7.1.1 Intermolecular Interactions

Intermolecular interactions can to a large extent be described using classical concepts, in particular based on the multipole expansion of the interaction between the charge distributions of molecular constellations, of which details can be found in [1, 2]. Here we follow a more descriptive approach signifying the most important players in the game. If molecules contain a charge, we have the Coulomb (or charge–charge) interaction between the molecules, given by the potential

7.1

where Q₁ and Q₂ represent the charge on molecules 1 and 2 and r is the distance between the centers of the molecules. The quantity ε = ε_rε₀ represents the permittivity of the medium with ε_r the relative permittivity (also often denoted as the dielectric constant) and ε₀ the permittivity of vacuum. For vacuum ε_r = 1, and for ambient air ε_r is very close to 1. Generally, ε_r ≥ 1 and for organic solvents typically ranges from about 3 to 40, while for H₂O, ε_r ≅ 78 (at 20 °C). The permittivity for a range of solvents is given in Appendix B.

For Coulomb interactions in (aqueous) ionic solutions, we have to add a few further considerations. Because ionic solutions have to be neutral as a whole, once, say, positive ions are present, also negative ions must be present, and since positive ions attract negative ions, while positive (negative) ions repel each other, around every ion in solution a quickly fluctuating atmosphere of counterions and co‐ions is created. This leads to screening of the Coulomb interactions. In the simplest model, the solvent is considered as a continuum with permittivity ε, while the ions with number density n_j are considered as charged hard spheres with charge ez_j (unit charge e, valency z_j) and effective radius a. The Poisson equation ∇²ψ = −ρ/ε connects the (electrostatic) potential ψ around an ion to the charge density ρ = Σ_jz_jen_j. The number density of ions n_j is dictated by the Boltzmann equation n_j = n_j,0 exp(−z_jeψ/kT) with n_j,0 as their bulk concentration. Solving the combined equation – the Poisson–Boltzmann equation – in a linearized form for a spherical surface, that is, for an ion, leads to the Debye–Hückel model with the potential:

7.2

where k represents Boltzmann′s constant and T the temperature. The charge density around an ion is then given by

7.3

Both the potential and the charge distribution contain a factor exp(−κr), which is the result of the screening of the ion by its ionic atmosphere that renders the Coulomb interactions with ions outside this screening layer virtually negligible. Hence, λ_D = 1/κ is normally addressed as the screening or Debye length. For a 1–1 electrolyte in H₂O at 25 °C, we obtain κ⁻¹ = 3.04 × 10⁻¹⁰n₀^−1/2 with n₀ mol l⁻¹. So, for n₀ = Σ_jn_j,0 = 0.01 mol l⁻¹, the Debye length is κ⁻¹ ≅ 3.0 nm, while for n₀ = 0.1 mol l⁻¹, the result is κ⁻¹ ≅ 0.96 nm. Note that for water there is a limiting value for κ⁻¹, determined by its dissociation equilibrium. At pH = 7, n(H⁺) = n(OH⁻) = 1 × 10⁻⁷ mol l⁻¹, and this leads to κ⁻¹ ≅ 680 nm. As it is assumed that z_jeψ < kT, the magnitude of the potential ψ should be less than 25 mV (≈kT at 300 K) for univalent ions, but practically the model is reasonably accurate up to 50–80 mV.

More frequently a molecule is neutral (has no charge) but is polar or apolar, that is, either has or has not a dipole moment. A dipole moment implies a separation of the center of the positive and negative charges (protons and electrons) Q within a molecule. If this separation distance is given by d, the dipole moment μ is defined by Qd. The dipole moment μ is thus in units of C m, but often the old unit “Debye” (D) is used, where 1 D corresponds to 1 unity charge 0.21 Å apart, or equivalently 3.336 × 10⁻³⁰ C m. For H₂O, for example, μ = 1.85 D. The dipole moment for a range of solvents is given in Appendix B.

Of course, in a mixture molecule 1 can be charged and apolar, and molecule 2 neutral but polar in which case a force arises between a dipole μ₂ and charge Q₁. We consider two options: the orientation of the dipole is fixed at angle ϑ or the dipole is freely rotating. Evidently, the latter option is the most realistic. The (charge–dipole) potentials for these two options, for a distance r ≫ d, are

7.4

respectively. In the latter case, the interaction is thus temperature dependent.

Thus, if both molecules are neutral but polar, the potential can be quantified by the Keesom (or orientation) interaction, given for freely rotating dipoles by

7.5

which is again a temperature‐dependent quantity. However, when a molecule approaches another molecule, its electric field influences the charge distribution of the other molecule and vice versa. This leads to polarization of both molecules. Generally, an electric field E induces a dipole αE in a charge distribution with polarizability α. Using α′ = α/4πε₀, one has the polarizability volume, proportional to the volume of the molecule with a proportionality factor dependent on the definition of size of a molecule [3]. For example, for H₂O with α′ ≅ 1.46 ų, using α′ = r³, we obtain a radius r of 1.13 Å, while the radius σ/2 as calculated from the van der Waals (vdW) constant b = 2πN_Aσ³/3 = 0.0305 l mol⁻¹ is 1.45 Å. The polarizability can be calculated from the molar refraction R_m ≡ V_m(n² − 1)/(n² + 2) = αN_A/3ε₀, where N_A is Avogadro's number, V_m the molar volume, and n the refractive index. For a range of solvents, R_m values are given in Appendix B. According to Maxwell's equations, the refractive index n is given by n² = ε_rμ_r with ε_r and μ_r as the relative electric and magnetic permittivity, respectively. Since for a dielectric μ_r = 1, at high frequency in the optical region, ε_r = n². The interaction between a polar molecule 1 and a polarizable molecule 2 is the Debye (or induction) interaction, given by

7.6

Finally, even if both molecules are neutral and apolar, the instantaneous charge distributions of the electrons interact, leading to the London (or dispersion) interaction, approximately given by

7.7

where hν is a characteristic energy and in the last step the frequently made geometric mean approximation is invoked. While in the approximation yielding this expression ν represents a characteristic oscillation frequency of the electrons, in practice one often approximates hν as the ionization energy of the molecule. Generally, this is a poor approximation, typically too large by a factor of 1.5–4 as compared with estimates obtained by eliminating hν in favor of α [2].

For neutral molecules, the total potential is often taken as the sum of w_ori, w_ind, and w_dis, together labeled as van der Waals (vdW) interactions. As these contributions all have a r⁻⁶ dependency, they are collectively denoted by C/r⁻⁶. For identical molecules the dipole moment of molecule 1 polarizes molecule 2 and vice versa, and hence this contribution should thus be multiplied by a factor of 2. Further, note that the work done is calculated for constant T and V (or T and P), so that the Helmholtz energy F (or Gibbs energy G) is obtained. For a potential w ∼ 1/T, one easily shows that the internal energy U = ∂(F/T)/∂(1/T) is given by 2F. Different authors may discuss either F or U, leading to different numerical values. Contrary to what one might expect, the London interactions are often dominating¹ (Table 7.1). Only for highly polar molecules, such as CH₃Cl, H₂O, and (CH₃)₂CO, the Keesom interactions constitute a significant part of the total interaction. A similar conclusion (actually even a bit stronger) can be reached on the basis of the Lifshitz theory (see Section 7.1.2).

Finally, we remark that at close proximity molecules repel each other. This repulsion is due to overlap of the electron orbitals and often called the Born repulsion. From quantum mechanics one can rationalize an exponential dependence, although in practice it is often described by a power law. Hence, we have w_rep = bexp(−r/ρ) or w_rep = B/rⁿ where (b,ρ) and (B,n) are parameters. The value of r/ρ is typically between 12 and 16, while n generally ranges from 9 to 15, where a value of n = 12 is often taken for mathematical convenience. Combining the Born repulsion and the vdW attraction, we obtain the Lennard‐Jones potentialw = B/r¹² − C/r⁻⁶. It can be written as w(r) = ε[(r₀/r)¹² − 2(r₀/r)⁶] = 4ε[(σ/r)¹² − (σ/r)⁶], where ε is the depth of the potential at the equilibrium distance r₀ and σ = 2^−1/6r₀ is often denoted as the (Lennard‐Jones) diameter.

A more detailed review is given in [2]. A broad description dealing with both intermolecular and surface interactions is given in [4], and more recently in [5].

7.1.2 Mesoscopic Interactions: Continuum Aspects

For coatings the interaction of molecules with surfaces is at least as important as the intermolecular interactions, and the discussion of these effects is based on the considerations given above. Let us start, similarly as for molecules, with the electrostatic interaction [6, 7]. A surface may get charged as ions in solution adsorb preferentially at the surface of solids. Alternatively, molecules may adsorb and acquire a charge; for example, amines R-NH₂ may become protonated to yield R-NH₃⁺. Finally, oxides typically have a hydroxylated surface Me-OH, which can deprotonate resulting in Me-O⁻ + H⁺ or protonate to Me-OH₂⁺, depending on the pH. In all cases this leads to a charged surface and an associated double layer. The thermodynamic work necessary to create this double layer is , where ψ is the (electrical) potential difference between the two phases, Q is the charge of the surface, and ψ₀ is the surface potential. A relatively simple model of the double layer is due to Gouy and Chapman, in which the surface charge is smeared out uniformly, the ions are considered as point particles with charge ze (z valency, e unit charge) and the solvent as a continuous medium with permittivity ε. As before, the electrostatic interaction between the charges present in the system is described by the Poisson equation ∇²ψ = −ρ/ε, where ρ = Σ_jz_jen_j is the charge density, while the number density of ions n_j is given by the Boltzmann equation n_j = n_j,0 exp(−z_jeψ/kT) with n_j,0 as their bulk concentration. For a flat surface the solution of the combined Poisson–Boltzmann equation connecting the charge density ρ (hidden in κ) with the electrostatic potential ψ is

7.8

where the linearization in the last step can be made if zeψ/kT ≪ 1. Electroneutrality requires that the surface charge σ is the negative of the total space charge (the charge in the liquid adjacent to the surface), that is,

7.9

where in the last step the Poisson equation is inserted. Making use of

7.10

as obtained from the Poisson–Boltzmann equation solution process, leads to

7.11

In the absence of specific adsorption, the Gibbs energy per unit area reads

7.12

where for the first and second steps, Poisson's equation and partial integration are used, respectively. The final result upon integration becomes

7.13

for low potentials. Note that the sign is negative because the double layer forms spontaneously.

The Gouy–Chapman model contains a number of assumptions that will affect correspondence with reality. The permittivity ε is assumed to be constant but close to the surface the electric field strength is large, leading to a substantial lowering of the value for ε (Figure 7.1). Moreover, contrary to what is assumed, ions have a finite size radius a, while the surface charge in reality is discrete and not smeared out uniformly. Finally, ions may interact with the surface through other than Coulomb forces. Taking all of this together, it was suggested by Stern [8] that the double layer should be separated in two parts. The first layer consists of more or less immobile hydrated ions adsorbed at the surface and they are limited in their distance of closest approach to the surface by their size. This immobile layer of ions with thickness δ is normally called the Stern layer. The second layer is denoted as the diffuse Gouy layer. This implies that the surface potential ψ₀ should be (re‐)interpreted as the potential ψ_δ at the distance δ. Actually, some further considerations due to Grahame [9] and Bockris et al. [10] lead to a more sophisticated picture of the Stern layer (see [11, 12] for a general discussion). The potential at the surface of the solid is determined by potential‐determining ions adsorbed at the surface of the solid. For oxides this is frequently the H⁺ ion, while for pure metals specific (negatively charged), anions adsorb preferentially to metal surfaces (e.g. Cl⁻ ions form a stable complex with a Au surface). The water molecules in this layer show a clear orientation with respect to the surface, thereby lowering their permittivity (typically ε_r ≈ 6). Slightly further outward nonspecific counterions with their hydration shell adsorb. The water molecules in these shells still have a lower permittivity as compared with that of free water molecules (typically ε_r ≈ 30). Still further outward we find the diffuse layer. For a detailed discussion of the Stern layer on metal surfaces, see [13].

When two plane surfaces approach each other, both their charge distributions and electrical potentials change. The charge distributions near both surfaces start to overlap when the distance between the plates is x ≤ 2κ⁻¹ (Figure 7.2a), and this leads for a bulk concentration n₀ of monovalent ions to a force f per unit area A given by

7.14

in case we have low potentials. The first term in the disjoining pressure Π(x) is addressed as the osmotic pressure and the second as the Maxwell stress. To calculate this force, Eq. 7.8 has to be solved and as this relation is complicated, one uses commonly two (constant) boundary conditions that furnish bounds on Π(x). The first one is constant surface potential and the second one is constant surface charge density. For x ≫ λ_D both conditions lead to the same result, but for x ≅ λ_D the constant potential case leads to a smaller force (Figure 7.2b). To interpret this expression, consider the midplane for two identical surfaces where the last term is zero. The terms in curly brackets just indicate the increase in ion concentration between the planes leading to a contribution, the disjoining pressure, to the total osmotic pressure.

