Shapes of lipid monolayer domains: Solutions using elliptic functions

Solid lipid monolayer domains surrounded by a fluid phase at an air-water interface exhibit complex shapes. These intriguing shapes can be understood in terms of a competition between line tension and long-range dipole-dipole interaction. The dipolar energy has recently been relevant to a negative line tension and a positive curvature energy at the boundary, and a corresponding shape equation was derived by the variation of the approximated domain energy (Phys. Rev. Lett. 93, 206101 (2004)). Here we further incorporate surface pressure into the shape equation and show that the equation can be analytically solved: the curvature of the domain boundary is exactly obtained as an elliptic function of arc-length. We find that a circular domain can grow into bean-and peach-like domains with pressure, i.e., dipping and cuspidal transitions of circle by compression. The comparison with the experimental observation shows nice agreement.     

Hydrodynamic boundary conditions: An emergent behavior of fluid–solid interactions

By using the Onsager principle of minimum energy dissipation, the hydrodynamic boundary conditions at the fluid–solid interface are shown to be the natural emergent behavior of microscopic interactions that lead to the interfacial tension and the tangential friction at the fluid–solid interface [T. Qian, C. Qiu, P. Sheng, J. Fluid Mech. 611 (2008) 333]. This is satisfying because the equations of motion, e.g., the Stokes equation, and the hydrodynamic boundary conditions can now be derived from a unified framework. The resulting continuum hydrodynamic formulation yields predictions for immiscible two-phase flows that are in quantitative agreement with molecular dynamic simulations. In particular, the classical problem of the moving contact line is resolved. We also show results on the moving contact line over chemically patterned surfaces which exhibit striking nanoscale characteristics as well as sub-quadratic dependence of the moving contact line dissipation on its average velocity.     

Surface Tension of Multi-phase Flow with Multiple Junctions Governed by the Variational Principle

We explore a computational model of an incompressible fluid with a multi-phase field in three-dimensional Euclidean space. By investigating an incompressible fluid with a two-phase field geometrically, we reformulate the expression of the surface tension for the two-phase field found by Lafaurie et al. (J Comput Phys 113:134–147, 1994) as a variational problem related to an infinite dimensional Lie group, the volume-preserving diffeomorphism. The variational principle to the action integral with the surface energy reproduces their Euler equation of the two-phase field with the surface tension. Since the surface energy of multiple interfaces even with singularities is not difficult to be evaluated in general and the variational formulation works for every action integral, the new formulation enables us to extend their expression to that of a multi-phase (N-phase, N\geqslant2N\geqslant2) flow and to obtain a novel Euler equation with the surface tension of the multi-phase field. The obtained Euler equation governs the equation for motion of the multi-phase field with different surface tension coefficients without any difficulties for the singularities at multiple junctions. In other words, we unify the theory of multi-phase fields which express low dimensional interface geometry and the theory of the incompressible fluid dynamics on the infinite dimensional geometry as a variational problem. We apply the equation to the contact angle problems at triple junctions. We computed the fluid dynamics for a two-phase field with a wall numerically and show the numerical computational results that for given surface tension coefficients, the contact angles are generated by the surface tension as results of balances of the kinematic energy and the surface energy.     

Dynamics of multicomponent vesicles in a viscous fluid

We develop and investigate numerically a thermodynamically consistent model of two-dimensional multicomponent vesicles in an incompressible viscous fluid. The model is derived using an energy variation approach that accounts for different lipid surface phases, the excess energy (line energy) associated with surface phase domain boundaries, bending energy, spontaneous curvature, local inextensibility and fluid flow via the Stokes equations. The equations are high-order (fourth order) nonlinear and nonlocal due to incompressibility of the fluid and the local inextensibility of the vesicle membrane. To solve the equations numerically, we develop a nonstiff, pseudo-spectral boundary integral method that relies on an analysis of the equations at small scales. The algorithm is closely related to that developed very recently by Veerapaneni et al. [81] for homogeneous vesicles although we use a different and more efficient time stepping algorithm and a reformulation of the inextensibility equation. We present simulations of multicomponent vesicles in an initially quiescent fluid and investigate the effect of varying the average surface concentration of an initially unstable mixture of lipid phases. The phases then redistribute and alter the morphology of the vesicle and its dynamics. When an applied shear is introduced, an initially elliptical vesicle tank-treads and attains a steady shape and surface phase distribution. A sufficiently elongated vesicle tumbles and the presence of different surface phases with different bending stiffnesses and spontaneous curvatures yields a complex evolution of the vesicle morphology as the vesicle bends in regions where the bending stiffness and spontaneous curvature are small.     

