國 立 交 通 大 學
應用化學系碩士班
碩士論文
(1) 延遲時間的特徵值表示法
Representations of Time Lag by Eigenvalues
(2) 以擴散過程延遲時間及平均首度經過時間探討成核誘導
期及膠體穩定性
Study of Induction Time in Nucleation and Stability in
Colloids via Time Lag and Mean First Passage Time in
Diffusion
研 究 生:葉于榮
指導教授:陳振興 博士
(1) 延遲時間的特徵值表示法
Representations of Time Lag by Eigenvalues
(2) 以擴散過程延遲時間及平均首度經過時間探討成核誘導期及膠
體穩定性
Study of Induction Time in Nucleation and Stability in Colloids via
Time Lag and Mean First Passage Time in Diffusion
研究生:葉于榮 Student:Yu-Jung Yeh
指導教授:陳振興 博士 Advisor:Dr. Jenn-Shing Chen
國立交通大學
應用化學系碩士班
碩士論文
A Thesis
Submitted to M. S. Program,
Department of Applied Chemistry
National Chiao Tung University
in Partial Fulfillment of the Requirements
for the Degree of Master
in
Applied Chemistry
July 2011
(1)
(2)
(1) Representations of Time Lag by Eigenvalues
(2) Study of Induction Time in Nucleation and Stability in Colloids
via Time Lag and Mean First Passage Time in Diffusion
Student Yu-Rung Yeh Advisor Dr. Jenn-Shing Chen
M. S. Program, Department of Applied Chemistry
National Chiao Tung University
Abstract
In this thesis, the eigenvalue and Green’s function representations for
the time lag of first and second moments were formulated. The Green’s
function mentioned above is the one subject to the boundary conditions
on both ends being absorbing. The homogeneous nucleation and the
coagulation of colloids were discussed with the help of diffusion. Time
lag and mean first passage time were employed to interpret the induction
time in nucleation and the stability of colloids.
The time lag of the first and of second moments will decrease as a
result of the properties of Sturm-Liouville operator. We have derived the
kinetic equations of homogeneous nucleation in the discrete number of
particle coordinate, followed by solving in the Laplace domain. In this
way, time lag, mean first passage time, and their corresponding second
moments can be obtained. The formulas were tested in the problem of
condensing water vapor. The results show that induction time for vapor
condensation decreases with increasing vapor pressure.
The stability of colloids is commonly expressed by stability ratio. We
attempted to interpret the stability with the viewpoint of diffusion via the
parameters, relative time lag and relative mean first passage time. It is
indicated that relative mean first passage time matches with stability ratio
quite well. The relation between the barrier height and the stability ratio
is also discussed by applying the method of steepest descent, to obtain an
approximate formula. Furthermore, a linear equation was proposed to
calculate the critical coagulation concentration from known parameters.
...I ...III ...IV ...V ...VIII ...XI 1.1 ...1 1.2 – ...3 1.3 ...5 1.4 ...9 1.5 ρ(x, t) Jout(t) ...11 1.6 ...13 1.7 ...16 1.8 ...19 1.9 ...28 ...37 A1.1 x t t t t ( , )d 0 2 2
ρ
∫
∞ ∂ ∂ t t x t t ( , )d 0 3 3 2ρ
∫
∞ ∂ ∂ ...40 A1.2 f y y x x K x D h x h d d ) ( ) ( ) ( 1 0∫
∫
...42A1.4 K x K y G x y G y x x y y x h h d d ) , , ( ) , ( ) ( ) ( 2 0 0
∫ ∫
< ...49 2.1. ...54 2.2. ...56 2.3. ...60 2.4. ...65 2.5. ...69 2.6. ...71 2.7. ...74 2.8. ...81 ...88 A 2.1 (2-14) ...90 A 2.2 (2-41) ...92 A2.3 ...94 3.1. ...100 3.2. ...102 3.3. ...105 3.4. ...108 3.5. ...112 3.6. ...1203.8. ...142 3.9. ...155 ...159 A3.1 (3-41) T22(0)=1 ...162 A3.2 ...163 A3.3 ...167 ...175 ...183
