2. Mathematical model

Here we obtain the governing equations from a consideration of the system's mechanical equilibrium. The reader interested in the variational approach and results may go directly to § 5. Figure 1 depicts schematically the equilibrium configurations of sessile and pendent systems. In these a sessile (pendent) liquid drop is placed upon (hanging under) a pre-stretched, circular membrane that has been clamped along its circumference at a circular boundary of radius $r_0$.

Figure 1.

Schematic of a pre-stretched, circular membrane (grey) with (a) a sessile drop (blue) and (b) a pendent drop (blue) at equilibrium. The reference (unstretched), intermediate (pre-stretched) and current (equilibrium) configurations of the membrane are also shown. (c) Free-body diagram of a membrane element at equilibrium with normal $\hat {\boldsymbol {n}}$, mechanical tension $T_i^m$ ($i=s,\phi$) and membrane–air ($\gamma _{sv}$) and membrane–liquid ($\gamma _{sl}$) surface tensions indicated. See main text for details.

We assume that the membrane is chemically homogeneous and has a smooth surface. We model the membrane material as a homogeneous, isotropic, incompressible, hyperelastic material (Green & Adkins Reference Green and Adkins1960, p. 148). A hyperelastic material is an elastic material for which there exists a strain energy function (Ogden Reference Ogden1997, p. 206) that relates the stress and strain fields. We admit large deformations of the membrane, and so distinguish between the membrane's reference, intermediate and current configurations, which we now define.

In its reference configuration, for both sessile and pendent systems in figure 1, the membrane is flat and force free. This is indicated in figure 1 by a thick horizontal grey line, on which we have indicated a few representative material points: $P_0$, $G_0$, $C_0$ and $Q_0$. The membrane is then stretched uniformly and isotropically in its own plane by a stretch ratio $\lambda _0>1$ and clamped along the circular boundary, as shown in figure 1 by a dashed horizontal line connecting the two supports. The material points previously at $P_0$, $Q_0$, $G_0$ and $C_0$ now occupy positions at $P^I$, $Q^I$, $G^I$ and $C^I$, with $G^I$ coinciding with $G_0$ as the membrane's centre does not move when stretched uniformly. At this stage, the membrane is supposed to be in its intermediate configuration. A drop is then placed at or hung from the centre of the stretched flat membrane, so that the membrane deforms under the combined action of drop's weight and solid and fluid surface tensions, and finally achieves equilibrium. The equilibrated shape of the membrane is denoted by a solid line, and we label this as the membrane's current configuration. The material points we were following now lie at $P$, $Q$, $G$ and $C$ in figure 1. Note that $P$ and $Q$ coincide with their locations in the intermediate configuration as the membrane was pinned. Finally, the material point at $C$ lies on the three-phase contact line where the drop first meets the membrane. This material point, during the course of the membrane's deformation, went from its reference (unstretched membrane) location at $C_0$ to its intermediate (pre-stretched membrane) location $C^I$ before arriving its current (equilibrium) location at $C$.

The coordinate systems employed are shown in figure 1. The origin $O$ always lies at the drop's apex. As is standard in large-deformation elasticity, we locate material points in the reference and current configurations through different sets of coordinates, in our case the radial coordinates $R$ and $r$, respectively. Thus, the material point at $C$ in figure 1 is located at $r = a$, while in the reference configuration the same material point was at $R = A$. In general, if a material point is at the radial location $r$, then in the reference configuration it lay at a distance $R$. Note that for the drop only the current configuration is relevant.

2.1. Governing equations

We derive the equations governing the equilibrium of the drop and the membrane in turn, considering sessile and pendent systems depicted in figure 1 simultaneously. We restrict ourselves to axisymmetric systems.

In figure 1 the membrane at equilibrium spans $0\leq r\leq r_0$, with $r$ being the radial coordinate. At equilibrium, the three-phase contact line forms a contact circle of radius $a$. This allows us to divide the membrane into two parts: the wet and dry membranes that lie between, respectively, $0\leq r_{wm}\leq a$ and $a\leq r_{dm}\leq r_0$. In the unstretched (reference) state, the wet and dry parts of the membrane occupied, respectively, $0\leq R_{wm}\leq A$ and $A\leq R_{dm}\leq R_0$, where the radial coordinates $R_{wm}, R_{dm}$ and $R_0$ locate in the reference configuration material points currently at $r_{wm}$, $r_{dm}$ and $r_0$, respectively (see figure 1).

The equation governing the equilibrated shapes of drops is obtained from the Young–Laplace equation and is given by (Bashforth & Adams Reference Bashforth and Adams1883)

(2.1)\begin{equation} \frac{ z_d'' }{ ( 1 + z_d'^2 )^{ 3 / 2 } } + \frac{ 1 }{ r_d } \frac{ z_d' }{ ( 1 + z_d'^2 )^{ 1 / 2 } } = j \left( -\frac{ 2 }{ b } + \frac{ \rho g z_d }{ \gamma } \right), \end{equation}

where $j = \pm 1$ for sessile/pendent drops, $z_d( r_d )$ is the profile of the drop in figure 1, prime $(^\prime )$ always denotes differentiation with respect to relevant radial coordinate, in this case $r_d$, $\rho$ is the density of the liquid, $\gamma$ is the liquid–air surface tension, $b$ is the radius of curvature at the apex of the drop and $g$ is the acceleration due to gravity. Both $\rho$ and $\gamma$ are taken to be uniform. Every solution to (2.1) must satisfy the initial conditions

(2.2a,b)\begin{equation} z_d = 0 \quad \text{and}\quad z_d' = 0\quad \text{at } r_d = 0, \end{equation}

which follow from our definition of the coordinate system, and axisymmetry of the drop along with its smooth profile. While Bashforth & Adams (Reference Bashforth and Adams1883) obtained (2.1) and (2.2) to describe axisymmetric drops on a rigid surface, the formulation is agnostic to the type of substrate as long as it permits axisymmetric solutions. Finally, the triple point where the three-phase contact line penetrates the $rz$ plane lies at $\{ a, \, z_d( a ) \}$.

