentation with specially designed SRF non-accelerating cavities [6M7] has
clearly shown that there are no fundamental limits to the peak electric fields on a niobitnn
surface up to EMT, = 200 MV/m. The theoretical limit on accelerating cavity performance
is therefore dependant on the cavity’s magnetic fields. The DC critical magnetic field is
thus related with the metal temperature as

H.(T)=H.(0)[1—(%]] (1-3)

C

where H¢(0) is the critical magnetic field at T=0 K.

In RF conditions, the requirements are relaxed somewhat, as the penetration of the
magnetic field into the RF surface requires nucleation of a flux line, which requires a
finite amount of time. The nucleation time has been determined to be such that the
complete shielding of magnetic fields can persist to fields higher than the critical field, up
to a limit termed the superheating critical field, Hsh. In niobium, the superheating critical
field is estimated to be approximately Hsh = 2300 [8] oersted (or simply Oe).

In typical SRF accelerating cavities, Hm], = 2300 De corresponds to accelerating
gradients of 50 to 60 MV/m. Given that, accelerators with niobium cavities generally
operate at Eace = 10-20 MV/m, the need for further improvement is clear.

1.3 Limitations of SRF Cavities Through Thermal Breakdown

Thermal breakdown is a phenomenon in which the temperature of part or the entire
RF surface exceeds the critical temperature, thereby becoming normal conducting and
rapidly dissipating all of the stored energy in the cavity fields. The field at which
breakdown occurs depends upon multiple factors; including thermal conductivity of the
bulk niobium and heat transfer from the niobium to liquid helium bath. There are many
processes, which, by acting either independently or simultaneously, can cause thermal
breakdown of the cavity. Some of these are listed below.

1.3.1 Thermal Breakdown Due to Defects

Material defects are the most commonly understood cause of thermal break down in
SRF cavities. In this, the break down is considered localized, where a small defect in the
RF surface dissipates power more readily than surrounding superconducting walls.

Breakdown occurs when the power dissipation overwhelms the ability of the Nb to

conduct away the heat to surrounding liquid helium acting as the coolant. Thus the
temperature of the defect rises above the critical temperature of the bulk Nb, making it
normal conducting and eventually leading to thermal breakdown. It has been shown in
previous studies [9] that the local magnetic field just before thermal break down around a

hemispherical defect, to a first order approximation, can be given as:

Hm = ’M (1_4)
rde

Here, H ,b is the thermal breakdown magnetic field, k is the thermal conductivity of the

Nb, Tc is critical temperature of Nb, Tb is the bulk temperature of the liquid helium, ‘rd’ is
the defect radius and Rd is the electrical surface resistance of the defect. From this
relation, it appears that the defect in the Nb skin is thermally isolated from the cooling
effects of the surrounding liquid helium and the breakdown magnetic field is totally
dependant only upon the thermal conductivity of the bulk Nb.

1.3.2 Global Thermal Instability (GT1)

Another type of thermal breakdown usually attributed to high surface magnetic field
regions is global thermal instability (GT1) “0]. In GTI, the heating is nearly uniform over
all of the high surface magnetic field regions. GTI occurs because, even without localized
defects, the RF surface retains some uniform residual RF resistance. GTI is initiated
when the power dissipation due to residual resistance raises the temperature enough such
that the exponentially growing BCS surface resistance becomes dominant, causing a
thermal runaway process, which leads to a thermal breakdown. In this kind of thermal
breakdown, both the Nb thermal conductivity as well as Nb-He interface heat transfer

coefficient determines the maximum achievable fields before quenching.

1.3.3 Thermal Breakdown Due to Field Emission

Field emission (FE) is the process of electrons coming out of the Nb surface in the
presence of high surface electric fields. These emitted electrons impact elsewhere on the
cavity surface, heating the surface, and thereby increasing the surface resistance. This
increases power diSsipation of the cavity, as well as adding to the load on the
refiigeration plant. In the extreme, FE heating of the cavity walls can lead to thermal
breakdown, as mentioned in the last section. Acceleration of emitted electrons draws
power out of the electromagnetic fields, which would otherwise be available for
acceleration of the particle beam. Eventually, as fields are raised, the power dissipation
into FE related processes limits the attainable fields in the cavity.

1.4 Thermal Magnetic Interactions in SRF Cavities

As discussed in the last section, the heat removal from the inner surface (RF surface)
of the SRF cavity is via conduction through the niobium and convection to the liquid
helium surrounding the cavity. The operation of low frequency (f < ~300 MHz) cavities
as well as low field-testing of high frequency SRF cavity is usually performed in liquid
He I at around 4.2 K. However, the operation of high frequency cavities as well as high
field-testing of low frequency cavities is done in He-II at around 2.0 K, where the helium
is in its super-fluid state. Through a simple thermal circuit, as shown in figure 1.2, the

influence of different thermal variables on the critical magnetic field can be understood.

L
H n T, m r, Rx
e MERE”:
///. q ——> T

Figure 1.2: Heat Transfer from RF surface of SRF cavity to the surrounding liquid
helium as depicted through electrical analogy.

8

Here, T, the RF surface temperature due to various phenomena discussed in the last
section, is created through joule heating, k(T) is the temperature dependant thermal
conductivity of the niobium, e is the thickness of the niobium, Ti is the temperature of the
interface surface of the Nb exposed to the surrounding liquid helium, Rk is the bulk
thermal resistance of the liquid helium when it is in normal state, however, it is
commonly called the “Kapitza resistance” when liquid helium is in its super-fluid state,
i.e. below 2.17 K and, Than, is the temperature of the bulk liquid helium.

Figure 1.2 shows an element of the cavity wall in contact with He and the electrical
analog of the heat flow path fi'om the RF surface to the He bath. Through this electrical
analogy, it is clear that heat current, q, has two distinct heat resistances in its path. One is
characterized by the bulk material property of niobium i.e. thermal conductivity k, and
the other is characterized by the heat transfer coefficient h or by, of the liquid helium
depending upon its state.

For the case of a defect free cavity, the influence of R. (or ‘l/hk) and k on the
accelerating field, Eacc. can be investigated as done by others [1”, through some simple
thermal-magnetic simulations as shown in figure 1.3. Here, Eacc is directly proportional to
peak surface magnetic fields (Bpk). Consider that the heat flux q in W/m2 is solely
produced by Joule heating at the inner surface wall and is uniquely due to the RF surface

resistance R,, then it can be expressed [I I] as :

q = 1/ (21102)RsBz (1-4).

where, 110 is the magnetic permeability, B, expressed in Tesla, is the magnetic flux density
on the RF surface of the cavity and, R. is the surface resistance in ohms, which, itself is a
function of RF surface temperature Ts as given by equation (1-2). The use of this
equation on the high surface magnetic field regions, where the most intense heating is
taking place, implies that the magnetic fields are Bpk.

Using equation 1-4 along with 1-D thermal model of figure 1.2 for a defect free SRF
cavity, the maximum achievable magnetic fields (Bpk) are plotted as a function of the RF
surface temperature for the two sets of values of thermal conductivity and Kapitza
conductance. These constant property simulations clearly Show the effect of increased k
and hk on the higher allowable magnetic field, which is then limited by the critical
magnetic fields only. Another important observation can be made from these simulations
is that for a defect free cavity, the maximum RF surface temperature rise due to applied
magnetic fields is on the order of a few hundred milli-Kelvin only. This observation helps
in understanding the operating range of temperature dependant thermal conductivity of

Nb, as explained in the next section.

10

 

 

 

 

 

 

 

 

 

 

200 Bentical 1.oe+oo
130 ‘:-:f\—_ “ ‘ — ‘T‘L 1.05-01
160 —~ -_ ~\e - —4
_ _ _ » 1.05432
140 I k—.28,h—.24~
A ~ 1.05433
1- 120 I —k=.06;h=.08“
E100 ‘b‘r- -—-—- fl: *— ——- —- ———~~~1.0E—04A
V _ w _ ‘ JIL___§:fi ___ _____ __¢ m
m 23 -.. _ # iM/cmeL i J 1.0E-05E
40 -L- __ _7_h._(QVV_/CIT12[KL_ #. .H 105069,
20 _ _ __ 9“,; _u R,“ -~ roe-07¢”
0 . . '__. . 1.0E—O8
1.5 2 2.5 3 3.5 4
T3 (K)

Figure 1.3: Thermal-magnetic simulations with constant thermal properties, k and
h, for the two cases are shown. About five times increase in k and three times
increase in h resulted in almost 50% improvement in applied magnetic fields.
Parameters used for these simulations are Rm = 5 nohms, e = 3 mm, Tb = 2 K, f =
1.3 GHz, RRR = 230. Rise in surface resistance (right axis) due to RF surface
temperature increase is also plotted.

1.4.1 Effect of Thermal Conductivity on SRF Cavities

At the operating temperatures of the cavities, when Nb is in its superconducting state,
its thermal conductivity varies dramatically with temperature and is governed by the
interplay of both of the heat carriers, i.e. electrons and phonons. Figure 1.4 shows a
typical curve for Nb thermal conductivity at low temperatures where a local maxima is
clearly visible below ~ 3 K. The thermal conductivity of Nb above 4 K is well known and
correlates with its RRR value (k = RRR/4 @ 4.2 K) [9], however, below ~ 3 K, in the
phonon conduction region, this correlation is not well understood. In this phonon
conduction regime, the microstructure of the bulk Nb, characterized by strains and lattice

imperfections or grain size and grain boundaries, plays a vital role in determining the

11

final shape of its thermal conductivity curve. A more thorough treatment of the thermal

conductivity at low temperatures is given in chapter 2.

 

 

 

 

1000 '7 " ' r I T l T r “T‘ __
3%- . RRR = 400 33
500 A RRR = 250 D A q
1i- I RRR = 90 a f .
- no 4.” «ii
C
O
z; 200 - .’ A I
'5 L
._ o
'9 0 ‘ *
g 100 -- 'e d
o : 0 fl :
. i
0 P “ . ' "i
E ' ‘9 o P I "
E 50 n . . “ ' I 1
g ‘ ‘0‘. 0.. .
I- . he ‘ ' d
20 r 4 ‘1‘” r
I. 4
I..
_____. l - run“ I A" a r a l
101 2 5 T(K) 10

Figure 1.4: Temperature dependant thermal conductivity clearly shows the
formation of a phonon peak at ~ 2 K. Adapted from Padamsee et al '9'.

Much emphasis has been placed in the past on the techniques for improving thermal
conductivity of the bulk Nb for temperatures above ~ 4 K to reduce the effect of
‘localized defects’ on the performance of the SRF cavities. In the case of cavities with a
defect, which is of small size (on the order of micrometers) as compared to the wall
thickness (~ 2-4 mm), a normal conducting defect causes an intense localized RF heating

spot resulting ultimately in the thermal quench of the cavity. Due to poor thermal

12

conductivity of Nb, this localized heating remains thermally isolated from the surrounded
pool bath of liquid helium. Thus, in case of thermal breakdowns due to defects, it is
estimated [9] that since the temperature in the neighborhood of the defect is between the
bath temperature and Tc, it is the high-temperature (4 to 9.2 K) thermal conductivity of
Nb that is the most important and which will have the strongest effect on thermal
breakdown. However, for defect fiee cavities, as has been discussed earlier, the total RF
surface temperature rise is on the order of few hundreds of mK. Thus, the phonon thermal
conductivity regime (< 3 K) of Nb plays the dominant role in determining the quench
field of the defect free high frequency cavities. Nevertheless, for both of the cases, i.e.
high frequency defect free cavities (< 3 K) or the cavities with defects (> 4 K), improving
the thermal conductivity of Nb remains a paramount objective. The improved thermal
conductivity at temperatures above ~ 4 K comes fiom improved purity of the metal
quantified through its RRR value. Bulk Nb purity has improved greatly, from RR of 10-
30 to RR > 250, in the last twenty years through improved purification methods, e.g.
high vacuum electron beam melting ['2].

Niobium can be further purified of interstitial oxygen by solid-state gettering [“1’ “2'.
In gettering, the Nb is heated to ~1400 C, with exposure to either yttrium or titanium
vapor. The vapor is absorbed into the surface of the Nb; the higher affinity towards 02 of
Y or Ti (compared to Nb) effectively removes the oxygen from the bulk Nb. Oxide layer
at the outer surface is removed through acid etching, leaving purified bulk Nb. Typically,
material with a RRR of 250 can be purified to a RRR of 500 through this process.
Increasing the grain size or reducing the grain boundaries can also improve the

microstructure of the bulk Nb, thus improving the phonon thermal conductivity of the

13

Nb. Annealing the Nb at temperatures above ~ 1400 C in a high vacuum fumace creates
large grain size resulting in the formation of a “phonon peak” in the thermal conductivity
of Nb, as seen in figure 1.4 [9].
1.4.2 Effect of Heat Transfer Through Liquid Helium on SRF Cavities

The other resistive parameter in the thermal circuit, as shown in figure 1.2, is the Rk,
i.e. resistance to heat transfer through the liquid He. It has been shown “2] that in the case
of defects on the RF surface of the cavity, the heat transfer coefficient at the Nb-He
interface does not play as strong a role as the thermal conductivity in determining the
quench field. This may be the reason that little attention has been given in the past to this
particular aspect for the improvement of performance in SRF cavities. Although, a lot of
information has been accumulated in the understanding of this mode of heat transfer in
other spheres of scientific research, such as quench dynamics of superconducting
magnets, relatively little attention has been paid toward this mode for the purpose of
improving the performance of SRF cavities. Moreover, the importance of the heat
transfer to the pool bath of liquid helium increases, as the thermal conductivity improves
with purity, or as the Nb wall thickness is decreased, or if there are no defects such as that
shown through simple thermal-magnetic simulations presented in figure 1.3. Therefore,
it seems quite logical to explore various channels of pool boiling heat transfer, both at 4.2
K as well as at temperatures below 2.1 K, for the improvement of the cavity performance.
For this purpose, basic understanding of heat transfer in liquid He is of primary

importance and is presented in chapter 3.

l4

2 Thermal Conductivity at Low Temperatures

To understand how thermal conductivity depends upon temperature, especially at low
temperatures, it is useful to understand the basic mechanisms for energy transport
through materials. There are three basic modes of energy transport (phonon, electron, and
molecular), and hence heat conduction, through a solid. These are described below.

2.1 Phonon Conduction

Lattice vibrational energy transport, also known as phonon conduction, occurs in all
solids, both dielectrics and metals. This mode dominates in nonmetallic crystals, some
inter-metallic compounds and even exists in superconducting metals, such as Nb, below
their transition temperatures. For single crystals at low temperatures, this mode of heat
conduction can be effective, equaling or exceeding the conduction by pure metals.
Phonon thermal conductivity may be mathematically expressed as

kt=1/3 CVVA (2-1)
where, Cv is the lattice heat capacity of the material, v is the average propagation velocity
of phonons, which travel at the speed of sound, and A is the mean free path of the
phonons between collisions or before scattering from the obstacles. The variability of
these parameters with respect to temperature is illustrated in figure 2.1 "4]. The peak in
phonon conductivity for Nb (T6 = 9.2 K) usually occurs near 2K, whereas for some non-

metals it could be as high as about 30 K.

15

 

(a) c v (b)

 

 

 

 

 

 

 

(I
>—

(c’ (d)

 

 

 

 

T 1' —
Figure 2.1: Thermal conductivity by phonon mechanism "4'

2.2 Electronic Conduction

Energy transport in metals is dominated by electron motion. Metals, of course, also
have a lattice structure and hence experience a lattice contribution to the thermal
conductivity, as well. However, thermal conductivity in pure metals is due primarily to
the “free” conduction electrons, those that are only loosely bound to the atoms so that
they wander readily throughout the crystal lattice and thus transfer thermal energy.

Mathematically, electronic conduction of thermal energy may be expressed as
k6 = 1/3 C; ve A, (2-2)
where, C; is the electronic heat capacity of the material per unit volume, v, is the mean

velocity of electrons also known as Fermi velocity, and A, is the mean free path of the

electrons between collisions or before scattering fiom the obstacles. The variability of

16

these parameters with respect to temperature is illustrated in figure 2 [14]. For pure metals,
the maximum in the electronic contribution to thermal conductivity, as shown in figure

2.2, appears roughly in the temperature range of 20 to 100 K.

(a) (b)

 

 

 

 

<l
y

(c) (d)

 

 

 

 

Figure. 2.2. Thermal conductivity by electron mechanism "4'
2.3 Molecular Conduction
Molecular motion, such as noted in organic solids and fluids, is another energy
transport mechanism [14]. This characteristic disorder, as well as lattice imperfections of
organic solids, introduces resistances to heat flow. Even though the thermal energy
carriers are the same, where phonons dominate over the electrons, the path to carry

thermal energy (i.e. how disorderly molecules are chained and tangled together) in this

17

mode is what makes it different from the other two. Accordingly, the disordered
dielectrics such as glass and polymeric plastics are the poorest solid conductors of heat.
Since our current topic concerns structural (ordered) material, we shall ignore this energy
transport mechanism.

2.4 Thermal Conduction in Niobium

Since thermal conduction in Nb has contributions from both of the first two
mechanisms, namely, from electrons and lattice vibrations, the overall thermal
conductivity of Nb is addition of both the terms given in equation 2-1 and 2-2, and is
given as:

k = k, + k, (2-3)

Below its transition temperature Tc, Nb metal enters into its superconducting state
where electrons transform into Cooper pairs and are thus neither able to scatter phonons
nor accept thermal energy to carry through. Therefore, below Tc, as shown in the right
curve of figure 2.3 “5], the contribution of phonon conduction (equation 2-1) begins to
increase. As the temperature is further lowered, more electrons freeze into Cooper pairs
and are unavailable to conduct heat. Thus the thermal conductivity of Nb declines
rapidly. At about 3 K or so, a local minimum in thermal conductivity is usually observed
where the contribution of the two thermal conductivities, Ike and kt, becomes comparable
to each other. The “phonon peak” is seen with further decreasing temperature due to

[9], as shown in left curve of

increased contribution of the phonon conduction mechanism
figure 2.3. This phonon peak in thermal conductivity of Nb, as the name suggests, is due
to the large mean free path of the phonons before getting scattered. The magnitude of this

peak depends upon the absence of electron-phonon scattering mechanism in the

18

superconducting state of Nb, the mechanism which otherwise exists at these temperatures
for pure metals. Thus, for Nb, reducing the phonon scattering mechanisms, such as
increasing the grain size as well as other scattering mechanisms, as discussed below, can

form a local peak in the thermal conductivity near 2 K, as shown in Figure 2.3 [9].

