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
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.
Gross Interface Geometry |
Surface Type |
Deform.
Behavior | Loading Process |
Film Constant | Author(s) |
Year
Publ. |
Ref. No. |
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 p_{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 (F_{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
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
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,
where
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
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 R is separated from the flat in a dynamic fashion, so that the minimum spacing D is a function of time. Denoting the instantaneous vertical spacing between the sphere surface and the flat as
When a net external force
where F_{m} is the “meniscus” force, which accounts for the effect of the pressure drop across the curved free surface of the liquid meniscus and F_{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 F_{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
This pressure field causes an associated deformation field [29]
In the above equation,
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 (U_{S}) and surface energy U_{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
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 H 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 a 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 H (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.
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
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:
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.