Thermal Diffusion Shock Waves

in a Linear Temperature Field

and

Comparison of Ultrasonic Distillation to Sparging of Liquid Mixtures

by Hyeyun Jung

M.Sc., Brown University 2007 B.Sc., Seoul National University 2004

A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Physics at Brown University

PROVIDENCE, RHODE ISLAND

May 2012 © Copyright 2012 by Hyeyun Jung This dissertation by Hyeyun Jung is accepted in its present form by The Department of Physics as satisfying the dissertation requirement for the degree of Doctor of Philosophy.

Date

Gerald Diebold, Ph.D., Advisor

Recommended to the Graduate Council

Date

James Valles, Ph.D., Reader

Date

Xinsheng Sean Ling, Ph.D., Reader

Approved by the Graduate Council

Date

Peter M. Weber, Dean of the Graduate School

iii Vita

Hyeyun Jung was born on November 29, 1980 in Daegu, Republic of Korea. She com- pleted her secondary education at Wonhwa Girls High School in Daegu in February of 1999. She then attended Seoul National University. She obtained a Bachelor of Science in Chemical Engineering and Physics in February of 2004. In September 2004, she began graduate studies in the Department of Physics at Brown University. She carried out her doctoral research under the supervision of Dr. Gerald J. Diebold. Her publications include:

1. Hyeyun Jung, Vitaly Gusev, Hyoungsu Baek, Yaqi Wang, and Gerald J. Diebold, "Ludwig-Soret Effect in a Linear Temperature Field: Theory and Experiments for Distributions at Long Times", Physics Letters A 375(19), 1917-1920, 2011

2. Hye Yun Jung, Han Jung Park, Joseph M. Calo and Gerald J. Diebold, "Com- parison of Ultrasonic Distillation to Sparging of Liquid Mixtures", Analytical Chemistry 82, 10090-10094, 2010

iv Acknowledgements

I would like to especially thank my thesis advisor Prof. Gerald J. Diebold for his kind guidance during my graduate study. He has patiently waited for me until I learned how to do research and supported me in every aspect of my research. He has given me great opportunities to carry out a variety of exciting research projects in his group. From him, I learned how fun and exciting doing science can be, of his wide and deep insight in seeing scientific problems and how important it is not to lose a sense of humor and to smile while doing science.

I also would like to thank Dr. Gusev E. Vitaly and Dr. Hyoungsu Baek for their collaboration in mathematics. My thanks also go to Ken Talbot and Tim Pimental in the machine shop since they helped me in making various experimental equipment. I am grateful for Alfred Tente for helping me to explore electronics. I won’t be able to forget your kind support, Al!

I would like to thank Prof. John B. Marston for giving me an opportunity to work with him for one year before joining Dr. Diebold’s group. During this year, I could experience the depth and the beauty of theoretical physics. I also would like to thank Prof. James Valles and Prof. Xinsheng S. Ling for kindly serving on my thesis committee.

v My labmates made my graduate life very enjoyable and more colorful. I thank Dr. Shougang Wang, Dr. Theron J. Hamilton, Dr. Guohua Cao, Dr. Clifford Frez, Dr. Cuong K. Nguyen, Dr. Hanjung Park, Binbin Wu, Yanan Liu, Katherine Phillips and Charles Beyrouthy for their friendship!

I also would like to give many thanks to my postdoctoral advisor Dr. Christopher Endres for his kind consideration in allowing me to write a thesis while conducting research with him and the excellent Johns Hopkins research teams. I also would like to thank friends who helped my proofreading and who cannot be listed due to limited space and to give special thanks to Dr. Mark Smith for helping me to thoroughly proofread the second half of my thesis! I would like to give special thanks to Dr. Heekyoung Ko, Chunwoo Kim and Rozita Jalali who encouraged and supported me while I was writing my thesis. I appreciate everyone who helped directly or indirectly to get this work done.

My most important acknowledgment is to my beloved family God has given me: my parents, grandmother, my three siblings, Yumi, Jiyun and Minyoon, and brother- in-law Junho and my nephew and niece Junghyun and Jungwon, who have filled my life with joy and love. Especially, my mother’s support, sacrifice and endless love have brought me this far. Her hard work, persistence, and positive thinking in achieving her dream always inspired me to live the same. I thank my lovely sister and brother, Jiyun and Minyoon, since their presence and love brought so much happiness during my graduate study.

Above all, I thank my God that He has led me to Brown to meet wonderful professors and friends and given me a wonderful opportunity to grow up there.

vi To my God and my family

vii Abstract of “Thermal Diffusion Shock Waves in a Linear Temperature Field and Comparison of Ultrasonic Distillation to Sparging of Liquid Mixtures”by Hyeyun Jung, Ph.D., Brown University, May 2012

The Ludwig-Soret effect, also known as thermal diffusion, refers to the separation of mixtures in a temperature gradient. Thermal diffusion is governed by a pair of coupled differential equations which reduce to a nonlinear partial differential equa- tion when the temperature profile is specified. Here two solutions are given to the partial differential equation describing thermal diffusion in a linear temperature field where the components are constrained in space. The first solution considers thermal diffusion without the effects of mass diffusion and shows the underlying motion of the components of the mixture to be that of shock waves. The second solution is an exact solution of the Ludwig-Soret equation that includes both the effect of the thermal gradient and mass diffusion. An additional solution is found for the problem of thermal diffusion in unbounded space. A new experimental method was devel- oped to monitor distributions of components of the mixture in a linear temperature field based on probing a cell containing fluorescent nanoparticles with a confocal microscope. The nanoparticles were chemically synthesized and labeled with a flu- orophore that absorbed 488 nm radiation and fluoresced at a 520 nm peak. The temperature gradient in the cell was generated by cooling one surface of the cell, a sapphire plate, with flowing water and electrically heating the other surface, which was an indium tin oxide coated glass plate. The dynamics of the separation of the mixture was recorded by monitoring fluorescence from the particles with the scan- ning confocal microscope. Data were fitted to a new numerical solution to the full partial differential equation for thermal diffusion with mass diffusion included. The method developed here is shown to provide Soret parameters, including the thermal diffusion factor and the Soret coefficient, based on either a single recording of the terminal density fraction profile, or by fitting the density profile at several times with the results of numerical integration. Ultrasonic distillation was investigated. Experiments were carried out to verify the recently reported, perfect separation of ethanol from water by ultrasonic dis- tillation. Ultrasonic distillation refers to the application of intense ultrasound to a liquid resulting in the formation of an ultrasonic fountain that generates both mist and vapor. Here, the composition of the vapor and aerosol above an ultra- sonic fountain was determined as a function of irradiation time and compared with the results of sparging for five different solutions. The experimental apparatus for determining the efficiency of separation consists of a glass vessel containing a piezo- electric transducer driven at either 1.65 or 2.40 MHz. Dry nitrogen was passed over the ultrasonic fountain to remove the vapor and aerosol. The composition of the liquid solutions remaining in the apparatus were recorded following irradiation us- ing gas chromatography, refractive index measurement, nuclear magnetic resonance, or spectrophotometry as diagnostics for the concentrations of the components of the mixtures. Experiments were carried out with ethanol-water and ethyl acetate- ethanol solutions, cobalt chloride in water, colloidal silica, and colloidal gold. The data show that ultrasonic distillation produces separations that are somewhat less complete than what is obtained using sparging. No evidence for the perfect separa- tion of ethanol from water-ethanol mixtures was found. Contents

I Thermal Diffusion Shock Waves in the Linear Temperature Field 1

1 Thermal Diffusion and Non-equilibrium Thermodynamics 2

1.1 History of the Soret Effect(Thermal Diffusion) ...... 2

1.1.1 Three Postulates on Non-equilibrium Thermodynamics . . . .4

1.2 Conservation Laws ...... 6

1.2.1 Mass Flow and Chemical Reactions ...... 6

1.2.2 General Conservation Laws ...... 8

1.2.3 Conservation of Mass ...... 9

1.2.4 Equation of Motion ...... 10

1.2.5 Energy Transport Equation ...... 12

1.3 The Assumption of Local Equilibrium ...... 13

1.3.1 First Law Heat Flux ...... 15

1.3.2 Second-law Heat Flux ...... 17

1.3.3 Entropy Law and Entropy Balance Equation ...... 18

ix 1.4 The Linear Laws ...... 21

1.5 Onsager Theory ...... 22

1.6 Soret Effect ...... 25

2 Analytical Solution for the Dynamics of the Soret Effect 28

2.1 Equation of Motion For a Linear Temperature Field ...... 29

2.1.1 Exact Solution Using Hopf-Cole Transformation ...... 31

2.1.2 Solution without Mass Diffusion ...... 37

2.1.3 Shock Waves with Moving Coordinates in the Infinite Domain without Boundary ...... 40

2.1.4 Long Time (Terminal) Solution for Soret Equation ...... 44

3 Experiments and Results 47

3.1 Different Techniques to Measure the Soret Coefficient ...... 47

3.1.1 The Standard Soret Cell ...... 48

3.1.2 The Beam Deflection Technique ...... 50

3.1.3 Thermal Diffusion Forced Rayleigh Scatting Technique (TDFRS) ...... 52

3.2 New Experimental Method Based on the Use of a Confocal Microscope 57

3.2.1 Suspension with Soret Effect ...... 57

3.2.2 Experimental Setup ...... 57

3.2.3 Data Acquisition ...... 59

x 3.2.4 Confocal Microscope ...... 60

3.2.5 Numerical Solution using Weighted Essentially Non-Oscillating Schemes ...... 62

3.2.6 Preparation of Fluorescent Silica Nanoparticles ...... 67

3.3 Results and Analysis ...... 68

3.3.1 Experimental Data ...... 68

3.3.2 Results and Analysis ...... 69

3.4 Discussion ...... 71

4 Summary of Thermal Diffusion Shock Waves in a Linear Temperature Field 74

II Comparison of Ultrasonic Distillation to Sparging of liquid Mixtures 78

5 Introduction 79

5.1 Review of Reported Perfect Separation of Ethanol from Water Using Ultrasonic Distillation ...... 80

5.2 Introduction to Ultrasonic Atomization ...... 84

5.2.1 Capillary Wave Hypothesis ...... 85

5.2.2 Cavitation Hypothesis ...... 88

6 Experiments and Results 92

6.1 Experimental Setup and Procedures ...... 92

xi 6.2 Analysis Methods ...... 96

6.2.1 NMR Sensitivity ...... 97

6.2.2 Spectrophotometer ...... 98

6.2.3 Gas Chromatography ...... 99

6.2.4 Refractometer ...... 104

6.3 Experiments ...... 106

6.3.1 Ethanol-Water Mixture Experiment ...... 106

6.3.2 Ethyl Acetate-Ethanol Mixture Experiment ...... 109

6.3.3 Experiments with Aqueous Solutions of Cobalt Chloride (CoCl2)111

6.3.4 Two Colloidal Suspension Experiment ...... 111

6.3.5 Vapor Pressure Change of Ethanol Acetone Mixture ...... 112

7 Discussion 114

8 Summary of Comparison of Ultrasonic Distillation and Sparging of Liquid Mixtures 119

A Thermal Diffusion Data Analysis Matlab Code 121

A.1 Code to Obtain the Soret Coefficient and Total Concentration . . . . 121

A.2 Code to Achieve Numerical Density Profile at Different Times Using Soret Coefficient and Total Concentration Obtained from Long Time Density Profile ...... 123

B Image Formation Theory 128

xii B.1 Abbe’s Theory of Image Formation ...... 128

B.2 Diffraction Theory and a Lens as a Diffraction Medium ...... 129

C NMR principle 133

C.1 T1 Processes ...... 135

C.2 T2 Processes ...... 136

C.3 Chemical Shift and Spin-Spin Coupling ...... 136

C.4 J(spin-spin) Coupling ...... 137

xiii List of Tables

r 3.1 Coefficients ak,l ...... 65

r 3.2 Optimal Weights Ck ...... 66

6.1 Nuclear properties of interest ...... 97

6.2 Ratio of the amount of vapor occupying r plates in GC column ...... 100

6.3 Performance comparison among the main GC detectors ...... 103

6.4 Comparison of absorption measurements of aqueous CoCl2 solutions be- fore and after ultrasonic or nitrogen sparging distillation ...... 110

6.5 Comparison of absorption measurements of colloidal solutions before and after ultrasonic or nitrogen sparging distillation ...... 111

xiv List of Figures

2.1 Periodic temperature profile and its derivatives ...... 30

2.2 Periodic density profile corresponding to periodic temperature profile . . 33

2.3 Density fraction c versus coordinate ξ at different times with an initial uniform density profile ...... 36

2.4 Density fraction c versus coordinate ξ at different times with an initial Gaussian density profile ...... 37

2.5 Phase plot for an initially uniform density in space ...... 39

2.6 Change in the initially Gaussian density fraction in space ξ over time at an infinite linear temperature domain ...... 42

2.7 Terminal density distribution for c0 = 0.4 for different α values ...... 45

2.8 Terminal density distribution for c0 = 0.04 for different α values . . . . . 46

3.1 Standard Soret Cell ...... 48

3.2 Sketch of an elementary Soret cell with the beam deflection technique . . 51

3.3 Thermal Diffusion Forced Rayleigh Scattering Technique (TDFRS) . . . 52

3.4 Two heterodyne detection schemes ...... 54

3.5 Experimental setup ...... 58

3.6 Photograph of brass cell and ITO glass plate ...... 59

xv 3.7 Photograph of the confocal microscope and the cell ...... 60

3.8 Principle of the confocal microscope ...... 61

3.9 Movement of fluorescence labelled nanoparticles under the imposition of linear temperature field ...... 68

3.11 Comparison of the experimental light intensity data to numerical integration 73

5.1 Experimental setup in the report of a perfect ethanol separation from ethanol-water solution using ultrasonic atomization ...... 81

5.2 Vapor-liquid equilibrium diagram of ethanol-water solution using the ul- trasonic atomization and energy required for separation vs ethanol con- centration in solution at different temperatures ...... 82

5.3 Capillary wave formation mechanism and the dispersion curve of para- metric decay waves ...... 83

5.4 Ultrasonic distillation setup used by Kirpalani et al...... 84

6.1 Apparatus used for the ultrasonic distillation ...... 93

6.2 Commercial 25 mm diameter ceramic piezoelectric transducer assemblies 94

6.3 Ultrasonic fountain formation ...... 95

6.4 Experimental setup for Sparging ...... 96

6.5 Elution progression in a column composed of plates ...... 101

6.6 Flame ionization detector ...... 102

6.7 Schematic diagram of an Abbe refractometer ...... 104

6.8 Ethanol mole fraction of ethanol-water solutions determined by NMR measurements after sparging and ultrasonic distillation ...... 106

6.9 Comparison of ultrasonic distillation to nitrogen sparging of water-ethanol solutions ...... 107

xvi 6.10 Composition change of ethyl acetate fraction in the remaining solutions of ethyl acetate and ethanol mixture in the process of ultrasonic or nitrogen sparging distillation ...... 109

6.11 EtOHAcetoneVPchange ...... 112

7.1 A breaking droplet off from the liquid surface where surfactant is enriched on the surface ...... 115

B.1 Geometry to calculate the Fraunhofer diffraction equation of rays passing through a lens ...... 130

B.2 A phase difference by a lens ...... 130

B.3 Abbe’s Image formation theory ...... 132

C.1 Nuclear spin-spin interaction ...... 137

C.2 Nuclear magnetic energy level for a two-spin system ...... 138

xvii Part I

Thermal Diffusion Shock Waves in the Linear Temperature Field

1 Chapter 1

Thermal Diffusion and Non-equilibrium Thermodynamics

“It is common practice in reviewing a subject to begin with a brief state- ment of the nature of the phenomena involved and their physical interpre- tation. This is not possible for thermal diffusion. Although it is easy to describe thermal diffusion phenomenologically, no one has succeeded in giving a simple physical explanation of thermal diffusion; the elementary theories proposed have been either incorrect in essential points or else almost as complicated as the rigorous Chapman-Enskog kinetic theory. The reason is that thermal diffusion is a secondary effect in the sense that its very existence depends on the nature of molecular collisions, whereas the existence of the other transport properties, (i.e. viscosity, heat con- ductivity, and ordinary diffusion) depends on the occurrence of collisions and only secondarily on their nature.”(Chapman, 1962) [3].

1.1 History of the Soret Effect(Thermal Diffusion)

If a drop of blue ink is put into a cup of water, it diffuses outwardly in the water over time. This occurs because of an initial inhomogeneity of composition. When the

2 concentration of each component of a mixture is not homogeneous under constant temperature and pressure, each component moves to form homogeneity of compo- sition in space within a mixture (i.e. until there is no net flux). This is called “ordinary” or “mass” diffusion. However, a homogeneous mixture can become inho- mogeneous in space as a result of a temperature gradient. The mass flux caused by a temperature gradient is called ‘thermal’ diffusion.

Thermal diffusion was first discovered in a liquid solution by Ludwig [47] in 1856. Using an inverted U-tube containing sodium sulphate solution, he heated one column with boiling water and the other he cooled with ice. After a few days, he found that the cold column had more sodium sulphate when compared with the heated column. This phenomenon was examined more thoroughly about 20 years later by Soret in 1872, thus thermal diffusion in liquids came to be called the ‘Soret’ effect. Soret placed 2 cm wide and 20 cm long cylindrical tube vertically, cooling the lower part and heating the upper part. He filled the tube with sodium sulphate solution and after 55 days measured a higher solute concentration at the lower part of the tube. Regardless of how long the tube was left, the concentration never became uniform. He did experiments with mixtures of sodium and lithium chloride, potassium nitrate and copper sulphate, all showing a spatial inhomogeneity in the solute concentration.

Gas phase thermal diffusion was first predicted theoretically by Enskog [17] in 1911 and independently by Chapman [9] in 1912 before it was demonstrated exper- imentally by Chapman and Dootson [10]. They electrically heated a hot bulb at approximately about 230 ◦C and kept a cold bulb at 10 ◦C, which contained a mix- ture of hydrogen and carbon dioxide. These two containers were connected by a thin horizontal tube. It was found that heavier carbon-dioxide molecules move towards the cold side, with a difference in composition of about 3%. The gas mixture was analyzed by a means of a katharometer which measures the thermal conductivity

3 of gases or mixture of gases with a high sensitivity. The amount of separation was

proportional to log(Th/Tc), where Th and Tc are the absolute temperatures of the hot and cold sides respectively.

Just as the temperature gradient induces a concentration gradient in the mixture with initially no density variation in space, a concentration gradient of components can cause a temperature gradient in the mixture with initially uniform temperature in space. This phenomenon was reported by Swiss physicist L. Dufour [15] in 1873 in the paper entitled ‘On the diffusion of gases through porous partitions and the accompanying temperature changes’.

1.1.1 Three Postulates on Non-equilibrium Thermodynam-

ics

Thermal diffusion can be classified as one of the transport phenomena which refers to a flux of matter, energy, or some other quantity of the system. Transport phenomena are often described by irreversible or non-equilibrium thermodynamics while classical reversible thermodynamics only describes an equilibrium system where there is no net flux. Thomson first attempted to describe the irreversible processes with thermo- dynamics in order to explain various thermo-electric phenomena in 1854 [36, 37, 38].

Non-equilibrium thermodynamics is based on three postulates: local equilibrium, the phenomenological linear law, and time reversal symmetry, which means that a physical law is invariant under time reversal [11].

In a nonequilibrium system, thermodynamic quantities cannot be homogenous in time or over entire space. The idea of a local equilibrium postulates that even in

4 nonequilibrium systems, each element with an infinitesimally small volume element forms a system in equilibrium with itself. In other words, thermodynamic quantities of the system in irreversible processes can be described as a function of the state variables of systems in equilibrium. For example, the local pressure p(R, t) and temperature T (R, t) in nonequilibrium systems can be defined by the same functions, p(ρ, E) and T (ρ, E), where ρ = ρ(R, t) and E = E(R, t) from measurements on equilibrium systems.

The phenomenological linear law states that the fluxes (or currents) Jα are linear functions of the conjugate forces Yα. In other words, any flux Jα and the conjugate force has the relation X Jα = LαβYβ, (1.1)

where Lαβ is the phenomenological coefficient.

Time reversal symmetry states that if time is reversed, the motion can be traced back [20]. This leads to Onsager’s reciprocal relation, which states that the phe- nomenological coefficient matrix in Eq. 1.1 is symmetric. The phenomenological linear law combined with time reversal symmetry can be used to derive the two phenomenological transport equations that govern thermal diffusion.

The general theory for irreversible processes supplements the two fundamental laws of the thermodynamics: the law of conservation of energy and the entropy law. In the following section, these two laws will be formulated to derive the entropy production rate. Using the entropy production rate expression, the phenomenological linear equations will be derived to describe a nonequilibrium system with the help of Onsager’s reciprocal relations.

5 1.2 Conservation Laws

In this section, the conservation laws for general quantities including mass, momen- tum (i.e. equation of motion), and energy will be derived. Combined with a local equilibrium postulate in the following section, all of these conservation laws will be used to derive an entropy production equation for phenomenological linear expres- sion for transport phenomena. In this section, which is taken from Ref. [20], the fundamental non-equilibrium thermodynamics will be reviewed to describe the Soret effect.

1.2.1 Mass Flow and Chemical Reactions

Consider ith molecule in a microscopically large but macroscopically small region of

fluid. When ρi and ui are the density and velocity for the component i, respectively, the velocity of the center of mass u can be defined by

r r X X J = ρu = Ji = ρiui, (1.2) i=1 i=1

where Ji is the mass current density of the ith component relative to the external coordinate. The mass density current (mass flux) of component i relative to the local

center of mass, ji can be expressed as

ji = ρi(ui − u) = ρiu − Ji. (1.3)

Eq. 1.3 shows that the motion of ith component Ji consists of convective motion

ρiu and diffusive motion of ith component, ji. Also it shows that all components of

6 ji are not independent. r X ji = 0. (1.4) i=1

The system with s chemical reactions may be characterized as

r X νikAi = 0 where k=1,2, ... s, (1.5) i=1

where Ai is the chemical symbol of ith component and νik is the number of grams of ith component produced per gram of reaction k. When the dkmi is defined as the mass change of ith component due to the kth reaction for dt time interval, the total mass change of ith component for dt is

s X dmi = dkmi. (1.6) k=i

Before discussing the chemical reactions, the following need to be defined: the progress variable λk for the kth reaction as the grams reacted per gram of origi- nal reactants and νik as the number of ith component produced per mass of reaction k. Therefore, the progress variable can now be related with the stoichiometric coef- ficient as

dkmi = mdλk. (1.7) νik

Here dλk is independent of component i. Summing Eq. 1.7 over k followed by dividing by total volume, the density change of ith component from all chemical reactions are found to be

s X dρi = ρ νikdλk where i=1,2,...,r. (1.8) k=1

where r is the number of chemical component. Eq. 1.8 shows that the conservation

7 of mass can be expressed as

r X νik = 0 where k=1,2, ... s. (1.9) i=1

1.2.2 General Conservation Laws

Before discussing conservation of mass, momentum and energy, general conservation laws can be reformulated . For any extensive property W of the system, W is define as W per unit mass. The time rate of change of ω in an arbitrary volume V is

dW Z ∂ = (ρW )dV. (1.10) dt V ∂t

dW dt can consist of two parts: convection due to JW (the actual flux of a related

quantity defined as ρW u), and production due to φW (an internal source per unit time and volume such as chemical reaction). Then the time rate of change in the total quantity of W within V becomes

Z ∂ Z Z (ρW )dV = − JW · dA + φW dV, (1.11) V ∂t A V

where the vector dA with a magnitude dA is defined as an infinitesimal element of area and is taken to be perpendicular to the element of surface dA and pointing outward from the volume V .

