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

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3.7 Dimensionless consumption rate R1 of sphere 1 versus the −1 dimensionless intersphere center-to-center distance d( a1 + a2 ) to sphere 2 for sphere 2 inverse dimensionless reactivity −1 λ 2 ( = D( k2a 2 ) ) = 50 , and a sink-to-sink radius ratio ( 1 / 2 ) a a = γ of 0.10. The different curves refer to selected dimensionless, surface −1 inverse sphere 1 reaction rate coefficients λ1( = D( k1a1 ) ) of 0.02, 0.1, 0.5, 1, 5, 10 and 50 assigned from the uppermost curve in descending order.…………………………………………...………... 81 3.8 Dimensionless consumption rate R1 of sphere 1 versus the −1 dimensionless intersphere center-to-center distance d( a1 + a2 ) to sphere 2 for sphere 2 inverse dimensionless reactivity −1 λ 2 ( = D( k1a1 ) ) = 1, and a sink-to-sink radius ratio ( 1 / 2 ) a a = γ of 1.0. The different curves refer to selected dimensionless, surface −1 inverse sphere 1 reaction rate coefficients λ1( = D( k1a1 ) ) of 0.02, 0.1, 0.5, 1, 5, 10 and 50 assigned from the uppermost curve in descending order.………………………………………...…………... 82 3.9 Dimensionless consumption rate R1 of sphere 1 versus the dimensionless intersphere center-to-center distance ( ) 1 − d a1 + a2 to sphere 2 for sphere 2 inverse dimensionless reactivity −1 λ 2 ( = D( k2a 2 ) ) = 0. 02 , and a sink-to-sink radius ratio ( 1 / 2 ) a a = γ of 10. The different curves refer to selected dimensionless, surface −1 inverse sphere 1 reaction rate coefficients λ1( = D( k1a1 ) ) of 0.02, 0.1, 0.5, 1, 5, 10 and 50 assigned from the uppermost curve in descending order.…………..……………………………………….... 83 3.10 Dimensionless consumption rate Ω1 of sphere 1 generated from the bispherical coordinate system versus the dimensionless intersphere −1 center-to-center distance d( a1 + a2 ) for two infinitely reactive spheres with inverse dimensionless reactivity of sphere 1 −1 −1 λ 1( = D( k1a1 ) ) and sphere 2 λ2 ( = D( k2a 2 ) ) , and a sink-to-sink radius ratio ( 1 / 2 ) a a = γ of 10………………………………………... 84 3.11 Constant concentration 0 c c contour curves for two diffusioncontrolled sinks. Sphere 2 the smaller sink ( 2 1 = a ) is shown at the center. Only the lower portion of the larger sink 1 ( 1 10 ) is shown at the top. The center-to-center separation distance is taken near the distance for the minimum reaction for sphere 1 .……………………………………………… = a −1 d( a1 + a2 ) = 1. 7 R1 85 viii
3.12 Constant concentration 0 c c contour curves for two diffusioncontrolled sinks. Sphere 2 the smaller sink ( 2 1 = a ) is shown at the center. Only the lower portion of he larger sink 1 ( 1 10 = a ) is shown at the top. The center-to-center separation distance −1 d( a1 + a2 ) = 1. 01 is taken near closest approach contact for maximum recovery of R1 from the minimum………...……………... 86 4.1 Dimensionless consumption rate of the diffusion-limited sink R1 −1 versus the dimensionless intersphere distance d( a1 + a2 ) between the sink and source for a sink source radius ratio ( 1 / 2 ) a a = γ of 0.10. The different curves refer to the dimensionless sphere 2 surface strength α ( = c2 / c0 −1) of –1, -0.5, 0, 0.2, 0.4, 0.6, and 0.8 assigned from the lowest curve in ascending order…………………...……….. 100 4.2 Dimensionless consumption rate of the diffusion-limited sink R1 −1 versus the dimensionless intersphere distance d( a1 + a2 ) between the sink and source for a sink source radius ratio ( 1 / 2 ) a a = γ of 1. The different curves refer to the dimensionless sphere 2 surface strength α ( = c2 / c0 −1) of –1, -0.5, 0, 0.2, 0.4, 0.6, and 0.8 assigned from the lowest curve in ascending order……………………………. 101 4.3 Dimensionless consumption rate of the diffusion-limited sink R1 −1 versus the dimensionless intersphere distance d( a1 + a2 ) between the sink and source for a sink source radius ratio ( 1 / 2 ) a a = γ of 10. The different curves refer to the dimensionless sphere 2 surface strength α ( = c2 / c0 −1) of –1, -0.5, 0, 0.2, 0.4, 0.6, and 0.8 assigned from the lowest curve in ascending order…………………………..... 102 5.1 Reaction probability Pint for a first order penetrable sink and a zeroth order source versus the dimensionless center to center distance d /( a1 + a2 ) between the source and sink, for a sink to source radius ratio ( 1 / 2 ) a a = γ of 0.1, extracellular to internal sink diffusivity ratio ( / int ) D D = ε of 0.1 and sink Thiele Modulus ) / ( T a1 kint Dint = φ of 0.1, 0.5, 1, 5, 10 and 50. The lower solid lines are the exact reaction probability, equation (5.41), for an internally reactive sink, and the dashed lines refers to the approximate, sink surface reaction probability Psur , equation (5.45), with an effective inverse −1 dimensionless reaction rate coefficient λ1eff ( = ε /( φT coth φT −1) ) of 30, 1.22, 0.320, 0.025, 0.011, and 0.002……………………………... 128 ix
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: 3.3 Dimensionless consumption rate
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 and 58: G ( 1) n ( 0) ( 0) G0 K = Gn + , (2
Page 59 and 60: 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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Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
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
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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 157 and 158:
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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.
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[84] Torquato, S., “Diffusion and
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