Asymmetric rectified electric fields between parallel electrodes: Numerical and scaling analyses

Recent computational and experimental work has established the existence of asymmetric rectified electric fields (AREFs), a type of steady electric field that occurs in liquids in response to an applied oscillatory potential, provided the ions present have different mobilities [Hashemi et al., Phys. Rev. Lett. 121, 185504 (2018)]. Here we use scaling analyses and numerical calculations to elaborate the nature of one-dimensional AREFs between parallel electrodes. The AREF magnitude is shown to increase quadratically with applied potential at low potentials, increase nonlinearly at intermediate potentials, then increase with a constant rate slower than quadratically at sufficiently high potentials, with no impact at any potential on the spatial structure of the AREF. In contrast, the AREF peak location increases linearly with a frequency-dependent diffusive length scale for all conditions tested, with corresponding decreases in both the magnitude and number of sign changes in the directionality of AREF. Furthermore, both the magnitude and spatial structure of the AREF depend sensitively on the ionic mobilities, valencies, and concentrations, with a potential-dependent peak AREF magnitude occurring at an ionic mobility ratio of $D_{-}/D_{+}⪅5$. The results are summarized with approximate scaling expressions that will facilitate interpretation of the steady component for oscillatory fields in liquid systems.

Figure 1

Illustration of two-ion model: harmonic oscillations $\left[y_i\left(t\right)\right]$ of two oppositely charged isolated ions in response to a sinusoidal electric field for different $\delta =D_{-}/D_{+}$ values. The generated electric field at an arbitrary location $y=y_f$ due to these harmonic oscillations is asymmetrical and has a nonzero time average for $\delta \ne 1$. The dotted black curves denote the oscillations of the center of charge.

Figure 2

Schematic diagram of two parallel electrodes and not to scale comparison of different length scales. AREF varies over a diffusive length scale of ${\ell }_D=\sqrt{D/f}$ which is several orders of magnitude higher than the characteristic length scale of the Debye layer (${\kappa }^{-1}$).

Figure 3

Representative time variations of the electric field ($\tilde{E}$) (a), (b) and free charge density ($\tilde{\rho }$) (c), (d) at $\tilde{y}=0.04$ for different values of ${\mathrm{\Phi }}_0$ and ${\mathcal{L}}_D$. For all plots, $\delta =3$, $\kappa H=2600$.

Figure 4

Representative spatial distribution of AREF ($\langle \tilde{E}\rangle$) and time average free charge density ($\langle \tilde{\rho }\rangle$) at micron scale (a), (b) and close to the electrode surface (c), (d) for different values of ${\mathrm{\Phi }}_0$. For all plots, ${\mathcal{L}}_D=0.17$, $\delta =3$, $\kappa H=2600$.

Figure 5

(a) Illustrative example of AREF distribution with its peak magnitude (${\langle \tilde{E}\rangle }_{\mathrm{peak}}$) and the corresponding peak location (${\tilde{y}}_{\mathrm{peak}}$) for $\delta >1$. (b) AREF peak magnitude versus ${\mathrm{\Phi }}_0$ for different values of ${\mathcal{L}}_D$. (c) Effect of ${\mathrm{\Phi }}_0$ on the power-law exponent $a$ (${\mathcal{L}}_D=0.24$). For all plots, $\delta =3$, $\kappa H=2600$.

Figure 6

Effect of ${\mathcal{L}}_D$ on AREF behavior. (a) Spatial distribution of AREF for different ${\mathcal{L}}_D$ values (${\mathrm{\Phi }}_0=10$). (b) Peak location of AREF (${\tilde{y}}_{\mathrm{peak}}$) versus ${\mathcal{L}}_D$. (c) Peak magnitude of AREF (${\langle \tilde{E}\rangle }_{\mathrm{peak}}$) versus ${\mathcal{L}}_D$. (d) Effect of ${\mathrm{\Phi }}_0$ on the power-law exponent $b$. For all plots, $\delta =3$, $\kappa H=2600$.

Figure 7

Effect of $\delta$ on AREF behavior. (a) Spatial distribution of AREF for different $\delta$ values (${\mathrm{\Phi }}_0=10$, ${\mathcal{L}}_D=0.2$, $\kappa H=2600$). (b) Peak magnitude of AREF (${\langle \tilde{E}\rangle }_{\mathrm{peak}}$) versus $\delta$ for different conditions; markers: red circles, ${\mathrm{\Phi }}_0=10$, ${\mathcal{L}}_D=0.2$, $\kappa H=2600$; blue squares, ${\mathrm{\Phi }}_0=10$, ${\mathcal{L}}_D=0.14$, $\kappa H=2600$; green triangles, ${\mathrm{\Phi }}_0=10$, ${\mathcal{L}}_D=0.2$, $\kappa H=822$; dashed black curves are empirical fits; cf. Sec. 4. (c) Normalized peak magnitude of AREF $[{\langle \tilde{E}\rangle }_{\mathrm{peak}}/max\left({\langle \tilde{E}\rangle }_{\mathrm{peak}}\right)]$ versus $\delta$ for different ${\mathrm{\Phi }}_0$ values (${\mathcal{L}}_D=0.2$, $\kappa H=2600$). (d) ${\delta }_{\max }$ versus ${\mathrm{\Phi }}_0$ (${\mathcal{L}}_D=0.2$, $\kappa H=2600$); dashed black curve is an empirical fit; cf. Sec. 4.