We now consider the equilibrium of the membrane. Because the membrane deforms axisymmetrically, the principal stresses and stretches in the current configuration are in the meridional (along the arc length $s$), circumferential (along the azimuthal angle $\phi$) and normal (along the unit normal $\hat {n}$) directions of a deformed membrane element (see figure 1 c). Referring to figure 1(c), the balances of forces along the meridional and circumferential directions of a membrane element in the current configuration are (Green & Adkins Reference Green and Adkins1960, p. 151)

(2.3a,b)\begin{equation} T'_s + \frac{ T_s - T_\phi }{ r } = 0 \quad \text{and} \quad \kappa_s T_s + \kappa_\phi T_\phi = \tilde{p}, \end{equation}

respectively, where $T_s( s )$ and $T_\phi ( s )$ are the total principal tensions per unit edge length along the $s$ and $\phi$ directions, respectively, prime $(^\prime )$ now denotes differentiation with respect to the radial coordinate $r$, $\tilde {p}$ is the normal pressure acting on a membrane element and

(2.4a,b)\begin{equation} \kappa_s = \frac{ z'' }{ ( 1 + z'^2 )^{ 3 / 2 } }\quad \text{and}\quad \kappa_\phi = \frac{ 1 }{ r } \frac{ z' }{ (1 + z'^2)^{ 1 / 2 } } \end{equation}

are principal curvatures along the $s$ and $\phi$ directions (Kühnel Reference Kühnel2015, p. 95), where $z( r )$ describes the membrane's equilibrium (current) profile in the $rz$ plane. The normal pressure

(2.5)\begin{equation} \tilde{p} ={-}j \left( \frac{ -2 \gamma }{ b } + \rho g z \right) \end{equation}

is the pressure exerted on the membrane by a fluid column in the drop and has contributions from both the Laplace pressure and the liquid's weight. We emphasize that the total principal tensions $T_s( s )$ and $T_\phi ( s )$ are the sum of the mechanical tension in the membrane and the surface tensions associated with the membrane–liquid and membrane–air interfaces (see the derivation in Appendix A). The mechanical tension is really the internal or Cauchy tension in the membrane that arises from the action of one part of the membrane on the other across a dividing surface (see Love Reference Love1927, Note B).

To complete the description of the membrane's deformation we need to relate the tensions in the membrane to the stretch it experiences. In a hyperelastic membrane the principal mechanical tensions may be obtained in terms of a strain energy density function $W$ (Green & Adkins Reference Green and Adkins1960; Haughton Reference Haughton2001) so that, following Long, Shull & Hui (Reference Long, Shull and Hui2010), we may write

(2.6a,b)\begin{equation} T_s = \frac{ t_0 }{ \hat{ \lambda }_\phi } \frac{ \partial W }{ \partial \hat{ \lambda }_s } + \gamma_{ m } \quad \text{and} \quad T_\phi = \frac{ t_0 }{ \hat{ \lambda }_s } \frac{ \partial W }{\partial \hat{ \lambda }_\phi } + \gamma_{ m }, \end{equation}

where $t_0$ is the thickness of the membrane in the reference configuration, $\hat { \lambda }_s$ and $\hat { \lambda }_\phi$ are the total principal stretches with respect to the reference configuration in the $s$ and $\phi$ directions, respectively, and $\gamma _{ m }$ is the membrane surface tension that is assumed to be uniform and isotropic. Equation (2.6) expresses in mathematical terms the split of the total tension in terms of the mechanical tension and the surface tensions mentioned above. For an isotropic, incompressible and hyperelastic material, the strain energy density function has the following form (Ogden Reference Ogden1997, p. 220):

(2.7)\begin{equation} W = W( I_1, I_2), \end{equation}

where $I_1$ and $I_2$ are, respectively, the first and second principal invariants of the Cauchy–Green deformation tensor $\boldsymbol { F } \boldsymbol { F }^{ T }$, with $\boldsymbol { F }$ being the deformation gradient (Ogden Reference Ogden1997, p. 218) of the material. The principal invariants for the membrane may be expressed in terms of its principal stretches by

(2.8a,b)\begin{equation} I_1 = \hat{ \lambda }_s^2 + \hat{ \lambda }_\phi^2 + ( \hat{ \lambda }_s \hat{ \lambda }_\phi )^{ {-}2 } \quad\text{and}\quad I_2 = \hat{ \lambda }_s^{ {-}2 } + \hat{ \lambda }_\phi^{ {-}2 } + ( \hat{ \lambda }_s \hat{ \lambda }_\phi)^2, \end{equation}

where the principal stretch along unit normal $\hat {n}$ is set to $( \hat { \lambda }_s \hat { \lambda }_\phi )^{ -1 }$ in order to satisfy the incompressibility condition. The principal stretches $\hat { \lambda }_s$ and $\hat { \lambda }_\phi$ are, in turn, given by

(2.9a)\begin{equation} \hat{\lambda}_s = \lambda_0 \lambda_s =\lambda_0 \left\{ \left( \frac{ \mathrm{ d } r }{ \mathrm{ d } R^I } \right)^2 + \left( \frac{ \mathrm{ d } z }{ \mathrm{ d } R^I } \right)^2 \right\}^{1/2} \end{equation}

and

(2.9b)\begin{equation} \hat{ \lambda }_\phi = \lambda_0 \lambda_\phi = \lambda_0 r / R^I, \end{equation}

where $\lambda _0$, we recall, is the uniform, in-plane, isotropic stretch relating the reference to the intermediate configuration in figure 1, $R^I$ is the radial coordinate in the intermediate configuration of a material point of the membrane currently at $r$ and

(2.10a,b)\begin{equation} \lambda_s = \sqrt{ ( \mathrm{ d } r / \mathrm{ d } R^I )^2 + ( \mathrm{ d } z / \mathrm{ d } R^I )^2 } \quad \text{and}\quad \lambda_\phi = r / R^I \end{equation}

are principal stretches in the meridional and circumferential directions, respectively, that relate the membrane's intermediate and current configurations.

Now, when the stretches in the membrane are large, i.e. $\hat { \lambda }_s , \hat { \lambda }_\phi \gg 1$, we have $I_1 \approx \hat { \lambda }_s^2 + \hat { \lambda }_\phi ^2$ and $I_2 \approx ( \hat { \lambda }_s \hat { \lambda }_\phi )^2$ and, thus, $I_2 \gg I_1$. Under such circumstances, we proceed by retaining the dependence of $W$ on $I_2$ alone. Then, following Long & Hui (Reference Long and Hui2012), we restrict ourselves to a class of hyperelastic membranes with $W = \hat { W }( I_2 )$, henceforth called membranes made of $I_2$ materials, or $I_2$ membranes for brevity.

The principal tensions (2.6) become equal in membranes with $I_2$ materials, i.e.