1.0

 

0.5 -

k8 /kl'1

 

 

 

 

0 Tire 0.5 1-0

Figure 2.3. Measured thermal conductivity for a Nb sample as compared with a
theoretical prediction. Steep decrease in electronic thermal conductivity is clearly
seen below Tc. The curve on the right shows the rise in the theoretical prediction of
phonon thermal conductivity when scattering of phonons from electrons is
substantially reduced in the superconducting state as the temperature is decreased.
This is the cause of phonon peak visible in the curve on the left ‘9'.

2.5 Phonon Scattering Mechanisms

We have seen the importance of phonon heat conduction in Nb, which is unique
among pure metals due primarily to its superconductivity below Tc. It is interesting to
look at various phonon scattering mechanisms specifically in superconductor metals like

Nb, which generally occur in dielectric solids. There are many characteristic phonon

19

scattering mechanisms that limit the mean free path of the phonons and hence limiting the
thermal conductivity of the material. Since scattering increases the resistance, therefore it
is ofien related in terms of a resistance parameter W “51’“6], where W at l/k.
2.5.1 Phonon-Electron Scattering
This scattering mechanism results in thermal resistance on the order of WE ~ l/TZ.
The mechanism is Significant in metals at low temperatures. This is the main cause of
reduction in thermal conductivity of Nb in the temperature range from just below Tc
down to around 3 K.
2.5.2 Boundary (Grain or Specimen) Scattering
Boundary scattering, such as grain boundary or specimen boundary, results in thermal
resistance on the order of W3 ~ l/T3 at very low temperatures, where the mean free path
becomes comparable to crystal dimensions. In temperature ranges below ~1K the
specimen size effects play a dominant role in determining the thermal conductivity of the
material.
2.5.3 Impurity Scattering
Scattering of phonons due to interstitial impurities present another mechanism of
resistance to thermal energy and is on the order of WD ~ 1/T3/2.
2.5.4 Dislocation Scattering
Scattering of phonons due to dislocation of atoms in the material or due to other

defects results in thermal resistance on the order of Wdis ~ 1/T2.

20

3 Liquid Helium and Its Pool Boiling Theory

3.1 Pool Boiling Heat Transfer in He—I

He-I, or normal He, is like an ordinary fluid with state properties that can be
described by empirical relations. The general characteristics of He-I Show rather low
thermal conductivity and large volumetric heat capacity, which suggests that conduction
heat transport is of little Significance to the overall heat transfer.

The traditional approach of empirical relations to the interpretation of heat transfer is
best suited for dealing with the He—I. The real phenomena of heat transfer through a
heated surface into the fluid are quite complex and therefore are difficult to model from
the first principles. A specific problem requires a solution of a complex set of equations,
which are only treatable in the simplest geometries. Therefore, engineering problems are
scaled on the basis of dimensionless variables, which are functions of the properties of
the system. Usually, through the use of these dimensionless variables, fitting them to the
experimental data develops functional relations. In general, for the problems of heat
transfer, one of the important dimensionless parameters to consider is the Nusselt
number, Nu. It represents a dimensionless heat transfer coefficient, defined by the
relationship

Nu = hL/k (3-1)
where h = q/(Ts-Tb), the heat transfer coefficient of the surface, T, and T1, are the local
surface and bath temperatures, k is the thermal conductivity, and L is a characteristic
length in the problem. In pool boiling, L is the characteristic length of the heater, either

its diameter or width as appropriate.

21

In the case of pool boiling heat transfer, the two relevant dimensionless numbers are
the Grashof number (Gr) and the Prandtl number (Pr). The Grashof number indicates the
ratio of buoyancy forces relative to the viscous forces. It is represented by the

relationship

_ 3
Gr :gfl(T. T.)L

v2 (3-2)

 

Where, g is the acceleration due to gravity, [5 is the bulk expansivity, and v is the
kinematic viscosity. The Prandtl number contains most of the fluid properties and is
given as:

Pr = v/a = nCp/k (3'3)
where, a is the thermal diffusivity, n is the viscosity and, Cp is the specific heat of the
fluid.

In pool boiling heat transfer, when the dominant mechanism is natural convection, the
Nusselt number is a firnction of these two numbers,
Nu = (p(Gr) \v (Pr) (3-4)
where, (p and \y are functions determined by empirical correlations of data.
Most empirical correlations for natural convection are given in terms of the Rayleigh

number, which is Simply the product of the Grashof and Prandtl numbers, i.e.,

_ 3
Ra= GrPr= ng: Tb” (35)

(1V

 

Simple correlations can be written in the form
Nu = C Ran (3-6)
where ‘C’ and ‘n’ are empirically determined parameters. The coefficient 11 depends

mostly on the geometrical and flow conditions. Generally, n = 1/4 when flow is laminar

22

and n = 1/3 for turbulent flow in a vertically oriented heater in an open bath. The critical
Rac is defined to distinguish between the two regimes. For flat plates, the transition
between pure conduction and convection occurs for Rac = 103, while the transition
between laminar and turbulent convection heat transfer usually occurs for Rac = 109.
These relationships are only valid when no phase change process is taking place in the
system. For an extensive review of natural convection, see Gebhart et a1 [4”.
3.1.1 Regimes of Pool Boiling Heat Transfer in He—I

There are typically three regions of heat transfer in pool boiling of He-I as shown in
figure 3.1 “7]: (1) natural convection, (2) nucleate boiling, and (3) film boiling. At the
lowest heat fluxes up to a few mW/cmz, heat is transferred by natural convection. Since
no phase change is evident, density-driven mechanisms drive this heat transfer regime.
The empirical correlation given in equation 3-6 can be used to estimate the surface
temperature differences. As the heat flux is increased, bubbles of helium vapor begin to
form locally at preferred sites on the heated surface, which are typically surface
imperfections. Hysteresis in the heat transfer curve is caused by activation or deactivation
of these imperfections by the low contact angle He-I. As the heat flux is further
increased, the nucleation sites get fully activated with each site containing a bubble. This
is the stage when any further increase in heat flux only serves to accelerate the rate of
bubble growth and detachment. This in turn causes macroscopic turbulence where the

heat transfer is near its maximum.

23

 

 

 

 

. I. I
film
0 ' boiling
i.“
8
3 no:
‘3', i
.....L .
- ' '

 

ATIKJ

Figure 3.1 Typical heat transfer relationship for pool boiling He-I "7'.

At still higher heat fluxes, the bubble size and frequency of detaching get so high that
they become unstable and coalesce into continuous vapor films. The corresponding value
of heat flux at this stage is known as peak nucleate boiling heat flux and the regime of
nucleate boiling changes to film boiling. In the film-boiling regime, a vapor layer
insulates the fluid from the surface and the cavity surface temperature difference is an
order of magnitude higher than with nucleate boiling. The hysteresis in this regime of
heat transfer is associated with the stability of a vapor film below a higher-density liquid.

3.1.2 Factors Affecting Pool Boiling Heat Transfer in He-I.
As mentioned earlier, there are numerous factors that affect heat transfer

characteristics. For example, surface orientation has a profound effect on heat transfer

24

behavior. Variation of the surface orientation with respect to the gravitational force can
cause significant changes in the heat flux and the minimum fihn boiling heat flux. The
highest values for both these quantities occur with the surface facing upward, because the
buoyancy force aids bubble detachment “7].

The thermodynamic state of the liquid helium bath is also an important parameter in
the heat transfer process. The bath temperature has a significant affect on various heat
transfer parameters, especially the peak nucleate boiling heat flux q. Similarly, the bath
pressure affects these values, particularly when considering the sub-cooled or
supercritical state. These variables can be taken into account through the changes in the
helium properties with temperature and pressure “7].

Variation in geometry can also have a profound effect on the heat transfer. Many
engineering systems consist of channels, tubes, or other complex geometries, which are
vastly different from the open “infinite bat ” configuration. Due to the limited coolant
volume associated with such geometrical limitations, difference in heat transfer can be
expected. Some of the physical phenomena that can occur include heat-induced natural
circulation and vapor locking in narrow channels ['7].

The surface condition of the heat transfer sample can affect both the peak heat flux q
and the ATs in the nucleate boiling regime. The mechanism by which surface preparation
affects the heat transfer characteristics, which also is the main aim of this research, is
believed to be associated with the number of available nucleation sites. Improvement of

pool boiling heat transfer in He-I through surface preparation holds significant potential

in its application in the SRF cavity design. As discussed above, surface preparation

25

affects the heat transfer through the nucleation sites; therefore it is important to
understand nucleate boiling heat transfer in more detail.
3.1.3 Nucleate Boiling Heat Transfer in He-I

In this mode of heat transfer, the helium is in two phases, liquid and vapor, mixed.
Hence the free convection theory discussed previously is no longer valid. This phase of
heat transfer is characterized by the formation of small bubbles at the nucleation site due
to small (mW/cmz) increases in the heat flux. The bubbles, when detached from the
surface, rise above to the free surface thus carrying away the heat. The surrounding
colder liquid quickly occupies the space left by the bubbles on the surface and thereby
cooling the surface. AS the heat flux is increased, essentially all of the nucleation sites are
active. This condition is quite different from that of natural convection because it is
controlled mostly by the hydrodynamics of bubble growth and detachment rather than
convection in the liquid “71’ [42].

The conditions under which bubble nucleation sites get activated are governed by
nucleation theory. According to this theory, in order to have a bubble grow at a
nucleation site, there must exist two conditions. Firstly, there must be a thin boundary
layer of superheated liquid near the heat transfer surface interface, whose thickness may

be approximated as,

5 ~ kATS/ q (3-7)
where, k is the thermal conductivity of the liquid, AT, = Ts-Tb is the allowable superheat
of the surface above the bath temperature, and q is the applied heat flux.

Typically, 5 is of the order of 1-10 pm for the He-I near its normal boiling point. The

reasons for the requirement of such a superheated boundary layer are envisaged in the

26

liquid near the interface being sub cooled due to hydrostatic head (Ap = pgh). So, the
local temperature must rise above ambient before saturation conditions can be attained.
Secondly, and probably more important, the surface tension of the liquid must be
overcome in order for a vapor bubble to have a positive radius of curvature, as can be
seen through the following equation:

20
Pv - P, =7 (3-7a)

here p, is the vapor pressure inside the bubble, p, is the saturation pressure of the
surrounding liquid and r is the radius of the bubble. Van Sciver et a1 [17], through some
simplifying assumptions, has found the value of critical radius, rc, of the bubble to be
0.0164 um at saturation temperature and pressure conditions with ATs = 0.3. A bubble
greater than rc would be expected to grow while one smaller would collapse. This implies
local surface imperfections must exist. These imperfections are necessary to provide
preferential regions where bubbles can form. The imperfections are usually in the form of
grooves or slots that allow bubble formation with a negative curvature, thus taking

advantage of surface tension to stimulate the bubble nucleation.

27

 

,_
I!
x"

o 240 grit ground

  

 

 

 

{I 1' 6009rrlground
' {1. u 0.3pm mirror
AMP ' / ° . 3209mm
N ' ire-19M“ 0 2409nliamed
g Roughul .. -:-I ' I gygfmimd
3 SW“ , g ,i gmgw . mi" 1' 0pm
/ .i ' l v roughness 5 pm
5' .01 . / - ‘ roughness1 ,U m -
. I- .- polished -
e o urrorepared
otyciean
0
m1 - -

Figure 3.2: Nucleate boiling heat transfer to He 1. Variation in the data is seen
through a difference in surface roughness where increasing surface roughness
lowers the surface super heat. Plotted empirical fit is by Schmidt and adapted from
Van Sciver "7'.

Though some engineering correlations do describe the nucleate boiling regime quite
satisfactorily, especially those given by Kutateladze and Schmidt, there is wide variation
in experimental data owing to differences in sample preparation, surface material and
sample orientation. Based on the hydrodynamic arguments presented above, the different
surface preparations would yield much different results. Figure 3.2 is a compilation ['71 of
nucleate pool boiling data for a flat copper surface facing upward as a function of heat
flux. The empirical fit used by Schmidt is also plotted in figure 3.2. Due to its
conservative form it is more popular for engineering applications. The data vary over at

least half an order of magnitude in AT for a given q, with the smoothest surfaces

28

apparently allowing larger superheat. This is consistent with the arguments presented
earlier based on the nucleation theory. Therefore, a rough surface is preferable for
increasing the heat transfer by providing preferential nucleation sites for the bubbles
formation.

3.2 Pool Boiling Heat Transfer in He-II

High field measurements on SRF cavities are routinely done at temperatures below
2.1 K where the state of liquid helium is super fluid. The appearance of super-fluidity
only at low temperatures suggests that quantum effects exists on a macroscopic scale,
whose character can be explained through the Bose-Einstein condensation model.

Therefore, He-II is considered to be an ideal fluid to aid in understanding quantum

 

    

 

 

 

effects.

7
106E-1-solldW
105 ' He I Critical Point?

E 1° F 3 T"; a

4: :

g 10 , Hell 1,
3 t 1
§1oooE a
5 100g 1
10;r a
1:..1[LLILAI....IIII.1....11...:

0 1 2 3 4 6 6

Temperature (K)

Figure 3.3: Phase diagram of Liquid Helium

As discussed earlier, liquid helium can exist in either of the two phases. Normal
helium or He—I exists above T}, whereas super fluid He-II exists below T1. The A-line as

shown in figure 3.3 of the phase diagram, separates these two liquid states. The transition

29

between these two states of the liquid is associated with a change of slope of the entropy.
Thus, the A-transition is a second-order phase transition and has no latentheat associated
with it. The transport properties of He-II are therefore best described through a Two-
Fluid Model. This model assumes the He-II to be comprised of two interpenetrating
fluids: normal fluid, which contains all the viscosity in the liquid, and the super fluid,
which has no viscosity. The normal fluid component of He-II, which must be

distinguished from He-I, is assumed to behave as an ordinary liquid. It is described by

having density p“, viscosity 11,, and specific entropy Sn. On the contrary, the super-fluid

component has density ps, no viscosity (115:0) and no entropy (Ss=0). The overall

properties of the fluid must be comprised of a linear combination of the two components.

p = pn + p. (3-8)
And, since the super-fluid has no entropy, it follows that the total entropy (S) of the

He-II can be written in terms of the product of the normal fluid components alone:
S = PS = pnsn (3-9)

The two-fluid model has successfully explained many unusual phenomena peculiar to
He-II such as the fountain effect, thermo-mechanical effect, and the second sound in He-
II, the details of which are given in reference ['7]. However, it must be realized that this is
just a hypothesis and the existence of super—fluid and normal fluid components may not
be taken too literally.

Unlike common fluids, the heat transfer from a solid boundary to He-II occurs at a
very thin interface. In this process, surface heat transfer is more controlled by the

interfacial character, including the properties of the solid rather than that of the bulk He-

30

II alone. In general, there are two regimes of interface surface heat transfer in He-II. At
low AT, no boiling occurs and the heat transfer is controlled by a phenomenon called
Kapitza conductance. At high AT, and for heat fluxes greater than q‘, the surface is
blanketed by a film of He-I, or vapor, or both, depending upon the bath conditions of the
He-II. In this region, primarily the character of this film controls the heat transfer. As
shown in figure 3.3, if the state of He-II corresponds to point 1, then the surface will be
blanketed by the helium vapor. If it corresponds to point 2 with q not very high, then the
surface will be blanketed by He-I surrounded by He-II. If it corresponds to point 2 with a
heat flux greater than q. then the surface would be blanketed by both He-I and vapor. For
operation of SRF cavities, thermal magnetic simulations have shown that due to the
characteristically small temperature rise, this research concerns itself with the first
problem, that is of heat transfer directly from the solid surface into the liquid He-II, or
Kapitza conductance.

3.3 Kapitza Conductance

In 1941, Kapitza [43] observed the first evidence of thermal boundary resistance (R3,)
at the interface between solid and He-II. During his experiment to study the flow of heat
around a copper block immersed in the liquid He-II, he observed that within the He-II the
temperature gradients were negligible. However, a considerable temperature difference
occured between the copper block and the He-II. Analogous to conventional heat transfer
coefficient, by definition, the Kapitza conductance (reciprocal of boundary resistance) is

defined in the limit where q and AT, are vanishingly small, that is,

1 .
— = hko = lrrn —q—- (3_10)

RBd ””0 ATs

31

here, the 0 subscript refers to the limit as AT, —> 0. This quantity, hko, has a quite strong

temperature dependence going as Tn with n varying anywhere from approximately 2 to 4.
A more general definition of Kapitza conductance, for small AT, is to simply relate it to

the finite values of q and AT,:
hko = q / ATS (3-11)

It is important to note that the equation 3-11 is valid only for experiments with low
values of heat flux (approximately from 10 mW/cm2 to 40 mW/cmz) and the T, remains
smaller than the super-fluid transition temperature [3 5 1. However, for many practical
engineering applications of He-II involving large heat fluxes, large AT, (51‘ bath) may be
encountered resulting from large heat flux imposed on the surface. It follows that the

Kapitza conductance for finite AT, is larger than him by the magnitude of this expression

[17]

 

 

 

2 3
3AT AT lAT
h AT=h1+— ‘+ 3 +— s
"( ’ "° 271(15] 4(1),] ""2’

where, for AT/Tbath z 0.5, the bracketed quantity is approximately 2. This relation is an

analytic expansion of a more general equation, equation 3.13, to describe heat transfer

rate in the Kapitza regime with an exponent n = 4. It is given as:

q = “(Tsn “72") (3.13)

here a and n are adjustable parameters. This equation obtained fi'om phonon radiation
laws “7] has close resemblance to photon radiation. The expansion given in equation 3-12

makes the initial assumption of an explicit T3 dependence of hko. However, experimental

32

observations vary considerably from this exact form, obeying power laws varying
between T2 and T4.

