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Register a new userThermodynamic equilibrium of binary mixtures on curved surfaces
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We study the global influence of curvature on the free energy landscape of two-dimensional binary mixtures confined on closed surfaces. Starting from a generic effective free energy, constructed on the basis of symmetry considerations and conservation laws, we identify several model-independent phenomena, such as a curvature-dependent line tension and local shifts in the binodal concentrations. To shed light on the origin of the phenomenological parameters appearing in the effective free energy, we further construct a lattice-gas model of binary mixtures on non-trivial substrates, based on the curved-space generalization of the two-dimensional Ising model. This allows us to decompose the interaction between the local concentration of the mixture and the substrate curvature into four distinct contributions, as a result of which the phase diagram splits into critical sub-diagrams. The resulting free energy landscape can admit, as stable equilibria, strongly inhomogeneous mixed phases, which we refer to as antimixed states below the critical temperature. We corroborate our semi-analytical findings with phase-field numerical simulations on realistic curved lattices. Despite this work being primarily motivated by recent experimental observations of multi-component lipid vesicles supported by colloidal scaffolds, our results are applicable to any binary mixture confined on closed surfaces of arbitrary geometry.
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Motivated by recent experimental work on multicomponent lipid membranes supported by colloidal scaffolds, we report an exhaustive theoretical investigation of the equilibrium configurations of binary mixtures on curved substrates. Starting from the Julicher-Lipowsky generalization of the Canham-Helfrich free energy to multicomponent membranes, we derive a number of exact relations governing the structure of an interface separating two lipid phases on arbitrarily shaped substrates and its stability. We then restrict our analysis to four classes of surfaces of both applied and conceptual interest: the sphere, axisymmetric surfaces, minimal surfaces and developable surfaces. For each class we investigate how the structure of the geometry and topology of the interface is affected by the shape of the substrate and we make various testable predictions. Our work sheds light on the subtle interaction mechanism between membrane shape and its chemical composition and provides a solid framework for interpreting results from experiments on supported lipid bilayers.
Recent studies have highlighted the sensitivity of active matter to boundaries and their geometries. Here we develop a general theory for the dynamics and statistics of active particles on curved surfaces and illustrate it on two examples. We first show that active particles moving on a surface with no ability to probe its curvature only exhibit steady-state inhomogeneities in the presence of orientational order. We then consider a strongly confined 3D ideal active gas and compute its steady-state density distribution in a box of arbitrary convex shape.
We present a systematic study of how vortices in superfluid films interact with the spatially varying Gaussian curvature of the underlying substrate. The Gaussian curvature acts as a source for a geometric potential that attracts (repels) vortices towards regions of negative (positive) Gaussian curvature independently of the sign of their topological charge. Various experimental tests involving rotating superfluid films and vortex pinning are first discussed for films coating gently curved substrates that can be treated in perturbation theory from flatness. An estimate of the experimental regimes of interest is obtained by comparing the strength of the geometrical forces to the vortex pinning induced by the varying thickness of the film which is in turn caused by capillary effects and gravity. We then present a non-perturbative technique based on conformal mappings that leads an exact solution for the geometric potential as well as the geometric correction to the interaction between vortices. The conformal mapping approach is illustrated by means of explicit calculations of the geometric effects encountered in the study of some strongly curved surfaces and by deriving universal bounds on their strength.
Conforming materials to rigid substrates with Gaussian curvature --- positive for spheres and negative for saddles --- has proven a versatile tool to guide the self-assembly of defects such as scars, pleats, folds, blisters, and liquid crystal ripples. Here, we show how curvature can likewise be used to control material failure and guide the paths of cracks. In our experiments, and unlike in previous studies on cracked plates and shells, we constrained flat elastic sheets to adopt fixed curvature profiles. This constraint provides a geometric tool for controlling fracture behavior: curvature can stimulate or suppress the growth of cracks, and steer or arrest their propagation. A simple analytical model captures crack behavior at the onset of propagation, while a two-dimensional phase-field model with an added curvature term successfully captures the cracks path. Because the curvature-induced stresses are independent of material parameters for isotropic, brittle media, our results apply across scales.
We study the thermodynamic stability of fluid-fluid phase separation in binary nonadditive mixtures of hard-spheres for moderate size ratios. We are interested in elucidating the role played by small amounts of nonadditivity in determining the stability of fluid-fluid phase separation with respect to the fluid-solid phase transition. The demixing curves are built in the framework of the modified-hypernetted chain and of the Rogers-Young integral equation theories through the calculation of the Gibbs free energy. We also evaluate fluid-fluid phase equilibria within a first-order thermodynamic perturbation theory applied to an effective one-component potential obtained by integrating out the degrees of freedom of the small spheres. A qualitative agreement emerges between the two different approaches. We also address the determination of the freezing line by applying the first-order thermodynamic perturbation theory to the effective interaction between large spheres. Our results suggest that for intermediate size ratios a modest amount of nonadditivity, smaller than earlier thought, can be sufficient to drive the fluid-fluid critical point into the thermodinamically stable region of the phase diagram. These findings could be significant for rare-gas mixtures in extreme pressure and temperature conditions, where nonadditivity is expected to be rather small.
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