The comparison between the analytical solution performed by Cameron  and the numerical simulation code.
For the last years, there has been a tremendous effort towards the development of Micro-Electro-Mechanical System (MEMS) for a wide variety of applications in aerospace, automotive, biomedical, computer, agricultural industries, electronic instrumentation, industrial process control, biotechnology, office equipment, and telecommunications. MEMS devices integrate chemical, physical, and even biological processes in micro-scale technology packages.
Stiction (a subtraction of ‘static friction’) in micro-system technology has been a problem ever since the advent of surface micromachining in the eighties of the last century. As the overall size of the machine is reduced, the capillary and surface tension force of liquid become large, which induce stiction rendering the devices to fail or malfunction. In particular, stiction forces created between moving parts that come into contact with one another, either intentionally or accidentally, during operation are a common problem with micro-mechanical devices. Stiction-type failures occur when the interfacial attraction forces exceed restoring forces. Consequently, the surfaces of these parts either temporarily or permanently adhere to each other, causing device malfunction or failure.
Several approaches to address the stiction between two opposing surfaces have been presented in the various literatures [1-4]. The basic approaches to prevent stiction include increasing surface roughness (topography) and/or lowering solid surface energy by coating with low surface energy materials. This includes self-assembled molecular (SAM) coatings, hermetic packaging and the use of reactive materials in the package .
Other attractive technique to tackle the stiction problem is by inserting a lubricant into the region around the interacting devices to reduce the chance of stiction-type failures. As is well-known, many MEMS devices include moving (sliding/rolling) surfaces and thus it is necessary to apply a lubricant between the contacting surfaces to reduce friction and wear. However, a significant barrier to the development of MEMS lubrication is the problem of achieving effective tribological performance of their moving parts. This is because the lubricant behavior is different at micro-scale compared to macro-scale. At the macroscopic level, it is well accepted that the boundary condition for a viscous fluid at a solid wall is no-slip, i.e. the fluid velocity matches the velocity of the solid boundary. While the no-slip boundary condition has been proven experimentally to be accurate for a number of macroscopic flows, it remains an assumption that is not based on physicals principles. At micro-scale level, certain phenomena must be taken into account when analyzing liquid flows such as a slip condition at solid wall boundaries.
As a consequence of the MEMS technology revolutionary application to many areas, it is possible for scientists to observe the boundary slip on micro/nano-meter scale. A variety of techniques are now available that are capable of probing lubricant flow on micro-scales and are therefore suitable for the investigation of boundary conditions. There are three techniques so far for detecting the boundary slip: nano-particle image velocimetry (NPIV) , atomic force microscope (AFM) [7-9] and surface force apparatus (SFA) . The NPIV technique is a direct observation method with a measurement precision depending on the size of the nano-particles but with poor moderate accuracy. The AFM and SFA are indirect observation techniques based on the assumption that boundary slip takes place precisely on the interface of liquid and solid. These methods need a high accuracy boundary slip model to infer the slip velocity. Boundary slip has been observed not only for a hydrophobic surface [6, 7, 10] but also for a hydrophilic surface [8, 9]. Therefore, the slip evidence has been generally accepted and for certain cases the no-slip boundary condition is not valid.
There is a large body of literature dealing with the analysis of lubricant slip flow based on the analytical and numerical solution of molecular dynamic simulations [11, 12], Lattice-Boltzman [13, 14], and the Reynolds equation [15-23]. The accurate description of slip at the wall is very difficult and still remains a subject of intensive research. The so-called Navier slip model and the critical shear stress model are usually used to describe a boundary slip. In fact, nearly two hundred years ago Navier  proposed a general boundary condition that incorporates the possibility of fluid slip at a solid boundary. Navier's proposed boundary condition assumes that the velocity,
In micro-scales such as MEMS, the boundary condition will play a very important role in determining the lubricant flow behavior. Control of the boundary condition will allow a degree of control over the hydrodynamic pressure in confined systems and be important in lubricated-MEMS. To prevent a stiction, in a controlled way, one is able to enhance, a hydrophobic/hydrophilic behavior of surfaces. If one surface is hydrophobic (slip) and the other is hydrophilic (no-slip) the sliding velocity or displacement between the surfaces is accommodated by shear at the hydrophobic surface (the lubricant is kept in the contact by the hydrophilic surface). In this way wear of the surfaces is prevented and the surfaces are able to move because stiction is prevented.
