Open access peer-reviewed chapter

Solid-Liquid-Solid Interfaces

By Jeffrey L. Streator

Submitted: November 11th 2014Reviewed: September 21st 2015Published: December 16th 2015

DOI: 10.5772/61572

Downloaded: 1894

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 [17]. 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 (Δp)gravity=ρgR, where ρis the water mass density, gis the gravitational acceleration, and Ris the approximate radius of the droplet. By comparison, the change in pressure across the free surface of the droplet is given by (Δp)surface tension=2γ/R, where γis the liquid surface tension. Thus, the ratio of gravitational effects to surface tension effects is equal to ρgR2/2γ. For water at room temperature (and ambient pressure), one has ρ= 1000 kg/m3 and γ= 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.

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 GeometrySurfaceTypeDeform.
Behavior
Loading ProcessFilmConstantAuthor(s)Year
Publ.
Ref.No.
flat on flatsmoothelasticquasi-staticvolumeZheng and Streator200428
flat on flatroughelastic-plasticquasi-staticpressureDel Rio et al.200819
flat on flatroughelasticquasi-staticpressureWang and Regnier201537
flat on flatroughelasticquasi-staticpressurePeng et al.200921
flat on flatroughrigidquasi-staticpressure or volumede Boer and
de Boer
200718
flat on flatroughelastically hardquasi-staticpressurede Boer200717
flat on flatroughelasticquasi-staticpressurePersson200820
flat on flatroughelasticquasi-staticvolumeStreator and Jackson200934
flat on flatroughelasticquasi-staticvolumeStreator200933
flat on flatroughelasticquasi-staticvolumeRostami and Streator201535
flat on flatroughelasticquasi-staticvolumeRostami and Streator201536
sphere on flatsmoothelasticquasi-staticpressureMen et al.200924
sphere on flat, and sphere on spheresmoothrigidquasi-staticvolumeRabonivich et al.200523
sphere on spheresmoothelastically softquasi-staticpressureButt et al.201022
sphere on spheresmoothelasticquasi-staticvolumeZheng and Streator200330
sphere on spheresmoothelasticquasi-staticvolumeZheng and Streator200731
flat on flatsmoothrigiddynamicn/a (flooded)Roemer et al.201515
flat on flat or
sphere on flat
smoothrigiddynamicvolumeCai and Bhushan20079
sphere on flatsmoothrigiddynamicn/a (flooded)Streator200625

Table 1.

Recent research on the topic of the liquid-mediated adhesion

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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 Δpacross the free surface is given by

Δp=pap=γ(1R1+1R2).E1

Where pais the ambient pressure, pis the film pressure, and R1,2 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 (R2) that exists in the plane of the figure at each free surface point is much smaller than the other principal radius of curvature (R1), 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 R1 that exists in the plane of minimum horizontal diameter. Assuming R1 is sufficiently larger than R2that 1/R1 may be neglected, the pressure drop in Eq. (1) becomes

Δp=γR2E2

Figure 1.

Profile of an axisymmetric liquid film between rigid, parallel plates. (a) The liquid wets both surfaces leading to a concave film shape. (b) The liquid wets neither surface, leading to a convex film shape. (c) The liquid wets one surface but not the other, with the wet surface closer to complete wetting than the non-wet surface is to complete non-wetting, leading to a convex film. (d) The liquid wets one surface, but not the other, with the non-wet surface closer to complete non-wetting than the wet surface is to complete wetting, leading to a convex film.

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 R2is 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

h=R2(cosθ1+cosθ2)E3

so that

Δp=γ(cosθ1+cosθ2)hE4

where θ1,2are 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 [8]

γSV=γSL+γLVcosθE5

where γSV, γSL, and γLVare the surface energies per unit area of the solid–vapor, solid–liquid, and liquid–vapor interfaces, respectively. Note that γLVis the same as the surface tension of the liquid γ.

Figure 2.

Geometrical relationship between plate spacing and radius of curvature of free surface for an assumed circular profile. Without loss of generality, the liquid is shown here as wetting both surfaces.

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 (Fad) can be expressed as

Fad=(p)πR12+2πR1γ=γ(cosθ1+cosθ2)h πR12+2πR1γE6

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 R1is much greater than R2, 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 FΔpand the second term be denoted by Fγ. Then Fγ/FΔp=2 R2/R1. Thus, to the extent that 1/R1 can be neglected in Eq. (1), which leads to Eq. (2), the force contributed by the free surface 2πR1γmay be neglected in Eq. (6).

Figure 3.

Sources of force exerted on the upper section of the fluid (dividing line chosen at the plane of minimum diameter).

