Deformation-dependent gel surface topography due to the elastocapillary and osmocapillary effects

Abstract

Actively tuning surface topography is crucial for the design of smart surfaces with stimuli-responsive friction, wetting, and adhesion properties. This paper studies how elastocapillary deformation and osmocapillary phase separation can lead to rich deformation-dependent surface topography in polymeric gels. In a purely elastic material, stretching always flattens the surface due to the Poisson effect. We show that stretching can roughen the surface due to the elastocapillary and osmocapillary effects. The roughening can be tuned by the gel stiffness, the gel osmotic pressure, the deformation mode, and the initial amplitude of surface roughness. The rich deformation-dependent behavior of gel surface topography points to a new direction in designing smart surfaces.

---

1. Introduction

Surface topography governs many surface behaviors such as friction, wetting, adhesion, and transparency.^1–7 Smart surfaces that actively control surface behavior through surface topography have been extensively studied. For example, swelling a polymeric gel expands surface features,^8–12 and stretching a soft material deforms the surface topography with the bulk.^13–17 Although deforming a piece of material is simple, predicting the deformation-dependent surface topography is not.^12,14,18 By elastic deformation alone, stretching the material flattens surface topography due to the Poisson effect. However, if the material is sufficiently soft, surface tension can deform the surface topography, causing elastocapillary deformation.^14,17–22 If the material is sufficiently swollen with a solvent, surface tension can pull the solvent out from the material bulk, leading to osmocapillary phase separation.^12,23–25 While the elastocapillary and osmocapillary effects on a static surface are relatively well studied, how they affect surface topography during deformation remains unclear. Understanding the deformation-dependent elastocapillary and osmocapillary effects is crucial to predicting the deformation-dependent surface topography and designing novel smart surfaces.

In general, a deformation can be decomposed into a volumetric and an incompressible part. This paper focuses on the surface topography change under incompressible deformation. The change under volumetric deformation have been recently reported.¹² We perform novel finite element simulation that incorporates coupled elastocapillary and osmocapillary effects. We use a sinusoidal profile to reveal the length dependence of the elastocapillary and osmocapillary effects. The simulation shows that the elastocapillary and osmocapillary effects can lead to non-monotonic surface roughening and flattening with deformation. Here roughening or flattening means an increase or decrease in the amplitude of surface undulation. The findings point to new possibilities to program complex surface responses. Our findings also echo a recent experimental study that found PDMS gels can either roughen or flatten under stretch depending on the elastic moduli.¹⁴ The paper attributed the observation to strain-dependent surface tension and elastocapillary deformation alone. Our study shows that similar phenomena could happen under constant surface tension when both the elastocapillary and osmocapillary effects are considered.

2. Governing mechanisms for the deformation-dependent gel surface topography

A polymeric gel consists of a polymer network infiltrated with a solvent. The polymer network leads to a solid-like elastic behavior;²⁶ the solvent leads to a liquid-like capillary behavior, which dominates the gel surface at high solvent content;²⁷ and the absorption of the solvent by the polymer network leads to an osmotic behavior.²⁸ In the absence of external load, the gel surface topography is governed by the competition between elasticity, capillarity, and osmosis. Elasticity resists any deformation thus maintaining the stress-free surface topography. Capillarity tends to minimize the surface area by flattening the surface topography. The competition between elasticity and capillarity to deform the polymer network is known as elastocapillary deformation (Fig. 1a).²⁰ Here capillarity is governed by the Laplace pressure γκ, where γ is surface energy and κ is the sum of the two principal curvatures of the surface. Elasticity is governed by the shear modulus μ of the gel. The elastocapillary effect is governed by the dimensionless number γκ/μ, where γκ ≫ μ leads to significant elastocapillary deformation. On the other hand, osmosis absorbs the solvent that covers surface asperities thus roughening the surface topography. The competition between osmosis and capillarity to move the surface solvent is known as osmocapillary phase separation (Fig. 1b).²³ Here osmosis is governed by the osmotic pressure Π. The osmocapillary effect is governed by the dimensionless number γκ/Π, where γκ ≫ Π leads to significant osmocapillary phase separation. Due to the elastocapillary and osmocapillary effects, the equilibrium surface topography generally deviates from the stress-free surface topography.