Using Gauss’s theorem, Eq. 1.11 gives a general conservation equation,

∂(ρW ) + ∇ · J − φ = 0. (1.12) ∂t W W

8 1.2.3 Conservation of Mass

Consider a system of n components where s reactions occur. The continuity equation of ith component can be derived from Eq. 1.12. Then W represents the total mass of ith component, and W becomes ρi/ρ or ci, the mass fraction of ith component.

The current density JW of mass of ith component becomes

JW = ρW ui = ρiui = Ji. (1.13)

The internal source of the mass production of ith component by the chemical reaction

dρi Ps dλk is given as φW = dt = ρ k=1 νik dt . Substitution of this expression into Eq. 1.12 yields s ∂ρi X dλk + ∇ · (ρiui) = ρ νik . (1.14) ∂t k=1 dt Summing Eq. 1.14 over all components makes the right hand side reduce to zero due to conservation of mass. With the definition of current density given by Eq. 1.2, Eq. 1.14 yields the law of conservation of mass.

∂ρ + ∇ · (ρu) = 0. (1.15) ∂t

In order for the total mass to be conserved, the internal source φW in Eq. 1.12 should

vanish. Since W can be set as 1, JW becomes ρu. Then, the law of conservation of mass can be derived from Eq. 1.12. Using the substantial derivative definition,

d ∂ = + u · ∇. (1.16) dt ∂t

Eq. 1.15 can be expressed as

dρ + ρ∇ · u = 0. (1.17) dt

9 Eq. 1.14, Eq. 1.3, and the use of ρi = ρci yield

s ∂ci X dλk ∂ρ ρ + ρu · ∇ci + ∇ · ji − ρ νik = −ci − ci∇ · (ρu). (1.18) ∂t k=1 dt ∂t

From Eq. 1.15, it can be seen that the right hand side vanishes. With the help of Eq. 1.16, Eq. 1.18 becomes

s dci X dλk ρ + ∇ · ji = ρ νik . (1.19) dt k=1 dt

1.2.4 Equation of Motion

When there is a surface force acting on the area (dA) of a volume (V ), the total R surface force is A σ · dA.

The stress tensor σ with nine components is defined as

σ = −pI + η (I is the 3 × 3 identity matrix), (1.20)

where η is a viscosity.

σij(i, j = x, y, z) is defined as a force per unit area in the i direction applied on the area which is normal in the j direction. The three diagonal components are called the normal stresses while the six off-diagonal components are called the shear stresses, which vanish at equilibrium. At equilibrium, the shear stress vanishes and the normal stress becomes constant −p regardless of orientation

σxx = σyy = σzz = −p. (1.21)

10 In the nonequilibrium state, the pressure P , the normal force per unit area applied by the fluid, is given by 1 P = − (σ + σ + σ ), (1.22) 3 xx yy zz as a sum of the diagonal components of the tensor σ.

Consider a fluid system with an arbitrary volume V in a fixed coordinate. The total force exerted on an arbitrary volume V can be decomposed into two parts with the help of the law of conservation of mass and the definition of substantial derivative Eq. 1.16.

Z du Z dρ Z ρ dV = dρu − u dV = dρu − ρu∇ · udV V dt V dt V Z dρu Z ∂ρu = − ∇ · (ρuu) + u · ∇(ρu)dV = − ∇ · (ρuu)dV. V dt V ∂t (1.23)

From the last expression, it is found that the total force is the sum of the com- ponent acting on the volume and the component exerted on the surface area, i.e. R R V ∇ · (ρuu)dV = A ρuu · dA. The external force consists of a volume force due to external force X acting on V and a surface force σ acting on the surface of the fluid,

n X Z Z Z Z ρiXidV + σ · dA = ρXdV + σ · dA, (1.24) i=1 V A V A

where Xi is the force acting on component i per unit mass, X is the total force per unit mass and σ is stress tensor equal to the negative of the pressure tensor with nine components. Equating Eq. 1.23 and Eq. 1.24 yields the equation of motion

du ρ = ρX + ∇ · σ. (1.25) dt

11 1.2.5 Energy Transport Equation

An equation rate of change of energy in a volume V can also be derived. The total energy change rate is the sum of the energy change rate in an arbitrary volume V of fluid and the energy flowing through its surface A. The total rate of energy change

R ∂ 2 in a volume V is V ∂t [ρ(E + (1/2)u )]dV , where E is the specific thermodynamic internal energy for the fluid.

As the energy flows through its surface A, the energy of volume V decreases. The

R ∂ 2 energy current comprises a convection current density A ∂t [ρ(E + (1/2)u )]u · dA R and conduction current density A jE · dA. jE includes energy flow due to diffusion and pure heat flow q0 which is the heat flux related to the first thermodynamic law. With the help of the Gauss’ theorem, the total energy change rate for a volume V of fluid becomes

Z Z ∂ 2 2 [ρ(E + 1/2u )]dV + (jE + ρ(E + 1/2u )u) · dA V ∂t A

Z ∂ 2 2 = [ [ρ(E + 1/2u )] − ∇ · [jE + ρ(E + 1/2u )u]]dV. (1.26) V ∂t

dW The work rate dt done on the volume V also consists of two parts: the work rate R Pn R done by volume force V i=1 ui · ρiXidV and work rate by surface force A u · σ · dA.

With the help of Eq. 1.2 and the Gauss theorem, the rate that work is done on a volume V becomes

Z n X ∂ 2 2 { ji · Xi +ρu·X+∇·(u·σ)− [ρ(E +1/2u )]−∇·[jE +ρ(E +1/2u )u]}dV = 0. V i=1 ∂t (1.27) For an arbitrary volume V, the integration can vanish. With the definition of the

12 substantial derivative, i.e. Eq. 1.16, the energy transport equation can be obtained

n X ∂ 2 2 ji · Xi + ρu · X + ∇ · (u · σ) − ∇ · jE = [ρ(E + (1/2)u )] − ∇ · [ρ(E + (1/2)u )u] i=1 ∂t dE du = ρ + ρu · . (1.28) dt dt

The quantity ∇ · (u · σ) can be expressed as:

X ∂ X ∇ · (u · σ) = uασαβ β ∂xβ α X X ∂σβα X X ∂uα = uα + σβα α β ∂xβ α β ∂xβ = u · (∇ · σ) + σ : ∇u. (1.29)

Using Eq. 1.29, the transport equation becomes

r dE X du ρ = σ : ∇u + ji · Xi − ∇ · jE + u · (ρX + ∇ · σ − ρ ). (1.30) dt i=1 dt

With the help of the equation of motion Eq. 1.25, the transport equation is expressed as: r dE X ρ = σ : ∇u + ji · Xi − ∇ · jE. (1.31) dt i=1

1.3 The Assumption of Local Equilibrium

As stated in the introduction, the local equilibrium postulate assumes that irre- versible transport system forms equilibrium states locally within an infinitesimally small region and can be described as a function of local state variables of a system in equilibrium. Based on the conservation laws for mass, momentum, and energy, the local equilibrium postulate will be applied to establish a phenomenological expression

13 for the general irreversible process using entropy balance equation.

According to Gibbs’ relation, the entropy S of a system in equilibrium can be expressed in terms of differential internal energy as

r X dE = T dS − pdV + µidmi, (1.32) i=1

where p is the pressure and µi is the partial specific Gibbs free energy of ith chemical

component and mi is the mass of ith component. For an arbitrary extensive quantity W with a volume element dV , two quantities need to be defined. One is the specific quantity W as the extensive quantity W per unit mass

W = W/m. (1.33)

¯ Another is the partial specific quantity Wi for ith component

! ¯ ∂W Wi = , (1.34) ∂mi T,p,ml

where the locally defined temperature T and pressure p and the masses ml of the other r − 1 components are held constant while taking the derivatives.

The partial specific quantity is related to the specific quantity as

r X ¯ ρW = ρiWi. (1.35) i=1

In the same way, the partial specific quantities for internal energy, enthalpy, volume, ¯ ¯ ¯ ¯ entropy, and Gibbs free energy can be defined as Ei, Hi, Vi, Si, and µ¯i.

The thermodynamics of a nonequilibrium system can be found by differentiating

14 Eq. 1.32 with respect to mi, which gives

¯ ¯ ¯ Ei = T Si − pVi +µ ¯i. (1.36)

Since the relation between the enthalpy and internal energy is known as

¯ ¯ ¯ Hi = Ei + pVi, (1.37)

Eq. 1.36 is found to be ¯ ¯ Hi = T Si +µ ¯i. (1.38)

1.3.1 First Law Heat Flux

The energy transport equation can be re-derived using the postulate of local equilib- rium. The total energy of the ith chemical component with volume dV with partial specific internal energy and kinetic energy in local equilibrium state can be written as Z r   X ¯ 1 2 ρi Ei + ui dV. (1.39) V i=1 2 The rate of total energy change can be written as,

Z r ∂ X  ¯ 2 [ ρi Ei + 1/2ui ]dV (1.40) V ∂t i=1

Pr ¯ Based on Eq. 1.35, i=1 ρiEi is equal to ρE. This, together with Eq. 1.3 gives the kinetic energy term as

r r r 1 X 2 1 2 1 X 2 1 X 1 2 1 2 ρiui = ρu + ρi ji · u + ρi 2 ji ≈ ρu . (1.41) 2 i=1 2 2 i=1 ρi 2 i=1 ρi 2

15 j2 The last term i becomes negligible in a slow diffusion process in local equilibrium ρi Pr while the second term vanishes since i=1 ji = 0. Therefore, it is recognized that the total energy change, Eq. 1.40, under local equilibrium is the same as that of the equilibrium system without the assumption of local equilibrium,

Z ∂ h i ρ(E + (1/2)u2) . (1.42) V ∂t

The energy flux through the surface dA consists of an energy current due to mass

 ¯ 2 flow ρi Ei + 1/2ui ui of ith component under local equilibrium and a pure heat

R 0 flux A q · dA which appears in the first law of thermodynamics the energy transport equation can be rederived

Z " r # 0 X  ¯ 2  q + ρi Ei + 1/2ui )ui · dA. (1.43) A i=1

Using Eq. 1.2, energy current due to mass flow is shown to be a combination of

2 ¯ convection energy current ρ(E + (1/2)u )u and diffusion heat flow jiEi. The total energy current through the surface dA under local equilibrium conditions becomes

Z r ! 0 X ¯ 2 q + jiEi + ρ(E + 1/2u )u · dA. (1.44) A i=1

Comparing this equation with Eq. 1.3, it is shown that conduction density is com- posed of a pure heat flux and the energy current due to diffusion.

r 0 X ¯ jE = q + jiEi. (1.45) i=1

Since Eq. 1.45 is based on the first law of thermodynamics and the assumption of local equilibrium, q0 is called the first-law heat flux.

16 1.3.2 Second-law Heat Flux

In the previous section, the internal energy flux jE was separated into a diffusive term of internal energy and conductive term of first law heat flux. Since there are

several forms of heat flow for an open system, jE can be rewritten based on the second law of thermodynamics and can also be applied to derive entropy balance equation in the next subsection.

¯ ¯ ¯ Since it is already known from Eq. 1.37 that Hi = Ei +pVi, jE can be rearranged in terms of enthalpy instead of the internal energy,

r r r r r 0 X ¯ 0 X ¯ ¯ 0 X ¯ X ¯ X ¯ jE = q + jiEi = q + ji(Hi − pVi) = (q − pVi) + jiHi = q + jiHi. i=1 i=1 i=1 i=1 i=1 | {z } q (1.46) Pr ¯ Here, q is called the second-law heat flux. The term i=1 pVi is work flux with the volume increase caused by diffusion under constant temperature and pressure.

Using Eq. 1.38, jE can be reexpressed as

r r ! X X ¯ q + jiµ¯i + jiT Si . (1.47) i=1 i=1

Combining Eq. 1.46 and Eq. 1.31 gives the energy transport equation in terms of q,

r r r dE X X ¯ X ρ = σ : ∇u − ∇ · q − ∇ · ji · µi − ∇ · jiT · Si + ji · Xi. (1.48) dt i=1 i=1 i=1

17 1.3.3 Entropy Law and Entropy Balance Equation

A change in entropy dS is the sum of two terms

dQ dS = deS + diS (deS = T ), (1.49)

where deS is the entropy provided to the system from outside and diS the entropy production within the system at the absolute temperature T at which heat is provided to the closed system. The Second Law of Thermodynamics states that the change in

dQ entropy S of the system must be greater than T in the closed system. Therefore,

Si can be characterized as

dSi ≥ 0, (1.50)

where dSi is zero for a reversible and isolated system and positive for an irreversible system.

Each component in Eq. 1.49 can be rewritten in terms of continuous functions in space as Z S = ρSdV , (1.51) V

dSe Z = − JS · dA, (1.52) dt A

dSi Z = φSdV ≥ 0. (1.53) dt V

Here S is the specific entropy, JS is the entropy flux from the surroundings, and φS is the internal entropy production rate per unit time and volume. Eq. 1.50 leads to

φS ≥ 0. Eq. 1.51 and Eq. 1.49 yield the conservation equation for entropy.

Z ∂ρS ! + ∇ · JS − φS dV = 0. (1.54) V ∂t

18 Since the volume is arbitrary, the integrand must vanish so that Eq. 1.54 yields the general conservation equation Eq. 1.12 for specific entropy. Using Eq. 1.15 and the definition of substantial derivative, Eq. 1.12 for specific entropy S can be expressed as dS ρ + ∇ · (J − ρSu) − φ = 0. (1.55) dt S S jS is defined as the quantitity JS − ρSu. Now jS includes the current density of the entropy due to diffusion only and to heat flow since convective current density of entropy, ρSu, is subtracted. The entropy production rate per unit volume, φS can be rewritten in the form of Φ/T , where T is the local temperature. Then, under local equilibrium, Eq. 1.55 becomes

dS Φ ρ = − ∇ · j . (1.56) dt T S

The rate of production of specific entropy in local equilibrium can be obtained after

dV dividing Eq. 1.32 with m and differentiating it with time. Note, dt can be written 1 dρ as − ρ2 dt so that r dS ρ dE p dρ ρ X dxi ρ = − − µi . (1.57) dt T dt ρT dt T i=1 dt

dρ The quantity dt in Eq. 1.17 is −ρI : ∇u. Substitution of Eq. 1.20, Eq. 1.48,

19 dρ and Eq. 1.19 into Eq. 1.57 together with the expression for dt yields

r dS 1 1 1 X ρ = (σ + pI): ∇u − ∇ · q − ji · (∇µi − Xi) dt T T T i=1 r r r 1 X ¯ 1 X X dλj − ∇ · jiT Si − ρ µiνij T i=1 T i=1 j=1 dt r ! r q X ¯ 1 1 X ¯ = −∇ · + jiSi − 2 q · ∇T − ji · (∇µi − Xi + Si∇T ) T T T | {z } i=1 i=1 0 | {z } ∇T µi jS r r 1 1 X X dλj − (σ + pI) : ∇u − ρ µiνij . (1.58) T | {z } T j=1 i=1 dt −η | {z } −Aj

The first term in Eq. 1.58 corresponds to entropy flux jS. The entropy flux jS con- sists of flux due to pure heat flow and entropy flow due to diffusion of components. The entropy production rate should satisfy two requirements. First, the entropy pro- duction rate should be zero under condition of thermodynamic equilibrium. Second requirement is that the entropy production rate should be invariant under a Galilean transformation.

The last four terms in Eq. 1.58 correspond to various entropy production rates divided by the absolute temperature T. The last four terms of the entropy production rate can defined as φ1, φ2, φ3, and φ4. It may be noted that all these four components are given as the product of a flux and a force. In φ1 and φ2 the heat current density

1 0 q and the diffusion current densities ji are fluxes and T ∇T and −∇T µi are forces.

In φ3 and φ4 viscous pressure tensor −η and the so-called chemical affinity −Aj are

dλj fluxes, the gradient of velocity ∇u and chemical reaction rate dt are forces. The quantity Φ in Eq. 1.56 is in the form of

X Φ = JiFi. (1.59) i

20 The flux and force in Eq. 1.59 with indices i are conjugated parameters. Then Ji is the conjugated flux to the force Fi.

1.4 The Linear Laws

It is known empirically that for a large class of irreversible phenomena, irreversible flux is linearly proportional to the thermodynamic forces [14]. In a one dimensional system, it can be generally written as

df J = −ζ . (1.60) dx

As examples, Fourier’s law for heat conduction expresses that heat flow is linearly proportional to the temperature gradient, and Fick’s law establishes the linear rela- tion between mass flow and the concentration gradient.

dT J = −k , (Fourier’s Law) (1.61) q dx

dV I = −κA · , (Ohm’s Law) (1.62) dx dc J = −D i , (Fick’s First Law) (1.63) i dx where Jq, I, and Ji are heat flux, electrical current, and the mass flux of component i. k, κ, and D are the coefficients of thermal, electrical conductivity and diffusion respectively. Using Fourier’s law and the law of conservation of energy, the heat diffusion equation can be derived. Note first the following

Z ∂q Z Z Z dV = − qv · dA = − ∇ · qvdV = − ∇ · J dV . (1.64) ∂t q

21 Eq. 1.61 and q = ρcpT give ∂T k ∂2T = 2 , (1.65) ∂t ρcp ∂x

which is known as the heat equation. Here, ρ is the density, cp is the heat capacity at constant pressure, and k is the diffusivity or thermometric conductivity. In the ρcp same way, the Fick’s second law for mass diffusion can be obtained as

∂c ∂2c i = D i . (1.66) ∂t ∂x2

However, there are cross phenomena in which non-equilibrium processes interfere with one another such as the Soret effect and the Dufour effect. The Soret effect is mass flow which depends linearly on the temperature gradients(besides concentration gradients), while Dufour effect is heat flow which depends linearly on concentration gradients(besides the temperature gradients). The linear laws can be applied to most thermodynamic systems except nonlinear systems such as a system with chemical reactions. These “Linear Laws” can generally be written as

X Ji = LikFk,, (1.67) k

where Ji and Fk are the fluxes and thermodynamic forces which appear in the form of Eq. 1.59. The quantities Lik are called the phenomenological coefficients and Eq. 1.67 is called the phenomenological equation.

1.5 Onsager Theory

Lars Onsager introduced theory to explain non-equilibrium thermodynamics which is now referred as Onsager theory. Onsager theory postulates the time reversal

22 symmetry. Consider an adiabatically insulated system under no external magnetic

field. The state of the system can be described by thermodynamic parameters Aα.

0 The deviations of these parameters from their equilibrium values, Aα are defined as

0 aα equal to Aα − Aα. At the equilibrium the entropy is at its maximum, and the state variable aα becomes zero. Therefore, the deviation ∆S of the entropy from its equilibrium state value can be expressed as

1 X X 1 ∆S = gαβaαaβ. (1.68) 2 α β T

It is assumed that the constant coefficients gαβ obey the following relation for all α and β,

gαβ = gβα. (1.69)

Taking the substantial time derivative of the entropy on Eq. 1.68 yields

1 X X daα 1 X X daα Φ = gαβ aβ + aαgαβ . (1.70) 2 α β dt 2 α β dt

Eq. 1.70 is valid both for isolated and open systems. Interchange of the dummy variables α and β and usage of Eq. 1.69 gives Φ as

X X daα Φ = gαβ aβ. (1.71) α β dt

The flux Jα can be defined as da J = α , (1.72) α dt

and define its conjugated force Fα as

∂S X Fα = T = gαβaβ, (1.73) ∂aα β

23 then Eq. 1.71 becomes X Φ = Jα Fα. (1.74) α

Here it is shown that the entropy production rate can be expressed as the product of fluxes and forces.

It can be assumed that the flux of the thermodynamic parameter can be described by linear phenomenological equations as

daα X = Mαγaγ. (1.75) dt γ

Using Eq. 1.73, Eq. 1.75 can be expressed as

daα X Jα = = LαβFβ, (1.76) dt β

P −1 where the phenomenological coefficient Lαβ is γ Mαγgγβ which is independent of the forces. Eq. 1.76 is the second postulate for the general theory of irreversible processes. The second postulate states that the linear relations can be written for the set of conjugated fluxes and forces which appear in Eq. 1.74. Onsager has shown that if the linear relations are written as in Eq. 1.76 with fluxes and forces given

by Eq. 1.72 and Eq. 1.73 and if thermodynamic parameter aα is an even function of the velocities of the molecules in the system, the matrix of the phenomenological coefficients is symmetric. This is known as the Onsager reciprocal relation stated as

Lαβ = Lβα. (1.77)

24 1.6 Soret Effect

Here the analysis of thermal diffusion will be done, which is the cross phenomena of mass diffusion caused by thermal gradient, based on equations derived in the previous sections. Consider isotropic fluids, with r components, where there are no chemical reaction and no external forces [14]. In the absence of external forces, the pressure is uniform in the system. Then, the third and the fourth term in Eq. 1.58

vanish. Eliminating jr using Eq. 1.4, Φ in the entropy production rate Eq. 1.58 for this system becomes

r−1 ∇T X 0 0 Φ = −q · − ji · ∇T,P (µi − µr). (1.78) T i=1

With the help of the Gibbs-Duhem relation, the following is obtained

r X 0 cjδµj = 0 (p and T constant). (1.79) j=1

Using Eq. 1.79, Φ becomes

r−1 ∇T X 0 −q · − ji · ∆ij∇T,P µj. (1.80) T i,j=1

∆ij is defined as, cj ∆ij = δij + (i, j = 1, ..., r − 1) (1.81) cr

where δij is 1 when i = j; otherwise it is 0. For isothermal and isotropic system,

0 Pr 0c the gradients of the chemical potentials ∇µj(T, p, cl) can be written as i=1 µmi∇ci 0 c ∂µj where µ = ( )T,p,c . Therefore, Eq. 1.78 becomes jl ∂cl m6=l

r−1 ∇T X c Φ = −q · − ji · ∆ijµjl∇cl. (1.82) T i,j,l=1

25 By comparing this with Eq. 1.76, the phenomenological equation for the fluxes and thermodynamic forces can be seen to reduce to

r−1 ∇T X c q = −Lqq − Lqi∆ij(µjl∇cl), (1.83) T i,j,l

r−1 ∇T X c jk = −Lkq − Lki∆ij(µjl∇cl). (k = 1, ..., r − 1) (1.84) T i,j,l

The Onsager reciprocal relation Eq. 1.77 states Lqi = Liq and Lki = Lik. The coeffi-

cients Lqq and Lki are related to the heat conductivity and the diffusion coefficients,

respectively. The coefficients Lkq characterize the phenomenon called thermal diffu- sion or the Soret effect in liquids, the flow of matter caused by a thermal gradient.