1-1 ...1 1-2 h2 / 6D ± h2 / √90D ...36 2-1 G n n* ...57 2-2 n=1 n=5 ...58 2-3 n=1 n=m ...63 2-4 n = 1 n=4 ...69 2-5 n = 1 n = m ...73 2-6 n* n = m ...81 2-7 P/Psat ...87 3-1 ...109 3-2 ...112 3-3 1-1 ...122 3-4 1-1 h = r/a r a ...123 3-5 2-2 ...124 3-6 2-2 h = r/a r a ...125 3-7 3-3 ...126
3-9 Hamaker A = 0.7×10-20 J 1-1 ...128 3-10 Hamaker A = 1.2×10-20 J 1-1 ...129 3-11 Hamaker A = 1.7×10-20 J 1-1 ...130 3-12 Hamaker A = 2.2×10-20 J 1-1 ...131 3-13 A = 2×10-20 J ...134 3-14 1-1 Hamaker ...135 3-15 logW C ...136 3-16 ...140
3-17 logWapproximation kT h V( 0) (3-46) ion approximat W (3-44) ...141 3-18 c.c.c. log(C/molL-1) = -2.219 h = r/a r a ...142 3-19 (3-80) (3-81) ...146 3-20 (3-84) (3-85) ...148 3-21 3-13 ...154
2-1 P/Psat = 5 ...86 3-1 3-3 3-8 ...121 3-2 Hamaker ...121 3-3 (3-85) ...151 3-4 ...152
1.1
(time lag)
(absorbing)
Q(t)
1-1
1-1 1 ( n = 200)1-1
J
ss(steady state)
Q(t)
J(t)
1-1
J
sst
L 2 ss L) ( lim ) ( limQ t t t J t t→∞ = →∞ −(1-1)
(1-1)
Daynes
2Daynes
(permeability)
(time lag)
(diffusion coefficient)
(absorption coefficient)
Barrer
3,4Goodknight
5(porous
media)
(porosity)
(tortuosity)
Frisch
6-7(solubility)
Leypoldt
Gough
10Siegel
11-12Chen
13Siegel
11(Dirac delta function)
14
(breakthrough curve)
15 16(higher mome-
nts)
ssρ
(Green’s function)
1.2
–
Q(t)
(1-1)
(1-1)
Daynes
2(1-1)
Q(t)
J
out(t)
∫
′ ′ = t t t J t Q 0 out( )d ) ((1-2)
Q(t)
) (t Q∫
′ ′ ′ ∂ ∂ ′ − ′ ′ = t t t t J t t t J t 0 out 0 out( ) ( )d(1-3)
(1-1)
t
L(1-4)
(1-3)
Q(t)
) ) ( ( lim ss L J t Q t t t − = ∞ →(1-4)
) d ) ( ) ( ( lim ss 0 out out J t t J t t t tJ t t t∫
′ ′ ′ ∂ ∂ ′ − − = ∞ → ss 0 out ss out d ) ( ) ( lim J t t J t t J t J t t t∫
∞ ∞ → ′ ∂ ∂ + − = ss 0 out( )d J t t J t t∫
∞ ∂ ∂ =(1-5)
(1-5)
(1-1)
(1-5)
∂function)
t = 0
t → ∞
(first moment)
(1-5)
(2) L t ss 0 out 2 (2) L d ) ( J t t J t t t∫
∞ ∂ ∂ =(1-6)
(variance)
17( )
2 L (2) L out ) ( ) ( t t J t J t Variance ss − = ∂ ∂(1-7)
σ
Variance =σ
(1-8)
(second moment)
1.3
1.4
(advection)
D(x) K(x)
ρ 8 ) , ( ) ( ) , ( ) ( ) ( x t t x K t x x x K x D xρ
ρ
∂ ∂ = ∂ ∂ ∂ ∂(1-9)
x
t
x
0 ) ( ) 0 , (x =δ
x−x0ρ
δ
(x−x0)(Dirac delta function)
(1-9)
(Laplace transform)
) ( ) , , ( ˆ ) ( ) , , ( ˆ ) ( ) ( 0 s x x0 s x x0 x K s x x x x K x D x ∂ = − − ∂ ∂ ∂ ρ ρ δ(1-10)
^
ρˆ(x,x0,s)=L{ρ(x,x0,t)}L
(activity)
) ( ) , ( ) , ( x K t x t x a = ρ(1-11)
) ( ) , ( ˆ ) , ( ˆ x K s x s x a = ρ(1-12)
(1-10)
ρˆ(x,x0,s) aˆ(x,x0,s) ) ( ) , , ( ˆ ) ( ) , , ( ˆ ) ( ) ( a x x0 s K x sa x x0 s x x0 x x K x D x ∂ = − − ∂ ∂ ∂ δ(1-13)
∞ → t 0 ) , , ( lim ) , , ( ˆ lim 0 0 0 = →∞ = → sa x x s t a x x t s
(1-13)
s → 0
) ( ) 0 , , ( ˆ ) ( ) ( a x x0 x x0 x x K x D x ∂ =− − ∂ ∂ ∂ δ(1-14)
(Green’s function)
18 aˆ(x,x0,0) x x K x D x ∂ ∂ ∂ ∂ − ( ) ( )G(x,y)
(1-14)
) ( ) , ( ) ( ) ( G x x0 x x0 x x K x D x ∂ =− − ∂ ∂ ∂ δ(1-15)
0 ) , ( ) , 0 ( x0 =G h x0 = G(1-16)
(1-15)
δ
(
x
−
x
0)
(unit step
function)
( ) ( 0) 0 0 dx u x x x x x − = −∫
δ
(1-15)
0
x
) ( ) , ( ) 0 ( ) 0 ( ) , ( ) ( ) ( 0 0 0 0 G x x u x x x K D x x G x x K x D x − − = ∂ ∂ − ∂ ∂ =(1-17)
0 0 0 1 0 ) ( x x x x x x u > < = −(1-18)
) ( ) , ( ) 0 ( ) 0 ( 0 0 0 A x x x G x K D x = ∂ ∂ = ) ( ) ( ) ( ) ( ) ( ) ( ) , ( 0 0 0 x K x D x x u x K x D x A x x G x − − = ∂ ∂
(1-19)
(1-19)
x = 0
x = h
(1-16)
(1-18)
A(x
0)
∫
−∫
− = − h h x x K x D x x u x x K x D x A x G x h G 0 0 0 0 0 0 d ) ( ) ( ) ( d ) ( ) ( ) ( ) , 0 ( ) , (∫
∫
− = h x h x x K x D x x K x D x A 0 d ) ( ) ( 1 d ) ( ) ( 1 ) ( 0 0 0(1-20)
A(x
0)
∫
∫
= h h x x x K x D x x K x D x A 0 0 d ) ( ) ( 1 d ) ( ) ( 1 ) ( 0(1-21)
A
(x
0)
(1-19)
G(x,x0)(1-19)
0
x
G(x,x0)∫
∫
