Recent research on the topic of the liquid-mediated adhesion

## Abstract

Interfaces comprised of a liquid interposed between two solids in close proximity are common in small-scale devices. In many cases, the liquid induces large and undesired adhesive forces. It is of interest, therefore, to model the way in which forces are developed in such an interface. The following chapter presents several models of liquid-mediated adhesion, considering the roles of surface geometry, liquid surface tension, elastic deformation, surface roughness, and surface motion on the development of interfacial forces.

### Keywords

- Capillary film
- liquid-mediated adhesion
- liquid bridge

## 1. Introduction

Phenomena related to the wetting of solid–solid interfaces are of technological importance. When two surfaces are in close proximity, the presence of a liquid film may cause the surfaces to stick together. Such liquid-mediated adhesion can negatively affect the operation of micro/nanoscale systems [1–7]. The interfacial liquid film, which may be present due to condensation, contamination, or lubrication, may experience large concave curvatures at the liquid-vapor interface and large negative pressures. These negative pressures give rise to large adhesive forces, which can have a potentially deleterious effect on the performance of small-scale devices.

In this chapter, we will discuss the behavior of an interface comprised of a liquid interposed between two solids. Throughout this chapter, we are concerned with the role of liquid films in regimes where gravitational effects are negligible, which generally implies that the vertical length scale is small. As an illustration, it can be easily shown that the change in pressure due to gravity within a near-hemispherical water droplet (resting on a horizontal surface) from just within the top of the free surface to the bottom of the droplet is given by * ρ*= 1000 kg/m

^{3}and

*= 0.0727 N/m, so that for a radius of 1.0 mm, we have a ratio of about 0.07, meaning that the change in pressure due to gravity is only about 7% of that due to surface tension. Moreover, it is seen that the relative effects of gravity decrease in proportion to the square of the droplet radius. In general, the smaller the vertical scale, the less important are the effects of gravity in comparison to those of surface tension.*γ

Of particular interest in this chapter is the topic of liquid-mediation adhesion, a mechanism by which the liquid film pulls inward on the solid surfaces. We consider the effects of liquid surface tension, liquid viscosity, surface geometry, surface roughness, surface elasticity, and surface motion on the development of adhesive forces in the interface. Our approach to discussing the recent literature on the topic of liquid-mediated adhesion is to organize things according to several basic characteristics: gross interface geometry (flat or curved), surface topography (smooth or rough), structural properties (rigid or deforming), meniscus type (constant-volume or constant-pressure) and separating process (quasi-static or dynamic). In this context, Table 1 categorizes recent research that is particularly relevant to the subject of this chapter. It is noted that an entry of “volume” under the “film constant” heading means that the volume of the liquid bridge is held fixed during the separation process, while an entry of “pressure” indicates that the liquid is assumed to remain in thermodynamic equilibrium with its vapor during the separation process.

flat on flat | smooth | elastic | quasi-static | volume | Zheng and Streator | 2004 | 28 |

flat on flat | rough | elastic-plastic | quasi-static | pressure | Del Rio et al. | 2008 | 19 |

flat on flat | rough | elastic | quasi-static | pressure | Wang and Regnier | 2015 | 37 |

flat on flat | rough | elastic | quasi-static | pressure | Peng et al. | 2009 | 21 |

flat on flat | rough | rigid | quasi-static | pressure or volume | de Boer and de Boer | 2007 | 18 |

flat on flat | rough | elastically hard | quasi-static | pressure | de Boer | 2007 | 17 |

flat on flat | rough | elastic | quasi-static | pressure | Persson | 2008 | 20 |

flat on flat | rough | elastic | quasi-static | volume | Streator and Jackson | 2009 | 34 |

flat on flat | rough | elastic | quasi-static | volume | Streator | 2009 | 33 |

flat on flat | rough | elastic | quasi-static | volume | Rostami and Streator | 2015 | 35 |

flat on flat | rough | elastic | quasi-static | volume | Rostami and Streator | 2015 | 36 |

