The inverse problem of capillary filling, as defined in this work, consists in determining the capillary radius profile from experimental data of the meniscus position $l$ as a function of time $t$. This problem is central in diverse applications, such as the characterization of nanopore arrays or the design of passive transport in microfluidics; it is mathematically ill posed and has multiple solutions; i.e., capillaries with different geometries may produce the same imbibition kinematics. Here a suitable approach is proposed to solve this problem, which is based on measuring the imbibition kinematics in both tube directions. Capillary filling experiments to validate the calculation were made in a wide range of length scales: glass capillaries with a radius of around $150\mu \mathrm{m}$ and anodized alumina membranes with a pores radius of around $30\mathrm{nm}$ were used. The proposed method was successful in identifying the radius profile in both systems. Fundamental aspects also emerge in this study, notably the fact that the $l(t)\propto t^{1/2}$ kinematics (Lucas-Washburn relation) is not exclusive of uniform cross-sectional capillaries.

• Received 12 December 2013

DOI:https://doi.org/10.1103/PhysRevLett.112.134502