When the discovery was first made, it was thought that this resistance (Rgd=l/hk,)
was due to some special property of super-fluid helium. Only later was it realized that
such a Kapitza resistance, exists at the interface between any pair of dissimilar materials.
In non-boiling He-H, the heat transfer coefficient is all the result of Kapitza conductance.
Although the Kapitza conductance is an experimentally defined quantity, the
considerable theoretical work aimed at understanding this complex phenomenon helps in
analyzing different aspects associated with the transfer of heat that can have practical
applications on SRF cavities.

Unlike the conventional concepts such as treating the convection heat transfer
coefficient through the properties of the fluid only, such as its thermodynamic state and
its velocity profile, understanding of the Kapitza conductance is obtained through the
properties of interacting phonons of both the liquid helium as well as the solid. Phonons
can be visualized as quantized lattice vibrations, similar to photons in EM wave theory.
Their effect on the Kapitza conductance can be accounted for through:

0 Number of carriers (phonons) incident on the interface

0 Energy carried by each phonon.

0 Probability that each phonon is transmitted across the interface.

Different models have been theorized to interpret the physical understanding of this
phenomenon. Each of these theories distinguish themselves as to how they model the
phonon transmission probability, that is, the total fraction of the energy transmitted across

the interface. Enhanced phonon transmission implies an increase in Kapitza conductance.

33

3.3.1 Acoustic Mismatch Model (AMM)

In 1952, Khalatnikov “91 presented a model approximating what is now known as
acoustic mismatch model (AMM), to explain the thermal resistance at the boundary
interface. The acoustic mismatch is extremely large at the interface between helium and
most ordinary solids because the density and sound velocity of helium are very much less
than for other materials. However, Mazo and Onsager in 1955, independent of
Khalatnikov’s model, gave the modern form of acoustic mismatch model, which takes
care of some other channels of phonon interactions through internal reflections and are
discussed later in the section. Although quite a comprehensive review of the AM model

[131,119]

is available , only some of its salient features are presented here.

 

(helium)

(solid)

 

 

 

Figure 3.4: Incident and transmitted phonon angles, related according to Snell’s
law. If the velocities and angles of the incident and transmitted phonons are C1“, 0...,

Cam] and 0,...” respectively, then sin Gin/C... = sin emu/Cm, (Adapted from Swartz et
al. )

34

o In this model, it is assumed that the phonon system is governed by continuum
acoustics.

o The interface is treated as a plane and, as shown in Figure 3.4, the phonons are
treated as plane waves in a continuous medium.

0 The phonons transmit specularly without any scattering at the interface.

0 The interface has no intrinsic property but merely joins the two grains. As an
example, the AM model gives the phonon energy transmission coefficient tAB in
material A incident normal to the interface with material B as:

tAB : 4Z A Z 3 2
(Z A + Z 3)

 

here, Z = pc is the acoustic impedance (analog of Fresnel equations), with c and p being
the speed of sound and mass density, respectively.
3.3.2 Diffuse Mismatch Model (DMM)

While the AMM was able to explain the existence of Kapitza resistance, the value of

R1, (or 1/hk) calculated was between 10 and 100 times larger than typical experimental

values measured at l to 2 K. This indicates that phonons are able to cross the interface

more easily than expected. A crucial assumption made in AMM is that no scattering

occurs at the interface. However, various phonon scattering measurements reveal that

significant phonon scattering takes place especially for high frequency phonons (> 100

GHz) at helium-solid interface. This scattering of phonons leads to a significant reduction

of the thermal boundary resistance by opening up new channels for heat transport. This

35

effect of phonon scattering resulted in Diffuse Mismatch Model, first presented by

Swartz in 1987. The salient features of this model are ['8]:

o Assumes that the transmittance of phonons on the interface is a diffuse phenomenon
as oppose to specular.

. All phonons striking an interface from both sides lose memory of from where they
came.

0 The fraction of energy transmitted is independent of the structure of the interface.

. Scattering probability into a given side is proportional to the phonon density of states
on that side and the density of states are proportional to (Dz/03; where 0) is the
frequency of the phonons and c is the speed of sound.

0 Since the Speed of sound in helium is on the order of 20 times lower than in most
solids, the density of relatively low-frequency phonon states in helium is on the order
of 104 higher than in most solids. This results in almost only forward scattering of the
phonons incident on the interface from solid to the liquid helium.

3.3.3 Phonon Radiation Limit (PRL)

The Kapitza resistance has always been found to be lower than predicted by the
AMM, often by orders of magnitude, as shown in figure 3.5. This had led to the
formulation of the so-called phonon radiation limit, sometimes referred to as the perfect
match model, was first proposed by Snyder [36] in 1970. The salient features Of this limit
are “8]:

o PRL is not a new model but a limit at which the maximum possible (phonon-

dominated) thermal boundary conductance can be achieved.

36

0 AS an extension of AMM, it is assumed that all the phonons from the solid side (a
side with lower phonon density) are transmitted. From the helium Side (with higher
phonon density), precisely enough phonons are transmitted to satisfy the principle of
detailed balance (i.e. no more than all the phonons fiom the solid can be transmitted)
and second law of thermodynamics.

0 Although this is the maximum transport that is allowed by the principle of detailed
balance, no physical mechanism is described that allows precisely the correct number
of phonons to transmit from the side with higher phonon density.

0 PRL is not a model but simply a limit, however, due to closer theoretical agreement
between the results of this limit and the diffuse mismatch model, the two are
sometimes considered equivalent especially in problems involving helium.
Experimentally, there is a large variation in the Kapitza resistance values even for the

same material as seen in Figure 3.6. The difference in the theoretical predictions of

Kapitza resistance by the two models is also shown with the upper solid line

corresponding to AMM and the lower corresponding to DMM with more than two orders

of magnitude.

37

 

 

 

1000
acoustic mismatch
' a
r— rm-"
NE 100 r- \s,
U \
“‘ \
g i \ x b
J“ y).
x \
I—J \ .
‘L 1 ‘. _. Ad-
"0 ' ‘1 x e
m a
03 10 i- " c—
P
diffuse mismatch
- J - I 1

 

 

 

0.1 1 .0
Temperature (K)

Figure 3.5: Experimental data of various researchers shown as a,b,c,d and, e
(adapted from Swartz et al. '18') for the measurement of Kapitza boundary
resistance (RM=RK) between copper and He-II, multiplied by 'I‘3 to remove the
strong temperature dependence. Note the drop in RMT" with temperature above 0.1
K. The upper solid line is the prediction of AMM and the lower one is the prediction
of DMM, almost two orders of magnitude different. The kink in the curve b is the
result of irreproducibility of Kapitza resistance even on the same sample from run
to run, and does not reflect any physical phenomenon.

3.4 Factors Affecting Kapitza Resistance

As mentioned earlier, the SRF cavities are routinely made from pure Nb which
becomes a superconductor at temperatures below T0 = 9.2 K. The metal has a
characteristic Debye temperature of Go =275 K, which is above average for common
elements used in cryogenics “7]. Debye theory of materials has been very successful in
predicting the low temperature heat capacities of various solids by treating the solids as
infinite elastic continuum and considering the excitement of all possible standing waves

38

in the material. With this in mind, it is easier to remain focused on the present research
while analyzing the trend of nrunerous experimental studies done to categorize the effect
of solids on Kapitza resistance. In many of the experiments, the effect of damaging or
oxidizing the surface of the solid has been ambiguous. It has been even observed [201.121]
that the lowest Kapitza resistances corresponded to samples with the least damaged
surfaces, contrary to the present understanding. Such trends imply that the boundary
resistance to dense, soft (low-Debye-temperature) solids will be much closer to the
prediction of AMM than will be the boundary resistance to light, hard (high-Debye-
temperature) solids. Thus, the trend is found ['8] independent of whether the solid is
metallic or dielectric. This conclusion is also confirmed through another analysis [22],
when a plot of reduced boundary resistance, let us say P (reduced logarithmically so that
a value of 1 corresponded to AMM and a value of 0 corresponded to DMM), is made
against T01), the product of temperature and the Debye temperature of the solid. This plot
is reproduced in Figure 3.6 [22]. As seen in this plot, for a given solid, RBdT3 drops from
near the AMM prediction at low temperatures toward a very low value at high
temperature. By plotting the reduced boundary resistance as a firnction of T09, all the
data follow approximately the same curve. This is the same conclusion as reached in the
case described above where the boundary resistance to solids with high Debye
temperatures, such as sapphire or LiF, will drop relative to the AMM at lower
temperatures than will the boundary resistance to softer solids, such as lead or mercury.
Niobium is clearly shown to be on the middle to lower half of the plot indicating its
inclination to follow the prediction of DMM as compared to AMM, especially at

temperatures around 1.8 to 2.1 K.

39

 

 

 

   

 

 

 

1.0 l I l 7 T
T 00* Hg \\ 1
0.8 ’- CMN‘”v in >Pb u -
0.6 i c"'\ ‘ a .
P {- Pb \ cu 1
0.4 ’- w “i
' SKZIN M ‘
0.2 - Co NI 5": .
O L- - - - 000:1:M'
1 a 1'0 50 160 300 1000 3000

100 [10),

Figure 3.6: Plot of the reduced thermal boundary resistance P as a function of T01).
The stars indicate measurements between the solid and 3He. The data is by several
researchers and compiled by Swartz et al "8'.

The analysis of much of the existing Kapitza resistance data often involves the
proposal of a new transmission mechanism or “channel”. No single channel can be
responsible for all the observations, rather, for a given observation, several channels may
contribute. For different interfaces, different channels may be important. Some of the
commonly known channels are discussed in the subsequent paragraphs.

3.4.1 Effect of Boundary Layer

At the interface between He and a solid, due to strong van der Waals attraction forces,
the (first few mono layers) boundary layer thickness is around 30 A. The pressure in this
boundary layer is very large on the order of tens of atmosphere. Therefore, this boundary
layer is thought “8] to be acting as an anti-reflective coating. For long-wavelength

phonons, i.e. at temperatures less than 0.1 K (the wavelength in the layer is much longer

40

than the thickness of the layer), the effect of a dense layer would be small. There is no

dTOp in Kapitza resistance, R1,, and the effect of the boundary layer remains small. For

shorter-wavelength phonons, however, i.e. at temperatures more than about 0.5 K,

(wavelength in the layer comparable to four times the layer thickness), the layer could

greatly enhance the transmission. Hence the drop in Kapitza resistance is quite evident.
3.4.2 Effect of Pressure

AS we know, the speed of sound in helium is a strong function of pressure. Thus,
AMM would predict great pressure dependence for Kapitza resistance. However, as seen
from Figure 3.7 “8], R, has pressure dependence only at temperatures less than 0.5 K. At
temperatures above 0.5 K, the R1, is almost independent of pressure. Close similarity of
Kapitza resistance of 3 He plotted in figure 3.8 with that of 4He in figure 3.6 is also quite
evident.

This effect of the pressure independence of Kapitza resistance at temperature above a
few tenths Kelvin has also been explained by observing that the pressure in the interfacial
layer (due to van der Waals attraction) is very large and is independent of the pressure of
the bulk. At low temperatures, where the boundary layer is unimportant, the pressure
dependence is unaffected by the layer, but at temperatures where the boundary layer is

important, pressure dependence is reduced.

41

 

 

 

 

I I J I. I I I I T I I I
200 L‘A’i’i’u '
O 1p51
A 95 psi
I 39509
O .
‘0. . + SOIId
ah ‘0
150 - 'q.
g I ’
«E - “I. 5
U
ex . ,
n" 100 - I
’—
3 I I
o: 5
i-
. I
f... O .. .
50 _ ’ I I. 1|. 1.
. 01 .0 .
s .r 3 ’
1- .l
‘ n a
i
o .l l l l l I l l h I 1 1 1 I
0.06 0.1 0.2 0.4 0.7
T(K)

Figure 3. 7: Measured Kapitza resistance between electro-polished copper and 3He,
multiplied by T" to remove strong temperature dependence. The pressure
dependence is seen well below the 0. 5 K, but near 0. 5 K and above, the Kapitza
resistance is nearly pressure independent. Adapted from Swartz et al "8'.

3.4.3 Effect of Attenuation
The effect of attenuation is to broaden the critical cone and lower the predicted value
of R1,. In the case of Kapitza resistance, with reference to AMM, if the phonon from the
helium side is incident upon the interface from outside the critical cone (see figure 3.4),
the phonon will be totally internally reflected. An evanescent wave is generated in the
solid to satisfy the boundary condition. An evanescent wave is an electromagnetic wave
observed in total internal reflection, undersized wave-guides, and in periodic dielectric

hetero-structures. While wave solutions have real wave number k, k for an evanescent

42

mode is purely imaginary. Though the amplitude of this evanescent wave decreases

exponentially with respect to distance within the solid, the coupling of energy transfer

across the interface does take place. Thus the total internal reflected phonon can end up

being transmitted. Physically, this effect is accounted for by including classical phonon

attenuation in the model. As much as factor of three reductions in R, has been

approximated due to attenuation. It is this effect that results in modified AM model.

3.4.4 Effect of Conduction Electrons

In addition to the effects of electrons in metals attenuating phonons and thereby

broadening the critical cone, other significant effects of electrons on thermal boundary

resistance are through:

flignetic Field Effects: The effect of the conduction electrons in superconductors
can be expected at a Kapitza interface. When the metal is in a superconducting state,
the electrons play little or no role. The application of a magnetic field to drive the
superconductor “normal”, allows the electrons to contribute to the thermal transport
process across the interface. However, due to extremely small electronic contribution,
direct measurement of this effect still remains inconclusive.

Effect of Oxide Layer: Halbritter et al [231424] has suggested a relationship between
the observed RF losses at non-ideal metal surface interfaces and the observed phonon
scattering at such interfaces. The coupling arises because electrons can be localized
in, for example, oxide layers at the metal surface. Phonons incident on these localized
electrons cause them to vibrate and generate RF fields, which interact with other
localized electrons. The result is strong diffuse scattering and therefore, a reduction in

Kapitza resistance. Some models have also been proposed [25] for the microscopic

43

interactions of trapped individual and coupled electrons in thin oxide layers. This also
may relate directly to diffuse phonon scattering.
3.4.5 Effect of Localized States at the Interface

All non-acoustic channels invoked to couple phonons across a boundary require
excitations at the interface, which couple phonons on both sides of interface. If a surface
contains a large density of defects or loosely bound impurities, then their excitations can
couple the phonons on both sides of the interface through surface waves. It has been
shown by Gel’fgat and Syrkin [26] that a layer of loosely bound impurities on the surface
could increase phonon transport across the interface by coupling through surface waves.
Some models have also been proposed [271’ [28] by assuming the defects as localized
masses and springs. These defects and loosely bound impurities promote the coupling
between bulk phonons and the surface acoustic waves, which couple to the phonons in
the helium. However, the model has not been worked out quantitatively in three
dimensions due to complications introduced by the inclusion of phonon attenuation and
boundary layers.

3.4.6 Effect of Surface Roughness

The acoustic mismatch theory (AMM) assumes perfect, planner interfaces. There
exists experimental evidence [29] to support this theory whereby the in-situ cleaving of a
LiF crystal at 1 K created a nearly ideal surface. The experimental result demonstrated
excellent agreement with AMM. However, real surfaces are usually quite rough on the

scale of hundreds of A, i.e. the typical phonon wavelength in solids at 1 K.

44

4.5 9111-111 r-rv TTTTT-rrr-rrrrrrrrrrrrrrwq
4 1

 

Lirlrmlrrr Llellllelillll 1111'4

 

 

.5
1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

 

 

 

 

 

 

 

 

 

 

Temperam (K)
Sample a; RRR Surface Treatment 03m; (1" ”0' R “(one It 1W) R K .1 1.. 1,. K
g 1 1.);170 ce (~301un) 1.111 0.4 10.7 x T '3'“ 1.320
11! 21.1;178 5" 0.35 110.25 21.3 11 T '4‘" 1.902
- _-
s 31.]; 34? A+CE (~3Dum) 1.3 t 0.4 111.1 a T '3'” 1.5911
111 4 15); 64? A+CE + EP 0.2 1’ 0.1 19.1 11 1'3"“ 2.288

 

Figure 3.8: A Kapitza resistance of the four samples, along with their surface
treatment and corresponding roughness data are shown in the chart. The graph
shows the relative effect of surface roughness on R, with temperature variation. CE
stands for chemical etch, EP means electropolishing and, A stands for annealing.

(Adapted from Amrit et. A]. [3”)

One of the effects of surface roughness is to couple bulk phonons (or other
excitations) on either side of the interface through Rayleigh (or surface) waves. The
phonons with wavelengths comparable to the characteristic dimension of the roughness
are most strongly scattered, hence, lowering the Kapitza resistance. Since the wavelength

of the phonons increase as the temperature is lowered, a larger length scale surface

45

 

roughness start interacting in the scattering of phonons. This is seen as an increase of
Kapitza resistance with lowering of temperature.

A variation of Kapitza resistance with temperature must be expected. This effect is
evident in some of the recent experiments conducted by Amrit et al. [301’ [3” between 1.5
K and 2.1 K on well-characterized superconducting Nb samples of different surface
roughness with He-II. As clearly seen from figure 3.8 (graph), the effect of an increase in
surface roughness results in a decrease in R1,.

The wavelength of phonons in liquid helium is on the order of ~ 4/T nm within the
temperature range of 1.5 to 2.1 K “8]. Now on a macroscopic length scales, the ordinary
surface contains large number of micro-length scale roughnesses superimposed on the
larger scale roughnesses such as seen in figure 3.9. However at micro-length scale levels,
we see even smaller (~nm) length scale roughnesses are superimposed on the micro
length scale roughnesses. Thus, based upon these observations, it can be argued that
increasing the exposed surface area of Nb to He-II at a macro level statistically results in
more of the nano-length scale roughnesses. Therefore, increased surface index (SI) of the
surface can lead to reduced Kapitza resistance. SI, here, is defined as the ratio of exposed

surface area to that of the projected area.

