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

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[97] Weisz, P. B., “Diffusion and Chemical Transformation”, Science, 129, 433-440 (1973). [98] Wood, B. D., and S. Whitaker, “Diffusion and Reactions in Biofilms”, Chemical Engineering Science, 53, 397-425 (1998). [99] Zhang, T. C. and P.L. Bishop, “Evaluation of Tortuosity Factors and Effective Diffusivities in Biofilms”, Water Research, 28, 2279-2287 (1994). [100] Zheng, L. and Yee C. Chiew, “Computer Simulation of Steady State Diffusion-controlled Reaction Rates in Dispersions of Static Sinks: Effect of Sink Sizes”, Journal of Chemical Physics, 93, 2658-2663 (1990). [101] Zhou, H. X., “Enhancement of Protein-Protein Association Rate by Interaction Potential: Accuracy of Prediction Based on Local Boltzmann Factor”, Biophysical Journal, 73, 2441–2445 (1997). [102] Zoia, G., Ph.D. Thesis, “Transport Across Interfacial Resistance and Diffusion-Limited Reactions in Heterogeneous Systems”, University of Notre Dame (1998). [103] Zoia, G. and W. Strieder, “Competitive Diffusion into Two Sinks with a Finite Surface Reaction Coefficient”, Journal of Chemical Physics, 108, 3114 – 3118 (1998). [104] Zoli, M., and L. F. Agnati, “Wiring and Volume Transmission in the Central Nervous System: The Concept of Closed and Open Synapses” Progress in Neurobiology, 49, 363–380 (1996). 163
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DIFFUSION INTERACTIONS FOR A PAIR O
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Nyrée McDonald bispherical coordin
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TABLE OF CONTENTS FIGURES………
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FIGURES 1.1 Two spheres of radius a
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3.3 Dimensionless consumption rate
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3.12 Constant concentration 0 c c c
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5.5 Reaction probability Pint for a
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B.1 The bispherical coordinates (μ
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a Radius of sphere 1 1 a Radius of
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K Matrix elements for the probabili
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η Bispherical coordinate θ1 θ2 S
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CHAPTER 1 HISTORICAL REVIEW OF DIFF
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compounds (Siebel and Charcklis, 19
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the end, Tsao (2001) claimed that t
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1990), permeable (Torquato and Avel
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4 that for site volumes fractions 1
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internal reaction in sphere 1 and c
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directly to the concentration and t
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CHAPTER 2 DIFFUSION REACTION FOR A
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2.2 Mutualism Interaction Between a
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P 2πa Dc 2 π 1 0 = ⎡∂u ⎤ si
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P ( cosΘ1) n ( n+ 1) r1 1 = n d 1
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2.4 Mutualism-like Problem: Analyti
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h 10 ∑ m 1 ∞ = 1+ K = 1− K 0m
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where and ( 2) ( 2) h = Q + h K , n
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They can be calculated exactly for
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Similarly, the coefficient h can be
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coefficients f1n. Since h1n was act
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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
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Figure 3.11: Constant concentration
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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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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
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This Pint is the ratio of the amoun
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integral is reduced to 2 π times t
Page 135 and 136: ( cosθ 1) Pn 1 m ⎛ m + n⎞⎛ r
Page 137 and 138: ′ i particularly for n = 0, n ′
Page 139 and 140: where int D D = ε , Γ ( φ ) = I
Page 141 and 142: T ( i+ 1) ( i) n = T n ( ) ( i) ( i
Page 143 and 144: where ε is the ratio of external t
Page 145 and 146: equation (5.14) and u the external
Page 147 and 148: Figure (5.7) where ( γ = 0. 1 and
Page 149 and 150: approach [ d a + a ) = 1], small Th
Page 151 and 152: Figure 5.2: Reaction probability Pi
Page 159 and 160: Figure 5.10: Reaction probability P
Page 161 and 162: APPENDIX A ANALYTICAL FORMS OF Fnm
Page 163 and 164: APPENDIX B TWO SPHERES IN BISPHERIC
Page 165 and 166: with center at the origin, and the
Page 167 and 168: c ( r ) = c (r far from either sphe
Page 169 and 170: 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
Page 175 and 176: The rate is scaled by the Smoluchow
Page 177 and 178: 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: [84] Torquato, S., “Diffusion and
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