Figure 8 recalls the experiments of Blake, et al [19] to illustrate how the theory can predict contact angle saturation, whereby $cos\theta$ does not increase with voltage squared until reaching perfect wetting at $cos\theta =+1$ (and non-equilibrium beyond), but instead switches regime to limit its rise. Because these authors obtained the data for a PET film without additional texture, comparisons of our model with those data are only qualitative. However, they suggest that saturation begins as the voltage rises across the transition between regime II and IV. Because hysteretic regimes I and II have $cos{\theta }^{+}<cos{\theta }^{-}$, they possess a substantial energy barrier, calculated in reference [1], which promotes contact line pinning until saturation occurs. If a goniometry experiment could keep the drop in place in regime IV despite the absence of pinning, regime IV would eventually give way to hysteretic regime V at $\xi \simeq 1.18$, then to regime VI at $\xi \simeq 1.34$ (inset of Figure 8). However, because regimes V and VI have out of equilibrium advancing angles, they cannot be observed.

Gupta, et al [20] conducted experiments in which they quantified hysteresis on a hydrophobic solid. Once again, because the latter’s microscopic texture is unknown, comparisons with our model are merely instructive. As Figure 9 suggests, these authors observed a stable drop until the transition from regime II to IV, at which point contact angle saturation occurred. In principle, greater voltages should raise $cos\theta$ further toward perfect hydrophily in subsequent regimes IX and VI (inset). However, the absence of an energy barrier in non-hysteretic regime IV made it impractical to keep the drop steady and explore those regimes at larger $U^2$. On the other hand, the sudden removal of the barrier near $\xi \simeq 0.641$ allowed the drop to undergo a recession that the substantial barrier in hysteretic regimes I and II had hitherto prohibited. With effective values of $ϵ$, $\alpha$ and $b^{*}$ shown in Figure 9, the model of Equation (32) captures essential features of the data, with the exception of the apparent threshold $\xi \simeq 0.06$ that electrowetting had to exceed before affecting the advancing angle.

Herbertson, et al [21] created a patterned solid surface featuring cylindrical posts in an array of equilateral triangles. Because each of their unit cells possessed three adjacent cavities, their arrangement was topologically different from the square packing of columns with four neighbors that we modeled. In addition, their design included ridges of height $h/2$ joining every two posts along parallel lines, which amounted to two and a half cross-sections separating each cavity to its neighbors, thereby reducing $\lambda$. In spite of these differences, it is instructive to compare our predictions to their data, in which they raised voltage from zero until saturation, then progressively brought it back down again (Figure 10). As these authors suggested, we estimate the underlying scale of $\xi$ by fixing $(1/2){\upsilon }_0{K}_s/\left(H{\gamma }_{\ell g}\right)\simeq 7{10}^{-5}{\mathrm{V}}^{-2}$. Otherwise, we adopt values of $\alpha$, $\lambda$, $ϵ$, and $h^{*}$ calculated from their triangular lattice, including ridges and, for compatibility with our model, we prescribed an effective $b^{*}$ from Equation (13).

Figure 10 compares our predictions to data. As these authors raised $\xi$ from zero, the textured solid conformed to regime III in $0<\xi <0.63$, then regime V in $0.63<\xi <1.07$, while the advancing angle adjusted to a progressively smaller equilibrium value at each new voltage. Both regimes are deemed ‘metastable’ [4,5], as they cannot spontaneously achieve complete cavity wetting upon recession (Table 1). Since they are also hysteretic, they oppose a substantial energy barrier to retracting the contact angle [1]. Consequently, because the subsequent transition to regime II required a sharp increase in $\theta$ at $\xi \simeq 1.07$, the small drop, barely deformed by gravity, likely could not recede. Instead it never transitioned to regime II and likely remained ‘frozen’ in its current state of regime V (dash-dotted line). Then, as $\xi$ was subsequently lowered, the contact patch area and $cos\theta$ also remained nearly invariant. In this stand-down, the metastability and energy barrier of regime V prevented the system to explore its equilibrium receding angle found in Table 1. Instead, because returning values of $cos\theta$ were located between the two ‘primary curves’ of $cos{\theta }^{-}$ and $cos{\theta }^{+}$, they exhibited a behavior called ‘return-point’ hysteresis [8].