The coefficients Lqi characterize the reciprocal phenomenon called the Dufour effect which is heat flow caused by concentration gradients.

1 For the binary mixture system with r = 2, ∆11 becomes . The phenomenolog- c2 ical equations for the binary mixture system become

c ∇T (µ11∇c1) q = −Lqq − Lq1 , (1.85) T c2

c ∇T (µ11∇c1) j1 = −L1q − L11 . (1.86) T c2

It is possible to replace the phenomenological coefficients with a new set of coeffi- cients, L κ = qq , Thermal Conductivity (1.87) T L D00 = q1 , Dufour coefficient (1.88) ρc1c2T L D0 = 1q , Thermal diffusion coefficient (1.89) ρc1c2T L µc D = 11 11 . Diffusion coefficient (1.90) ρc2

26 Substituting the coefficients from Eq. 1.87 to Eq. 1.90 into Eq. 1.85 and Eq. 1.86 yields the following coupled transport equations,

00 c q = −κ∇T − ρ1TD µ11∇c1, (1.91)

0 j1 = −ρc1c2D ∇T − ρD∇c1. (1.92)

By using the conservation of mass and conservation of energy, the phenomenological equations change into new forms,

∂T ρc = −∇ · q = ∇ · (κ∇T + ρ TD00µc ∇c ) , (1.93) p ∂t 1 11 1

∂c1 j 0 = −∇ · = ∇ · (c1c2D ∇T + D∇c1). (1.94) ∂t ρ 1

These are the two governing transport equations for thermal diffusion.

27 Chapter 2

Analytical Solution for the Dynamics of the Soret Effect

As shown in the previous chapter, thermal diffusion of a binary mixture system is governed by two transport equations:

∂T ρc = ∇ · (κ∇T + D00∇c ), (2.1) p ∂t 1

∂c 1 = ∇ · (c c D0∇T + D∇c ), (2.2) ∂t 1 2 1

kg J where ρ is the density of the material ( m3 ), cp is the specific heat capacity ( kg·K ), κ W 00 is the thermal conductivity ( m·K ) (found in the Fourier heat equation), and D is the m2 coefficient of the Dufour effect ( sec·K ). The Dufour effect is the formation of a density gradient caused by the temperature gradient. D0 is the thermal diffusion coefficient

m2 m2 ( sec·K ) and D is the ordinary mass diffusion coefficient ( sec ) seen in Fick’s law. c1

and c2 are the density fractions of species in a binary mixture which is normalized to 1. The analytical and numerical solutions for the dynamics of the Soret effect in

28 a sinusoidal temperature field were investigated in Ref. [12, 27]. Analytic solution of the Soret effect in a linear temperature field for a one dimensional system will be discussed in the following section.

2.1 Equation of Motion For a Linear Temperature

Field

To get the solution of Eq. 2.2 in a one dimensional temperature field, consider

two infinitely large parallel slabs for which the temperature difference is T0 across a distance L along the z axis.

T T (z, t) = − 0 z + T . (2.3) L h

Since Dufour effect is considered to be negligible, Eq. 2.1 becomes the Fourier heat equation. For a one dimensional temperature field system, Eq. 2.2 becomes

∂c(z, t) D0T ∂ dT ∂2c(z, t) = − 0 (c(z, t)[1 − c(z, t)] ) + . (2.4) ∂t D ∂z dz ∂z2

The dimensionless variables ξ and θ are defined as Kz and K2Dt respectively, where

1 K is L . After replacing c2 with 1 − c1, the subscript 1 can be dropped from the

density fraction c1. Then Eq. 2.4 becomes

∂c(ξ, θ) D0T ∂ dT ∂2c(ξ, θ) = − 0 (c(ξ, θ)[1 − c(ξ, θ)] ) + . (2.5) ∂θ D ∂ξ dξ ∂ξ2

For convenience in dealing with boundary conditions at ξ = 0 and 1 in the following section, the temperature field can be expanded to form a periodic triangular temper-

29 temperature profile T

-1 0 1/2 1 3/2 2  -1/2 dT first derivative of the temperature d

-1 -1/2 0 1/2 1 3/2 2

second derivative of the temperature d 2T d 2

-1 -1/2 0 1/2 1 3/2 2

Figure 2.1: Periodic linear temperature field and its first and second derivative

0 D T0 ature field, as shown in Fig. 2.1. The thermal diffusion factor α is defined as D . ∂c(ξ,θ) ∂c(ξ,θ) When ∂ξ and ∂θ are written as cξ and cθ respectively, Eq. 2.5 becomes

cθ = [αc(1 − c)Tξ + cξ]ξ. (2.6)

With c¯ = c − 1/2, Eq.2.6 becomes, in symmetric form:

1 c¯ = [α(¯c2 − ) +c ¯ ] = φ , (2.7) θ 4 ξ ξ ξ

where φξ is a flux function. It is of note that Eq. 2.7 for the density fraction c¯ is equivalent to c¯θ = 2αc¯c¯ξ +c ¯ξξ which is Burgers’ equation. Burgers’ equation is a fundamental partial differential equation seen in fluid mechanics. The last term in Eq. 2.7 describes mass diffusion. When it vanishes, Burgers’ equation becomes the inviscid Burgers’ equation whose solution can have propagating discontinuities known as shock waves.

30 2.1.1 Exact Solution Using Hopf-Cole Transformation

For solution of Eq. 2.5, the Hopf-Cole transformation is used:

1 ∂ c¯(ξ, θ) = lnV (ξ, θ). (2.8) α ∂ξ

Using Eq. 2.8, the flux equation, Eq. 2.7 becomes

c¯θ = φξ (2.9) where

1 φ = α(¯c2 − ) +c ¯ 4 ξ V α = ξξ − . (2.10) αV 4

When partial differentiation is taken to c in Eq. 2.8, the flux equation becomes

c¯θ = φξ 1 V = ( ξ ) α V θ 1 V = ( θ ) α V ξ 1 V = ( ξξ − α/4) . (2.11) α V ξ

From Eq. 2.11, Vθ can be expressed as

1 V ( θ ) = φ + f(θ) α V α 1 V = − + ξξ + f(θ). (2.12) 4 α V

31 which is rearranged as

2 Vθ − (f(θ) − α /4)V = Vξξ, (2.13) where f(θ) is a function of θ. It is possible to define a new function W (ξ, θ) according to

θ R (f(θ0)−α2/4)dθ0 V (ξ, θ) = W (ξ, θ)e 0 . (2.14)

Eq. 2.13 then becomes an ordinary diffusion equation

Wθ = Wξξ. (2.15)

Using both periodicity in the temperature field as shown in Fig. 2.1 and the law of conservation of mass, boundary conditions can be determined at ξ = 0 and 1. There is an initial condition as well.

The heat diffusion problem with a specified initial condition and two boundary conditions is expressed as

Wθ = Wξξ, 0 < ξ < 1, θ > 0

W (ξ, 0) = f(ξ), 0 < ξ < 1

W (0, θ) = g(θ), θ > 0

W (1, θ) = h(θ), θ > 0, (2.16) which has the following solution[8]

Z 1 W (ξ, θ) = [ϑ(ξ − ξ0, θ) − ϑ(ξ + ξ0, θ)]W (ξ0, 0)dξ0 0 Z θ −2 ϑξ(ξ, θ − τ)W (0, τ)dτ 0 Z θ +2 ϑξ(ξ − 1, θ − τ)W (1, τ)dτ. (2.17) 0

32 Figure 2.2: Periodic density profile corresponding to periodic temperature profile

P∞ −1/2 −ξ2/(4θ) where ϑ is defined as ϑ(ξ, θ) = m=−∞ K(ξ + 2m, θ), where K(ξ, θ) = (4πθ) e and m is an integer. To determine the solution, consider the boundary conditions at ξ = 0 and 1.

As it was shown in Fig. 2.1, the first derivative of the temperature is the periodic square function dT k=∞ = X −2u(ξ − 2k) + 2u(ξ − 2k + 1), (2.18) dξ k=−∞ where u is the heaviside step function and k is an integer.

Using c¯ = c − 1/2 and Eq. 2.6, a more symmetric form of the Soret equation is found to be 1 c¯ = [−α(¯c2 − )T +c ¯ ] . (2.19) θ 4 ξ ξ ξ

Eq. 2.20 can be integrated around the infinitesimally narrow area near the bound- aries ξ = 0 and ξ = 1. At the boundary ξ = 0,

Z  Z  d d 2 1 cdξ¯ = [−α(¯c − )Tξ +c ¯ξ]ξdξ. (2.20) dθ − − dξ 4

33 As  goes to zero, the left hand side becomes zero since c(ξ) is a continuous function.

1 0 = [−α(¯c2 − )T +c ¯ ] 4 ξ ξ − 1 1 = [−α(¯c2 − )T +c ¯ ] − [−α(¯c2 − )T +c ¯ ] R 4 ξ Rξ L 4 ξ Lξ 1 = − α + αc¯2 + αc¯2 +c ¯ − c¯ . (2.21) 2 R L Rξ Lξ

Knowing cR = cL and cRξ = cLξ, the derivative values of c at the boundary ξ = 0 can be obtained as:

2 c¯Rξ = (−c¯R + 1/4)α,

2 cLξ = (¯cL − 1/4)α. (2.22)

In the same way, the first derivatives of c at ξ = 1 can be determined as:

2 c¯lξ = (−c¯l + 1/4)α,

2 c¯rξ = (¯cr − 1/4)α. (2.23)

When the boundary values are applied to Eq. 2.9, the flux φ becomes 0. Substitution of φ = 0 into Eq. 2.12 yields

Vθ = αf(θ)V (2.24) which implies

θ0 α R f(θ0)dθ0 V (ξ, θ) = [V (ξ, θ = 0)e 0 ]|ξ=0,1. (2.25)

From the conservation of density fraction, it follows that

Z 1 c0 − 1/2 = c¯(ξ, θ)dξ 0 1 Z 1 ∂ = lnV dξ, (2.26) α 0 ∂ξ

34 so that V (1, θ) = eα(c0−1/2) where c = R 1 c(ξ, θ)dξ. (2.27) V (0, θ) 0 0

From Eq. 2.26 and Eq. 2.24, Eq. 2.14 at the boundaries becomes

θ R (f(θ0)−α2/4)dθ0 V (ξ, θ)|ξ=0,1 = W (ξ, θ)e 0

θ α(c −1/2)ξ R f(θ0)dθ0 = V (0, θ)e 0 e 0 , (2.28) hence

α2 α(c0−1/2)ξ+ θ W (ξ, θ) = V (0, θ)e 4 |ξ=0,1

α2 α(c0−1/2)ξ+ θ = e 4 |ξ=0,1, (2.29)

where V (0, θ) is unity.

Now consider the initial condition θ = 0. Since Eq. 2.8 and Eq. 2.14 yield

Z ξ Z ξ 1 V 0 c¯(ξ0, θ = 0)dξ0 = ξ dξ0 0 0 α V Z ξ 1 W 0 = ξ dξ0, (2.30) 0 α W

thus

ξ R c¯(ξ0,θ=0)dξ0 W (ξ, 0) = e 0 . (2.31)

According to Eq. 2.17 which is the solution to the heat diffusion equation with two boundary conditions and the initial condition of a uniform density profile, Eq. 2.17

35 Figure 2.3: Density distribution versus coordinate ξ at several different times with an initially, spatially uniform density distribution with a value of c = 0.3 and a thermal diffusion factor of α = 100. The dimensionless θ values are 7 × 10−5 (−); 4 × 10−3 (−−); and 1 (− · −). It is shown that the change of c starts from the each boundary forming shock wave fronts that propagate in opposite directions. (From Ref. [28].)

becomes

Z 1 W (ξ, θ) = [ϑ(ξ − ξ0, θ) − ϑ(ξ + ξ0, θ)]W (ξ0, 0)dξ0 0 Z θ 2 α τ −2 ϑξ(ξ, θ − τ)e 4 dτ 0 Z θ α2 α(c0−1/2) τ +2e ϑξ(ξ − 1, θ − τ)e 4 dτ, (2.32) 0

Pm=∞ where ϑ is the theta function defined as ϑ(x, t) = m=−∞ K(x + 2m, t). Fig. 2.3 shows the density fraction profile at different times for the initial uniform density distribution W (ξ, 0) = e(c0−1/2)ξ. Fig. 2.4 shows the density change for an initially

q −1 −(ξ−1/2)2/σ2 Gaussian density distribution with c(ξ, 0) = c0[σ (π)erf(1/2σ)] e which

36 Figure 2.4: The change in density profile versus coordinate ξ with a Gaussian dis- tribution as the initial profile. Values of c0, α, and σ used are taken to be 0.3, 100, and 0.17. The dimensionless times θ are 1 × 10−4 (−); 2 × 10−3 (−−); 5 × 10−3 (− · −); and 1 (··· ). Just as the initially constant density distribution shown in Fig. 2.3, the initially Gaussian density distribution moves from each boundary forming counter-propagating shock waves. In other words, the interaction of the thermal diffusion waves with the boundaries creates the shock wave fronts. Regardless of the initial density profile, the steady states at very long time are the same with the same values of α and c0. (From Ref. [28].)

is normalized to c0. Figs 2.3 and 2.4 shows that the final density fraction distributions

for identical values of c0 and large α form identical, nearly rectangular square waves regardless of the initial distributions.

2.1.2 Solution without Mass Diffusion

Ignoring dissipative effects dependent on cξξ permits determination of the underlying characteristics of time development of c. The following shows that without the mass

diffusion term (i.e. cξξ), density shock waves are formed. A new flux function f(c, ξ)

can be defined which can be expressed as f(c, ξ) = −c(1 − c)Tξ. After redefining τ

37 as αθ, the thermal diffusion equation becomes

∂c df = − , (2.33) ∂τ dξ

which can be expressed as

∂c ∂f ∂c ∂f + + = 0. (2.34) ∂τ ∂c ∂ξ ∂ξ

Using Eq. 2.34 with the substantial time derivative of c(ξ, τ), Eq. 2.35,

∂c ∂c ∂ξ dc + − = 0, (2.35) ∂τ ∂ξ ∂τ dτ

the characteristic equations are built:

∂ξ ∂f(ξ, τ) dT = = −α(1 − 2c) , (2.36) ∂τ ∂c dξ

dc ∂f(ξ, τ) d2T = − = αc(1 − c) . (2.37) dτ ∂ξ dξ2

These equations describe the formation of shock waves. For the periodic triangular temperature field shown in Fig. 2.1, these characteristic equations become

∂ξ = α(1 − 2c)[u(ξ) − u(ξ − 1)], (2.38) ∂τ

dc = αc(1 − c)δ(ξ) − c(1 − c)δ(ξ − 1). (2.39) dτ where u is the Heaviside function.

Consider the initial, uniformly distributed density profile with c(ξ, τ = 0) = c0. At the moment of switching on the temperature field, the density fraction increases without bound at the boundary of ξ = 1 and decreases at ξ = 0. Based on Eq. 2.36,

38 Figure 2.5: The phase plot follows Eq. 2.38 and Eq. 2.39. The plot shows that the each phase point moves at a speed of dz/dτ which is proportional to the density difference to c = 1/2. The arrows show the magnitude and the direction of the speed. Shock wave fronts are formed immediately at the boundaries on the imposition of the temperature field. The shock front shift is determined after averaging the area enclosed by diffusion waves and the first shock front, which is shown as triangles here. (From Ref. [28].)

the shock wave fronts move at a different velocities. For a density fraction c > 1/2, the shock moves to the left, and, for a density fraction c < 1/2, the front moves to the right with a velocity proportional to α(1−2c). In Fig. 2.5, the change of position of each point on the shock wave front at ξ = 1 is shown as the upper triangle S2 for

the density value c > 1/2 and as the lower triangle S1 for c < 1/2. The change in location of the shock wave fronts after a time dτ can be obtained by determining the

rectangular area with a moving front whose height ranges from c0 to 1 and whose area is equal to the average area change during shock front movement. The average area change of the shock front is given as R 1 dcdξ = R 1 α(1 − 2c)dcdτ during the time c0 c0

interval dτ based on Eq. 2.36. The area change is found to be αc0(1−c0)dτ and is also

39 equivalent to the value of S2 − S1. The subsequent location of the rectangular shock

− − wave front after a time dτ is −α(1−c0)vshdτ which moves a distance of −vshdτ along

− ξ where vsh is the shock velocity. By equating the rectangular area and average area

− change of the shock front, the shock wave velocity is found to be vsh = αc0. In the same way, the velocity of the shock wave front originating from ξ = 0 can be obtained.

R c0 The area change in phase diagram at ξ = 0 is 0 α(1 − 2c)dcdτ = αc0(1 − c0)dτ.

Given the shock wave front with a vertical range of 0 to c0, the shock wave velocity

+ at ξ = 0 is determined to be vsh = α(1 − c0). From the above analysis for shock wave front movement, the profile for the shock wave front as a function of time and space is

c = 1 − (1 − c0)u(ξ + 1/2 − αc0τ) − c0u(ξ − 1/2 − α(1 − c0)τ), (2.40) where u is the Heaviside function.

2.1.3 Shock Waves with Moving Coordinates in the Infinite

Domain without Boundary

Consider a linear temperature field in a finite region extended into the infinite do- main. By changing variables from (ξ, θ) into the moving coordinate variables set (ζ, θ), the thermal diffusion problem is extended into the infinite domain, where ζ = ξ − αθ. Insertion of the following derivative relation into Eq. 2.6

∂ ∂ ∂ ∂ ∂ = , = + α , (2.41) ∂ξ ∂ζ ∂θ ∂θ ∂ζ gives

2 cθ = α(c )ζ + cζζ . (2.42)

40 1 Vζ 1 ∂ With the original definition of c¯(ζ, θ) as α V = α ∂ζ lnV , Eq. 2.42 yields

1 ∂ ∂ α ∂ V ∂ ∂ 1 V lnV − ( ζ )2 − ζ = 0. (2.43) α ∂ζ ∂θ α2 ∂ζ V ∂ζ ∂ζ α V thus

Vθ − Vζζ = g(θ)V, (2.44) where g(θ) is a function of θ. The same procedure is repeated for the case of the finite

θ R g(θ0)dθ0 domain using variable V (ζ, θ) = U(ζ, θ)e 0 which gives the ordinary diffusion equation as Eq. 2.15

Uθ = Uζζ , (2.45) except that the initial condition is changed into an infinite domain,

∞ α R c(ζ0,0)dζ0 U(ζ, 0) = e −∞ . (2.46)

Consider Eq. 2.45 on the infinite domain −∞ < ζ < ∞ with the initial value U(ζ, 0) at θ = 0. The separation of the variables in the form U(ζ, θ) = Z(ζ)T (θ) reduces the heat equation to 1 ∂ 1 ∂2 T (θ) = Z(ζ). (2.47) T (θ) ∂θ Z(ζ) ∂ζ2

Since both sides of the equations are independent, they can be equated to the con- stant value −λ2. The solutions for each side of Eq. 2.47 are

T (θ) = Ae−λ2t Z(ζ) = a(λ)cos(λζ) + b(λ)sin(λζ), (2.48)

where A, a(λ) and b(λ) are unspecified constants. The general solution of Eq. 2.45 is in the form with the superposition of Z(ζ)T (θ)

Z ∞ U(ζ, θ) = e−λ2θ[a(λ)cos(λζ) + b(λ)sin(λζ)]dλ. (2.49) 0

41 Figure 2.6: Density fraction versus coordinate ξ at different times given an initial Gaussian distribution of density fraction along an infinite linear temperature domain. The values of α and σ are taken to be 100 and 0.1, respectively. The values of the dimensionless time θ are 1×10−5(-); 2×10−3(−−); 7×10−3(−·−); 1.3×10−2(−··−); and 2 × 10−2(···). Due to absence of boundaries and thermal diffusion interactions, no shock wave fronts are formed. The right-most curve is a self similar wave. (From Ref. [28].)

The unknown coefficients a(λ) and b(λ) can be determined to describe the initial condition. From Eq. 2.49, the initial condition can be written as

Z ∞ U(ζ, 0) = [a(λ)cos(λζ) + b(λ)sin(λζ)]dλ −∞ < ζ < ∞. (2.50) λ=0

Using the Fourier transform with a known initial condition U(ζ, 0) gives

1 Z ∞ 1 Z ∞ a(λ) = U(ζ0, 0)cosλζ0dζ0 b(λ) = U(ζ0, 0)sinλζ0dζ0. (2.51) π ζ0=−∞ π ζ0=−∞

Substitution of equations of Eq. 2.51 into Eq. 2.49 gives

42 ∞ ∞ 1 Z 2 Z U(ζ, θ) = e−λ θ U(ζ0, 0)cosλ(ζ − ζ0)dλ (2.52) π λ=0 ζ0=−∞ Z ∞ 0 2 1 0 − (ζ−ζ ) 0 = U(ζ , 0)e 4θ dζ . (2.53) [4παθ]1/2 ζ=−∞

Introduction of a new variable β = √ζ into Eq. 2.52 gives 2 θ

∞ 1 Z √ 0 2 U(β, θ) = √ U(2 θβ0, 0)e−(β−β ) dβ. (2.54) π −∞

The difference in evolution of the Gaussian initial distribution in Fig. 2.4 where c is confined in space and Fig. 2.6, where c is not confined in space, can be attributed to the presence or absence of boundary conditions. That is, the nonlinear interaction of the density wave with the boundaries at ξ = 0 and ξ = 1 gives rise to the shock wave front for spatially restricted c.

Regardless of the initial density distribution, the terminal density distributions shown in Figs 2.3 and 2.4 are identical for the same α value and the same value of

R 1 the total density fraction, i.e. 0 c(ξ, 0)dξ [34]. This is shown in section 2.1.4 by solving for the terminal solution of c by setting cθ as zero. Eq. 2.6 becomes then an ordinary partial differential equation which depends only on a single parameter α. The result of differentiation of Eq. 2.54 at infinite θ is independent of θ as noted in √ Ref. [63]. With the definition of the variable ζ = ξ −αθ and the variable β = ζ/2 θ, √ Eq. 2.54 determines the wave speed as ξ/θ = α + 2β/ θ which approaches α as θ becomes large.