′ ′ ′ − − ′ ′ ′ = x x x x K x D x x u x x K x D x A x x G 0 0 0 0 0 d ) ( ) ( ) ( d ) ( ) ( 1 ) ( ) , ((1-22)
(1-18)
u(x, x
0)
G(x, x
0)
′ ′ ′ − ′ ′ ′ ′ ′ ′ =∫
∫
∫
x x x x x x K x D x x K x D x A x x K x D x A x x G d ) ( ) ( 1 d ) ( ) ( 1 ) ( d ) ( ) ( 1 ) ( ) , ( 0 0 0 0 0 0 0 x x x x > <(1-23)
′ ′ =
∫
∫
∫
∫
∫
∫
− − h x x h h x x h x DK x DK x DK x DK x DK x DK x x G d 1 d 1 d 1 d 1 d 1 d 1 ) , ( 0 0 0 1 0 0 1 0 0 0 0 x x x x > <(1-24)
1.4
8 ) (x ssρ
8 t t x x K t x x x K x D x ∂ ∂ = ∂ ∂ ∂ ∂ ( , ) ) ( ) , ( ) ( ) (ρ
ρ
(1-25)
D
K
x = 0
ρ
(0,t)=ρ
0x = h
ρ(h,t)=0t = 0
ρ(x,0)=0(1-25)
8∫
= h ss x x K x D J 0 0 d ) ( ) ( 1ρ
(1-26)
8∫
∫
= h h x ss dx x K x D dy y K y D x K x 0 0 ) ( ) ( 1 ) ( ) ( 1 ) ( ) (ρ
ρ
(1-27)
(1-24)
G(x, x
0)
x
0x
G(x, x)
∫
∫
∫
′ ′ = h h x x x DK x DK x DK x x G 0 0 d 1 d 1 d 1 ) , ((1-28)
(1-27)
(1-28)
ρ
ss(x)G(x, x)
∫
= x ss dy y K y D x x G x K x 0 0 ) ( ) ( 1 ) , ( ) ( ) (ρ
ρ
(1-29)
(1-29)
G(x, x)
1.5
ρ( tx, ) Jout(t)) , ( tx ρ
(1-15)
) ( ) , ( ) ( ) ( G x x0 x x0 x x K x D x ∂ =− − ∂ ∂ ∂ δ(1-15)
0 ) , ( ) , 0 ( x0 =G h x0 = G(1-16)
) , ( ) ( ) , ( ) ( ) ( a x t t x K t x a x x K x D x ∂ ∂ = ∂ ∂ ∂ ∂(1-30)
) 0 ( ) , 0 ( t 0 K a =ρ
, a(h,t)=0(1-31)
A 1.3
(A 1-31)
a(x, t)
0 0 0 ) , ( ) 0 ( d ) , ( ) , ( ) , ( = ∂ ∂ + ∂ ∂ − =∫
y h y x G y D y t y y y x G t x aρ
ρ
(1-32)
) , ( txρ
0 0 0 ) , ( ) 0 ( ) ( d ) , ( ) , ( ) ( ) , ( = ∂ ∂ + ∂ ∂ − =
∫
y h y x G y D x K y t y y y x G x K t xρ
ρ
ρ
(1-33)
A 1.3
-J
out(t)
(1-9)
x
h
∫
∂ ∂ = ∂ ∂ − ∂ ∂ → h x h x t y t y x K t x x x K x D x K t x x x K x D ( , )d ) ( ) , ( ) ( ) ( ) ( ) , ( ) ( ) ( limρ
ρ
ρ
∫
∂ ∂ = ∂ ∂ − − h x y t t y x K t x x x K x D t J ( , )d ) ( ) , ( ) ( ) ( ) ( outρ
ρ
(1-34)
K t x x x K x D t x J( , ) ( ) ( ) ρ( , ) ∂ ∂ − = Jout(t)x = h
(1-34)
D(x) K(x)
D
K
∫
∂ ∂ + = ∂ ∂ − h x y t t y DK DK t J K t x x d ) , ( 1 1 ) ( ) , ( out ρ ρ(1-35)
x
h
∫
∫
∫
+ ∂ ∂ = + − h x h y h x dy z t t z DK y DK t J x K t x h K h d ) , ( 1 d 1 ) ( ) ( ) , ( ) ( ) ( outρ
ρ
ρ
(1-36)
0 ) (h = ρK(x)
∫
∫
∫
+ ∂ ∂ = h x h y h x dy z t t z DK x K y DK x K t J t x, ) ( ) ( ) 1 d ( ) 1 ( , )d ( outρ
ρ
(1-37)
(1-37)
x = 0
ρ
(0,t)=ρ
0∫
∫
∫
+ ∂ ∂ = h h y h dy z t t z DK K y DK K t J 0 0 out 0 d ) , ( 1 ) 0 ( d 1 ) 0 ( ) (ρ
ρ
(1-38)
) ( out t J∫
∫
∫
∫
∂ ∂ − = h h h y h y DK K dy z t t z DK K y DK K t J 0 0 0 0 out d 1 ) 0 ( d ) , ( 1 ) 0 ( d 1 ) 0 ( ) (ρ
ρ
(1-39)
∫
h y DK K 0 0 d 1 ) 0 (ρ
(1-26)
J
ss 8 ) ( out t J∫
∫
∫
∂ ∂ − = h h h y y DK dy z t t z DK J t J 0 0 ss out d 1 d ) , ( 1 ) (ρ
(1-40)
(1-40)
1.6
(1-44)
(1-47)
(1-5)
J
out(t)
(1-40)
ss 0 out L d ) ( J t t J t t t
∫
∞ ∂ ∂ =(1-5)
∫
∫
∫
∫
∞ ∂ ∂ − ∂ ∂ = 0 0 0 ss ss d d 1 d ) , ( 1 1 t y DK dx y t t y DK J t t J h h h xρ
∫
∫ ∫
∫
∞ ∂ ∂ − = h h x h t t ydx t y t DK y DK J 0 0 2 2 0 ss d d ) , ( 1 d 1 1 1ρ
(1-41)
(A 1-5)
( , )d ss( ) 0 2 2 y t t t y tρ
=−ρ
∂ ∂∫
∞(1-26)
J
ss(1-41)
∫
∫
∫
∫
− = h h x ss h h dx y y DK y DK x DK t 0 0 0 0 L - ( )d 1 d 1 1 d 1ρ
ρ
∫
∫
= h h x x y y DK 0 ss 0 d )d ( 1 1ρ
ρ
(1-42)
A 1.2
(A 1-10)
(
)
∫ ∫
= h x y y x DK t 0 ss 0 0 L d ( )d 1 1ρ
ρ
(1-43)
) ( ss xρ
(1-27)
∫
∫
∫
= h x x x y DK x x G x K y DK t 0 0 0 0 0 L d d 1 ) , ( ) ( d 1 1ρ
ρ
∫
= hK x G x x x 0 d ) , ( ) ((1-44)
19∑
= i i i i x v y w y x Gλ
) ( ) ( ) , ((1-45)
wi
(eigenfunction)
vi
(adjoint
operator)
wi
vi
(orthogonal
function)
wi
vi
20 = ≠ =∫
x w x v x x nn mm h m n 1 0 d ) ( ) ( ) ( 0ψ
(1-46)
) (x ψ(weighting function)
K(x)
L t =∫
∑
h i i i i x x v x w x K 0 d ) ( ) ( ) ( λ∑
= i λi 1(1-47)
(1-44)
(1-47)
1.7
1.6
(1-53)
(1-54)
J
out(t)
(1-40)
J
out(t)
(1-6)
