2.1 Free energy at equilibrium: emergence of Cahn–Hilliard framework with height-dependent surface tension

Consider a nanometric thin liquid film sitting on a solid substrate, as shown in figure 1. When the film is perturbed, it can become unstable if the solid prefers to be in contact with the gas phase; in this configuration the uniform film has the lowest interfacial area, so dewetting lowers the bulk energy of the system at the expense of increasing its interfacial energy. We can write the energy of a half drop in one dimension as

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}=\int _{0}^{x_{0}}\unicode[STIX]{x1D719}(h,h_{x})\,\text{d}x, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D719}$ is the free energy density, $x=0$ marks the centre of the drop and $x=x_{0}$ is where the liquid, solid and gas meet, i.e. the contact line. At equilibrium, the variation of the free energy is zero, i.e. $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}=0$ :

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FF}\left(\int _{0}^{x_{0}}(\unicode[STIX]{x1D719}(h,h_{x})-Ph)\,\text{d}x\right)=0, & \displaystyle\end{eqnarray}$$

where the operator $\unicode[STIX]{x1D6FF}$ indicates variation of the functional and the Lagrange multiplier $P$ is introduced to enforce mass conservation. Expanding the above equation leads to

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{0}^{x_{0}}\left({\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}h}}-P\right)\unicode[STIX]{x1D6FF}h\,\text{d}x+\int _{0}^{x_{0}}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}h_{x}}}\unicode[STIX]{x1D6FF}h_{x}\,\text{d}x+\unicode[STIX]{x1D719}|_{x_{0}}\unicode[STIX]{x1D6FF}x_{0}=0. & \displaystyle\end{eqnarray}$$

Using integration by parts, we can rewrite the second integral in (2.3), leading to

(2.4) $$\begin{eqnarray}\displaystyle \displaystyle \unicode[STIX]{x1D6FF}\int _{0}^{x_{0}}(\unicode[STIX]{x1D719}(h,h_{x})-Ph)\,\text{d}x & = & \displaystyle \int _{0}^{x_{0}}\left[{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}h}}-{\displaystyle \frac{\text{d}}{\text{d}x}}\left({\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}h_{x}}}\right)-P\right]\unicode[STIX]{x1D6FF}h\,\text{d}x\nonumber\\ \displaystyle & & \displaystyle \left.+\left({\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}h_{x}}}\right)\unicode[STIX]{x1D6FF}h\right|_{0}^{x_{0}}+\unicode[STIX]{x1D719}|_{x_{0}}\unicode[STIX]{x1D6FF}x_{0}=0.\end{eqnarray}$$

We can further use the Taylor expansion for the boundary terms and write $\unicode[STIX]{x1D6FF}h|_{x_{0}}=-\unicode[STIX]{x1D6FF}x_{0}(\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}x)|_{x_{0}}$ .

Figure 1.

Schematic of a liquid film on a solid substrate; films with a height less than approximately $100~\text{nm}$ can become unstable to infinitesimal perturbations if the liquid does not completely wet the substrate; in such situations, the instabilities grow and the film dewets from the substrate, forming small droplets that can then coarsen to form larger drops; the shape of the small drops is governed mainly by surface tension, leading to a spherical cap shape with a macroscopic Young contact angle $\unicode[STIX]{x1D703}_{Y}$ ; deviations from the spherical cap shape can occur due to the intermolecular forces very close to the contact line. If the height of the drop becomes comparable to the capillary length $l_{\unicode[STIX]{x1D6FE}}=\sqrt{\unicode[STIX]{x1D6FE}/(\unicode[STIX]{x1D70C}g)}$ , gravitational force becomes important, levelling the drop and forming a puddle shape; gravity sets the upper limit $h_{\ast }=2l_{\unicode[STIX]{x1D6FE}}\sin (\unicode[STIX]{x1D703}_{Y}/2)$ for the puddle height.

At equilibrium, the integrand as well as the boundary terms need to be independently zero. The integrand represents the Euler–Lagrange equation (Yeh, Newman & Radke Reference Yeh, Newman and Radke1999; Starov, Velarde & Radke Reference Starov, Velarde and Radke2007; Arfken, Weber & Harris Reference Arfken, Weber and Harris2013)

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle \left[{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}h}}-{\displaystyle \frac{\text{d}}{\text{d}x}}\left({\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}h_{x}}}\right)\right]=P. & \displaystyle\end{eqnarray}$$

The boundary condition at $x=x_{0}$ then becomes

(2.6) $$\begin{eqnarray}\displaystyle & \displaystyle \left.-h_{x}\left({\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}h_{x}}}\right)\right|_{x_{0}}+\unicode[STIX]{x1D719}|_{x_{0}}=0. & \displaystyle\end{eqnarray}$$

This equation is known as the transversality condition or the Augmented Young equation. Note that the boundary terms at $x=0$ automatically cancel out due to the symmetry conditions.

We therefore have two constraints on the free energy as we approach the contact line: (1) it should satisfy the Augmented Young equation (2.6), and (2) it should recover the solid–gas free energy at the contact line to allow for a continuous variation of the energy. Our objective therefore is to find the functional form of the free energy, whose minimizer at equilibrium satisfies these two conditions. Considering the macroscopic contributions to the free energy, i.e. gravity and interfacial energies, we can write the free energy as

(2.7) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}_{M}=\int _{0}^{x_{0}}\left({\displaystyle \frac{1}{2}}\unicode[STIX]{x1D70C}gh^{2}+(\unicode[STIX]{x1D6FE}_{sl}-\unicode[STIX]{x1D6FE}_{sg})+\unicode[STIX]{x1D6FE}\sqrt{1+h_{x}^{2}}\right)\text{d}x, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}gh^{2}/2$ represents the gravitational potential energy, $\unicode[STIX]{x1D6FE}_{sl}$ and $\unicode[STIX]{x1D6FE}_{sg}$ represent the solid–liquid and solid–gas interfacial energies, and the term $\unicode[STIX]{x1D6FE}\sqrt{1+h_{x}^{2}}$ represents the liquid–gas interfacial energy contribution. Neglecting gravity, the Euler–Lagrange equation (2.5) will simplify to the usual Laplace equation for the pressure jump across an interface. We can now proceed to check whether the macroscopic free energy density satisfies the two constraints given above for equilibrium.

As the film height goes to zero, $\unicode[STIX]{x1D719}$ must be equal to the solid–gas interfacial energy; in the above representation of the free energy, we have subtracted this contribution, so we require $\unicode[STIX]{x1D719}(h=0)=0$ . Therefore, as we move close to the contact line, we have $(\unicode[STIX]{x1D6FE}_{sl}-\unicode[STIX]{x1D6FE}_{sg})+\unicode[STIX]{x1D6FE}\sqrt{1+h_{x}^{2}}|_{x_{0}}=0$ . Here we consider the true partial-wetting regime, where a droplet is surrounded by a dry solid substrate. The same arguments can be applied for the case of pseudopartial wetting, where the droplet is surrounded by a precursor film of height $h_{f}$ ; in this case, the height at the contact line goes to $h_{f}$ instead of zero.

The Augmented Young equation further requires the following:

(2.8) $$\begin{eqnarray}\displaystyle & \displaystyle \left.-h_{x}\left(\unicode[STIX]{x1D6FE}{\displaystyle \frac{h_{x}}{\sqrt{1+h_{x}^{2}}}}\right)\right|_{x_{0}}+\unicode[STIX]{x1D6FE}\left.\sqrt{1+h_{x}^{2}}\right|_{x_{0}}+(\unicode[STIX]{x1D6FE}_{sl}-\unicode[STIX]{x1D6FE}_{sg})=0. & \displaystyle\end{eqnarray}$$

Note that this equation can be simplified to give us the Young equation, i.e. $(\unicode[STIX]{x1D6FE}_{sl}-\unicode[STIX]{x1D6FE}_{sg})+(\unicode[STIX]{x1D6FE}/\sqrt{1+h_{x}^{2}})|_{x_{0}}=0$ , which in turn simplifies to $\unicode[STIX]{x1D6FE}\cos \unicode[STIX]{x1D703}_{Y}=(\unicode[STIX]{x1D6FE}_{sg}-\unicode[STIX]{x1D6FE}_{sl})$ , where $\unicode[STIX]{x1D703}_{Y}$ is the macroscopic Young contact angle (Young Reference Young1805).

