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

DIFFUSION INTERACTIONS FOR A PAIR OF REACTIVE SPHERES A Dissertation Submitted to the Graduate School **of** the University **of** Notre Dame in Partial Fulfillment **of** the Requirements **for** the Degree **of** Doctor **of** Philosophy by Nyrée V. McDonald, B.S., M.S. William Strieder, Director Graduate Program in Chemical and Biomolecular Engineering Notre Dame, Indiana November 2005

- Page 2 and 3: DIFFUSION INTERACTIONS FOR A PAIR O
- Page 4 and 5: DEDICATION This is for my daughter,
- Page 6 and 7: 3.2 Competitive Interaction: Two Si
- Page 8 and 9: 2.5 Constant concentration contour
- Page 10 and 11: 3.7 Dimensionless consumption rate
- Page 12 and 13: 5.2 Reaction probability Pint for a
- Page 14 and 15: 5.8 Reaction probability Pint for a
- Page 16 and 17: TABLES 5.1 The results show the per
- Page 18 and 19: f Equation coefficient of sphere 2
- Page 20 and 21: s Modified Bessel function of the f
- Page 22 and 23: ACKNOWLEDGMENTS Special thanks to t
- Page 24 and 25: Marshall, 1990; Converti et al., 19
- Page 26 and 27: problems focused on diffusion to tw
- Page 28 and 29: the size and reactivity of the inte
- Page 30 and 31: In the past most efforts to underst
- Page 32 and 33: Hassan, 1985; Teixeira et al., 1994
- Page 34 and 35: from the set of linear equations. T
- Page 36 and 37: d a1 a2 Figure 1.1: Two spheres of
- Page 38 and 39: cells have to be before they intera
- Page 40 and 41: The intermediate product is consume
- Page 42 and 43: and the dimensionless center-to-cen
- Page 44 and 45: condition of the Legendre polynomia
- Page 46 and 47: with the substitution of (2.20) int
- Page 48 and 49: and ( 0) K = K nm nm . (2.30) Then
- Page 50 and 51: () () ∑ ∞ h1 n i i = Qn + K nmh
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It is important to recognize that e

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So, by eliminating all for m > 0 an

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∑ ∞ = 1n n n= 0 G h u , 1 d1 a1

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G ( 3) n ( 2) ( 2) G2 K = Gn + , (2

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4πa Dc = 1 m R m . (2.71) ( 1+ λ1

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λ 1 the slope of the curve is alwa

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intermediate product (growth factor

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Figure 2.2: Reaction probability P

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Figure 2.4: Reaction probability P

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Figure 2.6: Constant concentration

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Finally, the bispherical coordinate

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Note, boundary conditions (3.2), (3

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where Pn is the Legendre polynomial

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1 ∫ − 1 is P ( z) P ( z) n m 2

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where and Λ are given by equation

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If equation (3.25) is written for n

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3.4.2 Diffusion Limited Concentrati

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where = − − ∑ ∞ κ cosh μ

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3.5 Results And Discussion Competit

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of the size ratio / ] a = γ = 0.1,

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eactivity 1 λ and radius a there a

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less and less of the larger sink 1

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arising from possible size and reac

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Figure 3.2: Dimensionless consumpti

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Figure 3.4: Dimensionless consumpti

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Figure 3.6: Dimensionless consumpti

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Figure 3.8: Dimensionless consumpti

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Figure 3.10: Dimensionless consumpt

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Figure 3.12: Constant concentration

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In this chapter the exact steady st

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subtracting the last scenario from

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-∞

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such that the concentration in the

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2 sinh μ1 d(cosθ ) = d(cosη) . (

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μ → ∞ and μ → ∞ and all c

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Figure 4.1: Dimensionless consumpti

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Figure 4.3: Dimensionless consumpti

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species. Therefore, commensalism is

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2 D c = 0 (5.1) ∇ ext where D is

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the dimensionless center-to-center

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of the first kind of half-integer o

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5.4 External Concentration The dime

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The final set of coefficients b , a

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5.5 Internal Reaction Probability:

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P β 0 int = − g10 . (5.35) d1 No

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is accomplished by setting the effe

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and ( ) 0 Q = 1. (2.43) n They are

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epresents P and a solid line repres

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difference is − 4. 48% and forε

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Figure 5.1: Reaction probability Pi

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Figure 5.3: Reaction probability Pi

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Figure 5.5: Reaction probability Pi

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Figure 5.7: Reaction probability Pi

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Figure 5.9: Reaction probability Pi

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TABLE 5.1 PERCENT ERROR FOR THE EXA

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x d ⎧1 d = ⎨ m! dx ⎩ x dx [

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2 ⎡ ⎤ 2 1 ∂ ⎛ ∂ψ ⎞ 1

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⎡ 1 ⎤ m cos cosh μ − cosη e

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c0 − ci 0 = c 0 2 [ − ( n + 1 2

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and Also, sinη sinh μ −1 1− c

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∂c 1 = − sinh μ1 ∂μ 2 −

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1 ∫ − 1 2δ nm Pn ( z) Pm ( z)

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−μ μ=−μ2 μ=-∞ z=-f f sinh

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REFERENCES [1] Abramowitz, M. and S

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[23] Fradkov, V. E., Glickman, M. E

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[49] Morse, P.M. and H. Feshbach, M

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[73] Smoluchowski, M. “Zusammenfa

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[97] Weisz, P. B., “Diffusion and