46

 

 

 

. ' i" \ i

‘ I wavinoss ,1

// \\ height
.41' \‘Qs

i" Peaks \\

.r' ' A \. Nano

 
  

- Roughness

valleys
spacrng

Figure 3.9: Pictorial display of surface characteristics with associated different
levels of length scales.

3.4.7 Effect of Titanium Getter Material
As discussed earlier in chapter 1, gettering is a technique in which usually Ti is
evaporated on the surface of the Nb to about a l um thick layer. Since Ti has a greater
affinity to the interstitial contents (e.g. H, C, 0), than does Nb, all of these impurities get
attracted towards the surface. After removing this layer from the surface, the metal left
behind is much purer with a higher RRR value. The influence of a layer of foreign atoms
close to the surface in the solid has also been studied [30]. Here, a distinction must be

made between the case of foreign atoms, where Ti has been deposited acting as a solid

47

layer of almost 30 um deep into the metal, with those of loosely bound foreign atoms, as

discussed in section 3.4.5 above.

 

 
  

 

 

 

16
Ti atoms present in Nb near
14 _
' \\ Ti Layer etched to 5 pm
A , Q MK
1:- \
3 12 - o .9 ‘Trhfl
‘x " ‘Q-o-°---o---o-Q‘°.o__-°*Fwo
N P
E b
3' 10
(o 1—
"x _
a: L
Ti Layer Etched to 30 pm
8 P A
6 h l 1 r 1 I l 1 J I r l 1 l I 1 I I 1 1 r l I 1 I I
1.5 1.6 1.7 1.8 1.9 2 2.1

Temperature (K)

Figure 3.10: Effect of Ti atoms present in Nb near the interface on Rk (Adapted
from J. Amrit et a1 I30').

In this case, the effect of the Ti layer resulted in an ahnost twofold increase in the
value of R1,. After etching the surface to 30 pm, as shown in the figure 3.10 [30], RI,
decreased and became comparable to that of the initial sample within the limits of

experimental error.
3.4.8 Effect of Deposited Thin Films on Kapitza Conductance
Phonon scattering experiments have been performed to analyze the effect of surface
roughness on the Kapitza conductance. The study has concluded “81 that surface

roughness, even at very low amplitude, causes strong geometric and Rayleigh scattering,

48

suggesting that surface roughness could be a very important transmission channel in the
Kapitza conductance. It has been observed that diffuse phonon scattering is directly
related to enhanced transmission of phonons across the interface, increasing the observed
Kapitza conductance. Thin films of varying thicknesses (usually of the order of few A to
tens of A) of hydrogen, deuterium, or neon were deposited on a silicon surface [321’ [331’ [34]
to analyze the effect of diffuse phonon scattering at the interface. At low enough
temperatures (minimum temperature of measurement was 50 mK), no significant
scattering was observed. However, as the temperature was increased, the scattering from
the films increased, often with sharp onset. Depending upon the type of thin film, island
or continuous, the dominant phonon wavelength at the onset temperatures of diffuse
scattering in all cases corresponded to the characteristic dimension of the film, whether it
was based on island diameter or film thickness. Other types of films studied included
oxides, ion-implanted regions, and metal thin films. The scattering rates, in all these
experiments, reached a limit when the film thickness was no more than a few times the
dominant phonon wavelength in the fihn (at a given temperature). This work clearly

shows that the phonon scattering from geometric imperfections, such as roughness,

adsorbates, or thin deposited films, is very large, causing increased Kapitza conductance.

49

4 Experimental Apparatus

4.1 Cryogenic Dewar

The most commonly used technique of testing the SRF cavities is through a long
cylindrical cryogenic Dewar filled with liquid helium, whose bath temperature is
controlled through pressure regulation. Usually the liquid helium is supplied to the testing
Dewar through transfer lines at ~ 4.3 K. A schematic diagram of a cryogenic Dewar is
shown in figure 4.1, along with the lid assembly carrying sample housing.

The lid assembly for the experimental setup acts as a testing stand, which is inserted
into the cryostat Dewar. The lid contains the receptacles for filling with liquid helium,
warm gas return and feed-throughs for connecting various sensors on or around the cavity
for instrumentation purposes. To minimize thermal losses inside the cryostat, there are a
number of layers of insulation surrounding the Dewar. The outermost insulation layer is
of a vacuum jacket along with another layer of liquid nitrogen shield to minimize the heat
leaks to the interior. A layer of super insulation (multiple layers of aluminized myler) is
placed in the room temperature and liquid nitrogen gap, as well as the liquid nitrogen as
liquid He gap to further reduce the heat leaking inside the tank. The inside of the Dewar
is filled with liquid helium to the desired level. Nucleate boiling heat transfer coefficient
measurements are performed at temperatures ~ 4.2 K and 1 atmosphere pressure in He-I.
For the measurements of Kapitza conductance, the pressure on the liquid helium is
gradually lowered by pumping out He gas to about 23.8 torr so as to bring the bath
temperature at 2.0 K. Below 2.175 K, the liquid helium becomes He-II, a super fluid. The
quantity of the He in the Dewar is monitored using a superconducting wire as a level

sensor.

50

          

.Wl‘.
' 1

e,
,1.
1V

HERE

Super
hnnflafion

Figure 4.1: Schematic diagram of the cryostat Dewar with experimental setup

hooked up on a lid assembly inside the Dewar.

51

4.2 New Lid Insert Assembly

Since the cleaning requirements for SRF cavities are critical, to avoid contamination
to the already existing vacuum line used for testing the SRF cavities, a new lid insert
assembly was built for measuring the thermal conductivity and the interface heat transfer
coefficient of Nb samples as shown in figure 4.2. The assembly has various ports and
feed-throughs mounted on the top of the lid. The lid, the sample housing assembly is
mounted on long vacuum tube through a six-way cross Conflat flange and a zero length
reducer. Baffle plates are also seen in figure 4.2 for radiation shielding and better
insulation from the top end of the Dewar. The description of each component on the lid

assembly is discussed in the subsequent paragraphs.

52

 

Figure 4.2: New lid assembly carrying two sample holder assemblies.

53

4.3 Dewar Lid Top Plate

The lid of the cryogenic Dewar has a ~ 2.5 cm thick stainless steel base plate with
various ports and receptacles attached to it. The top View of the plate is shown in figure
4.3. The vacuum port, shown in the center of the figure, provides not only a line
connection to the vacuum pump but also a port for electrical feed-through for sensors and
heaters that are mounted on the sample holder assemblies inside the vacuum chamber.
Essentially everything outside this vacuum port on the lid is on the liquid helium side of
the cryostat Dewar. Provision of the heater port, as Shown in figure 4.4, on the helium
side allows quick boil off of left over helium to warm up the Dewar after completion of
the experiment. To physically see the liquid helium inside the Dewar, a viewing port is
also provided on the lid top plate. A port for the pressure transducer is provided to
monitor the helium vapor pressure inside the Dewar. Some of the ports have been
intentionally left blank for future expansion / needs.

4.4 Thermal Radiation Shields

On the underside of the lid, there are seven thermal radiation shields, as shown in
figure 4.2 and 4.5, which are made of 6061T6 Aluminum alloy. The shields have access
holes drilled through them to provide space for passing electrical wiring, and pipelines

for vacuum, helium fill, and gas return.

54

 
 
 
  
    
 

PRESSURE ELECTRICAL FEED-
TRANSDUCER HROUGH
HELIUM FILL AND GAS

RETURN RECEPTACLES
STAINLESS STEEL
BASE PLATE

VIEWING
WINDOW

VACUUM AND

HELIUM WARM
WIRING
FEED THROUGH UP HEATER

Figure 4.3: Dewar lid top plate with various ports and receptacles

55

4.5 Sensors and Wiring on Liquid Helium Side

A specially built [391

multi-connector electrical feed through, as shown in figure 4.4, is
mounted on the lid plate to accommodate all the interconnection wiring requirements on
the helium side of the Dewar. Standard ribbon cables and connectors, as shown in figure
4.5, are used inside the Dewar to carry electrical signals from the sensors to the feed
through at the top of the lid. Three platinum temperature sensors having temperature
measurement range from room temperature to 30 K are mounted on the thermal shields
for estimation of heat leaks through the lid. They are located on the top, middle and
bottom shields as Shown in figure 4.5. Usually the top shield sensor varies in temperature
from 300 K to 285 K, indicating good thermal insulation. The bottom shield temperature
sensor remains below its calibration limits most of the time when the liquid helium level
is closer to it, but occasionally rises to ~ 35 K as the liquid He level drops to the bottom
of the tank. To measure the level of liquid helium, a He level sensor is also mounted on
one of the stainless steel support rods used for holding the thermal shields in place. This
is a 30 inches long (29 inches effective length) He level sensor supplied by American
Magnetics, Inc. Other than the temperature sensors on the radiation shields, there are two
Ge temperature sensors to measure the bath temperature of the bulk liquid helium, one in

the open bath conditions and the other is inside the stainless steel tube, closer (~ 1-2 cm

away) to the sample interface surface as shown in figure 4.6.

56

Multi-connector
feed through . .. Vacuum gage

connector

Dewar warm up
heater port

Dewar pressure
transducer

 

 

Figure 4. 4: Top plate of lid assembly has various ports and feed-through assemblies.
Multi-connecter electrical feed-through is a custom made assembly designed and
assembled at NSCL 13’ '

I‘d

Ribbon cables _

, Platinum sensor
I

 

Figure 4.5: Ribbon cables from the multi-connector feed-through on the top plate
are passing through the radiation shields. A platinum sensor is shown on the lower
shield.

57

Ge sensor placed closer
to the sample inside the
tube (tube not shown)

Ge sensor in open
bath condition

 

Figure 4.6: Helium level and temperature sensors are placed in liquid helium.

58

4.6 Housing for Sample Holder Assemblies

At the very bottom of the vacuum pipe on the lid of figure 4.2, there are two housings
for sample holder assemblies mounted on a six-way cross Conflat flange. A close up
view for this part of the lid assembly is shown in figure 4.7. The set up can be extended
for additional two housings on this flange, however, for the present study, only two of
them are used for testing two Nb samples at a time. A zero length reducer is used to
connect 3 3/8” Conflat flange housing with the 2 3/4” Six way cross Conflat flange.

4.7 Sample Holder Assembly
The sample holder assembly, shown in figure 4.7, consists of a standard 3 3/8” Conflat
flange with an opening in the middle of the flange for a thin stainless steel tube (2.5 cm
diameter). The full view of this assembly is redrawn in figure 4.8 (a). The one end of the
tube is welded flush with the top end of the Conflat flange whereas the other end has a
knife-edge surface to cut through the Nb sample for sealing against the vacuum leakage.
There are six symmetric blind tap drill holes on the sealing end side of the Conflat flange
for attaching the six threaded support rods for holding the sample base plate, and are
shown in figure 4.8 (b). The stainless steel base plate holds an insulating circular disc of
G-10, a thin foil heater, a 5 mm thick copper disc and the cylindrical Nb sample, all in the
same order of arrangement. The six stainless steel threaded rods, as also shown‘in figure
4.9, on the base plate are then hand tightened against the bottom end of the Conflat flange
in the holes provided for them. In this configuration, the knife-edge of stainless steel tube

cuts deep enough into the Nb sample to seal the vacuum effectively.

59

 

 

Sample holder

assembly

Zero length Housings for

reducer sample holder
assemblies

 

 

 

Six-way cross
Conflat flange

/’

 

Figure 4.7: Cut away view of the housings for sample holder assemblies mounted on
the six-way cross conflate flange.

60

AAAMM/‘x/
Sample holder assembly

7. qty 2 shown typical J, _.
' Germainlum \it“ THII
. * sensor In LHe

\ To vacuum pump
N \—Nb sample ._

 

 

 

gr
———‘II

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 000 Sample holder
*1 ' I._ assmbly

 

 

 

1.565
' Tube containing
Stainless steel support rods. Qty
six per sample holder assembly 0190 ”'19 with knife edge
Ge sensor In
vacuum gl/l/B/O/ Three carbon
/ sensors
Niobium sample ' G-1O disc
0.100 0. 240
Heater \ \ 3, I:
Um, m ,nches Stainless steel base
1.875 plate
(b)

Figure 4.8: The two housing (a) for the two sample holder assemblies (b) are
connected to a common vacuum line. This sample holder assembly is used to
measure thermal conductivity as well as the interface heat transfer coefficient of the
sample simultaneously in one experimental setting.

61

Despite several thermal cycles of the assembly down to 1.6 K from 300 K and back
again over a period of several months, the sealing remained effective, maintaining the
vacuum pressure down to mtorr vacuum pressure levels.

4.8 Sample Heater

A thin (~ 0.3 mm) foil heater, as shown in figure 4.9 is supplied by Minco® and is
pasted on a copper disc (5 mm thick) to provide a homogeneous heat flux, into the Nb
sample. Typical voltage (~100 mV) and current (~0.5 mA) measurements for power
calculations are made with the help of digital meters with resolution of 1 mV and 1 LIA
respectively. As mentioned in the last section, the power leads to the heater need special
attention for minimizing the heat leakage into the sample through their high thermal
conductor copper wires. Cold sinking the wires at several locations all along the length of
the vacuum tube provides adequate thermal isolation. However, closer to the heater end,
the nearest cold sink is provided at least 5 cm away from the heater, just to avoid the heat
from the heater when it is switched on for test, flowing back through the wires into the
cold sink. Error induced due to such heat leaks are analyzed in appendix C.

4.9 Temperature Sensors and Wiring in the Vacuum

There is a wide range of sensors available in the temperature range of 1.5 K and
above. Carbon sensors (100 ohm 1/8 W Allen Bradley resistors) are the most commonly
available and the least expensive. Carbon sensors are remarkably good in low
temperature measurements. Since their resistance increases exponentially at low
temperatures, their sensitivities are better than most of the sensors available for cryogenic
use. Typical sensitivity values of a carbon sensor in comparison with a germanium sensor

are listed in appendix E.

62

9 s "i
Imimln. l.|.l.i.l.‘.m '

Nb sample Cu disc G-lO disc

Thin foil
. . . heater &
Conflat flange ‘ . ; wires
with hollow tube

Base plate with
six rods

 

Figure 4.9: Unassembled components of cylindrical sample holder assembly

Each sample has an attachment of three co-linear carbon sensors as shown in figure
4.10 for cylindrical sample and figure 4.11 for flat samples. There is also one 4-wire
factory calibrated germanium (Ge) sensor on one of the two samples for in situ
calibration of the carbon sensors. Before mounting each carbon sensors, their outer
varnish is peeled off and its cylindrical body is made slightly flat with emery cloth on one
edge to increase its contact area with the Nb sample. Each sensor is mounted with the
help of a Stycast epoxy using a very delicate and careful procedure to ensure that
adequate pressure is maintained on the sensor through the curing time of the epoxy. The
usual room temperature curing time for Stycast epoxy is ~14 hrs. A good bonding of the
sensors with the sample is an important and essential step in the sample preparation for

testing at cryogenic temperatures. The carbon sensors are attached with a thermally

63

insulating 45 cm long 36 AWG manganin wire (provided by LakeShore®). Beyond this
45 cm insulating length of the manganin wire, the signal from the sensors inside the
vacuum pipe to the top of the lid is carried through 28 AWG multiple stranded copper
ribbon cables. Also, to avoid spurious heat leaks into the test samples, the signal cables
for all the sensors as well as for the heaters, are anchored at several places along the

length of the vacuum tube to provide adequate heat sinking.

  
   
  

Manganin wires

Heater wire

Stainless steel
Tube

Figure 4.10: Assembled Nb samples in a sample holder assembly. Three carbon

sensors with manganin wires are also visible. Sensor C1 is closer to the heater end of
the Nb sample and sensor C3 is closer to the He interface.

   

64

The electrical resistance, R, of the carbon sensors is measured with the help of
LakeShore® instrument, employing a four lead method, and is recorded through custom
made LabView® data acquisition soflware. This also helped in avoiding carbon sensor’s
self-heating problem where the instrument employs constant current source of 10 LIA for
resistance measurements. However, for temperatures below 1.9 K, the resistance of a
typical carbon sensor becomes more than 7500 (2, an upper limit for the Lakeshore
instrument® to measure. To resolve such a situation, a resistor with about 10 k0
resistance is attached in parallel with the temperature measuring carbon sensor, thus
limiting the current going into the sensor and reducing the overall resistance of the
measurement. With this technique, temperature measurements down to 1.6 K are made
without any significant loss of sensitivity.

4.10 Calibration of Carbon Sensors

Calibration of the carbon sensors is carried out before every test. Thus, when no
power is applied to the heater, the He pressure is varied at various settings and the
resistance values of carbon sensors are compared against the temperature of Ge sensor in
vacuum. By adjusting the gas return valve so as to raise the internal pressure of the
Dewar, raising the temperature of the bath to obtain calibration points above 4.2 K. For
safety reasons the maximum temperature reading needed for a useful calibration data
remain below ~ 4.5 K, corresponding to a pressure of ~ 980 torr. Two or three
temperature data points in between them are sufficient for reliable calibration. Sufficient
settling time is allowed at each calibration temperature data point for steady state to be
achieved for all sensor readings. A similar procedure is applied for calibration between

4.2 K to 2.0 K where about 10 intermediate calibration data points are sufficient to obtain

65

a reasonable calibration fit having a typical standard deviation of better than 0.1 mK at T,
of 2.0 K.

4.11 Flat Plate Sample

The sample holders Shown in figure 4.10 are for measuring both the thermal
conductivity as well as the heat transfer coefficient in the same experiment. However, to
measure only the thermal conductivity of the Nb sample, a much simpler sample holder is
used.

