2D-Sintering Kinetics of Two Model Fluids as Drops Jean-Damien Muller, † Mosto Bousmina, ‡,§ and Abderrahim Maazouz* ,†,§ LMM/IMP, UMR CNRS #5223, UniVersite ´ de Lyon, INSA de Lyon, Ba ˆ timent Jules Verne, 69100 Villeurbanne, France, Canada Research Chair on Polymer Physics and Nanomaterials, Department of Chemical Engineering, CREPEC, LaVal UniVersity, Sainte-Foy Que ´ bec, G1K 7P4 Que ´ bec, Canada, and Hassan II Academy of Science and Technology, Rabat, Morocco ReceiVed October 23, 2007; ReVised Manuscript ReceiVed January 8, 2008 ABSTRACT: The aim of the present work was to examine the effect of the rheological behavior of polymer liquids on the kinetics of the sintering process between two drops of the same liquid put in close vicinity of a solid surface. Nine samples of three different natures have been used: three poly(dimetylsiloxane)s (PDMS) of various viscosities, two polybutenes (PB) of different viscosities and four Boger fluids. The sintering experiments were carried out with a PTFE (polytetrafluoroethylene) substrate at room temperature. The surface tension of the samples was measured with the sessile drop method and the sintering process was captured by a CCD camera and the variation in time of the neck radius during was compared to Bellehumeur et al. model. The effects of viscosity, surface tension and relaxation time are discussed. 1. Introduction When two drops of a liquid are brought into close contact, they coalesce, leading to a bigger drop. Coalescence may occur in emulsions with drop-matrix morphology during flow or at rest after cessation of flow. Such a process is very complex and involves the drainage of the thin film of the matrix entrapped in the gap between the two drops. Another simple coalescence process occurs during the sintering of two drops in a film-free state as illustrated in Figure 1. Such a situation can be generated by depositing two liquid drops in close vicinity on a solid substrate. Among the popular models for describing such film- free sintering process in Newtonian liquids was proposed by Frenkel: 1 Here, x, r, t, γ, and η represent the neck radius, the initial particle radius, the time of sintering, the surface tension, and the viscosity of the liquid, respectively. Some refinements of the previous model were given by Eshelby. 2 The approach was then extended to account for the viscoelastic effects. A simple modification of Frenkel model was proposed by Lontz 3 who introduced a simple relaxation process involving one relaxation time: Pokluda et al. 4 modified the model by taking into account the change in particle radius during the coalescence process. Coalescence in liquids obeying convected Maxwell model was proposed by Bellehumeur et al. 5 Under the quasi-steady state approximation, the Bellehumeur et al. model may be written as follows: Here η 0 is the zero-shear viscosity and r 0 the uniform initial radius of the drops. K 1 and K 2 are geometrical parameters given by θ is designed as the sintering angle as illustrated on Figure 1.  is the Maxwell coefficient that takes the value +1 for the upper convected Maxwell model (UCM). It was found that the steady-state UCM model does not describe well the sintering process of all viscoelastic materials. An effort was thus made to develop a model in which the transient rheology was taken into account. 6,7 It was concluded that the influence of viscoelasticity was limited to the first step of the coalescence process. For long times, the transient model converges to the Newtonian solution. This can be explained by the following three steps: (1) at short time, the behavior is * Corresponding author. Telephone: +33 4 72 43 63 32. Fax : +33 4 72 43 85 15. E-mail address: abderrahim.maazouz@insa-lyon.fr. † LMM/IMP, UMR CNRS #5223, Universite ´ de Lyon, INSA de Lyon. ‡ Canada Research Chair on Polymer Physics and Nanomaterials, Department of Chemical Engineering, CREPEC, Laval University. § Hassan II Academy of Science and Technology. x r ) ( 3 2 γt ηr ) 0,5 (1) x r ) ( 3 2 γt ηr ) 0,5 1 1 - e (-t/λ) (2) Figure 1. Geometrical illustration of the sintering process of two drops at (a) t ) 0 s and (b) t > 0 s. The parameters r, x, and θ represent the radius, the neck radius, and the sintering angle, respectively. 8(λK 1 θ′) 2 + ( 2λK 1 + η 0 r 0 γ K 1 2 K 2 29 θ′ - 1 ) 0 (3) K 1 ) sin θ (1 + cos θ)(2 - cos θ) (4) K 2 ) 2 -5/3 cos θ sin θ (1 + cos θ) 4/3 (2 - cos θ) 5/3 (5) 2096 Macromolecules 2008, 41, 2096-2103 10.1021/ma702349s CCC: $40.75 © 2008 American Chemical Society Published on Web 02/22/2008