JKR Model Research Articles

Analyzing the vibrational response of an AFM cantilever in liquid with the consideration of tip mass by comparing the hydrodynamic and contact repulsive force models in higher modes

This paper investigates the vibration of a tapping-mode Atomic Force Microscope (AFM) cantilever covered with two whole piezoelectric layers in a liquid medium. The authors of this article have already modeled the vibration of a cantilever immersed in liquid over rough surfaces. Five new ideas have been considered for improving the results of the previous work. Mass and damping of a cantilever probe tip have been considered. Since the probe tip of an AFM cantilever has a mass, which can itself affect the natural frequency of vibration, the significance of this mass has been explored. Also, two hydrodynamic force models for analyzing the mass and damping added to a cantilever in liquid medium have been evaluated. In modeling the vibration of a cantilever in liquid, simplifications are made to the theoretical equations used in the modeling, which may make the obtained results different from those in the real case. So, two hydrodynamic force models are introduced and compared with each other. In addition to the already introduced DMT model, the JKR model has been proposed. The forces acting on a probe tip have attractive and repulsive effects. The attractive Van der Waals force can vary depending on the surface smoothness or roughness, and the repulsive contact force, which is independent of the type of surface roughness and usually varies with the hardness or softness of a surface. When the first mode is used in the vibration of an AFM cantilever, the changes of the existing physical parameters in the simulation do not usually produce a significant difference in the response. Thus, three cantilever vibration modes have been investigated. Finally, an analytical approach for obtaining the response of equations is presented which solves the resulting motion equation by the Laplace method and, thus, a time function is obtained for cantilever deflection is determined. Also, using the COMSOL software to model a cantilever in a liquid medium, the computed natural frequencies have been compared.

Simulating the AFM-based biomanipulation of cylindrical micro/nanoparticles in different biological environments

Although remarkable attempts have already been made to automate the AFM-based manipulation of biological cells, there are still many ambiguous points regarding the interactions between biological nanoparticles and manipulation tools, especially in different biological environments. As micro/nanoparticles of different shapes are placed in different biological environments, thus, it is necessary to accurately investigate the effects of the target nanoparticle’s geometry and material as well as the impact of the biomanipulation environment on the values of critical force and time of AFM-based manipulation. Therefore, in this article the cylindrical contact models of Hertz, JKR and viscoelastic model are developed for use in the AFM-based manipulation process conducted within different biological environments. Then, these models are applied to simulate the manipulation of gold and yeast nanoparticles in various environments such as air, water, alcohol and plasma. By comparing the results obtained in this paper, it is concluded that the movement of particles in different biological environments starts later and by a force of higher magnitude relative to the air medium. Also, it is found that the critical force and time of manipulation increase considerably for cylindrical particles relative to the spherical ones. At the end, the results are compared with the existing empirical results and their reliability is verified.

Lubrication of Individual Microcontacts by a Self-Assembled Alkyl Phosphonic Acid Monolayer on α-Al2O3(0001).

We report on the tribological behavior of a self-assembled alkyl phosphonic acid monolayer on the microscale using the colloidal probe technique. Friction-load data and adhesion forces were measured with borosilicate glass particles on uncoated and octadecylphosphonic acid (ODPA) coated α-Al2O3(0001) surfaces. A significant decrease in friction force was observed after surface coating, while the adhesion force was only moderately reduced. We assume the lubrication effect of the ODPA self-assembled monolayer (SAM) to be close to the maximum obtainable of alkyl phosphonic acids in the studied system due to the high molecular order which was confirmed by vibrational sum-frequency generation. At small loads, a nonlinear dependence of friction force to load was maintained after surface coating. However, a shift from a contact behavior well described by the DMT model toward the JKR model occurred that is possibly related to the altered elastic properties of the coated surface. With increasing load, a linear friction-load behavior was observed on the coated samples. Molecular plowing and adhesive interactions were identified as responsible mechanisms. In all friction experiments, we could not detect any wear neither of the colloidal probes nor at the surfaces of uncoated and coated samples. This proves the high wear resistivity of the studied ODPA SAM.