Putting the Augmented Young equation and the continuity constraint together leads to the following conclusion:

(2.9) $$\begin{eqnarray}\displaystyle & \displaystyle \left.\left(\unicode[STIX]{x1D6FE}{\displaystyle \frac{h_{x}^{2}}{\sqrt{1+h_{x}^{2}}}}\right)\right|_{x_{0}}=0. & \displaystyle\end{eqnarray}$$

The only way for this equation to be satisfied is if $h_{x}(x_{0})=0$ ; if we substitute $h_{x}(x_{0})=0$ back in the Augmented Young equation, we arrive at $(\unicode[STIX]{x1D6FE}_{sl}-\unicode[STIX]{x1D6FE}_{sg})+\unicode[STIX]{x1D6FE}=0$ ; using the Young equation, this leads to $\unicode[STIX]{x1D6FE}(1-\cos \unicode[STIX]{x1D703}_{Y})=0$ , indicating that $\unicode[STIX]{x1D703}_{Y}=0$ . In other words, the liquid needs to completely wet the surface for the constraints on the free energy to be satisfied. An alternative way of arriving at this conclusion is to consider a uniform flat film; the energy of this film is then written as $\int ((1/2)\unicode[STIX]{x1D70C}gh^{2}+(\unicode[STIX]{x1D6FE}_{sl}-\unicode[STIX]{x1D6FE}_{sg})+\unicode[STIX]{x1D6FE})\,\text{d}x$ ; as we thin down the film and its height goes to zero, we need to recover the solid–gas interfacial energy, therefore we require $(\unicode[STIX]{x1D6FE}_{sl}-\unicode[STIX]{x1D6FE}_{sg})+\unicode[STIX]{x1D6FE}=0$ , which leads to the same result.

The macroscopic contributions to the free energy are therefore insufficient to model the partial-wetting regime; we need to incorporate the physics at the nanoscale close to the contact line. As the height of the liquid film becomes small, the liquid–solid and liquid–gas interfaces start to interact with each other, leading to an additional contribution in the free energy, commonly known as the disjoining pressure (de Gennes Reference de Gennes1985). We can therefore write the free energy in the following form:

(2.10) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D6E4}_{M}+\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D707}}=\int _{0}^{x_{0}}\left({\displaystyle \frac{1}{2}}\unicode[STIX]{x1D70C}gh^{2}+(\unicode[STIX]{x1D6FE}_{sl}-\unicode[STIX]{x1D6FE}_{sg})+\unicode[STIX]{x1D6FE}\sqrt{1+h_{x}^{2}}+\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}(h)\right)\text{d}x, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}(h)$ represents the intermolecular interactions close to the contact line. We now can check whether this form of the free energy satisfies the continuity constraint and the Augmented Young equation. The continuity of the free energy dictates

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x1D6FE}_{sl}-\unicode[STIX]{x1D6FE}_{sg})+\unicode[STIX]{x1D6FE}\left.\sqrt{1+h_{x}^{2}}\right|_{x_{0}}+\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}(h=0)=0. & \displaystyle\end{eqnarray}$$

The Augmented Young equation leads to the following:

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle \left.-h_{x}\left(\unicode[STIX]{x1D6FE}{\displaystyle \frac{h_{x}}{\sqrt{1+h_{x}^{2}}}}\right)\right|_{x_{0}}+\unicode[STIX]{x1D6FE}\left.\sqrt{1+h_{x}^{2}}\right|_{x_{0}}+(\unicode[STIX]{x1D6FE}_{sl}-\unicode[STIX]{x1D6FE}_{sg})+\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}(h=0)=0. & \displaystyle\end{eqnarray}$$

This equation represents the so-called Derjaguin–Frumkin equation (Starov et al. Reference Starov, Velarde and Radke2007), relating the Young angle to the microscopic forces close to the contact line: $\unicode[STIX]{x1D6FE}\cos \unicode[STIX]{x1D703}_{Y}=\unicode[STIX]{x1D6FE}\cos \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}+\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}(h=0)$ , where $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}$ represents the microscopic angle at the contact line and we have used $h_{x}|_{x_{0}}=\tan \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}$ and the Young equation $\unicode[STIX]{x1D6FE}_{sg}-\unicode[STIX]{x1D6FE}_{sl}=\unicode[STIX]{x1D6FE}\cos \unicode[STIX]{x1D703}_{Y}$ (Yeh et al. Reference Yeh, Newman and Radke1999; Starov et al. Reference Starov, Velarde and Radke2007; Starov Reference Starov2010). The above two constraints on continuity of the energy and the augmented Young equation lead to $h_{x}(x_{0})=0$ , or $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}=0$ . Putting $h_{x}(x_{0})=0$ back into either of the above equations leads to $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}(h=0)={\mathcal{S}}$ , where ${\mathcal{S}}\equiv \unicode[STIX]{x1D6FE}_{sg}-\unicode[STIX]{x1D6FE}_{sl}-\unicode[STIX]{x1D6FE}$ is known as the spreading parameter. Therefore, we arrive at the conclusion that while the macroscopic Young angle ( $\unicode[STIX]{x1D703}_{Y}$ ) can be non-zero, the microscopic angle at the contact line ( $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}$ ) needs to be zero to satisfy the energetic constraints; this regime is commonly known as pseudopartial wetting, where a macroscopic liquid drop is surrounded by a microscopic precursor film (Heslot et al. Reference Heslot, Cazabat, Levinson and Fraysse1990; Brochard-Wyart et al. Reference Brochard-Wyart, Di Meglio, Quere and de Gennes1991; Sharma Reference Sharma1993b ). To describe the true partial-wetting regime, in which a liquid droplet sits on a dry surface without a precursor film (de Gennes et al. Reference de Gennes, Brochard-Wyart and Quéré2004), additional physics is needed.

Surface tension at a liquid–gas interface arises due to the collective interactions between the liquid and gas molecules (Israelachvili Reference Israelachvili2011; Marchand et al. Reference Marchand, Weijs, Snoeijer and Andreotti2011). In the case of a uniform thin liquid film of nanometric thickness on a solid substrate, where liquid–gas and liquid–solid interfaces come very close to each other, the collective interactions also give rise to an excess free energy in addition to the interfacial tensions. For a nearly uniform thin film, this excess energy simply depends on the height of the film as represented by $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}(h)$ (Israelachvili Reference Israelachvili2011). Close to the contact line, however, the interfaces are not parallel, and therefore the collective intermolecular interactions between solid, liquid and gas molecules lead to a slope-dependent excess free energy, i.e. $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}(h,h_{x})$ for a liquid wedge sitting on a solid substrate (Hocking Reference Hocking1993; Wu & Wong Reference Wu and Wong2004; Dai, Leal & Redondo Reference Dai, Leal and Redondo2008). Using density functional theory, this excess free energy can be represented as a non-local integral of all the interactions (Keller & Merchant Reference Keller and Merchant1991; Merchant & Keller Reference Merchant and Keller1992; Getta & Dietrich Reference Getta and Dietrich1998; Snoeijer & Andreotti Reference Snoeijer and Andreotti2008); however, here we use a simplified local approximation of the excess free energy, which has been shown to agree well with the non-local formulations (Bauer & Dietrich Reference Bauer and Dietrich1999; Bonn et al. Reference Bonn, Eggers, Indekeu, Meunier and Rolley2009). We can therefore write the free energy as

(2.13) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}=\int _{0}^{x_{0}}\left({\displaystyle \frac{1}{2}}\unicode[STIX]{x1D70C}gh^{2}+(\unicode[STIX]{x1D6FE}_{sl}-\unicode[STIX]{x1D6FE}_{sg})+\unicode[STIX]{x1D6FE}\sqrt{1+h_{x}^{2}}+\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}(h,h_{x})\right)\text{d}x. & \displaystyle\end{eqnarray}$$

Continuity of the free energy leads to the following constraint at the contact line:

(2.14) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x1D6FE}_{sl}-\unicode[STIX]{x1D6FE}_{sg})+\unicode[STIX]{x1D6FE}\left.\sqrt{1+h_{x}^{2}}\right|_{x_{0}}+\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}(h,h_{x})|_{x_{0}}=0. & \displaystyle\end{eqnarray}$$

The augmented Young equation further becomes

(2.15) $$\begin{eqnarray}\displaystyle & \displaystyle \left.-h_{x}\left(\unicode[STIX]{x1D6FE}{\displaystyle \frac{h_{x}}{\sqrt{1+h_{x}^{2}}}}+{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}}{\unicode[STIX]{x2202}h_{x}}}\right)\right|_{x_{0}}+\unicode[STIX]{x1D6FE}\left.\sqrt{1+h_{x}^{2}}\right|_{x_{0}}+(\unicode[STIX]{x1D6FE}_{sl}-\unicode[STIX]{x1D6FE}_{sg})+\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}(h,h_{x})|_{x_{0}}=0. & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Based on the derivations of Dai et al. (Reference Dai, Leal and Redondo2008), we propose to decompose the vdW interactions, $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}(h,h_{x})$ , into two parts: a height-dependent part, and a slope-dependent part: $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}=(\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707},1}(h)+\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707},2}(h))-\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707},2}(h)\sqrt{1+(h_{x})^{2}}$ ; using the long-wave approximation, this form further simplifies to $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}=\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707},1}(h)-(1/2)\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707},2}(h)(h_{x})^{2}$ ; note that for parallel interfaces, $h_{x}=0$ , we recover the original height-dependent potential. An alternative way to arrive at the proposed form for the vdW forces is based on simple symmetry arguments: $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}$ cannot be linearly dependent on $h_{x}$ as it should have the same sign everywhere around the droplet; therefore it must be a function of even powers of $h_{x}$ , leading, in the simplest case, to the proposed gradient-squared form above.

