Capillary Filling in Open Rectangular Microchannels with a Spatially Varying Contact Angle

Capillary flow in microchannels is important for many technologies, such as microfluidic devices, heat exchangers, and fabrication of printed electronics. Due to a readily accessible interior, open rectangular microchannels are particularly attractive for these applications. Here, we develop modifications of the Lucas-Washburn model to explore how a spatially varying contact angle influences capillary flow in open rectangular microchannels. Four cases are considered: (i) different uniform contact angles on channel sidewalls and channel bottom, (ii) contact angles varying along the channel cross section, (iii) contact angle varying monotonically along the channel length, and (iv) contact angle varying periodically along the channel length. For case (i), it is found that the maximum filling velocity is more sensitive to changes in the wall contact angle. For case (ii), the contact angles can be averaged to transform the problem into that of case (i). For case (iii), the time evolution of the meniscus position no longer follows the simple square-root law at short times. Finally, for case (iv), the problem is well described by using a uniform contact angle that is a suitable average. These results provide insights into how to design contact-angle variations to control capillary filling and into the influence of naturally occurring contact-angle variations on capillary flow.