## Abstract

Adhesive or repulsive forces contributed by both meniscus and viscous forces can be significant and become one of the main reliability issues when the contacting surfaces are ultra smooth, and the normal load is small, as is common for micro/nano devices. In this study, both meniscus and viscous forces during separation for smooth and rough hydrophilic and hydrophobic surfaces are studied. The effects of separation distance, initial meniscus height, separation time, contact angle and roughness are presented. Meniscus force decreases with an increase of separation distance, whereas the viscous force has an opposite trend. Both forces decrease with an increase of initial meniscus height. An increase of separation time, initial meniscus height or a decrease of contact angle leads to an increase of critical meniscus area at which both forces are equivalent. An increase in contact angle leads to a decrease of attractive meniscus force but an increase of repulsive meniscus force (attractive or repulsive dependent on hydrophilic or hydrophobic surface, respectively). Contact angle has a limited effect on the viscous force. For asymmetric contact angles, the magnitude of the meniscus force and the critical meniscus area are in between the values for the two angles. An increase in the number of surface asperities (roughness) leads to an increase of meniscus force; however, its effect on viscous force is trivial. A slightly attractive force is observed for the hydrophobic surface during the end stage of separation though the magnitude is small. The study provides a fundamental understanding of the physics of the separation process and it can be useful for control of the forces in nanotechnology applications.

## 1. Introduction

Adhesion due to condensation of water from the ambient or the presence of a thin liquid film on hydrophilic contacting surfaces has been studied extensively in various biological and technological applications, such as the forces developed for attachment by insects, spiders and lizards to various surfaces (Gorb 2001), stiction of an atomic force microscope (AFM) tip to a sample during interaction, magnetic storage devices, micro/nano devices and fuel injectors in automobiles (Bhushan 1996, 1999, 2002, 2005, 2007). The primary mechanism responsible for the adhesion/stiction is the formation of micro menisci. The repulsion in the case of hydrophobic contacting surfaces has not been studied to the best of our knowledge in the past even though this phenomenon is also common in nature.

Menisci form around the contacting and near-contacting asperities due to surface energy effects in the presence of a thin liquid film (Adamson 1990; Israelachvili 1992; Bhushan 1999, 2002, 2003, 2005). Pendular rings are formed on contacting asperities and liquid bridges are formed on near-contacting asperities. (A small amount of liquid at the point of contact between the solid surfaces is often called a pendular ring.) For two hydrophilic surfaces, concave-shaped menisci are formed, and for two hydrophobic surfaces convex-shaped menisci are formed. For a hydrophilic surface, the lower pressure inside the meniscus, that is, negative Laplace pressure, results in an intrinsic attractive force, called the meniscus force, acting on the interfaces. For hydrophobic surfaces, a repulsive meniscus force will act. When separation of two surfaces is required, the viscosity of the liquid causes an additional attractive force, rate-dependent viscous force during separation. Meniscus and viscous forces govern the break of a meniscus bridge. The resultant force, adhesive or repulsive, is highly dependent on the formed meniscus area, contact angles, number of menisci, separation time, and surface tension and viscosity of the liquid.

For hydrophilic surfaces, many studies on meniscus forces have been carried out based on identical contact angles with static contact configurations. Fisher (1926) analysed the mean curvature of axisymmetric menisci and the volume of trapped liquid and forces due to pendular rings between identical spheres by using interpolation in the solutions to satisfy the boundary conditions. Woodrow *et al.* (1961) solved the Laplace–Young equation under given initial conditions to calculate the meniscus profile and meniscus forces between identical spheres. An increase of adhesion force was generally observed when a thin liquid film was introduced at the contact interface either through adsorption or by deposition. Orr *et al.* (1975) solved the Laplace–Young equation for axisymmetric menisci between a sphere and a flat surface. In their analysis, profiles of pendular rings had been calculated and expressed in terms of elliptical integrals, and the corresponding enclosed volumes and capillary forces were reported based on static contact configurations. Surface roughness, properties of contacting solids, such as layers, Young's modulus, hardness and film thickness (RH), have also been introduced in many studies. Tian & Bhushan (1996) studied the micro-meniscus effect based on a multi-asperity contact model for homogeneous solids. Peng & Bhushan (2001) and Cai & Bhushan (2006) studied the meniscus effects for rough-layered contact models and examined the dependence of meniscus force on layer properties.