The slip situation, however, can be controlled to obtain a positive effect by surface technology. Coating and texturing technologies can be used to engineer large slip. In practice, a large slip can be made using super-hydrophobic surfaces. Such surfaces can be manufactured by grafting or by deposition of hydrophobic compounds on the initial surface at a certain zone. Super-hydrophobic surfaces were originally inspired by the unique water-repellent properties of the lotus leaf. It is the combination of a very large contact angle and a low contact-angle hysteresis that defines a surface as super-hydrophobic. Implementing the slip property (hydrophobicity) on a surface in a wide range of application for the mechanical components is of great challenge by numerous authors recently. In published works [15-23], both experimentally and numerically, slip surface is able to reduce friction force at the contacting surfaces and finally reduce energy consumption, increase component's life-time and reduce economic and environmental costs.
1.2. Problem statement
In classical liquid lubrication it is assumed that surfaces are fully wetted and no-slip occurs between the fluid and the solid boundary. In MEMS, this wetting is actually an unwanted process because it can encourage the occurrence of stiction and as a result micro-parts can not be moved . It is expected that slip can reduce the friction and improve the load support. However, with respect to the engineered slip pattern, the choice of slip zone on a certain surface must be taken carefully in relation to such tribological performances. In other words, an inappropriate slip zone pattern on a certain surface or the election of inappropriate surface containing a slip situation may lead to the deterioration of the lubrication performance. How to control the boundary slip in the application of a lubricated-MEMS is one of the challenging tasks in the future. This chapter will explore the provision of a new lubrication model based on the continuum approach for moving parts in MEMS in order to improve the tribological performance of lubricated contacts. In MEMS, by lubrication, low friction force and high load support are the goals which want to be achieved. The artificial slip surface will be introduced as one of the solutions to improve the lubrication performance of MEMS so as MEMS with a longer life-time can be obtained. The term "artificial slip surface" is used to address a non-homogeneous engineered slip/no-slip pattern, i.e. a surface consisting of a slip zone and a no-slip zone.
2. Research methods
Full film lubrication of lubricated contact is often described by the Reynolds theory . According to the classical Reynolds theory, no-slip boundary is assumed and the convergent geometrical wedge is one of the most important conditions to generate hydrodynamic pressure. Therefore, the lubrication model for lubricated MEMS will be an extension of the classical lubrication theory. This means that modeling the lubricant through very narrow gap, normally modeled by assuming no-slip at the boundaries will be modified by introducing a boundary slip.
2.1. Modified reynolds equation
The classical Reynolds equation that is valid under no-slip condition can be generalized for taking into account slip conditions. It is then possible, for any film height distributions, to calculate the pressure distribution and the shear rate profile. The model of lubrication presented here is based on the fact that slip of the lubricant will exist at the interface of a lubricated sliding contact. Thus, a boundary slip is employed both on the moving and stationary surface, see Figure 1. The proposed lubrication model with slip leads to a modified Reynolds equation as presented in Eq. (1).
The physical meanings of the symbols in Eq. (1) are as follows:
Eq. (1) is derived by following the usual approach to deduce the Reynolds equation from the Navier-Stokes system by assuming classical assumptions except that boundary slip is applied both on the stationary surface and moving surface as depicted in Figure 1.
Eq. (1) can be derived by considering the equilibrium of an element of fluid.
and in this case are either constants or functions of
This velocity is used to compute the flow rate,
When full film lubrication is assumed, the entire load
The friction force
The simulation results will be presented in dimensionless form, i.e. for the dimensionless pressure, for the dimensionless load support (where
2.2. Solution method
The modified Reynolds equation, Eq. (1) is discretized over the flow using the finite volume method, and is solved using the tridiagonal matrix algorithm (TDMA), . By employing the discretization scheme, the computed domain is divided into a number of control volumes using a grid with uniform mesh size. The grid independency is validated by various numbers of mesh sizes. An assumption is made that the boundary pressures are zero at both sides of the contact.
A numerical simulation is conducted to investigate the possible application so as a boundary slip can be beneficial to achieve a high load support and low friction force. In order to maximize the performance of lubrication, the boundary conditions (slip zones,
The parametric analysis is performed using a developed computer code to investigate the effect of various slip parameters on the lubrication performances (load support, friction force, and friction coefficient). A parametric study is conducted with the variation of slip parameters (slip zone and slip length) over a large range of values considering different performance parameters. The design variables and the objective function are referred to as the optimization variables. The design variables are independent quantities which are varied in order to achieve the optimum design. The objective function is the dependent variable that is maximized, i.e. the load support. In the present study, the design variables are slip zones as indicated in Figure 1. The algorithm used in the present study is depicted on Figure 2.