Fig. 1 (a) The competition between elasticity and capillarity results in elastocapillary deformation. (b) The competition between osmosis and capillarity results in osmocapillary phase separation. (c) Elasticity leads to surface flattening under stretch. (d) Capillarity is weakened by the lower curvature under stretch, thus roughening the surface. (e) Osmosis is strengthened by the tensile hydrostatic stress that increases Π under stretch, thus roughening the surface.

When the gel is externally loaded, elasticity, osmosis, and capillarity respond to the deformation differently. Consider a uniform stretch (Fig. 1c). Elasticity elongates the surface topography with the gel bulk and the Poisson effect flattens the surface. The elongation of the surface topography decreases the surface curvature κ. Consequently, the Laplace pressure γκ that flattens the surface is reduced, and the surface roughens (Fig. 1d). On the other hand, the tensile stress during stretch increases the osmotic pressure Π that resists phase separation.²⁹ Consequently, the volume of solvent between asperity reduces, which also roughens the surface (Fig. 1e). The competition between elasticity, capillarity, and osmosis evolves with deformation, which can lead to complex deformation-dependent surface topography.

3. Finite element simulation of coupled elastocapillary and osmocapillary effects

This paper uses finite element simulation to study the coupled elastocapillary and osmocapillary effects in deformation-dependent surface topography. The elastocapillary and osmocapillary effects are characterized by elastocapillary length γ/μ and osmocapillary length γ/Π.^20,23 To cleanly study the length dependence of the elastocapillary and osmocapillary effects, we study a sinusoidal surface profile, which only involves two length scales, the wavelength L₀ and the peak-to-peak amplitude h₀ in the stress-free state (Fig. 2a). Then the equilibrium surface amplitude h without external load is completely determined by three dimensionless numbers:When the gel is uniformly deformed, the deformation is governed by the principle stretches, λ₁,λ₂,λ₃. For incompressible deformation, λ₁λ₂λ₃ = 1, only two stretches are independent. Take λ₁ along the direction of the sinusoidal wave and λ₂ along the direction perpendicular to the surface (Fig. 2a), the deformation-dependent surface amplitude change can be written as:

Fig. 2 (a) We study the evolution of a sinusoidal profile under uniform deformation. (b) Thanks to the symmetry and periodicity, only half a period in 2D needs to be simulated. (c) The surface is simulated as a beam with pretension that matches the surface tension γ pulled towards the gel by the osmotic pressure Π.

We perform finite element simulation to investigate the effects of each of the four dimensionless numbers in eqn (2).

A commercial finite element package, ABAQUS, is used for the simulation. We perform all simulations in the plane strain condition (λ₃ = 1), then transform the results to generalized plane strain conditions with different λ₃.^30,31 Due to the symmetry and periodicity of a sinusoid, we only model half a period (Fig. 2b). We consider the common case that the surface amplitude is much smaller than the sample thickness and set the gel thickness is to be larger than 50h₀. Then the bottom boundary negligibly affects the simulation. The gel bulk is modeled with neo-Hookean elasticity. 2D quadratic hybrid plane-strain element, CPE8H, is used to discretize the simulation domain. We only study the equilibrium state after the complete poroelastic relaxation. Consequently, there is no solvent flux, and the osmotic pressure is uniform everywhere. On the other hand, since the gel modulus depends on the solvent content,³² the near-surface solvent migration can lead to spatial variation in the gel modulus. For common low-roughness surfaces, the deformation and solvent migration to flatten the asperities are small. Consequently, we neglect the deformation-dependent gel modulus in the current study.