Revisit the adhesive contact between a nanoscale rigid sphere and an elastic half-space: considering the effect of sphere size

The current paper revisits the adhesive contact between a rigid nanoscale sphere and an elastic half-space. Using analytical solution for traction and a modified numerical scheme, a new simulation is proposed. The adhesive contact is simulated faster and more easily than previous researches. The effect of sphere size is investigated. A modified spherical JKR model and spherical DMT model are proposed. The results are compared with spherical JKR, spherical DMT, and rigid sphere model. The load–approach and contact radius–load relations can be predicted by the modified spherical JKR model for sphere radius larger than 50ɛ. Finally, approximation equations for pull-off force vs. Tabor parameter and jump-in distance vs. Tabor parameter are proposed.

Investigation of work of adhesion of biological cell (human hepatocellular carcinoma) by AFM nanoindentation.

In this study, we presented an investigation of mechanical properties by AFM nanoindentation on human hepatocellular carcinoma cells treated with fullerenol for 24, 48 and 72 h. AFM nanoindentation was routinely applied to investigate the morphology and biomechanical properties of living carcinoma cells, and adhesion phenomena (negative force) were detected in the obtained force-displacement curves. Conventionally, Hertz contact model has been widely used for determination of cell elasticity, however this contact model cannot account for adhesion. Alternatively, JKR contact model, as expected for adhesion circumstance, has been applied to fit the obtained force-displacement curves. In this investigation, we have derived both the work of adhesion and the elastic modulus of biological cells (human hepatocellular carcinoma) under fullerenol treatment. The results show that the chosen JKR model can provide better fitting results than Hertz contact model. The results show that both Young’s modulus and work of adhesion exhibit significant variation as the treatment time increases. The calculated mechanical properties of elastic modulus and work of adhesion can be used as an effective bio-index to evaluate the effects of fullerenol or other anticancer agents on cancer cells and thus to provide insight into cancer progression in the treatment.

Effects of surface tension on the adhesive contact between a hard sphere and a soft substrate

In the present paper, the adhesive contact between a rigid sphere and a compliant elastic substrate with surface tension is analyzed. Using the minimum potential energy principle and considering both the effects of surface tension and adhesion, explicit solutions are obtained for the contact radius and the indent depth for the case without external loading. It is found that surface tension evidently alters the pressure distribution in the contact region and tends to decrease both the contact radius and the indent depth. The present model establishes a bridge between the JKR model for adhesive contact problems and the Young-Laplace's law of surface tension.

Mechanical Properties of Differentiating Stem Cells on Peptide Nanofibers

Mechanical properties of differentiating stem cells on peptide nanofibers Here we investigate changes in Young's moduli of mesenchymal stem cells while they are differentiating to osteogenic lineage on bioactive peptide nanofiber. We measured the elasticity of cells on hydrogels made of peptide amphiphiles that can be used to manipulate the cells through bioactive peptide sequences. Peptide nanofibers can mimic the extracellular matrix both in physical shape and biochemical activity. In this study, the peptide nanofibers presented osteoinductive Glu-Glu-Glu (EEE) sequence and Lys-Lys-Lys (KKK) sequence. Cells were analyzed on peptide-coated glass coverslips at days 1, 3, 5, 7, 14, 21 and 28 of osteogenic differentiation by measuring 50 μm x 50 μm and 16 x 16 points and lines force maps on cells. 5.2 μm diameter gold colloidal microsphere probes were used for AFM measurements and the force maps were converted to Young's modulus maps by applying JKR model. Peptide nanofibers provide a three-dimensional environment to cells and reduce the stiff glass substrate effect on Young's moduli value. Osteogenic differentiation of stem cells cultured in normal growth medium and in osteogenic growth medium was also compared. Differentiation of mesenchymal stem cells to osteoblasts was detected by analyzing mechanical properties during osteogenic differentiation.

Multi-asperity elliptical JKR model for adhesion of a surface with non-axially symmetric asperities

Surfaces can present high levels of topographic asymmetry and, therefore, theories based on the assumption of symmetry cannot be effectively employed.A new multi-asperity adhesion model that assumes that asperities are not perfectly hemispherical is presented here, this model is based on the elliptical JKR model for a single asperity. The adhesion between a soft tissue with asperities greatly asymmetric and a polymer was modelled, the predicted adhesion forces were successfully validated against experimentally obtained data. Moreover, simulations with a simpler model, which assumes symmetrical asperities, have been also carried out; these results were significantly different from those obtained, using both the newly developed model and those determined experimentally. This highlights the importance of the model presented in this work.