3. Key results
The behavior of traditional (no-slip) hydrodynamic lubrication between the opposing surfaces can be estimated by a classical form of the Reynolds equation. The derivation of the classical Reynolds equation is based on the assumption of no-slip between the lubricant and the surfaces. In the classical Reynolds lubrication, the mechanism to generate a pressure is due to the convergent wedge effect. An artificial slip surface presented here is designed to be able to carry the external load during lubrication even if the wedge effect is not present. This situation is very beneficial in designing lubricated-MEMS which exhibits parallel gaps.
In this chapter, there are two main investigations. At first, the study is conducted in order to validate the developed numerical scheme. It assures that the numerical method used can be employed for solving other hydrodynamic characteristics. The no-slip case of lubricated contact is of main interest due to the availability of the analytical solution. Secondly, the study will be extended to explore the effect of the slip zone of the artificial slip surface on pressure, load support, friction force, and friction coefficient. The comparison between the modified sliding contact containing an artificial slip surface and the traditional one is conducted in order to describe the benefit of the use of an artificial slip pattern quantitatively.
3.1. No-slip condition
The modified Reynolds equation (Eq. (1)) is the governing equation for the fluid lubrication system containing a boundary slip. If the slip coefficient, , is set to zero, Eq. (1) reduces to the classical Reynolds equation. In this section, in order to validate the developed computer code containing a numerical scheme using finite volume method combined with tridiagonal matrix algorithm (TDMA), the classical Reynolds equation (no-slip condition) is solved numerically for calculating the pressure distribution, and finally the friction in a lubricated sliding contact as depicted in Figure 1 and Table 1, respectively. These results are compared with the analytical solution based on the work of Cameron  as follows:
for the pressure distribution where
for the friction force per unit width.
In Figure 3 the numerical results obtained with TDMA as well as analytical results for the dimensionless pressure distribution along the bottom wall of the contact are shown alongside those obtained from the Reynolds approximation. The wedge ratio
The comparison between the dimensionless friction force
|Analytical solution ||0.77|
|Numerical prediction [present study]||0.77|
3.2. Artificial slip surface
In MEMS, liquid lubrication has generally been omitted due to high hydrodynamic friction force that occurs in fluid film. Compared with a solid coating, stiction prevention using liquid lubrication is less practical. However, recent studies have demonstrated that it is possible for Newtonian liquids to slip along very smooth solid walls  and this result may make liquid lubricants for MEMS devices feasible. The main advantage of a liquid lubricant over a solid lubricant is that they generally produce no-contact shear stresses. Unfortunately, a stiction-type failure due to a large shear and capillary forces occurs. In [15, 16], a lubrication model of low load contacts was proposed to reduce such stiction or friction. The idea behind that work was how to use a lubricant that does not wet one of the solid surfaces. It was found that a half-wetted bearing generates a significant friction reduction compared to a traditional bearing.
In order to reduce stiction, two principal methods are available, chemical and physical modification of the surfaces. To generate wall slip, in the chemical approach, the chemical composition of the surface is altered. In the physical approach, the surfaces are roughened to decrease the effective contact area .
In practice, the slip zone of the artificial slip surface can be prepared from (super)hydrophobic surface which uses chemical properties as well as micro- and nano-structures in order to achieve a high level of friction force reduction. The main characteristic of (super)hydrophobic surfaces is the slip length. Extensive studies have confirmed that the chemical treatment of the surfaces generates a slip length in the order of 1 µm , while longer slip length up to 100 µm can be obtained through a combination of a hydrophobic surface with textured structure [20, 31, 32]. In the present study, it will be shown by the computational analysis that a longer slip length applied on the slip zone of the artificial slip surface leads to a greater friction force reduction in combination with an improved load support.
3.2.1. Beneficial surface of slip
Recently, the use of an engineered slip surface has become popular with respect to lubrication, since this type of surface enhancement would give a better tribological performance. The great challenge for an engineered slip surface from the perspective of a numerical simulation is choosing the optimal slip zone geometry with respect to the lubrication performance. Two engineered slip surface modes were used currently: homogeneous slip surface (i.e. slip applied over the whole surface) and artificial slip surface (i.e. surface consisting of slip zone and no-slip zone). It can be noted that term “artificial slip surface” was sometimes also called as heterogeneous slip/no-slip surface [17, 18] and mixed slip surface . The first study to mention using a homogeneous slip was dedicated by Spikes [15, 16] who numerically studied the effect of slip profiles on friction. The author pointed out that by introducing the half-wetted bearing having a homogeneous slip boundary on one of the surfaces, a reduced friction can be obtained. Subsequently, an experimental study was published in  confirming the finding of [15, 16]. However, in addition to the friction reduction, it was shown that a homogeneous slip surface usually has a negative effect, i.e. the decrease in the load support. If the lubricated contact exhibits a perfect slip property, it was found that the fluid load support was only half of that without slip [115, 19, 21, 23]. Clearly, this is unwanted effect with respect to the lubrication. Therefore, to date, an artificial slip surface has become of great interest by some researchers [17-19, 21, 23] with the focus of how to balance the slip effect on the load support and friction.