The gel surface is modeled with a layer of quadratic beam elements, B22. We ensure the beam behaves like the surface tension through three features:³¹ (1) a uniform prestress is applied so that the tension in the beam matches the surface tension γ, (2) the tensile stiffness of the beam is so low that the deformation in the simulation negligibly affects the prestress and the cross-section Poisson's ratio is set to 0 so that deformation does not change the cross-section area. Then surface tension γ is independent of deformation. (3) The bending stiffness of the beam is so low that the beam layer negligibly affects the deformation of the gel. For simplicity, we assume the surface tension of the gel is identical to that of the solvent, which is valid for gels of high solvent content.^12,27 In gels of low solvent content, the gel surface stress can be different from the solvent surface tension and can be strain-dependent.^14,33,34

To simulate coupled elastocapillary deformation and osmocapillary phase separation, the beam and the gel are pulled towards each other by a uniform pressure that matches the magnitude of the osmotic pressure Π (Fig. 2c). The “hard contact” interaction in ABAQUS is used to prevent the beam from penetrating the gel. We model the deformation-dependent osmotic pressure Π using the incompressible neo-Hookean model:²⁹

σ₂ = μ(λ₂² − 1) − Π₀ + Π. (3)

Here Π₀ and Π are the initial and current osmotic pressure. Assuming the surface is stress-free, σ₂ = 0 and recall the incompressibility constraint λ₁λ₂λ₃ = 1, we have:The incompressible neo-Hookean law is valid to derive the osmotic pressure change because (1) we assume the amplitude of the surface topography is much smaller than the thickness of the gel. Consequently, osmocapillary phase separation on the surface negligibly affects the bulk solvent content. And (2) we neglect solvent evaporation from the gel surface. The local deformation near the surface can be compressible due to solvent migration. We will first study the incompressible case in Sections 4 and 5 and discuss the effect of compressibility in Section 6.

4. The elastocapillary effect: γ/μL₀

When osmotic pressure Π is sufficiently large (γ/Π₀L₀ ≪ 1), the solvent will be completely absorbed into the gel bulk. Then there is no osmocapillary phase separation. Elastocapillary deformation alone determines the deformation-dependent surface topography. Consider a fixed h₀/L₀ = 0.2 under plane strain deformation, eqn (2) simplified to h/h₀ = G(λ₁;γ/μL₀), which can be easily studied by simulating a group of h(λ₁)/h₀ curves with various γ/μL₀. Our simulations show that when the capillary effect is weak (γ/μL₀ < 1, Fig. 3a), the surface is little deformed before stretching (left half of the subfigure with h/h₀ = 0.88) and stretching the surface leads to flattening. This is expected from the Poisson effect. When the capillary effect is strong (γ/μL₀ > 1, Fig. 3a), the surface is significantly flattened before stretching (left half of the subfigure with h/h₀ = 0.06), and stretching the surface roughens the surface. This is because the capillary flattening is weakened.

Fig. 3 (a) Surface flattens with stretch when elasticity dominates over capillarity and roughens with stretch when capillarity dominates over elasticity. (b) Surface roughens with stretch for large γ/μL₀ and small tensile stretch λ₁. (c) Elasticity monotonically flattens while capillarity monotonically roughens the surface with stretch.

To quantitatively illustrate the deformation-dependent surface flattening in Fig. 3a, we plot h/h₀ = G(λ₁;γ/μL₀) in Fig. 3b. It shows that a larger γ/μL₀ results in more significant surface flattening (smaller h/h₀) at all stretch levels. When the capillary effect is weak (γ/μL₀ = 0.02), the surface monotonically flattens with stretch (decreasing h/h₀) similar to an elastically deformed surface. However, when the capillary effect is strong (γ/μL₀ = 0.2 or 2), the surface roughens with stretch for small deformation (increasing h/h₀ for λ₁ near 1), then flattens with stretch for large deformation (λ₁ far from 1, either tensile or compressive). The range of roughening behavior broadens as γ/μL₀ increases. We do not simulate λ₁ < 0.85 because a crease forms at the bottom of the trough. Crease-induced surface topography change is a different topic that we will not investigate. Readers may refer to existing ref. 30, 31 and 35.

