A phase field model for failure in interconnect lines due to coupled diffusion mechanisms

We describe a diffuse interface, or phase field model for simulating electromigration and stress-induced void evolution and growth in interconnect lines. Microstructural evolution is tracked by defining an order parameter, which takes on distinct uniform values within solid material and voids, and varying rapidly from one to the other over narrow interfacial layers associated with the void surfaces. The order parameter is governed by a form of the Cahn-Hilliard equation. An asymptotic analysis demonstrates that the zero contour of order parameter tracks the motion of a void evolving by coupled surface and lattice diffusion, driven by stress, electron wind and vacancy concentration gradients. Efficient finite element schemes are described to solve the modified Cahn-Hilliard equation, as well as the equations associated with the accompanying mechanical, electrical and bulk diffusion problems. The accuracy and convergence of the numerical scheme is investigated by comparing results to known analytical solutions. The method is applied to solve various problems involving void growth and evolution in representative interconnect geometries.     

A finite element method for free surface flows of incompressible fluids in three dimensions. Part II. Dynamic wetting lines

To date, few researchers have solved three‐dimensional free surface problems with dynamic wetting lines. This paper extends the free surface finite element method (FEM) described in a companion paper [Cairncross RA, Schunk PR, Baer TA, Sackinger PA, Rao RR. A finite element method for free surface flows of incompressible fluid in three dimensions. Part I. Boundary fitted mesh motion. International Journal for Numerical Methods in Fluids 2000; 33 : 375–403] to handle dynamic wetting. A generalization of the technique used in two‐dimensional modeling to circumvent double‐valued velocities at the wetting line, the so‐called kinematic paradox, is presented for a wetting line in three dimensions. This approach requires the fluid velocity normal to the contact line to be zero, the fluid velocity tangent to the contact line to be equal to the tangential component of web velocity, and the fluid velocity into the web to be zero. In addition, slip is allowed in a narrow strip along the substrate surface near the dynamic contact line. For realistic wetting line motion, a contact angle that varies with wetting speed is required because contact lines in three dimensions typically advance or recede at different rates depending upon location and/or have both advancing and receding portions. The theory is applied to capillary rise of static fluid in a corner, the initial motion of a Newtonian droplet down an inclined plane, and extrusion of a Newtonian fluid from a nozzle onto a moving substrate. The extrusion results are compared with experimental visualization. Copyright © 2000 John Wiley & Sons, Ltd.     

Discontinuous Galerkin finite element method applied to the coupled unsteady Stokes/Cahn-Hilliard equations

Two-phase flows driven by the interfacial dynamics are studied by tracking implicitly interfaces in the framework of the Cahn-Hilliard theory. The fluid dynamics is described by the Stokes equations with an additional source term in the momentum equation taking into account the capillary forces. A discontinuous Galerkin finite element method is used to solve the coupled Stokes/Cahn-Hilliard equations. The Cahn-Hilliard equation is treated as a system of two coupled equations corresponding to the advection-diffusion equation for the phase field and a nonlinear elliptic equation for the chemical potential. First, the variational formulation of the Cahn-Hilliard equation is presented. A numerical test is achieved showing the optimal order in error bounds. Second, the variational formulation in discontinuous Galerkin finite element approach of the Stokes equations is recalled, in which the same space of approximation is used for the velocity and the pressure with an adequate stabilization technique. The rates of convergence in space and time are evaluated leading to an optimal order in error bounds in space and a second order in time with a backward differentiation formula at the second order. Numerical tests devoted to two-phase flows are provided on ellipsoidal droplet retraction, on the capillary rising of a liquid in a tube, and on the wetting drop over a horizontal solid wall.     

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列举了自然界中的种种现象以证明最小作用量的普适性. 同时运用最小作用量原理阐述了表 面浸润现象的本质,并指出能量提法与微分提法对于浸润现象的等价性关系. 运用最小作用 量原理合理地解释了诸如荷叶的超疏水、水黾水面的自由跑跳、液滴的定向运动以 及毛细黏 附等物理现象. 本文也对最小作用量原理的哲学意义作了一些探讨.     

Comparison of a Rapidely Converging Phase Field Model for Interfaces in Solids with the Allen-Cahn Model

We compare two phase field models for interfaces in elastic solids carrying low surface energy. One model has hybrid properties of a Hamilton-Jacobi and a parabolic equation, the other is the Allen-Cahn model. For vanishing width of the interface we construct asymptotic solutions of second order for the hybrid model and of first order for the Allen-Cahn model. These constructions show that the width of the diffused interface necessary for tracking accurately the sharp interface can be chosen much larger for the hybrid model than for the Allen-Cahn model, and that moreover the hybrid model can describe interfaces with nonlinear kinetic relation. This explains why numerical simulations based on the hybrid model are considerably more effective. These simulations are discussed in the last section.     