For the more important case of spherical surfaces as occurring in colloidal systems, the double layer repulsion based on the Poisson–Boltzmann equation yields more complicated results. For spherical particles with radius a, an approximate analytical solution based on the potential ψ = ψ₀ a exp(r − a)/r as discussed for ions is available using the linearized Poisson–Boltzmann equation (Debye–Hückel approximation). This leads to the charge Q and Gibbs energy G, given by

7.15

For the full Poisson–Boltzmann equation, only a numerical solution is available [14]. Several approximate solutions have been offered, both for symmetric and nonsymmetric electrolytes for which an overview is given by Lamm [15]. Approximate but accurate expressions for ψ(r) in spherical and cylindrical geometry are presented by Tuinier [16]. A good approximation for the electrostatic repulsive potential Φ_rep between two equally sized particles of radius a [17] at the closest distance D between the surfaces (that is, a distance between the centers r = 2a + D) for constant surface potential ψ₀ is given by

7.16

where, if required, the term ln[1 + exp(−κD)] can be further approximated by exp(−κD) for D ≫ κ⁻¹. Similar expressions for particles with different sizes and different surface conditions have been given by [18]. For the case of symmetrical electrolytes with number density (concentration) n₀ = Σ_jn_j,0, Stokes [19] offered a relatively simple approximation for the surface charge density σ reading

7.17

Limiting the approximation to two terms, a maximum deviation of the exact result of about 26% results (the error peaks at κa = 0.1 and zeψ₀/kT = 6, [14]). Generally the deviation is less than 10%. For small particle diameter values, say, below 10 nm, the contribution of the second term cannot be neglected as compared with that of the first term, as it attains the same order of magnitude or, depending on size, is much larger than the first term. Hence, using for small particles the Grahame equation – essentially the first term of Eq. 7.13, σ = 8zen₀t/κ(1−t²) – provides an estimate for σ that is significantly different from the Stokes equation, the full Eq. 7.13.

Finally, we mention the zeta potentialζ, that is, the potential difference across the mobile part of the double layer (assuming that the innermost part is immovably attached to the particle). This is, equivalently, the potential at the plane of shear. Although not necessarily the case, the position of the zeta plane is often taken at the boundary of the Stern layer. In that case the particle radius in Eq. 7.13, hidden in κ, is an effective radius, being the minimum distance to which the center of a hydrated ion can approach the particle. The zeta potential depends on the pH and the type and concentration of the electrolyte used, which should be quoted upon reporting a measured zeta potential. Figure 7.3 shows the zeta potential for a few oxides as a function of pH. Converting ζ to σ and vice versa can be simply done according to Hückel theory [20] if κa < 2.

Also for the vdW interaction expressions with respect to surfaces are required. The conventional approach is using Hamaker's theory, which, assuming simple pairwise additivity of the interaction potentials, integrates the vdW interaction φ_vdW = −C_AB/r⁶ of one molecule A with a semi‐infinite flat solid of molecules B with number density ρ_B using the volume element dV = 4πr²dr according to

7.18

For the interaction per unit area A between two planar surfaces, one obtains

7.19

where H = π²C_ABρ_Aρ_B is the Hamaker constant. Typical values for H in water are 0.3–1 × 10⁻²⁰ J for hydrocarbon particles, 0.5–5 × 10⁻²⁰ J for oxide and halide particles, and 5–30 × 10⁻²⁰ J for metal particles. Values for the Hamaker constant H for several specific compounds are given in Table 7.4. For the vdW interaction of two spheres of radius a at a distance D = r−2a between the surfaces, Hamaker obtained [6, 21]

7.20

which can be approximated, if required, for D/a ≪ 1 by w_att = −H[L/D +2ln(D/L)]/12 where L = a + 3D/4 [17]. The further approximation to w_att = −Ha/12D is generally poor, unless D/a is really small. When deriving this expression, Hamaker neglected retardation, which is questionable for a ≥ 100 nm. The vdW attraction between macroscopic bodies has been studied experimentally, and it appears that the theoretical results agree very satisfactorily with experiments [11, 22]. With Hamaker's theory, also the surface energy of solids γ can be estimated (see Section 7.3.2). Obviously, at very small distances also the Born repulsion has to be taken into account [23]. Fowkes [24] argued that, in view of the discrete nature of molecules, the interactions should not be integrated, but summed. This leads to similar expressions, but with considerable differences in magnitude.

In case materials 1 and 2 are separated by another material 3, an approximate value for H₁₃₂ is given by H₁₃₂ ≅ H₁₂ − H₃₂ − H₁₃ + H₃₃, based on summing the contributions for the various material combinations. For a symmetrical configuration, H₁₃₁ ≅ H₁₁ + H₃₃ − 2H₁₃ ≅ H₃₁₃. Further approximating H_ij with H_ij ≅ (H_iiH_jj)^1/2 yields H₁₃₂ ≅ (√H₂₂ − √H₃₃)(√H₁₁ − √H₃₃), or for a symmetrical configuration, H₁₃₁ ≅ (√H₁₁ − √H₃₃)². These approximation rules should only be used when the London forces are dominant. vdW interactions between composite particles in a liquid are discussed by [25].

7.1.3 Lifshitz Theory

A more complete description of vdW interactions is given by the Lifshitz theory (see [26]; [4, 27, 28] provide introductions). By considering the (complex) permittivity ε over the complete frequency range ω, one can calculate the complete macroscopic vdW interaction between two bodies (including all many‐body effects within the two bodies), and dealing correctly with the effect of intermediate substances. Retardation can be accounted for as well, but this is often not done. The most important result is that the Hamaker expression remains valid. In brief, the (nonretarded) Hamaker constant between two half‐spaces 1 and 2 separated by a third medium 3 is given by

7.21

where

7.22

The star in the sum indicates that the term with m = 0 has to be multiplied by ½. The zero frequency term m = 0 corresponds to the Debye and Keesom interactions, while the terms m ≠ 0 relate to the London forces. Some analysis shows that the m = 0 term contributes only limitedly. Because Δ₁₃Δ₁₂exp(−x) ≤ 1, the logarithmic term can be expanded in a power series and integrated term by term. Hence, one obtains

7.23

Since the summation over s converges rapidly, usually only a few terms have to be included. Further simplification is obtained by considering the dielectric response. The real and imaginary components ε′ and ε″ of the dielectric response function ε(ω) = ε′(ω) + iε″(ω) are coupled via the Kramers–Kronig transform:

7.24

so that real function ε(iω) with imaginary frequency iω given by

7.25

Because ε‐data over the complete frequency range are usually unavailable, one considers normally only the most important frequency regimes, which are in the UV region. For this one uses a set of delta functions

7.26

with f_j the oscillator strength for the absorption frequency ω_j. Based on a few measurements and the functional dependency of ε on ω, one then calculates the interactions. Substitution of Eq. 7.22 in Eq. 7.21 leads for dielectrics to the interpolation functions:

7.27

The second term on the rhs represents the contribution from the microwave region, but Hunter warns explicitly against including it [27] as it may become “numerically dangerous,” as he calls it. This is because it easily becomes the dominant contribution in the UV region where it has no right to exist. The third term on the rhs represents the UV region. The parameter χ_j in this term introduces damping, but is often neglected. The constants C_j and ω_j can be determined from a fit using the relation n(ω)² − 1 = [n(ω)² − 1]ω²/ω_j² + C_j. Hence plotting n(ω)² − 1 versus [n(ω)² − 1]ω² should yield a straight line. Such a plot is usually called a Cauchy plot. For a first estimate, apart from the zero frequency term, a single‐term expression can be employed, in which case C_j reduces to n² − 1. Table 7.2 provides a few examples. For comparison, water is included in this table, but even for a first estimate, a somewhat more complex description for its behavior is needed as for water the microwave region does contribute. For metals one typically uses

7.28

The parameter ω_e represents the plasmon frequency of the electrons, typically in the range 25–30 × 10¹⁵ Hz.

Altogether, although the (complete) vdW theory is complex, it is capable of delivering relatively accurate estimates for the vdW interaction, particularly relevant for inorganic fillers [31]. Apart from its basic interest, Lifshitz–van der Waals theory also forms (part of the) basis for the so‐called acid–base (AB) interactions model for interfaces (see Section 7.3.3).

7.1.4 The Derjaguin Approximation

Generally the calculation of the vdW interaction energy between two macroscopic bodies is complex and can be done exactly only for simple geometrical shapes. Derjaguin proposed an approximate way to do such a calculation for arbitrary shapes. He related the energy per unit area between two planar semi‐infinite surfaces w(r), separated by a gap of width r, to the energy W(D) between two bodies of arbitrary shape, where D denotes the distance of closest approach. His approximation reads

7.29

where A(x) is the cross‐sectional area at distance x, the integration is over the entire surface of the solid, and the last step is valid for axially symmetric configurations only.

To illustrate this calculation, let us calculate the interaction between two equal‐sized spheres of radius a. For this configuration, we have x(r) = D + 2a − 2(a² − r²)^1/2 so that dx = 2r(a² − r²)^−1/2dr or 2rdr = (a² − r²)^1/2dx. If the range of interaction is significantly smaller than the radius a, we need to consider only the caps of the spheres with contributions for small values for r. In this case we can approximate 2rdr = (a² − r²)^1/2dx by 2rdr ≅ adx and

7.30

Using the Hamaker expression w(x) = −H/12πx², we directly obtain W(D) = −Ha/12D, in agreement with the approximate form of Eq. 7.20. The Derjaguin approximation is valid if the characteristic decay length of the surface force is small as compared to the curvature of the surfaces. If valid, it generally separates the interaction energy W(D) in a geometrical factor and the distance dependent factor w(x).

7.1.5 Mesoscopic Interactions: Molecular Aspects

Up to now liquids have been treated largely as a continuum, but at very short distances (a few molecular diameters) between two surfaces, this approximation is not adequate. Moreover, we have until now only dealt with vdW forces and electrostatic forces, while for mesoscopic systems solvation, hydration, and hydrophobic and steric interactions are also important. We follow closely the discussion as given in [29].

Let us start with the solvation forces. Generally, a dissolved molecule in a solvent, particularly in water, is surrounded by a shell of solvent molecules tightly bound to the dissolved molecule. For molecules confined between two parallel plates, this solvation shell plays a role leading to a force between two such plates, which as a function of the distance x is given by f(x) = f₀ cos(2πx/d) exp(−x/x₀), with d the (effective) diameter of the dissolved molecule and x₀ a “decay constant” (Figure 7.4a, [32]). This shape for f(x) has been confirmed experimentally (Figure 7.4b, [33]).

Hydration forces describe the repulsive force between hydrophilic surfaces in water. They are typically of short range (≈1 nm) and deal with the energy required to remove the hydration (water) layer or the surface‐adsorbed species, probably due to strong charge–dipole, dipole–dipole, or hydrogen bonding interactions [34]. Many aspects of hydration forces are still poorly understood.

Hydrophobic interactions deal with the forces between hydrophobic molecules or surfaces in water. Hydrophobic surfaces attract each other, and generally two force components are distinguished [35]. A short‐range attraction decaying approximately exponentially with a decay length of 1–2 nm is attributed to a change in water structure upon approach of the surfaces. The second, long‐ranged contribution extends out to some 100 nm in some cases and is not well understood.

Finally, we have steric interactions, which are relevant for polymer‐coated surfaces used for the stabilization of dispersions by steric repulsion. Steric interactions are thus of high importance for technical applications. To discuss this effect briefly, we recall the image of a polymeric chain as a freely jointed chain of n′ Kuhn segments with length l′ and any angle between sequential segments allowed. This leads to a random coil with root mean square end‐to‐end distance R₀ = l′n′^1/2, or, equivalently, a radius of gyration R_g = R₀/6^1/2. As an example, take M = 10⁵ g mol⁻¹ and the molar mass of a segment as M₀ = 100 g mol⁻¹, which leads to n′ = 10³. Using l′ = 1 nm, we obtain R_g = 1(10³)^1/2/6^1/2 = 13 nm. The Kuhn length represents an effective length and is related to the real length of the polymer chain by the characteristic ratio C, in such a way that R₀² = Cnl² where n is the number of backbone bonds with (average) length l (see Section 2.1.4). The parameter C thus characterizes the chemical structure at hand. We also recall that the “true” or Flory radius R_F of the polymer depends on the solvent, so that R_F = αR_g with the solvent expansion factor α, dependent on the type of solvent and the theta temperature θ of the polymer in that solvent [2].