The boundary conditions at ξ = 0 and 1, imply that the deposition of density is unbounded over time which seems non-physical. However, the boundary condition

R 1 0 0 α2 W = exp[α 0 c(ξ , 0)dξ + 4 θ] at ξ = 1 is constrained to give a finite value of R 1 0 c(ξ, 0)dξ = c0 because it contains this source term in its own expression. The boundary value of W acts as a source term of density fraction controlled by the

43 integrated value of c over space multiplied by α.

From Figs. 2.3 and 2.5, it can be seen that square shock waves propagate towards each other until they meet, at which point they form a stationary profile at infinite time. As is shown in section 2.1.4, counteracting shock waves are formed at the boundary immediately when the temperature gradient is applied. Figs. 2.3 and 2.5 show that the mass diffusion term acts to filter out the high spatial frequency components of thermal diffusion waves resulting in smoothing of the density fraction profile.

2.1.4 Long Time (Terminal) Solution for Soret Equation

Consider the terminal solution for Eq. 2.6. In the long time limit, the density profile

dc will be time independent, resulting in dθ = 0, leaving the flux φ as a constant k. Therefore, Eq. 2.6 becomes

∂c αc(1 − c) + = k. (2.55) ∂ξ

Eqs. 2.22 and 2.23 require that the value of k should be zero at the boundaries ξ = 0 and 1, giving Z c 0 Z ξ dc 0 02 0 = α dξ , (2.56) cB c − c 0

where cB is the density fraction at ξ = 0. From Eq. 2.56, the terminal distribution becomes 1 c(ξ, ∞) = . (2.57) 1 + 1−cB eαξ cB

44 

Figure 2.7: Terminal density distribution for c0 = 0.4 for α = 1(−−); α = 10(− · −); and α = 100(−). (From Ref. [34].)

From the conservation of density fraction

Z 1 0 0 c(ξ , ∞)dξ = c0, (2.58) 0

cB, the density fraction at ξ = 0 is found to be

1 − e−αc0 c = , (2.59) B 1 − e−α

which in turn gives cE, the density fraction at ξ = 1

1 − e−αc0 cE = . (2.60) eα(c0−1) − e−αc0

Figs. 2.7 and 2.8 show several terminal density fraction distributions for different α values and initial concentrations. At large values of α, the Soret effect produces an almost perfect separation with a square shaped terminal density profile.

45 

Figure 2.8: Terminal density distribution for c0 = 0.04 for α = 1(−−); α = 10(−·−); and α = 100(−). (From Ref. [34].)

As shown in Fig. 2.3, the counter-acting propagations at large α pair off in nearly square waves propagating in opposite directions until they meet at which point they become stationary. In the limiting case where the effects of the mass diffusion are neglected, as shown in Eq. 2.39, two counter-propagating shock waves are formed instantaneously upon application of the temperature field. Therefore, the fundamental evolution of the density fraction distribution can be described by the motion of a pair of shock waves where mass diffusion damps out the highest spatial frequency components of the waves.

It is of note, that the terminal density distributions of Fig. 2.3 and Fig. 2.7 are similar for the same large value of α (α = 100) which indicates that the steady state of the exact solution of the thermal diffusion equation Eq. 2.32 agrees qualitatively with the terminal density distribution solution Eq. 2.57.

46 Chapter 3

Experiments and Results

3.1 Different Techniques to Measure the Soret

Coefficient

There are two different techniques to measure the Soret coefficient: The first is essentially a convectionless systems and the second uses convective coupling. Two categories of experimental methods for the convectionless system will be investigated: first using a standard simple Soret cell that creates a linear temperature field, and the other utilizing a sinusoidal temperature field created by overlapping of two laser beams. The technique using convective coupling, has been reviewed in Ref. [59].

47 Figure 3.1: Standard Soret Cell. (From Ref. [59].)

3.1.1 The Standard Soret Cell

In a standard Soret cell, a liquid mixture is held between two thermally conducting metal housings, as shown in Fig. 3.1. Using a thermostatic water circulator, the housings produce a linear temperature gradient inside the cell.

The system is heated from above to prevent convection. Small holes in a PVC brace placed between the plates allow the sampling of minute amounts of liquid. The concentration difference between the top and the bottom samples, ∆c, can be determined with a high resolution refractometer with a sensitivity of 1 part in 105 or densitometer with a sensitivity of 1 part in 106(g/cm3) after comparing with calibration curves. In the steady state approximation, cθ eventually becomes zero. Thus, Eq. 2.6 for the steady state approximation can be written as

∆c = −ST c0(1 − c0)∆T. (3.1)

Local temperature measurement at the location of sampling port using a thermocou- ple is preferred over a measurement of ∆T between the two plates. Sampling over

48 time gives the time evolution of ∆c/∆T given by

−n2 t ∆c 8 X e θ = −ST c0(1 − c0)[1 − 2 2 ], (3.2) ∆T π n odd n

where θ is the relaxation time defined as

a2 θ = . (3.3) π2D

The phenomenological equation for thermal diffusion can be written in terms of flux function Γ,

∂c ∂Γ −ρ = ∂t ∂z ∂ ∂c T = − [ρD − ρD0c(1 − c) 0 ]. (3.4) ∂z ∂z L

where z is a distance measured from bottom plate, T0 is a temperature difference between the plates, and L is a separation of the plates. Bierlein[6] and Groot[13] solved Eq. 3.4 with the identical initial condition of uniform density distribution and the boundary conditions: The flux Γ vanishes at z = 0 and 1. For convenience, dimensionless variables can be defined as:

z ξ = , L

α = ST T0, (3.5)

. With the tangential approximation,

∼ c(1 − c) = f(c) = c0(1 − c0) + C(c − c0), (3.6)

0 0 where C stands for f (c0) = 1 − 2c0 ≡ f (c) = 1 − 2c0, mathematical manipulation

49 yields the time evolution of the density fraction as

1 4α 2 c(ξ, θ) = c [1 − α(1 − c )( − ξ) − (cosπξ)e−π θ]. (3.7) 0 0 2 π2

dc The direct differentiation of Eq. 3.7 with respect to ξ gives dξ as[6],

dc 4 2 = −αc (1 − c )[1 − (sinπξ)e−π θ]. (3.8) dξ 0 0 π

However, this mathematical manipulation has a drawback that its validity is re-

1 stricted only to time θ > 3π2 .

With a curve fitting procedure, ST = DT /D and D in Eq. 3.2 can be determined.

Therefore, DT can be also obtained as a byproduct. Drawbacks of this technique include loss of samples and the challenge in monitoring the real time change in density fraction[59].

3.1.2 The Beam Deflection Technique

In the beam deflection technique, the vertical cell walls shown in Fig. 3.2 must be made of optically high quality glass. In the absence of an index of refraction gradient, the beam passes the cell horizontally. The refractive index n is a function of density and temperature. Therefore, the existence of a vertical temperature and

dn a concentration gradient cause an index of refraction gradient dξ defined as

dn dn dc dn = + T . (3.9) dξ dc dξ 0 dT

50 Figure 3.2: Sketch of an elementary Soret cell that uses a beam deflection technique. When the deflected beam hits the BDU (Beam Deflection Unit) at position B, the deflection distance AB gives the index of refraction gradient. (From Ref. [59].)

where T0 is the temperature difference between the two slabs in Eq. 2.3. Eq. 3.8 combined with Eq. 3.9 gives

dn ∂n 4 2 dn = −αc (1 − c ) [1 − (sinπξ)e−π θ] + T . (3.10) dξ 0 0 ∂c π 0 dT

dn The index of refraction gradient dξ can be determined from the measurement of dn the angular deflection, Θ = l dξ , of a laser beam, where l is the length of the cell. The thermal diffusion factor α can be obtained from the slope of the semilog plot of ∆(θ) = Θ(∞) − Θ(θ) versus θ where ∆(θ = ∞) is the measurement at the steady state[23]. The optical beam deflection technique was first used to measure the Soret coefficient for macromolecules[22, 23].

Usually for liquids, the thermal diffusivity is two orders of magnitude larger than the diffusion coefficient. This result in a relatively instantaneous formation of a temperature gradient after the thermostatic baths are switched on, which induces a first fast variation of the index of refraction and a first deflection of the beam,

51 Figure 3.3: A TDFRS experimental setup. (From Ref. [73].)

followed by slow formation of a concentration gradient. The kinetics of the slow component of the beam yield the diffusion relaxation time, which leads to the value of DT /D. Once the steady state is reached and the thermostatic bath is switched off, the concentration gradient will slowly disappear whereas the temperature gradient dissipates quickly. The time needed for the deflection angle to return to zero gives the isothermal diffusion coefficient D.

3.1.3 Thermal Diffusion Forced Rayleigh Scatting Technique

(TDFRS)

The overlapping of two phase coherent light beams (typically from the same laser) creates a periodic sinusoidal temperature gradient [59]. A TDFRS experimental setup is shown in Fig. 3.3. An argon-ion laser (λ = 488nm) is expanded and split

52 into two beams of equal intensity and intersected within the sample at an angle θ. The sinusoidal temperature gradient formed through interference causes a periodic concentration gradient due to thermal diffusion. This also changes the index of refraction. A Pockels cell acting as a half wave plate rotates the polarization of the beam by 90o. The two orthogonal beams do not form the interference grating in the sample. The piezo actuator allows phase adjustment of the grating and diffraction for heterodyne detection. The fringe spacing in the grating, d = 2πq−1 is determined as 4π θ q = sin . (3.11) λ 2

Small values of the angle θ of a few degrees result in fringe spacings of a few µm. The diffraction efficiency of the grating is measured using a HeNe laser (λ = 632.8 nm) through Bragg diffraction.

Two heterodyne detection schemes are shown in Fig. 3.4. Generally, the detected signal S contains both homodyne and heterodyne signals

iϕ 2 2 S(ϕ) ∝ |Ec + Ese | + Einc

2 2 2 = Es + 2EsEccosϕ + Ec + Einc, (3.12)

where Ec, Es, and Einc are the electric field amplitudes of the coherent background, diffracted beam and incoherent background, respectively, and ϕ is the phase shift between signal and the reference. Homodyne (Shom) and heterodyne (Shet) signals are defined as

1 S = [S(ϕ) + S(ϕ + π)] − S ∝ E2, hom 2 b s 1 S = [S(ϕ) − S(ϕ + π)] ∝ 2E E cosϕ. (3.13) het 2 s c

53 Figure 3.4: Two heterodyne detection schemes: phase matching of the two beams can be achieved by (a) phase control of the reference beam or (b) phase control of the diffracted beam. (From Ref. [73].)

3.1.3.1 Phenomenological Model

When the z-axis is perpendicular to the optical axis and within the plane formed by two beams, the periodic intensity within the grating is

iqz I(z, t) = I0 + Iq(t)e . (3.14)

With this, one-dimensional heat equation can be expressed as

2 ∂T (z, t) ∂ T (z, t) αλ = Dth 2 + I(z, t), (3.15) ∂t ∂z ρcp

where αλ, ρ, and cp are the optical absorption coefficient, the density, and the specific heat at constant pressure, respectively. Without consideration of the energy loss due

54 to luminescence, Eq. 3.15 is solved by:

iqz T (z, t) = T0 + Tm(t) + Tq(t)e ,

αλI0 Tm(t) = t, ρcp t α Z 0 λ 0 0 −(t−t )/τth Tq(t) = dt Iq(t )e , (3.16) ρcp −∞

where T0 is the initial temperature and Tm(t) is the initial increase of the mean temperature in the sample, which reaches a steady state caused by the heat loss from the cuvette to the laboratory surroundings.

The stationary modulation depth of the temperature grating is

−1 δT = αλI0τth(ρcp) . (3.17)

−1 −1 When parameters are given as αλ =4 cm , I0 = 0.2 W cm , τth = 20 µs, ρ =

−3 −1 0.87 gcm , and cp = 1.7 J(gK) , a modulation depth becomes 11 µK. For weak perturbations, c(z, t)[1 − c(z, t)] ≈ c0[1 − c0] and a solution of Eq. 2.4 is given by

iqz c(z, t) = c0 + cq(t)e , Z t 2 0 0 −(t−t0)/τ cq(t) = −q DT c0(1 − c0) dt Tq(t )e . (3.18) −∞

Modulation of the refractive index from the temperature and concentration grating is given by

iqz n(z, t) = n0 + nq(t)e , ∂n ∂n n (t) = ( ) T (t) + ( ) c (t), (3.19) q ∂T p,c q ∂c p,T q

∂n ∂n where ( ∂T )p,c can be determined by interferometers[42] and ( ∂c )p,T can be determined

55 by either a scanning Michelson interferometer[4] or an Abbe-refractometer. Since

Iq(t) in Eq. 3.14 and Eq. 3.16 is not specified, the formalism can be applied to holographic gratings with arbitrary time dependent amplitudes. The heterodyne diffraction efficiency at t = 0 after normalization to the thermal signal is

A −t/τth −t/τ −t/τ ζhet(t) = 1 − e − [τ(1 − e − τth(1 − e )], (3.20) τ − τth where the amplitude factor A is defined as

∂n ∂n A = ( ) ( )−1S c (1 − c ). (3.21) ∂c p,T ∂T p,c T 0 0

By fitting Eq. 3.20 to the measured heterodyne signal, the Soret coefficient ST and the decay time of the thermal and concentration grating τth and τ can be obtained respectively as fit parameters. Nonlinear least square procedure is usually imple- mented with the algorithm of Marquardt. After the grating vector q is determined, the thermal diffusivity Dth, the mutual coefficient D, and the thermal diffusion co- efficient DT can be obtained from the following relations[58]:

1 Dth = 2 , q τth 1 D = , q2τ

DT = ST · D. (3.22)

(3.23)

56 3.2 New Experimental Method Based on the Use

of a Confocal Microscope

3.2.1 Suspension with Soret Effect

A material with a large Soret coefficient was chosen here to observe the shock wave effect under the most favorable conditions. Normally Soret coefficients in both gas and liquid mixtures are on the order of 10−3 to 10−5K−1, but colloidal systems have 100-10,000 times bigger Soret coefficients compared to gas and liquid mixtures[11].

Since ferromagnetic nanoparticle colloids are well-known to have a strong Soret effect, magnetic nanoparticle F e2O3 dissolved in hexane was first used as the sus- pension of interest to record the thermal diffusion. The absorption of the suspension could be recorded experimentally. However, it was difficult to separate the original absorption signal from the interference signal arising from the multiple reflections at the interface between the glass slides. To resolve the reflection issue, fluorescent labeled silica nanoparticles were used.

3.2.2 Experimental Setup

In this experiment, the dynamics of the density profile of binary mixture in a cell was recorded using a confocal microscope, configured to record to demonstrate thermal diffusion in a linear temperature field. Two circular sapphire windows, each with 23 mm diameter and 1 mm thickness, were mounted on the top and the bottom of a brass housing with dimension of 25 mm× 25 mm× 12 mm. Cold water from a temperature-controlled chiller flowed over the top surface of the bottom sapphire

57 water out cooling block water in

sapphire sample

electrode electrode indium tin oxide layer glass plate 20V, 0.5 A DC power supply Objective lens 10X, 0.3 NA

Collimator Dichroic mirror Laser source

Lens

Pinhole

Photomultiplier Tube Computer

Figure 3.5: Experimental setup: The aqueous fluorescent silica nanoparticle suspen- sion is placed between the electrically heated glass and the brass cooling block. A laser light (wavelength of 488 nm) reflected by a dichroic mirror excites the suspen- sion. Emitted light (532 nm) from the suspension passes through a dichroic mirror and is detected by a photomultiplier tube.

window of a brass housing block. The cell containing the suspension of interest was placed between a bottom sapphire of the cooling brass housing and a glass plate coated with electrically conducting Indium Tin Oxide (ITO) with dimensions of 25 mm× 50 mm× 0.5 mm. A glass plate coated with ITO (CB-50IN-1105 from Delta- technologies Inc.) through which an electrical current passed acted as a heat at the bottom of the cell. The spacer contacted the bottom sapphire window of the cooling brass housing and the ITO coated glass plate. The spacer made of either polystyrene spheres or a thin rubber disk with its center removed formed the cell with heights of either 96 µm or 300 µm. The top sapphire window of the brass housing block was used to position the scanning area of the confocal microscope. The brass housing and the ITO coated glass forming the cell were held together by two pairs of clamps attached to a specially designed confocal microscope stage as shown in Fig. 3.7. The

58 Figure 3.6: Photograph of brass cell and ITO glass plate

current flowed through two copper electrodes attached with conducting epoxy to the surface of ITO plate, and the cell was heated with 40 W from a DC power supply.

3.2.3 Data Acquisition

The density change of the cell was monitored by a laser scanning confocal micro- scope (Zeiss, Model LSM 510-Meta). The data was acquired by lateral scans over a rectangular area of 450 µm by 25 µm. The lateral resolution was 0.88 µm × 0.88 µm. A vertical scanning height of 120 µm was recorded for the polystyrene sphere spacer and 400 µm for the thin rubber disk. These heights were measured along the direction of the temperature gradient field, with a vertical interval of 4.8 µm. The wavelength of excitation of the microscope was 488 nm with 0.2 % of the 30 mW total intensity of the Argon laser. Fluorescence from the colloidal silica nanoparticles in the cell was recorded after passing through a long pass filter with a cutoff wave- length of 505 nm. The objective lens used was a Plan-Neofluar 10 ×, with NA of 0.3. The pinhole size used was 72 µm. Time to scan an entire stack was approximately

59 Figure 3.7: Photograph of the confocal microscope and cell

20 sec.

3.2.4 Confocal Microscope

3.2.4.1 The Principle of a Confocal Microscope

The confocal microscope was invented by the mathematician Marvin Minsky in 1957. A diagram of a confocal microscope is shown in Fig. 3.8. A light is focused on a spot in the focal plane of the specimen by an objective lens after being reflected by a dichroic mirror. Photons originating from the focal spot are refocused by a second lens and pass through a pinhole aperture placed in front of a detector. Most

60 Figure 3.8: Principle of the confocal microscope. (From Ref. [24].)

of the photons from below (red line,--) and above (blue line,--) the focal plane are blocked by the pinhole aperture and are not collected by the detector.

3.2.4.2 Rayleigh Criterion

Since an objective lens is a circular aperture, the image of a point object forms an Airy disc pattern rather than a point. Thus, the diffraction effect sets a limit on the resolution of an object[69]. The intensity of a diffraction pattern for a circular

61 aperture with an acceptance angle θ is given by

2J (kasinθ) 2J (x) I(θ) = I ( 1 )2 = I ( 1 )2, (3.24) 0 kasinθ 0 x

where k is a wave number, a is the radius of the aperture and J1 is the Bessel function of the first kind of order one. The Rayleigh criterion says that two point sources can be resolved when the central maximum of the Airy disc from one of the point sources lies on the first minimum of the Airy disc of the second point source. The minimum

3.83 λ λ of the sinc function appears at x = kasinθ = 3.83 or sinθ ≈ ka = 1.22 2a = 1.22 d . Therefore the limit of resolution of the confocal microscope d in the air is given by

0.61λ d = . (3.25) NA

Objects are better resolved as the wavelength of the illuminated light of the micro- scope decreases and NA increases.

3.2.5 Numerical Solution using Weighted Essentially Non-

Oscillating Schemes

The thermal diffusion factor, α, can be calculated by fitting the experimental steady state density profile with Eq. 2.57 using least squares fitting. With the obtained α, the density profile can be calculated at intermediate time points and compared with the experimental data. Discontinuities in derivatives of the thermal field cause oscillations in the solution when implementing the finite difference method. To resolve this, Weighted, Essentially Non-Oscillatory (WENO) schemes are used. A review of WENO schemes in the following section is taken from Ref. [33].

62 3.2.5.1 Introduction to WENO schemes

WENO schemes are used to approximate hyperbolic conservation laws of the type

ct + divf(c) = 0, (3.26)

or with a forcing term g(c, ξ, t) on the right hand side, where c = (c1, . . . , cm), f = (f1,..., fd), ξ = (ξ1, . . . , ξd) and t > 0. WENO schemes are based on ENO (Es- sentially Non Oscillatory) schemes. ENO schemes were first introduced by Harten, Osher, Engquist, and Chakravarthy[29] in the form of cell averages. ENO schemes choose the “smoothest” stencil from several candidates to calculate the fluxes at cell boundaries to a high order accuracy and to eliminate spurious oscillations near discontinuities. WENO schemes utilize a convex combination of all the candidate stencils whose weights determine the contribution of this stencil to the final approx- imation of the numerical flux. The weights are defined to achieve a high order of accuracy in both smooth regions and near discontinuities. Near discontinuities, the stencils containing the discontinuities are assigned a near-zero weight. An rth-order ENO scheme leads to a (2r − 1)th-order WENO scheme.

For the ordinary differential equation given by

dc = L(c), (3.27) dt where L(c) is a discretization of the spatial operator, the third-order Total Variation Diminishing (TVD) Runge-Kutta scheme was implemented as

c(1) = c(n) + ∆tL(c(n)), (3.28)

63 3 1 1 c(2) = c(n) + c(1) + ∆tL(c(1)), (3.29) 4 4 4 1 1 2 c(n+1) = c(n) + c(2) + ∆tL(c(2)). (3.30) 3 3 3

For a one dimensional system, Eq. 3.26 becomes

ct + f(c)ξ = 0. (3.31)

The space can be discretized into uniform intervals of size ∆z and zj can be defined as j∆z. The spatial operator L(u) can be approximated as −f(c)ξ and takes the form 1 L = − (fˆ − fˆ ). (3.32) ∆ξ j+1/2 j−1/2

To assume f 0(c) ≥ 0 for all c in the range of interest, a general flux including f 0(c) < 0 can be split into two parts either globally or locally,

f(c) = f +(c) + f −(c), (3.33) where df(c)+/dc ≥ 0 and df(c)+/dc ≤ 0. The global Lax-Friedriches (LF) flux splitting was used with 1 f ± = (f(c) ± αc), (3.34) 2 where α = max|f 0(c)| in the whole range where c is relevant.

ˆ+ ˆ− With the positive and negative numerical fluxes, fj+1/2 and fj+1/2 respectively, ˆ+ ˆ− ˆ+ f(c) can be expressed as fj+1/2 + fj+1/2. The rth-order fj+1/2 can be constructed as

r−1 ˆ+ X r fj+1/2 = ak,lfj+k−r+1+l, (3.35) l=0

for the given r candidate stencils by Sk = (ξj+k−r+1, ξj+k−r+2, . . . , ξj+k). The coeffi-

64 r k l = 0 l = 1 l = 2 2 0 -1/2 3/2 1 1/2 1/2 3 0 1/3 -7/6 11/6 1 -1/6 5/6 1/3 2 1/3 5/6 -1/6

r Table 3.1: Coefficients ak,l

r cients ak for r = 2 and 3 are given in the Table 3.1.

