Ion Diffusion - Revision history

Abdehait: /* Importance of Ion Diffusion Study */

2013-09-04T01:04:34Z

Importance of Ion Diffusion Study

Abdehait: /* Importance of Ion Diffusion Study */

2013-09-04T01:02:24Z

Importance of Ion Diffusion Study

Abdehait: /* Importance of Ion Diffusion Study */

2013-09-04T01:01:37Z

Importance of Ion Diffusion Study

Abdehait: /* Importance of Ion Diffusion Study */

2013-09-03T20:36:03Z

Importance of Ion Diffusion Study

Abdehait: /* long */

2013-09-03T20:35:17Z

long

Abdehait: /* long */

2013-09-03T20:34:32Z

long

Abdehait: /* long */

2013-09-03T20:33:03Z

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Abdehait: /* long */

2013-09-03T20:32:28Z

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Abdehait: /* long */

2013-09-03T20:30:25Z

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Abdehait: /* long */

2013-09-03T20:25:59Z

long

← Older revision		Revision as of 01:04, 4 September 2013
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−	<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} \right)} \times \left [ 1- \sum_{m=0}^\infty \sum_{i=0}^m \frac{(r^2/4D_Tt)^m(r_0^2/4D_Tt)^i} {m!i!} ~~\left( 1-~~ e^{- ((r_0^2+r^2)/4D_T t)} ~~\right)~~ \right]</math>	+	<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} \right)} \times \left [ 1- \sum_{m=0}^\infty \sum_{i=0}^m \frac{(r^2/4D_Tt)^m(r_0^2/4D_Tt)^i} {m!i!} e^{-((r_0^2+r^2)/4D_T t)}\right]</math>

@@ Line 51: / Line 51: @@
-<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} -\frac{r^2}{4D_T t} \right)} x \sum_{m=0}^\infty \frac{(r^2/4D_Tt)^m} {m!}  \left( 1- e^{-r^2/4D_T t} \sum_{i=0}^m \frac{(r_0^2/4D_T t)^i}{i!} \right)</math>
+<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} -\frac{r^2}{4D_T t} \right)} \times \sum_{m=0}^\infty \frac{(r^2/4D_Tt)^m} {m!}  \left( 1- e^{-r^2/4D_T t} \sum_{i=0}^m \frac{(r_0^2/4D_T t)^i}{i!} \right)</math>
 which becomes:
-<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} \right)} x  \left [ 1- \sum_{m=0}^\infty \sum_{i=0}^m \frac{(r^2/4D_Tt)^m(r_0^2/4D_Tt)^i} {m!i!}  \left( 1- e^{- ((r_0^2+r^2)/4D_T t)} \right) \right]</math>
+<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} \right)} \times  \left [ 1- \sum_{m=0}^\infty \sum_{i=0}^m \frac{(r^2/4D_Tt)^m(r_0^2/4D_Tt)^i} {m!i!}  \left( 1- e^{- ((r_0^2+r^2)/4D_T t)} \right) \right]</math>

@@ Line 51: / Line 51: @@
-<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} -\frac{r^2}{4D_T t} \right)} X \sum_{m=0}^\infty \frac{(r^2/4D_Tt)^m} {m!}  \left( 1- e^{-r^2/4D_T t} \sum_{i=0}^m \frac{(r_0^2/4D_T t)^i}{i!} \right)</math>
+<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} -\frac{r^2}{4D_T t} \right)} x \sum_{m=0}^\infty \frac{(r^2/4D_Tt)^m} {m!}  \left( 1- e^{-r^2/4D_T t} \sum_{i=0}^m \frac{(r_0^2/4D_T t)^i}{i!} \right)</math>
 which becomes:
-<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} \right)} X  \left [ 1- \sum_{m=0}^\infty \sum_{i=0}^m \frac{(r^2/4D_Tt)^m(r_0^2/4D_Tt)^i} {m!i!}  \left( 1- e^{- ((r_0^2+r^2)/4D_T t)} \right) \right]</math>
+<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} \right)} x  \left [ 1- \sum_{m=0}^\infty \sum_{i=0}^m \frac{(r^2/4D_Tt)^m(r_0^2/4D_Tt)^i} {m!i!}  \left( 1- e^{- ((r_0^2+r^2)/4D_T t)} \right) \right]</math>

