1 Wetting of Surfaces and Interfaces: a Conceptual ... · Fig. 1.2 The interrelationships between...

Jarl B. Rosenholm

Abstract

Owing to the focus on molecular engineering of intelligent materials, growing in-terest has been focused on the specific interactions occurring at molecular dis-tances from a surface. Advanced experimental techniques have been developed in-cluding instruments able to measure directly interactions at nanometer distancesand to identify special structural features with comparable perpendicular and lat-eral resolution. With this new information at hand, the theories of molecular in-teractions have been re-evaluated and developed further to encompass specific in-teractions such as Lewis and Brønsted acidity and basicity. However, the new the-ories are based on critical approximations, making the upscaling to macroscopiccondensed systems uncertain. Therefore, the aim of this study was to evaluatesome recent models by introducing macroscopic work functions (cohesion, adhe-sion, spreading and immersion) of wetting of solid surfaces within the proper con-ceptual thermodynamic (macroscopic) framework. The properties of binary andternary systems are discussed with the focus on four non-ideal inorganic (SiO2

and TiO2) model substrates. The results obtained after applying the recent andmore traditional models for dispersive and specific (polar) interactions are com-pared with those utilizing simplifying assumptions. The sources of uncertaintiesare sought, e.g. from the contribution of surface pressure determined from con-tact angle and adsorption isotherms. Finally, the influence of chemical and struc-tural heterogeneities and also external stimuli on wetting is briefly discussed.

1.1Introduction

Owing to the focus on molecular engineering of intelligent materials, growinginterest has been focused on the specific interactions occurring at molecular dis-tances from a surface. Advanced experimental techniques have been developedincluding instruments able to measure directly interactions at nanometer dis-

1

Colloids and Interface Science Series, Vol. 2Colloid Stability: The Role of Surface Forces, Part II. Edited by Tharwat F. TadrosCopyright © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 978-3-527-31503-1

1Wetting of Surfaces and Interfaces:a Conceptual Equilibrium Thermodynamic Approach

tances and identify special structural features with the same perpendicular andlateral resolution. Consequently, the theories developed for the dispersive inter-action of hydrocarbons has been re-evaluated and developed further to encom-pass specific interactions such as both Lewis and Brønsted acidity and basicity.However, the new theories are based on molecular properties for which the up-scaling to macroscopic condensed systems includes a number of critical approx-imations.

This chapter takes the opposite, conceptual approach. The basic thermody-namic functions are chosen as the basic framework by introducing macroscopicwork functions (cohesion, adhesion, spreading and immersion) of wetting ofsolid surfaces. The properties and wetting processes characterizing binary andternary systems are discussed with reference to recent molecular models andmore traditional models for dispersive and specific (polar) interactions includingsurface pressure and adsorption approaches. The Lewis and Brønsted acid–baseinteraction is carefully separated, since they diverge both in strength and dis-tance. This fact is frequently disregarded. Care is taken to use systematic andclear indexing.

A number of new simplifying experimental procedures to utilize the modelsdeveloped are suggested. Again, the different frameworks that the models repre-sent are kept apart and later compared mutually with key properties. In orderto facilitate the comparison of the data presented, the analysis is focused onfour non-ideal solid samples of silica and titania at equilibrium. Finally, the in-fluence of chemical and structural heterogeneities and also external stimuli onwetting is placed within the same thermodynamic conceptual framework.

1.2Thermodynamic Reference Parameters

The state of a system is defined by its internal energy (U or E), which equalsthe sum of heat (Q) and work (W). For a spontaneous reaction to occur U is ex-pected to be negative. Using the conventions for heat and work we may writethe differential equation

dU � dQ � dW �1�

It should be noted that U is a state function and is thus dependent solely onthe initial and final state (dU��U), whereas the work (dW) is dependent onthe path between these two states. IUPAC recommends that both are positivewhen there is an increase in energy of the system. For reversible processes theheat exchange is customarily exchanged for the entropic work (dQ = TdS). Thechange of work is defined as the product of two conjugative properties, thechange being given by the extensive property multiplied by its intensive conju-gated pair (Table 1.1) [1]. All the work functions are exchangeable and must beconsidered when defining the internal energy.

1 Wetting of Surfaces and Interfaces: a Conceptual Equilibrium Thermodynamic Approach2

Selecting the most typical set of parameters for a system containing twohomogeneous phases (� and �) separated by one flat interface, we obtain for thedifferentials of the internal energy (U) of each phase

dU � TdS� PdV � �dA� �dn �2�

As shown, the change is directed only to the extensive properties, while the in-tensive variable are kept constant (dW = –PdV+�dA+�dn). The differentials ofthe intensive state variables have been omitted as stated specifically by theGibbs-Duhem relationship:

SdT � VdP � Ad�� nd� � 0 �3 a�

The Helmholtz energy (F or A), the enthalpy (H) and the Gibbs free energy (G)are all related to the internal energy:

dF � d�U � TS� � �SdT � PdV � �dA� �dn �3 b�

dH � d�U � PV� � TdS� VdP � �dA� �dn �3 c�

dG � d�H � TS� � �SdT � VdP � �dA� �dn �3 d�

Each of them considers different dependences on the working state variables, T,P, V and S. An extended evaluation of these relationships is presented else-where [1].

If more than one component (liquid mixture) or if more than one surface (crys-tal facets) is accounted for, a summation over these variables must be consideredseparately. Since non-relaxed solid surfaces are included in the considerations, it isadvisable to distinguish specifically the relaxed surface tension of the liquid com-ponents denoted �/mN m–1 (force) from the (strained) surface energies of the sol-ids denoted �/mJ m–2 (energy). Assuming that a pure liquid (L) is placed in con-tact with a smooth and homogeneous solid surface (S), maintaining the tempera-ture and pressure and composition constant, we may, in the absence of other workfunctions, derive the Young equation [2] in the following way (Fig. 1.1):

1.2 Thermodynamic Reference Parameters 3

Table 1.1 Typical internally exchangeable conjugated extensive and intensivework variables [1].

Process Extensive variable Intensive variable

Thermal Entropy TemperatureP–V Volume PressureChemical Amount Chemical potentialSurface Surface area Surface tension (energy)Electric Amount of charge Electric potentialMagnetic Magnetization Magnetic field strength

dGSL � �i�idAi � �SVdA� �LV�dA cos�SL� � �SLdA �4 a�

where the subscript i represents V= vapor, L = liquid and S = solid. At equilib-rium:

dGSL�dA � �GSL�A � 0 � �SV � �LV cos�SL � �SL �4 b�

Hence under these circumstances (but only then) the work function equals theGibbs free energy per unit area with an opposite sign.

The change in free surface energy may also be expressed by the Dupré equa-tion [3] for the work of adhesion:

dGSL�dA � �GSL�A � �GsA � �WA � �SL � �LV � �SV �5�

The contact angle can be determined graphically, geometrically by assumingthat the drop is represented by a hemisphere and by deriving it from the expres-sion for the Laplace pressure [4]. Note that the pinning of the three-phase con-tact line (tpcl) may be represented by the frictional surface tension vector direct-ed perpendicular to the surface.

Combining the Young and Dupré equations, the work done at the interfacemay be defined as four key wetting (work) functions (omitting the differentialsign):

Cohesion: WC � CLL � 2�LV or CSS � 2�SV �6 a�

Adhesion: WA �WSL � �SV � �LV � �SL � �LV�cos�SL � 1� �6 b�

Spreading: WS � SSL � �SV � �LV � �SL �WSL � CLL � �LV�cos�SL � 1� �6 c�

Immersion: WI � ISL � �SV � �SL � �LV cos�SL �6 d�


Fig. 1.1 The contact angle of a sessile drop on an ideally smooth andhomogeneous surface is defined by the vectorial stress laid upon thethree-phase (solid–liquid–vapor) contact line (tpcl, Young equation).The transverse component (–�LVsin�SL) may be considered torepresent the frictional pinning of the tpcl.

However, as shown, the work functions are defined for the separation of the in-terconnected phases (work done by the system), while the Gibbs free energy foradhesion is usually defined as uniting the surfaces. Hence for a spontaneousprocess they have opposite signs!

If the processes occurring at the sharp solid–liquid interface (S) alone areconsidered, then the Gibbs dividing plane may be applied, being characterizedby a zero volume and a zero surface excess of the liquid. For a single surface,when the two homogeneous phases have been subtracted (Vs = 0) (� and � ), wefind [5]

dUs � dU � dU� � dU� � TdSs � �dA� �dns �7 a�

dFs � d�Us � TSs� � �SsdT � �dA� �dns �7 b�

dHs � d�Us � �A� � TdSs � Ad�� dns �7 c�

dGs � d�Hs � TSs� � �SsdT � Ad�� dns �7 d�

1.2 Thermodynamic Reference Parameters 5

Fig. 1.2 The interrelationships between the first- and second-order partialderivatives according to the working state variables V, T, P and S and thefree energies F and G on the one hand and U and H on the other forprocesses at the Gibbs surface.

Note that if the IUPAC recommendation for the surface enthalpy is followed,then neither the Gibbs free energy nor the enthalpy can be used to derive theYoung equation. If the PV work is excluded (Vs = 0), then it would appear to bemore appropriate to equate the internal energy and the enthalpy in order tomaintain the symmetry of the equations. However, if both the absolute valueand the difference in the surface pressure is taken to be opposite to that of thesurface tension (energy), then PdV terms and the VdP of the bulk systems cor-respond to the dA and the Ad terms of the surface system, respectively [6].

It should be particularly pointed out that a conceptual analysis of the hierar-chy of thermodynamic parameters (the Thermodynamic Family Tree [1]) bothfor bulk systems and for interfaces [6] reveals that the traditional “work” stateparameters (T–S and P–V) are only sensitive to dramatic changes, providing in-formation on the order of the phase transitions (Fig. 1.2). On the other hand,the next class of partial derivative parameters (heat capacities CP, CV and com-pressibilities KT, KS) and their cross-derivatives (their ratio �, cubic expansioncoefficient � and pressure coefficient �) are sensitive to higher order interac-tions, such as hydrogen bonding or Lewis acidity and basicity. In particular, theyprovide information on the extension of the interaction such as the cooperativityof molecular association (cf. lambda transitions). This has not been fully under-stood by those relating particular molecular properties to the macroscopic ther-modynamic network (Fig. 1.2).

1.3Wetting in Idealized Binary Systems

Viewed from the point of view of thermodynamics, the models for dispersive(hydrocarbon) interactions are usually based on the van der Waals gas law. Thevan Laar model for hydrocarbon liquids considers the components to be mixedin the ideal gaseous state and the non-ideality is averaged geometrically. Theseconsiderations form a base for the modeling of the dispersive interaction pa-rameters.

1.3.1Models for Dispersive Solid–Liquid Interactions

Traditionally, medium to long chain length (C6–C16) saturated hydrocarbon liq-uids have been utilized as standards for the fully dispersive (London) interac-tions with solids. Thus, the standard method of Zisman [7] relies on a range ofcontact angles measured for such hydrocarbons on a solid. When plotted ascos�L against the surface tension of the liquids, the extrapolation to cos�SL = 1for experimental points falling on a straight line gives the critical surface ten-sion (not surface energy), which is considered to correspond to the surface en-ergy of the solid (�SV��crit� lim�LV). In Fig. 1.3, cos�SL is plotted as a func-tion of the surface tension of seven probe liquids on silica [8].


It is seen that octane represents the critical liquid (�crit) for the hydrophobicsilica, but its surface tension is too small for the determination of the surfacetension of hydrophilic silica. Hexadecane may be considered to represent thecritical liquid for the hydrophilic silica. On the other hand, water seems to havea too high surface tension in order to comply with the trend of the other probeliquids on hydrophobic silica. It thus appears that there exist a frame within thesurface tensions and the surface energies are sufficiently close in order to pro-vide physically relevant information. Outside this range the results extractedmay be seriously distorted.

Instead of using the total surface tension, Fowkes subtracted a dispersive partand evaluated the excess as the surface pressure of a vapor film [9]. Later, Zettle-moyer, for example, identified this fraction as a polar (�LV

p =�LV –�LVd ) component

[10]. The polar interaction should be understood as a specific (molecularly arrest-ing) interaction without any specific nature (e.g. dipolar) in mind. This procedureenabled experimentalists also to use a broader range of probe liquids. Twostraightforward alternatives for averaging the work of adhesion have been pro-posed based on the dispersive component of the surface tension/energy [9, 10]:

WSL � 2��dSV�

dLV�1�2 geometric average �8 a�

WSL � �dSV � �d

LV arithmetic average �8 b�

1.3 Wetting in Idealized Binary Systems 7

Fig. 1.3 Zisman plot for silanized hydrophobic silica (spheres) and neathydrophilic silica (triangles). The probe liquids are in increasing order ofthe surface tension; octane, hexa-decane, �-bromonaphthalene, ethyleneglycol, diiodomethane, formamide and water. The contact angles have beendetermined in air using the Laplace approach (data partially from [8, 42]).

Combining these models with the previous definition of the work of adhesiongives the following expressions for the interfacial tension assuming that�LV��LV

d and �SV��SVd :

�SL � �SV � �LV � 2��dSV�

dLV�1�2 � ��d

SV ��d

LV�2 �9 a�

�SL � �SV � �LV � ��dSV � �d

LV� � 0 �9 b�

Owing to the square of the term in parentheses, the geometrically averagedinteraction is always attractive, but for the critical surface tension (�S

d � �Ld �

�crit) the interfacial surface tension vanishes. For the arithmetic mean the inter-facial tension is irrespectively zero. This is inherent to the definition.

In the Scatchard-Hildebrand and the Regular Solution models the solutioninteraction is documented as the solubility parameter (). For the relationshipwith the surface tension (��kV

1/3y) Hildebrand [11] suggested k= 0.0376 andy = 2.326, and Beerbower [12] arrived at k= 0.0714 and y= 2.000. The interfacialenergy in the last case thus be related to the solubility parameters as

�SL � �kSS � kLL�2 �10�

where ki =��

k

= 0,2763.Girifalco and Good [13] offered a geometric mean procedure involving the to-

tal surface tensions/energies corrected by a ratio (�). For London-van der Waalsinteractions it is represented by the ratio of arithmetic averaging to geometricaveraging of the work of adhesion. For non-specific interactions this ratio isclose to unity even when the substances are appreciably different, thus claimingequal validity for the geometric and arithmetic averaging procedures. Wu pre-sented a harmonic mean model for the dispersive and polar interactions equal-ing the ratio when the ionization potentials are replaced by the surface tension/energy components [14]. Neumann and Sell suggested an equation of statemodel with an exponential dependence on the surface tension/energy compo-nents [15]. These methods have found only limited use and are not dealt withhere.

Fowkes related the Hamaker constant to the GG ratio. For a solid–liquid con-tact he arrived at the following relationship [9]:

ASL � A2SV � 2�

ASV

ALV �A2LV �11 a�

Only when � is not too far from unity (or when the ionization potentials arenot too different), the equation reduces to

ASL � �ASV �ALV�2 �11 b�

in symmetry with geometric averaging of the dispersive interfacial energy (Eq.9 a). The van der Waals gas law constants have also been related to the molecu-


lar Hamaker constant and the surface tension for liquids and surface energiesfor solids [16–18]:

ALL � kA�LVd20 �12 a�

ASS � kA�SVd20 �12 b�

where kA�75.40 for van der Waals liquids, 100.5 for pure hydrocarbons and10.47 for semi-polar liquids. The equilibrium distance between the molecules inthe condensed medium (d0) has been reported to be 0.22 ± 0.05 nm [16] and0.13 < d0 < 1.7 ± 0.01 nm [17, 18]. It is therefore obvious that the specific (polar)interactions should be considered as an excess from the ideal gaseous state.

Owing to the direct relationship between the surface tension/energy and thework of cohesion (C) on the one hand and the Hamaker constant (A) on theother for fully or nearly dispersive substances, we can relate the work of cohe-sion and the Hamaker constant to the contact angle:

CSS � 0�25CLL�cos�SL � 1�2 �13 a�

ASS � 0�25ALL�cos�SL � 1�2 �13 b�

Conversely, the latter equation offers the possibility of estimating cos �L fromtabulated data.

Fowkes [9] developed a method for evaluating the work of adhesion from geo-metric averaging (G) of the dispersive components of the surface tension or sur-face energy. Combining the Young-Dupré and Fowkes geometric average modelsfor the work of adhesion gives the Young-Dupré-Fowkes (YDF) equation (�LV =�OV��OV

d ):

WSO�G� � �OV�cos�SO � 1� � 2��dSV�

dOV�1�2 �14 a�

�dSV�G� � 0�25��OV��cos�SO � 1�2 �14 b�

Rewritten in terms of cos �SO, we obtain

cos�SL � �1� 2��dSV�1�2��d

LV�1�2��LV� � �1� 2��dSV��d

LV��1�2 �14 c�

If cos �SL is plotted against [1/(�OVd )1/2] for fully dispersive liquids or (�LV

d )1/2/�LV

for polar liquids, a straight line should be obtained with slope 2(�SVd )1/2. For geo-

metric averaging the line should moreover cross the ordinate at –1. The latterrequirement is a crucial intrinsic standard to ensure that polar interactions donot seriously distort the slope providing the dispersive component of the solid.

In Fig. 1.4 it is shown that the expectation is fulfilled only for the silanizedhydrophobic silica sample [8]. The more polar the surfaces are the smaller isthe slope. For hydrophilic titania both octane and hexadecane have too small


surface tensions for the determination of but �-bromonaphthalene has a match-ing surface tension. As compared with Fig. 1.3 the requirement of matchingsurface tensions with surface energies become again evident. For some time thedeviation from –1 was used as a measure for the polarity of the solid surface oras an indication of the presence of a vapour film on the surface [9]. Thus water(and EG) seems to be unsuitable for the characterisation of hydrophobic silica.

Combining the Young-Dupré and the arithmetic average (A) equations for afully dispersive system, we obtain the Young-Dupré-Zettlemoyer (YDZ) equation(�LV = �OV��OV

d ) [10]:

WSO�A� � �OV�cos�SO � 1� � ��dSV � �d

OV� �15 a�

�dSV�A� � �OV cos�SO �15 b�

Obviously, the dispersive surface energy [�SVd (A)] equals the work of immersion

when �SO�0, since �SLd �0. The arithmetic and geometric surface energies

and the GG ratio for our model substrates (cf. Fig. 1.4) determined with theprobe liquids are collected in Table 1.2 [8].