To improve accuracy if the stencil contains a discontinuity, a weight ωk can be assigned to each candidate stencil Sk, k = 0, 1, . . . , r − 1, yielding

r−1 ˆ X r fj+1/2 = ωkqk(fj+k−r+1,...,fj+k ) k=0 r−1 r−1 X X r = ωkakff+k−r+1+l. (3.36) k=0 l=0

Weights need to adapt to the relative smoothness of f on each candidate stencil in order to make any discontinuous stencil have a zero weight. Then the weight for the stencil Sk can be defined as[46],

αk ωk = , (3.37) α0 + ··· + αr−1

where r Ck αk = p k = 0,. . . , r-1, (3.38) ( + ISk)

is a smoothness measurement of the flux function on the kth candidate stencil. 

−6 r and p values are set to be 10 and 2, respectively. Optimal weights Ck are given in

Table 3.2. For r = 3, ISk, k = 0, . . . , r − 1 are given as

13 1 IS = (f − 2f + f )2 + (f − 4f + 3f )2, (3.39) 0 12 j−2 j−1 j 4 j−2 j−1 j

65 r Ck k = 0 k = 1 k = 2 r = 2 1/3 2/3 r = 3 1/10 6/10 3/10

r Table 3.2: Optimal Weights Ck

13 1 IS = (f − 2f + f )2 + (f − f )2, (3.40) 1 12 j−1 j j+1 4 j−1 j+1 13 1 IS = (f − 2f + f )2 + (3f − 4f + f )2. (3.41) 2 12 j j+1 j+2 4 j j+1 j+2

3.2.5.2 Implementation of WENO Schemes for Thermal Diffusion

Fifth-order finite difference WENO was applied for space discretization while third- order total variation with diminishing Runge-Kutta time discretization was employed for time discretization. Between 40 and 200 uniformly discretized grid points were employed. The boundary condition was set assuming a periodic solution.

Taking the derivative of the temperature function T (in Eq. 2.6) with respect to ξ yields

dT 2ξ 2 2(ξ − 2) 2 = − e−(ξ/σ) − e−((ξ−2)/σ) . (3.42) dξ σ2 σ2

The discretization of the diffusion term was taken up to fifth-order in the ξ coordi- nate. The flux function was taken as f(c, ξ) = αc(1 − c)dT/dξ from the conservative form of Eq. 2.6. Despite the small number of grid points, e.g. 40 grid points were used in Fig. 3.11, there was no spurious oscillation in the numerical integration from the sharp transition of the temperature gradient near the thin boundary layer.

66 3.2.6 Preparation of Fluorescent Silica Nanoparticles

Fluorescent silica nanoparticles were synthesized following a modified version of Tan’s method[2]. Dye precursors were formed from the reaction of fluorescein isothio- cyanate and 3-aminopropyltriethoxysilane (APTS) in a 1:5 molar ratio for 12 hours in a sealed microwave tube. Fluorescent nanoparticles were prepared using a ternary microemulsion system by adding 40µl of the dye precursor solution and 120 µl of Tetraethyl orthosilicate (TEOS) to a water-in-oil micelle system. The micelle system was made from a mixture of 480 µl of the surfactant poly(oxyethylene) nonylphenyl ether (Igepal Co., Surfactant #520) and 240 µl of 30% ammonium hydroxide in 9 ml cyclohexane. The reaction then took place in a 20 ml glass vial sealed with a Teflon screw cap for 24 hours at room temperature. The synthesized fluorescent nanoparticles had a mean diameter of 45 nm and were isolated by centrifugation. In the experiments, a colloidal suspension was prepared at a concentration of 3 mg of fluorescent nanoparticles per 1 ml of distilled water.

3.2.6.1 Effect of photobleaching

The effect of photobleaching was examined by lateral scanning with the same ex- citation laser intensity used in the experiment corresponding to Fig. 3.11. There was no change in quantum efficiency after one hour of observation. The temperature gradient is along the z-axis or the optical axis of the confocal microscope.

67 Figure 3.9: Movement of fluorescence labelled nanoparticles under the influence of a linear temperature field.

3.3 Results and Analysis

3.3.1 Experimental Data

The experimental setup is described in Sec. 3.2.2. The experiment was performed with DC input power of approximately 40W . Both current and DC voltage were kept constant during the experiment. The nanoparticles accumulated on the heated surface over time. Fig. 3.9 shows the change in the fluorescence intensity profile of the 96µm cell as a function of time. Since the density of the suspension is proportional to its fluorescence, the normalized fluorescence intensity profile is the same as the normalized density profile.

68 Figure 3.10: Normalized light intensity of the 96µm cell versus position at six dif- ferent times collected by the confocal microscope. The cold side of the cell is on the left vertical axis of each plot.

3.3.2 Results and Analysis

As time progresses, the fluorescent nanoparticles accumulate toward the heated sur- face as shown in Fig. 3.10. In Fig. 3.10, each density profile shows the light intensity averaged over the rectangular area in the lateral direction at different times. The profile at t = 0 in Fig. 3.11 was taken prior to the application of the temperature gradient. The asymmetrical profile at t = 0 could have resulted from optical ab- sorption of the colloidal suspension and the scattering characteristics of the sapphire and ITO surfaces. The rounded edge at the boundary of the plot at t = 0 can be explained by the limit of optical resolution in the vertical direction along the tem- perature gradient. In addition, the profile at t = 0 is described by a square wave intensity convolved with the Gaussian spatial response function of the microscope.

69 The response function was determined from the t = 0 profile when there was no temperature gradient. Taking the derivative of the profile at t = 0 produced a Gaussian function whose full width at half maximum (FWHM) was determined to be 5µm. This FWHM is consistent with the axial optical section thickness (i.e. the optical resolution or a 1/e Gaussian parameter, approximately 4.8µm) of the micro- scope. This optical resolution of the microscope is affected by objective numerical aperture (NA), the excitation wavelength and pinhole diameter.

Thirty minutes after removing the temperature gradient along the optical axis, the density profile returned to the same uniform pattern seen prior to applying the temperature field. This redistribution is caused by mass diffusion.

The diffusion factor α was determined to be −5.6 ± 0.3 by least squares fitting of the terminal density profile at t = 48 min into Eq. 2.57 (repeated below)

1 c(ξ, ∞) = (3.43) 1 + 1−cB eαξ cB

convolved with the Gaussian response function.

Another experiment was conducted with the ITO coated glass inverted (ITO coated side facing down) and using a thin rubber disk with its center removed as a spacer to form a cell for the suspension. The depth of the cell was measured to be 300µm using a confocal microscope. The thermal diffusion factor obtained from terminal density fraction profile shown in the inset to Fig. 3.11 was α = −1.03±0.3. The temperature of the bottom side of ITO coated glass was measured using a thermocouple immersed in thermal grease. This yields a more accurate temperature gradient in the suspension than that from the calculation from the heat equation with a given input electric power. The calculated temperature difference across the cell

70 suspension using a heat diffusion equation was 12 K which leads to a temperature gradient of 4 × 104K/m. The temperature difference across the cell yields a Soret coefficient, ST , i.e. α/∆T , to be 0.09 /K. This ST value agrees well with the

−1 previously reported value of ST = 0.047 K for a silica colloidal suspension of 22 nm diameter silica nanoparticle [51, 62].

Data from Fig. 3.11 was produced at the smaller cell size and a higher tem- perature gradient compared to the inset of Fig. 3.11. The time constant τ was calculated using the cell dimension and the mass diffusion constant D obtained from the Einstein-Stokes law.

3.4 Discussion

Direct single measurement of the terminal density fraction is possible using a confocal microscope, and higher accuracy can be achieved with better optical resolution of the confocal microscope. The Soret coefficient is obtained by dividing the thermal diffusion factor by the temperature difference of the cell of interest. Therefore, a more exact way of measuring the temperature of the electrically heated glass surface is desirable for a more accurate Soret coefficient.

Generally the Soret parameter is strongly dependent on concentration and tem- perature. However, at lower concentrations the Soret parameter becomes indepen- dent of concentration, and so measurements of the Soret parameter at lower con- centrations are preferred. The high sensitivity of the confocal microscope to detect fluorescence enables precise measurement of the Soret parameter of fluorescent mix- tures at lower concentrations.

71 Another advantage of the present method is that it provides a simple way to determine the Soret parameter with a single measurement of the terminal density distribution. With a measured temperature difference across the cell suspension, fitting the terminal density profile to Eq. 2.57 convolved with a spatial resolution function gives the thermal diffusion factor α. There is a strong agreement between the experimental time evolution of the density profile and the numerically integrated density profile dynamics with the determined thermal diffusion factor using a single parameter fit from the terminal profile. With a confocal microscope it is easy to mea- sure the Soret parameter for colloidal suspensions that have fluorescence or sufficient optical and differential scattering. The easy implementation, direct measurement, and straightforward interpretation favor the use of the current confocal microscope method.

72 Figure 3.11: Data points from the experimental measurements are fitted with curves by numerical integration using the WENO method. A value of α = −5.6 is obtained from the terminal light intensity profile. The time variable is determined by Einstein- Stokes relation using the size of the cell and mass diffusion coefficient of water. Inset: Normalized light intensity at a steady state versus position for a cell with a dimension of 300 µm constructed with the thin rubber disk with its center removed. The curve is fitted using the method of least squares to the terminal distribution of nanoparticles from convolution of Eq. 2.57 with a Gaussian function characterizing the spatial resolution of the microscope. The Gaussian function is determined by taking the spatial derivative on the light intensity profile prior to the imposition of the temperature gradient. (From Ref. [34].)

73 Chapter 4

Summary of Thermal Diffusion Shock Waves in a Linear Temperature Field

Thermal diffusion refers to the separation of mixtures under a temperature gradient. The solutions of thermal diffusion were investigated in a linear temperature field along the z axis T T (z, t) = − 0 z + T . (4.1) L h

When the Dufour effect is negligible two phenomenological transport equations gov- erning thermal diffusion in a linear temperature field are

∂T ρc = ∇ · (κ∇T + D00∇c ) (4.2) p ∂t 1 and ∂c 1 = ∇ · (c c D0∇T + D∇c ), (4.3) ∂t 1 2 1

74 which reduce to

∂c(ξ, θ) D0T ∂ dT ∂2c(ξ, θ) = − 0 (c(ξ, θ)[1 − c(ξ, θ)] ) + . (4.4) ∂θ D ∂ξ dξ ∂ξ2

where the dimensionless variables ξ and θ are defined as Kz and K2Dt, respectively. Using the Hopf-Cole transformation

1 ∂ c¯(ξ, θ) = lnV (ξ, θ) (4.5) α ∂ξ

Eq. 4.4 can be expressed as

1 c¯ = (α(¯c2 − ) +c ¯ ) θ 4 ξ ξ 1 V = ( ξ ) α V θ 1 V = ( θ ) α V ξ 1 V = ( ξξ − α/4) (4.6) α V ξ where c¯ = c − 1/2. Integrating Eq. 4.6 with respect to ξ yields

2 Vθ − (f(θ) − α /4)V = Vξξ, (4.7)

where f(θ) is a function of θ. Substitution of a new function W (ξ, θ)

θ R (f(θ0)−α2/4)dθ0 V (ξ, θ) = W (ξ, θ)e 0 (4.8)

into Eq. 4.7 yields the ordinary diffusion equation Wθ = Wξξ. Two boundary condi- tions and the initial condition for the system where the components are constrained

75 in space give an exact solution as

Z 1 W (ξ, θ) = [ϑ(ξ − ξ0, θ) − ϑ(ξ + ξ0, θ)]W (ξ0, 0)dξ0 0 Z θ 2 α τ −2 ϑξ(ξ, θ − τ)e 4 dτ 0 Z θ α2 α(c0−1/2) τ +2e ϑξ(ξ − 1, θ − τ)e 4 dτ, (4.9) 0

Pm=∞ where ϑ is the theta function defined as ϑ(x, t) = m=−∞ K(x + 2m, t). Another solution was studied when the components are unbounded by changing variables from (ξ, θ) to a moving coordinate system (ζ, θ) where ζ = ξ − αθ, giving Burger’s

2 equation, cθ = α(c )ζ + cζζ . Using the same procedure as in the case of the bounded mixture, the solution was given as

∞ 1 Z √ 0 2 U(β, θ) = √ U(2 θβ0, 0)e−(β−β ) dβ. (4.10) π −∞

R ∞ 0 0 ζ α c(ζ ,0)dζ where β = √ and U(ζ, 0) = e −∞ . 2 θ

∂c In the long time limit where ∂θ = 0, the solution of Eq. 4.4 is given by

1 c(ξ, ∞) = . (4.11) 1 + 1−cB eαξ cB where 1 − e−αc0 c = . (4.12) B 1 − e−α

The solution of thermal diffusion without the effects of mass diffusion,

c = 1 − (1 − c0)u(ξ + 1/2 − αc0τ) − c0u(ξ − 1/2 − α(1 − c0)τ), (4.13) where u is the Heaviside function, shows the underlying motion of the components of the mixture to be that of shock waves.

76 An experimental method using a confocal microscope was developed to observe the density distribution change of fluorescence labelled nanoparticles suspended in water in a linear temperature field. A single direct measurement of the terminal flu- orescence signal provided the value of the thermal diffusion factor α. The numerical density solution of Eq. 4.4 at times other than the terminal time was in good agree- ment with the experimental results when the determined thermal diffusion factor was used.

In conclusion, shock wave like behavior of binary mixtures caused by thermal diffusion in a linear temperature field was shown in both theory and experiment.

77 Part II

Comparison of Ultrasonic Distillation to Sparging of liquid Mixtures

78 Chapter 5

Introduction

Bioethanol is a renewable energy source that is an alternative to petroleum. It is extracted from agricultural crops such as sugar or corn. The conversion of fermented biomass to pure alcohol involves ethanol separation from water. The energy efficiency of this separation determines the economic feasibility of bioethanol as an alternative fuel source.

Recently, perfect separation of ethanol from water was reported with ultrasonic atomization [39, 64]. This ultrasonic distillation refers to the formation of vapor and mist ejected from the fountain jet of a liquid which is irradiated with acoustic waves.

79 5.1 Review of Reported Perfect Separation of Ethanol

from Water Using Ultrasonic Distillation

The author in Ref. [64] used the experimental setup shown in Fig. 5.1. The ultra- sonic vibrator was composed of piezoceramic that vibrated at a 2.3 MHz resonant frequency. The piezoceramic material was 20 mm in diameter and was driven by electric input power of 20W . Air flowed into the apparatus at 25 l/min. The ethanol concentration of the mist EM (mol%) was calculated using the concentration and weight of the residue according to

E1W1 − E2W2 EM = , (5.1) W1 − W2

where E1 (mol%) and W1 (mol) were the concentration and the mass of the ethanol solution before atomization and E2 (mol%) and W2 (mol) were the ones after atom- ization, respectively. The ethanol concentration was determined by a gas chromato- graph (GC14A, Shimadzu) through sample injection using an auto-sampler system (AOC100, Shimadzu). The standard deviation of the ethanol measurement was 0.0094 mol% (0.024wt%) at .

Fig. 5.2(a) shows the experimental results that indicated a complete separation of ethanol and water at 10 ◦C with a several mol% ethanol-water solution. Fig. 5.2(b) shows that the electric input energy was equal to the vaporization energy of ethanol at 10 ◦C. At temperatures 30 ◦C and 50 ◦C, the separation of ethanol and water are incomplete.

Sato et al.[64] theorized that the unstable parametric decay of the capillary wave accounts for the effect of the selective ethanol separation. The ultrasonic vibrator

80 Figure 5.1: Experimental setup in the report of a perfect ethanol separation from ethanol-water solution using ultrasonic atomization. A 20 mm diameter piezoceramic transducer disk driven at 2.3 MHz was used. (From Ref. [64].)

produces a longitudinal wave which decays into two surface waves. These waves propagate in opposite directions along the surface of the liquid, generating capillary waves. Sato et al. assumed that the parametric decay may have been caused by a thermal fluctuation and claimed that high localization and accumulation of acoustic phonon occur, leading to ultrasonic atomization. However, the dispersion relation

q σ 3/2 for the capillary wave hypothesis, ω = ρ k , does not give a significant difference in the droplet size, which is known to be proportional to the surface wavelength, between water and ethanol.

Another group reported an efficient separation of ethanol using ultrasonic atomization[39]. Shown in Fig. 5.4, a 20 W sonicator (Honda Electronics) which has a maximum power output of 8 W/cm2 at 2.4 MHz was the ultrasound source. Air, at a rate of 20 ml/min, was introduced to a 1 l glass cylinder to carry the mist generated during the sonication into a collection outlet connected to a condenser. The condensed mist

81 Figure 5.2: (a)Vapor-liquid equilibrium diagram of ethanol-water solution using ul- trasonic atomization under 1013 hP a. Perfect separation of ethanol is shown in the entire concentration at lower temperature of 10 ◦C.(b)Energy required for separation vs ethanol concentration in solution at different temperatures. The energy required for the vaporization of the ethanol is shown in solid line based on the vaporization energy of ethanol 38.6 kJ/mol. (From Ref. [64].)

was analyzed with an Abbe refractometer (Leica Mark II Plus). 40 mol% ethanol and 35 mol% glycerol in water were obtained from 150 ml of 20 mol% of ethanol-water and glycerol-water solution at 25 oC subjected to ultrasound, respectively.

A high speed frame rate video of 2000 frames/sec captured the formation of cavitating bubbles near the threshold power input, at 7 W . Capillary rupture formed structural defects and created cavitation centers in the bulk liquid.

The author reports that the separation occurs only when the fountain jet is formed. A localized hot spot develops an interfacial area to vaporize alcohol into the bubble. The maximum temperature in the system was found at the center of

82 Figure 5.3: (a) Longitudinal waves traveling along the z-direction with the wave number kSUR are on the liquid surface. The surface motion has a frequency ω0 and wave number kSUR = 0. Surface acoustic waves are on the liquid surface shown as (b). (c) Dispersion curve (ω-k) of parametric decay. The wave number kSUR is along the surface, the longitudinal waves, P , is on the ω-axis. Longitudinal waves decays into two surface waves which propagates into opposite directions. The frequency of the surface waves are determined from the parallelogram OS+PS−. (From Ref. [64].)

the fountain jet. This phenomenon was attributed to the rapid quenching of the cavitating bubbles in the bulk liquid[39].

Kirpalani et. al. proposed microbubble interaction with the fountain jet in terms of surface molecular effects to explain the ultrasonic separation of ethanol-water. More ethanol-water bonds are assumed to be formed and a large number of water- water bonds are broken at concentrations of ethanol higher than 20 mol% leading to ethanol surface excess[1, 72]. Sato et al.[64] attributed the reported reduction of separation efficiency at a higher temperature to an increase of the vapor pressure of both ethanol and water above the solution. The increase of the vapor pressure leads

83 Figure 5.4: Experimental setup used in Ref. [39]

to an increase in the atomization rate of both the alcohol and the water, decreasing separation efficiency.

5.2 Introduction to Ultrasonic Atomization

When a liquid is exposed to vertical irradiation of sound pressure with an ultrasound frequency, it forms a periodic wave pattern. This is called a Faraday wave, named after Michael Faraday[18]. With a frequency above a critical value, the wave surface becomes unstable; this is called Faraday instability.

When the amplitude of acoustic pressure with ultrasonic frequency is above a certain threshold, the exposed liquid starts to atomize and mist, which is called ultrasonic atomization. The atomization of liquid by high frequency ultrasonic os- cillations in a fountain was discovered and characterized by Wood and Loomis in

84 1927[74].

Two hypotheses exist to explain ultrasonic atomization: the cavitation hypothesis and the capillary wave hypothesis.

5.2.1 Capillary Wave Hypothesis

The capillary wave hypothesis starts with Faraday instability. When the intensity increases, unstable oscillations cause aerosol droplets to be separated or atomized from the crests of the surface capillary waves. The formation of droplets is related to capillary or capillary-gravitation waves at the surface of the liquid.

When a liquid is exposed to ultrasonic acoustic pressure irradiation, measure- ments using stroboscopic light indicate that the period of the surface waves of the liquid is twice as long as the period of the vertical oscillation of the liquid. This was discovered by Faraday in 1831[18]. Rayleigh pointed out the analogy between the parametric nature of the “Faraday” waves excitation and the Melde effect.

Sorokin[68] derived the criteria threshold for the excitation of waves utilizing the Navier-Stokes equations and the continuity equation. For an incompressible viscous liquid exposed to vertical oscillations along y-axis given by the amplitude

ξ = ξ0cosω0t, these two equations can be written as:

∂v 1 = − gradP + ν∆v + (−g + ω2ξ cosω t)yˆ, ∂t ρ 0 0 0 divv = 0, (5.2)

where v = (vx, vy) = (u, v) and yˆ is a unit vector in the y direction. Eq. 5.2 is

85 satisfied by the following expressions:

∂ϕ ∂ψ u = − − , ∂x ∂y ∂ϕ ∂ψ v = − − , ∂y ∂x p ∂ϕ = − y(g − ω2ξ cosω t), (5.3) ρ ∂t 0 0 0

when the functions ϕ and ψ satisfy the two equations:

∆ϕ = 0, ∂ψ = ν∆ψ. (5.4) ∂t

The function η = f(x, t) indicates the displacements of the liquid surface. The free surface should fulfill the following boundary conditions:

p ∂v T − + 2ν − ∂2η∂x2 = 0, ρ ∂y ρ ∂v ∂u µ[ + ] = 0. (5.5) ∂x ∂y

v needs to be zero when y → ∞. A sinusoidal standing wave is formed on the surface of the liquid, and the displacement function can be written as η = ζ(t)coskx. The functions ϕ and ψ satisfying the Eq. 5.4 can be of the form:

1 m2 + k2 ϕ = − ζ˙(t)ekycoskx, k m2 − k2 2k ψ = − ζ˙(t)emysinkx, (5.6) m2 − k2

where the wave numbers k and m are determined by Eq. 5.4. For the frequency range which the wave number of the viscosity waves is large (i.e., k/m << 1), the

86 function ζ(t) becomes

¨ 2 ˙ 2 2 ζ(t) + 4νk ζ(t) + [ω − ω0ξ0kcosω0t]ζ(t) = 0, (5.7)

where ω2 = gk + (T/ρ)k3 and T is the surface-tension coefficient of the liquid.

