Electrochemical impedance and conductivity measurements in a heterogeneous Fe powder particle—electrolyte system with or without electrochemical reaction

Abstract

The participation of solid Fe powder particles in the transfer of charge through a heterogeneous system consisting of an electrolyte and conducting powder particles was studied by means of electrochemical impedance spectroscopy and conductivity measurements. Different behaviour was encountered in electroinactive Na₂SO₄ electrolyte without metal electrodeposition on the Fe particles and in NiSO₄ electrolyte where Ni electrodeposition occurs on the Fe particle surfaces. From impedance diagrams and the proposed model the formation of aggregates and chains of Fe particles is deduced. The important role of electrochemical reaction proceeding at the particle surface in the charge transfer behaviour in stirred heterogeneous systems was also demonstrated.

Appendix

1.1 Derivation of the total impedance Z _T of the system according to Figs. 9 and 10

The total impedance of the system is the sum of the electrolyte impedance Z _el , contact impedance Z _con , interfacial impedance Z as depicted in Fig. 8. Also the outer impedance Z _out of the system should be included in the expression for Z _T .

$$ Z_{T} =Z_{out} +Z_{el} +Z_{con} +Z $$

(A.1)

For the partial impedances Z _out , Z _el , Z _con , Z according Figs. 9 and 10 it can be written

$$ Z_{el} =R_{el} $$

(A.2)

$$ Z_{out} =i\omega L_{out} $$

(A.3)

$$\frac{1}{Z_{con} }={i}\omega C_{con} +\frac{1}{R_{con} }\quad \Rightarrow \quad Z_{con} =\frac{R_{con} }{(\omega {C}_{ con} R_{con} )^2+1}-{i}\frac{\omega C_{con} R_{con}^2 }{(\omega C_{con} R_{con} )^2+1} $$

(A.4)

$$ \frac{1}{Z}=\frac{1}{Z_{dl} }+\frac{1}{Z_{ct} +Z_{dif} }\quad \Rightarrow \quad Z=\frac{Z_{dl} (Z_{ct} +Z_{dif} )}{Z_{dl} +Z_{ ct} +Z_{dif} } $$

(A.5)

where

$$ Z_{dl} =-\frac{i}{\omega C_{dl} } $$

(A.6)

$$ Z_{ct} =R_{ct} $$

(A.7)

$$ Z_{dif} =O=\frac{1}{Y_0 \sqrt {{i}\omega } }\tanh \left( {B\sqrt {{i}\omega } } \right) $$

(A.8)

Z _dl is double-layer impedance, Z _ct charge transfer impedance and Z _dif diffusion impedance.

Applying the Moivre theorem ${(a+bi)^{1/n}=r^{1/n}\left( {\cos \frac{2l \pi +\varphi }{n}+{i}\sin \frac{2l\pi +\varphi }{n}} \right)}$

where ${l=0,1,2\ldots (n−1);\quad r=({a}^2+{b}^2)^{1/2};\quad \tan \varphi =\frac{b}{a}}$ and if

$$ \tanh \left( {x\pm iy} \right)=\frac{\sinh (2x)\pm \sin (2y)}{\cosh(2x)+\cos (2y)} $$

${\sqrt {i} }$ can be rewritten as ${\frac{1+i}{\sqrt {2} }}$ and then

$$ \frac{1}{Y_0 \sqrt {i\omega } }=\frac{\sqrt {2} }{Y_0 \sqrt {\omega } (1−i)}=\sqrt {\frac{2}{\omega }} \frac{1}{2Y_0 }+{i}\sqrt {\frac{2}{\omega }} \frac{1}{2Y_0 } $$

(A.9)

$$\tanh \left( {i^{1/2}B\sqrt {\omega } } \right)=\tanh \left( {(1+i)B\sqrt {\frac{\omega }{2}} } \right)=\frac{\sinh \left( {2B\sqrt {\frac{\omega }{2}} } \right)+ i\sin \left( {2B\sqrt {\frac{\omega }{2}} } \right)}{\cosh \left( {2B\sqrt {\frac{\omega }{2}} } \right)+\cos \left( {2B\sqrt {\frac{\omega }{2}} } \right)} $$