The augmented Young equation can thus be written as

(2.16) $$\begin{eqnarray}\displaystyle & \displaystyle \left.-h_{x}\left((\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707},2}){\displaystyle \frac{h_{x}}{\sqrt{1+h_{x}^{2}}}}\right)\right|_{x_{0}}+\unicode[STIX]{x1D6FE}\left.\sqrt{1+h_{x}^{2}}\right|_{x_{0}}+(\unicode[STIX]{x1D6FE}_{sl}-\unicode[STIX]{x1D6FE}_{sg})+\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707}}(h,h_{x})|_{x_{0}}=0. & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Simplifying this equation leads to the following relation: $\unicode[STIX]{x1D6FE}\cos \unicode[STIX]{x1D703}_{Y}=[\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707},2}(h=0)]\cos \unicode[STIX]{x1D703}_{\unicode[STIX]{x1D707}}+[\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707},1}(h=0)+\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707},2}(h=0)]$ . Combining the augmented Young equation (2.16) and the continuity constraint (2.14) leads to the following condition at the contact line:

(2.17) $$\begin{eqnarray}\displaystyle & \displaystyle \left.\left((\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707},2}){\displaystyle \frac{h_{x}^{2}}{\sqrt{1+h_{x}^{2}}}}\right)\right|_{x_{0}}=0. & \displaystyle\end{eqnarray}$$

To satisfy this condition at the contact line, we either need to have $h_{x}(x_{0})=0$ , leading to a zero microscopic angle as before, or $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707},2}(h=0)=\unicode[STIX]{x1D6FE}$ , which does not constrain the value of the microscopic contact angle; this allows the microscopic angle to naturally arise as part of the solution; this outcome is consistent with the predictions of non-local density functional theory (Snoeijer & Andreotti Reference Snoeijer and Andreotti2008). The augmented Young equation (2.16) simplifies to $\unicode[STIX]{x1D6FE}\cos \unicode[STIX]{x1D703}_{Y}=\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707},1}(h=0)+\unicode[STIX]{x1D6FE}$ ; having $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707},1}(h=0)={\mathcal{S}}$ therefore leads to the Young equation $\unicode[STIX]{x1D6FE}\cos \unicode[STIX]{x1D703}_{Y}=\unicode[STIX]{x1D6FE}_{sg}-\unicode[STIX]{x1D6FE}_{sl}$ .

Allowing for slope dependence of the vdW forces therefore leads to a form of the free energy that satisfies all the constraints of the partial-wetting regime. Using the long-wave approximation, i.e. $\sqrt{1+(h_{x})^{2}}\approx 1+(1/2)(h_{x})^{2}$ , the total Helmholtz free energy can be written as

(2.18) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}=\int _{0}^{x_{0}}\left(f(h)+{\displaystyle \frac{1}{2}}\unicode[STIX]{x1D705}(h)(h_{x})^{2}\right)\text{d}x, & \displaystyle\end{eqnarray}$$

which, interestingly, resembles the Cahn–Hilliard formulation (Cahn & Hilliard Reference Cahn and Hilliard1958). The free energy here is divided into a bulk contribution, $f(h)=(1/2)\unicode[STIX]{x1D70C}gh^{2}-{\mathcal{S}}+\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707},1}(h)$ , and an interfacial contribution with a height-dependent interfacial tension, $\unicode[STIX]{x1D705}(h)=\unicode[STIX]{x1D6FE}-\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707},2}(h)$ . The above constraints on $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707},1}$ and $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707},2}$ , therefore, lead to $f(h=0)=0$ and $\unicode[STIX]{x1D705}(h=0)=0$ , which allow for compactly supported droplets sitting on a dry solid substrate (Benzi et al. Reference Benzi, Sbragaglia, Bernaschi and Succi2011; Cueto-Felgueroso & Juanes Reference Cueto-Felgueroso and Juanes2012; Pahlavan et al. Reference Pahlavan, Cueto-Felgueroso, McKinley and Juanes2015). Note that the height-dependent interfacial term leads to a new nonlinear term of the Kadar–Parisi–Zhang (KPZ) type (Kardar, Parisi & Zhang Reference Kardar, Parisi and Zhang1986); similar forms have also been recently proposed in the context of active suspensions (Stenhammar et al. Reference Stenhammar, Tiribocchi, Allen, Marenduzzo and Cates2013; Wittkowski et al. Reference Wittkowski, Tiribocchi, Stenhammar, Allen, Marenduzzo and Cates2014). The idea of an order-parameter-dependent interfacial tension has been proposed in the context of binary alloys (Cahn Reference Cahn1961) and polymer blends (de Gennes Reference de Gennes1980), and the dependence of surface tension on height has also been recently proposed in the context of capillary waves on thin liquid films (MacDowell, Benet & Katcho Reference MacDowell, Benet and Katcho2013; MacDowell et al. Reference MacDowell, Benet, Katcho and Palanco2014). In the context of moving contact lines, Shikhmurzaev (Reference Shikhmurzaev1997, Reference Shikhmurzaev2007) has proposed that the idea of a dynamic surface tension close to the contact line can resolve the moving contact-line singularity and lead to a natural emergence of the dynamic contact angle (Sibley et al. Reference Sibley, Nold, Savva and Kalliadasis2015); both these features are also reproduced in our model (Pahlavan et al. Reference Pahlavan, Cueto-Felgueroso, McKinley and Juanes2015).

2.2 Thin-film model: generalized disjoining pressure

Having defined the free energy, we can write the evolution equation for the height of the liquid film in the form of a mass-conservative gradient flow as (Hohenberg & Halperin Reference Hohenberg and Halperin1977; Cross & Hohenberg Reference Cross and Hohenberg1993; Bray Reference Bray2002)

(2.19) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}t}}={\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}}\left({\mathcal{M}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}}\left({\displaystyle \frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}}{\unicode[STIX]{x1D6FF}h}}\right)\right), & \displaystyle\end{eqnarray}$$

where ${\mathcal{M}}$ represents the mobility, and the variational derivative of the free energy is defined as $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D6FF}h=\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}/\unicode[STIX]{x2202}h-\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x[\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}/\unicode[STIX]{x2202}(h_{x})]$ (Anderson, McFadden & Wheeler Reference Anderson, McFadden and Wheeler1998).

An alternative way to arrive at the same thin-film model is to start from the Navier–Stokes equations of motion and simplify them using the lubrication approximation:

(2.20) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}\left({\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}\right)=-\unicode[STIX]{x1D735}P+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D64F}-\unicode[STIX]{x1D70C}g\boldsymbol{e}_{z}, & \displaystyle\end{eqnarray}$$

in which $\unicode[STIX]{x1D70C}$ is the liquid density, $\boldsymbol{u}=(u,w)$ is the velocity field, $P$ is the isotropic liquid pressure, $\unicode[STIX]{x1D64F}$ is the stress tensor, and the last term in the equation represents the gravitational force. The stress tensor can be further decomposed into two parts, $\unicode[STIX]{x1D64F}=\unicode[STIX]{x1D749}+\unicode[STIX]{x1D648}$ , where $\unicode[STIX]{x1D749}$ is the deviatoric stress and $\unicode[STIX]{x1D648}$ represents the stress due to external body forces, taken to be zero in this work.

We non-dimensionalize the equations in the following way: $\tilde{x}=x/L$ , $\tilde{z}=z/H$ , $\tilde{h}=h/H$ , $\tilde{u} =u/U$ , $\tilde{w}=w/(\unicode[STIX]{x1D716}U)$ , $\tilde{t}=t/(\unicode[STIX]{x1D707}L^{4}/\unicode[STIX]{x1D6FE}H^{3})$ , $\tilde{P}=P/(\unicode[STIX]{x1D6FE}H/L^{2})$ and $\tilde{\unicode[STIX]{x1D749}}=\unicode[STIX]{x1D749}/(\unicode[STIX]{x1D6FE}H^{2}/L^{3})$ , where $\unicode[STIX]{x1D716}=H/L$ , and $H$ and $L$ represent a characteristic height and length, respectively. This leads to three dimensionless groups, $Re=\unicode[STIX]{x1D70C}UH/\unicode[STIX]{x1D707}$ , $\tilde{Ca}=Ca/\unicode[STIX]{x1D716}^{3}$ with $Ca=\unicode[STIX]{x1D707}U/\unicode[STIX]{x1D6FE}$ , and $Bo=(L/l_{\unicode[STIX]{x1D6FE}})^{2}$ , with $l_{\unicode[STIX]{x1D6FE}}=\sqrt{\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D70C}g}$ as the capillary length.

