Condensational Droplet Growth in Rarefied Quiescent Vapor and Forced Convective Conditions

Home , Knudsen number

A dissertation submitted to the

Graduate School

of the University of Cincinnati

in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

in the School of Dynamic Systems

of the College of Engineering

Sushant Anand

M.Tech, Indian Institute of Technology Kharagpur, 2005

B.Tech, Indian Institute of Technology Kharagpur, 2005

Committee Chair: Sang Young Son, Ph.D.

Frank M. Gerner, Ph.D.

Milind Jog, Ph.D.

Jason Heikenfeld, Ph.D.

ABSTRACT

Multiphase Heat transfer is ubiquitous in diverse fields of application such as cooling systems, micro and mini power systems and many chemical processes. By now, single phase dynamics are mostly understood in their applications in vast fields, however multiphase systems especially involving phase changes are still a challenge.

Present study aims to enhance understanding in this domain especially in the field of condensation heat transfer. Of special relevance to present studies is study of condensation phenomenon for detection of airborne nanoparticles using heterogeneous nucleation. Detection of particulate matter in the environment via heterogeneous condensation is based on the droplet growth phenomenon where seeding particles in presence of supersaturated vapor undergo condensation on their surface and amplify in size to micrometric ranges, thereby making them optically visible. Previous investigations show that condensation is a molecular exchange process affected by mean free path of vapor molecules (λ) in conjunction with size of condensing droplet

(d), which is measured in terms of Knudsen number (Kn=λ/d). In an event involving heterogeneous nucleation with favorable thermodynamic conditions for condensation to take place, the droplet growth process begins with accretion of vapor molecules on a surface through random molecular collision (Kn>1) until diffusive forces start dominating the mass transport process (Kn<<1). Knowledge of droplet growth thus requires understanding of mass transport in both of these regimes.

Present study aims to understand the dynamics of the Microthermofluidic sensor which has been developed, based on above mentioned fundamentals. Using continuum approach, numerical modeling was carried to understand the effect of various system parameters for improving the device performance to produce conditions which can lead to conditions abetting

condensational growth. The study reveals that the minimum size of nanoparticle which can be detected is critically dependent upon controlling wall geometry and size, wall temperature, flow rate and relative humidity of nanoparticle laden air stream. Droplet growths rates and sizes have been predicted based on different models. The efficacy of the device under various conditions has been measured in terms of its ability to activate nanoparticles of different sizes.

Since the condensation mechanism is dependent upon the Knudsen regime in which droplets are growing via condensation, special consideration was made to understand their behavior in large Knudsen number conditions. For this purpose, ESEM was used to study condensation on a bare surface. Droplet growth obtained as a function of time reveals that the rate of growth decreases as the droplet increases in size. The experimental results obtained from these experiments were matched with theoretical description provided by a model based on framework of kinetic theory. Evidence was also found which establishes the presence of submicroscopic droplets nucleating and growing in between microscopic droplets for partially wetting case.

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Dedicated

Shri Vishwamitra Ji Maharaj

for teaching me the essence of life

And also to

Dr. S. K. Anand and Mrs. Neelam Anand.

for being most wonderful parents,

and role models for hard work, persistence and inspiration

ACKNOWLEDGEMENTS

It is said – ‘Life is a great teacher’ and the remarkable thing about life is, no matter how bad a student one might be, life teaches us enough to avoid flunking once a while. I suppose by proclaiming the above self-evident truth that summarizes my journey towards writing this dissertation, I can successfully join the ranks of individuals who think they are being deeply philosophical and profoundly wise. Apart from philosophy, my five years of studentship has empowered me with much coveted technical skills such as plumbing, assembling furniture, computer repair and getting access to free experimental equipment to the extent I can write manuals for dummies on them. I once watched a documentary on

Green Berets and in all fairness to the navy seals, I believe they can probably withstand the rigors of a

PhD program.

As a long journey comes to an end, I have to offer my deepest gratitude to my advisor Dr. Sang

Young Son. Dr Son has been strong, supportive and incredibly brave to allow me carry my

(mis)adventures in his lab. It has been a tremendous journey which began with an empty room and to what has now grown to be state of the art lab with very “cool and happening stuff”. Between the tremendous number of things I have broken or crashed, he has held a remarkably cool demeanor worthy of a Himalayan monk, allowing me to learn much from my own mistakes – which I must say have been quite a many. And yet he has allowed me to pursue independent work with great freedom and demonstrated his faith in me from time to time. Very importantly, he has allowed me to have open and frank arguments with him and still given me freedom of thought to hold my own views. His sense of perfection and diligence to approach a problem is something I will try to follow during rest of my career and times.

In the same vein, I wish to thank my doctoral committee – Dr Frank Gerner, Dr Milind Jog and

Dr Jason Heikenfeld for agreeing to be part of my adventure. Without their support, I could not have done what I was able to do. Although I could not study anything course under him, Dr Gerner has provided me with encouraging words in our encounters beyond the classroom. Dr Jog has provided with unflagging v

encouragement and his generosity with time to have discussions with me on my work has had huge impact in completion of this work. Dr Heikenfeld has been a role model for us to follow and he has been very generous to us by granting us access with his lab facilities whenever we have requested.

I have had amazing learning experience while being a student at University of Cincinnati. The faculty in the department has provided me with tremendous graduate education and the staff in our department has had great role in alleviating any predicaments on my way. I would like to express my gratitude to these individuals for their support and assistance. Several individuals deserve special mention for their contributions to this dissertation. Mechanical Engineering Department is very fortunate to have a committed person as Mr. Patrick Brown to handle the financial aspects for students. Largely due to the support provided by Ms Rhonda Christman, dealing with companies to procure any instrument was a cakewalk. I have had fortune of working under Mr. Jeff Simkins at the UC Clean Room and his affableness and his desire to help has cleared many of the hurdles I faced during my work. Ms Luree

Blythe has been amazingly supportive and ever helping to any of our cause. A large portion of my research was carried at Advanced Materials Characterization Center at UC under the managing guidance of Dr Doug Kohls. Dr Doug has been very kind to overlook many of my mistakes and provide me with chance to experiment with his equipment.

An anonymous master once said ‘When I find myself fading, I close my eyes and realize my friends are my energy’. I suppose many of them have over period of time held a deeper suspicion on my objectives for being so nice to them. Holding this dissertation to be my holy book, I am ready to confess –

“I did it for free dinner and/or coffee”. Nevertheless, all my friends have known this for a while and they still have sustained me and at same time have been blind to my idiosyncrasies. Finding such unwavering friendships to stand by you in troubled times is tough always, something which even Google will not be able to find. So I can decisively proclaim myself to be a better search engine.

I have had fortune of working with Dr Jae Yong Lee for over a period of three years. To say that I feel intimidated by his grasp on research would be an understatement. Together I have shared many

moments of insightful discussions and good laughs with him. For me he is a sensei who has reciprocated my naggings to him with help in framing my thoughts and experiments. I may not be Luke Skywalker, but he certainly is Yoda to everyone in our lab. Without his support and care, I would not have been able to overcome setbacks during my graduate studies. For all his assistance I shall remain ever grateful.

This long and perilous journey would have been incomplete without the support of two key individuals – Rainy Shukla and Dr Ratandeep Singh Kukreja. Rainy has been a guiding light and the most influential force during these five years. Together, we had amazing times of fun and intellectual dribble – which I am sure I will have fond remembrance for in years to come. In Ratan, I have found an elder brother and someone I can conspire with. I shall ever remain thankful to god for sending these amazing two individuals in my life.

Over these last five years I have been extremely fortunate to have had great labmates. Their support and graciousness to have frank discussions has been invaluable to this research. For this I shall be ever thankful to Sriharsha Pulipaka, Prakash Rapolu, Hylic Foo, Huayan, Michael Martin and Dr Jin

Young Choi.

I also extend my gratitude to my friends Shashank Chauhan, Jaspal Saini, Prahit Dubey, Shubham

Gilda, Sandeep Kaur, Jasman Kaur, Dr. Babita Baruwati, Sanchit Saxena, Harmandeep Kaur, Anand

Balasubramaniam, Prachi Rojatkar, Shrikant Pattnaik, and Neeraj Pathak for being friends and for the support they have lent me during my stay in Cincinnati. I am also thankful to my friends in India, specifically Rishi Thaper, Ankit Jain, Ravinder Singh and Chetan Garg. They have been across seven oceans, and yet their wishes and their faith in me has kept me strong at all times.

I had the fortune of having two of my brothers with me – Karan and Saurabh. I feel pride that lack of my support in kitchen has helped Saurabh become the good cook he is today. He has been a pillar

(and a large one at that) in supporting me through thick and thin. Whenever my mind would get adrift in agitations of daily problems, Karan would pester me with his questions and remind me on smallness of those problems compared with his persistent questioning.

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In the long list of counting my blessings, perhaps the single biggest event to unfold in my life would be having my wife Shruti in my life. She has given me 1000 reasons to look forward for the future and also reassured me that she will be there whenever I need her the most eg., while choosing the right color of tie that may get along with a shirt.

Lastly, I reserve to wish thanks to my mom and my dad for they have endured a great deal.

Sitting miles away and yet ever present with me, they have taught me lessons of great patience and wisdom. Charles Darwin might have proclaimed “Survival of the fittest”, however I suppose even those who are unfit can survive and succeed if they are fortunate to have such wonderful human beings as their family.

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TABLE OF CONTENTS

ABSTRACT………………...……………………………………………………………….………...... … i

DEDICATED TO... .…… .…… .…………………………………………………..…………………… iv

ACKNOWLEDGEMENTS....……………………………………………………..…………………… v

TABLE OF CONTENTS ………………………...………………………………….…...... …...... ix

LIST OF FIGURES ………………………………………….…………………...... ………...……… xiii

LIST OF TABLES …….…………………………………………..………………………………….. xvii

NOMENCLATURE…….……………………………………………………………….…………… xviii

Chapter 1: INTRODUCTION

1.1. Overview ………………………………………………………………………………………… 1

1.2. In this study ……………………………………………………………………………………… 5

Part 1: DROPLET GROWTH ON NANOPARTICLES IN CONVECTIVE FLOW CONDITIONS

Chapter 2: LITERATURE REVIEW

7.1. Nature of Condensation ………………………………………………………………………… 8

7.2. Droplet Growth kinetics on submicroscopic particles ………………………….……..….……. 12

7.3. Particle Codnensation Devices ………………………………………………..……………….. 15

Chapter 3: FUNDAMENTALS OF HETEROGENEOUS CONDENSATION

3.1. Condensation and Nucleation ………………………………………………………………... 20

Chapter 4: DROPLET GROWTH DYNAMICS ON NANOPARTICLES IN A VAPOR-GAS

MIXTURE …………………………………………………………………………………………...... 24

Chapter 5: CONDENSATION ON NANOPARTICLES IN MICROTHERMOFLUIDIC

SENSOR

5.1. Overview ……………………………………………………………………………………... 29

5.2. Design A: Experimental Setup and Device Preparation ………………...... 30

5.2.1. Condensation Chamber Preparation …………………………………...... 32

5.2.2. Computational modeling of thermofluidic sensor ………………………...... 33

5.2.3. Discussion of Results ………………………...... 38

5.2.3.1. Effect of Relative Humidity ………………………...... 38

5.2.3.2. Effect of Wall Heating ……………………….……………...... 40

5.2.3.3. Effect of Flow Rate ………………………...... 43

5.2.3.4. Effect of Chamber Size (Condensation Chamber Diameter) ……..…………... 44

5.3. Design B: Experimental Setup and Device Preparation ……..………...... 45

5.3.1. Condensation Chamber Preparation ………………………...... 45

5.3.2. Computational modeling of thermofluidic sensor ………………………...... 46

5.3.3. Discussion of Results ………………………...... 49

5.3.3.1. Effect of Flow Rate on Saturation and droplet growth ...... 50 x

5.3.3.2. Effect of Wall Heating ...... 57

5.3.3.3. Effect of Relative Humidity of inlet fluid ...... 60

5.3.3.4. Effect of Temperature of Incoming Fluid stream ...... 63

Chapter 6: CONCLUSIONS ...... 66

Part 2: DROPLET GROWTH IN RAREFIED ENVIRONMENT

Chapter 7: DROPLET SIZE MEASUREMENT IN RAREFIED ENVIRONMENT ...... 69

Chapter 8: LITERATURE REVIEW

8.1. Droplet Growth Kinetics on flat surfaces ...... 71

8.2. Accommodation Coefficient and its importance ...... 72

8.3. ENVINRONMENTAL SCANNING ELECTRON MICROSCOPY ...... 73

Chapter 9: EXPERIMENTAL STUDIES ON DROPLET GROWTH MEASUREMENT

9.1. EXPERIMENTAL SETUP ...... 75

9.2. SUBSTRATE PREPARATION ...... 78

9.3. EXPERIMENTAL METHODOLOGY ...... 78

9.3.1. Method of condensation ...... 80

9.3.2. Effect of Beam Heating ...... 81

9.4. RESULTS: DROPWISE GROWTH FOR SINGLE DROPLETS ...... 83

9.5. RESULTS: METHODOLOGY FOR DIAMETETIC DROPLET GROWTH ANALYSIS ..... 87

9.6. RESULTS: ADVANCING CONTACT ANGLE DURING DROPWISE CONDENSATION

…………………………………………………………………………………………... 92

9.7. RESULTS: DROPWISE GROWTH ALONG WITH COALESCNCE MECHANISM

………………………………………………………………………………………..…. 93

Chapter 10: DROPLET GROWTH MEASUREMENT: THEORETICAL BASIS ...... 97

Chapter 11: DROPLET GROWTH: SUBMICROSCOPIC VISUALIZATION ...... 103

Chapter 12: CONCLUSIONS ...... 106

Chapter 13: FUTURE WORK ...... 107

REFERENCES ...... 108

APPENDIX

A.1. Example ImageJ Macro for Droplet Diameter Measurement ...... 121

A.2. Methodology for measurement advancing contact angle using LBDSA plugin ...... 122

A.3. Material Properties and relations ...... 124

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LIST OF FIGURES

1. Figure 1. Formation of Clouds via heterogeneous condensation …………………….………..…. 2

2. Figure 2. Size Scale of nano-micro entities ……………………….……………………….….…. 3

3. Figure 3. Pathway of lung damage by particulate matter ingestion ………………………..…….. 3

4. Figure 4. Molecular Exchange phenomenon during vapor deposition process……………….… 20

5. Figure 5. Nucleation rate as a function of supersaturation for different contact angle ………..…. 22

6. Figure 6(a). Regional heat and mass transfer and (b). The conceptual drawing of mini

thermofluidic sensor for nanoparticle detection ………………………………..…….….….….. 31

7. Figure 7. Schematic of the model considered for computational modeling ……….….….…….. 34

8. Figure 8. Mesh Independence Test. Saturation Ratio versus Axial Location at the centerline (r=0)

……………………………………………………………………………………….…..…….... 36

9. Figure 9. Variation of (a) Saturation Ratio and (b) Droplet growth at centerline (r=0) with varying

relative humidity of incoming air flow ………………………………………………...…….…. 39

10. Figure 10. Variation of (a) Saturation Ratio and (b) Droplet Growth at centerline (r=0) with

varying base temperature at the wall …………………………………………………...…….…. 41

11. Figure 11. Droplet Heating at centerline along tube length ……………………….…..…….…. 42

12. Figure 12. Variation of (a) Saturation Ratio and (b) Droplet Growth at centerline (r=0) with

varying volume flow rate …………………………………………………………….……...….. 43

13. Figure 13. Variation of (a) Saturation Ratio and (b) Droplet Growth at centerline (r=0) with

varying diameter of the condensation chamber …………………………………….……….…. 44

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14. Figure 14. Schematic of the model considered for computational modeling for Design B….…. 46

15. Figure 15. Saturation Ratio Contours inside the device Heating (Tw=328 K,, RHin=0.5, Ti=298 K)

at r/R=0 ………………………………………………………………..…………………….….. 53

16. Figure 16. Variation of Saturation Ratio (a) for varying flow rate (Tw=328 K, RHin=0.5, Tin=298

K) at r/R=0 (b) at different radial locations along axial for flow rate of Re=139 (Tw=328 K,

RHin=0.5, Tin=298 K) ……………………………………………..…………………….…..…... 54

17. Figure 17. Variation of (a) Temperature at r/R=0 (b) Volume fraction of Saturation Ratio inside

the minichannel for flow rate of Re=139 (Tw=328 K, RHin=0.5, Tin=298 K) ………..…...….. 55

18. Figure 18. (a) Droplet growth and (b) associated droplet heating at r/R=0 for varying flow rate

(Tw=328 K, RHin=0.5, Tin=298 K) ……………………………………..………….…...….….. 56

19. Figure 19. Variation of (a) Saturation Ratio (b) associated droplet heating for varying wall

temperature (Re=139, RHin=0.5, Tin=298 K) at r/R=0 …………………………………….….. 58

20. Figure 20. Droplet growth for varying wall temperature (Re=139, RHin=0.5, Tin=298 K) at

r/R=0 ……………………………………………..………………………………………….….. 59

21. Figure 21. Variation of (a) Saturation Ratio (b) Temperature for varying relative humidity at inlet

(Re=139, Tw=328 K, Tin=298 K) at r/R=0 ……………………………………………..….….. 61

22. Figure 22. Variation of (a) droplet heating (b) associated droplet growth for varying relative

humidity at inlet (Re=139, Tw=328 K, Tin=298 K) at r/R=0 ……………………………...….. 62

23. Figure 23. Variation of (a) Saturation Ratio (b) Temperature for varying temperature of inlet

fluid stream (Re=139, Tw=328 K, RHin=0.5) at r/R=0 ………………………….………...….. 64

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24. Figure 24. Variation of (a) droplet heating (b) associated droplet growth for varying temperature

of inlet fluid stream (Re=139, Tw=328 K, RHin=0.5) at r/R=0 …………………………….….. 65

25. Figure 25. Mean free path as a function of chamber pressure …………………………….……. 69

26. Figure 26. Experimental Setup for Condensation Observation in ESEM …………………….. 75

27. Figure 27. Saturation Vapor Pressure – Temperature chart for ESEM ………………...….….. 76

28. Figure 28. Copper stage for Observing droplet condensation on an inclined surface ……..….. 78

29. Figure 29. Temperature Profile during Condensation Process …………………………….….. 80

30. Figure 30. Example of Temperature Profile during Condensation Process (Pressure. 4.4 Torr). 81

31. Figure 31. Thermocouple tip ……………………………………..………….……….....….….. 82

32. Figure 32. Beam Heating with Beam Voltage for line mode and spot mode ………………….. 82

33. Figure 33. Beam Heating with respect to spot mode (25 keV) …………………………….…... 83

34. Figure 34. Condensed droplets on Si surface (4.2 Torr) (Constant Substrate Temperature.-0.5 oC,

Varying vapor pressure mode) ………………..…………………………………………….…... 86

35. Figure 35. Droplet on a surface ………………..…………………………………………….….. 87

36. Figure 36. Droplet diameter growth for individual droplets occurring in same frame with different

initial diameters at 4.2 Torr (Constant substrate temperature, varying vapor pressure mode) …. 89

37. Figure 37. Droplet growth for individual droplets ………………..…………………….….….. 90

38. Figure 38. Droplet growth for individual droplets evaluated for overall growth ……...………. 91

39. Figure 39. Condensed droplets on Si surface using side view (4.2 Torr) ……....………..…….. 92

40. Figure 40. Average contact angle of droplets measured during their isolated and coalescence

growth regime ………………..………………………………………………………….….. 93

41. Figure 41. Condensing droplet at 4.2 Torr visualized at 90o inclined surface (Constant substrate

Temperature, Varying vapor pressure mode) ………………..………………………….….. 94

42. Figure 42. Droplet Growth rate at 4.2 Torr (Constant Substrate Temperature, Varying vapor

pressure mode) ………………..……………………………………….……………..….….. 95

43. Figure 90. Droplet on a surface ……………………………..………………………….….. 98

44. Figure 44. Theoretical and experimental droplet growth analysis ………………..……..... 101

45. Figure 45. Image Visualization at 6500X between visible droplets ………………..…..... 104

46. Figure 46. Image Visualization at 12,000X between visible droplets ………………..…... 105

47. Figure 47. Image Visualization at 25,00X between visible droplets ………………..……. 105

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LIST OF TABLES

1. Table 1: Boundary Conditions for Design A prototype ………………………….…………………. 34

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Nomenclature a Radius of Droplet [m]

