Flat Capillary

In the case of a meniscus in a flat capillary, the integration constant, C, is determined from the condition at the capillary center  

This equation describes an equilibrium profile of the meniscus in flat capillaries. Let us consider the solution of Equation 2.23 in more detail. This equation determines the liquid profile in three different regions  

Note that Equation 2.42 coincides with Equation 2.3, but we keep this equation for convenience. In the following text, we consider only macroscopic capillaries. In these capillaries, the radius, H, is much bigger than the range of action of surface forces. Let the radius of the action of surface forces be t = 10 cm = 10(X) A = 0.1 p = 100 nm, that is, at ft t, Il(ft) = 0. In this case. Equation 2.23 can be rewritten at ft as Equation 2.41 with boundary conditions 

FIGURE 2.13 Profile of a meniscus in a flat capillary. 1 — a spherical part of the meniscus of curvature p, 2 — transition zone between the spherical meniscus and flat films in front, 3 — flat equilibrium liquid film of thickness h. Further in the text, the liquid profile inside the transition zone will be considered in more detail. 

We refer to such capillaries as thick capillaries. In the case of aqueous solutions, Y 70 dyn/cm, 10 erg d hence, 1 lO cm 10 cm. Otherwise, the capillary is referred to as a thin capillary, that is, the capillary is thin if its thickness H is in the range t H // , where t is the radius of the disjoining pressure action. According to the definition of thin capillaries, these capillaries are still big enough when compared with the radius of the action of surface forces, tj. If the capillary radius is compared with the radius of action of surface forces, t  

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With respect to carbon membranes, the molecular sieving carbon membranes, produced as unsupported flat, capillary tubes, or hollow fibers membranes, and supported membranes on a macropo-rous material are good in terms of separation properties as well as reasonable flux and stabilities, but are not yet commercially available at a sufficiently large scale, because of brittleness and cost among other drawbacks [3,6],... 

The optically flat capillaries are coated with Sigmacote. 

The cells tethered to the surface of the optically flat capillary are observed using a Nikon Optiphot (or similar) microscope with a lOOx oil immersion phase contrast objective using light >950 nm to image the cells (18). 

One key question that remains is what is the validity of these macroscopic models when we scale the size of our spherical particle down to nanoscale dimensions The liquid-liquid interface can no longer be modeled as flat (capillary waves need to be considered), and additional small-scale effects, such as discrete rather than continuous wetting of the spherical nanoparticle by the liquid molecules, need to be taken into account. Can this be reflected in line tension ... 

Fig. 9.4 Geometry of the Gahwiller experiment. A flat capillary is placed between S and N poles of a magnet. The flow velocity is directed along -z and due to condition a Cb the shear rate has only dvjdx component. The magnetic field is fixed along y and the cell with a liquid oystal may be rotated about the z-axis by 90°...

Figure 2. Flow in a flat capillary with weak anchoring at the capillary walls. The director orientation deviates from the direction of the magnetic field due to flow alignment. Upper part velocity profile, lower part vector field of the director for d>o = 90°. <1 ...

In the case of a meniscus in a flat capillary, the integration constant, C, is determined from the following condition at the capillary center, h (H) = -< , which gives C = P H, where H is the half-width of the capillary (see Section 2.3). In the case of equilibrium droplets, the constant should be selected using a different condition at the droplet apex, h = H h (H) = 0 (see Section 2.3), which results in C = Y H- P H. An alternate way of selection of the integration constant, C, is by using the transversality condition (2.21). 

In the case of equilibrium liquid drops and menisci (see Section 2.3), they are supposed to be always at equilibrium with flat films with which they are in contact with in the front. Only the capillary pressure acts inside the spherical parts of drops or menisci, and only the disjoining pressure acts inside thin flat films. However, there is a transition zone between the bnlk liquid (drops or menisci) and the thin flat film in front of them. In this transition zone, both the capillary pressure and the disjoining pressure act simultaneously (see Section 2.3 for more details). A profile of the transition zone between a meniscus in a flat capillary and a thin a-fllm in front of it, in the case of partial wetting, is presented in Figure 2.5. It shows that the liquid profile is not always concave but changes its curvature inside the transition zone. Just this peculiar liquid shape in the transition zone determines the static hysteresis of contact angle (see Chapter 3)... 

FIGURE 2.18 Complete wetting case. Schematic representation of a circular capillary meniscus (1), transition zone (2), wetting films (3) in a flat capillary. Continuation of a spherical meniscus (broken line) does not intersect either the solid walls of the capillary or the thin liquid film of thickness hg in front of the meniscus. The radius of the curvature of the meniscus, Pg, is smaller than the half-width H. 

The approach that we use in the preceding expression can also be utilized in the case of a concave meniscus in a capillary, or in the case of drops in partial wetting, that is, forming a finite contact angle, 0 with the equilibrium films on a solid substrate. In Figure 2.31, we qualitatively show the form of distribution of the excess forces/(jc), normal to the substrate for a meniscus in a flat capillary with a width of 2H (Figure 2.31a, curve 1, > 0) and for a drop (Figure 2.31b,... 

This form of the preceding equation is identical to Equation 2.47 (meniscus in a flat capillary) and Equation 2.55 (droplet on a flat substrate) deduced in Section 2.3. However, there are substantial differences between Equation 2.47, Equation 2.55, and Equation 2.264 the lower limit of integration in these equations, which corresponds to the thickness of the uniform film, is substantially different in each of them. 

DYNAMIC ADVANCING CONTACT ANGLE AND THE FORM OF THE MOVING MENISCUS IN FLAT CAPILLARIES IN THE CASE OF COMPLETE WETTING... 

FIGURE 3.9 Schematic presentation of the profile of the meniscus in a flat capillary, (a) at equihhrium, (b) velocity of motion below the critical velocity the only minimum on the hquid profile (c) velocity of motion is above the critical velocity, and 2 are thickness of the first minimnm and maximnm on the liquid profile (1) the spherical meniscus, (2) the transition zone, (3) flat equilibrium film. 

Let us consider the motion of the advancing meniscus in a flat capillary (Figure 3.9b and Figure 3.9c) from a state of equilibrium (Figure 3.9a). The zone of the flow in which the main hydrodynamic resistance is exerted takes in a region of the thicknesses of the layer h(x) above the surface of the substrate on the order of h. x-axis is directed along the capillary axis (Figure 3.9). 

For analyzing the equilibrium profile of a liquid in a flat capillary, we rewrite Equation 3.311 in the following form ... 

FIGURE 3.28 Schematics of the profile of the menisci in a flat capillary of the thickness 2H. (a) equUihrium state, no flow (h) a state of local equihhrium, when the flow is located in zone 2 ((1) the new quasi-equilihrium spherical menisci and quasi-equilibrium transition zone, (3) the equilibrium hquid film, which cannot be at the equilibrium with zone 1). 

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