Numerical simulations of capillary spreading of a particle-laden droplet on a solid surface

Abstract

We present a direct numerical simulation technique and some results for the capillary spreading of a particulate droplet on a solid surface which is of great importance in the industrial inkjet printing technology as an alternative to the conventional lithography process for precise particle delivery. Since the spreading of particulate droplets is quite complicated in nature, the present work focuses on 2D capillary spreading behavior with full consideration of hydrodynamic interactions as a preliminary study for the particle effect on spreading. To understand the micro-structural phenomena underlying the process, we present a finite-element based computational scheme by combining the level-set method for an accurate interface description with the interfacial tension and the equilibrium contact angle, and the fictitious-domain method for suspended particles with implicit treatment of the hydrodynamic interactions. We investigated droplet spreading by capillary force in a Newtonian fluid and discussed effects of the presence of particles on the spreading behavior along with the particle movement. The amount of spread of a particulate droplet appears smaller than that of a homogeneous fluid droplet during the spread process and this reduced rate of spreading has been interpreted the results in conjunction with the distribution of the shear rate, the angular velocity of particles, and the kinetic energy.

Introduction

The present study is motivated by the necessity for the development of a numerical simulation technique to understand the spreading behavior of a particulate droplet, a droplet containing a large number of solid particles, on a flat surface. Application of a droplet on the solid surface is widely exploited in many existing and emerging technologies such as inkjet printing and spray coating. The particulate droplet behavior is of particular importance in industrial inkjet printing technology, as it provides an alternative to the conventional lithography process for the purpose of accurate particle delivery to a solid surface.

Numerous studies on droplet spreading on a solid surface have been reported. Fukai et al., 1993, Fukai et al., 1995 reported experimental and theoretical studies on the deformation of a droplet – the splat radius and height – impinging on a flat surface at a high Reynolds number (Re ≈ 4500) in an axisymmetric moving mesh. Pasandideh-Fard et al. (2002) performed numerical simulations on the droplet impact using a volume-of-fluid (VOF) method for various surface conditions. Gunjal et al. (2005) presented numerical and experimental studies on the dynamics of a drop impact on a flat surface and consecutive spreading, rebounding or splashing. Their VOF approach captured characteristic features of the process reasonably well. Khatavkar (2005) and Khatavkar et al. (2007) employed a diffuse-interface method to study capillary and low inertia spreading of a microdroplet on a flat or pre-patterned surface at a low Reynolds number.

Although a number of studies have focused on the spreading or impact of a liquid drop, their applicability to the particulate droplet case is obviously limited due to the presence of the hydrodynamic interactions between particles and fluid. In spite of increasing interest in this area, only a few experimental studies have been reported on the particle effect. Among them, Nicolas (2005) carried out an experiment on the spreading of a droplet of a neutrally buoyant suspension containing particles of size O(100) μm and reported a reduction of the spreading factor, i.e., the ratio of the spread drop radius to the original radius. He interpreted the reduced spread in terms of the viscosity increase, due to the presence of particles, using the effective viscosity concept such as the Krieger–Doherty (KD) relationship (Larson, 1999). On the other hand, Furbank and Morris (2004) examined the particle effect on the drop formation of particle suspensions. They reported that the pinch-off of drop is strongly accelerated in the case of the particle suspension, compared with a pure liquid drop with the same effective viscosity. This would indicate that the simple scaling of viscosity may not be adequate with respect to understanding the particulate drop behavior.

In this work, we developed a direct numerical simulation technique for spreading of the particulate drop by the capillary force on a flat surface in 2D and presented some preliminary results on evolutions of the droplet shape, the particle distribution and motion, and the flow field. The term ‘direct’ has been used regarding the simulation, since hydrodynamic and inter-particle interactions are fully considered in the individual particle level. To take the full hydrodynamic interactions into account, we employ the direct simulation scheme of Hwang et al. (2004a), a distributed Lagrangian-multipler (DLM) method, in the present work (Hwang et al., 2004a, Hwang et al., 2004b). The treatment of free surfaces, which is essential for the droplet spreading simulation, in the Eulerian frame is well suited to the direct simulation technique. Though restricted to the 2D capillary spreading problem, the problem in the present study is nevertheless very complicated. Besides the hydrodynamic interaction, it is necessary to consider accurate interface tracking of the droplet during spreading along with adequate treatments of the interfacial tension and the contact angle. For the purpose of the free surface problem, we used the level-set method (Chang et al., 1996) together with the continuous surface stress formulation to capture the interface and to assign the interfacial tension on the boundary. A virtual stress tensor has been introduced on the contact point to exert a force toward the equilibrium contact angle.

Using this numerical approach, we studied the spreading behavior of a particulate drop on a solid surface, in comparison with the homogeneous drop case, and discussed the effect of the presence of particles by examining the microscopic shear rate distribution, particle motions, and kinetic energies of drop fluids. The paper is organized as follows: first, in Section 2, we define the problem and introduce the basic governing equations. In Section 3 we explain numerical methods and implementation techniques. Then, in Section 4, we present results for the spreading of particulate droplets and the effects of the particle on the spreading behavior. Finally, we end up this paper with conclusions.