A Area [m2]

Cp Specific Heat capacity at constant pressure [J/(kg*K)]

d pore Pore Diameter of porous tube [m]

2 DAB Diffusion coefficient between species A and species B [m /s]

D flux Diffusive Flux from Wall (Evaporation Rate)

) Djk jk component of the multicomponent Fick diffusivity

Djk jk component of the multicomponent Maxwell-Stefan diffusivity matrix= DAB

G Gibbs Free Energy [W/m] h Heat Transfer Coefficient [W/(m2K)]

I Identity Matrix

J Nucleation Rate k Thermal Conductivity [W/(K•m)]

kBo Boltzmann Constant [J/K]

Kn Knudsen Number = laA∞

KR Kelvin Ratio

L Enthalpy of vaporization [kJ/kg]

lA∞ Vapor mean free path [m]

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m Mass of a molecule [kg] m& Rate of mass change [kg/s]

M Molar Mass [kg/mol] n Number of molecules p Pressure [Pa]

pA Partial pressure of vapor in the gas surrounding the droplet [Pa]

pS Saturation vapor pressure at temperature T [Pa]

pSA Saturation vapor pressure at droplet surface [Pa] = pTSd( )

P Porosity of porous tube q Heat flux vector [m/s]

Q Volume Flow Rate [m3/s] r Radial component of cylindrical coordinate system

R Radius of the Evaporation-Condensation Tube (ECT) [m]

RH Relative Humidity

Rg Gas constant [J/(mol•K)] rad Molecular radius [m]

S Saturation Ratio (or Relative Humidity)

T Temperature [K] u Velocity vector [m/s]

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V Velocity [m/s] v Velocity vector of species [m/s] x Mole Fraction

X Radius of Droplet [m] z Axial component of cylindrical coordinate system

Greek Symbols

αm Condensation coefficient

β Effective area constant

ρ Density [kg/m3]

µ Dynamic Viscosity [Pa*s]

υ Specific Volume [m3/kg]

φ Fuchs Correction Factor

σ Surface tension [N/m]

ω Mass fraction

Subscripts

0 Initial condition

∞ Conditions far away

A Species 1 (Vapor)

B Species 2 (Air)

c Continuum regime d Droplet/Liquid eff Effective i Inlet k Kinetic regime l liquid mix Mixture s Saturation condition t Total w Wall

Superscripts

* Conditions at droplet surface considering Kelvin Effect

M Multicomponent thermal diffusion coefficient

T Transpose of matrix

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CHAPTER - 1

INTRODUCTION

1.1 OVERVIEW

Phase change phenomenon has been a primary focus of scientific studies for decades. The physics behind phase transitions and associated thermal and flow processes occurring at macro scales are well understood. With emergence of micro and nanotechnology, it has been possible to scale down the systems to sizes matching the molecules contained in them. It has been demonstrated that physical size of systems has different impact on the physics of flows. The microscale phenomenon has been found to behave differently than the well established theories for macro scales [1, 2]. Pertaining to phase transitions, a great deal of research is being done to understand boiling heat transfer at microscales, while phase change involving condensation has received comparatively less attention.

Condensation in two-phase flows plays a major role in a number of technologies critical to diverse applications such as manned and unmanned space missions, fuel cells, phase and particle separators, sensors, and thermal management systems. On small scales, condensation represents the dominant phase change phenomena in phase heat transfer systems (e.g., Heat Pipe, Capillary Pumped Loop, and HVAC).

Due to the large surface to volume ratio in micro devices, the molecular effect influences microchannel flow motion and the related heat transfer phenomena including condensation. Capillarity is one of the major factors dominating thermal and mass transport phenomena in micro/minichannel systems. Since capillary force is proportional to the inverse power of channel hydraulic diameter, it plays much greater role in microsystems as compared to macroscale systems.

The behavior of condensation dynamics as described above is in fact widely observed in our surroundings beyond the realm of engineering devices. Condensation plays vital role in environmental events such as climate change or aerosol formation where condensation of vapor molecules on particulate matter is responsible for formation of clouds, haze and ice crystals via precipitation (Figure 1).[3] 1

Particulate Matter (1 nm - 5µm) Vapor Air P sat >1 Pd

Heterogeneous Condensation Process

Cloud Formation

Figure 1: Formation of Clouds via heterogeneous condensation

It was Coulier who showed that condensation occurs in unfiltered air more readily than filtered air upon adiabatic expansion in 1875.[3] The aerosol particles in unfiltered air were too small to be seen by naked eye. Since then, researchers have been trying to devise instruments for qualitative classification of airborne particulate matter. The earliest designs such as Wilsons Cloud Chamber and Aitkens condensation counters etc. aimed at analyzing effect of vapor condensation under different circumstances.

They were able to realize that supersaturation of vapor is a necessary step for a vapor to condense. They were able to identify that condensation occurred in various modes – condensation on neutral molecules

(or heterogeneous condensation), condensation on ions and condensation of vapor molecules on their own

(or homogeneous condensation). It was established that particulates if present in a saturated vapor, initiate heterogeneous condensation before the onset of homogeneous nucleation. Moreover, observation of visible mist formations in such devices was due to condensation which resulted in deposition of vapor on particulate matter and thereby altering particulates to sizes where they are no longer optically invisible.

Condensation on particulate matter is of special interest for many reasons.[4] Particulate matter is typically a motley of different components such as heavy metallic ions and compounds, diesel exhaust particles (DEP), soot, and organic particles (bacteria, viruses) etc.[5] Moreover, these particulate matter

2 comprise of a range of size distribution from nano-micron sizes (Figure 2) which further increases the potent power of these material in terms of their chemical activity.

0.001 0.01 0.1 1 10 100 µm Particle 1 10 100 1000 10000 100000 nm Diameter Diesel Smoke Combustion Products Beach Sand Tobacco Smoke Pollen Carbon Black Bacteria Paint Pigment Human Hair Virus Asbestos

Figure 2: Size Scale of nano-micro entities

Each of these components has potential to cause adverse health effects and as a result these have been center of epidemiological studies for last few decades.[6] The primary means by which these particulate matters enter human body is inhalation (Figure 3).

Figure 3: Pathway of lung damage by particulate matter ingestion [7]

Many of the constituting components are carcinogenic in nature and inhalation over a prolonged period of time is thus detrimental to health. Hence detection and differentiation of the causative components of particulate matter is necessary for proper understanding of their health effect. In recent years, numerous human studies have suggested that particles have greater adverse health effects as their size decreases; these effects may be more closely related to number concentration than to mass [8, 9].

Particles smaller than 100 nm, referred to as ultrafine particles (UFPs) comprise the vast majority in numbers of airborne particles. UFPs are particularly important in environmental health due to characteristics associated with concentration, surface area per unit mass and high pulmonary deposition efficiency [10]. Furthermore with current scientific trends of introducing specifically engineered nanoparticles such as carbon nanotubes, nanowires in daily usage items like house paints, clothes etc, the threat towards health has reached even inside our houses.[11, 12] Although the particle size is the most important parameter related to the toxicity and deposition upon inhalation, the chemical characteristics undoubtedly influence the biological effect upon adsorption in human body.[13] Therefore, these increasing concerns about the health impacts of UFPs demand more precise and personal monitoring for both environmental exposure such as that associated with exposures to air pollutants as well as many workplace exposures.

However, ultrafine particles (Dp = 1-100 nm) do not have sufficient scattering to be efficiently detected using optical methods with visible wavelength (400-700 nm). Extremely large optical source strengths are required to provide adequate signal-to-noise-ratios for even the most sensitive optical detectors, rendering this approach unfeasible for conventional applications, and thus inapplicable for incorporation into personal sensors or mobile devices. A method is therefore required to enhance the scattering properties of these particles. To overcome this predicament, condensation is the only technique available for detecting neutral atmospheric airborne particles that are too small for optical methods.

In fact, condensation of supersaturated vapor has been used for more than a century to grow ultrafine aerosol particles to sizes that can be detected optically.[14] Condensation Particle Counters

(CPC) use these principles for activation of nanoparticles to sizes where they become optically detectable.

Since the number concentration of the input particles is preserved, these artificially grown particles can are used for condensate counting to evaluate number concentration of ultrafine aerosol particles.

Condensate counting is accomplished by first passing the incoming particle stream through a supersaturated vapor environment. Under these conditions, the particles serve as sites for heterogeneous nucleation, and rapidly grow to near-uniform micrometer sized droplets which can be readily detected by optical means [15-17].

Despite progress in devising means to utilize our understanding of condensation phenomenon, there is a lack of pertinent explanation to describe physics behind these transformations. There is no satisfactory description which predicts the behavior of what happens when a gal molecule approaches the surface of a liquid or solid.[18] The inception of condensation, condensation rate (or droplet growth rate), mode of condensation dramatically change depending on the nature liquid/solid surface. When condensation surface area is smaller than mean free path of vapor, as is the case for condensation on environmental ultrafine particulate matter, then surface energy and morphology can significantly influence the rate of condensation. However, due to a fundamental lack of understanding of nanoscale condensation, the engineering design of condensation-based devices must rely upon intuition, which often gives biased result, instead of solid engineering design tools. Thus there is requirement of an extensive observation based study to build theories which can predict the various aspects of condensation.

1.2 IN THIS STUDY

In light of above discussion, the present research utilizes a combination of different methods in studying phase change heat transfer from continuum scales to nano-scales. The domain of study involves phase change occurring over airborne nanoparticles and flat surfaces; and water condensational growth via heterogeneous nucleation on them.

In order to gain a deep insight into the physics associated with heterogeneous nucleation, a novel miniature channel system has been developed at MicroThermofluidics Laboratory. Such a device can provide with an opportunity to carry a more in-depth study to provide insights in heterogeneous

5 condensation process. Present work provides evidence of successful operation of a device based on this principle wherein it is possible to observe heterogeneous condensation on very small number densities of incoming stream of nanoparticles.[15] Since droplet condensation and wall condensation effects play central role in development of the sensor, this work presents theoretical principles and experimental data pertaining to these phenomena. To better understand the complex issues related to the heat and mass transfer in the thermofluidic sensor and the associated parameters affecting droplet growth, a numerical approach was undertaken. Computational modeling was carried using COMSOL® for single phase and multi species.

Since most of the condensation phenomena initiates at submicroscopic scale where the size of droplet is comparable to the collision distance between vapor molecules, condensation behavior in this regime was studied experimentally in analogous conditions using Environmental Scanning Electron

Microscopy. Electron imaging instead of light optics was used as much higher spatial resolutions can be achieved through this method and visualization capabilities are comparable to the mean free path of vapor molecules inside the instrument. Moreover, Environmental Scanning Electron Microscope (ESEM) has capabilities of introducing phase change which can be then studied in absence of non-condensable gases and shear effect of vapor velocities. The novel approach has revealed significant details on how droplets nucleate and grow at submicroscopic and thus has been identified as having vast potential in studying many underlying phenomenon at submicroscopic scales.[19, 20]

PART 1:

DROPLET GROWTH ON NANOPARTICLES IN CONVECTIVE FLOW

CONDITIONS

CHAPTER - 2

LITERATURE REVIEW

Phase change accompanying conversion of a saturated or superheated vapor in presence of subcooled surfaces is one of the most common occurring phenomena in nature. The mode of phase change which follows such a transformation is dependent upon surface properties like as of contact angle and thermodynamic conditions of the system. The understanding of phase change dynamics requires a broad understanding of many associated phenomenon such as nucleation, droplet growth kinetics and various factors affecting the behavior of phase change phenomenon such as surface energy, surface morphology.

As such, these phenomenon were studied during the period of this study and a brief literature review is provided below.

2. 1. Nature of Condensation

Efforts to understand phase transitions from vapors into liquids have been made since decades. After it was understood that airborne particulate matter was responsible for precipitation of clouds, scientists have focused their attention on understanding the physics behind condensation on heterogeneous surfaces.

Volmer and Flood carried experiments on liquid to solid phase transitions on heterogeneous surfaces in

1934.[21] They proposed that for a vapor species to condense on a non-wetting plane surface, the vapor pressure should be higher than saturation vapor pressure at surface temperature. Rate of nucleation as observed in their experiments was described semiqunatitavely by the nucleation theory provided by

Becker-Doring.[22] The theory proposed by them provided a semi-quantitative estimate on rates of nucleation. It was established that a minimum energy barrier should be overcome before a species can condense on a surface.

Vonnegut (1947) observed the effect of seeding particle size on their capacity to cause nucleation in a supersaturated system.[23] He observed that 1 μm nuclei triggered phase change earlier than 10 nm

8 nuclei. Turnbull (1949) studied the effect of surface roughness and considered nucleation in conical and cylindrical cavities.[22] . He noted that thermal history has a large impact on the transformation of phases and he was able to find a relation between superheating and subcooling. Fletcher (1958) performed experiments for studying effects of particle sizes, and surface properties. [24] He then presented a theoretical model based upon surface energy characteristics of interacting species. The nucleation rate has been found to have an exponential dependence on free energy and temperature and is given by

c (1) JJ=−Δ0 exp() GkT / Bo

From his theory he was able to establish critical radius beyond which droplet growth would occur and critical free energy required for achieving it. The nucleation theory as established by Becker-Doring-

Fletcher is now referred to as Classical Nucleation Theory (CNT). Twomey performed experiments to test the CNT and found it to be successful matching with experimental observations.[25] Koutsky et al (1965) carried experiments on various substrates to study condensation on insoluble nuclei. [26] They established the condensation was strongly dependent on supercooling. Hydrophilic surfaces were found to be more favorable for condensation as compared to hydrophobic surfaces. Critical degree of supersaturation required to initiate nucleation on hydrophilic surfaces was much less than that required for hydrophobic surfaces. It was also found that at rough sites, the density of droplet formation was higher than compared to smooth surfaces. With surfaces having significant curvature such as insoluble particles, the presence of a curvature modifies the vapor pressure necessary for a species to condense on them [27, 28]. This is termed as ‘Kelvin Effect’ and it establishes that a minimum degree of supersaturation is required for condensation to occur on a particle [5]. Thus the dynamics of condensation on particles are governed by the size range of the particle in consideration.[29, 30]

Mahata and Alofs (1975) ascertained that adsorption plays a smaller role in condensation on surfaces. [31] Ching-Chen et. al (2000) used submicrometer size particles as condensation nuclei and passed them through supersaturated vapor. It was observed that the observations had significant deviation from theoretical predictions using CNT.[32]

It is well known that many phenomena of common occurrence such as spreading of liquids, formation of droplets or thin films are governed by surface free energy interaction between the liquid and solid surface. Phase change processes involving liquid solid interaction such as condensation explicitly depend upon these surface energy interactions. Accurate calculation of surface energies is a tedious task involving computation of intermolecular interactions, so contact angle serves as a good indicator of defining the dynamics of a liquid-gas-solid system. Based on contact angle the liquid forms on a surface, condensation can be described as filmwise, dropwise or mixed condensation.[33] For hydrophilic to super hydrophilic surfaces, a thin sheet of liquid film is formed upon condensation. This film offers an additional resistance for heat transfer between the solid surface and the gas and thereby limiting the amount of heat transfer that can take place across the wall. For hydrophobic surfaces there is instead a large bare surface in between condensing droplets which gives them a heat transfer coefficient which is 2-

10 times more than filmwise condensation.[34] With a large nucleation site density (~ 108/cm2 on smooth surfaces), it is possible to achieve high heat flux of 170-300 kW/m2 in dropwise condensation mode.[34],[35]

The extent of advantage that dropwise condensation has to offer makes it a more desirable mode of heat transfer as compared to filmwise condensation. However, dropwise condensation is notoriously tough to be maintained. Solutions to promote dropwise condensation primarily work on changing the surface energy of a solid surface by coating with chemicals such as dioctadecyl disulphide or oleic acid which give large contact angle (>120o).[36-38] Ion implantation techniques have also been used to promote dropwise condensation mode.[39, 40]

In general, dropwise condensation depends upon several factors such as substrate orientation, thermal properties of substrate material, steam velocity, surface characteristics, wall subcooling and presence of non-condensable gases.[41] These factors inherently affect several phenomenon associated with dropwise condensation such as droplet formation, size, nucleation density, drop size distribution and heat transfer coefficient. Understanding droplet formation and droplet growth kinetics is of utmost importance to develop a sound comprehension of vapor-liquid dynamics.

Based on previous experimental studies to observe the microscopic nature of droplet formation, two theories have been hypothesized to predict droplet formation. One hypothesis by Eucken treats dropwise condensation as a heterogeneous nucleation process with droplet embryos forming at nucleation sites while the intermediate area remains dry.[42, 43] This theory postulates that all of the condensation takes place on the drops, while no more than monolayer exists in between the droplets. Experimental observations have established that dropwise condensation indeed begins as nucleation process.[44, 45]

The other hypothesis put forward by Jakob postulates that condensation begins in a filmwise manner initially covering the whole surface.[46] This thin film ruptures after reaching a critical thickness (about 1

µm), where after droplets are formed. Meanwhile smaller droplets are drawn to adjacent droplets due to surface tension effect, while the droplets also grow simultaneously due to vapor-liquid transformation on them and coalescence. This model has also been supported by observations of others.[47] Yongji, S., et al. also noted that a thin film of condensate exists in the area between the droplets, and while the droplets may depart this thin film remains.[48]

Although the pathways of droplet formation are still under debate, the theory of droplet condensation as a heterogeneous nucleation process has become widely accepted. Based on this hypothesis, several models have been proposed to describe the heat transfer for dropwise condensation.

Theoretical models based on this model match to a great extent with the experimental observations.

McCormick and Baer proposed a model in 1963 assuming vapor condensation to take place on submicroscopic areas.[49] Gose.et al. proposed a model on which condensation occurs only on already formed droplets.[50] Several other models were then introduced incorporating effects of coalescence, sweeping droplets, drop-size distributions etc.[51] Griffith and Suk noted that surfaces having higher thermal conductivity have higher heat transfer rate for the same subcooling.[52] They noted that the definition of heat transfer coefficient inherently makes dropwise condensation as a function of thermal conductivity of base material. Mikic however attributed this effect on non-uniform heat flux distribution around droplets and proposed “constriction resistance” as an important parameter affecting heat transfer.[53] It was noted that dropwise condensation can be described if the drop-size distribution is

11 known. Several attempts have been made in this regard.[51, 54, 55] However the experimental limitations associated with using optical microscopes do not allow studying droplets smaller than 10 microns and as a result, the dropwise distributions are not available for the submicroscopic droplet regime. Obtaining this information is critical as Graham and Griffith noted that at least 50% of heat transfer took place through drops which were less than 10 micron in diameter.[56] Welch and Westwater further hypothesized that

97% of heat transfer was carried out by submicroscopic droplets or through bare area, while large droplets remained mostly absent in active heat transfer.[47]

2. 2. Droplet Growth kinetics on submicroscopic particles

Droplet growth studies primarily aim to investigate the subsequent interactions between condensate and a nucleate once a nucleation event has occurred either via homogeneous or heterogeneous process.

Looking at the history of these studies, the larger context of these studies has been to understand cloud microphysics and precipitation behavior of submicron particles. In addition, these studies are relevant to understand dynamics of material synthesis, aerosol formation, and particle counting instruments such as the ultrafine particle condensation device described in Part I of the present investigation. Consequently, there is a vast realm of research available on this topic and detailed analysis has been done by numerous authors.

The earliest studies pertaining to provide an insight into the phase change dynamics were carried by

Maxwell [57]. Based on Fick’s diffusion theory, Maxwell showed that the mass transfer related to transfer of a liquid molecule to vapor state is governed by diffusion process. This dynamics is equally applicable to the droplet growth process.[57, 58] Maxwell assumed that transport coefficient did not vary with temperature and neglected mean motion of gas and vapor which is often referred to as Stefan Flow [58].

The Stefan flow arises due to the cross diffusion of vapor and gaseous molecules to keep total pressure of the system as constant.