To measure only the thermal conductivity of an actual Nb sheet that is used for the
fabrication of SRF cavities, rectangular flat plate Nb samples are used as Shown in Figure
4.11. The sample holder assembly for these samples is simply a blank Conflat flange with
a blind tap drilled hole in the middle for the attachment of the sample plate. The thin foil
heater is pasted on the free end of the plate with three collinear carbon sensors (~ 1 cm
apart) are pasted on the middle section as shown in Figure 4.12. A calibrated Ge sensor is
pasted on the bottom side of the plate at the same axial position as the middle sensor. The
cold sink end of plate is attached with the Conflat flange with the help of a brass screw.
Apizon® N grease (a high thermal conductive cryogenic grease) is applied between the
mating surfaces to reduce contact resistance between the sample plate and the Conflat

flange.

66

Three carbon
sensors Thin foil

03 fr f .5131 heater

Calibrated
Brass Ge Sensor
screw

\ Conflat
flange

 

 

Rectangular Flat
plate Nb sample

Figure 4.11: Rectangular flat plate Nb sample (11 X 1.4 X 0.3 cm) attached to a 2 :’/4”
Conflat flange with the help of a brass screw. Three collinear carbon sensors are
placed to measure the thermal conductivity of the sample. (Drawing not to scale)

Heater

. IIIIIIII’IIIIII'M
_OO
‘ IIIIIIIIIIj‘

 

Conflat flange

Figure 4.12: Flat plate Nb sample is attached to a Conflat flange. Three collinear
carbon sensors (C1, C2 and C3) are placed to measure the thermal conductivity of
the sample. A thin foil heater on the free end of the sample is also shown.

67

4.12 Cold Shock and Leak Test

In order to verify the sealing of the vacuum system, especially through the improvised
sealing of the Nb sample with the sharp knife-edge stainless steel tube, the assembly must
be tested against cold shock and leaks. Since most of the thermal contraction and
expansion takes place above 100 K, liquid nitrogen provides a reliable cold shock test, as
shown in figure 4.13. During the cold shock test, the leak detector with a gas sensitive

filament detects any evidence of seal leaks.

 

 

   

Figure 4.13. Lid assembly is on a test stand and hooked up with a leak detector. The
housing assembly is immersed in a pool of liquid nitrogen for cold shock.

68

4.13 Steady State Measurement Technique

Traditionally, the experimental technique to measure Kapitza conductance is through
steady state measurements. The experimental arrangement, as described above, has the
advantage of measuring both the thermal conductivity as well as the Nb-He interface heat
transfer coefficient at the same time. The sample holder assembly in figure 4.5(b) has a
resemblance to the one used by Mittag [35] except some improvement in the design to
reduce the parallel heat losses. The helium bath temperature T, in the helium-sample
chamber is stabilized to within 14 Pa by using the Dewar pressure control valve. T1, is
measured with the help of the factory calibrated germanium sensor installed inside the
tube. The temperature distribution along the sample is measured with the three carbon-
resistance thermometers, C1 to C3 with Cl being closer to the heater end as shown in
figure 4.10 and 4.12. Another calibrated germanium sensor is placed next to C2 on the
sample in vacuum to provide accurate in situ calibration of the carbon sensors. The
distance between C3 and the interface surface is 5 mm.

Measurements of both k and h at a given bath temperature, typically consists of
recording all three temperatures and the corresponding heater power. The residual
calibration errors are minimized by the method discussed in section 4.1.3 and the sample
interface temperature is estimated by linear extrapolation as also employed by Mittag [35].
Heat leaks into liquid helium through the six rods via the base plate and the lG-lO disc
from the heater as well as heat leaks from the heater current wires, as shown in appendix
C, are estimated to be less than 1%. Heat leaks along the sensor leads were estimated to

be negligible. Also, heat leaks through the stainless steel tube sealing the vacuum from

the liquid He, as shown in appendix D, are estimated to be negligible (approx 0.110 % at

69

2 K). The, reduction in the area of the sample surface exposed to liquid helium is also

accounted for in the data deduction.

70

5 Results and Analysis

Test results from two types of samples, i.e. cylinders and flat rectangular plates, are
presented here. The cylindrical samples, 81 and S2, were taken fiom a long Nb rod of
small grain supplied by Tokyo Denkai with an ingot RRR of 232. Measurements of both,
the thermal conductivity and the Kapitza conductance are reported for these two
cylindrical samples. Wah Chang supplied the two rectangular flat plate samples, F1 and
F2, for thermal conductivity measurements to evaluate the possibility of partial
crystallization in the “as received” Nb. The supplier for these samples reported a RRR
value of 300. (Specification sheets from each lot of the samples are given in appendix F).

5.1 Cylindrical Sample Preparation

A series of tests were conducted on the two Nb cylindrical samples, to evaluate the
effect of various surface and bulk conditions. Table 5.1 lists the treatments of each test
performed on these samples in chronological order. Sample conditions that may have an
effect on the bulk material properties, such as plastic deformations and heat treatment or
annealing, their thermal conductivity results, are analyzed. Also, those conditions that
affect the Nb surface interface with He, such as BCP etching, mechanical polishing,
surface roughness, and annealing, are also analyzed. Here, the word “BCP” stands for
Buffer Chemical Polish, a chemical etching procedure routinely employed on SRF
cavities to remove surface contaminations and improve surface texture. A standard BCP
solution comprised of three chemicals i.e. HF, H3PO4, and HNO3 in a mixture ratio of
1:1:2 ['3]. Figure 5.1 shows the general surface conditions of sample 81 and $2. The
rough surface of sample 82 can be qualitatively compared with that of relatively

smoother surface of sample S1. As discussed earlier, titanification is a gettering process

71

 

usually employed on the SRF cavities to improve the purity of the Nb. This should result
in increased RRR value of the Nb by removing the interstitial impurities. Since the
process takes place at moderate temperatures of ~1300 °C to 1400 °C, the grain Size of
the Nb also increases significantly. Subsequently, the outer contaminated surface, ~45 to

50 pm, is removed through regular BCP etching.

Table 5.1: Surface and bulk preparation of the two cylindrical niobium

 

 

 

 

 

 

 

 

 

samples
Date Sample Surface Process
Nov Flat & clean ' .
81 Machine cut & llght BCP etch (10-20 pm)
2005 surface
Nov
82 Flat surface Machine out
2005
May Mechanical. polish to a mirrored surface & light
S1 Flat smooth
2006 BCP etch (5-6 pm)
Cut Stn'face with stainless steel knife-edge through
May Rough on the .
SZ mechanical hand press deep enough for SI > 3.
2006 order of 1-2 mm _
nght BCP etch (5-6 pm)
J ~ 3% of total length plastically deformed axially &
une
2006 81 Flat smooth mechanical polish to mirrored surface and light BCP
etch (10-15 pm)
June 32 Rough on the BCP etch cleaning & heat treatment at 750 °C for 2
2006 order of 1-2 mm hrs
Oct Mirror polished & titanification at 1300 °C for 2 hrs
81 Flat smooth
2006 and 1200 °C for 4 hrs & BCP etch (~50 pm)
Oct 82 Rough on the Titanification at 1300 °C for 2 hrs and 1200 °C for 4
2006 order of 1-2 mm hrs & BCP etch (~50 11m)

 

 

 

 

 

 

72

_'_v—--- 7r ‘1. ‘

A 9
IILLLIfLLIJJLIlLlILIIIJLLI.

 

  

 

 

r
1.); 11: Hi.

Figure 5.1: Two cylindrical samples 81 (upper view) and 82 (lower View) are shown.
82 sample has a rough surface (~ 1-2 mm) as compared with smooth surface of
sample 81.

73

5.2 Flat plate Nb Sample Preparation
Each of the flat plate samples is identical to the other and has undergone the same
heat treatment (750 °C for 2 hours) process as that for cylindrical sample 82, prior to its

June 2006 test. Table 5.2 lists the preparation details of these samples prior to each test.

 

Table 5.2: Bulk preparation of two flat plate niobium samples

 

 

 

 

 

Date Sample Surface Process
March F 1 Flat Ordinary cleaning
2006 smooth
March F2 Flat Ordinary cleaning
2006 smooth
July F1 Flat BCP etch cleaning & heat treatment
2006 smooth at 750 °C for 2 hrs
July F2 F lat BCP etch cleaning & heat treatment f
2006 smooth at 750 °C for 2 hrs

 

 

 

 

 

 

74

5.3 Thermal Conductivity Measurements
5.3.1 Repeatability Analysis
(a) Comparison at Different Bath Temperatures

Thermal conductivity measurements of Nb sample 81 tested in November 2005, is
shown in figure 5.2 for 3 different bath temperatures i.e. at 1.6 K, 2.0 K and 2.1 K. Here,
values of thermal conductivity for the sample are given for each of the two pair of sensors
measuring temperature rise (i.e. pair 1 corresponds to sensor 1 and 2 and pair 2
corresponds to sensor 2 and 3) and are plotted as a function of average temperature of
each pair of sensors. In this nomenclature, sensor 1 is closest to the heater. Though the
error bars have been placed only on the result at 2.0 K to avoid clutter, the figure shows
reasonable repeatability through overlap of measurement values at three different bath
temperatures. Below ~ 3 K, the hump in thermal conductivity is due to phonon
conduction where the maxima occurring at ~ 2 K is known as ‘Phonon peak’ and is

discussed earlier in chapter 2.

75

 

 

 

 

 

 

 

 

 

 

 

A 81(1-2) at 1.6K
x 81(2-3) at 1.6K
“ e 31(1-2) at 2.0K
1' i I 81(2-3) at 2.0K
‘ l O S1(1-2) at 2.1K
- 81(2-3) at 2.1K

 

 

 

 

 

 

Thermal Conductrvrty
W/cm/K
l—I-l
*‘k
i—ir-l

"Z$<

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.1
1 Avg. Temp. (K) 10

Figure 5.2: Thermal conductivity of sample 81 at three different bath temperatures,
as shown in the legend, measured in Nov 2005. Error bars are on the graph with
bath temperature of 2.0 K.

 

76

(b) Comparison of Both Samples

Thermal conductivity measurements of both the Nb samples, 31 and 82, tested in
November 2005, are shown in figure 5.3 to demonstrate the repeatability of the
measurements. Again, the error bars have been suppressed to avoid clutter and for clarity
of the comparison. Clear formation of a phonon peak at around ~2 K in both of the
cylindrical samples is evident. As expected, there is no significant difference in the
thermal conductivity measurements of the two samples as both are from the same bulk
material. This comparison in figure 5.3 provides reasonable confidence in repeatability of
the experimental setup where two identical but independent samples provide identical
results. There is slightly more dispersion in the data of the two samples below ~ 3 K, in
the phonon conduction regime where microstructure of the bulk Nb plays a dominant role
in determining its thermal conductivity. A closer look on the figure 5.3 reveals that this
dispersion in the data is more pronounced within the two pair of sensors i.e. (1-2) and (2-
3) on the same sample, than within the two samples. Since no measurements for the
microstructure of the samples have been done, therefore, no certain explanation can be
provided. However one possible source for the variation in the microstructure within the
sample could be due to the stresses induced within the vicinity of stainless steel tube
sealing against the Nb surface. This effect has also observed by Mittag [35] in this kind of

experimental apparatus.

77

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 T ‘7 i ’ V ' J

i’“‘ ‘ i i j ‘ * '
g _ _ _ _ i _ __,_
§ 4 .3102)“
SE T >3 ism-2)—
c_) 0 f? - s1(2-3) -
‘° 3 a(ax x"!
E p“.x... x82(2-3) i
q; ,L. ,, .7 - .f “w - ._L -_ .w. . .
.C
[.—

0.1
1 Av. Temp. (K) 10

Figure 5.3: The phonon peak is shown near average temperature of 2 K in thermal
conductivity measurements of the two samples, 81 and 82, in November 2005 tests.

(T b=2 K)-

78

5.3.2 Validation Comparison with Literature Data
The comparison of the results with data from the literature [9] for thermal conductivity
of Nb with different RRR values is given in figure 5.4. Here, the results of the sample S2
is shown in the figure for a test carried out prior to November 2005 [40]. Close agreement
with data from the literature “2’ suggests validity of the experimental design and

execution.

 

 

 

   

 

 

 

1000 f, A; T ' I if I I 1“"?3‘”
"If—lg. ; e RRR = 400 J
500.- ‘ RRR = 250 D l 1‘
‘- l RRR = 90 0' 1‘,
i + RRR = 232 .. 1‘ 4'
ii ' ‘0 E
5200 r ,0: -‘
.2 . o 1‘
'5 . a 3
a 1
73-100:- ‘11 a .11
9 " at ' '1
1b . all
E 50, 1
. -i
g *_—+—‘.
k u
20 '.
. Sample 2 1
: .wm... .. - -1 ‘ ‘__.-' e l I
101 2 5 T(K) 10

Figure 5.4: Variation of thermal conductivity with temperature for sample 82 is
shown along with data "0'. (Literature data adapted from Padamsee "21).

79

5.3.3 A New Trend in the Data
(a) Effect of Plastic Deformations

Measurements of the thermal conductivity performed in May and June 2006 are
compared in figure 5.5 for sample Sl. As noted fiom table 1, prior to the June test,
sample 81 was intentionally compressed axially to induce a nominal axial strain (AI/L)
of ~3% in the sample because of plastic deformation. The absence of a phonon peak and
a nearly 80% reduction in thermal conductivity at 2 K in the June measurements on S l , in
contrast with the values from its May 2006 test, are considered to be the effect of plastic
deformation on the thermal conductivity of Nb. As expected, this effect is insignificant at
temperature above ~ 3 K, where the electronic contribution to thermal conductivity of Nb

starts to dominate and no significant difference in the results from the two tests is evident.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 I , 4* 7 - j ”7%,; “If, , 3"- 4. TT L;T_;:L7‘iffi

a + g -_ -: 3139..-]:— —.—.: —
b l--~ »- A. ‘
'5- A e -- - e—n _ — _
g E Iggy ' '2 T
E g x" . S1(2-3)May
2 0.1 _ f. 'i x S1(1-2)May
(B F‘”—' 'T” ’——‘ * ‘ —“
E, . S1(1-2)Jun
E _ _ - S1(2-3)Jun

0.01
1 Avg.Temp. (K) 10

May. Unstrained sample
June: 3% axial strain

Figure 5.5: Phonon peak is clearly absent in the thermal conductivity measurements
of sample 81 in the tests carried out after 3% strains induced in the sample (June
2006 test) as compared with its tests performed in May 2006

80

b) Effect of Low Temperature Heat Treatment

For sample 82, the absence of a phonon peak in the measurements of May 2006, in
contrast with those of November 2005, can be understood as surface roughness
preparation protocol, as described in table 5.1 and shown in figure 5.1, with the stainless
steel blade pressing against the surface, causing inadvertent strains in the bulk Nb, similar
to those induced intentionally in S 1.

Low temperature annealing (750 °C for 2 hrs) on 82, intended to recover the phonon
peak in the Nb and shown in figure 5.6, is insufficient, although S2 demonstrates a small

increase (~30%) in thermal conductivity over the temperature range measured.

A SZ(1-2)June

x 82(2-3)June
+ 82(1-2)May
- 82(2-3)May

Thermal Conductivity

 

1 Avg. Temp. (K) 10
May: Axially strained sample
June: Heat treated

Figure 5.6. The phonon peak did not reappear after it was lost during the
compression process in preparation of (May test) sample SZ. Low temperature heat
treatment (June test) has small effect on the thermal conductivity of the niobium.

81

This is possibly due to knocking off some of the hydrogen impurities at the annealing
temperatures causing a slight increase in the RR value of the sample. The same effect of
low temperature annealing is also observed in thermal conductivity measurements done
on the two flat plate samples. The initial measurements on flat samples done in March
2006 did not reveal a phonon peak in the as received material, possibly due to partial re-
crystallization in the material. After annealing them at 750 °C for 2 hours, no significant
change in their thermal conductivities was observed for most of the temperature range
measured. However, a slight improvement below ~ 3 K can still be observed due to the
low temperature heat treatment. The result for F1 is presented in figure 5.7(a) and those
of F2 is in figure 5.7(b). Close agreement of these results increases the confidence in the
experimental results.

Based on these observations, it can be argued that low temperature annealing is
insufficient to recover the lost phonon peak in the thermal conductivity curve of Nb with

RRR ranging from 200 to 300.

82

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10 L : 7' : Q : : : ; : a: : ; ' ; L ; W; L
e:_5:_- * _‘_-_L j; '5 :1: E :i e
g ...... - - - _ _ _ ._ _ -
.> T T v T 7 T ‘ T _" T"'“‘" “"' “"T’ 'T A
5 l
1:: 1 . j“: 7:“: «4+ : : "2 :5“: ~—
E g - L L-__T 4:" j11:10-2va
7, Q g 2 g ;_ 4:, :f f‘” “ I F1(2-3)July
g B L ”7‘3“! L AF1(1-2)Mar
15 01 .- "[ eF1(2-3) Mar
1 Ave Temp. (K) 10
(a)
10

A

. F2(1-2) July

Thermal Conductivity
o W/cm/K

 

,1 I F2(2-3) July
A F2(1-2) Mar
0 F2(2-3) Mar
0.01
1 Avg. Temp. (K) 10
(b)

Figure 5.7 (a) and (b): The low temperature heat treatment (July 2006) has slightly
increased the thermal conductivity of flat samples in phonon conduction regime,
from their previous test in March 2006 on the as received samples.

83

(c) Effect of Titanification
As mentioned earlier, during the titanification process the temperature of the sample
is raised to ~l300 °C in high vacuum conditions. At these moderate temperatures,
sufficient annealing within the Nb takes place to relieve internal strains in the material.
Also, the grain size of Nb grows from the order of pm to m. As shown in figure 5.8 for

sample 81 and figure 5.9 for sample 82, this process resulted in the recovery of phonon peaks in
the thermal conductivity for the two Nb samples that were lost due to induced strains. There is
slightly more recovery of phonon peak for sample S2 after titanification than it was in the as
received sample. This could be due to the fact that the sample S2 had already under gone low
temperature annealing with slight increase in its thermal conductivity below 3 K, as mentioned in
section (b) above. Possibly, this second annealing at moderate temperatures augmented the
reformation of phonon peak slightly more than it was in the as received sample.

