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

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1 ∫ − 1 2δ nm Pn ( z) Pm ( z) dz = , (B.32) 1+ 2n Integrating the first term is done by considering two separate cases, • when j = 2 n + 1, n = 0, 1, 2,... , is odd and m = 2k , k = 0, 1, 2,... , is even (the m sum begins at j + 1, or. 2 n + 2 and the i sum goes from P to P1 ), • and when j = 2n , n = 0, 1, 2,... , is even and m = 2 k + 1, k = 0, 1, 2,... , is odd (the m sum begins at 2 n + 1, and the i sum goes from up to P ). Pm −1 0 Finally the last term is directly integrated using the orthogonality condition of Legendre polynomials. After some rearranging the resulting reaction rate of the diffusion-limited sphere 1 ( c 0 ) in the presence of sphere 2 is 1 = ⎛ ⎜ ⎜ = ⎜ + ( )× ⎜ ∑ ⎜ ⎜ ⎝ ∞ 1 R 1 1 sinh μ1 W j μ1 2 j= 0 ⎧ ⎪ ⎨ ⎪ ⎪ ⎩ ⎡ ( μ1) cosh⎢ 1 ⎣ ⎡ sinh⎢ ⎣ ( μ + μ ) ( μ + μ ) 1 ( c − c ) ⎛ 1 ⎞⎤ 2 ⎜ j + ⎟⎥ + ⎝ 2 ⎠⎦ c ⎛ 1 ⎞⎤ 2 ⎜ j + ⎟⎥ ⎝ 2 ⎠⎦ W j 2 0 j 0 152 ⎫ W ( μ 2 ) ⎪ ⎬ ⎪ ⎪ ⎭ ⎞ ⎟ ⎟ ⎟ , ⎟ ⎟ ⎟ ⎠ m−1 (B.33)
The rate is scaled by the Smoluchowski diffusion-limited single sphere solution 4π a Dc ). ( 1 0 153
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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
Page 123 and 124: Figure 4.2: Dimensionless consumpti
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
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 dP d cos ∑( ) ≥ − − m 2
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 and 186: [84] Torquato, S., “Diffusion and
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