The influence of non-matching liquids on the determination of surface ener-gies is obvious. With octane and in most cases hexadecane the surface energy isindependent of the properties of the solids, equalling the surface tension of the


Fig. 1.4 YDF-plot of cos �SL of the probe liquids (see Fig. 1.3) plottedagainst the reduced surface tension of hydrophobic silica (spheres)and hydrophilic titania (squares). The frames encompass the probeliquids having matching surface tensions with the surface energiesof the solids [8, 42].

(completely wetting) liquid. Only ABN and DIM provide a surface energy whichreflects the expectations. On the other hand all probe liquids give a comparable(low) surface energy of the hydrophobic silica. Zettlemoyer [10] compared thearithmetic mean with the geometric mean for a more polar metallic mercurysurface and found that the former method was much more sensitive to showsalient polarities of unsaturated and halogenated hydrocarbons unavailable withthe geometric averaging (Table 1.3).

Obviously, if one component is (nearly) zero, the geometric average vanisheswhereas the arithmetic remains significant. This insensitivity of the geometricmean to respond to specific interactions has also been widely noted using theRegular Solution model [19].


Table 1.2 Arithmetic (A) and geometric (G) averaging of the dispersive sur-face energy �d

SV and the ratio � for four solid model surfaces with octane,hexadecane, �-bromonaphthalene and diiodomethane [8].

�dSV (A) (mJ m–2) �d

SV (G) (mJ m–2) � (�)

Octane

SiO2–hydrophobic 22.2 22.2 1.00SiO2–hydrophilic 22.2 22.2 1.00TiO2–hydrophobic 22.2 22.2 1.00TiO2–hydrophilic 22.2 22.2 1.00

Hexadecane


�-Bromonaphthalene


Diiodomethane


The work of immersion offers a sensitive mean to determine the dispersivecomponent of the surface tension/energy. If we relate the Gibbs free surface en-ergy from the Gibbs-Helmholtz equation with the surface tension/energy, wemay write

�GsSi � �Si � �Hs

Si � T ��GsSi��T �P � �Hs

Si � T ��Si��T �P�n �16�

where i = V, L. Since �GI =�GSL/A= –ISL =�SL–�SV, the equation takes the form

�HsI � �Gs

I � T�d�GsI�dT� �17 a�

�HsI � �SL � T ��SL��T �P�n � �SV � T ��SV��T �P�n �17 b�

�HsI � �SL � �SV � T ��SL � �SV��T �P�n �17 c�

where �HIs =�HSL

s /A= (HSLs –HSV

s )/A. Substituting Young’s equation, we obtain

�HsI � ��LV cos�SL � T ��LV cos�SL��T �P�n �18 a�

�HsI � ��LV cos�SL � T cos�SL��LV��P�n � T�LV��cos�SL��T�P�n �18 b�

�HsI � ISL � T cos�SL��LV��T �P�n � T�LV��cos�SL��T�P�n �18 c�

It is therefore possible to relate the enthalpy of immersion to the temperaturedependence of the surface tension of the test liquid and the change of its con-tact angle with the solid. In Fig. 1.5, the immersion enthalpy is calculated usingEq. (18b) and compared with those determined using calorimetry [20].

However, as discussed in the Introduction, the relationships introduced corre-spond to the two phases in equilibrium with the interface, which is different fromthe process occurring solely at the interface. For hydrocarbons on hydrophobicsurfaces (implied for all equations), the temperature dependence of the surfacetension is usually small [21] {(–�SLV

s )d = [�(�LVd )/�T]P,n�–0.1 mJ m–2 K–1}, which

simplifies the calculation.We may use the geometric averaging of the interfacial energy to express the

Gibbs free energy of immersion:


Table 1.3 Interfacial mercury–organic interfacial tension (mN m–1) calculatedusing arithmetic and geometric averaging [10].

Liquid Geometric average Arithmetic average

Hexane 0 0Benzene 0 13Toluene 0 17Bromobenzene 0 251,2-Dibromomethane 0 29Aniline 0 34

�GsI � �GSL�A � �ISL � �SL � �SV � �LV � 2��d

SV�dLV�1�2 �19 a�

�HsI � �LV � 2��d

SV�dLV�1�2 � T��LV��T �P�n � 2

�d

SV��d

LV��T �P�n� 2�d

LV��d

SV��T �P�n �19 b�

The expression is again dramatically simplified if the arithmetic mean is ap-plied for �LV��LV

d :

�GsI � �GSL�A � �ISL � �SL � �SV � �LV � ��d

SV � �dLV� �20 a�

��GsI�d � ��d

SV �20 b�

�HsI � ��d

SV � T ��dSV��T�P�n� �20 c�

For a dispersive probe liquid (O) on low-energy surfaces, we find at room tem-perature that [10]

(�HIS)d � –�SV

d + 298(–0.07) � –�SVd + 21 (21)

Note that Eq. (21) states that the temperature dependence (entropy contribution)is constant when considering the dispersive surface energy of the solid.

Van Oss et al. [17, 18] introduced an extended scale for the non-specific inter-action by choosing halogenated hydrocarbons as probe (oil, O) liquids. The mostpopular of them, diiodomethane (DIM; �OV

LW��OV = 51 mN m–1) and �-bromo-naphthalene (ABN; �OV

LW��OV = 44 mN m–1) have a cohesive energy (COO = 2�OVLW/

mN m–1) which is about double that of dispersive probe hydrocarbon liquids(18 <�OV

d < 26 mN m–1) [22]. This reduces the contribution of the specific interac-tion to only a fraction of its “polar” value and distorts the scaling published pre-viously in the literature. The designation Lifshitz-van der Waals (LW) compo-nent emphasizes the fact that the contribution is considered to include thesemi-polar (Debye and Keesom) interactions of the slightly acidic halocarbons.As discussed, these are largely erased in geometric averaging, but remain ob-servable using arithmetic averaging [10].


Fig. 1.5 Immersion enthalpy for Teflonin various alkanes is determinedcalorimetrically and calculated usingEq. (18b) (from [20], with permission).

ISL

mJ�m2

However, as discussed in relation to Figs. 1.3 and 1.4 the high surface tensionof ABN and DIM are superior for high energy surfaces, while O and HD are ap-plicable to low energy surfaces. In the past most model studies were made onpolymers with a rather low surface energy. In conclusion, we may define twodifferent alternative scales for the surface tension/energy:

�LV � �dLV � �

pLV and �SV � �d

SV � �pSV �22 a�

�LV � �LWLV � �AB

LV and �SV � �LWSV � �AB

SV �22 b�

It should be pointed out that maintaining the traditional designation “polar” fornon-dispersive interactions does not mean that they are solely dipolar in origin,but non-specific in a more general chemical sense.

1.3.2Contribution from the Surface Pressure of (Gaseous) Molecules and Spreadingof Liquid Films

The ideal solid surface model assumes a smooth homogeneous surface struc-ture, resembling that of a liquid. The model is suitable for the definition of thewetting of (mainly) hydrophobic surfaces. In reality, the surface of semipolar orpolar surfaces is always covered by an adsorbed, condensed layer of vapor whichreduces the surface energy considerably [9, 10, 23–25]. As shown in Fig. 1.6 thecontribution of the surface pressure is quite dramatic on polar surfaces, butsmall on hydrophobic surfaces. We therefore consider the influence of an ad-sorbed vapor layer in equilibrium with its own liquid (drop). The surface pres-sures are defined as

L�L� � �LV � �L�L� � �oL � �L�L� � 0 �23 a�

S�L� � �SV � �S�L� �� oS � �S�L� �23 b�

The contribution of an adsorbed vapor layer to the surface tension of the liquidsmay be considerable, but has been found to be negligible for our model systems[8]. In the previous equations the vapor (V) denotes the adventitious adsorptionof ever-existing vapor from the environment. Hence in practice the work of ad-hesion represents the displacement of V by L. This vapor dramatically lowersthe surface energies. The Dupré equation takes the form {note the designationWS(L)L = W[S(L)]L, i.e. preadsorbed (L)}:

WS�L�L � �S�L� � �L�L� � �SL � �SV � S�L� � �LV � �SL �24 a�

WS�L�L �WSL � S�L� �� S�L� �WSL �WS�L�L �24 b�


We now consider the Young equation for these two limiting states:

�S�L� � �SL � �LV cos�S�L�L � �SV � S�L� � �SL �25 a�

�SV � �SL � �LV cos�SL �25 b�

S�L� � �LV cos�SL � cos�S�L�L� �25 c�

since �LV��L(L). Obviously, the surface pressure of polar liquid vapors on thesolid surface can be determined from the change in contact angle on adsorptionof the liquid vapor replacing the air. As shown in Fig. 1.6, this contribution canbe substantial for polar surfaces [8].

A more consistent value for the surface pressure may be determined fromthe adsorption isotherm of liquid vapors on evacuated (powder) samples. Forthe surface pressure of the monomolecular film we may write [10]

0S�L� � �0

S � �S�L� � RT�

�L�mon�

�L�0

�Ldln�PL�P�L� �26 a�

S�L� � �RT�MLAw� ��

�L�mon�

�L�0

wLdln�PL�P�L� �26 b�


Fig. 1.6 The cos �SL measured in air (filled symbols) and in saturatedprobe liquid vapour (open symbols) on hydrophobic (inversted triangles)and hydrophilic (squares) titania, plotted as a function of the surfacetension of the probe liquids [8, 42].

where S(L)0 and �S

0 (0 = reference state vapor free surface in vacuum),�L = surface excess (�wL/MLAw), PL

� = (partial) pressure (� = at standard pres-sure = 1 bar), Aw = specific surface area of the sample, wL = weight ratio of adsor-bate to substrate and ML = molar mass of adsorbent. Note that for monolayercoverage Pmon corresponds to �L,mon. It should be noted that this equation is re-stricted to the monolayer adsorption and it should be revised for multilayer andinfinitely thick films equaling the work of spreading. For an infinitely thick (du-plex) film, we find [10]

S0SL � �0

S � �L�L� � �SL � RT�

�

�L�0

�Ldln�PL�P�L� �27�

These two states are illustrated in Fig. 1.7.The liquid close to the tpcl can therefore microscopically be divided into three

distinct regions: the bulk liquid, the unstable transition region and the vapor film.The thickness of the liquid collar has been related to the work of spreading [26]:

h � �3�LVd20�2SSL�1�2 � �3ALL�2kASSL�1�2 �28�

where d0 again denotes the mean distance between the molecules, 0.18< d0

< 0.26 nm, in the substance. The development of a contact line tension (�10–11

–10–10 J m–1) contributes to the macroscopic (�SL –�SV) surface energy balanceonly for very small droplets, i.e. the line tension is of the order /�SL (�10–11

J m–1/10–2 J m–2�1 nm). However, line tensions up to 10–5 J m–1 have been re-ported [27], corresponding to experimentally verified liquid collars in the milli-meter range [29]. The contribution of the line tension can be introduced as a cor-rected contact angle over the Young contact angle:

cos��SL � cos�SL � � ��LV� � cos�SL � � �rc�LV� �29�

where �LV is the surface tension of the liquid and � is the curvature of the con-tact line [27] or the local radius of curvature rc of the contact line [28]. A lineardependence is thus expected of cos��SL upon the local curvature of the contactline. In kinetic considerations, a hydrodynamic model is usually applied for theunstable phase condensed liquid halo whereas a molecular kinetic model is ap-plied on the gaseous vapor film [30].

The difference between these two integrals is represented by the shaded area(van der Waals loops) [10]. Subtraction gives the work of spreading on a vapor-covered surface:

SS�L�L � S0S�L� � 0

S�L� � �S�L� � �L�L� � �SL � RT�

�

�L�mon�

�Ldln�PL�P�L� �30�


The amount adsorbed can also be evaluated from a simple experiment. Thesample plate is hooked on to a balance and placed in an almost closed samplecell above, but not in contact with the liquid. Then the change in weight ismonitored as a function of time. The surface of the sample plate is known(A = Aw*wS, specific surface area times the weight), as is the molecular surfacearea of water (�L = 3VL/rLNA = A/Nm

S ). The number of molecules at the saturatedsurface is Nm

S = nSmA=A/�NA and the surface excess for monomolecular coverage

is �L(mon) = Nms /ANA = nS

m/A. Multiplying by RT, we may therefore calculate thesurface pressure of the liquid vapor, S(L) = �SV –�S(L), from gravimetric data. Al-ternatively, the amount adsorbed on a flat surface can be determined utilizing,e.g., ellipsometry [29].

Introducing the work of cohesion, CL(L)L = 2�L(L)�CLL, we find for the work ofspreading of the bulk liquid:

SS�L�L �WS�L�L � CL�L�L � �S�L� � �L�L� � �SL �31 a�

SS�L�L � �SV � S�L� � �LV � �SL �31 b�

SS�L�L � SSL � S�L� �31 c�

These equations apply since the original reference state (�S0) is subtracted from

the equation. Following the formalism introduced, the work of adhesion of thewetting liquid in equilibrium with its own vapor WS(L)L may thus be subdividedinto the work of adhesion (WSL) in the absence of an adsorbed film and the sur-face pressure of the vapor film on the solid [S(L)]:

WS�L�L � �L�L�cos�S�L�L � 1� � �LV�cos�SL � 1� � S�L� �32 a�

SS�L�L � �L�L�cos�S�L�L � 1� � �AV�cos�SL � 1� � S�L� �32 b�


Fig. 1.7 The three-phase-contact line can in reality be subdivided intothree regions, bulk liquid, condensation range and gaseous molecular film.Plotted as the surface excess against the relative pressure, the phasecondensation is represented by the van der Waals loops. In most casesonly the adsorption (bold line) is observed.

At the critical point of complete wetting (cos�SL = 1), the work of adhesionequals the difference between the work of cohesion of the liquid and the surfacepressure of the film [WS(L)L = CLL –S(L)]. Then, the negative surface pressureequals the work of spreading: SS(L)L = –S(L).

Table 1.4 collects the work of immersion, adhesion and spreading when thereference state was a vapor-free surface in vacuum. The contribution from thesurface pressure of adsorbed vapor films is, depending on the system, consider-able (unequal probe liquids) or predominant (most polar surfaces). It is there-fore not surprising that wetting vapors and liquids are used to lower the surfaceenergy when machining, drilling or grinding polar solids. Note that W0

SL�I0

SL+�LV and S0SL� I0

SL–�LV (cf. Eq. 6 b–d).Since the work of spreading from the monomolecular film to a condensed

surface is zero [SS(L)L = 0, Eq. (30); �SL = 0], obviously the surface pressure equalsSSL

0 = S(L)0 .

The surface pressure of the probe liquids used as examples in Fig. 1.6 weredetermined from the contact angle difference in air and under saturated vapor.The results of these calculations are collected in Table 1.5.

For TiO2 (anatase) the following saturation surface pressures/work of adhe-sion have been found [25]: 46/86 (n-heptane), 108/154 (n-propyl alcohol) and196/340 (water). Although the surface pressure of the monomolecular film(Eq. 26) is only a fraction of this saturation value (Eq. 27), the values reportedin Table 1.4 are almost negligible. ABN and DIM produce, however, a signifi-cant surface pressure on hydrophilic TiO2. As shown, the vapor surface pressureof the polar liquids is considerably greater on all sample surfaces. The surfacepressure on hydrophilic TiO2 is greatest, whereas the other varies in an irra-tional way from positive to negative values. The latter are again due to�S(L)L <�SL. The influence of adventitious vapors competing for the adsorptionsites is obvious, i.e. S(L)� S(L)

0 .Combining the Young and Dupré equations with Fowkes’ geometric averaging

gives the Young-Dupré-Fowkes (YDF) equation including the contribution fromthe surface pressure:

WS�L�L � �L�L�cos�S�L�L � 1� � 2��dSV�

dLV�1�2 � S�L� �33 a�

cos�S�L�L � �1� 2��dS�

LL�1�2 � �S�L��L�L� �33 b�

Hence the deviation of the slope of the line through the experimental pointsplotted as a function of 1/�L(L) (Fig. 1.8) may be interpreted in part as a contri-bution from the surface film pressure. As an intrinsic consistency test the lineshould pass through –1 at 1/�L(L) = 0.

In the absence of a surface pressure the equation equals the YDF equation(14 c).

Note that the contact angle measured in air and in saturated probe liquid va-por differ considerably in particular for hydrophilic surfaces. This observation



Table 1.4 Work of immersion, adhesion and spreading (mJ/m2) for silicaand titania-liquid vapor pairs determined from vacuum [24].

Solid Vapor I0SL W0

SL S0SL

SiO2 Water 316 388 244SiO2 n-Propanol 134 158 110SiO2 Benzene 81 110 52SiO2 n-Heptane 59 79 38TiO2 Water 300 370 228TiO2 n-Propanol 114 138 90TiO2 Benzene 85 114 56TiO2 n-Heptane 58 78 38

Table 1.5 Surface pressure of octane, hexadecane, �-bromonaphthalene,diiodomethane, ethylene glycol and formamide and water on different solidsubstrates calculated from the measured contact angle in air and insaturated vapor (Eq. 25c) [8, 42].

�S(L) �S(L)

Octane Ethylene glycol

SiO2–hydrophobic 0.0 SiO2–hydrophobic 3.3SiO2–hydrophilic 0.0 SiO2–hydrophilic 5.7TiO2–hydrophobic 0.0 TiO2–hydrophobic –4.6TiO2–hydrophilic 0.0 TiO2–hydrophilic 7.1

Hexadecane Formamide

SiO2–hydrophobic 0.2 SiO2–hydrophobic –6.8SiO2–hydrophilic 1.7 SiO2–hydrophilic 7.0TiO2–hydrophobic 0.0 TiO2–hydrophobic 3.1TiO2–hydrophilic 0.0 TiO2–hydrophilic 12.7

�-Bromonaphthalene Water

SiO2–hydrophobic –0.7 SiO2–hydrophobic 8.9SiO2–hydrophilic –1.1 SiO2–hydrophilic 6.3TiO2–hydrophobic 1.5 TiO2–hydrophobic 19.1TiO2–hydrophilic 6.4 TiO2–hydrophilic 31.7

Diiodomethane

SiO2–hydrophobic –3.2SiO2–hydrophilic 0.0TiO2–hydrophobic 0.3TiO2–hydrophilic 6.3

was discussed previously (Fig. 1.6). However, the plot for hydrophobic silicabreaks into two parts at ABN, the low energy liquids extrapolating to octane atcos�SL = 1 and the high-energy liquids, including water to the expected intercept–1 at 1/�LV = 0. The surface pressure obviously does not explain the deviation ofhydrophilic silica from this point, but should rather be devoted to specific inter-actions.