Unexpectedly, a slight decrease in the thermal conductivity above ~3 K is also
noticeable in both the samples, ~20 % in sample 1 and ~8 % in sample 2 at 4.2 K, as
compared to their values before annealing. This is understandable since the purification
process through titanification relies on the diffirsion of oxygen in the bulk to the surface
of niobium where they are captured by titanium. The diffusion rate of oxygen at these

[45 1, and the titanification process is optimized to treat 2

temperatures is ~ 1 mm per hour
to 3 mm thick Nb cavities. However, the cylindrical samples have 30mm diameter. It
seems quite reasonable that much longer time than the 6 hours, namely (15 mm / l.

mm/hr) 15 hrs [46], for the cylindrical samples were needed for complete purification.

84

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

i:::f:::_:{51531~1i—~£
2 ———————r~——————.~ ———
E __E A __E ______ _ A MEIEi-é_—___
2 e e e ~§- ~ ~-~—{~I~~~—— -———~
E . L L - r - MEAL L4-__ __.__ _ -2 - - _
z: ,-"'
t; 0-1 I — _" — J’ : : _ : ' 'E_' ;:' _: ;‘L:’::__—
g 1 TL 5, ,, L. '3 #-_.-,:_ A * # ”i
3 - . A $1 May 06; as received Z
E - 81 June 06; 3% strained :
0
f. 9 S1 Oct 06; titanified -

0.01
0 1 2 3 4 5
Sample S1 Avg. Temp. (K)

Figure 5.8: Moderate temperature annealing during titanification process resulted
in the reformation of the phonon peak (Oct 06) after it was lost due to induced
strains (June 06) in the as received Nb sample 81 (May 06)

85

 

 

1 .-_____L _____ A- _W __-___,.

f I 82 Nov 05; as received To
9 82 May 06; strained 1
L A 82 Oct 06; titanified

-
fl
__- _ -_-._, _ 10+...
I

1
y . _—¥_ A 7. 7 , 7 7 -— ,,__,_ _.. ,_+ A} - I _‘ *.———4
I

 

 

 

T
l

 

 

 

 

 

 

 

 

 

Thermal Conductivity (WIcm/K)

 

 

 

 

 

 

0.1

 

o 1 2 3 4 5
Sample 82 Avg. Temp. (K)

Figure 5.9: Moderate temperature annealing during titanification resulted in the
reformation of phonon peak (Oct 06) after it was lost due to inadvertent strains
induced (May 06) in the as received Nb sample 82 (Nov 05)

86

5.4 Nucleate Boiling Heat Transfer Measurements

As discussed in chapter 3, in the nucleate boiling regime, heat transfer is governed
primarily through the nucleation sites on the heater surface. The rough surface provides
nucleation sites of preferential locations for the bubbles to form more readily than on a
smoother surface. As is generally understood, the surface roughness varies with the type
of process performed on the surface. The effect of surface roughness due to different
processes, as mentioned in table 5.1, on the nucleate boiling heat transfer has been
studied both qualitatively and quantitatively on each of the two cylindrical samples, 81
and S2. For quantitative analysis, optical microscope has been used for the surface
roughness measurements. However, when the surface roughness increased beyond the
optical resolution of the microscope, results are qualitatively discussed. The qualitative
analysis is based on either visual observations or wherever possible, a use of improvised
depth gauge is made in order to estimate the surface character.

5.4.1 Surface Roughness Measurements

Figure 5.10 and 5.11 shows the surface roughness measurements on sample 81 and
82 respectively prior to their November 2005 test. Also, the data provides the value of
surface index as well. From table 5.1 recall that both of the samples, 81 and 82, had a
machine cut surface, however, the sample 81 went an additional step of a light BCP etch
as is usually performed on the SRF cavities for surface texture improvements. Visually
comparing the surfaces of sample 81 and 82, sample 82 obviously appeared rougher than
the SI .

The surface roughness parameters for the two samples obtained from the optical

microscope can be compared with each other. By comparing their average roughness

87

parameter, Ra, for instance, at a first glance it appears that S1 with Ra = 9.87 pm is at
least two times rougher than S2 with Ra = 4.57 pm. This is a misleading result, although
both of the measurements have been made with the same magnification and on the same
length scales with the same sampling size. The reason for such an unexpected result is
basically the limited optical resolution of the microscope for the rougher sample. The
amount of valid data, as shown in the figures 5.11, obtained by the microscope for a
visually rough sample S2 is 31.2% only as compared with relatively smoother surface of
sample 81 having 88.4% of valid data and shown in figure 5.10. Thus, the statistical
parameters so computed for sample S2 are not the true representative of the actual surface
and thus are unreliable and can’t be compared with the more reliable data of sample 81. It
is in such situations that nucleate boiling results are discussed in qualitative terms. More
detailed information about surface roughness parameters is given in appendix G.
Estimates of surface index on the Sample S2 after it has been made further rough
prior to its May 2006 test were made through an improvised depth gauge. In this, the
thick depth rod of the gauge was replaced by a custom made thin pin having tip diameter
not more than 0.65 mm. Since the surface of sample S2, as shown in figure 5.1, looks like
a 2-D horizontal grid made of grooves / valleys while the mountain or the peaks
appearing on the intersections of these valleys in the vertical dimension. Thus by
estimating the width and the length of the grooves along with the height of each of the
mountain like structure on the surface and a known diameter of the surface, provided a

reasonable value of surface index (SI) > 3 for the sample S2.

88

Histogram

 

 

 

   
   
  
  
   
  
  
  

 

 

 

Mag: 2.5 X Date: 01/25/2006
M dc: VSI - . . .
0 Time. 14.26.01 v
3030——
“ Surhce Area:
- Actual/V Smiles Statistics:
2500—
: 6.988 mmZ data Ran 9.87 um
2000: Lateral Area: Rq: 12.79 um
: 4146 m2 Guassian R1: 89.42 um
‘0 : Rt; 103.32 um
E 1500 _ Surfiice Index:
1&1]: 1'685523 Sci-I! Pmbrs:
: Size: 368 X 238
500: Sampling: 6.82 um
0 ‘ 4 mm
I I l I l I I I I F I T T I I I T T I l l l l l l l I I I I
-0.05 -0.04 —0.02 0.00 0.02 0.04 0.05
Title: N'BSl-centerview Height Valld Data II 88.4 %
Invalid Data - 11.6%

Figure 5.10: Histogram as well as different surface roughness parameters is shown

for sample 81.

89

Histogram

 

 

 

 

 

 

 

 

 

 

 

Mag 2.5 X Date: 01l25l2006
Mode: VS! Time: 14:16:20 V
14003 Surfice Area: Smfice Statistics:
: 2.412 mm2 Actual M 4.57 um
um: data \ Rq: 525 um
- Lateral Area:
KID—— R: 46.10 um
a 1.471 mmZ
in j Rt 54.8Sum
E an: Surhce Index: GuaSSian _J
-l f "A
5130—“ 1.639924 \ 3"" ’mm‘
2 Size: 368 X238
41]): I Sampling: 6.8211111
2130—:
0': um
[IrferIIYYIIIfifififiII[rlr]rllrril[
-40 -30 -20 -1 0 0 1 0 20 30 40
Title: NbSZ-Center Wew Height Valld Data = 312 %
Invalld Data " 68.8%

Figure 5.11: Histogram as well as different surface roughness parameters is shown
for sample 82. Note that the valid data is only 31.2 % in this measurement.

90

5.4.2 Comparison of Machine Cut and BCP etched Surfaces

The effect of surface roughness on nucleate boiling heat transfer in liquid helium is
well known and has been discussed in chapter 3. Also recall that the empirical correlation
provided by Schmidt “7] and shown in figure 3.3 is conservative with most experimental
data demonstrating lesser superheat than the one predicted by the correlation. The same
observation can be made in comparison of the test results for the two samples, S1 and S2,
obtained in November 2005 and shown in figure 5.12. Here S1, with improved surface
texture through BCP etching is visually smoother than S2. As expected, this resulted in

more superheat than 82, but is still less than that predicted by the empirical correlation.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

—Schm'dt
A 0.1
"E 9 S1
0 I
E #— whim 32
v I." I
U' 001 .U-y/
0.(X)1
0.1 1
T"T" (K) Tb = 4.2 K

Figure 5.12: Nucleate boiling heat transfer measurements for the two cylindrical
samples made in November 2005 tests at bath temperature of 4.2 K. Schmidt
correlation is the same as the one in figure 3.3.

91

5.4.3 Effect of Mechanical Polishing

The effect of mechanical polishing on the sample S1 is shown in figure 5.13 by
comparing its nucleate boiling results in three different tests. Both the May and the June
06 tests are of mechanically polished surface of sample S1. This resulted in increased
superheat for the same amount of heat flux for the sample 81 as compared with its
relatively rough surface in Nov 05 test. As expected, this increase in the superheat is due
to the effect of mechanical polishing on the nucleate boiling characteristics of liquid
helium for smoother surfaces. The results of mechanically polished surface (May and
June 2006) are closer to the conservative correlation provided by Schmidt “7]. Also note
the variation of the result within the mechanically polished surfaces. This is attributed to
slight differences in the surface preparations for the two tests. Differences in etch rates /
duration during the BCP etching or variations within the mechanical polishing procedure

may cause such variations in the nucleate boiling heat transfer.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.1 :- #_ _.
__ ;____ L 77:, A". i f ‘

A . +"+:f/:’ + Nov05
NE 001 + +7QO/x/x‘ 0 June 06 -
E’- + l’ 5‘ Schmidt T i

+ ‘X x May06

0.001 1 I 1 l l

0.1 Ts'Tb(K) 1

Sample 81

Figure 5.13: Effect of mechanical polishing on sample 81 has resulted in an increase
of superheat for the same applied heat flux.

92

5.4.4 Effect of Surface Index (SI)

From table 5.1 and the figure 5.1, it is noted that surface preparations on S2 for May
06 test resulted in SI > 3. Considerable decrease in superheat for the same heat flux
resulted for its June 06 measurements, shown in figure 5.14, as compared with its
November 05 result. This decrease is more prominent at low heat fluxes, where few
bubbles are formed and natural convection effects still play a significant role in
transferring the energy, as compared with those at relatively higher heat fluxes where
increased numbers of bubbles are formed on the surface thus increasingly removing the
energy from the surface. The same trend is seen, though less pronounced, in the June 06
test before which the 82 was BCP etched and heat-treated. At this time there is no
quantitative explanation for the increase in superheat in sample S2 measured in the June
06 test as compared with the May 06 tests. The BCP etch done prior to the heat treatment,
due to the change in surface texture, could be one possible explanation for this

observation.

93

A May 06
I June 06

0 Nov 05
— Schmidt

 

0.01 0.1 1

Sample 82

Figure 5.14: Decrease in superheat due to increase in SI of sample 82 is seen in May
06 measurements as compared with Nov 05.

94

5.4.5 Overall Effect of Surface Roughness

The overall effect of surface roughness on the interface heat transfer can be realized
qualitatively from the figure 5.15 where the results from all of the measurements made
during this study for both samples 81 and S2 have been presented on a common scale.
These data tend to confirm the trend already discussed in chapter 3 where an increase in

surface roughness decreases the surface superheat for a given heat flux or allows more

heat flux for a given temperature rise on the surface.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

01 :. L ; 7,. ': : ‘ : A ,‘7: fig.
4:1- 3L _:.. :i- _1:._ * L“ _ A_:__—E. qurpizbfii
- f; Increasing , . 32 May06
f - roughness . $2 Jun06
E a $2 Nov05
\ 0.01 ___ '11:- L:
s : , _.4“ :L‘ZH' _ + 31 NOV 05
0' T: : L: , _ _ 0 S1 Jun06
A x _.. -_ _ a draw m1 "3 —Schmidt
x S1 May06
0.001 1 1 1 1 1111
0.01 0.1 1
Ts"Tb (K)

Figure 5.15: The effect of increasing surface roughness is to reduce the superheat

for a given heat flux.

95

5.5 Kapitza Conductance Measurements
Recall from chapter 3 that nano-meter length scale surface roughness plays an important
role in increasing the Kapitza conductance. At a macroscopic level, larger scale
roughness carry more number of smaller scale roughness superimposed onto them. Thus
increased SI of the surface may lead to enhanced Kapitza conductance.

5.5.1 Comparison with Literature Data

The BCP etched sample S1, having a cleaner surface than $2, has Kapitza
conductance at 2 K that is about two times as that of S2, as shown in figure 5.16, for the
test carried out in November 2005. The Kapitza conductance measurements of S1 are in
reasonable agreement with the values provided in the literature by Bousson et al. [38] for
Nb having comparable RRRs and from the same supplier. This provides confidence in the

validity of the current experimental results.

96

e 81
I 82
——Bousson

 

m (chm2 K)
O

 

0.01
1.6 1.7 1.8 1.9 2 2.1 2.2

Tb (K)

Figure 5.16: Kapitza conductance measurements for the two cylindrical samples, 81
and 82, carried out in November 2005, are compared with the one given by S.
Bousson et al. ”8'.

97

5.5.2 Repeatability

The measurement of Kapitza conductance of the sample S1 before and after plastic
deformation, as shown in figure 5.17, does not reveal any significant difference. This is

likely due to the surface preparation for the two tests being identical on sample S1, i.e.

mechanical polish and a light BCP etch.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 .. - -g ,,,,, ,2, -+-H,A_:.A -g_gl E
:.:::7:::::::::1-:l
* * i ii 1:3;1111
i 7 f i A v__ V i v _ _ ;
Q ——#r i —— 7 7w #fiflr— — 1 '37
NE I .I i
g - f -4 n .2 J _ __
E 0'1 _:.-__:1:'7: L —:_:—‘ 5::
_E 7‘ ____ : a"? -_-__- . # _ '#
IS1May06
_~ _ _ __ 0 S1 June06
0.01
1.5 1.7 1.9 2.1

T0 (K)

 

Figure 5.17: Kapitza conductance measurements of the sample 81 before and after
the plastic deformations reveal no significant difference.

98

5.5.3 Effect of Surface Index

In order to evaluate the effect of large SI on Kapitza conductance, the surface
roughness of sample 82 was increased such that its surface index became greater than 3.
A sharp stainless steel edge surface was pressed against the Nb with the help of a hand
press. Several deep (1-2 mm) cuts both in a horizontal as well as in vertical directions,
like a 2-D grid on a flat surface and shown in figure 5.1, resulted in an overall SI > 3.
Results of Kapitza measurements of sample 82 as shown in figure 5.18 can be compared
for its May 06 test (SI >3) with those of its Nov 05 test (Machine cut surface). Although
there is ~64% increase in by. in May 06 measurements from its Nov 05 values and brought
the values closer to the data by Bousson et a1. [38], however, this is still below the values
expected proportionately to SI > 3. This increase in hk can possibly be due to the BCP
etching of the surface that took place prior to the May 06 testing and thus simply
increased the values of sample S2 closer to the literature data and the data of sample S1
for its Nov 05 test that was shown in figure 5.17.

One possible explanation could be the effect of surface strains on the Kapitza
conductance that might have been inadvertently induced on the surface during the
roughness process described above. More discussion on the issue is made in subsequent
sections and the next chapter.

5.5.4 Effect of Low Temperature Annealing

The same sample surface on 82, after annealing (June 06) at 750 °C for 2 hours,

resulted in a nearly doubling in its Kapitza conductance values at 2 K, as shown in figure

5.18. This result may provide some help in understanding the effect of low temperature

99

annealing on the Kapitza conductance by relieving some of the induced surface strains

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

during the process.
11; o SZJune06 :_
C 0 82 May06
Z - $2 Nov05 7
A 0
(3‘ 1 —Bousson etal 4,
E
g .
x
.c
0.01
1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

Tb (K)

Figure 5.18: The Kapitza conductance of sample 82 surface with SI > 3 (May 06)
and the effect of low temperature annealing with SI > 3 (June 06) is compared with
the machine cut surface (Nov 05). Literature data is adapted from Bousson et al. '38“.

100

5.5.5 Effect of Moderate Temperature Annealing

The moderate temperature annealing employed during the titanification process
relieves the stresses not only inside the Nb but also on the surface as well. This surface
effect on the Kapitza conductance is shown in figure 5.19 for sample 81 and in figure
5.20 for sample S2. The greater than two fold increase in Kapitza conductance at 2 K for
sample 1 from its June test is possibly due to the moderate temperature annealing that
occurred during the titanification process. For sample S2, the increase in Kapitza
conductance is about ~30% as compared to its June test when its value was already

increased through low temperature annealing as discussed in section 5.5.4 above.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 T v 7 a—VE
_ ** _‘*—_ ‘ _-- ‘ j ‘_.__‘
" m
5 01. ___ - «» +__
E __—— .. ——4h 3 _._____ ”_“y 77#—;
E L—4—9 2:27:— - A S1 Oct06 -
o 81 June 06
IS1 May06
0.01
1.5 1.7 1.9 2.1
TblK)

 

 

 

Figure 5.19: Kapitza conductance for sample 8] comparison before (May and June
06) and after titanification (Oct 06).

101

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 7 :
9 . j.'
E x x . ___,"_ _ {
€01- « _ " f .e _»
'c 1“ : " 1kg??? ._
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0 SZJuneOG - ~~ -~ - — - -7 , _ _..
0 SZMay06
. $2 NOV05 1.6 1.7 1.8 1.9 2 2.1 2.2
— S. Soussori Tb (K)

 

 

Figure 5.20: Kapitza conductance increase history for sample 82 through all the
processes especially after titanification (Oct 06).