The interaction depends on whether the polymers are adsorbed (bonded via vdW forces) or grafted (covalently bonded) at the surface, on the surface coverage and on the solvent quality. Entropic repulsion between particles is favored by a high surface coverage and good solvent conditions. For adsorbed molecules, the maximum grafting density is roughly R_g⁻² and the interaction extends to a distance of about 2R_g from the surface. For grafted molecules, the grafting density can be much higher and ranges from Γ ≅ 1 × 10¹⁶ m⁻² to Γ ≅ 1 × 10¹⁸ m⁻². When such grafted polymers have a grafting density Γ ≫ R_g⁻², we call the layer a (molecular) brush. In this case the layer thickness is about L₀ ≅ nl^5/3Γ^1/3. For Γ ≪ R_g⁻², the grafted polymers are called mushrooms. For low grafting density (Γ < R_g⁻²), the repulsive force per unit area Π(x) in a good solvent between two coated surfaces is given by [36]:

7.31

For high grafting density the pressure can be approximated by [37]:

7.32

Hence, the pressure Π decreases with distance x, but increases with surface coverage Γ (Figure 7.5a) and temperature T (Figure 7.5b).

Finally, we indicate two other effects that can mediate the interactions between polymer‐covered surfaces. First, the intersegment force resulting in an attractive interaction between polymer segments and favored by poor solvent conditions (T < θ). For good solvent conditions, the effect is repulsive (Figure 7.5b). Second, the bridging force results if the polymer bound to one of the surfaces can attach to the other surface, leading to an attractive force. This force is favored by low surface coverage and high molar mass. Both forces can cause flocculation of dispersions (see Section 10.2).

Rather extensive and highly readable reviews of intermolecular and interfacial interactions have been given by Israelachvili [4], Fennell Evans and Wennerström [38], Butt and Kappl [5], Russel et al. [39], and Fleer et al. [40].

7.3.1 Surface Thermodynamics

The interface between two phases obviously is not infinitesimally thin, although actually it is thin in almost every respect (except very close to the critical point). The surface thermodynamics of an infinitesimal thin interface has been dealt with by Gibbs. Nevertheless, Guggenheim [48], following van der Waals, discussed the thermodynamics of an interphase, that is, an interface with a finite thickness. Although from a physical point of view, this “finite thickness” approach is straightforward, the Gibbs approach or “infinitesimal thickness” approach is more often used, since detailed information about surfaces is (was) generally scarce. Here, we limit ourselves mainly to the Gibbs approach. The starting point is the Gibbs expression for the internal energy U of a system at equilibrium with a volume V, surface area A, and variable number of components n_j, reading

7.37

where S and μ_j represent the entropy and chemical potential of component j, respectively, while, as usual, T denotes the temperature and P the pressure. The quantity γ is the interface energy, but if one of the phases involved is a gas, often called the surface energy. It is the energy required to create a unit of surface area reversibly. Applying a Legendre transform with respect to temperature T to obtain the Helmholtz energy F leads to

7.38

while a transform with respect to both temperature T and pressure P to obtain the Gibbs energy G leads to

7.39

so that for a single component system, the interface energy γ is given by

7.40

From this expression we see that for the experimentally accessible constant conditions T,V (or T,P), the interface energy γ is actually a Helmholtz (or Gibbs) energy. Denoting the two phases by α and β and the interface itself by σ, the total potential X is the sum X^(α) + X^(β) + X^(σ), where X denotes either U, F, or G. For a single component system applying the Gibbs dividing surface so that V^(σ) = 0, we have n^(σ) = 0. Then we obtain for the Helmholtz energydF^(σ) =  − S^(σ) dT + γ dA, leading to, using Euler's theorem, F^(σ) = γA. Hence,

7.41

In this case, the specific surface internal energy u^(σ) – the analogue of the specific internal energy for the bulk – is given by

7.42

For a multicomponent system, we have, similarly,

7.43

and

7.44

For the excess amount of component j, we have , where the superscript (1) refers to component 1, usually the solvent, and for which the interface is chosen to be at the Gibbs dividing plane (Γ₁⁽¹⁾ = 0). This leads to , so that with , where a_j (c_j) represents the activity (concentration), we obtain by differentiating Eq. 7.37 and subtracting Eq. 7.32

7.45

known as the Gibbs adsorption equation. This equation describes how the change in surface tension dγ of a solution can be calculated once the adsorption Γ_j⁽¹⁾(c_j) is known. The function Γ_j⁽¹⁾ is conventionally addressed as the adsorption isotherm, for which various expressions are in use, depending on the precise conditions [11, 29]. The most important adsorption isotherm is probably the Langmuir isotherm [29]. In view of its importance, we briefly discuss its background.

Consider a surface as a plane with a certain density of surface sites. The adsorption process can be represented as a chemical reaction where a solute molecule (B) dissolved in the bulk “reacts” with an empty surface site (S) to an occupied adsorbed site (A), that is, B + S ⇆ A. We also assume that the solute molecules at the surface do not interact. Since the solute concentration is low, we approximate the activity a by concentration c = N_B/V, where N_B is the number of dissolved molecules in the volume V. We denote the fraction of occupied surface sites by θ = N_A/N_max, where N_A is the number of occupied surface sites and N_max the total number of surface sites, corresponding to monolayer coverage. Hence, N_S/N_max = 1 − θ. The equilibrium constant K for this adsorption “reaction” is

7.46

usually called the Langmuir adsorption isotherm. The amount adsorbed is Γ ≡ Γ₂⁽¹⁾ = Γ_maxθ, with Γ_max a proportionality constant representing monolayer coverage.³ Integration of the Gibbs adsorption equation using the Langmuir isotherm leads to the Szyszkowski equation describing the amount of adsorbed material at low concentration often rather well.

We note that for a solid–gas interface the Gibbs adsorption equation also describes the influence of the vapor phase components that can react with the surface of the solid. For example, an oxide like SiO₂ can have a siloxane surface with  ≅ 0.26 N m⁻¹ in the absence of H₂O or a silanol surface with γ_SiOH ≅ 0.13 N m⁻¹ in the presence of water (Figure 7.8). This transition is reversible, albeit slow with a rate dependent on temperature and humidity of the vapor [49]. The difference Δγ =  − γ_SiOH can be calculated from , if the adsorption isotherm Γ⁽¹⁾(c) is known. As almost all metals do have an oxide layer, this adsorption effect is also present to a smaller or larger extent for most metal substrates. The interaction is not necessarily of a covalent nature but may also be due to adsorption of a component j from the gas phase with partial pressure P_j (the adsorbed component may also be an impurity in the solid state segregating to the interface but this is less likely at room temperature). This again leads to a change in γ, normally denoted as the film pressure (or, confusingly, as the spreading pressure) π and is similarly given by

7.47

The film pressure π = γ_S − γ_SV represents the difference in surface energy of the solid γ_S with respect to vacuum and the surface energy γ_SV in contact with vapor, leading to an adsorbed layer [50]. Often a subscript e is added to π, indicating that the surface at hand should be in equilibrium with the vapor. As we are dealing with equilibrium thermodynamics anyway, we do not adhere to this convention. Finally, we note (i) that the adsorption process as described above for solid surfaces occurs also at liquid surfaces and (ii) that for apolar solids and liquids, the magnitude of π is often considered to be negligible [51], but at occasion is called to the rescue for explanation (see Section 7.3.2).

The basic reason for a surface to have an excess energy is the discontinuity in density at the interface (or, more generally, for an interface a concentration gradient of at least one of the components). There is a balance of repulsive and attractive interactions between molecules in the bulk of the condensed phase, but at the interface the number of nearest neighbors is reduced, so that γ corresponds to the work needed to bring a molecule to the surface (Figure 7.9). A simple nearest‐neighbor broken bond model provides an order‐of‐magnitude estimate. As example, we estimate the surface energy of the cyclohexane–air interface. Cyclohexane has a vaporization enthalpy Δ_vapH = 30.5 kJ mol⁻¹, density ρ = 773 kg m⁻³ and molar mass M = 84.16 g mol⁻¹. Using for simplicity a lattice‐like structure with a cubic representative volume element (RVE) of molecules with six nearest neighbors, we obtain for the energy per bond 30.5/6 = 5.1 kJ mol⁻¹. For the volume of the RVE with lattice constant a, we obtain a³ = M/ρN_A = 1.8 × 10⁻²⁸ m³, so that γ = 26 mJ m⁻², to be compared with the experimental value of γ = 25 mJ m⁻². This good agreement is probably fortuitous, but the order of magnitude is generally correct.

Moreover, liquid surfaces are dynamic. The number of vapor molecules hitting area A at pressure P is given by [52] and in equilibrium the same number of molecules reaches the surface from the liquid side. Consider, for example, the surface of water at 25 °C. Using the vapor pressure of water at room temperature P = 3168 Pa, the molar mass  = 18 g mol⁻¹ ≈ 3 × 10⁻²⁶ kg and the area per molecule A ≈ 10 Ų = 10 × 10⁻²⁰ m², we obtain τ⁻¹ = 1.1 × 10⁷ s⁻¹. Hence, the mean residence time of a water molecule at the surface is approximately 0.1 µs.

For liquids the surface energy is often called the surface tension, which in principle indicates a surface stress. The terms “surface tension,” “surface stress,” and “surface energy” are often used indiscriminately in the literature. Since all thermodynamic work is given by the product of a “generalized displacement,” for surface work the increase in area dA, and a “generalized force,” for surface work called surface stress Ψ (a force per unit length in the surface), we have for a single component, single phase system δW = Ψ dA. For liquid surfaces upon increasing the area the average distance between molecules at the surface does not change, which usually (but confusingly) is called a plastic increase dA_pla of the surface area. For solid surfaces upon increasing the area, the average distance between molecules at the surface does change so that an elastic increase dA_ela of the surface area is possible. Hence, in general the increase in area is given by dA = dA_ela + dA_pla. Since for a single component, single phase system F = Aγ, one can easily derive Ψ = ∂F/∂A = γ + A∂γ/∂A, conventionally addressed as the Shuttleworth equation. The second term in this equation is usually interpreted as the increase in surface stress due to the elastic increase in surface area, that is, the surface elasticity. This equation has been heavily attacked and defended in the last decade or so (for an in‐detail discussion, see, e.g. [53]). A large part of the confusion is due to the use of different reference configurations [54, 55]. Note that in the Gibbs equation, both γ and dA refer to coordinates in the deformed state (Euler coordinates). If we adopt a coordinate system in the undeformed state (Lagrange coordinates), the actual area A of the strained surface is given by A = A_L(1 + ∑_je_jj) with e_ij as the surface strains and A_L as the (invariant) measure of the surface in Lagrange coordinates.⁴ The term ∑_je_jj reduces to dA/A_L for isotropic surfaces. Obviously, the surface energy value is independent of the description used, and we have γ_LA_L = γA and therefore γ_L = γ(1 + dA/A_L). If we stretch the surface, we have dγ_L = ∑_ijg_ijde_ij with the surface stress components g_ij. Equivalently, g_ij = ∂γ_L/∂e_ij, reducing for isotropic surfaces to Ψ = ∂γ_L/∂A. Substituting γ_L, we obtain g_ij = γδ_ij + ∂γ/∂e_ij (where δ_ij is the Kronecker delta), reducing to Ψ = γ + A_L∂γ/∂A for isotropic surfaces. As A/A_L = 1 + ∑_je_jj ≅ 1, we can write as well Ψ = γ + A∂γ/∂A. A detailed discussion is given in [57] and indicates that the partial derivatives have to be taken at constant temperature T, chemical potential μ, and concentration n. While the surface energy is obviously always positive for stability reasons, the surface stress may be positive (tensile) or negative (compressive). For metals the surface stress is tensile with an order of magnitude similar to that of the surface energy, while for covalently bonded materials, it is compressive. For example, for Al {111} u^(σ) ≅ 0.71 J m⁻² and g ≅ 2.32 N m⁻¹, while for Si {111} u^(σ) ≅ 1.83 J m⁻² and g ≅ −0.63 N m⁻¹ [58]. It should be stated that this information is due, as most information is, to calculations for u^(σ), although for metals the change in lattice constant for nanometer‐sized particles confirms the sign and order of magnitude of the surface stress. For polymers there seems to be very limited information available, but one could expect low surface stresses due to relaxation.

