Regular Article
Correlations of surface free energy and solubility parameters for solid substances

https://doi.org/10.1016/j.jcis.2019.02.074Get rights and content

Abstract

Hypothesis

Both the surface free energy (γ) and solubility (δ) parameters of substances are related to their cohesive energies which are decided by intermolecular interactions, and there should be some intrinsic relationships between the two parameters. Understanding of the γδ correlations is of great fundamental and practical importance. Several empirical γδ equations have been proposed so far, but their application to solids is limited. This is because the molar volume (V~) as a parameter exists in these equations while the V~ of solids is commonly hard to be obtained. Hence, the development of γδ equations without the parameter V~ is essential for solids.

Method

The γ and δ data of 21 solids including polymers and layered solid materials were chosen, and possible γδ relationships were systematically explored using the parameter data of solids by a trial and error fitting method.

Finding

Six γδ equations without the parameter V~ are proposed. The γ parameters include total (γt), dispersive (γd), and polar (γp) ones, and the δ parameters include the Hildebrand parameter (δt) and the Hansen dispersive (δd), polar (δp), and hydrogen-bonding (δh) ones. Interestingly, the so-obtained V~-free γδ equations are also valid for most liquids including nonpolar and polar ones. These γδ equations can provide a way to estimate non-measurable parameters from measurable parameters for solid materials, which is beneficial to the application of the characteristic parameters (γ and δ) for solid material engineering.

Introduction

The surface free energy (γ) and solubility parameter (δ) are two important characteristic parameters of substances [1], [2], [3], [4], [5], [6], which are closely related to many phenomena such as adsorption, adhesion, wetting, dissolution, and dispersion occurring at interfaces or in bulk phases. The total γ of a substance (liquid or solid), denoted as γt, is defined as half of the free energy change due to cohesion of the substance in vacuo [3],γt=-12ΔGcohwhere ΔGcoh is the free energy change of cohesion in vacuo. The total δ of a substance, denoted as δt, is defined as the square root of the cohesive energy density of the substance [2],δt=EcohV~1/2where Ecoh is the molar cohesive energy, and V~ is the molar volume. The solubility parameter concept was first introduced by Hildebrand and Scott [7], so called Hildebrand solubility parameter.

The γt and δt both arise from the intermolecular interactions, and thus can be divided into different components. The γt is commonly divided into dispersive (non-polar) and polar (non-dispersive) components [1]. The former results from the London dispersion forces, and the latter results from the dipole-dipole, induced dipole, and hydrogen bonding interactions. Thus, the γt of a substance can be expressed as:γt=γd+γpwhere γd and γp are the dispersive and polar components of γt, respectively. Hansen split the total Ecoh into three components: dispersive (Ecoh,d), polar (Ecoh,p), and hydrogen-bonding (Ecoh,h) components [2], [7]. The Ecoh,d arises from the contribution of the London forces, the Ecoh,p from the dipole forces, and the Ecoh,h from the hydrogen-bonding, induced dipoles, and probably other interactions. Accordingly, the δt is extended to three components, i.e., dispersive (δd), polar (δp), and hydrogen-bonding (δh) parameters. The relation between the Hildebrand solubility parameter (δt) and the Hansen solubility parameters (δd, δp, and δh) is,δt2=δd2+δp2+δh2

Because both γ and δ parameters are related to cohesive energies that are decided by intermolecular interactions, some intrinsic relationships should exist between the two parameters. Research on the relationships can provide better understanding of the nature of substances. Great effort has been focused on the establishment of the relationship [2], [7], [8], [9], [10], and several empirical (or semiempirical) equations have been proposed, such as [7], [9], [10]:γt=0.0715V~1/3[δd2+0.632+(δp2+δh2)]δd2+δp2=13.8V~-1/3γtδd2=13.2V~-1/3γdδt2=16.1V~-1/3γt

In these relation equations between γ and δ parameters (called “γδ equations” for short), the units of γ, δ, and V~ are mJ/m2, MPa1/2, and ml/mole, respectively. Notably, the parameter V~ exists in the γ–δ equations, which results in a limitation when applied to solid substances [11]. This is because the V~ of solids is commonly hard to be obtained. Recently, Jia and Shi [11] proposed a relation between the δd and γd without V~ as:γdδd2/3=0.555+0.132γd

This equation is valid for most liquids and common polymers. This suggests that, most likely, there also exist V~-free relations among other characteristic parameters, which are necessary to be explored because understanding of the relations is of great fundamental and practical importance.