The non-monotonic behavior can be interpreted as the competition between the elastic flattening and capillary roughening. To separate the elastic and capillary contributions on the surface deformation, we divide the total change in surface profile amplitude, h − h₀, into two parts: the capillary part h − h_e and the elastic part h_e − h₀. Here h_e is the stretch-dependent surface profile amplitude without capillary effect (i.e., γ/μL₀ = 0). Then h_e − h₀ represents the contribution due to elastic deformation, and h − h_e represents the additional deformation induced by capillarity. Fig. 3c shows that elastic flattening due to the Poisson effect strengthens monotonically with stretch in both tensile (h_e − h₀ < 0) and compressive (h_e − h₀ > 0) directions. In contrast, the capillary flattening weakens monotonically with stretch in the tensile direction (approaches 0) because stretching reduces the local curvature. For small γ/μL₀, the elastic part dominates over the capillary part at any stretch λ₁ (the absolute value of the red curve dominates over the blue curves), leading to monotonically flattening with stretch. For larger γ/μL₀, the capillary part dominates over the elastic part for small deformation (λ₁ near 1), leading to stretch-dependent roughening. However, the elastic part still dominates over the capillary part for large deformation (λ₁ far from 1), leading to stretch-dependent flattening.

5. Osmocapillary effect: γ/Π₀L₀

When osmotic pressure is sufficiently small, the solvent can be pulled out from the gel bulk by capillarity, leading to osmocapillary phase separation.²³ Deforming the gel bulk impacts osmocapillary phase separation in two ways: (1) stretching the surface reduces the local curvature thus weakening the capillary effect, and (2) tensile stress increases the osmotic pressure thus strengthening the osmotic effect. Since both the weakening of the capillary effect and the strengthening of the osmotic effect roughen the surface, stretching always roughens the surface in the presence of osmocapillary phase separation (Fig. 4a). Since both effects reduce the amount of osmocapillary phase separation, there exists a critical λ_dry where osmocapillary phase separation disappears for λ₁ > λ_dry. Then the surface deformation is purely elastocapillary, which has been discussed in Section 4. Based on the same argument, compression increases the amount of osmocapillary phase separation. There exists a critical λ_wet where Π = 0 and the solvent is squeezed out of the surface. When the solvent completely covers the surface, the liquid surface is completely flattened by capillarity.

Fig. 4 (a) Under plane strain condition, osmocapillary phase separation reduces with stretch. Osmocapillary phase separation disappears for λ₁ > λ_dry and inundate the surface for λ₁ < λ_wet. Here γ/μL₀ = 0.2, γ/Π₀L₀ = 2.5. (b) Under a fixed γ/μL₀, a larger γ/Π₀L₀ shifts λ_dry and λ_wet to larger stretches. (c) Under a fixed γ/Π₀L₀, a larger γ/μL₀ flattens the surface and widen the gap between λ_dry and λ_wet.

The deformation-dependent osmocapillary phase separation in Fig. 4a is governed by two dimensionless numbers, the same elastocapillary number γ/μL₀, and the osmocapillary number γ/Π₀L₀. We first plot h(λ₁)/h₀ with the fixed γ/μL₀ = 0.2 and varying γ/Π₀L₀ (Fig. 4b). It shows that a larger γ/Π₀L₀ corresponds to a larger amount of initial osmocapillary phase separation at λ₁ = 1 and shifts both λ_dry and λ_wet to larger stretches (Fig. S2a, ESI†). The surface roughness is independent of γ/Π₀L₀ once phase separation disappears (λ₁ > λ_dry) or inundates the surface (λ₁ < λ_wet). Next, we plot h(λ₁)/h₀ with fixed γ/Π₀L₀ = 10 and varying γ/μL₀ (Fig. 4c). It shows two trends: (1) a larger γ implies stronger elastocapillary flattening, thus lowering h/h₀ when the surface is stretched (λ₁ > 1). (2) According to eqn (4), a smaller μ results in a weaker stretch dependence of Π, thus a larger gap between λ_dry and λ_wet (Fig. S2b, ESI†).

