Stability and folds in an elastocapillary system
We examine the equilibrium and stability of an elastocapillary system to model drying-induced structural failures. The model comprises a circular elastic membrane with a hole at the center that is deformed by the capillary pressure of simply connected and doubly connected menisci. Using variational and spectral methods, stability is related to the slope of equilibrium branches in the liquid content versus pressure diagram for the constrained and unconstrained problems. The second-variation spectra are separately determined for the membrane and meniscus, showing that the membrane out-of-plane spectrum and the in-plane spectrum at large elatocapillary numbers are both positive, so that only meniscus perturbations can cause instability. At small elastocapillary numbers, the in-plane spectrum has a negative eigenvalue, inducing wrinkling instabilities in thin membranes. In contrast, the smallest eigenvalue of the meniscus spectrum always changes sign at a pressure turning point where stability exchange occurs in the unconstrained problem. We also examine configurations in which the meniscus and membrane are individually stable, while the elastocapillary system as a whole is not; this emphasizes the connection between stability and the coupling of elastic and capillary forces.
Elastic deformations induced by capillary forces have been identified as leading causes of pattern collapse in miniature electronic devices and sensors [10, 17]. These microstructures are more prone to collapse when miniaturized because adhesion and capillary forces become comparable to elastic forces when there is a high surface area to volume ratio . Capillary driven collapse poses major problems for micro-fabrication techniques that are based on wet etching. In particular, microelectronic systems are commonly fabricated through wet lithography where structures often experience permanent deformation and stiction upon drying, significantly limiting the design and operating conditions .
Elastocapillary systems have been extensively studied over the past two decades [25, 7, 32, 16, 19]. Aggregation, coalescence, and self-assembly of filaments and flexible fibres [13, 21, 9, 32], failure of microelectronic devices [26, 33], and capillary wrinkling of elastic membranes [20, 42] are applications where system configurations and structures are determined by elastic-capillary force interactions. The foregoing studies are mostly concerned with systems where equilibrium configurations are always stable (or assumed to be). However, mechanical stability is central to applications in which preventing structural failure upon drying is crucial.
Recently, a few studies have focused on the mechanical stability of elastocapillary systems. Giomi & Mahadevan  examined the equilibrium and stability of minimal surfaces spanning deformable frames. Subjecting a circular frame to spatial perturbations, they approximated instability modes and the critical elastocapillary numbers corresponding to the primary and secondary buckling of the frame into elliptical and twisted structures. Taroni & Vella  identified multiple equilibria in an elastocapillary system related to the aggregation of paint-brush bristles where the stable solutions for a given liquid content were determined through a temporal stability and dynamic analysis.
Mastrangelo & Hsu  took a different approach to determine stability in elastocapillary systems. Their approach hinges on a pervasive theory, known as catastrophe theory  in nonlinear dynamics, stating that stability exchanges only occur at folds and branch points on equilibrium branches . While this has not been generally proved for all mechanical systems, the idea has been extensively examined for purely capillary [29, 43, 37, 2, 3] and purely elastic  problems. In this context, Maddocks  established a theory for systems where equilibria are described by a continuous functional of a single function with prescribed boundary conditions (from a Hilbert space). This theory relates the stability of constrained and unconstrained variational problems to the shape of equilibrium branches with no branch point where stability exchanges occur only at simple folds.
Determining the stability of an elastic structure deformed by the Laplace pressure or contact line force of a meniscus is more challenging than determining its equilibria. Elastic and capillary parts for equilibrium states can be decoupled and determined separately by imposing the proper boundary conditions where the meniscus and structure meet. However, the stability of elastic and capillary parts alone is not sufficient to deduce the stability of elastocapillary systems for which the control-parameter role is particularly important.
