a) To show that the function w(r, t) = r c (r, t) satisfies a diffusion equation for the spatial variable r, we compute the partial derivatives taking into account the fact that the variables r and t are independent. Using the diffusion equation for the matter concentration c (r, t), the partial time derivative of w(r, t) can be written as,
\frac{∂w(r, t)}{∂t} = \frac{r ∂c (r, t)}{∂t} = D \Bigl(r\frac{∂^2c (r, t)}{∂r^2} +2 \frac{∂c (r, t)}{∂r}\Bigr) .
The second-order partial spatial derivative is given by,
\frac{∂^2w (r, t)}{∂r^2} = \frac{∂}{∂r} \Bigl(\frac{∂}{∂r} (r c (r, t))\Bigr) = \frac{∂c (r, t)}{∂r} + \frac{∂}{∂r} \Bigl(r\frac{∂c (r, t)}{∂r}\Bigr)
= r \frac{∂^2c (r, t)}{∂r^2} +2 \frac{∂c (r, t)}{∂r}
Thus,
\frac{∂w(r, t)}{∂t} = D \frac{∂^2 w(r, t)}{∂r^2}
b) The partial derivatives of the function w(r, t) have to be recast in terms of the partial derivatives of the function f (η) where η (r, t) = r^2/Dt.^32 The partial derivatives of the function η (r, t) = r^2/Dt are,
\frac{∂η}{∂t} = – \frac{r^2}{Dt^2} = – \frac{η}{t} and \frac{∂η}{∂r} = \frac{2r}{Dt} = \frac{2η}{r}
Since f (η) = w(r, t), the first-order partial derivatives of the function w(r, t) are recast
in terms of the first-order derivative of the function f (η) as,
\frac{\partial w}{\partial t} =\frac{df}{d\eta } \frac{\partial \eta }{\partial t} = – \frac{r^2}{Dt^2} \frac{df}{d\eta } = – \frac{\eta }{t} \frac{df}{d\eta } .
\frac{\partial w}{\partial r} =\frac{df}{d\eta } \frac{\partial \eta }{\partial r} = \frac{2 r}{Dt} \frac{df}{d\eta } = \frac{2\eta }{r} \frac{df}{d\eta }
The second-order partial derivative of the function w(r, t) is recast in terms of the second-order derivative of the function f (η) as,
\frac{\partial^2 w}{\partial r^2} = \frac{\partial}{\partial r}\biggl(\frac{2r}{Dt}\frac{df}{d\eta } \biggr) =\frac{2}{Dt} \frac{df}{d\eta }+ \frac{2r}{Dt} \frac{d^2f}{d\eta ^2} \frac{\partial \eta }{\partial r} = \frac{2\eta }{r^2} \frac{df}{d\eta } + \frac{4\eta ^2}{r^2}\frac{d^2f}{d\eta ^2}
Thus, the diffusion equation becomes,
-\frac{\eta }{t} \frac{df}{d\eta } = \frac{2D\eta }{r^2} \frac{df}{d\eta } + \frac{4D\eta ^2}{r^2} \frac{d^2f}{d\eta ^2}
Using the definition of the dimensionless function η, this differential equation is recast as,
4η\frac{d^ 2 f}{dη^2}+ (η + 2)\frac{df}{dη} = 0
Here, we introduce a function g (η) as the derivative of f (η) with respect to η,
g (η) = \frac{df}{dη}
Thus, the differential equation becomes,
4 η \frac{dg (η)}{dη} + (η + 2) g (η) = 0
which is recast as,
\frac{dg (η)}{g (η)} = – \biggl(\frac{1}{4} + \frac{1}{2\eta } \biggr) dη
When integrating from η_0 to η we find,
\int_{g_0}^{g(\eta )}{\frac{d\acute{g}(\acute{\eta } ) }{\acute{g}(\acute{\eta } ) } } = \int_{\eta _0}^{\eta }{\biggl(\frac{1}{4}+\frac{1}{2\acute{\eta } } \biggr) } d\acute{\eta }
where g_0 = g (η_0) . The solution is,
\ln \biggl(\frac{g(\eta )}{g_0}\biggr) = -\frac{1}{2} \ln \Bigl(\frac{\eta }{\eta _0}\Bigr) -\frac{1}{4} (\eta -\eta _0)
which is recast as,
\ln \Biggl(\frac{g(\eta )\eta ^{1/2}}{g_0 \eta ^{1/2}_0}\Biggr) = -\frac{1}{4} (\eta -\eta _0)
and implies that,
g (η) = \frac{A}{η^{1/2}} \exp \bigl(-\frac{\eta }{4}\bigr)
where the constant A= g_0 η^{1/2}_0 \exp (η_0/4) .When integrating from η_0 to η we find,
f(\eta ) = \int_{\eta _0}^{\eta }{\acute{g}(\acute{\eta } )d\acute{\eta } } = A \int_{\eta _0}^{\eta }{\frac{1}{\acute{\eta }^{1/2} } }\exp \Bigl(-\frac{\acute{\eta } }{4}\Bigr) d\acute{\eta }
Using the change of variable,
\nu =\frac{\eta ^{1/2}}{2} = \frac{r}{2\sqrt{Dt} } then d\nu = \frac{d\eta }{4\eta ^{1/2}}
and taking into account the definition h (ν) = f (η) = w(r, t), we can recast the solution as,
w (r, t) = h (ν) = B \int_{\nu _0}^{\nu }{\exp (-\acute{\nu }^2 )d\acute{\nu } }
where the constant B = 4 A. For ν_0 = 0, the solution is the error function h (ν) = erf (ν) multiplied by a constant.