6. The effect of Poisson's ratio

The two above sections illustrate the competition between the stretch-induced flattening caused by the Poisson effect and the stretch-induced roughening caused by elastocapillary and osmocapillary effects. So far, we have assumed the polymer network to be incompressible with a Poisson's ratio ν = 0.5. In general, the Poisson's ratio of a gel depends on the swelling ratio, as can be illustrated by the Flory–Rehner model (Section S3, ESI†). According to the Flory–Rehner model, polymer networks are nearly incompressible, i.e. shear is much easier than volumetric deformation, in most conditions even if free solvent exchange is allowed (Fig. S3, ESI†). This contradicts with much lower Poisson's ratios measured through some indentation tests.^36–39 Such contradictions are expected because the Flory–Rehner model has been shown to severely underestimate the volumetric elasticity of the polymer network.⁴⁰ In the lack of an accurate compressible constitutive model for polymer networks, here we use the classical compressible neo-Hookean model to qualitatively illustrate the effect of network compressibility.⁴¹ We tune the Poisson's ratio between ν = 0 to 0.5 and repeat the simulations for the elastocapillary and osmocapillary cases. Since the Poisson effect is responsible for the stretch-induced flattening and the capillary effect is responsible for the stretch-induced roughening, lowering the Poisson ratio will reduce the flattening effect. In the limit of ν = 0, the surface will monotonically roughen with stretch (Fig. 5a). On the other hand, a lower Poisson's ratio makes the gel more deformable under surface tension, thus rendering the surface flatter before stretching. The flatter surface makes osmocapillary phase separation easier to absorb, thus lowering λ_dry (Fig. 5b).

Fig. 5 (a) A lower Poisson's ratio decreases elastic effect, thus flattening the surface at the undeformed state and suppresses the stretch-induced flattening. (b) A lower Poisson's ratio flattens surface at smaller stretch, thus shifting λ_dry to a smaller stretch.

7. The effect of out-of-plane deformation: λ₃

So far, we have been concerned with the plane strain deformation where λ₃ = 1. The results of plane strain deformation can be transformed into any generalized plane strain deformation with a uniform λ₃ ≠ 1.^30,31 We derive the transformation for osmocapillary phase separation as follows (Fig. 6). We take the reference state as the stress-free state with no surface energy, thus no elastocapillary deformation or osmocapillary phase separation. After applying a macroscopic plane strain stretch and relaxing the surface under surface tension γ and osmotic pressure Π, the bulk material will have an inhomogeneous field of F^(PE) and there will be some osmocapillary phase separation. Consider a unit thickness of material in the reference state, the free energy after deformation is:Here α,β = 1, 2. The first integration integrates over the deformation plane. S is the length of the surface in the deformation plane, and A is the area of phase separation in the deformation plane.

Fig. 6 A generalized plane strain deformation (GPE) can be decomposed into a uniaxial deformation (UA) followed by a plane strain deformation (PE).

Consider the same piece of material undergoes a generalized plane strain deformation. The total deformation of the gel can be decomposed into a uniaxial deformation F^(UA) followed by a plane strain deformation F^(PE), F^(GPE) = F^(PE)F^(UA). We assume the uniaxial deformation is uniform with no capillary effect. Then the following plane strain deformation relative to the intermediate state is governed by the same physics as the plane strain deformation relative to the reference state discussed above. The only difference is that after F^(UA), the unit thickness in the reference state elongates to λ₃ in the intermediate state and any in-plane dimension in the intermediate state is shrunk by compared to the reference state. Then eqn (5) should be modified as:

The last term f(λ₃) involves a few terms that depend on λ₃ but not the in-plane deformation or phase separation. Since λ₃ is a constant during the generalized plane strain deformation, f(λ₃) is a constant that can be dropped from the analysis. Comparing eqn (5) and (6), we see that the energy minimization of W_GPE with μ^(GPE), γ^(GPE), Π^(GPE) under a macroscopic stretch λ^(GPE)₁ is equivalent to the energy minimization of W_PE with μ^(PE), γ^(PE), Π^(PE) under a macroscopic stretch λ^(PE)₁ through the following transformation:Here directly follows from F^(GPE) = F^(PE)F^(UA).