During separation of two surfaces, both meniscus and viscous forces operate inside the meniscus. The latter is significant especially when menisci have a larger size and the separation time is short. Also, asymmetric contact angles and division of meniscus (which are the real cases) can significantly affect the magnitudes and behaviour of both meniscus and viscous forces during separation. Though many studies have been carried out on meniscus forces and meniscus profiles, studies on the separation process are rare. Fortes (1982), Carter (1988), Gao (1997) and Stifter *et al.* (2000) investigated the meniscus force–distance relationship which is one of the important relationships during separation. Though the distance dependence of meniscus forces was well presented, these studies were confined to a purely static meniscus force analysis since no viscous force and separation time were considered. Chan & Horn (1985) calculated the viscous force due to viscous dissipation for the case of a sphere moving normally to a flat surface with some separation by considering Reynolds' lubrication equation. The force equation derived is suitable for an infinite wetted region. Matthewson (1988) modified the viscous force equation to be applicable to a finite wetted region, but his assumption of an infinite break point leads to divergence when integrating over separation. The equation derived by taking limit operation based on an infinite break point is not accurate to estimate viscous force when the break of a meniscus bridge occurs at a distance comparable to meniscus height. More recently, Cai & Bhushan (2007*a*) developed a model to study the meniscus and viscous forces during separation of two hydrophilic smooth surfaces with symmetric contact angles. These models were further extended by Cai & Bhushan (2007*b*) to investigate asymmetric contact angles by integrating a moving boundary technique into the previous numerical simulation to capture the liquid–solid interface differences between the two sides of a meniscus. It was found that both meniscus and viscous forces are closely dependent on the separation distance, initial meniscus height, separation time, surface tension and viscosity. The contact angles significantly affect the break distance of a meniscus. The magnitude of meniscus force can be largely reduced by choosing proper asymmetric contact angles. ‘Force scaling’ effects are found to be true for both meniscus and viscous forces when one larger meniscus is divided into a large number of identical micro-menisci. Meniscus force is proportional to the number of divisions whereas viscous force is proportional to the inverse of the number of divisions (1/*N*). During separation, either meniscus or viscous force could be a dominant one at a given separation time. Viscous force becomes dominant at a relatively larger meniscus area with larger initial meniscus height. The results show that viscous force is more likely to become a dominant force for a liquid with high viscosity at larger contact angles at given conditions. Though comprehensive studies have been performed on meniscus and viscous forces for hydrophilic surfaces, the study of the forces for hydrophobic surfaces has not been carried out to the best of our knowledge. The effect of hydrophobicity and hydrophilicity of surfaces (which is common in nature and is seen frequently in applications) on the forces is still needed to better understand their role and control of the forces.

In the present work, a comprehensive study of both meniscus and viscous forces is carried out for smooth and rough hydrophilic and hydrophobic surfaces. The role of these two forces is evaluated during a separation process. The effects of the parameters, such as separation distance, initial meniscus height, separation time and contact angles, as well as roughness are presented. The study provides a fundamental understanding of the forces and it can be useful for control and use of these forces in nanotechnology applications.

## 2. Approaches and analyses

The separation of two hydrophilic and hydrophobic smooth/rough surfaces with meniscus bridges having symmetric/asymmetric contact angles is presented. The two surfaces are assumed to be rigid. Concave and convex arc shapes are assumed for meniscus curves for hydrophilic and hydrophobic surfaces, respectively. The meniscus bridge is assumed to be in equilibrium, and the liquid is incompressible. Thermal effects are considered to be negligible. Body force and inertia of the liquid are neglected, which has been justified, for example, by Cameron & McEttles (1981). The pressure is constant on a vertical cross-section plane, whereas it varies along radial direction through the meniscus bridge in the process of separation. Based on these assumptions, Reynolds' lubrication theory is applicable during the separation. Since a separation is usually done within a very short time, the evaporation of liquid is assumed to be negligible.

### (a) Forces during separation due to meniscus

In the study we consider separation of two smooth and a rough against smooth surfaces with a liquid film between as shown in figures 1 and 2*a*, respectively. Figure 1*a*,*b* shows the configuration of liquid–solid interface for hydrophilic and hydrophobic surfaces, respectively, and figure 2*a* shows the distribution of a number of identical spherical asperities *N* on a flat surface with radius *R* for each asperity. For the purpose of comparison, the separation of two smooth surfaces with a number of identical menisci is also presented as shown in figure 2*b*.