ss 0 out 2 ) 2 ( L d ) ( J t t J t t t∫
∞ ∂ ∂ =(1-5)
∫
∫
∫
∫
∞ ∂ ∂ − ∂ ∂ = 0 0 0 ss 2 ss d d 1 d ) , ( 1 1 t y DK dx y t t y DK J t t J h h h xρ
∫
∫ ∫
∞ ∂ ∂ − = h h x dx y t t t y t DK 0 0 2 2 2 0 d d ) , ( 1 1ρ
ρ
(1-48)
∫
∞ ∂ ∂ 0 2 2 2 d ) , (y t t t tρ
∞ - ∞
t
(A 1-8)
(1-33)
t
0 0 0 ) , ( ) 0 ( ) ( d ) , ( ) , ( ) ( ) , ( = ∂ ∂ + ∂ ∂ − =∫
z h z y G z D y K z t z z z y G y K t yρ
ρ
ρ
(1-33)
(1-33)
(1-48)
(1-33)
t
) 2 ( L t z t t z y x t z y G y K t t DK h h x h d d d d ) , ( ) , ( ) ( 1 1 0 0 0 2 2 2 0
∫
∫ ∫ ∫
∂ ∂ − ∂ ∂ − = ∞ρ
ρ
x y x t t z t t z y G y K DK h h x h d d d d ) , ( ) , ( ) ( 1 1 0 0 0 0 3 3 2 0
∫
∫ ∫
∫
∂ ∂ = ∞ρ
ρ
(1-49)
(1-49)
t
(A 1-8)
∫
∞ = ∂ ∂ 0 ss 3 3 2 ) ( 2 d ) , (z t t z t tρ
ρ
(1-49)
) 2 ( L t K y G y z z z y x DK h h x h d d )d ( ) , ( ) ( 1 2 0 0 ss 0∫
∫ ∫
=ρ
ρ
(1-50)
(1-50)
(A 1-10)
) 2 ( L t z K x G x y y y x DK h x h d )d ( ) , ( ) ( d 1 2 0 0 ss 0 0∫
∫
∫
=ρ
ρ
(1-51)
) ( ss yρ
(1-27)
∫
∫
= h h x ss dx x K x D dy y K y D x K x 0 0 ) ( ) ( 1 ) ( ) ( 1 ) ( ) (ρ
ρ
(1-27)
) ( ss yρ
(1-27)
(1-51)
∫
x z DK 0 d 1 ) 2 ( L t y x dz DK dz DK y K y x G x K z DK h h h h y x d d 1 1 ) ( ) , ( ) ( d 1 2 0 0 0 0 0 0∫
∫
∫
∫
∫
=ρ
ρ
x y z DK z DK z DK y x G y K x K h h h h y x d d d 1 d 1 d 1 ) , ( ) ( ) ( 2 0 0 0 0
∫ ∫
∫
∫
∫
= x y y x x y G y x G y K x K h h d d ) , , ( ) , ( ) ( ) ( 2 0 0∫ ∫
< =(1-52)
(1-52)
K x K y G x y G y x x y y x h h d d ) , , ( ) , ( ) ( ) ( 2 0 0∫ ∫
<(A 1-58)
x y y x x y G y x G y K x K h h d d ) , , ( ) , ( ) ( ) ( 2 0 0∫ ∫
< x y x y G y x G y K x K x y y y G x x G y K x K h h h h d d ) , ( ) , ( ) ( ) ( d d ) , ( ) , ( ) ( ) ( 0 0 0 0∫ ∫
∫ ∫
+ =(A 1-58)
(A 1-58)
A 1.4
(A 1-58)
(1-52)
= ) 2 ( L t K x K y G x x G y y y x h h d d ) , ( ) , ( ) ( ) ( 0 0∫ ∫
x y x y G y x G y K x K h h d d ) , ( ) , ( ) ( ) ( 0 0∫ ∫
+(1-53)
x y y v y w x v x w y K x K h h i i i i i i i i d d ) ( ) ( ) ( ) ( ) ( ) ( 0 0∫ ∫
∑
∑
=λ
λ
x y x v y w y v x w y K x K h h i i i i i i i i d d ) ( ) ( ) ( ) ( ) ( ) ( 0 0∫ ∫
∑
∑
+λ
λ
∑
∑
+ = 1 2 1(1-53)
(1-54)
1.8
(1-9)
(first order reaction)
-) , ( ) , ( ) ( ) ( ) , ( ) ( ) ( x t t t x x R x K t x x x K x D x
ρ
ρ
ρ
∂ ∂ = − ∂ ∂ ∂ ∂(1-55)
R(x)
(rate coefficient)
0 ) 0 , (x = ρ ρ(0,t)=ρ0 ρ(h,t)=0(activity)
) ( ) , ( ) , ( x K t x t x a = ρ(1-56)
(1-55)
) , ( ) ( ) , ( ) ( ) ( ) , ( ) ( ) ( a x t t x K t x a x K x R t x a x x K x D x ∂ ∂ = − ∂ ∂ ∂ ∂(1-57)
(1-57)
(Laplace transform)
) , ( ˆ ) ( ) , ( ˆ ) ( ) ( ) , ( ˆ ) ( ) ( a x s R x K x a x s K x sa x s x x K x D x ∂ − = ∂ ∂ ∂(1-58)
s K s a 0 ) 0 ( 1 ) , 0 ( ˆ = ρ
,
aˆ(h,s)=0(1-59)
s
Lx
) ( ) ( d d ) ( ) ( d d x K x R x x K x D x Lx =− +(1-60)
(1-58)
) , ( ˆ ) ( ) , ( ˆ x s K x sa x s a Lx =−(1-61)
) , ( ˆ x s a s ) , ( ) , ( ˆ x s P x s a s =(1-62)
(1-61)
s
saˆ(x,s)= P(x,s)[
ˆ( , )]
) ( ) , ( ˆ x s K x s sa x s a s Lx =− ) , ( ) ( ) , (x s K x sP x s P Lx =−(1-63)
(1-61)
(1-63)
aˆ(x,s) ) , (x s P P(x,s) ) , ( ˆ x s aP(x,s)
s = 0
L + ′′ + ′ + = ( ,0) 2 ) 0 , ( ) 0 , ( ) , ( 2 x P s x P s x P s x P(1-64)
) 0 , (x P ) ( ) , ( lim ) , ( ˆ lim ) 0 , ( ss 0sa x s a x t a x x P t s = = = ∞ → →
(1-65)
(1-65)
(1-64)
L + ′′ + ′ + = ( ,0) 2 ) 0 , ( ) ( ) , ( 2 ss P x s x P s x a s x P(1-66)
) , (x s P(1-66)
[
( )]
0 lim ) , ( lim 0 0 = → = → sP x s s sass x s(1-67)
∂ ∂ + = ∂ ∂ → → ( , ) lim ( , ) ( , ) lim 0 0 ssP x s s P x s s sP x s s =lims→0P(x,s)=ass(x)(1-68)
[
( , )]
lim 2 ( , ) ( , ) 2 ( ,0) lim 2 2 0 2 2 0 s sP x s s sP x s s s P x s P x s = ′ ∂ ∂ + ∂ ∂ = ∂ ∂ → →(1-69)
(1-66)
(1-69)
(1-63)
LxP(x,s)=−K(x)sP(x,s)1.