Looking at cell mechanics with atomic force microscopy: experiment and theory.

This review reports on the use of the atomic force microscopy in the investigation of the mechanical properties of cells. It is shown that the technique is able to deliver information about the cell surface properties (e.g., topography), the Young modulus, the viscosity, and the cell the relaxation times. Another aspect that this short review points out is the utilization of the atomic force microscope to investigate basic questions related to materials physics, biology, and medicine. The review is written in a chronological way to offer an overview of phenomenological facts and quantitative results to the reader. The final section discusses in detail the advantages and disadvantages of the Hertz and JKR models. A new implementation of the JKR model derived by Dufresne is presented.

Adhesion and Wetting of Nanoparticles on Soft Surfaces

We study adhesion of spherical and cylindrical nanoparticles on soft (gel-like) substrates using a combination of the molecular dynamics simulations and theoretical calculations. The substrate deformation is obtained as a function of the gel shear modulus, nanoparticle size, surface tension of nanoparticles and substrate, and work of adhesion. It was shown recently that the classical JKR model can only be applied to describe nanoparticle adhesion on relatively stiff substrates. In this so-called adhesion regime the deformation of the substrate is determined by balancing the elastic energy of indentation and the work of adhesion between a nanoparticle and a gel. However, in the case of soft gels when substrates undergo moderate deformations the depth of the indentation produced by a nanoparticle is determined by the surface tension of the gel and the work of adhesion (the wetting regime). We present an analytical model describing crossover between adhesion and wetting regimes. In the framework of this model a crossover between different interaction regimes is controlled by a dimensionless parameter γs(GRp)−2/3W–1/3, where γs and G are the surface tension and shear modulus of the gel, W is the work of adhesion between gel and nanoparticle, and Rp is the nanoparticle radius. Nanoparticle adhesion regime corresponds to small values of this parameter, γs(GRp)−2/3W–1/3 ≪ 1, while the wetting regime takes place at γs(GRp)−2/3W–1/3 ≫ 1. We applied our model to obtain work of adhesion between silicon substrates and silica microspheres and to obtain surface tension of silicon gels from particle indentation experiments.

A unified model of hysteresis and relaxation in geophysical materials

A physical model of stress-strain dynamics and long-time relaxation (slow time) in structured media such as the rock is suggested. It is based on an adhesion mechanism (JKR model) for inter-grain contacts which implies the surface force potential with a barrier. This model results in a unified description of the classical nonlinearity, stress-strain hysteresis, and logarithmic relaxation law for sound velocity and, hence,for the frequency of nonlinear resonance in samples of structured materials. Estimates of a characteristic volume of interacting contacts give close values for a variety of consolidated materials. Propagation of waves in such materials is briefly considered. For a weak (linear) testing wave, the logarithmic relaxation occurs if the classical quadratic nonlinearity is added to the stress-strain relation.

Large scale simulation of red blood cell aggregation in shear flows

Aggregation of highly deformable red blood cells (RBCs) significantly affects the blood flow in the human circulatory system. To investigate the effect of deformation and aggregation of RBCs in blood flow, a mathematical model has been established by coupling the interaction between the fluid and the deformable solids. The model includes a three-dimensional finite volume method solver for incompressible viscous flows, the combined finite-discrete element method for computing the deformation of the RBCs, a JKR model–Johnson, Kendall and Roberts (1964–1971) (Johnson et al., 1971) to take account of the adhesion forces between different RBCs and an iterative direct-forcing immersed boundary method to couple the fluid–solid interactions. The flow of 49,512 RBCs at 45% concentration under the influence of aggregating forces was examined, improving the existing knowledge on simulating flow and structural characteristics of blood at a large scale: previous studies on the particular issue were restricted to simulating the flow of 13,000 aggregative ellipsoidal particles at a 10% concentration. The results are in excellent agreement with experimental studies. More specifically, both the experimental and the simulation results show uniform RBC distributions under high shear rates (60–100/s) whereas large aggregation structures were observed under a lower shear rate of 10/s. The statistical analysis of the simulation data also shows that the shear rate has significant influence on both the flow velocity profiles and the frequency distribution of the RBC orientation angles.