The intricacy behind droplet growth emanates from the conjugated nature of thermal and mass transport associated with the phenomenon. Essentially, the transport and the exchange process occur at

12 the molecular level [59]. In an environment suitable for condensational growth, a nucleated droplet may interact with condensate molecules via random molecular collision process. However as the droplet size steadily increases in relation to vapor molecule size, there is accumulation of vapor concentration on the droplet surface which establishes concentration gradients giving rise to the diffusion phenomenon [57].

There is thus a dependence of growth mechanism upon the relative size of the droplet with collision distance between vapor/air molecules. To consider this distinction in theoretical investigations, the

Knudsen number (Kn) is introduced where it is defined as the ratio between the size of the droplet and mean free path of vapor molecules in the environment [29, 30, 59].

Droplet growth theories have thus been based on treating the growth mechanism in two separate regimes – kinetic growth regime and diffusion dominated continuum regime. Correction factors are usually introduced as a way to preserve the continuity of mathematical description of droplet growth.

Notable among these theories is Mason [60], Fuchs and Sutugin [59], Barrett et. al. [61], Kulmala [62].

Mason formulated the droplet growth model based on the principles illustrated by Maxwell[60, 63].

The equations of mass transfer by Maxwell were simplified under the assumption of isothermal vapor and diffusion coefficient independent of temperature and space. Mason considered that energy transfer between condensing vapor and surrounding vapor essentially takes place via thermal conduction. So using energy balance, he gave the equation relating the conjugate nature of thermal and mass transport for droplet growth. The simplified equations given by Mason did not contain effect of variation of saturation pressure due to curvature effect or more known as Gibbs-Thompson or Kelvin Effect. Moreover, consideration of molecular regime was absent in his analysis.

Fuchs and Sutugin [59] recognized the difference in nature of growth regime in terms of relation between droplet size and mean free path of vapor molecules surrounding the droplet. To compensate for the discontinuity that can exist for droplets nearing the mean free path size scale, they introduced the idea of employing a correction factor. However, their droplet growth model did not include the contribution of

Stefan flow and thermal diffusion.

An extensive framework on droplet growth was considered by Barrett and Clement.[61] They considered the stefans diffusion, which had had previously been neglected in earlier analysis by Maxwell,

Fuchs and Mason and incorporated the effect of radiation while considering the heat transfer from the droplet to the surrounding environment. Using linear approximations along with Clausius-Clapeyron

Equation, they were able to present a simplified expression of droplet growth. Moreover, they attributed droplet growth regime as molecular, transition and continuum regime.

The mass flux expansion undertaken by Barrett and Clement had been done under the assumption that Kelvin Effect had a negligible effect on droplet growth and the transport coefficients were independent of temperature dependence. Kulmala et. al. made several adjustments to incorporate various effects which had not been included previously.[62, 64, 65] Kulmala [65] assumed the pressure at the droplet surface to include the Gibbs Thompson effect and subsequently improvised droplet growth derived by Barrett to yield equations in the continuum and transition regime. In his work, Kulmala was instrumental in providing for energy calculations which included many effects such as conduction, thermal diffusion and Dufour effect. The droplet heating equation was slightly modified to give the thermal conductivity of binary mixture as temperature dependent

A separate attempt has concentrated on providing a unified theory of droplet growth applicable at arbitrary Knudsen numbers by considering solutions provided by solving Boltzmann Equations. Shankar

[58] solved for the Boltzmann Equation of particle dynamics using moment method to solve for maxwellian distribution of vapor and inert gas molecules in an isothermal environment around a droplet maintained at temperature. While he did not include the energy transfer, it could be found out by solving higher order terms of the Boltzmann equation framework laid down by him.

Similar to Shankar’s approach of using Loyalka [66] used the BGK model of molecules and considered a spherical droplet in an infinite expanse of isothermal vapor. They argued that Fuchs and

Sutugin’s approach was particularly valid for low vapor concentrations and vapor-vapor molecular collisions were not included in their solutions. These effects were incorporated in their simplified model.

A more elaborate framework of droplet growth was presented[67-69]. In the latter models, effects of

14 microscopic temperature and density jumps at the vapor-liquid interface were included and a model was presented over a wide range of Knudsen numbers.[68]

Like Shankar and Loyolka’s approach, many works have been carried to understand droplet growth behavior. For sake of brevity, not all of those have been included in the present review. Notable among these attempts have been by Gajewski et al.[70], Yoshida et al.[71], Margilevskii [72], Sitarski et al.[73],

Young[74], Qu and Davis[75] and recently Nadykto[76].

2. 3. Particle Condensation Devices

The interest in supersaturated vapor as means to observe microscopically small sized matter to optically detectable limits is more than a century old. The dust counters used by Aitkens were one of the first commercialized device and of importance to academic research in its approach to classify pollutants in environment.[77] The dust counters by Aitkens were essentially cloud chambers using sudden adiabatic expansion of vapor and were precursors of most modern techniques and are still in use, particularly for their capability to produce very large supersaturation ratios.[78] Numerical simulations by

Vohra et. al. have shown them to be capable of condensing as small as 5 nm particles.[79] In their work they observed that ability to detect particles is critically depending upon establishing a minimum supersaturation ratio which is dependent upon particle size, concentrations and device parameters such as geometry and expansion times. Despite the success of adiabatic expansion condensation devices, there were some limitations on their use in field. Their large sizes implied that they cannot be used as a standalone portable device.

The next generation of the condensation particle devices were based on diffusion principles incorporating fluids with lower threshold of vaporization which allows operation of device in conditions which ensure that supersaturation conditions are fully met in them. This allowed them to operate where unlike adiabatic expansion devices they could intake aerosol flow continuously. Sinclair and Hoopes [80] built a device based on thermal cooling mechanism wherein aerosol was mixed with alcohol vapor diffusing from walls to grow particles detectable by classical photometry. In their device which they

15 called as ‘condensation flow counter’, aerosol was passed through a tube containing alcohol at room temperature, so as to saturate it. The saturated aerosol was then passed through a second tube kept at a much lower temperature of -20 oC. Using this mechanism they were able to measure aerosols in size ranges from 2-100 nm. However, the whole mechanism required an extensive setup and residence time for aerosol from beginning till the end was up to 14 seconds, which is considerable amount of time to be put on the field tests. Moreover, the choice of their condensing medium were limited to alcohols as water freezes at 0 oC.

A new condensation counter based on butanol as condensing medium was developed by Agarwal and Sem in 1980.[81] In principle, their device was similar to Sinclair and Hoopes design, as the incoming aerosol was saturated first and then passed along a cooler tube to cause butanol to condense on particles. However, the first step involving saturation of air-vapor mixture was achieved at slightly higher temperature of 25-40 oC. This allowed the condensing tube to be operated at moderately higher temperature of 10 oC. Butanol was chosen particularly because as compared to other vaporizing alcoholic medium, it was found to have smallest ability to absorb water vapor from incoming air.

Several works have been undertaken to study the mechanism responsible for the working of these devices. Clement proposed an elaborate framework of theoretical basis on conditions which can lead to supersaturation conditions in devices incorporating evaporation process. [82] Coupled heat and mass transfer equations were solved with aerosol size distributions for mixtures where pressure changes remain nearly constant. Based on his developed theories, Clement identified that both cooling and heating mechanisms can be used to cause conditions which favor aerosol growth. He identified Lewis number as critical basis for deciding conditions which can enhance or decimate heterogeneous conditions. He conjectured that for water vapor-air mixtures, cooling of the mixture may result in more evaporation and that thermal diffusion may be negligible in vapor-gas mixtures. This model was later simplified to yield one-dimensional ordinary differential equation,[83] and it was proposed that water-vapor and air mixture behaved differently under conditions of evaporation and condensation. In further studies using the same methodology, Barrett and Fissan [84] showed that for water-vapor and air mixture, the aerosol growth

16 may be impeded by latent heat release and if the condensing tube is chosen with saturated water at cooler temperatures, most of the cooling would actually occur on the wall. Barrett and Baldwin undertook simulations [85] for the case where the incoming hot air-vapor mixture was passed through a condenser tube at lower temperatures and showed that the location of maximum saturation is determined by Lewis number and latent heat of the working fluid. For air-water vapor mixture maximum saturation occurs near wall, while for butanol-propanol systems it occurs at the tube axis.

Numerical simulations were performed by Ahn and Liu [86] for studying effect of sampling flow rate, condenser wall temperature and different carrier gases (air, argon and helium). They commented that activation efficiency of aerosols is not affected by sampling flow rate as long as supersaturation exists inside the condensing tube. Droplet growth processes were modeled using Fuchs Equation to predict condensation behavior in commercialized version of Agarwal and Sem’s condensation device. The study was later extended to include affects of pressure and temperature difference between saturator and condenser region.[87] The authors concluded that pressure affected the performance of the device and that temperature difference between saturator and condenser are very critical parameters for working of the device. Experimental studies were later performed by the authors to study affects of pressure and flow rates on working of Agarwal and Sem’s device and large discrepancies were found at low pressures and low flow rates in the numerical predictions and observed phenomenon.[88] Kim et. al. also performed numerical simulations using conjugated diffusion and convection equations at low vapor concentrations with varying vapor-gas thermodynamic properties and showed that vapor loss decreases by increasing tube diameter. Similar to these methods, numerous other works have been carried to understand conditions which can aid or oppose formation of condensational droplets inside these devices. Notable among them are works by Jacobson et. al. [89],

A refined model of Agarwal and Sem’s condensation device was introduced by Stolzenburg and

McMurry [90], to detect particles smaller than 20 nm by introducing sheath flow as means to concentrate aerosol flow to the center of the tube and avoid losses of aerosol sticking to the walls. A simplified numerical scheme using Graetz solution was applied to the air-vapor mixture to predict conditions inside

17 the device. Vohra and Heist [91] introduced another form of device in form of flow diffusion chamber comprising of a cold saturation chamber, followed by a hot preheater unit and a condenser section called nucleation chamber. The residence time was kept in seconds and isopropanol was used as working fluid.

Similar to Stolzenburg and McMurry, simulations were provided using Graetz solution to explain the working principle of the device. The devices by Agarwal, Stolzenburg, Vohra and others were based on laminar flow. Chuang et al. [92] designed an alternative form of condensation counter which incorporating alternating regions of hot and cold temperatures on a wetted porous column. They noted that although this design provided a lighter and more portable version of the device, it was not particularly suitable for accurate assessments of wide range of nuclei sizes. This design was further improved by

Roberts and Nene [16], who established it as device capable of producing steady supersaturation in range of 0.13-3%. Supesaturation was critically found to depend upon flow rate, pressure and temperature.

Sioutas et al. [93] proposed a device based on turbulent flow regimes where the flow rate was as high as 110 lpm, although the condenser section was operated at 3.5-7 lpm. Mavliev and Wang to design a turbulent mixing type using flow rates of 3 lpm and were able to detect particles as small as 3 nm.[94-

96] The design principles were similar to the laminar flow models where in hot air-vapor mixture was introduced into a condensing tube at much colder temperatures. Dibutyl phthalate was used as working fluid.

Most of the devices discussed above were based on using organic fluids as working medium, primarily because they have significantly higher thermal diffusivities as compared to mass diffusion and are easier to evaporate. Water based condensation devices were tested but were not particularly as successful as compared to organic alcohol derived condensation devices. Although there have been many efforts to use water as the condensing fluid and the concept of water CPS was developed in 1981,[97] the water based condensation device was not commercially available until recently. Mavliev went on to develop a condensation device where water was the working fluid [98]. The water vapor-air mixture was saturated at high temperature and the mixture was introduced into a cooling tube through a nozzle. The choice of water was recognized as more environment friendly and less toxic to handle as compared to

18 alcohol based devices. Biswas et al. [99] showed development of a continuous flow condensation device using water with ability to detect as small as 5nm nanoparticles. Although the structure of the device was similar to those preceding it, and it comprised of a saturator and a growth tube, the mechanism of achieving supersaturation was considerably different [100, 101]. The condensing section of the device was held completely wetted at warmer temperature as compared to the incoming air-vapor mixture. They proved that the device worked more efficiently as compared to the cold wall type or mixing type devices.

Simulations were provided using Graetz solution to predict behavior of temperature and vapor conditions inside the device.

The above review shows that while the fundamentals of droplet growth have been studied and devices incorporating mechanism of nanoparticle estimation by heterogeneous nucleation have been developed, these systems are noticeably large in size and portability in their operations is a considerable challenge. It is highly desirable to develop condensation particle counters that are miniscale and highly portable so that they can be effectively employed to gain real-time measurements. For the application of portable sensors, the system volume and mass should be small and light enough to be carried by an individual with sufficient sensor-operation hours. As a result of reducing the size of condensation channel, the portable condensation particle counter can be smaller. Furthermore, the mini condensation channel enhance the power management of system that is powered by battery for longer operation.

A numerical framework to find optimized conditions for development of the device being developed at University of Cincinnati was carried by Deepak et. al. [102] and many aspects related to species transport were discussed for the case of saturated wall at uniformly elevated temperature. The above mentioned review covers some of the most potent areas which are associated with the current research.

The breadth and depth of areas associated with current research encompass areas of study which are spread out in many fields of science.

CHAPTER - 3

FUNDAMENTALS OF HETEROGENEOUS CONDENSATION

3. 1. Condensation and Nucleation

In general, the initiation of vapor-liquid or vapor-solid transformation process is through formation of nuclei. Thermodynamically, presence of a surface lowers the energy barrier for nucleation to occur as compared to nucleation of vapor/liquid in its own environment. A molecular collision between vapor and solid surface is a perpetual phenomenon which exists even when the system is thermodynamically in equilibrium. However ‘nucleation’ event requires establishment of non-equilibrium which give rise to natural forces resulting in atomic gradients and mass transport effects.[103]

A system is in non-equilibrium when there are thermal or concentration gradients. These concentration gradients result in sticking of molecules to the solid surface, and is accompanied by processes such as adsorption, surface diffusion, chemical binding or intermolecular bindings at the surfaces. The key determinates in establishing non-equilibrium in systems are substrate characteristics

(surface roughness, surface energy) and substrate temperature and vapor saturation.[103, 104] These deterministic processes are shown in Figure 4, which shows physical description of molecular description from perspective of molecular exchange phenomenon.

Deposition()or Condensation Desorption(or Evaporation)

γ lv Pv Pl γ sl θ γ sv

θ R

Figure 4: Molecular Exchange phenomenon during vapor deposition process[103]

In a vapor system, the non-equilibrium state can be stated in terms of its partial pressure in relation to the saturation pressure at a particular temperature. The ratio of two quantities (Saturation Ratio or

Relative humidity) is given by

SppT= As() (2)

When the partial pressure of the available vapor is more than the amount of vapor that can be held at a given temperature, the state of the system is referred to as supersaturation. Experiments carried as early in 1949 by Twomey [26] established that supersaturation required to nucleate a surface is dependent upon its surface energy. In this regards, the concept of nucleation has been well studied and the ability of a surface to nucleate can be given in terms of nucleation rate (Eqn. 3).[24]

⎛⎞−ΔGc JJExp= 0 * ⎜⎟ ⎝⎠kTBo v 3 2 c 8πσ fmxM(), ()ll υ N A Where Δ=G 2 3ln{}[]S 33⎡⎤ (3) ⎛⎞1−−−mx 32 ⎛⎞⎛⎞xm xm ⎛ xm−⎞ fmx(),1=+⎜⎟ +x⎢⎥23 − ⎜⎟⎜⎟ + +3mx ⎜ − 1⎟ ⎝⎠ggg⎣⎦⎢⎥ ⎝⎠⎝⎠ ⎝g⎠ mxRr==cosθ and / c 2σ Where r c = ()MNkTSllυ A Bovln[]

A plot of Eqn. 3 is shown in Figure 5 in terms of nucleation rate for surfaces with different contact angle. From Figure 5 it is clear that high surface energy surfaces (with lower contact angle) have lower critical supersaturation ratio as compared to low surface energy particles and thus get nucleated quickly.[105] Thus lower energy surfaces require higher degree of subcooling to initiate condensation upon them. Thus for the same degree of subcooling, condensation rate will differ for surfaces with different surface energies. Thermodynamically, this is given in terms of High surface energy particles have lower critical supersaturation ratio as compared to low surface energy particles and thus get

nucleated quickly and would generally require lower power consumption to reach their critical

supersaturation ratio.

Nucleation Rate vs Contact Angle Nucleation Rate vs Contact Angle 1010 1025

107 1019 Θ=20o Θ=45o Æ 4 o Æ L 10 Θ=90 L 13 1 o 1 10 - Θ=120 - s s 2 2 - - m m H 10 Θ=180o H 107 J J

0.01 10 S=1.78 S=1.08 S=1.48

10-5 10-5 80 100 120 140 160 180 200 220 0 50 100 150

Pv kPa Æ q Degrees Æ

Figure 5: NucleationH L rate as a function of supersaturation for differentHL contact angle

Previous investigations with diesel particles in form of carbon black with different surface energies show the same trend concerning the influence of surface characteristics on the nucleating parameter i.e. critical supersaturation ratio [19]. As expected lower contact angle carbon surfaces require lesser supersaturation to nucleate as compared with higher contact angle coated carbon surfaces.

However for surfaces with curvature e.g. droplets and particles as per the Gibbs-Thompson effect, the presence of curvature modifies the saturation vapor pressure of condensing liquid. The Gibbs-

Thomspon effect is more pronounced for droplets/particles which are smaller than 0.1 µm. Thus while a supersaturation ratio of SR may be sufficient to result in nucleation on a plane surface, for condensation to occur on a submicroscopic particle, the partial pressure of vapor must be larger the droplet saturation vapor pressure for condensation to occur on it. For a particle/droplet with diameter Dp, this ratio is

defined as Kelvin Ratio (KR) and given by

⎛⎞2σ M KR== p* p T exp A ASd() ⎜⎟ (4) ⎝⎠ρdgdRTa

For given conditions and a SR> KR, droplet growth occurs for particles larger than critical diameter

(Dp).[5] Since for given thermal conditions inside the device the Saturation Ratio will be fixed, there will 22 be a minimum sized nanoparticle which can be detected under those conditions. From this equation it can be estimated that at a temperature of 310 K, a 10 nm particle requires a minimum supersaturation of 22% while a 20 nm particle gets activated at 10% supersaturation.

CHAPTER - 4

DROPLET GROWTH DYNAMICS ON NANOPARTICLES IN A VAPOR-

GAS MIXTURE

The original research by Maxwell was intended for study of vapor-liquid transformation in case of

evaporating substances. Using Fick’s diffusion theory Mawell proposed that mass transfer from liquid to

vapor state is governed by diffusion process. This dynamics is equally applicable to the droplet growth

process, where the vapor transfer to a droplet can be written as [57]

2 maDddXcA= 4πρBA (5)

Here, the temperature dependent diffusion coefficient at any general temperature is given as

1.5 DDTTAB= AB∞∞() (6)

DiffvAB∞ denotes diffusion coefficient at a temperature T∞ and can be calculated using Chapman-

Enskog theory and given by [106]

1.5 2 ()RTg ∞ 11 DAB∞ =+ (7) 1.5 2 MM π pradradtA()+ B A B

Above equation under the assumption of constant vapor transfer and isothermal vapor at temperature

* T∞ can be integrated from Ra→∞( , ) where ρρρ→ ( AA, ∞ ) to give

* mcA=−4πaDBA∞∞(ρρA) (8)

Eqn. 8 is fundamental basis of most of the attempts since Maxwell to explain the droplet growth

dynamics. During the integration of Eqn. 5 to obtain Eqn. 8, Maxwell assumed that transport coefficient

did not vary with temperature and neglected mean motion of gas and vapor which is often referred to as

Stefan Flow [58]. The Stefan flow arises due to the cross diffusion of vapor and gaseous molecules to

keep total pressure of the system as constant.

Eqn. 8 provides an accurate assessment for droplets whose size is larger than the mean free path of

the vapor molecules surrounding them. Since fundamentally the condensation process is a molecular

process, the transport mechanism is influenced by the relative size of the droplet in comparison to the

surrounding vapor molecules and the mean free path of diffusing species. This behavior is captured by

characterizing different transport mechanisms on basis of Knudsen Number [107]. Eqn. 8 illustrates that

growth does not depend on the rate of random molecular collisions but on the rate of diffusion of

molecules to the droplet surface. Eqn. 8 can be rephrased in terms of droplet diameter to give

da DM⎛⎞p p = AB A ⎜⎟A∞ − d (9) dtρA aRgd⎝⎠TT∞

At the other spectrum, when droplet sizes are much less than the mean free path length ( Kn →∞), the rate of growth of drople t is governed by the rate of random molecular collision of vapor molecules.

