Capillary rise in a microchannel of arbitrary shape and wettability: hysteresis loop

Capillary rise in an asymmetric microchannel, in which both contact angle (wettability) and open angle (geometry) can vary with position, is investigated based on free-energy minimization. The integration of the Young-Laplace equation yields the general force balance between surface tension and gravity. The former is surface tension times the integration of cos θ(u) along the contact line, where θ(u) depicts the local difference between contact angle and open angle. The latter comes from the total volume right underneath the meniscus. For the same channel height, multiple solutions can be obtained from the force balance. However, the stable height of capillary rise must satisfy stability analysis. Several interesting cases have been studied, including short capillary, truncated cone, hyperboloid, and two different plates. As the tube length is smaller than Jurin's height, the angle of contact will be tuned to fulfill the force balance. While only one stable state is seen for divergent channels, two stable states can be observed for convergent channels. Three regimes can be identified for the plot of the stable height of capillary rise against the channel height. The higher height dominates in the short channel regime, while the lower height prevails in the tall channel regime. However, both solutions are stable in the intermediate regime. Surface Evolver simulations and experiments are performed to validate our theoretical predictions. Our results offer some implications for water transport to the tops of tall trees. A small bore at the uppermost leaf connected to a larger xylem conduit corresponds to a convergent channel, and two stable heights are possible. The slow growth of the tree can be regarded as a gradual rise of the convergent channel. Consequently, the stable height of capillary rise to the top of a tall tree can always be achieved.