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

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In the past most efforts to understand the N sink problem focused on the diffusion-limited case. This is an extreme case of the diffusion reaction scheme where the reaction rate is instantaneous and the diffusion rate is finite. Little has been done for the nondiffusion-limited N sink problem even for the simplest case of equisized spheres. Torquato and Avellaneda (1991) studied the permeable N sink problem with first order reaction for both the time dependent and steady state cases. Lu et al., (2002) calculated the steady state rate constant for a cluster of equisized sinks with first order reaction occurring at the surface of the sinks. The author determined that when the reactant concentration is very high the effective reaction rate is less dependent on the volume fraction of sinks in the system. Monte Carlo Simulations were carried out for clusters of monodispersed spheres with internal reactions from diffusion to reaction-limited conditions Riley (1995) determined for the reaction-limited case the overall rate constant is independent of the cellular arrangement (non-overlapping, overlapping and clusters). Weiss (1986), Richards (1986) and Torqauto (1991) studied polydispersion for the finite sink case. In all of these papers the work has focused on competitive interaction. Even though in real systems such as remediation there are other relevant types of particle interactions taking place. Using this approach we have learned that the Smoluchowski (1916) solution cannot be accurately applied to dense N particle systems, the sink-sink interaction is relevant for volume fractions >1%. In these systems the particle interactions become more complex and must be considered (Fradkov et al., 1996 and Mandyam et al., 1998). Recent work by Zoia and Strieder (1998) has shown 8
4 that for site volumes fractions 10 when the two-sphere problem is explicitly − < included in the N-sink problem, it could improve the bounds on the effective reaction rate. 1.3 Cell-Cell Interaction Bailey and Ollis (1986) outline a number of different types of cellular interaction with competition and mutualism described as the two extremes. On the one hand, competition implies that the presence of the second cells results in a negative effect on the growth rate, i.e. the consumption rate, of both cell while mutualism implies that the two cells are mutually beneficial and hence this improves the growth rates of each species. From published experimental results (Cooney and McDonald, 1995; Manz, 1999; Raskin et al., 1996; Rauch, 1999; Stewart et al., 1995 and 1997) it is clear that many if not all of these types of interaction are occurring within a given system. For instance in a biofilm, a highly organized multiphasic system composed of cells, water and metabolic products (Costerton and Stewart, 2001 and De Beer et al., 1994 and 1997), microbial growth is encouraged by the introduction of a substrate (Cao and Alaerts, 1995), this may stimulate competition between like species for say oxygen (Raskin et al., 1996) or mutualism as a consortium of microbes co- metabolizes hazardous waste (Wanner and Gujer, 1986). There have been many attempts to solve the biological N-particle system with numerical simulations while neglecting the importance of cellular interaction (Rodriguez, 1983; Beg and 9
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: 1990), permeable (Torquato and Avel
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
Page 79 and 80: ∑( ) ∞ − j ⎛ j + n⎞⎛ j
Page 81 and 82:
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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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
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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:
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
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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,
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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