Eq. 5.7 can be expressed as an equation of the Mathieu type in canonical form when δ = 2νk2 and when ζ(t) = e−κτ H(τ) where the dimensionless time variable τ is defined as τ = (ω0/2)t:

∂2H + [a − 2qcos2τ]H = 0. (5.8) ∂τ 2

In this equation,

2ω a = ( )2 − κ2, ω0

q = 2ξ0k, 2δ κ = . (5.9) ω0

The solution of Eq. 5.8 can be written in terms of a Mathieu function. In case of a >> q, q > 0, the solution of Eq. 5.7 can be expressed in terms of first order Mathieu’s functions[52]:

(µ−κ)τ ζ(t) ≈ e [C1(q)Ce1(τ, q) + S1(q)Se1(τ, q)]. (5.10)

The characteristic exponent can be approximated as

1 a ≈ 1 + q − q2, 1 8 1 b ≈ 1 − q − q2. (5.11) 1 8

87 The instability zone from the condition |µ| > κ is determined by the inequality

s s 2 2δ 2 2ω 2 2 2δ 2 1 − (2ξ0k) − 4( ) < ( ) < 1 + (2ξ0k) − 4( ) , (5.12) ω0 ω0 ω0 from which the threshold value of the excitation amplitude is[68]

2δ δ ξthreshold = = . (5.13) ω0k ωk

2π 2 3 In the limit of small λ = k , ω = gk + (T/ρ)k reduces to Kelvin’s formula describing the phenomenon that capillary waves become predominant. Hhe half-

q3 2πT wavelengths of capillary waves λ are calculated as λ = ρf 2 where f is the ultra- sound frequency.

To verify the capillary wave hypothesis, the length of capillary waves, λk, was measured by Lang[41] in the excitation frequency range from 10 to 800 kHz. He found a constant relationship D = 0.34λk, where the diameters D is the diameter of the most frequently encountered fog droplets. Experiments supporting the capillary hypothesis were conducted under low and below-threshold sound levels.

5.2.2 Cavitation Hypothesis

The cavitation hypothesis was first proposed by Söllner. It states that the liquid is atomized by hydraulic shocks from the implosion of cavitation bubbles near its surface. In other words, small bubbles form droplets and mist when they collapse near the surface when they oscillate with periodic sound irradiation. The cavitation hypothesis is usually applied to liquids that are exposed to high frequency (> 100 kHz) and high intensity ultrasound systems[39].

88 The presence of sonoluminescence proves the presence of cavitation in a lumines- cent area in ultrasonic atomization[16]. In megacycle range, atomization of different liquids becomes independent of the frequency or the characteristics of the liquid[32]. Atomization in an ultrasonic fountain occurred when cavitation zone was formed in the actual jet of the fountain and all acoustical energy transferred to the fountain jet was dissipated mainly in the region in which cavitation is observed during the atomization process[16].

Consider the bubble dynamics in an acoustic field. The probability of forming a

nucleus in a unit time in a unit volume is proportional to (−Wcr/kT ) where Wcr is the work of formation of the critical nucleus. The work to form the critical nucleus to determine the growth of the bubble is found by the following method[19]. The total work to form a bubble of radius r is equal to:

2 3 W = 4πr σ + (4/3)πr (p − pV )

≈ 4πr2σ + (4/3)πr3p (5.14)

where σ is the surface tension. The maximum value of work which makes the bubble increase and decrease above and below the critical radius rcr is given by Wcr =

3 2 16πσ /3p for rcr = 2σ/p. In Ref. [19], the formation rate of critical nuclei takes the form: dn NkT 16πσ3 = exp[−(∆f + )/kT ]. (5.15) dt h 3p2

where N is the number of the molecules and ∆f is the free activation energy for the liquid molecules to move toward and from the surface of the bubble. By substituting

dn/dt = 1/t, the magnitude of the liquid strength p0 is obtained as

16π σ3 p = −[ ]1/2. (5.16) 0 3 kT ln(NkT t/h) − ∆f

89 The behavior of stable nuclei in an acoustic field can be derived as follows. The nucleus is assumed to have a constant gas content during its motion, and its radius to be small compared to a wavelength in the liquid. The pressure in the liquid is the sum of the hydrostatic and acoustic pressure: PL = PA − P0sinωt. The bubble containg gas with a radius R0 is under a pressure PG = PA + 2σ/R0. The kinetic energy of the coupled liquid is the sum of the surface-tension work, the gas pressure work, and the liquid-pressure work at infinity:

Z R 2 3 3 3 2 4πR [P0sinωt − PA + (PA + 2σ/R0)R /R0] − 8πRσdR = 2πρR (dR/dt) , R0 (5.17) which acquires the form of motion equation under isothermal gas motion as:

d2R 2R[P sinωt − P + (P + 2σ/R )R3/R3] = 4σ + 3ρR(dR/dt)2 + 2ρR2 . (5.18) 0 A A 0 0 dt2

A nucleus with a shape of a solid spherical particle that is not wetted by the liquid has no gas in the cavity formed at the moment the liquid is torn away from the solid particle. This makes the gas work term zero. Then, the equation of motion becomes

d2R 2R[P sinωt − P ] = 4σ + 3ρR(dR/dt)2 + 2ρR2 . (5.19) 0 A dt2

The size of gas-filled bubbles increases instantaneously upon the lowering of the pressure, while an empty cavity is created around the solid sphere if the negative pressure is sufficiently large[61].

Observations of cavitation in ultrasonic fountains have been made extensively[16]. Two types of sonoluminescence are intimately related to the cavitation process. Lit- tle or no fog is generated for the first type that appears at the fountain base. Above a certain temperature threshold, sonoluminescence moves up in the fountain jet and

90 forms the second type of sonoluminescence which involves cavitation that usually accompanies vigorous atomization. This process was also observed using ultra-high- speed photographic equipment[66]. Cavitation nuclei were reported to be of the order of 1 mm, growing into larger cavitation regions. The relation among the vapor pressure, gas content, static pressure, and the atomization capacity favors the cav- itation theory[21]. The atomization capacity A, is proportional to p/ση where p is the vapor pressure, σ the surface tension, and η is the dynamic viscosity coefficient. The hypothesis of periodic hydraulic shocks produced by cavitating waves on the surface of the liquid accounts for the correlation between average atomized droplet size and capillary wavelength while explaining the aforementioned atomization ca- pacity dependence on p/ση[7]. The approximate prediction of the parameters of the front of shock waves formed by cavitation indicated a strong enough effect to excite standing capillary waves of finite amplitude on the liquid surface, consistent with the cavitation wave hypothesis.

91 Chapter 6

Experiments and Results

6.1 Experimental Setup and Procedures

As shown in Fig. 6.1, the cell consisted of a glass cylinder mounted on an atomization circuit board. The glass cylinder was 41 mm diameter and approximately 25 cm high. It had two side branches of glass tubes of diameter 1 cm for sample extraction and nitrogen flow. The first side branch was 2-3 cm from the bottom of the main cylinder and was above the level of the liquid. It was used for flowing nitrogen into the cylinder and sample extraction. This branch was angled downward to allow liquid sampling using a syringe with an attached needle. The second branch allowed nitrogen to exit the cylinder along with the mist and vapor mixture created by ultrasonic distillation.

Fig. 6.2 shows commercial atomization circuit boards with two ultrasound fre- quencies of 1.65 MHz and 2.40 MHz (APC International, Model 50-1011) which were used in this experiment. The circuit board assembly has a piezoceramic of diameter 25 mm and contains supporting electronics. The maximum power of the

92 Figure 6.1: A vessel used for the ultrasonic distillation is made of a 41 mm diameter glass cylinder. A piezoelectric transducer driven at 2.4 MHz was located at the bottom of a vessel. Copper cooling coils wrapped around the bottom outside of the cylinder. Two ports were located in the side of the vessel for liquid sampling and nitrogen inflow and as an outlet for mist, vapor, and nitrogen. A 30-50 ml volume of solution was placed in a vessel for distillation. (From Ref. [35].)

circuit board was 29 W . The power to the atomization unit was adjusted to create sufficient mist and vapor from the ejected capillary wave water jet. The height of the capillary wave was about 5 cm above the water surface level as shown in Fig. 6.3(b).

To dissipate heat from the cell, a 1/16 inch diameter copper cooling tube was used. A temperature controlled chiller supplied water through this tube which was wrapped around the bottom of the bottom of the cell.

A volume of 40 ml of the liquid mixture of interest was used for the experiments. The liquid mixtures included ethanol-ethyl acetate, ethanol-water, gold colloid and cobalt chloride aqueous ionic solutions.

93 Figure 6.2: Commercial 25 mm diameter ceramic piezoelectric transducer assemblies (From Ref. [57].)

To analyze the composition of the extracted mixture sample as distillation pro- gressed, gas chromatography was used for the ethyl acetate and ethanol mixtures, a UV spectrometer was used for the gold colloid and cobalt chloride aqueous mixtures and an index refractometer and nuclear magnetic resonance spectrometer were used for the ethanol-water mixtures.

To determine if ultrasonic distillation achieves better separation than the con- ventional sparging distillation method, ultrasonic distillation and nitrogen sparging experiments were conducted. Ultrasonic distillation experiments used ultrasonic at- omization to distill a liquid, as previously described. For the nitrogen sparging experiment, a fine diameter tube (inner diameter less than 1 mm) made of stainless steel or teflon with a fine diameter was inserted into the cell with the end of the tube at the bottom of the cell below the level of the liquid. Either a glass graduated cylinder or the ultrasonic distillation vessel without the application of ultrasound

94 Figure 6.3: Ultrasonic fountain formation. (a) transducer off, (b) transducer on with flowing nitrogen, and (c) transducer on without flowing nitrogen (From Ref. [57].)

was used as a cell for sparging. The solution was distilled by flowing high purity nitrogen at 2 l/min through this tube. The experimental setup for sparging is shown in Fig. 6.4.

Two ways to monitor the concentration change during distillation were imple- mented. For the first method, several liquid mixtures with different concentrations were distilled until a fixed amount of liquid remained. The concentrations of the liq- uid components were measured before and after distillation. For the second method, the liquid mixture was sampled as distillation proceeded and the height of the liquid mixtures decreased. The concentrations of the liquid components were determined from the samples, which were a few µl in volume.

95 Figure 6.4: Experimental setup for Sparging for experimental method I. A flowmeter is shown as well (From Ref. [57].)

6.2 Analysis Methods

For refractive index measurements, a few µl of liquid sample was extracted using a micro syringe, and placed on the prism surface of an Abbe refractometer (Fisher Scientific, Model 13-964).

For NMR analysis, 25 µl of extracted sample was mixed with 600 µl of 99.9 atom % D-6 acetone (Sigma Aldrich, 444863) in an NMR tube and was placed in a 300 MHz NMR spectrometer (Bruker, Inc., Model DPX Advance) to determine the mole fraction of the sample constituents.

96 A capillary gas chromatograph (Varian, Model CP 3900 with a Factor Four VX- 1ms column) with flame ionization detection was used to determine the mass fraction of organic solvents.

For the analysis of ionic and colloidal solutions, a conventional spectrophotometer (Varian Cary, Model 50) was used.

6.2.1 NMR Sensitivity

Table 6.1: Nuclear properties of interest[53]

Isotope Natural Abundance(%) Spin Sensitivity1 Frequency (MHz)2

1H 99.98 1/2 1.0 500.0 19F 100.0 1/2 0.83 470.2 29Si 4.7 1/2 0.078 99.3 31P 100.0 1/2 0.066 202.3 13C 1.1 1/2 0.0159 125.6 2H 0.015 1 0.00964 76.7 15N 0.365 1/2 0.001 50.6 1 The sensitivity relative to protons. 2 The resonance frequency in a 11.7 T magnetic field.

The principles of NMR are explained in the Appendix C. Since the energy differ- ence between the two spin states is quite small under an external magnetic field, the two spin states are almost equally populated at room temperature. The population of the lower energy state exceeds that of the higher energy state by only about 0.001 %. Both the absorption intensity and the NMR sensitivity are proportional to the number of nuclei absorbing RF energy. Since the energy difference between spin

97 states (∆E = hν) is proportional to the gyromagnetic ratio (γ), the sensitivity of nuclei also depends on the magnitude of γ. The natural abundance of the spin-active nucleus under observation also affects the sensitivity of NMR. Table 6.1 shows the natural abundance, spin quantum number, and sensitivity for some selected nuclei.

The sensitivity of NMR can be increased by increasing the field strength B0. At a field strength of 14.1 T , the energy difference between spin states is on the order of 10−4 kJmol−1. With the most sensitive high field instruments, tens to thousands of micrograms of sample are required[30].

6.2.2 Spectrophotometer

A spectrophotometer measures an absorbance by passing a collimated light beam of wavelength λ through a slab of material. The absorbance A of a sample is defined as the logarithmic ratio of the transmitted light intensity I passing through the solvent

plus solute sample to the transmitted light intensity I0 passing through pure solvent,

I A = −log10 . (6.1) I0

A follows Beer’s Law: A = lc (6.2)

where  is molar absorptivity of the solute at wavelength λ, l is the length of the light path through the material and c is the concentration of the solute of interest.

A spectrophotometer is less expensive than most other fundamental measurement instruments used in chemical analysis. It is characterized by high versatility, sensi- tivity, and precision. It can be used for determining almost all chemical elements over a wide range of concentrations, from macroquantities by means of differential

98 spectrophotometry to trace quantities ranging from 10−6 − 10−8 %. Spectrophoto- metric methods are considered the most precise instrumental methods of chemical analysis[48].

6.2.3 Gas Chromatography

Gas chromatography (GC) is one way to separate mixtures of volatile compounds. It is useful since it does not require a large amount of sample. It separates the mixture by using the difference in the adsorption of components in the gas phase onto a column.

The speed at which the sample mixture moves along the column depends on two factors: the flow rate of the carrier gas and the degree to which the vapor phase is adsorbed. Substances differing in their adsorption can be separated by using the different speeds of progression through the column, causing them to reach the far end of the column at different times.

The concentration of a component in equilibrium between the gas phase and the stationary phase in the column depends on the vapor pressure. Thus gas chromatog- raphy is similar to fractional distillation since both of them separate the mixture using vapor pressure differences. The most important part of GC is the column, which is a long tube packed with a permeable adsorbent.

The progression of the sample through the column can explained by “Plate The- ory” proposed by Martin and Synge[50]. They imagined a column divided along its length into a number of separate zones within which the vapor and the stationary phases are in equilibrium. Each zone is called a “theoretical plate”.

99 Maximum number of incremental volumes δV (n) occupied by the vapor serial number of plate (r) 0 1 2 3 4 5 6 0 1 1 q p 2 q2 2qp 3 q3 3q2p 3qp2 4 q4 4q3p 6q2p2 4qp3 p4 5 q5 5q4p 10q3p2 10q2p3 5qp4 p5 6 q6 6q5p 15q4p2 20q3p3 15q2p4 6qp5 p6

Table 6.2: Ratio of amount of vapor occupying r plates after passing n incremental volumes δV , assuming that all of the vapor occupies the zeroth plate when n = 0. (From Ref. [45].)

At the beginning of the column, the first plate contains all of a particular sam- ple(Fig. 6.5a). An incremental volume of carrier gas δV carries a portion of the sample that is in the vapor phase to the second plate where it equilibrates with the stationary column phase (Fig. 6.5b). Meanwhile, the remaining sample in the sta- tionary phase in the first plate equilibrates with the clean carrier gas. This process repeats (Fig. 6.5c) as more increments of carrier gas pass by and slowly the front of the sample moves along the column. The sample amount in the first plate decreases until it depletes, and the whole zone moves along the column (Fig. 6.5d). The ratio of the sample of any plate that moves into the next plate by passage of a single incremental volume, δV of the carrier gas is defined by

cδV p = Hc(a + αb) δV = , (6.3) H(a + αb) where H=height equivalent to the theoretical plate (cm), a=cross-sectional area of column occupied by the gas phase (cm2), b=cross-sectional area of column occupied by the stationary phase(cm2),

100 Figure 6.5: Elution progression in a column composed of plates(From Ref. [45].)

α=partition coefficient defined as the ratio of the weight of solute per gram of sta- tionary phase to the weight of solute per cc of gas at column temperature, c=concentration of vapor in the gas phase(g/cc), δV =incremental volume of carrier gas(cc).

After a given number of incremental volumes (n) has been passed, w(n, r), i.e. the mass distribution of vapor in the rth plate is given as

n! w(n, r) = prq(n−r), 1 ≤ r ≤ n (6.4) r!(n − r)! with q = 1 − p. Some example results are shown in Table 6.2.

101 The mean value of w(n, r), r¯ and the variance, σ2 are defined by

Z r¯ = rw(r)dr Z σ2 = (r − r¯)2w(r)dr. (6.5)

For a binomial distribution at large n, these approach[49]

r¯ = np,

σ2 = npq. (6.6)

When n → ∞, r¯ becomes large and w(r) can be shown to be the Gaussian distribution[45]

1 w(r) = √ exp[−(r − r¯)2/2σ2]. (6.7) 2πσ

6.2.3.1 Flame Ionization Detector(FID)

+

200 V - Electrometer Top of Resistor amplifier and Column recorder

Figure 6.6: Flame ionization detector (From Ref. [70].)

102 Table 6.3: Performance comparison among the main GC detectors. (From Ref. [25].)

Detectors Sensitivity1 Detection Limit2 Dynamic Linear Range Selectivity

Gas Density Balance 1,000 2·10−5 5,000 N Thermal Conductivity 10,000 2·10−6 50,000 N Electron Capture 800 2·10−11 2,000 Y Flame Ionization 1·10−5 2·10−9 1,000,000 N Thermoionic 2·10−2 3·10−13 10,000 Y Photometric 4·10−2 2·10−9 none Y Photoionization 1·107 Y Helium 10,000 Y 1 In mL × mL/mg for concentration detectors (except ECD, mA × mL/mg), in C/mg for mass flow detectors(FID, TID). 2 In mg/mL for concentration detectors, in mg/sec for mass flow detectors.(Reprinted from Analytical Chemistry, 43, 113A (1971).)

The flame ionization detector (FID) uses hydrogen as a carrier gas which is burned in a small metal jet, such as a hypodermic needle cut square shown in Fig. 6.6. The jet usually forms the negative electrode of the detector and the other electrode is made of a piece of brass or platinum wire mounted near the tip of the flame. The potential difference applied between the electrodes is 200 V . Ionization of substances in the flame gives rise to a signal that can be recorded. The pure hydrogen or hydrogen/carrier gas mixture forms a small background signal which may be offset electrically. The mechanism of ion production is not understood clearly, but for organic compounds, the ions are assumed to originate from carbon aggregates which ionize relatively easily (4.3 eV ). For organic compounds, the amount of carbon in the substance determines response of the detector.

FID does not detect the following substances because of their high ionization

103 potentials:

the noble gases, H2, O2, N2, SiCl2, SiF4, H2S, SO2, COS, NH2, NO, NO2,

N2O, CO, CO2, H2O.

Table 6.3 shows the sensitivity and linear dynamic range of FID compared to other detectors.

6.2.4 Refractometer

Refractometers usually operate based on the concept of the critical angle φcrit which is an angle of an incident ray above which total internal reflection occurs[70].

Figure 6.7: Schematic diagram of an Abbe refractometer(From Ref. [65].)

The Abbe refractometer requires only a few drops of a sample held by capillary action in a thin space between the refracting prism and an illuminating prism. It

104 adjusts the critical-ray boundaries to be at the intersection of a pair of cross hairs by rotating the refracting prism until the telescope axis makes the exit angle of the ray to be normal to the air interface of the prism. The index of refraction is then determined directly from a scale associated with the prism rotation.

The Abbe refractometer usually has two Amici compensating prisms, geared to rotate in opposite directions. An Amici prism consists of two different kinds of glass designed to produce a considerable amount of dispersion but without producing angular deviation of light corresponding to the sodium D line. Incorporation of the Amici prisms compensates for the dispersion within the sample so as to produce the same result with white light as would be obtained if a sodium D line were used for illumination. The net dispersion can be from zero to some maximum value in either direction after passing through two counter-rotating Amici prisms. This is achieved when the boundary between light and dark fields becomes sharp. The dispersion of a sample is not always exactly compensated. The most precise results can be achieved with illumination by a sodium arc when the Amici prism is set at zero dispersion[65].

The Abbe refractometer has a precision of ±0.0001 and requires temperature control variation of ±0.2oC. For this, water is circulated from a thermostatically controlled bath through the prism housing.

105 Figure 6.8: Ethanol mole fraction of ethanol-water mixtures was determined by NMR after (a) sparging and (b) ultrasonic distillation with resonance frequency of 1.6 MHz and at 22oC. Curve (c) shows ethanol mole fractions of the starting solutions before distillation and was used as a calibration curve. (From Ref. [35].)

6.3 Experiments

6.3.1 Ethanol-Water Mixture Experiment

For the ethanol-water mixtures, 99.8 % ethanol (Sigma Aldrich, spectrophotometric grade) and distilled water were used.

Method I compares the composition before and after distillation for 10 different volume fractions of an ethanol-water solution after a certain amount of the total vol- ume decreases. Mixtures of 40 ml total volume were placed in a vessel for ultrasound distillation and in a 41 mm diameter graduated cylinder for nitrogen sparging. They were distilled until the total volume decreased by 25 %. The remaining solution from experimental method I was analyzed with the nuclear magnetic resonance. The re- sult shown in Fig. 6.8. The H resonance peak for ethanol was 1.11 ppm and for water

106 Figure 6.9: Comparison of ultrasonic distillation to nitrogen sparging of water- ethanol solutions using experimental method I conducted at 22 ◦C. The ethanol composition of the remaining solution was determined from measurement of the in- dex of refraction after the (a)sparging and (b)ultrasonic distillation driven at the resonance frequency of 2.4 MHz. The top curve (c) is the calibration curve taken from the starting liquid mixtures before distillation and was used as a reference. The inset shows the change in composition of mixtures from experimental method II at a temperature of 10 ◦C. The upper curves correspond to solutions of 0.95 ethanol volume ratio. The lower curves correspond to solutions of 0.5 ethanol volume frac- tion. Sparging data are shown as 4 and ultrasonic data are shown as . (From Ref. [35].)

was 3.09 ppm, and the peak area was integrated to determine the mole fraction ratio of water and ethanol. As shown in Fig. 6.8, the remaining solution from nitrogen sparging process showed smaller ethanol mole fraction compared to that of ultrasonic distillation for solutions with initial ethanol mole fractions less than 0.5. At larger initial ethanol volume fractions, the difference between the sparging and ultrasonic methods was very small. This result shows the better separation efficiency of ni- trogen sparging distillation of ethanol, which is the more volatile component than water.

107 The remaining solutions were reanalyzed with an Abbe refractometer. Before the distillation experiments, calibration was carried out to determine the refractive index versus ethanol volume fraction in ten ethanol-water mixtures ranging from 0.05 to 0.95 and used as a reference for the solutions to be distilled (Fig. 6.9). The accuracy of the measurements was ±0.0001 in index of refraction.

The vertical line of any ethanol volume fraction consistently shows that the re- maining solution of the sparging distillation has a lower ethanol volume fraction. Therefore, the sparging distillation is more efficient than ultrasonic distillation in separating ethanol, which is the more volatile component, from water.