The above equations of motion then need to be supplemented by boundary conditions at the wall and at the liquid–gas interface. At the wall, we impose a Navier slip boundary condition: $\tilde{u} |_{\tilde{z}=0}=\unicode[STIX]{x1D6FD}\unicode[STIX]{x2202}\tilde{u} /\unicode[STIX]{x2202}\tilde{z}|_{\tilde{z}=0}$ , where $\unicode[STIX]{x1D6FD}=b/H$ is the non-dimensional slip or extrapolation length (Granick, Zhu & Lee Reference Granick, Zhu and Lee2003; Neto et al. Reference Neto, Evans, Bonaccurso, Butt and Craig2005; Lauga, Brenner & Stone Reference Lauga, Brenner and Stone2007); we also assume no penetration at the wall: $\tilde{w}|_{\tilde{z}=0}=0$ . At the liquid–gas interface, one must satisfy the kinematic boundary condition $\tilde{w}_{s}=(1/\tilde{Ca})\unicode[STIX]{x2202}\tilde{h}/\unicode[STIX]{x2202}\tilde{t}+\tilde{u} _{s}\unicode[STIX]{x2202}\tilde{h}/\unicode[STIX]{x2202}\tilde{x}$ , where $\tilde{u} _{s}$ and $\tilde{w}_{s}$ are the velocity components at the interface. The stress boundary condition at the interface can be written as

(2.21) $$\begin{eqnarray}\displaystyle & \displaystyle ([P]\unicode[STIX]{x1D644}-[\unicode[STIX]{x1D64F}])\boldsymbol{\cdot }\boldsymbol{n}=(\unicode[STIX]{x1D6FE}{\mathcal{K}}+\unicode[STIX]{x1D6F7})\boldsymbol{n}, & \displaystyle\end{eqnarray}$$

where $[-]$ represents the jump across the interface, $\boldsymbol{n}=(-h_{x},1)/\sqrt{1+h_{x}^{2}}$ is the unit vector normal to the interface, $\boldsymbol{t}=(1,h_{x})/\sqrt{1+h_{x}^{2}}$ is the unit vector tangent to the interface, ${\mathcal{K}}=-\unicode[STIX]{x1D735}_{s}\boldsymbol{\cdot }\boldsymbol{n}=h_{xx}/(1+h_{x}^{2})^{3/2}$ represents the curvature with $\unicode[STIX]{x1D735}_{s}=(\unicode[STIX]{x1D644}-\boldsymbol{n}\otimes \boldsymbol{n})\boldsymbol{\cdot }\unicode[STIX]{x1D735}$ . The jump in the liquid pressure and stress across the interface is represented by the terms on the right-hand side of the equation: the first term gives the Laplace pressure jump due to the interface curvature, $\unicode[STIX]{x1D6F7}$ arises due to the intermolecular interactions between the solid–liquid and liquid–gas interfaces and only becomes relevant when these two interfaces are closer than a few nanometres (Bonn et al. Reference Bonn, Eggers, Indekeu, Meunier and Rolley2009; Israelachvili Reference Israelachvili2011).

Using the above non-dimensionalization and long-wave approximation, $\unicode[STIX]{x1D716}\ll 1$ , and negligible inertia, $\unicode[STIX]{x1D716}Re\ll 1$ , and surface tension-dominated flow, $\tilde{Ca}=O(1)$ , the $x$ -momentum equation reduces to $0=-\unicode[STIX]{x2202}\tilde{P}/\unicode[STIX]{x2202}\tilde{x}+\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D70F}}_{zx}/\unicode[STIX]{x2202}\tilde{z}$ . Further, assuming $\unicode[STIX]{x1D716}^{3}Re\ll 1$ , the $z$ -momentum equation further simplifies to $0=-Bo-\unicode[STIX]{x2202}\tilde{P}/\unicode[STIX]{x2202}\tilde{z}$ . The slip and kinematic boundary conditions remain unchanged under the lubrication approximation. The stress boundary condition, however, simplifies further; the tangential component becomes $[\tilde{\unicode[STIX]{x1D70F}}_{zx}]=0$ and the normal component reduces to $[\tilde{P}]=-\tilde{h}_{\tilde{x}\tilde{x}}+\tilde{\unicode[STIX]{x1D6F7}}$ . The deviatoric stress is a function of the strain rate $\unicode[STIX]{x1D750}=(\unicode[STIX]{x1D735}\boldsymbol{u}+\unicode[STIX]{x1D735}\boldsymbol{u}^{\text{T}})/2$ , and for a Newtonian liquid can be simply written as $\unicode[STIX]{x1D70F}_{ij}=\unicode[STIX]{x1D707}(\unicode[STIX]{x2202}u_{i}/\unicode[STIX]{x2202}x_{j}+\unicode[STIX]{x2202}u_{j}/\unicode[STIX]{x2202}x_{i})$ . Using this definition and integrating the $z$ -momentum equation, and then applying the normal stress boundary condition, we can write the liquid pressure as follows: $\tilde{P}=Bo(\tilde{h}-\tilde{z})-\tilde{h}_{\tilde{x}\tilde{x}}+\tilde{\unicode[STIX]{x1D6F7}}$ .

Integrating the continuity equation and using the kinematic boundary condition, we arrive at $(1/\tilde{Ca})\unicode[STIX]{x2202}\tilde{h}/\unicode[STIX]{x2202}\tilde{t}+\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\tilde{x}\int _{0}^{\tilde{h}}\tilde{u} \,\text{d}\tilde{z}=0$ .

Replacing the $x$ -velocity component in the above equation we therefore arrive at the evolution equation for the height of the film:

(2.22) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}\tilde{h}}{\unicode[STIX]{x2202}\tilde{t}}}={\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\tilde{x}}}\left\{\left[{\displaystyle \frac{\tilde{h}^{3}}{3}}+\unicode[STIX]{x1D6FD}\tilde{h}^{2}\right]{\displaystyle \frac{\unicode[STIX]{x2202}\tilde{P}}{\unicode[STIX]{x2202}\tilde{x}}}\right\}. & \displaystyle\end{eqnarray}$$

Comparing (2.22) above and (2.19) in the main manuscript, it is evident that they are equivalent if $\tilde{P}=\unicode[STIX]{x1D6FF}\tilde{\unicode[STIX]{x1D6E4}}/\unicode[STIX]{x1D6FF}\tilde{h}=\text{d}\tilde{f}/\text{d}\tilde{h}-\sqrt{\tilde{\unicode[STIX]{x1D705}}}(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\tilde{x})(\sqrt{\tilde{\unicode[STIX]{x1D705}}}\tilde{h}_{\tilde{x}})$ , where $\tilde{\unicode[STIX]{x1D6E4}}=\unicode[STIX]{x1D6E4}/(\unicode[STIX]{x1D6FE}H^{2}/L^{2})$ , $\tilde{f}=f/(\unicode[STIX]{x1D6FE}H^{2}/L^{2})$ , and $\tilde{\unicode[STIX]{x1D705}}=\unicode[STIX]{x1D705}/\unicode[STIX]{x1D6FE}$ ; we therefore find the disjoining pressure to be

(2.23) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\unicode[STIX]{x1D6F7}}(\tilde{h},\tilde{h}_{\tilde{x}},\tilde{h}_{\tilde{x}\tilde{x}})={\displaystyle \frac{\text{d}\tilde{\unicode[STIX]{x1D719}}_{\unicode[STIX]{x1D707},1}}{\text{d}\tilde{h}}}+\sqrt{\tilde{\unicode[STIX]{x1D719}}_{\unicode[STIX]{x1D707},2}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\tilde{x}}}\left(\sqrt{\tilde{\unicode[STIX]{x1D719}}_{\unicode[STIX]{x1D707},2}}\tilde{h}_{\tilde{x}}\right), & \displaystyle\end{eqnarray}$$

where $\tilde{\unicode[STIX]{x1D719}}_{\unicode[STIX]{x1D707},1}=\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707},1}/(\unicode[STIX]{x1D6FE}H^{2}/L^{2})$ and $\tilde{\unicode[STIX]{x1D719}}_{\unicode[STIX]{x1D707},2}=\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D707},2}/\unicode[STIX]{x1D6FE}$ , and $\tilde{\unicode[STIX]{x1D6F7}}$ is a generalized disjoining pressure that depends not only on the film height, but also on its slope and curvature. Similar ideas for a generalized disjoining pressure have been proposed in the past: by integrating the intermolecular interactions in a liquid wedge, Miller & Ruckenstein (Reference Miller and Ruckenstein1974), Hocking (Reference Hocking1993) and Indeikina & Chang (Reference Indeikina, Chang, Durban and Pearson1999) derived a slope-dependent disjoining pressure, which was later generalized by Wu & Wong (Reference Wu and Wong2004), who incorporated the interactions with the molecules of the gas phase and showed that it leads to the appearance of a higher-order curvature term in the disjoining pressure. Snoeijer & Andreotti (Reference Snoeijer and Andreotti2008) compared these results with the predictions of Keller & Merchant (Reference Keller and Merchant1991) and Merchant & Keller (Reference Merchant and Keller1992), and showed that the disjoining pressures do not recover the correct macroscopic Young contact angle. Dai et al. (Reference Dai, Leal and Redondo2008) later showed that the form derived by Wu & Wong (Reference Wu and Wong2004) does not recover the classical Lifshitz formulation (Dzyaloshinskii, Lifshitz & Pitaevskii Reference Dzyaloshinskii, Lifshitz and Pitaevskii1961) in the limit of parallel interfaces, and derived a new consistent form for the disjoining pressure. The functional form of disjoining pressure we have proposed here in (2.23) resembles that of Dai et al. (Reference Dai, Leal and Redondo2008), and simplifies to a height-dependent form for parallel interfaces, recovering the classical height-dependent Lifshitz theory for parallel interfaces.