Figure 10 illustrates the important fact that values of $cos\theta$ predicted by the equilibrium theory cannot always be achieved in practice. In other words, changing the voltage can occasionally induce regime transitions requiring a reversal of the contact line that the energy barrier of hysteresis may forbid. If ‘metastable’ regimes III and V are involved, some transitions can similarly be forbidden, as they are predicated on an initial state that these regimes cannot spontaneously reach without applying extrinsic energy [1].

In this context, we illustrate similar limitations of the equilibrium theory by comparing its predictions with the data of Krupenkin, et al [12], who created a texture consisting of very slender microscopic posts on a square lattice with relatively large pitch and, unlike solids in Figure 2, Figure 3, Figure 4 and Figure 5, a thin dielectric layer covering all features at a constant distance w from their exposed surface. Because this layer sets the electrostatic energy, its uniform thickness produces different capacitances and predictions than in Section 3, Section 4, Section 5, Section 6 and Section 7. Because $w\ll H$, the resulting dimensionless square voltages $\xi \equiv (1/2){\upsilon }_0{K}_s{U}^2/\left(w{\gamma }_{\ell g}\right)$ are also much larger than those in Figure 8, Figure 9 and Figure 10. To illustrate how the theory is applied to this case, Appendix B summarizes how predictions are updated.

Contrary to what Krupenkin, et al [12] surmised from their observations of a linear relation of $cos\theta$ vs $(1-ϵ)$, the drop began in hysteretic and ‘metastable’ regime V rather than the non-hysteretic dry Cassie-Baxter state (regime VI). As $cos\theta$ increased with $\xi$, the contact angle decreased, therefore causing the constant-volume drop to advance. The drop then experienced a sudden transition to regime II (Figure 11), which should have resulted in a rapid reduction of $cos\theta$ or, equivalently, a higher $\theta$ leading to recession. However, the hysteretic nature of regimes V and II erected an energy barrier that pinned the contact line and prevented $\theta$ from reversing course. In addition, the transition threw the system out of equilibrium, as the $cos\theta <-1$ outcome of Equation (32) is impossible. Shortly afterwards, the rising square voltage caused yet another transition to regime IV, once again out of equilibrium ($cos\theta >+1$).

As Figure 11 shows, the theory, which adopts the actual textured geometry and Young contact angle that Krupenkin, et al [12] reported, predicts the rise of $cos\theta$ with $\xi$ in regime V and the sudden transition that they observed. However, it cannot capture with certainty $cos\theta$ behind the transition, as the system deviates sharply from equilibrium. Lastly, although the theory predicts remarkably well the transition value of $\xi$ for the relatively short pitch of $b+d=1.05\mu$m that Krupenkin, et al [12] staged (Figure 11, left), it fares less well for the sparse forest of slender posts on the much larger $4\mu$m pitch (Figure 11, right). Because cavities were deep and wide at $4\mu$m, they were unlikely to possess only states of complete filling or complete drainage, thus challenging the basic assumption of the Ising theory.

The following supporting information can be downloaded at the website of this paper posted on Preprints.org.

This Appendix derives lumped-parameter capacitances of the central cell, accounting for possible electrostatic interactions with unlike nearest neighbors, as illustrated in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6. As mentioned earlier, capacitances and lengths are made dimensionless with $K_s{\upsilon }_0(A_0+A_s)/H$ and H, respectively, and denoted by an asterisk^*. Results are presented in a form that is not indeterminate as $ϵ\to 0$, so they may be programmed without singularity. Relations among geometrical parameters are found in Equations (10)-(13).

Because case 2 couples nearby cavities through pillars (Figure 3), it is the most algebraically-complicated configuration, and therefore we consider it first. Its principal source of electrostatic energy arises from the capacitance between the bottom of the cavity at voltage U and the reference voltage. To capture the contribution of pillars in case 2, we split their capacitance into a piece below the base and two complementary pieces $C_q^{*}/(1-p_1)$ and $C_q^{*}/p_1$ in series, where and $p_1$ is a parameter that we match to a wide range of homogeneous simulations with $n=0$. At their junction, the two pieces are connected to parts of the wetted pillar walls belonging to the unit cell by a capacitance of area $4b^{*}{h}^{*}$ across the effective distance $p_2{b}^{*}$, where $p_2$ is a parameter fitted to simulations with $n=0$. If nearby cavities are dry ($n\ne 0$), the electric field in pillars within the central unit cell is drawn toward parts of these pillars belonging to as many as n edge cells. Such interaction has a coupling capacitance of area $h^{*}{b}^{*}$ and an effective distance $p_3{b}^{*}$, where $p_3$ is fitted to simulations. Pillars touching dry edge cells are modeled as before by splitting $C_q^{*}$ into two parts. However, there is no longer a capacitance $C_w^{*}$ connecting their junction to any of the n dry cavities. Contributions from n pillars are connected in parallel as shown in Figure 3.

