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

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G ( 3) n ( 2) ( 2) G2 K = Gn + , (2.67) 1− K ( 2) ( 2) with Q defined in equation (2.35) and K by equations (2.22) and (2.36). 2 ( 2) 2n ( 2) 22 Then by induction for all n the concentration is described by ∑ ∞ n= ( n) ( n) 2n Gn Qn r 1 u = ( n) , < d1 . (2.68) 0 1− K a ( ) n n Higher order terms of G are developed similar to Q (2.42) as n G ( i+ 1) ( i) n = G n nn + ( i) ( i ) Gi K in () i ( 1− K ) ii , 1 ( ) n i ≥ 0 (2.69) ( 0) (i) (i) with the initial value, G , defined by (2.58). The quantities K and Q for n equations (2.68) are determined from equations (2.22) and (2.40) – (2.43). 2.5 Results and Discussion Spherical sink and spherical source mutualism-like interaction model is solved with the bispherical expansion, for first order consumption at the sink surface and a reactant generated at the source surface, with Fickian diffusion in the inert bulk phase. Using the method of elimination outlined above the reaction 36 nm n
probability P (2.55), and the concentration u profile (2.68) in the neighborhood of sphere 1 is completely specified. Both of the equations include a single infinite sum, which converges for all values of the dimensionless parameters γ, λ1 and d1. The infinite sum is calculated as a set of nested continued fractions, equations (2.40) – (2.43), generated from a system of linear equations. Then these types of equations are easily solved as nested loop algorithm on a desktop computer. The reaction probability P, equation (2.55) contains a source term, sink term and a source-sink interaction series. The first factor comes from the monopole reactant concentration arising from the zeroth order reactant generation at the surface of the source. The monopole source concentration at any point from the center of the source is expressed as c m 2 σ 2a 2 = . (2.70) dD If the sink is immersed in the same medium with the uniform concentration c at a center to center distance d from the source then the single sphere solution for a finite surface reaction rate is the well-known Smoluchowski diffusion-controlled reaction rate ( 4π Dc ) (Cukier, 1986 and Weiss, 1986) divided by the correction factor ( 1 a 1 m 1 + λ ) for the finite surface reaction rate (Reid, 1952). Then for larger sink-source separation distances i.e., the monopole sink reaction rate is 37 m d
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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………
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: G ( 1) n ( 0) ( 0) G0 K = Gn + , (2
Page 61 and 62: ate constant λ 50 . The figures 2.
Page 63 and 64: Figure 2.2 - 2.4 shows the monotoni
Page 65 and 66: Θ1 d Figure 2.1: Two spheres of ra
Page 67 and 68: Figure 2.3: Reaction probability P
Page 69 and 70: Figure 2.5: Constant concentration
Page 71 and 72: CHAPTER 3 COMPETITIVE INTERACTION B
Page 73 and 74: a2, respectively. Any point externa
Page 75 and 76: The overall reaction rate at sphere
Page 77 and 78: where Λ kn ⎛ ak = ⎜ ⎝ d n
Page 79 and 80: ∑( ) ∞ − j ⎛ j + n⎞⎛ j
Page 81 and 82: and ( ) 1 − 1− L ( i+ 1) () i (
Page 83 and 84: i→∞ ( n+ i) ( ∞) v1 n = lim M
Page 85 and 86: 1953). The reaction rate for this c
Page 87 and 88: a1 sinh μ1 h μ = , (3.47) cosh μ
Page 89 and 90: ispherical coordinate system holds
Page 91 and 92: eaction rate remains virtually unch
Page 93 and 94: on the x = 0 line with the larger s
Page 95 and 96: contact (d=a1+a2) to far apart (d>>
Page 97 and 98: Figure 3.1: Dimensionless consumpti
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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Figure 4.2: Dimensionless consumpti
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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
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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
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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
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where c is the concentration of the
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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
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[11] Clesceri, L., A. E. Greenberg,
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[37] Lifshitz, E. M. and L.P. Pitae
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[60] Regalbuto, M. C., Strieder, W.
Page 185 and 186:
[84] Torquato, S., “Diffusion and
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