Stationary Shapes Research Articles

Helical states of nonlocally interacting molecules and their linear stability: a geometric approach

The equations for strands of rigid charge configurations interacting nonlocally are formulated on the special Euclidean group, SE(3), which naturally generates helical conformations. Helical stationary shapes are found by minimizing the energy for rigid charge configurations positioned along an infinitely long molecule with charges that are off-axis. The classical energy landscape for such a molecule is complex with a large number of energy minima, even when limited to helical shapes. The question of linear stability and selection of stationary shapes is studied using an SE(3) method that naturally accounts for the helical geometry. We investigate the linear stability of a general helical polymer that possesses torque-inducing nonlocal self-interactions and find the exact dispersion relation for the stability of the helical shapes with an arbitrary interaction potential. We explicitly determine the linearization operators and compute the numerical stability for the particular example of a linear polymer comprising a flexible rod with a repeated configuration of two equal and opposite off-axis charges, thereby showing that even in this simple case the nonlocal terms can induce instability that leads to the rod assuming helical shapes.

Memory for motion and spatial location is mediated by contralateral and ipsilateral motion processing cortex

Memory and perception have been associated with common sensory cortical activity. However, previous studies have only investigated memory and perception effects associated with a single feature (i.e., spatial location or color). The aim of the present functional magnetic resonance imaging (fMRI) and transcranial magnetic stimulation (TMS) study was to assess whether memory for multiple (two) features would produce sensory cortical activity that mirrored perceptual processing of the same features. During encoding, moving or stationary abstract shapes were presented to the right or left of fixation. During retrieval, shapes were presented at fixation and participants classified each item as previously in motion or stationary within the right or left visual field. Memory for items in motion, regardless of spatial location, produced fMRI activity in perceptual motion processing region MT+. Memory for motion and spatial location produced contralateral and ipsilateral fMRI activity in perceptual motion processing sub-region MT. Following TMS to MT, memory for motion was impaired, but performance did not differ between the contralateral and ipsilateral visual fields. The present results are consistent with previous findings in that memory for motion produced fMRI activity in MT+ and was impaired following TMS to MT. However, memory for motion and spatial location produced contralateral and ipsilateral fMRI and TMS effects, deviating from the primarily contralateral perceptual processing organization of MT. The present evidence suggests that during memory for motion and spatial location only motion information is coded in motion processing cortex, while previous findings suggest spatial location information is coded in earlier extrastriate cortex.

Stationary shapes of confined rotating magnetic liquid droplets

We study the family of steady shapes which arise when a magnetic liquid droplet is confined in a rotating Hele-Shaw cell and subjected to an azimuthal magnetic field. Two different scenarios are considered: first, the magnetic fluid is assumed to be a Newtonian ferrofluid, and then it is taken as a viscoelastic magnetorheological fluid. The influence of the distinct material properties of the fluids on the ultimate morphology of the emerging stationary patterns is investigated by using a vortex-sheet formalism. Some of these exact steady structures are similar to the advanced time patterns obtained by existing time-evolving numerical simulations of the problem. A weakly nonlinear approach is employed to examine this fact and to gain analytical insight about relevant aspects related to the stability of such exact stationary solutions.

Numerical analysis of tips in viscous flow

When a drop (or gas bubble) is placed in a strong viscous flow (e.g., a shear flow), it develops very sharp tips at its ends. Sharp tips are also formed when a viscous fluid is withdrawn from the neighborhood of its interface with the ambient air or with another fluid (selective withdrawal). However, it is observed frequently that there exists a critical flow strength above which the drop transitions toward a "jetting state" in which a jet comes out from the tip. In this paper, we look numerically for stationary drop shapes, both globally and close to the tip, which we study with very high resolution. To this end we use a boundary integral method to solve the axisymmetric flow equations for arbitrary viscosity ratios in the inertialess (Stokes) limit. Stationary states are solved for using Newton's method. This permits us to find both stable and unstable steady states and to investigate the nature of the jetting transition. The critical parameters for this transition are in reasonable agreement with slender-body theory. Excellent agreement is found with our earlier experiments in the selective withdrawal geometry [S. Courrech du Pont and J. Eggers, Phys. Rev. Lett. 96, 034501 (2006)], for which the viscosity of the phase inside the tip is negligible. We describe a scale invariance of the experimental interface profiles away from the tip. Then we investigate the highly curved tip region not considered previously with comparable precision. We find that the shape near the tip is universal, i.e., independent of the outer flow and of the geometry of the system (drop or selective withdrawal). While the tip curvature becomes extremely large, it always remains finite if surface tension is present.

