General Shape Equation Research Articles

Droplet dynamics driven by electrowetting

Even though electrowetting-on-dielectric (EWOD) is a useful strategy in a wide array of biological and engineering processes with numerous droplet-manipulation applications, there is still a lack of complete theoretical interpretation on the dynamics of electrowetting. In this paper we present an effective theoretical model and use the Onsager variational principle to successfully derive general dynamic shape equationsfor electrowetting droplets in both the overdamped and underdamped regimes. It is found that the spreading and retraction dynamics of a droplet on EWOD substrates can be fairly well captured by our model, which agrees with previous experimental results very well in the overdamped regime. We also confirm that the transient dynamics of EW can be characterized by a timescale independent of liquid viscosity, droplet size, and applied voltage. Our model provides a complete fundamental explanation of EW-driven spreading dynamics, which is important for a wide range of applications, from self-cleaning to novel optical and digital microfluidic devices.

An integral case of the axisymmetric shape equation of open vesicles with free edges

The equilibrium shapes of lipid vesicles are governed by the general shape equation which is derived from the minimization of the Helfrich free energy and can be reduced to the Willmore equation in a special case. The general shape equation is a high-order nonlinear partial differential equation and it is very difficult to find analytical solution even in axisymmetric case, which is reduced to a second-order ordinary differential equation. In the traditional axisymmetric shape equation, the turning radius is the variable. Here we study the shape equation by choosing the tangential angle as the variable. In this case, the Willmore equation is reduced to the Bernoulli differential equation and the general solution is obtained conveniently. We find that the curvature in this solution is discontinuous in some cases, which was not noticed previously. This solution can satisfy the boundary conditions for an open vesicle with free edges.

Exact solutions of the general equilibrium shape equations in a general power model of free energy for DNA structures

The aim of this paper is to generalize the results of the Feoli's formalism (A. Feoli et al., Nucl. Phys. B 705, 577 (2005)) for DNA structures. The exact solutions of the general equilibrium shape equations for a general power model of free energy are investigated using the Feoli's formalism. The free energy of B- to Z-DNA transition is also calculated in this formalism.

B- TO Z-DNA TRANSITION PROBED BY THE FEOLI'S FORMALISM FOR A KIRCHHOFF MODEL

In this paper, we are going to use the techniques of quantum field theory to describe the free energy of DNA structures. The exact solutions of the general equilibrium shape equations for DNA structures in a Kirchhoff model by using the Feoli's formalism [A. Feoli, V. V. Nesterenko and G. Scarpetta, Nucl. Phys. B705, 577 (2005)] are investigated. The free energy of the B- to Z-DNA transition is also calculated in this formalism.

Anisotropic surface effects on the formation of chiral morphologies of nanomaterials

Nanomaterials with chiral morphologies hold promise for a wide diversity of technologically important applications in such fields as micro/nano-electromechanical systems and medical engineering. Understanding the mechanisms underlying the formation of chiral morphologies of natural and synthesized materials remains an issue of crucial significance. In this study, a refined Kirchhoff rod model taking into account anisotropic surface effects is employed to describe quasi-one-dimensional nanomaterials with complicated spatial morphologies. It is shown that anisotropic surface stresses can induce the formation of rich morphologies of nanomaterials. A general shape equation of nanowires is derived by the variational method of energy. Thereby, the effects of anisotropic surface properties, bulk elastic properties and cross-sectional sizes on the chiral morphologies of nanomaterials are quantitatively investigated, and the conditions for the formation of binormal nanohelices are given. The physical mechanism addressed in this study is verified by our recent experiments on tuning the twisting chirality of polymer lamellae via surface treatments. Our analysis suggests that one can design and adjust the morphology of synthesized nanohelices by tailoring or functionalizing their surfaces during fabrication. This study is also helpful in interpreting the formation of such artificial and biological chiral materials as the flagella of bacterial and self-assembled helical ribbons.

Periodic-cylinder vesicle with minimal energy

We give some details about the periodic cylindrical solution found by Zhang and Ou-Yang in [1996 Phys. Rev. E 53 4206] for the general shape equation of vesicle. Three different kinds of periodic cylindrical surfaces and a special closed cylindrical surface are obtained. Using the elliptic functions contained in mathematic, we find that this periodic shape has the minimal total energy for one period when the period–amplitude ratio β ≈ 1.477, and point out that it is a discontinuous deformation between plane and this periodic shape. Our results also are suitable for DNA and multi-walled carbon nanotubes (MWNTs).

