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

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intermediate product (growth factor, toxin or nutrient) with zeroth order kinetic rate constant. The intermediate diffuses through a fully permeable medium; such as a biofilm or biological tissue, it may become trapped on the surface of the sink and be consumed with first-order kinetics (nutrient uptake or receptor binding). The bispherical expansion is coupled with a matrix elimination technique to produce a completely convergent series for the dimensionless reaction rate of the sink and the same technique generated the concentration around the sink. Both are completely convergent series with a single infinite sum. The dimensionless reaction rates of the sink is graphically displayed as a function of the sink-source center to center separation distance for a wide range of sink-source size ratio, and sink reactivities from diffusion to reaction-controlled. Concentration contour plots are provided for the infinitely reactive sink in the presence of the source. The ratio of sink reaction rate to source generation rate displayed in the probability plots are harmonic for dimensionless center to center separation distances larger than 2. The probability approaches the diffusion-controlled limit ( λ = D / k a ) → 0 ) and goes to zero when the sink has no strength ( λ → ∞ ). 1( 1 1 When the sink-source size ratio ( γ → ∞ ), dimensionless inverse reaction r ate ( λ → ∞ ) and the two spheres are in contact [ d a + a ) = 1] the reaction 1 ( 1 2 probability goes to unity. Additionally when the source is an order of magnitude larger than the sink and the sink is relatively reactive ( λ 1) the maximum reactivity does not always occur at closest approach. The concentration profiles illustrate the blocking effect that occurs when the two spheres are too close for the sink to consume much of the product created at the source. 42 1 < 1
Θ1 d Figure 2.1: Two spheres of radius a1 and a2, spherical coordinates ( 1 1 ,θ r and 2 2 ,θ r ) separated by center-to-center distance d. 43 r1 a1 a2 r2 Θ2
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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
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
Page 61 and 62: ate constant λ 50 . The figures 2.
Page 63: Figure 2.2 - 2.4 shows the monotoni
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 and 112: for all points external to the sink
Page 113 and 114: 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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Figure 4.2: Dimensionless consumpti
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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 155 and 156:
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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