Diffusion–reaction modelling of modulated hydrogen loading

Figures & data

Figure 1. Schematic display of the solute density, $\rho (x,t)$, inside the sample during hydrogen charging (a) and subsequent modulation (b). (a) Applying a constant electrode potential $U=U_0$ for a sufficient time interval leads to a homogeneous hydrogen concentration ${\rho }_{\mathrm{eq}}$ inside the sample. Before equilibrium is reached, the difference ${\rho }_{\mathrm{eq}}-{\rho }_s$ results in a current density $j_{\mathrm{s}}\left(t\right)$, Equation (10(10) $j_{\mathrm{s}}=\frac{j_{\mathrm{ex}}}{{\rho }_{\mathrm{eq}}}({\rho }_{\mathrm{eq}}-{\rho }_{\mathrm{s}}).$(10) ), of solutes through the surface, where ${\rho }_s$ is the time-dependent density underneath the surface. (b) After reaching equilibrium, the electrode potential $U\left(t\right)$, Equation (2(2) $U\left(t\right)=U_0+\hat{U}\sin \left(\mathrm{\omega t}\right),$(2) ), is modulated with respect to $U_0$ giving rise to an oscillating variation of the solute density imposed by the surface, ${\rho }_{\mathrm{eq}}\left(t\right)={\rho }_0+\hat{\rho }\sin \left(\mathrm{\omega t}\right)$, Equation (11(11) ${\rho }_{\mathrm{eq}}\left(t\right)={\rho }_0+\hat{\rho }\sin \mathrm{\omega t}={\rho }_0(1-\frac{\mathrm{zF}}{\mathrm{RT}}\hat{U}\sin \mathrm{\omega t}),$(11) ). As a result of the modulation, a time-dependent density profile, $\rho (x,t)$, inside the sample evolves, which is determined by the reaction- and diffusion-limitation of the process, Section 2.3. The difference between ${\rho }_{\mathrm{eq}}\left(t\right)$ and ${\rho }_s\left(t\right)=\rho (x=l,t)$ leads again to a current density $j_{\mathrm{s}}\left(t\right)$ of particles through the surface (Equation (12(12) $j_{\mathrm{s}}\left(t\right)=\frac{j_{\mathrm{ex}}}{{\rho }_0}\left({\rho }_{\mathrm{eq}}\right(t)-\rho (l,t\left)\right),$(12) ) and boundary condition according to Equation (14(14) ${D\frac{\mathrm{\partial }\rho (x,t)}{\mathrm{\partial }x}|}_{x=l}=\kappa \left({\rho }_{\mathrm{eq}}\right(t)-\rho (l,t\left)\right),$(14) )). (c) In comparison to (b) the situation for the entirely diffusion-limited case (boundary condition according to Equation (16(16) $\rho (x=l,t)={\rho }_0+\hat{\rho }\sin \left(\mathrm{\omega t}\right)$(16) )). Note that figure parts b and c pertain to the special case of the diffusion–reaction model with the initial condition, Equation (18(18) $\rho (t=0)={\rho }_0^{\mathrm{ini}}.$(18) ), $\rho (t=0)={\rho }_0^{\mathrm{ini}}={\rho }_0$.