This mechanism was first discussed by Fuchs [59]. From kinetic theory, the droplet growth mechanism can be given as

M 2*A maKm= 4πα () pp AA∞ − (10) 2π RTg ∞

The rate of growth of particle when the initial particle size is less than the mean gas free path is governed by the rate of random molecular collision of vapor molecules. Using kinetic theory, the rate of droplet growth in this regime is given by

* da αmAM ()pp A∞ − A = (11) dt ρAaNmkT2π ABo

Since the nature of the condensation growth below the mean free path and above the mean free path is different, a discontinuity can exist for droplets nearing the mean free path size scale. To compensate for this effect, a correction factor is often employed [5, 87]

1+ Kn φ = 2 (12) 14++()0.337+ 4αm 3 Knαm Kn 3

With these assumptions, the rate of droplet growth in continuum regime needs to be modified to

give

da DMAB A ⎛⎞pA∞ pd =⎜⎟− φ for Dd >λ (13) dtρ A aRgd⎝⎠TT∞

The condensation process is accompanied by release of latent heat of vaporization which can result in droplet heating to take place. When the condensation rate is low, heat transfer to the surroundings vap or

gives sufficient time for the droplet surface temperature to transfer does not differ much from the ambient

vapor temperature surrounding the droplet. However under conditions of rapid condensation, the dissipation of heat to the surroundings is not sufficient and droplet growth is accompanied by droplet heating. The increase in droplet temperature results in increase of saturation vapor pressure at droplet

surface. This leads to decrease in condensation rate at the droplet surface. Due to the coupled nature of mass and energy transport related to droplet growth, the mass flux and energy flux equations associated with droplet growth process need to be solve simultaneously. The energy transfer process and the role of

different modes of energy transfer during the condensation process has been studied by various researchers and is explained along with discussion of their proposed droplet growth models.

A complete description of energy exchange using kinetic theory in a binary mixture can be provided by [108]

TT δT kTBo ⎡⎤nBA D kT nAB D kT q=−+++knhnhABAAABBBvvv⎢⎥()AB −+vv() BAr −+vq (14) δr nmD⎣⎦AAB mDBAB

Here, it describes that conduction, thermal diffusion and energy flux carried by diffusing vapor and air molecules effect the energy exchange between different species in a binary mixtur e. The term

nhAAvvA+ nh BB Bincludes the contribution of Dufour Effect (dependence of heat flux on concentration gradient). By neglecting Dufour effect and thermal diffusion, a much simpler form of droplet heating can be obtained as [5, 87]

mLc TTd = ∞ + (15) 4πakAB

Thus if the thermodynamic conditions are known around a particle, droplet growth can be predicted with help of Eqn(8)-Eqn(15). The conjugated and non-linear form of Eqn. 15 makes it very difficult to solve. A simplified empirical expression based on matching experimental data was given by Hinds[5]

2 ()6.65+•+• 0.345TTS∞∞ 0.0031( − 1) TTd = ∞ + (16) 1++() 0.082 0.000782 •TS∞ *

Here, T is in Celsius.

However, it should be noted that this expression is based under the assumption that droplet

condensation begins as soon as the saturation ratio becomes more than 1. Since the droplet growth does

not begin until a critical supersaturation ratio (or Kelvin Ratio) is reached, using Eqn. 15 in solving the

conjugated problem of droplet growth will lead to over prediction of droplet heating and negative solution

from Eqn. 9 and Eqn. 11. This error can be fixed by recognizing that droplet heating will begin

simultaneously once droplet formation is initiated which is when S>KR.

2 (6.65+•+• 0.345TTSKR∞∞ 0.0031 )( −) TTd = ∞ + (17) 1++() 0.082 0.000782 •TS∞ *

It is clear that solution of droplet growth requires simultaneously solving the mass flux and energy flux equations. By basing this knowledge on the data ob tained by CFD analysis, many effects which were inherently a part of CFD process get incorporated in droplet growth process. This eliminates inclusion of

27 many of these effects at a later stage, as many of the models make special arrangement to include these effects at the droplet growth stage. Thus effects such as variation of transport coefficients, thermal conductivity of the mixture with temperature get automatically included in droplet growth models because the CFD system already incorporates these assumptions. Moreover, since CFD already includes the effect of gaseous and vapor velocities, the effect of molecular diffusion on heat transfer from the droplet too gets incorporated and even this information can be coupled in any model regardless of the fact that many models were based under the assumption of a static isothermal vapor conditions. Under such conditions, the models were solved using finite difference schemes.

CHAPTER - 5

CONDENSATION ON NANOPARTICLES IN MICROTHERMOFLUIDIC

SENSOR

5.1. Overview

Enormous progress has been achieved since the development of the condensation technology that

was first reported by Aitken as early as 1889.[109] As a result, for now, the condensation particle

counters (CPCs) based on this principle are the most commonly used instruments for continuous

measurement of the number concentration of airborne UFPs. However, the smallest portable CPC

available in the market is approximately 5700 cc in total volume (29.2 cm × 14 cm × 14 cm) and weighs

1.7 kg, which is still too large and heavy to be worn. Particularly, the CPC generally requires many

operating systems to control liquid and vapor flow due to its principle that the creation of the controlled

supersaturation region is needed for the enlargement of particle size by condensation. The need for this

kind of fluid control systems has been a significant obstacle in miniaturizing the CPC for the application

to portable device with high mobility. This system for particle condensational growth, commonly referred

to as “particle condensation device” or “particle size magnifier”, is the most essential part of the CPC and

also occupies the largest amount of sensor volume.

As discussed in the literature section, most of the commercial continuous flow CPC devices have

used butanol or isopropanol as the condensing fluid because these require a slowly diffusing molecule for

preventing vapor depletion [17]. However, it is obviously preferable to use water as the working fluid for

many applications. Particularly, the selection of a safe substance is an important consideration for the personal sensor that may have direct contact with the human body. A water based condensation method with a continuous, laminar flow [100] was reported in 2005, and this technology is commercially available [110], [111].

In light of above discussion, a microthermofluidic sensor has been under development at University

of Cincinnati. This part of the study corresponds to the theoretical and numerical developments related to

the sensor design. The essential feature of the design is the small size of the condensation device (~ 3-5 cm) and it is shown that although the device has been successfully miniaturized, it works optimally to detect nanoparticles in air of size ranges upto 5 nm.

For the sensor, two designs have been studied. These are

a) Growth region cooled by natural convective means

b) Growth region maintained at wetted but constant wall temperature which is higher than

temperature of incoming air-vapor mixture.

These two designs are separately discussed below and various aspects related to working of design have thoroughly been studied.

5.2. Design A: Experimental Setup and Device Preparation

Droplet growth by condensation requires a supersaturated vapor condition. In the developed sensor,

supersaturation is achieved by the combined effect of vaporizing the working liquid and cooling of condensation chamber by natural convection (Figure 6(a)). The design of condensation chamber is

fundamental for sustaining supersaturated conditions. Two types of condensation chambers were

prepared. The first set was prepared using glass tubes with square cross-sections and patterned with

microfabricated v-grooved walls. This was done for visualization studies of evaporation-condensation

inside the chamber. A second set was made out of round copper tubes with grooved structure in their

inside face. Typical configuration of the device included inside diameter (or width) ~3 mm, condenser

length ~30 mm and inside glass capillary channel of diameter (or width) of 2 mm and inner width 500 μm leaving a gap of 500 μm between the channels. The characteristic details on the construction of the device are shown in Figure 6(b).

5.2.1. Condensation Chamber Preparation

(a)

10 mm Inner 40 mm Channel

Outer Channel

(b) Figure 6 (a) regional heat and mass transfer and (b) the conceptual drawing of mini thermofluidic sensor

for nanoparticle detection.

The interior wall design of the condensation chamber has a primary role to play in successful working for thermofluidic sensor. Since the wall is also cooled by means of natural convection, this

results in a non-uniform temperature distribution along the wall. As temperature drop occurs along the wall, the water vapor present in the chamber condenses along the wall. Liquid hold-up by the droplet growth of condensate liquid on the wall (film or dropwise condensation) is recognized as to be controlled for the optimal sensor operation. It is widely known fact that film condensation is the prominent mode of vapor to liquid phase change on hydrophilic surfaces. On the other hand, the primary mode of condensation on hydrophobic surfaces is the formation of droplets upon them.

In order to prevent liquid hold up in the nanoparticle flow channel, inside channel wall are given a grooved interface, with angles between the grooves satisfying the Concus-Finn condition

θ ≤−πβ2 (18)

It has been well established that capillary forces generated in grooves satisfying this condition result in spontaneous spread of the liquid.[112-114] Experiments conducted with groove pattern surface in the channel conclusively show that when Eqn. (18) is met, the liquid droplets spread along the surface even though the surface itself may be non-wetting in nature. It can clearly be observed that there is no visible droplet formation in the grooved surface area. However, small droplets in large numbers are clearly visible on the flat surface of the channel which is non-wetting in nature. The groove pattern on the wall was fabricated by micro broaching technology with 10 μm resolution.

The groove patterning on the wall can be extended to cover the complete condensation chamber.

Since the spacing between the wall of the condensation chamber and the nanoparticle supply channel acts like a reservoir for the liquid, the capillary action emanating when liquid comes in contact with the grooved surface results in spreading of the liquid inside the grooves to a height where the liquid pressure is balanced by the body forces. Thus although the wall is not wetted completely, this results in enhancement of surface area for liquid to evaporate. From Figure 6(b), it can be seen that a heater is deployed at the wall near the entrance of the nanoparticle supply channel. This results in heating of the

wall line through conduction which is then spread to the liquid present in the micro-grooves along the

wall. Since the air flow introduced in the chamber is at a lower temperature as compared to the wetted

wall, evaporation occurs from the liquid column in the micro-grooves of the wall. The combined

evaporation from the wetted groove and the capillary meniscus formed between the wall and the nanoparticle supply channel produces high supersaturation conditions inside the condensation chamber.

Thus the condensation chamber can be viewed as a single device comprising of both evaporator and condenser sections. Evaporation and condensation both occur on the wall, rates of which are dependent upon the thermodynamics of conditions inside the condensation chamber. In general, condensation is more likely to occur near the exit region of the chamber while evaporation is favored at the wall near the exit of the nanoparticle supply channel.

As is well known, the surface tension is a function of temperature, given by

σσ=−01 σ(TT − 0) (19)

Whereσ01,σ are surface tension coefficients. Since temperature drops along the channel length, this

would result in an increase of surface tension along the channel length. Unlike similar structures which

are used in heat pipe, the hot end in the thermodynamic sensor is constantly fed with fresh liquid from a liquid reservoir as shown in Figure 6(a). As a result the capillary pumping power of the grooves from the cold end to the hot end transports the condensed liquid from the condensation zone back to its source.

When this film reaches to the evaporation zone, it coalesces with the meniscus in that region and a spontaneous bridge is formed.

5.2.2. Computational modeling of thermofluidic sensor

To better understand the complex issues related to the heat and mass transfer in the thermofluidic

sensor and the associated parameters affecting droplet growth, a numerical approach was undertaken.

Computational modeling was carried using COMSOL® for single phase and multi species.

Figure 7 illustrates the geometry of the problem considered which is modeled as an axis-symmetric

2D problem. The system is assumed to be in steady state laminar flow with density taken as a function of

temperature and pressure. Moist air is considered as flowing through the nanoparticle supply channel

while water vapor diffuses inside from the wall and spacing between the wall and the nanoparticle supply line. The base of the wall is considered to be kept at an elevated temperature whereas the rest of the wall

is cooled by means of convection.

D2=1.0 mm r

1 7 2 D =1.5 mm 1 3 6 4 L=30 mm 5 Thickness = 0.5 mm Figure 7. Schematic of the model considered for computational modeling

Owing to the small gap between the nanoparticle supply channel wall and the condensation chamber

wall (0.5 mm), capillary forces result in formation of a meniscus between the two surfaces. Further, the microscale aspects of the liquid formation between the surfaces indicate present of extended meniscus in the gap region as well as the grooved region of the wall of condensation chamber. Since it was not

possible to model the extended meniscus region for this problem or to estimate its role in mass transport,

a simpler condition involving saturated vapor pressure at the reservoir boundary and the wall boundary

was chosen. Further to include the effects of condensation on the side walls, latent heat removal was

included. Thus heat and mass both diffuse from the wall to the inside of the condensation chamber. The

boundary conditions for the computational model are illustrated in Table 1.

Boundary Boundary Description Boundary Condition

1 Axis (r=0) Axis symmetry

2 Outlet (z=30 mm) Outflow

Inner wall of condensation V=0 (No Slip Boundary Condition) 3

channel (r=1.5 mm) ωω= A (T )

Outer wall of condensation q”= -h*(T-298) – L*(Diffusive mass flux of water 4 chamber (r=2.0 mm) vapor at the boundary)

Base of Condensation 5 T=Tw channel(z=0, r=1.5 to 2.0 mm)

Gap between outer condensation V=0 (No Slip Boundary Condition)

6 channel and inner channel (z=0, T=Tw

r=1.0 to 1.5 mm) ωω= A (Tw )

V=Vi Incoming Air Inlet (z=0, r=0 to 7 T=Ti 1.0 mm)

ω = ω Ai,

Table 1: Boundary Conditi ons for Design A prototype

The domain consi dered is then solved for multispecies (Air and Water Vapor) for mass, momentum,

energy and species conservation. Unstructured meshes with triangular meshes were used to find grid

independent solutions. Figure 8 displays the mesh independent test with volumetric flow of 5*10-6 m3/s, temperature of incoming air at 298 K, convection coefficient of 10 W/m2K and temperature of water vapor diffusing inside the chamber from reservoir and base of copper wall was 333 K. The mesh independent solution shows the plot of ‘saturation ratio’ along the axial location at the centerline of the chamber (r=0). From the figure it is apparent that although an approximate solution can be obtained over all grid sizes, the precision can be improved by increasing the mesh quality. However, beyond a certain mesh size, the solution is not affected by the number of grid elements. All the solutions were obtained for

49679 elements.

1.4

1.2

0.8

0.6 Saturation Ratio 5257 0.4 12790 19535 0.2 32375 49679 76218 0 0 0.005 0.01 0.015 0.02 0.025 0.03 Axial Location at the Centerline (m) Figure 8. Mesh Independence Test: Saturation Ratio versus Axial Location at the centerline (r=0)

The equations for which the system is solved are

Continuity Equation:

1 ∂(ρρrV)∂() V rz+=0 (20) r∂∂r z

Radial Momentum Equation:

∂∂()ρρVV() V 11∂∂p μ ⎡ ⎛⎞rV∂∂2 V V⎤ rz r rrr Vr +=−+⎢ ⎜⎟ +−22⎥ (21) ∂∂rzρ ∂ rρ⎣rrrzz∂∂∂⎝⎠ ⎦

Axial Momentum Equation:

∂∂()ρρVV() V 11∂∂p μ ⎡ ⎛⎞rV∂∂2 V⎤ zz z zz Vr +=−+⎢ ⎜⎟ +2 ⎥ (22) ∂∂rzρ ∂ zρ⎣rrrz∂∂∂⎝⎠ ⎦

Energy Equation:

⎛⎞∂∂()ρρTT() krT∂∂⎛⎞ ∂2T

ρCVpr⎜⎟+= Vz ⎜⎟ +k 2 (23) ⎝⎠∂∂∂∂rzrrr⎝⎠∂z

Mass Diffusion Equation:

∂∂ω ωωD ∂∂()rrω ∂ ∂2 VVA +=AAB A +DA (24) rz∂∂rzrr ∂AB ∂z2

The droplet growth dynamics as laid in Eqn. (9) and Eqn. (11) were transformed into space

coordinates by assuming that the incoming nanoparticles travel in the minichannel at a constant average

velocity of U m/s.

Thus, tzUdtdzU=→=//

And Eqn. (9) and Eqn. (11), can thus be rewritten as

* da α mAMp( A∞ − p A) U = for D < λ (25) dz p ρ AaNmkT2π ABo

da Diffv M ⎛⎞p p AB AA∞ d U =>⎜⎟− φ for Dp λ (26) dzρ A aRgd⎝⎠TT∞

Various physical properties such as viscosity, density are further calculated using a mixing law based on weighted mole fractio n of the two species.

ηmix= xx airηη air+ vapor vapor (26)

Where η is a property variable (Thermal conductivity, Viscosity, Specific heat capacity, Molar

weight).

Inclusion of latent heat removal effect necessitates knowledge of vapor mass flux condensing on the

wall whose value is not known beforehand. Since condensing liquid limits the evaporative flux diffusing

away from the wall, an iterative scheme was employed to get this information. The methodology followed

to solve the problem can be summarized as

a) Assuming diffusive flux of vapor=0, solve for Eq. (9)-(13)

b) Integrate diffusive flux over boundary 3, to obtain diffusive flux Dflux1 under these conditions.

c) Update N0, by substituting Dflux0=0.5*(Dflux0, prior + Dflux1)and solve for equations again.

d) Repeat process (b) and (c) until, (Dflux0- Dflux1)/Dflux0<0.0001

e) Store the solution and solve for Eqn. (9) and Eqn. (11) along the axis using thermodynamic

conditions at that location for an inlet nanoparticle size of 10 nm. Droplet heating by rapid

condensation is considered by including Eqn. (16) in the solution scheme and Fuchs correction

factor given by Eqn. (12) is included to avoid discontinuity near mean free path.

5.2.3. Discussion of Results

As mentioned before, the primary aim of the design aspects of thermofluidic sensor is sustainability of supersaturation conditions inside the condensation chamber. Various factors can influence the thermofluidic transport inside the condensation chamber. The primary parameters of concern are relative humidity of incoming air and temperature of the water vapor diffusing inside the chamber, volumetric flow rate and chamber diameter.

5.2.3.1. Effect of Relative Humidity

The variation of saturation ratio at the centerline (axis) with moist air carrying water vapor at different relative humidities (RH=0%, 10%, 50% and 100%) is shown in Figure 9(a). The critical supersaturation ratio or Kelvin ratio for particles corresponding to 10 nm seeding particle size are also plotted for the corresponding cases. Supersaturation conditions are produced inside the condensation chamber owing to the mixing between water vapor diffusing from the side wall and the reservoir at an elevated temperature with incoming air at lower temperature. The variation of saturation ratio inside the condensation chamber shows that for all of the cases, the saturation ratio is more than Kelvin ratio for most part of the condensation chamber. Particle growth is initiated as soon as the saturation ratio exceeds the threshold value of Kelvin ratio. The seeding particles are nucleated instantaneously and the particles grow at growth rates given by Eqn. (9) and Eqn. (11). The residence time of the particles in the tube at

38 volumetric flow rate of 5*10-6 m3/s is 18.85 milliseconds which is sufficient for nanoparticles to grow from nano sizes to micron sizes.

1.6

1.4

1.2

0.8

Saturation Ratio 0.6 Kelvin Ratio 0.4 RH=0% RH=10% 0.2 RH=50% RH=100% 0 0 0.005 0.01 0.015 0.02 0.025 0.03 Axial Location at the Centerline (m)

4.5 RH=0%

4 RH=10% RH=50% 3.5 RH=100%

2.5

1.5 Droplet Diameter (µm)

0.5

0 0 0.005 0.01 0.015 0.02 0.025 0.03 Axial Location at the Centerline (m)

(b)

Figure 9. Variation of (a) Saturation Ratio and (b) Droplet growth at centerline (r=0) with varying relative

humidity of incoming air flow

From the figure it is clear that under these conditions for droplet condensation to occur, a certain

degree of humidity in air is required for growing nanoparticles to micron sized particles. This becomes

more evident in Figure 9(b), where droplet growth is shown along the axis at the centerline. The droplet

growth is initiated immediately where Saturation ratio exceeds Kelvin Ratio. Since a more vapor laden air

reaches this threshold earlier, the droplet growth begins sooner resulting in bigger droplets at the exit. It

should be noted that as droplets increase in size along the device length, the partial pressure required to

induce evaporation decreases owing to increase of surface curvature.

