Keywords

Surface tension; interface; surface free energy; adhesion; measurement

2.1 Introduction

Adhesion is a phenomenon that takes place at the interfaces of adherends and adhesives. An understanding of the forces that develop at the interfaces is essential in the selection of the right adhesive, proper surface treatment of adherends, and effective and economical processes to form bonds. This chapter is devoted to the discussion of the thermodynamic principles and the work of adhesion that quantitatively characterize surfaces of materials.

2.2 What is an Interface?

Two solid or liquid phases in contact have atoms/molecules on both sides of an imaginary plane called the interface. The interfacial particles differ energetically from those in the bulk of each phase because of being on the phase boundary and interactions with the particles of the other phases. The composition and energy vary continuously from one phase to the other throughout the interface. This region has a finite thickness, usually <0.1 μm [1].

2.3 Surface Tension

The molecules of a liquid are held together by attractive forces. The sum of all attractive forces on any molecule present in the bulk of a liquid averages zero. The net force (also known as cohesion force) on a surface molecule is a non-zero quantity in the direction towards the bulk (Figure 2.1). This is the force that must be counteracted to increase the surface area; the energy consumed by this process is called surface energy. The unbalanced forces on the interface cause it to contract to a minimum surface area value. Therefore, water droplets are spherical because a sphere has the minimum surface area of all shapes for a given volume. Surface tension and surface free energy of a liquid are equal while the same is not true for a solid surface.

Surface tension is the work required to increase the area of a surface isothermally and reversibly by a unit amount. Surface tension (γ) is expressed as surface energy per unit area and alternatively as force per unit length. Surface tension of liquids can be measured directly and expressed in the units of work or energy per unit area (erg/cm²), which is then simplified to dyne/cm (erg/cm²=dyne cm/cm²=dyne/cm). Chapters 2 and 3 review methods for the measurement of surface tension. The challenge has been to find methods to determine the surface tensions of solid surfaces (see Chapter 3).

Surface tension of polymers can be divided into two components, polar (γ^p) and dispersion (γ^d), to account for the type of attraction forces at the interfaces [2]. The chemical composition of the surface determines the relative contribution of each component to surface tension. The polar component is comprised of various polar molecular interactions including hydrogen bonding, dipole energy, and induction energy, while the dispersion component arises from London dispersion attractions. The attractive forces (van der Waals and London dispersion) are additive, which results in the surface tension components being additive: γ=γ^p+γ^d.

2.4 Surface Free Energy

A hypothetical example is used to describe the concept of surface free energy. Suppose a box with a sliding cover is filled with a liquid (Figure 2.2). The sliding cover is assumed to have no interfacial tension with the liquid. If the cover is slid back to uncover a surface area of dA, the necessary reversible work will be (γdA). For a pure substance, the increase in the free energy of the system at constant temperature and pressure is defined by Eq. (2.1).

$\mathrm{d}G=\gamma \mathrm{d}A$ (2.1)

(2.1)

The total free energy of the system is comprised of the energy of the bulk liquid and the surface liquid. The latter is equal to the surface free energy per unit area (G_S) multiplied by the surface area as shown in Eq. (2.2). Combining Eqs. (2.1) and (2.2) results in Eq. (2.3), which illustrates that the free surface energy of a pure substance is equal to its surface tension.

$\mathrm{d}G=G_{\mathrm{S}}\mathrm{d}A$ (2.2)

(2.2)

$G_{\mathrm{S}}={\left(\frac{\mathrm{d}G}{\mathrm{d}A}\right)}_{T,P}=\gamma$ (2.3)

(2.3)

In a reversible system, the heat (q) associated with the elimination of the new surface can be related to entropy (S) or surface entropy (Eq. (2.4)), where S_S represents surface entropy per unit area. Equation (2.5) is a thermodynamic relationship applied to the liquid surface in which T represents absolute temperature. Equation (2.6) is obtained by substituting for G_S from Eq. (2.3).

