Variational treatment of electrostatic interaction force in atomic force microscopy

In this paper we introduce the mathematical model for the electrostatic interaction force between an atomic force microscope (AFM) tip and a sample surface. We formulate the electrostatic potential problem in Sobolev spaces and find the corresponding weak solution in terms of the integral potential, which can be approximated numerically by generalized Fourier series and used to find the interaction force between an AFM tip and a sample surface. The formulation of the problem in a weak (Sobolev) space setting allows us to determine the force for AFM tips of arbitrary shape. Efficiency of the method is illustrated using numerical examples for the spherical and tetrahedral AFM tips.      

Phase-Locked Loop design applied to frequency-modulated atomic force microscope

The atomic force microscope (AFM) introduced the surface investigation with true atomic resolution. In the frequency modulation technique (FM-AFM) both the amplitude and the frequency of oscillation of the micro-cantilever must be kept constant even in the presence of tip–surface interaction forces. For that reason, the proper design of the Phase-Locked Loop (PLL) used in FM-AFM is vital to system performance. Here, the mathematical model of the FM-AFM control system is derived considering high order PLL. In addition a method to design stable third-order Phase-Locked Loops is presented.     

Modeling of Electromechanical Contact on a Microscopic Lengthscale

This paper treats the simulation of the contact mode of an atomic force microscope (AFM). The contact forces between the tip and the sample surface is calculated via the van derWaals force in a Finite Element approach, the integration of the contact force a FEM environment is discussed and numerical examples are presented. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermoelastic damping effect analysis in micro flexural resonator of atomic force microscopy

In the design of high-Q micro/nano-resonators, dissipation mechanisms may have damaging effects on the quality factor (Q). One of the major dissipation mechanisms is thermoelastic damping (TED) that needs an accurate consideration for prediction. Aim of this paper is to evaluate the effect of TED on the vibrations of thin beam resonators. In particular, we will focus on cantilever beam resonator used in atomic force microscopy (AFM). AFM resonator is actually a cantilever with a spring attached to its free end. The end spring is considered to capture the effect of surface stiffness between tip and sample surface. The coupled governing equations of motion of thin beam with consideration of TED effects are derived. In general, there are four elastic equations that are coupled with thermal conduction equation. Based on accurate assumptions, these equations are simplified and the various boundary conditions have been used in order to validate the computational procedure. In order to accurately determine TED effects, the coupled thermal conduction equation is solved for the temperature field by considering three-dimensional (3-D) heat conduction along the length, width and thickness of the beam. Weighted residual Galerkin technique is used to obtain frequency shift and the quality factor of the thin beam resonator. The obtained results for quality factor, frequency shift and sensitivity change due to thermo-elastic coupling are presented graphically. Furthermore, the effects of beam aspect ratio, stress-free temperature on the quality factor and the influence of the surface stiffness on the frequencies and modal sensitivity of the AFM cantilever with and without considering thermo-elastic damping effects are discussed.     

Nonlinear dynamic analysis of a V-shaped microcantilever of an atomic force microscope

This paper is devoted to investigate the nonlinear behaviors of a V-shaped microcantilever of an atomic force microscope (AFM) operating in its two major modes: amplitude modulation and frequency modulation. The nonlinear behavior of the AFM is due to the nonlinear nature of the AFM tip–sample interaction caused by the Van der Waals attraction/repulsion force. Considering the V-shaped microcantilever as a flexible continuous system, the resonant frequencies, mode shapes, governing nonlinear partial and ordinary differential equations (PDE and ODE) of motion, boundary conditions, frequency and time responses, potential function and phase-plane of the system are obtained analytically. The governing PDE is determined by employing the Hamilton principle. Subsequently, the Galerkin method is utilized to gain the governing nonlinear ODE. Afterward, the resulting ODE is analytically solved by means of some perturbation techniques including the method of multiple scales and the Lindsted–Poincare method. In addition, the effects of different parameters including geometrical one on the frequency response of the system are assessed.     