Figure 8

Effect of $\kappa H$ on AREF behavior. (a) Spatial distribution of AREF for different $\kappa H$ values (${\mathrm{\Phi }}_0=10$, ${\mathcal{L}}_D=0.24$, $\delta =3$). (b) Peak location of AREF (${\tilde{y}}_{\mathrm{peak}}$) versus $\kappa H$ for two different values of ${\mathcal{L}}_D$ (${\mathrm{\Phi }}_0=10$, $\delta =3$). (c) Peak magnitude of AREF (${\langle \tilde{E}\rangle }_{\mathrm{peak}}$) versus $\kappa H$ for four different electrolytes of $\delta =3$ (${\mathcal{L}}_D=0.24$), NaCl ($\delta =1.52$, ${\mathcal{L}}_D=0.23$), KOH ($\delta =2.7$, ${\mathcal{L}}_D=0.32$), and NaOH ($\delta =3.95$, ${\mathcal{L}}_D=0.29$) (${\mathrm{\Phi }}_0=10$). (d) Effect of ${\mathrm{\Phi }}_0$ on the power-law exponent $c$ (${\mathcal{L}}_D=0.24$, $\delta =3$).

Figure 9

Effect of the ion valences ($z_{+}$ and $z_{-}$) on the AREF behavior. (a, b) Spatial distribution of AREF for different $z_{+},z_{-}$ combinations and two values of $\delta$. (c) Normalized peak magnitude of AREF (${\langle \tilde{E}\rangle }_{\mathrm{peak}}^{z_{+},z_{-}}/{\langle \tilde{E}\rangle }_{\mathrm{peak}}^{1,-1}$) for different $z_{+},z_{-}$ combinations and $\delta =1.5$. Parameters: ${\mathrm{\Phi }}_0=10$, ${\mathcal{L}}_D=0.17$, $\kappa H=2600$ for $z_{+}=\left|z_{-}\right|=1$ ($H=25\mu \mathrm{m}$, $1\mathrm{mM}$ electrolyte).

Figure 10

Numerically observed peak location of AREF (${\tilde{y}}_{\mathrm{peak}}$) versus the predictions of the empirical formula $K{\mathcal{L}}_D=K\sqrt{\overline{D}/f}/H$ for conditions that ${\mathcal{L}}_D<1/3$. The correction coefficient is $K\approx 0.83$. Range of dimensional parameters ($\approx 5000$ data points): $0.1\le {\varphi }_0\le 100k_{B}T/e$, $1\le f\le 30000\mathrm{Hz}$, $2\le H\le 100\mu \mathrm{m}$, $0.01\le \delta \le 100$, $5\times {10}^{-10}\le D_{+}\le 5\times {10}^{-9}{\mathrm{m}}^2/\mathrm{s}$, ${10}^{-5}\le c_{\infty }\le 2\times {10}^{-2}\mathrm{M}$.

Figure 11

Oscillatory behavior of AREF for different values of ${\mathcal{L}}_D$. Filled black and empty red circles are the points that AREF passes zero with positive and negative slopes, respectively. Parameters: ${\mathrm{\Phi }}_0=10$, $\delta =3$, $\kappa H=2600$.

Figure 12

Bifurcation diagrams of AREF for ${\mathcal{L}}_D$ (a) and $\delta$ (b) as varying parameters. Filled black and empty red circles are the points that AREF passes zero with positive and negative slopes, respectively. Parameters: ${\mathrm{\Phi }}_0=10$, ${\mathcal{L}}_D=0.2$ (b), $\delta =3$ (a), $\kappa H=2600$.

Figure 13

Numerically observed peak magnitude of AREF (${\langle \tilde{E}\rangle }_{\mathrm{peak}}$) versus the predictions of the empirical formula ${\langle \tilde{E}\rangle }_{\mathrm{peak}}\approx {\mathrm{\Phi }}_0^a{\mathcal{L}}_D^b\gamma {\left(\kappa H\right)}^c$ for conditions that ${\mathcal{L}}_D<1/3$. (a) Low to moderate voltages ($0<{\mathrm{\Phi }}_0\le 20$): fitting is performed using low-voltage data (${\mathrm{\Phi }}_0<5$, $K\approx 0.19$). (b) Moderate to high voltages ($20\le {\mathrm{\Phi }}_0\le 100$): Fitting is performed using all data points ($K\approx 2\times {10}^{-4}$). Range of dimensional parameters ($\approx 5000$ data points): $0.1\le {\varphi }_0\le 100k_{B}T/e$, $1\le f\le 30000\mathrm{Hz}$, $2\le H\le 100\mu \mathrm{m}$, $0.01\le \delta \le 100$, $5\times {10}^{-10}\le D_{+}\le 5\times {10}^{-9}{\mathrm{m}}^2/\mathrm{s}$, ${10}^{-5}\le c_{\infty }\le 2\times {10}^{-2}\mathrm{M}$.