(1-63)
s
→0
(1-66)
(1-67)
− = → → ( , ) ( ) lim ( , ) lim 0 0P x s K x sP x s L s s x 0 ) ( ss x = a Lx(1-70)
0 ) ( , ) 0 ( ) 0 ( 0 ss ss = a h = K a ρ
(1-71)
(1-70)
(1-71)
(A 1-40)
0 0 ss( ) (0) ( , ) = ∂ ∂ = y y x G y D x aρ
(1-72)
(1-70)
(1-72)
A 1.3
2.
(1-63)
s
s
→0
(1-66)
(1-68)
∂ ∂ − = ∂ ∂ → → ( , ) ( ) lim ( , ) lim 0 0 s P x s K x ssP x s L s s x ) ( ) ( ) 0 , (x K x ass x P Lx ′ =−(1-73)
0 ) 0 , ( , 0 ) 0 , 0 ( = ′ = ′ P h P(1-74)
(1-72)
ass(x)∫
− = ′ x hG x y K y ass y y P 0 d ) ( ) ( ) , ( ) 0 , ((
)
0 0 ( , ) ( ) ( , ) d ) 0 ( = ∂ ∂ − =∫
hG x y K y G y z y z Dρ
(1-75)
3.
(1-63)
s
s
→0
(1-66)
(1-69)
∂ ∂ − = ∂ ∂ → → ( , ) ( ) lim ( , ) lim 2 2 0 2 2 0 s P x s K x s sP x s L s s x ) 0 , ( ) ( 2 ) 0 , (x K x P x P Lx ′′ =− ′(1-76)
0 ) 0 , ( , 0 ) 0 , 0 ( = ′′ = ′′ P h P(1-77)
(1-76)
(1-75)
P′(x,0)∫
′ − = ′′ x hG x y K y P y y P 0 d ) 0 , ( ) ( ) , ( 2 ) 0 , ((
)(
)
0 0 0 0 ( , ) ( ) ( , ) ( ) ( , ) d d ) 0 ( 2 = ∂ ∂ =∫ ∫
z h h y z z G z K z y G y K y x G Dα
α
ρ
(1-78)
) ( ss x a P′(x,0) P′′(x,0) ) ( out t J h x t x a x h K h D t J = ∂ ∂ − = ( ) ( ) ( , ) ) ( out(1-79)
(Laplace domain)
h x s x a x h K h D s J = ∂ ∂ − = ( ) ( ) ˆ( , ) ) ( ˆ out
(1-80)
(1-5)
(1-6)
ss 0 out L d ) ( d d J t t J t t t∫
∞ =(1-5)
ss 0 out( )d lim J t t J t t t t∫
′ ′ ′ ∂ ∂ ′ = ∞ → ′ ′ ′ ∂ ∂ ′ =∫
→ ss 0 out 0 d ) ( lim J t t J t t s t s L ) ( ˆ ) ( ˆ d d lim out out 0 sJ s s J s s s − = →(1-81)
ss JL
[ f(x) ]
f(x)
ss 0 out 2 (2) L d ) ( d d J t t J t t t∫
∞ =(1-6)
) ( ˆ ) ( ˆ d d lim out out 2 2 0 sJ s s J s s s→ =(1-82)
(1-80)
Jˆout(s)(1-81)
ˆ ) ( ˆ d d lim out L s J s s t = − → ∂ ∂ − ∂ ∂ − − = = = → h x h x s s x a x h K h D s s x a x h K h D s s ) , ( ˆ ) ( ) ( ) , ( ˆ ) ( ) ( d d lim 0 h x s h x s s x a s x s x a s s x = → = → ∂ ∂ ∂ ∂ ∂ ∂ − = ) , ( ˆ lim ) , ( ˆ lim 0 0 h x s h x s s x P x s x P s x = → = → ∂ ∂ ∂ ∂ ∂ ∂ − = ) , ( lim ) , ( lim 0 0 h x ss h x x a x x P x = = ∂ ∂ ′ ∂ ∂ − = ) ( ) 0 , (
(1-83)
(1-72)
ass(x)(1-75)
P′(x,0)(
)
h x y h x z h y x G y x y z y G y K y x G z x t = = = = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ =∫
0 0 0 L ) , ( d ) , ( ) ( ) , ((1-84)
(L’Hôspital’s rule)
f(a) = g(a) = 0
) ( ) ( lim ) ( ) (a g a f x g x f a x→ = ′ ′
(1-84)
G(0, y) = G(h, y) = G(x, 0) = G(x, h) = 0
(1-84)
) , ( d ) , ( ) ( ) , ( lim 0 0 L z x G y z y G y K y x G t h z h x∫
→→ =(1-85)
22 = ) ( ) ( ) ( ) ( ) ( ) ( ) , ( 1 2 2 1 y W y f x f y W y f x f y x Gy x y x ≤ ≥
(1-86)
f
1f
2(homogeneous solution)
W
f
1f
2(Wronskian)
22) ( d d ) ( d d ) ( ) ( ) ( 2 1 2 1 y f y y f y y f y f y W =
(1-87)
(1-86)
(1-85)
) ( ) ( ) ( d ) ( ) ( ) ( ) ( ) ( ) ( ) ( lim 2 1 0 2 1 2 1 0 L z W z f x f y z W z f y f y K y W y f x f t h z h x∫