(2.11)\begin{equation} T_s = T_\phi = 2 t_0 \hat{ \lambda }_s \hat{ \lambda }_\phi \frac{ \mathrm{ d } \hat{ W } }{ \mathrm{ d } I_2 } + \gamma_m =: T; \end{equation}

thus, under large stretches the membrane experiences an equibiaxial tension given by $T( s )$. Such a scenario may arise when the pre-stretch $\lambda _0 \gg 1$ or drop volumes are moderate/large. In the latter case, the effect of gravity must be considered. With the membrane under equibiaxial tension (2.11), the force balance along the meridional direction (2.3a) provides

(2.12)\begin{equation} T = 2 t_0 \hat{ \lambda }_s \hat{ \lambda }_\phi \frac{ \partial \hat{ W } }{ \partial I_2 } + \gamma_m = \text{constant}, \end{equation}

i.e. the tension $T$ in the membrane is uniform. This also suggests that the product of the principal stretches is the same everywhere, although the individual stretches need not be uniform (Long & Hui Reference Long and Hui2012). We note that the choice of $I_2$ membranes is a simplifying but not a limiting assumption. The present framework has been utilized to investigate a wider class of non-Hookean membranes (Nair Reference Nair2022) in which $T_s \neq T_\phi$.

We now adapt the above derivation to the equilibrium of wet and dry membranes in the presence of a drop, as in figure 1. Combining (2.5), (2.4) and (2.11), the force balance along the normal direction (2.3b) yields the equation governing the profiles of the wet $(z_{wm})$ and dry membranes $(z_{dm})$ as, respectively,

(2.13)\begin{equation} T_{wm} \left\{ \frac{ z_{wm}'' }{ ( 1 + z_{wm}'^2 )^{ 3 / 2 } } + \frac{ 1 }{ r_{wm} } \frac{ z_{wm}' }{ ( 1 + z_{wm}'^2 )^{ 1 / 2 } } \right\} ={-}j \left( -\frac{ 2 \gamma }{ b } + \rho g z_{wm} \right) \end{equation}

and

(2.14)\begin{equation} T_{dm} \left\{ \frac{ z_{dm}'' }{ ( 1 + z_{dm}'^2 )^{ 3 / 2 } } + \frac{ 1 }{ r_{dm} } \frac{ z_{dm}' }{ ( 1 + z_{dm}'^2)^{ 1 / 2 } } \right\} = 0, \end{equation}

where $T_{wm}$ and $T_{dm}$ are the uniform, equibiaxial tensions in the wet and dry membranes. To obtain (2.14), we have utilized the fact that the dry membrane is free from external loads.

Before we specify the boundary conditions, we will establish the local geometry at the triple point and the forces acting there, as displayed in figure 2. Balance of forces along the $r$ and $z$ directions provides, in turn,

(2.15a)\begin{equation} T_{dm}\cos\theta_{dm}-\gamma\cos\theta_d-T_{wm}\cos\theta_{wm}=0 \end{equation}

and

(2.15b)\begin{equation} T_{dm}\sin\theta_{dm}+j\gamma\sin\theta_d-T_{wm}\sin\theta_{wm}=0, \end{equation}

where the drop angle $\theta _d$, the wet membrane angle $\theta _{ wm }$ and the dry membrane angle $\theta _{ dm }$ are the angles made at the triple point with the horizontal by the drop, the wet membrane and the dry membrane, respectively. The drop angle $\theta _d$ is also sometimes referred to as the apparent contact angle in the literature (Liu et al. Reference Liu, Liu, Jagota and Hui2020). We note that (2.15) is analogous to Neumann's triangle involving the balance of surface tensions in a system consisting of three fluid phases (de Gennes et al. Reference de Gennes, Brochard-Wyart and Quéré2004).

Figure 2.

Local geometry and the forces acting at the triple point of a membrane at equilibrium with (a) a sessile drop and (b) a pendent drop.

We may now write the boundary conditions for the wet membrane as

(2.16a,b)\begin{equation} z_{wm}' = 0 \ \text{at} \ r_{wm} = 0 \quad \text{and}\quad z_{wm} = z_d \ \text{at} \ r_{wm} = a, \end{equation}

where the latter follows from the membrane's profile $z_{wm}(r_{wm})$ in the $rz$ plane passing through the triple point $\{ a, \, z_d( a ) \}$. The boundary conditions for the dry membrane are

(2.17a,b)\begin{equation} z_{dm} = z_d \quad \text{and}\quad z_{dm}' = \tan \theta_{dm} \quad \text{at} \ r_{dm} = a. \end{equation}

This completes the formulation of the drop–membrane system based purely upon balance of forces. At present, the interfacial interaction was modelled through surface tension forces, and we will revisit this in § 5. Note also that the drop's volume is computed after solving for the equilibrium shapes of the drop and the membrane, and is not an a priori constraint imposed upon the system.

2.2. Non-dimensionalization

We now non-dimensionalize the governing equations. We set

(2.18a–d)\begin{equation} r_d / r_0 = \bar{r}_d, \quad z_d / r_0 = \bar{z}_d, \quad a / r_0 = \bar{a}\quad \text{and} \quad b / r_0 = \bar{b}, \end{equation}

where $r_0$ is the radius of the clamped boundary (see figure 1). With this (2.1) and (2.2) become

(2.19)\begin{equation} \frac{ \bar{z}_d'' }{ ( 1 + \bar{z}_d'^2 )^{ 3 / 2 } } + \frac{ 1 }{ \bar{r}_d } \frac{ \bar{z}_d' }{ ( 1 + \bar{z}_d'^2 )^{ 1 / 2 } } = j ({-}2 \beta + \omega^2 \bar{z}_d ) \end{equation}

and

(2.20a,b)\begin{equation} \bar{z}_d = 0 \quad \text{and} \quad\bar{z}_d' = 0 \quad \text{at } \bar{r}_d = 0, \end{equation}

respectively, where $\beta := 1 / b$ is the non-dimensionalized curvature of the drop at its apex and

(2.21)\begin{equation} \omega = r_0 / l_c, \quad \text{with} \ l_c = \sqrt{\gamma/\rho g} \end{equation}

being the capillary length associated with the liquid forming the drop.