Information about the detailed form of the disjoining pressure becomes essential in studying many phenomena such as moving contact lines (Gogotsi et al. Reference Gogotsi, Libera, Güvenç-Yazicioglu and Megaridis2001; Sibley, Savva & Kalliadasis Reference Sibley, Savva and Kalliadasis2012; Sibley et al. Reference Sibley, Nold, Savva and Kalliadasis2015), the final stages of coalescence of drops (Yiantsios & Davis Reference Yiantsios and Davis1991; Leal Reference Leal2004; Zeng et al. Reference Zeng, Zhao, Tian, Tirrell, Leal and Israelachvili2007; Li Reference Li2016) or in applications such as solidification (Tao, Yeckel & Derby Reference Tao, Yeckel and Derby2016). Here, we further show the consequences of the form of the disjoining pressure on the instability and dewetting of thin liquid films in the partial-wetting regime.

5.1 Coarsening phase diagram

The drops formed after the dewetting process are in a metastable state; they can still communicate through the ultrathin films connecting them and thus coarsen in time to lower the overall energy of the system. Multiplying both sides of (2.19) by $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D6FF}\tilde{h}$ , integrating in space and using the no-flux boundary conditions, we can easily show (see also Thiele et al. (Reference Thiele, Velarde, Neuffer and Pomeau2001b ), Glasner & Witelski (Reference Glasner and Witelski2003)):

(5.1) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\text{d}\unicode[STIX]{x1D6E4}}{\text{d}\tilde{t}}}=-\int {\mathcal{M}}\left({\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\tilde{x}}}\left({\displaystyle \frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}}{\unicode[STIX]{x1D6FF}\tilde{h}}}\right)\right)^{2}\text{d}\tilde{x}\leqslant 0, & & \displaystyle\end{eqnarray}$$

demonstrating that the evolution governed by the thin-film equation leads to a decreasing free energy and, therefore, a thermodynamically admissible system.

Figure 10.

Space–time diagrams showing the evolution of drop height of the classical thin-film equation (constant surface tension); the results of the height-dependent case are similar. The simulations are done in a domain of size $100\unicode[STIX]{x1D706}_{m}$ and the frames shown here show a zoom-in of a size $40\unicode[STIX]{x1D706}_{m}$ . In all the cases, a line coming to an end indicates an Ostwald-ripening event, in which a smaller drop feeds into larger neighbouring drops through the ultrathin film that connects them. (a) $\tilde{h}_{0}=4\unicode[STIX]{x1D6FF}$ , representing the classical spinodal regime, where the number of drops is approximately set by the most unstable wavelength $\unicode[STIX]{x1D706}_{m}$ ( $\tilde{h}_{max}=35\unicode[STIX]{x1D6FF}$ ). (b) $\tilde{h}_{0}=10\unicode[STIX]{x1D6FF}$ representing a mixed-mode instability regime in between spinodal and nucleation regimes; here, fewer drops than the spinodal regime form and the distance between the formed drops seems to be random ( $\tilde{h}_{max}=100\unicode[STIX]{x1D6FF}$ ). (c) $\tilde{h}_{0}=25\unicode[STIX]{x1D6FF}$ , which is deep into the nucleation regime, showing the formation of very few drops ( $\tilde{h}_{max}=700\unicode[STIX]{x1D6FF}$ ). (d) $\tilde{h}_{0}=4\unicode[STIX]{x1D6FF}$ and including thermal fluctuations ( $T=50\,^{\circ }\text{C}$ , i.e. $\unicode[STIX]{x1D70E}=0.039$ ) ( $\tilde{h}_{max}=25\unicode[STIX]{x1D6FF}$ ); thermal noise enhances the lateral motion of the drops; the coarsening mechanism, however, still seems to be dominated by Ostwald ripening.

Figure 10 shows space–time diagrams of the dewetting of thin liquid films of different thickness and their coarsening over time. The simulations presented here are conducted on a large domain of size $100\unicode[STIX]{x1D706}_{m}$ and the frames shown are a zoomed-in view of a window of size $40\unicode[STIX]{x1D706}_{m}$ ; these simulations show the results corresponding to the classical model with a constant surface tension; the results of the height-dependent surface tension model are very similar. In all cases we start with a uniform film of thickness $\tilde{h}_{0}$ on which random perturbations of amplitude $10^{-4}\tilde{h}_{0}$ are imposed. While all the initial thicknesses are within the linearly unstable regime (see figure 5), they lead to entirely different morphologies.

For small initial film height of $\tilde{h}_{0}=4\unicode[STIX]{x1D6FF}$ , we observe the characteristics of the spinodal dewetting, a term originally coined by Mitlin (Reference Mitlin1993) due to its similarity to the spinodal decomposition in binary mixtures (Cahn Reference Cahn1961). In this regime, the distance between the drops is the same as the most unstable wavelength, i.e. we get ${\approx}100$ drops in a domain of size $100\unicode[STIX]{x1D706}_{m}$ (Diez & Kondic Reference Diez and Kondic2007). The coarsening here proceeds through Ostwald ripening (Ostwald Reference Ostwald1897), or collapse of smaller drops at the expense of growth of larger ones (Glasner & Witelski Reference Glasner and Witelski2003).

As the initial film height increases to $\tilde{h}_{0}=10\unicode[STIX]{x1D6FF}$ , we can immediately see in figure 10(b) that the number of drops formed upon dewetting and their spacing no longer follows the predictions of the linear stability analysis, i.e. fewer drops form and their spacing becomes random. This behaviour has signatures of nucleation dewetting within the linearly unstable regime (Thiele, Velarde & Neuffer Reference Thiele, Velarde and Neuffer2001a ; Thiele et al. Reference Thiele, Velarde, Neuffer and Pomeau2001b ; Diez & Kondic Reference Diez and Kondic2007) and shows a mixed-mode instability, i.e. a behaviour in between spinodal and nucleation regimes. Such a transition has also been reported in phase separation dynamics within the Cahn–Hilliard framework (Novick-Cohen Reference Novick-Cohen1985). Figure 11 shows corresponding 2D simulations of dewetting in this regime; in contrast with the spinodal regime, where holes appear around the same time with a uniform spacing, here the nucleation process begins with formation of a hole, which then laterally expands and forms a rim behind it (Seemann, Herminghaus & Jacobs Reference Seemann, Herminghaus and Jacobs2001b ; Fetzer et al. Reference Fetzer, Jacobs, Münch, Wagner and Witelski2005; Seemann et al. Reference Seemann, Herminghaus, Neto, Schlagowski, Podzimek, Konrad, Mantz and Jacobs2005; Bäumchen & Jacobs Reference Bäumchen and Jacobs2010; Bäumchen et al. Reference Bäumchen, Marquant, Blossey, Münch, Wagner and Jacobs2014). In this mixed-mode regime, the growth rate is still large enough that shortly after growth of a hole, the depression behind the dewetting rim can lead to rupture; this behaviour leads to the satellite-hole formation observed in the experiments (Becker et al. Reference Becker, Grun, Seemann, Mantz, Jacobs, Mecke and Blossey2003; Neto et al. Reference Neto, Jacobs, Seemann, Blossey, Becker and Grün2003) and in the 2D simulations of figure 11. Figure 12 illustrates this nucleation process in a space–time diagram of a one-dimensional (1D) simulation of the height-dependent surface tension model with $\tilde{h}_{0}=10\unicode[STIX]{x1D6FF}$ .

Figure 11.

