Recent research on the topic of the liquid-mediated adhesion
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.
- Capillary film
- liquid-mediated adhesion
- liquid bridge
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 , where is the water mass density, is the gravitational acceleration, and is the approximate radius of the droplet. By comparison, the change in pressure across the free surface of the droplet is given by , where is the liquid surface tension. Thus, the ratio of gravitational effects to surface tension effects is equal to . For water at room temperature (and ambient pressure), one has
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|
|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 , the pressure drop across the free surface is given by
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 is the same at every point of the free surface. Thus, the free surface profile is in the shape of a circle. Using this result leads to the geometrical relationship depicted in Figure 2, by which one concludes that
where are the liquid–solid contact angles. A liquid is considered to “wet” a given surface if its contact angle (measured from solid–liquid interface to the solid-vapor interface) is less than 90 degrees, and a liquid is considered to be non-wetting if its contact angle is greater than 90 degrees. Complete wetting is associated with a contact angle of 0 degrees and complete non-wetting corresponds to a contact angle of 180 degrees. While Figure 2 shows a case for which the liquid wets both upper and lower surfaces, Eq. (3) holds for each of the configurations depicted in Figure 1. Depending on both signs and relative magnitudes of the cosine terms in Eq. (4), the film pressure may be greater than, equal to, or less than atmospheric pressure. For the case of Figure 1a, where both surfaces are wet by the liquid, both cosine terms are positive and the film pressure is sub-ambient. In contrast, for the case of Figure 1b, where neither surface is wet by the liquid, both cosine terms are negative and the film pressure exceeds the ambient pressure. When one of the surfaces is wet by the film while the other is not (Figures 1c and 1d), the sign of the film pressure depends on the relative magnitudes of the two cosine terms. When the contact angle associated with the surface that is wet by the liquid is closer to zero than the other contact angle is to 180 degrees (Figure 1c), then the film shape is concave and the film pressure is sub-ambient. On the other hand, if the contact angle associated with the surface that is wet by the liquid departs from zero degrees more than the opposing surface departs from 180 degrees (Figure 1d), then the film shape is convex and the film pressure is greater than ambient.
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 
where , , and are the surface energies per unit area of the solid–vapor, solid–liquid, and liquid–vapor interfaces, respectively. Note that is the same as the surface tension of the liquid .
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 (
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 is much greater than , it can be shown that the force contribution to the pressure drop dominates the force contribution due to the free surface. Let the first term on the far right-hand side be denoted by and the second term be denoted by . Then =2 . Thus, to the extent that 1/
Suppose now that the liquid film has a fixed volume . To a good approximation, this volume may be expressed as , where the deviation from a cylindrical geometry has been ignored. Then the adhesive force may be written as:
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 
where is the saturation pressure at the given temperature, is the pressure of the vapor just outside of the liquid film, is the molar volume, is the Kelvin radius,
Using this result in Eq. (2) gives
Now, suppose the chemical species in question is water, so that the ratio represents the relative humidity. Then Eq. (9) states that, at thermodynamic equilibrium, the radius of curvature of any free surface of the liquid film is determined by the relative humidity. For example, taking properties of water at room temperature and assuming a relative humidity of 95%, we have
so that, from Eq. (10),
If we take the contact angles to be zero, then, from Eq. (4) and Eq. (11), . Now, for this mathematically idealized case of perfectly parallel plates, 21 nm would be the only spacing for which a liquid film could exist at thermodynamic equilibrium at 95% relative humidity. On the other hand, if the surfaces were curved, even slightly, then there would be a range of humidity values for which a film could be sustained at thermodynamic equilibrium. It should be noted that, in practical situations, the establishment of thermodynamic equilibrium may require a considerable amount of time, such as hours, or even days. In the interim, the adhesive forces will be dictated by the current amount of liquid within the interface.
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  is to assume that the liquid flow is governed by the Reynolds equation of lubrication .
where and and are constants of integration (i.e., independent of
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
Now, the adhesive force is just given by
For a fixed liquid volume , which is approximated by , we arrive at
The above equation shows that adhesive force grows in proportion to the rate at which the plates are being pulled apart and is quite sensitive to the value of separation. Small separations require a much larger separating force than what is required at larger separations for the given separating rate. One caveat, however, is that there is a practical limit as to the magnitude of negative pressure that can be sustained during the separation of surfaces. Whereas thermodynamic equilibrium suggests that the liquid will cavitate once its absolute pressure approaches zero (e.g., ), it has also been found that, in certain cases, the liquid film may achieve absolute pressures that are negative [7, 12–15]. For example, an analytical model was developed  for the dynamic vertical separation of opposing plates and, based on fitting with experimental data, a tensile strength of 35 kPa was found for a mineral oil. In any case, if one denotes
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 . When the radial width of the liquid film is sufficiently small compared to the radius of the sphere, the slope of the sphere at the location of the free surface may be taken as horizontal. In this case, the pressure within the film will be given by Eq. (4) with replaced by , the film thickness at the radial location of the free surface. Additionally, the sphere contour can be approximated well by that of a paraboloid, so that
The force of adhesion is obtained by multiplying this pressure difference by the cross-section area of the liquid bridge (, giving (after re-arrangement)
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 . In this case the value of appearing in Eq. (21) would be determined directly by the relative humidity, via Eqs. (3) and (9).
