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Register a new userMeasuring Gaussian rigidity using curved substrates
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The Gaussian (saddle splay) rigidity of fluid membranes controls their equilibrium topology but is notoriously difficult to measure. In lipid mixtures, typical of living cells, linear interfaces separate liquid ordered (LO) from liquid disordered (LD) bilayer phases at subcritical temperatures. Here we consider such membranes supported by curved supports that thereby control the membrane curvatures. We show how spectral analysis of the fluctuations of the LO-LD interface provides a novel way of measuring the difference in Gaussian rigidity between the two phases. We provide a number of conditions for such interface fluctuations to be both experimentally measurable and sufficiently sensitive to the value of the Gaussian rigidity, whilst remaining in the perturbative regime of our analysis.
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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.
Liquid crystal elastomers/glasses are active materials that can have significant metric change upon stimulation. The local metric change is determined by its director pattern that describes the ordering direction and hence the direction of contraction. We study logarithmic spiral patterns on flat sheets that evolve into cones on deformation, with Gaussian curvature localized at tips. Such surfaces, Gaussian flat except at their tips, can be combined to give compound surfaces with GC concentrated in lines. We characterize all possible metric-compatible interfaces between two spiral patterns, specifically where the same metric change occurs on each side. They are classified as hyperbolic-type, elliptic-type, concentric spiral, and continuous-director interfaces. Upon the cone deformations and additional isometries, the actuated interfaces form creases bearing non-vanishing concentrated Gaussian curvature, which is formulated analytically for all cases and simulated numerically for some examples. Analytical calculations and the simulations agree qualitatively well. Furthermore, the relaxation of Gaussian-curved creases is discussed and cantilevers with Gaussian curvature-enhanced strength are proposed. Taken together, our results provide new insights in the study of curved creases, lines bearing Gaussian curvature, and their mechanics arising in actuated liquid crystal elastomers/glasses, and other related active systems.
Within mean-field theory we study wetting of elastic substrates. Our analysis is based on a grand canonical free energy functional of the fluid number density and of the substrate displacement field. The substrate is described in terms of the linear theory of elasticity, parametrized by two Lame coefficients. The fluid contribution is of the van der Waals type. Two potentials characterize the interparticle interactions in the system. The long-ranged attraction between the fluid particles is described by a potential $w(r)$, and $v(r)$ characterizes the substrate-fluid interaction. By integrating out the elastic degrees of freedom we obtain an effective theory for the fluid number density alone. Its structure is similar to the one for wetting of an inert substrate. However, the potential $w(r)$ is replaced by an effective potential which, in addition to $w(r)$, contains a term bilinear in $v(r)$. We discuss the corresponding wetting transitions in terms of an effective interface potential $omega(ell)$, where $ell$ denotes the thickness of the wetting layer. We show that in the case of algebraically decaying interactions the elasticity of the substrate may suppress critical wetting transitions, and may even turn them first order.
Cracks in thin layers are influenced by what lies beneath them. From buried craters to crocodile skin, crack patterns are found over an enormous range of length scales. Regardless of absolute size, their substrates can dramatically influence how cracks form, guiding them in some cases, or shielding regions from them in others. Here we investigate how a substrates shape affects the appearance of cracks above it, by preparing mud cracks over sinusoidally varying surfaces. We find that as the thickness of the cracking layer increases, the observed crack patterns change from wavy to ladder-like to isotropic. Two order parameters are introduced to measure the relative alignment of these crack networks, and, along with Fourier methods, are used to characterise the transitions between crack pattern types. Finally, we explain these results with a model, based on the Griffith criteria of fracture, that identifies the conditions for which straight or wavy cracks will be seen, and predicts how well-ordered the cracks will be. Our metrics and results can be applied to any situation where connected networks of cracks are expected, or found.
rigidPy is a Python package that provides a set of tools necessary for studying rigidity and mechanical response in spring networks. It also includes suitable modules for generating new realizations of networks with applications in glassy systems and protein structures. rigidPy is available freely on GitHub and can be installed using Python Package Index (PyPi). The detailed setup information is provided in this paper, along with an overview of the mathematical framework that has been used in developing the package.
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