Debye-Hückel approximation

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Debye-Hückel approximation

For plane:

$\displaystyle \phi$ ⁽ⁱⁿ⁾(z) = $\displaystyle \sigma$ _aexp(- $\displaystyle \kappa$ d /2)exp( $\displaystyle \kappa$ z)/ $\displaystyle \epsilon$ $\displaystyle \kappa$

for z $\leq$ - d /2, and

$\displaystyle \phi$ ^(out)(z) = $\displaystyle \sigma$ _bexp( $\displaystyle \kappa$ d /2)exp(- $\displaystyle \kappa$ z)/ $\displaystyle \epsilon$ $\displaystyle \kappa$

for z $\geq$ d /2.

For cylinder:

$\displaystyle \phi$ ⁽ⁱⁿ⁾(r) = $\displaystyle \sigma$ _aI₀( $\displaystyle \kappa$ r)/ $\displaystyle \epsilon$ $\displaystyle \kappa$ I₁( $\displaystyle \kappa$ a)

for r $\leq$ a, and

$\displaystyle \phi$ ^(out)(r) = $\displaystyle \sigma$ _bK₀( $\displaystyle \kappa$ r)/ $\displaystyle \epsilon$ $\displaystyle \kappa$ K₁( $\displaystyle \kappa$ a)

for r $\leq$ b.

For sphere:

$\displaystyle \phi$ ⁽ⁱⁿ⁾(r) = $\displaystyle \sigma$ _aa²sinh( $\displaystyle \kappa$ r)/ $\displaystyle \epsilon$ ( $\displaystyle \kappa$ a - 1)r cosh( $\displaystyle \kappa$ a)

for r $\leq$ a, and

$\displaystyle \phi$ ^(out)(r) = $\displaystyle \sigma$ _bexp(- $\displaystyle \kappa$ r)/ $\displaystyle \epsilon$ (1 + $\displaystyle \kappa$ b)r exp( $\displaystyle \kappa$ b)

for r $\geq$ b.

Toan T Nguyen 2004-09-06