Daniel M. Anderson and Geoffrey B. McFadden, ACMD
In classical models of solidification there is typically a solid phase and a liquid phase, separated by an infinitely thin solidification front. This interface has associated with it, for example, surface energy, and is the location of latent heat release during solidification. Solutions to such models in conjunction with experimental results provide a wealth of insight into the understanding of physical phenomena. However, since the interface position must be determined as part of the solution, often only solutions with very special interface shapes (e.g. planar, spherical, nearly planar, or nearly spherical) can be obtained. Solutions become exceedingly difficult to obtain as the interface morphology becomes more complicated.
Phase-field models use a different approach in which a sharp interface is replaced by continuous variations which are measured by a new variable, the phase field. This variable is governed by a partial differential equation over the entire domain and is coupled to other variables such as temperature or concentration. The position of the interface is determined by the value of the phase-field variable -- the interface location does not need to be tracked explicitly. Consequently, phase-field models are easier to implement computationally than sharp-interface models, especially when complex interface morphologies are present.
Contact-line, and, in particular, moving contact-line problems form a large class of free-boundary problems and are a subject of considerable technological importance. The behavior of a liquid column in a capillary tube, the spreading of droplets on heated substrates, and wetting and coating flows in the production of thin films are examples. Contact lines are also present in most solidification and crystal growth systems where the melt, its solid, and a third phase (e.g. an ambient gas in containerless systems or a container wall in directional solidification systems) meet. At NIST, such problems arise in determining the spreading behavior of solder on a metallic substrate, or the penetration of a fluid into a porous medium such as concrete. These are multi-phase systems with complex interface morphologies which pose significant difficulties when formulated as sharp-interface systems. A phase-field description, however, has the potential to overcome these difficulties.
As a first step in formulating phase-field models for these systems, the investigators plan to develop a phase-field model for a static contact line representing the junction of two liquid phases and a solid phase. There exist many studies of contact-line phenomena using sharp-interface models; these include treatments based on continuum mechanics, statistical mechanics and molecular dynamics. A phase-field approach provides an alternate description wherein the transition from the macroscopic length scales characteristic of the sample size to the length scales of the tri-junction region are affected by the inclusion of gradient energy terms to produce diffuse rather than sharp interfaces.
Promising results from the static contact-line model will suggest further studies, including the treatment of moving contact lines with the incorporation of fluid flow into the model, the treatment of tri-junctions in systems undergoing phase transitions, and the treatment of chemical reactions between the phases, such as those occurring in soldering phenomena.