Meniscus and viscous forces are present when separating two surfaces with liquid-mediated contacts. Meniscus force *F*_{m} is contributed by both Laplace pressure and the surface tension around the circumference. The magnitude of the meniscus force depends on liquid properties and the size of the meniscus which relates to the liquid volume and interface geometry. The strength of the viscous force depends not only on the properties of the liquid and the size of the meniscus, but also on the separation time and initial gap between the two bodies. An external force which is larger than the resultant force is needed to make an initial separation occur. During separation, if the meniscus force is larger than the viscous force, then the externally applied force to separate two surfaces depends on the meniscus forces, and vice versa. If the viscous force becomes larger than the meniscus force, the meniscus will eventually break slowly even without an increase in the applied force. However, the time taken to separate the two surfaces would be long.

#### (i) Meniscus force

Meniscus forming between the two flat surfaces due to surface tension *γ* results in pressure difference Δ*p*, which is often referred to as capillary or Laplace pressure, and is given by the Laplace equation (Adamson 1990)(2.1)The pressure difference Δ*p* can be negative or positive depending on the surface properties. A hydrophilic surface results in a negative pressure difference, whereas a hydrophobic surface leads to a positive one. In the equation, *γ* is the surface tension of liquid; *r*_{1} is the meniscus radius as shown in figures 1 and 3; and *r*_{2} is another radius of the meniscus in the orthogonal plane to *r*_{1} (not shown in the figures). 1/*r*_{1}+1/*r*_{2} equals a so-called Kelvin radius at equilibrium. Based on thermodynamic law, *r*_{k} can be obtained from the Kelvin equation (Thomson 1870)(2.2)where *V* is molar volume; is the gas constant; *T* is the absolute temperature; *p* is the ambient pressure; and *p*_{0} is the saturated vapour pressure at *T*. For the case , *r*_{1}=*r*_{k} and Δ*p*=*γ*/*r*_{1}. The meniscus force can be obtained by integrating the Laplace pressure over the meniscus area and adding the surface tension effect acting on the circumference of the interface (Fortes 1982)(2.3)where *Ω* is the meniscus area; *x*_{n} is the meniscus radius; and *θ*_{1,2} in the second term on the r.h.s. corresponds to the contact angle of the liquid on the surfaces being pulled. The consideration for this situation is that the two surfaces are being pulled apart by an external force within a short time, thus the forces on both sides of the liquid bridge can be different during the pull for asymmetric contact angles. For initial meniscus radius *x*_{n0}≫*r*_{1}, one may neglect the surface tension contribution in equation (2.3) (second term on the right). If does not hold, *r*_{2} may be replaced with the difference between *x*_{c}, the centre coordinate of the meniscus curve, and *r*_{1}. Thus, (Stifter *et al.* 2000). For given conditions, one can readily obtain *r*_{k} using equation (2.2).

For the separation of two smooth flat surfaces with geometry configurations as shown in figure 1, a meniscus height *h* can be calculated, *h*=*r*_{k}(cos *θ*_{1}+cos *θ*_{2}), thus meniscus height is related to meniscus force. For a circular meniscus, the meniscus force can be calculated using(2.4)

For the separation of two rough surfaces (figure 2*a*) with geometry configurations as shown in figure 3, the meniscus force is the superposition of the separation of a number of single sphere-on-flat surface. The meniscus force for sphere-on-flat surface can be calculated using the equation (Orr *et al.* 1975; Bhushan 2002)(2.5)where the index *i* represents the *i*th location of separation.

