Controlled cracks are useful in a wide range of applications, including stretchable electronics, microfluidics, sensors, templates, biomimics, and surface engineering. Here we report on the spontaneous formation of hierarchical crack patterns in metal (nickel) films sputter deposited on soft elastic polydimethylsiloxane (PDMS) substrates. The experiment shows that the nickel film generates a high tensile stress during deposition, which is relieved by the formation of disordered crack networks (called primary cracks). Due to the strong interfacial adhesion and soft substrate, the cracks can penetrate into the PDMS substrate deeply. The width and depth of the primary cracks both increase with increasing film thickness, whereas the crack spacing is insensitive to the film thickness. The film pieces dividing by the primary cracks can fracture further when they are triggered by an external disturbance due to the residual tensile stress, resulting in the formation of fine crack networks (called secondary cracks). The width and spacing of the secondary cracks show different behaviors in comparison to the primary cracks. The morphological characteristics, growth behaviors, and formation mechanisms of the primary and secondary cracking modes have been discussed in detail. The report in this work could provide better understanding of two distinct cracking modes with different sizes and morphologies.

Lee-Yang and Fisher zeros are crucial for the study of phase transitions in the grand canonical and the canonical ensembles, respectively. However, these powerful methods do not cover the isothermal-isobaric ensemble ($NPT$ ensemble), which reflects the conditions of many experiments. In this work we present a theory of the phase transitions in terms of the zeros of the $NPT$-ensemble partition functions in the complex plane. The proposed theory provides an approach to calculate all the partition function zeros in the $NPT$ ensemble, which form certain curves in the thermodynamic limit. To verify the theory we consider Tonks gas and van der Waals fluid in the $NPT$ ensemble. In the case of Tonks gas, similarly to the Lee-Yang circle theorem, we obtain an exact equation for the zero limit curve. We also derive an approximated limit curve equation for van der Waals fluid in terms of the Szegö curve. This curve fits numerically calculated zeros and correctly describes how the phenomenon of phase transition depends on the temperature.

A combinatorial approach is used to study the critical behavior of a $q$-state Potts model with a round-the-face interaction. Using this approach, it is shown that the model exhibits a first order transition for $q>3$. A second order transition is numerically detected for $q=2$. Based on these findings, it is deduced that for some two-dimensional ferromagnetic Potts models with completely local interaction, there is a changeover in the transition order at a critical integer $q_c\le 3$. This stands in contrast to the standard two-spin interaction Potts model where the maximal integer value for which the transition is continuous is $q_c=4$. A lower bound on the first order critical temperature is additionally derived.

We study pattern formation, fluctuations, and scaling induced by a growth-promoting active walker on an otherwise static interface. Active particles on an interface define a simple model for energy-consuming proteins embedded in the plasma membrane, responsible for membrane deformation and cell movement. In our model, the active particle overturns local valleys of the interface into hills, simulating growth, while itself sliding and seeking new valleys. In one dimension, this “overturn-slide-search” dynamics of the active particle causes it to move superdiffusively in the transverse direction while pulling the immobile interface upward. Using Monte Carlo simulations, we find an emerging tentlike mean profile developing with time, despite large fluctuations. The roughness of the interface follows scaling with the growth, dynamic, and roughness exponents, derived using simple arguments as $\beta =2/3$, $z=3/2$, and $\alpha =1/2$, respectively, implying a breakdown of the usual scaling law $\beta =\alpha /z$, due to very local growth of the interface. The transverse displacement of the puller on the interface scales as $\sim t^{2/3}$ and the probability distribution of its displacement is bimodal, with an unusual linear cusp at the origin. Both the mean interface pattern and probability distribution display scaling. A puller on a static two-dimensional interface also displays aspects of scaling in the mean profile and probability distribution. We also show that a pusher on a fluctuating interface moves subdiffusively leading to a separation of timescale in pusher motion and interface response.

Single-file Brownian motion in periodic structures is an important process in nature and technology, which becomes increasingly amenable for experimental investigation under controlled conditions. To explore and understand generic features of this motion, the Brownian asymmetric simple exclusion process (BASEP) was recently introduced. The BASEP refers to diffusion models where hard spheres are driven by a constant drag force through a periodic potential. Here we derive general properties of the rich collective dynamics in the BASEP. Average currents in the steady state change dramatically with the particle size and density. For an open system coupled to particle reservoirs, extremal current principles predict various nonequilibrium phases, which we verify by Brownian dynamics simulations. For general pair interactions we discuss connections to single-file transport by traveling-wave potentials and prove the impossibility of current reversals in systems driven by a constant drag and by traveling waves.