Electrohydrodynamic Model of Vesicle Deformation in Alternating Electric Fields

We develop an analytical theory to explain the experimentally observed morphological transitions of quasispherical giant vesicles induced by alternating electric fields. The model treats the inner and suspending media as lossy dielectrics, and the membrane as an impermeable flexible incompressible–fluid sheet. The vesicle shape is obtained by balancing electric, hydrodynamic, bending, and tension stresses exerted on the membrane. Our approach, which is based on force balance, also allows us to describe the time evolution of the vesicle deformation, in contrast to earlier works based on energy minimization, which are able to predict only stationary shapes. Our theoretical predictions for vesicle deformation are consistent with experiment. If the inner fluid is more conducting than the suspending medium, the vesicle always adopts a prolate shape. In the opposite case, the vesicle undergoes a transition from a prolate to oblate ellipsoid at a critical frequency, which the theory identifies with the inverse membrane charging time. At frequencies higher than the inverse Maxwell-Wagner polarization time, the electrohydrodynamic stresses become too small to alter the vesicle's quasispherical rest shape. The model can be used to rationalize the transient and steady deformation of biological cells in electric fields.

Ground-Based Vibration Response of a Spinning, Cyclic, Symmetric Rotor With Gyroscopic and Centrifugal Softening Effects

This paper is to study ground-based vibration response of a spinning, cyclic, symmetric rotor through a theoretical analysis and an experimental study. The theoretical analysis consists of three steps. The first step is to analyze the vibration characteristics of a stationary, cyclic, symmetric rotor with N identical substructures. For each vibration mode, we identify a phase index n and derive a Fourier expansion of the mode shape in terms of the phase index n. The second step is to predict the rotor-based vibration response of the spinning, cyclic, symmetric rotor based on the Fourier expansion of the mode shapes and the phase indices. The rotor-based formulation includes gyroscopic and centrifugal softening terms. Moreover, rotor-based response of repeated modes and distinct modes is obtained analytically. The third step is to transform the rotor-based response to ground-based response using the Fourier expansion of the stationary mode shapes. The theoretical analysis leads to the following conclusions. First, gyroscopic effects have no significant effects on distinct modes. Second, the presence of gyroscopic and centrifugal softening effects causes the repeated modes to split into two modes with distinct frequencies ω1 and ω2 in the rotor-based coordinates. Third, the transformation to ground-based observers leads to primary and secondary frequency components. In general, the ground-based response presents frequency branches in the Campbell diagram at ω1±kω3 and ω2±kω3, where k is phase index n plus an integer multiple of cyclic symmetry N. When the gyroscopic effect is significantly greater than the centrifugal softening effect, two of the four frequency branches vanish. The remaining frequency branches take the form of either ω1+kω3 and ω2−kω3 or ω1−kω3 and ω2+kω3. To verify these predictions, we also conduct a modal testing on a spinning disk carrying four pairs of brackets evenly spaced in the circumferential direction with ground-based excitations and responses. The disk-bracket system is mounted on a high-speed, air-bearing spindle. An automatic hammer excites the spinning disk-bracket system and a laser Doppler vibrometer measures its vibration response. A spectrum analyzer processes the hammer excitation force and the vibrometer measurements to obtain waterfall plots at various spin speeds. The measured primary and secondary frequency branches from the waterfall plots agree well with those predicted analytically.