$\mathrm{d}q=T\mathrm{d}S=T{S}_{\mathrm{S}}\mathrm{d}A$ (2.4)

(2.4)

${\left(\frac{\mathrm{d}{G}_{\mathrm{S}}}{\mathrm{d}T}\right)}_{\mathrm{P}}=-S_{\mathrm{S}}$ (2.5)

(2.5)

$\frac{\mathrm{d}\gamma }{\mathrm{d}T}=-S_{\mathrm{S}}$ (2.6)

(2.6)

The total surface energy (E_S in Eq. (2.7)) can be calculated by applying the enthalpy relationship (H_S) with Gibbs free energy and entropy to the liquid surface.

$H_{\mathrm{S}}=E_{\mathrm{S}}=G_{\mathrm{S}}+T{S}_{\mathrm{S}}$ (2.7)

(2.7)

Equation (2.8) is the result of substitution from Eq. (2.6) into Eq. (2.7).

$E_{\mathrm{S}}=G_{\mathrm{S}}-T\frac{\mathrm{d}\gamma }{\mathrm{d}T}$ (2.8)

(2.8)

Surface tension of most liquids decreases with increasing temperature in a linear manner. A well-known equation (Eq. (2.9)) defining the relationship between temperature and surface tension has been attributed to EÖTVÖS [3].

$\gamma V^{2/3}=k(T_{\mathrm{c}}-T)$ (2.9)

(2.9)

V is the molar volume, k has the same value for most liquids (2.1 erg/K), T_c is the critical temperature of the liquid, and T is the liquid temperature. The expectation is that surface tension of a liquid approaches zero at its critical temperature. There are other equations that express the behavior of liquids as a function of temperature [1].

2.4.2 Work of Adhesion

The work of adhesion is defined as the reversible thermodynamic work required to separate the interface from the equilibrium state of two phases to a separation distance of infinity. Equation (2.10) shows the work of adhesion for a liquid–solid combination. This definition is attributed to the French scientist A. Dupre.

$W_{\mathrm{a}}={\gamma }_{\mathrm{L}}+{\gamma }_{\mathrm{S}}-{\gamma }_{\mathrm{SL}}$ (2.10)

(2.10)

γ_L is the surface energy (tension) of the liquid phase, γ_S is the surface energy of the solid phase, γ_SL is the interfacialsurface tension, and W_a is the work of adhesion. A rise in the interfacial attraction results in an increase in the work of adhesion. Equation (2.10) can be rewritten to determine the work of cohesion (W_c) when the two phases are identical (γ_L=γ_S and γ_SL=0) and no interface is present as shown in Eq. (2.11) for a solid phase.

$W_{\mathrm{a}}=2{\gamma }_{\mathrm{S}}$ (2.11)

(2.11)

2.5 Contact Angle (Young’s Equation)

Most liquids wet solid surfaces and exhibit a contact angle. A contact angle in a static system can be measured at equilibrium. Figure 2.3 illustrates the contact angle in an ideal system where the solid surface is homogeneous, smooth, planar, and rigid. The interfacial tensions designated as γ represent equilibrium values at the point that three phases intersect. The subscripts L, S, V denote liquid, solid, and vapor phases. The γ° designation is used to indicate that the solid surface must be in equilibrium with the liquid’s saturated vapor; that is, a film of the liquid is absorbed on the solid surface. Young [6] described Eq. (2.12) without presenting a proof; it has been since proven by different researchers [7–9].

${\gamma }_{\mathrm{LV}}\cos \theta ={\gamma }_{\mathrm{SV}}^{\mathrm{o}}-{\gamma }_{\mathrm{SL}}\text{Young}\text{'}\text{s}\text{equation}$ (2.12)

(2.12)

One route to proving Young’s equation is using the Gibbs free energy of the wetting, proposed by Poynting and Thompson [10]. After the liquid droplet forms the meniscus and reaches equilibrium, the variation in Gibbs free energy is zero. An assumption in Eq. (2.13) is the neglect of the gravitational force.