Thus even though the saturation conditions deteriorate as we progress along the tube, the decrease is

not sufficient to cause reverse evaporation from condensing droplet to the vapor. However, this does lead

to decrease in growth rate (dD/dt) along the tube length. It should also be noted that although the device is

not efficient in nucleating 10 nm particles, the conditions inside the chamber even when the incoming air

is moisture free can nucleate particles more than 20 nm in size. As water vapor diffuses into the

condensation chamber and mixes with the moist air, temperature of the mixture increases along the tube.

This results in increasing saturation vapor pressure along the tube whereas at the same time more vapor is

added from the surrounding wall leading to increase in vapor content. Eventually the increase in

saturation vapor pressure tends to overcome the effect of added water vapor. Thus the saturation profile

so obtained shows increase of saturation ratio inside the chamber where after it starts to decline. If the

incoming air itself is laden with moisture, it further helps in enhancing the supersaturation conditions

inside the condensation chamber. It results in longer time period of exposure of nanoparticles to

supersaturated air and also increases the peak supersaturation conditions inside the condensation chamber.

5.2.3.2. Effect of Wall Heating

The amount of heat to be supplied to the condensation device for vaporizing the liquid content in the reservoir section and the wetted liquid column is requisite to obtain proper conditions for saturation. To gage this behavior, temperature conditions at the reservoir and the base of the thin copper wall was used as a parameter of interest. From Figure 10(a), it is clear that increasing the heat flux to water increases the

40 efficacy of the condensation device. It results in higher amount of water vapor available for diffusion across the channel. It also shows when the incoming air is completely dry, a minimum temperature of around 65 oC is required to sustain conditions in which saturation ratio exceeds critical threshold value of

Kelvin Ratio for nucleating 10 nm nanoparticles.

1.8 Kelvin Ratio

1.6 Tw= 333 K Tw=338 K 1.4 Tw= 343 K

1.2

0.8

Saturation Ratio 0.6

0.4

0.2

0 0 0.005 0.01 0.015 0.02 0.025 0.03 Axial Location at the Centerline (m)

(a)

4 Tw=333K

3.5 Tw=338K Tw=343K 3

2.5

1.5 Droplet Diameter (µm) 1

0.5

0 0 0.005 0.01 0.015 0.02 0.025 0.03 Axial Location at the Centerline (m)

(b)

Figure 10. Variation of (a) Saturation Ratio and (b) Droplet Growth at centerline (r=0) with varying base

temperature at the wall 41

From Figure 10(a) it is clear that when the working fluid is kept at 60 oC, the supersaturation conditions are still produced inside the condensation device, but these conditions are not amiable to growth of particles below 10 nm. This can be clearly seen in Figure 10(b) which shows droplet growth at the centerline along the tube length for different wall temperatures. It is clear that a higher wall temperature results in higher diffusion from the wall and hence higher growth rate.

More noticeably, the initiation of droplet growth is spurred by increasing the wall temperature.

However, this increment is accompanied by an increase of droplet heating (Figure 11). As wall heating is increased, the droplet heating increases owing to increase in average temperature of the air-vapor mixture and thus resulting in lesser heat transfer between condensing droplet and surrounding vapor. It can be proven that this effect can result in evaporation of condensing droplet and thus decrease in droplet size if the channel length is increased much further. Thus droplet heating imposes a restriction on the maximum size of droplet which can be visualized at the exit of the channel.

323 Droplet Temperature (Tw=338K, RH=0%)

Temperature (Tw=338K, RH=0%)

Droplet Temperature (Tw=343K, RH=0%) 318 Temperature (Tw=343K, RH=0%) ∆Td=Td-T

313 ∆Td=Td-T

308 Temperature (K)

303

298 0 0.005 0.01 0.015 0.02 0.025 0.03 Axial Location at the Centerline (m)

Figure 11. Droplet Heating at centerline along tube length

5.2.3.3. Effect of Flow Rate

The variation of saturation ratio at the centerline of the condensation device with varying volumetric flow rate for dry air for base temperature of 65 oC and incoming relative humidity of 0% is shown in

Figure 12(a).

1.4

1.2

0.8

0.6 Saturation Ratio

0.4

VolVol=3.33 = 3.33e-6 m2/s m3/s 0.2 VolVol=3.33 = 5.00e-6 m2/s m3/s VolVol=3.33 = 8.33e-6 m2/s m3/s 0 0 0.005 0.01 0.015 0.02 0.025 0.03 Axial Location at the Centerline (m)

(a)

3.5 VolVol=3.33 = 3.33e-6 m2/s m3/s

Vol=3.33Vol = 5.00e-6 m2/s m3/s 3 VolVol=3.33 = 8.33e-6 m2/s m3/s

2.5

1.5

Droplet Diameter (µm) 1

0.5

0 0 0.005 0.01 0.015 0.02 0.025 0.03 Axial Location at the Centerline (m)

(b)

Figure 12. Variation of (a) Saturation Ratio and (b) Droplet Growth at centerline (r=0) with varying

volume flow rate 43

Increasing flow rate results in decrease of residence time and consequently lesser time available for the

water vapor to mix with incoming air. As a result, for lower flow rates supersaturation is achieved more

quickly and particles growth is higher (Figure 12(b)). However as the mixed vapor progresses

downstream it is exposed to hot water vapor diffusing from the walls for more time at lower flow rates.

This results in much larger increase of temperature along the centerline for lower flow rates and as a

result the drop in saturation vapor pressure is larger in these cases. From the figure, it is clear that an optimized flow rate for the condensation device is obtained for volumetric flow rate of of 5*10-6 m3/s.

5.2.3.4. Effect of Chamber Size (Condensation Chamber Diameter)

The size and the geometry of the condensation chamber affect the distribution of vapor content inside the device. For the same volumetric flow rate, the flow velocity is altered by the diameter of the condensation chamber. A larger diameter chamber results in decrease of flow velocity and provides more amount of time for better mixing of incoming air with vapor.

This results in an earlier onset of supersaturation conditions inside the chamber. Figure 13(a) shows the variation of saturation ratio at the centerline of the condensation device for condensation chambers of different diameters for a volume flow rate of 5*10-6 m3/s, base and wall temperature of 60 oC, and relative

humidity of incoming air of 10%. The gap between the condensation chamber and the incoming

nanoparticle minichannel has been fixed at 0.5 mm. From Figure 11(b) it is clear that although 10nm

particles are activated earlier and condensation process is initiated for condensation chamber of diameter

3 mm - 9 mm. However as diameter is further increased, the effect of velocity decrease is offset by

increase of time required for vapor to diffuse radially from the wall and the gap. As a result, the vapor is

distributed sparsely in the chamber. This results in saturation conditions which are not amicable to initiate

nucleation on a 10 nm nanoparticle and condensation is not initiated. From Figure 13(a) and Figure 13(b)

it is clear that a 3 mm diameter chamber provides optimum conditions which can lead to required vapor

conditions inside the chamber.

1.4

1.2

0.8

0.6 Saturation Ratio

0.4 Channel Diameter =3 mm Channel Diameter =6 mm

0.2 Channel Diameter =9 mm

Channel Diameter =12 mm 0 0 0.005 0.01 0.015 0.02 0.025 0.03 Axial Location at the Centerline (m)

(a)

Channel Diameter =3 mm 2.5 Channel Diameter =6 mm

Channel Diameter =9 mm

2 Channel Diameter =12 mm

1.5

1 Droplet Diameter (µm)

0.5

0 0 0.005 0.01 0.015 0.02 0.025 0.03 Axial Location at the Centerline (m)

(b)

Figure 13. Variation of (a) Saturation Ratio and (b) Droplet Growth at centerline (r=0) with varying

diameter of the condensation chamber

5.3. Design B: Experimental Setup and Device Preparation

5.3.1. Condensation Chamber Preparation

From previous design, it became apparent that supersaturation in a minichannel with water as working fluid is enhanced by warm wetted walls. For this reason, the design was modified. A heater element is wrapped all around the chamber to provide heat at constant rate to the chamber and it is covered with an insulating tape.

5.3.2. Computational modeling of thermofluidic sensor

The geometry of the minichannel device is shown in Figure 14. The diameter of the minichannel is 3 mm, while the length is taken as 3 cm. As we will show later, a length of 3 cm is sufficient to produce conditions which allow sizable growth of nanoparticles in the minichannel. The minichannel is modeled as an axis-symmetric domainc. Moist air with known vapor content given by relative humidity RHi and at temperature Tin is considered as flowing through the inlet. The minichannel wall is maintained as fully

wet wall with water as working fluid, at a temperature (Tw) which is higher than temperature of incoming

o vapor-air mixture (Tin). Usually the wall temperature is in range of 40-65 C and the temperature of

incoming air is in range of room temperature (25 oC). Under the assumption of moderate evaporation

rates, the saturated liquid on the wall can be considered to be in thermal equilibrium with saturated vapor

just above the wall. The system is assumed to be in steady state laminar flow with density taken as a

function of temperature and pressure.

Inlet air flow Axis z T i ID RHi r Q

Lcon

Wetted wall during the

whole operation, Tcon >Ti

Condenser & Evaporator part

Figure 14. Schematic of the model considered for computational modeling for Design B

Since the wall is considered in saturated state at known temperature, this considerably simplifies the

conjugated problem of thermal and mass transfer for the minichannel. Thus instead of considering the

multiphase flow, the physics of the problem is governed by diffusion of multispecies inside the

minichannel and the interaction of diffusive components with incoming advection flow. The movements

of diffusing species relative to net diffusive flow plays strong role in condensation process and cannot be

neglected. In addition, the Fuchs and Sutugin model does not include this effect, its effect can be

incorporated in calculating droplet growth by including it during calculation of the thermodynamic field

inside the minichannel. For this reason, the CFD model presented in current work employed Stefan’s

correction in the mass transport equations. The multicomponent diffusion coefficients are functions of

localized temperature, species and concentration and these dependencies were accommodated in current

work (Eqn. (31)) for more accurate results.

The multispecies components are be modeled together as a homogeneous mixture and the domain

considered was solved for multispecies (Air and Water Vapor) for mass, momentum, energy and species

conservation using a finite element based package (Comsol ®). The equations for which the system is

solved are

Continuity Equation:

∇⋅()ρu =0 (27)

Momentum Equation:

⎛⎞T 2 ρμμuu⋅∇ =−∇p +∇⋅⎜⎟ ∇u +() ∇ u −() ∇⋅ uI (28) ⎝⎠( ) 3

Energy Equation:

ρCTkp ()u ⋅∇ =∇⋅ ( ∇T) (29)

Mass Diffusion Equation:

⎡⎤n ∇ω j ∇⋅⎢⎥ρωiiu − ρω MD∑ ij =0 (30) ⎣⎦⎢⎥j=1 M j

Where, for a binary mixture

2 ωω21− ω2 D11 == Diffvij D12 Diffvij (31) xx12**xx12

Furthermore, with the assumptions of fully developed velocity profile at the intlet, the boundary conditions can be expressed as:

At the inlet 2Qr⎛⎞2 V= ⎜⎟1− 2 , TT= i , ω = ωA (Ti ) AR⎝⎠ECT

xM⋅ SpT⋅ Where, Ai, A iSi( ) ωAi()Tx==, Ai, xMAi,,⋅+−⋅ A()1 x Ai M B pt

At the axis Axis symmetry

(r=0) (32) At the wall In the condenser section:

(r=R) * V=0 (No Slip Boundary Condition), T=Tw ,ω = ωA (Tw )

xM* ⋅ pT( ) Where, ω**Tx==AA , Sw Aw() **A xMAA⋅+−⋅()1 x A M B pt

Outlet Outflow

Physical properties such as viscosity, density, thermal conductivity need to incorporated in localized

fashion so as to include effects of variation of species and temperature in a localized environment. This

estimation was done by using a mixing law based on weighted mole fraction of the two species.

ηmix= xx Aηη A+ B B (33)

Where η is a property variable (thermal conductivity, viscosity, specific heat capacity, molar

weight). The details on the variable properties are provided in the appendix section of this work.

Upon solving the numerical model, thermodynamic information at desired locations such as along the centerline of the minichannel and along the length of the axis was exported by including large number of data points (2000). This thermodynamic data was then employed to solve the modified droplet growth models (Eqn. (9) and Eqn. (11) which were modified from time coordinates to space coordinates using

tz= /V. The modified equations then are given as

da α Mp⎛⎞p V = mdAA ⎜⎟∞ − (34) dz 22π R ⎜⎟ g ⎝⎠TTAd∞

da DM⎛⎞p p V = AB AA⎜⎟∞ − d φ (35) dzρ A aRgd⎝⎠TT∞

The resulting system of equations was then solved along with Eqn. (16) using finite difference scheme with 2000 grid points.

5.3.3. Discussion of Results

In general, incoming air enters in unsaturated form in the minichannel at a lower temperature as

compared to the side wall of the minichannel. Also, the concentration of vapor species in incoming air is

lesser than concentration at wall, where the water vapor is fully saturated. Thus transfer of energy and

vapor takes place from the wall towards the center of the minichannel. The rate at which energy and mass

transfers from the wall is largest near the entrance of the minichannel where the temperature and

concentration difference is largest. The temperature and concentration gradient then declines along the

minichannel as the diffusing vapor mixes with incoming air to increase the vapor concentration inside the

minichannel and subsequently there is little evaporation from the walls.

The strength of energy transfer versus mass transfer is given through Lewis Number which gives the

ratio of thermal diffusion by mass diffusion. For water vapor the Lewis number is less than one, energy

transfer takes place slowly as compared to mass transfer. It is this particularly characteristic of water

vapor which allows attainment of high supersaturation environment inside the minichannel. As discussed

previously, saturation ratio in a localized environment is a function of vapor concentration and inversely

proportional to the saturated vapor pressure. Near the entrance of the minichannel, the localized

temperature obtained by mixing of diffusing vapor from the wall and the incoming cooler air results in

temperature increment of the incoming air. However the temperature increment is small as compared to

increase in vapor concentration. Thus saturation vapor pressure is lower but the partial pressure of vapor

is higher as compared to incoming air. This results in rapid build-up of supersaturation conditions inside

the minichannel.

In this section, the result of numerical calculations for the system of equations (Eqn. (9) – Eqn. (11))

with boundary conditions (Eqn. (32)) are discussed. The distribution of water vapor species inside the

minichannel is given in terms of saturation ratio. The distribution of saturation ratio along with associated

droplet heating is used to explain the dynamics of droplet growth. The main parameters of interest are

volume flow rate of incoming moist air, temperature of moist air, wall temperature and saturation ratio of incoming air. Although knowledge of vapor distribution in the entire minichannel is of interest, but the effect of various parameters discussed above is particularly tested at the centerline (r/RECT=0) of the minichannel. For droplet growth calculations, particle with diameter of 20 nm is used as a seed particle.

The results obtained from the CFD analysis are presented from Figure 15-24 for the various parameters discussed below.

5.3.3.1. Effect of Flow Rate on Saturation and droplet growth

The flow rate of the incoming air-vapor stream determines the residence time of convective flow inside the minichannel which directly determines how well the mixing between the incoming cold stream and the diffusing vapor and energy takes place. Since the incoming flow is fully developed, velocity boundary layer formation near the inlet of the minichannel decreases the convective flow velocity near the wall. This allows for the vapor to diffusive towards the centerline although the strength of diffusive flux from the wall is orders of magnitude smaller than the convective flow current of the incoming stream. The incoming air interacts with developing thermal and concentration boundary which start to

50 develop at the inlet of the minichannel. This interaction imparts axial momentum to diffusive vapor flow, which continues to diffuse towards the centerline as the flow is carried to the downstream. Figure 16 shows the effect these dynamics have on the vapor distribution inside the minichannel. The flow rate has been defined in terms of Reynolds number of incoming flow for a volumetric flow rate of 0.2 liters per minute to 0.8 liters per minute.

Increasing the flow rate decreases the residence time for vapors to mix while simultaneously suppressing the thermal and concentration boundary layer development. Thus it reduces the energy exchange between interacting species. Thus at lower flow rates, the onset of supersaturation ratio arises much more quickly, and this onset is delayed as the flow rate increases. From Figure 16(a), it can be observed that the saturation ratio increases progressively along the channel, reaching a peak and thereafter decreases. At the downstream of the minichannel, temperature increases continuously on account of energy influx from the sidewalls (Figure 17(a)). Since the saturation vapor pressure is exponential function of the temperature, small rise in temperature results in higher saturation vapor pressure at a location. This detrimental effect is compensated by increase in diffusing species from the sidewall.

However, depending upon the flow rate a point is reached when the detrimental effect overtakes the positive influence of increase in vapor concentration. At this juncture, the saturation conditions inside the minichannel start to deteriorate.

Figure 16(b) shows the saturation ratio profile at different radial locations along the axis of the minichannel. As can be seen, the saturation ratio profile has a much more rapid development near the wall and develops slowly at the centerline of the minichannel. Since the wall is already saturated, it is expected that very rapid mixing with incoming air-vapor stream near the wall will result in sharp increment in saturation ratio. However, the initial cooling is rapidly overtaken by increasing energy influx from the wall and as a result the temperature near the wall is much higher and decreases progressively away from the wall. This results in deterioration of saturation profile away from the wall. By the time energy and vapor are transferred to the center, the temperature is much lower as compared to the wall and concentration is very high. This results in highest supersaturation at the centerline of the minichannel.

Aside from above discussed factors, increasing the Reynolds flow changes the dynamics of relative humidity distribution inside the minichannel (Figure 16). Figure 17(b) shows the variation of volumetric distribution of vapor in different range inside the minichannel. From Figure 16(a), it can be deduced that even low flow rate of Re=92 is sufficient to cause peak saturation ratio of 1.5 at the centerline of the minichannel. Thus this may lead us to conclude that low flow rates are better in establishing highest supersaturation conditions inside the minichannel. However Figure 17(b) reveals that at Re=92, most of the volume of minichannel has supersaturation in range of 1-1.1. As Reynolds number increases, the mixing provides much more rapid cooling and this effect spreads further inside deeper regions of the minichannel. As a result, the volume of region with increasing saturation ratio range increases progressively as flow rate is increased. But then, this increase is limited by degradation of locally available vapor concentration. From the graph we can ascertain that a flow rate in range of Re=93-232 is optimum for producing requisite conditions of supersaturation inside the minichannel.

These dynamics of vapor distribution and saturation have extensive impact on nucleation and growth of nanoparticles at different locations inside the minichannel. The growth curves of 20 nm nanoparticle flowing along the inlet of the air-vapor mixture for different flow Reynolds Number are shown in Figure 18(a). The growth of the nanoparticle begins at the location where the saturation profile exceeds the minimum saturation required to nucleate that particle size. The growth of droplet by condensation takes place concurrently with increment in droplet heating due to latent heat release. (Figure

3(a) and Figure 18(b)). The accommodation coefficient of molecules is taken as 0.4 [5] which makes the condensation process much slower in this mode and hence the droplet heating is far lower as compared to continuum regime. It can be seen that the maximum increase in droplet temperature is nearly similar (~

6.5 K) for Re in range of 93-232. As can be seen, droplets at low Reynolds number grow to biggest size on account of increased residence time. At Re=92, the residence time for droplets is 64 ms, for Re =139 the residence time is 42 ms and for Re=371, the residence time is 16 ms. It is clear that droplet growth is a very fast process and in less than a second, droplet can grow from 20 nm to more than a micron size, once the particle is nucleated.