Simulation of Atomic Force Microscopy for investigating BaTiO3 and LiMn2O4 nanostructures based on Phase Field Approach

Atomic Force Microscopy (AFM) probes the surface features of specimens using an extremely sharp tip scanning the sample surface while the force is applied. AFM is also widely used for investigating the electrically non-conductive materials by applying an electric potential on the tip. Piezoresponse Force Microscopy (PFM) and Electrochemical Strain Microscopy (ESM) are variants of AFM for different materials. Both PFM and ESM signals are obtained by observing the displacement of the tip when applying electric fields during the scanning process. The PFM technique is based on converse piezoelectric effect of ferroelectrics and the ESM technique is based on electrochemical coupling in solid ionic conductors. In this work, two continuum-mechanical formulations for simulation of PFM and ESM are discussed. In the first model, for PFM simulation, a phase field approach based on the Allen-Cahn equation for non-conserved order parameters is employed for ferroelectrics. Here, the polarization vector is chosen as order parameter. Since ferroelectrics have highly anisotropic properties, this model accounts for transversely isotropic symmetry using an invariant formulation. The polarization switching behavior under the electric field will be discussed with some numerical examples. In the simulation of ESM, we employ a constitutive model based on the work of Bohn et al. [8] for the modeling of lithium manganese dioxide LiMn₂O₄ (LMO). It simulates the deformation of the LMO particle according to an applied voltage and the evolution of lithium concentration after removing a DC pulse. The modeling results are compared to experimental data. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear dynamics of tapping mode atomic force microscopy in the bistable phase

Nonlinear dynamics of amplitude modulation atomic force microscopy (AFM) is studied employing a reduced-order model based on a differential quadrature method (DQM). The AFM microcantilever is assumed to be operating in the dynamic contact or tapping mode while the microcantilever tip being initially located in the bistable region. We have found that the DQM is capable of precise prediction of the static bifurcation diagram and natural frequencies of the microcantilever. We have used the DQM to discretize the partial-differential equation governing the microcantilever motion and a finite difference method (FDM) to calculate limit-cycle responses of the AFM tip. It is shown that a combination of the DQM and FDM applied, respectively, to discretize the spatial and temporal derivatives provides an efficient, accurate procedure to address the complicated dynamic behavior exhibited by the AFM probe. The procedure was, therefore, utilized to study the response of the microcantilever to a base harmonic excitation through several numerical examples. We found that the dynamics of the AFM probe in the bistable region is totally different from those in the monostable region.     

Analysis of nanoindentation experiments by means of atomic force microscopy

Nanoindentation is quite a common method for local material characterization. Values for hardness and Young's modulus can be determined directly from the recorded data. Essential for the correct determination of the material parameters is the precise measurement of the actual indentation depth of the indenter. The indenter measures the current depth by means of a Wheatstone bridge which correlates the indentation depth to a change in voltage. A possible tool for the verification of the recorded indentation depths is Atomic Force Microscopy (AFM). AFM is able to scan an area of indents for almost any surface. The deflection of the tip is measured by a laser spot reflected from the surface of the cantilever. The difference in height between the surface and the indent can directly be read off from the plotted image. However, using an AFM only allows us to measure the depth of the permanent indentation depth after unloading the indenter. Nevertheless, correlation between the remaining indentation depths measured by the explained methods allows for a first assessment of the correctness of the online recorded depth-data by the nanoindenter. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal boundary control by a force at one end of a string with a given force mode at the other end

We consider the problem of boundary control by a force applied to one end of a string in the case of a given force mode at the other end. The problem is studied in the sense of the generalized solution of the corresponding mixed initial-boundary value problem in the Sobolev space. We also solve the problem of choosing an optimal boundary control in the set of all admissible controls. The generalized solution of the mixed initial-boundary value problem is constructed in closed form, and its uniqueness is proved.     