sphere on flat | smooth | elastic | quasi-static | pressure | Men et al. | 2009 | 24 |

sphere on flat, and sphere on sphere | smooth | rigid | quasi-static | volume | Rabonivich et al. | 2005 | 23 |

sphere on sphere | smooth | elastically soft | quasi-static | pressure | Butt et al. | 2010 | 22 |

sphere on sphere | smooth | elastic | quasi-static | volume | Zheng and Streator | 2003 | 30 |

sphere on sphere | smooth | elastic | quasi-static | volume | Zheng and Streator | 2007 | 31 |

flat on flat | smooth | rigid | dynamic | n/a (flooded) | Roemer et al. | 2015 | 15 |

flat on flat or sphere on flat | smooth | rigid | dynamic | volume | Cai and Bhushan | 2007 | 9 |

sphere on flat | smooth | rigid | dynamic | n/a (flooded) | Streator | 2006 | 25 |

## 2. Models of solid surfaces bridged by a liquid

### 2.1. Liquid film between smooth, rigid, parallel flats

#### 2.1.1. Static and quasi-static conditions

Consider the problem of a continuous liquid film that is at static equilibrium between two rigid, parallel flats in close proximity as shown in Figure 1. In this idealized case, the liquid forms an axisymmetric configuration, so that any horizontal cross section is circular. Because the liquid is in static equilibrium, the entire film must be at a single pressure. Per the Young-Laplace equation [8], the pressure drop

Where _{a}is the ambient pressure, * p*is the film pressure, and

R

_{1,2}are the principal radii of normal curvature of the free surface at any given point on the free surface. Since we are dealing with small vertical spacing, it is reasonable to assume the radius of curvature (

R

_{2}) that exists in the plane of the figure at each free surface point is much smaller than the other principal radius of curvature (

R

_{1}), which lies in a plane that is perpendicular to the plane of the figure as well as perpendicular to the tangent plane to the free surface at the point in question. In Figure 1, we have chosen to illustrate the value of

R

_{1}that exists in the plane of minimum horizontal diameter. Assuming

R

_{1}is sufficiently larger than

R

_{2}that 1/

R

_{1}may be neglected, the pressure drop in Eq. (1) becomes

Moreover, owing to the fact that the liquid film, being continuous and in static equilibrium, must experience a uniform pressure, one may conclude that the radius of curvature

so that

where

The value of the contact angle for a particular case is determined by a local thermodynamic equilibrium among the three relevant interfaces, which can be expressed in the Young-Dupree equation [8]

where

For a concave film shape (Figures 1a and 1c) the sum on the right-hand side of Eq. (4) is positive, yielding a positive pressure drop relative to atmospheric pressure. Thus, in terms of gauge pressure, the pressure within the film is negative. One important consequence is that the liquid exerts a force that pulls inward on the two plates so that the force exerted on either of the plates may be considered the force of adhesion due to the presence of the film. With reference to Figure 3, this adhesive force (_{ad}) can be expressed as

The first term on the right-hand side is the contribution to the adhesive force arising from the pressure drop across the free surface, while the second term is the adhesive force exerted by the free surface itself. Note that the total force exerted on the bottom of this upper section of the liquid film is simply transmitted to the upper plate, so the force given by Eq. (6) is indeed the adhesive force. Now under the assumption that _{1} can be neglected in Eq. (1), which leads to Eq. (2), the force contributed by the free surface

Suppose now that the liquid film has a fixed volume

This equation shows that under the conditions of fixed liquid volume the adhesive force is inversely proportional to the square of the film thickness.