(a) (b)

(e) (f)

(g)

Figure 15 Saturation Ratio Contours inside the device Heating (Tw=328 K,, RHi=0.5, Ti=298 K) at r/RECT =0

1.5

1.3

1.1

(a) Re= 93 0.9 Re= 139 Re= 185 Saturation Ratio Re= 232 0.7 Re= 278 Re= 324 Re= 371 0.5 0 0.01 0.02 0.03 Length Along Axis (m)

1.7

1.5

1.3

1.1 (b) 0.9 r/R=0 r/R=0.25 Saturation Ratio 0.7 r/R=0.50 r/R=0.75 r/R=0.9 0.5 0 0.01 0.02 0.03 Length Along Axis (m)

Figure 16 Variation of Saturation Ratio (a) for varying flow rate (Tw=328 K, RHi=0.5, Ti=298

K) at r/RECT =0 (b) at different radial locations along axial for flow rate of Re=139 (Tw=328 K,

RHi=0.5, Ti=298 K)

Re= 93 328 Re= 139 Re= 185 Re= 232 323 Re= 278 Re= 324 Re= 371 318

313 (a)

308 Temperature (K) Temperature

303

298 0 0.01 0.02 0.03 Length Along Axis (m)

0.4 1.0<=SR<1.1 1.1<=SR<1.2 0.35 1.2<=SR<1.3 1.3<=SR<1.4 0.3 1.4<=SR<1.5 1.5<=SR 0.25

0.2 (b) 0.15

0.1 Volume Fraction (v/V) Fraction Volume 0.05

0 90 140 190 240 290 340 390 Re

Figure 17 Variation of (a) Temperature at r/RECT =0 (b) Volume fraction of Saturation Ratio

inside the minichannel for flow rate of Re=139 (Tw=328 K, RHi=0.5, Ti=298 K)

7 Re= 93 Re= 139 6 Re= 185 Re= 232 5 Re= 278 Re= 324 Re= 371 4

3 (a)

2 Droplet Diameter Diameter Droplet (µm) 1

0 0 0.01 0.02 0.03 Length Along Axis (m)

8 Re= 93 Re= 139 7 Re= 185 Re= 232 6 Re= 278 Re= 324 Re= 371 5

-T (K) -T (b) d

T 3

0 0 0.01 0.02 0.03 Length Along Axis (m)

Figure 18 (a) Droplet growth and (b) associated droplet heating at r/RECT =0 for varying flow

rate (Tw=328 K, RHi=0.5, Ti=298 K)

5.3.3.2. Effect of Wall Heating on Saturation Profile

The effect of increasing temperature of the wall on saturation profile at the centerline of the minichannel is shown in Figure 19(a). As expected, increasing wall temperature increases saturation conditions inside the minichannel. The amount of saturated vapor available at the wall is directly a function of wall temperature. Thus when wall is heated, more amount of vapor is available to be distributed inside the minichannel. As a result, the location at which the critical supersaturation is achieved progresses towards the upstream when wall temperature is increased. The strength of diffusive flux also increases and so the increase in saturation conditions becomes sharper as the temperature is increased. However, increasing wall temperature also results in increment of the species temperature. This increment in temperature inside the channel results in steeper decline once the positive effects of vapor concentration wear off inside the minichannel.

Thus Increment of wall temperature results in higher amount of saturation ratio which enhances the condensation process. Since the residence time of the particles is same, this results in much higher release of latent heat of vaporization by condensing species. As can be seen from Figure 19(b), this causes much higher droplet heating for higher wall temperatures. Owing to these factors, the droplet growth is spurred by increase in wall temperature; however the increase becomes smaller as the wall temperature is increased. The droplet diameter at the exit of the minichannel is 3 µm when wall temperature is 328 K.

When the wall temperature is increase by 5 K, the droplet size increases by 22% (Figure 20). A further increase by 5 K increases the final droplet size by 13% and when wall temperature is raised further by 5

K, the increase in final droplet size is by 8%. The decreasing rate of diameter final size with increasing the wall temperature is attributed to increase in droplet heating by condensation process.

1.7

1.5

1.3

1.1 (a)

0.9 Tw= 323K Saturation Ratio Saturation Tw= 328K 0.7 Tw= 333K Tw= 338K 0.5 0 0.01 0.02 0.03 Length Along Axis (m)

10 Tw= 323K 9 Tw= 328K 8 Tw= 333K 7 Tw= 338K 6

-T (K) -T (b) d 4 T 3

0 0 0.01 0.02 0.03 Length Along Axis (m)

Figure 19 Variation of (a) Saturation Ratio (b) associated droplet heating for varying wall temperature (Re=139, RHi=0.5, Ti=298 K) at r/RECT =0

5 Tw= 323 K 4.5 Tw= 328 K 4 Tw= 333 K 3.5 Tw= 338 K 3

2.5

1.5

Droplet Diameter Diameter Droplet (µm) 1

0.5

0 0 0.010.020.03 Length Along Axis (m)

Figure 20 Droplet growth for varying wall temperature (Re=139, RHi=0.5, Ti=298 K) at r/RECT =0

5.3.3.3. Effect of Relative Humidity on Saturation Profile

The diffusion of the saturated vapor from the wall is based on the concentration gradient of the vapor species inside the minichannel. When the incoming air-vapor mixture saturation increases, larger quantity of vapor is available to mix with the diffusing air-vapor components. As a result, saturation conditions inside the channel increase. The variation of saturation profile at the centerline with relative humidity is given in Figure 21 (a). Figure 21(a) clearly shows that increasing relative humidity at the inlet is advantageous as increment, as at the same location higher vapor concentration is available. Thus initiation of nucleation and condensation proceeds earlier when the relative humidity at the saturation is increased. Moreover, enhancement of vapor concentration does not significantly change the thermal conditions inside the minichannel. The amount of vapor even when incoming air is fully saturated at 25 oC is insufficient to change the mixture physical properties significantly. As a result, temperature profile remains unaltered while the vapor concentration increases (Figure 21(b)).

It should be noted that even in the worst conditions when the incoming air is fully unsaturated, the critical conditions for initiation of nucleation are met. Although the location of this event is delayed, it occurs roughly at halfway through the channel axis. Even then, once the droplet nucleation is initiated, it rapidly progresses in its diameter to reach micron size scales. As before for the case of increment in wall temperature, presence of large supersaturated conditions result in rapid condensation process which increases the amount of droplet heating (Figure 22(a)). For a fully unsaturated case, the droplet heating is of order of 2.8 K, while for fully saturated inlet air-vapor mixture the maximum droplet heating increases up to 6 K. The marginal increase in droplet diameter size at the exit decreases by increase of RH on account of increased droplet heating, but this decrease in droplet diameter gain is much slower as compared to effect of increasing wall temperature (Figure 22(b)). Nucleation and condensation for fully saturated incoming fluid begins almost less than a third of the minichannel, while for worst scenario of

RH = 0, the droplet diameter begins to increase after almost half of the channel length.

1.6

1.4

1.2

0.8 (a) 0.6 RH= 0 RH= 0.25

Saturation Ratio 0.4 RH= 0.5 0.2 RH= 0.75 RH= 1 0 0 0.01 0.02 0.03 Length Along Axis (m)

323 RH= 0 RH= 0.25 318 RH= 0.5 RH= 0.75 RH= 1 313

(b) 308 Temperature (K) Temperature 303

298 0 0.01 0.02 0.03 Length Along Axis (m)

Figure 21 Variation of (a) Saturation Ratio (b) Temperature for varying relative humidity at inlet (Re=139, Tw=328 K, Ti=298 K) at r/RECT =0

7 RH= 0 RH= 0.25 6 RH= 0.5 RH= 0.75 5 RH= 1

-T (K) -T (a)

d 3 T

0 0 0.01 0.02 0.03 Length Along Axis (m)

4.5 RH= 0 4 RH= 0.25 3.5 RH= 0.50 RH= 0.75 3 RH= 1.00 2.5

2 (b)

1.5

1 Droplet Diameter Diameter Droplet (µm)

0.5

0 0 0.010.020.03 Length Along Axis (m)

Figure 22 Variation of (a) droplet heating (b) associated droplet growth for varying relative humidity at inlet (Re=139, Tw=328 K, Ti=298 K) at r/RECT =0

5.3.3.4. Effect of Temperature of incoming fluid stream on Saturation Ratio and Droplet Growth

Out of the discussed parameters, the temperature of the inlet fluid stream has the most significant impact on deciding the attainment of requisite conditions inside the minichannel. For studying this variation, the temperature of the wall was kept at 328 K and the incoming fluid stream temperature was varied from 298 K to 358 K, while the relative humidity of the incoming air was maintained at 0.5. It is noticeable that a 10 degree rise in temperature from 298 K to 308 K of incoming air decreases the maximum saturation ratio from 1.35 to 1.09 – a 20% drop (Figure 23(a)). Further increase in temperature decreases the saturation conditions to an extent where nucleation cannot happen. When the incoming air temperature is less than wall temperature, both the thermal and species transport occurs from the wall to the center of the minichannel. At inlet temperature of 328 K, the system gains isothermal status, but the vapor diffusion from wall to the center is still active because of difference in saturation condition between wall and the incoming fluid stream. As temperature of inlet increases further, inversion of thermal field takes place (Figure 23(b)). The temperature gradient takes place from the inlet stream towards the wall.

Even in this condition, vapor diffusion takes place from wall to the centerline because the concentration gradient has insignificant change by change of thermodynamic environment. The increase in temperature causes increase in saturation vapor pressure which decreases the saturation profile in the local environment.

In these conditions, droplet growth occurs only for cases of inlet temperature of 298 K and 308 K.

Thus a minimum of 20 K temperature difference is required between wall temperature and incoming stream to produce conditions where condensation can take place. For all the following cases, no condensation will occur. A 10 degree rise from 298 K to 308 K of the inlet air-vapor temperature decreases droplet diameter by 38% (Figure 24).

1.4 Tin= 298 K 1.3 Tin= 308 K Tin= 318 K 1.2 Tin= 328 K Tin= 338 K 1.1 Tin= 348 K Tin= 358 K 1

0.9 (a)

0.8

Saturation Ratio Saturation 0.7

0.6

0.5 0 0.01 0.02 0.03 Length Along Axis (m)

358

348

338

328 (b) 318 Tin= 298 K Tin= 308 K Temperature (K) Temperature Tin= 318 K 308 Tin= 328 K Tin= 338 K Tin= 348 K Tin= 358 K 298 0 0.01 0.02 0.03 Length Along Axis (m)

Figure 23 Variation of (a) Saturation Ratio (b) Temperature for varying temperature of inlet fluid stream (Re=139, Tw=328 K, RHi=0.5) at r/RECT =0

Tin= 298 K Tin= 308 K 4 Tin= 318 K Tin= 328 K Tin= 338 K Tin= 348 K 3 Tin= 358 K

-T (K) -T 2

d (a) T

0 0 0.01 0.02 0.03 Length Along Axis (m)

3.5 Tin= 298 K 3 Tin= 308 K

2.5

1.5 (b)

1 Droplet Diameter Diameter Droplet (µm) 0.5

0 0 0.01 0.02 0.03 Length Along Axis (m)

Figure 24 Variation of (a) droplet heating (b) associated droplet growth for varying temperature of inlet fluid stream (Re=139, Tw=328 K, RHi=0.5) at r/RECT =0

CHAPTER - 6

CONCLUSIONS

A novel minichannel device is presented which is capable of identifying presence of

nanoparticles in extremely small number densities in an environment. The device works on principle of

heterogeneous nucleation accompanied by droplet growth on ultrafine particles in an environment of

supersaturated vapor.

Two models for generation of supersaturation conditions were studied in present work. The first

model involves two concentric minichannels separated by a gap of 500 µm, and occupied by water using

capillary forces. The channel is heated near the base of the inlet to heat up water in the gap, while rest of

the channel is cooled by means of natural convection. The second model is based on a single minichannel

which is uniformly heated to an elevated temperature as compared to the inlet air. Computational

modeling was conducted to understand affect of various parameters such as volumetric flow rate,

humidity of incoming particle laden air and temperature of water vapor diffusing inside the condensation

chamber. It is shown that both of the designs can provide conditions which aid in supersaturation

development inside the minichannel

In general it was found that optimum sensor design for the convectively cooled channel can be

obtained for volumetric flow rate corresponding to 3.3*10-6 to 8.33*10-6 m3/s, channel diameter of 3 mm, water reservoir temperature of around 55 to 65 oC, and cooling by means of natural convection.

Depending upon controllable parameters such as wall heating, flow rate, temperature and relative

humidity of incoming fluid stream, the vapor distribution inside minichannel possesses nucleation

capability to nucleate particles as small as 2 nm – 100 nm. Numerical growth studies reveal that a 20 nm

can be grown to 3-4 micron in size in less than a period of 60 ms for both designs. Increasing flow rate

decreases residence time of vapor inside the minichannel and leads to imperfect energy exchange between

incoming colder fluid stream and vapor diffusing from the warm wall. These two opposing forces govern

66 the fate of droplet growth inside the minichannel and it is expected that droplet growth will vary according to radial location of the minichannel.

For the design involving uniformly heated wall, inlet flow Reynolds number in range of Re = 93

– 232 is optimized range for operation of the minichannel considered in this study. Increasing wall temperature and inlet relative humidity favor conditions which enhance droplet condensation and cause larger droplet growth. However, the minichannel performance is severely limited by the temperature of incoming air. A minimum temperature difference between incoming stream and wall temperature is necessary to produce conditions in which nanoparticles can be nucleated and allowed to grow to an optimal size. An optimum range of temperature difference for ensuring proper attainability of supersaturation conditions inside the minichannel is around 30-40 K.

In general it can be concluded that supersaturation conditions can be generated using water as working fluid by using a warm wetted column and mixing it with stream of cold air-vapor mixture. In this manner, the design involving uniformly heated wet wall offers better conditions as compared to the convectively cooled minichannel.

PART 2:

DROPLET GROWTH IN RAREFIED ENVIRONMENT

CHAPTER - 7

DROPLET SIZE MEASUREMENT IN RAREFIED ENVIRONMENT

The pressure of the environment plays crucial role in the transport processes associated with the phase change process. Depending upon the pressure, the intermolecular distance may vary, this in turn affects the rate at which collisions may take place between different molecules. A larger number density of particles means frequent collisions, and other forces such as diffusion may arise. On the other hand in the rarefied environment, the transport processes are completely kinetic controlled. Thus the transport behavior may change depending upon the chamber pressure and this is best captured through relating the size of a condensing droplet with mean free path of vapor molecules surrounding it. Figure 25 shows the affect of chamber pressure on the mean free path of the vapor molecules. In normal atmospheric conditions, the mean free path is in range of ~ 20nm, which is substantially lower than the size of condensing droplets.

22 20 18 Æ

L 16 m m

H 14 l 12 10

4 5 6 7 8 9 10 Pressure Torr Æ

Figure 25: Mean free path as a function of chamber pressure HL Thus it becomes virtually impossible to observe any change of transport behavior when droplet size becomes from smaller to larger than the mean free path. On the other hand, observing condensation behavior in rarefied environment allows capture of droplet sizes below and larger than the mean free path.

As an example, the mean free path of vapor molecules at 4.2 Torr is 22 µm and this has allowed us to

capture droplets whose size is much smaller than the mean free path and which grow to size much larger

than the mean free path.

In present studies, an experimental approach is used to study the physics behind droplet growth on a

partially wetting surface. Superheated vapor at low pressures of 4-5 torr was condensed on subcooled

silicon surface with static contact angle as of 60o in absence of non-condensable gases, and the

condensation process monitored using Environmental Scanning Electron Microscope (ESEM) with

submicroscopic spatial resolution. The condensation process was analyzed in the form of size growth of

isolated droplets for before a coalescence event ended the regime of single droplet growth.

Thus it becomes critical to understand the droplet growth behavior in the molecular region and the

continuum region. Such studies on the transport phenomenon are particularly difficult to visualize at

ambient pressures owing to much smaller mean free path of vapor molecules. However study of such

processes in rarefied environment provides opportunity to observe the transport mechanisms during

nucleation and growth process of condensate at large Knudsen numbers. Lowering pressure increases the

mean free path of the vapor molecules and hence the characteristic size of the minimum droplet size is

increased. At 760 torr and 17 oC the mean free path (λ) of water vapor is 0.11 µm, while at same temperature but at 4.2 torr MFP reaches 21.3 µm. Thus if there is an effect of mean free path on interfacial phase change kinetics, it is rendered visually noticeable in these conditions. Our research has led us to establish a methodology for systematic study of phase change processes such as condensation with ESEM through which studies on condensation in molecular flow regime can be investigated.

CHAPTER - 8

LITERATURE REVIEW

8.1. Droplet Growth Kinetics on flat surfaces

Phase change processes involving liquid solid interaction such as condensation explicitly depend upon these surface energy interactions. Based on surface free energy of a solid surface, condensation can be described as filmwise, dropwise or mixed condensation.[33] After initiation of nucleation, the role of surface energy gains more prominence during the growth stage. The manner in which the vapor deposition occurs depends upon the bonding of depositing molecules to the substrate.[115] The balance of surface energy with interfacial energy determines the shape of the nuclei and nature of growth during later stages.[116] The deposition studies in materials industry have resulted in establishing a broad categorization of vapor deposition behavior in terms of three modes – the island mode or the Volmer-

Weber mode, The 2D layer growth or Frank-van der Merwe mode and island-layer or Stranski-Krastanov growth.[104] On high energy surfaces, the deposition of atoms results in monolayer formation first and the minimization of free energy dictates that atoms adhere much strongly to the substrate rather than bonding with each other. This results in 2D layer growth, because further interaction leads to formation from monolayer to multilayer on top of each other. For low energy surfaces, condensing atoms nucleate on substrate and tend to adhere strongly to each other as compared to bonding with substrate leading to formation of islands or droplets.[104] Surfaces with intermediate values of surface energy tend to form monolayer first and thereafter the growth proceeds in island/drop fashion.

Since dropwise condensation is a transient process where surface coverage of liquid on a substrate increases with time, studying droplet growth kinetics is vital to predict the associated transient nature of heat transfer. Different growth laws have been reported previously for describing growth of droplets with and without the coalescence. Initial condensation experiments on silanized surfaces with impinging water vapor showed the dependence as Rαt1/4 for single droplet growth.[117, 118] This was also supported by 71 theoretical and numerical simulations which attributed this growth owing to diffusion of submicroscopic droplets under effect of concentration gradient.[119] Later analysis showed that the dependence however to be Rαt1/3 which was subsequently verified when the experiments were carried under gradual introduction of vapor.[120-123] Experiments using non-polar liquid with very low contact angle have shown these drops growing to be linearly with time.[124] It is clear that our understanding on mechanism and factors affecting droplet growth during condensation is not established fully.

Previous research for water vapor condensation suggests that at higher pressures drop conduction is the limiting resistance, while at the lower pressures interfacial heat transfer gains more importance.[51,

125] A droplet growth model was presented by Umur based on kinetic theory wherein heat transfer through conduction was considered. Using the theory it was proved that interfacial heat transfer coefficient decreases by decreasing the operating pressure. Further the steam side heat transfer coefficient was found to decrease with decreasing pressure. This was supported by experimental observations.[126,

127] However many aspects of vapor-liquid phase change remain unresolved owing to limitations of experimental setups such as spatial resolution, contamination issues, presence of non-condensable gases etc.

8.2. Accommodation Coefficient and its importance

The vapor deposition and subsequent condensation begins as a molecular process and the first nuclei may be composed of only few molecules. Further accretion of molecules to the condensing island/drop gives rise to intermolecular forces between condensed vapor molecules giving rise to the capillary forces.

From this juncture, the shape of the nuclei affects the saturation vapor pressure above the condensing vapor, which in return affects the vapor deposition rate and hence the shape of the condensate. Although the field has been well studied for a period of over several decades to deterministically represent the growth stage, the descriptive theories have not been entirely inclusive of the surface energy component.

The binding probability of a vapor molecule to a substrate is given in terms of ‘condensation accommodation coefficient’ αc. From theoretical perspective, description of condensation mechanism is

incomplete without knowing how many molecules in fact stay bonded to the substrate surface.

Unfortunately, there is no means to predict this very important factor. Irrefutably, the sticking probability

depends upon the surface energy of the substrate material and thus discerning its role is critical in

developing a comprehensive theory of condensation process.

8.3. ENVINRONMENTAL SCANNING ELECTRON MICROSCOPY

Graham and Griffirth [56] noted that at least 50% of heat transfer took place through drops which were less than 10 micron in diameter. However the experimental limitations associated with using optical microscopes do not allow studying droplets smaller than 10 microns. Hence submicroscopic condensation remains open to discussion. To overcome above mentioned limitations an Environmental Scanning

Electron Microscope (ESEM) was used. ESEM is a relatively new development in the field of SEM operations. It combines the ability of Scanning Electron Microscope (SEM) to resolve objects at submicroscopic resolution with ability to perform dynamic experiments like condensation and evaporation by using water vapor as inside gas at lower pressures (2-20 Torr).