Surface nucleation in the freezing of gold nanoparticles

We use molecular simulation to calculate the nucleation free energy barrier for the freezing of a 456 atom gold cluster over a range of temperatures. The results show that the embryo of the solid cluster grows at the vapor-surface interface for all temperatures studied and that the usual classical nucleation model, with the embryo growing in the core of the cluster, is unable to predict the shape of the free energy barrier. We use a simple partial wetting model that treats the crystal as a lens-shaped nucleus at the liquid-vapor interface and find that the line tension plays an important role in the freezing of gold nanoparticles.     

LJ模型的热力学特性、相平衡及用于真实分子的研究

结合描述硬球固体Helmholtz自由能的自由体积方法与描述硬球固体径向分布函数的拟合的分析表达式与一阶热力学摄动理论,用于描述Lennard Jones(LJ)固体的Helmholtz自由能.按照一个修正的WCA方法将LJ势分为短程排斥部分与长程吸引部分,将文献中一个用于求取液相的等价的硬球直径的简单的迭代法扩展到固相,用于求取固相的等价的硬球直径.在固体Helmholtz自由能的计算中,使用200壳层,以便获得精确的结果.体相LJ液体的热力学特性由一个最近提出的状态方程求取.该方法很好地描述了LJ固体的过量Helmholtz自由能与状态方程,满意地描述了Lennard Jones模型的相平衡;通过选取合适的LJ势参数,能很好地描述了真实分子的融化曲线.     

Modelling of the influence of dihedral angle,volume fractions,particle size and coordination on the driving forces for sintering of dual phase systems

The equilibrium shape of solid particles in an aggregate immersed in a liquid or in a gas results from the minimization of interface energy. A model is developed for expressing the dependence of the solid–solid and solid–second phase interface areas on the system parameters: phase volume fractions, dihedral angle, particle size and coordination. The model aims at allowing quantitative assessment of the role of these parameters on the driving force for sintering. The representative volume element is a cone of which the apex angle accounts for the average particle coordination. In order to comply with the uniformity of interface curvature, the solid–second phase interfaces are described using the mathematics of the Delaunay surfaces. The results are compared with the solutions obtained by approximating the interface shape by the revolution of an arc of circle around the cone axis. This approximation does not involve a significant loss of precision.     

Density functional theory of vapor to liquid heterogeneous nucleation: Lennard–Jones fluid on solid substrate

Density functional theory has been applied to investigate the vapor to liquid heterogeneous nucleation on a flat solid surface, by invoking a model free energy density functional along with an exponential density model. The effects of supersaturation of the vapor and the strength of the solid-fluid interaction on the nucleation barrier have been investigated for Lennard–Jones fluid with 12–6 fluid–fluid and 9–3 solid–fluid interaction model. The spinodal decomposition of vapor has been observed at higher supersaturation or at higher strength of the solid–fluid interaction. The shape, density profile and the free energy of formation of droplets of any arbitrary size have been obtained in this work.     

Adhesion of fluid vesicles at chemically structured substrates

The adhesion of fluid vesicles at chemically structured substrates is studied theoretically via Monte Carlo simulations. The substrate surface is planar and repels the vesicle membrane apart from a single surface domain γ , which strongly attracts this membrane. If the vesicle is larger than the attractive γ domain, the spreading of the vesicle onto the substrate is restricted by the size of this surface domain. Once the contact line of the adhering vesicle has reached the boundaries of the γ domain, further deflation of the vesicle leads to a regime of low membrane tension with pronounced shape fluctuations, which are now governed by the bending rigidity. For a circular γ domain and a small bending rigidity, the membrane oscillates strongly around an average spherical cap shape. If such a vesicle is deflated, the contact area increases or decreases with increasing osmotic pressure, depending on the relative size of the vesicle and the circular γ domain. The lateral localization of the vesicle's center of mass by such a domain is optimal for a certain domain radius, which is found to be rather independent of adhesion strength and bending rigidity. For vesicles adhering to stripe-shaped surface domains, the width of the contact area perpendicular to the stripe varies nonmonotonically with the adhesion strength.     

Theory of monolayers with boundaries: exact results and perturbative analysis

Domains and bubbles in tilted phases of Langmuir monolayers contain a class of textures known as boojums. The boundaries of such domains and bubbles may display either cusplike features or indentations. We derive analytic expressions for the textures within domains and surrounding bubbles, and for the shapes of the boundaries of these regions. The derivation is perturbative in the deviation of the bounding curve from a circle. This method is not expected to be accurate when the boundary suffers large distortions, but it does provide important clues with regard to the influence of various energetic terms on the order-parameter texture and the shape of the domain or bubble bounding curve. We also look into the effects of thermal fluctuations, which include a sample-size-dependent effective line tension.     

