Surface morphological response of crystalline solids to mechanical stresses and electric fields

Surface morphological evolution under the action of external fields is a fascinating topic that has attracted considerable attention within the surface science community over the past two decades. In addition to the interest in a fundamental understanding of field-induced nonlinear response and stability of surface morphology, the problem has been technologically significant in various engineering applications such as microelectronics and nanofabrication. In this report, we review theoretical progress in modeling the surface morphological response of stressed elastic solids under conditions that promote surface diffusion and of electrically conducting solids under surface electromigration conditions. A self-consistent model of surface transport and morphological evolution is presented that has provided the basis for the theoretical and computational work that is reviewed. According to this model, the surface morphological response of electrically conducting elastic solids to the simultaneous action of mechanical stresses and electric fields is analyzed. Emphasis is placed on metallic surfaces, including surfaces of voids in metallic thin films.Surfaces of stressed elastic solids are known to undergo morphological instabilities, such as the Asaro–Tiller or Grinfeld (ATG) instability that leads to emanation of crack-like features from the surface and their fast propagation into the bulk of the solid material. This instability is analyzed theoretically, simulated numerically, and compared with experimental measurements. The surface morphological evolution of electrically conducting, single-crystalline, stressed elastic solids under surface electromigration conditions is also examined. We demonstrate that, through surface electromigration, a properly applied and sufficiently strong electric field can stabilize the surface morphology of the stressed solid against both crack-like ATG instabilities and newly discovered secondary rippling instabilities; the effects of important parameters, such as surface crystallographic orientation, on the surface morphological response to the simultaneous action of an electric field and mechanical stress also are reviewed. In addition, electromigration-driven surface morphological response is analyzed systematically, focusing on the current-driven surface morphological evolution of voids in metallic thin films; this analysis has been motivated largely by the crucial role of void dynamics in determining the reliability of metallic interconnects in integrated circuits and has led to the interpretation of a large body of experimental observations and measurements. The electromigration-driven translational motion of morphologically stable voids, effects of current-driven void dynamics on the evolution of the electrical resistance of metallic thin films, and current-driven void–void interactions also are reviewed. Furthermore, theoretical studies are reviewed that demonstrated very interesting current-driven nonlinear void dynamics in stressed metallic thin films, including the inhibition of electromigration-induced instabilities due to the action of biaxial tensile stress, and stress effects on the electromigration-driven translational motion of morphologically stable voids.Complex, oscillatory surface states under surface electromigration conditions have been observed in numerical studies. In this report, emphasis is placed on void surfaces in metallic thin films, for which stable time-periodic states have been demonstrated. It is shown that increasing parameters such as the electric-field strength or the void size past certain critical values leads to morphological transitions from steady to time-periodic states; the latter states are characterized by wave propagation on the surface of a void that migrates along the metallic film at constant speed. The transition onset corresponds to a Hopf bifurcation that may be either supercritical or subcritical, depending on the symmetry of the surface diffusional anisotropy as determined by the crystallographic orientation of the film plane. It is also shown that, in the case where the Hopf bifurcation is subcritical, the simultaneous action of mechanical stress leads the current-driven void morphological response to the stabilization of chaotic attractors; in such cases, as the applied stress level increases, the void dynamics is set on a route to chaos through a sequence of period-doubling bifurcations. The observation of current-driven chaotic dynamics in homoepitaxial islands also is discussed.     

High-temperature field evaporation and its connection with surface ionization

High-temperature field evaporation of metals and alloys and its connection with surface ionization are considered. The main parameters of the evaporation process (dependence of the evaporation rate on the emitter temperature and on the electric field at the emitter surface, the charge of the ions being evaporated and its temperature dependence, kinetic parameters of the evaporation process, as well as the state of the emitter surface under simultaneous action of high fields and temperatures) are analyzed. The similarity and the difference between field evaporation at high temperatures and surface ionization in a strong electric field are determined.     

