Adhesion models: From single to multiple asperity contacts

Advances in Colloid and Interface Science 168 (2011) 210–222

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Advances in Colloid and Interface Science j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c i s

Adhesion models: From single to multiple asperity contacts Polina Prokopovich a,c,d,⁎, Victor Starov b a

Institute of Medical and Biological Engineering, School of Mechanical Engineering, University of Leeds, Leeds, LS2 9JT, United Kingdom Chemical Engineering Department, Loughborough University, Loughborough, LE11 3TU, United Kingdom c Welsh School of Pharmacy, Cardiff University, King Edward VII Avenue, Cardiff, CF10 3NB, United Kingdom d Institute of Medical Engineering and Medical Physics, School of Engineering, Cardiff University, Cardiff, United Kingdom b

a r t i c l e

i n f o

Available online 25 March 2011 Keywords: Adhesion Meniscus force Multi-asperity adhesion models JKR DMT

a b s t r a c t This review presents a summary of the current adhesion models available to date, between real rough surfaces, starting from single asperity models and expanding to multiple asperity contacts. The focus is made on multi-asperity contact interactions. Both van der Waals and contact mechanics approaches have been considered and relevant adhesion models are reviewed and discussed. The influence of the meniscus forces on adhesion has been considered, along with a summary of the various meniscus models. The effect of surface geometry, its topography and environmental conditions on meniscus action are also discussed along with its integration into multi-asperity adhesion models. © 2011 Elsevier B.V. All rights reserved.

1. Introduction The problem of contact and adherence between elastic solids has a long history [1–6]. However, interest in adhesion processes has increased immensely during the past 10–15 years. Approaches to control the force of adhesion between elastic materials are required in a number of industrial and medical applications [7–13]. The interest is stimulated by a broad occurrence of adhesion in everyday life, ranging from walking and driving to holding a sheet of paper and attaching a band-aid [14]. Contact and non-contact molecular interactions between two bodies, the extent of the contact zone and the force of adhesion play an important role in many natural and industrial processes such as cell–cell and cell–material interactions [15,16], microfluidics [17], drug delivery systems [7–10] and medical prostheses [12,13]. Adhesion also has a significant influence on friction and wear performance of contacting surfaces [17–21]. Adhesion is important in magnetic storage devices, in micro/nanomechanical systems (MEMS), biological micro-electromechanical systems (bioMEMs) and other miniaturised devices. Important characteristics, such as efficiency, power output and the steady state operation of these systems and devices are fundamentally affected by adhesion properties [22,23]. For such miniaturised devices, where the contact between a pair of meshing teeth is of the order of a few to thousandths of micrometres squared, with separations of a few nanometers, interfacial effects become more and more important and dominate over inertia and gravitation. Adhesion plays a decisive role ⁎ Corresponding author at: Welsh School of Pharmacy, Cardiff University, Redwood Building, King Edward VII Avenue, Cardiff, CF10 3NB, United Kingdom. E-mail addresses: [email protected], [email protected] (P. Prokopovich), [email protected] (V. Starov). 0001-8686/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cis.2011.03.004

in wear and frictional performance of a wide range of materials, including medical devices [9,10,24,25]. Adhesion is equally important and occurs widely in the natural environment. For example, various organisms, like lizards, geckos and insects use their attachment mechanisms to adhere to and detach from both biotic and abiotic surfaces. This capacity of those creatures to reversibly adhere to surfaces is referred to as “smart adhesion” [26– 31]. The surfaces of real solid materials exhibit some irregularities, irrespective of the preparation methods they have undergone. The highest points on the surface are called asperities, while the lowest points are referred to as valleys. When two nominally flat surfaces are put into contact, the latter occurs between pairs of asperities on opposite surfaces due to surface roughness. As a result, the real area of contact is much smaller than the apparent contact area. The proximity of asperities results in adhesion forces due to interatomic interactions. The adhesion of those asperities contributes to frictional forces, when two surfaces slide along each other. Repeated surface interactions at the interfaces can lead to the formation of wear particles and failure of the device [32]. Therefore, modelling of multi-asperity adhesion contact interactions is crucial in understanding friction and wear behaviour of materials and preventing the possible failure of different mechanisms. 2. Single asperity adhesion models The classical sphere–sphere or sphere to flat plane approaches are widely used as adhesion models in systems with a single asperity. A number of techniques have been used to fundamentally predict interactions of ideal spherical surfaces (perfectly smooth) based on van der Waals adhesion (Hamaker, 1937) or through surface energy

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based approaches such as Johnson-Kendall-Roberts (JKR) [33], Derjaguin-Muller-Toporov (DMT) [34] or Maugis adhesion theories [35]. These are based on two fundamentally different methods: (i) utilisation of the attractive van der Waals forces, leading to adhesion; (ii) the method, which includes the DMT, JKR and Maugis models, considers the elastic response of the interacting bodies. Adhesion, based on van der Waals forces is reviewed in Section 1.1 while adhesion models, based on contact mechanics are discussed in Section 1.2. 2.1. Adhesion models based on van der Waals approach 2.1.1. van der waals forces Molecules of a fluid at the interface with another medium are subject to adhesive forces. These forces are in addition to the cohesive forces between molecules themselves, which occur in the bulk of the fluid, away from the interface. Therefore, the molecular organisation (packing) near another medium is different from that in the bulk. The term, ‘adhesion force’, in this context is different from that used in other sections below. At the moment, it is a general term that describes a range of forces acting at a close proximity to interfaces. According to Israelachvili [36], the short-range forces, which arise during a close approach between two surfaces (5–10 molecular solvent diameters apart) determine the adhesion between such surfaces and the properties of colloidal dispersions. At those distances, the interactions become sensitive to the molecular structure of both the liquid and the surfaces. There are intermolecular forces, such as London–van der Waals force, electrical double layer forces, solvation or hydration forces, hydrophobic and steric forces that all usually act on a scale of a few nanometres. The attractive force, at close particle- or asperity-substrate separations, is the van der Waals interaction amongst these forces. There are two approaches available in the literature to calculate the van der Waals forces between surfaces; the macroscopic (Hamaker (1937) [3]; Derjaguin and Landau (1933) [37]) and the microscopic (Lifshitz (1956) [4] and Dzyaloshinskii (1961) [5]). The first method is based on pair-wise additivity, where the influence of neighbouring atoms on the interaction between any pair of atoms is ignored. The overall interaction force represents the sum of the dispersion energies for all contributing atoms. The attractive force between a sphere and a flat surface is determined in [38]: FW ðdÞ = −