For liquids, we regain the usual description δW = γ dA where the surface energy γ (J m⁻²) equals the surface stress Ψ (N m⁻¹). The surface tension is relatively easily measured, for example, with the Wilhelmy plate technique. In this technique, typically a thin roughened Pt plate is partially immersed in the liquid, and γ can be calculated from the force balance between gravity, buoyancy, and surface tension knowing the contact angle or assuming θ = 0°. A variation is the du Noüy ring technique in which a ring is used, thereby minimizing the amount of liquid required (but compromising accuracy). Many other techniques are available (see, e.g. [27, 30]). We mention here only the pendant drop technique where the volume of a drop (or a number of drops) is measured flowing out a capillary tube. Many details on surface tension measurements are provided by Lyklema [59]. For solids it is imperative to distinguish between Ψ and γ, and measuring Ψ (or γ for that matter) is in general a difficult task since different methods lead to different values for Ψ. Solid surfaces are usually not in equilibrium (the relaxation might be kinetically hampered), are usually not really flat (it will have a certain roughness and/or microfissures), and may be chemically and/or structurally inhomogeneous (for example, a (micro‐)phase separated polymer (blend) or a polycrystalline metal). Although the stresses, tensions, and energies discussed have all the same dimension (see Eq. 7.40), they refer to different entities and usually are numerically different. For liquids surface tension and surface Helmholtz energy are numerically equal, as long as relaxation is fast. Moreover, one would expect that surface energy refers to surface internal energy, similarly as energy refers to internal energy, but this is usually not the case. We use the designation surface tension only for liquids and employ for solids surface energy (adding the label Helmholtz only when required) and surface stress, according to the situation at hand.

Since, for a liquid, the surface energy represents also a surface stress, liquid surfaces tend to contract to minimum surface area (actually this is the reason the surface energy for liquids is often called the surface tension). Remember that for the interface between two pure fluid phases at constant temperature:

7.48

and thus

7.49

One can show that generally ∂V^(β)/∂A ≡ (R₁⁻¹ + R₂⁻¹), where R₁ and R₂ are the principal radii of curvature of the interface. Hence, for curved surfaces in equilibrium a pressure difference exists denoted as the Laplace pressure, and given by ΔP = γC ≡ γ(R₁⁻¹ + R₂⁻¹), where C is the (total) curvature. To illustrate this, consider a cylinder and a sphere. For the former, R₁ = R_cylinder and R₂ = ∞ and hence ΔP = γ/R_cylinder, while for the latter R₁ = R₂ = R_sphere and thus ΔP = 2γ/R_sphere. This pressure difference can be considerable: while for an air bubble with radius R = 1 mm in water, the pressure difference is only ΔP = 144 Pa, for a bubble with radius R = 10 nm, ΔP = 144 bar. Curvature is counted positive if the surface is curved toward the liquid (that is, the liquid side is convex and the air side is concave). Hence, for a spherical liquid drop in its vapor C₁ = C₂ = 1/R and thus ΔP > 0, and the pressure inside the drop is larger than in the vapor. For a bubble in a liquid, C₁ = C₂ = −1/R and hence ΔP < 0. In that case the pressure inside the bubble is larger than in the liquid.

7.3.2 Representative Behavior

The surface tension γ of liquids ranges typically from 10 to 50 mJ m⁻², while for water γ ≅ 72 mJ m⁻², or, equivalently, 72 N m⁻¹. Appendix B lists values for γ for a range of liquids. As shown in the previous section, a simple nearest‐neighbor model is capable of estimating the order of magnitude. However, one can do somewhat better by invoking some considerations as given by Israelachvili [4] based on the vdW attraction between molecules. Using Hamaker's theory the surface energy γ can be estimated as

7.50

where H is the Hamaker constant (see Section 7.1.2) and d₀ is a distance parameter related to the approach of the surfaces. As the parameter d₀ is an effective distance between planes that are actually containing atoms, its value is smaller than the distance of closest approach for atoms. As examples, let us calculate γ for He and Teflon. Taking for He, d₀ = 1.6 Å and H = 5.7 × 10⁻²² J leads to γ = H/24πd₀² = 0.29 mJ m⁻², to be compared with the experimental value γ = 0.12 – 0.35 mJ m⁻². For Teflon, one can estimate d₀ = 1.7 Å and H = 3.4 – 6.0 × 10⁻²⁰ J, leading to 16–28 mJ m⁻², while the experimental value is about 20 mJ m⁻². In both cases, agreement is satisfactory in view of the crude model used. In fact, taking a “universal” value of d₀ = 0.165 nm yields γ ≅ H/24π(0.165 nm)² mJ m⁻², providing estimates that appear to be accurate within about 20% (Table 7.4), except for strong hydrogen‐bonding compounds (the last six compounds in Table 7.4).

With increasing temperature the surface tension decreases continuously until at the critical temperature the surface tension vanishes. This temperature dependence is for many liquids remarkably well described by

7.51

where n is a characteristic exponent, T_cri the critical temperature, and γ₀ the (fictive) surface tension at 0 K. A value of n = 11/9 ≅ 1.222 can be rationalized [2, 48]. A recent compilation of data yields for 85 normal liquids n = 1.249 with a sample standard deviation of 0.054 [60].

The values for the surface energy of solids vary widely. We address briefly the surface energy of covalently bonded solids, vdW solids, ionically bonded solids, and metals. But first we note that upon cleaving a solid, the direction of cleaving is important. This means that different surfaces (e.g. the {100} and {110} planes) have different surface energies. Moreover, the surfaces created generally show relaxation (show distortions in the vertical, z‐direction) and often also reconstruction (show also distortions in the lateral, x‐ and y‐directions). These phenomena significantly change the magnitude of the surface energy as estimated from ideal surfaces that are the result of cleaving only. These transitions may be slow and depend on temperature and environment. Finally, surfaces can hydrate, again influencing the surface energy. As an example, we quote in Table 7.5 simulation results on the spinel MgAl₂O₄ [61]. From these results one clearly observes the influence of anisotropy (increasing values for the sequence {100}, {110}, and {111}), the relaxation effect and the hydration effect on the surface energy. Note that a reversed order occurs for the {100} and {111} surfaces after hydration.

Covalent materials have covalent bonds that are strong and directional. They occur, for example, in C (diamond), B₄C, and SiC. Hence, the cohesive energy, the energy required to break all bonds, is high and ranges from 300 to 700 kJ mol⁻¹. Estimates for the surface energy based on nearest‐neighbor interactions generally suffice as a first‐order estimate, resulting in values of the order of a 1–10 J m⁻². For example, for polycrystalline B₄C such an estimate yields ≅12 J m⁻² [62]. For {111} diamond, Haiss [58] estimated 5.4 J m⁻², while a value of 2.8 J m⁻² is obtained for a relaxed surface. In this case surface relaxation is almost entirely limited to the first layer, which contracts by about 30%. For these materials relaxation seems to be of lesser importance and reconstruction hardly of any importance, thereby rendering relatively high surface energies.

vdW solids show the opposite behavior. In these materials the bonding is due only to vdW interactions. Noble gas crystals provide an example, but also in solid polymers the bonding between the molecules in the solid is due to vdW interactions. Cohesive energies typically range from 10 to 40 kJ mol⁻¹. Their bonding energy and thus their surface energies can be estimated from pair potentials, such as the Lennard‐Jones potential, and yield typically values of 20–70 mJ m⁻². For pure polymeric materials, γ is a function of the molar mass M, often described by the Legrand–Gaines relation [63] reading γ = γ₀ − k/M^2/3 with γ₀ representing the surface energy at infinite mass and k a constant, typically a few hundred mJ m⁻² mol^−2/3. This relation can be rationalized by free volume arguments (see Section 9.3.5), although this interpretation is not generally accepted.

For ionic solids the attraction is mainly given by the Coulomb potential, and this again yields a rather high cohesive energy, ranging from 600 to 1500 kJ mol⁻¹. Hence, also rather high surface energies result, typically 0.1–0.5 J m⁻². As for these materials surface relaxation is significant, the surface energies are low as compared to those of covalent solids. For inorganic glasses simple additivity rules for the constituting components yield rather satisfactory results [64]. As a first approximation, this scheme can also be used for polycrystalline materials.

Metals do form a bit of an exemption in this list. Although for metals bonding has to be described quantum mechanically, many attempts have been made to model the properties of metals using pair potentials with the semi‐empirical Morse potential being the favorite. The cohesive energy of metals ranges from 100 to 800 kJ mol⁻¹, somewhat lower than of that of covalent and ionic solids, but their surface energy is still relatively high, typically 1–3 J m⁻². Note that metals generally react easily with oxygen and water, so that almost any metal is covered with an oxide layer. As said before, this not only influences their surface energy but also can result in a surface energy value that depends on the environment.

7.3.3 Wetting of Ideal Surfaces

Turning now to wetting, consider a droplet on an ideal solid surface, that is, a solid surface that is flat and chemically homogeneous and does not react with the liquid. The various contact configurations that experimentally occur, primarily characterized by the contact angle θ, are displayed in Figure 7.10. When 0° < θ < 90°, the surface is (partially) wetted, while for 90° < θ < 180°, the surface is (partially) nonwetted. Complete wetting is given by θ = 0°. This behavior can be derived from an energy balance between the solid (S), liquid (L), and vapor (V) using the principle of virtual displacements [65, 66]. Equilibrium can be obtained only when the atmosphere is saturated with the vapor of the liquid and therefore interfaces are labeled SL, LV, and SV, respectively. If A_XY denotes the area between phase X and phase Y, and the specific work done to create a unit of XY interface is γ_XY, the work to create a droplet on a substrate (Figure 7.11a), neglecting pressure effects, can be written as

7.52

At equilibrium the surface Helmholtz energy is minimized so that we require γ_LV(dA_SL + dA_LV) − γ_SLdA_SL = 0. Using dA_LV/dA_SL = cos θ leads to Young's equation [67] (Figure 7.11), reading

7.53

For γ_SL < γ_SV, we have 0 < θ < 90°, while for γ_SL > γ_SV, we have 90° < θ < 180°. Since cos θ cannot be larger than 1, γ_SV ≥ γ_LV + γ_SL implies total wetting (θ = 0°). Generally, the work of adhesion W_adh between two materials is the work needed to separate one interface into two surfaces. It is the negative of the Gibbs energy change and defined by W^adh = −ΔG_ij^adh = γ₁ + γ₂ − 2γ₁₂, where γ₁ (γ₂) denotes γ_SV for material 1 (2) and γ₁₂ the interface energy. Similarly the work of cohesion of a single substance is defined as W^coh = −ΔG₁^coh = 2γ₁, as γ₁₂ = 0. Combining for a solid–liquid system Young's equation with the expression for W^adh, we obtain the Young–Dupré equationW^adh = γ_SV(1 + cos θ) [68], which indicates that neither γ_SV nor γ_SL can be larger than the sum of the other two surface energies. The consequence of this restriction is the prediction of complete wetting for γ_SV > γ_SL + γ_LV and full dewetting when γ_SL > γ_SV + γ_LV. The lack of a solution to the Young–Dupré equation indicates that there is no equilibrium configuration with a contact angle between 0 and 180°. The difference S = γ_SV − (γ_SL + γ_LV) is generally addressed as the spreading coefficient and can be interpreted as the difference W_SL^adh − W_LL^coh. For spontaneous wetting, S > 0 and for θ > 0°, S < 0. Padday [69] has suggested that the relation S = −ρgh² can be used to directly determine S by measuring the height h of a large drop when the cap has flattened, given the mass density ρ (g represents the acceleration of gravity). The transition from a drop with a spherical cap to a pancake‐like drop has been dealt with in [70].

Young's equation can also be derived from the force equilibrium between the horizontal components of the various surface tensions. A similar equilibrium exists for the vertical components, but since the stiffness of a solid is generally very high, hardly any deformation takes place in the vertical direction. Hence, this deformation is generally only detectable for soft solids (like gels): a drop of water put on a fresh paint, after it evaporates, indeed leaves a circular ridge on the paint. The “ridge effect” at the contact line of the liquid on the substrate surface of extremely low elastic modulus materials has been studied in quite some detail (see, e.g. [71] and for a wider context [72]).

