Adhesion models based on intermolecular forces

While bonding of two Si needle-like substrates at room temperature is due to presence of intermolecular forces (e.g., VdW forces or capillary forces) between them, an overview of adhesion models for two interacting surfaces based on these intermolecular forces is presented in this section.

In general, adhesion is tendency of similar or dissimilar particles or surfaces to cling to one another. A force which is required to separate adhered particles or surfaces from each other is called force of adhesion. The adhesion between two particles or surfaces at their contact points at room temperature without involvement of any intermediate layer is generally due to intermolecular forces, such as VdW forces, hydrogen bridging, electrostatic forces, and capillary forces acting between them [31, 32, 108]. The adhesive force between two adhered surfaces due to intermolecular forces can be typically obtained through following adhesion modeling approaches [109]:

I. Single-asperity approaches where the adhesive force between single elements (asperities) of surfaces is considered.

II. Multi-asperity approaches where the total adhesive force between two surfaces is obtained by summing of adhesive forces of all contributed asperities. In these approaches, surface roughness is presented as a set of asperities.

I. Single-asperity approaches

The classical sphere-sphere and sphere-flat plane approaches are broadly employed to describe the adhesion in the single asperity approaches. Over the last few decades, various techniques have been developed to describe and predict the adhesion based on VdW forces and capillary forces using the single-asperity approach. The attraction of two solid non-deformable macroscopic bodies based on London dispersion forces was predicted for the first time in 1937 by Hamaker [110, 111]. Hamaker used the London theory (pairwise interaction between atoms of neighboring bodies) and described sum of interactions between atoms of neighboring bodies through the Hamaker’s constant [112]. His model was extended later by Lifshitz, who considered the problem of London dispersive forces in terms of generation of an electromagnetic wave by an instant dipole in a material being absorbed by a neighboring material [113]. In 1990, Rumpf developed a more suitable model for nano-scale region based on interaction of a single hemispherical asperity with a larger spherical particle, along a line normal to the surface which connects their centers [114]. Rumpf’s model describes the total VdW forces based on the interaction of an adhering particle in contact with an asperity, as well as the noncontact force between the adhering particle and a flat surface separated by the asperity [109]. Later, in 2000, Rabinovich developed a more accurate model capable of predicting adhesion force between a particle and an asperity on a surface by taking both asperity height and breadth into account [32]. For deformable bodies, attractive VdW forces between two elastic spheres was introduced in 1971 by Johnson, Kendal and Roberts (JKR) [115], and later in 1975 by Derjaguin, Muller and Toporov (DMT) [116]. Both JKR and DMT models altered the Hertz equation [117] by taking surface energies of involved surfaces into consideration. The JKR model assumes surface forces only act over the contact area and cause a deformation at the contact point. Whereas, DMT model accounts for non-contact forces in the vicinity of the contact area [118]. Later, in 1977, Tabor compared results obtained by both JKR and DMT models and found an explanation for their inconsistency by introducing a non-dimensional parameter called Tabor’s parameter (𝜇𝑇), which is defined as value depends on the “neck” height [119, 120]. It was found that if the “neck” height is large (𝜇_𝑇 ≥ 1), noncontact forces of attraction can be neglected, and the adhesion force acquires the value predicted by the JKR theory. However, in cases where the “neck” height is small (𝜇_𝑇 ≤ 1), the

noncontact forces of attraction must be considered, and the adhesion force acquires the value predicted by the DMT model. In 1992, Maugis analytically analyzed the adhesion results obtained from both JKR and DMT approaches using the Dugdale model [121] and explained the latter in term of range of action of surfaces’ forces [122]. He developed a more composite and accurate approach, which is a continuous transition regime between JKR and DMT limits and can be applied to any materials with both high and low adhesions. He showed that the JKR model is proper for large spheres with high surface energies and low Young’s moduli, and the DMT model is appropriate for small spheres with low surface energies and high Young’s moduli [110]. In 1998, Greenwood and Johnson presented an alternative to the Maugis theory by combining two Hertzian pressure distributions to suppress the drawback of the Maugis theory (the shape of the gap in his theory involves elliptical integrals, which is inconvenient when an extension to viscoelastic spheres is considered) [123]. They showed that their results were very close to those obtained by the Maugis model, but the gap shape was only involved the elementary functions (the most commonly used mathematical functions (e.g., sin, cos, and tan functions), exponentials, and logarithms [124]).