For the separation of two smooth flat surfaces with *N* number of identical menisci, or separation of two rough surfaces with *N* number of identical spherical asperities arbitrarily distributed on a flat surface without fully occupying the total surface area, one can expect a maximum meniscus force(2.6)where *F*_{m} is equation (2.4) for flat smooth surfaces and equation (2.5) for rough surfaces, and this also applies to the case discussed below. For the case of *N* number of menisci or the spherical asperities fully occupying the total surface area, the maximum meniscus and viscous forces can be determined using equation (2.4) or (2.5) with a proper radius *x*_{n} or *R*_{n}. Here, for instance, for a flat surface with an area 2*R*×2*R*, given that the number of asperities is *N*=*n*×*n*, the radius for each meniscus is *x*_{n}/*n* for a flat smooth surface, and *R*/*n* for a single asperity, the maximum meniscus force is

(2.7)

#### (ii) Viscous force

Viscous force occurs due to the viscosity of the liquid when separating two bodies within a short time. One may ignore viscous force for an infinitely long separation time *t*_{s}. However, an infinitely long separation time is not practically feasible. Thus, characterization of the relevant viscous force is needed in order to properly estimate the total force needed to separate two surfaces from a liquid-mediated contact. In the derivation of the viscous force, we assume that Reynolds' lubrication theory applies to the process of separation. The pressure inside the meniscus bridge consists of horizontal pressure gradients, whereas the pressure is constant in any vertical plane inside a meniscus bridge, and at the outside of a meniscus ring *r*=*r*_{b} (liquid–air interfacial boundary, which is exposed in ambient), , the ambient pressure.

For separation of two smooth flat surfaces for a liquid with kinematic viscosity *η*, the equation for the viscous force has been derived by Cai & Bhushan (2007*a*) using Reynolds' lubrication equation with cylindrical coordinate system for the separation of two flat surfaces (e.g. Hocking 1973):(2.8)where *h* is the separation distance. Integrating the equation above and applying the boundary condition, *p*(*r*_{b})=*p*_{b}, the pressure difference at arbitrary radius *r* within a meniscus can be obtained as(2.9)From the above equation, one can obtain the maximum pressure difference occurring at the centre of a meniscus, and the minimum on the outmost boundary. An average pressure difference is one-half of the summation of the two:(2.10)The viscous force thus may be calculated by using the average pressure difference based on the above equation. Thus, the viscous force can be expressed as(2.11)By integrating the above equation, one can obtain(2.12)where *h*_{s} is the break point and *t*_{s} is the time to separate two bodies.

For the separation of a sphere and a flat surface, the viscous forces can be determined using the lubrication equation for separation of a sphere and a flat surface (Cai & Bhushan 2007*a*)(2.13)where *D* is separation, *H*(*r*) is meniscus height at radius *r* and . At the outside boundary *r*=*r*_{b}, and *p*(*r*_{b})=*p*_{0}. Integrating equation (2.13) and applying the boundary conditions gives(2.14)The viscous force can be found by integrating Δ*p* over the meniscus area:(2.15)*H*(*r*_{b}) changes with separation and needs to be solved instantly. For *R*≫*r*, conservation of volume leads to(2.16)where *r*_{0} and *D*_{0} are initial meniscus radius and gap, respectively. Substituting equation (2.16) into equation (2.15) and integrating at both sides, the viscous force can be obtained as(2.17)where *t*_{s} is separation time and *D*_{s} is the distance when separation occurs. The separation occurs when a meniscus neck radius equals zero; further integrating equation (2.17) gives(2.18)

For the separation of two smooth flat surfaces with *N* number of identical menisci, or separation of two rough surfaces with *N* number of identical spherical asperities arbitrarily distributed on a flat surface without fully occupying the total surface area, one can expect a maximum viscous force(2.19)where *F*_{v} is equation (2.12) for flat smooth surfaces and equation (2.18) for rough surfaces, and this also applies to the case discussed below. For the case of *N* number of menisci or the spherical asperities fully occupying the total surface area, the maximum viscous force is

(2.20)

### (b) Meniscus curvature

It is well known that viscosity starts to drop above a certain shear stress and the liquid becomes plastic and can only support a constant stress, known as the limiting shear strength, at higher strain rates (Bhushan 1996). If we assume the meniscus breaks at the break point, one may consider that point occurs at infinite distance. However, this may lead to an overestimation of the real viscous force since a meniscus bridge may break very quickly when it is small and the meniscus radius is comparable to its height. Thus, the break point should be determined to give a reasonable estimate of the viscous force. In this paper, the break distance is assumed to be the distance corresponding to a zero meniscus neck thickness during separation. The instant meniscus and viscous forces given in equation (2.3) and (2.17) depend on the solving of the break point and meniscus radius *x*_{n} which in turn rely on initial and boundary conditions, and the instant meniscus curvature needs to be calculated during the process of separation. The meniscus profile can be found by solving the Laplace–Young equation as done by Orr *et al.* (1975), who expressed the meniscus profile in terms of elliptical integrals. For simplicity, a concave arc-shaped meniscus is assumed to account for meniscus curve due to hydrophilic surfaces, and a convex arc-shaped curve for hydrophobic surfaces. For a concave arc-shaped meniscus, we have applied a simple approach to effectively capture the curvature during separation and avoid the complexity of handling elliptic integrals (Cai & Bhushan 2007*a*). For a hydrophobic surface, one may apply a similar approach with slight modifications to characterize the meniscus curvature.