@@ Line 29: / Line 29: @@
 in cylindrical coordinate:
-<math> n(x,y,z,t) = \int_0^\infty r'dr' \int_0^{2 \pi}d\theta' \int_{-\infty}^\infty dz' \int_{-\infty}^t dt' \frac {\beta(x',y',z',t')}{4 \pi (t-t')^{2/3} D_T D_L^{1/2}}  . \exp \left (-\alpha (t-t') - \frac{r'^2 +r^2- 2rr'cos(\theta - \theta')}{4D_T (t-t')} - \frac{[z-z' - v_d (t-t')]^2}{4D_L (t-t')}\right ) </math>
+<math> n(r,z,t) = \int_0^\infty r'dr' \int_0^{2 \pi}d\theta' \int_{-\infty}^\infty dz' \int_{-\infty}^t dt' \frac {\beta(x',y',z',t')}{4 \pi (t-t')^{2/3} D_T D_L^{1/2}}  . \exp \left (-\alpha (t-t') - \frac{r'^2 +r^2- 2rr'cos(\theta - \theta')}{4D_T (t-t')} - \frac{[z-z' - v_d (t-t')]^2}{4D_L (t-t')}\right ) </math>
@@ Line 42: / Line 42: @@
-<math> n(x,y,z,t) = s e^{-\alpha t - \frac{-(z- v_d t)^2/4D_L t}{(4 \pi t)^{3/2} D_L^{1/2} D_T}} \int_0^{r_0} r'dr' \int_0^{2 \pi} d\theta' e^ {\frac{r'^2 +r^2- 2rr'cos \theta'}{4D_T t}} </math>
+<math> n(r,z,t) = s e^{-\alpha t - \frac{-(z- v_d t)^2/4D_L t}{(4 \pi t)^{3/2} D_L^{1/2} D_T}} \int_0^{r_0} r'dr' \int_0^{2 \pi} d\theta' e^ {\frac{r'^2 +r^2- 2rr'cos \theta'}{4D_T t}} </math>
-Using intergal tables <math> 1/\pi \int d\theta e^{\pm x cos\theta} = \sum_{m=0}^\infty \frac{(x/2)^{2m}}{(m!)^2} </math> , then integrating over r' by parts:
+Using integral tables <math> 1/\pi \int d\theta e^{\pm x cos\theta} = \sum_{m=0}^\infty \frac{(x/2)^{2m}}{(m!)^2} </math> , then integrating over r' by parts:
 <math> \int_0^a x^m e^{-x} dx = m! ( 1- e^{-a} \sum_{i=0}^m \frac {a^i}{i!}) </math> so the solution looks like:

← Older revision		Revision as of 20:35, 3 September 2013
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−	~~=long=~~

	<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} -\frac{r^2}{4D_T t} \right)} X \sum_{m=0}^\infty \frac{(r^2/4D_Tt)^m} {m!} \left( 1- e^{-r^2/4D_T t} \sum_{i=0}^m \frac{(r_0^2/4D_T t)^i}{i!} \right)</math>		<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} -\frac{r^2}{4D_T t} \right)} X \sum_{m=0}^\infty \frac{(r^2/4D_Tt)^m} {m!} \left( 1- e^{-r^2/4D_T t} \sum_{i=0}^m \frac{(r_0^2/4D_T t)^i}{i!} \right)</math>

← Older revision		Revision as of 20:34, 3 September 2013
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−	<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} \right)} X \left [ 1- \sum_{m=0}^\infty \sum_{i=0}^m \frac{(r^2/4D_Tt)^m(r^2/4D_Tt)^i} {m!i!} \left( 1- e^{- ((r_0^2+r^2)/4D_T t)} \right) \right]</math>	+	<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} \right)} X \left [ 1- \sum_{m=0}^\infty \sum_{i=0}^m \frac{(r^2/4D_Tt)^m(r_0^2/4D_Tt)^i} {m!i!} \left( 1- e^{- ((r_0^2+r^2)/4D_T t)} \right) \right]</math>

← Older revision		Revision as of 20:33, 3 September 2013
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−	<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} \right)} X \left 1- \sum_{m=0}^\infty \sum_{i=0}^m \frac{(r^2/4D_Tt)^m(r^2/4D_Tt)^i} {m!i!} \left( 1- e^{- ((r_0^2+r^2)/4D_T t)} \right) \right]</math>	+	<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} \right)} X \left [ 1- \sum_{m=0}^\infty \sum_{i=0}^m \frac{(r^2/4D_Tt)^m(r^2/4D_Tt)^i} {m!i!} \left( 1- e^{- ((r_0^2+r^2)/4D_T t)} \right) \right]</math>

← Older revision		Revision as of 20:32, 3 September 2013
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−	<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} \right)} X \left 1- \sum_{m=0}^\infty \sum_{i=0}^m \frac{(r^2/4D_Tt)^m(r^2/4D_Tt)^i} {m!i!} \left( 1- e^{-~~r^2/4D_T t} e{~~ (r_0^2+r^2/4D_T t)} \right) \right]</math>	+	<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} \right)} X \left 1- \sum_{m=0}^\infty \sum_{i=0}^m \frac{(r^2/4D_Tt)^m(r^2/4D_Tt)^i} {m!i!} \left( 1- e^{- ((r_0^2+r^2)/4D_T t)} \right) \right]</math>

← Older revision		Revision as of 20:30, 3 September 2013
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−	<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} \right)} X \left [ \sum_{m=0}^\infty \frac{(r^2/4D_Tt)^m} {m!} \left( 1- e^{-r^2/4D_T t} ~~\sum_{i=0}^m \frac~~{(r_0^2/4D_T t)~~^i}{i!~~} \right) \right]</math>	+	<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} \right)} X \left 1- \sum_{m=0}^\infty \sum_{i=0}^m \frac{(r^2/4D_Tt)^m(r^2/4D_Tt)^i} {m!i!} \left( 1- e^{-r^2/4D_T t} e{ (r_0^2+r^2/4D_T t)} \right) \right]</math>

← Older revision		Revision as of 20:25, 3 September 2013
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−	<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} \right)} X \sum_{m=0}^\infty \frac{(r^2/4D_Tt)^m} {m!} \left( 1- e^{-r^2/4D_T t} \sum_{i=0}^m \frac{(r_0^2/4D_T t)^i}{i!} \right) </math>	+	<math> n(r,z,t) = \frac{s}{(4 \pi D_L t)^{1/2} } e^{\left( -\alpha t - \frac{(z-v_d t)^2}{4D_L t} \right)} X \left [ \sum_{m=0}^\infty \frac{(r^2/4D_Tt)^m} {m!} \left( 1- e^{-r^2/4D_T t} \sum_{i=0}^m \frac{(r_0^2/4D_T t)^i}{i!} \right) \right]</math>