Electrostatic assist of liquid transfer between plates and cavities

Roll-to-roll printing processes require formation and stretching of a liquid bridge to transfer liquid from one surface to another. Since inadequate liquid transfer can produce defects that are detrimental to printed products, electric fields are sometimes applied to enhance transfer, a method known as electrostatic assist (ESA). Because the physical mechanisms underlying ESA are not well-understood, we examine here the influence of electric fields on liquid transfer in two model geometries, both of which involve liquid bridges with moving contact lines. The bridges are axisymmetric and confined between two electrodes, one of which is flat and moves vertically upward, and the other which either is flat or has a cavity and is stationary. An electric field is applied in the axial direction, both perfect and leaky dielectric liquids are considered, and the governing equations are solved with the Galerkin finite-element method. For liquid transfer between two flat plates, application of an electric field stabilizes the liquid bridge. This allows more time for the contact line to retract on the less wettable surface and leads to an increase in liquid transfer to the more wettable surface. Tangential stresses due to surface charge can significantly enhance liquid transfer, even to the less wettable surface if the tangential stresses point toward that surface. For liquid transfer between a flat plate and a cavity, the electric field increases the pressure gradient near the contact line on the cavity wall, causing the contact line to slip and more liquid to be transferred from the cavity. Notably, the effect is more pronounced for a deep cavity, resulting in a larger percentage of liquid transferred compared to a shallow cavity. In contrast to the case of liquid transfer between two flat plates, surface charge does not have as significant an influence on liquid transfer due to the way the cavity and electric field modify the interface shape near the contact line. The results of this work illustrate the physical mechanisms through which electric fields can improve liquid transfer, and they provide guidance for optimizing ESA in industrial printing processes.

Figure 1

(a) Schematic of a gravure printing process with electrostatic assist (ESA). Ink is transferred from a cavity to a substrate, sometimes with the help of an electrostatic potential difference. (b) Schematic of a stretching liquid bridge.

Figure 2

Schematic of model geometries considered in this work. A liquid bridge between (a) two flat surfaces and (b) a flat surface and a cavity. The cavity bottom is located at $z=0$.

Figure 3

Relationship between the transfer ratio and the wettability difference for different values of $\chi$ at (a) $\text{Ca}=0.1$ and (b) $\text{Ca}=0.5$. Here, ${\theta }_{\text{top}}={60}^{\circ }$, and ${\theta }_{\text{bottom}}$ varies from ${40}^{\circ }$ to ${80}^{\circ }$. The wettability difference $\mathrm{\Delta }{\theta }_r={\theta }_{\text{bottom}}-{\theta }_{\text{top}}$.

Figure 4

(a) Time evolution of the contact line on the top and bottom plates for $\chi =0$ (solid line) and 40 (dashed line). (b) Final bridge shapes for $\chi =0$ (solid line) and 40 (dashed line). The bridge breakup times for $\chi =0$ and 40 are 5.4 and 8.3, respectively. The values of the other parameters are $\text{Ca}=0.5,{\theta }_{\text{top}}={60}^{\circ },{\theta }_{\text{bottom}}={80}^{\circ }$, and $\mathrm{\Delta }{\theta }_r={20}^{\circ }$.

Figure 5

Pressure contours and bridge shapes for the cases in which (a), (c) $\chi =0$ and (b), (d) $\chi =40$ at (a), (b) $t=0.6$ and (c), (d) $t=5$. The bridge breakup times for $\chi =0$ and 40 are 5.4 and 8.3, respectively. The values of the other parameters are $\text{Ca}=0.5,{\theta }_{\text{top}}={60}^{\circ },{\theta }_{\text{bottom}}={80}^{\circ }$, and $\mathrm{\Delta }{\theta }_r={20}^{\circ }$.

Figure 6

Electrostatic potential contours and bridge shapes for the case in which $\chi =40$ at (a) $t=0.6$ and (b) $t=5$. The values of the other parameters are $\text{Ca}=0.5,{\theta }_{\text{top}}={60}^{\circ },{\theta }_{\text{bottom}}={80}^{\circ }$, and $\mathrm{\Delta }{\theta }_r={20}^{\circ }$.

Figure 7

Schematic illustrating how surface charge and the associated tangential stresses can increase liquid transfer to the top plate. The electric field points in the negative $z$-direction and the tangential stresses point in the positive $z$-direction.

Figure 8

Transfer ratio vs wettability difference for cases in which $\chi =0(\square )$; $\chi =40,q_o=0,K^{*}=0(◯)$; $\chi =40,q_o=0.5,K^{*}=0(△)$; $\chi =40,q_o=0.5,K^{*}=1(▽)$; and $\chi =40,q_o=0,K^{*}=1(◃)$. Here, ${\theta }_{\text{top}}={60}^{\circ }$, and ${\theta }_{\text{bottom}}$ varies from ${40}^{\circ }$ to ${80}^{\circ }$. The values of the other parameters are $\text{Ca}=0.5$ and $\text{Pe}=1000$.