 A H R4 2dW ; 1 + R 6d2W ðdW + RÞ3

ð1Þ

where AH is Hamaker constant, which is defined as AH = π2Cρ1ρ2; where C is the attractive interaction strength and i =1,2 is the density of the molecules in the solid (1 or 2); dW is the separation distance, R is the radius of curvature of the asperity. It is usually adopted that asperities are hemispherically tipped [25,39]. Therefore, the latter equation can also be employed to predict the attractive force of an asperity on a flat plane. An asperity is, therefore, modelled as a sphere with a radius equivalent to the radius of curvature of the asperity tip. An atomic structure is neglected and bodies are treated as continuous media, according to the above macroscopic approach. Eq. (1) gives a reasonable qualitative dependence on dW and R. However, calculations of the Hamaker constant, according to this approach, can be both qualitatively and quantitatively incorrect. Lifshitz et al. microscopic theory [4] agrees well with all known experimental data obtained from attractive force measurements between smooth glass surfaces [40]. 2.1.2. The effect of surface roughness on micro and nanoscale adhesive forces Numerous efforts have been made to model van der Waals and electrostatic interactions between a single asperity or a particle and a surface, on both nano- and macro-scales.

211

The Derjaguin-Landau-Verwey-Overbeek (DLVO) [41,42] theory is the most accepted and utilised theory which describes those forces between an ideally regular and smooth spherical solid in liquid media. However, real surfaces contain geometrical irregularities and morphological heterogeneities. Note, the DVLO theory can only be applied to simple systems such as in the case where only the van der Waals forces determine the total interaction (i.e. interactions in vacuums, non-polar wetting films on surfaces and particle interactions in nonpolar media) [43]. A number of researchers have reported the deviation of experimentally measured forces from the ones predicted using DLVO theory [42–47]. The cause of these discrepancies has been linked to nonidealities in the geometry and roughness of the particles and surfaces. As a consequence, many attempts have been made to model these effects [48–50]. Several authors have incorporated roughness into theoretical models of van der Waals interactions [38,47,51,52]. Roughness has been described using model surfaces, such as hemispheres or cone-shape asperities [38,47], fractal surfaces [53–56] and waves of sinusoidal asperities [57]. In these cases, adhesion forces were calculated by summing up the interactions between asperities on the particle and the surface. It was observed that there was a significant deviation in the case of rough surfaces, compared to perfectly smooth surfaces. These publications support the requirements of a “true” representation of geometry and roughness of contacting surfaces for accurate predictions of adhesion forces. There are mainly two different approaches, involving van der Waal's forces, which include surface roughness. One approach integrates the effect of each asperity over the distance between the two bodies. The other approach assumes that, for rough surfaces, only a fraction of the surface is contributing to the contact. Therefore, the results of the pure van der Waals theory are reduced by using a coefficient that depends on both the density and the geometry of an asperity. The effect of the surface roughness is negligible for separation distances which are one order of magnitude greater than the size of the asperity, as shown by Sparnaay (1983) [58]. Sparnaay calculated the non-retarded, van der Waals interaction between conical asperities and a smooth flat plate and demonstrated that roughness was significant when the size of the asperity was more than 10% of the separation distance. All those researchers modelled the interaction between a sphere and a plane as that between a flat and a rough surface. In some cases, the asperities are placed on the flat surfaces, whilst in others, the asperities are located on the spherical particle. For example, Suresh and Walz (1996) [49] developed a set of analytical equations, which described van der Waals and electrostatic interactions for a rough spherical particle interacting with a smooth surface. They modelled the van der Waals force through a pair-wise additive method, under both retarded and non-retarded conditions. This model predicted a larger repulsive force than that estimated by the classical DLVO theory at large separation distances. However, at closer separations, it simulated a greater van der Waals attraction than that predicted by the DLVO theory. The model predictions have been found to be in good agreement with the experimental data obtained during studies of the DLVO force between PSL colloids and BK-7 glass slides, using total internal reflection microscopy (TIRM). One main limitation of those models is the assumption that the asperities are evenly distributed on the surface and that the asperities are of a constant shape. These limitations have been removed in a theory proposed by Bhattacharjee et al. (1998) [47]. The theory employs a surface element integration technique to model DLVO interactions between surfaces containing “morphological heterogeneity” or roughness. Predictions based on later models, deviated significantly from those for smooth surfaces, particularly at very close separations, reiterating

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the findings of Sparnaay (1983) [58]. Bhattacharjee et al. (1998) [47] also modelled a situation in which both the flat surface and the particles contain surface imperfections. It has been assumed that the asperity radii of curvature are normally distributed. However, many materials present asperities with radii of curvature that are not normally distributed [59,61]. Cooper and various co-workers [38,52,62] developed theories, based on a pair-wise additive approach, to account for the effect of surface roughness on the van der Waals forces. The influence of the surface was considered as an uneven factor and thus, this reduced the size of the surface involved in the interaction. Consequently, the force of a rough surface was reduced, compared to a flat one. Asperities, which were not in contact, were neglected. These models were also validated using AFM measurements. In the case of smooth spherical colloids, Cooper et al. (2000) [38] used the Derjaguin's approximation to calculate the force. However, this approximation cannot be applied to the asperities, as their radii are comparable in size with the separation distance. Cooper et al. (2000) [52] studied the adhesion of micron-scale particles to substrates, in systems where chemical reactions are occurring. Their model and experiments showed a substantial effect of surface roughness on the adhesive force. Changes in roughness of the substrate were found to change the interaction force by nearly 90% [38]. A year later, Cooper et al. (2001) [62] extended their van der Waals analysis [38,52]. They took into account the effects of particle and substrate surface morphology and mechanical properties. Their model described the interaction between asymmetrical particles and surfaces. The authors found that this more accurately predicted the adhesion force, than predictions based upon an ideal van der Waals model. Another work by Czarnecji and Dabros (1980) [51] seemed to support the approach of a proportional reduction of the adhesive force when roughness is considered as the ratio between the interaction energy of a rough particle with a smooth particle. Statistical analysis of the surface irregularities was utilised in order to obtain such a correction factor. Their approach is valid for separation distances which are much larger than the asperity height. Therefore, it is applicable for non-contact dispersion forces and not adhesion forces. As the size of the solid is reduced from micro- to nano-scale, the surface topography and its geometry play an increasingly important role in the adhesion behaviour. In the nano-scale region, the adhesion forces are described by the Rumpf's model (1990) [64]. This is based on an interaction of a single hemispherical asperity, with a much larger spherical particle, along a line normal to the surface which connects their centres. This model includes two terms that describe the total van der Waals interaction. The first term represents the interaction of the adhering particle in contact with the asperity. The second term describes the “noncontact” force between the adhering particle and a flat surface, separated by the asperity. For both interactions, Rumpf derived the following equation [40]: FRumpf =