Dynamic simulation of particle packing influenced by size, aspect ratio and surface energy

The multi-sphere method and JKR model are used in the discrete element method simulation to investigate the effect of the particle size, aspect ratios, and cohesiveness on the packing structure, characterized by porosity, radial distribution function (RDF), coordination number and contact geometry. In the absence of cohesive force, the porosity is nearly invariable with fixed aspect ratio, regardless of the size of the particles. In contrast, as surface energy increases, the porosity increases with decreasing particle size and increasing aspect ratio. The RDF results show that the number of peaks for different aspect ratios changes and show trends similar to the relaxation algorithm, expected for the finer particles. In the case of finer, cohesive particles, the most novel outcome of contact analysis is the existence of single contact, attributed to the formation of a cage structure, which has not been previously reported. The peak position and the width of the contact distributions are affected by higher surface energy because fewer contacts are required to achieve the mechanical equilibrium. Another interesting observation is that higher porosity does not always imply fewer contacts for particles with non-zero aspect ratios and high surface energies. The analysis of the distribution of the contact vector angles is found to better explain increased porosity in spite of higher coordination numbers. The results presented shed light on the packing density and structure, revealing features not easily discerned via experiments, and confirming the important role of the cohesion and aspect ratio in packing.

Quantitative Mapping of the Elastic Modulus of Soft Materials with HarmoniX and PeakForce QNM AFM Modes

The modulus of elasticity of soft materials on the nanoscale is of interest when studying thin films, nanocomposites, and biomaterials. Two novel modes of atomic force microscopy (AFM) have been introduced recently: HarmoniX and PeakForce QNM. Both modes produce distribution maps of the elastic modulus over the sample surface. Here we investigate the question of how quantitative these maps are when studying soft materials. Three different polymers with a macroscopic Young's modulus of 0.6-0.7 GPa (polyurethanes) and 2.7 GPa (polystyrene) are analyzed using these new modes. The moduli obtained are compared to the data measured with the other commonly used techniques, dynamic mechanical analyzer (DMA), regular AFM, and nanoindenter. We show that the elastic modulus is overestimated in both the HarmoniX and PeakForce QNM modes when using regular sharp probes because of excessively overstressed material in the samples. We further demonstrate that both AFM modes can work in the linear stress-strain regime when using a relatively dull indentation probe (starting from ~210 nm). The analysis of the elasticity models to be used shows that the JKR model should be used for the samples considered here instead of the DMT model, which is currently implemented in HarmoniX and PeakForce QNM modes. Using the JKR model and ~240 nm AFM probe in the PeakForce QNM mode, we demonstrate that a quantitative mapping of the elastic modulus of polymeric materials is possible. A spatial resolution of ~50 nm and a minimum 2 to 3 nm indentation depth are achieved.

Adhesion and Friction Behaviors of Probes Approaching Magnetic Disk Surfaces

In this study, we focused on the adhesion and friction behaviors of probes approaching magnetic disk surfaces. In order to observe these behaviors, a micro-tribotester was newly developed to evaluate the friction and adhesion forces of the ultrathin lubricant film on magnetic disks. The contact area between the spherical glass surface and the disk surface was measured using light intensities and Newton's rings and approximated by the JKR model. An adhesion force corresponding to a pull-off force acted on the interface between the glass surface and the disk surface. The friction coefficients on disks with different lubricant thicknesses were measured and the coefficient was found to decrease as the lubricant thickness increased. The pull-off force gradually increased as lubricant thickness increased.

Adhesive Elastic Contact of Rough Surfaces with Power-Law Axisymmetric Asperities

The contact model for rough surfaces with power-law axisymmetric asperities in the presence of adhesion is developed. The extended JKR adhesive contact model for power-law axisymmetric asperities, denoted as JKR-n, developed by Zheng and Yu (2007,“Using the Dugdale Approximation to Match a Specific Interaction in the Adhesive Contact of Elastic Objects,” J. Colloid Interface Sci., 310, pp. 27–34) is utilized to investigate the adhesive contact of rough surfaces. The JKR-n adhesive contact model generalizes the most adopted JKR model for spherical objects of n = 2. This work compares the effect of surface roughness on the adhesion force for rough surfaces with various power-law axisymmetric asperities. It is found that shapes of the asperities influence the pull-off forces greatly during the separation of rough surfaces. A general adhesion parameter that includes the shape index of asperities is proposed, and it can be used to characterize the adhesion performance of rough surfaces.