Let *H* and *M* denote the geometry shapes of the upper boundary and a meniscus as shown in figure 3. *H* and *M* can be chosen as needed(2.21)and if the meniscus shape is an arc, *M* may be expressed as(2.22)The geometry configurations satisfy a set of boundary conditions(2.23)One more condition is needed to fully constrain the problem and uniquely determine the meniscus curvature instantly. For incompressible fluid, conservation of volume gives(2.24)The volume can be found by integrating the whole area enclosed by *H*, *M* and the two coordinate axes, and the magnitude of *V*_{0} can be calculated from initial conditions. The instant values of *x*_{ni}, *r*_{i}, *x*_{ci} and *y*_{ci} can be determined with the boundary conditions and conservation of volume. Correspondingly, the instant meniscus curvature and meniscus force can be calculated. For separation of two parallel flat surfaces, one sets *y*_{h}=*h*, otherwise, the shape function *y*(*x*) for *H* should be defined. The case of *y*_{h}=*h* corresponds to a constant separation speed.

## 3. Results and discussion

Separation of two hydrophilic and hydrophobic surfaces relying on symmetric and asymmetric contact angles with various initial meniscus heights is analysed with respect to meniscus and viscous forces. The effects for a hydrophilic smooth surface with symmetric and asymmetric contact angles have been studied by Cai & Bhushan (2007*a*,*b*), and some results will be presented here for comparison. The two forces are calculated based on the curvatures during separation. For simulation purposes, we assume that a meniscus breaks at a zero meniscus neck radius though this is not true practically and it may break at a certain critical separation distance (Singh *et al.* 2006), and the forces disappear at the break point. The force needed to overcome a meniscus force is the force occurring at the beginning, whereas the force needed to overcome a viscous force (due to viscosity) equals the force resulting from the break of a meniscus bridge (which occurs at the break point). In the analyses for the separation of two flat surfaces, the dimensionless meniscus and viscous forces are defined as follows to eliminate effects of the contact angles due to surface property, and the liquid surface tension and viscosity for the purpose of comparison:(3.1)(3.2)The effect of a rough surface based on the numerical model is also presented. In the analyses, liquid bridges formed from water are evaluated. The meniscus curvatures presented as examples are generated based on an initial *h*_{0}=2 nm for separating two surfaces. An initial meniscus radius *x*_{n0} of 100 nm is used in the study except for the rough surface case which has a nominal area 100×100 μm^{2} with a larger initial meniscus height of 100 nm. The forces calculated during separation are based on various initial meniscus heights from 2 to 6 nm. Contact angles 0°, 90° and 180° corresponding to 0.001°, 89.999° for hydrophilic and 90.001° for hydrophobic surface, and 179.999° respectively to avoid singularities. The separation time *t*_{s} used in the study is 0.1 μs related to the real separation time of a diesel fuel injector.

### (a) Effects of contact angles for hydrophilic and hydrophobic surfaces

Figure 4 shows examples of instant meniscus curvatures during the separation of two liquid-mediated surfaces for hydrophilic (figure 4*a*) and hydrophobic surfaces (figure 4*b*). It is shown that for a given set of contact angles and a given initial meniscus height, asymmetric contact angles lead to a faster break of meniscus for both hydrophilic and hydrophobic surfaces. Hydrophobic surfaces with asymmetric contact angles have the shortest break distance as compared to the corresponding hydrophilic surfaces. This difference will eventually affect the forces as we can see from the data to be presented later.