We next non-dimensionalize the equations governing the wet and dry membranes. We set

(2.22a–c)\begin{equation} r_k / r_0 = \bar{r}_k, \quad z_k / r_0 = \bar{z}_k \quad\text{and} \quad T_k / \gamma = \bar{T}_k, \end{equation}

where the subscript $k = wm$ or $dm$, as appropriate, and we recall that $\gamma$ is the drop's surface tension. Thus, for the wet membrane, (2.13) and (2.16) are non-dimensionalized to obtain

(2.23)\begin{equation} \bar{T}_{wm} \left\{ \frac{ \bar{z}_{wm}'' }{ ( 1 + \bar{z}_{wm}'^2 )^{ 3 / 2 } } + \frac{ 1 }{ \bar{r}_{wm} } \frac{ \bar{z}_{wm}' }{ ( 1 + \bar{z}_{wm}'^2 )^{ 1 / 2 } } \right\} ={-}j ({-}2\beta + \omega^2 \bar{z}_{wm} ) \end{equation}

and

(2.24a,b)\begin{equation} \bar{z}_{wm}' = 0 \text{ at } \bar{r}_{wm} = 0 \quad \text{and} \quad \bar{z}_{wm} = \bar{z}_d \quad \text{at } \bar{r}_{wm} = \bar{a}, \end{equation}

respectively. Similarly, for the dry membrane, (2.14) and (2.17) we find, in turn,

(2.25)\begin{equation} \bar{T}_{dm} \left\{ \frac{ \bar{z}_{dm}'' }{ ( 1 + \bar{z}_{dm}'^2 )^{ 3 / 2 } } + \frac{ 1 }{ \bar{r}_{dm} } \frac{ \bar{z}_{dm}' }{ ( 1 + \bar{z}_{dm}'^2 )^{ 1 / 2 } } \right\} = 0 \end{equation}

and

(2.26a,b)\begin{equation} \bar{z}_{dm} = \bar{z}_d \quad \text{and} \quad \bar{z}_{dm}' = \tan \theta_{dm} \quad \text{at}\ \bar{r}_{dm} = \bar{a}. \end{equation}

Finally, the balance of forces (2.15) at the triple point is non-dimensionalized to yield

(2.27a)\begin{equation} \bar{T}_{dm}\cos\theta_{dm}-\cos\theta_d-\bar{T}_{wm}\cos\theta_{wm}=0 \end{equation}

and

(2.27b)\begin{equation} \bar{T}_{dm}\sin\theta_{dm}+j\sin\theta_d-\bar{T}_{wm}\sin\theta_{wm}=0. \end{equation}

3. Membrane pre-tension

We provide an outline of our solution procedure in Appendix B. It is important to note that the tensions $\bar {T}_{wm}$ and $\bar {T}_{dm}$ in the membrane are related to the system's equilibrium geometry and, so, are independent of the elastic properties of the membrane. The membrane's material affects the amount of pre-stretch that needs to be provided, as we discuss next.

The solution algorithm given in Appendix B finds the system's equilibrium for a choice of apex drop curvature $\beta$, contact circle radius $\bar {a}$ and current wet membrane tension $\bar {T}_{wm}$. At the same time, in typical experiments the membrane is stretched and affixed to a boundary prior to the drop's placement. This membrane pre-tension is measurable in experiments and it is expected that the pre-tension should be kept the same in order to compare results across different experiments. We will thus relate the pre-tension to the system's equilibrium configuration, which is again quantifiable in experiments.

To compute the pre-tension we need to select a particular form for the strain energy density function. We employ the Mooney–Rivlin strain energy function (Holzapfel Reference Holzapfel2000, p. 238):

(3.1)\begin{equation} W = \frac{ \mu }{ 2 } C_1 ( I_1 - 3 ) + \frac{ \mu }{ 2 } ( 1 - C_1 ) ( I_2 - 3 ), \end{equation}

where $\mu$ is the shear modulus and $0 \leq C_1 \leq 1$ is a material constant. For reasons listed previously, we focus here upon membranes made of $I_2$ materials for which $W = \hat {W}( I_2 )$, where $I_2 \approx ( \hat { \lambda }_s \hat { \lambda }_{ \phi } )^2$. To this end, we set $C_1 = 0$, so that

(3.2)\begin{equation} W = \hat{W}( I_2 ) = \frac{ \mu }{ 2 } ( I_2 - 3 ), \end{equation}

which is the strain energy function that we will utilize.

We now combine the membrane stretches (2.9) with the membrane tension (2.12) and non-dimensionalize to find the non-dimensional tensions in the wet and dry membranes as

(3.3)\begin{equation} \bar{T}_{wm} = \alpha_0\lambda_0^2 \frac{\bar{r}_{wm}}{\bar{R}_{wm}^I} \sqrt{ 1 + \bar{z}_{wm}'^2 } \frac{ \mathrm{ d } \bar{r}_{wm} }{ \mathrm{ d } \bar{R}_{wm}^I } + \bar{\gamma}_{ sl } + \bar{\gamma}_{ sv } \end{equation}

and

(3.4)\begin{equation} \bar{T}_{dm} = \alpha_0\lambda_0^2 \frac{\bar{r}_{dm}}{\bar{R}_{dm}^I} \sqrt{ 1 + \bar{z}_{dm}'^2 } \frac{ \mathrm{ d } \bar{r}_{dm} }{ \mathrm{ d } \bar{R}_{dm}^I } + 2 \bar{\gamma}_{ sv }, \end{equation}

respectively, where

(3.5a–d)\begin{align} \bar{ R }^I_k = R_k^I / r_0 \quad (k =dm, wm) , \quad \alpha_0 = \mu t_0 / \gamma , \quad \bar{\gamma}_{ sl } = \gamma_{ sl } / \gamma \quad \text{and} \quad \bar{\gamma}_{ sv } = \gamma_{ sv } / \gamma , \end{align}

with $\gamma _{ sl }$ and $\gamma _{ sv }$ being the surface tensions associated with the membrane–drop and membrane–air interfaces. We recall that the superscript ‘$I$’ indicates the intermediate configuration in figure 1. When deriving $\bar {T}_{wm}$ and $\bar {T}_{dm}$ above, we set the membrane surface tension $\gamma _m$ in (2.12) equal to $\gamma _{ sl } + \gamma _{ sv }$ for the wet membrane and $2 \gamma _{ sv }$ for the dry membrane.