c) The matter concentration c (r, t) is given by,
c (r, t) = \frac{w(r, t)}{r} = \frac{B}{r} \int_{\nu _0}^{\nu }{\exp (-\acute{\nu }^2 )d\acute{\nu } }
where ν (r, t) and we choose ν_0 (r, t) = ν (r_0, t) . Thus, the conductive matter current density scalar at the electrode of radius r_0 is written as,
j_r(r_0,t) = -D \frac{∂c (r, t)}{∂r}\mid _{r=r_0}
= \frac{BD}{r^2_0} \int_{\nu _0}^{\nu }{\exp (-\acute{\nu }^2 )} d\acute{\nu } \mid _{r=r_0} – \frac{BD}{r^2_0} \exp (-\nu ^2)\frac{\partial \nu }{\partial r} \mid _{r=r_0}
where we used the fact that the upper integration bound ν (r, t) is a function of r. In this relation, the integral vanishes since ν evaluated at r_0 is ν_0 , which means that the upper and lower integration bounds are equal. Taking into account that,
\exp (-\nu ^2) \frac{\partial \nu }{\partial r} \mid _{r=r_0} = \exp\Bigl(-\frac{r^2}{4Dt}\Bigr)\frac{\partial}{\partial r} \Bigl(\frac{r}{2\sqrt{Dt} }\Bigr) \mid _{r=r_0}
= \frac{1}{2\sqrt{Dt} } \exp \Bigl(-\frac{r^2_0}{Dt}\Bigr)
we find,
j_r(r_0,t) = -\frac{B}{2r^2_0} \sqrt{\frac{D}{t} } \exp\biggl(-\frac{r^2_0}{4Dt}\biggr)
In the limit where the radius of the electrode is negligible, i.e. r_0 = 0 , the scalar conductive matter current density is given by,
j_r(0,t) \lim _{r_0\rightarrow 0} j_r(r_0,t) = \lim _{r_0\rightarrow 0}\Biggl(-\frac{B}{2r^2_0}\sqrt{\frac{D}{t}}\biggl(1-\frac{r^2_0}{4Dt}\biggr) \Biggr) = \frac{B}{8 \sqrt{D} t^{3/2}}
d) The stationary state is reached in the limit where t→∞. In the stationary limit,
\lim _{t\rightarrow \infty }\nu (r_0,t) = \lim _{t\rightarrow \infty }\frac{r_0}{2\sqrt{Dt} } =0
and initially,
\nu _0 = \lim _{t\rightarrow 0}\nu (r_0,t) = \lim _{t\rightarrow \infty }\frac{r_0}{2\sqrt{Dt} } =\infty
Thus, in the stationary state, the general expression for the scalar conductive matter current density obtained in c) reduces to,
j_r(r_0,\infty ) = -D \frac{∂c (r,∞)}{∂r} \mid _{r=r_0} = -\frac{BD}{r^2_0} \int_{0}^{\infty }{\exp (-\nu ^2)}D\nu
The error function erf (x) is defined as,
erf (x) =\frac{2}{\sqrt{\pi } } \int_{0}^{x}{\exp (-\nu ^2)} d\nu
and erf (∞) = 1. Thus, the scalar conductive matter current density is recast as,
j_r(r_0,\infty )=-\frac{\sqrt{\pi }BD }{2r^2-0} erf (∞)==-\frac{\sqrt{\pi }BD }{2r^2-0}= -\frac{Dc^*}{r_0}
where
B = \frac{2r_0 c^*}{\sqrt{\pi } } .