离散型织构表面液滴的铺展及其接触线的力学特性分析

针对离散型织构表面上液滴的铺展过程,采用数值模拟和润湿性实验相结合的方法,引入织构润湿因子θ*,得到了不同类型的离散型织构对固体表面润湿性的影响,在此基础之上分析了液滴铺展过程中接触线的力学特性,以期从微观界面力学的角度解释微织构对液滴铺展过程的促进作用.研究表明:离散型织构增大了液滴铺展过程中的固-液接触面积,位于铺展前沿的液体分子部分浸润织构内部,导致液面曲率和液滴内部的拉普拉斯压力增大,相邻离散型织构间的液体获得了额外的驱动力和能量,铺展速度加快,平衡接触角减小;槽状离散型织构对表面润湿性的影响程度最大,液滴在其上铺展过程具有各向异性特性.另外数值仿真分析表明,接触线的钉扎效应与固体表面粗糙度的大小和微织构类型密切相关,表面粗糙度越大,钉扎效应越明显,其中槽状织构对接触线的钉扎作用还具有方向性.      

Finite element framework for describing dynamic wetting phenomena

The finite element simulation of dynamic wetting phenomena, requiring the computation of flow in a domain confined by intersecting a liquid–fluid free surface and a liquid–solid interface, with the three‐phase contact line moving across the solid, is considered. For this class of flows, different finite element method (FEM) implementations have been used in the literature, and in some cases, these produced apparently contradictory results. In the present paper, a robust framework for the FEM simulation of dynamic wetting flows is developed, which, by consistently adhering to the FEM methodology, leaves no room for ad hoc ‘optional’ variations in the numerical handling of these flows. The developed approach makes it possible to conduct a convergence study, assess the spatial resolution required to achieve a preset accuracy and provide the corresponding benchmark calculations. This analysis allows one to identify numerical artefacts, which had previously been interpreted as physical effects, and demonstrates that suppressing numerical errors using a ‘strong’ implementation of a boundary condition creates bigger and less detectable errors elsewhere in the computational domain. We provide practical recommendations on the spatial resolution required by a numerical scheme for a given set of non‐dimensional similarity parameters and give a user‐friendly step‐by‐step guide specifying the entire implementation, which allows the reader to easily reproduce all presented results including the benchmark calculations. It is also shown how the developed framework accommodates generalizations of the mathematical model accounting for additional physical effects, such as gradients in surface tensions. Copyright © 2011 John Wiley & Sons, Ltd.     

Diffusion of suspended particles in an isotropic turbulence field

A model for turbulent motion is proposed which makes it possible to evaluate the pulsation characteristics and the diffusion coefficients of the dispersed phase and also makes it possible to describe the effect of the suspended particles on the turbulence of the dispersing medium. Specific calculations are made for the situation when the undisturbed turbulent field is isotropic.The diffusion of an admisture having inertia in a turbulent stream has been studied previously on the assumption that the three-dimensional turbulence characteristics have practically no effect on the behavior of the suspended particles, so that the random motion of the latter is described by ordinary differential equations containing the natural independent variable the motion travel time [1–4]. In many cases this assumption is incorrect and the corresponding theory is obviously deficient. For example, a fundamental result of this theory, asserting that the turbulent diffusion coefficients of the particles and of the fluid moles are equal for a long diffusion time, is obviously incorrect if the relative motion of the particles is significant [5].     

梯度润湿表面上液滴定向迁移及合并行为的格子Boltzmann模拟

采用改进的格子Boltzmann方法,对梯度润湿性表面上液滴的定向迁移及合并行为进行了数值模拟,该模型在精度和稳定性上都有很大改善,同时,研究了梯度润湿性表面上液滴定向迁移和合并的动力学特性,并对液滴尺寸及润湿梯度对液滴动力学特性的影响规律进行了分析。数值结果表明,液滴在梯度润湿性表面运动时会发生形变,且动态接触角逐渐减小。润湿梯度对液滴定向迁移行为有显著影响,润湿梯度越大,液滴左右侧接触线位移越大,润湿长度增加越快。但是液滴尺寸对接触线位移影响较小。润湿梯度对液桥宽度基本无影响,但对液滴初始合并时间有显著影响。     