 Rpart AH ; 1+ 2 6z0 ð1 + R =z0 Þ

ð2Þ

where AH is the Hamaker constant, Rpart and R are the radii of the adhering particle and asperity, respectively, and z0 is the distance of the closest possible approach between surfaces (approximately 0.3 nm). The main limitation of this model is that the centre of the hemispherical asperity is required to be at the surface. Xie (1997) [65] developed two geometrical models, where van der Waals forces were modified and taking into account only the asperity radius. In the first model, Rumpf's approach was developed further. However, the interaction of the particle with the asperity on the surface was ignored. According to the second model, which was referred to as a “sandwich model”, the asperity was assumed to be a

small particle positioned between two larger surfaces. In the second model, the surface can be assumed to be smooth, with the asperity radii smaller than 10 nm. It has been shown in [66] that the Rumpf model does not accurately describe the surface on the nano-scale level. Rabinovich [42–67] developed a more accurate model that is capable of quantitatively predicting adhesion forces between a particle and an asperity on the surface. The mathematical representation of this model is as follows: 3

2 FRab

AH Rpart 6 6 = 6 6z0 41 +

1 58:144Rpart Rq λ1

+

7 1 7  2 7; 1:817Rq 5 1+ z0

ð3Þ

where λ1 is the asperity to asperity distance; Rq is the RMS surface roughness. Several surface roughness parameters, such as: asperity height and breadth were introduced into the model. This allowed more realistic representations of the surface topography and a quantitative prediction of adhesion force. This turned out to be two orders of magnitude greater than that obtained using the previous models, based on Rumpf's approach. Adhesion force has been found to be very sensitive to small variations in surface roughness. A significant decrease in adhesion force was found, for a variety of samples, with only a small increase in surface roughness of 1–2 nm [66,67]. Rabinovich's model was developed using only van der Waals forces. However, the proposed geometry can be extended to other adhesion models, based on contact mechanics. An initial decrease in the adhesion force was found, with an increase in roughness. This was followed by a gradual decrease in the adhesion force, when only the interactions during contact between the particles and asperities were considered and the contributions from the interaction of the particles with the underlying solid surface were ignored [63]. In this latter publication the interaction between a smooth glass particle and glass substrate, with roughness values ranging from 50 to 400 nm, was measured. Recently, Jaiswal et al. (2009) [68] presented a new method which models adhesion at the nano-scale. The authors extended their existing microscale adhesion approach [38,52,62] and incorporated the exact geometry and the precise surface morphology of the interacting bodies, using continuum models. They described precisely the spatial configurations of the adhering surfaces by superimposing the roughness and geometry models for the particle and the substrate. The interacting surfaces were then discretized and the adhesion force between the two surfaces was calculated, using Hamaker's additive approach, based on van der Waals interactions. The latter study showed that the continuum models are applicable for particles with sizes down to approximately a few tens of nanometers. The above-mentioned theoretical models, have been developed to account for the effects of surface roughness on van der Waals forces, for non-contact interactions. These models do not consider elastic deformation of the surfaces. The latter can lead to underestimation of adhesion. Once an asperity is in contact with the surface, additional factors must be taken into account. The most significant of these is the asperity deformation. This aspect and associated models are discussed in Section 2.2. 2.2. Adhesion models based on JKR, DMT and Maugis models Hertz (1882) [69] developed an adhesionless contact model for a circular point contact. He described the deformation versus contact radius as: δHerz =

a2 ; R

ð4Þ

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where a is an asperity contact half-width and R is the radius of the asperityand the relationship between the force and the contact radius is as follows: 4Ea3 ; 3R

FHertz =

ð5Þ

 2  2 −1 1 2 where E is the reduced Young modulus E = 1−υ + 1−υ where E1 E2 E1, E2 are the Young moduli, ν1, ν2 are the Poisson ratios of those materials, respectively. All adhesion models were advanced further using the Hertz approach. Initially, adhesion was considered by Bradley [1] and Derjaguin [2]. They independently studied the role of surface forces in the adhesion of spherical particles. The main difference between these two models is that Bradley included the full attractive and repulsive contributions to the interaction potential. Derjaguin considered the attractive part of the potential, but assumed that the repulsive contribution is steep, that the surfaces cannot interpenetrate. Derjaguin (1934) [2] was the first to consider the adhesion of spherical, elastic particles and made allowances for their deformation and the action of surface forces. Johnson (1958) [4] studied the interaction between the elastic spheres and allowed for the surface forces. He demonstrated that the pressure distribution across the contact surface differs from that obtained by Hertz and that it can be represented in the form of a superposition of solutions found by Hertz and Boussinesq [70]. Later, Johnson, Kendall and Roberts (JKR) [33] studied the interaction between elastic spheres and used the pressure distribution found by Johnson [4]. The JKR model assumed that the shortrange surface forces act only within the contact area (Fig. 1a). The JKR model comprises a deformation contribution, according to the Hertz model and contains an adhesion component, due to surface energy Δγ, defined in [71] as: FJKR =

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4Ea3 − 8πa3 ΔγE; 3R

ð6Þ

tot tot tot where: Δγ = Δγtot 1 + Δγ2 , Δγ1 and Δγ2 are the total surface energy of the two materials . The deformation according to the JKR model can be expressed as:

δJKR =

a2 2 − R 3

rffiffiffiffiffiffiffiffiffiffiffiffi πaΔγ : E

ð7Þ

The authors deduced that the force of adhesion is determined by the following equation: Fcð JKRÞ =

3 πRΔγ: 2

ð8Þ

An alternative approach was suggested by Derjaguin and colleagues (DMT) [34,72–74]. The authors introduced the non-contact forces, i.e., the forces of molecular attraction, which act across the gap between the two bodies (Fig. 1b). It was assumed that the molecular attraction forces do not change the shape of the particle, corresponding to the solution of the Hertz problem. The DMT model assumes the contact radius, according to the Hertzian theory and considers the action of the van der Waals forces along the contact area. In this case, Eqs (7) and (6) become: 2

δDMT =

a ; R

FDMT =

4Ea3 −2πRΔγ; 3R

respectively.