$\mathrm{d}G=0$ (2.13)

(2.13)

$\mathrm{d}G={\gamma }_{\mathrm{LV}}\mathrm{d}{A}_{\mathrm{LV}}+{\gamma }_{\mathrm{SV}}^{\mathrm{o}}\mathrm{d}{A}_{\mathrm{SV}}+{\gamma }_{\mathrm{SL}}\mathrm{d}{A}_{\mathrm{SL}}=0$ (2.14)

(2.14)

In Eq. (2.14), dA represents small incremental increases or decreases in the surface or interfacial area. Changes in the interfacial areas are given by Eqs. (2.15) and (2.16). Any increase in the solid–liquid interface is countered by a decrease in the solid–vapor interface. Substitution from these two equations into Eq. (2.14) will yield Young’s equation (Eq. (2.12)).

$\mathrm{d}{A}_{\mathrm{SL}}=-\mathrm{d}{A}_{\mathrm{SV}}$ (2.15)

(2.15)

$\begin{array}{l}\mathrm{d}{A}_{\mathrm{LV}}=\cos \theta \mathrm{d}{A}_{\mathrm{SL}}\hfill \\ {\gamma }_{\mathrm{LV}}(\cos \theta \mathrm{d}{A}_{\mathrm{SL}})+{\gamma }_{\mathrm{SV}}^{\mathrm{o}}(-\mathrm{d}{A}_{\mathrm{SL}})+{\gamma }_{\mathrm{SL}}\mathrm{d}{A}_{\mathrm{SL}}=0\hfill \end{array}$ (2.16)

(2.16)

${\gamma }_{\mathrm{LV}}\cos \theta ={\gamma }_{\mathrm{SV}}^{\mathrm{o}}-{\gamma }_{\mathrm{SL}}$ (2.12)

(2.12)

The difference between the equilibrium surface energy of solid–vapor and solid–liquid is sometimes called adhesion [11]. Note that the work of adhesion and adhesion tension involves the solid–vapor equilibrium rather than that of the solid–liquid:

$A_{\mathrm{SLV}}={\gamma }_{\mathrm{SV}}^{\mathrm{o}}-{\gamma }_{\mathrm{SL}}={\gamma }_{\mathrm{LV}}\cos \theta$

Most surfaces have heterogeneous composition and are not perfectly smooth. Wetting of such a surface may reach equilibrium or remain in a metastable state. In the case of an ideal surface, the addition or removal of a small volume of liquid from the drop will result in the advancement or recession of the drop. The contact angle will return to its equilibrium value. In the case of a surface that may contain roughness and heterogeneity, there is a delay in the movement of the liquid drop in response to the addition or removal of liquid. This phenomenon, called hysteresis, requires a revision of the definition of contact angle.

The contact angle, formed as a result of the addition of liquid, is referred to as the advancing angle. The angle formed by the removal of liquid is called the receding angle. The contact angle of a liquid on a real surface is measured in both advancing and receding modes. Typically, after the addition or removal of the liquid, there is a delay followed by a sudden motion in the drop of the liquid. The maximum angle for the advancing mode and the minimum angle for the receding mode are defined, respectively, as advancing and receding contact angles.

Harkins and Livingston [12] have proposed a correction to Young’s equation when the surface of the solid carries a film of the liquid’s vapor. The surface energy of a solid surface that contains an adsorbed vapor layer (γ_SA) is less than that of a “clean” surface. This concept has practical significance because clean surfaces tend to adsorb the ambient vapors and oils and must therefore be protected prior to the application of adhesive. Harkins and Livingston’s correction, known as spreading coefficient (π_E), is shown in Eq. (2.17), thus resulting in Eq. (2.18) after substitution into Eq. (2.12).

