Films, layers, and droplets: The effect of near-wall fluid structure on spreading dynamics

We present a study of the spreading of liquid droplets on a solid substrate at very small scales. We focus on the regime where effective wetting energy (binding potential) and surface tension effects significantly influence steady and spreading droplets. In particular, we focus on strong packing and layering effects in the liquid near the substrate due to underlying density oscillations in the fluid caused by attractive substrate-liquid interactions. We show that such phenomena can be described by a thin-film (or long-wave or lubrication) model including an oscillatory Derjaguin (or disjoining or conjoining) pressure and explore the effects it has on steady droplet shapes and the spreading dynamics of droplets on both an adsorption (or precursor) layer and completely dry substrates. At the molecular scale, commonly used two-term binding potentials with a single preferred minimum controlling the adsorption layer height are inadequate to capture the rich behavior caused by the near-wall layered molecular packing. The adsorption layer is often submonolayer in thickness, i.e., the dynamics along the layer consists of single-particle hopping, leading to a diffusive dynamics, rather than the collective hydrodynamic motion implicit in standard thin-film models. We therefore modify the model in such a way that for thicker films the standard hydrodynamic theory is realized, but for very thin layers a diffusion equation is recovered.

Figure 1

A sketch of the molecular configurations in a cross section through the contact line region of a drop of liquid which exhibits strong layering effects. The contact angle is $\theta \approx 0$. We identify three distinct regions, Areas 1–3, where the adsorption takes three distinct values due to the fact that when the substrate adsorption is low, the influence of molecular packing becomes important. The dashed line gives the corresponding effective film height. In Area 1, the amount adsorbed on the substrate is low, so the effective film height $h\ll \sigma$, the diameter of the molecules.

Figure 2

A plot of the binding potential in Eq. (13), calculated in Ref. [13] using DFT. The global minimum is labeled “0”, as this state corresponds to Area 1 in Fig. 1, where the film thickness is almost zero. The local minima at larger $h$ are labeled “1”, “2”, “3”, and “4” and represent one, two, three, and four layers of molecules, respectively.

Figure 3

A plot of binding potential $g_3\left(h\right)$ in Eq. (14) with $a=0.01,b=\pi /2,c=1,k=2\pi ,d=0.02$ (solid red line), $d=0$ (dotted black line), and $d=-0.02$ (dashed blue line).

Figure 4

A plot of binding potential $g_4\left(h\right)$ in Eq. (15) with $a=0.01,b=\pi /2,c=1,k=2\pi ,d=0.02$ (solid red line), $d=0$ (dotted black line), and $d=-0.02$ (dashed blue line).

Figure 5

A phase plane diagram when the binding potential is $g_2$ in Eq. (13) and ${\gamma }_{lv}=0.51$, which shows all possible equilibrium solutions of our thin-film equation as the maximal drop height is varied, with the Lagrange multiplier having the value $\lambda =9.8\times {10}^{-4}$.

Figure 6

A time sequence of drop profiles for a liquid drop spreading on a substrate, with binding potential $g_1$ given by Eq. (11) with $a=2.756\times {10}^{-8}$ and $b=5.453\times {10}^{-5}$. The parameters in the initial condition are chosen as $h_b=0.1081,C=60$, and $E=10$. The $t=0$ profile is centered at $x=200$ and then spreads until reaches equilibrium. The diffusive mobility coefficient ${\overline{\alpha }}_0=0$.

Figure 7

A time sequence of drop profiles for a liquid drop spreading on a substrate, with binding potential $g_2$ given by Eq. (13). The initial condition is the same as for Fig. 6 and also ${\overline{\alpha }}_0=0$. The $t=0$ profile is centered at $x=200$ and then spreads.

Figure 8

A comparison of the final equilibrium droplets with binding potentials $g_1$ and $g_2$, corresponding to Figs. 6 and 7. The equilibrium contact angle $\theta$, the surface tension ${\gamma }_{lv}$ and the initial condition are the same for both cases.

Figure 9

The time evolution of the free energy (16) as a droplet spreads to equilibrium with binding potentials $g_1$ and $g_2$, corresponding to Figs. 6 and 7. The equilibrium contact angle $\theta$, the surface tension ${\gamma }_{lv}$, and the initial condition are the same for both cases.