Figure 2. Amplitude (related to $\hat{\rho }/{\rho }_0$) and phase angle ${\varphi }_0$ of the modulating part of the relative fraction $n\left(t\right)$ of loaded species in dependence of frequency-parameter $M\propto \sqrt{\omega }$ (plate: Equation (30(30) $M=\sqrt{\frac{2l^2\omega }{D}},$(30) ); cylinder: Equation (42(42) $M=\sqrt{\frac{2r_0^2\omega }{D}},$(42) )) for parameter (a) L = 1 and (b) $L\to \mathrm{\infty }$ (plate: Equation (25(25) $L=\frac{\mathrm{\kappa l}}{D},$(25) ); cylinder: Equation (40(40) $L=\frac{\kappa r_0}{D}$(40) )). Amplitude of plate: black dash-dotted line, Equation (28(28) $\begin{array}{rl}n\left(t\right)& =1-\sum _{n=1}^{\mathrm{\infty }}\frac{2L^2}{{\overline{\text{β}}}_n^2({\overline{\text{β}}}_n^2+L^2+L)}[\frac{{\rho }_0-{\rho }_0^{\mathrm{ini}}}{{\rho }_0}-\frac{1}{2}\frac{\hat{\rho }}{{\rho }_0}\frac{M^2{\overline{\text{β}}}_n^2}{{\overline{\text{β}}}_n^4+\frac{1}{4}{M}^4}]\exp (-\frac{D{\overline{\text{β}}}_n^2}{l^2}t)\\ & +2\frac{\hat{\rho }}{{\rho }_0}\frac{L}{M}\frac{C}{\sqrt{A^2+B^2}}\sin (\mathrm{\omega t}+{\varphi }_0)\end{array}$(28) ); amplitude of cylinder: black dotted line, Equation (41(41) $\begin{array}{rl}n\left(t\right)& ={\displaystyle 1-\sum _{n=1}^{\mathrm{\infty }}\frac{4L^2}{{\overline{\text{β}}}_n^2({\overline{\text{β}}}_n^2+L^2)}[\frac{{\rho }_0-{\rho }_0^{\mathrm{ini}}}{{\rho }_0}-\frac{1}{2}\frac{\hat{\rho }}{{\rho }_0}\frac{M^2{\overline{\text{β}}}_n^2}{{\overline{\text{β}}}_n^4+\frac{1}{4}{M}^4}]\exp (-\frac{D{\overline{\text{β}}}_n^2}{r_0^2}t)}\\ & +2{\displaystyle \frac{\hat{\rho }}{{\rho }_0}\frac{L}{M}{\left\{2\left(\right(J_1^{\mathrm{Re}}{)}^2+\left(J_1^{\mathrm{Im}}{)}^2\right)\right\}}^{1/2}\times \{\frac{1}{2}{M}^2\left(\right(J_1^{\mathrm{Re}}{)}^2+\left(J_1^{\mathrm{Im}}{)}^2\right)}\\ & {+L^2\left(\right(J_0^{\mathrm{Re}}{)}^2+\left(J_0^{\mathrm{Im}}{)}^2\right)+\mathrm{ML}\left[J_1^{\mathrm{Im}}\right(J_0^{\mathrm{Re}}-J_0^{\mathrm{Im}})-J_1^{\mathrm{Re}}(J_0^{\mathrm{Re}}+J_0^{\mathrm{Im}}\left)\right]\phantom{\frac{1}{2}}\}}^{-1/2}\\ & \times \sin (\mathrm{\omega t}+{\varphi }_0)\end{array}$(41) ); ${\varphi }_0$ for plate: red dashed line, Equation (32(32) ${\varphi }_0=\arctan \frac{A}{B}+0\text{or}\pi$(32) ); ${\varphi }_0$ for cylinder: red solid line, Equation (45(45) ${\varphi }_0=\arctan \frac{L[-J_1^{\mathrm{Im}}(J_0^{\mathrm{Re}}-J_0^{\mathrm{Im}})+J_1^{\mathrm{Re}}(J_0^{\mathrm{Re}}+J_0^{\mathrm{Im}}\left)\right]-M\left(\right(J_1^{\mathrm{Re}}{)}^2+\left(J_1^{\mathrm{Im}}{)}^2\right)}{L\left[J_1^{\mathrm{Re}}\right(J_0^{\mathrm{Re}}-J_0^{\mathrm{Im}})+J_1^{\mathrm{Im}}(J_0^{\mathrm{Re}}+J_0^{\mathrm{Im}}\left)\right]}+0\text{or}\pi$(45) ); limiting case (b): Equations (48(48) ${\varphi }_0=\arctan \frac{\sin M-\sinh M}{\sin M+\sinh M}$(48) ), (49(49) ${\varphi }_0=\arctan \frac{-J_1^{\mathrm{Im}}(J_0^{\mathrm{Re}}-J_0^{\mathrm{Im}})+J_1^{\mathrm{Re}}(J_0^{\mathrm{Re}}+J_0^{\mathrm{Im}})}{J_1^{\mathrm{Re}}(J_0^{\mathrm{Re}}-J_0^{\mathrm{Im}})+J_1^{\mathrm{Im}}(J_0^{\mathrm{Re}}+J_0^{\mathrm{Im}})}.$(49) ), (50(50) $\hat{n}=\sqrt{2}\frac{\hat{\rho }}{{\rho }_0}\frac{1}{M}\frac{\cosh M-\cos M}{\sqrt{{\sin }^{2}M+{\sinh }^{2}M}}$(50) ), and (51(51) $\hat{n}=2\frac{\hat{\rho }}{{\rho }_0}\frac{1}{M}\frac{\sqrt{2\left(\right(J_1^{\mathrm{Re}}{)}^2+\left(J_1^{\mathrm{Im}}{)}^2\right)}}{\sqrt{\left(\right(J_0^{\mathrm{Re}}{)}^2+\left(J_0^{\mathrm{Im}}{)}^2\right)}}.$(51) ). Parameters: $D=3\times {10}^{-11}$ m${}^2$/s; $l=r_0={10}^{-3}$ m; $\kappa =3\times {10}^{-8}$ m/s (a), $\to \mathrm{\infty }$ (b).