The transformation eqn (7) allows us to study various loading conditions with arbitrary λ₃. Most common loading conditions can be represented by the relation λ₃ = 1/λ^α₁. For example, α = 0 corresponds to the plane strain deformation (λ₃ = 0) and α = −1 corresponds to equibiaxial deformation (λ₃ = λ₁). Fig. 7a lists a few common deformation modes and their corresponding α. In these cases, eqn (4) becomes:

Eqn (8) implies that the larger the α, the slower the Π increases with λ₁; Π is deformation independent when α = 1; and Π decreases with λ₁ for α > 1, (Fig. 7b). As discussed in Fig. 4c, the deformation-dependent Π governs the gap between λ_dry and λ_wet. For an incompressible network with fixed γ/Π₀L₀ = 10, γ/μL₀ = 0.2, we have performed generalized plane strain simulations for various α. The result can be represented by a phase diagram of the surface states (Fig. 7c). It shows that for α < 1, the generalized plane strain cases are qualitatively identical to the plane strain case discussed in Section 5. Lowering α narrows the gap between λ_dry and λ_wet. For α > 1, however, λ_dry and λ_wet are flipped because the elongation of the surface feature is driven by a compressive stress state. For example, in the case of uniaxial deformation in x₃ (α = 2), the elongation of the surface feature in x₁ is driven by the compression in x₃. Then Π decreases with λ₁, and more solvent is squeezed out as the surface features elongate.

Fig. 7 (a) Various deformation modes and the corresponding α in the constraint λ₃ = 1/λ^α₁. (b) α affects the Π − λ₁ dependence. (c) The phase diagram of the surface states. “Dry”, “osmocapillary”, and “wet” mean none, partial, and complete solvent coverage of the surface. Here γ/Π₀L₀ = 10, γ/μL₀ = 0.2.

8. The effect of stress-free amplitude: h₀/L₀

Under the assumption of shallow amplitude h₀/L₀ ≪ 1, the elastocapillary effect is linear and h(λ₁)/h₀ is independent of h₀/L₀.¹⁴ However, the osmocapillary effect is intrinsically nonlinear due to the moving boundaries of phase separation.²⁵ Consequently, h(λ₁)/h₀ is expected to show a strong dependence on h₀/L₀ for any amplitude in the presence osmocapillary phase separation. To illustrate this, we vary h₀/L₀ and simulate the evolution of h/h₀ for a purely elastocapillary case (γ/μL₀ = 0.2, γ/Π₀L₀ ≪ 1) and an osmocapillary case (γ/μL₀ = 0.2, γ/Π₀L₀ = 10) under the plane strain condition and assume incompressible network. In the elastocapillary case (Fig. 8a), h(λ₁)/h₀ shows weak dependence on the amplitude h₀/L₀ when h₀/L₀ < 1. Yet, a large amplitude of h₀/L₀ > 1 is significantly harder to flatten. In the osmocapillary case (Fig. 8b), increasing h₀/L₀ significantly enhances the initial flattening (lower h/h₀ at λ₁ = 1) and delays the disappearance of osmocapillary phase separation at λ_dry. λ_wet is determined by setting Π = 0 in eqn (4), thus independent of amplitude h₀/L₀. The change in the initial flattening and λ_dry is significant even between the cases that have nearly identical elastocapillary behaviors (h₀/L₀ = 0.02 and 0.2). The strong amplitude dependence of osmocapillary phase separation is easy to understand (Fig. 8c): the competition between the osmotic and capillary effects dictates the surface radius of curvature to be the osmocapillary length γ/Π,²³ thus the larger the h₀/L₀, the larger the amount of osmocapillary phase separation, and thus the more difficult it is to dry the surface.

Fig. 8 (a) Elastocapillary deformation is independent of amplitude for shallow surface with h₀/L₀ ≪ 1 and causes less flattening for h₀/L₀ > 1. (b) Osmocapillay phase separation extends to higher stretches with larger surface amplitude h₀/L₀. (c) A Larger amplitude leads to a larger amount of osmocapillary phase separation.