The inset of Fig. 6.9 shows the result of experimental method II both for ul- trasound and nitrogen sparging distillation. Two solutions with ethanol volume fractions of 0.5 and 0.95 were used. The latter solution is near the azeotrope point of an ethanol-water mixture; that is, it is near the point at which the ratio of the con- stituents in the vapor of the mixture is the same as that in the liquid. The mixtures of 40 ml were placed in an ultrasound apparatus and a 41 mm graduated cylinder, and were distilled until 1/2 of the total volume remained. Several 10 µl samples were extracted at seven different heights during this distillation process. The experiment was conducted at 10 ◦C. The mixture of 0.5 volume fraction showed slightly more ethanol evaporation in the sparging experiment than in the ultrasound distillation process. The solution of 0.95 volume fraction shows no composition change both for nitrogen sparging and ultrasound distillation even after half of the solution was dis- tilled. This means there was no significant ethanol separation in either process. Even if the accuracy of measurement of the volume fraction is low, a steeper downward trend of the graph should have been observed for ethanol separation from the mix- tures. Considering the fact that experimental method II is a result of the integrated effect of distillation, slightly better ethanol separation was observed in sparging dis-

108 

Figure 6.10: Composition change of ethyl acetate fraction in the remaining solution after sparging (4) and ultrasonic distillation () for three different ethyl acetate and ethanol mixtures. The Rayleigh curves are shown as solid lines. Time progresses from right to left, indicated by the arrow below the x-axis, as the total volume of the mixture in the vessel decreases. (From Ref. [35].)

tillation for 0.5 ethanol volume fraction mixture than for ultrasound distillation. There was no significant change in mixture composition near the azeotrope for both sparging and ultrasonic distillation. This experiment was conducted under the same conditions reported in Ref. [64] but the result of perfect ethanol separation from water could not be reproduced. Perfect ethanol separation was not observed and there is no evidence that ultrasound distillation has a better efficiency in separating more volatile component. In fact, the opposite was true in our experiments.

6.3.2 Ethyl Acetate-Ethanol Mixture Experiment

Three different mass ratios of ethanol and ethyl acetate mixtures were tested using experimental method II. Fig. 6.10 shows how the ethyl acetate mass fraction changed

109 Concentration Original solution Ultrasound Sparging 0.0301 M 0.227cm−1 0.234 cm−1 0.257cm−1 0.0904 0.427 0.437 0.522 0.181 0.734 0.760 0.937 0.301 1.133 1.161 1.467 0.422 1.534 1.637 2.027

Table 6.4: Comparison of absorption measurements of five different aqueous CoCl2 solutions before and after ultrasonic or nitrogen sparging distillation while either nitrogen sparging or ultrasonic distillation progressed. The mass ratio of solutions was determined by a gas chromatograph with a flame ionization detec- tor. Two solutions with ethyl acetate mass ratios of 18 % and 50% are below the azeotrope, where ethyl acetate is more volatile. A solution with a 90 % ethyl ac- etate mass ratio is above the azeotrope, where ethanol is more volatile. Below the azeotrope, the two solutions showed a lower mass ratio of ethyl acetate, which is the more volatile component in the remaining solution. Above the azeotrope, the remaining solution showed a greater mass ratio of ethyl acetate, which is less volatile than ethanol in nitrogen sparging distillation compared to ultrasonic distillation. This means sparging distillation is more efficient in separating the more volatile component than ultrasonic distillation. Sparging distillation results are compared with their theoretical prediction by the Rayleigh distillation curve in Fig. 6.10. The sparging distillation and the Rayleigh prediction curves are in general agreement. This agreement suggests that the sparging process allowed sufficient time for equi- librium of the mixture constituents to be achieved between the vapor in the bubbles and in the surrounding liquid.

110 6.3.3 Experiments with Aqueous Solutions of Cobalt Chlo-

ride (CoCl2)

Five aqueous ionic CoCl2 solutions in deionized water with different CoCl2 molar concentrations were tested using the first experimental method. The initial 40 ml volume decreased to 30 ml during distillation processes. The absorption spectrum using a UV spectrophotometer was measured at 510 nm to determine the concentra- tion of the CoCl2 in solutions before and after distillation. Table 6.4 shows that the remaining solutions after the sparging distillation are more concentrated in CoCl2 than after ultrasonic distillation, indicating that the sparging method is more effi- cient in separating water, which is the more volatile component.

6.3.4 Two Colloidal Suspension Experiment

Solution Original Ultrasound Sparging Gold 0.163cm−1 0.254 cm−1 0.303cm−1 Gold 0.317 0.383 0.531 Silica 0.186 0.233 0.330 Silica 0.276 0.339 0.559

Table 6.5: Comparison of absorption measurements of gold and silica colloidal solu- tions before and after ultrasonic or nitrogen sparging distillation

Two aqueous silica colloidal suspensions with different concentrations of 40-50 nm diameter particles (Nissan Chemical Co., Snowtex-20L), and two aqueous gold colloidal suspensions with 20 mg/l and 10 mg/l of 70 - 120 nm diameter parti- cles (Purest Colloid, Inc.) were distilled using experimental method I. The initial solution volume of 40 ml decreased to 20 ml after the distillation process. A UV spectrophotometer measured the attenuation of the light at 520 nm to determine concentration of the colloids. Attenuation was dominated by scattering for the silica

111 Total vapor pressure change 545

540

535

530

525

520

515 vapor vapor pressure(mmHg)

510

505 0 20 40 60 80 100 time (min)

Figure 6.11: Vapor pressure of a mixture of ethanol and acetone as a function of time as the ultrasound unit is turned on and off.

colloids and absorbance of the plasmon resonance for the gold colloids. Table 6.5 shows that the sparging distillation is more efficient at separating the more volatile component, water, from the colloidal suspension than ultrasonic distillation.

6.3.5 Vapor Pressure Change of Ethanol Acetone Mixture

An experiment was conducted to test if the mist from ultrasonic atomization con- sisted of fine droplets or vapor. The cell containing an ethanol and acetone mixture was evacuated to a pressure of approximately 500 mmHg. The electronic circuit board containing the ultrasound source was switched on and off several times. When the ultrasound unit was turned on, the total pressure of the cell increased. When the ultrasound unit was turned off, the pressure of the cell decreased. This procedure was repeated as shown in Fig. 6.11. Since the system was not completely sealed, the total pressure of the cell increased gradually over time. However, the cyclic increase

112 and decrease of the total pressure when the unit was turned on and off confirmed that vapors were created from the ultrasonic atomization process.

113 Chapter 7

Discussion

The effects of the ultrasound frequency and intensity, temperature, pressure, and composition of the liquid have been investigated by Beyer[5]. Two hypotheses have been suggested to explain the ultrasonic atomization process. The first is the cap- illary wave mechanism[44] where small droplets ejected from the capillary water jet by strong irradiation of sound pressure form mist and vapor. The second hy- pothesis is the cavitation mechanism, where periodic ultrasound pressure creates bubbles, and vapor and mist are created through hydraulic shocks when the bubbles collapse at the surface of liquid. Both hypotheses are supported by experimental evidence[16, 21, 61, 67]. Besides investigating the mechanism itself, separation char- acteristics of a mixture using ultrasonic atomization have been investigated[64].

Enrichment of surfactants in the collected mixture after ultrasonic atomization was first reported in Ref. [60]. The author used Triton X-100 (an octylphenyl non- ionic surfactant) and D-glucose as a reference nonsurfactant solute dissolved in wa- ter. The enrichment of the surfactant in the droplets of the ultrasonically generated

114 Figure 7.1: A breaking droplet off from the liquid surface where surfactant is en- riched. (From Ref. [60].)

aerosol was up to 10 times higher than the atomized solution for low concentrated solutions. To prevent the decomposition of components due to sonication, the so- lution was saturated with carbon dioxide. The glucose concentration was measured

using a UV spectraphotometer after a reaction with K3F e(CN)6.

The author constructed a model of aerosol formation to explain the enrichment of surfactant concentration based on the surface tension. Let C be the concentration of the surface-active solute in the bulk of the liquid and Γ be the quantity of the adsorbed solute per unit surface area. A droplet of radius r which is tearing from the surface has an average concentration of solute C¯ per unit volume of

4 4 C¯ = (C πr3 + Γ4πr2))/( πr3). (7.1) 3 3

The enrichment factor can be defined as K = C/C¯ and is

C/C¯ = 1 + 3Γ/(rC). (7.2)

115 This was generalized to the form

C/C¯ = 1 + A/(C1−n) (7.3) to fit their empirical data. They found n = 0.5 and A = 1.862 × 10−2M −0.5 for their mixture. This predicts the more significant enrichment of surfactant with smaller diameter droplets and smaller surfactant concentration in solution. Two other experiments[54, 71] corroborate the theory with greater enrichment of surfac- tant in the collected mist and decreased surfactant concentration in the solvent. The proposed theoretical model of ultrasonic distillation in Ref.[60] leads to the conclu- sion that ideal conditions for separation of surfactants using ultrasound is to find properties that concentrate the surfactants at the solution-vapor interface.

Ethanol enrichment on the surface of the solution was predicted theoretically. An article[76] corroborates this result by calculating the enrichment of alcohol in water based on the Gibbs equation. The surface excess of alcohol in alcohol-water mixture is given by the Gibbs adsorption equation

water 1 ∂γ Γalcohol = − ( ) (7.4) RT ∂lnaal

where γ is the surface tension and aalcohol the activity of alcohol. Experimental surface tension data and vapor pressure found in the literature were used to calculate the adsorption isotherms for six alcohols: methanol, ethanol, 1-propanol, 2-propanol, tert-butanol, and 1-butanol. All of the alcohol-water mixtures showed surface excess of the alcohols .i.e. the maximum value in the adsorption isotherm indicating the formation of monolayer coincides with the minimum in the excess partial molar volume of the solutes. The author concludes that the hydrogen bond network in water induces the surface excess of the alcohol. Measurement of surface ethanol

116 concentration using neutron and X-ray grazing incidence reflection[43] showed an ethanol excess at the surface of an ethanol/water solution. The x-ray reflection technique was used to determine the thickness of the ethanol layers. Small angle x-ray scattering [75, 77, 78] determined the particle size distribution of water and ethanol mists produced by high intensity ultrasound. The water droplets turned out to be of micron size and the ethanol droplets to be of nanometer size. Water fog produced by ultrasonic atomization was visible through scattering of visible radiation while ethanol fog produced under the same conditions showed no visible fog.

Enrichment of ethanol in the vapor phase compared to that in the atomized ethanol-water solution was observed as reported in [39, 64, 77]. However, perfect separation of ethanol from an ethanol-water mixture subjected to intense ultrasound pressure at a low temperature of 11oC was not observed. This was based on refractive index and NMR measurements. This was discordant with the results reported in [64, 77].

Compared to ultrasonic distillation, nitrogen sparging distillation consistently showed a better separation for the more volatile component of the mixture, as mea- sured by NMR, refractive index, gas chromatography and a spectrophotometer. The volatile components for each mixture were water in aqueous gold colloid, silica col- loid, and cobalt chloride ionic solution and ethyl acetate below and ethanol above azeotrope in ethyl acetate-ethanol solution.

The Rayleigh curve was calculated and predicted well the experimental nitrogen sparging distillation curve. The significant agreement between the nitrogen sparging curve and Rayleigh curve justifies the nitrogen sparging methodology.

The previous experiments with surfactants in water[60, 71] showed efficient sepa-

117 ration of surfactant with ultrasonic atomization. However, the surfactant experiment is a special case where the solute concentration is much higher at the liquid-vapor interface compared to those with bulk solvents.

Even though perfect separation of the more volatile component was not observed with ultrasonic atomization, unique characteristics in separation of the surfactant enriched in the surface of the liquid mixture suggests further research investigation of ultrasonic atomization as a separation method.

118 Chapter 8

Summary of Comparison of Ultrasonic Distillation and Sparging of Liquid Mixtures

Ultrasonic distillation refers to the formation of vapor and mist ejected from the fountain jet of a liquid which is irradiated with acoustic waves. Experiments were conducted to investigate the recently reported perfect separation of ethanol from wa- ter using ultrasonic distillation[64]. Five different solutions were distilled using the ultrasonic distillation method and compared with the results from the sparging dis- tillation method. Ultrasonic distillation experiments were conducted using a glass cell mounted above a piezoelectric transducer driven at either 1.65 MHz or 2.40 MHz. Dry nitrogen was flowed over the ultrasonic fountain to remove the mist va- por generated from atomization. Sparging distillation was conducted by flowing high purity dry nitrogen through a very fine diameter tube at the bottom of the solution in an ultrasonic distillation cell or a graduated cylinder. Two kinds of experimental

119 methods were conducted. Method I compared the composition before and after dis- tillation for different volume fractions of solution after a certain amount of the total volume decreased. Method II monitored the concentration at different heights during the distillation process. Experiments were carried out for ethanol-water and ethyl acetate-ethanol solutions, cobalt chloride in water, colloidal silica and colloidal gold. The concentration of the remaining liquid mixture was determined by gas chro- matograph, Abbe refractometer, nuclear magnetic resonance spectroscopy, or UV spectrophotometer. Sparging distillation consistently showed the better efficiency in separating the more volatile component in a mixture compared to ultrasonic distil- lation. No evidence was observed for perfect ethanol separation from water at low temperature.

120 Appendix A

Thermal Diffusion Data Analysis Matlab Code

A.1 Code to Obtain the Soret Coefficient and To-

tal Concentration

1 function gettingAlpha() 2 close all,clear,clc, pack; 3 global xdata1 Gydata Gxdata co; 4 %x data range corresponding to the single periodic temperature cycle 5 %is divided into 40 grid 6 N=41;interval=2/(N-1); 7 GN=13; 8 conv_center=interval*6; 9 %gaussian filter size is of 1.8 grid points. 10 sigma=1.8*interval; 11 12 %x data range with positive slope has the half of the x data range 13 %which is 20 14 N=21; 15 xdata=0:interval:interval*(N-2); 16 Gxdata=0:interval:interval*(GN-2); 17

121 18 %Gydata stands for the gaussian filter 19 Gydata=2./(sigma*sqrt(pi))*(exp(-((Gxdata-conv_center)./sigma).^2)); 20 %xdata1 is x data range after the convolution 21 xdata1=0:interval:interval*(N+GN-4); 22 23 normdatafile=’Mar_10_2008_afternooon6_15_norm.txt’ 24 B=load(normdatafile); 25 B=B’; 26 [B_row B_column]=size(B) 27 %column means number of data points, row-running time 28 for i=1:B_row 29 min_value=min(B(i,:)); 30 %substract the offset value of the fluorescence signal for 31 %normalization 32 B(i,:)=B(i,:)-min_value; 33 end 34 % normalizaing both exp_data and numerical data 35 for i=1:B_row 36 sum_signal(i)=0; 37 for j=1:B_column 38 sum_signal(i)=sum_signal(i)+B(i,j); 39 end 40 B(i,:)=B(i,:)/sum_signal(i); 41 end 42 %final_data2 is the data set to be used as a long time profile 43 file2=B;row2=B_row;column2=B_column; 44 final_data2=file2(row2-1,:); 45 46 i=1:column2; 47 final_data1(1,i)=final_data2(1,i); 48 49 min_value=min(final_data1); 50 51 i=1:column2; 52 final_data(1,i)=final_data1(1,i)-min_value; 53 ydata1=final_data; 54 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 55 z=linspace(0,2,40); 56 x=z(1:N-1); 57 %To compensate the mismatch of the data points after the convolution 58 %between the actual experimental data and numerical fitting curve, 59 %four more zero data points were added to the actual data set. 60 zmatrix=[0 0 0 0]; 61 convY=horzcat([0],ydata1,zmatrix); 62 63 beta0 = [1 1]; 64 [beta,r,J] =modified_nlinfit(x, convY, @fun, beta0); 65 66 y1=fun(beta,x); 67 ci=nlparci(beta,r,J) 68 figure; 69 plot(convY,’r’);hold on; 70 plot(convY,’.’,’markerEdgeColor’,’r’);hold on; 71 plot(y1,’b’);hold on;

122 72 plot(y1,’.’,’markerEdgeColor’,’b’);hold on; 73 plot(Gydata*0.012,’g’);hold on; 74 plot(Gydata*0.012,’.’,’markerEdgeColor’,’g’);hold on; 75 76 y1 77 2./(sigma*sqrt(pi)) 78 A=beta(1) 79 alpha=beta(2) 80 co=-(1/alpha*log((1+1/A*exp(-alpha))/(1+1/A))) 81 return 82 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 83 function [convY]=fun(beta, x) 84 global xdata1 Gydata Gxdata sigma co; 85 A=beta(1); 86 alpha=beta(2); 87 y=(1./(1+A*exp(alpha*x))); 88 convY=conv(Gydata,y); 89 return

A.2 Code to Achieve Numerical Density Profile

at Different Times Using Soret Coefficient

and Total Concentration Obtained from Long

Time Density Profile

1 function Lweno() 2 clc;clear all;close all; 3 4 global N z dTdz alpha em; 5 N =40; 6 interval=49; 7 sigma = 0.001; 8 %x data range is set from 0 to 2*pi instead of 0 to 2. 9 %This changed alpha value became 5.56 from 1.77 (=5.56/3.14). 10 %alpha = 1.7796; 11 alpha=5.5595 12 em = 1; 13 xt = linspace(0, 2, N); 14 z = xt(1:(N-1)); 15 dTdz = dTdzf(sigma); 16

123 17 figure(100); 18 plot(dTdz); 19 C = 0.0011*ones(1, N-1); 20 21 Ct = C; 22 dz = z(2)-z(1); 23 dt = dz*dz/10; 24 25 Tmax = 3500; 26 Ck=Ct; 27 for k=1:Tmax 28 msg = sprintf(’elapsed time = %f,%d’, k*dt,k); 29 disp(msg); 30 31 C1 = C + dt*L(C); 32 C2 = (3/4)*C + (1/4)*C1 + (1/4)*dt*L(C1); 33 Cn = (1/3)*C + (2/3)*C2 + (2/3)*dt*L(C2); 34 Ck=vertcat(Ck, Cn); 35 C = Cn; 36 % figure(1); 37 % plot(C); 38 % ylim([0,1]); 39 % drawnow; 40 end 41 42 figure; 43 for k=1:interval:Tmax 44 i=1:N-1; 45 q=k/Tmax; 46 h=plot(z(i),Ck(k,i),’.’,’markerEdgeColor’,[q,0,q]);hold on; 47 h=plot(z(i),Ck(k,i),’Color’,[q,0,q]);hold on; 48 end 49 50 figure; 51 plot(z,Ck(Tmax,:),’Color’,’red’);hold on; 52 %The values of A=166.2 and alpha=5.5595 are obtained from the 53 %long time fluorescence signal profile. This plot is to compare 54 %the analytical long time solution with numerical solution at long time 55 plot(z,1./(1+166.2* exp(5.5595 * z)),’Color’,’green’);hold on; 56 57 A=Ck; 58 Ck=A’; 59 file_format=’’; 60 format1=’%8.7f’; 61 for k=1:1:N-1 62 file_format=sprintf(’%s %s ’,file_format,format1); 63 end 64 enter_string=’\n’; 65 file_format=sprintf(’%s %s’,file_format,enter_string); 66 67 file_name=sprintf(’nonlinear_c00011_sigma0001_alpha_%d_imax%d_kmax_%d_interval%d_dt_dividedBy10.txt’,alpha,N,Tmax,interval); 68 fid0=fopen(file_name,’wt’); 69 fprintf(fid0,file_format,Ck); 70 fclose(fid0);

124 71 72 return 73 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 74 function Ln = L(C) 75 76 global N z; 77 dz = z(2) - z(1); 78 79 for j=1:N-1 80 Ln(j) = (1/dz)*( fphat(C, j) - fphat(C, j-1) - fmhat(C, j) + fmhat(C, j-1)); 81 end 82 83 Ln = Ln + d2Cdzdz(C); 84 return 85 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 86 function dTdz = dTdzf(sigma) 87 global z; 88 dTdz = erf(z/sigma); 89 dTdz = dTdz - erf((z-1)/sigma); 90 dTdz = dTdz + erf((z-2)/sigma); 91 figure;plot(z,dTdz,’.’);hold on;plot(z,dTdz); 92 ylim([-1,1]);hold on; 93 return 94 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 95 function fc = fpj(C, ji) 96 global dTdz alpha em; 97 j = J(ji); 98 fc = 0.5*(alpha*C(j)*(1-C(j))*dTdz(j) + em*alpha); 99 return 100 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 101 function fc = fmj(C, ji) 102 global dTdz alpha em; 103 j = J(ji); 104 fc = 0.5*(alpha*C(j)*(1-C(j))*dTdz(j) - em*alpha); 105 fc = -fc; 106 return 107 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 108 function q = qk3(k, f1, f2, f3) 109 switch(k) 110 case 1, 111 q = (1/3)*f1 + (-7/6)*f2 + (11/6)*f3; 112 113 case 2, 114 q = (-1/6)*f1 + (5/6)*f2 + (2/6)*f3; 115 116 case 3, 117 q = (1/3)*f1 + (5/6)*f2 + (-1/6)*f3; 118 end 119 return 120 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 121 function fhat_j = fphat( C, j ) 122 q(1) = qk3(1, fpj(C, j-2), fpj(C, j-1), fpj(C, j) ); 123 q(2) = qk3(2, fpj(C, j-1), fpj(C, j), fpj(C, j+1) ); 124 q(3) = qk3(3, fpj(C, j), fpj(C, j+1), fpj(C, j+2) );

125 125 126 w = wk(1, C, j); 127 128 fhat_j = w(1)*q(1); 129 fhat_j = fhat_j + w(2)*q(2); 130 fhat_j = fhat_j + w(3)*q(3); 131 return 132 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 133 function fhat_j = fmhat(C, j) 134 q(1) = qk3(1, fmj(C, j-2), fmj(C, j-1), fmj(C, j)); 135 q(2) = qk3(2, fmj(C, j-1), fmj(C, j), fmj(C, j+1)); 136 q(3) = qk3(3, fmj(C, j), fmj(C, j+1), fmj(C, j+2)); 137 138 w = wk(-1, C, j); 139 140 fhat_j = w(1)*q(1); 141 fhat_j = fhat_j + w(2)*q(2); 142 fhat_j = fhat_j + w(3)*q(3); 143 return 144 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 145 function [IS0 IS1 IS2] = smooth(pm, C, j) 146 jm2 = J(j-2); 147 jm1 = J(j-1); 148 jp1 = J(j+1); 149 jp2 = J(j+2); 150 151 if (pm ==1) 152 fjm2 = fpj(C, jm2); 153 fjm1 = fpj(C, jm1); 154 fj = fpj(C, j); 155 fjp1 = fpj(C, jp1); 156 fjp2 = fpj(C, jp2); 157 elseif (pm==-1) 158 fjm2 = fmj(C, jm2); 159 fjm1 = fmj(C, jm1); 160 fj = fmj(C, j); 161 fjp1 = fmj(C, jp1); 162 fjp2 = fmj(C, jp2); 163 else 164 disp(’smooth : Error’); 165 end 166 167 IS0 = (13/12)*(fjm2 - 2*fjm1 + fj)^2; 168 IS0 = IS0 + (1/4)*(fjm2 -4*fjm1 + 3*fj)^2 ; 169 170 IS1 = (13/12)*( fjm1 - 2*fj + fjp1 )^2; 171 IS1 = IS1 + (1/4)*( fjm1 - fjp1 )^2 ; 172 173 IS2 = (13/12)*(fj - 2*fjp1 + fjp2 )^2; 174 IS2 = IS2 + (1/4)*(3*fj - 4*fjp1 + fjp2)^2 ; 175 return 176 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 177 function w = wk(pm, C, j) 178 [IS0 IS1 IS2] = smooth(pm, C, j);

126 179 180 eps = 10e-6; 181 alp(1) = 0.1/(eps + IS0)^2; 182 alp(2) = 0.6/(eps + IS1)^2; 183 alp(3) = 0.3/(eps + IS2)^2; 184 185 alp_sum = alp(1) + alp(2) + alp(3); 186 187 w(1) = alp(1)/alp_sum; 188 w(2) = alp(2)/alp_sum; 189 w(3) = alp(3)/alp_sum; 190 return 191 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 192 function cpp = d2Cdzdz(C) 193 global z N; 194 195 dz = z(2) - z(1); 196 invdz2 = (1/dz)*(1/dz); 197 198 for i=1:N-1 199 cpp(i) = C(J(i-1)) - 2*C(i) + C(J(i+1)); 200 cpp(i) = invdz2*cpp(i); 201 end 202 return 203 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 204 function j = J(j) 205 globalN; 206 if(j<1) 207 j = j + N - 1; 208 elseif (j>= N) 209 j = j - N + 1; 210 else 211 j = j; 212 end 213 214 return 215 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

127 Appendix B

Image Formation Theory

B.1 Abbe’s Theory of Image Formation

Abbe assumed that an object acts like a periodic grating and diffracts an incident plane wave into a superposition of plane waves. These waves are brought to a focal plane of the lens. These spots act as a source of Huygens’ spherical wavelets and produce an image on the image plane by interference[40]. Since diffraction happens twice, first at the object image plane and second at the second focal back plane, Abbe’s theory of image formation is called double diffraction.