Suppose now that the liquid film has a fixed volume Vo. To a good approximation, this volume may be expressed as Vo=πR12h, where the deviation from a cylindrical geometry has been ignored. Then the adhesive force may be written as:

Fad=γ(cosθ1+cosθ2)h2 VoE7

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]

lnpspv=γ(1R1+1R2)VmRT=γRKVmRTE8

where psis the saturation pressure at the given temperature, pvis the pressure of the vapor just outside of the liquid film, Vmis the molar volume, RKis the Kelvin radius, Ris the universal gas constant, and Tis the absolute temperature. Assuming, as before, R1R2, and then isolating R2, one obtains

R2=RK=γVmRTlnpvpsE9

Using this result in Eq. (2) gives

pap=RTlnpvpsVmE10

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

R2=(0.0727Nm)(0.018 kg/mol)/(.00100m3/kg)8.314JmolK(293 K)ln(0.95)=10.5 nmE11

so that, from Eq. (10),

pap=RTlnpvpsVm=(0.0727Nm)10.5 nm=6.9 MPaE12

If we take the contact angles to be zero, then, from Eq. (4) and Eq. (11), h=2R2=21 nm. 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 [9] is to assume that the liquid flow is governed by the Reynolds equation of lubrication [10].

1rr(rh3pr)=12μht E13

where ris the radial coordinate measured from the center of the axisymmetric film cross-section. Assuming that the gap, h, is uniform (i.e., independent of r), the above equation can be integrated twice to give

p(r,t)=3μh˙h3+c1(t)ln(r)+co(t) E14

where h˙htand c1(t)and co(t)are constants of integration (i.e., independent of 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 R1, as before, denote the inner radius of the droplet, we obtain (in terms of gauge pressure):

p(r,t)=γ(cosθ1+cosθ2)h+3μh˙h3(r2R12) E15

Now, the adhesive force is just given by

Fad=0bp(r,t)2πrdr=γ(cosθ1+cosθ2)hπR12+32πμh˙h3R14 E16

For a fixed liquid volume (Vo), which is approximated by Vo=πR12h, we arrive at

Fad=0bp(r,t)2πrdr=γ(cosθ1+cosθ2)h2Vo+32μh˙h5 Vo2E17

The above equation shows that adhesive force grows in proportion to the rate h˙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., [11]), it has also been found that, in certain cases, the liquid film may achieve absolute pressures that are negative [7, 1215]. For example, an analytical model was developed [15] 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 pcav as the cavitation pressure (relative to atmospheric pressure), then the maximum possible adhesive force can be written as:

Fad=pcavπR12 E18

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.

Figure 4.

Liquid film between inclined surfaces: (a) non-equilibrium configuration and (b) equilibrium configuration.

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 (b)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 hreplaced by h(b), 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

h(b)=D+b22R E19

This gives

pap=γ(cosθ1+cosθ2)h(b)=γ(cosθ1+cosθ2)D+b22R E20

The force of adhesion is obtained by multiplying this pressure difference by the cross-section area of the liquid bridge (πb2), giving (after re-arrangement)

Fad=2πRγ(cosθ1+cosθ2)(1Dh(b))=2πRγ(cosθ1+cosθ2)1+2RDb2E21

Several studies have considered the role of relative humidity on the adhesion between a sphere and a flat (or sphere on sphere) [9, 1723], 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 h(b)appearing in Eq. (21) would be determined directly by the relative humidity, via Eqs. (3) and (9).

Figure 5.

Liquid film between a rigid sphere and a rigid flat.

2.3.2. Dynamic separation

Now we consider the forces that arise when a sphere of mass mand radius Ris separated from the flat in a dynamic fashion, so that the minimum spacing Dis a function of time. Denoting the instantaneous vertical spacing between the sphere surface and the flat as h(r,t), we have D=h(0,t). When the wetted radius bis much smaller than the sphere radius, one gets

h(r,t)=D(t)+r22R E22

When a net external force F(i.e., an applied force less the sphere weight) acts on the sphere (positive upward), the governing equation becomes [25]

md2Ddt2=FFmFv E23

where Fmis the “meniscus” force, which accounts for the effect of the pressure drop across the curved free surface of the liquid meniscus and Fvis 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

p(r,t)=p(b,t)3μR(1h(r,t)21h(b,t)2)1DdDdtE24

where μis the liquid viscosity. Note that the wetted radius  bis 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, p(b,t)is given by Eq. (4). Thus, integrating over the meniscus area gives [25]

Fad=Fm+Fv =2πRγ(cosθ1+cosθ2)(1Dh(b,t))+6πμR2(1Dh(b,t))21DdDdtE25

where Fadis the adhesive force. Direct integration of the film thickness profile (19) provides the liquid volume:

V=πb2D+πb44R=πR[h(b,t)2D2]E26

Assuming the meniscus volume is fixed, we set V=Voand express the film thickness at the free surface as

h(b,t)=D2+VoπRE27

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

Fad=Fm+Fv =2πRγ(cosθ1+cosθ2)(1DD2+VoπR)+6πμR2(1DD2+VoπR)21DdDdtE28

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

0ts(FFm)dt=0DsFvdt= 0DsFvdtdDdD=6πμR2ln[Ds(Do+Do2+VoπR)2Ds2+VoπRDo(Ds+Ds2+VoπR)2Do2+VoπR]E29

where tsis the time to completely separate the surfaces, Dsis the distance at which the sphere and flat are considered completely separated, and Dois the sphere-flat spacing at the beginning of the separation process. It was argued in [27] that even in the case of initial solid-solid contact, the value of Doshould 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 [27]. Now the distance of complete separation Dsis taken as infinite in [27], and finite in [9]. For Ds, Eq. (29) can be simplified [27], giving

Iv=6πμR2ln[(Do+Do2+VoπR)24DoDo2+VoπR]=6πμR2ln[(Do+Do2+VoπR)24DoDo2+VoπR]E30

where Ivis the viscous impulse. Applying Eq. (30) to the case of a sphere interacting with a surface that is initially covered with a thin film of uniform thickness ho, the above is approximated by [27]

Iv=6πμR2ln(ho/2Do) E31

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” (Fpulloff).

For example, when the externally applied force increases at a constant rate (F˙), the pull-off force takes the form:

Fpulloff=Fm+2F˙Iv E32

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]

Iv=6πμR2ln[0.1(6πμR2)32mDo2F˙] E33

where mis 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 E1,2and ν1,2are the elastic moduli and Poisson ratios of bodies 1 and 2, respectively and His 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 Δprepresent the pressure drop across the free surface of the liquid (from outside to inside), the film pressure (gauge) can be expressed as

p(r)={Δprb0r>bE34

This pressure field causes an associated deformation field [29]

u(r)=u1(r)+u2(r)={4ΔpbπE0π/21r2b2sin2ψdψrb4ΔpbπE0π/2b2cos2ψ1r2b2sin2ψdψr>bE35

Figure 6.

A liquid film bridging two elastic half-spaces.

In the above equation, u1(r)and u2(r)are the normal surface displacements of bodies 1 and 2, respectively (each positive toward the opposing body), u(r)is the total displacement, and the reduced modulus Eis given by 1/E(1ν12)/E1+(1ν22)/E2. 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 H.At any radial position within the wetted film, the film thickness can be expressed as

h(r)=Hu(r)E36

Using Eq. (35), the volume of the liquid bridge (Vo), which is assumed to be fixed, is given by

Vo=ob2πrh(r)dr=πHb2163EΔpb3E37

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

UE=12obΔpu(r)2πrdrE38

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

UE=83EΔp2b3E39

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

US=π(Ro2b2)(γSV1+γSV2)+πb2(γSL1+γSL2)+ALVγE40

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

UT=US+UE=π(Ro2b2)(γSV1+γSV2)+πb2(γSL1+γSL2)+83EΔp2b3E41

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

UT=πRo2(γSV1+γSV2)πb2γ(cosθ1+cosθ2)+83EΔp2b3E42

A stable equilibrium corresponds to the minimization of the free energy UTunder the constraint of constant liquid volume Vo. Let us now introduce dimensionless quantities:

η2ΔpbEHE43
UT*UTπRo2(γSV1+γSV2)πEH3/2Vo1/2E44
Γγ2(cosθ1+cosθ2)2Vo4E'2H5E45

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

UT*2η23π1(8/3π)η2Γπ11(8/3π)ηE46

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

hmin=Hu(0)=H2ΔpbE=H(1η)E47

So that ηprovides a dimensionless measure as to the degree of surface approach. When η= 0, the surfaces are at the original separation throughout, and when η= 1 the surfaces come into point contact. For a stable equilibrium, the dimensionless energy achieves a local minimum with respect to η. Thus, a necessary (but not sufficient) condition for stable equilibrium is that dUT*/dη=0, which yields the following requirement:

η2(12πη)2(183πη)=16ΓπE48

The solution space of Eq. (48) is shown in Figure 7. An investigation of d2UT*/dη2at the various equilibrium values of ηreveals that values to the left of the peak (η<0.5577)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 Γgreater than 0.0134. This result implies that solid-solid contact must occur whenever Γ>0.0134. 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).

Figure 7.

Equilibrium configurations for two half-spaces bridged by a liquid film.