ð9Þ

ð10Þ

Fig. 1. Regions of surface forces action.

In the absence of any deformation, the maximum value for the adhesion force, according to the DMT model, is located at the point of contact: Fc ðDMT Þ = 2πRΔγ:

ð11Þ

Tabor [49–75] analysed in detail the results obtained in [33] and [34,72–74] and explained the latter in terms of the range of action of the surface forces. He showed that the adhesion force should vary from 1.5πRΔγ to 2πRΔγ, depending on the height of the “neck” in the profile determined in [33]. Based on this, he introduced the nondimensional parameter, μT, called Tabor's parameter, which varies depending on the value of the “neck” height. According to [75,76], for μT ≥ 1, when the “neck” is high, the non-contact forces of attraction can be ignored and the adhesion force acquires the value of 1.5πRΔγ, which is found in [33]. If the “neck” height is low, the noncontact forces of attraction cannot be ignored and the adhesion force, in this case, becomes equal to 2πRΔγ, the value found in [34,72–74].

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Therefore, the Tabor's parameter is generally used to differentiate between the JKR and DMT models. Later, Muller, Derjaguin and Toporov [77] once again considered the interaction between an elastic sphere and a hard plane, assuming the Hertzian profile for the contact. Retaining the main content of the previous works [34,72–74], the authors also treated the problem of pressure discontinuity at the contour of pressure surface. They resolved this problem by introducing the short-range forces of repulsion. Hughes and White [78,79], as well as in [80,81] further developed the theoretical study of interactions between elastic bodies. In [78,79] the deformations in the contact zone of convex bodies have been taken into account more rigorously, and the nonlinear integral equation, describing normal components of displacement, has been solved. In [80,81] the internal deformations of a crystal lattice have been studied. It was shown that, at small distances and weak forces of attraction, the deformation can be unstable and the surfaces can come into direct contact in a jump-wise manner. In subsequent years, researchers concentrated their attention on the verification of the conclusions of the Hertz and JKR theories, using both numerical solutions of the integral equation for the normal component of displacement and various laws of interaction between the surfaces [82–84]. Results obtained in those publications actually confirmed the conclusions of previous works [75,76] described above. These concluded that the Hertz's and DMT theories [34,72–74] are valid for the “hard” sphere, whereas the results of JKR are true for the “soft “sphere [33]. An intermediate model was proposed by Maugis in 1992 [35]. This takes into account the effect of surface forces in a ring zone surrounding the Hertzian region of contact, between a pair of hemispherical asperities (Fig. 1c). Thus, the model can be used for any type of contacting materials for both high and low adhesion. According to the Maugis model, the adhesion is determined by the parameter λ:  λ = 2σ0

R πΔγE2

1 = 3

;

ð12Þ

where σ0 is a constant adhesion stress. The relationship between the Tabor parameter, μT, and the Maugis parameter, λ, is as follows: λ = 1:1570 μT :

ð13Þ

λ is often referred to as the transition parameter. The JKR model applies at λ N 5 and the DMT model is valid at λ b 0.1. Values between 0.1 and 5 correspond to the transition regime between the JKR and DMT models. Fig. 2 shows force-penetration depth variation for a pair of asperities for a range of values of the Maugis transition parameter, λ. Note, that whilst Maugis's model applies to the full range of asperity compliance, those of the JKR and DMT models represent the extremes of the whole spectrum. The normalised parameters of an asperity contact using a halfwidth, force and deformation were introduced, in order to compare these three models.

a=

F=

a πR2 Δγ E

F ; πRΔγ

!1 = 3 ;

ð14Þ

ð15Þ

Fig. 2. Force vs penetration depth for the DMT, JKR and Maugis models for different λ values. Redrawn from [188].

δ=

δ π2 RΔγ2 E2

!1 = 3 :

ð16Þ

The quantitative summary of the DMT, JKR and Maugis models is shown in Table 1. The Maugis correction to the Hertz problem solution is expressed implicitly via parameter m. This is the ratio of the half-width of the asperity contact, a, and an outer radius, c, at which the gap between the surfaces reaches δt (i.e., where the adhesive stress no longer acts). In 1995, Maugis [85] extended the JKR theory. This was valid for elastic spheres, with both small and large contact radii. Using the exact expression for the sphere profile, he revealed that the contact radius, under zero load, is large and that the particle/asperity radius varies to the 3/4 power rather than 2/3 power required by the JKR theory. Maugis' extension to the JKR model was in good agreement with experimental results [86–89]. Carpick and co-authors [90] simplified the analysis by introducing a numerical approximation and providing a load-displacement curve-fit method for nondimensional data. They derived the following equation in a nondimensional form: a = a0

α+

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2 = 3 1−F =Fc ; 1+α

ð17Þ

where α is the transition variable, α = 1 gives the JKR relationship, α = 0 corresponds to the DMT relationship and the range 0 b αb1 characterises the transition zone in between. The latter equation is referred to as a generalised transition equation and applies to the adhesion model for the JKR and MT limits and in between. Carpick's method allows a convenient and easy comparison of analytical solution, with experimental data obtained from indentation or AFM. Later, a full analytical solution was given by Schwarz [91]. Table 1 DMT, JKR and Maugis models quantitative comparison. Model

Normalised equations

DMT JKR

2 F = a3 −2;pδffiffiffiffiffi= pffiffiffiffi ffi a F = a3 −a 6a; δ = a2 −2 36a

Maugis

2 1 = λa 2

hpffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffii pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffii 2h

m2 −1 + m2 −2 arctan m2 −1 + 4λa 1−m + m2 −1 arctan m2 −1 ; 3 j k pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 m2 −1 + m2 arctan m2 −1 ; F = a −λa pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 m2 −1 δ = a − 4λa 3

