Pendant capsule elastometry

Hegemann, Jonas; Knoche, Sebastian; Egger, Simon; Kott, Maureen; Demand, Sarah; Unverfehrt, Anja; Rehage, Heinz; Kierfeld, Jan

doi:10.1016/j.jcis.2017.11.048

Cited by 42 publications

(72 citation statements)

References 45 publications

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“…Typical artificial micrometer-sized capsules have shell thicknesses of h ∼ 10nm and are made from soft materials with bulk Young moduli E ∼ 0.1 GPa. This results in 2D Young's moduli Y = Eh ∼ 1N/m and bending moduli κ ∼ Eh 3 ∼ 10 −16 Nm ∼ 2 × 10 4 k B T in agreement with elastometry measurements [53]. For R 0 = 10µm, typical Föppl-von Kármán numbers (2) are γ ∼ 10 6 − 10 7 .…”

Section: Discussionsupporting

confidence: 87%

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Shallow shell theory of the buckling energy barrier: From the Pogorelov state to softening and imperfection sensitivity close to the buckling pressure

Baumgarten

Kierfeld

2019

Phys. Rev. E

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We study the axisymmetric response of a complete spherical shell under homogeneous compressive pressure p to an additional point force. For a pressure p below the classical critical buckling pressure pc, indentation by a point force does not lead to spontaneous buckling but an energy barrier has to be overcome. The states at the maximum of the energy barrier represent a subcritical branch of unstable stationary points, which are the transition states to a snap-through buckled state. Starting from nonlinear shallow shell theory we obtain a closed analytical expression for the energy barrier height, which facilitates its effective numerical evaluation as a function of pressure by continuation techniques. We find a clear crossover between two regimes: For p/pc 1 the post-buckling barrier state is a mirror-inverted Pogorelov dimple, and for (1 − p/pc) 1 the barrier state is a shallow dimple with indentations smaller than shell thickness and exhibits extended oscillations, which are well described by linear response. We find systematic expansions of the nonlinear shallow shell equations about the Pogorelov mirror-inverted dimple for p/pc 1 and the linear response state for (1 − p/pc) 1, which enable us to derive asymptotic analytical results for the energy barrier landscape in both regimes. Upon approaching the buckling bifurcation at pc from below, we find a softening of an ideal spherical shell. The stiffness for the linear response to point forces vanishes ∝ (1 − p/pc) 1/2 ; the buckling energy barrier vanishes ∝ (1 − p/pc) 3/2 ; and the shell indentation in the barrier state vanishes ∝ (1 − p/pc) 1/2 . This makes shells sensitive to imperfections which can strongly reduce pc in an avoided buckling bifurcation. We find the same softening scaling in the vicinity of the reduced critical buckling pressure also in the presence of imperfections. We can also show that the effect of axisymmetric imperfections on the buckling instability is identical to the effect of a point force that is preindenting the shell. In the Pogorelov limit, the energy barrier maximum diverges ∝ (p/pc) −3 and the corresponding indentation diverges ∝ (p/pc) −2 . Numerical prefactors for proportionalities both in the softening and the Pogorelov regime are calculated analytically. This also enables us to obtain results for the critical unbuckling pressure and the Maxwell pressure.

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Section: Discussionsupporting

confidence: 87%

“…Because the dimensionless energy barrierĒ B can also be expressed directly by solutions of the shallow shell equations atF = 0 via Eq. (12), alsoĒ B will only depend on p/p c , see our main results (33) and (53) below. In particular,Ē B does not depend on the Poisson number ν in shallow shell theory.…”

Section: A Exact Analytical Resultsmentioning

confidence: 71%

Shallow shell theory of the buckling energy barrier: From the Pogorelov state to softening and imperfection sensitivity close to the buckling pressure

Baumgarten

Kierfeld

2019

Phys. Rev. E

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“…Another problem arises if compressive stresses τ s < 0 or τ φ < 0 occur, which can give rise to wrinkle formation and require a different effective constitutive relation in the compressed region [32,34]. We have checked explicity that such compressive stresses do not occur for capsules stretched at a liquid-liquid interface, i.e., σ > 0.…”