Fractional Levy motion has been derived from its generalized Langevin equation via path integrals in earlier works and has since proven to be a useful model for nonlocal and non-Markovian processes, especially in the context of nondiffusive transport. Here, we generalize the approach to treat tempered Lévy distributions and derive the propagator and diffusion equation of truncated asymmetrical fractional Levy motion via path integrals. The model now recovers exponentially tempered tails above a chosen scale in the propagator, and therefore finite moments at all orders. Concise analytical expressions for its variance, skewness, and kurtosis are derived as a function of time. We then illustrate the versatility of this model by applying it to simulations of the turbulent transport of fast ions in the TORPEX basic plasma device.

Fractal structures are very common in the phase space of nonlinear dynamical systems, both dissipative and conservative, and can be related to the final state uncertainty with respect to small perturbations on initial conditions. Fractal structures may also appear in the parameter space, since parameter values are always known up to some uncertainty. This problem, however, has received less attention, and only for dissipative systems. In this work we investigate fractal structures in the parameter space of two conservative dynamical systems: the standard nontwist map and the quartic nontwist map. For both maps there is a shearless invariant curve in the phase space that acts as a transport barrier separating chaotic orbits. Depending on the values of the system parameters this barrier can break up. In the corresponding parameter space the set of parameter values leading to barrier breakup is separated from the set not leading to breakup by a curve whose properties are investigated in this work, using tools as the uncertainty exponent and basin entropies. We conclude that this frontier in parameter space is a complicated curve exhibiting both smooth and fractal properties, that are characterized using the uncertainty dimension and basin and basin boundary entropies.

Clonal growth plants are abundant in both terrestrial and marine ecosystems. Some marine species are particularly important since they provide essential ecosystem services in the shores of all continents except Antarctica. For the appropriate modelization of clonal growth the discretization of the growth direction angle has to be carefully treated to correctly describe the dynamics of the meadow. Specifically, determining the minimum number of growth directions is important to maximize the efficiency of numerical simulations. We show that the presence of neutral modes in the growth direction angle tends to make the distribution of the growth directions uniform in the presence of noise, allowing us to choose the minimal number of discrete angles compatible with the branching. We also show that the formation of spatial patterns induces small differences in the population density within these different growth directions.

Multilayer networked systems are ubiquitous in nature and engineering, and the robustness of these systems against failures is of great interest. A main line of theoretical pursuit has been percolation-induced cascading failures, where interdependence between network layers is conveniently and tacitly assumed to be symmetric. In the real world, interdependent interactions are generally asymmetric. To uncover and quantify the impact of asymmetry in interdependence on network robustness, we focus on percolation dynamics in double-layer systems and implement the following failure mechanism: Once a node in a network layer fails, the damage it can cause depends not only on its position in the layer but also on the position of its counterpart neighbor in the other layer. We find that the characteristics of the percolation transition depend on the degree of asymmetry, where the striking phenomenon of a switch in the nature of the phase transition from first to second order arises. We derive a theory to calculate the percolation transition points in both network layers, as well as the transition switching point, with strong numerical support from synthetic and empirical networks. Not only does our work shed light on the factors that determine the robustness of multilayer networks against cascading failures, but it also provides a scenario by which the system can be designed or controlled to reach a desirable level of resilience.

The conformational flexibility and dynamics of unfolded peptide chains is of major interest in the context of protein folding and protein functioning. The rate with which amino acids at different positions along the peptide chain meet sets an upper speed limit for protein folding. By using single-molecule photo-induced energy transfer spectroscopy, we have systematically measured end-to-end and end-to-internal site contact formation rates for several intrinsically disordered protein fragments (IDPs) (11 to 41 amino acids) and have also determined their hydrodynamic radius using dual-focus fluorescence correlation spectroscopy. For interpreting the measured values, we have developed a Brownian dynamics model (based on bead-rod chain dynamics in a thermal bath including hydrodynamic interactions) which quantitatively reproduces all measured data surprisingly well while requiring only two fit parameters. The model provides a complete picture of the peptides' dynamics and allows us to translate the experimental rates and radii into molecular properties of the peptides: We find a persistence length of $l_P=5.2\pm 1.9\text{Å}$, a hydrodynamic radius of $a=3.5\pm 0.7\text{Å}$ per amino acid, and that excluded volume effects play an important role in the dynamics of IDPs.