疏液壁面上亲液杂质对滑移特性的影响

纳米流动系统具有高效、经济等优势,在众多领域具有广泛的应用前景.因该类系统具有极高的表面积体积比,致使界面滑移效应对流动具有显著影响.论文采用非平衡分子动力学模拟方法,研究了疏液壁面表层混入少量亲液原子时纳米通道内液体的滑移特性,并基于分子动理论解释了其影响机制.研究发现,亲液杂质(均布或集中)对液体法向密度振荡程度影响较弱,但会显著改变壁面附近类固体层的分布和液体滑移规律;随亲液杂质占比增加,液体类固体现象更趋明显,壁面处液体接触密度也线性增大,但通道内液体平均速度逐渐降低,滑移长度也迅速减小;相比于集中的亲液杂质,均匀分布亲液杂质对滑移的弱化效应更强.如当亲液杂质占比为28%时,其滑移长度比单纯疏液表面的降低率从50%扩大至56%.基于分子动理论的分析发现,亲液杂质会导致杂质原子附近第一液体层内的原子发生跃迁的能垒加大,即弱化了液体原子的流向跃迁行为,从而降低了滑移量;相比于集中杂质,均匀分布的杂质还会降低固液间的非公度性,致使滑移特性破坏更严重.     

Zero mass error using unsteady wetting–drying conditions in shallow flows over dry irregular topography

A wetting–drying condition (WDC) for unsteady shallow water flow in two dimensions leading to zero numerical error in mass conservation is presented in this work. Some applications are shown which demonstrate the effectiveness of the WDC in flood propagation and dam break flows over real geometries. The WDC has been incorporated into a cell centred finite volume method based on Roe's approximate Riemann solver across the edges of both structured and unstructured meshes. Previous wetting–drying condition based on steady‐state conditions lead to numerical errors in unsteady cases over configurations with strong variations on bed slope. A modification of the wetting–drying condition including the normal velocity to the cell edge enables to achieve zero numerical errors. The complete numerical technique is described in this work including source terms discretization as a complete and efficient 2D river flow simulation tool. Comparisons of experimental and numerical results are shown for some of the applications. Copyright © 2004 John Wiley & Sons, Ltd.     

Conformal curvature flows: From phase transitions to active vision

In this paper, we analyze geometric active contour models from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new feature-based Riemannian metrics. This leads to a novel edge-detection paradigm in which the feature of interest may be considered to lie at the bottom of a potential well. Thus an edge-seeking curve is attracted very naturally and efficiently to the desired feature. Comparison with the Allen-Cahn model clarifies some of the choices made in these models, and suggests inhomogeneous models which may in return be useful in phase transitions. We also consider some 3-dimensional active surface models based on these ideas. The justification of this model rests on the careful study of the viscosity solutions of evolution equations derived from a level-set approach.     

Interface motion by interface diffusion driven by bulk energy: justification of a diffusive interface model

We construct an asymptotic solution of a system consisting of the partial differential equations of linear elasticity theory coupled with a degenerate parabolic equation, and show that when a regularity parameter tends to zero, this solution converges to a solution of a sharp interface model, which describes the phase interface in an elastically deformable solid moving by interface diffusion. Therefore, the coupled system can be used as diffusive interface model. Differently from diffusive interface models based on the Cahn–Hilliard equation, the interface diffusion is solely driven by the bulk energy, hence the Laplacian of the curvature is not part of the driving force. Also, no rescaling of the parabolic equation is necessary. Since the asymptotic solution does not solve the system exactly, the proof is formal.     

A Variational Model of Disjoining Pressure: Liquid Film on a Nonplanar Surface

Variational methods have been successfully used in modelling thin liquid films in numerous theoretical studies of wettability. In this article, the variational model of the disjoining pressure is extended to the general case of a two-dimensional solid surface. The Helmholtz free energy functional depends both on the disjoining pressure isotherm and on the shape of the solid surface. The augmented Young–Laplace equation (AYLE) is a nonlinear second-order partial differential equation. A number of solutions describing wetting films on spherical grains have been obtained. In the case of cylindrical films, the phase portrait technique describes the entire variety of mathematically feasible solutions. It turns out that a periodic solution, which would describe wave-like wetting films, does not satisfy Jacobi’s condition of the classical calculus of variations. Therefore, such a solution is nonphysical. The roughness of the solid surface significantly affects liquid film stability. AYLE solutions suggest that film rupture is more likely at a location where the pore-wall surface is most exposed into the pore space, and the curvature is positive.     

Relative permeabilities of a near critical binary fluid

Relative permeabilities were measured at very low interfacial tensions (IFT) for two-phase mixtures of methanol and hexane flowing through Clashach sandstone. These two components pass from a two- to a single-phase system as the temperature is increased above the critical solution temperature (CST). The interfacial tension between the coexisting phases approaches zero as the solution reaches miscibility. The phase behaviour of methanol and hexane mixtures has been well characterised allowing the calculation of relative permeabilities, saturations and capillary numbers. Flow data are reported for four different temperatures in the two-phase region (i.e., four values of IFT and capillary number). The capillary desaturation curve (CDC) for the strongly wetting methanol rich phase is also presented. In addition to the novel technique presented for measurement of relative permeability, the results indicate that relative permeabilities approach straight line functions very near the critical point. Furthermore, desaturation of the wetting phase was found to be dependent on the capillary number which, in turn, depends on the location of the mixture on the fluid phase diagram and the proximity to the critical temperature.     