If the increase in Kapitza conductance is compared for the two samples together, as
shown in figure 5.21, it is shown that after the titanification process both the samples (81
Oct 06 and 82 Oct 06) attained comparably close values for the Kaptiza conductance.
This is despite the fact that both the samples have surface roughness drastically different
from each other. It is recalled from figure 5.1 that sample S1 has a smooth and plane
surface where as sample S2 has a rough surface (on the order of ~1-2 mm) resulting in SI
> 3. Although, the increase for the sample S2 (S2 Oct 06) is about 4 times from its initial
state (Nov 05), this result suggests that surface effects due to titanification could be more

stronger on increasing the kapitza conductance of the Nb samples than from the simple

102

area increase. The exact nature of these effects that resulted in the increase of Kapitza
conductance after titanification is still undetermined. Surface strains relieved during
moderate temperature annealing in titanification process could be one possible reason.
Other factor that may result in this unexpected result could be from the distribution of
crystal orientation (<111> or <110>) on the surface. The crystal size growth during the
titanification process may result in substantially different distribution of crystal
orientation closer to the surface as compared to their distribution before the titanification
process. If the distribution of crystal orientations on the surface were more random than a
uniform distribution of the same crystal orientation, the overall surface would be rough at
the scale lengths of nanometer, the scale lengths, which are also comparable to the
phonon wavelengths of He-II at these temperatures. This understanding may help explain
the observation of comparably closer kapitza conductance values for the two samples
after titanification despite their apparent differences in surface roughness at macro level.
Nevertheless, this is the fiiture work required for the subsequent studies in completely

understanding the Kapitza conductance.

103

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

- +~~— ____ -X_X_.1r_
.. x I I
g - >12 *2?“ firm—r —* a---”
N I
3013* '.'¥J—xr* *.X x 1:
E of ‘ - 32 Oct 06 :
E xs1 Oct 06 _
“’7 I 82 June 06 7
. S1June06 “
x 31 May 06 ‘“
A 82 May 06
0.01 .
1.5 1.7 1.9 2.1
Tb (K)

Figure 5.21: Comparison of Kapitza conductance values of the two samples, 81 and
82, after titanification (Oct 06).

104

6 Discussion

Several test for the thermal conductivity and Nb-He interface heat transfer
measurements have been carried out. Thermal conductivity measurements are made on
two different shapes of Nb samples i.e. Cylinder and flat plate. These measurements
show not only close agreement with each other but also with the already published
literature data to authenticate the results both for their repeatability as well as validity.
Nb-He interface heat transfer measurements are done on two cylindrical samples 81 and
82. At temperatures 4.2 K, the nucleate boiling heat transfer characteristics are analyzed
for various surface conditions. Similarly, at temperatures below 2.17 K, i.e. below T1
when He is in super-fluid state, the Kapitza conductance measurements are made for
various surface conditions.

6.1 Thermal Conductivity Above 3 K

As discussed in chapter 2, theoretical predictions require RRR value of Nb at 4.2 K
to be directly proportional to its thermal conductivity at that temperature. This is given
as:

RR = 4 k

Table 6.1 has been compiled to verify the existence of such a correlation in the
measurements carried out in this study as well as that of the data in the literature [91’ [45].
The data from this table is then plotted in Figure 6.1 to show the variation of RRR values
with thermal conductivity at 4.2 K. The least square fit to this data provides the slope of

the data as ~ 4.08, which is reasonably close to the expected value of 4 within the

experimental error.

105

Table 6.1: RRR values of Nb samples against the measured values of
thermal conductivity at 4.2 K.

 

 

 

 

 

 

 

 

 

 

 

 

 

Measured K
Nb Vendor
(WImIK) at Measured by
Samples RRR
4.2 K
81 232 62
$2 232 60 A. Aizaz in this
F1 390 98 research
F2 390 95
Pad1 400 95
Pad2 250 62 Padamsee et. al. "’1
Pad3 90 22
Bouch1 38 14
Bouch2 270 53
Bouch3 194 45 Boucheffa et. al. "51
Bouch4 186 48
Bouch5 100 30

 

 

 

 

 

 

106

450
400 r
350 Least square fit

300 ,_ RRR = 4.08 k _ /
250 -__,_ /

200

150 ~—-——

100 ~ /

50 ,

0 r r I 1 fl
0 20 4O 60 80 100 120

Thermal Conductivity k (WImIK)

 

 

 

 

RRR

 

 

 

 

Figure 6.1: A linear correlation (least square fit) is shown for the variation of RRR
values with those of thermal conductivity at 4.2 K. The (A )data points in the plot
are from current research.

107

6.2 Thermal Conductivity Below 3K

The thermal conductivity of Nb below its transition temperature is atypical of
pure metals because of the superconductivity of Nb. At temperatures less than Tc,
phonon conduction starts to play a significant role along with the electron conduction of
thermal energy. This role of phonons keeps increasing, as we keep moving down away
from the transition temperature, where increasing numbers of electrons are formed up
into Cooper pairs. Thus, below ~3 K, the phonons become the dominant energy carriers
and the microstructure of the bulk Nb becomes very important in determining the shape
of thermal conductivity curve.

Although the thermal conductivity of Nb above 3 K is well correlated with its
RRR value, as demonstrated above, the same is not true when the temperature is below 3
K. As discussed earlier, it is in this range of temperatures when the thermal conductivity
of Nb is dominantly governed by phonon conduction. Thus the history of microstructure
of the material becomes important. Higher purity (high RRR) Nb doesn’t necessarily
mean presence of phonon peak in the thermal conductivity at 2.0K. Even small amount of
plastic deformations (~ 3% of strains) induced at any stage of material processing can
cause the loss of phonon peak, as demonstrated in this research. This results in significant
reduction (about 80 % at 2 K) of the thermal conductivity. This is especially important
when the SRF cavities are routinely fabricated through the deep drawing process in
which plastic deformations are deliberately induced to shape the cavity to the design
requirements. Moderate temperature annealing, as demonstrated in this research, is a
viable remedy to recover back the lost phonon peak in order to achieve higher gradients

in the SRF cavities.

108

 

 

The major advantage of the improved phonon peak in the thermal conductivity is
for high frequency defect free cavities operating at 2 K. To realize the significance of the
phonon peak on the improved performance of SRF cavities, figure 1.3 is redrawn in this
section for illustrative purposes. Recall that the simulations in this figure are with
constant thermal parameters, i.e. constant thermal conductivity and the Kapitza
conductance. To a first order approximation the simulations shown in figure 1.3 provides
a good performance indicator for SRF cavities with lost phonon peak as compared with
the recovered phonon peak. Also, since the total temperature variation on the RF surface
is on the order of few hundreds of mK, the constant property approximation is not a bad

choice.

 

 

 

 

 

 

 

 

 

 

 

 

 

igg IIEJLcritical 1'05““
L44;- _ _ ’ ’v- 1.0E-01
.60 ,_ 7::A\ by
140 I —k=.28;h=.24~ '
A ~ 1.0503
n- 120 —k=.06;h=.08”‘
E 100 —» —~ m - —— 1.05-04A
V -_ h- _n_§.= i _. w
m 23 d i, g _k: (W/CW/KLW _ - ICE-05.5
40 -.__ ___ h:__gw/cm2/K) 41.05069
20 _, ___.w ,z”' R # roe-07¢;
K S
0 w . . f 1.0E-08
1.5 2 2.5 3 3.5 4
Ts (K)

Figure 1.3: (Redrawn for illustration purposes) Thermal-magnetic simulations with
constant thermal properties, k and h, for the two cases are shown. About five times
increase in k and three times increase in h resulted in almost 50% improvement in
applied magnetic fields.

109

Recall that figure 1.3 shows the two cases of selected thermal parameters, k and
hk, for which the applied magnetic field, B, is displayed against the calculated RF surface
temperature T,. The lower curve (thick line) is shown for a case of thermal conductivity
without the phonon peak and the upper curve (thin line) is with phonon peak. Also note
that the corresponding values of Kapitza conductance so chosen for the simulations have
also been duly demonstrated in this study, which are close to the two cases of Kapitza
conductance i.e. before and afier moderate temperature annealing. The simulations shown
in figure 1.3 reveals that applied magnetic field rose from 105 mT to 170 mT. Thus ~50%
improvement in the applied magnetic field is realizable due to the presence of phonon
peak and improved Kapitza conductance. Also both of these objectives, namely the
phonon peak as well as improved kapitza conductance, can be attained through one single
process of moderate temperature annealing of the Nb cavities.

For future research, moving forward in this direction, it is possible that much
higher phonon peak is realizable than the one demonstrated in this research either through
high temperature annealing (~1800 to 2000 °C), which may have a draw back of
considerable reduction in the yield strength of Nb, or for longer durations of annealing at
moderate temperatures. Also, since it’s the strains in the microstructure of the Nb, which
determines the presence of the phonon peak, it would be interesting to analyze the effect
of unstrained single crystal / large grain Nb material on the size of phonon peak as
compared with the polycrystalline Nb. Not to limit here, the effect of annealing of large

grain / single crystal material may also be a fruitful option to explore.

110

 

 

6.3 Nucleate Boiling Heat Transfer

Operation of low frequency SRF cavities is carried out at 4.2K, the boiling
temperature of He at atmospheric pressure. The present study has demonstrated, at least
qualitatively, the effect of surface roughness on the nucleate boiling heat transfer. Almost
55 % reduction in superheat at heat flux of 10 mW/cm2 has been shown due to the large
SI > 3. This effect of large SI seems to be more pronounced only in the low heat flux
regime when there are few bubbles on the surface and convection heat transfer effects
still play important role in energy transfer from the surface. However, this result is still
inconclusive due to the calibration range limitation in the experimental setup above 4.5
K. More experiments need to verify this observation over the extended calibration ranges
to span the whole nucleate pool boiling regime of liquid helium.

In order to understand the effect of thermal parameters on the defect free low
frequency cavities operating at 4.2 K, simulations similar to those discussed above for
high frequency cavities, is presented in figure 7.2. Since the thermal conductivity at these
temperatures is well correlated with the RRR of the Nb, this value is held constant for
one particular value of RR. However, for heat transfer coefficient two empirical
correlations are used to see the effect of improved heat transfer coefficient through
surface roughness. Schmidt’s conservative correlation is used for low heat transfer
coefficient (q = ATS“), whereas Kutateladze’s correlation (q = 5.8 ATS“) provides
almost six times more heat flux for the same amount of superheat “7], same as for a rough

surface.

111

 

 

300 MHz Cavity at 4.2 K
RRR=230 Rros=5n0hms

200 7.0E-08
Bcritical /

180
160 \ A Q _ , A — 6.0E-08

14o —,%—-‘ "’ " 7!. ‘ ‘ #Y . 5.0E-08
120 / \

E 100 _.. // // ‘R. L 4.0E-08

m _ — 3.0E—08

80 / / —Schmidt

60 7 / ————Kutateladze ” 2-0E'08

40 Bcn't

20 // _“_RS_Rtaxis ~1.0E-08
0 V , I . 0.05+00

4.2 4.3 4.4 4.5 4.6
q= 5.8(Ts-Tb)2'5 Kutateladze T. (K) q=(Ts-Tb)2‘5 Schmidt

 

 

 

 

 

 

Rs (Ohms)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 6.2: Thermal-Magnetic simulations for a defect free 300 MHz cavity at 4.2 K.
Variation in surface resistance R, is also plotted (Right axis) against the surface
temperature T,.

The simulations shown in figure 6.2 indicate that for a defect free 300 MHz cavity,
even with a conservative heat transfer relation of Schmidt, which is for smoother
surfaces, no fundamental limit on the thermal breakdown is found other than from the
critical magnetic field at 138 mT. Although the RF surface temperature rise above
ambient at this magnetic field is 0.28 K, the total change in R, due to this temperature rise
is ARs ~8.3 nOhms. The improvement due to high heat transfer coefficient through
surface roughness is shown through reduced Ts, which consequently results in lesser rise

in R, (ARS ~4.1 nOhms) as compared with low heat transfer coefficient. Thus, this

112

 

improvement of high heat transfer coefficient through 100 % reduction in AR,, may result
in achieving proportionally higher Q0 of the cavity at the break down fields. However this
heat transfer coefficient has limited role in increasing the break down fields since the
performance is shown limited by the critical magnetic fields.

In reality, low frequency cavities hardly reach closer to such a high magnetic field
levels at 4.2 K. One possible explanation for the divergence of simulations fi'om the real
cavities comes from the caution that was placed in chapter 1 while using the relation for
R, in equation 1-2. Recall that the binding energy A(O) was assumed as constant with the
condition that the T, < T,/2. However, operation of cavities at 4.2 K is so close to TJZ
that the departure of the relation for R, from the reality may be a significant effect. Also,
since the physical size of the low frequency cavities is much larger than the high
frequency cavities, there is more likelihood of having surface defects in the low fequency
cavities causing thermal breakdown due to defects at much lower fields than the one
predicted for the case of defect free cavity in the simulations. As discussed in chapter 1,
for the case of cavities with defects, it’s the thermal conductivity of the Nb that
determines the quench field of the cavity and heat transfer coefficient has only a
negligible effect.

6.4 Kapitza Conductance

The study has revealed a significant effect of annealing on the improvement of
Kapitza conductance. At least 150 % increase in the Kapitza conductance has been
demonstrated in this study at 2.1 K due to moderate temperature annealing. As discussed
earlier, the effect of increased SI on Kapitza conductance doesn’t seem to play a strong

role as compared with the effect of annealing of the samples. Among other factors,

113

 

 

possibly removal of surface strains that might have been induced during the roughness
process, has resulted in the increased Kapitza conductance after annealing. Also, since it
is the nano-length scale roughness that is more important, crystal orientations on the
surface could be one possible parameter that have changed due to annealing.

The effect of improved Kapitza conductance on the performance of SRF cavities
have already been discussed in the above section while discussing the importance of
phonon peak in the thermal conductivity of Nb. Also note that as the thermal conductivity
of Nb is improved due to enhanced phonon peak, the importance of Kapitza conductance
on the performance of SRF cavities becomes even more prominent. Even with such a
substantial improvement of Kaptiza conductance after annealing has been demonstrated,
yet the data found in this study for the Kapitza conductance is on the lower end of the
wide spread of the data available in the literature for Nb. Hence room for further
improvement in the Kapitza conductance still exists. Also, further study is required to
point out the exact parameter that changed during the process of annealing and resulted in

the improvement of Kapitza conductance.

114

 

7 Conclusions

7.1 Effect of Phonon Peak in Nb Thermal Conductivity

The performance of the defect free high frequency SRF cavities operating at
temperatures below 2 K is greatly influenced by the thermal conductivity of Nb close to
the operating temperatures. At these temperatures, phonons are the dominant energy
carriers of energy and the microstructure of the bulk Nb becomes very important in
determining the shape of thermal conductivity curve. This dissertation revealed that the
strains induced through plastic deformations reduced the thermal conductivity of Nb by
80% at 2 K. This resulted in the loss of phonon peak. Low temperature annealing (at 750
°C for 2 hours) found to be insufficient to recover the lost phonon peak. However,
titanification done at ~1300 °C restored the lost phonon peak near to its as received
values.

The actual cavities are fabricated through a deep drawing process in which plastic
deformations are induced to shape the cavity according to the design requirements. The
absence of phonon peak in such cavities may cause their performance degradation.
Simple numerical computations performed in this thesis reveal that the effect of recovery
of the lost phonon peak along with improved Kaptiza conductance resulted in ~50 %
improvement in the applied magnetic fields for the case of 1.3 GHz cavity operating at 2
K. Thus the need for titanification of cavities is found to be the single most necessary
step in the preparation of high frequency cavities resulting in improved performance

through enhanced phonon peak and significantly increased Kapitza conductance.

115

7.2 Effect of Surface Roughness on Nucleate Boiling Heat Transfer Coefficient

Surface rouglmess is found to increase the heat transfer by reducing the superheat of
Nb surface in its pool boiling nucleate heat transfer regime. The effect of the large
surface index is more pronounced in convection heat transfer regime of liquid helium
than in its nucleate boiling regime. More experiments are needed to verify this
observation over extended calibration ranges to span the whole nucleate pool-boiling
regime of liquid helium.

The effect of the nucleate boiling heat transfer coefficient is found to decrease the
AR, of the cavity by 100 % thereby proportionally increasing the Q0 of the cavity at the
break down magnetic fields. But it plays relatively small role in improving the break
down magnetic fields since the fields are limited by the critical magnetic field. However,
as discussed in the last section, the use of constant A(O) may be a poor approximation at
4.2 K in relation 1-2 for R,. This may lead to a different analysis and thus needs more
theoretical research.

7.3 Effect of Kapitza Conductance

The effect of the large surface index (increased surface area) on Kapitza conductance
remained inconclusive possibly due to other effects influencing the heat transfer. These
include, but are not limited to, the effect of surface strains, BCP etching, or the
distribution of crystal orientation on the surface on the Kaptiza conductance. Although,
both the low as well as the moderate temperature annealing has shown a consistent
increase in Kapitza conductance, however, the exact cause of this increase is still
undetermined. More research in this direction is needed to find the exact cause of

increase in Kapitza conductance after annealing.

116

 

 

7.4 Looking Ahead: Large Phonon Peak and Large Kapitza Conductance

Back in 2001-2002, before the first step towards this research was taken, the role of
the thermal parameters seemed to be exhaustively investigated and well known to the
SRF community. Not much room for improvement in the performance of SRF cavity was
talked about through improvement in thermal parameters, namely the thermal
conductivity of Nb and its interface heat transfer coefficient with He. In this study, effort
has been made to change this view by opening up new challenges for fiirther work in this
direction.

Enhancement of phonon peak in polycrystalline Nb could possibly be realizable
either through high temperature (1800 to 1900 0C) annealing or for a longer duration at
moderate temperatures. The use of unstrained single crystal / large grain Nb also seems to
be a promising material to look for a considerably more enhanced phonon peak.
Sometimes the lattice imperfections even in these single crystal / large grain material may
not show the enhanced phonon peak. Moderate temperature annealing of such material
may be necessary to bring the phonon peak to prominence.

Similarly, the domain of Kapitza conductance still pose many unanswered questions
for which further research is needed. As discussed earlier, the effect of surface strains and
/ or crystal orientations on the surface need more attention to understand the effect of

increased Kapitza conductance, as seen in this study through titanification process.