128 B.2 Diffraction Theory and a Lens as a Diffraction

Medium

When the incident plane wave passes through the small aperture compared to R0 as shown in Fig. B.1, secondary spherical wavelets are generated through diffraction. The Fraunhofer diffractive wavelets at point (µ, ν) in the observation plane can be given as

ZZ eikR G(µ, ν) = f(x, y) dxdy aperture R −ikR0 ZZ e −ik (µx+νy) ≈ f(x, y)e R0 dxdy aperture R0 ˜ = F(f(x, y)) = f(−kµ/R0, −kν/R0), (B.1)

q R = L2 + (µ − x)2 + (ν − y)2 q 2 2 2 = R0 − 2µx − 2νy + x + y , (B.2)

(x2+y2) and f(x, y) is an aperture function under the condition of 2 << 1 which is the 2R0 Fraunhofer condition. Also,

˜ ˜ F (−kµ/R0, −kν/R0) = F (−kx, −ky). (B.3)

It is of note that the diffraction pattern is a Fourier transformation of the original aperture function.

x2+y2 There is a phase difference of −ik for a planar incident wavelet where f0 is f0 the focal length of the lens and under the condition x2 + y2 << 1.

129 y 

(, ) (x, y) R  x Ro

L

Aperture(lens) Observation plane

Figure B.1: Geometry to calculate the Fraunhofer diffraction equation of rays passing through a lens(From Ref. [55].)

(x,y)

fo

2 2 2 2 2 x  y   ik ( ( fo  x  y )  fo )  ik 2 fo Figure B.2: A Phase difference of rays converging on the focal point of a lens(From Ref. [55].)

130 Consider the lens as an aperture and the diffraction patterns on the focal plane, i.e., L = f0. Given the angle between the axis (L) and the diffraction line (R),

L, R, and R0 can be set all equal to f0. The total phase difference is the sum of the phase difference due to Fraunhofer diffraction and the lens is given in Fig. B.2. The diffraction pattern on the focal plane with the aperture function f(x, y) becomes

ZZ eikR G(µ, ν) = f(x, y) dxdy aperture R ZZ e∆Ψ = f(x, y)dxdy aperture f0 −ikf0 ZZ e −ik (µx+νy) ≈ f(x, y)e f0 dxdy aperture f0 e−ikf0 = F(f(x, y)) f0 −ikf0 e ˜ = f(−kµ/R0, −kν/R0) (B.4) f0 (B.5) where

q 2 2 2 2 2 x + y ∆Ψ = −ik f0 + (µ − x) + (ν − y) − ik 2f0 µx + νy ≈ −ikf0 + ik f0 (B.6)

The Fourier transformation of the aperture function appears as a diffraction pattern on the back focal plane behind the lens. This is the first Fourier transform in the image formation. The spherical Huygen’s wavelets form an interference pattern on

131 x µ x'

y ν y' fo D

lens Back focal plane Observation plane Object G(µ,ν ) H (x', y') ~ ~ f (x, y) F(x, y) F(x, y)

Figure B.3: One zeroeth order and two first order beams are diffracted here. The dotted line on the observation plane shows the image when all the orders are passed through a lens. The objective aperture acts as a low pass filter(From Ref. [55].)

the observation plane as[55]

ZZ ikD 0 0 e −ik µ x0−ik ν y0 H(x , y ) = G(µ, ν) e D D dµdν focalplane D −ikf0 −ikD ZZ e µ ν e k µ f0 x0+k ν f0 y0 = f˜(k , k ) e f0 D f0 D dµdν focalplane f0 f0 f0 D e−ikf0 e−ikD f f f f f = ( 0 )2f˜(− 0 x0, − 0 y0) = const.f(− 0 x0, − 0 y0).(B.7) f0 D k D D D D

Here images on the observation plane are inverted after being magnified by a factor of D [69]. Because of the limited size of the aperture, an objective aperture behaves f0 like a low pass filter. That is, only lower orders of diffracted beams can be collected.

132 Appendix C

NMR principle

Nuclear Magnetic Resonance (NMR) is a phenomenon which a set of nuclei of certain atoms exhibits when they are exposed to a static magnetic field and the secondary oscillating magnetic field. It is a quantum mechanical phenomenon related to nucleon spin.

The nucleus magnetic moment is defined as

e µ = g L, (C.1) 2Mc where g is the gyromagnetic ratio, L is the angular momentum of the spin, M is the proton mass, e is the charge of electron and c is the velocity of light in free space.

Under the presence of the magnetic field H, the torque τ exerted on the spin magnetic moment µ is τ = µ × H. (C.2)

133 With the help of the Newton’s law for rotational motion, dp/dt = τ, Eq. C.1 gives

dL/dt = −g(e/2Mc)H × L. (C.3)

Based on Eq. C.3, L precesses with the angular velocity

ω0 = −g(e/2Mc)H, (C.4) regardless of the angle between µ and H. This is called the Larmor precession frequency. The spin dipole precesses at the Larmor frequency.

When the direction of an externally applied magnetic field H is set along the z-axis, the z-component of the angular momentum vector L is quantized into 2S+1 spin states from S, S-1, ..., -S+1, -S for a given maximum spin quantum number S and can be expressed as e µ = g m . (C.5) z 2Mc ~

The energy of these spin states are equal in the absence of the external magnetic field, i.e. the states are degenerate. The number of atoms in each spin state is equivalent at thermal equilibrium. However, the interaction with the external magnetic field splits these degenerate energy levels into 2S+1 different levels. The potential energy of a magnetic moment µ under the magnetic field H is

U = −µ · H = µzH = −g(e~/2Mc)mH (C.6) where m can take any values of S, S-1, ..., -S+1, and -S. Nuclei with a spin of one half such as 1H,13C, and 19F have two possible spin states: m=1/2 or m=-1/2. When a particle in the lower state with spin m=1/2 absorbs a photon with energy

134 corresponding to the energy difference of the two spin states, the particle transits from the lower to the higher energy state with spin number m=1/2.

When the population for spin up (m = 1/2) and spin down (m = −1/2) energy state are set as N(1/2) and N(-1/2), the transition between these two states will stay in equilibrium if

N(1/2)W (1/2 → −1/2) = N(−1/2)W (−1/2 → 1/2) (C.7) where W (1/2 → −1/2) is the transition probability per unit time from m=1/2 to m=-1/2 state. Since this probability follows Boltzman distribution, it can be expressed as W (up → down) = P exp(U(m = −1/2)/kT ), where P is a constant[56].

C.1 T1 Processes

The z component of magnetization µz is the same as the equilibrium magentization

µ0 subjected to the external magnetic field B0. When the particles absorb the elec- tromagnetic energy with Larmor frequency, the net magnetization can be changed to zero. The process for the spin system to return back to equilibrium state is called spin-lattice relaxation and its time is denoted as T1.

t/T1 µz = µ0(1 − e ) (C.8)

The peak area is proportional to the number of a given type of spin. This is the reason why NMR is so sensitive in detecting small population differences[31].

135 C.2 T2 Processes

When the spin precesses around the applied magnetic field, the net magnetization dephases due to slightly different Larmor frequencies for each of the spin packets since they experience slightly different magnetic fields. The spin-spin relaxation time, which is the time for the transverse magnetization µxy to return to the equilibrium state, is

−t/T2 µxy = µxy0 e (C.9)

Since T1 is always greater than T2, the net transverse magnetization goes to zero before the longitudinal magnetization goes back to µ0 along the z axis[31].

C.3 Chemical Shift and Spin-Spin Coupling

Electrons surrounding the proton of interest are always slightly diamagnetic and oppose the applied external field B0 with the electron precession. Protons under different electronic environments show slightly different diamagnetic properties and the resonance frequency is shifted from the reference frequency. The frequency shift divided by the reference frequency is called the “chemical shift” since it happens due to electrons which characterize chemical nature of the element shifting the resonance frequency compared to that of free protons. It is influenced by the distribution of electrons in the chemical bonds of the molecule. The expression for the chemical shift is

6 δ = (ν − νREF )/νREF × 10 (C.10)

where ν0 is the resonance frequency of the free proton[31]. Consider the chemical shift of ethyl alcohol. The protons are distributed to three different chemical envi-

136 Figure C.1: Nuclear spin-spin interaction (a) in the HF molecule and (b) in a CH2 group. The antiparallel orientation of nuclear and magnetic moments is considered as a low-energy state. (From Ref. [26].)

ronments, CH3,CH2 and OH. The proton resonance signal splits into three different values because electrons in these three different environments have slightly different diamagnetic properties. The ratio of the intensities of each resonance frequency peak is 1:2:3 corresponding the number of protons.

C.4 J(spin-spin) Coupling

If ethyl alcohol is considered, there is not only a difference in resonance frequency but also a difference in the multiplicity of the signals. This happens by a magnetic interaction between individual protons which is transmitted by the bonding electron through which the protons are indirectly connected. This is called spin-spin coupling or J-coupling. This is independent of the external magnetic field. [26] Consider the spin-spin interaction of the two nuclei, A and X. The energy, E, of the spin-spin

137 Figure C.2: These are the nuclear magnetic energy levels for a two-spin system: (a) without spin-spin coupling for J = 0; (b) with spin-spin coupling for J > 0. With the convention that the antiparallel arrangement of the nuclear moment is chosen as the low energy state, the eigenvalues E2 and E3 decrease, while eigenvalues E1 and E4 increase. Only the spectral lines of nucleus A are shown. (From Ref. [26].)

interaction between A and X can be expressed as

E = JAX IAIX (C.11)

where IA and IX are the nuclear spin vectors of nuclei and JAX is the scalar coupling constant between A and X nuclei. There are four possible states when the external

field B0 is present.

With no spin-spin coupling, i.e. J = 0, only one resonance signal is expected as shown in Fig. C.2a. The eigenvalues of the spin-spin coupling system are either decreased or increased depending on the relative orientation of nuclear moments. Since antiparallel orientation of the nuclear moments is chosen as the lower energy

138 state, the transitions A1 and A2 are no longer the same and induce splitting of the spectral resonance line into a doublet. The multiplicity for n equivalent nuclei X (spin 1/2) is n + 1 with peak ratios following Pascal’s triangle and the same as the coefficients of the expansion of the equation (x + 1)n. Therefore, the triplet for the

CH3 group that is affected by CH2 group is expected to be seen with a peak ratio of

1 : 2 : 1. In the same way, CH2 splits into a quartet with a peak ratio of 1 : 3 : 3 : 1 by the CH3 group. As for water, only one resonance spectral line is seen[26].

139 Bibliography

[1] M. Ashokkumar, R. Hall, P. Mulwaney, and F. Greizer. J. Phys. Chem. B, 101:10845, 1997.

[2] R. P. Bagwe, C. Yang, L. R. Hilliard, and W. Tan. Langmuir, 20:8336, 2004.

[3] D. R. Bates and Immanuel Estermann. Advances in Atomic and Molecular Physics, volume 2. Academic Press INC., 1966.

[4] A. Becker, W. Köhler, and B. Müller. Ber. Bunsenges. Phys. Chem., 99:600, 1995.

[5] R. T. Beyer. Nonlinear Acoustics. US Dep. of the Navy, 1974.

[6] James A. BIERLEIN. The Journal of Chemical Physics, 23:10, 1955.

[7] Yu. Ya. Boguslaviskii and O. K. Éknadiosyants. Physical mechanism of the acoustic atomization of a liquid. Soviet Physics - Acoustics, 15:14, 1969.

[8] J. R. Cannon. The One Dimensional Heat Equation, volume 2. Addison Wesley, New York, London, 1984.

[9] S. Chapman. Philos. Trans. A, 211:433, 1912.

[10] S. Chapman and F. W. Dootson. Phil. Mag., 33:248, 1917.

140 [11] Sorasak Danworaphong. Laser Induced Thermal Diffusion Shock Waves. PhD thesis, Brown University, 2004.

[12] Sorasak Danworaphong, Walter Craig, Vitaly Gusev, and Gerald J. Diebold. Phys. Rev. Lett., 94:095901, 2005.

[13] S. R. de Groot. Physica, 6:699, 1942.

[14] S. R. de Groot and P. Mazur. Non-equilibrium Thermodynamics. Dover Publi- cation, Inc, New York, the dover edition, 1984.

[15] L. Dufour. Pogg. Ann., 148:490, 1873.

[16] O. K. Éknadiosyants. Soviet Physics-Acoustics, 14(1):80, 1968.

[17] D. Enskog. Phys. Z., 12:533, 1911.

[18] M. Faraday. Philos. Trans. Roy. Soc., 121:299–318, 1831.

[19] J. C. Fisher. J. Appl. Phys., 19(11):1062, 1948.

[20] Donald D. Fitts. Nonequilibrium Thermodynamics. McGraw-Hill Inc., US, 1962.

[21] E. L. Gershenzon and O. K. Éknadiosyants. Sov. Phys. Acoust., 10:127–132, 1964.

[22] M. Giglio and A. Vendramini. Physical Review Letters, 34:561, 1975.

[23] M. Giglio and A. Vendramini. Physical Review Letters, 38:26, 1977.

[24] Carl Zeiss MicroImaging GmbH. Lsm 510Mk4 and lsm 510 meta laser scanning microscopes.

[25] Georges Guiochon and Claude L. Guillemin. Quantitative Gas Chromatography for Laboratory Analysis and On-line Process Control. Elseviere Science Publish- ers B.V., New York, 1988.

141 [26] Herald Gunther. NMR Spectroscopy: Basic Principles, Concepts, and Applica- tions in Chemistry. John Wiley and Sons, New York, 1994.

[27] V. Gusev, W. Craig, R. LiVoti, S. Danworaphong, and G. J. Diebold. Phys. Rev. E, 72:041205, 2005.

[28] Vitalyi Gusev, Binbin Wu, and Gerald J. Diebold. Dynamics of thermal diffusion in a linear temperature field. accepted to Journal of Applied Physics.

[29] A. Harten, B. Engquist, S. Osher, and S. Chakravarthy. J. Comp. Phys., 71:231, 1987.

[30] Andrew Wallace Hayes. Principles and Methods of Toxicology. Informa Health- care USA, Inc., New York, 2008.

[31] Joseph P. Hornak. Basics of nmr, interactive learning software. 2011.

[32] B. I. Il’in and O. K. Éknadiosyants. Soviet Physics-Acoustics, 12(3):310, 1966.

[33] G. S. Jiang and C. W. Shu. J. Comp. Phys., 126, 1996.

[34] H. Jung, V. Gusev, H. Baek, Y. Wang, and G. J. Diebold. Ludwig-soret effect in a linear temperature field: Theory and experiments for distributions at long times. Physics Letters A, 375:1917–1920, 2011.

[35] Hye Yun Jung, Han Jung Park, Joseph M. Calo, and Gerald J. Diebold. Com- parison of ultrasonic distillation to sparging of liquid mixtures. Anal. Chem., 82:10090–10094, 2010.

[36] W. Thomson (Lord Kelvin). Proc. Roy. Soc.Edinburgh, 3:225, 1854.

[37] W. Thomson (Lord Kelvin). Proc. Roy. Soc.Edinburgh, 21:123, 1857.

[38] W. Thomson (Lord Kelvin). Math. Phys. Papers, 1:232, 1882.

142 [39] D. M. Kirpalani and F. J. Toll. J. Chem. Phys., 117:3874–3877, 2002.

[40] Miles V. Klein. Optics. John Wiley and Sons, Inc., New York, London, Sydney, Toronto, 1970.

[41] R. J. Lang. J. Acoust. Soc. Am., 34(1):6, 1962.

[42] W. B. Li, P. N. Segre, R. W. Gammon, and J. V. Sengers. J. Chem. Phys., 101:5058, 1994.

[43] Z. X. Li, J. R. Lu, D. A. Styrkas, R. K. Thomas, A. R. Rennie, and J. Penfold. Mol. Phys., 80:925–939, 1993.

[44] J. Lighthill. Waves in Fluids. Cambridge University Press, Cambridge, 1987.

[45] Anthony Blair Littlewood. Gas chromatography : principles, techniques, and applications. New York, Academic Press, 1962.

[46] X-D. Liu, S. Osher, and T. Chan. J. Comp. Phys., 115:200, 1994.

[47] C. Ludwig. Sitzber. Akad. Wiss. Vien Math.-naturw, page 539, 1856.

[48] Zygmunt Marczenko and Maria Balcerzak. Separation, Preconcentration, and Spectrophotometry in Inorganic Analysis. Elsevier Science, B.V., Amsterdam, The Netherlands, 2000.

[49] H. Margenau and G. M. Murphy. The Mathematics of Physics and Chemistry. Van Nostrand, Princeton,New Jersey, 2nd ed. edition, 1956.

[50] A. J. P. Martin and R. L. M. Synge. BioChem. J., 35:1358, 1941.

[51] S. Mazzoni, R. Cerbino, A. Vailati, and M. Giglio. Eur. Phys. J. E, 15:306, 2004.

143 [52] N. W. McLachlan. Theory and Application of Mathieu Functions. Moscow, 1953.

[53] P. Mirau. Solid-State NMR of Polymers. Rapra Technolgy LTD., United King- dom, 2001.

[54] S. Nii, K. Matsuura, T. Fukazu, M. Toki, and F. Kawaizumi. Chem. Eng. Res. and Design, 84:412–415, 2006.

[55] University of New South Wales. Advanced optics.

[56] G. E. Pake. Am. J. Phys, 18:438, 1950.

[57] Han Jung Park. Photoacoustic Effect and Sonoluminescence Generated By Laser Initiated Exothermic Chemical Reactions. PhD thesis, Brown University, 2011.

[58] A. Perronace, C. Leppla, F. Leroy, B. Rousseau, and S. Wiegand. J. Chem. Phys., 116(9):3718, 2002.

[59] Jean K. Platten. Journal of Applied Mechanics, 73:5–14, 2006.

[60] D. Rassokhin. J. Phys. Chem. B, 102:4337–4341, 1998.

[61] N. A. Roi. Sov. Phys. Acoust., 3(1):1, 1957.

[62] R. Rusconi, L. Isa, and R. Piazza. J. Opt. Soc. Am., 21:605, 2004.

[63] P. L. Sachdev. Nonlinear Diffusive Waves. Cambridge University Press, Cam- bridge, 1987.

[64] M. Sato, K. Matsuura, and T. Fujii. J. Chem. Phys, 114:2382–2386, 2001.

[65] David P. Shoemaker, Carl W. Garland, and Joseph W. Nibler. Experiments in Physical Chemistry. McGraw-Hill, Inc., New York, 5th edition edition, 1989.

[66] M. G. Sirotyuk. Sov. Phys. Acoust., 8:165, 1962.

144 [67] K. Söllner. Trans. Far. Soc., 32:1532–1536, 1936.

[68] V. I. Sorokin. Soviet Physics Acoustics, 3:281, 1957.

[69] H. N. Southworth. Introduction to Moden Microscopy. Wykeham Publications (London) LTD, Springer-Verlag New York INC., London and Winchester, 1975.

[70] R. Stock and C.B.F. Rice. Chromatographic Methods. Chapman and Hall LTD, London, 2nd ed. edition, 1957.

[71] H. Takaya, S. Nii, and F. Kawaizumi. Ultrasonics Sonochemistry, 12:483–487, 2005.

[72] A. Tuulmets and S. Salmar. Ultrason. Sonochem., 8:209, 2001.

[73] Simone Wiegand and Werner Kohler. Thermal Nonequilibrium Phenomena in Fluid Mixtures, volume 584. Springer, Berlin; New York, 2002.

[74] R. W. Wood and A. L. Loomis. Philos. Mag., 4:417–437, 1927.

[75] Y. Yano, A. Kumagai, T. Limima, Y. Tomida, T. Miyamoto, and K. Matsuura. J. Chem. Phys, 125:174705, 2007.

[76] Y. F. Yano. J. Colloid Interface Sci., 284:255–259, 2005.

[77] Y. F. Yano, K. Matsuura, T. Fukazu, F. Abe, A. Wakisaka, H. Kobara, K. Kaneko, A. Kumagai, Y. Katsuya, M. Okui, and M. Tanaka. J. Appl. Crys- tall., 40:318–320, 2007.

[78] Y. J. Yano, K. Matsuura, T. Fukazu, F. Abe, A. Wakisaka, K. Kaneko, A. Ku- magai, U. Katsuya, and M. Tanaka. J. Chem. Phys, 127:031101, 2007.

145