Figures 5 and 6 show dimensionless (identified with ^{*}) and dimensional meniscus and viscous forces versus relative separation *Δ* (from initial distance *h*_{0} to *h*_{0}+*Δ*) for separating two parallel surfaces from various initial meniscus heights *h*_{0}*=*2–6 nm with symmetric and asymmetric contact angles for both hydrophilic and hydrophobic surfaces. The dimensionless figures (right column of figure 5, and figure 6*a*,*c*) presented here are for the purpose of generalization and for general use since the effects of liquid and surface properties have been eliminated. One can obtain the appropriate force magnitude from these figures for various surfaces and liquids by simply multiplying the surface tension, viscosity and contact angle effects. Since the dimensionless and dimensional figures show the same trends, we will mainly discuss the dimensional one for brevity.

For the case of hydrophilic surfaces, the meniscus forces are attractive. The results show that the asymmetric contact angles (one of them kept fixed, here, *θ*_{2}=60°) lead to a larger meniscus force, and the smaller the other contact angle, the larger the meniscus force (figure 5). It is noted that the asymmetric contact angles play a major role in the quick break of a meniscus. It is observed that for *θ*_{1}=0° and *θ*_{2}=60° there is a smaller break distance as compared to the case *θ*_{1}=*θ*_{2}=60°. For a given set of asymmetric contact angles, the effect of initial meniscus height *h*_{0} on the break distance *Δ* is insignificant as shown in figure 6*b*. For the case of hydrophobic surfaces, repulsive meniscus forces are observed in general as shown in figure 5. A slight attractive force is observed at the later stage of separation; however, the magnitude is small. The attractive effect disappears if one of the contact angles equals 180° (zoomed figures in figure 5*a*). This is believed to be the effect of the second term on the r.h.s. of equation (2.4), the force due to surface tension of the liquid on the circumference of the solid–liquid interface. The results show that the asymmetric contact angles (one of them kept fixed, here, *θ*_{2} equals 120°) lead to a larger value of the absolute meniscus force in magnitude, and the larger the other contact angle, the larger the meniscus force (figure 5), which is different from the hydrophilic situation. Again, it is observed that the asymmetric contact angles play a major role in the quick break of a meniscus. *θ*_{1}=180° has a much smaller break distance as compared to the other case. Also, the effects of a given set of asymmetric contact angles on the break distance are more significant than an initial meniscus height *h*_{0}, which has the same trend as for hydrophilic surfaces (figure 6*d*). An intersection is observed in figure 6*c* for *θ*_{1}=180° and *θ*_{2}=120°. This is due to the multiplication factors varying with various initial meniscus heights. These observations may be useful for the design of travel distance of two surfaces to achieve optimal size of a device. For both hydrophilic and hydrophobic surfaces, the effect of contact angle on viscous force is insignificant as shown in figure 6*b*,*d*.

Figure 7 summarizes the effects of contact angles on both forces for both hydrophilic and hydrophobic cases. The l.h.s.s of figure 7*a*,*b* show the effects on meniscus and viscous forces for fixed *θ*_{2} equals 60° and various *θ*_{1} from 0 to 90°. The r.h.s.s of figure 7*a*,*b* show the effects to these forces for fixed *θ*_{2} equals 120° and various *θ*_{1} from 90 to 180° for a set of various *h*_{0}=2–6 nm. It is observed that the contact angles have a large effect on the absolute magnitudes of meniscus forces for hydrophilic surfaces: the larger the *θ*_{1}, the smaller the meniscus forces. And a larger *θ*_{1} can also help decrease the effects of initial *h*_{0} on the magnitude of meniscus forces. For hydrophobic surfaces (r.h.s.s of figure 7*a*,*b*), the trends are opposite. Though asymmetric contact angles largely affect meniscus forces, the effects on viscous forces are trivial (figure 6*b*). As compared to figure 7*a*,*b*, figure 7*c*,*d* shows the effects on both forces for various *θ*_{1} and *θ*_{2} from 0 to 180° for a fixed *h*_{0}=2 nm. The results show that the increase of both or any one of the two contact angles leads to a noticeable decrease of the absolute magnitudes of meniscus forces for the hydrophilic case as shown on the l.h.s. of figure 7*c* but an increase of the absolute magnitudes of meniscus forces for the hydrophobic case as shown on the r.h.s. of figure 7*c*. Again, the effects on viscous forces are small (figure 7*d*). From the analysis, an increase in contact angle leads to a decrease of attractive meniscus force but an increase of repulsive meniscus force (attractive or repulsive dependent on hydrophilic or hydrophobic surface, respectively). The contact angle has limited effect on the viscous force. For asymmetric contact angles, the magnitude of the meniscus force is in between the values for the two angles.