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Greenwood [83] and Feng [92,93] analysed the adhesion behaviour over the complete range of the Tabor's parameter and assumed a surface force law based on the Lennard-Jones 6–12 potential law. An interesting analysis has been made by Greenwood and Johnson [94]. The authors presented a map delineating the regions of applicability of the different theories and they showed that the actual domain of applicability of the JKR theory is much wider than it was earlier assumed to be. Recently, Yao and others [95] repeated the numerical calculations, but used an exact sphere shape, instead of the usual paraboloidal approximation. They found that the pull-off force could be less than one-tenth of the JKR value and modified the Johnson and Greenwood map correspondingly. However, all the abovementioned theories assume that the surface is perfectly smooth, which is not always easy to achieve in real life. Even the most engineered surfaces, regardless of the preparation method, possess some finite surface roughness (nanoscale roughness). The existence of nanoscale roughness (as small as RMS roughness equal to 1 nm) is known to dramatically reduce the force of adhesion between surfaces or between a particle/asperity and a surface. The latter is due to a decrease in the real area in contact and an increase in the distance between the bulk surfaces [66,67]. It is obvious that adhesion is a function of roughness and depends on the geometry, size and number of asperities distributed over the surface. Therefore, there is a need for an appropriate multi-asperity adhesion model which accounts for both asperity deformation and stretching when a pull-off force is applied. This will be discussed in Section 4.

less than the capillary constant), the Laplace equation can be written as:

3. Influence of meniscus force on adhesion

Π = Πw + Πe + Πs ;

3.1. Summary of existing models of meniscus force

where Πw is the pressure from the van der Waals forces acting between the film and the substrate; Πe is the ionic electrostatic component; Πs is the structural component, resulting from the water molecules having a different orientation and structure in the film, compared to the bulk liquid. In the case of thin, flat liquid films only, the disjoining/conjoining pressure determines the film thickness. However, in the case of a curved or rough surface, the disjoining pressure and capillary pressure act simultaneously. Starov [107,108] showed that in cases of partial wetting, this simultaneous action results in the existence of non-flat equilibrium liquid shapes. Mate (1992) [109] showed that for a liquid meniscus at equilibrium, with a liquid lubricant film, the capillary pressure of the meniscus is equal to the disjoining-pressure in the film. As a result of this, the meniscus force was defined as:

Meniscus forces (or capillary forces) play an important role in adhesion, especially at the nanoscale. Formation of liquid menisci, which can form around the contact area of two neighbouring asperities or particles, results in an increase in the adhesion force. In the case of nanoscale contacts, the contribution of a capillary force becomes comparable to the normal load [96,97]. Formation of meniscus forces can be caused by a number of different phenomena: capillary condensation, liquid menisci [98] and capillary bridges [99]. The formation of a meniscus for a single m asperity is shown in Fig. 3, where Rm 1 and R2 are the principal radii of curvature of the meniscus and s is its height. The Laplace equation [100] relates the curvature of a liquid interface to the pressure difference, Δp, between the two fluid phases. When gravitation is negligible (for small menisci with the length scale

  1 1 Δp = γ m + m ; R1 R2

m where γ is surface tension of the liquid/air interface and Rm 1 and R2 are the principal radii of curvature. The Kelvin equation [101] describes the capillary condensation and it relates the actual vapour pressure, p, to the curvature of the surface of the condensed liquid [102,103]:

RT ln

  p 1 1 = γVm m + m p0 R1 R2

ð19Þ

where R is the molar gas constant equal to 8.314 J mol− 1K− 1; T is the temperature; p0 is the saturation vapour pressure; Vm is the molar volume of the liquid. The important phenomenon contributing to the meniscus force is the disjoining/conjoining pressure. It was first introduced by Derjaguin [104,105] and is defined as the negative derivative of the Gibbs free energy per unit area, with respect to the film thickness. The disjoining/conjoining pressure can be expressed as that force per unit area that the molecules on the surface of a liquid film experience compared to the molecules on the surface of a bulk liquid. These forces act to thicken or to thin the liquid film, depending on whether they are attractive or repulsive. The disjoining/conjoining pressure has three main components [104–106]:

Mate

Fm

Fig. 3. Meniscus formation between an asperity and a flat surface.

ð18Þ

= Acont

AH 6πd3W

ð20Þ

! ð21Þ

where Acont is the flooded contact area, AH is Hamaker constant, dw is the separation distance. Note, the latter expression is valid in cases where van der Waals forces are the only forces responsible for the disjoining/conjoining pressure action. There are two important issues that deserve discussion, concerning capillary forces. Both the magnitude of the capillary force on adhesion and the environmental conditions, such as relative humidity, which allows a meniscus to form and enhance adhesion, need to be predicted and taken into account. A number of authors have investigated the effect of relative humidity on condensation in thin capillaries and the validity of Kelvin's equation at various conditions [110–125]. An early discussion of those phenomena, related to adhesion, was presented by Coelho and Harnby [126–129]. These authors predicted the pressure inside a meniscus, utilising Kelvin's and the Laplace equations and developed criteria for meniscus formation, in terms of the vapour pressure and the BET constants.

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Later, Fisher and Israelachvili (1981) [130] showed that these criteria overestimated the critical relative humidity. Butt et al. (2006) demonstrated that values of capillary forces can also be used to determine the geometry of the contact surfaces [131]. The basic meniscus models for different geometries, environmental conditions and separation distances have been extended in [132– 134]. These models allow more accurate predictions of the meniscus force and its influence on adhesion. 3.2. Influence of geometry and surface topography on meniscus force Marmur (1993) and De Lazzer et al. (1999) [132,133] discussed capillary adhesion for a variety of probe geometries, including spherical, conical and parabolic. The Marmur and de Lazzer models were developed further by Sirghi et al. (2000) [135]. Sirghi studied the incorporation of local curvature and suggested an analytical solution for capillary adhesion forces, using an approximation for spherical particle geometry and a symmetrical water meniscus at thermodynamic equilibrium. Sirghi et al. (2000) [135] compared the theoretical predictions with experimental results, measuring adhesion forces between a silicon nitride AFM tip and a platinum-covered quartz sample, with roughness in the order of 20 nm. Farshchi-Tabrizi et al. (2006) [134] calculated the meniscus force, based on the two sphere model which includes surface roughness and took into account different AFM probe shapes. The authors showed that the precise geometry has a crucial influence on the dependence of the adhesion force with humidity. Fisher and Israelachvili (1981) [130] investigated adhesion between smooth mica surfaces, in the presence of water and cyclohexane vapour, and directly measured the relative contribution of solid–solid interactions and the capillary effect on the total adhesion force. Rabinovich et al. (1991) [136] further explored this area in various semi-miscible liquids via direct measurements of the force of adhesion between fused quartz filaments. Other authors [137] have explored the effect of surface contaminants on measured forces and critical relative humidity. Experimental investigations were performed in [138,139] to study the influence of surface hydrophobicity and specific surface groups at the onset of a capillary force. However, the change in roughness, possibly associated with those surface modifications, was not considered by the authors. While Quon et al. (2000) [140] measured adhesion forces between gold-coated mica crossed cylinders, modified by alcohol or methyl self-assembled monolayers. The authors observed a considerable effect of surface roughness at nanometer scale, approximately 2-nm roughness, on the adhesion forces. Rabinovich et al. (2002) [141] studied the influence of capillary forces on adhesion between a smooth particle and a surface with nanoscale roughness. The authors found that nanoscale roughness significantly lowered the capillary forces. They derived the following equation for capillary force:

Rab

Fm


= 4πRγ cosðϑÞ 1−

 H ; 2r cosϑ

ð22Þ

ments. They found that the meniscus force is the main source of the observed difference in pull-off forces. Kim et al. (2008) [144] measured the maximum adhesion force exerted by the menisci at nanoscale contacts, under various humidity conditions, using atomic force microscopy (AFM) at. The authors explained the adhesion results in terms of the contact deformation, via continuum contact mechanics, incorporating the Laplace pressure and surface energy. They found that, although the pull-off force varied significantly with humidity, the equivalent work of adhesion was invariant. The authors reported that an increase in humidity altered the nature of adhesion from a compliant contact with a localised, intense meniscus force, to a stiff contact with an extended, weak meniscus force. Many researchers [113,145–149] reported pull-off force instabilities observed at a low relative humidity of between 20% and 40%. The force discontinuity was caused by the minimum film thickness of water required to form a capillary neck. At low humidity, the water film thickness was too small to form a capillary neck with an asperity on hydrophilic surface. He et al. (2001) [149] introduced three different adhesive forcehumidity regimes (Fig. 4). At low humidity the van der Waals regime took place (Fig. 4, regime I) while at mid-RH a capillary force dominated (Fig. 4, regime II). At high humidity a mixed repulsiveattractive regime worked (Fig. 4, regime III). They also found that roughness reduced the magnitude of pull-off forces and the asperity size dispersion widened the force instability profile. Also Pakarinen et al. (2006) [114] and Butt et al. (2006) [131] reported that the van der Waals force was much smaller than the capillary force, under high relative humidity conditions (RH N 30%). Alternatively, Halsey and Levine [111], found that three different relationships exist between the dependence of the adhesive force on the amount of fluid present. These were determined by the ratio of the surface roughness and the volume of fluid added. One extreme was characterised by the predominance of the meniscus forces. In addition, the fluid was accumulated only around one or a few asperities. The other extreme involved a situation where the fluid thickness was greater than the average distance between neighbouring asperity peaks and the roughness did not have any additional affect on the adhesion force. Chen et al. (2008) [150] proposed a numerical model to estimate adhesive force under various relative humidity conditions. This model was based on both geometrical constraints and the force equilibrium of the water/air film. Like other authors, Chen also found that the capillary pressure force was a major component of the adhesive force at mid humidity levels (N30% RH). Their predicted adhesive force was in good agreement with experimental results found in the literature [151,153]. Tagawa et al. (2004) [152] studied microtribological properties of diamond-like carbon (DLC) films in the presence of water molecules. The authors observed the greatest adhesion at water coverage 2–3 monolayers, which corresponds to a relative humidity of 70–80%. The abrupt increase in adhesion was explained by the generation of a meniscus from the condensed water between an asperity tip and DLC surface.

where H is the separation distance between the average surface plane and the bottom of the adhering particle, which is equal to 1.817Rq; r is the radius of the meniscus. Recently, meniscus forces for different geometries and roughness have been summarised in the extensive review by Butt and Kappl (2009) [142]. 3.3. Effect of environmental conditions (RH) on meniscus force Grobelny et al. (2006) [143] studied adhesion forces between a gold sphere and flat gold substrate using AFM in different environ-

Fig. 4. Dependence of adhesion force on humidity. Redrawn from [149].

P. Prokopovich, V. Starov / Advances in Colloid and Interface Science 168 (2011) 210–222

Matsuoka et al. (2002) [153] developed a method/apparatus to study the effect of meniscus forces on surface/adhesion forces between solid surfaces. Following on McFarlane and Tabor (1950), Matsuoka et al. (2002) [153,154] measured the adhesion forces between mica surfaces in undersaturated vapours of various liquids (water, alcohols and hydrocarbons) in an atmosphere-controlled chamber. The adhesion forces changed gradually from those measurements under dry conditions to macroscopic meniscus forces with increasing relative vapour pressure. In the case of water, the surface forces gradually increase, while for the other liquids they decrease. The authors reported that the constant values of the surface forces, in the region of high relative vapour pressure, correspond to the macroscopic meniscus force.

4. Multi-asperity adhesion models 4.1. Modelling of real surfaces All real surfaces are uneven and the contact between two such surfaces is via the asperities. Various models, relating to the contact of rough surfaces, have been proposed in [155–159]. In general, there are two types of multi-asperity models. In the uncoupled model, the contribution of each asperity is estimated independently from others and the overall adhesion is the sum of all single contributing asperities. In the coupled model, the surface is considered in its entirety and thus, this is more complex to solve [160]. This review has focused predominantly on the un-coupled multi-asperity adhesion models, in section 1.7. The coupled multiasperity contact models are rarely used, due to their mathematical complexity. Instead of a random distribution of asperity heights, periodic interface profiles (sinusoidal) are usually considered with this approach. Only a few coupled contact models are available in the literature. Some of these are based on the Green's function method [161], and others [162] on applying a series techniques, or on a complex potential method [163]. The initial idea of multi-asperity contacts was proposed by Archard [164]. However, a profound refinement of this idea was due to Greenwood and Williamson [156], who modelled the roughness of the surface as an ensemble of identical spherical asperities, with randomly distributed heights. This model has been widely employed, despite some of its limitations, such as the assumption of perfectly hemispherical asperity and constant asperity tip radius. Later, Bush, Gibson and Thomas [165,166] considered a rough surface, with a random anisotropic distribution of asperity radii. Traditionally, these contact models assume a Gaussian distribution for the heights of the surface asperities. However, in many engineering applications, surfaces do not follow a Gaussian distribution. It is recognised that common machining processes, such as grinding and turning and, in miniature systems (MEMS), produce non-Gaussian surfaces with an asymmetric asperity heights distribution [167–169]. The modelling of the interaction between two rough surfaces, as the contact between a perfectly flat surface and a rough surface, whose asperity height is the sum of the asperity height of each surface, has been proposed by Greenwood and Tripp [170]. This approach has been accepted and validated. However, the assumption that both surfaces present normally distributed asperity height cannot describe all situations, as several materials have not shown this type of distribution. Other authors have proposed different types of distributions for the asperity heights. Kotwal and Bhushan [171] used the Pearson's system of frequency curves in order to represent asymmetric distribution. McCool [172,173] introduced the Weibull probability density function to model the distribution of asperity heights. This distribution is suitable for representing asymmetric engineering surfaces on a wide range of scales (from nano to macro scales) [174–176].

