# Short-Time Dynamics of Partial Wetting

Antony Orth

### Introduction

As the title implies, this paper[1] discusses the initial dynamics of a drop wetting a surface which it only partially wets. In contrast, the dynamics of a completely wetting system have been investigated and it has been found that the scaling of the spreading radius obeys simple dimensional arguments [2]. Additionally, it has been observed that the dynamics of droplet coalescence is virtually indistinguishable from the scenario of perfect wetting, likely due to the formation of precursor films which effectively mimic the environment of a second droplet [3].

### Dimensional Analysis

The simple dimensional arguments which dictate the scaling of the spreading radius <math>r(t)</math> are as follows: The sole length scale in the problem (in addition to <math>r(t)</math>) is the initial radius of the drop <math>R</math>. The other relevant dimensional quantities are the surface tension <math>\gamma</math>, the viscosity and density of the fluid <math>\mu</math> and <math>\rho</math>, respectively, and the time after initial contact between the two surfaces involved <math>t</math>. To find the behaviour of the non-dimensional spreading radius <math>r/R</math>, we construct a list of the non-dimensional groups. The Buckingham-<math>\pi</math> theorem [4] (not terribly deep, but useful nonetheless) tells us that with 3 fundamental physical units (length, time and mass), and <math>n</math> physical parameters, that there are <math>n-3</math> dimensionless variables which describe the system. For a partially wetting drop, we list the the physical quantities involved: <math>R</math>, <math>r</math>, <math>t</math>, <math>\theta_{eq}</math>, <math>\rho</math>, <math>\gamma</math> and <math>\mu</math>. Suppose we define a nondimesional spreading radius <math>\frac{r}{R}=f(q_1,q_2,...,q_j)</math> where <math>j=6-3=3</math> and <math>f</math> is some non-dimensional function of the dimensionless quantities <math>q_i</math>. If we limit ourselves to a inertial regime where viscous forces are negligible, then we have only 2 dimensionless quantities <math>q_1</math> and <math>q_2</math>. Observe that <math>\theta_{eq}</math> is naturally dimensionless, so that we only have one more dimensionless parameter to find which turns out to be <math>q = \frac{t}{\sqrt{\rho R^3/\gamma}}</math>, hence the assertion on the second page of the paper that <math> \frac{r(t)}{R}=f(\frac{t}{\sqrt{\rho R^3/\gamma}},\theta_{eq})</math>. Likewise, in a viscous regime where intertial effects is subdued, we have (for possibly a different function <math>f</math>): <math> \frac{r(t)}{R}=f(\frac{t}{\sqrt{\mu R/\gamma}},\theta_{eq})</math>.

Nefeli's comment: What is <math>\gamma</math>? Other than that, I am a big fan of dimensional arguments.

### Experiment

The experiment performed by the authors involved expelling a drop of liquid quasi-statically onto silicon wafers treated with various chemicals in order to vary the equilibrium contact angle of the liquid on the surface. The equilibrium contact angles <math>\theta_{eq}</math> spanned a complete spectrum from complete wetting to no wetting (<math>\theta_{eq}=0^{\circ}</math> to <math>180^{\circ}</math>). The key measured parameter here is the spreading radius <math>r(t)</math> which is obtained by analysing photographs taken by a high-speed camera (67,000 fps or <math>\sim 15\mu s</math> per frame). Along with a given <math>\theta_{eq}</math>, the authors also independently varied the drop radius <math>R</math>. The resulting plots, when non-dimensionalised (that is, plotting <math>r/R</math> against <math>\frac{t}{\sqrt{\rho R^3/\gamma}}</math>) collapse onto unique curves for each <math>\theta_{eq}</math>. This implies that we are indeed in a regime dictated heavily by inertial forces (and capillary forces); the authors also comment that no such collapsing of the <math>r</math> vs. <math>t</math> curves occur for viscous scaling, corroborating the result that viscous effects are negligible.

Nefeli's comment: I'm curious to know what were the surface treatments involved. Did they render the silicon more hydrophilic or hydrophobic?

### Results and Discussion

The authors then proceed to fit the scaled curves to a power law <math>r/R\propto (t/\tau)^{\alpha}</math> as is suggested by the known case for complete wetting when <math>\alpha=1/2</math> [5]. <math>\alpha</math> is seen to agree with the complete wetting result <math>\alpha=1/2</math> for <math>\theta_{eq}=0^{\circ}</math>. The dynamics slow down (ie. <math>\alpha</math> decreases) as <math>\theta_{eq}</math> increases, which at first may seem fairly reasonable, since the spreading radius will plateau at a smaller value. However, the authors also observed that the non-dimensional time at which this power-law spreading behaviour ceases is essentially independent (perhaps very weakly dependent on) of <math>\theta_{eq}</math>. It is interesting to note that the total time over which initial-wetting dynamics occur is therefore completely determined by the time scale <math>\sqrt{\rho R^3 / \gamma}</math>, whereas the speed and acceleration of the process is governed by <math>\theta_{eq}</math> (ie. the chemistry of the surface).

The authors present a simple energy conservation method to gain some further insight into the system. Consider a drop in an intertial regime. Energy conservation requires:

<math>\int_V\frac{\rho}{2}|\vec{u}(\vec{x},t)|^2dV + \gamma A(t) - \pi r(t)^2\cos(\theta{eq})=\gamma A(0)</math>

where <math>A(t)</math> is the surface area of the drop/atmosphere interface, V the volume of the drop and <math>\vec{u}</math> the interal velocity field on the drop. With a length scale set by the surface tension and intertial forces <math>l_{ci}\approx(\gamma t^2/\rho)^{1/3}</math> and the the scale of the velocity field inside the drop near the contact line <math>v\approx dr/dt</math>, we can estimate a relationship from the previous equation. We get <math>\frac{t}{r}\frac{dr}{dt}\sim(F(\theta_{eq})+\cos{\theta_{eq}})^{1/2}</math> which agrees with the observed behaviour by leading to a power-law relationship with <math>\alpha</math> only depending on <math>\theta_{eq}</math>.

The authors conclude by remarking that a more detailed theoretical analysis is needed in order to truly understand the physics behind the partial wetting process. They also point out that there may infact be a shorter time-scale involved in this process: The log-log plot of the spreading radius is essentially flat and then enters the power-law regime as the same dimensionless time, regardless of equilibrium contact angle. This is quite a curious observation, but unfortunately, resolving a smaller time scale may require too fast a camera!

Nefeli's comment: I find it very interesting the observation that ultimately, wetting is independent of the equilibrium contact angle. However, what I'm lacking in the description of this paper is outlook: why is wetting of treated silicon important? What are the potential applications?

Zachwg's comment: This is really strange. So does it turn out that there is some universal constant for <math>l_{ci}\approx(\gamma t^2/\rho)^{1/3}</math> above which spreading stops? That seems unlikely to me...