Nucleation dewetting for $\tilde{h}_{0}=10\unicode[STIX]{x1D6FF}$ for the constant surface tension model. In this regime, holes randomly appear and start expanding laterally, forming a growing rim behind them ( $\tilde{h}_{max}=30\unicode[STIX]{x1D6FF},54\unicode[STIX]{x1D6FF},70\unicode[STIX]{x1D6FF},120\unicode[STIX]{x1D6FF}$ , respectively). The depression behind the dewetting rim leads to the formation of satellite holes. The dewetting rims subsequently become unstable and lead to fingering and pinch off to form droplets (see Reiter & Sharma (Reference Reiter and Sharma2001), Bäumchen et al. (Reference Bäumchen, Marquant, Blossey, Münch, Wagner and Jacobs2014) for experimental details).

As we keep increasing the initial film thickness further to $\tilde{h}_{0}=25\unicode[STIX]{x1D6FF}$ (figure 10 c), we move deeper into the nucleation regime, where fewer drops form; in this case five drops formed in random locations in a domain of size $40\unicode[STIX]{x1D706}_{m}$ . In the nucleation regime two time scales compete: the time scale associated with the growth of linearly unstable modes and the time scale associated with the dewetting front (Thiele et al. Reference Thiele, Velarde and Neuffer2001a ; Diez & Kondic Reference Diez and Kondic2007; Snoeijer & Eggers Reference Snoeijer and Eggers2010). As we showed before, the growth rate of the most unstable mode $\unicode[STIX]{x1D6FD}_{m}$ scales as $\tilde{h}_{0}^{-5}$ , leading to significant decreases in the growth rate for thicker films, thereby favouring growth via nucleation.

Since the dynamics of the coarsening process is slaved to the flux through the ultrathin equilibrium films ( $\tilde{h}_{m}\approx \unicode[STIX]{x1D6FF}$ ) connecting the drops and the mobility scales as $\tilde{h}^{3}$ , the coarsening dynamics are extremely slow. Two time scales are involved in this process: a fast time scale over which individual coarsening events happen, and a slow time scale between the individual events (Glasner & Witelski Reference Glasner and Witelski2003). A full numerical simulation of the thin-film equation therefore can be computationally very expensive. Glasner & Witelski (Reference Glasner and Witelski2003) used the separation of time scales in this problem to reduce the governing partial differential equation, equation (2.19), to a system of ordinary differential equations for the droplet pressures and locations. Coarsening can proceed via two general mechanisms: coalescence of droplets or Ostwald ripening (Glasner & Witelski Reference Glasner and Witelski2005). Their mathematical model allowed them to simulate the coarsening of a very large number ( $O(10^{5})$ ) of drops. They observed a scaling of $N\sim \tilde{t}^{-2/5}$ for the number of drops in time and also observed a transition from Ostwald-ripening-dominated (capillary-driven drainage and collapse) coarsening to coalescence-dominated coarsening as they increased the total mass of liquid in the domain.

Figure 12.

Space–time diagram of the nucleation process in the height-dependent surface tension model with $\tilde{h}_{0}=10\unicode[STIX]{x1D6FF}$ in a domain of size $20\unicode[STIX]{x1D706}_{m}$ (each snapshot is shifted up by $10\unicode[STIX]{x1D6FF}=1$ ). The lateral expansion of holes, growth of the rim behind the hole, and the subsequent instability of the dip behind a growing rim can all be seen in this illustration.

In contrast with the findings of Glasner & Witelski (Reference Glasner and Witelski2005), we do not observe such a transition from Ostwald-ripening-dominated to coalescence-dominated coarsening (figure 10). The change in the instability mode from spinodal to nucleation leads to the formation of fewer drops as the total mass of the liquid in the domain increases, i.e. as the initial film height increases: the large distance between the drops favours coarsening through Ostwald ripening rather than coalescence. The reason Glasner & Witelski (Reference Glasner and Witelski2005) observe such a transition is that they keep the number of drops within a given domain fixed as they increase their mass; this naturally leads to wider drops that progressively get closer to each other, therefore favouring coalescence.

5.2 Influence of thermal fluctuations

Some experiments on thin-film dewetting (Limary & Green Reference Limary and Green2002, Reference Limary and Green2003) and nanoparticle growth in thin films (Meli & Green Reference Meli and Green2008; Woehl et al. Reference Woehl, Park, Evans, Arslan, Ristenpart and Browning2014) have reported a crossover from diffusion-dominated to coalescence-dominated behaviour. A potential source of this transition could be thermal noise, which may play a dominant role in thin-film systems (Mecke & Rauscher Reference Mecke and Rauscher2005; Grün et al. Reference Grün, Mecke and Rauscher2006; Fetzer et al. Reference Fetzer, Rauscher, Seemann, Jacobs and Mecke2007b ; Willis & Freund Reference Willis and Freund2009; Belardinelli et al. Reference Belardinelli, Sbragaglia, Gross and Andreotti2016; Diez, González & Fernández Reference Diez, González and Fernández2016). To explore this possibility, we generalize the thin-film equation (2.19) by incorporating thermal noise:

(5.2) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}\tilde{h}}{\unicode[STIX]{x2202}\tilde{t}}}={\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\tilde{x}}}\left(\tilde{h}^{3}{\displaystyle \frac{\unicode[STIX]{x2202}\tilde{P}}{\unicode[STIX]{x2202}\tilde{x}}}\right)+\unicode[STIX]{x1D70E}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\tilde{x}}}(\tilde{h}^{3/2}\tilde{\unicode[STIX]{x1D709}}(\tilde{x},\tilde{t})), & \displaystyle\end{eqnarray}$$

in which $\tilde{P}=\text{d}\tilde{f}/\text{d}\tilde{h}-\sqrt{\tilde{\unicode[STIX]{x1D705}}}\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\tilde{x}(\sqrt{\tilde{\unicode[STIX]{x1D705}}}\unicode[STIX]{x2202}\tilde{h}/\unicode[STIX]{x2202}\tilde{x})$ , with $\tilde{\unicode[STIX]{x1D705}}=\unicode[STIX]{x1D705}/\unicode[STIX]{x1D6FE}$ and $\tilde{f}=f/(\unicode[STIX]{x1D6FE}H^{2}/L^{2})$ , where $H$ and $L$ are characteristic height and length scales. The second term on the right-hand side of the equation represents the thermal fluctuations (Davidovitch, Moro & Stone Reference Davidovitch, Moro and Stone2005; Mecke & Rauscher Reference Mecke and Rauscher2005; Grün et al. Reference Grün, Mecke and Rauscher2006) with $\unicode[STIX]{x1D70E}=\sqrt{k_{B}T/\unicode[STIX]{x1D6FE}H^{2}}$ , where $k_{B}$ is Boltzmann’s constant, $T$ represents absolute temperature and $\tilde{\unicode[STIX]{x1D709}}$ represents a spatiotemporal Gaussian white noise: $\langle \tilde{\unicode[STIX]{x1D709}}\rangle =0$ and $\langle \tilde{\unicode[STIX]{x1D709}}(\tilde{x},\tilde{t})\tilde{\unicode[STIX]{x1D709}}(\tilde{x}^{\prime },\tilde{t}^{\prime })\rangle =\unicode[STIX]{x1D6FF}(\tilde{x}-\tilde{x}^{\prime })\unicode[STIX]{x1D6FF}(\tilde{t}-\tilde{t}^{\prime })$ with $\unicode[STIX]{x1D6FF}$ as the Dirac delta function and $\langle .\rangle$ implying ensemble average over realizations of the noise.

Introducing thermal noise to the system leads to enhanced lateral motion of the drops, as seen in figure 10(d), and this can promote coalescence. Our simulations for the stochastic thin-film equation (5.2) within the spinodal regime, however, show the dominance of Ostwald ripening despite the enhanced lateral motion of the drops. The crossover from the diffusion-dominated to coalescence-dominated behaviour observed in the experiments (Limary & Green Reference Limary and Green2002, Reference Limary and Green2003) therefore could be a consequence of the dimensionality of the problem, i.e. 2D in the experiments versus 1D in the simulations shown in figure 10. In two dimensions, each drop can be surrounded and interact with multiple other drops, whereas in one dimension each drop only interacts with its two neighbours. To investigate this, we have conducted 2D nonlinear simulations, which point to the dominance of coalescence events at short times and Ostwald ripening at long times, as shown in figure 13, which corresponds to the height-dependent surface tension model; similar results are obtained for both the classical and stochastic models. Our observations are in agreement with the 2D simulations of Glasner (Reference Glasner2008) using a reduced-order model that suggest the dominance of the coalescence mechanism.

Figure 13.

Two-dimensional simulations of coarsening dynamics at early and late times for the height-dependent surface tension model with $\tilde{h}_{0}=2\unicode[STIX]{x1D6FF}$ ( $\tilde{h}_{max}=8.5\unicode[STIX]{x1D6FF}$ in the colourbar). At early times, coalescence events, as highlighted by the dashed white circles, are dominant. At late times (once the metastable drops have formed), however, Ostwald ripening, i.e. capillary drainage and collapse of drops, becomes the key coarsening mechanism; note that this latter process is diffusion-dominated and therefore much slower than coarsening through coalescence events.