→→ =∫
= h y y W y f y f y K 0 1 2 d ) ( ) ( ) ( ) (∫
= hK y G y y y 0 d ) , ( ) ((1-88)
∫
∑
= h i i i i y y v y w y K 0 d ) ( ) ( ) (λ
∑
= i λi 1(1-89)
(1-80)
Jˆout(s)(1-82)
) ( ˆ ) ( ˆ d d lim out out 2 2 0 (2) L s J s s J s s t s − = → ∂ ∂ − ∂ ∂ − − = = = → h x h x s s x a x h K h D s s x a x h K h D s s ) , ( ˆ ) ( ) ( ) , ( ˆ ) ( ) ( d d lim 2 2 0 h x s s sa x s x = → ∂ ∂ ∂ ∂ ∂ − = ) , ( ˆ lim 2 2 0 h x s s P x s x = → ∂ ∂ ∂ ∂ ∂ − = ) , ( lim 2 2 0h x ss h x x a x x P x = = ∂ ∂ ′′ ∂ ∂ − = ) ( ) 0 , (
(1-90)
(1-72)
ass(x)(1-78)
P′′(x,0)(
)(
)
h x y h x z h h y x G y x y z z G z K z y G y K y x G x t = = = = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ =∫ ∫
0 0 0 0 ) 2 ( L ) , ( d d ) , ( ) ( ) , ( ) ( ) , ( 2α
α
(1-91)
) , ( d d ) , ( ) ( ) , ( ) ( ) , ( 2 lim 0 0 0 ) 2 ( Lα
α
α G x y z z G z K z y G y K y x G t h h h x∫ ∫
→ → =(1-92)
(1-86)
(1-92)
) ( ) ( ) ( d d ) ( ) ( ) ( ) ( ) , ( ) ( ) ( ) ( ) ( 2 lim 2 1 0 0 2 1 2 1 0 ) 2 ( Lα
α
α
α
α W f x f y z W f z f z K z y G y K y W y f x f t h h h x∫ ∫
→ → =∫ ∫
= h h K y K z G y z z y y W y f z f 0 0 2 1 ( ) ( ) ( , )d d ) ( ) ( ) ( 2∫ ∫
> = h hK z G z y z y K y G y z z y 0 0 d d ) , ( ) ( ) , , ( ) ( 2(1-93)
(1-93)
A 1.4
(A 1-58)
x y y x x y G y x G y K x K h h d d ) , , ( ) , ( ) ( ) ( 2 0 0∫ ∫
< x y x y G y x G y K x K x y y y G x x G y K x K h h h h d d ) , ( ) , ( ) ( ) ( d d ) , ( ) , ( ) ( ) ( 0 0 0 0∫ ∫
∫ ∫
+ =(A 1-58)
A 1.4
(1-93)
= ) 2 ( L t K x K y G x x G y y y x h h d d ) , ( ) , ( ) ( ) ( 0 0∫ ∫
x y x y G y x G y K x K h h d d ) , ( ) , ( ) ( ) ( 0 0∫ ∫
+(1-94)
x y y v y w x v x w y K x K h h i i i i i i i i d d ) ( ) ( ) ( ) ( ) ( ) ( 0 0∫ ∫
∑
∑
=λ
λ
x y x v y w y v x w y K x K h h i i i i i i i i d d ) ( ) ( ) ( ) ( ) ( ) ( 0 0∫ ∫
∑
∑
+λ
λ
∑
∑
+ = i i i i 2 2 1 ) 1 (λ
λ
(1-95)
1.7
1.8
1.9
(first order reaction)
-) , ( ) , ( ) ( ) ( ) , ( ) ( ) ( x t t t x x R x K t x x x K x D x
ρ
ρ
ρ
∂ ∂ = − ∂ ∂ ∂ ∂(1-96)
) , (x tρ
=(1-96)
) , ( ) ( ) , ( ) ( ) ( ) , ( ) ( ) ( a x t t x K t x a x K x R t x a x x K x D x ∂ ∂ = − ∂ ∂ ∂ ∂(1-98)
R(x)
(rate constant)
a(x, t)
(1-98)
t → ∞
0 ) ( ) ( ) ( ) ( d d ) ( ) ( d d ss ss x −R x K x a x = a x x K x D x(1-99)
L
x ) ( ) ( d d ) ( ) ( d d x R x K x x K x D x Lx =− +(1-100)
) ( ) ( ) (x K x u x u Lx i =λ
i i(1-101)
K(x)
(weighting function)
ui
(0) = u
i(h) = 0
λ
ii
ui
λ
i(1-101)
) ( ) ( ) ( ) ( ) ( ) ( d d ) ( ) ( d d x u x K x u x K x R x u x x K x D x i + i =λi i −(1-102)
(1-102)
Sturm-Liouville
[
( ) ( ) ( )]
( ) 0 ) ( d d ) ( ) ( d d + − = x u x K x R x K x u x x K x D x i λi i(1-103)
Sturm-Liouville
1.
D(x) > 0
K(x) > 0
R(x) > 0
ui
(0) = u
i(h) = 0
(1-103)
Sturm-Liouville
23(positive
definite operator)
242.