Diffuse-interface Modeling of Two-phase Flow for a One-component Fluid in a Porous Medium

The diffuse-interface (DI) model for the two-phase flow of a one-component fluid in a porous medium has been presented by Papatzacos [2002, Transport Porous Media 49, 139–174] and by Papatzacos and Skjæveland [2004, SPE J. (March 2004), 47–56]. Its main characteristics are: (i) a unified treatment of two phases as manifestations of one fluid with a van der Waals type equation of state, (ii) the inclusion of wetting, and (iii) the absence of relative permeabilities. The present paper completes the presentation by including the implementation of wetting in the general case of a mixed-wet rock. As a result of this implementation, some statements are made about capillary pressure, confirming similar statements by Hassanizadeh and Gray [1993, Water Resour. Res. 29, 3389–3405]. As an application of the model, we show that relative permeabilities depend on the spatial derivatives of the saturation.     

A comprehensive study on the behavior of a rigid block on an oscillating ground with friction,elastic and viscous forces

This paper investigates the behavior of a non-linear mechanical model where a block is driven by an oscillating ground through Coulomb friction, a linear viscous damper and a linear spring. The governing equation is solved analytically for different partial configurations: friction only, friction with viscous damping, friction with a linear restoring force, and for the complete model. Using dimensionless groups, the analysis of the block motion provides a comprehensive set of information on the motion regime (stick, stick-slip or permanent sliding), on the dominant energies or forces, on the resonance and on the amplification of the ground oscillation by the system. The limit between the stick-slip regime and the permanent slipping regime is found either analytically or numerically. It is also shown that there exists a set of parameters for which the friction force, the viscous dissipative force and the elastic restoring force are equal.     

Substrate elastic deformation due to vertical component of liquid-vapor interfacial tension

Young’s equation is a fundamental equation in capillarity and wetting,which reflects the balance of the horizontal components of the three interfacial tensions with the contact angle(CA).However,it does not consider the vertical component of the liquid-vapor interfacial tension(VCLVIT).It is now well understood that the VCLVIT causes the elastic deformation of the solid substrate,which plays a significant role in the fabrication of the microfluidic devices because of the wide use of the soft materials.In this paper,the theoretical,experimental,and numerical aspects of the problem are reviewed.The effects of the VCLVIT-induced surface deformation on the wetting and spreading,the deflection of the microcantilever,and the elasto-capillarity and electroelasto-capillarity are discussed.Besides a brief review on the historical development and the recent advances,some suggestions on the future research are also provided.     

Comparative study of two lattice Boltzmann multiphase models for simulating wetting phenomena: implementing static contact angles based on the geometric formulation

Wetting phenomena are widespread in nature and industrial applications. In general, systems concerning wetting phenomena are typical multicomponent/multiphase complex fluid systems. Simulating the behavior of such systems is important to both scientific research and practical applications. It is challenging due to the complexity of the phenomena and difficulties in choosing an appropriate numerical method. To provide some detailed guidelines for selecting a suitable multiphase lattice Boltzmann model, two kinds of lattice Boltzmann multiphase models, the modified S-C model and the H-C-Z model, are used in this paper to investigate the static contact angle on solid surfaces with different wettability combined with the geometric formulation (Ding, H. and Spelt, P. D. M. Wetting condition in diffuse interface simulations of contact line motion. Physical Review E, 75(4), 046708 (2007)). The specific characteristics and computational performance of these two lattice Boltzmann method (LBM) multiphase models are analyzed including relationship between surface tension and the control parameters, the achievable range of the static contact angle, the maximum magnitude of the spurious currents (MMSC), and most importantly, the convergence rate of the two models on simulating the static contact angle. The results show that a wide range of static contact angles from wetting to non-wetting can be realized for both models. MMSC mainly depends on the surface tension. With the numerical parameters used in this work, the maximum magnitudes of the spurious currents of the two models are on the same order of magnitude. MMSC of the S-C model is universally larger than that of the H-C-Z model. The convergence rate of the S-C model is much faster than that of the H-C-Z model. The major foci in this work are the frequently-omitted important details in simulating wetting phenomena. Thus, the major findings in this work can provide suggestions for simulating wetting phenomena with LBM multiphase models along with the geometric formulation.