Effect of ac electric field on the dynamics of a vesicle under shear flow in the small deformation regime

Vesicles or biological cells under simultaneous shear and electric field can be encountered in dielectrophoretic devices or designs used for continuous flow electrofusion or electroporation. In this work, the dynamics of a vesicle subjected to simultaneous shear and uniform alternating current (ac) electric field is investigated in the small deformation limit. The coupled equations for vesicle orientation and shape evolution are derived theoretically, and the resulting nonlinear equations are handled numerically to generate relevant phase diagrams that demonstrate the effect of electrical parameters on the different dynamical regimes such as tank treading (TT), vacillating breathing (VB) [called trembling (TR) in this work], and tumbling (TU). It is found that while the electric Mason number ($\text{Mn}$), which represents the relative strength of the electrical forces to the shear forces, promotes the TT regime, the response itself is found to be sensitive to the applied frequency as well as the conductivity ratio. While higher outer conductivity promotes orientation along the flow axis, orientation along the electric field is favored when the inner conductivity is higher. Similarly a switch of orientation from the direction of the electric field to the direction of flow is possible by a mere change of frequency when the outer conductivity is higher. Interestingly, in some cases, a coupling between electric field-induced deformation and shear can result in the system admitting an intermediate TU regime while attaining the TT regime at high $\text{Mn}$. The results could enable designing better dielectrophoretic devices wherein the residence time as well as the dynamical states of the vesicular suspension can be controlled as per the application.

Figure 1

Schematic of vesicle under ac electric field and shear flow.

Figure 2

Pure shear: (a) vesicle inclination angle in TT regime vs viscosity ratio and (b) the phase diagram showing different regimes for viscosity ratio vs Ca, in the absence of electric field ($\text{Mn}=0,\text{Ca}=1,▵=0.2$).

Figure 3

(a) Transmembrane potential ($V_{\mathrm{mem}}$, black lines) and interface charge ($Q_c$, red lines) variation with frequency for ${\sigma }_r=10$ (solid) and ${\sigma }_r=0.1$ (hollow), (b) Normal stress ($N$, black lines) and tangential stress ($S$, red lines) variation with frequency for ${\sigma }_r=10$ (solid) and ${\sigma }_r=0.1$(hollow) ($\zeta ={10}^{-7},C_{\mathrm{mem}}=50,{\epsilon }_r=1$).

Figure 4

Electric torque ($T^{\mathrm{el}}$) on a deformed vesicle in TT state, (a) variation of the electric torque with frequency (for $\psi =\pi /4$) for ${\sigma }_r=10$ (solid spheres), ${\sigma }_r=0.1$ (hollow spheres) for a vesicle inclined at $\psi =pi/4$ (b) electric torque at different inclination angles at $\text{Mn}=10$, $\omega ={10}^6$ for ${\sigma }_r=10$ (solid spheres), ${\sigma }_r=0.1$ (hollow spheres) ($\eta =3,\text{Ca}=1,\zeta ={10}^{-7},{\epsilon }_r=1,▵=0.2,C_{\mathrm{mem}}=50$).

Figure 5

(a, b) Variation of inclination angle with strength of applied electric field in TT regime for ${\sigma }_r=10$ and ${\sigma }_r=0.1$, respectively; (c) Variation of inclination angle with frequency of applied electric field in TT regime for ${\sigma }_r=10$, ${\sigma }_r=0.1$ at electric field strength $\text{Mn}=20$ ($\text{Ca}=1,\eta =3$).

Figure 6

Vesicle orientation under the effect of ac field in shear flow at fixed $\text{Mn}$. (a–f) ${\sigma }_r=10$, (g–l) ${\sigma }_r=0.1$. In each set $\omega$ increases as ${10}^3,{10}^5,{10}^7$ from left to right ($\eta =3,\text{Mn}=10,\text{Ca}=1,\zeta ={10}^{-7},{\epsilon }_r=1,▵=0.2,C_{\mathrm{mem}}=50$).

Figure 7

Vesicle orientation under the effect of ac field in shear flow at fixed $\omega$. (a–f) ${\sigma }_r=10$, (g–l) ${\sigma }_r=0.1$. In each set $\text{Mn}$ increases as $0.001,10,1000$ from left to right ($\eta =3,\omega ={10}^7,\text{Ca}=1,\zeta ={10}^{-7},{\epsilon }_r=1,▵=0.2,C_{\mathrm{mem}}=50$).

Figure 8

TR to TU transition as a function of time for ${\sigma }_r=10$: (a, b) TR motion and corresponding shape deformation at $\eta =9.3$; (c, d) transition from TR to TU motion and corresponding shape deformation at $\eta =9.4$ ($\text{Mn}=0.1,\omega ={10}^6,\text{Ca}=1,\zeta ={10}^{-7},{\epsilon }_r=1,▵=0.2,C_{\mathrm{mem}}=50$).

Figure 9

TR to TU transition as a function of time for ${\sigma }_r=0.1$: (a, b) TR motion and corresponding shape deformation at $\eta =10.6$; (c, d) transition from TR to TU motion and corresponding shape deformation at $\eta =10.7$ ($\text{Mn}=0.1,\omega ={10}^6,\text{Ca}=1,\zeta ={10}^{-7},{\epsilon }_r=1,▵=0.2,C_{\mathrm{mem}}=50$).

Figure 10

Phase diagram for transition between dynamic states (a) $\eta =10,{\sigma }_r=10$, ($\mathrm{b})\eta =12,{\sigma }_r=10$, (c) $\eta =10,{\sigma }_r=0.1$, and (d) $\eta =12,{\sigma }_r=0.1$. TR (yellow region), TT (green region), TT in intermediate frequency for ${\sigma }_r<1$ (purple), TU (blue region) (${\epsilon }_r=1,▵=0.2,C_{\mathrm{mem}}=50,\text{Ca}=1,\zeta ={10}^{-7}$).

Figure 11

$\eta$ vs $\text{Ca}$ phase diagram for transition in dynamic states (TT: green region, TR: yellow region, TU: blue region) for ${\sigma }_r=10$. (a–c) $\text{Mn}=0.01$, (d–f) $\text{Mn}=0.1$, and (g–i) $\text{Mn}=1$. In each set $\omega$ varies as ${10}^2,{10}^5,{10}^7$ ($▵=0.2,C_{\mathrm{mem}}=50,\zeta ={10}^{-7}$).

Figure 12

$\eta$ vs $\text{Ca}$ phase diagram for transition in dynamic states (TT: green region, TR: yellow region, TU: blue region) for ${\sigma }_r=0.1$. (a–c) $\text{Mn}=0.01$, (d–f) $\text{Mn}=0.1$, and (g–i) $\text{Mn}=1$. In each set $\omega$ varies as ${10}^2,{10}^5,{10}^7$ ($▵=0.2,C_{\mathrm{mem}}=50,\zeta ={10}^{-7}$).