  The spontaneous motion of liquid droplets on solid surfaces is the result of
an unbalanced surface tension force, which is sometimes called the "Marangoni
effect". This can be triggered by either a difference in surface temperature or
a heterogeneity in the topography or chemistry of the surface passively or
actively. The imbibition of liquid within capillary tubes, horizontal ice
wicking on either hydrophilic or hydrophobic substrates, and inkjet printing
for example are just some classic illustrations of where the Lucas-Washburn
equation can predict droplet behavior characteristics fairly well. In contrast,
this study reveals an example of droplet behavior not previously studied that
is not well-predicted by the Lucas-Washburn equation, namely the motion of
ferrofluid droplets in the presence of uniform magnetic field. When a
ferrofluid droplet is horizontally exposed to an external uniform magnetic
field on a biphilic surface tension gradient in the shape of a wedge, it
appears to violates the Lucas-Washburn equation which predicts that droplet
travel distance should scale with the square root of time (i.e. l~t^(1/2)).
Rather, our experimental results suggest that the movement of the ferrofluid
droplet is slower following the relationship, (l~t^(1/3)). Furthermore, due to
the relatively high viscosity of water-based ferrofluid droplets, we observed
that at the beginning of the motion, the visco-capillary effect dominates the
effects of the magnetism, and the droplets tend to follow the well-known
relationship, (l~t^(1/10)). This initial stage of droplet spreading is known as
"Tanner's Law".