9. Conclusion

In this paper, we have performed systematic finite element analyses of how a sinusoidal surface deforms with stretch in the presence of the coupled elastocapillary and osmocapillary effects. When both the elastocapillary and osmocapillary effects are weak (γ/μL₀ ≪ 1, γ/Π₀L₀ ≪ 1), the surface flattens with stretch due to the Poisson effect. Elastocapillary deformation alone can lead to surface roughening with stretching (γ/μL₀ ∼ 0.1 or higher, γ/Π₀L₀ ≪ 1). However, such roughening is limited to relatively small deformation. For large tension and compression, the Poisson effect dominates over the capillary effect thus resuming the flattening. Osmocapillary phase separation also leads to surface roughening with stretching (γ/Π₀L₀ ∼ 1 or higher with any γ/μL₀). The osmocapillary roughening happens over a limited stretch range bounded by λ_dry and λ_wet. Outside this range, the surface is either dry, thus purely elastocapillary, or inundated under exuded solvent, thus completely flat. The range between λ_dry and λ_wet is strongly affected by the out-of-plane constraint (λ₃) and the surface amplitude (h₀/L₀). In common loading conditions such as uniaxial and equibiaxial deformations, osmocapillary phase separation leads to much stronger stretching-dependent roughening than elastocapillary deformation. The richness of deformation-dependent gel surface topography can be used to design novel stimuli-responsive surfaces with tunable surface topography.

Conflicts of interest

There are no conflicts to declare.

