Surface tension of an ideal solid: What does it mean?

Abstract

The surface tension (ST) of an ideal (rigid, smooth, and inert) solid surface is usually calculated from a set of two equations: the Young equation and an additional equation that expresses the correlation between the individual STs of two phases and their interfacial tension. The present discussion suggests that this calculated ST may not be an appropriate characteristic of wettability of a solid surface. The reasons include the nonmeasurability of this ST and theoretical aspects related to the rigidity and inertness of an ideal solid. Instead, it is suggested to measure the contact angles of a set of a few standard liquids, from which a ‘wettability index’ can be calculated by properly averaging the wettability indices calculated for each of standard liquid. The optimal identity of these liquids and the optimal weights of the averages should be found by experimentation.

Introduction

Characterization of the wettability of solid surfaces is of utmost importance in many fields of engineering, science, and daily life [∗1, ∗2, 3, 4, 5, 6]. The goal of wettability characterization is to assign a quantitative measure to a solid surface that should be a characteristic, intrinsic property of the solid, independent of the liquid or fluid used. Wettability is usually assessed by measuring the contact angle (CA) that a drop of a liquid makes with the solid in air. In some cases, a few liquids are necessary to get meaningful results [7]. Other methods for CA measurement also exist, such as the Wilhelmy plate [8].

As is well known, the fundamental equation that correlates the CA, $\theta$, with the surface tensions (STs) of the involved phases is the Young equation [9].

$$\text{cos}{\theta }_Y=\frac{{\sigma }^S-{\sigma }^{SL}}{{\sigma }^L}$$

Here, ${\theta }_Y$is the Young CA (calculated from the Young equation), ${\sigma }^S$is the (so-called) ST of the solid surface, ${\sigma }^{SL}$is the interfacial tension of the solid–liquid interface, and${\sigma }^L$is the ST of the liquid. The main target of the calculation using Eq. (1) is finding the value of ${\sigma }^S$, which is believed to be a characteristic property of the solid surface. However, to be such a characteristic property, the solid surface must be smooth, rigid, insoluble, and nonreactive (usually referred to as an ideal solid) to assure that wetting of the solid depends only on the physicochemical interaction between the solid and the liquid. If the solid is rough, for example, the roughness strongly affects the CA; therefore, such a CA cannot characterize the solid material itself.

A most fundamental issue regarding the Young equation stems from the impossibility to measure solid-related STs. This is so because all ST measurement methods require the interface to be deformable. This implies that Eq. (1) inherently contains two unknowns, ${\sigma }^S\text{and }{\sigma }^{SL}$. Therefore, once a CA is properly measured [1], and the ST of the liquid is known, one additional equation is needed to solve for ${\sigma }^S\left(\text{and }{\sigma }^{SL}\right)$. The additional equation is a correlation between ${\sigma }^S\text{and}{\sigma }^L\text{and}{\sigma }^{SL}$. For a summary of the available correlations, refer the study by Marmur and Valal [3].

This methodology has been used for the last fifty years or so, despite some fundamental open questions regarding the meaning and validity of the concept of ST of a solid. The need for a second equation, without being able to directly measure the solid-related STs, makes it difficult to assess the credibility of the calculated ST of a solid surface as an intrinsic characteristic. For example, it was shown by Marmur and Valal [3] that the correlations used in the literature for ${\sigma }^{sl}$do not work well for liquid–liquid systems, for which all components can be measured and checked. Thus, there is no objective means to support the validity of the correlations of the type ${\sigma }^{sl}={\sigma }^{sl}({\sigma }^s,{\sigma }^l)$. Moreover, the theoretical definition of ST [9,10] must be differently interpreted for solids and liquids because of differences in the behavior of the stress components within the interface. This is so, as will be discussed in the following paragraphs, because the rigidity and inertness of the solid annul the work and mass transfer terms in the thermodynamic equilibrium condition. It is important to mention at this point that interesting pioneering work was done on elastic solids [11]. However, the Young equation does not account for elasticity, and most of the solid characterization studies follow the Young equation approach.

Thus, the purpose of this communication is to discuss the difficulties associated with the concept of ST of an ideal solid and suggest a possibility to avoid them. First, we attempt to clarify the confusion regarding the concepts of specific surface energy vs. ST. Then, ST of a liquid is discussed as an introduction to the debated concept of ST of a solid. Finally, the latter is discussed, and practical implications are presented.