Durability of polycrystalline ferroelectric ceramics in a constant electric field

The kinetics of mechanical fracture and electric breakdown of polycrystalline ferroelectric ceramics under simultaneous action of an electric field and mechanical load have been investigated. The procedures for determining the activation characteristics of both processes from only the fracture kinetics in one of them have been developed.     

A phase field method for simulating morphological evolution of vesicles in electric fields

A phase field method is developed to investigate the morphological evolution of a vesicle in an electric field, taking into account coupled mechanical and electric effects such as bending, osmotic pressure, surface tension, flexoelectricity, and dielectricity of the membrane. The energy of the system is formulated in terms of a continuous phase field variable and the electric potential. The governing equations of the phase field and the electric field are solved using the Galerkin weighted residual approach. The validation and robustness of the algorithm are verified by direct comparisons of the obtained numerical solutions with relevant experimental results. The morphological evolution of an axisymmetric vesicle under an electric field is studied in detail. The results demonstrate that the present method can simulate complex morphological evolutions of vesicles under coupled mechanical–electrical fields.     

Analysis of a combined influence of substrate wetting and surface electromigration on a thin film stability and dynamical morphologies

A PDE-based model combining surface electromigration and wetting is developed for the analysis of morphological stability of ultrathin solid films. Adatom mobility is assumed anisotropic, and two directions of the electric field (parallel and perpendicular to the surface) are discussed and contrasted. Linear stability analyses of small-slope evolution equations are performed, followed by computations of fully nonlinear parametric evolution equations that permit surface overhangs. The results reveal parameter domains of instability for wetting and non-wetting films and variable electric field strength, nonlinear steady-state solutions in certain cases, and interesting coarsening behavior for strongly wetting films.     

Investigation of piezoelectric effect in lithium tantalate crystals by high-resolution X-ray diffractometry

Lithium tantalate crystals under the action of an external electric field have been studied by x-ray diffractometry. Both single-domain samples and crystals with a regular domain structure have been investigated. In the case of a single-domain crystal, the angular position of the Bragg peak varies under the action of an external electric field due to the reverse piezoelectric effect. According to this variation, the corresponding tensor component of piezoelectric constants can be calculated. In the crystal with a regular domain structure, a surface relief appears under the action of an external electric field. Piezoelectric constants can also be calculated using this relief.     

Change in the intensity of light scattering in a magnetite colloid under the simultaneous action of electric and magnetic fields

The change in the intensity of light scattering in a colloidal solution of magnetite particles in kerosene under the simultaneous action of coaxial and mutually perpendicular electric and magnetic fields has been studied. It has been found that, at certain strengths of mutually perpendicular fields, the change in the scattering intensity is maximal. The effect observed has been interpreted on the basis of the Rayleigh-Debye-Gans anisotropic approximation within the model of orientation of aggregates of magnetite particles under the joint action of the fields.     

Electric strength of rocks at a pulsed voltage under high pressures at high temperatures

The electric strength of rocks (granite, limestone, and sandstone) for the first time has been measured under the simultaneous effect of the pressure up to 35 MPa and temperature up to 120°C in the system of rod-rod electrodes arranged on one sample surface and point-plane electrodes in the liquid medium of a drilling agent. With the simultaneous increase in pressure and temperature, the electric strength of rocks for point-plane electrodes continuously increases (especially rapidly in the pressure range of 10–24 MPa and temperature range of 35–85°C), while for rod-rod electrodes arranged on the same sample surface, the electric strength varies with a maximum at pressures of 5–12 MPa and temperatures 20–35°C.     

EHD convection in dielectric micropolar fluid layer

The onset of instability in a layer of dielectric micropolar fluid under the simultaneous action of an AC electric field and temperature gradient has been investigated. The dispersion relation has been derived and various critical values of non-dimensional Rayleigh number in the fluid layer have been determined. The influence of micropolar viscosity and electric Rayleigh number on the onset of convection has been analyzed. Thermal Rayleigh number has been computed for various values of electric Rayleigh number for the onset of instability. The stabilizing and destabilizing effects of electric Rayleigh number, micropolar viscosity and Prandtl number have been discussed.     