$\begin{array}{ll}{\gamma }_{\mathrm{SV}}^{\mathrm{o}}\hfill & ={\gamma }_{\mathrm{SA}}-{\pi }_{\mathrm{E}}\hfill \\ {\gamma }_{\mathrm{LV}}\cos \theta \hfill & ={\gamma }_{\mathrm{SA}}-{\pi }_{\mathrm{E}}-{\gamma }_{\mathrm{SL}}\hfill \end{array}$ (2.17)

(2.17)

${\gamma }_{\mathrm{SA}}={\gamma }_{\mathrm{LV}}\cos \theta +{\gamma }_{\mathrm{SL}}+{\pi }_{\mathrm{E}}$ (2.18)

(2.18)

The spreading coefficient can be measured by a technique developed by Paddy. This method assumes that a sessile drop, when successively increased in volume, reaches some constant maximum height (h) for a given solid–liquid system, provided the system is aged to reach equilibrium. This maximum height is related to the spreading coefficient by Eq. (2.19), in which ρ is the density of the liquid and g is the gravitational acceleration.

${\pi }_{\mathrm{E}}=-\frac{\rho g{h}^2}{2}$ (2.19)

(2.19)

Finally, by substituting for the interfacial tension from the modified Young’s equation (Eq. (2.12)) into the work of adhesion equation (Eq. (2.20)) for a solid–liquid system, the equation for the work of adhesion can be simplified to Eq. (2.21), also known as Young–Dupre’s equation.

$W_{\mathrm{a}}={\gamma }_{\mathrm{LV}}+{\gamma }_{\mathrm{SV}}^{\mathrm{o}}-{\gamma }_{\mathrm{SL}}$ (2.20)

(2.20)

$W_{\mathrm{a}}={\gamma }_{\mathrm{LV}}(1+\cos \theta )$ (2.21)

(2.21)

This means that the work of adhesion can be calculated by measuring the contact angle and the surface tension of the liquid.

2.6 Effect of Temperature on Surface Tension

An important variable affecting surface tension is temperature, which has practical value during the adhesion bonding of plastics. Surface tension of both adhesive and polymer is influenced by temperature. Guggenheim’s equation (Eq. (2.22)) is applicable to liquids comprised of small molecules [13] and polymers. In this equation, γ₀ is surface tension at T=0 K and T_c is the critical temperature (K) of the liquid. The values of γ₀ and T_c values can be determined by fitting a line to the surface tension data as a function of temperature. According to the Guggenheim equation, surface tension decreases as temperature increases [14]. The rate of decrease of surface tension as a function of temperature is 0.1 dynes/°C cm for liquids with small molecules [13,15].

$\gamma ={\gamma }_0{\left(1-\frac{T}{T_{\mathrm{c}}}\right)}^{11/9}$ (2.22)

(2.22)

2.7 Surface Tension Measurement

Surface tension measurement techniques are divided into methods for solids and liquids. There are two modes for measuring surface tension of liquids: static and dynamic. Values reported in the literature are often static surface tensions of liquids. Tables 2.1–2.3 present a brief description of the common techniques for surface tension measurement of liquids and solids. Some of these methods are described in further detail.

Several standards specify methods for measuring contact angles for different applications (Table 2.4). These methods provide procedures for the comparison of surface energy of differing industrial materials.

2.7.1 Measurement for Liquids: du Nouy Ring and Wilhelmy Plate Methods

The du Nouy ring and Wilhelmy plate methods (Figure 2.4) are the two most frequently used techniques for measuring surface tension at a liquid air interface or interfacial tension at a liquid–liquid interface. Only the du Nouy method can be applied to measure interfacial tension. Both of these techniques are based on pulling an object with a well-defined geometry of the surface of liquids and measuring the pull force. These techniques are also known as pull-force methods. In the Wilhelmy method, a plate is the pull object, while in the du Nouy technique, a ring is used. These techniques are named after the two scientists who conducted some of the early research in the field of surface tension measurement. In 1863, Wilhelmy [16] described the measurement of capillary constants without a detailed calculation of surface tension. Lecomte du Nouy illustrated the shortcomings of the past surface tension determination methods in a paper published in 1919 [17].