A simple relation for spherical liquid caps (so that the effect of gravity is neglected) reads tan(θ/2) = h/a (Figure 7.11b), valid for θ < 90°. The contact angle θ can be measured not only by using this configuration but also, if recorded, from the volume and either the height or diameter of the drop. In both cases circularity of the drop is assumed. The typical droplet volume used is 3–10 µl. Measuring θ within 2° error is relatively easy, but obtaining an accuracy better than 0.5° is difficult ([73], see also [59]). Measurement of θ is nowadays usually fully instrumented, and θ is calculated from an image taken of the droplet resting on the surface. In many cases, one not only measures the static contact angle, that is, the contact for a fixed volume of liquid, but also the advancing (receding) contact angle θ_A (θ_R), for which the volume of the droplets is increasing (decreasing) during the measurement (Figure 7.12). Often, one actually measures a contact angle right after the drop volume has been increased. Another way to determine θ is by employing the static Wilhelmy (plate) technique. If γ_L is known in advance, the contact angle can be calculated from the exerted force f = Lγ_L cos θ upon a plate with circumference L. A much more accurate way is to use the dynamic Wilhelmy method. In this method a thin plate of the material to be tested is partially immersed in a liquid and either further immersed or withdrawn at a rate of typically 0.01–0.1 mm min⁻¹. From immersion force‐position results, θ_A can be calculated, while from the withdrawal data, θ_R can be obtained. An accuracy of about 0.5° is obtainable (Figure 7.13). Tadmor [74] has suggested, based on pinning arguments for the contact line, that the (thermodynamic) equilibrium contact angle θ_E can be calculated from θ_A and θ_R according to

7.54

It is advisable in wetting research to report the static as well as the advancing and receding contact angles to properly characterize the solid–liquid combination investigated.

We neglected gravity, but for large drops gravity does affect the overall shape of the drop. For gravitation to be negligible, the variation in hydrostatic pressure inside a drop should be negligible compared with the pressure excess due to surface tension. We thus require for a spherical liquid cap with radius R that RΔρg ≪ 2γ/R, where Δρ is the density difference between the liquid and air. Solving for R the inequality becomes R ≪ a with a the relevant length scale of the system, called the capillary constant a = (γ/ρg)^−1/2 (or, alternatively, the capillary length a⁻¹ = (γ/ρg)^1/2). Note that some other authors include a factor 2^1/2 in the expression for a. At room temperature we obtain for water a⁻¹ = 2.7 mm and for hexane a⁻¹ = 1.7 mm, corresponding to drop volumes of 20 and 5.1 µl, respectively, so that in many wetting experiments, the effect of gravity can be neglected. Note, however, that for hexane a drop of 10 µl – an often used value in measuring contact angles – already has a diameter of 2.6 mm, so that the influence of gravity on the overall drop profile probably should be taken into account when drop profile analysis is used to measure θ. A direct measurement of the slope at the contact line should not be affected by gravity though [67]. Extensive reviews of contact angle measurements are given in [75–77], while concise discussions are provided in [11, 30].

7.3.4 Estimating Surface Helmholtz Energy

Measurement of the contact angle with various liquids on a substrate also provides the possibility to determine the surface energy of the substrate. The first and simplest attempt to do so was by Zisman, introducing the concept of critical surface tensionγ_cri (i.e. the critical surface energy) by plotting cos θ versus γ_LV for a range of liquids, extrapolating to cos θ_A = 1 (θ = 0°) and defining γ_cri = γ_LV(cos θ_A = 1) (for a review by Zisman himself, see [78]). When cos θ_A = 1, Young's equation yields for the spreading coefficient S = γ_SV − (γ_SL + γ_LV) = 0, representing a state of equal values for the cohesive interactions in the solid and liquid and the adhesive interaction between solid and liquid. Zisman considered γ_cri mainly as an empirical estimate for γ_SV, right so as γ_SV only equals γ_LV when γ_SL ≅ 0. Moreover, it became clear that a linear extrapolation of cos θ is only allowed under rather restricted circumstances (see below). Table 7.6 provides data for some liquids that are commonly in use, chosen on the basis of (high) stability, (easy) purifiability, (low) viscosity, and (low) volatility, as only in this case a well‐defined liquid surface tension is obtained. This table also shows values for γ_cri so obtained for various solids. Note that a relatively narrow range for γ_cri is obtained.

A number of alternative theories have been developed to assess γ_SV, as will be discussed below. Each of them combines in some way Young's equation with an approximation for γ_SL on the basis of γ_SV and γ_LV. All theories suppose that the surface Helmholtz energy γ can be written as a sum of contributions, that is,

7.55

where the superscript indicates the contributions taken into account (d = dispersion, i = induction, p = polar, A = acid, B = base), and assesses these contributions in its own way. As different probing liquids have different contributions for each of the components, using a range of liquids with known properties provides an estimate for the surface energy of an unknown substrate. The method is sometimes labeled as the surface tension components (STC) approach.

The initial approach to estimate γ_SV along these lines is due to Fowkes [24, 79], who assumed that the contributions between like interactions prevail and can be described by the geometric mean combining rule. Fowkes considered mainly the dispersion (London) contribution and making the approximations ΔΦ_j^vdW ≅ ΔU_j^coh ≅ ΔG_j^coh and using the geometric mean combining rule, he obtained ΔG_ij = (ΔG_i^cohΔG_j^coh)^1/2. This leads to

7.56

Consequently, using Young's equation, the contact angle θ is given by

7.57

abbreviating γ_LV (γ_SV) by γ_L (γ_S). Hence, this shows that γ_cri should be obtained from a plot of cos θ versus γ_L^−1/2 instead of versus γ_L. This clarifies why often a curved Zisman plot is obtained [79] (another explanation suggested is that for the film pressure π, the assumption π = 0 is not warranted [80]).

One more extended line of attack is due to Good, Girifalco, and Elbing [81], who took into account dispersion and polar interactions but explicitly excluded hydrogen bonding effects. Note that polar in this connection means the contributions of polarizable polar molecules and hence means taking into account Keesom and Debye forces. In this scheme one assumes for the interface energy the (geometric mean) combination rules γ_ij^d = (γ_i^dγ_j^d)^1/2 and γ_ij^p = (γ_i^pγ_j^p)^1/2. For the contact angle, one similarly obtains

7.58

A clear discussion on this approach can be found in [82]. Table 7.7 provides the two contributions for a few liquids and the predicted critical surface tension for a few solids. Actually, γ_ij^d initially also contained a so‐called interaction factor φ to account for polar interactions and thus for apolar interactions, φ should be φ ≅ 1.0, reasonably well supported by experiments. It will be clear that one needs at least two liquids with known components γ_L^d and γ_L^p to determine γ_S^d and γ_S^p. Using two liquids, the method has been advocated by Owens and Wendt and Kaeble [83, 84]. Wu [85] suggested using the harmonic mean 1/γ_ij = 1/γ_i + 1/γ_j, which is an allowed alternative for the dispersion contribution, but not for the polar contribution [86]. Writing the two equations in matrix form, matrix inversion yields the solution for the two components required. It is advisable though, to use several probing liquids. Dividing Eq. 7.45 by 2(γ_S^d)^1/2 and plotting γ_L(1 + cos θ)/2(γ_S^d)^1/2 versus (γ_L^p/γ_L^d)^1/2, one should obtain a straight line with slope (γ_S^p)^1/2 and intercept (γ_S^p)^1/2. Using two liquids this procedure provides the same result as the analytical solution, but using more than two liquids allows for an error estimate. Negative slopes indicate an error in the θ‐ or γ‐data. Alternatively, one uses a generalized matrix inversion method [87].

The method outlined above may be difficult or impossible to apply for high energy surfaces where the contact angle cannot be measured as the liquids spread on the substrate. As an alternative Schultz et al. and Melrose [88] have indicated a method, nowadays often referred to as the two‐liquids method, in which a substrate (S) is immersed in one liquid (usually a hydrocarbon (H) like n‐hexane or n‐hexadecane) and another is used as probing liquid (usually water (W)). Using for hydrocarbon γ_SH = γ_S + γ_S − 2(γ_S^dγ_H^d)^1/2 and for water γ_SW = γ_S + γ_W − 2(γ_S^dγ_W^d)^1/2 − γ^p, subtracting the two expressions and using Young's equation one obtains γ_W(1 + cos θ) − γ_H = 2(γ_S^d)^1/2[(γ_W^d)^1/2 − (γ_H^d)^1/2] + γ^p. Here γ^p represents the polar part of the substrate. By plotting γ_W(1 + cos θ) − γ_H versus [(γ_W^d)^1/2 − (γ_H^d)^1/2], one should obtain a straight line with slope 2(γ_S^d)^1/2 and intercept γ^p. Interpreting γ^p as 2(γ_S^pγ_W^p)^1/2, γ_S^p can be easily calculated.

A more recent approach, still following the additive scheme, due to van Oss, Good, and Chaudhury (vOGC), runs as follows [86]. All vdW interactions are taken collectively, that is, including London, Keesom, and Debye contributions, as described by the Lifschitz theory for vdW interactions (LW). The geometric mean combination rule is still used for this contribution, so that we have γ_ij^LW = (γ_i^LWγ_j^LW)^1/2. The short‐range interactions, like hydrogen bonding, are described via AB interactions (using the Lewis definitions of acid = electron acceptor, labeled ⊕, and base = electron donor, labeled ⊖).⁵ In this scheme, a compound having no AB interactions is called apolar. Note that in this connection polar means the presence of a donor–acceptor interaction, that is, a partial charge transfer. A compound can also act either as donor or as acceptor, in which case it is called (acidic or basic) monopolar. Finally, compounds that can act both as donor and acceptor are labeled bipolar. Examples are, respectively, C₁₆H₃₄, CHCl₃, CH₃OCH₃, and H₂O. While hexadecane is apolar, CHCl₃ is acidic monopolar, CH₃OCH₃ is basic monopolar, and H₂O is bipolar. These AB (acceptor–donor) interactions are supposed to take care of all the non‐vdW effects. In essence, acceptors interact only interact with donors and vice versa. Hence, contrary to the vdW interactions, they are asymmetric with respect to the components and the geometric mean rule cannot be used. Instead, one assumes γ^AB = 2(γ^⊕γ^⊖)^1/2 so that we have

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The total interfacial energy is γ_ij = γ_ij^LW + γ_ij^AB and for the contact angle one obtains

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Data for several liquids are given in Table 7.8 [86]. In this case one needs at least three liquids with known components γ_L^LW, γ_L^⊕, and γ_L^⊖ to determine γ_S^LW, γ_S^⊕, and γ_S^⊖. Writing the three equations similarly as for the previous method in matrix form, matrix inversion yields the solution for these three components. Again, it is advisable to use more liquids and employ a generalized matrix inversion to obtain the components required [87].

There are a few issues with the vOGC method. The first is the values of γ_L^LW, γ_L^⊕, and γ_L^⊖ to be used for water. Conventionally, one uses the data as given in Table 7.8, based γ_L^LW = 21.8 mJ m⁻², as given first by Fowkes [79], and divides γ_L^AB (arbitrarily) in equal parts yielding γ_L^⊕ = γ_L^⊖ = 25.5 mJ m⁻², leading to solid surfaces being rather basic. However, such an arbitrary choice has to be made since the scheme inherently contains the freedom to choose one parameter [91]. Della Volpe and coworkers [90] have argued for a set reading γ_L^LW = 26.2 mJ m⁻², γ_L^⊕ = 48.5 mJ m⁻², and γ_L^⊖ = 11.2 mJ m⁻², based on the simultaneous minimization of fitting errors using 10 liquids and 14 solid polymers. This leads to acidity and basicity of solid surfaces much more in line with chemical notion.⁶ Second, objections have been put forward that the scheme is not mathematically consistent in the sense that, if data for solids as calculated from the advised values for liquids are supposed to be given, the values calculated for the liquids are different from the advised data [92]. Moreover, sometimes negative values are obtained, for which the physical reason is unclear, to say the least. Białopiotrowicz [87] has shown, by using simulated error‐free data and assessing the influence of errors by adding increasingly larger errors, that both phenomena are artifacts that should not occur when accurate data are available. Third, we turn to the matter of surface enthalpy versus surface Helmholtz energy. Describing γ_SV as a sum of contributions is in principle wrong, as an addition scheme can be only rationalized for the enthalpy ΔH, but not for the entropy ΔS. For many phenomena, though, there exists an enthalpy–entropy correlation that might come to the rescue (see, e.g. [93]). If ΔS = fΔH, with f a constant for all the processes involved, an addition scheme will still be effective. Indeed ΔS tends to be a fraction of ΔH, although the value of f is different for liquids and solids [94, 95]. A clear discussion of the matter is given in [96, 97].

