The characteristic equation giving the eigenvalues for Eq. 24 is ( X r n l ,~ -1) (Xm2.2 -1) -m1.2 m2.1 X2 = 0 ml,l f ((ml.1-m2,2I2 + 4m1,2 rn2,1)1/2 2(m1,1 m2,z -m1.z m2.d which yields X = (25) and the eigenvectors are In order to construct a composition path in the (Cl -&)-plane, one chooses a Composition near the boundary, then evaluates Eqs. 25 and 26 from the definitions of the mi,j and the relations ci,t = Ci + c, -=-( C1.m Cl,d 3 ) ' C2,mCz,d ci erties and base velocity on dynamic contact angles, by plunging a gelatin-subbed polyester tape into various liquids. The effect of the third phase, usually air, was studied by replacing the air with immiscible oils. As there is no readily accessible characteristic length, only two dimensionless ratios of forces can be involved. By dimensional analysis the dynamic contact angle should be correlated with the capillary number, Nca = p V/a, a physical properties number, Npp = gp4/pa3, and ratios of the physical properties of two fluids, and perhaps to the static contact angle.

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In any given coating system the maximum coating speed occurs at that point when, for a slight increase in web speed, a coating can no longer be made, or when ribs (evenly spaced downweb lines) start occurring in the coated web. Similarly, at a given speed, the minimum wet thickness is that below which a coating cannot be made, or ribs start occurring. These two limitations are really one and the same and are known as the low flow limit of coatability. In slide coating one or more coating solutions are metered and fed through slots onto an inclined plane, where they flow down in laminar motion and then jump across a small gap onto an upward-moving web. Pulling a vacuum under the coating bead, as suggested by Beguin (1954), increases the ease and range of coatability. Deryagin and Levi (1 964) were the first to point out that when one exceeds the maximum coating speed air is entrained, and the dynamic contact angle reaches 1 80°.Garin and Vachagin (1 972) studied the minimum coating thickness on a slide coater. Their results can be expressed asTallmadge et al.( 1 979) also studied the limits of coatability, and their results can be given aswhere a = 0.13-0.17, and b = 0.3-0.4. onto the web. His results may possibly be applicable to a slide coater. His low flow limits of coatability can be written as (4)Higgins and Scriven (1980) modified Ruschak's theory to take into account viscous pressure drop, but the general conclusions are much the same. ExperimentalThe Newtonian liquids tested and their properties are listed in Table 1. The stainless steel slide coating head was inclined 30° to the horizontal, and formed an included angle with the base of 70°. In all cases the base was polyester with a surface layer of gelatin. All the coatings were made at room temperature, approximately 24OC. The coated width was 11.4 cm.In each test the liquid flow rate of the coating was fixed at a given value, the web speed a t which the base moved was adjusted down to obtain a good coating, and then the speed was increased until a good coating could no longer be maintained. The highest speed a t which there was good coating with no ribbing was the maximum coating speed. As the web (base) speed increased the upstream meniscus, pinned to the lip of the slide a t one end, became more and more extended. Ribs would sometimes form before the onset of air entrainment. At times, one or both edges would neck in to give a thicker, narrower coating. In other cases the bead would break because the liquid could not bridge the gap uniformly, and one or more rivulets would be coated. And in some cases air entrainment showed up as dry patches interspersed with coated areas. Chatter marks were sometimes observed and were disregarded. With the higher surface tension liquids-the aqueous solutions-many of the coatings de-wet shortly beyond the bead. This type of breakdown after the bead region was disregarded in determining the limits

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The wettability of polymeric tape surfaces such as polystyrene, polyethylene, and polypropylene does not significantly affect air entrainment velocities. However, rougher surfaces, as suggested by Scriven ( I 982), show significantly higher air entrainment velocities.Air entrainment velocities are related to maximum coating speeds in coating operations and have been studied using flexible tapes plunging vertically into pools of liquid by Perry (1 967), Burley and Kennedy (1 976 a, b), Kennedy and Burley (1977), Burley and Jolly (1982, 1984), and Gutoff and Kendrick (1982). These studies indicate that liquid viscosity has the most significant effect on the air entrainment velocity, which decreases with the 0.67-0.78 power of the viscosity. Gutoff and Kendrick found no significant effect on surface tension, but Burley and his coworkers believe that air entrainment velocities increase with the 0.38 power of surface tension. The wettability of the polymers they studied did not vary enough to determine surface effects. This work was carried out to study the effects of surface wettability and roughness on air entrainment velocities.A number of plastic and paper tapes were slit I6 rnm wide and tested in the apparatus shown in Figure 1. The tape first contacted grounded tinsel, and then passed over grounded metal rollers to reduce any static charges (Burley and Jolly, 1984) before plunging vertically into the pool of liquid. Various aqueous solutions and pure organic liquids were used. The tape velocity was increased (using a variable speed D C motor) until air entrainment was observed for the particular side of the tape studied. The velocity a t this point was determined by measuring the length of tape passing through the bath in a fixed time interval 10-30 s, and was found to vary between 0.07-1 .O m/s at air entrainment.To compare the tape surface roughness and wettability effects, all air entrainment velocity data were normalized to that for the polystyrene tape for each particular solution. The root mean square roughnesses of the tapes were measured on a Dektak surface Profilometer. These results are tabulated in Table 1. Although the data show more scatter than one would like, it is very clear that the rough surfaces of the uncoated sides of the paper tapes and the rough surface polyethylene coated paper show very significantly higher air entrainment velocities than do the smoother tapes. The air entrainment velocities seem to increase with root mean square roughness. This agrees with the suggestion of Scriven (1982) that near the dynamic wetting line with rough surfaces, air can "escape" through the valleys between peaks in the surface.Although not as clearly shown, the data also indicate that surface wettability, which varies from the more wettable polysty-