Here we investigate single-cell stochastic gene expression kinetics in a minimal coupled gene circuit with positive-plus-negative feedback. A triphasic stochastic bifurcation is observed upon increasing the ratio of the positive and negative feedback strengths, which reveals a strong synergistic interaction between positive and negative feedback loops. We discover that coupled positive-plus-negative feedback amplifies gene expression mean but reduces gene expression noise over a wide range of feedback strengths when promoter switching is relatively slow, stabilizing gene expression around a relatively high level. In addition, we study two types of macroscopic limits of the discrete chemical master equation model: the Kurtz limit applies to proteins with large burst frequencies and the Lévy limit applies to proteins with large burst sizes. We derive the analytic steady-state distributions of the protein abundance in a coupled gene circuit for both the discrete model and its two macroscopic limits, generalizing the results obtained by Liu et al. [Chaos 26, 043108 (2016)]. We also obtain the analytic time-dependent protein distribution for the classical Friedman-Cai-Xie random bursting model [Friedman, Cai, and Xie, Phys. Rev. Lett. 97, 168302 (2006)]. Our analytic results are further applied to study the structure of gene expression noise in a coupled gene circuit, and a complete decomposition of noise in terms of five different biophysical origins is provided.

We study analytically and numerically aspects of the dynamics of slope selection for one-dimensional models describing the motion of line defects, steps, in homoepitaxial crystal growth. The kinetic processes include diffusion of adsorbed atoms (adatoms) on terraces, attachment and detachment of atoms at steps with large yet finite, positive Ehrlich-Schwoebel step-edge barriers, material deposition on the surface from above, and the mechanism of downward funneling (DF) via a phenomenological parameter. In this context, we account for the influence of boundary conditions at extremal steps on the dynamics of slope selection. Furthermore, we consider the effect of repulsive, nearest-neighbor force-dipole step-step interactions. For geometries with straight steps, we carry out numerical simulations of step flow, which demonstrate that slope selection eventually occurs. We apply perturbation theory to characterize time-periodic solutions of step flow for slope-selected profiles. By this method, we show how a simplified step flow theory with constant probabilities for the motion of deposited atoms can serve as an effective model of slope selection in the presence of DF. Our analytical findings compare favorably to step simulations.

By employing molecular dynamics simulations we investigate the fluctuation of surface nanobubbles immersed in liquid phase. Our simulation results indicate that in comparison with the surrounding liquid and nanobubble interior, the vapor-liquid or gas-liquid interface of nanobubbles always exhibits the largest compressibility, demonstrating the enhanced fluctuation for nanobubble interfaces. We also find that vapor surface nanobubbles and gas surface nanobubbles exhibit different fluctuation behaviors. For vapor nanobubbles that appear in overheated pure liquid, both density fluctuation and interface fluctuation are independent on the external pressure since the internal pressure remains constant at a given temperature. For gas nanobubbles that appear in gas supersaturated solution, the density fluctuation monotonously decreases with the increase of gas concentration, while the interface fluctuation shows a nonmonotonic variation. Departure from the intermediate gas concentration with the minimal interface fluctuation would enhance the fluctuation, which may finally lead to nanobubble destabilization. Finally, our simulation results indicate that the complicated interface fluctuation of surface nanobubbles comprises two different modes: interface deformation and interface oscillation, both of which display similar trends as that of the combined interface fluctuation.

We study random sequential adsorption (RSA) of a class of solids that can be obtained from a cube by specific cutting of its vertices, in order to find out how the transition from tetrahedral to octahedral symmetry affects the densities of the resulting jammed packings. We find that in general solids of octahedral symmetry form less dense packing; however, the lowest density was obtained for the packing built of tetrahedra. The densest packing is formed by a solid close to a tetrahedron but with vertices and edges slightly cut. Its density is ${\theta }_{\max }=0.41278\pm 0.00059$ and is higher than the mean packing fraction of spheres or cuboids but is lower than that for the densest RSA packings built of ellipsoids or spherocylinders. The density autocorrelation function of the studied packings is typical for random media and vanishes very quickly with distance.

We demonstrate with experiments that wrinkling in stretched latex sheets occurs over finite strains, and that their amplitudes grow and then decay to zero over a greater range of applied strains compared with linear elastic materials. The wrinkles occur provided the sheet is sufficiently thin compared to its width, and only over a finite range of length-to-width ratios. We show with simulations that the Mooney-Rivlin hyperelastic model describes the observed growth and decay of the wrinkles in our experiments. The decrease of wavelength with applied tension is found to be consistent with a far-from-threshold scenario proposed by Cerda and Mahadevan in 2003. However, the amplitude is observed to decrease with increasing tensile load, in contrast with the prediction of their original model. We address the crucial assumption of collapse of compressive stress, as opposed to collapse of compressive strain, underlying the far-from-threshold analysis, and test it by measuring the actual arc-length of the stretched sheet in the transverse direction and its difference from the width of a planar projection of the wrinkled shape. Our experiments and numerical simulations indicate a collapse of the compressive stress and reveal that a proper implementation of the far-from-threshold analysis is consistent with the nonmonotonic dependence of the amplitude on applied tensile load observed in experiments and simulations. Thus, our work support and extend far-from-threshold analysis to the stretching problem of rectangular hyperelastic sheets.