(1-103)
R(x)=0 iλ
) rxn ( iλ
Birkhoff
25(1-103)
R(x)K(x)
i iλ
λ
(rxn) >R(x) = 0
R(x) > 0
R(x)K(x)
R(x)
1.9
L t =∑
i λi 1(1-104)
) 2 ( L t =∑
+∑
i i i i 2 2 1 ) 1 (λ
λ
(1-105)
iλ
λ
iR(x)
D
K
) , ( ) , ( 2 2 t x t t x x Dρ
ρ
∂ ∂ = ∂ ∂(1-106)
0 ) , 0 (ρ
ρ
t = ρ(h,t)=0 ρ(x,0)=0(1-107)
Crank
1(1-106)
∑
∞ = − − − = 1 2 2 2 0 0 sin exp( ) 2 ) 1 ( ) , ( n t h n D x h n n h x t xπ
π
π
ρ
ρ
ρ
(1-108)
2 2 2 h n D nπ
λ
=(1-109)
(1-107)
Jout(t)h x t x x D t J = ∂ ∂ − = ( , ) ) ( out
ρ
∑
∞ = − + = 1 2 2 2 0 0 2 (cos )exp( ) n t h n D n h D h Dπ
π
ρ
ρ
(1-110)
h D Jss = ρ0(1-111)
(1-110)
ss 0 out L d ) ( d d J t t J t t t
∫
∞ = t t h n D h n D n h D t J 2 n (cos ) exp( ) d 1 0 1 2 2 2 2 2 2 0 ss∫
∑
∞ ∞ = − − =ρ
π
π
π
∑
∞∫
= ∞ − − = 1 0 2 2 2 2 2 2 d ) exp( ) (cos 2 n t t h n D t h n D nπ
π
π
(1-112)
4 4 2 4 0 2 2 2 d ) exp(π
π
n D h t t h n D t − =∫
∞(1-113)
(1-113)
(1-112)
∑
∞ = − = 1 4 4 2 4 2 2 2 L 2 (cos ) n D n h h n D n tπ
π
π
1 2π
h∑
∞ − = − + − + = L 2 2 2 2 2 2 4 1 3 1 2 1 1 1 2
π
D h D h D h 6 12 2 2 2 2 2 = =π
π
(1-114)
12 ) 2 ( 1 ) 1 2 ( 1 2 1 2 2π
= − −∑
∞ = n n n 27 ss 0 out 2 (2) L d ) ( d d J t t J t t t∫
∞ = t t h n D h n D n h D t J 2 n (cos ) exp( ) d 1 0 1 2 2 2 2 2 2 0 2 ss∫
∑
∞ ∞ = − − =ρ
π
π
π
∑
∞∫
= ∞ − − = 1 0 2 2 2 2 2 2 2 d ) exp( ) (cos 2 n t t h n D t h n D nπ
π
π
(1-115)
6 6 3 6 0 2 2 2 2 2 d ) exp(π
π
n D h t t h n D t − =∫
∞(1-116)
(1-116)
(1-115)
∑
∞ = − = 1 6 6 3 6 2 2 2 (2) L 2 (cos ) 2 n D n h h n D n tπ
π
π
) (cos 1 4 1 4 4 2 4π
π
n n D h n∑
∞ = − = − + − + = L 4 4 4 4 4 2 4 4 1 3 1 2 1 1 1 4π
D h2 4 4 4 2 4 180 7 720 7 4 D h D h = =
π
π
(1-117)
27 720 7 ) 2 ( 1 ) 1 2 ( 1 4 1 4 4π
= − −∑
∞ = n n n D h t 6 2 L = 2 4 ) 2 ( L 180 7 D h t = 2 2 2 h n D nπ
λ
= D h n D h Dn h t n n n n 6 1 1 2 1 2 2 2 1 2 2 2 1 L =∑
=∑
=∑
= ∞ = ∞ = ∞ =λ
π
π
(1-118)
+ =∑
∑
∞ = ∞ = 1 2 2 1 (2) L 1 1 n n n n tλ
λ
+ =∑
∑
∞ = ∞ = 2 4 1 4 4 2 1 2 2 2 1 1 n n D n h n D hπ
π
2 4 2 4 2 4 180 7 90 36 D h D h D h + = =(1-119)
(1-106)
tL =h2 6D tL(2) =7h2 180D2 ) ( 1 out ss t J t J ∂ ∂
) ( 1 ) ( out ss t J t J t P ∂ ∂ =
(1-120)
1 d ) ( 0 =∫
∞ t t P(1-5)
t
> =< =∞∫
tP t t t t 0 L ( )d(1-121)
> <tt
(1-6)
> =< =∫
∞ 2 0 2 (2) L t P(t)dt t t(1-122)
(variance)
(standard deviation)
µ
σ
(
)
∫
∞ > < − = 0 2 d ) (t t P t tµ
(
)
∫
∞ > < + > < − = 0 2 2 d ) ( 2t t t P t t t∫
∫
∫
∞ ∞ ∞ > < + > < − = 0 2 0 0 2 d ) ( d ) ( 2 d ) (t t t tP t t t P t t P t 2 2 ) 2 ( L −2< > +< > =t t t2 L ) 2 ( L (t ) t − =
(1-123)
2 L ) 2 ( L (t ) t − = =µ
σ
(1-124)
D h tL = 2 6 tL(2) =7h4 180D2 D h2 90 1 =σ
(1-125)
D h 6 2 D h2 90 1 =σ
1-2
1-2 h2 / 6D ± h2 / √90D D h t 6 2 >= < D h2 90 1 =σ
1.
J. Crank, The Mathematics of Diffusion, 2nd ed. (Oxford University
Press, London, 1975), pp. 50-51.
2.
H. Daynes, Proc. Roy. Soc. A 97, 286 (1920).
3.
R. M. Barrer, J. Phys. Chem. 57, 35 (1953).
4.
R. M. Barrer and E. Strachan, Proc. Roy. Soc. A 231, 353 (1955).
5.
R. C. Goodknight and I. Fatt, J. Phys. Chem. 65, 1709 (1961).
6.
H. L. Frisch, J. Phys. Chem. 61, 93 (1957).
7.
H. L. Frisch, J. Phys. Chem. 62, 401 (1958).
8.
J. S. Chen and J. L. Fox, J. Chem. Phys. 89, 2278 (1988).
9.
J. S. Chen and F. Rosenberger, J. Phys. Chem. 95, 10164 (1991).
10.
J. K. Leypoldt and D. A. Gough, J. Phys. Chem. 84, 1058 (1980).
11.
R. A. Siegel, J. Phys. Chem. 95, 2556 (1991).
12.
W. Liang and R. A. Siegel, J. Chem. Phys. 125, 044707 (2006).
13.
J. S. Chen and W. Y. Chang, J. Chem. Phys. 106, 8022 (1997).
14.
Y. K. Zhang and N. M. Kocherginsky, J. Membr. Sci. 225, 105 (2003).
15.
R. A. Siegel and E. L. Cussler, J. Membr. Sci. 229, 33 (2004).
16.
X. Ye, L. Lv, X. S. Zhao, and K. Wang, J. Membr. Sci. 283, 425
17.
D. C. Montgomery and G. C. Runger, Applied Statistics and
Probability for Engineers (John Wiley & Sons, 1994), p. B-9.
18.
C. R. Wylie and L. C. Barrett, Advanced Engineering Mathematics, 6
thed. (McGraw-Hill, New York, 1995), p. 173.
19.
G. H. Weiss, Adv. Chem. Phys. 13, 1 (1967).
20.
E. Zauderer, Partial Differential Equations of Applied Mathematics,
2
nded. (John Wiley & Sons, New York, 1989), p. 447.
21.