References

1. B. N. Persson, Theory of rubber friction and contact mechanics, J. Chem. Phys., 2001, 115(8), 3840–3861.
2. D. E. Packham, Surface energy, surface topography and adhesion, Int. J. Adhes. Adhes., 2003, 23(6), 437–448.
3. K. Fuller and D. Tabor, The effect of surface roughness on the adhesion of elastic solids, Proc. R. Soc. London, Ser. A, 1975, 345(1642), 327–342.
4. M. Ramiasa, et al., The influence of topography on dynamic wetting, Adv. Colloid Interface Sci., 2014, 206, 275–293.
5. S. M. Lößlein, F. Mücklich and P. G. Grützmacher, Topography versus chemistry–How can we control surface wetting?, J. Colloid Interface Sci., 2022, 609, 645–656.
6. X. Yao, et al., Adaptive fluid-infused porous films with tunable transparency and wettability, Nat. Mater., 2013, 12(6), 529–534.
7. Y. Lin, et al., Surface roughness and light transmission of biaxially oriented polypropylene films, Polym. Eng. Sci., 2007, 47(10), 1658–1665.
8. L. Chen, et al., Thermal-responsive hydrogel surface: tunable wettability and adhesion to oil at the water/solid interface, Soft Matter, 2010, 6(12), 2708–2712.
9. D. Liu, et al., (Photo-) thermally induced formation of dynamic surface topographies in polymer hydrogel networks, Langmuir, 2013, 29(18), 5622–5629.
10. J. Comelles, et al., Microfabrication of poly (acrylamide) hydrogels with independently controlled topography and stiffness, Biofabrication, 2020, 12(2), 025023.
11. Y. Hu, J.-O. You and J. Aizenberg, Micropatterned hydrogel surface with high-aspect-ratio features for cell guidance and tissue growth, ACS Appl. Mater. Interfaces, 2016, 8(34), 21939–21945.
12. J. Zhu, C. Yang and Q. Liu, Experimental characterization of elastocapillary and osmocapillary effects on multi-scale gel surface topography, Soft Matter, 2023, 19(45), 8698–8705.
13. S. Zhao, et al., Mechanical stretch for tunable wetting from topological PDMS film, Soft Matter, 2013, 9(16), 4236–4240.
14. N. Bain, et al., Surface tension and the strain-dependent topography of soft solids, Phys. Rev. Lett., 2021, 127(20), 208001.
15. S. C. Truxal, Y.-C. Tung and K. Kurabayashi, High-speed deformation of soft lithographic nanograting patterns for ultrasensitive optical spectroscopy, Appl. Phys. Lett., 2008, 92(5), 051116.
16. P. Goel, et al., Mechanical strain induced wetting transitions between anisotropic and isotropic on polydimethylsiloxane (PDMS) films patterned by optical discs, Appl. Surf. Sci., 2015, 356, 102–109.
17. N. Lapinski, et al., A surface with stress, extensional elasticity, and bending stiffness, Soft Matter, 2019, 15(18), 3817–3827.
18. C.-Y. Hui, et al., How surface stress transforms surface profiles and adhesion of rough elastic bodies, Proc. R. Soc. A, 2020, 476(2243), 20200477.
19. B. Andreotti, et al., Solid capillarity: when and how does surface tension deform soft solids?, Soft Matter, 2016, 12(12), 2993–2996.
20. R. W. Style, et al., Elastocapillarity: Surface tension and the mechanics of soft solids, Annu. Rev. Condens. Matter Phys., 2017, 8, 99–118.
21. D. Paretkar, et al., Flattening of a patterned compliant solid by surface stress, Soft Matter, 2014, 10(23), 4084–4090.
22. A. Jagota, D. Paretkar and A. Ghatak, Surface-tension-induced flattening of a nearly plane elastic solid, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2012, 85(5), 051602.
23. Q. Liu and Z. Suo, Osmocapillary phase separation, Extreme Mech. Lett., 2016, 7, 27–33.
24. R. W. Style, et al., The contact mechanics challenge: Tribology meets soft matter, Soft Matter, 2018, 14(28), 5706–5709.
25. J. Zhu and Q. Liu, The osmocapillary effect on a rough gel surface, J. Mech. Phys. Solids, 2023, 170, 105124.
26. M. Rubinstein and S. Panyukov, Elasticity of polymer networks, Macromolecules, 2002, 35(17), 6670–6686.
27. A. Chakrabarti and M. K. Chaudhury, Direct measurement of the surface tension of a soft elastic hydrogel: exploration of elastocapillary instability in adhesion, Langmuir, 2013, 29(23), 6926–6935.
28. M. Quesada-Pérez, et al., Gel swelling theories: the classical formalism and recent approaches, Soft Matter, 2011, 7(22), 10536–10547.
29. W. Hong, et al., A theory of coupled diffusion and large deformation in polymeric gels, J. Mech. Phys. Solids, 2008, 56(5), 1779–1793.
30. W. Hong, X. Zhao and Z. Suo, Formation of creases on the surfaces of elastomers and gels, Appl. Phys. Lett., 2009, 95(11), 111901.
31. Q. Liu, et al., Elastocapillary crease, Phys. Rev. Lett., 2019, 122(9), 098003.
32. J. Li, et al., Experimental determination of equations of state for ideal elastomeric gels, Soft Matter, 2012, 8(31), 8121–8128.
33. Q. Xu, et al., Direct measurement of strain-dependent solid surface stress, Nat. Commun., 2017, 8(1), 555.
34. Q. Xu, R. W. Style and E. R. Dufresne, Surface elastic constants of a soft solid, Soft Matter, 2018, 14(6), 916–920.
35. X. Guan, et al., Rate-dependent creasing of a viscoelastic liquid, Extreme Mech. Lett., 2022, 55, 101784.
36. Y. Hu, et al., Using indentation to characterize the poroelasticity of gels, Appl. Phys. Lett., 2010, 96(12), 121904.
37. Y. Hu, et al., Indentation of polydimethylsiloxane submerged in organic solvents, J. Mater. Res., 2011, 26(6), 785–795.
38. S. Cai, et al., Poroelasticity of a covalently crosslinked alginate hydrogel under compression, J. Appl. Phys., 2010, 108(11), 113514.
39. Y. Hu, et al., Poroelastic relaxation indentation of thin layers of gels, J. Appl. Phys., 2011, 110(8), 086103.
40. Z. Shao and Q. Lin, Independent characterization of the elastic and the mixing parts of hydrogel osmotic pressure, Extreme Mech. Lett., 2023, 64, 102085.
41. A. N. Gent, Engineering with rubber: how to design rubber components, Carl Hanser Verlag GmbH Co KG, 2012.

---

Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4sm00139g

---