Figure 3. Density of loaded species $\rho (x,t)$, Equation (34(34) $\rho (x,t)={\rho }_{\mathrm{unmod}}(x,t)+{\rho }_{\mathrm{mod}}^{\mathrm{transient}}(x,t)+{\rho }_{\mathrm{mod}}^{\mathrm{periodic}}(x,t),$(34) ), with respect to ${\rho }_0$ in dependence of x (normalised to the half thickness l of the plate) for various values $\mathrm{\omega t}$ (see legend in part (a1)). Surface located at $x/l=1$. Initial condition ${\rho }_0^{\mathrm{ini}}={\rho }_0$, i.e. homogeneous density inside the plate in equilibrium with surface. Modulation amplitude $\hat{\rho }/{\rho }_0=0.1$. Parameters: $D=3\times {10}^{-11}$ m${}^2$/s; $l={10}^{-3}$ m; $\kappa =3\times {10}^{-5}$ m/s (for $L={10}^3$) or $3\times {10}^{-8}$ m/s (for $L=1$). (a1) $L={10}^3$: M = 5 ($\omega =5.4\times {10}^{-4}$ 1/s, black colour) and M = 20 ($\omega =6.0\times {10}^{-3}$ 1/s, red colour). (a2) Without transient part, i.e. ${\rho }_{\mathrm{mod}}^{\mathrm{periodic}}(x,t)$, Equation (35(35) ${\rho }_{\mathrm{mod}}^{\mathrm{periodic}}(x,t)=2\hat{\rho }L\sqrt{\frac{P_1^2+P_2^2}{R_1^2+R_2^2}}\sin (\mathrm{\omega t}+{\varphi }_0)$(35) ), exclusively; same parameters as for (a1). (b) M = 2 ($\omega =6.0\times {10}^{-5}$ 1/s): $L={10}^3$ (black colour) and L = 1 (red colour). (c) Periodic part ${\rho }_{\mathrm{mod}}^{\mathrm{periodic}}(x,t)$ for $L={10}^3$ (black colour, with M = 10) and L = 1 (red colour, with M = 5). (Note legend in (c): solid line corresponds to $\mathrm{\omega t}=3\pi /4$, unlike the other figure parts).