The contact angle affects the critical meniscus area as well, as shown in figure 8*a*,*b*. It is observed that a decrease of contact angle leads to an increase of critical meniscus area. For the given contact angle sets, the asymmetric contact angle pair leads the critical meniscus area to move to a larger value. This is expected since the decrease of one of the contact angles results in a larger meniscus force, and thus a larger meniscus area is needed for the viscous force to match the meniscus force. For asymmetric contact angles, the critical meniscus area is in between the values for the two angles.

### (b) Effects of separation distance, separation time and initial meniscus height for hydrophilic and hydrophobic surfaces

For both hydrophilic and hydrophobic surfaces, the effects of separation to meniscus and viscous forces can be observed in figures 5 and 6. Figure 5 shows dimensionless and dimensional meniscus forces versus relative separation *Δ* for separating two parallel surfaces from various initial meniscus heights *h*_{0}*=*2–6 nm. The dimensionless one is for the purpose of generalization. Figure 6 shows dimensionless and dimensional viscous forces versus relative separation *Δ* for separating two parallel surfaces in the same conditions. Meniscus force decreases with an increase of separation distance, whereas the viscous force has an opposite trend. Both forces decrease with an increase of initial meniscus height. Attractive and repulsive meniscus forces are observed for hydrophilic and hydrophobic surfaces, respectively. In either case, both types of meniscus and viscous forces change rapidly at the beginning. This trend is the same for both the dimensionless and dimensional results. The larger rate of change in the force at the beginning of separation is due to the larger change rate in volume at the beginning and becomes gradual thereafter (figure 4), and a larger volume change rate leads to a relatively larger decrease of pressure difference (separation at a constant speed).

Initial meniscus height *h*_{0} affects both meniscus and viscous forces for either hydrophilic or hydrophobic surface, which can be observed from figures 5 to 8. It is shown that a lower meniscus height leads to a larger meniscus force (attractive or repulsive) and viscous force. The increase of *h*_{0} leads to a decrease of the magnitudes of these forces. The dimensionless and dimensional results have the same trend. This is because at a fixed meniscus area, a higher *h*_{0} results in a larger Kelvin radius and lower absolute pressure difference Δ*p*, and thus lower meniscus force (attractive or repulsive). It is no surprise to see this trend for viscous force since it is the inverse of the square of the meniscus height. An increase of initial meniscus height leads to the critical meniscus area to move to a larger value since viscous force decreases much faster than meniscus force with an increase of initial meniscus height. It is observed that *h*_{0} plays a significant role in the theoretical break point. Smaller *h*_{0} leads to a quick break of the meniscus. This is because the meniscus bridge with smaller *h*_{0} has a smaller liquid volume for a given initial meniscus area. Both meniscus and viscous forces disappear at the break point. As compared to hydrophilic surfaces, the effect of initial meniscus height *h*_{0} to the break of meniscus becomes less significant for hydrophobic surfaces (figure 6*b*,*d*). The initial meniscus height affects the critical meniscus area as well. An increase of initial meniscus height leads to an increase of critical meniscus area. This is because viscous force increases faster with the decrease of initial meniscus height than meniscus force.

Figure 8*a* shows the behaviours of each of the forces as a function of meniscus area for various initial meniscus heights for contact angles *θ*_{1}=*θ*_{2}=60° and *θ*_{1}=0°, *θ*_{2}=60°, and figure 8*b* shows the critical meniscus area as a function of separation time *t*_{s} for various initial meniscus heights for contact angles *θ*_{1}=*θ*_{2}=60° and *θ*_{1}=0°, *θ*_{2}=60°. Since viscous force is a function of the inverse of separation time *t*_{s}, an increase of separation time, initial meniscus height leads to an increase of critical meniscus area. For a given initial meniscus height and a separation time *t*_{s}, one can readily determine the dominating force from the figure during the separation process, based on meniscus size information.