We rewrite (3.3) and integrate both sides:

(3.6)\begin{equation} \int_{ 0 }^{ \bar{A}^I } \bar{R}_{wm}^I\,\mathrm{d}\bar{R}_{wm}^I=\frac{ \alpha_0 \lambda_0^2 }{\bar{T}_{wm}^m} \int_{ 0 }^{ \bar{a} } \bar{r}_{wm}\sqrt{ 1 + \bar{z}_{wm}'^2 } \,\mathrm{d}\bar{r}_{wm} \end{equation}

to obtain

(3.7)\begin{equation} (\bar{A}^I)^2 = \left( \frac{ 2 \alpha_r \lambda_0^2 }{\bar{T}_{wm}^m}\right) \int_{ 0 }^{ \bar{a} } \bar{r}_{wm}\sqrt{ 1 + \bar{z}_{wm}'^2 } \,\mathrm{d}\bar{r}_{wm}, \end{equation}

where $\bar{A}^I = A^I / r_0$ is the location of the triple point $\bar {a}$ in the intermediate configuration, and

(3.8)\begin{equation} \bar{T}_{wm}^m=\bar{T}_{wm}-\bar{\gamma}_{ sl }-\bar{\gamma}_{ sv } \end{equation}

is the mechanical wet membrane tension. Similarly, we rewrite (3.4) and integrate both sides:

(3.9)\begin{equation} \int_{ \bar{A}^I }^{ 1 } \bar{R}_{dm}^I\,\mathrm{d}\bar{R}_{dm}^I = \frac{ \alpha_0 \lambda_0^2 }{ \bar{T}_{dm}^m } \int_{ \bar{a} }^{ 1 } \bar{r}_{dm}\sqrt{ 1 + \bar{z}_{dm}'^2 }\, \mathrm{d}\bar{r}_{dm} \end{equation}

to find

(3.10)\begin{equation} (\bar{A}^I)^2 = 1 - \left( \frac{ 2 \alpha_0 \lambda_0^2 }{ \bar{T}_{dm}^m } \right) \int_{ \bar{a} }^{ 1 } \bar{r}_{dm}\sqrt{ 1 + \bar{z}_{dm}'^2 } \,\mathrm{d}\bar{r}_{dm}, \end{equation}

where, now,

(3.11)\begin{equation} \bar{T}_{dm}^m=\bar{T}_{dm}-2\bar{\gamma}_{sv} \end{equation}

is the mechanical dry membrane tension. Because the tensions $\bar {T}_{wm}$ and $\bar {T}_{dm}$ are independent of the membrane's elastic properties, so too are the mechanical tensions. Finally, equating (3.7) and (3.10) we compute the pre-stretch in the membrane to be

(3.12)\begin{equation} \lambda_0^2 = \alpha_0^{ {-}1 } \left( \frac{ 2\bar{\varOmega}_{wm} }{ \bar{T}_{wm}^m } + \frac{ 2\bar{\varOmega}_{dm} }{ \bar{T}_{dm}^m } \right)^{ {-}1 }, \end{equation}

where the non-dimensional surface areas of the wet and dry membranes are, respectively,

(3.13a,b)\begin{equation} \bar{\varOmega}_{wm} = \int_{ 0 }^{ \bar{a} } \bar{r}_{wm}\sqrt{ 1 + \bar{z}_{wm}'^2 } \,\mathrm{d}\bar{r}_{wm} \quad \text{and} \quad \bar{\varOmega}_{dm} = \int_{ \bar{a} }^{ 1 } \bar{r}_{dm}\sqrt{ 1 + \bar{z}_{dm}'^2 } \,\mathrm{d}\bar{r}_{dm}; \end{equation}

note that the pre-stretch depends upon the parameter $\alpha _0 = \mu t_0 / \gamma$, which is the scaled shear modulus of the membrane.

Now, from the definition (2.6) of a hyperelastic material and the constitutive relation (3.2), the non-dimensional, uniform, isotropic membrane pre-tension $T_0$ corresponding to a uniform, isotropic pre-stretch $\lambda _0 \gg 1$ is given by

(3.14)\begin{equation} \bar{T}_0 = \alpha_0 \lambda_0^2 + 2 \bar{\gamma}_{ sv}. \end{equation}

Combining (3.12) with (3.14), we obtain the non-dimensional membrane pre-tension as

(3.15)\begin{equation} \bar{T}_0 = \left( \frac{ 2 \bar{\varOmega}_{wm} }{ \bar{T}_{wm}^m } + \frac{ 2 \bar{\varOmega}_{dm} }{ \bar{T}_{dm}^m } \right)^{ {-}1 } + 2 \bar{\gamma}_{ sv }. \end{equation}

We note that the non-dimensional pre-tension $\bar {T}_0$ is also independent of the membrane's shear modulus $\mu$ and thickness $t_0$. As (3.14) shows, the effect of the membrane's properties is expressed through the pre-stretch $\lambda _0$.

We end this section with the following remark, which is revisited at the end of § 5. The solution algorithm given in Appendix B indicates that we need to specify three parameters, namely the drop curvature $\beta$ at its apex, the radius $\bar {a}$ of the contact circle and the wet membrane tension $\bar {T}_{wm}$ at equilibrium, in order to obtain a unique equilibrium configuration. From this the volume of the drop and the pre-tension in the membrane may be computed through (B4) and (3.15), respectively. However, it is more natural and useful to fix the drop's volume $\bar {V}$ and the wet membrane's pre-tension $\bar {T}_0$ when reporting results, and we do this below. Further, as a third parameter we employ the drop angle $\theta _d$, computed in Step I of the solution algorithm in Appendix B, as it is more accessible than the radius $\bar {a}$.

4. Equilibrium configurations: non-uniqueness

We now follow the methodology of the previous section to solve for the equilibrium of a sessile drop on a pre-stretched membrane. Figure 3 shows three possible equilibrium configurations for a given drop volume $\bar {V}$ and membrane pre-tension $\bar {T}_0$, obtained by fixing three values of the drop angle $\theta _d$. Although we display only three equilibria, in fact, there exist infinitely many equilibria, corresponding to every possible choice of $\theta _d$.

Figure 3.

Equilibrium configurations corresponding to three choices of the drop angle $\theta _d$ when a sessile drop having non-dimensional volume $\bar {V} = 2$ is placed upon a pre-stretched nonlinear elastic membrane with non-dimensional pre-tension $\bar {T}_0 = 2$. The membrane surface tensions $\bar {\gamma }_{sv} = \bar {\gamma }_{sl} = 0.5$.

Now, in a typical experiment the membrane's pre-tension is first set and a drop of known volume is placed on the membrane. The system then achieves a unique configuration at equilibrium, which entails fixing a particular value of the drop angle $\theta _d$. Thus, it appears that we need one more condition to reduce the choice of the free parameters in our theory to two and force balance alone provides an incomplete description.

