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Limitations of Multicomponent Theories in Surface Thermodynamics and Adhesion Science

C. Della Volpe and S. Siboni*

Department of Civil, Environmental and Mechanical Engineering (DICAM) University of Trento, Via Mesiano, Trento, Italy

Abstract

Multicomponent theories are widely used in the study of surface thermodynamics and adhesion phenomena, and certainly provide a powerful tool to better understand and rationalize the processes involved. Nevertheless, since they are based on strong assumptions about the mathematical description of molecular interactions and postulate also that the properties of a material are fully characterized by a small number of fixed parameters, multicomponent models are necessarily approximate and show a series of limitations that in some cases may yield results of difficult or questionable interpretation. In this review some of the main problems encountered in applying multicomponent theories are illustrated. Experimental and theoretical evidence in the literature is also considered, which could question the appropriateness of the multicomponent approach as a general method for the analysis of this kind of phenomenon.

Keywords: Acid-base interactions, dispersion interactions, surface free energy, surface tension, work of adhesion, multicomponent theories, contact angle

1.1 Introduction

Many phenomena, at different scales, may affect the strength of the adhesion between two material surfaces, such as mechanical interlocking, irreversible deformations around the contact region, molecular interactions at the interface, adsorption and interdiffusion. In spite of its great complexity, however, it has been largely recognized [1-3] that in many cases the adhesion strength is proportional to the work of adhesion Wadh at the interface, a thermodynamic quantity defined as the Gibbs free energy per unit area needed to reversibly separate two surfaces originally in contact and create two distinct surfaces. For two interacting phases 1 and 2 Dupré equation (Figure 1.1) relates the work of adhesion to the interfacial free energy γ12 and the surface free energies γ1 and γ2 of the materials involved:

since the definition of Wadh implies the destruction of a unit interface area (γ12) and the concurrent creation of unit area free surfaces (γ1 and γ2).

If phases 1 and 2 are a solid S and a liquid L and the destruction of interface occurs in the presence of the liquid vapor V, the work of adhesion becomes Wadh = γS +γLV − γSL and is related to the Young equilibrium contact angle θ by the Young-Dupré equation (Figures 1.2 and 1.3):

where:

by Young equation, while πe = γS − γSV denotes the equilibrium spreading pressure of the liquid vapor on the solid surface [4]. The term πe describes the free energy reduction per unit area of the solid surface due to adsorption of liquid vapor on the pure solid surface at equilibrium and for low energy surfaces it is usually considered negligible whenever the equilibrium contact angle is significantly higher than zero [5], thus making it possible the determination of Wadh by the only measurement of θ and the surface free energy (surface tension) of the liquid, γLV. If the surface free energy of the solid is high, and particularly when the liquid spreads on the solid surface, πe cannot be ignored and the contact angle does not allow the determination of the work of adhesion. The latter can then be obtained [3] by measuring the free energy variation per unit area that occurs when a vapor adsorbs and forms a liquid film on the solid surface (for high specific area materials, such as powders or fibers).

Figure 1.1 Definition of the work of adhesion Wadh between phases 1 and 2 (in vacuo), as the Gibbs free energy per unit area needed to separate the two phases.

Figure 1.2 Thermodynamic interpretation of Young equation γSV = γLV cosθ + γSL. A slight translational displacement (thick arrow) of the LV interface (thick line) along the solid surface (thin line) results in small changes dASL, dASV = -dASL and dALV = dASL cosθ of the SL, SV and LV interface areas, respectively. At equilibrium the variation γSL dASL + γSV dASV + γLV dALV of the total free energy must vanish for any dASL, and implies Young equation.

Figure 1.3 Alternative interpretation of the Young equation as a balance of surface forces per unit length at the solid-liquid-vapor triple line (solid arrows) along the solid surface. Double arrows provide a pictorial illustration of the solid surface free energy in vacuo, γS, and of the spreading pressure πe= γS − γSV attributable to vapor adsorption on the solid surface at equilibrium.

In principle the work of adhesion is determined by the interactions between the two phases at a molecular level, so that the understanding of adhesion phenomena and that of intermolecular interactions have developed together. It is worthy to recall the fundamental steps of such a development, that finally resulted in the full recognition of acid-base interactions as a key element in adhesion phenomena.

In 1873 the van der Waals equation [6] for real gases introduced the qualitative concept of attractive forces between molecules, that Mie in 1903 [7] and Grüneisen in 1908 [8] proposed to model by a bimolecular potential of the form W(r) = A/rn –B/rm, dependent on the intermolecular distance r and appropriate constants A, B, n, m. In the early 1920s molecular interactions involving permanent dipoles were investigated by Debye [9, 10] and Keesom [11, 12], and in 1928 the need for a repulsive contribution to the molecular interactions led to Lennard-Jones and Dent proposal of the “6-12 potential” [13]. London’s work in 1930 [14] basically completed the picture of bimolecular interactions by the description, on a quantum mechanical ground, of the induced dipole-induced dipole attractive interaction between molecules not endowed with a permanent dipole, although it was necessary to wait until 1948 to arrive at a correct description of the retardation effects on the interaction, thanks to the contribution of Casimir and Polder [15]. Based on the assumption of purely bimolecular forces, in 1937 Hamaker made the first attempt to calculate the interaction between macroscopic bodies [16] and defined what is now commonly known as the Hamaker constant, the fundamental parameter to quantify the strength of such an interaction. The bimolecular approximation holds only for very dilute media, with low molecular number densities, so that Hamaker’s approach turns out to be rather inaccurate for condensed phases. This limitation of the theory was overcome in 1955 by Lifshitz [17], whose model has its roots in Casimir description of London interactions. In Casimir vision, that dates to 1948 [18], the interaction is not thought as a force at a distance between molecules, but as a local action of electromagnetic fields; the same approach, applied to the quantum analysis of the electromagnetic field between two neutral conductive parallel plates in a vacuum allowed Casimir to predict the occurrence of an attractive force between plates (Casimir effect). Lifshitz applied Casimir’s approach to the case of two dielectric parallel surfaces with possibily dissimilar permittivities separated by a third dielectric medium in between, thus developing a theory where macrobodies are treated as continua and their interaction is determined from macroscopic properties that can be experimentally measured (dielectric permittivity as a function of frequency for sinusoidal electromagnetic waves through the medium) [19]. The theory by Parsegian and Ninham in the 60s [20], based on some sort of “modified additivity approach” able however to overcome the problems of the bimolecular approximation, led essentially to the same results. One point that makes the bimolecular model particularly unsuitable to describe the forces between condensed phases is that such a model strongly overestimates the dipole-dipole interaction, since the relatively high density implies that dipoles are very close and not free to rotate, so that the minimum energy corresponds to a complex configuration of dipoles which does not correspond to a minimum energy of a simple pairwise interaction. The geometric shape of molecules may also play an important role, owing to the smallest mean distance of dipoles; finally, thermal motion is not very effective in randomizing the dipole orientations, especially for dipole moments of significant strength. Nowadays the standard form of Lifshitz theory is usually revisited to incorporate McLachlan’s approach, that extends the original two-body London description to consider the role played by the various electromagnetic absorption frequencies of the phases and eventually model the effect of a solvent in the interaction [21-23]. By using the Lifshitz-McLachlan theoretical framework, Good and coworkers suggested in 1966 [24, 25] that for polar molecules the contribution of purely dipole-dipole interactions...