Recent research on the topic of the liquidmediated adhesion
Abstract
Interfaces comprised of a liquid interposed between two solids in close proximity are common in smallscale 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 liquidmediated 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
 liquidmediated 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 liquidmediated 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 liquidvapor 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 smallscale 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 nearhemispherical 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 liquidmediation 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 liquidmediated 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 (constantvolume or constantpressure) and separating process (quasistatic 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  quasistatic  volume  Zheng and Streator  2004  28 
flat on flat  rough  elasticplastic  quasistatic  pressure  Del Rio et al.  2008  19 
flat on flat  rough  elastic  quasistatic  pressure  Wang and Regnier  2015  37 
flat on flat  rough  elastic  quasistatic  pressure  Peng et al.  2009  21 
flat on flat  rough  rigid  quasistatic  pressure or volume  de Boer and de Boer 
2007  18 
flat on flat  rough  elastically hard  quasistatic  pressure  de Boer  2007  17 
flat on flat  rough  elastic  quasistatic  pressure  Persson  2008  20 
flat on flat  rough  elastic  quasistatic  volume  Streator and Jackson  2009  34 
flat on flat  rough  elastic  quasistatic  volume  Streator  2009  33 
flat on flat  rough  elastic  quasistatic  volume  Rostami and Streator  2015  35 
flat on flat  rough  elastic  quasistatic  volume  Rostami and Streator  2015  36 
sphere on flat  smooth  elastic  quasistatic  pressure  Men et al.  2009  24 
sphere on flat, and sphere on sphere  smooth  rigid  quasistatic  volume  Rabonivich et al.  2005  23 
sphere on sphere  smooth  elastically soft  quasistatic  pressure  Butt et al.  2010  22 
sphere on sphere  smooth  elastic  quasistatic  volume  Zheng and Streator  2003  30 
sphere on sphere  smooth  elastic  quasistatic  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 quasistatic 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 YoungLaplace equation [8], the pressure drop
Where
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 YoungDupree equation [8]
where
For a concave film shape (Figures 1a and 1c) the sum on the righthand 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 righthand 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 wellknown 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 quasistatic) 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
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
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 nonequilibrium state owing to the greater freesurface 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 freesurface curvature at left and right ends, thereby yielding the same pressure drop. The twodimensional depiction of Figure 4, of course, obscures the required reconfiguration 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 sphereflat configuration is of interest in its own right and as an important part of a rough surface contact model, in which contributions from various asperityasperity 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 quasistatic 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 crosssection 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
When a net external force
where
where
where
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
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 “pulloff force” (
For example, when the externally applied force increases at a constant rate
A modified approach is needed to analyze the “fullyflooded” 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
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 semiinfinite 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 (
Using Eq. (35) and carrying out the integration gives
Now the surface energy consists for energy contributions from the solidvapor, solidliquid, and liquidvapor 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 halfspaces (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
Note that the expressions for the elastic strain energy and surface energy are the same as those for the two halfspaces, so that the total free energy is still given by Eq. (41). In addition to nondimensional 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 solidsolid contact, the pressure drop is given by Eq. (51), but with the gap at the freesurface 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
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 solidsolid 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
2.6. Liquid film between contacting rough, elastic surfaces
Adhesive forces arising due to the presence of a liquid film between rough, elastic (or elasticplastic) 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 threedimensional 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 solidsolid 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 subambient. 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 multiscale 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.
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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