217

The main limitations of the Greenwood-Williamson model are the assumptions of a constant asperity tip radius and the Gaussian distribution of the asperity height. The importance of the distribution of asperity curvature radius on contact pressure was highlighted by Onion and Archard [177]. This suggests that the model of Greenwood and Williamson can underestimate this value. One model that considers asperities with a distribution of both height and radius of curvature is that of Whitehouse and Archard [178]. However, it assumes a relationship between asperity height and its radius of curvature, which is controversial. It has been shown in [59], that some materials do not follow any relationship between asperity height and curvature radii. In order to take into account, the distribution of the radius of asperity curvature, Bush, Gibson and Thomas [165] employed the description of surface asperity height and radius of curvature proposed by Nayak [179]. This model assumes that the asperity heights are normally distributed, without taking into consideration the actual distribution for the radius of curvature. In addition, the simplification of the BGT model recently proposed by Greenwood [39], contains the same assumption. Prokopovich and Perni [59] developed a multi-asperity adhesion model based on the JKR and DMT models for a single asperity. The contact of two rough surfaces was modelled as that of a smooth and an equivalent rough surface. The validity of this approximation has been verified earlier [154,165,177]. In this model, the authors accounted for variability of both the asperity heights and asperity curvature radii on the surface of the material, without assuming a specific distribution for either the asperity heights or the curvature radii, in contrast to Bush, Gibson and Thomas [165] and Onions and Archard [177]. The effect of each individual asperity is local and considered separately from other asperities. The cumulative effect is the summation of the actions of individual asperities for both the JKR and DMT theory. This model has been extended by introducing a meniscus action between asperities and its contribution to the overall adhesion force has been considered in [59]. In later work, the multi-asperity adhesion model has been applied to describe the adhesion of a soft tissue (aorta) to a biomaterial surface (PVC) [59]. For both surfaces, PVC biomaterial and aorta tissue, surface energy, elastic moduli and surface topography characteristics (asperity curvature radii and their heights) were determined. The contact interactions between soft tissue and a biomaterial surface were modelled, considering effects of a large number of asperities on the surfaces. Surface roughness features, such as asperity heights and curvature radii, were taken into account. In addition, the curvature radii of the asperities were measured along x and y axes and the geometrical average was obtained in order to estimate the radius of curvature. 4.2. Un-coupled multi-asperity adhesion model The situation of two surfaces in contact can be modelled as one uneven surface in contact with a perfectly flat one (Fig. 5). Under this assumption, the resulting contact force is the sum of the individual contributions of each asperity in contact, according to its own deformation. The validity of this assumption has been repeatedly proven in [170,180]. In order to take into account the variability of the height and curvature radius of each asperity, it is necessary to employ the probability ϕ(ζ) that an asperity has a height (or a curvature radius) between ζ and ζ + d ζ. ϕ(ζ) is called probability density function. Different statistical distributions can be employed to describe the population of asperity height and asperity curvature radius. They can be the normal (Gaussian) distribution:

ϕðζÞ =

1 ðμ−ζÞ2 pffiffiffiffiffiffi exp − 2σ 2 σ 2π

! ð23Þ

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Fig. 5. Schematic representation of the model. "Reprinted with permission from Prokopovich P, Perni S. Multiasperity contact adhesion model for universal asperity height and radius of curvature distributions. Langmuir, 2010; 26(22): 17028–17036. Copyright 2010 American Chemical Society."

where σ is the standard deviation and μ is the average of the peaks height distribution (root mean square of the composite rough surfaces) or asperity curvature radius Or the Weibull distribution: ϕðζÞ =

    pstat ζ pstst −1 ζ pstat exp − Δstat Δstat Δstst

ð24Þ

where pstat is the “shape parameter” and Δstat is “scale parameter”. The respective equations for the cumulative distribution function Φ(ζ) (probability of asperity height or curvature radius with a value up to ζ) are, for the Gaussian distribution: ζ

ΦðζÞ =

2

ðμ−ξÞ 1 pffiffiffiffiffiffi ∫ exp − 2σ 2 σ 2π −∞

! dξ

Fig. 6. Different statistical distributions: (a) probability density function (b) cumulative function.

ð25Þ

and for the Weibull distribution:   ζ pstat ΦðζÞ = 1− exp − Δstst

ð26Þ

Examples of frequency and cumulative distributions are shown in Fig. 6. For each asperity pair, the JKR (Eqs. (6) and (7)) or the DMT (Eqs. (9) and (10)) model applies and each set of equations represents a system that gives the relationship between adhesion force and asperity deformation. The total contact force for rough surfaces approaching a perfectly flat one (at a distance of d), is the sum of all the compressive forces generated by the asperities where its height (hi) exceeds d or are within their respective critical deformation δc, i [48,59]: δi Nδc;i

Fadh ðdÞ = ∑ fnðδi ; Ri Þ i

ð27Þ

where: δi = d−hi

ð28Þ

δc, i is the critical distance of asperity i according to equation, fn(δi, Ri) is the adhesion force for one asperity with curvature radius Ri and deformation δi based on the JKR or DMT model. The results of this approach have been presented by Prokopovich and Perni [59] and show that the adhesion force has the profile described in Fig. 7, whilst the contact area is not always linear with the applied load (Fig. 8) as for the other multi-asperity adhesion model, the BGT. The estimated adhesion force is also in good agreement with the experimental data. The JKR model generates real area of contacts, which are bigger than in the DMT model. The contact area for a single asperity, predicted by the JKR model, is the result of the Hertzian contribution plus a contribution due to the surface energy. This second term is, instead, neglected in the DMT model.

Fig. 7. Measured and predicted cumulative force distributions between PBT/glass (a) and PBT/silicone (b). ○ experimental ■ DMT based model ▲ JKR based model. Redrawn from [60].

