Breakup of an electrified viscoelastic liquid bridge

We study both numerically and experimentally the breakup of a viscoelastic liquid bridge formed between two parallel electrodes. The polymer solutions and applied voltages are those commonly used in electrospinning and near-field electrospinning. We solve the leaky-dielectric finitely extensible nonlinear elastic-Peterlin (FENE-P) model to describe the dynamical response of the liquid bridge under isothermal conditions. The results show that the surface charge screens the inner electric field perpendicular to the free surface over the entire dynamical process. The liquid bridge deformation produces a normal electric field on the outer side of the free surface that is commensurate with the axial one. The surface conduction does not significantly affect the current intensity in the time interval analyzed in the experiments. The force due to the shear electric stress becomes comparable to both the viscoelastic and surface tension forces in the last stage of the filament. However, it does not alter the elastocapillary balance in the filament. As a consequence, the extensional relaxation times measured from the filament exponential thinning approximately coincides with the stress relaxation time prescribed in the FENE-P model. The above results allow us to interpret correctly the experiments. In the experiments, we measure the filament electrical conductivity and extensional relaxation time for polyethylene oxide (PEO) dissolved in deionized water and in a mixture of water and glycerine. We compare the filament electrical conductivity with the value measured in hydrostatic conditions for the same estimated temperature. Good agreement was found for PEO dissolved in water $+$ glycerine, which indicates that the change in the filament microscopic structure due to the presence of stretched polymeric chains does not significantly alter the ion mobility in the stretching direction. Significant deviations are found for PEO dissolved in deionized water. These deviations may be attributed to the heat transferred to the ambient, which is neglected in the calculation of the filament temperature. We measure the extensional relaxation time from the images acquired during the filament thinning. The relaxation times obtained in the first stage of the exponential thinning hardly depend on the applied voltage. Little but measurable influence of the applied voltage is found in the last phase of the filament thinning.

Figure 1

Sketch of the problem's formulation.

Figure 2

Sketch of the experimental configuration.

Figure 3

Dependence of the solution shear viscosity $\eta$ upon the shear rate $\dot{\gamma }$ at $22{}^{\circ }\mathrm{C}$. The dashed and dotted lines correspond to the low-shear-rate and elastic instability limits, respectively.

Figure 4

Electrical conductivity as a function of the liquid temperature. The dashed lines are the second-degree polynomial fits to the experimental data.

Figure 5

Free surface contour $R$ (a), inner $E_n^i$ and outer $E_n^o$ normal components of the electric field (b), and tangential component $E_t$ of the electric field (c). The solid black and dashed blue lines correspond to the instants $t/t_c=46.83$ and $t/t_c=77.38$, respectively.

Figure 6

Isolines of the electric potential at the instants $t/t_c=46.83$ (upper image) and $t/t_c=77.38$ (lower image). The color scale indicates the values of the dimensionless electric potential ${\varphi }^{i,o}/V_0$.

Figure 7

(a) Streamlines at the instants $t/t_c=46.83$ (left) and $t/t_c=77.38$ (right). The color scales indicate the value of the stream function divided by $v_c^2{R}_i$. The arrows indicate the flow direction. The right-hand graph shows a zoomed in view of the filament end. (b) Axial velocity profile $w\left(r\right)$ at the instant $t/t_c=77.38$.

Figure 8

Free surface contour $R\left(z\right)$ (a), and magnitude of surface conduction $I_s^{\left(\mathrm{cd}\right)}\left(z\right)$ (b), and bulk ohmic conduction $I_b\left(z\right)$ (c). The solid black and dashed blue lines correspond to the instants $t/t_c=46.83$ and $t/t_c=77.38$, respectively. $I_c={[{\gamma }^3{R}_i/\left(\rho V_0^2\right)]}^{1/2}$ is the characteristic electric current.