GUESSmix-guided optimization of elution–extrusion counter-current separations

Rational strategies for the optimization of separations are vital to any chromatographic technique. In counter-current separations (CS), once a suitable solvent system is selected for a given separation, the operator is faced with the task of optimizing the separation through the manipulation of those adjustable operation parameters allowed for by the current CS technology. This study employed a mixture of 21 natural products of varying polarity, molecular mass, and functionality, termed the GUESSmix, as a tool to assess the effectiveness of optimization strategies. The behavior of the GUESSmix was observed in the hexane/ethyl acetate/methanol/water 4:6:4:6 (HEMWat +3) solvent system. The effect of operation parameters on both the elution and extrusion stages of a recently introduced CS methodology, termed elution–extrusion counter-current chromatography (EECCC), was investigated. The resulting chromatograms were plotted with K-based reciprocal symmetry plots (ReS and ReSS), which allow comparison of the K values of significant peaks and assessment of resolution of eluting compounds in the interval 0 ≤ K ≤ ∞. The operation parameters studied were: (1) the effect of temperature controlled water circulation around the centrifuge; (2) the combination of flow rate and revolution speed; (3) sample loading capacity; (4) the direction of rotation either agreeing with or opposing the direction of coil winding; (5) injection before equilibration, a practice that saves operator time and reduces solvent consumption. The GUESSmix was found to be a highly useful reference mixture to compare and contrast stationary phase retention volume ratios, resolution, K-values, peak shapes, and extrusion characteristics between CS experiments. EECCC is shown to be a robust technique that may be enhanced with appropriate temperature, rpm, flow rate, sample loading, direction of rotation, and injection timing. Plotting ReS[S] chromatograms enables systematic study of CS parameters, which are not reproducibly represented in retention time or volume plots.

Geometric approach to stationary shapes in rotating Hele–Shaw flows

We consider the geometry of a special family of curves associated with the motion of a two-fluid interface in a rotating Hele–Shaw cell. This family of stationary exact solutions with surface tension consists of interface shapes which balance exactly the competing capillary and centrifugal forces. The result is the formation of patterned structures presenting fingers that eventually assume a teardrop-like shape, and tend to be detached from the main body of the rotating fluid (occurrence of a pinch-off event). By using the vortex–sheet formalism, we approach the problem analytically, and show that the curvature of these particular curves can be very simply expressed as a function of the radial distance to the rotation axis. Motivated by this fact, and through a simple geometric interpretation, we show that the exact solutions for the rotating Hele–Shaw problem satisfy a first-order ordinary differential equation which is readily solved, so that the shape solutions are determined up to quadratures. The ability to probe a number of key morphological features of such solutions analytically is demonstrated, and a gallery of plots is provided to highlight these findings. The possibility of accessing a criterion to predict the occurrence of pinch-off is also discussed. Finally, we prove that the general solutions are dense, and use this fact to guarantee the very existence of the neat, highly symmetric physical patterns.

Morphological stability of electromigration-driven vacancy islands

The electromigration-induced shape evolution of two-dimensional vacancy islands on a crystal surface is studied using a continuum approach. We consider the regime where mass transport is restricted to terrace diffusion in the interior of the island. In the limit of fast attachment (detachment) kinetics a circle translating at constant velocity is a stationary solution of the problem. In contrast to earlier work [O. Pierre-Louis and T. L. Einstein, Phys. Rev. B 62, 13697 (2000)], we show that the circular solution remains linearly stable for arbitrarily large driving forces. The numerical solution of the full nonlinear problem nevertheless reveals a fingering instability at the trailing end of the island, which develops from finite amplitude perturbations and eventually leads to pinch off. Relaxing the condition of instantaneous attachment (detachment) kinetics, we obtain noncircular elongated stationary shapes in an analytic approximation that compares favorably to the full numerical solution.

Shape instabilities in vesicles: A phase-field model

A phase field model for dealing with shape instabilities in fluid membrane vesicles is presented. This model takes into account the Canham-Helfrich bending energy with spontaneous curvature. A dynamic equation for the phase-field is also derived. With this model it is possible to see the vesicle shape deformation dynamically, when some external agent instabilizes the membrane, for instance, inducing an inhomogeneous spontaneous curvature. The numerical scheme used is detailed and some stationary shapes are shown together with a shape diagram for vesicles of spherical topology and no spontaneous curvature, in agreement with known results.

Kaleidoscopic motion and velocity illusions

A novel class of vivid motion and velocity illusions for contrast-defined shapes is presented and discussed. The illusions concern a starlike wheel that, physically, rotates with constant velocity between stationary starlike inner and outer shapes but that, perceptually, shows pulsations, jolts backwards, jolts forwards, and accelerations, each time it aligns with the inner and outer shapes. These dynamic effects turn from one into another when contrasts in the image change gradually, and seem to be caused by colour assimilation and ambiguous figure-ground segregation.

