Effects of the electric field on the overall drop impact on a solid surface

Depositing drops on a solid surface without entrapping bubbles is desirable for many spray coating and printing applications. Tian et al. [J. Fluid Mech. 946, A21 (2022)] reported that an electric field can be applied to eliminate air bubble entrapments for neutral drops. Herein we provide a complete physical picture of the entire process of a drop impacting onto the solid surface under an external electric field. The electrohydrodynamic behavior during the drop impact is divided into three stages: the deformation of the drop in the electric field prior to contact, the initial contact of the drop with the substrate, and the rich postcontact phenomena including spreading, receding, jetting, and fragmentation. The results show that under the increasingly stronger electric fields, the modest drop oscillation transforms into a vertically stretched spindle. As the drop approaches the substrate, the electric stress at the south pole increases rapidly, which sharpens the bottom surface into a conical shape. The cone angle is determined by both the impact velocity and the electric field strength. After the contact, the surface electric stress tends to pull the drop upward, breaking up the drop, forming several jetting modes, and reducing the maximum spreading radius. The various drop deposition modes are summarized in a phase diagram, which sheds light on identifying appropriate electric fields for high-quality drop depositions without air bubble entrapments or jettings.

Figure 1

(a) Schematic of the experimental setup. (b) Illustration of the computational domain of a leaky dielectric drop in an electric field.

Figure 2

The impact sequences from experiment and simulation in different electric field strengths (${\mathrm{\Gamma }}_E$). The spreading and receding process with ${\mathrm{\Gamma }}_E=0$ (a), ${\mathrm{\Gamma }}_E=0.43$ (b), ${\mathrm{\Gamma }}_E=0.85$, and ${\mathrm{\Gamma }}_E=0.94$ (c). The disintegration of the drop with ${\mathrm{\Gamma }}_E=1.03$ (d). The drop impacts the substrate at $\mathrm{We}=5.1$. The experimental images and simulation images show a good agreement when ${\mathrm{\Gamma }}_E<1$. Scale bar: $1\mathrm{mm}$.

Figure 3

The electric field and flow field with different ${\mathrm{\Gamma }}_E$. Each unit figure consists of three parts: a superposition of the electric field strength $\mathit{E}$ and electric isopotentials $\mathrm{\Phi }$ (left half); a superposition of the polarization charge density ${\rho }_e$ and vector of electric stresses ${\mathit{f}}_e$ (at the drop interface); a superposition of the streamline of the velocity $\mathit{U}$ and the gauge pressure $p_g$ (right half). (a) ${\mathrm{\Gamma }}_E=0.43$, (b) ${\mathrm{\Gamma }}_E=0.68$, (c) ${\mathrm{\Gamma }}_E=0.94$. The drop impacts the substrate at $\mathrm{We}=5.1$. Scale bar: $2.5\mathrm{mm}$.

Figure 4

The variation in $D$ after entering the electric field varies with different ${\mathrm{\Gamma }}_E$. (a) $\mathrm{We}=5.1$. (b) $\mathrm{We}=8.4$. (c) $\mathrm{We}=20.2$. The moment of contacting as the reference time $t=0\mathrm{ms}$.

Figure 5

The shape of the drop at the moment of initial contact with different ${\mathrm{\Gamma }}_E$ and We. (a) The bottom cone angle ($\theta$). As the ${\mathrm{\Gamma }}_E$ increases, the $\theta$ decreases quickly. The instantaneous centroid velocity $U_C$ (b) and the bottom velocity of the drop $U_B$ (c) when the drop contacts the substrate with different ${\mathrm{\Gamma }}_E$.