THE FIRST AND SECOND VARIATION OF THE TOTAL ENERGY OF CLOSED DUPLEX DNA IN PLANAR CASE

DNA's shape mostly lies on its total energy F. Its corresponding equilibrium shape equations can be obtained by classical variation method: letting the first energy variation δ(1)F = 0. Here, we not only provide the first variation δ(1)F but also give the second variation δ(2)F in planar case. Moreover, the general shape equations of DNA are abstained and a mistake in Zhang et al., [Phys. Rev. E70, 051902 (2004)] is pointed out.

Comment on “General equilibrium shape equations of polymer chains”

In this Comment, we point out that the Euler-Lagrange equations, which are referred to as the general equilibrium shape equations presented by Zhang et al. [Phys. Rev. E 70, 051902 (2004)] are incorrect, along with equations derived from them. The correct equations are provided here and they are cross-checked using certain energy functions previously presented in the literature. Further, with the use of the correct equations, we present new numerical results, which for the values of the constants given by Zhang et al. do not give rise to the physical behavior observed for DNA by those authors. However, the correct equations can be consistent with sensible behavior for different values of the constants.

Shape and stability of two-dimensional lipid domains with dipole-dipole interactions

We study the general energy and shape of the two-dimensional solid monolayer domains with the dipole-dipole interactions. Compared with the domain energy without tilted dipole moments [M. Iwamoto and Z. C. Ou-Yang, Phys. Rev. Lett. 93, 206101 (2004)], the general dipolar energy is not only shape and size but also boundary orientation dependent. The general shape equation derived by this energy using variational approach predicts a circular solution and an equilibrium shape grown from this circle. In particular, the latter is composed of two branches: a translation-induced growth of all odd harmonic modes and a pressure-induced cooperative deformation by all even harmonic modes. The good qualitative agreement between our prediction and the experimental observations shows the validity of the present theory.

Elasticity and stability of a helical filament

We derive the general shape equations in terms of Euler angles for a uniform elastic rod with spontaneous torsion and curvatures and subjected to external force and torque. Our results based on an analytic formalism show that the extension of a helical rod may undergo a one-step discontinuous transition with increasing stretching force. This agrees quantitatively with experimental observations for a helix in a chemically defined lipid concentrate. The larger the twisting rigidity, the larger the jump in the extension. The effect of torque on the jump is, however, dependent on the value of the spontaneous torsion. In contrast, increasing the spontaneous torsion encourages the continuous variation of the extension. An "over-collapse" behavior is observed for the rod with asymmetric bending rigidity, and an intrinsic asymmetric elasticity under twisting force is found.

Equilibrium Shape Equation and Geometrically Permissible Condition for Two-Component Lipid Bilayer Vesicles

Equilibrium shapes of vesicles composed of a mixture of partially miscible amphiphiles are investigated. To take into account the influences of the composition, a simple phenomenological coupling between the co mposition and the curvatures, including the mean curvature and the Gauss curvature of the membrane surface, is suggested. By minimizing the potential functional, the general shape equation is obtained and solved analytically for vesicles with simple shapes. Besides, the geometrical constraint equation and geometrically permissible condition for the two-component lipid vesicles are put forward. The influences of physical parameters on the geometrically permissible phase diagrams are predicted. The close relations between the predictions and existing experimental phenomena published recently are shown.

ELASTICITY OF A HELICAL RIBBON WITH NONCIRCULAR CROSS SECTION

We derive the general shape equations in terms of Euler angles for an elastic model of uniform ribbon with noncircular cross section and vanishing spontaneous curvatures. We show that it has in general not a planar solution for a closed ribbon free of external force and torque. We study the conditions to form a helix with the axis along the direction of the applied force for a ribbon under external force and twisting. We find that if the bending rigidity is greater than the twisting rigidity, then no such helical rod can exist. Our stability analysis shows that a helical ribbon is in general stable or at least metastable under arbitrary force and torque. We find that the extension of the ribbon may undergo a discontinuous transition from a twisted straight rod to a helical ribbon. The intrinsic asymmetric elasticity of a helical ribbon under external torque is also studied.