Computing stationary free-surface shapes in microfluidics

A finite-element algorithm for computing free-surface flows driven by arbitrary body forces is presented. The algorithm is primarily designed for the microfluidic parameter range where (i) the Reynolds number is small and (ii) force-driven pressure and flow fields compete with the surface tension for the shape of a stationary free surface. The free surface shape is represented by the boundaries of finite elements that move according to the stress applied by the adjacent fluid. Additionally, the surface tends to minimize its free energy and by that adapts its curvature to balance the normal stress at the surface. The numerical approach consists of the iteration of two alternating steps: The solution of a fluidic problem in a prescribed domain with slip boundary conditions at the free surface and a consecutive update of the domain driven by the previously determined pressure and velocity fields. For a Stokes problem the first step is linear, whereas the second step involves the nonlinear free-surface boundary condition. This algorithm is justified both by physical and mathematical arguments. It is tested in two dimensions for two cases that can be solved analytically. The magnitude of the errors is discussed in dependence on the approximation order of the finite elements and on a step-width parameter of the algorithm. Moreover, the algorithm is shown to be robust in the sense that convergence is reached also from initial forms that strongly deviate from the final shape. The presented algorithm does not require a remeshing of the used grid at the boundary. This advantage is achieved by a built-in mechanism that causes a smooth change from the behavior of a free surface to that of a rubber blanket if the boundary mesh becomes irregular. As a side effect, the element sides building up the free surface in two dimensions all approach equal lengths. The presented variational derivation of the boundary condition corroborates the numerical finding that a second-order approximation of the velocity also necessitates a second-order approximation for the free surface discretization.

Dynamic model and stationary shapes of fluid vesicles

A phase-field model that takes into account the bending energy of fluid vesicles is presented. The Canham-Helfrich model is derived in the sharp-interface limit. A dynamic equation for the phase-field has been solved numerically to find stationary shapes of vesicles with different topologies and the dynamic evolution towards them. The results are in agreement with those found by minimization of the Canham-Helfrich free energy. This fact shows that our phase-field model could be applied to more complex problems of instabilities.

Morphomechanics: goals, basic experiments and models

Morphomechanics is a branch of developmental biology, studying the generation, space-time patterns and morphogenetic role of mechanical stresses (MS) which reside in embryonic tissues. All the morphogenetically active embryonic tissues studied in this respect have been shown to bear substantial mechanical stresses of tension or pressure. MS are indispensable for organized cell movements, expression of a number of developmentally important genes and the very viability of cells. Even a temporary relaxation of MS leads to an increase in the morphological variability and asymmetry of embryonic rudiments. Moreover, MS may be among the decisive links of morphogenetic feedback required for driving forth embryonic development and providing its regular space-time patterns. We hypothesize that one such feedback is based upon the tendency of cells and tissues to hyperrestore (restore with an overshoot) their MS values after any deviations, either artificial or produced by neighboring morphogenetically active tissues. This idea is supported by a number of observations and experiments performed on the tissue and individual cell levels. We describe also the models demonstrating that a number of biologically realistic stationary shapes and propagating waves can be generated by varying the parameters of the hyperrestoration feedback loop. Morphomechanics is an important and rapidly developing branch of developmental and cell biology, being complementary to other approaches.

The diffusive vesicle supply center model for tip growth in fungal hyphae

We propose the diffusive vesicle supply center model for tip growth in fungal hyphae. The model is based on the three-dimensional vesicle supply center (VSC) model [Gierz, G., Bartnicki-García, S., 2001. A three-dimensional model of fungal morphogenesis based on the vesicle supply center concept: J. Theor. Biol. 208, 151–164], but incorporates two aspects of a more realistic vesicle delivery mechanism: vesicle diffusion from the VSC and a finite rate constant for vesicle fusion with the cell membrane. We develop a framework to describe tip growth for a general class of models based on the vesicle supply center concept. Combining this with a method for calculating the steady state distribution of diffusive vesicles we iteratively solve for stationary cell shapes. These show a blunter tip than predicted by the original VSC model, which we attribute to increased forward-directed vesicle delivery via diffusion. The predicted distance between the VSC and the utmost tip of the cell is set by the ratio between the diffusion constant and the rate constant for vesicle exocytosis. Combined with the cell radius, these define the only dimensionless parameter for our model.