Quite a different method is the equation of state (EoS) approach, as advocated by Neumann et al. [98]. The authors noticed that the interaction factor φ, as introduced by Good and Girifalco [81], seems to be linearly related to the surface tension of the liquids used, that is, φ = −aγ_SL + b. This led to the notion that γ_SL = f(γ_SV,γ_LV), that is, γ_SL is a function of the surface Helmholtz energies of the solid and liquid only. The original expressions for cos θ leads to an expression with a denominator that might become zero for large values of γ_SV − γ_LV. Accordingly, the expression was changed to [99] φ = exp[−β(γ_LV − γ_SV)²] with β as an empirical parameter of which, unfortunately, the physical meaning is unclear [100], but when the expression was fitted to a large number of systems, leads to β = 1.247 × 10⁻⁴ (m² mJ⁻¹)². The EoS expression for cos θ reads

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It has been shown that there are many cases of disagreement between EoS predictions and experiments [91]. The EoS method has been under heavy attack (see, e.g. [101]), followed by fierce defense [102], but not to a discussion that led to a common view. An extensive overview of the EoS approach is given in [103].

A critical review of the methods described above has been given by [104]. These authors conclude that the CTS approach has its difficulties, that the EoS approach is based on shaky grounds, and that, as might have been expected, in principle the most fruitful way may be to follow the route indicated by Lyklema [94] based on solid thermodynamics. Unfortunately, this route is hampered by the unavailability of data required for many liquids and solids.

Finally, we note that Chibowski [105] proposed an expression to calculate the surface energy from θ_A and θ_R reading

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He assumed that, upon retracting the liquid, a film of the liquid is left. No further assumption about the film is made, and the hysteresis is assumed to be caused by the interactions between the liquid drop and the surface of the solid. However, the calculated values of γ_S depend on the type of the measuring liquid.

7.3.5 Wetting of Real Surfaces real surfaces"?>

So far, we considered ideal surfaces, that is, flat isotropic surfaces with surface energy as only characteristic. However, real surfaces often do have a certain, possibly anisotropic, roughness, characterized by the height and wavelength (distributions) of the surface irregularities. They can also be anisotropic in the crystallographic sense and inhomogeneous in chemical nature and are frequently contaminated. Moreover, a surface may be influenced by the interrogating liquid itself. All these effects influence the wetting behavior of solid surfaces.

Let us indicate the effect of roughness first. Here one can distinguish between two configurations: the Wenzel state and the Cassie–Baxter state (Figure 7.14). For moderately rough surfaces, a droplet is typically in the Wenzel state, for which the liquid is in complete contact with the rough surface below the droplet [106]. Roughness introduces a surplus of surface with respect to a flat surface. Denoting the ratio of the real surface area A_real by that of the projected area A_proj by r = A_real/A_proj implies replacing A_SL in Eq. 7.36 by rA_SL, resulting in

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This expression suggests that if θ < 90°, θ_W < θ, thus roughness enhances wetting, and if θ > 90°, θ_W > θ, and in this case roughness enhances dewetting, as has been observed experimentally.

Another aspect is that a surface may be flat but chemically inhomogeneous, containing areas of, say, two, different “materials” with surface fractions φ₁ and φ₂ = 1 − φ₁, each characterized by their own surface energy γ₁ and γ₂. As long as the typical dimensions of the heterogeneities are much smaller than the typical droplet size, the contact angle is given by cos θ_CB = φ₁cos θ₁+φ₂cos θ₂. If phase 2 is air, cos θ₂ = −1 and the result is cos θ_CB = φ_S cos θ + φ_S − 1, where φ_S is the surface fraction solid. For sufficiently rough surfaces, contact of a liquid with the surface indeed occurs only at the hillocks of the surface, and there are air pockets below the droplet. This is the Cassie–Baxter (or fakir) state [107]. Wherever the liquid is contacting the solid, the roughness of the contact areas r^* is influencing the contact angle in a similar way as for the Wenzel state. Including this effect leads to

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For a Cassie–Baxter state, θ_CB > θ always, also if θ < 90°. A transition from the Cassie–Baxter state to the Wenzel state can occur, depending on roughness, interrogating liquid, environment, and time. A simple criterion for this transition, valid for topographies described by a (average) wavelength λ and (average) amplitude a, states that the transition can take place when a > (λ/2π) tan θ, where θ is the intrinsic contact angle [108]. Although this criterion is designed for a simple (sinusoidal) surface having a single wavelength and without any discontinuous transitions, it displays that the Cassie–Baxter to Wenzel state transition depends on the spacing and amplitude of the roughness involved. However, time is on the Wenzel side and there are more options to assist the transition to the Wenzel state. For example, vibration may lead to protrusions of the liquid resting on the hillocks contacting the “bottom” of the corrugated surface. Once this happens, the transition from the Cassie–Baxter to the Wenzel state occurs quickly. Another mechanism that induces the transition to the Wenzel state is via diffusion of the entrapped gas below the droplet into the droplet. As an example, we quote from a systematic study to assess the effect of (double‐sized) roughness on the contact angles [109]. In this study the roughness factors, which determine the wetting properties of films, were reported, based on monolayers of well‐defined raspberry silica–silica nanoparticles, exhibiting a wide‐ranged and systematic variation of large and small particle sizes and their ratios. The advancing water contact angle reported did not depend on the particles sizes or their ratio, but the contact angle hysteresis (CAH) was strongly dependent on both. Apart from demonstrating clearly the boundaries of the existence of a Cassie–Baxter state as dependent on various size ratios, this study also supported the view that air below the droplet diffuses into the droplet in time, thereby inducing the transition from the Cassie–Baxter state to the Wenzel state.

Chemically inhomogeneous surfaces frequently arise due to the occurrence of (more) hydrophobic and (more) hydrophilic areas. In the case of an advancing drop, water is held back by the hydrophobic areas, leading to a higher contact angle as compared to the equilibrium contact angle θ_E. Reversely, for a receding drop, water is held back by the hydrophilic regions, leading to a lower contact angle (Figure 7.15). Thus, it appears that θ_A is generally more sensitive to the low surface energy species, while θ_R is more dependent on the high surface energy moieties present [110]. The contact of the solid surface with the interrogating liquid itself may gradually change the average composition at the surface, so that time‐dependent results are obtained. This is the basis for the dynamic recovery contact angle (DRCA) method to estimate rearrangements effects at the surface [111].

Anisotropy in roughness, anisotropy in microstructured surfaces, or preferential molecular orientation (stretched polymer or single crystal facet) as well as crystallographic anisotropy can lead to asymmetric sessile drops. An example of the influence of anisotropic machining roughness on the contact angle is given in [112] (Figure 7.16). In fact, anisotropic surface wetting can be induced by using tunable micro‐wrinkling of PDMS films [113]. A brief review on various experimental and theoretical approaches to the design, synthesis, and characterization of engineered surfaces demonstrating anisotropic wetting properties can be found in [114].

A small amount of contamination significantly influences the contact angle, as the liquid typically senses only the outermost layer of the surface. For example, a hydrophobic contaminant (oil, grease, etc.) may render a hydrophilic surface (metal oxide) largely hydrophobic. Hence, contact angle measurements are employed as an easy method to assess the cleanliness of surfaces. The matter is extensively treated in the book series Developments in Surface Contamination and Cleaning [115].

Finally, the temperature and treatment used may influence the surface. For example, in poly(fluoroalkyl acrylate)s the fluorocarbon side chains orient depending on the treatment given [116]. Using for the side chains CH₂F=CHCOO(CH₂)₈(CF₂)F as monomer, a poorly wettable surface is created in air at about 85 °C, a temperature nearly equal to the melting point (Figure 7.15a(2)). When in contact with water, the surface changes at the same temperature to a good wettable surface (Figure 7.15a(1)) with low receding angle (but, obviously, still a high advancing angle). The behavior is shown to be reversible for over 100 times. The change in wettability is caused by the orientation of the fluorocarbon chains that are normal to the surface in the low wettability state (Figure 7.15b(1)) and tilt to a small angle in the high wetting state (Figure 7.15b(2)).

Also the contact with the interrogating liquid itself may change the behavior [117]. For example, in polyurethane (PU) films in contact with water, the urethane groups reorient resulting in a large CAH. It is well known that poly(dimethyl siloxane)s (PDMS) have a water contact angle of typically about 110° and a CAH of less than 10°, due to heavy segregation at the surface. Therefore, by adding PDMS to a PU film, one might expect a decrease in hysteresis. By modifying the surface using mono‐ and bifunctional PDMS, this appeared to be true, but only if sufficiently long PDMS chains are used. Both the monofunctional PDMS₆₀-OH and the bifunctional PDMS₇₀-(OH)₂ are effective, the more so when cured at 40 °C instead of the more usual 80 or 120 °C. For a PU film containing PDMS₆₀-OH with in total 0.02–0.03 wt% Si, the advancing contact angle θ_A increased from about 80° to 105°, while the receding contact angle θ_R increased from about 50° to 85°. The CAH is thus close to 20°, to be compared with the original CAH of approximately 30°. X‐ray photoelectron spectroscopy indicated that for 0.03 wt% Si the surface concentration of Si within the first 5 nm is already about 18 at.%. Adding the relatively short PDMS₂₀-OH is ineffective in shielding the PU surface from water.

A similar affect was observed for fluorinated PU with silica particles incorporated [118]. For the original film with 50 wt% 0.9 µm silica particles, the contact angles θ_A and θ_R were 150° and 82°, respectively. Adding PDMS led to a CAH of about 45°. This type of work eventually enables to develop superhydrophobic films, as discussed in Chapter 12.

A highly readable overview on wetting phenomena is [119], while shorter general introductions are given in [65].

7.5.1 Basic Types of Emulsions

An emulsion is a dispersion of fluid particles (droplets) in a liquid matrix. One generally distinguishes between oil‐in‐water (O/W) and water‐in‐oil (W/O) emulsions (Figure 7.18). The liquids are evidently immiscible, but a surfactant, typically a molecule with a hydrophilic head and a hydrophobic tail, can provide kinetic stability of the emulsion by lowering the interfacial tension and stabilizing the interface by steric repulsion. Depending on the value of the interfacial tension γ between the two liquids, one can have normal or macro‐, mini‐ and microemulsions. In case of a finite value for γ, coarsening (larger emulsion droplets) is thermodynamically favorable since this lowers their Gibbs energy according to dG = γdA.

Macro‐emulsions, with a droplet size of typically ≈0.3–100 µm, are thermodynamically unstable as they have a γ value of 1–20 mN m⁻¹ and ΔG > 0. However, in practice they may be kinetically stable. Upon creating a very low interfacial tension (by using a surfactant and cosurfactant, typically 1–4%), the droplets may be more easily dispersed and broken up in very small particles with diameters, say, in the range 50–1000 nm, more typically 50–100 nm, leading to mini‐emulsions with a γ range of 0.1–1 mN m⁻¹. With a very high concentration of surfactant and cosurfactant (>10%), the droplets become so small (particle radius ≈ 10–30 nm) that the entropy due to thermal motion Δ_theS ∼ nkT, with n the number of droplets, becomes important. In that case ΔG = Δ_emuG − TΔ_theS lowers even further than that due to lowering of interfacial tension only. As a consequence, the resulting micro‐emulsions with γ ≅ 1 µN m⁻¹ are thermodynamically stable with ΔG < 0. They provide an interesting option for size‐controlled polymerization within a droplet. Microemulsions are reviewed in [121] (although, unfortunately, called nanoemulsions). The stability of emulsions is thus kinetically controlled for macro‐ and mini‐emulsions, either via electrostatic or via steric stabilization mechanisms, while micro‐emulsions are thermodynamically controlled.

Coagulation, that is, the process in which droplets stick together, is determined largely by the potential between two droplets. If it is sufficiently repulsive, the dispersion will be stable. Coagulation occurs because the barrier between the secondary minimum and primary minimum in the potential curve is sufficiently low so that within the available time a transition from secondary to primary minimum can take place (Figure 7.17a). The coagulation is called fast or slow, depending on whether the barrier is less than kT or much higher than kT. In addition, for dispersions the coagulation is termed reversible or irreversible depending on whether the depth of the primary minimum is comparable with kT or much greater than kT. Reversible coagulation is also called flocculation and can often be removed by stirring. Coalescence of liquid droplets, that is, merging to a larger droplet, is controlled by the (overall) attractive interaction and the details of the draining process of the liquid between the droplets. Deformation of liquid droplets, occurring more easily for low interface tension and high shear, is hindering coalescence as the draining becomes more difficult. Moreover, the disturbance of the surfactant distribution leads to Marangoni flow (see Section 10.1). Furthermore, since there is always some solubility of one component in the other, generally Ostwald ripening occurs (via diffusion), resulting in the growth of the larger droplets at the expense of the smaller droplets. Nevertheless, for having colloidal stability in a dispersion, the initial barrier should be sufficiently high. The coagulation process is rather complex, and for a brief review we refer to [122].

Finally, we note that the variety of surfactants used is large and various families can be distinguished. Table 7.9 shows a few of these types together with specific examples [123].

