Wetting considerations in capillary rise and imbibition in closed square tubes and open rectangular cross-section channels

Abstract

The spontaneous capillary-driven filling of microchannels is important for a wide range of applications. These channels are often rectangular in cross-section, can be closed or open, and horizontal or vertically orientated. In this work, we develop the theory for capillary imbibition and rise in channels of rectangular cross-section, taking into account rigidified and non-rigidified boundary conditions for the liquid-air interfaces and the effects of surface topography assuming Wenzel or Cassie-Baxter states. We provide simple interpolation formulae for the viscous friction associated with flow through rectangular cross-section channels as a function of aspect ratio. We derive a dimensionless cross-over time, T c , below which the exact numerical solution can be approximated by the Bousanquet solution and above which by the visco-gravitational solution. For capillary rise heights significantly below the equilibrium height, this cross-over time is T c  (3X e /2) 2/3 and has an associated dimensionless crossover rise height X c  (3X e /2) 1/3 , where X e =1/G is the dimensionless equilibrium rise height and G is a dimensionless form of the acceleration due to gravity. We also show from wetting considerations that for rectangular channels, fingers of a wetting liquid can be expected to imbibe in advance of the main meniscus along the corners of the channel walls. We test the theory via capillary rise experiments using polydimethylsiloxane oils of viscosity 96.0, 48.0, 19.2 and 4.8 mPa s within a range of closed square tubes and open rectangular cross-section channels with SU-8 walls. We show that the capillary rise heights can be fitted using the exact numerical solution and that these are similar to fits using the analytical visco-gravitational solution. The viscous friction contribution was found to be slightly higher than predicted by theory assuming a non-rigidified liquid-air boundary, but far below that for a rigidified boundary, which was recently reported for imbibition into horizontally mounted open microchannels. In these experiments we also observed fingers of liquid spreading along the internal edges of the channels in advance of the main body of liquid consistent with wetting expectations. We briefly discuss the implications of these observations for the design of microfluidic systems.

Capillary Penetration into Inclined Circular Glass Tubes

“…The equilibrium length tended to be slightly different to that predicted using the literature reference parameter values, but this could be corrected by using a scaling factor (∼1) for b (or equivalently (γ LV /ρ) scaling ). However, as previously reported 6 , for large radii tubes the data appeared to follow curves with much higher viscosity (or more To fit the experimental data, an increase in effective viscosity of a scaling =(η/ρ) scaling =2.628 is required. In this case, and all other viscosities and radii examined for vertical tubes, the viscogravitational solution could be fitted to the data.…”

“…14) are both proportional to b, and, if we now consider that the contact 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 indicates that: a) it cannot account solely for the increase in the scaled viscosity with radius as we proposed previously 6 , and b) that use of a static angle in the numerical and analytical fits should still be adequate to describe imbibition. However, the method above is solely experimental and does not attempt to include the effects of θ (t) in the governing equations or in the numerical fits.…”

“…One difficulty with this approach, as described in our previous work 6 , is that although the numerical fitting of eq. (1) to experimental data is possible using a constant contact angle, it necessarily requires the a parameter to be numerically adjusted (via an effective viscosity significantly larger than known value) to obtain fits that agree with the experimental data.…”

“…However, it is often assumed that the contact angle can be approximated to a constantθ=θ A (the advancing contact angle) and with this assumption eq. (1) can be fitted numerically to a set of experimental data 6 .…”

“…This was probably due to the effective viscosity compensating for the use of a fixed value of contact angle rather than including a time dependent early stage dynamic contact angle 20,6 . Here, we conclude that this effect is controlled by R/κ -1 (ϕ), where κ -1 (ϕ) is the capillary length taking into account the reduction in gravitational force due to the angle of inclination, and becomes less of an issue as a tube approaches the horizontal orientation.…”