Adhesion models based on VdW forces are suitable and applicable for dry or low humidity (RH < 20 %) environments where the influence of humidity or liquid medium between involved surfaces is very small, and VdW forces and capillary forces are comparable [28, 125]. However, at high humidity environments, the adhering force is strongly influenced by spontaneous formation of liquid meniscuses between two adhering surfaces [32]. A liquid meniscus formed in the middle of two hydrophilic solid surfaces causes a strong attractive force, which is called capillary forces. This liquid meniscus is usually formed by capillary condensation or by accumulation of adsorbed liquid. Under ambient conditions and between lyophilic surfaces, capillary forces dominate over other surfaces forces [126]. A plenty of work has been reported to model capillary forces between a spherical particle and a flat plane and between two spherical particles with different tips shapes. These models have been mainly developed using thermodynamic equilibrium and non-equilibrium approaches [127].

The thermodynamic equilibrium approach is based on direct force computation of the meniscus geometry either obtained by a numerical solution, so-called Laplace equation or approximated by a predefined geometrical profile such as circle (circular

25 approximation) or parabola (parabolic approximation) [128]. In contrast, the thermodynamic non-equilibrium approach is formed on so-called energetic method in which the energy is minimized for a given fixed liquid volume [129]. However, both approaches have shown equivalence results in term of total capillary adhesion forces [126, 128]. In both approaches, due to presence of unknown variables, such as radius of curvature of the meniscus, filling angle, and volume of the liquid enclosed, additional expressions have been proposed considering a circular approximation for the curvature of the liquid vapor interface [130–133] and for numerical computation of the curvature [134, 135]. In some approaches, symmetric configuration is assumed for the liquid bridge by considering equal contact angles [108, 136–138]. The circular approximation has shown strong validity where adhering bodies and volume of the enclosed liquid are very small. In this case, relinquishing the gravity is valid and the circular approximation is preferred for calculation of capillary forces [139].

An analytical solution for the Laplace-Young equation for axisymmetric menisci of enclosed liquid between a spherical body and a plane was first introduced by Orr et al. in 1975 [140]. Elliptic integrals were used to calculate enclosed volume of trapped liquids and their corresponding forces for all possible types of pendular rings and liquid bridges between a spherical body and a plane. It was shown that capillary attraction might become capillary repulsion when wetting is not perfect. In 1993, Marmur calculated the capillary adhesion force between tip of a particle and a flat surface by a numerical solution of his own developed equation as well as by his suggested approximate analytical equation [138]. He presented results for spherical, paraboloidal, and conical tips, and compared them with the classical approximate equation for the capillary adhesion force between a sphere and a flat surface. He found that the capillary adhesion force is independent of the position of the liquid-vapor interface, and geometry of the tip has a major impact on the adhesion force. In 1999, de Lazzer et al. considered particle-surface interactions due to capillary effects between a spherical particle, a paraboloidal particle, and a conical particle and a flat substrate by improving the formula given by Marmur for the analytical approximation [141]. They verified the range of applicability of the improved approach by comparison with numerical solutions of the capillary equation for particle-substrate interactions.

Additionally, they found that surface tension forces acting along the contact line to the total adhesion force (which had intentionally omitted in Marmur’s work) show to be of

a similar order of magnitude as capillary forces. Later in 2000, Sirghi et al. discussed Marmur and de Lazzer models further by considering incorporation of local curvature of the meniscus and suggested an analytical solution for capillary and surface tension forces using approximation of a spherical particle geometry and a symmetrical water meniscus at thermodynamic equilibrium [142]. Theoretical results were compared with forces measured between an AFM tip made of silicon nitride and a platinum sample covered by quartz with a roughness in the order of 20 nm. In 2002, Rabinovich et al.