Effect of three types of piezoelectric cantilever on the topography quality in the vicinity of rough surface in a fluid ambient

The use of higher vibration modes and different geometries of the atomic force microscopy (AFM) piezoelectric micro cantilever (MC) is affected by the surface topography quality in a liquid medium. Therefore, utilizing an appropriate MC geometry and vibration mode is of a great importance. This paper analyzes the effect of different types of AFM MCs on the surface topography quality in the noncontact and tapping modes in a liquid medium. The modified couple stress theory (MCS) in a liquid based on the Timoshenko beam theory is used in order to enhance the accuracy of MC dynamic modeling. In addition, the differential quadrature (DQ) method has been used to discrete the equations. Identification of environmental forces helps to measure the accurate MC vibration amplitude. Investigating the effect of geometric and force parameters on the MC vibration behavior leads to understanding the system to design it properly in a liquid medium. Based on the advanced dynamic modeling, the best MC geometry for the specific surface roughness has been determined in the liquid for the surface topography. Also, due to oscillating the MC near the sample surface, the effect of interaction forces between the sample surface and the MC, including van der Waals, contact and squeeze forces is analyzed in a liquid medium in addition to the hydrodynamic forces. Furthermore, due to the reduction of the MC amplitude caused by the squeeze force; the MC is angled in proportion to the horizontal surface.     

Effect of spherically symmetric mass flow from the surface of a particle on the force of interaction with a plane surface

A stationary velocity field of the flow of a gaseous medium generated by uniform radial injection from the surface of a spherical particle near a wall is considered in the Stokes' approximation. Bispherical coordinates are used to write the expression for the stream function. A formula is obtained for the force acting on the spherical particle when there is an arbitrary mass flow from its surface, generalizing earlier results /1, 2/. An expression for the force acting on the particle is obtained for the case of spherically symmetric injection from the surface of the particle, and asymptotic formulas at short and long distances from the wall are studied.

An analogous problem concerning the forces of interaction between two spherical particles of the same radius, when uniform injection of equal intensity takes place from their surfaces, is discussed. This is equivalent to the problem of the interaction of a spherical particle with a free surface. A general expression for the force of interaction, and its asymptotic forms for short and long distances, are obtained.     

Randomized approximation of Sobolev embeddings, III

We continue the study of randomized approximation of embeddings between Sobolev spaces on the basis of function values. The source space is a Sobolev space with nonnegative smoothness order; the target space has negative smoothness order. The optimal order of approximation (in some cases only up to logarithmic factors) is determined. Extensions to Besov and Bessel potential spaces are given and a problem recently posed by Novak and Woźniakowski is partially solved. The results are applied to the complexity analysis of weak solution of elliptic PDE.     

On the linearized theory op flow around bodies. Method of force sources

The method of force sources is proposed for solving linear problems related to the interaction between rigid bodies, and fluids, or gases. Method is based on the introduction of perturbation force sources into equation of motion of fluid media. Boundary conditions at the rigid body surface make it possible to reduce the problem of hydrodynamic reactions to an integral equation defining the function of force sources. Method is illustrated by the solution of three simple problems in the field of acoustics, and of viscous, and compressible media flow around bodies.

In the linearized theory of flow around rigid bodies, as well as in acoustics, an important part of the sound wave generation analysis concerns the determination of hydrodynamic reactions of the medium on moving, pulsating, or oscillating bodies. Such reactions make themselves felt as constant, or variable mechanical forces, such as drag and lift, or in the case of sound wave emitters, as the wave resistance. Various methods had been proposed for the computation of such forces, as for example, in the monographs [1 to 6].

Here, a different approach to the problem of determination of surface forces exerted by liquids and gases on the rigid body is proposed. By resorting to the formalism of the generalized functions it is possible to introduce into the equations of motion of fluid media a perturbation source in the form volume density of forces exercised by the body on the gas. The distribution of surface tension entering into the expression of this force is selected in such a manner as to satisfy boundary conditions at the body surface. It becomes possible with the use of this device to reduce the problem of determination of forces acting on the body surface to the solution of certain Integral equations. The proposed method is in all respects completely analogous to the well-known method of sources and sinks [1 to 1]. Both methods reduce the problem of interaction between body and gas to the solution of Integral equations. The method of sources and sinks, however, leads to an integral equation which describes the distribution of fictitious sources and sinks in the volume of the body having the density of the medium, while the method of force sources yields an integral equation which directly defines the distribution of mechanical forces over the surface of the body (*).