We study numerically the basic flow and linear stability of a capillary jet confined in a rectangular microchannel. We consider both the case where the interface does not touch the solid surfaces and that in which the jet adheres to them with a contact angle slightly smaller than ${180}^{\circ }$. Given an arbitrary set of values of the governing parameters, the fully developed (parallel) two-dimensional basic flow is calculated and then the growth rate of the dominant perturbation mode is determined as a function of the wave number. The flow is linearly stable if that growth rate is negative for all the wave numbers considered. We show that when the coflowing stream viscosity is sufficiently small in terms of that of the jet, there is an interval of the flow rate ratio $Q$ for which the jet adheres to the walls or not depending on whether the flow is established by decreasing or increasing the value of $Q$. When the distance between the interface and the channel wall is of the order of the jet radius, the jet is unconditionally unstable. However, for sufficiently small interface-to-wall distances, the viscous stress can dominate the capillary pressure and fully stabilize the flow. Our results suggest that the capillary modes are suppressed and the flow becomes stable when the jet adheres to the channel walls. The combination of the above results indicates that, under certain parametric conditions, stable or unstable jets can be formed depending on whether the experimenter sets the flow rate ratio by decreasing or increasing progressively the jet flow rate while keeping constant that of the outer stream. Our theoretical predictions for the stablity of a coflow in a rectangular channel are consistent with previous experimental results [Humphry et al., Phys. Rev. E 79, 056310 (2009)].

Under the influence of an external magnetic field H, the suspended ferromagnetic particles of a laminar ferrofluid flow try to be oriented along H through a relaxation mechanism. Turbulence affects the interaction between the magnetization of each suspended particle and the external field thereby leading to a large relaxation time and hence a slow relaxation process. This can be obtained by replacing viscous drag force with turbulent drag force in Brownian motion. We show that the total energy is an inviscid invariant in turbulent ferrofluids. Using two-point statistics we formulate an exact relation in the inertial zone of incompressible ferrofluid turbulence. This exact relation gives an accurate measure of the energy dissipation rate in a turbulent ferrofluid. We also show that $(\mathbf{u}\times \mathit{\omega }), (\mathbf{M}\times \mathbf{H}), (\mathbf{M}·\mathbf{\nabla })\mathbf{H}$, and $(\mathit{\omega }\times \mathbf{M})$ play the major role in energy cascading in turbulent ferrofluids.

In this paper, we show how plasma discharge capillaries can be numerically modeled as resistors within an RLC-series discharge circuit, allowing for a simple description of these systems, while taking into account heat and radiation losses. An analytic radial model is also provided and compared to the numerical model for plasma discharge capillaries at thermal equilibrium, with corrections due to radiation losses. Finally, diagnostic techniques based on visible spectroscopy of plasma emission lines are discussed both for atomic and molecular gases, comparing experimental results with numerical simulations and theoretical calculations.

Transmission line-based metamaterials are used to realize and model the conjugate-impedance matched superabsorbers. Here, we formulate an analytical-numerical approach for maximizing the effective absorption cross section of the metamaterial wormhole superabsorber, under the goal of minimizing the complexity of the structure. Analytical expressions for the gradient of the absorption cross section as a function of the structural parameters are derived. Numerical results showing enhanced absorption are obtained under three different optimization strategies: a ring-by-ring approach, a gradient-based optimization, and a mixed algorithm. The best results are achieved with the mixed algorithm, with which it is demonstrated that the optimal wormhole superabsorber significantly outperforms a black body-type absorber of a similar size. This study is of a particular interest for applications of the conjugate-impedance-matched superabsorbers as efficient harvesters of electromagnetic radiation.

In a recent paper, Silva et al. [Phys. Rev. E 99, 063002 (2019)] consider the deformation of vesicles in response to the acoustic radiation stresses of ultrasonic standing waves. An important part of the analysis concerns the quadrupole projection of the radial component of radiation stress on a spherical vesicle having a radius much smaller than the acoustic wavelength evaluated with a software-aided expansion approach. Here we note that their projection agrees with results previously found for compressible spheres in the corresponding limit [J. Acoust. Soc. Am. 69, 1499 (1981)] provided an error in Eq. (B5) of Silva et al. is corrected.