The construction details of an ESEM are widely available.[128, 129] The availability of additional

tools such as Peltier Cooler, Stress-Strain stage, micro injector and heater stage have extended the use of

ESEM to more dynamical measurements. Although the physics behind the condensation-evaporation in

ESEM has been investigated,[130-132] application of ESEM towards studying fluid dynamics is a more

recent phenomenon. Investigators have used ESEM to observe contact angles on fiber surfaces which is

particularly useful to observe micron sized droplets on fine fibers as conventional techniques cannot

obtain this information.[133-135] ESEM has also been used to view aggregation and development of

colloidal films.[136, 137] Expressions have been developed to obtain contact angles from the visual

images obtained through ESEM wherever a droplet can be measured.[138] More recently ESEM has been

used towards studying fluid flows in carbon nanopipes.[139-141] Only in recent times ESEM has been

applied to study dynamical growth of droplets on different surfaces. With respect to work done by Rossi

et al. simple image analysis to find contact angle of meniscus visible because of disparate contrast from

its surroundings was made. Yarin et al. also performed experiments using Carbon nanopipes, and

simultaneously a theory of fluid was developed in carbon nanopipes.[141] Lauri et al.[142] used ESEM to

study droplet size growth on flat surfaces. Heterogeneous nucleation theory was used and the droplet size

increment was measured and presented. Bhushan et al have used ESEM to observe transition effects of condensation and evaporation on patterned surfaces.[143-145] ESEM images were analyzed and droplet size measurements obtained from the analysis to present size growth during condensation and droplet diameter decrease during evaporation.

While the lateral resolution to image samples is higher than normal optical microscopes, the imaging technique of ESEM gives a poor time resolution of order of seconds (1-10 sec). ESEM operation requires

the chamber to be completely vented before introducing gas such as water vapor inside it. This effectively

eliminates the probability of any non-condensable gas present in the chamber. This is of vital importance

as experimental studies have shown that even small quantity of non-condensable gases results in

significant deterioration in heat transfer performance by obstructing contact between vapor and the solid

surface.[146] The vapor inside the chamber is maintained at a particular pressure which is user controlled

with help of computer. Further, the vapor inside carries negligible momentum and hence samples can be

imaged without any surface shear effect due to vapor velocity. The rarefied environment inside the ESEM

enhances the role of interfacial forces in vapor to liquid conversion.

CHAPTER - 9

EXPERIMENTAL STUDIES ON DROPLET GROWTH MEASUREMENT

9.1. EXPERIMENTAL SETUP

Figure 26: Experimental Setup for Condensation Observation in ESEM

The experimental setup used for the present studies is illustrated in Figure 26.

Experiments were carried inside Philips XL-30 Environmental Scanning Electron Microscope. It is equipped with vacuum pump and a vapor supply line which is linked with a glass bottle filled with DI water. At first, ESEM chamber is brought to a pressure approximately equal to 10-6 torr by the application of the vacuum pump. Then the vapor supply line is opened up. Because of the vacuum pressure inside the chamber and the supply line, the DI water in the glass bottle boils at room temperature and the vapor

75 floods the ESEM chamber. The amount of vapor inside the chamber is computer controlled such that user designated pressure is maintained inside the chamber without significant fluctuations.

For surface cooling, a Peltier cooler stage was mounted inside the ESEM. Since the saturation vapor pressure is related to saturation temperature, a dynamical process such as water condensation can be studied in ESEM by using the Peltier cooler in conjunction with controlling the pressure and consequently the Relative Humidity (RH) in the ESEM chamber. When RH>100%, condensation can be observed immediately. Thermodynamically, the saturation vapor pressure and temperature dependence for ESEM is shown in Figure 27. The saturation vapor pressure-temperature curve of ESEM differs from the standard curve because the temperature of the vapor Tv above the sample is different from the sample temperature Ts. The thermodynamic relation between the saturation vapor pressure and temperature for condensation to take place can be expressed by the Eqn (1) as given by Cameron and Donald[147]

0.5 ⎛⎞Tsub ppvS= ⎜⎟ ()Tsub (36) ⎝⎠Tv

Where, PS(Tsub) is saturation vapor pressure at temperature Ts.

Figure 27. Saturation Vapor Pressure – Temperature chart for ESEM

Since the vapor inside the ESEM chamber had originated from the boiling of water under low pressure,

and the water temperature was maintained at ambient, the vapor temperature is same as water temperature

in the glass bottle (around 17 oC). Thus with respect to the pressure inside the chamber (around 4.2 torr),

the vapor is in superheated condition.

The Philips XL30 ESEM has a K-Type thermocouple connector available which was used in our studies.

It was connected with the K-type thermocouple attached on Si surface and externally connected to a Data

Acquisition Device (Agilent 34970A). A LabViewTM program was used to monitor the thermocouple

temperature and save the data on computer. Initially to test the peltier readout, it was attached with the

peltier surface. Substantial temperature difference between the thermocouple and temperature shown by

the peltier cooler was observed. As a result it was decided to use K-type thermocouple as the basis for

calculating the saturation vapor pressure-temperature curve. According to the theoretical predictions, the

saturation temperature at 4.2 torr is -0.675 oC. However, using the K-type thermocouple, the temperature at which condensation at 4.2 torr of vapor pressure was found to be -0.5 oC. In absence of proper calibration of the thermocouple and the inherent uncertainty in K-type thermocouple (+/- 0.5 oC), this degree of temperature difference can be anticipated in the measurements.

Droplet growth was visualized in two ways with respect to substrate plane – from the top and from an almost vertical stance. For observing the substrate on the top, it was attached to the peltier surface using thermally conductive carbon tape. A special copper stage was prepared to rest on the peltier cooler to observe growth of droplets from side view to observe contact angle changes. The copper stage had an inclined surface of 80o as shown in Figure 28. The bare silicon wafer was placed on the inclined surface using thermally conductive carbon tape. In this arrangement, the thermocouple was attached with copper stage instead of Silicon surface to allow proper viewing.

The imaging of specimen inside the chamber was done using Gaseous Secondary Electron Detector

(GSED). The ESEM imaging involves an electron beam scanning across a given area. Electron beam parameters like dwell time and scan rate decide the amount of time taken to scan a given area visible on screen. These parameters were adjusted and a small area was selected to give a scan time of 2-10 seconds

with adequate clarity. The imaging detector feeds live images to the computer. A BNC video output was

used to divert the image signals to a Sony DVCAM, and live video recording was done. The timing of the

DVCAM was manually synchronized with computer to match temperature conditions with associated

imaging signal.

Figure 28. Copper stage for Observing droplet condensation on an inclined surface

9.2. SUBSTRATE PREPARATION

Bare Silicon wafer was used as condensation surface. A small piece (2 mm X 2 mm) of Silicon wafer was

prepared and cleaned thoroughly using RCA cleaning process to remove any organic or inorganic

contaminants present on the surface. Just before the experiment, the silicon wafer was dipped in Buffered

Oxide Etched Solution for 30 seconds to remove any naturally formed SiO2 layer on the wafer surface.

Through this method contact angles of around 62 ± 2o were obtained for water on the silicon surface. A

K-type thermocouple was planted on the Silicon surface using thermal epoxy to measure surface temperature. Due care was taken while application of the epoxy to the surface to prevent addition of large thermal mass while at same time maintaining proper contact between thermocouple tip and surface.

9.3. EXPERIMENTAL METHODOLOGY

Once the system was setup, the first step involved venting the chamber and filling it with pure water

vapor. An eight cycle pumpdown sequence between 0.5 torr to 9 torr was performed to make sure air is

properly vented out and chamber is filled with saturated vapor pressure in the end.[147] The presence of a

foreign surface in vicinity of metastable vapor lowers the energy barrier required to cause nucleation as

compared to homogeneous nucleation.[123] Characteristics such as contact angle, defects, and scratches

influence the nucleation capability on a surface. Thus, at the same pressure, dropwise condensation can occur at slightly higher temperature than predicted by the theoretical modified equation. It was noted that in rough regions along the edges of the substrate, condensation began at a still higher temperature than shown in Figure 27. Clearly, the micro-cavities or edges were sites of much lowered surface energy barrier which resulted in this phenomenon.

Once the pumpdown was performed, condensation can be initiated by manipulating chamber pressure and

peltier temperature which were also the controlling parameters. This condensation process was recorded

at different magnifications (250X, 500X, 1500X, 3500X and 6500X) and for accelerating voltage of 15

keV. The observation of submicroscopic droplets requires higher beam voltage for sharper contrast, thus

the beam voltage was increased to 25 keV for better visualization. Since the removal of condensing

droplets cannot be imposed inside the experimental setup, so condensation once triggered eventually

leads to complete surface coverage of the substrate. An area was selected randomly to observe a droplet

growth. After the droplet merged with another droplet, the focus of electron gun was moved to another

area to observe another droplet which had a higher probability of growing without merger because of

lesser population density of droplets around it. With this method, droplet growth histories were recorded

for different droplets at different stages of their growth. The choice of electron beam voltage was based

on determination of best contrast for visualization, while also minimizing any effect of beam heating.

Hence these experiments were carried at 15 keV.

Post processing results involved image analysis to determine droplet growth rate which was done using

ImageJ software.[148] The controlling parameters for the study were chamber pressure and the peltier

cooler temperature.

9.3.1. Method of condensation

As mentioned above, the condensation process can be triggered by controlling the pressure and temperature. In particular, there are two methods to achieve condensation:

Mode 1- Constant vapor pressure, varying substrate temperature. The pressure is fixed and the temperature is slowly decreased to initiate dropwise condensation

Mode 2- Constant substrate temperature, varying vapor pressure. The temperature of the peltier cooler is fixed and the pressure is slowly increased until condensation process starts.

We found that the droplet growth and coalescence results were affected to some extent by the method to achieve condensation. This happened because of its influence on the time required to achieve a steady state.

It was observed that temperature control was much more difficult in the first mode. This resulted in non-uniform droplet growth. On the other hand, the second mode gave a more uniform temperature profile and the time required to reach steady state was drastically reduced (Figure 29 and Figure 30).

Figure 29. Temperature Profile during Condensation Process

Figure 30. Example of Temperature Profile during Condensation Process (Pressure: 4.4 Torr)

In this mode, a more uniform model of coalescence and droplet growth was achieved which is discussed in the following sections.

9.3.2. Effect of Beam Heating

Since the mechanism of imaging in ESEM involves line scan of an electron beam over the surface, the impinging electrons can cause instantaneous heating of the spot at which impingement is occurring.

Although the beam current may be less and the electron beam diameter is of order of nanometers, this can result in a very high dosage of electrons concentrated over a very small area and hence high heat flux may ensue. If the beam heating is significant, it can affect the condensation process. To ascertain the effect of beam heating, a 100 μm junction diameter thermocouple wire (Figure 31) was inserted in the ESEM chamber and was exposed to electron beam under vacuum conditions (pressure ~10-6 torr).

100 μm

Figure 31. Thermocouple tip

Two modes are possible – spot mode and line scan mode and both of these modes were chosen to evaluate the effect of heating. It was found that beam heating was dependent upon several factors such as line time, beam voltage, mode type and spot size. Figure 32 and Figure 33 shows the effect of beam heating with line scan mode and spot mode (for spot size 3) on thermocouple tip.

Figure 32. Beam Heating with Beam Voltage for line mode and spot mode

Figure 33. Beam Heating with respect to spot mode (25 keV)

In general it was found that spot mode produced higher amount of heating and a larger spot size

produced larger amount of heating. A maximum of 0.5 K of heating was observed under the spot mode

for the current operating conditions. Increasing the spot size in general led to increase in temperature.

For imaging purposes, the map scan mode was used during the experiment which involved electron beam rastered across an entire chosen area. For current chosen conditions of 15 keV beam voltage and line time of 16.7 ms, it can be observed that beam heating is of order of 0.1 K. Increasing the beam voltage to 25 keV resulted in beam heating to the order of 0.3 K. It should be noted that current beam heating experiment was performed on a thermocouple tip which is of thermally and electrically conducting in nature.

On insulating materials, the effect of concentrated beam of electrons may be different from present observations. Under ESEM conditions the electron beam width is enhanced to the order of μm because of

collision events of electrons with vapor molecules inside.[149, 150] Thus the dosage of electrons reaching

a spot is greatly reduced and it can be expected that beam impingement would have negligible heating

effect.

9.4. RESULTS: DROPWISE GROWTH FOR SINGLE DROPLETS

As mentioned previously, understanding evolution of a single droplet can provide us with

understanding of heat transfer through the droplet which can then be used to evaluate over all heat

transfer. The droplet growth may occur by different mechanisms depending upon physical and

thermodynamic conditions such as.[123]

i) Direct phase change of vapor to liquid at the drop surface.

ii) Thermal release of interphase mass conversion can lead to Marangoni forces which trigger

temperature gradients inside the droplet. This can induce local subcooling at droplet periphery so

probability of vapor condensation increases along the periphery.

iii) Nucleation and growth of the droplets by means of surface diffusion. This has widely been

observed for water on several substrates. Experimental observations by others have led to a

generalized droplet growth model of D = ktn where D is the droplet diameter growing over a

period of time t and n and k are empirical constants. The value of n has been reported as 1/3 or

1/4 in cases where surface diffusion is a prominent mechanism.[120-123]

Irrespective of the mechanism involving how a droplet evolves, the condensational growth of droplet formation is a multi-stage process.[121] The first stage involves droplet growth of single and individual droplets wherein a number of nucleation sites may become active. This stage is marked by low surface coverage. As droplets grow in size, the inter-droplet distance decreases until the droplets meet together, where after the combined effect of surface tension and growth leads to coalescence of these droplets. This is referred to as second stage in the growth. Beyond this, in third stage more coalescence may occur and new droplets may be nucleated in the bare area between the droplets.

The essential aim of this study was to understand the first stage involving how an individual droplet evolves over time. However measuring droplet growth for single droplets over an extensive period of time is significantly difficult because the time for droplet interaction leading to coalescence is very short – lasting only to few number of frames. Droplet interaction depends upon the degree of subcooling with larger subcooling initiating an increased number of nucleation sites and hence increasing the population density of droplets on the surface. This then results in significant coalescence events and a much shorter

time period where a single droplet can evolve without interaction from neighboring droplets. Thus to record substantial period of growth devoid of the coalescence behavior is a challenge. Using a very low degree of subcooling the number of population sites can be controlled. Further, lower degree of

subcooling also decreases the droplet growth which can further help in observation of droplet evolution

over a larger period of time.

The temperature of the vapor inside the ESEM chamber is 17 oC, while the temperature at which condensation occurs is -0.675 oC. From this information and by usual convention, it can be affirmed that

the degree of subcooling is 17.675 oC. However, it should be noted that the presence of peltier cooler establishes a temperature gradient around the substrate. Around the substrate ~O(µm), the temperature of the vapor molecules is much closer to the actual peltier cooler. Thus the effective degree of subcooling is much lower when the peltier cooler is active. Accurate prediction of the degree of subcooling can be made by measuring temperature of vapor just above the peltier cooler surface. Since an additional thermocouple measurement cannot be incorporated in the experimental setup, prediction of effective degree of subcooling was not achievable. By controlling the peltier cooler temperature, we were able to locate and measure droplets growing on their own without merger for time periods in the range of 40-180 seconds.

Figure 34, shows a series of frame containing condensed droplets at 500X magnification on Silicon surface. The behavior of condensation pattern observed in Figure 34 is typical for non wetting surfaces because the bond between depositing molecules is stronger as compared to their attraction with the substrate.

Drop 2 Drop 2 Drop 1 Drop 1

Drop 3 (a) (b) t=0 s t=10.34 s

Drop 2 Drop 2 Drop 1 Drop 1

Drop 3 (c) Drop 3 (d) t=20.68 s t=31.02 s

Drop 2 Drop 1 Drop 1

Coalesced Drop

Drop 3 (e) (f) t=41.36 s t=51.70 s

Figure 34. Condensed droplets on Si surface (4.2 Torr) (Constant Substrate Temperature:-0.5 oC, Varying

vapor pressure mode)

9.5. RESULTS: METHODOLOGY FOR DIAMETETIC DROPLET GROWTH ANALYSIS

Image analysis of droplets such as shown in Figure 34 provides measurement of the diameter in terms of its base diameter (Figure 35).

Base Diameter( DBase )

R p d Droplet Height( H ) v p l θ

R R cosθ d Substrate d θ

RBase

Figure 35. Droplet on a surface

If RBase refers to the base radius of the droplet and R refers to the radius of curvature of the droplet, then a relation between these radii and their corresponding diameters can be expressed as

D ==2RDdBase and 2 RBase (37)

As can be observed from Figure (35), the base diameter (DBase) of a droplet sitting on a substrate is related to the radius of curvature of the droplet (Rd). This relation can be expressed as

RRdBase=⇒=sinθ DDBasesinθ (38)

Since the analyzed images give the estimated size of base diameter, Eqn. 38 was used to convert their representation in terms of base diameter to the radius of curvature. Thus the modified values of experimentally measured radii can be compared with the theoretical predictions of Eqn. 51, which is expressed in terms of the radius of curvature itself.

The height of the droplet sitting on a surface can be related to the radius of curvature of the droplet and the contact angle the droplet is making with the surface. This relation can be given by

HR=−d ()1cosθ (39)

Thus the base area of the droplet which is the substrate area covered by the droplet can be given by

222 Β=π RRBase =πθ d sin (40)

Since a droplet sitting on a surface is essentially a part of a sphere, the interfacial area and the

volume of the droplet can be expressed in terms of the radius of curvature and the contact angle as

2 Α=2π RHdd ⇒Α= 21cosπθR () − (41)

and

π HRH2 ()3 − Ω= d 3 (42) π R3 ()(2+− cosθθ 1 cos )2 ⇒Ω= d 3

Each droplet was identified with the initial peripheral diameter measured as the diameter of the droplet in the first frame of a series of frames where the droplet was observed to grow over a period of time. As mentioned previously, the time between successive growths is dependent upon the area and the scan rate of the electron beam. Thus growth record of each droplet was obtained using this technique. Figure (36) is

an indicative plot of droplet growth as a function of time for the three different droplets showed in Figure

34. The different droplets are identified based on the initial diameter.

Figure 36. Droplet diameter growth for individual droplets occurring in same frame with different initial

diameters at 4.2 Torr (Constant substrate temperature, varying vapor pressure mode)

The growth curve as shown in Figure 36 was found to be characteristic of droplets evolving under conditions under study. After careful analysis, many characteristics were revealed on the nature of droplets. As can be seen from Figure 36, the droplet diameter appears to grow linearly with time. A similar linear pattern was also observed when the data was plotted in terms of droplet area (D2) with time,

thereby giving an impression that D2 ∝ t . After careful consideration the perplexing nature of the

observed drop diameter growth pattern was resolved, when it was noted that the rate of diameter growth

was different for different droplets. As can be seen the rate of growth (dD/dt) was noted to decrease as the

diameter of the droplet increase.

The linear nature of diametrical growth for all the droplets gives a certain ambiguity to this observation.

The situation becomes clearer when the diameter was observed on a logarithmic scale as shown in Figure

37. From Figure 37, it becomes unambiguously clear that the rate of diametrical growth for smaller droplets is much higher than the rate at which larger droplets appear to grow. Moreover, the droplet diametrical growth rate appears to change as the droplet size reaches near the size of mean free path of vapor molecules in the chamber.

Figure 37. Droplet growth for individual droplets

However to understand the mechanism behind this behavior it is essential to identify an overall growth pattern where each of the droplet growth patterns can be fitted in. It is clear that droplet evolution observed for a period of 40 - 100 seconds was not enough to predict the mechanism by observing how one of the many single droplets was growing.

For this purpose, a scheme was employed to visualize droplets observed at different times and different stages of growth as a single droplet growth curve. From the records of droplet growth, the droplet with

90 smallest initial diameter (D01) analyzed was identified. Using polynomial fitting, a fourth order equation was fitted into this curve to obtain a relation between diameter and time. Then the droplet with next smallest initial diameter (D02) was chosen to fit into the polynomial curve of the earlier droplet for identifying the time t02 at which D02 would coincide. After identifying t02 the entire series of D02 was incremented in time by t02 to obtain the next curve. This process was repeated to interpolatively fit each individual droplet into the previous curve. For much larger droplets, where a pre-existing curve was not found to fit their diameters, an extrapolative scheme was used to identify the time increment with an understanding that this extrapolation scheme may not as accurate as the interpolative scheme. Using this scheme an overall droplet growth pattern could be observed as shown in Figure 38.