P. Prokopovich, V. Starov / Advances in Colloid and Interface Science 168 (2011) 210–222

219

Bhushan (2003) [184] proposed a model to obtain the total capillary force between contacting surfaces. In this model, all individual contacting and non-contacting asperities, where menisci are formed, were summed and the total capillary force was expressed as: Bhush

Fm

= 2πRt γð cosϑ1 + cosϑ2 ÞNðt Þ

ð29Þ

where Rt is radius of the contacting asperity tip and counter surface; γ is surface tension of the liquid film; θ1 and θ2 are the contact angles for the asperity tip and the counter surface; N(t) is the number of contacting and near contacting asperities where menisci build up and is a function of the residence time of the asperity tip at the counter surface. An alternative approach, based on thermal activation process, was shown by Bocquet (1998) [122] and Riedo et al. [119,184,185]. In the model, proposed by Bosquet [122], the number of capillary bridges which form between the solids, increases with the contact time, giving rise to an increase in adhesion force. Assuming the activation process, Bosquet estimated the time required to form a bridge of height h, and assumed an activation process t ðhÞ = tA exp

  ΔEðhÞ kB T

ð30Þ

where ΔE(h) is the free energy, T is the temperature, kB is the Boltzmann constant. Riedo et al. (2002) [185] observed similar effects as Bosquet found, at the nanoscale, using CrN [186] and diamond like carbon (DLC) films [184]. Riedo et al. (2002) [185] considered the length scale for capillary condensation and they introduced that the maximum asperity height where a liquid bridge can form after a time, t: Fig. 8. Relative contact area versus adhesion force for PBT/glass (a) and PBT/silicone (b). ● DMT based model □ JKR based model. Redrawn from [60]. max

ht 4.3. Meniscus contribution In the presence of a thin liquid film, such as a lubricant or an adsorbed water layer at the contact interface, curved menisci form around contacting and non-contacting asperities. As a result of surface tension, the attractive meniscus force arises from the negative Laplace pressure inside the curved meniscus [181]. Meniscus force action, different meniscus geometries and their effect on adhesion is detailed in Section 3 for the contact of a single asperity. For multiple menisci, several models have been developed to predict meniscus forces, using a statistical approach [182–184]. Gao et al. (1995) showed that the meniscus force increases as a function of liquid film thickness. In the meantime, he demonstrated that the meniscus force is strongly dependent on the geometry and distribution of surface features for a given film thickness. While most of the meniscus models consider equilibrium conditions, some researchers [98,184] developed a kinetic meniscus model to predict the time dependence of a meniscus force. Bhushan (1996) [184] found that the relative meniscus force reached a maximum at a rest time (called “equilibrium time”) and the growth rate of the relative meniscus force is inversely proportional to the liquid film viscosity. Chilamakuri and Bhushan (1999) [98] proposed a kinetic meniscus model to determine the capillary force as a function of time. This model considered liquid flow towards the contact zone, until equilibrium is attained. The driving forces for such flow were the Laplace pressure and the disjoining pressure. The limitation of the model is that it considered only those asperities which were fully immersed. However, meniscus bridge formation is not restricted to interfaces that are fully immersed in a liquid film. Meniscus bridges can form by condensation of liquids at preferential interstitial locations.

 
−1 t p ln 0 Aliq ρ = ln p tA

ð31Þ

where tA is the condensation time of a liquid monolayer, Aliq is the liquid bridge cross section, ρ is molecular density of the liquid. Taking into account surface roughness, the authors indicated that only a fraction of the total number of liquid bridges were formed at a given time. Therefore, the capillary forces become time dependent. The capillary force derived by Riedo can be written as: Riedo

Fm

=

  2πRt γð cosϑ1 + cosϑ2 Þ V ln A V λc ρAliq lnðp0 = pÞ

ð32Þ

where VA is the critical velocity corresponding to the condensation time of one liquid monolayer, λc is the typical width of the distribution of distances between the surfaces. Both authors [122,185] assumed menisci bridge formation to be the result of preferential condensation of liquid at the interface. This overcomes the energy barrier required for coating films to fully grow, coalesce and then to form menisci at the interface. The expression for the total number of asperities, where menisci bridges are formed, was derived starting from the Kelvin relationship. The total energy required for nucleating meniscus bridges by condensation of a certain volume of water [187] at the asperity tip–surface interface was also determined. Prokopovich and Perni [61] presented a multi-asperity adhesion model in the presence of a meniscus force. The authors estimated the meniscus force for each asperity and determined the overall meniscus force by summing the meniscus forces over all menisci formed at the interface biomaterial–soft tissue. The schematic model of contact interaction biomaterial–soft tissue is shown in Fig. 9.

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Fig. 9. Schematic of rough soft tissue surface in contact with biomaterial with a liquid film.

The resulting meniscus force for asperity i is defined as: Fmen;i =

γ ð cosϑ1 + cosϑ2 ÞAmen;i s

ð33Þ

Fig. 11. Contact area change as a function of adhesion force. ● lubricated case □ dry case. Redrawn from [61].

where s is the height of the meniscus given by: 5. Conclusion s = r ð cosϑ1 + cosϑ2 Þ

ð34Þ

Amen,i is the projected area of the meniscus liquid around asperity i. In case the hemispherical asperity is not in contact, this can be written as [185]: ð35Þ

Amen;i = 2πRi s

Various adhesion models have been discussed within this review. The majority of these models are based on single asperity contacts, and are used to describe ideal systems at both the nano- and microscale. Some attempts to include surface irregularities have been made. However, these approaches still do not adequately represent contacts between real surfaces. Therefore, in order to adequately represent a real life situation, more complex adhesion models, based on multiasperity contacts, have been developed and discussed in this review.

then the resulting meniscus force is the well known relation: Fmen;i = 2πRi ð cosϑ1 + cosϑ2 Þγ

ð36Þ

For asperities in contact the following relation was utilised: 2

Amen;i = Aprojected;i −πai

ð37Þ

Aprojected,i is the area projected from the asperity i from a height equal to ℓ + s. It has been shown in [61] that adhesion force is increased (Fig. 10) in the presence of meniscus action and that the static coefficient of friction is reduced by the presence of liquid menisci. The latter, is the result of the different relationship between contact force and real area of contact, when a liquid film is present compared to the dry case (Fig. 11).

Acknowledgment P Prokopovich acknowledges support from Welcome Trust, EPSRC UK and Arthritis Research UK. V Starov's research is supported by EPSRC UK and MULTYFLOW grant, EU. The authors thank Dr Richard Toon for his useful suggestions and helpful comments.

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[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] Fig. 10. Adhesion force versus separation distance with and without meniscus force. □ lubricated case ■ dry case. Redrawn from [61].

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