Nonlinear Shape Perturbations Induced by Vesicle Inclusions

ABSTRACT We analyse the effects that a rigid inclusion induces on the stationary shapes of an impermeable three-dimensional vesicle. Our study, performed via a numerical calculation, takes into account shapes which are not close to any reference configuration (neither spherical nor planar). The shape perturbations induced by the embedded inclusions are restricted within distances of the order of the inclusion size. Thus, inclusions do not interfere with global vesicle properties, such as budding transitions. The local character of the inclusion perturbation announces a fast distance decay of the membrane mediated elastic force between different proteins.

Shape and stability of a viscous thread

When a viscous fluid, like oil or syrup, streams from a small orifice and falls freely under gravity, it forms a long slender thread, which can be maintained in a stable, stationary state with lengths up to several meters. We discuss the shape of such liquid threads and their surprising stability. The stationary shapes are discussed within the long-wavelength approximation and compared to experiments. It turns out that the strong advection of the falling fluid can almost outrun the Rayleigh-Plateau instability. The asymptotic shape and stability are independent of viscosity and small perturbations grow with time as exp (C t(1/4)), where the constant is independent of viscosity. The corresponding spatial growth has the form exp [(z/L)(1/8)], where z is the down stream distance and L approximately Q(2) sigma(-2) g and where sigma is the surface tension divided by density, g is the gravity, and Q is the flux. We also show that a slow spatial increase of the gravitational field can make the thread stable.

Impermeability effects in three-dimensional vesicles

We analyse the effects that the impermeability constraint induces on the equilibrium shapes of a three-dimensional vesicle hosting a rigid inclusion. A given alteration of the inclusion and/or vesicle parameters leads to shape modifications of different orders of magnitude, when applied to permeable or impermeable vesicles. Moreover, the enclosed-volume constraint wrecks the uniqueness of stationary equilibrium shapes, and gives rise to pear-shaped or stomatocyte-like vesicles.

Stable vortex solitons in a vectorial cubic–quintic model

We investigate the stability of vectorial (two-component) vortex solitons of two types. Their stationary shapes are identical, but their stability (which is the most important issue for spinning solitons) is drastically different. These are solitons with vorticities (S,S) and (S,−S) in the two components. The analysis is performed in a vectorial cubic–quintic model, with the two components nonlinearly coupled by the incoherent cross-phase-modulation interaction, but we expect that the results are quite generic. The stability was investigated by means of computing eigenvalues of perturbations around the stationary solitons, as well as in direct simulations. We also report new analytical results for the well-known problem of the description of the stationary form of scalar solitons in media of this type. The analytical results explain the shape of the spinning solitons, and the strong dependence of their norm (power) on the vorticity, in both the 2D and 3D cases. In this paper we also give the first estimate of the physical characteristics (power and radius) of the stable solitons with different values of S, making use of recently measured values of the necessary nonlinear parameters. All the two-component solitons of type (S,−S) are unstable. In contrast, those of type (S,S) have their stability regions, the size of which strongly depends on S. An unstable soliton always splits into a set of separating zero-spin ones, in precise compliance with the azimuthal index of the most unstable perturbation eigenmode. Direct simulations demonstrate that stable solitons readily self-trap from arbitrary initial pulses which belong to their topological class.

Rotating Ferrofluid Drops

We study the stationary shapes and the rotational motion of drops of magnetic fluids floating in a non-magnetic liquid of equal density and spun up by an externally applied rotating magnetic field. For a sufficiently large magnetic susceptibility of the drop fluid transitions to non axial-symmetric shapes take place when the field amplitude is increased. We give a detailed theoretical account of the character of these shape bifurcations, of the resulting stationary drop forms, and of the slow rotational motion of the drop and compare our findings with results obtained in an experimental realization of the system. Quantitative agreement between theory and experiment can be obtained when saturation effects in the magnetization curve of the ferrofluid are taken into account. PACS numbers: 47.20.Hw, 47.55.Dz, 75.50.Mm