Recalling the Young-Dupré-Zettlemoyer (YDZ) equation, we may express thework of adhesion in terms of the arithmetic averaging:

WS�L�L � �L�L�cos�S�L�L � 1� � ��dSV � �d

LV� � S�L� �34 a�

cos�S�L�L � �1� �dSV � �d

LV � S�L��LV �34 b�

As an intrinsic consistency test the line should pass through –1 at 1/�L(L) = 0,which is obeyed by ABN, DIM, EG, FA and W. For hydrocarbons [�L(L)��LV��LV

d ] we have

cos�S�L�L � ��dSV � S�L��LV �34 c�

for which the line should pass through zero at 1/�L(L) = 0. This expectation is ful-filled for the O-HD-ABN branch of hydrophobic silica in Fig. 1.8. ABN is thusrepresented in both liquid series. Obviously Eq. (34 c) may be used to evaluate


Fig. 1.8 A comparison of the dependency cos �SL (in air, filled symbols) andcos �S(L)L (in saturated vapor, open symbols) for hydrophobic (spheres) andhydrophilic (triangles) silica on the inverse surface tension of the solid-probeliquid systems [8, 42].

the applicability of the arithmetic averaging for a range of probe liquids. When�LV��LV

d the YDZ Eq. (15 b) does not apply. We cannot deduce from Fig. 1.8 thepreference between Eqs. (33 b) and (34 c). Table 1.6 reports on the difference be-tween arithmetic and geometric averaging of �S

d using heptane (�H��Hd ) as

probe liquid.For polar solids, the arithmetic and geometric averaging provide divergent val-

ues for the dispersive component of the solids. One reason is obviously thefairly large surface pressure. However, the ratio between the geometric andarithmetic components, equaling the Girifalco–Good ratio (�), remains fairlyconstant in the range 0.6–0.8, being substantially above unity assumed for dis-persive interaction.

1.3.3Models for Specific Polar (Lewis) Interactions

Starting from the truly dispersive (London) interactions of hydrocarbons, thereis a broad range of molecular interactions of Lewis nature. However, as dis-cussed, the Debye and Keesom interactions diverge from the traditional van derWaals range to the Coulomb range when molecularly arrested or “frozen” uponadsorption on the surface sites [31].

In the classical treatment of surface interactions, the total contribution is sub-divided into a dispersive part and a polar part. The latter should be understoodas specific (molecularly arresting) interactions without any particular (e.g. dipo-lar) interaction in mind [10]:

�LV � �dLV � �

pLV �35 a�

�SV � �dSV � �

pSV �35 b�


Table 1.6 Values of �Sd and S(H) at 25 �C for a number of solid surfaces

using heptane as probe liquid [10].

Solid �S(H)0 �S

d (arithmetic mean) �Sd (geometric mean) �

Copper 29 49 60 0.82Silver 37 57 74 0.77Lead 49 69 99 0.70Tin 50 70 100 0.70Iron 53 73 108 0.68SiO2 39 59 78 0.76TiO2 (anatase) 46 66 92 0.72SnO2 54 74 111 0.67Fe2O3 54 74 107 0.69Graphite 56 76 115 0.66

Introducing arithmetic and geometric averaging of the dispersive componentfor the solid, we may now calculate the specific (polar) interaction as �LV ��OV

��OVd [10]:

�SVp (A,O) = �SV – �OV cos�SO (36 a)

�SVp (G,O) = �SV – 0.25(�OV)(cos�SO + 1)2 (36 b)

However, since the surface energy of the solid (�SV) is unknown, we cannot utilizethese simple equations. Instead, we may calculate the specific (polar) work of ad-hesion by using the dispersive surface energy of the solid determined with the ref-erence oils to calculate the polar work of adhesion (WSL

p ) in the following way:

WSL�I� � �LV�cos�SL � 1� �WdSL �Wp

SL �37 a�

WpSL�A� �WSL � �d

SV�A�O� � �dLV� �37 b�

WpSL�G� �WSL � 2�d

SV�G�O��dLV�1�2 �37 c�

Note that the separate determination of �SVd (O) using a fully dispersive liquid

(oil, O) is specifically indicated.Assuming that also the polar contributions are additive, the arithmetic aver-

aging available to resolve the polar component of the surface energy of the solidmay be represented as

�SV�A�p �WpSL�A� � �

pLV �38 a�

�SV�A�p � �LV cos�SL � �dSV�A�O� � �LV cos�SL � �OV cos�SO �38 b�

Although it has been considered unacceptable to apply geometric averaging toany interaction of a specific nature, we also consider this option (Eq. 37 c):

�SV�G� I�p � WpSL�G��2�4�p

LV �39�

If the dispersive surface tension of the probe liquid (L) and oil are almost equal(�LV

d ��OVd ��OV), we may also assume that the reference oil fully represents

the dispersive interaction and write considering Eq. (38b):

WpSL�II� �WSL �Wd

SL � �LV�cos�SL � 1� � �OV�cos�SO � 1� �40 a�

�pSV�G� II� � �p

SV�A� � �pLV�2�4�p

LV �40 b�

The specific (polar & AB) surface energy components (Eq. 38b) of hydrophobicsilica is plotted as a function of the surface tension of the probe oils (�OV) inFig. 1.9 (for details, see Ref. [42]).


As shown the specific component is quite consistent for all probe oils rangingfrom 0 to 5 mJ/m2. A slightly larger scatter is found for DIM.

Most new methods rely on a geometric (product) averaging of the directionalspecific interactions. When applying arithmetic averaging, maintaining the pre-vious symmetry, all terms cancel out, making it unavailable for this purpose.One may collectively express the Lewis interaction models representing mono-dentate acid–base bimolecular pairing with each (monodentate) site possessingboth an electrostatic and a covalent binding character:

Y � D1D2 � XA1 XB

2 � ZA1 ZB

2 �41�

where Y is a generalized property or a state variable (typically G, F or H), D isthe dispersive interaction, the A and B interaction between the molecular pairs1 and 2. Such models are, for example, those of Drago and Gutmann (see Table1.7).

As shown, none of these models considers in particular the non-specific inter-actions which in later developments of the models are subtracted from the gen-eralized property Y. Consequently, most of the scales refer to poorly solvatingsolutions (or gas mixtures) of acidic and basic molecules. The latter are basedon enthalpy alone, which was shown to be a tentative property and does not


Fig. 1.9 A comparison of the specific (polar and AB) component of thesurface energy of hydrophobic silica calculated with: Eq. (38b) (B–D),Eq. (39) (E–G), Eqs. (46) and (47) (H–K), Eqs. (49) and (50) (L–O) andEqs. (49) and (52) (P, Q), respectively plotted as a function of the surfacetension of the probe liquids (for details see text and [42]).

include the entropic contribution to the Gibbs free energy. As discussed in aprevious section, none of the properties are related to the second derivativessensitive to salient interactions such as the Lewis interactions.

In a recent paper, Peterson made a conceptual analysis of these types of mod-els and made a matrix transformation producing the following constants [39]:

X� � 1�

2�XA1 � XB

2 � and X� � 1�

2�XA1 � XB

2 � �42 a� b�Z� � 1�

2�ZA

1 � ZB2 � and Z� � 1�

2�ZA

1 � ZB2 � �42 c� d�

He concluded that the positive diagonal X+ and Z+ matrix terms represent theconcept of “like strengths attract their like”. The negative X– and Z– terms rep-resent the situation when “opposite sites attract each other”; the larger the dif-ference, the greater is the attraction. He suggested that all types of interactionscould contribute to the wetting phenomena. The analysis is interesting since itmakes it possible to evaluate the terms separately. However, depending on theinstrumental method utilized, the empirical constants obtained from fitting toexperimental data refer to enthalpic or free energy surface components.

1.3.4Partial Acid and Base Components

The method of van Oss et al. (vOCG) [17, 18] is related to the geometric deriva-tion procedure of the Lifshitz-van der Waals (LW) contribution [�SV

d (G) =�SV

LW(G)]. As for the division into dispersive and polar components, the LWforces are particularly considered and subtracted from the total surface tension/energy to give the specific acid–base (AB) component [�SV

p (G) =�SVAB(G)]. How-

ever, in the vOCG model, each probe molecule and surface are assigned bothacidic and basic sites (bidentacy), which interact with their counterparts inde-pendently. Also, the intrinsic interaction between these sites is allowed for.Hence for bidentate (one acidic and one basic site) probe liquids we find


Table 1.7 The D, X and Z terms of different models a) for specific interactions.

Property (Y) D term X term Z term Additional term

LogKAB N/A S(trength) � (soft) N/ALogKAB N/A S(trength) � (soft) s(teric)LogKAB N/A E(lectrostatic) C(ovalent) d(esolvation)–�HAB N/A E(lectrostatic) C(ovalent) D(ispersive)–�HAB N/A e(lectrostatic) c(ovalent) t(ransfer)–�GAB N/A A(cidic) B(asic) N/A

a) The models refer to the models of Gutmann et al. [33, 34], Maria and Gal[35], Handcock and Marciano [36], Drago and Wayland [32], Kroeger andDrago [37] and Della Volpe and Siboni [38].

�ABLV � �LV � �LW

LV � 2��LV��LV�1�2 �43 a�

�ABSV � �SV � �LW

SV � 2��SV��SV�1�2 �43 b�

where vOCG denote �LV�/�SV

� the acid (electron acceptor) component and �LV�/�SV

�

the base (electron donor) component of the surface tension/energy. For compar-ability with the literature values, the values in Table 1.8 were assumed to be rep-resentative. The AB work of cohesion is as usually twice this value, CLL

AB = 2�LVAB

and CSSAB = 2�SV

AB. For monodentate (one acid or one base site), the AB termvanishes for the respective component.

Similarly, the work of adhesion is taken as the sum of the LW and AB contri-butions, the latter being defined as [17, 18]

WABSL �WSL �WLW

SL � 2��SV��LV�1�2 � ��SV�

�LV�1�2� �44 a�

WABSL � �AB

SV � �ABLV � �AB

SL �44 b�

Again, if one of the AB pairs is monodentate, the term involving this surfacetension/energy vanishes. Combining the AB work of adhesion we obtain theAB component of the interfacial energy:

�ABSL � �AB

SV � �ABLV � 2��SV�

�LV�1�2 � ��SV�

�LV�1�2� �45 a�

�ABSL � 2��SV�

�SV�1�2 � ��LV�

�LV�1�2� � 2��SV�

�LV�1�2 � ��SV�

�LV�1�2� �45 b�

�ABSL � 2��SV �

��SV��

��SV �

��LV�� 45 c�

The last equation indicates that the AB interaction is repulsive if ��SV >��SV and��LV <��SV or if the reverse is true. In practice, the bimolecular bidentate interac-tion has the symmetry of a three-phase (non-specific) liquid contact (see below).

If the LW component of the solid has been determined with the LW probeoils (O) according to one of the methods indicated above, we may write accord-ing to the vOCG model for the work of adhesion for two bidentate AB probe liq-uids K and L [17, 18]

WABSK �WSK � 2�LW

SV �G�O��LWKV �1�2 � C

��SV�K� �D

��SV�K� �46 a�


Table 1.8 Surface tension components of the specific probe liquids (~ 20 �C) [17, 18].

Probe liquid �LV �LVLW �LV

AB �LV� �LV

�

Water 72.8 21.8 51.0 25.5 25.5Ethylene glycol 48.0 29.0 19.0 1.92 47.0Formamide 58.0 39.0 19.0 2.28 39.6

WABSL �WSL � 2�LW

SV �G�O��LWLV �1�2 � E

��SV�L� � F

��SV�L� �46 b�

where WSK =�KV(cos�SK + 1), WSL =�LV(cos�SL + 1), C= 2��KV, D = 2

��KV, E =

2��LV and F= 2

��LV. Since ��SV(K) =��SV(L) and ��SV(K) =��SV(L), the acid and

base components of the solid may then be obtained according to [17, 18]

��SV � �WAB

SK F �WABSL D��CF � DE� �47 a�

��SV � �WAB

SL C �WABSK E��CF �DE� �47 b�

The �ABSV values calculated from Eqs. (45) and (47) are included in Fig. 1.9. The

��SV and ��SV components are presented in Fig. 1.10.The problem with this method is that it may produce negative values for��S and

��S , which are squared artificially to positive numbers. In order to

control this problem, the intrinsic self-consistency check of recalculating WSKAB

and WABSL from the evaluated ��SV and ��SV using Eqs. (12 a) and (12 b), respec-

tively, must be applied! If either of the acid–base adhesions does not agree withthose calculated from the equations

WABSK � �cos�SK � 1� � 2�SV�G�O�LW�LW

KV �1�2 �48 a�

WABSL � �cos�SL � 1� � 2�SV�G�O�LW�LW

LV �1�2 �48 b�

they must be disregarded as intrinsically inconsistent.Apart from the problems related to the use of halogenated hydrocarbons as

LW references, the polar liquids utilized are of a predominantly basic character.It therefore seems rational to consider one or all of the liquid(s) as almost puremonodentate base(s). This is in accord with vOCG [17, 18], who state: “If eitherthe acidic or basic property is negligible and the other property is appreciable,the substance is termed monopolar”. The “degree of monodentacy” of the probeliquids is given in Table 1.9.

As shown, water has the largest AB component, but a zero base dominance.Glycerol has the second largest AB contribution, the largest absolute base com-


Table 1.9 Degree of basicity of the non-aqueous probe liquids suggested by vOCG [42].


AB �LV� �LV

� �LVAB (%) a) ��LV

�/�LV� (%) a)

Water 72.8 21.8 51 25.5 25.5 70.1 0.0Glycerol 64 34 30 3.92 57.4 46.9 93.2Ethylene glycol 48 29 19 1.92 47.0 39.6 95.9Formamide 58 39 19 2.28 39.6 32.8 94.2Dimethyl sulfoxide 44 36 8 0.50 32 18.2 98.4

a) �ABLV (%) = ��AB

LV ��LV� � 100 and ��LV��LV (%) = ��LV � ��LV��LV� � 100.


Fig. 1.10 The acid (left) and base (right) surface energy components ofhydrophobic silica calculated with the models defined in Fig. 1.9 plotted asa function of the surface tension of the probe liquids (for details see textand [42]).

ponent, but the second smallest base dominance. Ethylene glycol seems to havethe overall most favorable properties. With reference to the Girifalco–Good quo-tient it is expected that the arithmetic/total work of adhesion should not be toodifferent from the geometric work of adhesion (��1). The partial contribu-tions should then add up to the total values. However, the base components ex-ceed the AB-component and even equal the total surface tension (EG). Althoughmathematically consistent, the geometric averaging seems to produce rather un-physical values. Instead of becoming involved in the complicated vOCG aver-aging discussed above, a monodentacy is considered instead. Thus, since��KV �0 we may write the AB work of adhesion as [42]

WABSK �WSK �WLW

SK �WA�B�SK � 2�A

SV�K��KV�1�2 �49 a�

�ASV�K� � WA�B�

SK �2�4��KV �49 b�

The superscript A is used in order to distinguish the �ASV component from the

��SV component. The problem is, however, that no probe liquid has been re-ported with as pure acidity as the nearly pure basic solutions suggested for thevOCG model. Therefore, we choose another way of calculating the �B

SV(L) com-ponent. The acid–base contribution to the adhesion can be calculated by accept-ing that the LW probe (O) oil can fully represent the non-specific interaction (cf.Eq. 40 a). A matching of the LW component of the AB liquid (L) is then advisa-ble ��LW

LV � �LWOV � �OV�. In Eq. (38 b) it was assumed that �AB

LV � �LV � �LWOV. In

this case we may write [42]

WABSL �I� �WSL �WLW

SO � �LV�cos�SL � 1� � �OV�cos�SO � 1� �50 a�

WABSL �II� � �LV cos�SL � �LW

OV cos�LWSO � �AB

LV �50 b�

WABSL � 2��A

SV�K��LV�1�2 � ��BSV�L��LV�1�2� �50 c�

Rewritten in terms of the basic solid component, we find [42]

�BSV�L� � �WAB

SL � 2�ASV�K��LV�1�2�2�4��LV �50 d�

Depending on the way in which the work of adhesion is calculated (Eq. 50 aand b) for introduction into Eq. (50 d) the indexing is �B

SV(I) or �BSV(II). These A

and B components are compared with the corresponding parameters inFig. 1.10. As shown the values produced are rather consistent for each probeliquid, but with an enhanced spread for DIM, being dependent on the modelused. Yet another way is to derive a computational cos�AB

SL :

WABSL � �AB

LV �cos�ABSL � 1� �51 a�

WABSL � �LV�cos�SL � 1� � �LW

OV�cos�LWSO � 1� �51 b�


cos�ABSL � ��LV cos�SL � �LW

OV cos�SO��ABLV �51 c�

�BSV�III� � ��AB

LV �cos�ABSL � 1� � 2�A

SV�K��LV�1�2� 2�4��LV �51 d�

However, �BSV�III� (Eq. 51 d) equals �B

SV�II� (Eq. 50d). It should be noted that ifa faction of the surface were assigned both to the dispersive and the polar (AB)surface, Eq. (51c) would equal the well known Cassie equation for chemicalheterogeneous surfaces discussed below!

In the calculations the AB contribution of water has artificially been dividedinto two equal contributions (25.5 mN m–1, Table 1.4). The reference to water isrational since all acid–base scales have been related to some particular propertyof water. However, in order to better reflect the true balance between the acid–base character for water, a strongly weighted acid contribution has been sug-gested on experimental and theoretical grounds by a number of authors [40,41]. The most extreme balance suggested by Della Volpe and Siboni [38] is givenin Table 1.10.

As the most extreme case, we may again use the strongly weighted acid con-tribution suggested for water by Della Volpe and Siboni [38] and compare itspredominant acidity with the predominant basicity of the other vOCG probe liq-uids (Table 1.11).

Although the A/B balance is slightly changed also for ethylene glycol and for-mamide, their nearly pure basicity remains. The strongly dominant acidity of


Table 1.10 Weighted surface tension components of the polar probe liquidssuggested by Della Volpe and Siboni [38].


AB �LVa �LV

b

Water 72.8 21.8 51.0 65.0 10.0Ethylene glycol 48.0 31.4 16.4 1.58 42.5Formamide 58.0 35.6 22.6 1.95 65.7

Testing the values for intrinsic consistency the acid base components agree, butthe LW component for EG should read (31.6 mN/m) and for FA (35.4 mN/m),respectively.

