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

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problems focused on diffusion to two competing particles where the concentration infinitely far from the two spheres is fixed. Samson and Deutch (1977), Zoia and Strieder, (1998) and Strieder and Saddawi (2000) all use the bispherical coordinate system to resolve the impenetrable sink-sink problem. Samson and Deutch (1977) presented the exact solution for the concentration profiles of two equal sized (monodispersed) spherical sinks with infinitely fast surface reactions (diffusion control). Zoia and Strieder (1998) extended the work of Samson and Deutch (1977) by producing exact, asymptotic solutions of the sink-sink surface reaction problem for two identical sinks with the same first order reaction rate constant at the surface of both spheres. Their work included the development of the reaction rate as an asymptotic expansion in the dimensionless inverse reaction rate for the diffusion to reaction-controlled cases. Strieder and Saddawi (2000) provided an alternate solution for two monodispersed spheres with a first order reaction occurring at the surface of the spheres. These authors used an iterative technique to generate an exact; convergent series expression for the reaction rate of the two spheres. Tsao (2001) used the method of twin spherical expansion (Ross, 1968 and 1970) to solve the competitive diffusion-reaction problem for diffusion-limited to reaction-limited conditions for two penetrable sinks and two impenetrable sinks. His series expansion solution for the reaction rate for two impenetrable spheres of equal size was compared with the series solution for two permeable spheres of equal size. These rates are expressed in terms of the center-to-center distance. In 4
the end, Tsao (2001) claimed that the permeable sink can always be well represented by an impermeable sink model. Keep in mind that it is very unlikely that any two spheres are exactly the same size two key examples are: typical biological cells range is 1 – 100 micrometers (Alberts et al., 1998) and population models of ripening and coarsening where the size of the spheres are always changing (Lifshitz and Pitaevskii, 1981; Voorhees and Schaefer, 1987). Furthermore any insight arising from size difference cannot be illuminated within the context of these four studies. As seen in diverse cultures of microorganisms, variation in crystal growth rates, and coalesences of particles, size variation is very common. Work for two particles of varying sizes is limited (Voorhees and Schaefer, 1987; Tsao, 2002; Stoy 1989a, 1989b and 1991). Voorhees and Schaefer (1987), used bispherical coordinates (Morse and Feshbach 1953) where the two particles have constant concentrations, differ in size and are far apart, to correlate experimental results for coarsening. Here the authors have neglected to consider the impact proximity may have on the surface concentration of the two spheres, furthermore this method tends to an unphysical infinite reaction rate when the two spheres are in contact. Tsao (2002) obtained a solution for the two impermeable sink problem with first order reactions occurring at the surface of both spheres. Using the bispherical expansions (Ross, 1968 and 1970) he generated a series solution for any combination of diffusion-limited to reaction-limited conditions for the two sinks. Tsao (2002) determined that the interaction is significantly influenced by 5
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: compounds (Siebel and Charcklis, 19
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 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
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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
Page 109 and 110:
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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Page 125 and 126:
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
Page 141 and 142:
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:
Page 159 and 160:
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
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
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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
Page 183 and 184:
[60] Regalbuto, M. C., Strieder, W.
Page 185 and 186:
[84] Torquato, S., “Diffusion and
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