We may note that the method of force sources had to a certain extent been already used in papers [6 and 7] for the determination of sound radiation by means of point-force sources.     

On the variational formulation of a transmission problem for the Biot equations

In this article, a variational formulation for the transmission problem of the fluid–bone interaction is formulated. The formulation is based on a modified Biot system of equations for the cancellous bone together with a boundary integral equation formulation of the pressure in the water. Existence and uniqueness for the weak solution of the interaction problem are established in appropriate Sobolev spaces.     

Optimal boundary force control at one end of a string for a given displacement mode at the other end

We solve the problem of optimal boundary force control at one end of a string for the case of a given displacement mode at the other end. The problem is studied in the sense of a generalized solution of the corresponding mixed initial-boundary value problem in the Sobolev space. We also solve the problem of choosing an optimal boundary control from infinitely many feasible controls. The generalized solution of the mixed initial-boundary value problem is constructed in closed form, and the uniqueness of the solution is proved.     

Optimal boundary control by displacement at one end of a string under a given elastic force at the other end

The problem of optimal boundary control by displacement at one end of a string under a specified force mode at the other end is studied in the sense of a generalized solution of the corresponding mixed initial-boundary value problem from a Sobolev space. The problem of choosing an optimal boundary control from an infinite number of admissible controls is solved. A generalized solution of the mixed initial-boundary value problem is constructed explicitly and the uniqueness of the solution is proved.     

Global Solution and Blow-up for a Class of p-Laplacian Evolution Equations with Logarithmic Nonlinearity

The main goal of this work is to study an initial boundary value problem for a quasilinear parabolic equation with logarithmic source term. By using the potential well method and a logarithmic Sobolev inequality, we obtain results of existence or nonexistence of global weak solutions. In addition, we also provided sufficient conditions for the large time decay of global weak solutions and for the finite time blow-up of weak solutions.     

Surface-tension-driven flow in a slender cone

After a droplet has broken away from a slender thread or jetof liquid, the tip of the thread or jet recoils rapidly. Atthe moment of break-off, the tip of the thread/jet is observedto have the shape of a cone close to the bifurcation point.In this paper, we study the evolution of an ideal fluid whichis initially conical, where the only force acting on the fluidis due to surface tension. We find an asymptotic solution tothe problem in terms of the aspect ratio of the cone which isassumed to be small. Using a similarity transformation, whichis valid for small times after the bifurcation, we identifya rapidly oscillating non-linear wave which propagates awayfrom the tip, as observed in experiments.     

Variational analysis of fully coupled electro‐elastic frictional contact problems

We consider a mathematical model which describes the static frictional contact between a piezoelectric body and a foundation. The material behavior is described with a nonlinear electro‐elastic constitutive law. The novelty of the model consists in the fact that the foundation is assumed to be electrically conductive and both the frictional contact and the conductivity on the contact surface are described with subdifferential boundary conditions which involve a fully coupling between the mechanical and electrical variables. We derive a variational formulation of the problem which is in the form of a system coupling two hemivariational inequalities for the displacement and the electric potential fields, respectively. Then we prove the existence of a weak solution to the model and, under additional assumptions, its uniqueness. The proofs are based on recent results for inclusions of subdifferential type in Sobolev spaces (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Driving force acting on a running crack

The asymptotic analysis of the dynamic crack problem for the anti‐plane shear mode is provided. The field near the crack tip is studied in detail for a nonlinear elastic incompressible material whose stored energy behaves asymptotically as a power of the first invariant of the strain tensor at large strains. It is shown that the hardening parameter characterizes fully the singularity degree of the near‐crack‐tip field. Based on the latter knowledge the driving force acting on the crack tip is calculated. Possible scenarios of the crack propagation are discussed.