Table 1.11 Weighted surface tension components and the degree ofacidity or basicity of the polar probe liquids suggested by Della Volpeand Siboni [38].


AB �LVa �LV

b �LVAB (%) a) ��LV

a, b/�LVa, b (%) a)

Water 72.8 21.8 51.0 65.0 10.0 70.1 84.6Ethylene glycol 48.0 31.4 16.4 1.58 42.5 34.2 96.3Formamide 58.0 35.6 22.6 1.95 65.7 39.0 97.0

a) See Table 1.10.

water suggests that water may be taken as an acidic probe. The base componentis then obtained simply as (cf. Eq. 49b) [42]

WABSL �WB�A�

SL � 2�BSV�W��a

WV�1�2 �52 a�

�BSV�W� � WB�A�

SW �2�4�aWV �52 b�

The use of water is specifically denoted. We may then proceed to calculate thetotal and AB component of the surface energy [42]:

�SV � �LWSV � �AB

SV � �LWSV�O� � 2�A

SV�K��BSV�W��1�2 �53�

The �ASV and �B

SV components are compared with the corresponding parametersin Fig. 1.10, and the �AB

SV component is reproduced in Fig. 1.9 (for details, see[42]). The surface energy components calculated with each model introduced forthe model surfaces introduced in Fig. 1.3 are plotted as a function of the surfacetension of the probe liquids in Fig. 1.11. It is particularly rewarding to note thatthe mono-bi and mono-mono dentate models closely agree with the bi-bi den-tate vOCG model, when corrected for the DVS acid-base balance.

The geometric averaging principle can be maintained for total surface energiesand surface tensions by applying the GG ratio �. The work of adhesion is then

WSL � 2��SV�SV��LV�LV��1�2 � 2�SL��SV�LV�1�2 �54�

i.e. �SL � ��SV�LV�1�2. However, since the numerical value of the GG ratio ismostly unknown, it does not provide any advantages over the vOCG model. Wecan instead use the arithmetic equivalent for the separation of the acid (a) andbase (b) components from the total specific (polar) part of the surface tensionand surface energy (cf. Eq. 4) [42]:

�pLV � �LV � �d

LV � �aLV � �b

LV �54 a�

�pSV � �SV � �d

SV � �aSV � �b

SV �54 b�

Considering Eqs. (6 a) and (6 b), we may now define the work of acid–base inter-action:

WpSL �WSL �WLW

SL � �LV�cos�SL � 1� � �dSV�A�O� � �d

LV� �55 a�

WpSL � ��a

SV � �bLV� � ��b

SV � �aLV� �55 b�

WpSL � �

pSV � �

pLV � �

pSL � ��a

SV � �bSV� � ��a

LV � �bLV� � �

pSL �55 c�

Combining the last two equations, it is obvious that the arithmetic averagingbeing symmetric with the vOCG model predicts that the interfacial tension


(�pSL) is always zero at equilibrium. Thus, the arithmetic averaging does not lend

itself for calculations of interfacial energies. With reference to the Hard–SoftAcid–Base (HSAB) principle, we define that if one of the sites of the acid–basepair is absent the other cannot interact and the entire parentheses vanish. Thisprinciple for monodentacy is also applied in the vOCG model. Combining Eqs.(21 a) and (21 b) and rearranging we find an exceedingly simple expression forthe surface energy:

�SV � ��aSV � �b

SV� � �dSV�A�O� � �LV cos�SL � ISL �56�

This expression equals the work of immersion which agrees with Eq. (15 b) forfully dispersive liquids. Note that �b

SV � �SV. For dispersive systems, the equa-tion may be tested for consistency by extrapolating �LV cos�SL to �SL � 0:

lim��dLV cos�SL� � �d

LV � �dcrit � �d

SV �57�

Obviously a Zisman like plot is recovered. Replotted as the work of immersionagainst the surface tension of the liquids Neumann et al. have identified thesurface energy of the solid as the maximum crossing point of the lines definedby �SL � 0��LV � �SV� and �SV � constant ��LV � �SV� [13].

As shown the break point may, indeed, be identified for the hydrophobic sur-faces. For the hydrophilic surfaces the break point is less clear.


Fig. 1.11 The work of immersion for silica and titania (symbols as in pre-vious figures, air, filled symbols and saturated vapour, open symbols)plotted as a function of the surface tension of the probe liquid-solids pairs(from [42]).

When considering enthalpic relationships, the state of the surface film mustalso be kept in mind. For polar liquids adsorbing on polar surface sites on anotherwise hydrophobic surface, it has been found that the enthalpy of immer-sion may rise with the degree of vapor coverage (cf. Fig. 1.7) [10]. This meansthat the film has a greater affinity for the vapor molecules than the bare sur-face. The enthalpy of (mono-molecular) adsorption (�Hads

s ) may be related tothe enthalpy of adhesion (�HA

s ) of multilayer adhesion (or spreading):

�Hsads � �Hads�A � ��Hs

I�L� � �HsI�V�� 58 a�

�HsA � �Hs

ads � �Hsliq �58 b�

where the molar enthalpy of liquefaction is �Hm,liq (�Hliqs =�L�Hm,liq). The first

part (�Hadss ) relates to the adsorbed (gaseous) film while (�HA

s ) accounts alsofor the condensation of the multiplex film. All the terms are negative. The en-thalpy of immersion for the vapor covered and the precoated sample is:

�HsI�V� � �Hs

SL�A � �HSL �HSV��A �59 a�

�HsI�L� � �Hs

S�L�L�A � �HSL �HS�L��A �59 b�

The difference between immersion enthalpy of vapor and liquid covered sampleis:

�Hsads � ��Hs

I�V� � �HsI�L�� Hs

S�L� �HsSV��A �60 a�

The enthalpy of adsorption is typically determined utilizing the Clausius–Clapeyron relation for the isosteric heat of adsorption (Qst):

�H0ads �

� �L

�L�0

Qstd ln�L �61�

Hence the change in the immersion enthalpy may be expressed as

�HsI�V� � �Hs

I�L� � �L��Hm�ads � �Hm�liq� �62�

In this way it expresses the energy change of the adsorbate in moving from thebulk liquid to the solid surface if the solid is negligibly perturbed and lateral in-teractions are similar in the adsorbed film to those in the bulk liquid.

Fowkes and Mostafa [43] suggested relating the work of adhesion to the(exothermic) enthalpy per mole of acid–base adduct formation at the inter-face ��Hab� with a function f supposed to convert the enthalpic quantity�ab��Hab�� to the Gibbs free energy for AB interaction:

WABSL �WSL �WLW

SL � ��GABI � f �ab��Hab�� 63�


where �ab�� Ns�NAA� is the number of moles of accessible acid or base func-tional groups per unit area of the solid surface determined by colorimetric titra-tion or adsorption isotherms of the Lewis sites. Assuming that the temperaturedependence does not influence the division into LW and AB contributions, werecall Eqs. (19) and (20):

�GSI � �GS

SL�A � �HSI � T �d�GS

I ��dT �P�n � �GLWI � �GAB

I �64�

Then, introducing the Fowkes model:

�HABI � �HAB

SL �A � �ab�Hab � �1�f ��GABI �65 a�

�HABI � �GAB

I � T �d�GABI ��dT �P�n �65 b�

we may derive the condition for the equality of �GABI as [44]

1�f � 1� d�� ln GABI ��d ln T �P�n �66�

Alternatively, we assume that the arithmetic averaging of the surface energycomponents applies (�SL

AB = 0). Then, as shown in Eq. (65 a) we may write:

�GABI � �WAB

SL � ��ABSV � �HAB

I � T�d��ABSV ��dT�P�n �67�

We may thus rewrite the f-factor in the form:

f � ��ABSV��HAB

I � 1� �T d��ABSV ��dT �P�n��HAB

I �68 a�

f � 1� T��SABI ��HAB

I � �68 b�


Fig. 1.12 The Fowkes f factor as a function of temperature for bromoform–poly(methyl methacrylate) (circles), dimethyl sufloxide (DMSO)–poly(vinylchloride) (squares), DMSO–[polyethylene/poly(acrylic acid), 5%] (triangles)and DMSO–[polyethylene/poly(acrylic acid), 20%] (diamonds) (from [44],with permission)

Fig. 1.12 shows that the f factor is substantially less than unity in most casesand increases with temperature. Obviously, for these systems the prediction of astraightforward relationship does not exist between the enthalpy and free energyfor the acid–base interaction.

It should be noted that the Drago model [32, 33] refers to the enthalpy of for-mation of one-to-one molecular adducts in the gas phase and in poorly coordi-nating solvents. In the latter case, the enthalpy of the non-specific interactionsmust be subtracted to obtain �HAB

SL , which, according to Eq. (21), can be ap-proximated as

�HABI � �HI � �HLW

I � �HI � �dSV � 21 �69�

Alternatively, we may recall the enthalpy of immersion (Eq. 17):

�HABI � �AB

SL � T ��ABSL ��T �P�n � �AB

SV � T ��ABSV ��T �P�n �70 a�

�HABI � �AB

SL � �ABSV � T ��AB

SL � �ABSV ��T �P�n �70 b�

�HABI � ��AB

LV cos�ABL � T ��AB

LV cos�ABL ��T �P�n �70 c�

�HABI � ��AB

LV cos�ABL � T�AB

LV ��cos�ABL ��T �P�n � T cos�AB

L ��ABLV ��T �P�n

�70 d�

where cos�ABL may be derived in the way described above. This approach

seems, however, to be an unacceptably tedious approach. Douillard and Médout-Marère extended the vOCG division of the components to the enthalpic contri-bution [45]:

HsSL � Hs

SV �HsLV � 2�HLW

SV HLWLV �1�2 � 2�H�SVH�LV�1�2 � �H�SVH�LV�1�2� �71�

where the components of the heat of immersion are

�HsI � �Hs

SL �HsSV��A � �HLW

I � �HABI �72 a�

�HLWI � HLW

LV � 2�HLWSV HLW

LV �1�2 �72 b�

�HABI � HAB

LV � 2�H�SVH�LV�1�2 � �H�SVH�LV�1�2� �72 c�

If the immersion is first done in non-specific liquids, then �HLWI can be sub-

tracted from the total heat of immersion for acidic and basic probes to give theacid–base components. Alternatively, the acidic or basic probes are titrated tothe solid dispersed in the non-specific liquid displacing the LW molecules fromthe AB sites. Douillard and Médout-Marère suggested using the Fowkes f functionto convert the enthalpies further to surface energy components of the vOCG mod-el discussed previously. As shown above, the latter suggestion is bound to fail.


The conversion into arithmetic averaging maintaining the symmetry of thevOCG model is not possible, since the subtraction of the terms involving theacid–base components cancels the interfacial terms altogether. However, sincethe extensive experimental material analyzed by Drago is enthalpic in origin, wemay rewrite his E and C constants as partial enthalpies:

�HABI � �HI � �HLW

I � �HEASV HEB

LV� � �HCASV HCB

LV � �73�

In this way, all the accumulated data are made available for monodentate acid–base reactions immediately. Thus, the immersion is first done in pure LW liq-uids to give the �HAB

I contribution.Alternatively, we may, in line with the Douillard and Médout-Marère (DMM)

geometric model, apply the much simpler arithmetic averaging model:

HsSL � Hs

SV �HsLV � �HLW

SV �HLWLV � � �HA

SV �HBLV� � �HB

SV �HALV�� 74�

The components of the heat of immersion are by symmetry

�HsI � �Hs

SL �HsSV��A � �HLW

I � �HABI �75 a�

�HLWI � HLW

LV � �HLWSV �HLW

LV � �75 b�

�HABI � HAB

LV � �HASV �HB

LV� � �HBSV �HA

LV�� 75 c�

In all these cases, the state molecular gaseous film should be kept apart fromthe condensed liquid film including the enthalpy of condensation and the be-havior of the bulk liquid (see Fig. 1.7).

As discussed in the Introduction, rather than aiming for the free energies (Fand G) as done in the discussion above, one should relate the enthalpy to heatcapacity instead. For the free energies, all interactions are balanced against eachother and thence only a break point is recorded for free energies at first-orderphase transitions. For enthalpy this produces a sudden jump to a new level,which is sharper the more extensive the phase transition is. However, the sali-ent interactions are sensitively reflected only for the second order derivativeproperties, such as heat capacities, expansivities and compressibilities [1, 6].Consider the distribution of a probe between acidic (or neutral) solution (stateA) and basic surface sites (state B):

A� B� K � xB�xA � xB��1� xB� �76�

where the system is considered ideal, i.e. the activity coefficients have been setequal to unity. The heat capacity of such a system will first contain contribu-tions from the probe in each state and may be written as [46]

CintraP � xACA

P � xBCBP �77�


The latter is termed “intra” in order to distinguish it from another possible con-tribution, which can arise from shifts in the equilibrium populations of eachsite with temperature. If there is an enthalpy difference between states A and B(�HAB), then the equilibrium shift is obtained through

d�ln K��dT � d�ln K��dxB�dxB�dT � �HAB�RT2 �78�

After derivation and rearranging, we may write

dxB�dT � xAxB�HAB�RT2 �79�

Now, the heat adsorbed for this equilibrium shift will contribute and “inter-state” heat capacity, defined as [46]

CinterP � dH�dT � �dH�dxB��dxB�dT� � �HAB�dxB�dT� �80�

Insertion of �dxB�dT� gives

CinterP � xAxB��HAB�2�RT2 �81�

Hence the total heat capacity has been related to both Gibbs free energy (lnK)and the enthalpy change of the acid–base site binding (�HAB) [46]:

CP � CintraP � Cinter

P � xACAP � xBCB

P � xAxB��HAB�2�RT2 �82�

Corresponding relationships can also be written for the other second derivativesfrom the Gibbs free energy, i.e. the expansivity and the isothermal compressibil-ity [46]:

E � Eintra � Einter � xAEA � xBEB � xAxB��HABV �VAB��RT2 �83�

KT � K intraT � K inter

T � xAKAT � xBKB

T � xAxB��VAB�2�RT2 �84�

where the cubic expansion coefficient � � E�V and the isothermal compressibil-ity coefficient �T � KT�V [1].

The interstate contribution will be maximum at xA � xB � 1�2 (i.e.K � 1��GAB � 0) and its maximum will depend on (�HAB�2, ��HAB�VAB) or��VAB�2. When squared the inter-state contribution is always positive, regard-less of the sign of the enthalpy change. It must be acknowledged that in thissystem, xA and xB cannot be varied at will, except by changing the temperatureor pressure. The limiting cases where the inter-state contribution is small, i.e.xA � 0 or xB � 0, originate from either very large or very small equilibriumconstants. It is now understood that very weak (van der Waals) interactions willyield rather small contributions to Cinter

P . On the other hand, such interactionsare sensitively reflected in K inter

T . Intermediate Lewis acid–base interactions such


as hydrogen bonding with energies several times RT are expected to producegreater effects on Cinter

P . The expansivity �Einter� is expected to be sensitive toboth types of interaction. The real potential of �Cinter

P , Einter, K interT � measure-

ments may thus be fully appreciated as a selective emphasis on contributionswhich correspond to a particular range of interaction energies.

1.4Wetting in Idealized Ternary Systems

In order to rationalize the concepts, we describe the processes of wetting usinghalf-spheres having unit target area directed towards the dividing plane. Thework of cohesion and adhesion are illustrated in Fig. 1.13.

We consider specifically the three phases discussed previously, i.e. the liquids(K and L) and the solid (S) (Fig. 1.14). We may now easily derive the work ofspreading for each pair of phases, disregarding the third phase (vapor, V) at thethree-phase contact-line (tpcl). For non-condensed (vapor) phases the surfacetension is negligible. In the indexing the most condensed phase is written first:

WKL � �KV � �LV � �KL

SKL � �LV � �KV � �KL

�

SKL �WKL � CKK �85 a�

WSK � �SV � �KV � �SK

SSK � �SV � �KV � �SK

�

SKL �WSK � CKK �85 b�

WSL � �SV � �LV � �SL

SSL � �SV � �LV � �SL

�

SKL �WSL � CLL �85 c�

In all cases the upper phase is like a reversed process considered to spread onthe lower one, until the work of adhesion and work of cohesion are equal. Weshall make use of these binary systems when considering the phase equilibriumin three-component systems.

We may expand the optional work of adhesion in terms of the surface ten-sions of two liquids (K = 1 and L = 2) previously discussed in contact with a sol-id (S = 3), assuming that they are fully immiscible with each other (more con-densed phase first):

1.4 Wetting in Idealized Ternary Systems 37

Fig. 1.13 The work of cohesion (X = 1) represents theseparation of the same phase and the work adhesion(X = 2) two phases (half droplets) in contact, thusbringing them in contact with (their) vapor.

WKL � �KV � �LV � �KL �W12 � �1 � �2 � �12 �86 a�

WSK � �SV � �KV � �SK �W13 � �3 � �1 � �13 �86 b�

WSL � �SV � �LV � �SL �W23 � �3 � �2 � �23 �86 c�

When three phases are brought into contact, the situation is rendered muchmore complex. In addition to the binary contact area we have to consider athree-phase contact point (tpcp) (Fig. 1.15). Assuming that the outer curvedlines remain excluded from the considerations, the following options for theprocesses appear reasonable (no particular indexing order):

If only one third (phase 1) is separated, we find for the work of adhesion(process I)

W�I�123 � 2�1 � �2 � �3 � �12 � �13 � �123 �87 a�

W�I�123 �W12 �W13 � �123 �87 b�

When phase 1 is immersed in phases 2 and 3 the interfacial contacts 1–2 and1–3 remain and the work of adhesion is dramatically simplified (should be 3 in1 and 2, process II):

W�II�123 � �2 � �3 �88 a�

W�II�123 �W23 � �23 �88 b�


Fig. 1.14 The work of spreading may be expressed as the differencebetween the work of adhesion and the work of cohesion (Eq. 89). The liquidphase is described as a half droplet in contact with the solid being incontact with the vapor.

If all three phases are separated simultaneously, we find (process III)

W�III�123 � 2�1 � 2�2 � 23 � �12 � �13 � �23 � �123 �89 a�

W�III�123 �W12 �W13 �W23 � �123 �89 b�

The process considered is obviously of prime importance for the surface ten-sion–surface energy balance found. The energy balance at the tpcp should equalzero at equilibrium.

Two processes are offered as a standard for the work of adhesion in textbooks onsurface and colloid chemistry [16, 31]. The reason for the particular averagingscheme is probably to maintain the symmetry of the geometric averaging rule.