### (c) Roughness effects on meniscus and viscous forces

In the study of roughness effects, the number of asperities *N* used here ranges from 1 to 10^{4}, and we assume that these surface asperities are identical and have spherical shapes, and these asperities fully occupy the nominal flat area. Figure 9*a* shows the effects of the number of asperities on meniscus and viscous forces at different contact angles for both hydrophilic and hydrophobic rough surfaces. It is observed that the increase in the number of asperities leads to an increase of meniscus force (an increase of attractive meniscus force for hydrophilic surfaces and an increase of repulsive meniscus force for hydrophobic surfaces) for a given fixed nominal flat surface area (here 100×100 μm^{2}). As compared to meniscus force, the effect of the number of asperities on viscous force is trivial for both hydrophilic and hydrophobic surfaces. A noticeable decrease of viscous force is observed for *θ*_{1}=180° and *θ*_{2}=120°. This is believed to be due to the quick break of meniscus under the given condition.

For the purpose of comparison, meniscus and viscous forces for the separation of two smooth hydrophilic and hydrophobic surfaces with number of *N* identical menisci are also calculated as shown in figure 9*b*. The initial separation of two surfaces is the same as for the rough surface case. It is observed that for the study cases the attractive meniscus force slightly increases with the increase in *N* for hydrophilic smooth surfaces, whereas it slightly decreases with an increase in *N* for hydrophobic smooth surfaces. The rate change of force is gradual. Part of the reason is it may be due to the insignificant change of total meniscus area with *N* for smooth surfaces. A quick decrease of viscous force is observed with the increase of *N* for either hydrophilic or hydrophobic surface. This is because each meniscus has a smaller meniscus area at larger *N*, and the meniscus can be broken very quickly. As compared to the rough surface case, both forces are much larger for smooth surfaces at smaller number of *N*. This indicates that the introduction of a small number of asperities can help to reduce both forces significantly and thus reduce stiction.

## 4. Conclusions

Both meniscus and viscous forces during the separation of hydrophilic and hydrophobic smooth/rough surfaces with symmetric and asymmetric contact angles are calculated. The effects of the separation distance, initial meniscus height, separation time, contact angles and roughness are presented.

The results show that meniscus and viscous forces change at a rapid rate at the early stages of separation. The meniscus force decreases with an increase of separation distance, whereas the viscous force has an opposite trend. Both forces decrease with an increase of initial meniscus height. Also, larger initial meniscus height has a longer meniscus break distance. An increase of separation time, initial meniscus height or the decrease of contact angle leads to an increase of critical meniscus area at which both forces are equivalent. An increase in contact angle leads to a decrease of attractive meniscus force but an increase of repulsive meniscus force (attractive or repulsive dependent on hydrophilic or hydrophobic surface, respectively). For asymmetric contact angles, the magnitude of the meniscus force and the critical meniscus area are in between the values for the two angles. Though the contact angle significantly affects meniscus force, it only has a limited effect on the viscous force. A slightly attractive force is observed for the hydrophobic surface during the end stage of separation though the magnitude is small.

The combination effects of initial meniscus height and contact angles on both forces are summarized in table 1. At a fixed initial meniscus height, an increase of contact angle leads to a decrease of attractive meniscus force for a hydrophilic surface. An increase of both initial meniscus height and contact angle leads to a decrease of attractive meniscus force and vice versa. For a hydrophobic surface, an increase of contact angle leads to an increase of repulsive meniscus force. An increase of initial meniscus height and a decrease of contact angle lead to a decrease of repulsive meniscus force and vice versa. As compared to contact angle, the initial meniscus height dominates the effect on viscous force. An increase of initial meniscus height leads to a decrease of viscous force and vice versa.

For a rough surface, an increase in the number of surface asperities (roughness) leads to an increase of meniscus force; however, its effect on viscous force is trivial. As compared to a smooth surface, the introduction of a small number of asperities can help to reduce both forces.

This study provides a comprehensive analysis of meniscus and viscous forces during separation of two hydrophilic or hydrophobic surfaces from liquid menisci. It helps in better understanding the physics of both hydrophilic and hydrophobic phenomena. Control of forces may be achieved by proper manipulation of the surface properties. It is also useful for solving real technological problems by understanding the behaviour of these forces.

## Footnotes

One contribution of 7 to a Theme Issue ‘Nanotribology, nanomechanics and applications to nanotechnology II’.

- © 2008 The Royal Society