Diffusion Reaction Interaction for a Pair of Spheres - ETD ...

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DEDICATION This is for my daughter, Ayari, my favorite study partner. My grandmother, Ivy Johnson gave me the best mother and aunts anyone could ask for. These women all contributed to my success, they provided for me throughout my life and shaped most of my ideas. I thank my mother, her advice and love sustained my soul and kept me focused. My Aunt Nita took me to school most of my life and insisted that I be vigilant. She taught me to pursue my dreams with immense passion. Smiles, kisses, hugs and home cooked foods from Aunt Joycey were always abundant. My Aunt Sherry and Aunt Pearl always welcomed me home with my favorite dishes. Nicky, life would be boring without our weekend conversations, thanks for your advice and love. Finally, Louise McDonald, my paternal grandmother, gave me nothing but love. From these women I learned that I could do anything because they would allow no one to stand in my way. Thanks to them all I have completed my Ph. D. ii
TABLE OF CONTENTS FIGURES………………………………………………………………………... v TABLES………………………………………………………………………….xiv SYMBOLS……………………………………………………………………..... xv ACKNOWLEDGMENTS……………………………………………...………...xx CHAPTER 1: HISTORICAL REVIEW OF DIFFUSION AND REACTION IN POROUS MEDIA………………………..………………….1 1.1 Introduction……………………………………………. 1 1.2 Previous Work………………………………………….3 1.21 Two Particle System……………………………....3 1.2.2N Particle System……………………………….... 6 1.3 Cell-Cell Interaction……………….…………………...9 1.4 Diffusive Interaction in Two Particle System…………. 10 1.4.1 Overview of Two Particle Model……………...…10 1.4.2 Solution Method and Results….………………… 11 Figure..……………………………….….………………… 14 CHAPTER 2: DIFFUSION REACTION FOR A SPHERICAL SOURCE AND A SPHERICAL SINK………………………………….... 15 2.1 Introduction……......………………….……………….. 15 2.2 Mutualism Interaction for Spherical Source and Sink… 17 2.3 Bispherical Expansion……….………………………....20 2.4 Mutualism-like Problem: Analytical Solution..……….. 23 2.4.1 Dimensionless Reaction Probability……....…..… 23 2.4.2 Dimensionless Concentration……..………..…….32 2.5 Results and Discussion……………….....………..…….36 2.6 Summary and Conclusions……………...……...………41 Figures……………………………….….…….……………43 CHAPTER 3: COMPETITIVE INTERACTION BETWEEN TWO DIFFERENT SPHERICAL SINKS…………………………..... 49 3.1 Introduction……...………...……….………………….. 49 iii
Page 1 and 2: DIFFUSION INTERACTIONS FOR A PAIR O
Page 3: Nyrée McDonald bispherical coordin
Page 7 and 8: FIGURES 1.1 Two spheres of radius a
Page 9 and 10: 3.3 Dimensionless consumption rate
Page 11 and 12: 3.12 Constant concentration 0 c c c
Page 13 and 14: 5.5 Reaction probability Pint for a
Page 15 and 16: B.1 The bispherical coordinates (μ
Page 17 and 18: a Radius of sphere 1 1 a Radius of
Page 19 and 20: K Matrix elements for the probabili
Page 21 and 22: η Bispherical coordinate θ1 θ2 S
Page 23 and 24: CHAPTER 1 HISTORICAL REVIEW OF DIFF
Page 25 and 26: compounds (Siebel and Charcklis, 19
Page 27 and 28: the end, Tsao (2001) claimed that t
Page 29 and 30: 1990), permeable (Torquato and Avel
Page 31 and 32: 4 that for site volumes fractions 1
Page 33 and 34: internal reaction in sphere 1 and c
Page 35 and 36: directly to the concentration and t
Page 37 and 38: CHAPTER 2 DIFFUSION REACTION FOR A
Page 39 and 40: 2.2 Mutualism Interaction Between a
Page 41 and 42: P 2πa Dc 2 π 1 0 = ⎡∂u ⎤ si
Page 43 and 44: P ( cosΘ1) n ( n+ 1) r1 1 = n d 1
Page 45 and 46: 2.4 Mutualism-like Problem: Analyti
Page 47 and 48: h 10 ∑ m 1 ∞ = 1+ K = 1− K 0m
Page 49 and 50: where and ( 2) ( 2) h = Q + h K , n
Page 51 and 52: They can be calculated exactly for
Page 53 and 54: Similarly, the coefficient h can be
Page 55 and 56:
coefficients f1n. Since h1n was act
Page 57 and 58:
G ( 1) n ( 0) ( 0) G0 K = Gn + , (2
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probability P (2.55), and the conce
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ate constant λ 50 . The figures 2.
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Figure 2.2 - 2.4 shows the monotoni
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Θ1 d Figure 2.1: Two spheres of ra
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Figure 2.3: Reaction probability P
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Figure 2.5: Constant concentration
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CHAPTER 3 COMPETITIVE INTERACTION B
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a2, respectively. Any point externa
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The overall reaction rate at sphere
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where Λ kn ⎛ ak = ⎜ ⎝ d n
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∑( ) ∞ − j ⎛ j + n⎞⎛ j
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and ( ) 1 − 1− L ( i+ 1) () i (
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i→∞ ( n+ i) ( ∞) v1 n = lim M
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1953). The reaction rate for this c
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a1 sinh μ1 h μ = , (3.47) cosh μ
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ispherical coordinate system holds
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eaction rate remains virtually unch
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on the x = 0 line with the larger s
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contact (d=a1+a2) to far apart (d>>
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Figure 3.1: Dimensionless consumpti
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
Figure 3.11: Constant concentration
Page 109 and 110:
CHAPTER 4 EXACT SOLUTION FOR THE CO
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for all points external to the sink
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R 1 π π ⎡ ⎤ , ⎢⎣ ∂r1
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and −1⎛ f ⎞ μ = ⎜ ⎟ 1 si
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and the small number of infinite su
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4.4 Results and Discussion Bispheri
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4.5 Summary and Conclusions Exact a
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Page 125 and 126:
CHAPTER 5 DIFFUSION FROM A SPHERICA
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more concise expression for the int
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across the interface where the para
Page 131 and 132:
This Pint is the ratio of the amoun
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integral is reduced to 2 π times t
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( cosθ 1) Pn 1 m ⎛ m + n⎞⎛ r
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′ i particularly for n = 0, n ′
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where int D D = ε , Γ ( φ ) = I
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T ( i+ 1) ( i) n = T n ( ) ( i) ( i
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where ε is the ratio of external t
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equation (5.14) and u the external
Page 147 and 148:
Figure (5.7) where ( γ = 0. 1 and
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approach [ d a + a ) = 1], small Th
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Figure 5.2: Reaction probability Pi
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Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Figure 5.10: Reaction probability P
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APPENDIX A ANALYTICAL FORMS OF Fnm
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APPENDIX B TWO SPHERES IN BISPHERIC
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with center at the origin, and the
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c ( r ) = c (r far from either sphe
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B.3 Reaction Rate The reaction rate
Page 171 and 172:
where c is the concentration of the
Page 173 and 174:
and dP d cos ∑( ) ≥ − − m 2
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The rate is scaled by the Smoluchow
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Sphere 2 μ =−μ2 y u φ x η= η
Page 179 and 180:
[11] Clesceri, L., A. E. Greenberg,
Page 181 and 182:
[37] Lifshitz, E. M. and L.P. Pitae
Page 183 and 184:
[60] Regalbuto, M. C., Strieder, W.
Page 185 and 186:
[84] Torquato, S., “Diffusion and
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