With a homogeneous filling distribution in case 2, the configuration is strictly periodic, the energy of the central cell is unaffected by its neighbors and, because $n=0$, its dimensionless capacitance reduces to If instead $n\ne 0$, loss of periodicity compels us to distinguish how much current flows to the base of the central cell from how much flows to the base of its near neighbors. To that end, we transform the capacitance network in three steps shown on the bottom Figure 3. First, we define a dimensionless impedance as the inverse of the corresponding dimensionless capacitance, $Z^{*}\equiv 1/C^{*}$. From Figure 3, we find After executing the triangle-to-star transformation we find the equivalent impedance $Z_2^{*}$ by matching currents flowing through either side of the pillars, thus obtaining and, finally, the dimensionless capacitance of a unit cell in case 2 with arbitrary number $0⩽n⩽4$ of dry edge cavities, As expected, $C_2^{*}(n=0)=C_{2_0}^{*}$.

As Figure 2 shows, case 1 has the same elementary capacitances $C_d^{*}$, $C_p^{*}$, $C_q^{*}$, $C_w^{*}$, $C_{\ell }^{*}$ and parameters $p_1$, $p_2$, $p_3$ as in case 2 (Equations (A1)–A5). Its unit cell capacitance with near-neighbors of identical filling state ($n=0$) is If the filled central cavity in case 1 interacts with n dry neighbors, we invoke a transformation similar to case 2 to find the equivalent pillar impedance $Z_1^{*}$,

Unlike cases 1 and 2, the coupling of dry central cavities in cases 3 and 4 to filled nearest neighbors is no longer dominated by capacitances through pillars. Instead, field lines emanating from the bottom of the dry central cavity are drawn to the liquid interface across the two adjacent pillars without penetrating the latter significantly.

In the homogeneous case 3 with dry neighboring cavities, the central cell capacitance vanishes, When instead there are n neighboring wet cells, Figure 4 indicates a higher capacitance with coupling capacitance of area $d^{*}{h}^{*}$ and effective distance $p_4{d}^{*}$ where $p_4$ is fitted to simulations. In Equation (A15), the ratio $C_m^{*}/C_d^{*}=(\alpha -1)(1-h^{*})/\left(4p_4{K}_s{b}^{*}\right)$ is finite.

In the homogeneous case 4 without neighbors of different state ($n=0$), Figure 5 suggests If instead $n\ne 0$, then where is the capacitance of the dry cavity, with ratio $C_d^{*}/C_c^{*}=K_s{h}^{*}/(1-h^{*})$. In Equation (A18), $C_k^{*}$ is a coupling capacitance of area similar to $C_m^{*}$, but with different effective distance $p_5{d}^{*}$ that reflects the presence of a wet ceiling on the central cavity, such that $C_k^{*}/C_c^{*}=ϵ{(\alpha -1)}^2/\left[16p_5(1-ϵ)\right]$ in Equation (A18).

We fitted $p_1\simeq 0.929$ and $p_2\simeq 0.197$ to 36 or more homogeneous simulations ($n=0$) in cases 1, 2 and 4 with $0.12<b^{*}<2.2$, $0.25<d^{*}<3$, $0.11<h^{*}<0.86$ and $K_s=3.2$, thereby matching the lumped-parameter expressions of $C_{1_0}^{*}$, $C_{2_0}^{*}$ and $C_{4_0}^{*}$ to within relative errors $<2.6\%$, $<1.3\%$ and $<5.8\%$ in those respective cases. We then conducted 20 inhomogeneous simulations with $b^{*}=d^{*}=1$ and $h^{*}=1/2$ in all four cases with $0⩽n⩽4$ to least-squares fit $p_3\simeq 0.044$, $p_4\simeq 0.065$ and $p_5\simeq 0.10$ (Figure 6).

Finally, we expand expressions in Equations (19)–(20) to note how they converge as $ϵ\to 0$, and where and