Figure 9

Free surface contour $R\left(z\right)$ (a), and magnitude of the forces per unit volume exerted on a slice of the liquid bridge between $z$ and $z+dz$ (b) and (c). The labels in panels (b) and (c) are those displayed in Eq. (15): tensile force, surface tension force, and electric force due surface charge, polarization, and shear electric stress. The forces have been made dimensionless with the characteristic for per unit volume $p_c/R_i$. The left-hand and right-hand graphs correspond to the instants $t/t_c=46.83$ and $t/t_c=77.38$, respectively. All the forces have been made dimensionless with $p_c/R_i$.

Figure 10

Minimum diameter $d_{min}$ as a function of time. The solid line and circles correspond to the solution to the FENE-P and Oldroyd-B models respectively. The dashed line corresponds to the exponential decay (19) with ${\lambda }_e={\lambda }_s$. The arrow indicates the instant at which the voltage is applied. The squares corresponds to the solution to the FENE-P model for a nonelectrified liquid bridge ($V_0=0$).

Figure 11

Sequence of images of an experiment conducted with G/W-PEO2M, $V_o=1\mathrm{kV}$, and ${\tilde{R}}_r=1\mathrm{M}\mathrm{\Omega }$. The origin of time is taken as the instant for which $d_{min}=250\mu \mathrm{m}$.

Figure 12

(a) Filament length ${\ell }_f$, (b) voltage $V_f$, and (c) filament conductivity $K_f$ as a function of the minimum diameter $d_{min}$ for DIW-PEO2M. The circles and triangles correspond to two experiments conducted for ($V_o=1\mathrm{kV},{\tilde{R}}_r=1\mathrm{M}\mathrm{\Omega }$).

Figure 13

Filament temperature $T_f$ as a function of the minimum diameter $d_{min}$ for G/W-PEO2M (a) and DIW-PEO2M (b). The circles, up-triangles, down-triangles, and diamonds correspond to $(V_o=2\mathrm{kV},{\tilde{R}}_r=1\mathrm{M}\mathrm{\Omega }),(V_o=2\mathrm{kV},{\tilde{R}}_r=2\mathrm{M}\mathrm{\Omega }),(V_o=1\mathrm{kV},{\tilde{R}}_r=1\mathrm{M}\mathrm{\Omega }$), and ($V_o=1\mathrm{kV},{\tilde{R}}_r=2\mathrm{M}\mathrm{\Omega }$), respectively.

Figure 14

(a) Filament length ${\ell }_f$, (b) voltage drop $V_f$, (c) electric field $E_f$, and (d) conductivity $K_f$ as a function of the minimum diameter $d_{min}$ for G/W-PEO2M (left-hand panels) and DIW-PEO2M (right-hand panels). The circles, up-triangles, down-triangles, and diamonds correspond to $(V_o=2\mathrm{kV},{\tilde{R}}_r=1\mathrm{M}\mathrm{\Omega }),(V_o=2\mathrm{kV},{\tilde{R}}_r=2\mathrm{M}\mathrm{\Omega }),(V_o=1\mathrm{kV},{\tilde{R}}_r=1\mathrm{M}\mathrm{\Omega })$, and ($V_o=1\mathrm{kV},{\tilde{R}}_r=2\mathrm{M}\mathrm{\Omega }$), respectively. In the conductivity panel (d), the solid and open symbols correspond to the values measured in the course of the experiment and in hydrostatics, respectively.

Figure 15

(a) Voltage $V_f$ and (b) minimum diameter $d_{min}$ as a function of time for G/W-PEO2M (left-hand panels) and DIW-PEO2M (right-hand panels). The circles, triangles, and squares correspond to $(V_o=0\mathrm{kV},{\tilde{R}}_r=1\mathrm{M}\mathrm{\Omega }),(V_o=1\mathrm{kV},{\tilde{R}}_r=1\mathrm{M}\mathrm{\Omega })$, and ($V_o=2\mathrm{kV},{\tilde{R}}_r=1\mathrm{M}\mathrm{\Omega }$), respectively. The solid lines are the fit of (19) to the experimental data. The origin of time is taken as the instant for which $d_{min}=250\mu \mathrm{m}$.