Series	Chemical nature	Designation	Example
Igepon	Fatty acid amide of methyltaurine	TN, R = Palmityl
Aerosol	Alkylester of sulfosuccinic acid	OT, R = Octyl
Span	Fatty acid esters of anhydrosorbitols	60, R = Stearyl
Tween	Fatty acid ester and ethylene oxide esters of anhydrosorbitols	21, n = 4, R = Lauryl
Triton	Ethylene oxide ethers of alkyl benzene	X‐45, n = 5, R = Octyl
Hyamine	Alkylbenzene dimethyl ammonium salts	3500, R = C12-C16

7.5.2 The HLB Concept

In the past the selection of a surfactant, in this context usually addressed as emulsifier, was usually done on empirical grounds. Nowadays, the empiricism is captured by using the balance between the hydrophilic and lipophilic nature of the emulsifier, as quantified in the hydrophilic–lipophilic balance (HLB) value, and the selection of the emulsifier is conventionally based on this concept.

The HLB value is an empirical number between 0 and 20, where 0 represents the most lipophilic type and 20 the most hydrophilic type of emulsifier. The first one to design such a scale was Griffin [124], and he based his scale on C_m(EO)_n surfactants, copolymers of alkanes (C, the lipophilic groups or lipophiles), and ethylene oxides (EO, the hydrophilic groups or hydrophiles). The HLB value was given by ξ = W_EO/5 where W_EO is the mass percentage of hydrophiles in the C_m(EO)_n compound. Other surfactants were given an HLB value, as based on the comparability in properties with C_m(EO)_n. However, generalization appeared to be not so easy for a wide range of surfactants.

Later, Davies [125] created a scale for predicting the HLB values based on group contributions. The basic idea is that more hydrophiles (H) increase the HLB value, while more lipophiles (L) decrease it. Davies analyzed a range of experimental HLB values and correlated them to molecular functional groups of the surfactants, using the same contribution for all lipophilic groups. Table 7.11 provides the contribution to the HLB value for various functional chemical groups according to Davies. The expression for the HLB value ξ he obtained reads

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so that for a particular combination of lipophilic and hydrophilic groups, the surfactant is characterized as more hydrophilic for ξ > 7 and more lipophilic for ξ < 7. Here n_j and m_j represent, respectively, the number of hydrophilic and lipophilic groups of each type j in the surfactant. The value 7 is used to define an emulsifier that favors neither a W/O nor an O/W emulsion, in line with Bancroft's rule stating that the phase in which the emulsifier is more soluble tends to be the dispersion medium. Note that the contribution of an ethylene oxide group (EO, -(CH₂)₂-O-) is hydrophilic (1.3 – 2 × 0.475 = 0.35), while that of a propylene oxide group (PO, -(CH₂)₃-O-) is lipophilic (1.3 − 3 × 0.475 = −0.13). These values do not hold for very large molecules, for example, consisting of 200 EO groups, since the different parts of the same molecule become virtually independent. In practice, there is an upper limit to the HLB value of about 20 for high molar mass ethylene oxide derivatives, which in fact determined the maximum value on the scale.

There is an optimal HLB value of surfactants for different tasks, depending on the oil at hand, as each oil has its own optimal surfactant HLB value. The typical range for antifoaming agents is ξ = 2–3, for W/O emulsifiers ξ = 3.5–6, for wetting agents ξ = 7–9, for O/W emulsifiers ξ = 8–18, and for detergents ξ = 13–15. In the case of emulsification, the underlying logic takes into account the variability in hydrophobicity of the oils and the fact that the oil itself may contain a surfactant. Typical HLB values involved for various types of oils are ξ = 6 for vegetable oils, ξ = 8–12 for silicone oils, ξ = 10 for petroleum oils, ξ = 12 for typical ester emollients, and ξ = 14–15 for fatty acids and alcohols. The optimal procedure for finding the best surfactant system is to start with a nonionic surfactant. Generally, a mixture of nonionic surfactants performs better than a single nonionic surfactant because this leads to a lower γ, a particular useful feature for mini‐emulsions. For such a mixture one takes for the HLB value a weighted average of the various components. One starts to test performance with a series of surfactants with an HLB value varying between, say, 1 < ξ < 19. Subsequently, one tests the performance with much smaller range, so as to obtain the optimal HLB value. For an oil mixture, one takes the HLB value as the weighted average of the oil components, similarly as for surfactants. Finally, one checks whether surfactant mixtures with optimal HLB value perform better, if useful combined with ionic surfactants. Generally, long carbon chain surfactants perform the best.

Let us, as an example, calculate the HLB value calculation for two surfactants. The first is sodium oleate (CH₃(CH₂)₇CH=CH(CH₂)₇COONa) and the second is a commercial, often used surfactant called Span 60 (Table 7.9). Table 7.10 shows details of the calculation. From the values obtained, we conclude that sodium oleate (ξ ≅ 18.5) can act as a solubilizer and is a fair surfactant for O/W emulsions, while Span 60 (ξ ≅ 5.2) just acts as an emulsifier.

A significant improvement to the HLB concept was provided by Guo et al. [126]. These authors noticed that the Davies set of surfactants did not cover well many employed types, and, in particular, for surfactants with EO chain as the sole hydrophilic moiety, Davies' HLB values differ greatly from the accepted data. Contributions of frequently occurring chemical groups in the surfactant, such as the -CH₂-, EO, and PO groups, on the HLB value were overestimated, and these effects were corrected by using the effective chain length (ECL), as introduced earlier by Lin [127]. Moreover, the terminal -OH groups deviate from side -OH groups, while the sorbitan ring is considered as a special group. Finally, Guo et al. excluded F‐containing surfactants. The ECL values according to Guo et al. are, respectively,

where m_X indicates the number of X‐groups. These effective m_j values are to be used in Eq. 7.67 and their m‐dependencies are plotted in Figure 7.19. Their table of group contributions (Table 7.11), based on 224 nonionic surfactants, predicts rather reasonable HLB numbers with an average absolute deviation from accepted values [128] of less than 1.5. Hence, from a practical point of view, when using the HLB concept for nonionic surfactants, the advice is to use the data by Guo et al.

Finally, for completeness, we mention the use of the effective (or equivalent) alkane carbon number (EACN) in the empirical description of emulsions of complex oil mixtures such as crude oils. For linear alkane chains, this number just represents the number of carbon atoms, but for a complex mixture it characterizes the behavior of the mixtures by comparing it with the behavior of a linear chain alkane. Its value appears to be closely related to the HLB value [129], and the concept is capable of characterizing the stability of these complex mixtures rather satisfactorily [130]. A predictive model based on thermodynamics for use in enhanced oil recovery is available [131].

7.6.1 Unhindered Coagulation

Von Smoluchowski was the first to quantify the kinetics involved in coagulation [132]. The starting point of his calculation was the assumption that particles coagulate upon encounter. In this process diffusion by Brownian motion is the rate‐determining step, which we call diffusion limited coagulation (DLC). Here we summarize the careful discussion as given by Overbeek [21]. Considering a steady‐state situation, we have for the flux J of particles with concentration (particle number density) n whose centers pass through a sphere of radius r surrounding the central particle, as described by Fick's first law:

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with (effective) diffusion coefficient D′. Using n = n₀ for the bulk concentration at r = ∞, integration leads to

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We assume that an approaching particle coagulates with the central particle if it touches the surface of that central particle with radius R. It thus disappears from the dispersion as a separate particle, making n = 0 at r = R. Therefore

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As the central particle also experiences Brownian motion, the effective diffusion coefficient reads D′ = D_ij = D_i + D_j with D_i and D_j as the diffusion coefficients of particles i and j, respectively, or D′ = 2D if all initial particles have the same size. We also use R_ij = R_i + R_j. Hence, the initial rate of coagulation between particles of the same size is [133]

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where the factor is introduced to avoid double counting of collisions. Similarly, for collisions between particles i and j of different sizes, we have

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These equations apply only for the initial stage of the aggregation process. We define a particle created by aggregation of i equal‐sized particles as an i‐fold particle. If we denote the number density of particles at a certain stage by n₁ and that of i‐fold particles by n_i, the number of k‐fold particles increases by collisions of i‐fold and j‐fold particles where k = i + j and decreases by collisions with other k‐fold particles. Using this balance one can show that the total concentration N changes according to

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Here we used the approximation D_ijR_ij = DR{4 + [(a_i/a_j)^1/2 − (a_j/a_i)^1/2]²} ≅ 4DR, where a_j refers to the radius of a j‐fold particle. As the initial density is n₀, integration leads to

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where τ = 1/4πDRn₀ is the coagulation time in which the number density halves. This leads to

7.75

describing the evolution of the particle size distribution. Using the Stokes–Einstein relation between the diffusion coefficient D and viscosity η, given by D = kT/6πηa with a the particle radius, and assuming R = 2a because a particle disappears when it touches the central particle, we have τ = 3η/4kTn₀. For water at 25 °C, this leads to τ ≅ 2 × 10⁵/n₀ m³ s and with, say, n₀ = 10⁸ m⁻³, we have τ ≅ 2 × 10⁻³ s, and therefore, on the basis of this calculation, fast coagulation is expected. Of course, the steady‐state approximation is used, but one can show that generally the steady‐state is reached rather quickly, in fact for t ≫ R²/πD, or, equivalently using R = 2a, t ≫ 8φτ with φ the volume fraction. Von Schmoluchowski introduced a factor h, representing the probability that an encounter leads to coagulation, to accommodate for the, typically large, difference in rate as observed and as calculated. This factor was attributed to the presence of a barrier to coagulation and therefore h = 1 for rapid kinetics and h ≤ 1 for observed slow kinetics. Much later Davies [125] discussed semiquantitatively the reasons why the experimental rate is often so much smaller than the calculated, rapid rate, and we deal with his arguments in Section 7.6.2.

Assuming that coalescence occurs directly after coagulation, the total volume of droplets is conserved, and one obtains from the von Schmoluchowski model at any time the scaling relation:

7.76

The behavior of the droplet radius a(t) as calculated by the von Schmoluchowski theory for various initial radii a₀ is shown in Figure 7.20.

7.6.2 Hindered Coagulation

From the von Smoluchowski theory,we obtain for the rapid (or unhindered) rates

7.77

where φ_W and φ_O denote the volume fraction of water and oil in the W/O and O/W emulsions, respectively. Here the number density n₀ is replaced by the volume fraction φ as the values for the mass densities of water and oils are similar. These rates can be (considerably) reduced due to various reasons. For O/W emulsions double layer repulsion between charged oil droplets and nonionic components (contributing to hydrate barriers) play a role. For W/O emulsions the CH₂ groups of the emulsifier are forced into the water phase (breaking the water structure). Nevertheless, it is beneficial to use von Schmoluchowski theory to assess the influence of surfactants on predicted coagulation rates, as the type of emulsion one obtains after dispersing, either an O/W or a W/O emulsion, depends on the coagulation rates k of both types. Although emulsions are thus not absolutely stable, a large coagulation rate implies practical instability, while a small rate implies practical stability. Practical limits to the terms large and small are 10⁻²k (coalescence occurring within an hour) and 10⁻⁵k (emulsion stable for several months). A review on the stability of colloidal dispersions and emulsions is given by Overbeek in [134] as well as in [38].

Limiting ourselves to initial coagulation, the flux in the presence of a potential barrier Φ is given by [135]

7.78

where Stokes' law gives the friction coefficient f = 6πηa. As before, D′ = 2D (as the reference particle is nonstationary), but at the same time the relative displacement of two particles with potential Φ is also doubled, so that the effective friction coefficient becomes f/2. Meanwhile using the Stokes–Einstein equationfD = kT, the flux becomes

7.79

with as boundary conditions n = 0 at r = 2a and n = n₀ at r = ∞. The solution is

7.80

Comparing with J_fast = 4πn₀D₁₁R₁₁ = 4πn₀ × 2D × 2a = 16πn₀Da as obtained from Eq. 7.71, we see that the rate is slowed down by the factor W. The evaluation of the integral usually has to be done numerically, but, as the contributions to the integral mainly arise from the neighborhood of the maximum in the potential Φ_max, to a first approximation, we have W = (2κa)⁻¹exp(Φ_max/kT). Depending on the ratio Φ_max/kT, the factor W can thus be quite large. Carrying the analysis to later stages yields a complex theory for which we refer to the monograph by Russel et al. [39].

In the sequel therefore, a highly approximate approach to estimate the reduction in coagulation rate is given in which we assume that a barrier exists for coagulation. Then the reduction in rate can be described using a Boltzmann factor due to the presence of such a barrier, as indicated above. In practice droplets are stable if the exponent in this Boltzmann exponential expression leads to a reduction of approximately 10⁻⁶. We deal with O/W and W/O emulsions separately.