The extra condition that will ensure uniqueness of the equilibrium solution will now be found by minimizing the total free energy of the system. The governing equations obtained above will also be recovered as part of the process.

5. Energy minimization

We expect that the system's equilibrium configuration will minimize the system's total potential energy while keeping the drop volume and the unstretched membrane surface area constant for a given membrane thickness $t_0$. This is equivalent to the minimizing the Helmholtz free energy – free energy for short – of the mechanical system. Figure 4 shows a schematic of the system we are considering, but now with a coordinate system whose origin is at a fixed distance below the clamp. We revert to a dimensional framework in this section to make the derivation transparent. Quantities without subscript refer to the drop, while those with subscripts ‘1’ and ‘2’ pertain to the wet and dry membranes, respectively. The drop's profile is given by $z = z(r)$, while the profiles of the wet and dry membranes are defined by, respectively, $z_1 = z_1(r_1)$ and $z_2 = z_2(r_2)$. At the same time, with $k = 1$ or 2, we may express the current radial $(r_k)$ and vertical $(z_k)$ locations of a material point in terms of its radial location $R_k$ in the reference configuration as $r_k = {\hat r}_k(R_k)$ and $z_k = {\hat z}_k(R_k)$.

Figure 4.

Schematic of axisymmetric drop–membrane systems: (a) sessile drop; (b) pendent drop. The coordinate system is placed at the datum which is at a fixed distance below the level of the clamp at which the membrane is pinned.

The total potential energy of the system is given by

(5.1)\begin{align} E_T&=2{\rm \pi}\gamma\int_0^{a}r\sqrt{1+z'^2}\,\mathrm{d}r+j{\rm \pi}\rho g\int_0^{a}z^2r\,\mathrm{d}r-j{\rm \pi}\rho g\int_0^{a} z_1^2r_1\,\mathrm{d}r_1\nonumber\\ &\quad +2{\rm \pi}\int_0^{A}\left(R_1W_1(\hat{\lambda}_{s1},\hat{\lambda}_{\phi 1})+(\gamma_{sv}+\gamma_{sl})\hat{r}_1\sqrt{\hat{r}_1'^2+\hat{z}_1'^2}\right)\,\mathrm{d}R_1\nonumber\\ &\quad + 2{\rm \pi}\int_{A}^{R_0} \left(R_2W_2(\hat{\lambda}_{s2},\hat{\lambda}_{\phi 2})+2\gamma_{sv}\hat{r}_2\sqrt{\hat{r}_2'^2+\hat{z}_2'^2}\right)\,\mathrm{d}R_2, \end{align}

where we recall that $j=\pm 1$ for sessile/pendent drops, $W_k \ (k=1, 2)$ are the strain energy density functions of the wet and dry membranes, $\hat {\lambda }_{sk}=\sqrt {\hat {r}_k'^2+\hat {z}_k'^2}$ and ${\hat {\lambda }_{\phi k}=\hat {r}_k/r_k}$ are, respectively, the total meridional and circumferential stretches, the prime $(^\prime )$ denotes differentiation with respect to the independent variable, so that $z'=\mathrm {d}z/\mathrm {d}r, {\hat r}_k'=\mathrm {d}{\hat r}_k/\mathrm {d}R_k$ and ${\hat z}_k'=\mathrm {d}{\hat z}_k/\mathrm {d}R_k$, and $\gamma$, $\gamma _{sv}$ and $\gamma _{sl}$ are the surface energies at the liquid–air, membrane–air and membrane–liquid interfaces, respectively. In (5.1), the first integral is the total surface energy of the drop, the second and third integrals constitute the gravitational potential energy of the drop, while the fourth and fifth integrals include the total strain and surface energies of the wet and dry parts of the membrane, respectively. Note that the membrane's strain energy is defined per unit area of the reference configuration, but the surface energy of the membrane is per unit area of the current configuration. Finally, in (5.1) we have taken the surface energies numerically equal to the surface tensions, and hence employed the same notation. This is a simplifying and commonly employed assumption (Andreotti & Snoeijer Reference Andreotti and Snoeijer2020). Strain-dependent surface energies may easily be incorporated into our overall process in the manner of, for example, Liu et al. (Reference Liu, Liu, Jagota and Hui2020).

We wish to minimize $E_T$ while keeping the drop's volume constant at, say, $V_0$. This is done most expeditiously by considering the augmented functional

(5.2)\begin{equation} H_T = E_T +2{\rm \pi} p\left(j\int_0^{a}zr\,\mathrm{d}r-j\int_0^{A}z_1r_1\,\mathrm{d}r_1 -{V}_0\right), \end{equation}

where $p$ is a Lagrange multiplier, which will turn out to be the thermodynamic pressure. The unstretched membrane's surface area is kept constant by fixing $R_0$ in (5.1). Note that $H_T$ is the total free energy of the system.

Combining (5.1) and (5.2), rearranging terms and setting $\mathrm {d}r_1=\hat {r}_1'\,\mathrm {d}R_1$, we obtain

(5.3)\begin{align} H_T[z, \hat{r}_1, \hat{z}_1, \hat{r}_2, \hat{z}_2, p]&=\int_0^{a}(F+p G)\,\mathrm{d}r+\int_0^{A}(F_1+p G_1)\,\mathrm{d}R_1\nonumber\\ &\quad +\int_{A}^{R_0}F_2\,\mathrm{d}R_2-p V_0, \end{align}

where

(5.4a)$$\begin{gather} F(r, z, z')=2{\rm \pi}\left(\gamma r\sqrt{1+z'^2}+j\frac{\rho g}{2}z^2 r\right), \end{gather}$$

(5.4b)$$\begin{gather}G(r, z)=2{\rm \pi} j zr, \end{gather}$$

(5.4c)$$\begin{gather}F_1(r_1, \hat{r}_1, \hat{z}_1, \hat{r}_1', \hat{z}_1') =2{\rm \pi}\left\{R_1W_1+(\gamma_{sv}+\gamma_{sl} )\hat{r}_1\sqrt{\hat{r}_1'^2+\hat{z}_1'^2}-j\frac{\rho g}{2}\hat{z}_1^2\,\hat{r}_1\,\hat{r}_1'\right\}, \end{gather}$$

