Prediction of steady-state freeze front position during 3D printing of microstructures

Additive manufacturing with alloys at micro and meso-scales is an emerging technology with applications in printed, flexible or conformable electronics, solar energy and biomedical science. Among various additive manufacturing techniques, the recently introduced 3D-freeze-printing technique has the potential to revolutionize printed circuits, sensors and conformal wearable electronics. In 3D-freeze-printing, low melting point alloys are dispensed through a micro-scale nozzle on a cooled substrate and frozen simultaneously to create three-dimensional structures. The quality of the 3D printed structures relies on a continuous liquid-to-solid phase change of the printed filamentary structures through the propagation of a freeze-front. Thus to achieve stable printing of complex 3D structures, the study of freeze-front position is critical. In this paper, we present a thermal model to predict the steady-state freeze-front position during the freeze-printing process. Thermal modeling can aid in predicting parameter dependent process response and help achieve robust 3D printing with high accuracy and high throughput. Owing to the disparate length scale and nature of materials, quasi 1D energy equations are developed to model the printed structures and dispensing nozzle, while 2D energy equations are used to model the heat transfer from the liquid alloy reservoir. A finite volume method with a modified variable time-step approach is used for the discretization of governing differential equations to find the freeze front. The validity of this model was experimentally tested for three cases: vertical printed structures with and without a connected nozzle and a horizontally printed structure with a connected nozzle. It was shown that the model predicted the freeze-front position with high accuracy for various substrate temperatures and process conditions.