A deterministic model for line-contact partial elastohydrodynamic lubrication

A deterministic model for partial elastohydrodynamic lubrication (EHL) is presented in this paper. The modelling methodology adopts some of the concepts used in the stochastic modelling of partial EHL and some of the procedures for deterministic calculation of asperity pressures. The model is shown to be capable of simulating the basic process of asperity interaction and solid-to-solid contact within an EHL conjunction of rough surfaces. Deterministic results of transient partial EHL in line contacts are obtained when one pair or multiple pairs of asperities collide. The model may help to gain a fundamental understanding of the transient behaviour of asperity interactions in lubricated concentrated contacts of rough surfaces. Asperity pressures may be calculated more accurately than the conventional analyses under dry and static contact conditions. The work represents a first attempt in deterministic modelling of tribo-contacts operating in the mixed regime of micro-EHL and boundary lubrication. Future work will aim at developing more realistic models incorporating factors such as three-dimensional asperity contacts, asperity plastic deformation, thermal effects and the effect of tribo-chemistry.     

Numerical simulation of elastohydrodynamic lubrication for an elliptical contact

An elastohydrodynamic problem on elliptical contact is formulated and numerically solved. The mathematical model is described by a system of integro-differential equations and inequalities with boundary conditions. The direction of the rolling velocity vector with respect to the axes of the ellipse has a marked influence on the distribution of pressure and the lubricating film thickness in the contact area. The elastohydrodynamic contact parameters are determined depending on the compression force applied thereto. These results are useful for analyzing processes occurring in lubricated concentrated contacts.     

Power law fluid model incorporated into elastohydrodynamic lubrication theory of line contact

An algorithm is developed for the study of the infinitely long slider bearing in general form, considering the lubricant to be an incompressible power law fluid in isothermal conditions. The earlier works on this topic were considered by taking cavitation boundary conditions when a cylinder moves over a plane lubricated with a power law fluid and in EHL solution in a particular case, viz. pure rolling of a cylinder over an identical cylinder. We have considered a general solution including elastohydrodynamic lubrication (EHL) for different values of power law exponent. Deviation of values of central film thickness for different values of power law exponent from those for Newtonian lubricants are presented. The effects of the power law exponent on the central film thickness, minimum film thickness and load capacity are analysed. The effects of rolling and sliding velocities of contact surfaces are also analysed in terms of an equivalent radius of a cylinder moving over a moving plane. Film shapes and pressure distributions are also calculated numerically and presented graphically for various values of central film thickness considered in this paper. A number of observations obtained here with pseudoplastic nature of lubricants are in good agreement with the experimental results. The theoretical observations suggest the behaviour of common lubricants as pseudoplastic fluids in the cases of slowly moving surfaces and motion under heavy load.     

Integrated wear prediction model for cylindrical gear with variable hyperbolic circular arc tooth trace under mixed elastohydrodynamic lubrication

Cylindrical gear with variable hyperbolic circular arc tooth trace (VH-CATT) is a new type of gear. Sliding wear is the main mode of the surface failure of multiple mechanical parts. Both the lubrication state and contact temperature considerably influence wear characteristics, which may aggravate the transmission performance of gear pairs. Wear, contact temperature, as well as lubrication states are jointly explored. Therefore, an integrated wear prediction model was proposed through taking into account flash contact temperature and surface roughness of VH-CATT cylindrical gears in mixed elastohydrodynamic lubrication. According to the equivalent ellipse contact model of VH-CATT cylindrical gears and tooth surface equation, normal curvature and velocity relations for VH-CATT cylindrical gears were observed, and the normal meshing force was obtained through the consideration of load sharing coefficients and quality grades. Flash contact temperature was estimated by using the literature. This study proposes analytical solutions for investigating how various surface roughness, operation, and geometric parameters affect asperity contact ratio (ACR), asperity contact pressure (ACP), flash contact temperature (FCT), as well as wear depth (WD) related to driving gears. ACR, ACP, FCT, as well as WD initially decrease and then increase from engaging-in to engaging-out processes. The minimum occurs at the pitch point. The WD declines as module, cutter radius, and rotational velocity increase while augmenting when surface roughness and torques increase. The maximum and minimum wear depths in driving gears occur at the dedendum and pitch point, respectively. Its overall wear is reduced by 23.16 % compared to the wear of spur gears. The results are valuable for the studies of tooth pitting, wear resistance, and fatigue life improvement for VH-CATT cylindrical gear. These studies can provide verification data and references required for engineering designs and VH-CATT cylindrical gear operations.