On the scattering of a metallic cumulative jet with an intense electric current pulse passing through it

The extension of metallic cumulative jets, through which an intense electric current pulse is passed, is numerically simulated in the framework of the model of a uniformly elongating cylindrical rod. Emphasis is on the radial scattering of the jet, which is observed after it escapes from the interelectrode gap. The influence of the magnetic energy accumulated in the jet??s elements under the action of the current is elucidated. For central parts of cumulative jets generated by charges from 50 to 150 mm in diameter, the distributions of the material density and velocity along the radius of the jet immediately after the current cutoff are obtained. These distributions suggest that the surface layer of the jet may separate out and scatter under the action of the current with the central portion of the jet remaining solid. As the current passing through the jet grows, the thickness of its disintegrating layer, which acquires a radial velocity from the axis, increases. Critical currents causing separation of the jet??s surface layer and its collapse are determined.     

Specific features of the dielectric response of heterogeneous systems with a polar matrix containing electrically active inclusions

A theoretical model has been proposed which makes it possible to introduce the correction into the complex permittivity of electrically active condensed systems that contain a polar liquid matrix and low-dimensional solid particles with a developed electrically active surface. It has been established that there is an interfacial electrical interaction between surface charges of the solid component and polar molecules of the liquid matrix. The processes occurring during this interaction lead to the appearance of an intrinsic internal electric field in the system under investigation. The contribution from surface effects to the formation of the polarized state of the polar liquid medium has been investigated in terms of the proposed model. The possibility of controlling the processes of local change in the structure of the polar liquid component under the effect of the internal electric field by varying the electrically active specific surface area of the solid phase and the intrinsic dipole moment of molecules of the liquid matrix has been analyzed in the studied systems. The conditions providing for the appearance of a controlled potential gradient of the internal electric field have been determined for the dispersed systems under consideration. The parameter that makes it possible to evaluate the contribution from the interfacial electrical contact interaction to the dielectric response of the liquid component and the entire system has been introduced for the first time.     

A multidimensional pendulum in a nonconservative force field under the presence of linear damping

A nonconservative force field in the dynamics of a multidimensional solid is constructed according to the results from the dynamics of real solids occurring in the force field of the action of the medium. In this case, it becomes possible to generalize the equations of motion of a multidimensional solid in a similarly constructed field of forces and to obtain a complete list of, generally speaking, transcendental first integrals expressed through a finite combination of elementary functions. In the study, the integrability in elementary functions is shown for the simultaneous equations of motion of a dynamically symmetric fixed multidimensional solid under the action of a nonconservative pair of forces in the presence of the linear damping moment (the additional dependence of the force field on the tensor of angular velocity of the solid).     

Kinetics of failure of polycrystalline ferroelectric ceramics in mechanical and electric fields

The kinetics of mechanical failure and electrical breakdown in polycrystalline ferroelectric ceramics was studied under the simultaneous action of an electric field and a mechanical load. A kinetic approach is shown to be preferable as compared to the concepts that treat the failure and breakdown as critical phenomena. The mechanical failure and electrical breakdown are shown to be interrelated. It is found that a weak action of one of the fields retards failure caused by the other field and that the simultaneous action of these strong fields accelerates both the mechanical failure and electrical breakdown. Methods for determining the activation characteristics of both processes only from the failure kinetics in one of these processes are developed.     

The influence of the surface electron layer on the energy spectrum of photoelectrons

The influence of the self-consistent electric field of a surface electron layer on the energy spectrum of photoelectrons emitted under the action of a picosecond laser pulse on a metallic target was determined.     