2.3.2. Dynamic separation
Now we consider the forces that arise when a sphere of mass
When a net external force (i.e., an applied force less the sphere weight) acts on the sphere (positive upward), the governing equation becomes 
where is the liquid viscosity. Note that the wetted radius is itself a function of time. Assuming that the free surface of the liquid has the same shape as when the film is quasi-static, the pressure just inside the meniscus, is given by Eq. (4). Thus, integrating over the meniscus area gives 
Assuming the meniscus volume is fixed, we set and express the film thickness at the free surface as
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
where is the time to completely separate the surfaces, is the distance at which the sphere and flat are considered completely separated, and is the sphere-flat spacing at the beginning of the separation process. It was argued in  that even in the case of initial solid-solid contact, the value of should not be zero, but should be selected in accordance with the interface roughness. Eq. (29) suggests that the separation process for a viscous-dominated interaction is governed by the time integral of the viscous force, known as the viscous impulse, the value of which is a function only of the input parameters of the problem . Now the distance of complete separation is taken as infinite in , and finite in . For , Eq. (29) can be simplified , giving
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 , the pull-off force takes the form:
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 
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 and are the elastic moduli and Poisson ratios of bodies 1 and 2, respectively and
This pressure field causes an associated deformation field 
In the above equation, and are the normal surface displacements of bodies 1 and 2, respectively (each positive toward the opposing body), is the total displacement, and the reduced modulus is given by . It is assumed here that the bulk positions of the bodies are fixed, so that opposing surface points remote to the interface are maintained at a spacing of
Using Eq. (35), the volume of the liquid bridge , which is assumed to be fixed, is given by
The equilibrium configuration can be determined by considering the minimization of the free energy, which is comprised of elastic strain energy (
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 is the radius of an arbitrary control region that encloses the interface. It is reasonable to assume here that the film thickness at the free surface is sufficiently small compared to the meniscus radius that the energy contributions from the liquid-vapor interface are negligible. Then the total free energy is given by
Applying Eq. (5) to each surface and recalling that one obtains
A stable equilibrium corresponds to the minimization of the free energy under the constraint of constant liquid volume . Let us now introduce dimensionless quantities:
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 provides a dimensionless measure as to the degree of surface approach. When = 0, the surfaces are at the original separation throughout, and when = 1 the surfaces come into point contact. For a stable equilibrium, the dimensionless energy achieves a local minimum with respect to . Thus, a necessary (but not sufficient) condition for stable equilibrium is that , which yields the following requirement:
The solution space of Eq. (48) is shown in Figure 7. An investigation of at the various equilibrium values of reveals that values to the left of the peak correspond to stable equilibrium configurations, whereas those to the right of the peak, correspond to unstable equilibrium configurations. From the graph, we see that no equilibrium configurations exist for values of greater than 0.0134. This result implies that solid-solid contact must occur whenever . Further, once solid-solid contact occurs, the contact region grows without bound (i.e., the free energy, which now includes a solid-solid contribution, decreases monotonically with increasing contact radius).
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 , 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
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 (Eqs. 43 and 45), we introduce a dimensionless volume according to
and use a different form for the dimensionless free energy
This results in
Setting leads to the following necessary condition on for a stable equilibrium to be achieved
It can readily be shown that for , Eq. (58) reduces to Eq. (48), which is applicable to two half-spaces. This result is expected because for a finite liquid volume and a finite, non-zero value of surface spacing, an infinite value of sphere radius (which corresponds to a flat-flat interface) causes to vanish as per Eq. (55).
The solution space for Eq. (58) is plotted in Figure 9 for several values of dimensionless volume . As observed, the smallest chosen value of yields a curve that is quite close to the half-space solution (Figure 7). It turns out that stable equilibrium configurations exist without solid-solid contact for a range of values that depends on the value of . This relationship is summarized in Figure 10, which reveals regions with and without solid-solid contact within space. The boundary curve is given by . If, for a given , then there is no solid-solid contact, whereas for the spheres must experience contact.
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,
In cases where , the solids come into contact over some contact radius
where is the surface tension associated with the solid-solid interface.
The dimensionless formulation involves two additional ratios :
where is the well-known work of adhesion and is given by 
The equilibrium solution, for given values of and , is now expressed in terms of both and . In the case of solid-solid contact, no analytical expressions exist for the elastic strain energy, owing to the unknown contact solid-solid pressure distribution. Therefore, the equilibrium configurations and associated adhesive forces must be acquired through a numerical process .
It can be shown  that the advent of solid-solid contact introduces hysteresis, just as in the case of the JKR contact model , 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
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 is the film thickness at the location of the free surface. For a continuous liquid film in static equilibrium, the pressure throughout the film must be the same, so Eq. (66) suggests that the periphery of the liquid film is at a constant height. The overall equilibrium shape of the film will depend upon the details of the gap distribution within the surfaces, which itself will be modulated by surface deformation due to the tensile stresses exerted by the liquid film. In general, the equilibrium configuration of the liquid film will not be axisymmetric. However, for surfaces that are nearly flat aside from a small-scale roughness, we can expect that the equilibrium film shape will be nearly axisymmetric. Assuming an axisymmetric liquid film, one can describe the establishment of equilibrium as follows: First, suppose a quantity of liquid is placed upon a surface that will serve as the lower surface of the contacting pair. Then, the other surface is placed in contact with the lower surface (and the liquid film) under some external load
One numerical model of such an interface appears in . 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 . 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 within the wetted region is a good approximation to the spacing at the free surface so that pressure drop within the film is approximated by
Thus, the tensile force becomes
where is the radius of the wetted region. If one defines the adhesive force as the tensile contribution to the net force exerted on either solid body, then is the just the adhesive force (. An alternative definition for the adhesive force would be the value of the tensile external load required to maintain static equilibrium, or required to achieve a certain separation and separation rate. The latter definition views the adhesive force as the interfacial tensile force less the interfacial compressive force.
Sample results of the analysis are displayed in Figure 12, for the following input parameters: = 0.1 mm3, γ = 72.7 mN/m, σ = 0.4 μm, and
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.
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.