Figure 14.

Coarsening statistics; red circles and squares correspond to the classical model with $\tilde{h}_{0}=4\unicode[STIX]{x1D6FF}$ and $\tilde{h}_{0}=10\unicode[STIX]{x1D6FF}$ , respectively; blue diamonds show the results of the new model with $\tilde{h}_{0}=4\unicode[STIX]{x1D6FF}$ ; green stars correspond to the stochastic thin-film model with $\tilde{h}_{0}=4\unicode[STIX]{x1D6FF}$ . (a) Evolution of the number of drops in time shows the scaling $\tilde{t}^{-2/5}$ as suggested by Glasner & Witelski (Reference Glasner and Witelski2003). (b) Evolution of mean width of drops in time; as expected from conservation of mass and the scaling of the number of drops in time, the mean width scales as $\tilde{t}^{1/5}$ in time. (c) Evolution of the total energy in time shows a decrease in the energy with coarsening with a scaling of $\tilde{t}^{-1/5}$ , which is expected for the 1D case (Otto, Rump & Slepcev Reference Otto, Rump and Slepcev2006); the $\tilde{h}_{0}=10\unicode[STIX]{x1D6FF}$ case results in fewer drops upon dewetting, therefore the majority of the domain is covered by the equilibrium film, which means the initial energy is lower than the $\tilde{h}_{0}=4\unicode[STIX]{x1D6FF}$ case; at long times the dynamics becomes self-similar and both follow the same scaling. (d) Evolution of the bulk $E_{b}$ , interfacial $E_{i}$ and total $E_{t}=E_{b}+E_{i}$ energies for a typical simulation (here $\tilde{h}_{0}=4\unicode[STIX]{x1D6FF}$ and height-dependent surface tension model); at short times, creation of drops increases the interfacial energy, but the total energy is still lowered due to the reduction in the bulk energy; at long times, as drops coarsen, both the bulk and interfacial energy contribution decrease.

5.3 Coarsening statistics

The statistics of the coarsening process for (i) the spinodal regime, $\tilde{h}_{0}=4\unicode[STIX]{x1D6FF}$ (for both constant and height-dependent surface tension models as well as the classical stochastic thin-film equation with thermal noise) and (ii) for the mixed-mode instability regime $\tilde{h}_{0}=10\unicode[STIX]{x1D6FF}$ (for the constant surface tension model) are shown in figure 14. In the spinodal regime, the number of drops follows the scaling $N\sim \tilde{t}^{-2/5}$ in time (figure 14 a) as computed by Glasner & Witelski (Reference Glasner and Witelski2003); incorporating the height dependence of surface tension does not alter this scaling. Introducing thermal noise does not affect the scaling either – an observation that is consistent with recent studies (Nesic et al. Reference Nesic, Cuerno, Moro and Kondic2015). In the mixed-mode instability regime ( $\tilde{h}_{0}=10\unicode[STIX]{x1D6FF}$ ), fewer drops form, so our simulations have limited statistics and do not clearly reach the self-similar intermediate regime (Gratton & Witelski Reference Gratton and Witelski2009), but it seems that the results for this case are also in general agreement with the $\tilde{t}^{-2/5}$ scaling (figure 14 a). Due to mass conservation, it is easy to show that the mean width of drops should then follow the scaling $\tilde{t}^{1/5}$ , as shown in figure 14(b). An interesting feature of the variation of the mean width in time, particularly at late times, is the jumps observed as smaller drops shrink and feed into the larger drops.

As argued above, the thin-film evolution equation predicts a monotonically decreasing free energy for the system (equation (5.1)). We can further check this by looking at the variation of the numerically calculated total energy of the system $E_{t}\equiv \unicode[STIX]{x1D6E4}=\int \tilde{f}(\tilde{h})+(\tilde{\unicode[STIX]{x1D705}}/2)(\tilde{h}_{x})^{2}\,\text{d}\tilde{x}$ . In agreement with the predictions of Otto et al. (Reference Otto, Rump and Slepcev2006), we find a $\tilde{t}^{-1/5}$ scaling for the decrease of the free energy (figure 14 c). An interesting observation here is that for the thicker film of $\tilde{h}_{0}=10\unicode[STIX]{x1D6FF}$ , we observe a lower total free energy at early times that later converges to the $\tilde{t}^{-1/5}$ scaling. To understand the reason behind the difference at early times, it is instructive to look at the typical variation of the different components of the free energy during the entire dewetting process, as shown in figure 14(d). Here, $E_{b}=\int \tilde{f}(\tilde{h})\,\text{d}\tilde{x}$ is the bulk free energy and $E_{i}=\int (\tilde{\unicode[STIX]{x1D705}}(\tilde{h})/2)(\tilde{h}_{x})^{2}\,\text{d}\tilde{x}$ is the interfacial energy. As an initially uniform film becomes unstable, it must increase the interfacial energy to dewet and create droplets. This increase in the interfacial energy, however, is compensated by the reduction in the bulk free energy as the ultrathin films connecting the drops are in their near-equilibrium states. The drops connected by these ultrathin films, however, are only metastable and, after a long intermediate plateau state, the system starts lowering its total energy through coarsening, which lowers both the interfacial energy as well as the bulk free energy due to the creation of new ultrathin films. With this observation in mind, we can now go back to figure 14(c); here, the thicker film ( $\tilde{h}_{0}=10\unicode[STIX]{x1D6FF}$ ) leads to the formation of fewer drops upon dewetting, meaning lower interfacial and bulk free energies in the mixed-mode instability regime compared to the spinodal regime (formation of fewer drops means a higher fraction of the total surface is covered by the ultrathin films that have a lower bulk free energy). This is why, right after dewetting, the thicker film case $\tilde{h}_{0}=10\unicode[STIX]{x1D6FF}$ leads to a lower total free energy compared to the $\tilde{h}_{0}=4\unicode[STIX]{x1D6FF}$ case in the spinodal regime. At long times, however, as the coarsening sets in, the dynamics become self-similar and the total system energies per unit length of the substrate corresponding to the different regimes all collapse on top of each other.

5.4 LSW mean-field description of coarsening: self-similar drop-size distribution

In the late stages of a first-order phase transition of a binary alloy mixture, Ostwald ripening is quite common (Ostwald Reference Ostwald1897; Siggia Reference Siggia1979; Voorhees Reference Voorhees1985). The curvature dependence of the chemical potential leads to a flow from higher-curvature regions (smaller clusters) to lower-curvature regions (larger clusters). The result of this coarsening is a decrease in the total interfacial energy of the system. Lifshitz & Slyozov (Reference Lifshitz and Slyozov1961) were the first to place experimental measurements of Ostwald ripening within a consistent theoretical framework: they considered a spherical cluster in a supersaturated solution and developed a mean-field description for the evolution of the size of the cluster at long times, $\langle r(t)\rangle \sim t^{1/3}$ . This scaling is an intermediate-asymptotic behaviour (Barenblatt Reference Barenblatt1996) for a quasi-steady system in isolation, i.e. the interactions with other clusters are neglected, so it is expected to hold in the limit of dilute solutions. Lifshitz & Slyozov (Reference Lifshitz and Slyozov1961) originally assumed the transport in the medium to be diffusion-dominated; later, Wagner (Reference Wagner1961) independently studied the Ostwald-ripening process and considered the case where the attachment/detachment of particles from the clusters is the rate-limiting factor and derived a scaling $\langle r(t)\rangle \sim t^{1/2}$ . Apart from the mode of mass transport considered, the two theories are essentially the same and are known as the LSW model for coarsening (Kahlweit Reference Kahlweit1975). An alternative generalized view of the same problem is to consider all the clusters of size greater than $r_{c}\sim t^{\unicode[STIX]{x1D6FD}}$ to grow and all with a smaller size to shrink, where $\unicode[STIX]{x1D6FD}=1/3$ in the diffusion-dominated case and $\unicode[STIX]{x1D6FD}=1/2$ in the attachment/detachment-dominated case.

