Lattice Boltzmann investigation of droplet inertial spreading on various porous surfaces

The spreading of liquid drops on solid surfaces is a wide-spread phenomenon of both fundamental and industrial interest. In many applications, surfaces are porous and spreading patterns are very complex with respect to the case on smooth surfaces. Focusing on the inertial spreading just before the Tanner-like viscous regime, this work investigates the spreading of a low-viscosity droplet on a porous surface using lattice Boltzmann numerical simulations. The case of a flat surface is first considered, and it reveals a dependence on the solid equilibrium contact angle ${\theta }_s^{eq}$, which is in good agreement with published experimental data. We conducted numerical experiments with various surfaces perforated by a regular pattern of holes of infinite length. The results show that the global spreading dynamics is independent of the porosity morphology. Through the assumption that, for wetting, the pores can be regarded as surface patches with a contact angle of ${\theta }_{\mathrm{pore}}^{eq}={180}^{\circ }$, we deduce an effective equilibrium contact angle ${\theta }_{\mathrm{eff}}^{eq}$ on the porous surface from the Cassie-Baxter law. A spreading model is then proposed to describe both a prefactor and an exponent that are similar to a flat surface whose equilibrium contact angle is ${\theta }_{\mathrm{eff}}^{eq}$. This model compares satisfactorily with a large number of numerical experiments under varying conditions.

Figure 1

Scheme of a droplet spreading numerical experiment. A nonporous surface is used as an example. A: perspective view of the initial state. The solid surface is in the $xy$ plane, and both the $xz$ and $yz$ symmetry planes are shown. B1: definition of the droplet radius $R$. B2: definition of the spreading radius $r$.

Figure 2

Normalized spreading radius on a flat surface as a function of dimensionless time for various surface equilibrium contact angles. Each curve is fitted using a power law [Eq. (1)]. Insets are power law parameters as functions of the contact angle. Left inset: power law prefactor, right inset: power law exponent. In each inset, empty markers are experimental results from Bird et al. [45], full markers are values deduced from our numerical simulations, and lines are models of the power law prefactor and exponent, $C\left({\theta }_s^{eq}\right)=1.468-0.00631\times {\theta }_s^{eq}$ and $\alpha \left({\theta }_s^{eq}\right)=0.485-0.00102\times {\theta }_s^{eq}$, respectively. The theoretical duration of inertial spreading is underlined with a vertical line.

Figure 3

Pore lattices used. C-S: square lattice of circle-sectioned pores, C-CS: centered square lattice of circle-sectioned pores, S-S: square lattice of square-sectioned pores. A unit cell is underlined with dashed lines. Both the pore period $\lambda$ and pore size $d_{\mathrm{pore}}$ are shown for each case.

Figure 4

Perspective view of the solid phase. The square lattice of circular section pores; the pore diameter is $d_{\mathrm{pore}}=20\mathrm{l}.\mathrm{u}.$, and the pore lattice spatial period is $\lambda =30\mathrm{l}.\mathrm{u}.$ A small part of the solid phase is removed to facilitate the visualization.

Figure 5

Snapshot of a droplet spreading simulation on a porous surface. The square lattice of circular section pores; the pore diameter is $d_{\mathrm{pore}}=20\mathrm{l}.\mathrm{u}.$, the pore lattice spatial period is $\lambda =30\mathrm{l}.\mathrm{u}.$, and the intrinsic solid equilibrium contact angle is ${\theta }_s^{eq}={77}^{\circ }$.

Figure 6

Snapshot of droplet spreading simulation on a porous surface with two values of the intrinsic solid equilibrium contact angle, the solid phase being removed. The square lattice of circular section pores; the pore diameter is $d_{\mathrm{pore}}=20\mathrm{l}.\mathrm{u}.$, and the pore lattice spatial period is $\lambda =30\mathrm{l}.\mathrm{u}.$

Figure 7

Normalized radius of the wetted zone as a function of dimensionless time during droplet spreading on two porous surface pairs of porosity, $\epsilon =0.20$ and $\epsilon =0.44$, respectively, and with the solid equilibrium contact angles ${\theta }_s^{eq}={41}^{\circ }$ and ${\theta }_s^{eq}={112}^{\circ }$. Both $d_{\mathrm{pore}}$ and $\lambda$ in lattice units (l.u.).

Figure 8

Normalized spreading radius on various porous surfaces with various intrinsic solid equilibrium contact angles vs the prediction of the proposed effective spreading law with ${\theta }_{\mathrm{pore}}^{eq}={180}^{\circ }$. Pore lattices are square lattice of circle-sectioned pores (C-S), centered square lattice of circle-sectioned pores (C-CS), and square lattice of square-sectioned pores (S-S). Both $d_{\mathrm{pore}}$ and $\lambda$ in lattice units (l.u.). For each of the 12 pore lattices, 5 values of ${\theta }_s^{eq}$ were used (cf. Fig. 2), which means a total of 60 curves.