UC Berkeley

During intaglio (gravure) printing, a blade wipes excess ink from the engraved plate with the object of leaving ink--filled cells defining the image to be printed. That objective is not completely attained. Capillarity draws some ink from the cell into a meniscus connecting the blade to the substrate, and the continuing motion of the engraved plate smears that ink over its surface. That smear behind the cell delivers a feature lacking in sharpness. Even though the smear formation occurs at micron scale, it affects the functionality at the scale of micron size printed electronics. By examining the limit of vanishing capillary number ($Ca$, based on substrate speed), we reduce the problem of determining smear volume to one of hydrostatics. Using numerical solutions of the corresponding free boundary problem for the Stokes equations of motion, we show that the hydrostatic theory provides an upper bound to smear volume for finite $Ca$; as $Ca$ is decreased, smear volume increases. The theory explains why polishing to reduce the tip radius of the blade is an effective way to control smearing. As the motion continues, smear volume under the meniscus is printed as a tail behind the cell extending back toward the blade. In the limit of vanishing $Ca$, an inner and outer analysis of the meniscus printing problem shows that the meniscus rotates around the pinned contact line at the trailing edge of the cell, and the volume under the meniscus is used to coat a film of thickness decreasing linearly in time forming a tail shape. The tail lengthens with decreasing $Ca$, owing to the concomitant increase in smear volume; the opposite is true as $Ca$ approaches $O(1)$. For small $Ca$, it is computationally expensive to use the Stokes solver to show that the physical mechanism of tail formation described by the analysis as $Ca \to 0$ is correct. Lubrication theory, on the other hand, predicts tail formation mechanism closely for contact angles over the blade close to $\uppi/2$, and this motivates us to use the lubrication model for smaller $Ca$. With the continuing motion of substrate, we show the transition from the tip region (squeeze film) to the Landau-Levich film in the form of a tail shape extending back toward the bulk meniscus (static), and this agrees with the physical picture described by the analysis as $Ca \to 0$. The results contribute to the control and understanding of smear formation mechanism during gravure printing of electronics.