First we consider that a liquid (L) and a solid (S) initially in contact are sepa-rated from each other and brought into cohesive contact (Fig. 1.16):

W�IV�SLS � CLL � CSS � 2WSL �90�

Written in terms of interfacial tensions, this equation reduces to

W�IV�SL � 2�SL �91�

The second is the separation between two phases (S and K) initially in contactwith the medium (L) to form a contact with the two phases and the third phaseinternally.


Fig. 1.15 The work of adhesion represented by the separation of one tothree phases bringing them in contact with (their) vapor (Eqs. 91–93).

Converted into binary work of cohesion and adhesion (Fig. 1.17) [16, 31]:

W�V�SLK � CLL �WSK �WKL �WSL �92�

Written in terms of interfacial tensions:

W�V�SLK � �KL � �SL � �SK �93�

The latter process is illustrated in terms of the triangular sphere in Fig. 1.18.As shown, the latter two-phase W�IV�SL� and three-phase W�V�SKL� work

of adhesion correspond to only a fraction of the total work of adhesionW�III�PLK�. They therefore all represent different thermodynamic realities.


Fig. 1.16 The work of adhesion between a solidand a liquid represented by the separation of theinitially interconnected phases and joining eachphase to a unified phase, represented by the workof cohesion.

Fig. 1.17 The work of adhesion between liquid–liquid and solid–liquid phases represented by theseparation of the initially interconnected phasesand joining dispersion liquid to a unified phase,represented by the work of cohesion and the twodispersed phases represented by the work ofadhesion.

Fig. 1.18 The work of adhesion betweenliquid–liquid and solid–liquid phasesrepresented by the separation of the initiallyinterconnected phase (1 = L) and joining ofthe separated phases (2 = K and 3= S).

1.4.1Preferential Spreading at Three-component Interfaces

Dispersing a solid (S) and a liquid (K) in small amounts in a immiscible liquid(L) may lead to a full dispersion (rejection) of all the phases or an engulfment(preferential wetting) of the solid into the K liquid. The intervening situationwhen all phases partially wet each other is denoted a funicular state. In order todetermine these limiting states, we derive the ternary work of adhesion andspreading denoting the dispersion medium between the dispersed phases in thelower index. We select the traditional process (Eq. 86) and permute the liquid(K) and the dispersion medium (L) maintaining the solid (S).

With reference to the binary processes (Fig. 1.14), we may write the ternarywork of spreading and the work of adhesion in terms of interfacial tensions.For the first case, we find

SSLK � �SL � �LK � �SK

WSLK � �SL � �LK � �SK

�

SSLK �WSLK � 2�LK �94�

Considering L as the dispersed liquid and K as the dispersion medium, we find

SSKL � �SK � �KL � �SL

WSKL � �SK � �KL � �SL

�

SSKL �WSKL � 2�KL �95�

The ternary work of spreading may thus be expressed as the difference betweenthe ternary work of adhesion and the two times the interfacial tension betweenthe liquids (Fig. 1.19).

Likewise as for the binary case, the spreading coefficient is expected to bepositive (negative Gibbs free energy) for spontaneous preferential spreading tooccur. Three limiting cases can be distinguished:

1. The dispersed liquid (K) cannot spread on the solid (S) since the dispersionliquid (L) preferentially wets the particles, i.e. SSLK � 0, but SSKL � 0.

2. The dispersed liquid (K) partially forms (liquid bridges between) the solids (S)if both SSLK < 0 and SSKL � 0.


Fig. 1.19 The preferential wetting of two non-miscible liquids on a solidmay be expressed by the ternary work of spreading expressed in termsof interfacial tensions. The work of spreading represents the differencebetween the ternary work of adhesion and two times the interfacialtension between the liquids.

3. The dispersed liquid (K) preferentially wets the solid particles (S), displacingthe dispersion liquid (L), if SSKL � 0, but SSLK < 0.

Note that the interfacial tension between the liquids determine whether the ter-nary work of spreading has a positive or negative sign (Fig. 1.20).

We may measure directly the work of adhesion by the introduction of a ter-nary Young-Dupré equation:

WSLK � �KL�cos�SLK � 1� �96 a�

WSKL � �LK�cos�SKL � 1� �96 b�

In the former case the contact angle between the solid (S) and the droplet (K) ismeasured immersed in liquid (L) and in the latter case the liquids are reversed.Owing to density differences, one measurement is usually made from a sessiledrop and the other from a pendant drop.

The ternary work of adhesion can be related to the binary work of adhesionas discussed previously [8]:


Fig. 1.20 Preferential wetting of oil on hydrophobic particles dispersed inan aqueous dispersion. The particles are the more efficiently removed thesmaller the interfacial tension between the water and the oil is and thelarger the difference �SK � �SL grows.

WSLK � CLL �WSK �WSL �WLK �97 a�

WSKL � CKK �WSL �WSK �WKL �97 b�

The liquid (interfacial) tensions are measured as usual and the binary workof adhesion for the solid as

WSL � �LV�cos�SL � 1� �98 a�

WSK � �KV�cos�SK � 1� �98 b�

Since CLL �WLK � �SLK and CKK �WKL � �SKL, we may rearrange the equations:

WSLK � �KL�cos�SLK � 1�� KV�cos�SK � 1�� LV�cos�SL � 1�� SLK �99 a�

�KL cos�SLK � �KV cos�SK � �LV cos�SL �99 b�

WSKL � �LK�cos�SKL � 1�� LV�cos�SL � 1�� KV�cos�SK � 1�� SKL �99 c�

�LK cos�SKL � �LV cos�SL � �KV cos�SK �99 d�

This so-called Bartell-Osterhof equation [47] shows that the ternary contact angle(solid–liquid–liquid) may be related to the binary one (solid–liquid–vapor) in astraightforward way. It may be considered as a Cassie equation for a multicom-ponent system.

1.4.2Models for Dispersive Solid–Liquid–Liquid Interaction

When considering the two standard processes for the work of cohesion and ad-hesion, we introduce the geometric average of the dispersive component:

CdSS � 2��d

SV

��d

SV� and CdLL � 2��d

LV

��d

LV� �100�

WdSL � 2��d

SV�dLV�1�2 �101�

Then we may write the work of adhesion for the extended (SLS) binary systemdefined by the equation [8, 17, 18]

WdSLS � 2��d

LV�dLV�1�2 � ��d

SV�dSV�1�2 � 2��d

LV�dSV�1�2 �102 a�

WdSLS � 2��d

SV ��d

LV�2 � 2�dSL �102 b�

As shown, the extended binary system produces, as expected from the processconsiderations, a double dispersive interfacial tension. For both versions theequation shows that only when

��d

SV ��d

LV does WdSLS � 0. Otherwise Wd

SLS is


always positive. For the three-component system we may write for the work ofadhesion

WdSLK � 2��d

LV�dLV�1�2 � ��d

SV�dKV�1�2 � ��d

SV�dLV�1�2 � ��d

KV�dLV�1�2 �103 a�

WdSLK � 2��d

KV ��d

LV��d

SV ��d

LV� �103 b�

The equation indicates that the dispersive interaction is repulsive if�d

SV � �dLV � �d

KV or �dSV � �d

LV � �dKV.

Since � � AH��kAd20�, where kA � 75.40 for van der Waals liquids, 100.5 for

pure hydrocarbons and 10.47 for semi-polar liquids, we may also write the equa-tions in terms of Hamaker constants (AH) [16, 31]:

ASLK � ALL � ASK � ASL � ALK �104 a�

ASKL � AKK � ASL � ASK � AKL �104 b�

The same geometric averaging rules have been applied to these interfacial Ha-maker constants. Owing to the definition of the work of adhesion for three-com-ponent systems without the tpcl(p), the interfacial tension cannot be derived ina straightforward way.

The three-phase systems offer an interesting alternative to measure contactangles of, e.g., a solid (S) immersed in a hydrocarbon (oil, O). If a drop of water(W) is placed as a sessile drop on the solid immersed in the oil, we may writefor the ternary Young equation [8]

�SO � �SW � �OW cos�SOW �105�

We assume that the hydrocarbon interacts with the solid solely through Lon-don-van der Waals forces and write the interfacial energy in terms of theDupré-Fowkes equation:

�SO � �SV � �OV � 2��dSV�

dOV�1�2 �106�

On the other hand, the �SW component is assumed to be polar, hence the inter-action is both dispersive and specific (polar) in origin �WSW �Wd

SW �WpSW�

[48]:

�SW � �SV � �WV � 2��dSV�

dWV�1�2 �Wp

SW �107�

Inserted into Eq. (105), we obtain the Schultz equation [49]:

�WV � �OV � �OW cos�SOW � 2��d

SV��d

WV ��d

OV� �WpSW �108�


A Schultz plot of �WV � �OV � �OW cos�SOW against��d

WV ��d

OV is expectedto give a straight line with slope 2

��d

SV and intercept WpSW. The extraction of

WpSW can be improved by choosing octane as the immersion liquid, since its

surface tension equals the dispersion component of water. The accuracy of themeasurement is frequently fairly low, but it can be confirmed by measuring thecontact angle from the pendant hydrocarbon drop against the solid immersedin water [8, 48].

1.4.3Contribution from the Surface Pressure of a Monomolecular (Gaseous) Film

The preferential spreading of a liquid (K) dispersed in small amounts in an im-miscible liquid (L) on an equally dispersed solid (S) may lead to a phenomenonsimilar to preferential adsorption of one component from a mixed solvent.However, as discussed before, the state of the film considered should be speci-fied, i.e. whether it is a molecular vapor (given by the surface pressure), an un-stable intermediate or a bulk (immiscible) liquid (given by the work of spread-ing). Since �S�K� � �SV � �S�K�, it is realized that the anticipated process is cor-rect. It is obvious from Eqs. (31) and (32) that the spreading coefficient equalsthe negative surface pressure of a duplex film, i.e. when the liquid fully wetsthe solid surface as a duplex film. Exchanging the vapor (gas) for the liquid (K),we obtain the following permutative ternary spreading coefficients between K,component L (liquid) and S (substrate), i.e. SSLK and SSKL, respectively.

The preferential spreading of the liquid probe (K) on the solid (S) displacingthe dispersion liquid (L) may be assumed to occur via an intermediate statewhere both liquids are preadsorbed on the solid represented by the surface pres-sures before immersion in the other liquid. Neglecting the surface pressures onthe liquids, we obtain

�S�K�L � �SL � �S�K�L �109 a�

�S�L�K � �SK � �S�L�K �109 b�

We obtain for the work of adhesion for competing surface pressures (Eq. 12 a)

WS�K�L�LK � CLL �WS�L�K �WS�K�L �WLK �110 a�

� 2�LV � �SV � �KV � �S�L�K��SV � �LV � �S�K�L��LV � �KV � �LK��110 b�

WS�K�L�LK � �LK � �SL � �SK � �S�K�L � �S�L�K �111�

where CLL � CL�L� � 2�L�L� � �LV. We find the following work of spreading:

SS�K�L�LK �WS�K�L�LK � 2�LK � �SL � �SK � �LK � �S�K�L � �S�L�K �112�


The preferential spreading of the liquid (K) occurs in parallel with the retreat ofliquid L (negative spreading, opposite signs). Clearly, the preferential wettingmay be treated as preferential adsorption from a mixed solvent system to pro-duce a film pressure! For non-miscible liquids the basic phenomena is, how-ever,’ more favorably described by an adsorption isotherm.

1.4.4Models for Lewis (Polar) Solid–Liquid–Liquid Interaction

When considering the two standard processes for the work of cohesion and ad-hesion, we introduce the geometric average of the acid–base component:

WABij �Wij �WLW

ij � 2��i � j �1�2 � �� i ��j �1�2� 2��i

�� j � � �

�� i��j � �113�

Additionally, for bidentate (one acidic and one basic site) probe liquids, we find

�ABSi � �Si � �LW

Si � 2��Si� Si�1�2 � 2��Si

�� Si� �114 a�

�ABLi � �Li � �LW

Li � 2��Li� Li�1�2 � 2��Li�

Li� �114 b�

The symmetry rule also applies for the work of cohesion, being WABii � CAB

ii �2�AB

i . The work of adhesion and cohesion indicated above thus takes the form[8, 17, 18]

WABSLK �W�4�AB

SLK � CABSS �WAB

LK �WABSL �WAB

SK �115 a�

WABSLK � 2�2��SV

�� SV� � �

��LV�� KV� � �

�� LV��KV� � �

��SV�� LV�

� �� SV��LV� � �

��SV�� KV� � �

�� SV��KV�� 115 b�

WABSLK � 2��SV �

��LV��

�� SV �

�� LV� � �

��SV �

��KV��

�� SV �

�� KV�

� ��LV ��KV��

�� LV �

�� KV�� 115 c�

WABSLK � 2��KV �

��LV��

�� SV �

�� LV� � �

�� KV �

�� LV��

��SV �

��LV��115 d�

The LW and AB interactions in the three-component system may be written ina more illustrative way [8]:

WLWSLK �WLW

SK � 2��LWLV ��LW

SV ��LW

KV ��LW

LV � �116 a�

WABSLK �WAB

SK � 2��LV�� SV �

�� KV �

�� LV��2�� LV�

��SV �

��KV�

��LV�

�116 b�


In both cases the binary work of adhesion between the dispersed componentsmay be separated from the interaction between the medium liquid and the dis-persed components. Thus the LW component of the medium liquid interactswith (is multiplied with) the dispersed S and K components, while the interac-tion with itself is subtracted from the balance. In a similar way, the acidic siteinteracts with the basic sites of the dispersed S and K components, while the in-teraction between the acidic and basic sites of the liquid is subtracted from thebalance. Conversely, the basic sites of the liquid interact with the acidic sites ofthe S and K components, while the interactions with its own acidic sites aresubtracted from the balance.

An interesting opportunity to evaluate the work of ternary interaction is pro-vided by the atomic force microscope (AFM), utilized as such or as a colloidalprobe [50]. According to the Derjaguin-Muller-Toporov (DMT) theory [51] for asmall-radius solid (tip, T) interacting with a flat solid (S) in a liquid (L), the forceof adhesion is given by

FA � 2�RWTLS �117�

where R is the radius of curvature of the tip (or colloid). Since

WTLS � CLL �WTS �WTL �WSL �118�

on combining the equations we obtain

WTS � �FA�2�R� �WTL �WSL � CLL �119�

Now WTL and WSL may be determined from contact angle measurements andCLL � 2�LV. Using standard vOCG liquids, the surface energy components weredetermined for a number of solid substrates using an Si3N4 AFM tip [50]. How-ever, in the colloidal probe procedure a roughly spherical particle (ca. 1 �m) isglued on the cantilever and then just about any combination of T–L–S and T–Sinteractions can be measured.

1.5Adsorption from Solution

As indicated earlier, the preferential adsorption of a liquid component from amixture cannot be determined from wetting experiments, e.g. from the work ofspreading. However, the determination of the adsorption of probe molecules onthe sites offers a straightforward way to determine the molecular surface pres-sure. In this section we discuss the Lewis type and the Brønsted type of acid–base interaction separately, since the mechanism and energy involved differ. Weshall also distinguish between adsorption from an undefined medium resem-bling the gas adsorption. The difference is, however, that in a hydrocarbon solu-

1.5 Adsorption from Solution 47

tion the dispersive interaction with the surface is neutralized whereas it remainsin gas adsorption. Alternatively, a competing adsorption is considered separatelywhen probe molecules displace, e.g., solvent molecules from the surface sites.

1.5.1Determination of Lewis (Polar) Interactions with Surface Sites

Adsorption provides the proper mean to evaluate the surface states of the solid.In the first model the process involves two steps. First, the adsorption of probemolecules to the surface which is determined separately providing the numberof sites. Second, the ability to transfer electrons from the adsorbed basic absor-bate to the acidic surface sites provides the strength of the surface sites. Assum-ing that the adsorbed probe molecule is an almost pure Lewis base (Bs) reactingon the surface with acidic surface sites (As) to form an adduct (ABs) we maywrite the equilibrium and the equilibrium constant as [52]:

As � Bs � �AB�s � 1�KsB � Ks

ads ��asAB��as

AasB� � �xs

AB�xsA��f s

AB�f sAas

B� �120�

where a= activity, x= mole fraction, f = activity coefficient, s= surface and b= (equi-librium) bulk solution. We may now introduce the Hammett function (H0):

H0 � � log KsB � log�xs

B�xsAB� � � log�as

A� � log�f sB�f s

AB� �121 a�

H0 � pKsB � log�xs

B�xsAB� � p�as

A� � log�f sAB�f s

A� �121 b�

When xsAB � xs

B then H0 � pKs. Since a mole fraction ratio is considered, itmay be exchanged for any other concentration scale. A slight excess of acidicand basic indicator probes has been adsorbed on solids of opposite naturedispersed in a saturated hydrocarbon solvent. After equilibration, the indicatorsare desorbed using even stronger acids and bases. The amount acid and baseneeded for changing the color of the adsorbed indicator �xs

AB � xsA� gives the

number of sites and then H0 � pKs � pKa (indicator).The fraction of acidic surface sites (A) occupied by the basic probe molecules

(B) dispersed in indifferent oil (O) for low surface site occupancy (surface cover-age) may be related to the surface film pressure [53]:

�S�B�O � �S�K�L � �SO � �S�B�O � �RT�Am� ln�asAB�as

A� � �RT�Am��xsAB�xs

A��122�

where the molar surface area Am � NA� and � is the surface area occupied byeach B or rather each site area. The number of surface sites Ns

m � NA�nsAB�m

and the area occupied by one site � � A�Ns � wSAw�Nsm. The monolayer sur-

face excess is �m � �nsAB�m�A � Ns

m�ANA. In this calculation, it is assumedthat the solvent is a fully inert oil (O) and that there is no (surface or concentra-tion) potential against which the adsorption occurs. It may not be possible to


identify �nsA�m, the end-point, but rather the equivalence point where

xsAB�xs

A � 1. This can conveniently be identified, e.g. from spectroscopic mea-surements (spectral or color changes).

For ideal surfaces, the term in Eq. (121) involving activity coefficients can beomitted. The relationship with the energy exchange upon adsorption can beconfirmed with the Boltzmann equation:

nsAB � nb

B exp��S�B�O�RT� �123�

Introduced in Eq. (117) we obtain:

�S�B�O � �SO � �S�B�O � �RT�Am��S�B�O�RT� � ��S�B�O�Am� �124�

since nABs /nB

b = xABs /xB

b.