(5.4d)$$\begin{gather}G_1(\hat{r}_1, \hat{z}_1, \hat{r}_1')={-}2{\rm \pi} j\hat{z}_1\,\hat{r}_1\,\hat{r}_1' \end{gather}$$

and

(5.4e)\begin{equation} F_2(r_2, \hat{r}_2, \hat{r}_2', \hat{z}_2')=2{\rm \pi} \left(R_2 W_2+2\gamma_{sv} \hat{r}_2\sqrt{\hat{r}_2'^2+\hat{z}_2'^2}\right). \end{equation}

A necessary condition for the free energy functional $H_T$ to have a minimizer is that the first variation of $H_T$ must vanish due to all admissible perturbations in the fields $z(r)$, ${\hat r}_k(R_k)$ and ${\hat z}_k(R_k)$, with $k = 1, 2$. The end-points $r= r_1= R_1=0$, $r_2 = r_0$ and $R_2 = R_0$ of the system's domain remain fixed. This follows from the system's symmetry, the clamping of the membrane at its edges and because the length of the unstretched membrane is fixed. However, we permit perturbations in the location of the triple point, as well as in the values there of the fields $z(r)$, ${\hat r}_k(R_k)$ and ${\hat z}_k(R_k)$. This acknowledges the fact that the three-phase contact line is determined internally by the system and cannot be externally specified.

We first recall (Gelfand & Fomin Reference Gelfand and Fomin2000, Ch. 3) that a field may be perturbed independently in the interior and at the ends of its domain. Computing the first variation in the five fields $z(r), {\hat r}_k(R_k)$ and ${\hat z}_k(R_k)$ due to perturbations in the interior of the domain leads to a set of five Euler–Lagrange equations. From these equations we exactly recover the governing equations for the drop (2.1) and for the wet and dry membranes (2.13) and (2.14), respectively. The surface energies $\gamma$, $\gamma _{sl}$ and $\gamma _{sv}$ now appear in the form of surface tensions, consistent with the free-body diagram in figure 1(c). The constraint on the volume carries over as is and may be identified with (B4). The details of these calculations may be found in Nair (Reference Nair2022). Variational analysis thus independently confirms the validity of the governing equations (2.13) and (2.14). Moreover, we note that the energy minimization clearly shows the validity of (2.6) with the mechanical equilibrium of the membrane being determined by the total tension $T$ and not just the mechanical part $T^m$. This is consistent with previous work (Shanahan Reference Shanahan1985; Neukirch et al. Reference Neukirch, Antkowiak and Marigo2014; Liu et al. Reference Liu, Liu, Jagota and Hui2020).

Turning to the first variation in $H_T$ obtained from perturbations in the fields at the ends of their domains leads to

(5.5)\begin{align} &\left.\left(\frac{\partial F}{\partial z'}+p\frac{\partial G}{\partial z'}\right)\delta z\right|_{r=0}^{r=a}+\left.\left(\frac{\partial F_1}{\partial\hat{r}_1'}+p\frac{\partial G_1}{\partial\hat{r}_1'}\right)\delta\hat{r}_1\right|_{R_1=0}^{R_1=A}+\left.\left(\frac{\partial F_1}{\partial\hat{z}_1'}+p\frac{\partial G_1}{\partial\hat{z}_1'}\right)\delta\hat{z}_1\right|_{R_1=0}^{R_1=A}\ldots\nonumber\\ &\quad +\left.\frac{\partial F_2}{\partial\hat{r}_2'}\delta\hat{r}_2\right|_{R_2=A}^{R_2=R_0}+\left.\frac{\partial F_2}{\partial\hat{z}_2'}\delta\hat{z}_2\right|_{R_2=A}^{R_2=R_0}+\left.\left\{(F+p G)-\left(\frac{\partial F}{\partial z'}+p\frac{\partial G}{\partial z'}\right)z'\right\}\delta r\right|_{r=0}^{r=a}\ldots\nonumber\\ &\quad +\left.\left\{(F_1+{p}G_1)-\left(\frac{\partial F_1}{\partial\hat{r}_1'}+p\frac{\partial G_1}{\partial\hat{r}_1'}\right)\hat{r}_1'-\left(\frac{\partial F_1}{\partial\hat{z}_1'}+p\frac{\partial G_1}{\partial\hat{z}_1'}\right)\hat{z}_1'\right\}\delta R_1\right|_{R_1=0}^{R_1=A} \ldots\nonumber\\ &\quad +\left.\left(F_2-\frac{\partial F_2}{\partial\hat{r}_2'}\hat{r}_2'-\frac{\partial F_2}{\partial\hat{z}_2'}\hat{z}_2'\right)\delta R_2\right|_{R_2=A}^{R_2=R_0}=0, \end{align}

where the vertical bar ‘$|$’ signifies evaluation at the boundaries as indicated, e.g.

(5.6)\begin{equation} \left.\frac{\partial F}{\partial z'}\delta z\right|_{r=0}^{r=a} = \frac{\partial F}{\partial z'}(a, z(a), z'(a) ) \delta z(r(a)) - \frac{\partial F}{\partial z'}(0, z(0), z'(0)) \delta z(r(0)). \end{equation}

Now, the perturbations at the boundaries in (5.5) are not arbitrary. Indeed, as discussed above, because of symmetry, $\delta r|_{r = 0} = \delta \hat {r}_1|_{R_1 = 0} = \delta R_1|_{R_1 = 0}=0$. Further, as the membrane is pinned at $R_2= R_0$, we must have $\delta \hat {r}_2|_{r_2=r_0}=\delta \hat {z}_2|_{R_2=R_0}=\delta R_2|_{R_2=R_0}=0$. Here, we remind ourselves that $z_k(r_k)=\hat {z}_k(R_k)$ with $r_k=\hat {r}_k(R_k)$ for $k = 1$ and 2.