C. R. Wylie and L. C. Barrett, Advanced Engineering Mathematics, 6
thed. (McGraw-Hill, New York, 1995), p. 181.
22.
A. Jeffrey, Advanced Engineering Mathematics (Harcourt/Academic
Press, New York, 2002), p. 318.
23.
D. Zwillinger, Handbook of Differential Equations (Academic Press,
San Diego, 1989), p.59.
24.
C. E. Pearson, Handbook of Applied Mathematics: Selected Results
and Methods, 2
nded. (Van Nostrand Reinhold, New York, 1990),
p.319.
25.
G. Birkhoff and G. C. Rota, Ordinary Differential Equations, 4
thed.
(John Wiley & Sons, Singapore, 1989), p. 320.
26.
E. A. Kraut, Fundamentals of Mathematical Physics (Dover
publications, New York, 1995), pp. 296-297.
A1.1
x t t t t ( , )d 0 2 2ρ
∫
∞ ∂ ∂ t t x t t ( , )d 0 3 3 2ρ
∫
∞ ∂ ∂ dt t x t t n n n ) , ( 0 1 1ρ
∫
∞ + + ∂ ∂ ) , ( tx ρt = 0
ρ(x,t)=0n = 1
n = 2
∫
∫
udv=uv− vdu(A 1-1)
dt t x t t ( , ) 0 2 2 ρ∫
∞ ∂ ∂t
u
=
x
t
t
t
v
(
,
)
d
d
2 2ρ
∂
∂
=
d
u
=
d
t
,)
,
( t
x
t
v
ρ
∂
∂
=
x t dt t t ( , ) 0 2 2 ρ∫
∞ ∂ ∂∫
∫
∞ ∞ ∞ ∂ ∂ − ∂ ∂ = ∂ ∂ 0 0 0 2 2 d ) , ( ) , ( ) , ( x t t t t x t t dt t x t t ρ ρ ρ(A 1-2)
) , ( tx ρ ) ( ) ( ) , (x t a w x e ss x i t i i iρ
ρ
=∑
−λ +(A 1-3)
(A 1-2)
= − = ∂ρ
∞∑
λ
−λt(A 1-2)
∫
∫
∞ ∞ ∂ ∂ − = ∂ ∂ 0 0 2 2 d ) , ( ) , ( x t t t dt t x t tρ
ρ
) (x ssρ
− =(A 1-5)
(A 1-5)
1.7
(1-40)
(A 1-5)
(A 1-3)
x t dt t t ( , ) 0 3 3 2ρ
∫
∞ ∂ ∂ 2t
u
=
x
t
t
t
v
(
,
)
d
d
3 3ρ
∂
∂
=
d
u
=
2
t
d
t
,(
,
)
2 2t
x
t
v
ρ
∂
∂
=
dt t x t t t x t t dt t x t t ( , ) ( , ) 2 ( , ) 0 2 2 0 2 2 2 0 3 3 2ρ
ρ
ρ
∫
∫
∞ ∞ ∞ ∂ ∂ − ∂ ∂ = ∂ ∂(A 1-6)
(A 1-3)
(A 1-6)
0 e ) ( lim ) , ( 2 2 0 2 2 2 = = ∂ ∂ − ∞ → ∞∑
t i i i i t i x w a t t x t tρ
λ
λ(A 1-7)
(A 1-5)
(A 1-5)
(A 1-6)
) ( 2 ) , ( 0 3 3 2 x dt t x t tρ
=ρ
ss ∂ ∂∫
∞(A 1-8)
(A 1-8)
(A 1-8)
(1-46)
A1.2
f y y x x K x D h x h d d ) ( ) ( ) ( 1 0∫
∫
1.7
1.8
∫
= h x y y f u ( )d y x K x D v d ) ( ) ( 1 d =d
u
=
− f(y)dy,∫
= x y y K y D v 0 d ) ( ) ( 1 x y y f x K x D h x h d d ) ( ) ( ) ( 1 0∫
∫
h x h x h x h y y K y D y y f x y y f x K x D 0 0 0 d ) ( ) ( 1 d ) ( d d ) ( ) ( ) ( 1∫
∫
∫
∫
=(
)
∫ ∫
−h x y f x x y K y D 0 0 d ) ( -d ) ( ) ( 1(A 1-9)
(A 1-9)
(A 1-9)
=∫
∫
f y y x x K x D h x h d d ) ( ) ( ) ( 1 0∫ ∫
h x x x f y y K y D 0 0 d ) ( d ) ( ) ( 1(A 1-10)
A 1.3
G(x,y)
a( tx, ) ass(x)26 ) , ( ) , ( ) ( ) ( ) , ( ) ( ) ( y t t t y y R y K t y y y K y D yρ
ρ
ρ
∂ ∂ = − ∂ ∂ ∂ ∂(A 1-11)
) , (y tρ
D(y)
K(y)
R(y)
(rate coefficient)
ρ(0,t)=ρ0 ρ(h,t)=0 0 ) 0 , (y = ρ(activity)
) ( ) , ( ) , ( y K t y t y a = ρ(A 1-12)
(A 1-12)
) , ( ) ( ) , ( ) ( ) ( ) , ( ) ( ) ( a y t t y K t y a y K y R t y a y y K y D y ∂ ∂ = − ∂ ∂ ∂ ∂(A 1-13)
L
y ) ( ) ( d d ) ( ) ( d d y R y K y y y K y D y L =− +(A 1-14)
G(x,y)
x
(source point) y
(field point)
G(x,y)
) ( ) , ( ) ( ) ( ) , ( ) ( ) ( G x y R y K y G x y x y y y K y D y ∂ − =− − ∂ ∂ ∂ δ
(A 1-15)
0 ) , ( ) 0 , (x =G x h = G
(A 1-16)
(A 1-13)
(A 1-15)
a(y,t)(A 1-13)
(A 1-15)
) , ( ) , ( ) , ( ) ( ) , ( 2 2 t y a y K t y RKa t y a y DK t y a y DK ∂ ∂ = − ∂ ∂ ′ + ∂ ∂(A 1-17)
) ( ) , ( ) , ( ) ( ) , ( 2 2 y x y x RKG y x G y DK y x G y DK − =− − ∂ ∂ ′ + ∂ ∂δ
(A 1-18)
D(y)
K(y)
R(y)
D
K
R
) (DK ′