Formulación del problema

In this subsection, we explain the principle of image processing method used for predicting the mass drop in drying experiments. First, one has to mention that this technique is made possible because the transparent nature of the micromodels.

CHAPTER 2. MATERIALS AND METHODS 25

Figure 2.13: Light deviation by a meniscus. Here the case of a liquid ring located around a cylinder is shown as an example. The quasi-parallel light rays from the LED backlight are refracted when encountering the liquid meniscus interface. Bi- narization (and subsequent inversion of the gray levels) is therefore straightforward.

Also, the ombroscopy configuration used, with the micromodel positioned between the LED panel (which provides an almost parallel light) and the camera is the preferred one when it comes to detecting liquid-gas interfaces.

A typical image obtained with the CCD camera is shown in Figure 2.12a) (the images are saved in a 16-bit tiff format). On such an image, the menisci appear as the darkest regions. In fact, the curved menisci deviate the almost parallel light coming from the backlight, see Figure 2.13. The contrast between the dark menisci region and the bulk fluid region (liquid or air) is large enough so that applying a binarization operation is easy, leading to the image shown in Figure 2.12b, which is then inverted and filled, Figure 2.12c and d (inversion is necessary for the built-in Matlab function imfill to work).

Before explaining how to obtain the mass of liquid in the micromodel using such images, it must be mentioned that the very good spatial resolution allows us to perform precise measurements at the scale of the cylinders. Figure 2.14a shows a close-up on a liquid film that does not extent over all the cylinders of a spiral. Figure 2.14b shows the gray value profiles obtained along line 1 and 2 highlighted in Figure 2.14a. Such profiles show that the contrast is very good: a wide range of binarization threshold can be chosen to isolate the meniscus region. Also, quantitative information, on the width of the liquid bridges for instance, could be obtained from such images.

We now turn to the measurement of the liquid volume contained within the micromodel. The starting point is a binarized, filled image as the one shown in

CHAPTER 2. MATERIALS AND METHODS 26

Figure 2.14: a) Zoom on a liquid bridge; b) Light intensity profile along line 1 (in blue) and line 2 (in black).

Figure 2.12d. The total area of the white region, A, can be readily obtained by image processing (using the Matlab regionprops built-in function). The same algorithm can be used to detect any cylinder that is no longer wet by the fluid and thus appear as an isolated circle. Knowing the total number of cylinders in the pattern (661 in the case of the Fibonacci pattern), one can have a first estimate of the liquid volume in the micromodel: V = A × h − Nw× Vp where Nw is the measured

number of cylinders embedded in the liquid cluster and Vp = πhd2c/4 the volume of

a cylinder. Then, one has to note that the white region of area A seen in Figure 2.12d contains the total projected area of the menisci. This total projected menisci area, Am, can be obtained from the image shown in Figure 2.12c (before the filling

operation). In the menisci region, the fluid is not present over the full height h of the micromodel and this has to be taken into account. In fact, the formula above giving V over-estimates the liquid volume, whereas the alternate expression V = (A − Am) × h − Nw× Vp would underestimate it.

To estimate the amount of liquid in the menisci region, we used the software Surface Evolver to simulate the capillary shape of a liquid bridge located in between two cylinders. Surface Evolver and the way we used it will be presented in the following Chapter. For the moment, let us just consider a typical liquid bridge shape as shown in Figure 2.15,top and two particular outlines of this shape, Figure 2.15,bottom.

CHAPTER 2. MATERIALS AND METHODS 27

outline 1

outline 2

Figure 2.15: Top: typical liquid bridge capillary shape obtained from Surface Evolver (see next Chapter for details). The liquid bridge is trapped between two cylinders that appear in red and the liquid-gas interface appears as meshed with triangular elements. Bottom : two outlines of the capillary shape, at z = 0 (in black, bottom plane) and z = h/2 (mid-plane, in yellow).

Surface Evolver simulation; let’s denote it VSE. Let’s also introduce A1 and A2,

the areas enclosed by the outlines 1 and 2 shown in Figure 2.15,bottom (minus the projected areas of the two cylinders). Following the image processing procedure explained above, the two above-mentioned formulas to compute V would come down to measure a liquid volume A1× h (over-estimation) or A2× h (under-estimation).

The volume of gas in the meniscus region is A1 × h − VSE and the volume of the

region between the two outline is (A1 − A2) × h. Consequently, the liquid content

in the meniscus region is 1 − λ where λ = (A1h − VSE)/(h(A1− A2)). The value

λ can be computed from Surface Evolver simulation of the liquid bridge shape and post-processing of the obtained shape (to get A1 and A2). It depends on the spacing

w

 between the cylinders, the liquid volume trapped in the liquid bridge and on the

contact angle, as seen in Figure 2.16. Finally, in the general case introduced above, the volume of liquid V in the cluster is estimated to be A×h−(1−λ)Am×h−Nw×Vp,

taking λ equals to a single average value of 0.16.

This technique for measuring the liquid content in the micromodel was validated against mass measurements performed with the balance. In Figure 2.17, the balance data are compared to data obtained from image processing with and without the

CHAPTER 2. MATERIALS AND METHODS 28

Figure 2.16: λ values as a function of the liquid bridge volume used in Surface Evolver, for 3 different center-to-center spacing w

 between the cylinders (made

dimensionless with 2dc) : 0.7, 0.9 and 1.1 (marked with triangles, circles and squares

respectively) and for different contact angle, θ = 0, 10, 20 (red, blue and black symbols respectively). Note the liquid volume is made dimensionless by dividing by dp× h × w.

specific correction for the menisci region discussed above. If the latter is taking into account, data from the balance and from image processing are similar. Note that two different experiments were performed for this validation test: one with balance only and a second one with the backlight but no weighting, as already explained. Note that in the following, evaporation rate measurement from image processing will be mostly used for the experiments performed with the small Fibonacci spiral pattern.

CHAPTER 2. MATERIALS AND METHODS 29

Figure 2.17: Evolution versus time of the mass of ethanol in a large Fibonacci pattern micro-model. Dots: data from the balance. Circles and squares: data from image processing. Circles do not take into account the menisci region correction (ρl(A × h − Nw× πhd2c/4)) whereas squares do (rhol(A × h − (1 − λ)Am× h − Nw×

πhd2

Chapter 3

Numerical techniques

In this chapter, we present the two numerical tools that were used in the present work. First, we deal with the energy minimization software called Surface Evolver. We used it to simulate liquid-gas interface shapes obtained in several situations of interest in the present study, such as the case of a liquid ring around a cylinder, a liquid bridge located between two cylinders, a chain of liquid bridges, etc... For liquid bridges, the equilibrium interface shape obtained in Surface Evolver was then exported and used in COMSOL Multiphysics to compute the Stokes within the liquid and get access to the viscous flow resistance β.

3.1

Surface Evolver: presentation and a first test

case

Surface Evolver (SE), created by Kenneth A. Brakke1, is an open source inter- active computer program that minimizes a surface energy when subject to specific constraints [5]. The initial surface shape is described as consisting of vertices, direc- tive edges and facets. The surface energy can be linked to surface tension, gravity and other user-defined energies. Some constraints can be imposed: geometrical constraints on vertex position or integrated quantities such as target volumes or pressures. The minimization process is implemented by evolving the surface with an energy gradient descent method.

In this section we first examine the case of the computation of a sessile droplet shape by SE. This simple example will help us to explain how SE works, by recov- ering some basic results on the droplet mean interface curvature. More example and pieces of information can be found in the Surface Evolver user manual2_.