Theories of LSW type appear in many diverse phenomena, such as stability of emulsions (Imhof & Pine Reference Imhof and Pine1997; Taylor Reference Taylor1998; Bibette, Calderon & Poulin Reference Bibette, Calderon and Poulin1999; Solans et al. Reference Solans, Izquierdo, Nolla, Azemar and Garcia-Celma2005; Gupta et al. Reference Gupta, Eral, Hatton and Doyle2016), droplet-size distribution in liquid jet fragmentation (Eggers & Villermaux Reference Eggers and Villermaux2008), coarsening of granular clusters (Aranson & Tsimring Reference Aranson and Tsimring2006), phase separation in polymer blends (Geoghegan & Krausch Reference Geoghegan and Krausch2003), growth of silicone nanowires (Schmidt, Wittemann & Gösele Reference Schmidt, Wittemann and Gösele2010), quantum dots (Liu & Risbud Reference Liu and Risbud1990), growth of nanoparticles in colloidal solutions (Talapin et al. Reference Talapin, Rogach, Haase and Weller2001), and grain growth in thin films (Thompson Reference Thompson1990), or even loss of electrocatalyst coating in low-temperature fuel cells (Shao-Horn et al. Reference Shao-Horn, Sheng, Chen, Ferreira, Holby and Morgan2007). There is a close connection between the LSW theory and coarsening of drops connected by thin films. The drops formed upon dewetting are in a metastable quasi-equilibrium configuration, so one can treat a droplet connected to a near-equilibrium film in isolation from the other drops. The transport through the surrounding thin film is diffusion-dominated, since curvature becomes negligible in the film. Diffusion here is set by the interface potential and mobility reaches a constant value set by the film thickness, so the transport between droplets can formally be written as a diffusion equation similar to the LSW theory.

At long times, the clusters or drops follow a size distribution function $F(r,\tilde{t})$ , or equivalently $F(m,\tilde{t})$ , where $m=\int _{-\tilde{w}}^{\tilde{w}}\tilde{h}\,\text{d}\tilde{x}$ is the mass of a 1D drop, which satisfies the following continuity equation (Voorhees Reference Voorhees1985; Gratton & Witelski Reference Gratton and Witelski2009):

(5.3) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}F(m,\tilde{t})}{\unicode[STIX]{x2202}\tilde{t}}}+{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}m}}(F(m,\tilde{t})v(m))=0, & \displaystyle\end{eqnarray}$$

where $v(m)=\text{d}m/\text{d}\tilde{t}$ determines the flux of particles. The number of drops can then be related to the distribution function as $N=\int _{0}^{\infty }F(m,\tilde{t})\,\text{d}m$ . The problem then reduces to defining the flux $v(m)$ . Gratton & Witelski (Reference Gratton and Witelski2009) showed that an equivalent LSW mean-field model can be defined for a system of drops, where a drop is considered to be at a mean distance $\bar{L}(\tilde{t})=L/N(\tilde{t})$ from two neighbouring drops of mass $m_{\ast }$ ; since the mass transport between the drops takes place through the ultrathin films with nearly zero curvature connecting them, the transport through the films becomes purely diffusive and its dynamics can be approximated as $\unicode[STIX]{x2202}\tilde{h}/\unicode[STIX]{x2202}\tilde{t}\sim \unicode[STIX]{x2202}^{2}/\unicode[STIX]{x2202}\tilde{x}^{2}({\mathcal{V}}(\tilde{h}))$ , where $d{\mathcal{V}}/\text{d}\tilde{h}=\tilde{h}^{3}\,\text{d}^{2}\tilde{f}/\text{d}\tilde{h}^{2}$ (Glasner & Witelski Reference Glasner and Witelski2003). The mean flux between the drops can then be approximated as $v(m)=2({\mathcal{V}}(m_{\ast })-{\mathcal{V}}(m))/\bar{L}$ , where the mean-field potential is defined as ${\mathcal{V}}(m_{\ast }(\tilde{t}))=(1/N)\int _{0}^{\infty }{\mathcal{V}}(m)F(m,\tilde{t})\,\text{d}m$ . The non-locality in this mean-field description comes from the mean quantities $\bar{L}$ and $m_{\ast }$ , both of which evolve in time. At long times, we expect the distribution of the drop sizes to become self-similar, for which we can postulate $F(m,\tilde{t})=g(m/m_{\ast })/\tilde{t}^{\unicode[STIX]{x1D6FC}}$ and $m_{\ast }=c\tilde{t}^{\unicode[STIX]{x1D6FD}}$ . Using conservation of mass, one then arrives at the following ODE for the self-similar distribution function $g(z)$ (Gratton & Witelski Reference Gratton and Witelski2009):

(5.4) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\text{d}g}{\text{d}z}}=\left({\displaystyle \frac{27/2-8z^{3/2}}{4z^{5/2}+27z(1-\sqrt{z})}}\right)g, & \displaystyle\end{eqnarray}$$

where $0\leqslant z\equiv (m/m_{\ast })\leqslant 9/4$ , and the solution of the above equation can be obtained in the analytical form to be $g(z)=C(\sqrt{z}\text{e}^{2/(-3+2\sqrt{z})})/((3-2\sqrt{z})^{28/9}(3+\sqrt{z})^{17/9})$ with $C\approx 70$ as a normalization constant. From the conservation of mass, one can further find $m_{\ast }\approx 0.846\bar{m}$ (with $\bar{m}=M_{tot}/N(\tilde{t})$ being the total liquid mass divided by the instantaneous number of drops), which separates growing and shrinking drops, i.e. $v(m_{\ast })=0$ . This mean-field treatment is expected to hold in the dilute limit, where local interactions between drops can be neglected and each drop only interacts with a background field.

Figure 15.

Droplet mass distribution for the spinodal regime with $\tilde{h}_{0}=4\unicode[STIX]{x1D6FF}$ obtained from ensemble averaging of $10$ realizations in domains of size $250\unicode[STIX]{x1D706}_{m}$ with $z=m/m_{\ast }$ ; vertical lines show the error bars computed from the standard deviation of the $10$ realizations; red squares represent classical model with a constant surface tension; green diamonds represent the stochastic thin-film model; blue circles represent the new model wth height-dependent surface tension; the solid line represents the mean-field LSW theory (5.4); the dashed line represents the Smoluchowski distribution (5.5).

The normalized distribution of the droplet mass ( $\langle F(z,\tilde{t})/N(\tilde{t})\rangle >$ with $z=m/m_{\ast }$ ) in the self-similar intermediate regime, where the LSW model is applicable is shown in figure 15. The results of the simulations closely follow (5.4) (solid line) for small droplets. The close agreement we observe between the classical and new models, as well as in the presence of thermal fluctuations, indicates that in the self-similar coarsening regime the details of the thin-film interfacial dynamics are not critical. For larger-size droplets, i.e. $z>1$ ( $m>m_{\ast }$ ), however, we do observe deviations from the LSW model. While some deviations have also been observed by Gratton & Witelski (Reference Gratton and Witelski2009) in their dynamical system treatment of coarsening, and it is a known issue in the LSW model, there is a distinct skewness in the distribution of the droplets in the new model. The long tail observed in the distribution of droplet sizes in our model has also been reported in some experiments and Monte Carlo models of thin-film coarsening (Lo & Skodje Reference Lo and Skodje2000; Limary & Green Reference Limary and Green2002, Reference Limary and Green2003; Green Reference Green2003; Meli & Green Reference Meli and Green2008; Woehl et al. Reference Woehl, Park, Evans, Arslan, Ristenpart and Browning2014) and is typically associated with coarsening through coalescence. This type of drop-size distribution is modelled with the Smoluchowski equation (Smoluchowski Reference Smoluchowski1917; Sholl & Skodje Reference Sholl and Skodje1996; Lo & Skodje Reference Lo and Skodje2000; Eggers & Villermaux Reference Eggers and Villermaux2008):

(5.5) $$\begin{eqnarray}\displaystyle & \displaystyle g(z)=\left({\displaystyle \frac{\text{d}W(Wz)^{d(\unicode[STIX]{x1D6FC}+1-1/d)}}{\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D6FC}+1)}}\right)\text{e}^{-(Wz)^{d}}, & \displaystyle\end{eqnarray}$$

where $W=\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D6FC}+1+1/d)/\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D6FC}+1)$ with $\unicode[STIX]{x1D6E4}$ the gamma function, and $d$ represents the dimension of the system. When mass transport is dominated by diffusion around the periphery of the droplet, $\unicode[STIX]{x1D6FC}=3/2$ ; when diffusion away from the boundaries is the dominant transport mechanism, $\unicode[STIX]{x1D6FC}=1$ ; and when evaporation–condensation in the periphery of the drops is the main transport avenue, $\unicode[STIX]{x1D6FC}=1/2$ (Lo & Skodje Reference Lo and Skodje2000). Here, we find the best fit for the tail of the distribution in a 1D system of droplets is obtained for the periphery-diffusion-dominated case with $\unicode[STIX]{x1D6FC}=3/2$ . It is interesting, however, that the change in the tail of the drop-size distribution in our model is not associated with a change in the coarsening mechanism, i.e. Ostwald ripening remains the dominant coarsening mechanism (figure 10); perhaps, it is the lower lateral motion of the larger-size droplets that leads to the asymmetry in the distribution. While our 1D simulation results cannot be directly compared with the experiments, the long-tailed distribution in our model calls into question whether the reported distributions in the experiments are uniquely a signature of the coalescence-dominated coarsening, and this motivates further detailed experimental observations.