     

A study of a finite element model for the chemical mechanical polishing process

In this paper, relative velocity at a given point on the wafer was first derived. The revolutions of wafer and pad are assumed the same and the axisymmetric uniformly distributed pressure form is given. Thus, a 2D axisymmetric quasic-static model for chemical-mechanical polishing process (CMP) was established. Based on the principle of minimum total potential energy and axisymmetric elastic stress-strain relations, a 2D axisymmetric quasic-static finite element model for CMP was thus established. In this model, the four-layer structures including wafer carrier, carrier film, wafer and pad are involved. The von Mises stress distributions on the wafer surface were analysed, the effects of axial, hoop, radial and shear stresses to von Mises stress and the effects of axial, hoop, radial and shear strains to deformation of the wafer were investigated. The findings indicate that near the wafer centre, von Mises stress distribution on the wafer surface was almost uniform, then increased gradually with a small amount. However, near the wafer edge, it would decrease in a large range. Finally, it would increase dramatically and peak significantly at the edge. Besides, the axial stress and strain are the dominant factors to the von Mises stress distributions on the wafer surface and the wafer deformation, respectively.     

Two-dimensional random surface model for asperity contact in elastohydrodynamic lubrication

Relations for the asperity contact time fraction during elastohydrodynamic lubrication of a ball bearing are presented. The analysis is based on a two-dimensional random surface model and actual profile traces of the bearing surfaces are used as statistical sample records. The results of the analysis show that transition from 90% contact to 1% contact occurs within a dimensionless film thickness range of approximately 4–5. This thickness ratio is several times larger than that reported in the literature where onedimensional random surface models were used. It is shown that low-pass filtering of the statistical records will bring agreement between the present results and those in the literature.     

A non-Newtonian model based on Ree-Eyring theory and surface effect to predict friction in elastohydrodynamic lubrication

The traction behaviour of various lubricants is investigated in elastohydrodynamic regime using a tribometre that simultaneously allows the contact visualization and the friction measurement under controlled contact kinematics. First, the film formation capability is determined: although the base oil and the polymer, via their viscosity, govern the film thickness in EHL regime, the additives control the existence of a boundary film at low entrainment speed. Second, the relative role of the additives and of the base oil on the frictional behaviour is studied for various experimental conditions: this clearly demonstrates that the piezo-viscous response of the lubricant controls the Newtonian/non-Newtonian transition and the friction level. The role of thermal effects is discussed. A strong influence of the entrainment speed, over the frictional response is unexpectedly observed, even in full film regime. A rheological model based on Ree-Eyring theory supposing a heterogeneous flow due to adsorbed surface layers, is proposed to discuss the organization of the molecules within the contact and its effect on the frictional response of the contact.     

A measurement system for thin elastohydrodynamic lubrication films

An elastohydrodynamic lubrication (EHL) film measurement system using multi-beam interferometry is introduced in this paper. The measurement principle and the instrumentation are discussed. A simple and efficient method is suggested to obtain the fringe order of measured points. It is demonstrated that the presented measurement system can provide continuous measurement of lubricating films from nano to micro scales at a nano-level resolution, and can be used to investigate ultra-thin EHL films and tiny variations in EHL films. Translated from Tribology, 2006, 26(2): 150–153 [译自: 摩擦学学报]     

Analysis of rough line contacts operating under mixed elastohydrodynamic lubrication conditions

Heavily loaded machine elements, such as gears, usually operate in the mixed lubrication regime. Surface roughness has a significant effect on the pressure distribution, the subsurface stress field, and the friction coefficient. Based on the superposition of a dry rough and a fully flooded smooth contact, a mixed lubrication model has been developed. The roughness profile is assumed to be known. Surface deformation is calculated by taking into account the pressure distribution that is built up by asperity contacts, asperity interactions, and lubricant flow. Thermal and sliding effects are incorporated into the analysis. Non‐Newtonian lubricant behaviour is considered by using a power‐law rheological model. The pressure distribution, subsurface stress field, and friction coefficient were calculated from the model at several points along the contact path for an FZG type C gear pair. It was shown that a significant part of the load is carried by the contacting asperities. The position of the maximum shear stress is very close to the surface.     