For an O/W emulsion the double layer repulsion is important. If we denote the electrical potential at the maximum of the barrier by ψ, according to double layer theory, in many cases the repulsion is proportional to ψ², that is,

7.81

The reduction in coagulation rate thus follows:

7.82

where B is an empirical constant for which Davies [125] rationalized the value B ≅ 0.24 when ψ is given in mV. Repulsive interactions are also mediated by nonionic components, such as water, that tightly bind to the interface. This water must be displaced before coalesce can occur. The contribution ΣE_H of this displacement to the total energy barrier depends on the number of and type of hydrated groups on each surfactant molecule and the fraction θ of the interface covered. This leads to

7.83

For example, a single -OH group will bind with water molecules with ε_HB ≅ 4 kcal mol⁻¹. For two of such groups in near contact the energy required is thus about 2ε_HB−ε_W–W ≈ ε_HB.

For W/O emulsions the rate is not influenced by double layer repulsion but occurs via repulsion stabilization of surfactants with hydrophobic, generally -CH₂- containing, groups, and the reduction of the coagulation rate is thus controlled by the number of -CH₂- groups m in the nonionic molecule. Hence, the result is

7.84

where the factor two is due to that both droplets have a surfactant layer. For a single CH₂ group, the energy difference in the oil and water phase is estimated as [125] w = E_oil(-CH₂-) − E_water(-CH₂-) ≈ 1.3 kJ mol⁻¹.

Now consider a mixture of oil and water which, after agitation, initially will contain O/W and W/O fractions (Figure 7.21). The final preference for either an O/W or W/O emulsion is determined by the ratio of k_O/W and k_W/O. The question which emulsion type is the most stable, either an O/W or a W/O emulsion, should thus be considered by taking the ratio of the retardation factors for coagulation. For simplicity, let us focus on the barrier effects and write

7.85

Hence, for τ_O/W ≫ (≪) τ_W/O, an O/W (W/O) emulsion is expected. The first two terms on the RHS represent the effect of hydrophilic groups in the molecule, while the last term represents the effect of lipophilic groups in the molecule. Equivalently, we may write

7.86

As all L_j values in the HLB expression, Eq. 7.67, are equal to L ≅ 0.475 (see Table 7.10), we have

7.87

Comparing these expressions shows that, as Davies put it, the HLB system rests on a firm kinetic basis. This is somewhat of an overstatement since the hydrophilic group number for a charged group is Bψ²/wθ, which obviously is not really constant. Further, using Table 7.11 we have for an -OH group H_OH ≅ 1.9, corresponding with 2ε_HB − ε_W–W ≅ 2.4 kcal mol⁻¹, less than the estimated value of 2ε_HB − ε_W–W ≅ ε_HB ≅ 4.0 kcal mol⁻¹, indicating that probably more hydrogen bond energy is gained as estimated.

To conclude, we remark that although the theory using the HLB concept is crude, it provides some practical understanding. The analysis of the energy barrier that slows down the coagulation helps to rationalize the HLB value approach. For unknown surfactants, one can use as a first estimate the available tables using group contributions to assess the influence of surfactants on the stability of emulsions.

7.7.1 SCF Computations

Theoretical studies on polymer assembly comprise a range of models including relatively simple geometric models, discrete or continuum mean‐field calculations, and molecular modeling employing both atomistic and coarse‐grained simulations. Here we focus on the numerical solution of self‐consistent field (SCF) equations, which is a powerful technique applicable to the assembly and adsorption of polymers and polymer‐mediated interactions [138]. In this model, the excluded volume interaction is incorporated by using a potential field resulting from the configurations of all the other compounds. The potential field is in turn a function of the segment distribution, which must be determined self‐consistently. While the SCF method neglects fluctuations of the potential, it includes fluctuations in the single chain. Equilibrium is guaranteed, and the method is fast and is therefore suitable for complex systems.

While the method is not limited to a lattice [139], the derivation of the equations is more transparent on a lattice. The type of lattice used depends on the goal of the simulation (Figure 7.22). The polymer chain is made up of repeat units of length b, the lattice spacing. The goal is to compute the polymer chain distribution function (s,r,r′), which describes the probability that a polymer chain of segment length s starting at r′ is at position r. Often, the system has some sort of symmetry that reduces the problem to a one‐dimensional one (Figure 7.22). The distribution function can then be written in terms of z and z′, and (s,z,z′) is determined by the self‐consistent potential V(z) generated from all possible polymer configurations that are consistent with some set of boundary conditions. For example, for a polymer end‐grafted to a surface, the boundary condition imposed is that one end is attached to the surface, which is otherwise impenetrable. The set of polymer configurations is made finite by imposing a spatial lattice and discretizing the path of a chain from r(s) to a sequence of lattice positions labeled by {s_i}. Since it is assumed that the interaction energies (po1ymer–polymer, polymer–solvent, and polymer–surface) are evaluated in a mean‐field approximation, they depend only on the average density ρ(z) of the polymer and the solvent volume fraction in each layer. One can then write an equation for ψ(s,z,z′) inductively as [140]

7.88

That is, to arrive at z in s steps, a chain must have been at one of the adjoining layers on the previous step. The factors 1/6 and 2/3 are for a simple cubic lattice. If one imposes appropriate boundary conditions on ψ(1,z), then one can iteratively solve Eq. 7.88 for V(z) = f[ψ(s,z)] and ψ(s,z) = f[V(z)]. For chains end‐grafted to a planar wall, V(0) = ∞ and ψ(1,z) = δ_z,1. The monomer density ρ(z) for a given (s,z,z′) is the normalized weight for a Gaussian chain to travel from its origin z′ = 0 through the point z in s steps to some other endpoint z′, which it must reach in N−s steps. Scheutjens and Fleer [141] were the first to exploit the numerical solution of the SCF equations on a lattice for polymers in solution using Eq. 7.85, comprising essentially Flory–Huggins theory in a gradient field. The short‐range interactions are linked to the Flory–Huggins parameters, while electrostatic effects are taken into account via Gouy–Chapman theory. The system is considered to be incompressible and complex (mixtures of) molecules can be handled. It is possible to generate as many as N = 10⁵ conformations. Their method has been widely applied to study adsorption of polymers, block copolymers, and polyelectrolytes as well as polymers end‐grafted on a flat or curved surface or adsorbed on surfaces and several other phenomena such as micellization [142], diblock copolymer micellization, and depletion phenomena [143] (for further references, see [138]). Figure 7.23 provides a block scheme on how this theory is implemented in practice.

The SCF method enables one to quantify thermodynamic properties of simple molecules, surfactants, (block)(co)polymers, polyelectrolytes, and dendrimers. It can be used at interfaces, both solid–liquid and liquid–liquid, and has been used for the micellization of amphiphilic molecules, complexation, and/or encapsulation. However, only mean‐field equilibrium properties can be computed.

It should be noted that SF–SCF theory is a lattice‐based theory but that lattices are just a means to calculate fluid properties only [2]. Although the mean‐field approximation has its limitations, SCF theory can be remarkably accurate, the difference in computing time as compared with off‐lattice simulations can be 4–5 orders of magnitude, and the SCF method can at least be used to generate trends. This also explains the success of the method.

7.7.2 An Example: Surfactant Modeling

Surfactant modeling using SF–SCF theory has been used for surfactant self‐assembly in the bulk, predicting the critical micelle concentration (CMC), aggregation number, structure, and geometry of the micelles. Also surfactants at liquid–liquid interfaces predicting adsorption isotherms, adsorption profiles, and interfacial tension have been dealt with. Surfactant adsorption on solid–liquid interfaces, describing adsorption isotherms and adsorption profiles, has been addressed. Finally, surfactants in confinement, calculating the Gibbs energy of interaction, wetting by using surfactants to describe the interfacial tension, and colloidal interactions mediated by polymers have been studied.

As an example, we deal here briefly with the concentration profiles of two well‐known rather hydrophilic surfactants, Tween 20 and Tween 80 (Figure 7.24), near an interface (z = 0) between H₂O and C₆H₆. The concentration profiles near an interface (z = 0) and the associated interfacial tensions that follow from SCF computations are shown in Figure 7.25. These profiles do show a rather asymmetric shape with respect to the interface toward the H₂O side due to the relatively large hydrophilic head group as compared with the lipophilic tail.

Span 20, with a much lower HLB value of 8.6 (Figure 7.26), shows a much more narrow concentration peak at the interface, due to the more equal size of its hydrophilic and hydrophobic parts. The interfacial tensions for all three surfactants is shown in Figure 7.27 and shows the relatively large difference in rate of change of γ with surfactant concentration φ as well as in level for these three surfactants. As discussed in Section 7.3.1, the adsorption isotherm can be calculated from these profiles, and it will be clear that for similar HLB values (Tween 20 and 80) still rather different adsorption behavior is observed.

7.7.3 Another Example: Fluorine Segregation

Fluorine‐containing molecules typically accumulate at a polymer melt–vapor interface. Such low energy surfaces prevent subsequent adhesion of most, if not all, materials. A strategy to administer fluorinated groups at the interface is to chemically link sufficient fluorinated units to the polymer that forms the melt.

SCF theory was used to analyze the properties of partially fluorinated poly(methyl methacrylate) chains [144] in the vicinity of the polymer–vapor interface using an “united atom” description in which the methyl ester and perfluoroalkyl esters are linked onto a C–C backbone, whereas the vapor was modeled as free volume. Replacing -OCH₃ groups by -OCH₂C₆F₁₃ groups was used to vary the chain composition/architecture. A small fraction of fluorinated groups led to a relatively large drop of the surface tension, in agreement with experimental data. The surface composition was dominated by the large affinity of fluorine toward the polymer–vapor interface. A single fluorine enriched layer at the liquid–vapor interface, followed by a depletion zone, was found with all the fluorinated groups at one end of the chain (Figure 7.28a). When the five fluorinated segments are grouped together in the middle of the polymer chain (Figure 7.28b), oscillations arise due to the preference of the fluorinated moieties for the surface. The density profile of the surface ordering suggests that the system is close to a microphase separation transition. Microphase separation in the bulk can be observed when the fluorinated block is placed at the end of the polymer chain (Figure 7.28c). Here a completely ordered bulk with lamellae parallel to the surface is visible. Dual gradient SCF calculations [145] showed that the free surface does not remain featureless and lateral gradients in fluorine density are accompanied by height undulations of the free surface. Not only do lateral changes in surface composition exist, but also depressions in the free interface are generated with a depth comparable to or smaller than the size of the segments of the fluorinated segments. These predictions are consistent with some earlier AFM investigations on these fluorinated films.

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Further Reading

1. Adamson, A.W. and Gast, A.P. (1997). Physical Chemistry of Surfaces, 6e. New York: Wiley.
2. Barton, A.F.H. (1991). Handbook of Solubility Parameters. Boca Raton, FL: CRC Press.
3. Butt, H.‐J., Graf, K., and Kappl, M. (2006). Physics and Chemistry of Interfaces, 2e. Weinheim: Wiley‐VCH.
4. Butt, H.‐J. and Kappl, M. (2010). Surface and Interfacial Forces. Weinheim: Wiley‐VCH.
5. Fennell Evans, D. and Wennerström, H. (1994). The Colloidal Domain. Weinheim: Wiley‐VCH.
6. de Gennes, P.‐G., Brochard‐Wyart, F., and Quéré, D. (2004). Capillarity and Wetting Phenomena. New York: Springer.
7. Hansen, C.M. (2007). Hansen Solubility Parameters: a User's Handbook, 2e. Boca Raton, London: CRC Press.
8. Hiemenz, P.C. and Rajagopalan, R. (1997). Principles of Colloid and Surface Chemistry, 3e. Basel: Dekker.
9. Hunter, R.J. (1987). Foundations of Colloid Science, 1e. Oxford: Oxford University Press, (2001), 2e.
10. Israelachvili, J.N. (2006). Intermolecular and Surface Forces, reprinted 2e. Amsterdam: Academic Press.
11. Kruyt, H.R. ed. (1952, 1949). Colloid Science, vol. I and II. Amsterdam: Elsevier.
12. Lyklema, J. (1991–2005). Fundamentals of Interface and Colloid Science, vol. I–V. Amsterdam: Elsevier.
13. Russel, W.R., Saville, D.A., and Schomaker, W.R. (1989). Colloidal Dispersions. Cambridge: Cambridge University Press.
14. Verweij, E.J. and Overbeek, J.Th.G. (1948). The Stability of Lyophobic Colloids. New York: Elsevier.