Thus, for dilute solutions depletion measurements may be used.Calorimetry can also be used to determine the degree of adsorption. Figure

1.21 illustrates the amount of an adsorbed basic (probe) molecule on acidic sili-ca silanol (Si–OH) groups plotted against increasing basicity (left) and increas-ing acidity (right) of a number of solvents. The adsorption is considered to bedependent on the relative degree of (specific) interaction, being greatest fromneutral solvents. The strength of the basic solvent is plotted as �HB�A�

BL , the heatof interaction of the basic solvents with t-BuOH. The choice of this alcohol isdue to the assumption that it has acidic properties similar to those of the sur-face Si–OH sites of silica [43]. The strength of the acidic solvents is plotted as


Fig. 1.21 Schematic illustration how basic probe molecules adsorb on theacidic surface silanol (Si–OH) groups of silica. The surface excess isgreatest from neutral solvents (middle) but is reduced when the basicity(LB, left branch) or the acidity (LA, right branch) of the solvent moleculesincreases due to SiOH–solvent complexation (left) or probe–solventcomplexation (right).

�HA�B�AL , the heat of interaction of the solvents with ethyl acetate. It is considered

as an oxygen base which models the oxygen basicity of the basic probe.The left side of Fig. 1.21 shows that as the solvent becomes more basic (e.g.

aromatic or oxygenated substances) it forms acid–base complexes with the Si–OH sites. A smaller amount of basic probes is then adsorbed because of re-duced exothermicity of adsorption to the SiOH–solvent complex sites [43]. Simi-larly, the right side of Fig. 1.21 shows that as the solvent becomes more acidic(e.g. halogenated hydrocarbons) and forms acid–base complexes with the basicprobe molecule, less probe–solvent complex is adsorbed on the acidic Si–OHsurface sites of silica owing to decreased exothermicity of adsorption.

1.5.2Determination of Brønsted (Charge) Interactions with Surface Sites

A Brønsted acid–base interaction is activated if the Lewis interaction is strongenough, e.g. for hydrogen bonds a protolysis occurs. Then in water both theadsorbate (probe) and the adsorbent (solid substrate) become charged. TheBrønsted acidity and basicity thus interlink the Lewis electron acceptor and do-nor activity into true Coulomb charge interactions. Since the distance overwhich this interaction is active supersedes the extension of the van der Waalsinteractions by orders of magnitude, they should be kept apart. However, theconsiderations of proton and electrolyte distributions as a function of the dis-tance from the surface (given, e.g., by the DLVO theory) is not considered heresince the discussion is focused on the surface properties alone.

The clear difference between Lewis and Brønsted acid–base interactions has,however, not always been recognized when selecting molecules for surface prob-ing. In order to avoid complications, nearly ideal polymers are then used asmodel surfaces. However, in particular when using water as a vOCG probe liq-uid on inorganic polar surfaces, the Brønsted activity must be considered.

It may be difficult to detect proton transfer at surface sites if the surface areais not sufficient for detectable adsorption to occur. In the simplest form, the ad-sorption of a proton (acid) on a basic surface site may be described by the Ham-mett parameter �H0� [52]:

AHs � Bs � As � �BH��s �125 a�

1�Ksa � Ks

ads � �asA��as

BH��asAHas

B� � �asAxs

BH�asAHxs

B��f sBH�f s

B � �125 b�

where a = activity, x = mole fraction, f = activity coefficient, s = surface. Assum-ing the surface to be ideal (�A

s =1) we obtain:

H0 � � log Ksa � log�xs

B�xsBH� � � log�as

�A�H� � log�f sB�f s

BH� �126�

H0 � pKsa � log�xs

B�xsBH� � p�A�Hs � log�f s

B�f sBH�


where p(A)Hs represents the proton activity at the surface sites. At the equiva-lence point when xs

BH � xsB then H0 � pKs

a , the acidity constant.The reaction does not necessarily have to be in water. It is sufficient that a

proton exchange between the surface and the basic probe (indicator) moleculesoccur. Figure 1.22 illustrates the titration of titania (anatase and rutile) powdersdispersed in cyclohexane using n-butylamine as the base titrant for acidic sur-face sites and trichloroacetic acid as the acidic titrant for basic surface sites [54].

The indicator probe molecules chosen for the acidic surface sites have increas-ing, but low, pKs

a . They are all weaker bases than n-butylamine. The strength of thesurface sites is determined by H0 < pKs

a and the number of sites is determined bythe amount of n-butylamine consumed in order to reach the equilibrium point(color change of indicator). For the basic surface sites, indicators with a rather highpKs

a are used and trichloroacetic acid is used to desorb these indicators from thesurface until the equivalence point. As shown in Fig. 1.22, the titania sampleshave both acidic and basic sites which can be identified both in number and in(H0) strength.

In water, neglecting the activity coefficients (ideal surface conditions), theequation may be rewritten in the form

H0 � p�A�Hs � pKsa � log�xs

BH�xsB� �127�


Fig. 1.22 Number (per nm2) and strength of acidic and basic surface sitesof titania powders expressed as the Hammett function (H0); anatase(down triangles) and rutile (up triangles) both as delivered (broken line)and washed (full drawn line). For comparison, the surface charge density(left axis, full line) and zeta potential (right axis, broken line) are given foranatase (triangles) and rutile (circles) determined in 0.001 mol dm–3 NaClat 25 �C (from [54], with permission).

It is obvious that the Hammett function may be expressed as a corrected pHscale. However, in non-aqueous solvents the pH concept is not clear. All criticalvalues H0,max, pHPZC and pHIEP usually match (~6.2 for anatase and ~5.3 forrutile) fairly well, but as shown they diverge in the present case [54]. This is in-dicative of surface impurities. The corresponding equation may be written forthe adsorption of the hydroxyl ions on acidic surface sites. Since the surface ischarged in aqueous dispersions, it is customary to relate the pH of the solutionwith the surface concentration with the Boltzmann relation:

nsH � nb

H exp��F0�RT� �128�

where xsH�xb

H � nsH�nb

H.The previous equations do not make any explicit consideration of the proton

exchange equilibrium at the surface. According to the Partial Charge Model, thedegree of hydrolysis (h) of a cation can be estimated from [55]

h � 1��1� 0�41 pH��1�36z� NC��0�236� 0�08 pH�� 2�621� 0�02 pH� �M��

��M� �129 a�

where h = the number of protons spontaneously released by the coordinatedcomplex [M(OH2)N]z+ in solution, z = valency (charge number), NC = coordina-tion number and M

* = Mulliken type electronegativity of the metal. At pH = 0,the equation reduces to [55]

h � �1�36z� 0�24NC� � �2�621� M��M �129 b�

and at pH = 14 to [55]

h � �1�14z� 0�25NC� � �0�836�2�341� �M��M� �129 c�

The most important parameter is the formal valency (z, charge number) of themetal cation; NC and ��M� are of lesser importance. The type of coordinationcan be approximated as the z–pH dependence (Fig. 1.23).

Assuming initially that the metal maintains its coordination complex at thesurface, the ligands may reside in the oxo (M–O–), the hydroxo (M–OH) andthe aquo (M–OH2

+) form. The charging of the surface is then due to a singletype of (average metal) surface sites. The hydrolysis may then be expressed interms of the surface charge density:

�� F��H � �OH� � �F�wSAw��nH � nOH� � �nbH � nb

OH� �130�

where F = Faraday constant, wS = mass of the solid powder sample, Aw = specif-ic surface area, ni = acid or base added to the suspension and nb

i = acid or baseadded to the separated supernatant. Of course, other acids and bases may com-


pete for the adsorption sites. The point where the charges are neutralized(�0 � 0) is denoted the point of zero charge, pHPZC.

The pHPZC values can be used to determine the ratio of OH groups attachedto hydrolyzable surface species (metals) such as Al or Ti. The surface site disso-ciation can be written as [56]

M�OH�2 � M�OH�H� K intM�2 � M�OHH��M�OH�2 �131 a�

M�OH� M�O� �H� K intM�1 � M�O�H��M�OH �131 b�

where KX,m is the equilibrium constant for the metal oxide (M) and m = num-ber of protons. [H+,s] denotes the activity of the protons at the Brønsted surfacesites, which is related to the bulk proton activity ([H+,b]) through the Boltzmannrelation (Eq. 128):

H��s � H��b exp��F0�RT� �132�

where 0 = surface potential, R = gas constant and T = absolute temperature.At pHPZC, when 0 � 0, Eq. (132) states that [H+,s] = [H+,b], i.e. the intrinsic

constant, Kn.mint represents the proton equilibrium constant (acidity constant of

the surface sites) in a chargeless environment. They are assumed to be indepen-dent of the concentrations of the species and the surface potential.


Fig. 1.23 Predominance of aquo, hydroxo and oxo ligands coordinated tometal cations of formal charge z as a function of solution pH (from [55],with permission).

The site number �Ns�� and the total number of OH groups at the surface sites

(Nsm) is given by

Nsm � M�OH � M�OH�2 � M�O� �133�

Using these site number definitions, the surface charge density can be defined as

�� F�A�Nsm��M�OH�2 �M�O��M�OH � M�OH�2 � M�O��

�134�Introducing the equilibrium constants [56]:

�� F�Nsm�A��H��b exp��F0�RT��K int

M�2� � �K intM�1 exp��F0�RT��H��b��

�1� �H��b exp��F0�RT��K intM � � �K int

M�1 exp��F0�RT��H��b�� 135�

The total number of OH per nm2 can be determined by, e.g., titration or ad-sorption experiments. The site density is very dependent on the experimentalmethod and model of analyzing the data [56].

Formally, the charged surfaces are subdivided into non-polarizable and polar-izable surfaces. The polarizable surfaces do not share potential-determining ions(PDIs) with the liquid. Non-polarizable surfaces are characterized by one com-mon species for the surface matrix and the intervening solution. This is typicalfor most solids where potential-determining cations dissolve partially from thesurface, thus determining the surface charge (0). Assuming the metal oxidesurface to be fully polarizable (insoluble) at constant ionic strength (i.e. neglect-ing the ion contribution), we may relate the electrochemical potential to theinterfacial energy by adding the electrical work to the Gibbs-Duhem equation(Eq. 3 a). At constant T and P we obtain

�nsHd�H � ns

OHd�OH� � Ad�SL � Ns�d0 � 0 �136 a�

��sH � � s

OH�d�H � d�SL � ��d0 � ��F�d�H � d�SL � ��d0 � 0 �136 b�

where �� is the surface charge density. This is the Lippmann equation:

�d�SL � ��d0 � ��F�d�H � ��d0 � �� sH � �s

OH�d�H �136 c�

This quasi-thermodynamic relation defines an electrical and a chemical contri-bution to the interfacial energy. The difference between the solid and liquidphases may be varied by an externally applied electrical potential V. The electri-cal potential difference U � �V can be used to replace �0. Deriving the sur-face charge density with respect of the surface potential at constant chemicalpotential for the protons, �H:

��d�SL�dV�P�T�� L � ��S � ��L � F�� sH � � s

OH� �137�


This is the electrocapillarity equation (Fig. 1.23), which shows that whend�SL�dV � 0, then �s

OH � �sH and when d�SL�dV � 0, then � s

H � � sOH. A max-

imum surface energy occurs at pHPZC, when d�SL�dV � 0:

�d�SL�d0�P�T�� 0� � sH � �s

OH �138�

Thus, at the pHPZC, �� 0 and thence � sH � �s

OH as expected. In the absence ofspecifically adsorbing ions, �SL is at a maximum at pHPZC. A shift from this posi-tion due to the presence of specifically adsorbing ions is denoted the Esin-Markoveffect [57]. The chemical contribution may be evaluated assuming Langmuir iso-therm conditions for the chemical potential (e.g. for protons alone) [58]:

d�H � RTd ln��1� �� 139 a�

Again, assuming that only M–O– and M–OH2+ sites exist, we may write

� � ��max � ��2�max �139 b�

where �max � 2F�PZC and �PZC is the surface excess at pHPZC (�� 0 and0 � 0) and �H � �OH � 1�2�max. The equation shows that the interfacial ten-sion is maximum at pHPZC and that both chemical and charge factors contrib-ute to the decrease in surface energy from PZC. Since the adsorption causes adecrease in interfacial energy and since spontaneous dispersion of the systemoccurs for �SL � 0, a point of zero interfacial tension pHPZIT may be identified.In the presence of PDIs, two pHPZIT may be identified on both sides of themaximum. In the presence of electrolytes, a range (pH > pHPZIT) has beenidentified where the surface charge becomes saturated [58].

In addition to a complete account of electrolytes in the double layer providedby the Gouy-Chapman approach [16, 31], Stol and de Bruyn offer the followingsimplified solution to the integrated interfacial energy [58]:

��SL � 2�RT�F�� RT�F��2��max� �140�

where �� and �� are the charge densities of the anions and cations, respec-tively, in the diffuse part of the double layer.

Recalling equation (136 c) �d�H�RTd ln aH � 2�3RTdpH�, the Lippmann equa-tion may be rewritten in the form

d�SL � ��d0 � 2�3��RT�F�dpH �141�

Barthes-Labrousse and Joud derived two limiting conditions from this equation[48].

First, when the pH of the aqueous solution is close to pHPZC of the metaloxide surface, a parabolic dependence of the integrated �SL on pH is observed(Fig. 1.24):


��SL � �8RT�c1�1��d��pHPZC � pH�2 �142�

where c1:1 is the concentration of possible supporting aqueous 1 : 1 electrolytesolution and 1/�d is the Debye screening length of the Gouy-Chapman equa-tion. The ratio expresses the amount of ions per unit surface area.

Second, if it is assumed that the mineral acid or base used to adjust the pHdoes not influence the components (e.g. �SO and �OW� other than the solid incontact with water, they may derive the Young equation (cf. Eq. 106):

d�SW�dpH � ��OW�d cos�SOW��dpH � �OW sin�SOW�d�SOW�dpH� �143�

In the presence of electrolytes, �OW is, however, expected to change but is easilymeasurable. When the surface charge density is close to the maximum value(�max), the interfacial surface energy is linearly dependent on pH (cf. Eq. 141):

d�SW�dpH � 4�6�RT�F��max �144 a�

��max � ��F�4�6RT��WO�d cos�SOW�dpH� �144 b�

��max � �F�4�6RT��WO sin�SOW�d�SOW�dpH� �144 c�

Note that, in order to avoid the development (adsorption and spreading) of anaqueous surface film �S�W�, the measurements were made in a hydrocarbon(O) liquid. Hence �OW represents the interfacial tension between an aqueous so-lution (W) and a hydrocarbon (oil, O) and �SOW the contact angle of the sessileaqueous drop on the solid (S) immersed in the hydrocarbon.

The influence of the electrolyte concentration and of pH on the surface ten-sion of water and on its contact angle with the hydrophilic silica in air (Table1.12) may be related to the previous equations by replacing the oil medium byair:


Fig. 1.24 Electrocapillary curve (Eq. 137) and the corresponding variation ofthe contact angle ��SL� as a function of pH (Eq. 144c) in contact with anamphoteric metal oxide surface. The dependence around the maximum isparabolic. The maximum at pHPZC can be deduced from the positive andnegative linear slopes.

�SL

��c1�1� pH� � ��F�4�6RT��WV�d cos�SW�dpH� �145�

Assuming that pHPZC�2 and that the number of surface sites were Ns�A �1.5 nm2 [56], the surface charge density was calculated using Eq. (145) and thechange in surface energy using Eq. (142), where RT�c1�1��d� � kT�Ns�A�. Theresults were not realistic and are not listed in Table 1.12. The change in contactangle and the change in pH (Eq. 146b) also did not balance. Obviously, theremust be some other effects that are not included in the models offered. Onereason may be the enhanced surface pressure of electrolyte solution or that thepHPZC, assumed to be 2, is shifted upon electrolyte addition. Therefore, the cal-culations were repeated assuming the pHPZC to be 6, but this did not improvethe results. The underlying assumption that the concepts of a fully polarizablesurface can be applied is not supported by the experiments.

In the absence of a water film on the surface, the equations should apply alsoin the absence of the hydrocarbon liquid. The slopes are proportional to �max.Bain and Whitesides related the contact angle to the pKa values of the carboxylgroups in a film [60]. The model can be modified to apply to the surface M–OHgroups with a single acid constant:

cos�SL � cos�SL�PZC� � �NskT�A�LV� lncH��cH � Ka� �146 a�

Since pHPZC = pKa, the equation takes the form (cH << Ka)

�LVcos�SL � cos�SL�PZC� � 2�3kT�Ns�A��pH� pHPZC� �146 b�

Obviously, when pH = pHPZC = KM,1int = pKa then cos�SL � cos�SL�PZC� and

the contact angles of water showed the typical maximum. The derivative [(�LV

cos �SL) – (�LV(PZC) cos �SL(PZC))]/2.3kT(pH – pHPZC) should thus give thechange in the number of surface sites (Ns/A) produced by the change in pH


Table 1.12 Change in surface tension of the electrolyte solution and the(cosine) contact angle for the electrolyte solution on the hydrophilicsilica at pH 2, 6, 10 and 22 ± 1 �C (from [59], with permission).

[NaCl] (mol dm–3) Parameter Value

0 pH 2.05 6.33 10.03�WV (mN m–1) 72.43 72.31 72.57�SW (�) 46.47 36.88 38.02

0.1 pH 2.04 6.20 9.92�WV (mN m–1) 72.79 72.75 73.03�SW (�) 41.71 42.15 35.51

0.5 pH 2.04 6.38 10.04�WV (mN m–1) 73.48 73.55 73.68�SW (�) 45.31 47.99 40.30

(Table 1.12). Assuming that pHPZC�2 and that the number of surface siteswere originally (Ns/A)�4/nm2 the change in surface charge density calculatedusing Eq. (146b) is of the order of only 1–5%.

1.5.3Adsorption Isotherms for Competitive Interaction at Surface Sites

The adsorption isotherms discussed so far can be extended to include the dis-placement of e.g. previously adsorbed liquid (solvent, L) molecules by base mol-ecules (B) from solution:

�AL�s � Bb � �AB�s � Lb �147 a�

KL � �asABab

L��asALab

B� � �xsAB�xs

AL��xbL�xb

B��f sABf b

L�f sALf b

B� �147 b�

where KL � Ksads in previous equations.