In the present work, we study an elastocapillary problem where conformations are controlled by the liquid content, similarly to Kwon et al. . This is analogous to the problem of disconnected free surfaces considered by Slobozhanin . Complexities in elastocapillary problems arise because meniscus perturbations are neither pressure controlled nor volume controlled, as in purely capillary problems [23, 3]. Moreover, elastic structures and menisci in many practical elastocapillary systems have free boundaries [7, 34, 38, 16], which considerably complicate the stability analysis. Vogel  highlights two major difficulties for analyzing the quadratic forms arising from the second variation of systems with free boundaries: (i) The function space of perturbations is not necessarily a symmetric Hilbert space . Instead, the quadratic forms are naturally expressed in , and an additional analysis is required to link the arising operators to the corresponding operators in a symmetric space
In this paper, we examine the elastocapillary system shown in Fig. 1 and relate stability to the slope of equilibrium branches in pressure versus volume diagrams, similarly to Maddocks . This system is a model for drying-induced structural failures arising in practical applications, such as the stiction of micro-machined sensors and collapse of wood fibres upon drying. It comprises a circular elastic membrane, with a hole at the center, anchored above a rigid plate, trapping a prescribed volume of liquid. We examine membrane deformations caused by a meniscus at the hole as the liquid is slowly removed. Our approach is variational, so that linear stability is determined by the sign of the second variation. We demonstrate that there are configurations in which the meniscus and membrane are individually stable, while the elastocapillary system as a whole is not. This emphasizes the significance of instabilities arising from the coupling of elastic and capillary forces. This result can be interpreted as the equivalent of the Weierstrass–Erdmann condition  for the second variation, and it is relevant to applications where extrema are represented by non-smooth functions, such as for elastocapillary systems, threshold phenomena , and data visualization .
We consider an elastocapillary model comprising a circular elastic membrane with a hole at the center supported on the sidewall of a cylindrical cavity with rigid walls, trapping a liquid volume below the membrane and air volume between the bounding surface (dashed line in Fig. 1(a)) and membrane, as shown in Fig. 1. The cavity is open to the atmosphere from the top. A meniscus forms at the hole as the liquid is removed, resulting in a difference between the liquid pressure and atmospheric pressure , which causes the membrane to deform. Here, the membrane radius , hole radius , and cylinder hight are the model length scales that control the interplay between elastic and capillary forces. To determine the equilibria at a given , we consider an imaginary bounding surface that covers the cavity from the top. The system is completely isolated from the surrounding by the bounding surface and cylinder walls. The meniscus is initially a bubble, which can bridge the gap upon contact with the plate at the bottom of the cylinder, forming a free contact line with the plate. Assuming that all the dimensions are small compared to the capillary length, the gravity force is neglected. The membrane and meniscus are assumed axisymmetric in equilibrium and perturbed configurations.
2.1 Variational principle
Following the development of Neumann et al. , we apply a variational principle to determine stable equilibria for the system depicted in Fig. 1. Noting that the liquid volume is the control parameter in drying, the total internal energy is to be minimized subject to , maintaining fixed total entropy and mass. Here, the grand canonical potential is a suitable free-energy representation because it restricts the minimization to states that are already in thermal (constant uniform temperature) and chemical (constant uniform chemical potential) equilibrium. The grand canonical potentials for bulk phases and interfaces, respectively, are 
with , , , , and the surface tension, specific grand canonical potential, specific internal energy, specific entropy, and density of the respective phase. The superscripts and denote volume density and area density for bulk phases and interfaces. Note that the temperature and chemical potential can be regarded as the Lagrange multipliers associated with the entropy and mass in the foregoing constrained minimization of the total internal energy. Hence, stable equilibria minimize
where are interfacial surface areas. Here, the membrane strain energy is separately incorporated into the total energy to account for variable and anisotropic stresses. Neglecting the bending energy, we only consider the stretching part of the elastic strain energy in von Kármán’s theory for moderately large deflections