Surface tension driven flow on a thin reaction front

Surface tension driven convection affects the propagation of chemical reaction fronts in liquids. The changes in surface tension across the front generate this type of convection. The resulting fluid motion increases the speed and changes the shape of fronts as observed in the iodate-arsenous acid reaction. We calculate these effects using a thin front approximation, where the reaction front is modeled by an abrupt discontinuity between reacted and unreacted substances. We analyze the propagation of reaction fronts of small curvature. In this case the front propagation equation becomes the deterministic Kardar-Parisi-Zhang (KPZ) equation with the addition of fluid flow. These results are compared to calculations based on a set of reaction-diffusion-convection equations.     

基于有限体积虚拟区域方法的两相流动直接模拟

为提高计算效率，提出有限体积法离散下的虚拟区域颗粒两相流动直接模拟方法.在控制方程中加入相应的虚拟区域源项，保证了颗粒内部的刚体运动特性.该源项中含有颗粒信息部分及流体信息部分.在每次迭代后，对源项中的流体信息部分进行更新，从而更好地保证颗粒内速度的刚体分布.计算静止颗粒圆柱绕流及单个颗粒的沉降过程，验证了算法的准确性.     

A molecular model for the line tension of lipid membranes

The line tension of a symmetric, lipid bilayer in its liquid-crystalline state is calculated on the basis of a molecular lipid model. The lipid model extends the opposing forces model by an expression for the conformational free energy of the hydrocarbon chains. We consider a membrane edge that consists of a perturbed bilayer covered by a section of a cylinder-like micelle. The structural rearrangement of the lipids implies an excess free energy which we minimize with respect to the cross-sectional shape of the membrane edge, including both the micellar and the bilayer region. The line tension is derived as a function of molecular lipid properties, like the lipid chain length or the head group interaction strength. We also relate it to the spontaneous curvature of the lipid layer. We find the line tension to become smaller for lipid layers that tend to curve more towards the hydrophobic core. Our predictions for the line tension and their relation to experimentally derived values are discussed. Received 2 January 2000     

A simplified approximate analytical model for Rayleigh–Taylor instability in elastic–plastic solid and viscous fluid with thicknesses

A simplified theoretical model for the linear Rayleigh-Taylor instability of finite thickness elastic-plastic solid constantly accelerated by finite thickness viscous fluid is performed.With the irrotational assumption,it is possible to consider viscosity,surface tension,elasticity or plasticity effects simultaneously.The model considers thicknesses at rigid wall boundary conditions with the velocity potentials,and deals with solid elastic-plastic transition and fluid viscosity based on the velocity continuity and force equilibrium at contact interface.The complete analytical expressions of the amplitude motion equation,the growth rate,and the instability boundary are obtained for arbitrary Atwood number,viscosity,thicknesses of solid and fluid.The thicknesses effects of two materials on the growth rate and the instability boundary are discussed.     

Atomistic investigation of phase transition in solid argon induced by shock wave transmission

A molecular dynamics (MD) simulation is employed to study the phase transition process in argon induced by shock wave transmission. Deriving the relation between the shock and piston velocities, the theoretical equation of state for argon is presented. Also, argon equation of state is obtained by measuring the quantities directly from simulations to be able to detect the phase transitions. The phase transition is also detected by using argon phase diagram and free energy calculations. A comparison shows good agreement between the theoretical and MD results for the phase transitions. Based on these simulations, it is concluded that under a shock wave transmission with suitable energy, the solid argon experiences a phase transition from solid to liquid and another from liquid to supercritical fluid. By reflecting the shock wave back at the end of its passage, the whole argon may reach the supercritical state.     

微通道内不混溶气液两相二氧化碳界面动力学行为的格子Boltzmann研究

将高精度的二氧化碳状态方程与气液两相流格子Boltzmann方法中的伪势模型耦合,研究微通道内二氧化碳气液两相流动的界面动力学行为,包括二氧化碳气泡和液滴的分裂、合并、变形,以及气液两相二氧化碳在演化过程中的质量交换.研究发现：当分裂和合并行为达到平衡,并且两相之间不发生质量交换时流动达到稳态.稳态时的流型主要依赖于表面张力,惯性力,管道的润湿性,以及初始体积分数.当表面张力较大时,微通道内形成的二氧化碳气泡或液滴会收缩成圆形,此时二氧化碳气泡或液滴会堵塞微通道,形成段塞流;随着表面张力的减小,形成的气泡或液滴不容易收缩,在微通道内更容易发生变形,出现泡状流或环状流.当壁面润湿性为强疏水性时,二氧化碳在微通道中的流动为环状流,其它润湿性下,流型为段塞流.体积分数较小时,二氧化碳两相流动的流型为段塞流,体积分数较大时,流型为环状流.     

