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

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with the substitution of (2.20) into (2.18) two simple expressions are obtained, and ∑ ∞ 1 h 1n = + K nmh , 1m (2.21) m= 0 K nm = Λ γ 1m d F nm 2+ m 1 . (2.22) Please note that all h1n and Knm are convergent because the initial series for f1n and F were absolutely convergent. Applying equations (2.15) and (2.20) to equation (2.17), the probability becomes h10 P = . (2.23) d 1+ λ ) 1( 1 Equation (2.21) and (2.23) require that all h1n be known. Though direct matrix inversion can be used to evaluate equations (2.21) it is not recommended because the inversion techniques used in many computer algorithms can produce significant error for a large non-random matrix. An elimination technique is employed to avoid this error. Beginning with equation (2.21) for n=0 24
h 10 ∑ m 1 ∞ = 1+ K = 1− K 0m 1m 00 h , (2.24) this expression is used to eliminate h10 from the remaining equations of (2.21) such that for n ≥ 1 ∑ ∞ K ⎛ ⎞ n0 K n0 K 0m h = + + ⎜ + ⎟ 1n 1 h1m K nm , n ≥ 1. (2.25) 1− K 00 m= 1 ⎝ 1− K 00 ⎠ Simplifying equation (2.25) becomes where ∑ ∞ m= 1 ( 1) ( 1) h = Q + h K , n ≥ 1. (2.26) 1n Q K n ( 1) n ( 1) nm 1m nm ( 0) ( 0) Q0 K = Qn + , (2.27) 1− K ( 0) n0 ( 0) 00 ( 0) Q = 1, (2.28) n ( 0) ( 0) Kn0 K = Knm + , (2.29) 1− K 25 ( 0) 0m ( 0) 00
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: 2.4 Mutualism-like Problem: Analyti
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 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:
Page 101 and 102:
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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