When a quantity of a pure liquid of given chemical species is at thermodynamic equilibrium, the partial pressure of the vapor phase of the species is equal to the vapor pressure of the liquid phase for the given temperature. For a curved free surface, there is a small deviation in the vapor pressure from that corresponding to a planar free surface. This deviation is accounted for by the well-known Kelvin equation [8]

where * R*is the universal gas constant, and

*is the absolute temperature. Assuming, as before,*T

Using this result in Eq. (2) gives

Now, suppose the chemical species in question is water, so that the ratio

so that, from Eq. (10),

If we take the contact angles to be zero, then, from Eq. (4) and Eq. (11),

#### 2.1.2. Dynamic separation

The foregoing analysis is applicable to conditions of static (or quasi-static) equilibrium. Additional effects may arise from viscous interactions. Consider now a situation where the upper plate is pulled upward at a prescribed rate, while the lower plate is held fixed. One approach to analyzing such a situation [9] is to assume that the liquid flow is governed by the Reynolds equation of lubrication [10].

where * r*is the radial coordinate measured from the center of the axisymmetric film cross-section. Assuming that the gap,

*), the above equation can be integrated twice to give*r

where * r*).

To obtain the constants of integration, we assume that (1) the pressure just inside the free surface is that corresponding to the static case (see Eq. 4), and (2) the pressure is finite at * r =*0. Then letting

Now, the adhesive force is just given by

For a fixed liquid volume

The above equation shows that adhesive force grows in proportion to the rate _{cav} as the cavitation pressure (relative to atmospheric pressure), then the maximum possible adhesive force can be written as:

### 2.2. Liquid film between rigid, inclined surfaces

Consider the situation depicted in Figure 4, where there is a liquid film between two flat surfaces whose planes intersect. The configuration of Figure 4a is a non-equilibrium state owing to the greater free-surface curvature on the right than on the left, and the associated lower pressure (i.e., greater reduction in pressure compared to ambient). Thus, the fluid will flow from left to right, all the way up to the edge (Figure 4b) until achieving a configuration with equal free-surface curvature at left and right ends, thereby yielding the same pressure drop. The two-dimensional depiction of Figure 4, of course, obscures the required re-configuration that happens in three dimensions. In fact, the entire free surface must attain the same curvature, which means that liquid would find its way to both the front and back edges as well as the right edge.

### 2.3. Liquid film between a smooth, rigid sphere and a rigid flat

The sphere-flat configuration is of interest in its own right and as an important part of a rough surface contact model, in which contributions from various asperity-asperity liquid bridges are summed by viewing each pair as reflecting the interaction between a pair of spheres having the asperity curvatures.

#### 2.3.1. Static and quasi-static conditions

The interaction between a sphere and flat bridged by a liquid film, as illustrated in Figure 5, has been analyzed in [16]. When the radial width of the liquid film

This gives

The force of adhesion is obtained by multiplying this pressure difference by the cross-section area of the liquid bridge (

Several studies have considered the role of relative humidity on the adhesion between a sphere and a flat (or sphere on sphere) [9, 17–23], where, at thermodynamic equilibrium, the radius of the curvature of the free surface of the meniscus would be equal to the Kelvin radius, per Eq. (9). Such analysis is most appropriate for volatile liquids [24]. In this case the value of

#### 2.3.2. Dynamic separation

Now we consider the forces that arise when a sphere of mass * m*and radius

*is separated from the flat in a dynamic fashion, so that the minimum spacing*R

*is a function of time. Denoting the instantaneous vertical spacing between the sphere surface and the flat as*D

When a net external force

where _{m}is the “meniscus” force, which accounts for the effect of the pressure drop across the curved free surface of the liquid meniscus and _{v}is the “viscous” force, which arises from the deformation of the liquid bridge. It is assumed that any buoyancy forces are negligible. Following [26], the pressure field, as derived from the solution of the Reynolds equation (e.g., [10]), can be written as

where

where _{ad}is the adhesive force. Direct integration of the film thickness profile (19) provides the liquid volume:

Assuming the meniscus volume is fixed, we set

Using this result in Eq. (25) allows the force exerted by the liquid to be expressed in terms of the separation

In cases where the inertial term of Eq. (23) is negligible, the net applied load * F*is equated with the sum of the capillary and viscous forces

F

_{ad}. Moreover, in cases where the variation in the capillary force is small compared to the variation in the viscous force, Eq. (28) can be integrated to give [9, 27]

where

where

One important result of the above relationship is that the rate of applied loading determines the peak adhesive load developed during separation, which we label here the “pull-off force” (

For example, when the externally applied force increases at a constant rate

A modified approach is needed to analyze the “fully-flooded” case, where the sphere interacts with a sufficiently thick lubricant film that further increases to the film thickness have negligible impact on the adhesive force. In this case, Eq. (32) still holds, but the viscous impulse becomes [25]

where * m*is the mass of the sphere.