117

l
“V

8 Appendices

 

118

8.1 Appendix A Pool Boiling Heat Transfer

Boiling is a phase change process in which vapor bubbles are formed either on a
heated surface or in a superheated liquid layer adjacent to the heated surface. It differs
from evaporation at gas-liquid interfaces because it also involves creation of these
interfaces at discrete sites on the heated surface. Nucleate boiling is a very efficient mode
of heat transfer, and is used in various energy conversion and heat exchange systems.
Pool boiling refers to boiling under natural convection conditions, whereas in flow
boiling, liquid flow over the heater surface is imposed by external means.

Boiling is a complex process. Because the process is complex and there are so many
heater and fluid variables that interact, complete theoretical models have not been
developed to predict the boiling heat fluxes as a function of heater surface superheat.
Figure A.l shows, qualitatively, the boiling curve [44] (i.e. dependence of the wall heat
flux, q, on the wall superheat on a surface submerged in a pool of saturated liquid). The
wall superheat, AT, is defined as the difference between the wall temperature and the
saturation temperature of the liquid at the system pressure. The plotted curve is for a flat
plate or a horizontal wire to which the heat input rate is controlled. As the rate of heat
input to the surface is increased, natural convection is the first mode of heat transfer to
appear in a gravitational field.

At a certain value of the wall superheat (Point A), vapor bubbles appear on the heater
surface. This is the onset of nucleate boiling. The bubbles form on cavities or scratches
on the surface that contain preexisting gas/vapor nuclei. Alter inception, a dramatic
increase in the slope of the boiling curve is observed. In partial nucleate boiling,

corresponding to region II (curve AB) in Figure A. l , discrete bubbles are released from

119

 

 

randomly located active sites on the heater surface. The density of active sites and the
frequency of bubble release increase with increasing wall superheat.

The transition from isolated bubbles to fully developed nucleate boiling (region 111)
occurs when bubbles at a given site begin to merge in the vertical direction. Vapor
appears to leave the heater in the form of jets. The condition of jet formation also
approximately coincides with the merger of vapor bubbles at the neighboring sites. A
small change in the slope of the boiling curve can occur upon transition from partial to
fiilly developed nucleate boiling. The heat flux on polished surfaces varies with wall
superheat roughly as

q ~ ATm (A-l)

where m has a value between 2 and 3 “7].

120

 

 

o 6- G
/’“’“{Sa‘9€3°9 335%“;ch 03%” 0 9 53C

. ” ’ ’ ’ ’ ’ ’ ’
t‘W “3991111 transrtion region 111 regionIV region V
region 1 partial

 

at 3 fully developed transition film
conrvaistgfigln Egg? i nucleate boiling boiling boiling
/

\3\1v//.

oooooooooooooooooooooooooooooooooooooooo

10g :1 —>
x
"flit.” IF:

 

 

 

 

 

 

log AT -—->

Figure A.1: Typical boiling curve, showing qualitatively the dependence of the wall
heat flux, q, on the wall superheat, AT, defined as the difference between the wall
temperature, T“, and the saturation temperature, T,.., of the liquid. Schematic
drawings show the boiling processes in regions l—V. These regions and the transition
points A—E are discussed in the text. (Adapted from Burmeister '4‘").

The maximum or critical heat flux, qmax sets the upper limit of fully developed
nucleate boiling for safe operation of equipment. Afier the maximum heat flux is reached
most of the surface is rapidly covered with vapor. The surface is nearly insulated and the
surface temperature rises very rapidly. When the rate of heat input is controlled the
heater surface passes quickly through regions IV and V (see Figure A.1) and stabilizes at

pomt E. If the temperature at E exceeds the melting temperature of the heater material

121

the heater will fail (burn out). The curve ED (region V) represents stable fihn boiling, and
the system can be made to follow this curve by reducing the heat flux fiom qmax.

In stable film boiling, the surface is covered with vapor film, and liquid does not
contact the solid. On a horizontal surface the vapor release pattern is governed by Taylor
instability of the vapor-liquid interface. With a reduction of heat flux in film boiling, a
condition is reached when a stable vapor film on the heater can no longer be sustained.
Heat flux and wall superheat corresponding to the condition at which vapor film collapse
occurs are referred to as the minimum heat flux qmin, and the minimum wall superheat
ATmin, respectively. Upon collapse of the vapor fihn, the surface goes through regions IV,
IH, and 11 very rapidly and settles in nucleate boiling. Region IV, falling between
nucleate and film boiling, is called transition boiling, which is a mixed mode of boiling
that has features of both nucleate and film boiling. Transition boiling is unstable, since it
is accompanied by a reduction in the heat flux with an increase in the wall superheat. As
a result, it is difficult to obtain steady state data in transition boiling, except when the

heater surface temperature is controlled.

122

 

 

8.2 Appendix B Estimation of Measurement Errors
All measurements involve two types of errors or uncertainties. These are

0 Uncertainties arising from a random effects

0 Uncertainties arising from a systematic effects

Since the systematic effects have been dealt separately in appendix C and D, in this
section quantitative estimates of the uncertainty due to random effects are analyzed.

8.2.1 Thermal Conductivity Measurement Uncertainty

Thermal conductivity is approximated from the measurement of the temperature
difference between the two sensors when a known input electrical power is applied to the
heater. Mathematically,

k = (Q/A)*(l/AT) B-l
where, Q is the known electrical power input to the heater, 1 is the distance between the
two sensors, A is the cross-sectional area through which heat is conducted in the material
from the heater to the sensors and, AT is the difference of temperatures measured by the

two sensors. Thus, the uncertainty in thermal conductivity, k, can be expressed as

fatiflifieifiei ..
k Q 1 A AT ’

Each uncertainty in equation B-l arising from the individual parameters can be estimated

 

separately.
The uncertainty in the electric power to the heater, Q, can be estimated from the
measurements of both the applied voltage v and the corresponding current I as given in

equation B-3.

123

 

 

 

 

55401101

where 8,, and 81 are provided by the manufacturer of the power supply unit.

Similarly, the uncertainties associated with measuring the length between the sensors
and the cross sectional area through which heat is conducted, are related with the
measuring device, usually through a digital read out of a vernier caliper.

The uncertainty associated with the AT is given as

EAT :\/(8Tl)2 +(5T2 )2 B-4

where 877 and an are the uncertainties related with the measurement of the temperature

 

of any given pair of sensors, say sensor 1 and 2 respectively, through which the thermal
conductivity is measured. The primary source of uncertainty in the temperature
measurements of the carbon sensors is from interpolation polynomial used for the
calibration fit. Detailed accounts of this uncertainty and other contributory factors have
been documented earlier [39].
8.2.2 Interface Heat Transfer Measurement Uncertainty

The measurement of the heat flux across Nb-He interface is done using the following
equation:

q=Q/A=h(Ts-Tb) I B-5
where q is the heat flux through the interface, T, is the Nb surface temperature, and T, is
the bath temperature of liquid helium. As discussed in chapter 3, usually the heat transfer

in nucleate boiling regime of He-I is shown on a plot of heat flux applied against the

surface temperature rise above the ambient. Errors associated with this temperature rise

124

 

 

are only related with the uncertainty of finding the temperature rise of exposed Nb
surface above ambient i.e. AT,. The interface surface temperature (T,) of Nb is estimated
from the linear extrapolation of temperature T3 as measured by carbon sensor C3 and
temperature T2 as measured by carbon sensor C2. Refer figure 4.10 for the location of the
two sensors where C3 is shown closer to the interface. The linear extrapolation assumes
constant thermal conductivity between the sensors and its final form is given as.
T, = (1+Lr)T3 —(Lr)T2 B-6

where Lr is the ratio of distance 13, to that of distance 123. In this nomenclature 13, is the

distance between the carbon sensor C3 and the interface, and 133 is the distance between

carbon sensor C2 and C3.
Thus,
ATS = T, - Tb=(1 + Ll‘) T3 —(LI') T2 - Tb B-7

Now the uncertainty in AT, is given as

8M, =\/(1+ Lr)2 (an )2 + (73 - T2 )2(8L. )2 + (L02 (an )2 + (an, )2 8-8

where 8T2, 8T3, and 8Tb are the uncertainties associated with the measurements of

 

temperatures T2, T3, and Tb respectively and are discussed in section 7.2 above. Also, BL,

is the uncertainty associated with the measurement of Lr and is given as

 

2 2
Lr 135 [23

where 8.3,, and 8123 are the uncertainties associated with measuring the distance 13, and 123

respectively and are discussed in section 7.2 above.

125

 

 

8.3 Appendix C Parallel Heat Flow Losses

Instead of all of the heat going through the Nb sample from the heater, some of it may
leak into the base plate through the G-10 block. This is known as parallel heat flow and is
shown in figure C.1 along with its thermal circuit diagram. This parallel heat loss can be

estimated with the help of known thermal resistances of various components in each of

 

the thermal path. In steady state, we know '1
Qin = (Th ‘ Tb) / Rtot = A T / Rtot (C‘l)
where Th is the temperature of the heater.

The percentage of heat lost in the parallel path, QL, is then given by the ratio of the

 

 

resistances in the actual path to that of the resistances in the parallel path.
Mathematically, it is given by the equation
QL = Qp / QNb+Cu*100= (RNb+Cu / RP)*100
where, Rp is resistance of the parallel path and is given as
Rp = RG-10+ R81)“ R6 Rods+ RFlange
Table C1 shows that the heat loss from parallel path is less than 0.1% of the total heat

supplied by the heater.

 

126

Conflat

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

flan 0 ~ -
9 \L_ LH, L__]
C3
Nb
Stainless steel 02
rods Qty6 0 C1
Cu
[-“l 0-10 [L 7'
UBase plate ‘
(BP)
Nb + Cu
Qin Qout s"?
G-lO BP 6 Rods Flange __,_

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure C.1: Thermal circuit showing parallel heat leak.

Table C1: Percentage of heat loss calculation based on thermal circuit
given in figure C.l

Title Nb Cu G-10 BP Rods Flange

 

 

Resistance

 

(KM) 0.71 0.060 31.43 32.12 1410.3 23.92

 

 

 

 

 

Sub total

(KIW)

Heat Loss

0.77 1497.77

 

 

 

('lo) 0.05

 

 

 

 

127

8.4 Appendix D Estimation of Heat Leak into Stainless Steel Tube

Estimation of heat leak from the Nb sample into the stainless steel tube containing
liquid He at 2.0 K can be made through an assumption of infinite length, as shown in
figure D.1. Given the excellent cooling properties of super-fluid He, most of the heat
coming into the tube from the Nb sample would go be removed through convection into

the liquid helium. Therefore the assumption of infinite length fin is reasonable.

 
   

Conduction Heat
into the 8.3 Fin
(Qx+dx)

 
     
 
 
   
     
 

Infinite Length

  

Heat (Qx)
into the base
of the fin

 

Figure D.1: Estimation of heat leak into the stainless steel tube

128

 

Using the basic elemental approach to formulate the heat transfer equation by

neglecting radiation losses, as shown in figure D.l, gives

Qx _Qx+dx _ Qconv : O

Q.—(Q,. +5’gAx)—h..(2nr.>(T—T.)=o

 

 

dx
Canceling out the terms results into the desired differential equation [-1
d 26) h
, — s. (e) = 0
dx kt
where, @=(TS - T b) with boundary conditions as (a) 6) =0 atxzoo; and (b) -. j

 

(9:93 atx=0. Here T, is the base surface temperature of the fin and is approximately
equal to the temperature of the Nb-He surface interface and T, is the bath temperature.

Solving the above differential equation by applying appropriate boundary conditions, we

get

Qfin : Vhssksst P(T.'s _Tb)

where, h,,, k,s t and P are Kaptiza conductance, thermal conductivity thickness and the

perimeter of stainless steel tube, respectively.

 

Thus, using the average values of thermal conductivity of stainless steel and its
Kapitza conductance [37] at 2.0 K, the percentage of heat lost is estimated as :

K,, = 0.2 W/m/K, 11,, = 2*103 W/mz/K, thickness t = rz-rl = 0.001 m

i=0.112%

total

129

8.5 Appendix E Sensors Sensitivity Comparison

Table E1 shows a comparison between a typical carbon sensor with that of standard

germanium sensor [39].

 

 

 

 

 

 

 

 

Table E1: Comparison of sensitivity of carbon sensor with germanium [*1
I39! 4 .
sensor
Sensor Type Germanium Carbon g.
Temperature Coefficient Negative Negative
Monitor Display Resolution 100 m0 100 m 0 ij
Sensor Sensitivity at 4 k -117 Q / K -572 Q / K
2 K -930 Q /K -10538 Q /K
Monitor Display 4 K 854.0 ,1: K 174.0 ,u K
Resolution w.r.t. 100
2K 107.5 pK 9.4 ,uK
mQ

 

 

 

 

 

 

 

However, due to high sensor resistance, self-heating could be a possible problem,

especially at temperatures below 1.9 K. Usually a high valued resistor is placed at room

 

temperature in parallel with the carbon sensor to overcome this problem, but it comes
with a price of a small reduction (~ <10 % at 4.2 K) in the sensor sensitivity. This
decrease in sensitivity is insignificant to pose any concern for the steady state

measurements in the described experiment apparatus.

130

Specification Sheet for Nb Samples

8.6 Appendix F

 

 

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131

8.7 Appendix G Surface Roughness Parameters
8.7.1 Introduction
For the discussions that follow, Z(x,y) is the function representing the height of
the surface relative to the best fitting plane, cylinder, or sphere. Note that a used in the
following integral expressions implies that the integration is performed over the area of

measurement and then normalized by the cross-sectional area “A” of the measurement

[48].
L
”0200 y)dxdy E i? jy Z (x. y)dxdy
0 0

8.7.2 Ra, The Roughness Average
It is the arithmetic average of the absolute values of the surface height deviations

measured from the best fitting plane, cylinder or sphere. Ra is described by:

R. = jjJZ(x,y)|dxdy

Historically, Ra was one of the first parameters used to quantify surface texture.
Most surface texture specifications include Ra either as a primary measurement or as a
reference. Unfortunately, Ra may be misleading in that many surfaces with grossly
different features (e. g., milled vs honed) may have the same Ra, but function quite

differently [48].

132

 

 

 

 

 

1(3)

 

 

 

 

 

Profile

 

 

Absolute Value 91"ther

Figure G.l: Profiles are shown above for simplicity for the understanding of basic
definition of Ra. When evaluating the 3D parameters the various surface functions
are understood to apply to the complete 3D dataset. Adapted from Cohen

133

 

Ra only quantifies the “absolute” magnitude of the surface heights and is
insensitive to the spatial distribution of the surface heights. Ra is also insensitive to the
“polarity” of the surface texture in that a deep valley or a high peak will result in the
same Ra value. Despite its shortcomings, once a process for forming a surface has been
established, Ra may be used as a good monitor as to whether something may have

changed during subsequent production of the surface [48].

8.7.3 Rq, The Root Mean Square (rms) Roughness
It is the rms (standard deviation) or “first moment” of the height distribution, as

described by:

Rq = j I (Z (x. y))2 dxdy

 

Rq, or the ms of the surface distribution, is very similar to Ra and will usually
correlate with Ra. Since the surface heights are “squar ” prior to being integrated /
averaged, peaks and valleys of equal height / depth are indistinguishable. As for Ra, a
series of high peaks or a series of deep valleys of equal magnitude will produce the same
Rq value. The Rq value is also insensitive to the spatial distribution of the surface
heights, in that two very high peaks will contribute the same to Rq whether the peaks are
close to each other or separated over the measurement field. The Rq parameter is
typically used in the optics industry for specifying surface finish, since various optical
theories relating the light scattering characteristics of a surface to Rq have been

developed [48].

134

 

 

 

 

1.50

1.00

0.50

200

- 0.50

 

 

 

 

- 1.00

 

Profile

 

 

 

 

 

 

 

 

 

 

Figure G.2: Profiles for Rq are shown above for simplicity. When evaluating the 3D
parameters the various surface functions are understood to apply to the complete

3D dataset. Adapted from Cohen [48'

135

8.7.4 Rp, Rv, Rt, Rpm, Rvm, Rz

Rp, Rv, and Rt are parameters evaluated from the absolute highest and lowest points
found on the surface. Rp, the maximum peak height, is the height of the highest point.

Rv, the maximum valley depth, is the depth of the lowest point. Rt, the maximum height
of the surface, is found from Rp — Rv.

The Rpm, Rvm, and R2 parameters are evaluated from an average of the heights and
depths of a number of extreme peaks and valleys. Rpm the average maximum peak
height is found by averaging the heights of the ten (10) highest peaks found over the
complete 3D image. Rvm, the average maximum valley depth is found by averaging the
depths of the ten (10) lowest valleys found over the complete 3D image. R2, the average
maximum height of the surface is then found fi'om Rpm-Rvm. Note that in determining
the peaks and valleys, the analysis software eliminates a grid of 11 x 11 pixels around a
given peak / valley before searching for the next peak / valley, thus assuring that

localized, separated peaks / valleys are found [48].

 

 

Typical surface demonstrating (Ire various
peak and valley structures

 

 

 

Figure G.3: Peak and valley structure of a typical surface is shown. Adapted from
Cohen “8'

136

8.7.5 Application
Since Rp, Rv, and Rt are found from single points, they tend to be unrepeatable
measurements. Thus when using these three parameters, one must be sure to make
multiple measurements, consistent with the observed variations to obtain a statistically
significant result. If the surface measurements are filtered prior to analysis (e.g. with a
low pass filter), the results may be more repeatable. Rpm, Rvm and R2 may be used to
characterize the extreme features of a surface, with Rz being a nominal measure of the
“Peak-to-Valley” range of the surface. Typical applications for R2 may include sealing
surfaces and coating applications. Rpm may find application when considering surfaces
that will be used in a sliding contact application. Rvm may find application when valley
depths relating to fluid retention may be of concern such as for lubrication and coating

systems [48].

137

 

 

10.

ll.

12.

13.

14.

15.

l6.

17.

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