Assuming ideal conditions both at the surface and in the bulk (dilute) solu-tion, we may omit the term including the activity coefficients. If we additionallyintroduce xs

AL � 1� xsAB and xb

L�xbB � cb

L�cbB (molar concentrations) into the

equilibrium constant, then

KL � xsAB��1� xs

AB��cbL�cb

B� �147 c�

Rearrangement gives

xsAB � cb

BKL��cbL � cb

BKL� �148 a�

This is one form of the Langmuir adsorption isotherm where xsAB represents

the surface site occupancy (�) of equal non-communicating ideal surface sites.Note that c b

L has been maintained in the equation in order to keep KL dimen-sionless.

Two limiting cases can be anticipated:

1. If abB �� 1, i.e. when cb

L �� cbB, we find � � xs

AB � �cbB�cb

L�KL, representing di-lute solutions.

2. If abB �� 1, i.e. when cb

L �� cbB, we find � � xs

AB � 1, representing a mono-layer of adsorbate molecules.

The experimental data are linearized by inverting the equation:

1�xsAB � cb

L��cbBKL� � 1 �148 b�

A plot of 1�xsAB against 1�cb

B should result in a straight line with interceptequal to unity and a slope of cb

L�KL.


Recognizing that xsAB � ns

AB��nsAB�m (m = maximum number of surface sites)

and multiplying both sides by cbB, we obtain

cbB�ns

AB � cbL��ns

AB�mKLL� cbB��ns

AB�m �148 c�

which is the typical form used (Fig. 1.25). A plot of cbB�ns

AB against cbB is

expected to give a straight line with an intercept cbL��ns

AB�mKL and a slopeof 1��ns

AB�m. Since the total volume (V) is known for dilute solutions,ns

AB � niB � cb

BV and nbL � cb

LV . The number of surface sites is Nsm � NA�ns

AB�mand the area occupied by one site is � � A�Ns

m � wSAw�Nsm. In this calculation,

it is assumed that there is no (surface or concentration) potential against whichthe adsorption occurs.

The molar Gibbs free energy of adsorption is given by

�Gsm�ads � �RT ln KL � �Hs

m�ads � T�Ssm�ads �149�

For spontaneous adsorption, �Gsm�ads must be negative. �Ss

m�ads is also nega-tive since the motion in three dimensions is restricted to bound molecules intwo dimensions. Self-evidently, �Hs

m�ads must be negative, corresponding to anexothermic process.

Suppose that the solid is homogeneous and that the dispersion solvent (L)and the base (B) molecules adsorb on acidic surface sites (A) of equal averagemolar surface areas Am�� NA�, where � is the average area occupied by eachsite). Suppose, further, that both the bulk and the surface phases are ideal. Un-der these circumstances, at adsorption equilibrium, we find for the fractionaladsorption from dilute solutions of B in indifferent (non-adsorbing) liquid L

�S�B�L � �SL � �S�B�L � �RT�Am� ln�xsAB�xb

B� �150 a�

Correspondingly, we find for the fractional adsorption of L in non-adsorbing B

�S�L�L � �SL � �S�L�L � �RT�Am� ln�xsAL�xb

A� �150 b�

The latter equation serves as an illustration of the partial coadsorption of theindifferent dispersion medium. Obviously �xs

AB � xsA� � �xs

AB�max � �xsA�max. We

may therefore write formally for the replacement of adsorbed L by B in an indif-ferent media:

�S�B�L�L � �S�B�L � �S�L�L � �S�L�L � �S�B�L � RT�Am ln�xsABxb

L�xsALxb

B� �150 c�

We define the molar Gibbs free energy of adsorption of monodentate bases onmonodentate acid surface sites thus replacing preadsorbed solvent molecules interms of surface pressure for the adsorbed molecules:



Fig. 1.25 Linearized Langmuir adsorption isotherms for phenol,benzylamine and 3-phenylpropionic acid (3-PPA) on porous silica powderfrom cyclohexane. Fits using both the Freundlich and the Langmuirisotherms to draw the line through the experimental points for phenolare also shown (from [61], with permission).

�Gsm�ads � �RT ln Ks

ads � �RT ln��xsAB�xs

AL��xbL�xb

B�� 151 a�

�Gsm�ads � ��RT�Am��ln�xs

AB�xbB� � ln�xb

L�xsAL�� 151 b�

�Gsm�ads � ��S�B�L � �S�L�L � ��S�B�L � ��S�L�L �151 c�

Utilizing Boltzmann equation the overall Gibbs free energy can thus be sub-divided into its components.

The surface pressure can be determined �xsAB � ns

AB��nsAB�max � xs

AB��xsAB�max�

as a depletion from the solution or an adsorption on the surface with different(e.g. spectroscopic) techniques. We define a bidentate surface �SAB� with pre-dominantly acidic �SA�B� � As� and basic �S�A�B � Bs� interactions, respectively.Likewise, for a bidentate probe �PAB� we denote �PA�B� � PA� and �P�A�B � PB�.For a bidentate probe adsorbing from an indifferent solution onto a bidentatesurface we may then write:

�PB�b � �PA�b � As � Bs �PBA�s � �PAB�s �152 a�Kads ��xs

PBAxsPAB��xb

PBxbPA��as

AasB�� 152 b�

Assuming that the surface may be assumed ideal �asA � as

B � 1� we find for thefractional adsorption of PA and PB in the indifferent liquid (L):

�S�PB�L � �SL � �S�PB�L � �RT�Am� ln�xsPBA�xb

PB� �153 a�

�S�PA�L � �SL � �S�PA�L � �RT�Am� ln�xsPAB�xb

PA� �153 b�

The sum of these reactions in indifferent media gives:

�S�PA�PB�L � �S�PA�L � �S�PB�L � �S�L�L � �S�B�L � 2�SL �154 a�

�S�PA�PB�L � RT�Am�ln�xsPAB�xb

PB� � ln�xsPBA�xb

PA�� 154 b�

The molar Gibbs free energy of adsorption then takes the form:

�Gsm�ads � �RT ln Ks

ads � �RT ln��xsPBAxs

PAB��xbPBxb

PA�� 155 a��Gs

ads � ��S�PA�L � �S�PB�L � ��S�B�L � ��S�L�L �155 b�

where all terms may be determined experimentally as discussed previously. Themass balance is recovered when summing the contributions:

nP � nPA � nPB � �nsPBA � ns

PAB� � �nbPA � nb

PB� �156 a�

nP � nbPA�ks

PAB � 1� � nbPB�ks

PBA � 1� �156 b�

since ksPBA � �ns

PBA�nbPB� and ks

PAB � �nsPAB�nb

PA�.


There is another way of evaluating the competitive adsorption of basic moleculeswith the solvent molecules or with a competing base for the same surface sites[62–64]. The preferential adsorption ms

B (= surface excess, nsB, per unit mass of solid

substrate powder, wS) from a mixed solvent on a solid surface (specific surface areaAw) may also be quantified by considering the following limiting cases [65]:

1. The total amount of competing components (B and L = solvent) before ad-sorption: nO = nBO + nLO.

2. The total amount of competing components in bulk solution at adsorptionequilibrium: nb

T � nbB � nb

L.3. The total number of competing components adsorbed per unit mass of

solid at adsorption equilibrium: msAT � ms

AB �msAL, where ms

B � nsB�wS and

nB � wB�MB and cB � nB�V .

Note that nsAB � nBO � ns

B. Consequently, nO � �nbB � wSms

B� � �nbL � wSms

L�.Since nb

B�nbL � xb

B�xbL, the mass balance may be described by

nBO � nbLxb

B�xbL � wSms

B �157 a�

nLO � nbAxb

L�xbA � wSns

L �157 b�

Multiplication with xbL and xb

B, respectively, gives

nBOxbL � nb

LxbB � wSms

BxbL �158 a�

nLOxbB � nb

BxbL � wSms

LxbB �158 b�

Subtraction of Eq. (158 b) from Eq. (158 a) and considering that nbLxb

B � nbBxb

L yields

nBOxbL � nLOxb

B � wS�msBxb

L �msLxb

B� �159�

Substitution of xbL � 1� xb

B and nLO � nO � nBO and recalling that nBO � nOxBO

gives the equation for the surface excess isotherm [62–64]:

nO�xbB�wS � ms

BxbL �ms

LxbB �160�

where �xbB � xBO � xb

B, which is frequently determined as depletion, e.g. throughthe change in refractive index or by separate sampling using gas chromatography.The plot of nO�xb

B�wS against xB indicates, for positive values (i.e. whenxBO � xB�, a preferential adsorption of component B over L on the solid S. It thenfollows that a negative plot of nO�xb

L�wS indicates that the liquid component L isnegatively adsorbed and the surface phase is less rich than the bulk in L. Figure1.26 shows the subdivision of the composite isotherm into two general classes.

Provided that a linear section can be identified in the composite isotherm, thethickness of the adsorbed layer can be estimated in the following way. Exchang-ing xL � 1� xB in Eq. (160) gives


nO�xbB�wS � ms

B � �msB �ms

L�xbB �161�

Evidently the extrapolated value at xbB � 0 gives �ms

B�xL�1 and the extrapolatedvalue at xb

B � 1 gives ��msL�xB�1. Monolayer adsorption is indicated if [63]

�msB�crit��ms

B�max � �msL�crit��ms

L�max � 1 �162�

Assuming that the probe molecules change surface tension and that all depletedmolecules adsorb on the solid surface, the Gibbs adsorption equation can beused to determine the surface excess:


Fig. 1.26 Schematic linearization of composite adsorption isothermsshowing concentration regions of excess adsorption of one component overthe other (1 � B� 2 � L� with benzene–methanol (open circles) and ben-zene–ethanol (filled circles) as examples (from [64, 65], with permission).

s

�B � �nsB�A� � �ms

B�Aw� � �d�LV�d�B� �1�RT��LV�d� ln cbB� �163�

The influence of the composition on the surface energy ��SV� tension of themixed liquid may also be derived in the following way. Since ns

B � nBO � nB,the composite isotherm Eq. (160) may be written in the form

nO�xbB�A � �Bxb

L � �LxbB �164 a�

Introducing xbL � 1� xb

B, we obtain

nO�xbB�A � �B � ��B � �L�xb

B ��L�B xb

L �164 b�

��L�B � �nO�xb

B�AxbL� �164 c�

The latter equality assumes that the phase boundary is normalized to erase thesurface excess of the dispersion liquid and to account only for the surface excessof the solute probe (B). Replacing �

�L�B with the Gibbs adsorption equation [64]

�xbL�1�RT��d�S�B�L�d ln aB� � �B � ��B � �L�xb

B � �B � �maxxbB �165�

It is thus possible to quantify the preferential adsorption on the L/V surface bysimply measuring the surface tension of the liquid as a function of the compo-sition. Since nO�xB�A represents the change in surface tension with the baseprobe concentration (B), a positive deviation should indicate preferential adsorp-tion and a negative deviation competitive desorption of component (B). For U-shaped isotherms, the component with the lower surface tension (B) adsorbspreferentially on the liquid surface (positive surface excess). In dilute solutions,xB � 0 and hence �B � �

�L�B .

The model introduced may be combined with the adsorption equilibrium, i.e.when a liquid (L) in contact with the solid (S) is replaced by the basic compo-nent B [64]:

�S�B�L � �SL � �S�B�L � �RT�A��

aB

aB�1

�nO�xbB�xL�d ln ab

B �166�

� plot of �nO�xbB�xL� against ln ab

B may thus be graphically integrated to give�S�B�L.

We now have all the elements required to relate the key parameters of adsorp-tion to the molecular models described in Table 1.7: �Gs

ads � �RT ln KL shouldbe proportional to ln KAB of the Edwards, Maria and Gal, Handcock and Marcia-no models, and �Hads should equal �HAB of the Drago and Wayland and Kroe-ger and Drago models. The implicit condition is that the equilibrium constantrefers to adsorption from a fully dispersive solvent and that the enthalpy of wet-ting the solid with the dispersive solvent has been subtracted from the total en-thalpy of adsorption to give �HAB. No consideration has been given, however,


to whether the process is considered to occur solely at the interface or at an in-terface in equilibrium with the bulk solution. Tentatively, the molecular modelsrelate to interactions at surfaces, whereas the adsorption concerns an equilib-rium of the probe molecules between the bulk solution and the surface. Thethermodynamic function relating to the surface alone is the surface pressure.Moreover, since no PV work has been considered, the proper state functionwould, as shown, be the internal energy. If salient specific (AB) interactions areaimed for, the third=order derivatives from the free energies (F or G) shouldprobably be superior state functions for the correlation.

1.6Contributions from Surface Heterogeneities

The surface of a solid substrate differs considerably from that of a liquid in thatthe heterogeneities are not equilibrated by the rapid molecular motions. In real-ity the solid surface is not molecularly smooth, but consists of surface heteroge-neities, such as asperities, dislocations (steps, kinks adatoms and vacancies) anddifferent crystal habits (crystal planes) and other physico-chemical surface het-erogeneities (Fig. 1.27). At each heterogeneous site an energy is stored (e.g. asbroken bonds) providing the surface with specific binding sites, which influencethe wetting phenomena.

Two cases are considered which influence the properties of the system. First,the surface may be of a chemically heterogeneous character, for which theBrønsted interaction and adsorption isotherm are discussed. Self-evidently, thereis an even more extensive influence on the Lewis interaction sites, exemplified,e.g., by the dislocations found at the molecular level on the surface. Second, thesurface may at the macroscopic level be structurally rough, which influencesthe wetting for extremely hydrophobic and hydrophilic surfaces. In the lattercase, the effect of line tension must be considered.

1.6 Contributions from Surface Heterogeneities 65

Fig. 1.27 Solid surfaces may be discontinuous on both the molecular andthe macroscopic scale. At the molecular level the dislocations (steps, kinks,adatoms and vacancies) liberate energetic bonds, the number of whichdepends on the direction of cleavage.

1.6.1Non-ideal Solid–Liquid Brønsted (Charge) Interactions

Jolivet strongly criticizes the naivistic treatment of solid surfaces described pre-viously and claims that no group may in reality exhibit an amphoteric character,in spite of the fact that they may be positively or negatively charged as a func-tion of pH [55]. Instead, a single equilibrium occurs for each site with pKX,m

int =pHPZC. This means that the successive involvement of two protons on the samesurface site appears completely unrealistic. Instead, for the equilibria in Eq.(131) to apply, they must all be assigned to different surface groups. The latterview is fully possible, since the surfaces contain in reality imperfections repre-senting different crystal planes with the surface elements being involved in dif-ferent degrees of binding [66]. Following the multisite complexation (MUSIC)model and the formal bond valence concept (� � z�NC�NC � coordination num-ber), we extend our focus on silica and titania to include alumina in order to il-lustrate the capabilities of this model. For alumina, having the a valency z = +3and coordination numbers 4 and 6, we find the following sequence for � andthe formal charge � � n�� z�O2�� mz�H�� where for singly coordinated OHligands n = 1 and m = 1 [55, 56]:

Al�OH � � 3�4 � 3�4 � � 3�4� 2� 1 � �1�4 Al�4��OH1�4�

Al�OH � � 3�6 � 1�2 � � 1�2� 2� 1 � �1�2 Al�6��OH1�2�

However, there are also doubly and triply coordinated OH (n = 2, 3; m = 1) [55].The dissociation equilibria should therefore be written in the generalized form(cf. Eq. 132):

Aln�OH�n��1� Aln�O�n��2� �H� K intn�1 � �Aln�O�n��2��H��Aln�OH�n��1��

�167 a�

Aln�OH�n��2 Aln�OHn��1� �H� K intn�2 � �Aln�OH�n��1��H��Aln�OH�n��2 �

�167 b�For gibbsite, Al(OH)3, the large 001 faces (13.8 OH per nm2) are characterizedby doubly coordinated OH groups (n = 2) while the sides of the platelets (hk0faces) contain singly (9.6 OH per nm2) and doubly (4.8 OH per nm2) coordi-nated OH groups (Fig. 1.28).

For these groups, Jolivet [55] estimates (NC = 6) that for the dissociations

Al�OH1�2�2 Al�OH1�2� �H� pK int

1�2 � 10 �168 a�

Al2�OH0 Al2�O� �H� pK int2�1 � 12�3 �168 b�

Al2�OH�2 Al2�OH�H� pK int2�2 � �1�5 �168 c�


The pK1,mint of the [Al–OH]1/2–/[Al–O–] equilibrium (pK1,1

int = 23.88) is very highand the singly coordinated groups are therefore present only as Al–OH1/2– orAl–OH2

1/2+, depending on the pH. James and Parks report (C. P. Huang, PhDThesis, 1971) [67] for �-Al2O3 pK1,2

int = 7.89 (0.42 OH per nm2) and pK1,1int = 9.05

(0.39 OH per nm2) and for -Al2O3 pKa1int = 8.50 and pKa2

int = 9.70 (2.7 OH pernm2) in the presence of 0.1 mol dm–3 NaCl. Doubly coordinated groups existonly as Al2–OH within the normal pH range. For Al2O3, the following dissocia-tion constants pKn,1

int were calculated: pK 1,1int (–Al–OH1/2–) = 24, pK 2,1

int (–Al2–OH0)= 12.3 and pK 3,1

int (–Al3–OH1/2+) = 1.6. The increased acidity or weakenedstrength of the O–H bond with increased degree of coordination of the hydroxylligand is clearly reflected.

If one or more of the pKM,mint is outside the available pH range, only a single equi-

librium occurs where pHPZC = pKM,mint . Jolivet [55] also compared the experimental

pHPZC with those calculated with the MUSIC model: Al2O3 (z = 3, CN = 6, �= 1/2),pHPZC(calc.) = 9.1. This compares well with the experimentally found value (9.1)and with those reported by James and Parks for �-Al2O3 (pHPZC = 8.47) and for -Al2O3 (pKPZC = 9.10). According to Jolivet, we may write [55, 56]

�pK intX � 2p�H�� pK int

1�1 � pK int1�2 � 2p�H�� log��XOH�2 ��XO�� 169�

At the pHPZC where 0 = 0, [H+] = [H+,b] and [XOH2+] � [XO–], we find

pHPZC � 1�2�pK int1�1 � pK int

1�2� � 1�2�pKxint �170 a�

pHPZC�Al2O3� � 1�2�pK intAl � 1�2�pK int

Al�1 � pK intAl�2� � 9�1 �170 b�

Obviously, the pHPZC values compare favorably with the values found experi-mentally and calculated with the MUSIC model. The equilibria can also be writ-ten in the following way:

�pK intAl � pK int

1�1 � pK int1�2 � log��AlOH�2��AlOH�2 ��AlO�� 1�2 �171�

We may draw the following general conclusions regarding any single metal (M)surface sites [55]:

� If �pK intX � 4 (high), then [XOH] >> [XOH2

+] � [XO–] and the acid [XOH2+] is

much stronger than the acid [XOH] and the base [XO–] is much stronger than


Fig. 1.28 The hexagonal structure of gibbsite particles andthe dimensions given as assumed maximal cross-section(� = 0.90 nm), length (l = 0.78 nm), width (w = 0.45 nm) andvariable thickness (t). The flat surface is indexed 001 (n = 2)and the sides 010 and 001, the last two being characterizedby n = 1 and 2.

the base [XOH]. The predominant species is XOH and the number of ionizedspecies is very small.

