formación de grietas y fisuras.

contact line

Figure 3.1.4. Triple contact line formation. (A) The evaporation of the liquid film between the sapphire window(s) and the gas bubble leads to a triple contact line close to the critical temperature Tc. (B) The heat transfer through the copper cell wall leads to a rapid evaporation of

the thin liquid film and a rapids change in the contact angle.

The analysis of the gas-liquid interface demonstrates the existence of the differential recoil vapor force [Palmer, 1976]. Besides the spreading that takes place at the copper wall, there is a similar process of liquid evaporation at the two sapphire windows. The evaporation of the thin fluid film between the sapphire windows and the gas bubble leads to the triple contact line (Figure 3.1.4A). Although the two processes are similar the major difference consists in a much

higher heat transfer through the copper wall than through the sapphire window. As a result, the speed of contact line motion is smaller than the rate of change of the contact angle.

3.2. Methods

The principle of the measurement is as follows. In the presence of any liquid-gas interface along the z-axis in the phase object (cell) Ω, the incident rays are deflected from their original direction. The deformation of the wave front causes a displacement or a distortion of the shadow of the grid on plane Π. The wetting condition of the interface at the wall requires a concave curvature near the wall while the flat interface away from the contact line requires convex curvature. We assume that between these edges of the contact line a linear approximation of the shape is a reasonable simplification that allows the estimation of the film thickness. Using this wedge model, the shadow displacement depends on the angle of the interface, as shown qualitatively in Figure 3.2.1. If the gas bubble interface is flat (α = 0), then the incident ray I0,

which is parallel to the optical axes, remains parallel to the optical axis after it crosses the cell and reaches the convergent system L at I1’. This ray is further refracted and hits the CCD plane

at I1”. The ray tracing program we wrote uses a large number N (103 - 106) of equally spaced

incident rays and computes for each incident ray the position of the point of incidence on the CCD plane. If the refracted ray passed between the grid lines, then the incidence point on the CCD plane is marked “white”, meaning that the corresponding pixel is turned white. If it happened that the refracted ray hit a grid line its intersection with the CCD plane is marked “black”, meaning that the corresponding pixel is turned black. This way we get the out-of-focus image of the grid by ray tracing (Figure 3.1.3B). If the gas-liquid interface is tilted (α≠ 0), then the incident light suffers successive refractions at the gas-liquid interface (Figure 3.2.1, point I1),

liquid-sapphire interface (Figure 3.2.1, point I2), and, sapphire-air interface (Figure 3.2.1, point

I3). As a result, the emergent ray leaves the cell at the height z3 with an angle r3. Assuming that

the refraction angle r3 is small, the convergent system would finally form the image on the CCD

plane at I6. We use the same ray tracing algorithm to find the new out-of focus grid shadow. In

the example shown in Figure 3.2.1, if the gas-liquid interface is flat then the initial ray I0 hits a

gas liquid saphire saphire cell lens _focal plane CCD F C x x x 1 2 g z 0 I1 I 2 I3 I4 I₅ I6 r3 r3 α α I1' I1'' I 0 f t δ copper wall nl ng ns ∆z H s w

Figure 3.2.1. Ray tracing through the cell and optical system. The tilt angle of the gas-liquid interface is exaggerated for illustration purposes. An incident ray I0, passing at distance z0 from

the optical axes, would remain parallel to the optical axes if the interface is not tilted with respect to the sapphire windows, and hit the lens plane at I1’. The ray is further refracted through the

focus F and reaches the CCD at I1”. If the tilt angle α of the gas-liquid interface is non-zero, then

the ray starting at I0 would hit the CCD plane at I6, following successive refractions. The

distortion of the shadow depends on the tilt angle α of the interface and its thickness, t. We computed the film thickness by measuring the displacement of a grid line (∆z) and the width (δ) of the triple contact line.

In the presence of a tilted gas-liquid interface, the same incident ray I0 would pass

through the same point I5 in the grid plane, and hit the CCD at I6. The displacement of the grid

shadow (I1”I6) depends on both the tilt angle (α) and the thickness of the gas-liquid interface (t).

Since we always observed only single contact line motions, we assumed that only one gas-liquid interface is tilted (Figure 3.2.1), and the tilt angle is α. Let z0 be the height of an incident ray

parallel to the optical axes. At point I1 Snell’s law gives

ng sin α = nl sin r1, (3.2.1)

where ng is the refraction index of the gas phase, nlis the refractive index of the liquid and r1 is the refraction angle at I1. At the point I2, applying Snell’s law we get

nl sin (α -r1 ) = ns sin r2, (3.2.2)

where ns is the index of refraction of sapphire and r2 is the refraction angle at I2. The height of the next incidence point (I2) is