Equation (2.23) shows the elements required for the calculation of surface tension by the du Nouy ring method. In this equation, P_T is the total force on the ring, P_R is the weight of the ring, R is the radius of the ring, and γ_ideal is the ideal surface tension. In practice, a meniscus correction factor is required because the size and shape of the surface inside and outside the ring are not the same. Surface tension must, therefore, be corrected for the shape of the ring by a factor (f) as shown in Eq. (2.24). The correction factors have been determined and tabulated [15,18,19].

$P_{\mathrm{T}}=P_{\mathrm{R}}+4\pi R\cdot {\gamma }_{\mathrm{ideal}}$ (2.23)

(2.23)

$\gamma =f{\gamma }_{\mathrm{ideal}}$ (2.24)

(2.24)

The Wilhelmy method does not require a correction factor for the meniscus shape, although it does require correction if the plate is partially or completely submerged in the liquid. In Eq. (2.25), P_T is the total force on the plate being measured, P_P is the weight of the plate, p is the perimeter of the plate, and γ_ideal is the ideal surface tension. A buoyant force term must be added to or subtracted from the second part of the equation, depending on whether the plate is above or below the level of the free liquid. In Eq. (2.26), h is the height above or below the free liquid level, A is the cross-sectional area of the plate, and γ is the surface tension.

$P_{\mathrm{T}}=P_{\mathrm{P}}+p{\gamma }_{\mathrm{ideal}}$ (2.25)

(2.25)

$P_{\mathrm{T}}=P_{\mathrm{P}}+p\gamma +\rho ghA$ (2.26)

(2.26)

The total force (P_T) acting on the ring (du Nouy) or the plate (Wilhelmy) can be measured by a balance connected to either device. Substituting for the total force and the other parameters in Eq. (2.26) allows the value of surface tension (γ) to be calculated.

2.7.2 Measurement for Solids: Liquid Homolog Series

Surface tension of solid plastics is determined indirectly, usually by contact angle methods. A problem with the direct measurement of surface tension arises from the difficulty in the reversible formation of a solid surface. Table 2.3 shows a list of methods that can be applied to measure the surface tension of solids.

An alternative method uses a concept called critical surface tension, proposed by Fox and Zissman [4,20,21], to characterize the surface energy of solids. A cosine plot of the contact angle (cos θ) and liquid–vapor surface tension (γ_LV) yields a straight line for a homologous series of liquids (Figure 2.5). Nonhomologous liquids yield a curved line that may not be easily extrapolated. The intercept of the line at cos(θ) equal to one is defined as the critical surface tension of the polymer (γ_c). Values of 18 dynes/cm for polytetrafluoroethylene and 30 dynes/cm for polyethylene are obtained according to this procedure (Figures 2.5 and 2.6). Tables 2.5 and 2.6 present surface free energies of solids and surface tension of liquids.

Table 2.5

Surface Free Energy of Select Plastics

Plastic Material	Surface Free Energy, dynes/cm
Polytetrafluoroethylene	18–19
Polytrifluoroethylene	22
Polyvinylidene fluoride	25
Polyvinyl fluoride	28
Polypropylene	29
Polyethylene	30–31
Ionomer (low) polystyrene	33
Ionomer (high) polystyrene	37
Polymethyl methacrylate	38
Polyvinyl chloride	39
Cellulosics	42
Polyester	43
Nylon	46

Table 2.6

Surface Tension of Select Liquids

Liquid	Surface Free Energy, dynes/cm
n-Hexane	18
Alcohols	22
Cyclohexane	25
Toluene, xylene	29
Phenol	41
Aniline	43
Glycol	47
Formamide	58
Glycerol	63
Water	72

One can obtain a relationship (Eq. 2.27) between the critical surface tension and the solid–vapor surface tension by setting the contact angle to zero in Young’s equation (Eq. (2.12)). Critical surface tension is, therefore, smaller than solid–vapor surface tension. A Zissman plot for polymethyl methacrylate is given in Figure 2.7.

Surface energy of plastics decreases with increasing temperature. Consequently, a molten polymer has a lower surface than its solid form. Surface energies of a number of common polymers are presented in Table 2.7.

