Contributions to modeling and applications of superhydrophobic surfaces for self-assembly of biological materials

16 1.3.1. Introduction Droplet evaporation was extensively studied during the last three decades. Deegan et al. 7 studied the so-called “coffee stain” effect, the well-known phenomena related to the drying droplet of coffee on a hydrophilic surface, and derived analytically the averaged velocity profile along the z- axis (normal to the surface). Popov 8, 9 modelled the evaporation of a droplet of colloidal solutions upon wettable or not wettable surfaces. In particular, Popov 9 deduced a general model taking into account different values of CAs, humidity levels, interface droplet area and others important parameters. Hui et Al 10 used the finite element method to figure out the velocity field in a drying droplet. Afterwards they integrated the Marangoni effect in their model 11 that takes into account the shear stresses along the droplet-air interface due to the temperature gradient. A programming code in Matlab was written by me, inspired by these previous results, that uses the finite element method for the calculations of the flows inside the drying droplets, also on SHSs. The program allows solving a model in which there are some approximations, i.e. the gravity. The partial differential equations used for the system are eq. (14) for the evaporation and eq. (18) for the convective flows inside the droplet. I will explain how to couple the two equations for solving the change of phase between solid and air. I will also study the dynamics of evaporation of a drying droplet upon a hydrophilic and a SHS by considering the Popov’s model. 9 1.3.2. Convective flow inside a drying droplet. A numerical simulation by finite element method was made to obtain the evaporation flux profile along the free surface of a drying droplet of pure water upon a hydrophilic surface that affects the inner flow. The model was programmed in Matlab. As written in the previous section, the system can be considered with an axisymmetric geometry (figure 1.5). Aqueous vapor concentration changes and becomes steady rapidly respect to the time of evaporation and whenever the droplet shape changes. For this reason the evaporation was thought to be a quasi-steady-state process 12 and I solved the diffusion problem given by the gradient of vapor concentration between the free surface and the atmosphere. The eq. (14) becomes:  7 R. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel, T. A. Witten. “Capillary flow as the cause of ring stains from dried liquid drops”. J. F. Institute. Nature(1997), 389, 827-829; 8 Y. O. Popov. “Singularities, universality, and scaling in evaporative deposition patterns”. PhD student’s thesis. Department of Physics. University of Chicago (US), (2003); 9 Y. O. Popov. “Evaporative deposition patterns: spatial dimensions of the deposits”. Phys. Rev. E (2005), 71, 036313; 10 H. Hu and R. G. Larson. “Analysis of the Microfluid Flow in an Evaporating Sessile Droplet”. Langmuir (2005), 21, 3963-3971; 11 H. Hu and R. G. Larson. “Analysis of the Effects of Marangoni Stresses on the Microflow in an evaporating Sessile Droplet”. Langmuir (2005), 21, 3972-3980; 0 C  (19)