where , , and are the membrane in the referential configuration (undeflected state), axial forces, and nonlinear strains . Substituting Eqs. (1), (2), and (4) into Eq. (3) and omitting additive and multiplicative constants that do not affect the minimization,
is the functional to be minimized subject to , where
Note that , , , , , and are the membrane axial rigidity, Poisson ratio, membrane in-plane displacement
2.2 Equilibrium from first variation
To construct the increment of in Eq. (5) with respect to axisymmetric perturbations, perturbed states are represented by
accounting for the linear and nonlinear parts of the increment when the meniscus is a bridge. Note that the form of the functionals in Eqs. (6) and (7) demands to be continuous, so . Here, equilibrium and perturbed states are denoted by hatted and unhatted variables, respectively. Equation (12) is particularly important, because it admits perturbations that can displace the position of the bubble apex and hole edge along the -axis. We impose a simply supported boundary condition for the membrane at where , resulting in
The meniscus is assumed to be pinned to the hole edge at , where and , furnishing
Moreover, the meridian curve intersects the symmetry axis at , where it can only move vertically when the meniscus is a bubble, so
whereas the contact line with the plate can only move horizontally at when the meniscus is a bridge, so
where all the functional integrands are evaluated at the equilibrium. Here, primes operating on functionals denote the first Fréchet derivative  with respect to the function in the subscript, and is the inner product over the domain of the respective function. The last term in Eq. (23) must be replaced with when the meniscus is a bubble. Equilibria are the stationary points of the total energy where for arbitrary , , and , requiring
with boundary conditions
Here, and are the contact and dihedral angles that the interface forms with the plate and membrane, respectively. The last boundary condition only holds when the meniscus is a bridge, while the boundary term associated with when the meniscus is a bubble is always zero with . Equation (27) is the Young-Laplace equation, and Eqs. (28) and (29) are the in- and out-of-plane equations of equilibrium for membranes in von Kármán’s theory with the capillary pressure acting normal to the neutral plane. Note that Eqs. (30) and (31) demand , implying as , contradicting the assumptions of von Kármán’s theory . To resolve this issue, we undertake a scaling analysis in section 3 to simplify Eqs. (30) and (31).
2.3 Stability from second variation
The Jacobian matrices of the functional integrants in Eqs. (6) and (7) are not symmetric. Since analyzing the second variation for functionals of multiple functions with non-symmetric Jacobians is intractable , we simplify the problem by neglecting in-plane variations at the hole edge, prescribing . Accordingly, the second variation is
where , , and
Note that the boundary term at in Eq. (33) arises only when the meniscus is a bubble, so has only one boundary term at when the meniscus is a bridge. Furthermore, when the meniscus is a bubble, is not bounded due to the axial symmetry condition at (see Appendix A), and it is unsuitable for representing quadratic forms. To resolve this issue, using the mapping , the first integral and boundary terms in Eq. (33) are represented with respect to the normal variation , where (see Fig. 2(a) and Appendix A). Equation (33) then reduces to
for simply connected and doubly connected menisci, respectively. Here, , corresponding to the polar angle at the hole edge (Fig. 2(a)), and
A necessary (sufficient) condition for to have a minimum is that be non-negative (strongly positive) for an equilibrium solution . Here, strong positivity, referred to as nonlinear stability in the literature , must be distinguished from positive-definiteness. According to Vogel , does not imply a strict local minimum for constrained infinite-dimensional problems. Nevertheless, defining stable equilibria as those for which holds, Maddocks  derived sufficient conditions for the positive-definiteness of the second variation. Interestingly, Vogel  showed that these conditions are sufficient for Madoccks’ functional to have a strict minimum. Therefore, following Maddocks , we adopt Madoccks’ definition of stability to relate stability to the slope of equilibrium branches.
Our analysis differs from Madoccks’ theory in two respects: (i) The elastocapillary energy is a functional of multiple functions, the stationary points of which are represented by non-smooth functions. Moreover, has three separate spectra, which complicates the relationship between stability exchanges and equilibrium-branch turning points. (ii) The boundary condition at the hole edge, where the membrane and meniscus meet, is not prescribed. Here, perturbations are arbitrarily finite, posing the same difficulties as discussed by Myshkis et al.  and Vogel  for analyzing the stability of capillary surfaces with free contact lines.