As we known, unlike most (volatile) liquids, the γ and δ parameters of solids cannot be directly measured, and many indirect methods are thus used to estimate the parameter values [2], [4], [5]. The γ parameters of solids are commonly estimated using contact angle [12], [13], [14], [15], [16], [17], [18] and inverse gas chromatography (IGC) [19], [20], [21] methods. For the contact angle method, the contact angles of at least two probe liquids with known γ parameter values on a solid surface are first measured, and the γ parameter values of the solid are then estimated based on theoretical models such as Young’s equation, Owens-Wendt equation, Neuman’s equation of state, and Wu’s equation [1], [4], [22], [23], [24]. The IGC method is essentially based on the measurement of adsorption-desorption behavior of probe liquids (commonly alkanes) on a solid to estimate the γ (commonly γd) value of the solid [19]. The δ parameters of solids are commonly estimated by determination of some quantities relating to the solid-liquid affinity, such as solubility, swelling, and dispersion of solids in various liquids with different δ (a “solvent spectrum”) [2]. The δ parameters of liquids that exhibit the maximum affinity with a solid are considered to be those of the solid. Recently, a liquid phase exfoliation (LPE) method has been developed to estimate the δ parameters of carbon nanotube bundles and layered solid materials (graphite and transition metal dichalcogenides), and the so-obtained δ-values can be used to screen effective liquid media for LPE [5], [25], [26], [27], [28], [29], [30], [31] However, the reliability of the γ and δ data obtained from different methods needs to be examined. General γ–δ equations deduced from abundant data reported previously can be used to estimate the accuracy of γ and δ data determined or the consistency between γ and δ data for new solid materials [11]. Furthermore, the γδ equations can provide a way to estimate non-measurable parameters from measurable parameters.

In the current work, we systematically explored the possible relationships between γ (including γt, γd and γp) and δ (including δt, δd, δp, and δh) parameters from reported data for solids including common (nonpolar and polar) polymers and layered solid materials (graphite and transition metal dichalcogenides). Several γδ equations without the parameter V~ are developed for solids. Especially, the as-established V~-free equations are also valid for most liquids including nonpolar and polar liquids. These equations can be beneficial to the application of the characteristic parameters for solid material engineering such as the LPE of layered solid materials that has been received great attention recently [5], [13], [15], [25], [26], [28], [29], [30], [31].

Section snippets

Method

To establish the V~-free relations between γ and δ parameters for solids, the γ and δ data of 21 solids including polymers and layered solid materials were chosen, and possible γδ relationships were systematically explored using the parameter data of solids by a trial and error fitting method. To examine the applicability of the so-obtained V~-free relations for liquids, the γ and δ data of 47 liquids were chosen and linearly fitted using the relations.

Examination of the reliability of parameter data

To establish the V~-free relations between γ and δ parameters for solids, 17 common polymers with different polarity and 4 layered solid materials (graphite, MoS2, WS2 and BN) were chosen, because the values of their characteristic parameters (γt, γd, γp, δt, δd, δp, and δh) are available from the literature (Tables S1 and S2 in the Supplementary Material, SM). To examine the applicability of the so-obtained V~-free relations for liquids, 47 liquids including nonpolar liquids (such as alkanes

Conclusions

Surface free energy (γ) and solubility parameter (δ) are two important characteristic parameters of substances [1], [2]. Understanding of the correlations between γ and δ parameters is of great fundamental and practical importance. Several empirical γ–δ equations have been proposed [7], [9], [10], and the molar volume (V~) as a parameter exists in the equations, which limits their application to solid substances due to the fact that the V~ of solids is commonly hard to be obtained [11]. Only

Acknowledgements

This work is supported financially by the National Natural Science Foundation of China (Nos. 21573133 and 21872082) and the Independent Innovation Foundation of Shandong University in China (No. 2016JC032).

Conflicts of interest

The authors have declared no conflict of interest.

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