Thermodynamics of graphene

The 21st century has brought a lot of new results related to graphene. Apparently, graphene has been characterized from all points of view except surface science and, especially, surface thermodynamics. This report aims to close this gap. Since graphene is the first real two-dimensional solid, a general formulation of the thermodynamics of two-dimensional solid bodies is given. The two-dimensional chemical potential tensor coupled with stress tensor is introduced, and fundamental equations are derived for energy, free energy, grand thermodynamic potential (in the classical and hybrid forms), enthalpy, and Gibbs energy. The fundamentals of linear boundary phenomena are formulated with explaining the concept of a dividing line, the mechanical and thermodynamic line tensions, line energy and other linear properties with necessary thermodynamic equations. The one-dimensional analogs of the Gibbs adsorption equation and Shuttleworth–Herring relation are presented. The general thermodynamic relationships are illustrated with calculations based on molecular theory. To make the reader sensible of the harmony of chemical and van der Waals forces in graphene, the remake of the classical graphite theory is presented with additional variable combinations of graphene sheets. The calculation of the line energy of graphene is exhibited including contributions both from chemical bonds and van der Waals forces (expectedly, the latter are considerably smaller than the former). The problem of graphene holes originating from migrating vacancies is discussed on the basis of the Gibbs–Curie principle. An important aspect of line tension is the planar sheet/nanotube transition where line tension acts as a driving force. Using the bending stiffness of graphene, the possible radius range is estimated for achiral (zigzag and armchair) nanotubes.     

Domain shape relaxation and local viscosity in stratifying foam films

We studied the dynamics of two different types of domain shape relaxation in a stratifying foam film composed of an anionic polymer and cationic surfactant. Those films thin in stepwise fashion: circular domains of lower film thickness are formed, expand and coalesce until they cover the whole film surface. We found that the shape relaxation of coalescing domains is governed only by 2D dissipation, and the measurement of the time scales allows to determine the ratio between the driving force (line tension) and local film viscosity. Further, we analyzed the withdrawal of stripes and modeled it by a moving disc pulled by an external force. Here, 3D dissipation can not be neglected (Stokes paradox) and the equilibrium velocity depends logarithmically on the viscosity of the surrounding 3D air. The evaluation of both kinds of relaxation events yields the orders of magnitude of film viscosity and line tension. For the investigated system we found that the film viscosity is at least 30 times larger than the bulk viscosity, which can be explained by the local molecular ordering and strong interactions with film surfaces.     

A coupled solid–fluid method for modelling subduction

We present a novel dynamic approach for solid–fluid coupling by joining two different numerical methods: the boundary-element method (BEM) and the finite element method (FEM). The FEM results describe the thermomechanical evolution of the solid while the fluid is solved with the BEM. The bidirectional feedback between the two domains evolves along a Lagrangian interface where the FEM domain is embedded inside the BEM domain. The feedback between the two codes is based on the calculation of a specific drag tensor for each boundary on finite element. The approach is presented here to solve the complex problem of the descent of a cold subducting oceanic plate into a hot fluid-like mantle. The coupling technique is shown to maintain the proper energy dissipation caused by the important secondary induced mantle flow induced by the lateral migrating of the subducting plate. We show how the method can be successfully applied for modelling the feedback between deformation of the oceanic plate and the induced mantle flow. We find that the mantle flow drag is singular at the edge of the retreating plate causing a distinct hook shape. In nature, such hooks can be observed at the northern end of the Tonga trench and at the southern perimeter, of the South American trench.     

Raman spectroscopy for study of interplay between phonon confinement and Fano effect in silicon nanowires

A combined effect of doping (type and species) and size on Raman scattering from silicon (Si) nanowires (NWs) has been presented here to study interplay between quantum confinement and Fano effects. The SiNWs prepared from low doping Si wafers show only confinement effect, as evident from the asymmetry in the Raman line‐shape, irrespective of the doping type. On the other hand SiNWs prepared from wafer with high doping shows the presence of electron–phonon interaction in addition to the phonon confinement effect as revealed from the presence of asymmetry and antiresonence in the corresponding Raman spectra. This combined effect induces an extra asymmetry in the lower energy side of Raman peak for n‐type SiNWs whereas the asymmetry flips from lower energy side to the higher energy side of the Raman peak in p‐type SiNWs. Such an interplay can be represented by considering a general Fano‐Raman line‐shape equation to take care of the combined effect in SiNWs. Copyright © 2015 John Wiley & Sons, Ltd.