Phase field simulation of single bubble behavior under an electric field

Based on the Cahn-Hilliard phase field model, a three-dimensional multiple-field coupling model for simulating the motion characteristics of a rising bubble in a liquid is established in a gas-liquid two-phase flow. The gas-liquid interface motion is simulated by using a phase-field method, and the effect of the electric field intensity on bubble dynamics is studied without electric field, or with vertical electric field or horizontal electric field. Through the coupling effect of electric field and flow field, the deformation of a single rising bubble and the formation of wake vortices under the action of gravity and electric field force are studied in detail. The correctness of the results is verified by mass conservation, and the influences of different electric field directions and different voltages on the movement of bubbles in liquid are considered. The results show that the ratio of the length to axis is proportional to the strength of the electric field when the air bubble is stretched into an ellipsoid along the electric field line under the action of electrostatic gravity and surface tension. In addition, the bubble rising speed is affected by the electric field, the vertical electric field accelerates the bubble rise, and the horizontal direction slows it down.     

Cathodic deposition of silver on silver bromide at microelectrodes

The cathodic deposition of silver metal on single crystalline silver bromide leads to deposits with different morphologies depending on the cathode geometry and the current density. The deposition at an extended planar electrode results in the growth of silver whiskers. The deposition at a point (micro-)electrode results in dendritic deposits on the surface of the solid electrolyte. The morphological development of the deposits is studied with microelectrodes of different diameters. A characteristic change from dendritic to whisker growth at microelectrodes upon time is always found. As the reason for the change of the growth mode from dendritic to whisker-type, the changing electric field distribution around a growing surface deposit is discussed.     

Visco-potential flows in electrohydrodynamics

In this study we consider the problem of the interface motion under the capillary–gravity and an external electric force. The infinitely deep fluid layer is assumed to be viscous, perfectly conducting and the flow to be incompressible. The weak viscous effects are introduced using the Helmholtz–Leray decomposition and the visco-potential flow approach. The electric charge distributions above and on the free surface are considered. Finally, we derive some linearized analytical solutions for the free surface elevation shape under the localized pressure distribution and the combined action of the forces mentioned herein above.     

Method for modifying the structure and elemental composition of the solid surface during a high-voltage vacuum discharge

A new method for modifying the surface of a solid, which makes it possible to change effectively the structure and elemental composition of the surface with a high precision, is developed and tested experimentally. The method is based on the action of the plasma of a pulsed high-voltage vacuum discharge, the ion beam from the plasma, and the electron beam on a solid target. The emission and plasma parameters are observed in a pulsed electric field produced in the diode system to which a pulsed voltage with an amplitude of ～10³–10⁵ V and a duration from 10^?9 to 10^?5 s is applied.     

Influence of the electric field on the alignment of carbon nanotubes during their growth and emission

The problems of the electric field action on carbon nanotubes (CNTs) during their growth and under the electron field emission conditions are considered. The relations determining the growth rate of an extended structure under the action of the electric field are established. The relation connecting the angle of orientation of a CNT inclined to the substrate surface and the applied electric field is used for computing current-voltage characteristics of the cathode consisting of inclined CNTs. The degree of deviation of these characteristics from the Fowler-Nordheim classic dependence is determined, on the one hand, by the parameters characterizing the CNT spread over the angles of inclination and, on the other hand, by the value of the Young modulus characterizing the bending stiffness of a nanotube. It is shown that in zero external electric field, a certain effect on the CNT orientation can be produced by the CNT potential relative to the substrate, which is due to the effect of the contact potential difference.     

微管道内两相流数值算法及在电浸润液滴控制中的应用

叙述了一种模拟电介质电润湿(electrowetting on dielectric，EWOD)下的微液滴的运动的数值方法. 采用二阶投影法求解N-S方程和level set 函数，并利用零level set函数俘获液滴运动界面，在液体与固体接触的边界上，通过引入动态接触角表征电介质表面润湿性随电势的改变. 数值计算基于MAC网格，模拟了2维微管道内与固体壁面接触的变润湿性的两种液体的分界面形状、平板上的微液滴在不同电势作用下处于不同湿润性的形态，以及微管道内改变接触角液滴的运动变形过程等算例. 关键词： 电浸润 接触角 level set函数 投影法