Figure 38. Droplet growth for individual droplets evaluated for overall growth

From Figure 38, it can be seen that the interpolative scheme provides a good fit to an overall trend, while for larger diameters; the time increment provided by the extrapolation method may not be as accurate. These results can be improved by obtaining droplets growing in the intermittent size range,

91 however even in our record of over 100 single droplets with substantial growth, the droplets in between these size ranges were missing. The larger the droplet grows the probability of coalescence increases and when coalescence happens, there is a sudden jump in droplet growth due to which intermediate droplets could not be isolated. From Figure 37 and Figure 38, it can be seen that under the experimental conditions, a droplet took a prolonged time (~90 seconds) to grow from 1 µm to 25 µm.

The behavior of droplet diameter growth provides an insight into the vapor-phase transformation which can be understood by application of principles of kinetic theory of gases.

9.6. RESULTS: ADVANCING CONTACT ANGLE DURING DROPWISE CONDENSATION

The specially built copper stage was used to observe coalescence behavior and observe change in contact angle. Figure 39 shows an example of the droplet growth captured on the Silicon surface attached to the copper stage. The details on the methodology for these measurements is provided in Appendix A.3.

(a) (b) t=0 s t=4.54 s

Figure 39. Condensed droplets on Si surface using side view (4.2 Torr)

Contact Angle (Degrees) 75

70 0 204060 Time (Second)

Figure 40. Average contact angle of droplets measured during their isolated and coalescence growth

regime

From these measurements, the average advancing contact angle of droplets in coalescence free

regime was calculated as ~82.5o. The effects of coalescence have a marked change in the advancing contact angle. Each decrement of contact angle in Figure 40 shows a coalescence event. Depending upon the size of the merging droplets, the advancing contact angle can change from 82.5o to 60o – a change of

22.5o. Upon coalescence, the capillary forces within the droplet act to recover from the sudden effect of droplet merger. Since this phenomenon is accompanied by continued condensation process, the droplet peripheral diameter increase is supplemented by droplet height changes while the droplet tries to regain the advancing contact angle.

9.7. RESULTS: DROPWISE GROWTH ALONG WITH COALESCNCE MECHANISM

The specially built copper stage was used to observe coalescence behavior and observe change in contact angle. Figure 41 shows an example of the droplet growth captured on the Silicon surface attached to the copper stage.

Figure 41. Condensing droplet at 4.2 Torr visualized at 80o inclined surface (Constant substrate

Temperature, Varying vapor pressure mode)

Figure 42 shows the results of droplet diameter growth rate at 4.2 torr wherein substrate temperature was held steady and vapor pressure was increased to initiate condensation. An average diameter was computed based on visible droplets in a given frame as per equation given below

n D= Dn (43) ()avg frame ∑ i 1

The diameter and contact angle of each droplet was calculated using LBADSA plugin in ImageJ.[151]

Figure 42. Droplet Growth rate at 4.2 Torr (Constant Substrate Temperature, Varying vapor pressure

mode)

As seen, there was a sudden jump in droplet diameter when coalescence took place. Depending upon the number of coalescing events, the jump could be much higher. It was noticeable that after coalescence, the droplets continued to grow at same diametrical rate law as of a single droplet growth until the next coalescence took place. Another distinct pattern observable under present experimental situation was the birth of a new generation of droplets in between coalescence times. Even when the drops had been nucleated and were growing, new droplets nucleated at different period of times. A substantial decrease in the average diameter D is indicative of large number of new droplets which nucleate in a given ()avg frame frame. This is clear from Figure 42, where new droplet formation shows a decrease in average diameter, and a coalescence event led to increase in average diameter in an image frame.

By observing the condensation process on the specially built copper stage, it was observed that coalescence and the droplet growth were accompanied by a change in dynamic contact angle. Particularly

95 during the coalescence, the contact angle change depends upon the volume of merging droplets and degree of their separation. Surface forces acting on the droplet are active during all the time, and as such when droplets merge there exists a brief period during which the combined droplet tries to adjust its center of mass to satisfy those forces by overcoming frictional forces. As noted, the droplet edges are pinned down due to frictional forces due to which the combined droplet requires some time to return to its normal state.[152]

CHAPTER - 10

DROPLET GROWTH MEASUREMENT: THEORETICAL BASIS

The key to understand behavior of the observed droplet diameter growth is identifying the mechanics of vapor-liquid transformation. For steam-water transformation the most common occurring mechanism of this transformation involves diffusional growth. This conjecture is supported by high magnification images which show presence of nano sized droplets along the periphery of the growing micro-sized droplets. These were observed to merge into the larger droplet. Thus diffusion of nano-droplets due to capillary imbibition is clearly active during the process of dropwise growth. The rate of growth if this mode alone is active is given by D=kt.0.33.[120-123] However a large discrepancy was found when the recorded data from present studies was tried to fit with this growth law. It became clear that diffusional growth cannot solely account for the droplet growth pattern observed in present case.

The kinetics of vapor-liquid phase change behavior observed under present experimental studies can be explained by application of principles of kinetic theory of gases. The following analysis is based on dropwise condensation theory developed by Umur.[44] In the following analysis, the effects of disjoining pressure and extended meniscus around the droplet are neglected. Also in the accompanying theory, the nucleation characteristics of the surface are not discussed and it is assumed that the droplet has already nucleated.

A single droplet sitting on a surface at temperature Tl is considered and surrounded by vapor at temperature Tv and pressure at Pv (Figure 43). The vapor-liquid interface temperature is assumed to be Tin.

The volume and the exposed surface area of the droplet sitting on the surface with contact angle θ is given by

π R3 ()(2cos1cos+−θθ)2 Ω= (44) 3

Α=2π RH ⇒Α= 21cosπθR2 () − (41)

While the area of the droplet base is given by

222 Β=π RRBase =πθsin (40)

TPvv,

ψv ψl

TPll, θ TT= substrate l R

Figure 43. Droplet on a surface

At the droplet interface, impinging vapor molecules become a part of the droplet, whereas some liquid

molecules escape into the vapor region. The imbalance between influx and out flux of water molecules

gives the net rate of droplet growth or dissipation. In particular, the net mass flux rate at a surface is given

ψψ=−vlψ (45)

The condensation term can be given by

αmvP ψv = 2πTRvg M (46)

The most significant aspect of the above equation is the condensation accommodation coefficient.

Condensation coefficient (α m ) defines the probability of a water molecule to stick to the condensing

surface or in other words become a part of the condensate. Thus (1−α m ) signifies the molecules which are reflected back to the vapor state. Substantially lower values of condensation coefficient signify much larger interfacial resistance and a larger temperature jump.[153]

Because it is assumed that the vapor-liquid interface is at a temperature Tlv, it implies inclusion of a temperature jump condition at the interface. To analyze this problem, the evaporative component can be given by Boltzmann law.[154]

⎛⎞L ψ=−ψ fExp ls0 ⎜⎟ (47) ⎝⎠RTg v

The above two equations can be solved in conjunction with heat transfer accompanying the phase

change. If heat transfer is considered between vapor, condensing liquid and the substrate, then

qhdlvvlvd=Α()( TThTT − =Β lvl −) (48)

The interfacial heat transfer coefficient is related to the mass transfer occurring at the liquid-vapor interface and is given by

ψ L hlv = (49) TTvl− v

The drop side coefficient hd can further be calculated using the drop distribution function by

Mccormick and Baer.[49]

Dh 89D/2 xdx .2 Base d =≈=f θ ∫ 2 2 () (50) kl θθ0 ()Dx/2 −

The above equations can be combined to eventually lead to

c 22 ⎪⎪⎧⎫c ⎛⎞RR− kTlv( − T l) R −+Rf02()θ fsRRR ()⎨⎬ −+0 ln ⎜⎟c = t ⎩⎭⎪⎪⎝⎠R01− Rf()θρl L where ()()2cos1cos+−θθ2 f ()θ = 1 sinθθi f ()

()()1cos+ θθf (51) f ()θ = 2 2sinθ 1.5 2π kRM T2.5 2 −αm lg() v fs()= 2 2αmvPL 9.2 f ()θ = θ

The above solution is a mixed solution of linear and exponential terms. As mentioned previously, a

very critical parameter in determining the solution for the above equation is the condensation

accommodation coefficientαm . Various methods employed in literature give a wide range of suitable

values of αm .[155] Typically it is expected that in pure vapor conditions, αm is closer to 1, while at

lower pressures its value is expected to be lesser than 1.

Evaluating condensation coefficient during condensation experiments is a challenge owing to presence

of non-condensable gases and contaminations. However as explained previously, the rarefied

environmental conditions inside ESEM are conducive for such measurements. The experimental

technique along with the presented theory can be used to evaluate condensation coefficient. Although the

current setup and the method can be used to determine condensation accommodation coefficient, there

remains ambiguity regarding the true value on account of uncertainty in temperature measurement.

By adjusting αm and degree of subcooling (Tv-Tl), a fit was obtained to match the theoretical results with the experimental values (Figure 44).

100

1000

100

Mean Free Path ~ 21.3 µm

10 Diameter (µm)

Experiment Theoretical Prediction 0.1 0 200 400 600 800 1000 Time (seconds)

Figure 44. Theoretical and experimental droplet growth analysis

The fit was obtained for an accommodation coefficient of 0.7 and a degree of subcooling of 0.01 oC with an advancing contact angle of 82.5o. As has been stated before, the effective degree of subcooling for the present setup cannot be measured experimentally. Eqn. 51 illustrates the growth rate does not have a direct dependence upon the temperature of the substrate. Instead the growth rate is affected by the degree of subcooling given by temperature difference of the vapor and the substrate. Thus, the diametrical growth rate is not sensitive to the experimental temperature uncertainties of the substrate temperature.

The much lower value of the effective degree of subcooling for the theoretical fit reflects the conjecture that only the molecules near the substrate contribute in phase change process. The molecules far away from the peltier cooler are much more energetic than the molecules near the substrate surface and hence do not play active role in condensation process. This further indicates that the degree of temperature jump

101

near the liquid-vapor interface is negligible and thus the interfacial resistance is very small. Moreover, the temperature conditions inside the droplet are nearly isothermal with negligible temperature difference between the substrate surface (Tsub) and the interfacial temperature (Tlv) at the droplet perimeter.

From the theoretical principles leading to conformity with experimental observations, it becomes clear

that under the conditions of ESEM, the droplet growth occurs directly by mechanism of conversion of

vapor to the liquid medium. From Figure 44, it can be seen that droplet diameter growth rate observed

during the experiment evidently changes as the droplet diameter approaches the mean free path of vapor

molecules. However, this observed relevance of mean free path of vapor molecules on droplet growth is

not captured in the dropwise condensation theory by Umur and requires further investigation.

102

CHAPTER - 11

DROPLET GROWTH: SUBMICROSCOPIC VISUALIZATION

Droplets were found to nucleate at the same spot which reinforces the idea that condensation is a

heterogeneous nucleation phenomenon. As can be seen from Figure 45, submicroscopic droplets were found to be nucleating and growing in between larger droplets for our case.

As mentioned previously, most of present models on dropwise condensation are based on premise that condensation occurs primarily on the droplets with little or no condensation on portions of the surface between droplets. However it is clear from Figure 45, that this assumption needs a revisit. Whether the formation of submicroscopic droplets occurs everywhere at the very beginning of first wave of droplet formation cannot be postulated yet. However it can be seen that the ‘bare area’ is populated entirely by submicroscopic droplets of different sizes.

Even further, from Figure 45, presence of sub-microscopic droplets appearing, growing and then coalescing with the larger droplet can be observed. Some of the nanodroplets can be seen sitting ~ 100 nm from the larger droplet (Figure 46 and Figure 47). The role of surface tension appears certainly an important one to cause diffusion/migration of micro/nano droplets into the bigger droplet.

103

Submicroscopic Droplets

Figure 45. Image Visualization at 6500X between visible droplets

104

Figure 46. Image Visualization at 12,000X between visible droplets

Figure 47. Image Visualization at 25,00X between visible droplets

105

CHAPTER - 12

CONCLUSIONS

Dropwise condensation was studied in a superheated environment and rarefied vapor condition at as

low as of 4 torr using an Environmental Scanning Electron Microscope. Droplet growth of single droplets

was studied along with their coalescence behavior with other droplets. The analysis of experimental

observation reveals a mechanism where growth of droplets occurs directly by means of vapor to liquid

conversion on the droplet itself. Numerical simulations on droplet growth using a theoretical model based

on kinetic theory of gases provide a deeper understanding of interfacial mass transport under the

experimental conditions. In particular the diameter of the droplet growth was found to decrease as the

droplet diameter increases. The advancing contact angle has been found to in order of 82.5o for the

droplets when the static contact angle is 62o on plane silicon surface. The coalescence period brings a marked change in the advancing contact angle. The change in advancing contact angle during the coalescence period depends upon the size of the merging droplets. Large droplets merging together bring substantial change in contact angle. Accommodation coefficient, degree of subcooling and advancing contact angle are three most important factors which determine the growth behavior of the droplet.

Evidence was also found which establishes presence of submicroscopic droplets nucleating and growing in between microscopic droplets for partially wetting case.

106

CHAPTER - 13

FUTURE WORK

• Submicroscopic/Nanoscopic Droplet growth rate observation: Droplet growth measurement is key in

obtaining a complete insight into the mechanism of physics involved with condensational phase

change processes. The ESEM images will be recorded and analyzed in manner already developed in

present studies. These studies should be extended for for more extensive droplet growth

measurements for submicroscopic sized droplets.

• Submicroscopic/Nanoscopic Droplet size distribution: The analyzed images need to be analyzed to

obtain distribution of droplet sizes in a given area and observing how the surface coverage alters with

time. These observations are central for obtaining the overall behavior of interaction between surfaces

and liquid as effects such as coalescence behavior and effect of merger can more readily be accounted

for in this way.

• Correlating with empirical growth relations to estimate condensation accommodation coefficient, αc:

Analyzing the growth rate for different surfaces with same level of degree of subcooling will allow

quantification of parameters such as accommodation coefficient of vapor on different surfaces. For this,

surfaces with different contact angle (chemical or mechanical means) need to be prepared. Although

each coating material possesses different surface energy, static contact angle is a good indicator of

surface character. Since there is no known general method to predict the accommodation coefficient,

this information will vastly be useful to provide empirical correlation of accommodation coefficient

with contact angle.

• Relation between advancing contact angle and static contact angle: Present method can be used to

correlate the dynamic contact angle in form of advancing contact angle with static contact angle on the

surface by image analysis. Transient response of this variation during condensation process in

conjunction with theoretical development will lead to significant improvements.

107

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120

APPENDIX

A.1. Example ImageJ Macro for Droplet Diameter Measurement open("C:\\Users\\Sushant Anand\\Documents\\Research\\2010-02-23(2)\\a01\\a01.tif"); run("Median...", "radius=1 stack"); run("Enhance Contrast", "saturated=2 normalize normalize_all"); run("Subtract Background...", "rolling=100 stack");

DROPLET1 run("Enhance Contrast", "saturated=5 normalize normalize_all"); run("Bandpass Filter...", "filter_large=40 filter_small=3 suppress=None tolerance=5 autoscale saturate process"); run("Subtract Background...", "rolling=20 stack"); run("Enhance Contrast", "saturated=5 normalize normalize_all"); setThreshold(17, 255); run("Convert to Mask", " black");

DROPLET2 open("C:\\Users\\Sushant Anand\\Documents\\Research\\2010-02-23(2)\\a01\\a01a.tif"); setMinAndMax(29, 141); run("Apply LUT", "stack"); setMinAndMax(0, 164); run("Apply LUT", "stack"); setThreshold(33, 255); run("Convert to Mask", " black"); makeOval(356, 181, 144, 93); run("Analyze Particles...", "size=1-Infinity circularity=0.00-1.00 show=Nothing display clear include add stack");

121 A.2. Methodology for measurement advancing contact angle using LBDSA plugin

• Open Image File

• The droplets are marked with a ‘Number’

4 1 3 12 2 13 67 5 14 8 16 9

10 15 11

• Convert to 8-bit

• Find a frame and a droplet, and draw a line between the base

• Measure the inclination of the line

• Rotate the Frame by that angle

122

• Use LBDSA plugin, fit the curve

123 A.3. Material Properties and relations

Property Relation Reference

Saturation Vapor Pressure ⎛⎞⎛⎞T [1], [2] ⎡ ⎜⎟11.344⎜⎟ 1−− 1 ⎤ ⎛⎞373.16 ⎛⎞373.16 ⎝⎠⎝⎠373.16 ⎢−−7.90298××⎜⎟ 1+ 5.02808 Log10 ⎜⎟ −1− () 1.3816e −+ 7 × 10 ⎥ of Water [Pa] 100×10^⎢ ⎝⎠TT ⎝⎠ ⎥ ⎢ ⎛⎞⎛⎞373.16 ⎥ ⎜⎟−−3.49149⎜⎟ 1− 1 ⎢ ⎝ ⎝⎠T ⎠ ⎥ ⎣(8.1328eL− 3)× 10 + og10 ()1013.246 ⎦

Thermal Conductivity, 0.0022014×T 1.5 [3] , (T in Kelvin) Air [W/(m*K)] T +134.31

Thermal Conductivity, ⎛⎞1084.7×+TT1.5 11.8 ×13 6 [3] 0.0000013⎜⎟, (T in Kelvin) Water vapor [W/(m*K)] ⎝⎠T +160

Specific Heat Capacity, 968.76+ 0.067783× T , (T in Kelvin) [3]

Air [J/(kg*K)]

Specific Heat Capacity, 1687+ 0.53389×T , (T in Kelvin) [3]

Water vapor [J/(kg*K)]

124 Dynamic Viscosity, Air 0.5 [3] ⎛⎞2.53928× 10−5 ⎛⎞T , (T in Kelvin) ⎜⎟⎜⎟ [Pa*s] ⎝⎠1122+ TT⎝⎠ref

Dynamic Viscosity, 0.5 [3] ⎛⎞3.01472× 10−5 ⎛⎞T ⎜⎟⎜⎟, (T in Kelvin) Water vapor [Pa*s] ⎜⎟ ⎝⎠1673+ TT⎝⎠ref

Density, pM [4] air R T Air [kg/m3] g

Mair, Molecular Weight 0.0288 [4]

Air [kg/mol]

Density, pM [5] water R T Water vapor [kg/m3] g

Mwater, Molecular Weight 0.018 [5]

Water [kg/mol]

2 Diffusivity, DAB [m /s] [6] −−68 − 102⎛⎞patm ()−×+×2.775 10 4.479 10TT +× 1.656 10 ⎜⎟, (T in Kelvin) ⎝⎠p

125 Gas Constant, Rg 8.314 [7]

[J/(mol*K)]

Surface Tension, Water 0.955 [8] ⎛⎞T 0.132674⎜⎟ 1− [N/m] ⎝⎠647.13

32 Enthalpy of Vaporization, (−×−+×−−×−+× 0.0000614342()TTT 273.15 0.00158927 273.15 () 2.36418 273.15 2500.79) () 1000 [9] Water [J/kg]

126 References

1. Saturation Vapor Pressure of Water Vapor. Available from: http://hurri.kean.edu/~yoh/calculations/satvap/satvap.html.

2. Weiss, A., Algorithms for the calculation of moist air properties on a hand calculator.

3. Boukadida, N. and S.B. Nasrallah, Mass and heat transfer during water evaporation in laminar flow inside a rectangular channel -

Validity of heat and mass transfer analogy. International Journal of Thermal Sciences, 2001. 40(1): p. 67-81.

4. Air. Available from: http://en.wikipedia.org/wiki/Air.

5. Water Vapor. Available from: http://en.wikipedia.org/wiki/Water_vapor.

6. Bolz, R. and G. Tuve, Handbook of tables for applied engineering science. 1973.

7. Gas constant. Available from: http://en.wikipedia.org/wiki/Gas_constant.

8. Coker, A. and E. Ludwig, Ludwig's applied process design for chemical and petrochemical plants. Fourth ed. Vol. 1. 2007: Gulf

Professional Publishing.

9. Latent Heat. Available from: http://en.wikipedia.org/wiki/Latent_heat#Latent_heat_for_water.

127