General equilibrium shape equations of polymer chains

The general equilibrium shape equations of polymer chains are analytically derived in this paper. This provides a unified description for many models, such as the well-known wormlike chain (WLC) model, the wormlike rod chain (WLRC) model, carbon nanotubes, and so on. Using the WLC model, we find that the pitch-to-radius ratio of coils, 4.443, agrees with Z-DNA, and the pitch-to-radius ratio from WLRC agrees with the data of B-DNA qualitatively. Using the general shape equations, we discuss a chiral model in which the solutions of straight, helical, and circular biopolymers are given, respectively. We also find that the model suggested by Helfrich [Langmuir 7, 567 (1991)] is very appropriate to describe B-DNA (or other biopolymers) if we choose the four phenomenological parameters as A=50 nm , C=60 nm(2) , alpha=40 nm(3) , and beta=50 nm(2) .

New solutions of the shape equation

The general shape equation describing the forms of vesicles is a highly nonlinear partial differential equation for which only a few explicit solutions are known. These solvable cases are briefly reviewed and a new analytical solution which represents the class of the constant mean curvature surfaces is described. Pearling states of the tubular fluid membranes can be explained as a continuous deformation preserving membrane mean curvature.

Constraint for the Existence of Ellipsoidal Vesicles

Under the spontaneous curvature model of lipid bilayers, the constraints for the existence of equilibrium axisymmetric oblate and prolate ellipsoidal vesicles are obtained from the general shape equation. They degenerate either to the constraint for the existence of a spherical vesicle or to that of a circular cylindrical vesicle given by Ou-Yang and Helfrich [Phys. Rev. Lett. 59 (1987) 2486; 60(1988)120; Phys. Rev. A 39 (1989) 5280].

EQUIVALENCE OF THE TWO METHODS IN CALCULATING THE EQUILIBRIUM SHAPES OF CYLINDRICAL VESICLES

So far, two methods are often used in solving the equilibrium shapes of vesicles. One method is by starting with the general shape equation and restricting it to the shapes with particular symmetry. The other method is by assuming the symmetry and topology of the vesicle first and treating it with the calculus of variation to get a set of ordinary differential equations. The relationship between these two methods in the case of cylindrical vesicles, and a comparison of the results are given.

NEW APPROACH ON THE GENERAL SHAPE EQUATION OF AXISYMMETRIC VESICLES

The general Helfrich shape equation determined by minimizing the curvature free energy describes the equilibrium shapes of the axisymmetric lipid bilayer vesicles in different conditions. It is a nonlinear differential equation with variable coefficients. In this letter, by analyzing the unique property of the solution, we change this shape equation into a system of the two differential equations. One of them is a linear differential equation. This equation system contains all of the known rigorous solutions of the general shape equation. And the more general constraint conditions are found for the solution of the general shape equation.

Periodic cylindrical surface solution for fluid bilayer membranes.

The shape equation of cylindrical surfaces for lipid bilayer membranes expressed by the Gaussian maps of their cross sections is given. General cylindrical surface solution has been obtained for the case of \ensuremath{\Delta}p=0. This is the first case found that can be solved for the general shape equation. With this solution we show two kinds of periodic surface solutions to the general shape equation for fluid bilayer membranes. \textcopyright{} 1996 The American Physical Society.

Shape equations for axisymmetric vesicles: A clarification.

We derive the shape equations for axisymmetric vesicles and show that they are identical to the general shape equation [Ou-Yang Zhong-Can and W. Helfrich, Phys. Rev. A 39, 5280 (1989)] specialized to axisymmetry. We consider three different topologies (an axisymmetric membrane segment suspended between two circular rings and closed vesicles of spherical and toroidal topology.) We point out that the shape equations are independent of the variational method used.

Relation Between Shape Equations for Axisymmetric vesicles11The project supported in part by National Natural Science Foundation of China, the Grant LWTZ-1298 of the Chinese Academy of Sciences and the Grant of the National Commission of Science and Technology

In deriving the shape equation for axisymmetric vesicles from the Helfrich free energy the variation must be taken with respect to the contour. It is pointed out that the widely used Helfrich shape equation and the other one given by Peterson were derived by improper variations, hence both equations need to be examined. The validity of the two equations is discussed based on analysis of their relation revealed by two derived first integrals. A newly found exact solution for the general shape equation is studied and the relation of this exact solution of spherical topology with that of the Helfrich shape equation is discussed.