presented an approximate theoretical formula to predict magnitude and onset of the capillary adhesion between a smooth adhering particle and a surface with a roughness in nanometer scale [29]. Experimental adhesion values between 23 mN/m and 380 mN/m for glass microspheres interacting with different surfaces (e.g., silica substrates covered by 180 nm oxide layer, Si wafer with Root Mean Square (RMS) roughness of 0.2 nm, and PECVD silica boiled in a 1:1 mixture of hydrogen peroxide and ammonia for 6 hours producing an RMS roughness of 0.7 nm) were obtained via the AFM and were used to validate the theoretical predictions. It was found that adhesive capillary forces are significantly reduced in the presence of a nanoscale roughness. However, the critical relative humidity where capillary forces are first observed, increases by increasing the nanoscale roughness. In 2005, Rabinovich et al. compared theoretical formulas driven from the total liquid bridge energy (the energetic method) and the pressure difference across the liquid bridge (the Laplace method) to estimate capillary forces between a sphere and a plate due to a liquid bridge [143]. Theoretical estimations were compared with experimental measurements done by the AFM. It was shown that results obtained from both methodologies and the AFM agree and confirm capability of the total energy-based approach, despite thermodynamic nonequilibrium conditions of a fixed volume bridge rupture process. In 2007, Chau et al. presented a general model for simple geometries (e.g., planes, spheres, cones, and pyramids) to compute capillary forces due to water condensation [134]. The model allowed computation of capillary forces for non-axisymmetrical shapes with a meniscus fulfilling the Kelvin equation without assuming a profile for it. The model considered contact angles, relative humidity, temperature, and geometrical description of the object. The model was validated by comparing its results with existing theoretical and experimental results. It was found that capillary forces between a pyramid and a plane are significantly smaller than the ones between a cone (with the same aperture angle as the pyramid) and a plane, and this ratio does not significantly depend on humidity.

27 It was also found that tilting angle of an object (e.g., cone or pyramid) with respect to an interacting plane has a substantial effect on capillary forces. In 2008, Xue et al.

presented an improved capillary meniscus model for a deformable sphere on a rigid flat surface covering a large range of interference or compression (ω) values, from a non-contact (ω < 0) to a fully plastic contact (ω > 0) [144]. For the elastic contact, the Hertz solution was used to determine the real contact area, the deformed sphere profile, and the projected meniscus area. However, for the elastic-plastic contact range, a finite element-based spherical solution was employed to estimate the real wetted area and meniscus force. It was found that capillary meniscus force increases by increasing interreference in the elastic and the elastic-plastic regions resulting from the spherical deformation. It was also found that in the non-contacting range (ω < 0), the meniscus force increases with increasing of relative humidity. While in the contacting region (ω > 0), the meniscus force increases with decreasing of relative humidity. However, at a point contact (ω = 0), the meniscus force is independent of relative humidity. Additionally, it was observed that the adhesion force decreases with increasing of relative humidity due to reduction of solid surface energy by adsorption of vapor in the range of 0 < RH > 70 % [144]. It was suggested that solid-solid interaction should be considered besides attractive forces caused by capillary condensation to calculate the total adhesion force at intermediate relative humidity values. It was found that at high relative humidity values (RH > 70%), the solid-solid contribution is negligible, and the pull-off force is mainly given by the Laplace pressure contribution of the adhesion. In 2011, Payam et al. presented an analytical approach for computing capillary forces for sphere-sphere and sphere-plane geometries based on the energetic method [127]. The model considered spheres with both equal and non-equal radii, for both symmetric and asymmetric configurations at liquid-solid interfaces. Numerical analysis was used to investigate validity and efficiency of the derived model. Impacts of different parameters, such as humidity, distance between two spheres, radii of spheres, and contact angles on meniscus forces were studied.

Theoretical estimations were compared with experimental measurements and theoretical results based on the Rabinovich model [143] to show accuracy and precision of the model. The model showed a higher precision, generality, and applicability to predict situations including known volume and known embracing angle in analyses in comparison with the Rabinovich model.

II. Multi-asperity approaches

All real surfaces, regardless of their preparation conditions and how smooth they

appear, possess asperities, but at different scales. Thereby, interactions or contacts between two surfaces mainly occur through their asperities. Various adhesion models based on the convectional multi-asperity approaches have been so far presented to express interaction or bonding between two rough surfaces. These models can be generally categorized into two main groups [145–147]:

A. Uncoupled multi-asperity models B. Coupled multi-asperity models

In the uncoupled models, surface roughness is represented as set of asperities, frequently with statistically distribution parameters. The contribution of each asperity is considered locally and independently from others, and the total adhesion forces are obtained by adding up all single contributing asperities [145]. The coupled models are enhanced versions of the uncouple models in which effect of loading on one asperity on deformation of neighboring asperities is also considered [147].