It is emphasized here that Eqs. (28)-(33) presume the liquid film is not experiencing any cavitation. As discussed previously (see Eq. (18)), the potential development of a fully cavitated film would provide an upper bound for the adhesive force.

### 2.4. Liquid film between smooth, elastic flats

Figure 6 depicts a scenario when a liquid film interacts with two semi-infinite elastic bodies, where * H*is the uniform gap between the surfaces that exists in the absence of deformation. For this situation, the pressure within the liquid film causes elastic deformation of the half-spaces. Here we focus our attention on the case were the liquid film wets both surfaces such that they each experience a contact angle less than 90 degrees. This problem has been analyzed previously [28] and that work is summarized here. Letting

This pressure field causes an associated deformation field [29]

In the above equation, * H.*At any radial position within the wetted film, the film thickness can be expressed as

Using Eq. (35), the volume of the liquid bridge

The equilibrium configuration can be determined by considering the minimization of the free energy, which is comprised of elastic strain energy (_{S}) and surface energy _{E}* .*The elastic strain energy is simply given by the work done in creating the deformation field

Using Eq. (35) and carrying out the integration gives

Now the surface energy consists for energy contributions from the solid-vapor, solid-liquid, and liquid-vapor interfaces, so that

where subscripts 1 and 2 refer to the upper and lower surfaces, respectively and

Applying Eq. (5) to each surface and recalling that

A stable equilibrium corresponds to the minimization of the free energy

With these definitions, the dimensionless free energy can be expressed as

Note also that from Eqs. (35), (36) and (43), the minimum film thickness is given by

So that

The solution space of Eq. (48) is shown in Figure 7. An investigation of

Using Figure 7, one can determine the adhesive force. Letting the subscript “eq” identify values corresponding to a stable equilibrium configuration, it can be shown using Eqs. (35)–(37), (43), and (48), that

Then, the adhesive force is given by

### 2.5. Liquid film between smooth, elastic spheres

When a liquid bridges two elastic spheres [30], as illustrated in Figure 8, the situation is similar to the case of two elastic half-spaces (discussed above), but with an added feature due the surface curvature. The displacement profile is still given by Eq. (35), but the film thickness profile is now given by

where * R*is the composite radius of curvature, defined by

Note that the expressions for the elastic strain energy and surface energy are the same as those for the two half-spaces, so that the total free energy is still given by Eq. (41). In addition to non-dimensional parameters

and use a different form for the dimensionless free energy

This results in

Setting

It can readily be shown that for

The solution space for Eq. (58) is plotted in Figure 9 for several values of dimensionless volume

For equilibrium configurations that do not involve solid-solid contact, the pressure drop is given by Eq. (51), but with the gap at the free-surface given by

and the wetted radius given by (via solution of Eq. 54)

Thus, the adhesive force then becomes

The above force represents the external, separating force (over and above the weight of the sphere) required to maintain the spheres at the given configuration (i.e., with undeformed separation, * H*).