Complex variable formulation for non-slipping plane strain contact of two elastic solids in the presence of interface mismatch eigenstrain

In this paper, the problems of non-slipping contact, non-slipping adhesive contact, and non-slipping adhesive contact with a stretched substrate are sequentially studied under the plane strain theory. The main results are obtained as follows:(i) The explicit solutions for a kind of singular integrals frequently encountered in contact mechanics (and fracture mechanics) are derived, which enables a comprehensive analysis of non-slipping contacts. (ii) The non-slipping contact problems are formulated in terms of the Kolosov–Muskhelishvili complex potential formulae and their exact solutions are obtained in closed or explicit forms. The relative tangential displacement within a non-slipping contact is found in a compact form. (iii) The spatial derivative of this relative displacement will be referred to in this study as the interface mismatch eigenstrain. Taking into account the interfacemismatch eigenstrain, a new non-slipping adhesive contact model is proposed and its solution is obtained. It is shown that the pull-off force and the half-width of the non-slipping adhesive contact are smaller than the corresponding solutions of the JKR model (Johnson et al., 1971). The maximum difference can reach 9% for pull-off force and 17% for pull-off width, respectively. In contrast, the new model may be more accurate in modeling the non-slipping adhesion. (iv) The non-slipping adhesions with a stretch strain (S-strain) imposed to one of contact counterparts are re-examined and the analytical solutions are obtained. The accurate analysis shows that under small values of the S-strain both the natural adhesive contact half-width and the pull-off force may be augmented, but for the larger S-strain values they are always reduced. It is also found that Dundurs’ parameter β may exert a considerable effect on the solution of the pull-off problem under the S-strain.These solutions may be used to study contacts at macro-, micro-, and nano-scales.

S162012 ピンオンディスク試験による磁気ディスク上の超薄膜液体潤滑膜の摩擦力・凝着力の測定

In this study, we focused on the adhesion and friction behaviors of probes approaching magnetic disk surfaces In order to observe these behaviors, a micro-tribotester was newly developed to evaluate the friction and adhesion forces of the ultrathin lubricant film on magnetic disks. The contact area between the spherical glass surface and the disk surface was measured using light intensities and Newton's rings and approximated by the JKR model. An adhesion force corresponding to a pull-off force acted on the interface between the glass surface and the disk surface. The friction coefficients on disks with different lubricant thicknesses were measured and the coefficient was found to decrease as the lubricant thickness increased. The pull-off force gradually increased as lubricant thickness increased.

Study of the rheological behaviour of monomodal quartz particle beds under stress. A model for the shear yield functions of powders

A model is proposed that describes the yield locus of powders. The model is based on the adhesive contact of elastic spheres theory developed by Johnson, Kendall, and Roberts (JKR model). The proposed model improves the results by applying the Warren Spring Laboratory (WSL) equation and the Coulomb model. Only two parameters are needed in the present model to describe the yield locus of a powder: powder cohesion ( C) and the coefficient of friction ( μ) of the powder bed. The study verifies the validity of the new equation to describe the yield locus of different powders. The proposed model was used to calculate the cohesion and coefficient of friction of monomodal quartz particle beds with different average particle sizes. The study examines the influence of average particle size ( d s ) and bed compactness ( ϕ) on these parameters. Cohesion was observed to increase by the sixth power of compactness and inverse of the particle Sauter diameter. The bed coefficient of friction increased asymptotically with bed compactness, this relationship depending on particle size.

Determination of the Shear Modulus of Spin-Coated Lipid Multibilayer Films by the Spontaneous Embedment of Submicrometer-Sized Particles

A novel submicrometer particle embedment technique has been used to determine the shear modulus of 1,2-dipalmitoyl-sn-glycero-3-phosphotidylcholine (DPPC) lipid multibilayers. The depth of the spontaneous embedment of polystyrene and silica spherical particles of 200 nm nominal size has been determined from atomic force microscopy measurements on colloidal particles dispersed onto the surfaces of the DPPC multibilayers. The standard JKR model was used to relate the shear modulus of the lipid multibilayer films to the depth of embedment of the particles. The thus-determined modulus of the DPPC is within the range of reported literature values. Gold particles are also considered, and it is found that for the smallest particles (13 nm) complete engulfment by the DPPC multibilayer film takes place.