Behavior of charged and uncharged drops in high alternating tangential electric fields

The interaction of drops and electric fields occurs in many applications like electrowetting, electrospinning, atomization, but also causes unwanted effects like the aging of high-voltage composite insulators. Water drops are influenced by electric fields due to the polar properties of the water molecules. The behavior of the drops depends on several parameters like the orientation and strength of the electric field, drop volume, and frequency of the applied field. In addition, electric charges can influence the behavior of drops significantly. However, the impact of electric charges, including the interaction with the drop as well as the electric field strength, is far from being well understood. In this work, the impact of electric charges on the behavior of single sessile drops is investigated experimentally under well-defined conditions. The effects of the drop volume, electric field strength, field frequency, and electric charge of the drop are studied. The necessary amount of charge to change the behavior of drops, depending on the applied electric field and drop volume, is determined and different drop behavior regimes are identified. Depending on the boundary conditions, the drop oscillates with the same or double the frequency of the applied voltage. The different regimes are investigated for the first three oscillation modes. The obtained results will help to improve the understanding and to manipulate the behavior of uncharged and charged drops in strong electric fields.

Figure 1

One cycle of the first three modes for $n=2,3,4$ of uncharged drops. (a) Example of mode 1 ($n=2$) of a 20-$\mu \mathrm{l}$ drop at 27 Hz and $\widehat{E}=3.81\mathrm{kV}$/cm, (b) example of mode 2 ($n=3$) of a 30-$\mu \mathrm{l}$ drop at 23 Hz and $\widehat{E}=4.67\mathrm{kV}$/cm, and (c) example of mode 3 ($n=4$) of a 60-$\mu \mathrm{l}$ drop at 48 Hz and $\widehat{E}=7.37\mathrm{kV}$/cm. The videos can be found in the Supplemental Material [35].

Figure 2

Principle of acting forces depending on electric field strength and electric charge. Region I characterizes the behavior of charged and region II uncharged drops, adapted and reprinted with permission from [39].

Figure 3

Schematic of the specimen, similar to [40].

Figure 4

Schematic of the experimental setup.

Figure 5

Schematic of the drop charger [39].

Figure 6

Comparison of one cycle of an uncharged and a charged drop in resonance mode 1. (a) One cycle of an uncharged drop with a volume of $V=20\mu \mathrm{l}$ and a voltage frequency of 27 Hz at an electric field strength of $\widehat{E}=3.81$ kV/cm and (b) one cycle of a charged drop ($Q=0.646$ nC) with a volume of $V=20\mu \mathrm{l}$ and a voltage frequency of 27 Hz at an electric field strength of $\widehat{E}=4.42$ kV/cm. The videos can be found in the Supplemental Material [35].

Figure 7

Dimensionless frequency ratio depending on the characteristic ratio $\xi$ for a drop oscillating in mode 1 with $V=20\mu \mathrm{l}$.

Figure 8

Minimum dimensionless onset charge depending on the dimensionless drop volume represented by the Bond number.

Figure 9

Comparison of one cycle for an uncharged and a charged drop in resonance mode 2. (a) One cycle of an uncharged drop with a volume of $V=80\mu \mathrm{l}$ and a voltage frequency of 26 Hz at an electric field strength of $\widehat{E}=7.37$ kV/cm, (b) one cycle of a charged drop ($Q=2.76$ nC) with a volume of $V=80\mu \mathrm{l}$ and a voltage frequency of 26 Hz at an electric field strength of $\widehat{E}=2.58$ kV/cm, and (c) one cycle of a charged drop ($Q=3.36$ nC) with a volume of $V=80\mu \mathrm{l}$ and a voltage frequency of 26 Hz at an electric field strength of $\widehat{E}=7.37$ kV/cm. The videos can be found in the Supplemental Material [35].

Figure 10

Dimensionless frequency ratio depending on the characteristic ratio $\xi$ for a drop oscillating in mode 2 with $V=60\mu \mathrm{l}$. Dashed line shows the characteristic ratio ${\xi }^{*}$ for the change of the behavior, reprinted with permission from [39].

Figure 11

Minimum onset charge $Q^{*}$ for frequency change depending on the drop volume $V$ compared with $Q_{\max }$ as defined by Eq. (5) for mode 2 and different viscosities. Data for pure water are taken from [39]. Regime I includes drops with same frequency and regime II drops with double the frequency of the applied voltage. Dashed line visualizes the two regimes. The solid line limits region I due to the Rayleigh limit, therefore, the gray area includes nonphysical values.

Figure 12

Dimensionless onset charge $Q^{*}/Q_{\max }$ depending on the Bond number Bo for drops oscillating mode 2. Regime I includes drops with same frequency and regime II doubled frequency of the applied voltage. Dashed line visualizes the two regimes.

Figure 13

Comparison of one cycle for an uncharged and a charged drop in resonance mode 3. (a) One cycle of an uncharged drop with a volume of $V=60\mu \mathrm{l}$ and a voltage frequency of 48 Hz at an electric field strength of $\widehat{E}=7.37$ kV/cm, (b) one cycle of a charged drop ($Q=3.69$ nC) with a volume of $V=60\mu \mathrm{l}$ and a voltage frequency of 48 Hz at an electric field strength of $\widehat{E}=2.58$ kV/cm, and (c) one cycle of a charged drop ($Q=2.72$ nC) with a volume of $V=60\mu \mathrm{l}$ and a voltage frequency of 48 Hz at an electric field strength of $\widehat{E}=7.37$ kV/cm. The videos can be found in the Supplemental Material [35].

Figure 14

Dimensionless frequency depending on the characteristic ratio $\xi$ for a drop oscillating in mode 3 with $V=20\mu \mathrm{l}$. Gray area shows the transition regime.

Figure 15

Dimensionless onset charge $Q^{*}/Q_{\max }$ depending on the Bond number Bo for drops oscillating in mode 3. Regime I includes drops with same frequency and regime II with double the frequency of the applied voltage. Dashed line visualizes the two regimes.

Figure 16

Calibration of the Faraday cup. Comparison of applied and measured charges including the uncertainty of the measurements. The inset shows the range from $Q=1\times {10}^{-9}$ C to $Q=10\times {10}^{-9}$ C in detail.

Figure 17

Detected drop contour and contact angles.

Figure 18

Example of the analysis for a drop of a volume $V=20\mu \mathrm{l}$ and a frequency of $f=27\mathrm{Hz}$. (a) Determined contact angles of the drop depending on time for several cycles. ${\theta }_r$ and ${\theta }_l$ are the right and left contact angle, respectively. (b) Single-sided amplitude spectrum of the determined contact angle ${\theta }_r$. Signal consists of several signals of frequencies higher than $f=27.7\mathrm{Hz}$.

Figure 19

Mean measured surface potential of cleaned surface of 10 measurements.