In cases where * a*, with the contact region surrounded by an annulus of liquid. The presence of contact modifies the form of the free energy, which becomes [31]

where

The dimensionless formulation involves two additional ratios [31]:

where

The equilibrium solution, for given values of

It can be shown [31] that the advent of solid-solid contact introduces hysteresis, just as in the case of the JKR contact model [32], which applies to dry contact. Thus, the set of configurations that the interface would pass through when breaking the contact, such as during a controlled separation process, would be different from those experienced upon its formation. For example, the value of * H*at which the solid-solid contact is lost during a separation process is different from the value of

*that corresponds to the formation of solid-solid contact during an approach process. Put another way, there is a jump-on instability at a certain*H

*upon approach, where the interface goes suddenly from no contact to contact, as well as a jump-off instability upon separation (at a larger*H

*), where the interface proceeds suddenly from having a contact radius*H

*to having no solid-solid contact. One convenient experimental measure of the strength of an adhesive contact is the pull-off force, which can take on different values depending upon how the pull-off process is conducted. When the separation*a

*(which is defined by the minimum gap between the undeformed sphere contours) is specified and increased quasi-statically, the interface will reach a configuration that is unstable and then abruptly lose contact. The magnitude of external, separating force required to reach this point of instability during separation is defined as the pull-off force during a controlled separation process.*H

### 2.6. Liquid film between contacting rough, elastic surfaces

Adhesive forces arising due to the presence of a liquid film between rough, elastic (or elastic-plastic) surfaces have been the subject of several recent works [17, 19–21, 33–37]. Figure 11 depicts a situation where two rough, elastic surfaces are in contact in the presence of an intervening liquid film. Taking into consideration a three-dimensional geometry, the assumption here is that the liquid film is continuous, so that there are no regions of liquid completely encased within a zone of solid-solid contact. Now in the case where the liquid wets the surfaces (i.e., the contact angles are less than 90°), the free surface of the liquid is concave and the film pressure is sub-ambient. Assuming that the lateral dimensions are much greater than the liquid film thickness, the pressure drop across the free surface is given by

where * P*. As solid-solid contact is first formed at the mutual asperity peaks, the liquid film will be quickly squeezed out to a radius that is determined by the given liquid volume and the average gap between the surfaces within the wetted region. However, as capillary forces take effect, the elastic surfaces will further deform, thereby reducing the mean gap between the surfaces and causing an increase in the wetted radius. An increase in the wetted radius in conjunction with a decreasing interfacial gap will cause a greater tensile force

One numerical model of such an interface appears in [35]. Here it is assumed that the liquid film is axisymmetric and that deformation of the asperities is modeled according to the multi-scale contact model of [38]. Thus, the surface topography is characterized by its spectral content and algebraic formulas are applied to compute the effects of external and capillary forces on the average spacing within the interface. Another important assumption is that the mean spacing

Thus, the tensile force

where

Sample results of the analysis are displayed in Figure 12, for the following input parameters: ^{3}, γ = 72.7 mN/m, σ = 0.4 μm, and _{n}= 4 cm^{2}, where σ is the r.m.s. surface roughness of a 3D isotropic surface with a Gaussian height distribution, and _{n}is the nominal contact area of the interface (i.e., the projected area of the interface in Figure 11). Figure 12 shows the influence of external load on several contact parameters, including tensile force and contact area (Fig. 12a) as well as average gap and wetted radius (Fig. 12b). The tensile force is seen to grow steadily with increasing external load until approaching a critical load, where the rate of increase of tensile force with load approaches infinity. The attainment of a near vertical slope in the curve suggests that the interface is unstable: no equilibrium configurations could be found for values of external load beyond the critical value. Analogous results are found for the average gap, tensile radius and solid-solid contact area. Such behavior suggests interface collapse, whereby beyond the critical point, the surfaces come into complete and near complete contact [33–36]. By introducing certain dimensionless parameters, the results can be generalized. Let an adhesion parameter Γ be defined according to

and let the dimensionless versions of external load, tensile force, and liquid volume be defined respectively as

The results for dimensionless tensile force versus the adhesion parameter are depicted in Figure 13 at several values of dimensionless volume. This figure reveals that, for each dimensionless volume considered, there is a critical value of the adhesion parameter whereby the force curve becomes vertical, suggesting the onset of surface collapse.

## Acknowledgments

The author would like to thank the National Science Foundation (US) for support of this work and Amir Rostami, a graduate research assistant, for performing some calculations used herein.

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