Quantifying the dynamic spreading of a molten sand droplet using multiphase mesoscopic simulations

Upon coming into contact with a solid surface, a liquid droplet spreads rapidly during the early moments due to inertial/capillary effects before the viscous dissipation slows it down. The temporal evolution of the spreading radius depends on the viscosity of the liquid drop. For low-viscosity liquids, the spreading radius follows a power-law, whereas for higher viscosity liquids it scales linearly with time with additional logarithmic corrections. In this work, the spreading dynamics of molten sand is investigated at isothermal conditions. The molten sand is a mixture of Calcia, Magnesia, Alumina, and Silicate, commonly referred to as CMAS, and is characterized by large viscosity, density, and surface tension. The multiphase many-body dissipative particle dynamics (mDPD) model is carefully parameterized to simulate a highly viscous molten CMAS droplet at ${1260}^{\mathrm{o}}\mathrm{C}$. Three-dimensional (3D) simulations were carried out at different initial drop sizes and equilibrium contact angles. Despite its unique properties, the spreading behavior of molten CMAS is in good agreement with theory and experiments of viscous coalescence of drops. Importantly, the two distinct spreading regimes are observed in the mDPD simulations. Due to the large viscosity, a slower but a nonunique spreading rate is observed in the inertial regime. However, the spreading rate in the viscous regime is in agreement with Tanner's law. The spreading radius remains unaffected by the initial drop size and collapses onto a master curve under viscous time scaling in agreement with theory and experiments. For different equilibrium angles, the spreading rate is observed to be nearly identical in the inertial regime. This indicates a universal spreading behavior during the early stages of spreading unaffected by both the initial drop size and the equilibrium contact angle. The contact line velocity was also computed to assess its relation with the dynamic contact angle. The dynamic contact angle data collapse when plotted as a function of the capillary number, displaying a remarkable agreement with Hoffman's description of dynamic contact angle evolution.

Figure 1

Left: Experimental measurement of the evolution of temperature $T$ and wetting contact angle ${\theta }_{\mathrm{mean}}$ of a molten CMAS drop, where the molten temperature of the CMAS is $1260{}^{\mathrm{o}}\mathrm{C}$. Right: the setup of multiphase DPD simulations of a 3D CMAS droplet spreading dynamics (isothermal process) on a hydrophilic surface with equilibrium contact angle ${\theta }_{\text{eq}}={39}^{\circ }$.

Figure 2

(a) Setup of the DPP flow. The particles are colored by their velocities along the $x$-direction. (b) The mDPD and the exact solution of the flow at steady state. (c) Viscosity $\nu$ computed from the curve-fit to the mDPD solution for different cutoff distances $r_D$. The open star corresponds to $r_D=1.45$ and $\nu =29.093$ in mDPD units, i.e., the values used for the molten CMAS drops.

Figure 3

(a) Setup used in the thin liquid film method. (b) The distribution of $\sigma$ for different attraction parameter ($A$) values. (c) Mean values of $\sigma$ for different $A$. The open star corresponds to $A=-40$ and $\langle \sigma \rangle =9.287$ in mDPD units, i.e., the values value used for the molten CMAS drops.

Figure 4

Equilibrium contact angle is plotted as a function of the mDPD liquid-solid attraction parameter $A_{\text{ls}}$. The open stars correspond to $A_{\text{ls}}=-28,-30,$ and $-32$, i.e., the parameters used in the simulations.

Figure 5

The effect of (a) initial drop size $R$ and (b) equilibrium contact angle ${\theta }_{\text{eq}}$ on the spreading radius is shown on a log-log plot.

Figure 6

A cross-section view (left) and an aerial view (right) of the spreading dynamics of a 3D CMAS drop with a radius of $R=0.17\mathrm{mm}$ on a hydrophilic surface with ${\theta }_{\text{eq}}={39}^{\circ }$.

Figure 7

Nondimensional spreading radius as a function of time rescaled using the viscous timescale is shown on a log-log plot for (a) $R$ = 0.136 mm, 0.17 mm and 0.204 mm at ${\theta }_{\text{eq}}={55}^{\circ }$ and (b) ${\theta }_{\text{eq}}={36}^{\circ },{55}^{\circ }$, and ${70}^{\circ }$ at $R$ = 0.17 mm. The least-squares fit to the power law in the inertial and viscous regimes are shown in dashed black lines along with the exponent $\alpha$ of the power law.

Figure 8

Spreading rate as a function of (a) time and (b) nondimensional spreading radius $r/R$ is shown on a semilogarithmic plot. The dashed line in panel (b) corresponds to Eq. (15). The spreading data of pure glycerin drop ($R=0.5\mathrm{mm},\mu =1.12\mathrm{Pa}·\mathrm{s},\sigma =0.063\mathrm{N}/\mathrm{m}$, and $\rho =1262\mathrm{kg}/{\mathrm{m}}^3$) from Eddi et al. [8] is represented by the open circles.

Figure 9

Time evolution of the apparent contact angle ${\theta }_{\text{ap}}$.

Figure 10

(a) The apparent contact angle ${\theta }_{\text{ap}}$ is plotted as a function of the contact line velocity $u_{\text{cl}}$ for drops with different equilibrium contact angles ${\theta }_{\text{eq}}$. (b) The apparent contact angle ${\theta }_{\text{ap}}$ is plotted as a function of the capillary number Ca. The solid black curve is the model proposed by Kistler and is given by Eqs. (18, 19, 20).