The three-phase contact line is the circle defined by $r=a$, the materials points of which lie on the circle $R_1 = A = R_2$ in the reference configurations of the wet and dry membranes. At the triple point defined in the $rz$ plane by this contact circle, the profiles of the drop and the wet and dry membranes intersect. Thus, all admissible perturbations in the fields $z(r), {\hat r}_k(R_k)$ and ${\hat z}_k(R_k)$, with $k = 1,2$, must satisfy $r = a$, $R_1=R_2 = A$, $\hat {r}_1(R_1 = A)=\hat {r}_2(R_2 = A) = a$ and $z(r=a)=\hat {z}_1(R_1 = A)=\hat {z}_2(R_2 = A)$, so that

(5.7)\begin{equation} \delta r = \delta a, \ \delta R_1=\delta R_2 = \delta A, \delta\hat{r}_1(A)=\delta\hat{r}_2(A)=\delta a \ \text{and}\ \delta z(a)=\delta\hat{z}_1(A)=\delta\hat{z}_2(A). \end{equation}

Note that $\delta a$ denotes $\delta r$ at $r=a$, i.e. the perturbation in the radial location of the triple point in the current configuration, while $\delta A$ is the perturbation in the reference location of the contact circle. The two perturbations $\delta a$ and $\delta A$ are independent, because they are linked by the membrane's deformation field, which is itself perturbed. Thus, in (5.5) the perturbations $\delta z(0)$, $\delta \hat {z}_1(0)$, $\delta z(a)$, $\delta a$ and $\delta A$ are independent, so that the vanishing of the left-hand side of (5.5) for all admissible perturbations leads to the following boundary conditions:

(5.8a)$$\begin{gather} -\left.\left(\frac{\partial F}{\partial z'}+p\frac{\partial G}{\partial z'}\right)\right|_{0} =0 , \end{gather}$$

(5.8b)$$\begin{gather}-\left.\left(\frac{\partial F_1}{\partial\hat{z}_1'}+p\frac{\partial G_1}{\partial\hat{z}_1'}\right)\right|_{0}=0, \end{gather}$$

(5.8c)$$\begin{gather}\left.\left\{(F+p G)-\left(\frac{\partial F}{\partial z'}+p\frac{\partial G}{\partial z'}\right)z' +\left(\frac{\partial F_1}{\partial\hat{r}_1'}+p\frac{\partial G_1}{\partial\hat{r}_1'}\right) -\frac{\partial F_2}{\partial\hat{r}_2'}\right\}\right|_{a,A}=0, \end{gather}$$

(5.8d)$$\begin{gather}\left.\left\{\left(\frac{\partial F}{\partial z'}+p\frac{\partial G}{\partial z'}\right)+\left(\frac{\partial F_1}{\partial\hat{z}_1'}+p\frac{\partial G_1}{\partial\hat{z}_1'}\right)-\frac{\partial F_2}{\partial\hat{z}_2'}\right\}\right|_{a,A}=0 \end{gather}$$

and

(5.8e)\begin{align} &\left\{(F_1+p G_1)-\left(\frac{\partial F_1}{\partial\hat{r}_1'} +p\frac{\partial G_1}{\partial\hat{r}_1'}\right)\hat{r}_1' \right. \ldots \nonumber\\ &\quad \left.\left. -\left(\frac{\partial F_1}{\partial\hat{z}_1'}+p\frac{\partial G_1}{\partial\hat{z}_1'}\right)\hat{z}_1'- \left(F_2-\frac{\partial F_2}{\partial\hat{r}_2'}\hat{r}_2'-\frac{\partial F_2}{\partial\hat{z}_2'}\hat{z}_2'\right)\right\}\right|_{a}=0. \end{align}

From (5.8a) and (5.8b) we may recover the boundary conditions for the drop (2.2b) and the wet membrane (2.15a). Similarly, from (5.8c) and (5.8d), we obtain the horizontal (2.14a) and vertical (2.14b) force balances at the triple point, with $\bar {T}_{wm}=\bar {T}_{s1}$ and $\bar {T}_{dm}=\bar {T}_{s2}$. We note from the sketch in figure 4 that $z'=-j\tan \theta _d$, $z_1'=\tan \theta _{wm}$ and $z_2'=\tan \theta _{dm}$.

Finally, from (5.8e) we obtain

(5.9)\begin{equation} \left.\left(\frac{W_1}{\hat{\lambda}_{\phi 1}}-T_{s1}^m\hat{\lambda}_{s1}\right)\right|_{a}=\left.\left(\frac{W_2}{\hat{\lambda}_{\phi 2}}-\bar{T}_{s2}^m\hat{\lambda}_{s2}\right)\right|_{a}. \end{equation}

We recall from (3.2) and (2.11) that the strain energies for an $I_2$ membrane of the wet and dry parts, and the mechanical tensions in them are, respectively,

(5.10a,b)\begin{equation} W_k = \mu t_r \{ ( \hat{ \lambda }_{ sk } \hat{ \lambda }_{ \phi k } )^2 - 3 \} \quad \text{and} \quad T_{sk}^m=\hat{\lambda}_{\phi k}^{{-}1}(\mathrm{d} W_k/\mathrm{d}\hat{\lambda}_{sk}), \quad k = 1, 2, \end{equation}

where we have employed the definitions for $I_2$ and the mechanical tension. From the definitions of the circumferential stretches $\hat {\lambda }_{\phi k}$ and the continuity of ${\hat r}_k$ and $r_k$ at ${R_1 = R_2 = A}$, we have

(5.11)\begin{equation} \hat{\lambda}_{\phi 1}=\hat{\lambda}_{\phi 2} \quad \text{at} \ R_1=R_2 = A. \end{equation}

Combining the preceding two equations with (5.9) reduces it to

(5.12)\begin{equation} \hat{\lambda}_{s1}=\hat{\lambda}_{s2}\quad \text{at} \ R_1=R_2 = A; \end{equation}

thus, the meridional stretches in the wet and dry membranes must be continuous at the triple point $r=r_1=r_2=a$. This is the additional condition that was missing in the formulation of § 2.1, which led to a multiplicity of equilibrium solutions in § 4.

An immediate consequence of (5.12) is that the mechanical wet and dry membrane tensions at the triple point are equal, i.e.

(5.13)\begin{equation} \bar{T}_{wm}^m=\bar{T}_{dm}^m=\bar{T}^m , \end{equation}

and, because the wet and dry membrane tensions are uniform in their respective domains, therefore $\bar{T}^m$ is, in fact, the uniform mechanical tension throughout the membrane. Schulman & Dalnoki-Veress (Reference Schulman and Dalnoki-Veress2015), when modelling their experiments involving micro-droplets on thin, free-standing films, had assumed both a uniform mechanical tension as well as its continuity at the triple point. Here we provide theoretical support to those claims for membranes made of $I_2$ materials with large stretches.

The revised solution procedure is provided in Appendix C. Returning to the remark at the end of § 3, we now see that the system's equilibrium landscape may be generated by varying only two parameters, which are taken to be the apex drop curvature $\beta$ and contact circle radius $\bar {a}$ in the above algorithm. However, we prefer to report our results below in terms of the drop volume $\bar {V}$ and the membrane pre-tension $\bar {T}_0$, which are a more natural and practically useful set of control parameters. This may be accomplished by inverting the solutions found utilizing the procedure in Appendix C appropriately.