A. Uncoupled multi-asperity models

The preliminary idea of multi-asperity contacts was offered by Archard in 1957 [148]. However, an intense improvement of this idea was proposed in 1966 by Greenwood and Williamson, who modelled roughness of a surface as a group of identical spherical asperities with randomly distributed heights [149]. Their model known as “GW model” and has been broadly used irrespective of some of its limitations, such as assumption of perfectly hemispherical asperity and constant asperity diameter of curvature. Modeling of interaction between two rough surfaces as a contact between a perfectly flat surface and a rough surface whose asperities height is the sum of asperities heights of involving surfaces was first proposed by Greenwood and Tripp in 1970 [33]. This model has been accepted and validated by other researchers. However, the assumption that both surfaces present normal distributions of asperities heights is not valid for all situations since some materials do not reveal this type of distribution. In the same year (1970), Whitehouse and Archard developed another model, which took distributions of both asperities heights and radius of

29 curvatures into the account [150]. In 1975, Bush et al. developed an elastic contact model of an isotropic rough surface with a plane in which asperities were treated as elliptical paraboloids with randomly oriented elliptical contact areas [151]. The classical Hertz solution was used to describe deformation of asperities at contact points due to a contacting load. It was found that the contact area is approximately proportional to the contacting load, and the constant of proportionality is dependent on the Hertzian elastic modulus and the profile absolute mean slope. Later, in 1979, the same group presented an enhanced version of their model in which the load and the real contact area were derived as a function of separation (distance between the asperities and the plane) [152]. In 1987, Chang et al. developed an elastic-plastic asperity model to analyze the contact of rough surfaces [153]. The model considered a volume conservation of plastically deformed asperities and gave an expression to calculate plastic contact area for interferences (deformations) larger than the critical one. They found that the GW model [149] for the elastic contact and the plastic surface microgeometry model known as “PW model” developed by Pullen and Williamson in 1972 [154] for a purely plastic contact provide lower and upper limits of their given expression [153], respectively. They compared load-separation and load-contact area results obtained from their model with the ones obtained from the GW model and the PW model over a relatively broad range of a plasticity index (ψ). It was found that the PW model is only valid at high value of the plasticity index (ψ > 2.5) or at very high loads, and the GW model becomes invalid because of large contact area ratio resulting from merging of neighboring asperities and the violation of the assumption of noninteracting asperities. One year later, the same group (Chang et al.) presented an improved DMT adhesion model in conjunction with their elastic-plastic contact model [153] to study adhesion of contacting rough metallic surfaces [155]. A rough surface was modeled as a large number of asperities with a Gaussian height distribution.

Asperities were assumed to have spherical shapes near their summits with constant radius of curvatures. The elastic-plastic contact model was used to calculate the contact load, and the modified DMT adhesion model was used to study the adhesion force. It was shown that for clean surfaces, the adhesion is quite large even for relatively rough surfaces. Adhesion becomes negligible only when surfaces are contaminated or pressed with a very high external contacting load. In 1990, Stanley et al. used the Chang adhesion model [155] to study the adhesion between two rough surfaces covered with extremely thin layers of a lubricant [156]. It was found that when

an extremely thin layer of lubricant is applied to a solid surface, it forms a strong bond with the surface without tending to form menisci. Their analyses showed that although introduction of a lubricant on surfaces reduces the surface free energy, there can be an increase in the overall adhesive force relative to the dry case without formation of liquid meniscus bridges. Later, in 1996, Kotwal and Bhushan presented a statistical model for non-gaussian rough surfaces based on the GW model [157]. Real area of contacts, number of contacts, and contact pressures were calculated as probability density functions having different skewness (measure of degree of symmetry in a variable distribution [158]) and kurtosis (measure of degree of tailedness in a variable distribution [159]). It was found that optimum values of skewness and kurtosis depend on the applied contact load. A positive skewness value in a range of 0.3 - 0.7 (skew of the distribution to the right) and a kurtosis value larger than 5 (a leptokurtic distribution) significantly reduce the real area of the contact. In 2002, Adams et al. presented a multi-asperity model for contact and friction [147]. A combination of the frictional slip model developed by Hurtado and Kim [160] and the GW model [149] were used to

Im Dokument Silicon Needle-like Surfaces for Room Temperature Silicon-Silicon Bonding Applications (Seite 62-72)