化学机械抛光的研究进展

化学机械抛光简称CMP技术是迄今唯一的可以提供整体平面化的表面精加工技术,可广泛用于集成电路芯片、计算机硬磁盘、微型机械系统等表面的平坦化。介绍了半导体加工领域CMP技术的特点,重点叙述了CMP技术的发展历程、设备特性、研磨抛光耗材和应用领域的技术现状,指出CMP急待解决的技术和理论问题,并对其发展方向进行展望。     

The influence of transverse roughness in thin film, mixed elastohydrodynamic lubrication

The Spacer Layer Imaging (SLIM) method has been used to investigate the influence of transverse roughness on the thickness and shape of elastohydrodynamic (EHD) films. The effects of entrainment speed and lubricant viscosity on film distribution are shown for three distinct asperity heights over a wide range of lambda ratio (ratio of lubricant film thickness, separating two contacting surfaces, to their combined RMS roughnesses). Subsequently, the behaviour of film distribution for a range of mixed rolling–sliding conditions is also studied for both thin and thick film conditions. This study provides an estimate of how and when transverse asperities decompress and an indication of conditions under which these asperities cease to affect lubricant film formation.     

Estimating chemical mechanical polishing pad wear with compressibility

A polishing pad is an important component in a chemical mechanical polishing (CMP) system. Few investigations have specialized in the variation of the characteristics of the pad as it undergoes wear. All of the information concerning a pad, such as compressibility and pad life, comes from the manufacturer. No acknowledged standard or instrument exists for determining a pad’s quality. This study obtained the variation of the compressibility (K) of major types of pad (single-layer pad and composite pad) with polishing time (pad wearing) by theoretical modeling and real experiments. The K of single-layer or composite pads changes due to wear, and the trends in the K values of the two types of pads are exactly opposite. A finite element method (FEM) is used here to show that the variation in K strongly affects the polishing process. Theoretically, the compressibility of a pad can be used to judge whether the pad is good for polishing or not.     

Analysis of retaining ring using finite element simulation in chemical mechanical polishing process

During the chemical mechanical polishing process (i.e., CMP for short), it is expected to attain the requirement of global planarization. However, the stress concentration, which occurred when approaching the wafer edge, has resulted in over-grinding. The increasing material removal rate has also contributed to the wafer’s nonuniformity. In this paper, a retaining ring surrounding the wafer carrier was added to the conventional CMP mechanism in order to improve the over-grinding phenomenon and avoid the wafer sliding from the carrier as much as possible. The revolutions of the wafer and the pad were assumed to be the same, and the force forms of the carrier and the retaining ring were axisymmetric uniformly-distributed. In addition, when the principle of minimum potential energy was applied, a two-dimensional axisymmetric quasi-static finite element model for CMP including the carrier, the retaining ring, the film, the wafer and the pad could be established. Following this model, the von Mises stress distribution of the wafer surface without a retaining ring was analyzed to verify the model. The effects of the gap between the ring and the pad and the ratio of the ring load and the carrier load on the stress and the wafer’s nonuniformity were investigated. The results indicated that the von Mises stress distribution of the wafer surface was almost uniform near the wafer center, and the maximum value appeared near the edge. The value decreased as the edge was approached, but it rose again very close to the edge. Besides, the wafer’s nonuniformity would be prohibited while the gap and the load were designed within the certain range to supply the retaining ring.     

A general elastohydrodynamic lubrication analysis of artificial hip joints employing a compliant layered socket under steady state rotation

A general numerical methodology was developed in the present study to analyse the elastohydrodynamic lubrication problem of a compliant layered socket against a rigid ball under steady state rotation representing flexion and extension during walking, with particular reference to artificial hip joint replacements. The general numerical methodology consisted of using the Newton-Raphson method to solve the Reynolds equation, simultaneously with the full elasticity equation using the finite element method in combination with the fast Fourier transform technique. Two specific types of acetabular cup were considered, one with ultra-high molecular weight polyethylene used in current total hip joint replacements, and one with polyurethane proposed for compliant layered 'cushion form bearings' for future developments. The film thickness and the pressure distribution for both cups were obtained under a wide range of operating conditions. The predicted central or average film thicknesses within the contact conjunction were compared with those estimated from various simplified theories available in the literature. A simple analytical methodology was consequently established to estimate the lubricating film thickness in a compliant layered socket, based on the corresponding ball-on-plane model and the consideration of the curvature effect.     