Figure 4. Real (Re$\left(Z_{\mathrm{f}}\right)/\left(\hat{U}/\right(e\hat{\rho }\left)\right)$) and negative imaginary part (−Im$\left(Z_{\mathrm{f}}\right)/\left(\hat{U}/\right(e\hat{\rho }\left)\right)$) of complex impedance, Equation (59(59) $Z_{\mathrm{f}}\left(\omega \right)=\frac{\hat{U}}{e\hat{\rho }}\frac{M}{2\mathrm{lL\omega }}\frac{\sqrt{A^2+B^2}}{C}\exp (-i{\varphi }_Z).$(59) ), of planar electrode in Nyquist-plot presentation for various values of L, Equation (25(25) $L=\frac{\mathrm{\kappa l}}{D},$(25) ). The intersection with the real axis, i.e. −Im$\left(Z_{\mathrm{f}}\right)/\left(\hat{U}/\right(e\hat{\rho }\left)\right)$, marks the limit $\omega \to \mathrm{\infty }$, whereas −Im$\left(Z_{\mathrm{f}}\right)/\left(\hat{U}/\right(e\hat{\rho }\left)\right)\to -\mathrm{\infty }$ marks the limit $\omega =0$. For easy visualising, simple values in power of tens are used for the parameters, D (unit: m${}^2$/s), κ (unit: m/s), and l (unit: m). Variation of κ with D = 1 and l = 1: κ = 1 (L = 1, black solid), 10 (L = 10, blue dashed), ${10}^3$ ($L={10}^3$, blue dashed-dotted); variation of D with $\kappa =1$ and l = 1: D = 1 (L = 1, black solid), 10 (L = 0.1, black dashed), ${10}^3$ ($L={10}^{-3}$, black dashed-dotted). Relations for the $Z_{\mathrm{f}}$-values at the frequency limits $\omega =0$ and $\omega \to \mathrm{\infty }$ are given in Table 1.

Table 1. Amplitude $Z(=U_0/I_0)$, real part of amplitude Re$\left(Z_{\mathrm{f}}\right)$ and phase ${\varphi }_Z$ of the complex impedance, Equation (59(59) $Z_{\mathrm{f}}\left(\omega \right)=\frac{\hat{U}}{e\hat{\rho }}\frac{M}{2\mathrm{lL\omega }}\frac{\sqrt{A^2+B^2}}{C}\exp (-i{\varphi }_Z).$(59) ), and of the phase ${\varphi }_0$, Equation (32(32) ${\varphi }_0=\arctan \frac{A}{B}+0\text{or}\pi$(32) ), for $\omega =0$ and $\omega \to \mathrm{\infty }$ in the general case of diffusion and reaction limitation as well as in the special case of either reaction or diffusion limitation. Planar electrode; L, M according to Equations (25(25) $L=\frac{\mathrm{\kappa l}}{D},$(25) ) and (30(30) $M=\sqrt{\frac{2l^2\omega }{D}},$(30) ), respectively.

	$\omega =0$ (M = 0)			$\omega \to \mathrm{\infty }$ ($M\to \mathrm{\infty }$)
limitation	Diffusion and reaction	Reaction	Diffusion	Diffusion and reaction	Reaction	Diffusion ${}^{\left(\mathrm{a}\right)}$
		$L\to 0$	$L\to \mathrm{\infty }$		$L\to 0$	$L\to \mathrm{\infty }$
		$D\to \mathrm{\infty }$	$\kappa \to \mathrm{\infty }$		$D\to \mathrm{\infty }$	$\kappa \to \mathrm{\infty }$
${\varphi }_0$	0	0	0	$-\pi /2$	$-\pi /2$	$-\pi /4$
${\varphi }_Z$	$\pi /2$	$\pi /2$	$\pi /2$	0	0	$\pi /4$
$Z/\left(\hat{U}/\right(e\hat{\rho }\left)\right)$)	$1/\left(\mathrm{l\omega }\right)$	$1/\left(\mathrm{l\omega }\right)$	$1/\left(\mathrm{l\omega }\right)$	$1/\kappa$	$1/\kappa$	$1/\sqrt{\mathrm{\omega D}}$
Re$\left(Z_{\mathrm{f}}\right)/\left(\hat{U}/\right(e\hat{\rho }\left)\right)$	$(1+L/3)/\kappa$	1/κ	$l/\left(3D\right)$	$1/\kappa$	$1/\kappa$	$1/\sqrt{2\mathrm{\omega D}}$

${}^{\left(\mathrm{a}\right)}$ valid for $M/L=\sqrt{2\mathrm{\omega D}/\kappa }\ll 1$.