Statistical mechanics of thin spherical shells

We explore how thermal fluctuations affect the mechanics of thin amorphous spherical shells. In flat membranes with a shear modulus, thermal fluctuations increase the bending rigidity and reduce the in-plane elastic moduli in a scale-dependent fashion. This is still true for spherical shells. However, the additional coupling between the shell curvature, the local in-plane stretching modes and the local out-of-plane undulations, leads to novel phenomena. In spherical shells thermal fluctuations produce a radius-dependent negative effective surface tension, equivalent to applying an inward external pressure. By adapting renormalization group calculations to allow for a spherical background curvature, we show that while small spherical shells are stable, sufficiently large shells are crushed by this thermally generated "pressure". Such shells can be stabilized by an outward osmotic pressure, but the effective shell size grows non-linearly with increasing outward pressure, with the same universal power law exponent that characterizes the response of fluctuating flat membranes to a uniform tension.Comment: 16 pages, 6 figure

Phase behavior and morphology of multicomponent liquid mixtures

Multicomponent systems are ubiquitous in nature and industry. While the physics of few-component liquid mixtures (i.e., binary and ternary ones) is well-understood and routinely taught in undergraduate courses, the thermodynamic and kinetic properties of $N$-component mixtures with $N>3$ have remained relatively unexplored. An example of such a mixture is provided by the intracellular fluid, in which protein-rich droplets phase separate into distinct membraneless organelles. In this work, we investigate equilibrium phase behavior and morphology of $N$-component liquid mixtures within the Flory-Huggins theory of regular solutions. In order to determine the number of coexisting phases and their compositions, we developed a new algorithm for constructing complete phase diagrams, based on numerical convexification of the discretized free energy landscape. Together with a Cahn-Hilliard approach for kinetics, we employ this method to study mixtures with $N=4$ and $5$ components. We report on both the coarsening behavior of such systems, as well as the resulting morphologies in three spatial dimensions. We discuss how the number of coexisting phases and their compositions can be extracted with Principal Component Analysis (PCA) and K-Means clustering algorithms. Finally, we discuss how one can reverse engineer the interaction parameters and volume fractions of components in order to achieve a range of desired packing structures, such as nested `Russian dolls' and encapsulated Janus droplets.Comment: 16 pages, 11 figures + hyperlinks to 7 video

Thermal Excitations of Warped Membranes

We explore thermal fluctuations of thin planar membranes with a frozen spatially varying background metric and a shear modulus. We focus on a special class of D-dimensional “warped membranes” embedded in a d-dimensional space with d≥D+1 and a preferred height profile characterized by quenched random Gaussian variables $\{h_\alpha(q)\}$, $\alpha=D+1,...,d$, in Fourier space with zero mean and a power-law variance $\over{h\alpha(q_1)h_\beta(q_2)}$ $\sim \delta_{\alpha,\beta} \delta_{q_1,−q_2} q_1^{-d_h}$. The case D=2, d=3, with $d_h=4$ could be realized by flash-polymerizing lyotropic smectic liquid crystals. For $D\lt max\{4,d_h\}$ the elastic constants are nontrivially renormalized and become scale dependent. Via a self-consistent screening approximation we find that the renormalized bending rigidity increases for small wave vectors q as $\kappa_R \sim q^{−\eta_f}$, while the in-hyperplane elastic constants decrease according to $\lambda_R, \mu_R \sim q^{+\eta_u}$. The quenched background metric is relevant (irrelevant) for warped membranes characterized by exponent $d_h\gt 4−\eta^{(F)}_f (d_h\lt 4−\eta ^{(F)}_f)$, where $\eta^{(F)}_f$ is the scaling exponent for tethered surfaces with a flat background metric, and the scaling exponents are related through $\eta_u+\eta_f=d_h−D (\eta_u+2\eta_f=4−D)$.Molecular and Cellular BiologyPhysic

Mechanical Properties of Warped Membranes

We explore how a frozen background metric affects the mechanical properties of planar membranes with a shear modulus. We focus on a special class of “warped membranes” with a preferred random height profile characterized by random Gaussian variables h(q) in Fourier space with zero mean and variance $⟨| h(q)|^2〉\sim q^{−d_h}$ and show that in the linear response regime the mechanical properties depend dramatically on the system size L for $d_h\geq 2$. Membranes with $d_h=4$ could be produced by flash polymerization of lyotropic smectic liquid crystals. Via a self-consistent screening approximation we find that the renormalized bending rigidity increases as $\kappa R\sim L^{(d_h−2)/2}$ for membranes of size L, while the Young and shear moduli decrease according to $Y_R,\mu R \sim L^{−(d_h−2)/2}$ resulting in a universal Poisson ratio. Numerical results show good agreement with analytically determined exponents.Molecular and Cellular BiologyPhysic

Grundy dominating sequences and zero forcing sets

In a graph $G$ a sequence $v_1,v_2,\dots,v_m$ of vertices is Grundy dominating if for all $2\le i \le m$ we have $N[v_i]\not\subseteq \cup_{j=1}^{i-1}N[v_j]$ and is Grundy total dominating if for all $2\le i \le m$ we have $N(v_i)\not\subseteq \cup_{j=1}^{i-1}N(v_j)$. The length of the longest Grundy (total) dominating sequence has been studied by several authors. In this paper we introduce two similar concepts when the requirement on the neighborhoods is changed to $N(v_i)\not\subseteq \cup_{j=1}^{i-1}N[v_j]$ or $N[v_i]\not\subseteq \cup_{j=1}^{i-1}N(v_j)$. In the former case we establish a strong connection to the zero forcing number of a graph, while we determine the complexity of the decision problem in the latter case. We also study the relationships among the four concepts, and discuss their computational complexities