Section: B Hookean Elasticity and Alternative Constitutive Relationsmentioning

confidence: 99%

“…In Ref. 34 it has been explicitly shown how to use a Mooney-Rivlin relation to close the shape equations (5).…”

Section: E Shape Equationsmentioning

confidence: 99%

“…For the Mooney-Rivlin law it has been explicitly demonstrated in Ref. 34 how this constitutive relation can be used to close the shape equations (5). Because the shape equations (5) are closed by eliminating λ s and τ φ by using the constitutive relations for stresses and strains, nonlinear relations such as Mooney-Rivlin laws are considerably more difficult to handle and lead to a larger computational cost [34].…”

Section: B Hookean Elasticity and Alternative Constitutive Relationsmentioning

confidence: 99%

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Elastic capsules at liquid–liquid interfaces

2018

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We investigate the deformation of elastic microcapsules adsorbed at liquid-liquid interfaces. An initially spherical elastic capsule at a liquid-liquid interface undergoes circumferential stretching due to the liquid-liquid surface tension and becomes lens- or discus-shaped, depending on its bending rigidity. The resulting elastic capsule deformation is qualitatively similar, but distinct from the deformation of a liquid droplet into a liquid lens at a liquid-liquid interface. We discuss the deformed shapes of droplets and capsules adsorbed at liquid-liquid interfaces for a whole range of different surface elasticities: from droplets (only surface tension) deforming into liquid lenses, droplets with a Hookean membrane (finite stretching modulus, zero bending modulus) deforming into elastic lenses, to microcapsules (finite stretching and bending modulus) deforming into rounded elastic lenses. We calculate capsule shapes at liquid-liquid interfaces numerically using shape equations from nonlinear elastic shell theory. Finally, we present theoretical results for the contact angle (or the capsule height) and the maximal capsule curvature at the three phase contact line. These results can be used to infer information about the elastic moduli from optical measurements. During capsule deformation into a lens-like shape, surface energy of the liquid-liquid interface is converted into elastic energy of the capsule shell giving rise to an overall adsorption energy gain by deformation. Soft hollow capsules exhibit a pronounced increase of the adsorption energy as compared to filled soft particles and, thus, are attractive candidates as foam and emulsion stabilizers.

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Building Reconfigurable Devices Using Complex Liquid–Fluid Interfaces

et al. 2019

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imparts novel mechanical and functional properties to the macroscopic interface itself. These properties include size-and charge-selective pathways for molecules and particulates, shear and compressive moduli, and selective binding and reaction sites. Complex interfaces can then be used to generate complex, macroscopic, all-liquid materials with very high surface areas that have unique properties, such as compartmentalization, communication between compartments, and the ability to move and reconfigure. With the rapid growth in the study of complex materials made from interfacially structured liquids, there is a need to review the field from the funda-Liquid-fluid interfaces provide a platform both for structuring liquids into complex shapes and assembling dimensionally confined, functional nanomaterials. Historically, attention in this area has focused on simple emulsions and foams, in which surface-active materials such as surfactants or colloids stabilize structures against coalescence and alter the mechanical properties of the interface. In recent decades, however, a growing body of work has begun to demonstrate the full potential of the assembly of nanomaterials at liquid-fluid interfaces to generate functionally advanced, biomimetic systems. Here, a broad overview is given, from fundamentals to applications, of the use of liquid-fluid interfaces to generate complex, all-liquid devices with a myriad of potential applications. Figure 2.Interactions between particles attached to liquid-fluid interfaces. a) Dipole-dipole interactions between adsorbed particles due to asymmetric charge distributions near the surface of particles at interfaces. Reproduced with permission. [34] Copyright 1980, American Physical Society. b) Dipole-dipole interactions give rise to 2D colloidal crystals with large lattice spacings. Scale bar, 50 µm. Reproduced with permission. [36]

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Pendant capsule elastometry

Cited by 42 publications

References 45 publications

Shallow shell theory of the buckling energy barrier: From the Pogorelov state to softening and imperfection sensitivity close to the buckling pressure

Shallow shell theory of the buckling energy barrier: From the Pogorelov state to softening and imperfection sensitivity close to the buckling pressure

Elastic capsules at liquid–liquid interfaces

Building Reconfigurable Devices Using Complex Liquid–Fluid Interfaces

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