${\gamma }_{\mathrm{c}}=\underset{\theta \to 0}{\lim }({\gamma }_{\mathrm{LV}}\cos \theta )={\gamma }_{\mathrm{SV}}-{\gamma }_{\mathrm{SL}}$ (2.27)

(2.27)

Table 2.7

Solid Surface Energy Data for Common Polymers^a

Name	CAS Ref. No.	Surface Free Energy (SFE) at 20°C in mN/m	Temp. Coefficient SFE in mN/(m K)	Dispersive Contrib. of SFE in mN/m	Polar Contrib. of SFE in mN/m
Polyethylene (PE)–linear	9002-88-4	35.7	−0.057	35.7	0
Polyethylene (PE)–branched	9002-88-4	35.3	−0.067	35.3	0
Polypropylene (PP)–isotactic	25085-53-4	30.1	−0.058	30.1	0
Polyisobutylene (PIB)	9003-27-4	33.6	−0.064	33.6	0
Polystyrene (PS)	9003-53-6	40.7	−0.072	(34.5)	(6.1)
Poly-α-methyl styrene (PMS) (polyvinyltoluene [PVT])	9017-21-4	39.0	−0.058	(35)	(4)
Polyvinyl fluoride (PVF)	24981-14-4	36.7	–	(31.2)	(5.5)
Polyvinylidene fluoride (PVDF)	24937-79-9	30.3	–	(23.3)	(7)
Polytrifluoroethylene (P3FEt/PTrFE)	24980-67-4	23.9	–	19.8	4.1
Polytetrafluoroethylene (PTFE) (Teflon™)	9002-84-0	20	−0.058	18.4	1.6
Polyvinyl chloride (PVC)	9002-86-2	41.5	–	(39.5)	(2)
Polyvinylidene chloride (PVDC)	9002-85-1	45.0	–	(40.5)	(4.5)
Polychlorotrifluoroethylene (PCTrFE)	25101-45-5	30.9	−0.067	22.3	8.6
Polyvinylacetate (PVA)	9003-20-7	36.5	−0.066	25.1	11.4
Polymethylacrylate (polymethacrylic acid) (PMAA)	25087-26-7	41.0	−0.077	29.7	10.3
Polyethylacrylate (PEA)	9003-32-1	37.0	−0.077	30.7	6.3
Polymethylmethacrylate (PMMA)	87210-32-0	41.1	−0.076	29.6	11.5
Polyethylmethacrylate (PEMA)	9003-42-3	35.9	−0.070	26.9	9.0
Polybutylmethacrylate (PBMA)	25608-33-7	31.2	−0.059	26.2	5.0
Polyisobutylmethacrylate (PIBMA)	9011-15-8	30.9	−0.060	26.6	4.3
Poly(t-butylmethacrylate) (PtBMA)	–	30.4	−0.059	26.7	3.7
Polyhexylmethacrylate (PHMA)	25087-17-6	30.0	−0.062	(27.0)	(3)
Polyethyleneoxide (PEO)	25322-68-3	42.9	−0.076	30.9	12.0
Polytetramethylene oxide (PTME) (polytetrahydrofuran [PTHF])	25190-06-1	31.9	−0.061	27.4	4.5
Polyethyleneterephthalate (PET)	25038-59-9	44.6	−0.065	(35.6)	(9)
Polyamide-6,6 (PA-66)	32131-17-2	46.5	−0.065	(32.5)	(14)
Polyamide-12 (PA-12)	24937-16-4	40.7	–	35.9	4.9
Polydimethylsiloxane (PDMS)	9016-00-6	19.8	−0.048	19.0	0.8
Polycarbonate (PC)	24936-68-3	34.2	−0.04	27.7	6.5
Polyetheretherketone (PEEK)	31694-16-13	42.1	–	36.2	5.9

^awww.surface-tension.de/solid-surface-energy.htm, February 2013.

In summary, the experimental and analytical methods described in this chapter enable the reader to both measure and calculate surface energy of liquids and solids. Surface preparation techniques often change the surface energy of materials, which can be determined by the methods provided in this chapter.