Buckling of thermally fluctuating spherical shells: Parameter renormalization and thermally activated barrier crossing

Snapping elastic disks as microswimmers: swimming at low Reynolds numbers by shape hysteresis

We illustrate a concept for shape-changing microswimmers, which exploits the hysteresis of a shape transition of an elastic object, by an elastic disk undergoing cyclic localized swelling. Driving the control parameter of a hysteretic shape transition in a completely time-reversible manner gives rise to a non-time-reversible shape sequence and a net swimming motion if the elastic object is immersed into a viscous fluid. We prove this concept with a microswimmer which is a flat circular elastic disk that undergoes a transition into a dome-like shape by localized swelling of an inner disk. The control parameter of this shape transition is a scalar swelling factor of the disk material. With a fixed outer frame with an additional attractive interaction in the central region, the shape transition between flat and dome-like shape becomes hysteretic and resembles a hysteretic opening and closing of a scallop. Employing Stokesian dynamics simulations of a discretized version of the disk we show that the swimmer is effectively moving into the direction of the opening of the dome in a viscous fluid if the swelling parameter is changed in a time-reversible manner. The swimming mechanism can be qualitatively reproduced by a simple 9-bead model.

Finite Temperature Phase Behavior of Viral Capsids as Oriented Particle Shells

A general phase plot is proposed for discrete particle shells that allows for thermal fluctuations of the shell geometry and of the inter-particle connectivities. The phase plot contains a first-order melting transition, a buckling transition, and a collapse transition and is used to interpret the thermodynamics of microbiological shells.

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

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 p_{c}, 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/p_{c}≪1 the postbuckling barrier state is a mirror-inverted Pogorelov dimple, and for (1-p/p_{c})≪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/p_{c}≪1 and the linear response state for (1-p/p_{c})≪1, which enable us to derive asymptotic analytical results for the energy barrier landscape in both regimes. Upon approaching the buckling bifurcation at p_{c} from below, we find a softening of an ideal spherical shell. The stiffness for the linear response to point forces vanishes ∝(1-p/p_{c})^{1/2}; the buckling energy barrier vanishes ∝(1-p/p_{c})^{3/2}; and the shell indentation in the barrier state vanishes ∝(1-p/p_{c})^{1/2}. This makes shells sensitive to imperfections which can strongly reduce p_{c} 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/p_{c})^{-3} and the corresponding indentation diverges ∝(p/p_{c})^{-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.