FLOW-3D has many numerical modeling capabilities ideal for engineers interested in improving coating performance. Computational simulation is an excellent way to study the relative importance and effect of different physical processes affecting coating flows. In physical tests it is not always possible to separate processes or arbitrarily adjust the magnitudes of those processes. In the examples in this section, simulations are used to investigate details of several coating issues that are relevant to many coating applications. These include such things as rivulet formation, fingering, evaporation, contact-line movement over rough surfaces and fluid absorption.
The impact and absorption of a droplet onto a paper substrate can be studied with computational fluid dynamics software. In the FLOW-3D animation shown below, the drop is 40 microns in diameter, with an initial downward velocity of 300 cm/s. The substrate is paper and is 20 microns thick with a given porosity of 30%. The droplet surface is colored by pressure and fraction of fluid of the droplet is shown as it impacts the paper.
Fingering in Liquid Films
In FLOW-3D, dynamic contact lines are modeled directly without the need to specify dynamic contact angles or the location of the contact lines. This is accomplished by using a numerical model that includes all the dynamic forces affecting fluid in small control volumes. Static contact angles are used to characterize liquid-solid adhesive forces.
An application of the power of this approach is given by the fingering observed in liquid films flowing down an inclined surface. Experimental observations show that two distinct patterns of fingering occur. One pattern, corresponding to small static contact angles (i.e., highly wetting conditions), exhibits wedge shaped fingers whose top and bottom limits both move downward. The second pattern, corresponding to large static contact angles (i.e., poorly wetting conditions), is characterized by long fingers of nearly uniform width whose top most limits are not moving downward.
In the figure on the left, the results of two FLOW-3D simulations are compared with experiments (Silvi, N. and Dussan V, E.B., “On the rewetting of an inclined solid surface by a liquid,” Phys. Fluids, 28, p.5, 1985).The simulations used the assumption of depth-averaged flow, which is justified because of the thinness of the liquid films. The only difference between the two simulations is the value of the static contact angle. The computational results do an excellent job of reproducing all the essential features of the observed flows.
Contact Line Issues
L.M. Hocking has proposed [“A moving fluid interface on a rough surface,” J. Fluid Mech., 76, 801, (1976)] that contact lines are able to move over solid surfaces because microscopic irregularities in the surface induce flow structures that may be interpreted as “velocity slip” from a macroscopic point of view.
A computational investigation of this hypothesis is easily carried out using FLOW-3D. The test selected consists of a two-dimensional solid surface with a pattern of transverse, regularly spaced, rectangular slots. The slots are 2mm deep and 10mm wide, and spaced to have 10mm wide solid pieces between them. These dimensions are typical of scratches on relatively smooth surfaces. The static contact angle between liquid and solid was chosen to be 60°. Water is the working fluid. The test consisted of placing the rough surface on the bottom of a channel of height 15mm and driving water at an average 30cm/s through the channel. The top of the channel has a free-slip boundary.
Hocking’s assertion that micro-scale disturbances can be interpreted as a kind of velocity slip when looked at from the point of larger scales is supported by the computed velocity field. This is shown graphically in the x-y plot, which gives the computed horizontal velocity distribution in the layer of control volumes immediately above the surface. With further grid refinement, the velocity above the solid portions of the surface would tend to zero, but above the slots the velocity remains non-zero. Averaging this velocity over many roughness slots results in a non-zero horizontal velocity that could be interpreted as an effective slip.
When liquid droplets containing dispersed solid material dry on a solid surface they leave the solid material as a deposit. The pattern of this deposit has important implications for many printing, cleaning and coating processes. A classic example of one type of deposit is the “coffee-ring” problem in which a ring stain is formed along the perimeter of a patch of spilled coffee (see the photo on the left). This type of ring deposit develops as a consequence of surface-tension driven flows resulting from evaporation of liquid, particularly at the drop’s perimeter (Deegan, R.D., et al, Nature 389, 827, 1997).
Drying is a critical part of the coating process; a well-applied coating can be completely undone by drying defects. During drying, temperature and solute gradients can drive flow within the coating due to density and surface tension gradients, which can potentially destroy the coating quality. FLOW-3D‘s evaporation residue model allows users to simulate drying-induced flows and reduces time spent on costly physical experimentation.
Modeling Ring Formation
Marker particles pile up at contact line where the evaporation is greatest. Only every third flow vector is plotted near the contact line to reduce clutter. A requirement for ring formation is that the contact line at the edge of a drop must be pinned.
In the figure above, a FLOW-3D simulation shows that edge pinning occurs because of deposition at a contact line where evaporation is greatest. In this example, the fluid makes a 15° static contact angle with the bottom plate. The liquid is water initially at 20°C and evaporation takes place at the liquid surface under the assumption that the surrounding air has a saturation temperature of 4°C. The vertical height of the simulation is 15μm. Evaporation cools the liquid because of heat loss due to evaporation (color indicates temperature). At the same time the solid surface heats the liquid by conduction. Evaporation is greatest in the vicinity of the contact line causing liquid to flow towards the contact line to reestablish static conditions. The net result is a deposition of suspended solid at the liquid edge where the liquid is completely evaporating.
Modeling Drop Impingement
Simulation of drop impingement. Courtesy of UC Berkely. Dan Soltman and Vivek Subramanian, Inkjet-Printed Line Morphologies and Temperature Control of the Coffee Ring Effect, Langmuir; 2008; ASAP Web Release Date: 16-Jan-2008; (Research Article) DOI: 10.1021/la7026847
Ducks are generally recognized as having perfected a means of repelling water. A classic paper, by A.B.D. Cassie and S. Baxter (Trans. Faraday Soc. 40, 1546, 1944) explains that ducks accomplish this by a particular micro-structure of their feathers and not by some chemical coating. A duck’s feather consists of barbs on either side of a main shaft. Along the barbs there are fine fibers extending out on either side. On one side the fibers have notches while on the other side there are hooks. This arrangement allows an engagement of the fibers from neighboring barbs to bond together forming a connected structure.
The barb-fiber structure has a large amount of open space. The diameter of the fibers is about 8μm but the distance between adjacent, parallel fibers is about 5 diameters, center to center. Experiments show that the static contact angle for water on the solid portion of a duck’s feather is on the order of 100°. With this combination of a non-wetting static contact angle and regular micro-structure, water placed on the feathers will easily roll off without penetrating through them.