The big question with respect to the tribological performance of lubricated-MEMS emerges in accordance with at which wall boundary slip must be applied, at the stationary surface, moving surface, or both of them. Besides that, the types of slip zone pattern become also great issue. Therefore, a series of simulations were conducted with such boundaries to find the best possibility of slip boundary application in terms of load support. Investigations were made for four kinds of slip boundaries to find the best boundary slip in terms of tribological performance, i.e. (1) slip applied on both the stationary and moving surfaces is referred as 'condition 1', (2) slip applied on the stationary surface is referred as 'condition 2', (3) slip applied on the moving surface is referred as 'condition 3', and (4) no-slip condition applied on the both of surfaces is referred as 'condition 4'. Here, a homogeneous slip surface is employed for all slip conditions.
Figure 4 presents the effect of the wedge ratio
3.2.2. The optimum slip zone of the artificial slip surface
This section is intended to investigate the optimum slip zone of the artificial slip surface for several values of wedge ratios
In the following computations, as discussed in the previous section, for a high load support, it is considered that the artificial slip surface will take place on the stationary surface, whereas no-slip condition occurs on the moving surface. The parameter
Figure 7 shows the normalized representation of lubrication film pressure distributions as a function of wedge ratio which are predicted by the modified Reynolds equation (Eq. (1)). For slip configuration, the optimal slip zone
Figure 8 shows the dimensionless surface friction force
3.2.3. Effect of dimensionless slip length on lubrication performance
The hydrophobicity of a solid surface, as discussed in the previous section, is usually expressed in terms of a slip length, which quantifies the extent to which the fluid elements near the surface are affected by the surface energy and the surface geometry. The surface energy is an intrinsic property of a material that can be controlled by chemical treatment, such as etching approach and/or coat-on/cast approach. The surface roughness of a hydrophobic solid material can be tuned in order to increment its hydrophobicity and obtain a super-hydrophobic solid surface [33, 34]. In this section, from the numerical point of view, the effect of slip length on the lubrication behavior is studied. The dimensionless slip length is varied from 3 to 300.
Figure 9 shows the effect of slip zone of the artificial slip surface for several dimensionless slip length values on the dimensionless load support. As indicated in Figure 9, the increase in the slip length leads to an increase in the predicted load support. Generally, the larger the slip length at the optimized slip zone of the artificial slip zone, the higher the load support. However, when dimensionless slip length is larger than 30, the dimensionless load support is not affected significantly with the increase in the dimensionless slip length. So, the increase of the load support is not infinitely large. It can be deduced that there is no fluid load support for a lubricated-MEMS when it contains no-slip condition (
In Figure 10 the effect of slip zone of the artificial slip surface for several dimensionless slip length values on the dimensionless friction force is presented. As is well-known, the ability to control and manipulate friction force during sliding is extremely important key to prolong a life-time of lubricated-MEMS. Better understanding of the friction force phenomena at micro-scales is needed to provide designers and engineers the required tools and capabilities to control friction force and predict failure of lubrication in MEMS.
As can be seen in Figure 10 that the artificial slip surface leads to a reduction of the friction force for all dimensionless slip length. The friction force decreases with increasing the slip zone
The combined effect of slip zone parameter on load support and friction force can be better analyzed using the dimensionless friction coefficient. In the present study the dimensionless friction coefficient
4. Concluding remarks
Numerical results show that the hydrodynamics of a lubrication film confined between a moving no-slip surface and a stationary with an artificial slip surface differ significantly from that of a film confined between two no-slip surfaces. It is found that a homogeneous slip boundary on one surface produces a lower hydrodynamic pressure in a lubricated sliding contact at various conditions (slope incline, and slip length), resulting in a reduced load support which reduces the positive effect of slip on friction. However, if the surface is designed with an optimal artificial slip pattern (the slip zone is applied on 0.65 of contact length), even when there is no wedge effect, the load support has a maximum value. In addition, the friction force can decrease significantly. Therefore, it is very beneficial to make one of the contacting surfaces in lubricated-MEMS with an artificial slip surface for achieving ideal lubrication performance, i.e. reduced friction coefficient and increased load support.