The quadratic forms in Eqs. (38) and (39) demand , where perturbations satisfy the boundary conditions Eqs. (16)-(20). Since each bilinear term in Eqs. (38) and (39) is bounded, can be represented as [4, 46]
for simply connected and doubly connected menisci, where is the inner product  and are uniquely determined by the respective bilinear terms in Eqs. (38) and (39). Because perturbations are arbitrary at the hole edge, the spaces from which perturbations are selected are not symmetric, and, thus, are not generally self-adjoint. On the other hand, integrating Eqs. (38) and (39) by parts leads to
for simply connected and doubly connected menisci, where with double primes denoting the second Fréchet derivative and perturbations satisfying the boundary conditions
furnishing a symmetric for evaluating . Here,
subject to Eqs. (45)-(48) are regular Sturm-Liouville operators; thus, they are self-adjoint and Fredholm . Establishing a relationship between the spectrum of the barred operators on and those of unbarred operators on significantly simplifies the analysis, providing a setting to apply the well-studied Sturm-Liouville theory. Lemma 2.5 of Vogel  furnishes this relationship by stating that the barred operators have the same number of negative and non-positive eigenvalues as the corresponding unbarred operators. Moreover, the constant-volume constraint () can be written
3 Scaling analysis
Starting from Eq. (28) and taking , we begin by scaling all lengths with . Given and , where , , and , we find to balance all the terms in the in-plane equilibrium. Considering the bending and stretching parts of the strain energy in von Kármán’s theory for axisymmetric plates 
and noting that , , , and , where and are the bending rigidity and plate thickness, one infers when by comparing the energy scales and . For thin membranes, this justifies neglecting the bending energy compared to the stretching energy.
Given , follows from the out-of-plane equilibrium, where , , are the scaled hole radius, scaled capillary pressure and elastocapillary number. Noting that , the elastocapillary number corresponding to a specific state of the system in Fig. 1 can be estimated as , where and are the meniscus slenderness and aspect ratio. For example, when the meniscus is a bubble contacting the plate, before bridging the gap, as will be discussed elsewhere, and ; thus, at , which is the critical dihedral angle below which the bubble cannot be stably pinned to the hole edge , we have
4 Membrane profile
The stretching part of the strain energy leads to nonlinear equations of equilibrium, which can be solved numerically in most practical problems. Although the case considered in this paper, namely, axisymmetric plate with a hole at the center, has a series solution , expressions for the unknown coefficients are cumbersome. Therefore, we apply a variational approximation, as commonly used in the literature [6, 25], to construct a general solution for the membrane deflection.
Here, we approximate the boundary condition at as a free edge. Noting that the test function
is consistent with the boundary conditions and the plate stress distribution under tension at zero deflection , is obtained by minimizing the stretching energy, where
which is in agreement with the scaling relation for derived in section 3. Equation (55) contrasts with the test function that Mastrangelo & Hsu  used to describe the bending and stretching contribution to the overall deflection of beams. The approximation accuracy hinges on choosing appropriate test functions for bending- and stretching-dominated regimes. Figures 3 and 4 demonstrate a reasonable agreement between the variational approximation and numerically exact solutions of the membrane equilibrium, for a wide range of deflections.
5 Second variation spectra
We adopt the foregoing variational approximation in this section to determine the second variation spectra corresponding to the in-plane and out-of-plane equilibria.
5.1 In-plane spectrum
The spectrum of the membrane in-plane equilibrium is determined by
where , , and are the eigenvalue, eigenfunction, and weight function of . Non-dimensionalizing Eq. (57) furnishes
with and . The general solution of Eq. (57) is with , where and are the Bessel functions of first and second kind. A similar eigenvalue problem was derived by Timoshenko & Gere  for the buckling of circular plates under in-plane compressive loads. Imposing the boundary conditions furnishes
where is the th root of
From Eq. (60), unless at a given , corresponding to has only a trivial solution where and at . Here, 0 is the identically zero function, and the overdot denotes differentiation along equilibrium branches. Therefore, stability loss due to in-plane perturbations is not generally related to pressure turning points. Studying these instabilities, which are responsible for wrinkling in thin elastic membranes [14, 15, 31], is beyond the scope of the present work and will not be further elaborated upon.
Figure 5 shows the first four eigenvalues of the in-plane spectrum. Note that is accurate in the range to within 12% of computed values. A positive spectrum is ensured by . As water is removed from the elastocapillary system of Fig. 1, reaches its maximum value when the membrane touches the plate where . To ensure that always holds during drying, we require , leading to
based on the foregoing approximation of and the scaling relation . Consequently, guarantees that the in-plane spectrum is positive, and that the membrane profile can be accurately approximated by Eq. (55).
5.2 Out-of-plane spectrum
The spectrum of the membrane out-of-plane equilibrium is given by