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

beq=VoπH(183πηeq)E49
heq(beq)=H(12πηeq)E50
Δpeq=γ(cosθ1+cosθ2)heq(beq)E51

Then, the adhesive force is given by

Fad=πbeq2Δpeq=γ(cosθ1+cosθ2)VoH2(183πηeq)(12πηeq)E52

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

h(r)=H+r22Ru(r)E53

where Ris the composite radius of curvature, defined by 1/R=1/R1+1/R2. Thus, upon integrating, the liquid volume takes the form

Vo=πHb2163EΔpb3+πb44RE54

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 ηand Γ(Eqs. 43 and 45), we introduce a dimensionless volume according to

ΨVoRH2E55

and use a different form for the dimensionless free energy

UT*UTπRo2(γSV1+γSV2)πEH5/2R1/2E56

This results in

UT*=4Γπ[183πη(183πη)2+Ψπ]+23πη22(183πη)+(183πη)2+ΨπE57

Setting dUT*/dη=0leads to the following necessary condition on ηfor a stable equilibrium to be achieved

η2[23πη+(183πη)2+Ψπ]2[183πη+(183πη)2+Ψπ]=32ΓπE58

It can readily be shown that for Ψ=0, 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 Ψto vanish as per Eq. (55).

The solution space for Eq. (58) is plotted in Figure 9 for several values of dimensionless volume Ψ. As observed, the smallest chosen value of Ψyields 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 Γvalues that depends on the value of Ψ. This relationship is summarized in Figure 10, which reveals regions with and without solid-solid contact within ΨΓspace. The boundary curve is given by Γc=0.03007Ψ1.3955+0.01336 . If, for a given Ψ, Γ<Γcthen there is no solid-solid contact, whereas for ΓΓcthe 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

heq(beq)=H[23πηeq+(183πηeq)2+Ψπ]E59

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

beq={2VoπH[183πηeq+(183πηeq)2+VoπRH2]}12E60
.

Thus, the adhesive force then becomes

Fad=πbeq2Δpeq=2γ(cosθ1+cosθ2)VoH2[183πηeq+(183πηeq)2+VoπRH2][23πηeq+(183πηeq)2+Ψπ]E61

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 Γ>Γc, the solids come into contact over some contact radius 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]

UT=π(Ro2b2)(γSV1+γSV2)π(b2a2)(γSL1+γSL2)+πa2γS12E62

where γS12is the surface tension associated with the solid-solid interface.

The dimensionless formulation involves two additional ratios [31]:

mbaE63
ΦΔγγ(cosθ1+cosθ2)1E64

where Δγis the well-known work of adhesion and is given by [8]

Δγ=γSV1+γSV2γS12E65

The equilibrium solution, for given values of Γand Ψ, is now expressed in terms of both ηand m. 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 [31].

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 Hat which the solid-solid contact is lost during a separation process is different from the value of Hthat corresponds to the formation of solid-solid contact during an approach process. Put another way, there is a jump-on instability at a certain Hupon 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 ato 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.

Figure 8.

A liquid film bridging two smooth, elastic half-spaces with no solid-solid contact.

Figure 9.

Equilibrium configurations for two spheres bridged by a liquid film without solid-solid contact.

Figure 10.

Regions in Γ-Ψ space showing regions with and without solid-solid contact.

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, 1921, 3337]. 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

Δp= pap=γ(cosθ1+cosθ2)hfs E66

where hfsis 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 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 Ftand greater surface deformation. Hence, there is the possibility that the rate of increase of the capillary force (with expanding wetted radius) will exceed the rate of increase of the compressive force coming from the contacting asperities. In such a situation, the interface is expected to collapse, whereby the solid surfaces come into complete or nearly complete contact.

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 h¯within the wetted region is a good approximation to the spacing at the free surface hfsso that pressure drop within the film is approximated by

Δp=γ(cosθ1+cosθ2)h¯ E67

Figure 11.

A liquid film bridging two rough, elastic surfaces.

Thus, the tensile force (Ft)becomes

Ft=πb2γ(cosθ1+cosθ2)h¯ E68

where bis 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 Ftis the just the adhesive force (Fad). 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: Vo= 0.1 mm3, γ = 72.7 mN/m, σ = 0.4 μm, and An= 4 cm2, where σ is the r.m.s. surface roughness of a 3D isotropic surface with a Gaussian height distribution, and Anis 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 [3336]. By introducing certain dimensionless parameters, the results can be generalized. Let an adhesion parameter Γ be defined according to

Γγ(cosθ1+cosθ2)Voπ2An1/2Eσ3E69

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

P*PγVo(cosθ1+cosθ2)/σ2E70
Ft*FtγVo(cosθ1+cosθ2)/σ2E71
Vo*VtAnσE72

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.

Figure 12.

The effect of external load: (a) tensile force and contact area; (b) average gap and wetted radius.

Figure 13.

Dimensionless tensile force as a function of adhesion parameter for several dimensionless liquid volumes.

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.

© 2015 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Jeffrey L. Streator (December 16th 2015). Solid-Liquid-Solid Interfaces, Surface Energy, Mahmood Aliofkhazraei, IntechOpen, DOI: 10.5772/61572. Available from:

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