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

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() () ∑ ∞ h1 n i i = Qn + K nmh1m m= i . i of particular use in future derivations is the case n = i h 1 i Qi = ∑ m i ∞ = + 1 () i () i + 1− K K () i ii im h 1m , n ≥ , (2.38) n = i . (2.39) The forms of the coefficients of equations (2.38) and (2.39) are generalizations of (i) (i) the Q , K form (2.27) – (2.30), (2.35) and (2.36) for any integer i and n nm K Q ( i+ 1) ( i ) nm = K ( 0) nm nm K K = ( i+ 1) ( i) n = Q n ( ) ( i ) ( i) K im K + 1− K nm ni () i ii , (2.40) , (2.41) ( i) ( i) Qi K + 1− K ni () i ii , (2.42) 0 Q = 1. (2.43) n 28
They can be calculated exactly for any of the nonnegative integer indices i, m, and n. Equations (2.40) – (2.43) are similar to continued fractions, and are well suited to nested computer calculations. In order to calculate the reaction probability P of equation (2.23) it is necessary by using equations (2.38) and (2.39) to generate an expression for h . Using equation (2.38) for n = i = 0 ∑ ∞ m= 0 ( 0) ( 0) = + K Q h . (2.44) 10 0 0m 1m h and applying equation (2.39) for i = 0 to eliminate h from the right hand side of equation (2.44) ∞ ⎡ ( 0) ( 0) ⎤ ⎢Q0 + ∑ K 0m h1m ⎥ ∞ ( 0) ( 0) m= 1 ( 0) h = + ⎢ ⎥ 10 Q0 K 00 + ( 0) ∑ K 0m h1m . (2.45) ⎢ 1− K 00 ⎥ m= 1 ⎢ ⎣ ⎥ ⎦ ( 1) Rearranging and collecting the coefficients of with Q from equation (2.27) h1m ( 1) for n = 0 and K from equation (2.29) for n = 0 0m ∑ ∞ m= 1 () 1 () 1 10 = + K Q h . (2.46) 10 0 29 0m 1m h 0 10
Page 1 and 2: DIFFUSION INTERACTIONS FOR A PAIR O
Page 3 and 4: Nyrée McDonald bispherical coordin
Page 5 and 6: 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: where and ( 2) ( 2) h = Q + h K , n
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
Page 59 and 60: 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 99 and 100: Figure 3.3: Dimensionless consumpti
Page 101 and 102:
Figure 3.5: Dimensionless consumpti
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
Page 111 and 112:
for all points external to the sink
Page 113 and 114:
R 1 π π ⎡ ⎤ , ⎢⎣ ∂r1
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 123 and 124:
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 153 and 154:
Page 155 and 156:
Page 157 and 158:
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 and 186:
[84] Torquato, S., “Diffusion and
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