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

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subtracting the last scenario from the results in Chapter 3. Focusing on the third inequality ( c < c < c ) the dimensionless source strength is defined as 1 0 2 c2 α = −1, (4.5) c 0 where α is the nonnegative value. The surroundings contain no reactant for infinite source strength ( α → ∞ ), when the source has no strength ( α = −1), and α = 0 for equivalent source and surroundings concentrations. The overall dimensionless reaction rate at sphere 1, R , will depend on the dimensionless source strength, the dimensionless distance between the two spheres d 1 and the sink source radius ratio γ d = d / a , (4.6) 1 1 1 / a a = γ . (4.7) 2 The reactivity at sphere 1, R , the possibility that a reactant molecule from the 1 surface of the source or medium will diffuse to sphere 1 become trapped on the surface of the sphere 1 and be consumed, is expressed in spherical coordinates (r,θ,φ) as the integral of the normal derivative of c over the surface of sphere 1, 90 1
R 1 π π ⎡ ⎤ , ⎢⎣ ∂r1 ⎥⎦ 1 a1 0 −1 2 ( 4 a1Dc0 ) ( 2πa1 D) ∂c sin Θ1dΘ1 = ∫ r = . (4.8) The first term in equation (4.8) is the single sphere solution for an infinitely reactive sphere 1 fed only by the background concentration (no sphere 2 interaction). The ratio R is always positive; since both the source and surroundings feed the sink it may exceed unity 1 R 0 . (4.9) 4.3 Bispherical Coordinate System: Analytical Solution 4.3.1 Concentration Field 1 > In order to evaluate the reaction rate at the surface of sphere 1, R1, the Laplace equation (1) must be solved for the concentration. There are two well known solution forms for this two sphere problem, the bispherical expansion (Ross, 1968 and 1970) and the bispherical coordinate system (Morse and Feshbach, 1953). Earlier work by Tsao (2001 and 2002), McDonald and Strieder (2003 and 2004) , and others used the method of twin spherical expansion; however Voorhees and Schaefer (1987) developed a solution using Bispherical coordinates (μ,η,φ) that merits further study. The coordinates are bounded by; 91
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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
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
Page 111: for all points external to the sink
Page 115 and 116: and −1⎛ f ⎞ μ = ⎜ ⎟ 1 si
Page 117 and 118: and the small number of infinite su
Page 119 and 120: 4.4 Results and Discussion Bispheri
Page 121 and 122: 4.5 Summary and Conclusions Exact a
Page 125 and 126: CHAPTER 5 DIFFUSION FROM A SPHERICA
Page 127 and 128: more concise expression for the int
Page 129 and 130: across the interface where the para
Page 131 and 132: This Pint is the ratio of the amoun
Page 133 and 134: 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
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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.
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[84] Torquato, S., “Diffusion and
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