Figure 6

The interference fringes of the air film captured from the bottom-view camera and the substrate wetting map obtained numerically at different ${\mathrm{\Gamma }}_E$ of three typical contact modes. Annular contact mode: (a) a normal size Newton's ring retracting into an air bubble with ${\mathrm{\Gamma }}_E=0$; (b) a smaller Newton's ring retracting into a smaller air bubble with ${\mathrm{\Gamma }}_E=0.1$ and ${\mathrm{\Gamma }}_E=0.19$. Multiple contact mode: (c) Multiple concentric circular contact lines with toroid air bubbles entrapped underneath with ${\mathrm{\Gamma }}_E=0.18$ and ${\mathrm{\Gamma }}_E=0.22$. Center contact mode: (d) A single dark circular disk without Newton's ring and air bubble with ${\mathrm{\Gamma }}_E=0.43$. [(e)–(h)] The profiles of the bottom air film ($h\sim r$) evolving with time for above contact modes. Here drop impacts the substrate at $\mathrm{We}=8.4$. Scale bar: $300\mu \mathrm{m}$. The time interval is $20\mu \mathrm{s}$ in simulations.

Figure 7

(a) Schematic of the drop initial wetting on the substrate in three contact modes. The sequential evolution of air film radius $L_E$ (b) and drop spreading radius $R$ (c) with time for different electric field strengths with different impact velocities in simulations. $L_{E0}$ and $R_0$ both decrease as the ${\mathrm{\Gamma }}_E$ increase.

Figure 8

(a) Schematic of the drop spreading on the substrate. The sequential evolution of the dimensionless spreading radius $R/a$ [(b) and (d)] and free surface height $H/a$ [(c) and (e)] with time for different electric field strengths with different impact velocities in experiments. The vertical and horizontal dotted lines in (b)–(e) are the time mark of the maximum $R/a$ and the maximum height $H/a$ with ${\mathrm{\Gamma }}_E=0$. The maximum $R/a$ decreases with stronger ${\mathrm{\Gamma }}_E$. The $H/a$ shows a sharp falling when ${\mathrm{\Gamma }}_E\ge 0.68$ because of the breakup of the sessile drop surface.

Figure 9

Three modes of drop top surface rupture with different impact conditions. (a) A small droplet with diameter at least an order of magnitude smaller than the initial drop ejected during the spreading ($\mathrm{We}=5.1, {\mathrm{\Gamma }}_E=0.85$). (b) A large drop of few hundred micrometers in diameter produced by pinch-off of the stretched drop during the receding ($\mathrm{We}=8.4, {\mathrm{\Gamma }}_E=0.68$). (c) A continuous jetting during the receding ($\mathrm{We}=8.4, {\mathrm{\Gamma }}_E=0.94$). Scale bar: $1\mathrm{mm}$.

Figure 10

The corresponding three modes of drop top surface rupture in simulations. (a) A small droplet with diameter at least an order of magnitude smaller than the initial drop ejected during the spreading ($\mathrm{We}=5.1, {\mathrm{\Gamma }}_E=0.94$). (b) A large drop of few hundred micrometers in diameter produced by pinch-off of the stretched drop during the receding ($\mathrm{We}=8.4, {\mathrm{\Gamma }}_E=0.68$). (c) A continuous jetting during the receding ($\mathrm{We}=8.4, {\mathrm{\Gamma }}_E=0.94$). Scale bar: $2.5\mathrm{mm}$.

Figure 11

Four types of drop disintegration. (a) bottom disintegration in the precontact stage ($\mathrm{We}=5.1, {\mathrm{\Gamma }}_E=1.03$); (b) bottom disintegration in the initial contact state ($\mathrm{We}=5.1, {\mathrm{\Gamma }}_E=1.28$); (c) top disintegration in the initial contact state ($\mathrm{We}=5.1, {\mathrm{\Gamma }}_E=1.28$); (d) top disintegration during spreading in the postcontact stage ($\mathrm{We}=8.4, {\mathrm{\Gamma }}_E=1.03$). The moment of drop disintegration that occurs is marked as reference time $t_d=0\mathrm{ms}$ in this picture. Scale bar: $1\mathrm{mm}$.

Figure 12

Phase diagram for drop deposition modes with different ${\mathrm{\Gamma }}_E$ and We in experiments and simulations. The solid patterns for numerical simulations and the hollow patterns for experiments.