化学机械抛光机理研究

虽然CMP过程材料去除机理得到了广泛深入研究,取得了许多理论成果,但是每一种理论都存在着或多或少的缺陷,没有一种理论完全正确的解释了CMP的机理。在众多CMP材料去除机理里,比较得到认可的是抛光垫、颗粒和工件表面三体磨损理论和化学腐蚀理论[1~4]。但是表面质量较好的工件在CMP之后没有观察到划痕,三体磨损理论受到了极大的挑战。而化学腐蚀理论很难解释腐蚀速度和腐蚀后表面质量关系[5,6]。本文结合CMP过程的实际环境,实验得到了化学机械抛光的抛光机理。     

Hydrodynamic analysis of chemical mechanical polishing process

Chemical Mechanical Polishing (CMP) refers to a material removal process done by rubbing a work piece against a polishing pad under load in the presence of chemically active abrasive containing slurry. The CMP process is a combination of chemical dissolution and mechanical action. The mechanical action of CMP involves hydrodynamic lubrication. The liquid slurry is trapped between the work piece (wafer) and pad (tooling) forming a lubricating film. For the first step to understand the mechanism of the CMP process, hydrodynamic analysis is done with a semiconductor wafer. Slurry pressure distribution, resultant forces and moments acting on the wafer are calculated in typical conditions of the wafer polishing, and then nominal clearance of the slurry film, roll and pitch angles at the steady state are obtained.     

The elastohydrodynamic lubrication of line contacts with pseudoplastic fluids

A new Reynolds equation, obtained by using the pseudoplastic model to express the non-newtonian fluid property, is derived for line contact elastohydrodynanic lubrication problems. The Reynolds equation results in a non-linear equation of high order which is derived by using an approximate method. The Newton-Raphson technique is utilized to solve the simultaneous system of modified Reynolds and elasticity equations. The results in this paper show that on increase in the coefficient of non-linearity, the pressure spike and film thickness decrease, and they approach a hertzian situation. With respect to the various loads and/or velocity values, the behaviours of the pressure spike and film thickness for non-newtonian fluids are almost the same as those of a newtonian fluid.     

An integrated material removal model for silicon dioxide layers in chemical mechanical polishing processes

The material removal rate in CMP processes obeys Preston's equation, which can be expressed as a linear function of the applied areal power density under usual operating conditions. However, some experimental results have shown a nonlinear relationship between the CMP material removal rate and the applied areal power density, suggesting non-Prestonian behavior under certain operating regimes. Although the material removal rate is caused by the coupled effect of both mechanical and chemical actions in actual CMP processes, the treatment of the chemical action as a mere supplementary means of softening the surface makes it difficult to explain this non-Prestonian behavior. In this work, we propose an integrated material removal model for silicon dioxide during CMP based on a multiscale mechanical abrasion model coupled with the slurry chemical diffusion effects. The synergetic effects on the material removal mechanism due to both mechanical and chemical actions are incorporated in the model, and the total material removal rate is predicted by accounting for both effects. Consequently, the non-Prestonian behavior often shown in silicon dioxide CMP may be explained using the proposed model. The validity of the model is supported by comparing the predicted material removal rates with experimental values available in the literature.     

A computer model of mixed lubrication in point contacts

A numerical model of mixed lubrication is presented in this paper. The idea introduced here is that asperity contact may be viewed as a result of a continuous decrease in film thickness, so that the transition between contact and non-contact is continuous and the same mathematical model should work for both regions. The pressure over the thin films is assumed to obey the Reynolds equation, and the solution of the equation, under the condition of h→0, is expected to be the same as that predicted by the theory of elasticity. To achieve convergent and stable solutions, the left-hand side terms of the Reynolds equation are switched off when the local film thickness approaches zero, leading to a reduced Reynolds equation. Pressure distributions over the entire computation domain are thus obtained through solving a unified equation system without identifying hydrodynamic or asperity contact regions. Computations were conducted for several example cases and results show that convergent solutions are achievable on different types of roughness, over a wide range of λ ratios (0.01 to infinity), and for different slide-to-roll ratios (0.0–2.0).