Figure 10

Equilibrium drop profiles with binding potential $g_3$ in Eq. (14) as the parameter $d$ is varied and with $a=0.01,b=\frac{\pi }{2},c=1,k=2\pi$.

Figure 11

The normalized free-energy difference for droplet spreading under the influence of binding potential $g_3$ for various values of the parameter $d$. The expression of $g_3$ is given by Eq. (14), with $a=0.01,b=\frac{\pi }{2},c=1$, and $k=2\pi$.

Figure 12

A time sequence of drop profiles for a liquid drop spreading on a substrate, with binding potential $g_3$ given by Eq. (14), with $a=0.01,b=\frac{\pi }{2},c=1,k=2\pi ,d=0.02$. The initial condition is chosen as $h_b=0.2522,C=6,E=10$, and $x_f=200$. The corresponding free-energy time evolution is displayed as the solid (red) line in Fig. 11.

Figure 13

(a) A sequence of equilibrium droplet profiles with binding potential $g_3$ and varying $a$, as given in the legend. We also have $d=0.02$, and $h_b$ is set as the lowest positive minimum of the binding potential (for $a=0.01,h_b=0.2523$, for $a=0.02,h_b=0.2402$, for $a\ge 0.04,h_b=0.2282)$. (b) A plot of the time evolution of the normalized free-energy differences of the dynamics leading to the formation of the drops in (a). (c) The corresponding binding potentials.

Figure 14

(a) Equilibrium drop profiles with the binding potential $g_4$ in Eq. (15), as the parameter $d$ is varied. We take $a=0.01,b=\frac{\pi }{2},c=1,k=2\pi$, and $d=\{0.02,0,-0.02\}$. (b) The corresponding normalized free-energy difference for the spreading of the droplets.

Figure 15

(a) A sequence of equilibrium droplet profiles with binding potential $g_4$ and varying $a$, as given in the legend. We also have $d=0.02$, and $h_b$ is set as the lowest positive minimum of the binding potential (for $a=0.01,h_b=0.300$, for $a=0.02,h_b=0.2642$, for $a=\{0.04,0.05\},h_b=0.2402$, and for the remaining values of $a,h_b=0.2282)$. (b) A plot of the time evolution of the normalized free-energy differences of the dynamics leading to the formation of the drops in (a). (c) The corresponding binding potentials.

Figure 16

(a) The final equilibrium state obtained for the same volume of liquid on the surface undergoing both spreading and dewetting on a background film $h=h_0$. The binding potential used is $g_2$, with initial conditions that were evolved to reach these final equilibria given in (37) for the spreading situation and (38) for dewetting. (b) The corresponding evolutions of the normalized free-energy differences, noting that for spreading $F\left(0\right)=16.89$ and $F\left(\infty \right)=0.046$, whereas for dewetting $F\left(0\right)=0.44$ and $F\left(\infty \right)=0.15$.

Figure 17

The free energy over time during droplet spreading on a substrate with binding potential $g_2$ covered by a film of thickness $h_b\ne 0$ for various values of the diffusion coefficient ${\overline{\alpha }}_0$. The initial condition has $h_b=h_0=0.1081,C=60,E=10$, and $x_f=400$.

Figure 18

The free energy over time during droplet spreading on a totally dry substrate with binding potential $g_2$ for various values of the diffusion coefficient ${\overline{\alpha }}_0$. The initial condition has $h_b=0,C=60,E=10$, and $x_f=400$.

Figure 19

The free energy over time during droplet spreading on a totally dry substrate with binding potential $g_3$ for various values of the diffusion coefficient ${\overline{\alpha }}_0$. The initial condition is $h_b=0,C=6,E=10$, and $x_f=200$.

Figure 20

The free energy over time during droplet spreading on a totally dry substrate with binding potential $g_4$, for various values of the diffusion coefficient ${\overline{\alpha }}_0$. The initial condition is $h_b=0,C=6,E=10$, and $x_f=200$.

Figure 21

Equilibrium droplet profile with $g_3$ and with $x_f=200,d=0.02,a=0.3,h_b=0.2282$, discretizing with various grid spacings $dx$ to obtain the numerical solution, where $dx=\frac{x_f}{N}$ and where $N$ is the total number of grid points. The values used are $N=100,N=200,N=400,N=667,N=1000,N=2000,N=2500$, and $N=3333$. The last four of these are virtually indistinguishable.