Figure 9

Bridge shapes at times (a) $t=4$, (b) $t=6$, and (c) $t=8$. The breakup times for the cases with $q_o=0,K^{*}=0$ (solid line); $q_o=0.5,K^{*}=0$ (dashed line); and $q_o=0,K^{*}=1$ (dotted line) are 8.3, 6.9, and 8.0, respectively. The values of the other parameters are $\text{Ca}=0.5,\chi =40,{\theta }_{\text{top}}={60}^{\circ },{\theta }_{\text{bottom}}={80}^{\circ }$, and $\mathrm{\Delta }{\theta }_r={20}^{\circ }$.

Figure 10

Relationship between transfer ratio and wettability difference for different values of $\chi$ at (a) $\text{Ca}=0.1$ and (b) $\text{Ca}=0.5$. Here, ${\theta }_{\text{top}}={70}^{\circ }$, and ${\theta }_{\text{bottom}}$ varies from ${50}^{\circ }$ to ${90}^{\circ }$. The wettability difference $\mathrm{\Delta }{\theta }_r={\theta }_{\text{bottom}}-{\theta }_{\text{top}}$.

Figure 11

(a) Time evolution of the contact line on the top (flat plate) and bottom (cavity) surfaces for $\chi =0$ (solid line) and 40 (dashed line). (b) Final bridge shapes for $\chi =0$ (solid line) and 40 (dashed line). The bridge breakup times for $\chi =0$ and 40 are 2.8 and 3.9, respectively. The values of the other parameters are $\text{Ca}=0.5,{\theta }_{\text{top}}={70}^{\circ },{\theta }_{\text{bottom}}={90}^{\circ }$, and $\mathrm{\Delta }{\theta }_r={20}^{\circ }$.

Figure 12

Pressure contours and interface shapes for (a), (b) $\chi =0$ and (c), (d) $\chi =40$ at $t=0.4$. Panels (a) and (c) show the region near the contact line on the cavity wall, and panels (b) and (d) show the region near the flat plate. The values of the other parameters are $\text{Ca}=0.5,{\theta }_{\text{top}}={70}^{\circ },{\theta }_{\text{bottom}}={90}^{\circ }$, and $\mathrm{\Delta }{\theta }_r={20}^{\circ }$. The insets show the locations of the interface shapes.

Figure 13

(a) Cavity shapes for $\delta =1$ (right) and 0.5 (left). (b) Relationship between transfer ratio and wettability difference for different values of $\chi$ and cavity depths at $\text{Ca}=0.5$. Here, ${\theta }_{\text{top}}={70}^{\circ }$, and ${\theta }_{\text{bottom}}$ varies from ${50}^{\circ }$ to ${90}^{\circ }$.

Figure 14

Pressure contours and interface shapes for (a) the shallow cavity and (b) the deep cavity at $t=0.4$. Values of other parameters are $\text{Ca}=0.5,\chi =40,{\theta }_{\text{top}}={70}^{\circ },{\theta }_{\text{bottom}}={90}^{\circ }$, and $\mathrm{\Delta }{\theta }_r={20}^{\circ }$.

Figure 15

(a) Relationship between transfer ratio and wettability difference when (i) $q_o=0,K^{*}=0(\square )$; (ii) $q_o=0.5,K^{*}=0(△)$; (iii) $q_o=0,K^{*}=1(▽)$; and (iv) $q_o=0.5,K^{*}=1(◯)$. Here, ${\theta }_{\text{top}}={70}^{\circ }$, and ${\theta }_{\text{bottom}}$ varies from ${50}^{\circ }$ to ${90}^{\circ }$. (b) Final bridge shapes for different cases when ${\theta }_{\text{top}}={\theta }_{\text{bottom}}={70}^{\circ }$. The bridge breakup times for cases (i)–(iv) are 3.56, 3.64, 3.73, and 3.55, respectively. Values of other parameters are $\text{Ca}=0.5$ and $\chi =40$.

Figure 16

(a), (b) Electrostatic potential contours and (c), (d) electrostatic contribution to the tangential stress for liquid transfer between (a), (c) flat plates and (b), (d) a flat plate and a cavity at $t=0.6$. The values of the other parameters are $\text{Ca}=0.5,\chi =40,{\theta }_{\text{top}}={\theta }_{\text{bottom}}={70}^{\circ },q_o=0.5$, and $K^{*}=1$.