(A.10)

After combining Eqs. A.8–A.10 we obtain an expression for Z _dif

$$ Z_{dif} =\frac{\sqrt 2 \left[ {\sinh (2\alpha )-\sin (2\alpha )} \right]}{2Y_0 \sqrt \omega \left[ {\cosh (2\alpha )+\cos (2\alpha )} \right]}+{i}\frac{\sqrt 2 \left[ {\sinh (2\alpha )-\sin (2\alpha )} \right]}{2Y_0 \sqrt \omega \left[ {\cosh (2\alpha )+\cos (2\alpha )} \right]} $$

(A.11)

where ${\alpha =B\sqrt {\frac{\omega }{2}} }$ .

Then the expression for the total impedance Z _T can be derived as

$$ \begin{aligned} Z_{T}& =i\omega L_{out} +R_{el} +\frac{R_{con} }{{i}\omega C_{con} R_{con}+1} \\ & +\frac{{i}\left( {R_{ct} +Y_0 \sqrt {{i}\omega } +\tanh \left( {B\sqrt {{i}\omega } } \right)} \right)}{iY_0 \sqrt {{i}\omega } -R_{ct} C_{dl} Y_0 \omega \sqrt {{i}\omega } -\omega C_{dl}\tanh \left( {B\sqrt {{i}\omega } } \right)} \end{aligned} $$

(A.12)

$$ \begin{aligned} {Z}'_{T}&=R_{el} +\frac{R_{con} }{1+(\omega C_{con} R_{con} )^2}+ \\ &\frac{\frac{\sqrt {2} }{2}\frac{\left( {1+R_{ct} \sqrt {\omega } } \right)\left( {\sinh 2\alpha +\sin 2\alpha } \right)}{R_{ct} C_{dl} Y_0 \omega \left( {\cosh 2\alpha +\cos 2\alpha } \right)}}{\left[ {R_{ct} +\frac{\sqrt {2} }{2}\frac{\sinh 2\alpha -\sin 2\alpha }{Y_0 \sqrt {\omega } \left( {\cosh 2\alpha +\cos 2\alpha } \right)}} \right]^2+\left[ {\frac{\sqrt {2} }{2}\frac{\sinh 2\alpha +\sin 2\alpha }{Y_0 \sqrt {\omega } \left( {\cosh 2\alpha +\cos 2\alpha } \right)}-\frac{1}{\omega R_{ct} }} \right]^2} \end{aligned} $$

(A.13)

$$ \begin{aligned} Z''_{T} &=\omega L_{out} -\frac{\omega C_{con} R_{con}^2 }{1+\left( {\omega C_{con} R_{con} } \right)^2}- \\ &\frac{\frac{\sqrt {2} }{2}\frac{Y_0 \left( {\sinh 2\alpha +\sin 2\alpha } \right)\left( {2Y_0 R_{ct}^2 \omega -1} \right)+2R_{ct} \sqrt {\omega } \left( {\sinh ^2 2\alpha +\sin ^2 2\alpha } \right)}{C_{dl} Y_0^2 \omega ^{5/2}R_{ct} \left( {\cosh 2\alpha +\cos 2\alpha } \right)}}{\left[ {R_{ct} +\frac{\sqrt {2} }{2}\frac{\sinh 2\alpha -\sin 2\alpha }{Y_0 \sqrt {\omega } \left( {\cosh 2\alpha +\cos 2\alpha } \right)}} \right]^2+\left[ {\frac{\sqrt {2} }{2}\frac{\sinh 2\alpha +\sin 2\alpha }{Y_0 \sqrt {\omega } \left( {\cosh 2\alpha +\cos 2\alpha } \right)}-\frac{1}{\omega R_{ct} }} \right]^2} \end{aligned} $$

(A.14)