� If �pK intX � 4 (small), the XOH2

+ and XOH acids and XO– and XOH bases, re-spectively, have similar strengths. Then the number of charged groups XOH2

+

and XOH– is large.

These predictions seem, however, not to be met by a number of acidic and basicgroups listed by James and Parks [67], indicating a variable hydroxyl group den-sity for oxides. A more refined analysis of the MUSIC model shows that the dis-sociation equilibrium must be considered separately for each surface group.The pH at which the net charge is zero depends on the relative fractions ofeach type of group, and also on their respective pK int

1 . For many oxides, cancel-lation of the global charge may take place through compensation. Moreover, theinfluence of neighboring hydroxyl groups must be taken into account. The hy-droxyl groups decrease linearly with increase in temperature [68]. However,when the communication between the –OH groups ceases, the dependence ontemperature is strongly reduced. In this case the rehydroxylation becomes muchslower. In porous matrices doubly (geminal) and triply coordinated hydroxylgroups exist [66] which are not described by the MUSIC model and they areonly fractionally available for chemical reactions.

1.6.2Surface Energy of Coexisting Crystal Planes

Since the crystal habit of gibbsite is fairly symmetrical (Fig. 1.28), we may derivea simple model to resolve the surface energy for the two dominant crystalplanes, i.e. the surface (face = F) and the sides (edge = E) in equilibrium withthe saturated solution. The surface area, the volume and the thickness of theparticle may be expressed in terms of the width (w) and a constant k as follows:

A � 2kw2 � 6tw �172 a�

V � kw2t �172 b�

t � V�kw2 �172 c�

where k = (3/2)�

3 = 2.598. The density (�) and molar mass (M) of gibbsite areknown and we define the molar surface energy Gs

m of the particle as

Gsm � 2�3��2kw2�SV�F� � 6wt�SV�E��M �V� �173 a�

Introducing the thickness, the equation takes the form

Gsm � 2�3�2�kw2�SV�F�� 6�V�SV�E��kw��M �V� �173 b�


Keeping V and � constant, we derive the molar surface energy with respect to w:

dGsm�dw � 2�3�4�kw�SV�F�� 6�V�SV�E��kw2��M �V� �173 c�

Since also the second derivative is positive, the extreme point is a minimum.For this equilibrium, we set the free energy derivative equal to zero and reintro-duce the thickness:

4�kw�SV�F�� 6�V�SV�E��kw2�� 6t�SV�E� �174 a�

�SV�E� � 2kw�SV�F��3t �174 b�

We reintroduce this relationship into the original equation for the molar freeenergy and eliminate V:

�SV�F� � Gsmt��4M

�SV�E� � kwGsm��6M

�

� �SV�E��SV�F� � 2kw�3t � 3�46 �175�

The ratio has been found to agree with electron microscope-measured ratios. Forintermediate aging times, �SV�F� � 140 ± 24 mJ m–2 and �SV�E� = 483 ± 84 mJ m–2,which agree with experimentally determined values. It has been shown that theratio of a variable mixture of hydrophilic and hydrophobic particles was linearlyrelated to the contact angle determined with the Cassie model [69, 70]. Conse-quently, the measured average surface energy determined for gibbsite particlescan be subdivided into the surface energy contributions of each crystal plane know-ing the fraction of each partial surface. The opposite is also true: from the totalsurface energy the contributions of each crystal plane can be calculated.

1.6.3Competing Multi-site Adsorption

The nature of these surface sites has been characterized using attributes suchas polar, acid, basic, etc. In medium and high dielectric media, the surface sitesmay develop charges which are enhanced by surface reactions and isomorphicsubstitutions of the constituent atoms (ions). The driving force for the adsorp-tion is to neutralize the excess energy of these surface sites. For adsorption onthese surface sites a “generalized Langmuir equation” (GL) has been developedfrom the localized Langmuir (L) isotherm [71]:

nsA��ns

A�m � ��KGLnA�q��1� KGLnA�q�r�q �176�

The constants q and r are assumed to characterize the width of the distributionfunction and lie within the range 0–1. The constant q characterizes the distribu-tion widening in the direction of the lower adsorption energies and r representsthis widening towards higher adsorption energies. For q = r = 1 the equation is


transformed back into the local Langmuir equation for adsorption on homoge-neous solids. However, when q = 1 and r (= 1/k) is between 0 and 1, a general-ized Freundlich isotherm is obtained. This corresponds to an asymmetric, quasi-Gaussian energy distribution, with widening occurring in the direction of loweradsorption energies. When both q and r lie between 0 and 1, the equation de-scribes the Langmuir-Freundlich isotherm, having a symmetrical quasi-Gaus-sian energy distribution of the sites. Alternatively, the adsorption energy deter-mined with the Freundlich isotherm can be considered to be dependent on thedegree of surface coverage (the lateral interaction between the adsorbate mole-cules). Empirically, the Freundlich adsorption equation is expressed as [16, 31]

msAB � kF�nb

B�1�k �177 a�

ln msAB � ln kF � 1�k ln nb

B �177 b�

where kF and k are experimental constants. The logarithm of the amount ad-sorbed per unit mass of solid �ms

AB � nsAB�wS� is plotted against the logarithm

of equilibrium concentration �ln nbB� to give the constant 1/k as the slope and

the constant ln kF as the intercept. The enhanced fit over the Langmuir adsorp-tion isotherm is illustrated for phenol on silica in Fig. 1.24.

For uniform sites (k = 1), the Freundlich isotherm is comparable to the Lang-muir isotherm for dilute solutions:

Freundlich isotherm: nsAB � kF�nb

B�1�k � kF � nsAB�nb

B

Langmuir isotherm: nsAB � �ns

AB�m�nbB�nb

L�KL � KL � nsAB��ns

AB�m�nbB�nb

L��Consequently, the constants used are related to each others askF � KL��ns

AB�m�nbL��.

According to Perkel and Ullman, the adsorption saturation ��msAB�max� for

polymers fits the equation [72]

�wsAB�wS�max � kFMk �177 c�

The exponent can be related to the conformation that the polymer takes at thesurface: k = 0 (in plane), k = 1 (upright), k = 0.5 (tangled and intertwined), 0 <k < 0.1 (spherical threads).

The indifferent adsorption in excess of the Langmuir adsorption can be ac-counted for by setting the exponential constant of the Freundlich isothermequal to one (k = 1). The resulting isotherm is then called a Henry isotherm.The Langmuir-Henry isotherm then takes the form [61]

�msA�LH � �ms

A�exp � kHnA � �msA�LH�KLHnb

A��1� KLHnbA� �178�

The Freundlich equation is particularly usable for the characterization of theconcurrent adsorption on the different crystal planes. It may also be used for


characterizing the adsorption in multilayers. Since the degree of adsorptionmay continue infinitely with increasing nb

B, the equation is not suitable for highcoverages. Assume that the probe molecules first adsorbed form a monolayer(k= 0) on the surface. In such cases, the multilayer adsorption may be describedby the following equation:

msAB � �kF�m � kFn1�k

B �179�

The constant �kF�m represents the monolayer adsorption (k = 0) and the lastterm kFn1�k

B the multilayer adsorption.

1.6.4Structural Heterogeneities of the Surface

The dependence of the contact angle on chemical heterogeneities at the surfacewas studied by Cassie and Baxter [70]. As discussed previously, the contributionof known crystal planes to the average total surface energy is linearly dependenton their fractional surface area. This observation can be rationalized with thewell-known Cassie equation being related to the work of adhesion as

WSLV � ��G��A� � f1��SL1 � �SV� � f2��SL2 � �SV� �180�

where the function fi denotes the probability of finding (fractional) surface areawith the property i characterized by the contact angle �SLi. Applying the Youngequation, we may write for the contact angle

cos��SL� � f1 cos�SL1 � f2 cos�SL2 �181 a�

where ��SL� is the average contact angle. Multiplying each term by the surfacetension, one obtain the arithmetic surface energy dependence on the fractionalsurface energy of each diverging property. For porous surfaces, the fractionalporc area covered by the liquid (e.g. f2) may be accounted for by setting�SL2 � 0 to give

cos��SL� � f1 cos�SL1 � f2 �181 b�

Wenzel reported on the interdependence of wettability and surface roughnessfor polar surfaces as early as 1936 [73]. The so-called Wenzel equation can bederived from the Young equation [74]:

dG � i�idAi � �SVdA� �LVdA cos�SL � �SLdA �182�

where i represents V = vapor, L = liquid, S = solid. Realizing that the real surfacearea is much greater than the projected (Young) surface area q= Areal /Aprojected, weobtain the work of adhesion:


WSLV � ��G��A� � q��SL � �SV� � �LV cos��SL �183�

Introducing the Young equation ��SL � �SV� � ��LV cos�SL at equilibrium��G��A � 0� we obtain the well-known Wenzel equation:

cos��SL � q cos�SL �184�

where q � 1 relates the Young contact angle ��SL� to the effective contact angle��SL� of the real surface.

Four basic wetting behaviors of a corrugated surface can be identified:

1. the Imbition range3. the Wenzel range4. the Cassie range6. the Lotus range

and two transition zones (2) and (5) (Fig. 1.29).Above the limit of imbition, the liquid is modeled to be sucked into the po-

rous surface structure leaving the top of the asperities in contact with air [75].The surface area fraction in contact with air is denoted f1 � �s

SV. However, wealso have to consider the extended real surface by multiplying the solid surfaceby q, but subtracting the fraction of surface not in contact with the liquid ��s

SV�.Finally, as discussed above, we find a fraction of the surface area covered withliquid for which f2 � 1� �s

SV. The work of adhesion thus takes the form�cos�LV � 1�

WSLV� ��G��A� � q��SL � �SV� � �sSV��SL � �SV� � �LV�1� �s

SV� �185�

Introducing Young equation at equilibrium, we find

cos��SL � q cos�SL � �sSV cos�SL � �1� �s

SV� �186�


Fig. 1.29 Relationship between the effective contact angle �cos��SL� andthe ideal (Young) contact angle �cos�SL� can be divided schematically intofour basic ranges (1, 3, 4 and 6) and two transition zones (2 and 5). Theinfluence of the surface structure on wetting is illustrated for each range.

Therefore, if cos��SL is plotted against cos�SL for the imbition range, the lineextrapolates to 1� �s

SV with a slope �sSV when cos�SL � 0. The critical contact

angle for imbition is [75]

cos�Ic � �1� �s

SV��q� �sSV� �187�

which may be found as a break point above the cos� axis. Imbition occurswhen �SL � �c, i.e. the surface roughness is flooded by the liquid. For a flatsurface q = 1, i.e. cos�c � 1. Common for both the Wenzel and Imbitionranges is that �SL < 90�. When �SL � �c the rough surface remains dry aheadof the drop.

Considering the surface asperities as a powder applied as a layer on a platefor the wicking experiments, but assuming full contact of all surfaces with thewetting liquid, White derived an expression for the suction (Laplace like) pres-sure due to wetting [76]:

�P � �2��SV � �SL��Reff � � ��V � �L�gh � �2�LV cos�SL�Reff � � ��V � �L�gh

�188�

where Reff � 2�1� �S��SAw�S�, �S the density of the particles, Aw the specificsurface area of the particles and �S the volume fraction of the particles in thewetting space considered.

The transition from complete to partial wetting of the surface as a function ofchanging shape of the solid surface (slope of the surface asperities) was demon-strated by Wapner and Hoffman [77]. Their paper actually demonstrates howcertain topographical features may give rise to the birth of air pockets and there-by, for example, explains the formation of nanobubbles when such a surface iscovered by a liquid. This transition is opposite to the Wenzel range, but nowthe air pockets form below the drop on top of the asperities. Hence maintainingthe surface fractions, the same the Cassie range is defined by [78]

WSLV � ��G��A� � q��SL � �SV� � �SL��SL � �SV� � �LV�1� �SL� �189�

since cos�LV � �1 on the liquid-air pockets. Hence, at equilibrium, the Youngequation gives

cos��SL � q cos�SL � �1� �SL�cos�SL � 1� � ��SL � 1� � �SL cos�SL �190�

Therefore, if cos��SL is plotted against cos�SL for the imbition range, the lineextrapolates to �SL � 1 with slope �SL when cos�SL � 0. The critical contact an-gle for the Lotus surface is [79]

cos�Lc � ��SL � 1��q� �SL� �191�

Hence it is through the interdependence of topography and enhanced hydro-phobicity that a surface turns from “normal” hydrophobic to superhydrophobic.


Nature utilizes this phenomenon, for example, in self-cleaning plant leaves (theso-called Lotus effect) [78, 79]. Bico et al. assumed a discontinuity between theWenzel and Lotus (i.e. the Cassie) ranges; however, this was not supported bythe experimental results presented [79], but is more an artifact due to the as-sumption of only purely vertical surface asperities. For hemispherical asperities,Bico et al. derived the equation [78]

cos��SL � �1� �sB�cos�SL � 1�2 �192�

where �sB is the ratio of the base over the total surface area of the asperities.

This type of representation is in agreement with experimental findings andmay probably be used to represent the transition ranges (2) and (5). For exam-ple, in the transition range between the Imbition (1) and Wenzel (3) ranges aliquid collar may be found around the drop in the surface heterogeneities as de-scribed by Apel-Paz and Marmur [80].

Tsujii’s group has demonstrated that even molecular-scale topography contrib-utes to contact angle hysteresis [81]. Topographical characterization has to becarried out with high resolution and at scales of different lengths, which is char-acteristic for the surface structure [81]:

cos��SL � �L�l�D�2 cos�SL �193�

where L and l are the upper and lower limit step lengths, respectively, overwhich the fractal dimension was analyzed and D (2 < D < 3) is the fractal di-mension of the surface (Fig. 1.30). However, in their presentation they made afit only to the data within the Wenzel range, leaving the analysis of the Cassierange uncompleted [82].

Compared with artificial textured surfaces, the description of the form andshape of real surfaces with complex topography sets high requirements on sur-


Fig. 1.30 Plot of cos��SL against cos�SL for an alkylketene dimer (AKD)/dialkyl ketone (DAK) surface. The line drawn according to Eq. (193) ischaracteristic only for part of the surface structure (bold line) relating tothe Wenzel region (from [81], with permission).

face microscopy and especially image analysis. Obviously, root mean square(RMS) roughness or peak-to-valley height parameters may be regarded as onlyindicative when considering, for example, surface porosity or topography-cor-rected wetting behavior. Peltonen et al. applied a set of topographical parame-ters for the description of amplitude, spatial and hybrid properties of surfacesfor a versatile 3D surface characterization of sol–gel samples with differenttopography [83]. It was demonstrated that different sets of parameters describeand identify surfaces of different character. They also demonstrated the topogra-phy-dependent functionality of the studied surfaces. The 3D image data werecaptured by atomic force microscopy (AFM). The challenge to be met is toquantify a real surface not only by RMS roughness, but also, for example, bythe effective surface area, height asymmetry, surface porosity and the number,size and form of local maxima. In this way, the understanding of the role of to-pography in phenomena such as wetting, adsorption/precipitation and liquidpenetration can be considerably enhanced.

1.7Contributions from External Stimuli

Many wetting experiments are performed under uncontrolled conditions, e.g.under the influence of external electrostatic potentials and intensive light. Itmay therefore be of value to consider some not immediately obvious depen-dences of wetting on external stimuli. The discussion on the relationship be-tween the surface energy and the surface potential showed that electromagnet-ism may have a profound influence on the surface states of a solid. This is par-ticularly true for semiconductors, in which the electron distribution within thespace charge region may be considered to be a (negative) mirror image of theions in the aqueous double layer outside the surface.

1.7.1External Electrostatic Potential

For conductive polarizable or non-polarizable surfaces, the distribution of excesscations or anions near the surface is represented by the surface charge density.The Lippmann or electrocapillarity equation discussed previously shows that thesurface energy may be adjusted by applying an external electrostatic potential(E) over the surface (Eq. 137):

��SL ��

0d�SL � �

�

0��dV �194�

From the Young equation, we obtain

cos�SL � �1��LV��

0��dV� � �SV� �195�

1.7 Contributions from External Stimuli 75

Alternatively, the electrowetting can be derived by way of the minimum free en-ergy requirement for thermodynamic equilibrium conditions for a droplet of con-ducting liquid (e.g. water, W) on a solid (S) immersed in a insulating oil (O) [84]:

cos�SOW � cos�0SOW � ��r�0�V2�2�OW�� 196 a�

�OW cos�SOW � �SO � �SW � ��r�0�V2�2�� 196 b�

where �0SOW is the contact angle in the absence and �SOW that in the presence

of the applied external electrical potential ��V�, �r �� SW� the relative dielectricconstant of the conductive liquid close to the solid and � is the thickness of thedielectric layer. The latter is typically of the order of micrometers whereas sizeof the droplet is of the order of millimeters.

Kang identified the Maxwell stress (Fe) and the perpendicular force compo-nents acting at the three-phase contact line (tpcl) as (Fig. 1.31) [84]

Fe � ��r�0�V2�2�� cos ec�SOW �197 a�

Fex � ��r�0�V2�2�� 197 b�

Fey � ��r�0�V2�2�� cot�SOW �197 c�

It is interesting that the horizontal force component is independent of the con-tact angle. This very localized point force would pull the tpcl until it balanceswith the dragging force of surface tension. The macroscopic balance of the hori-zontal force components is given by the electrocapillary equation (Eq. 137).

In simple terms, the surface potential gradient is related to the surface chargedensity �� Ns

��A�

�d�dx� �� r�0 �198�

�V � � � 0 � � is the potential drop between two plates at a distance �

with a medium characterized by the relative dielectric constant �r and �V is the


Fig. 1.31 Electrostatic force and its influence of horizontal and verticalbalance of forces acting on the three-phase contact line (tpcl) (from [84],with permission).

externally applied potential. We may now proceed to relate the electrostaticpotential across the interface, being represented by the differential capacity ofthe double layer in the liquid phase using the electrocapillary equation:

CSL � �d��dV�P�T�� d2�SL�dV2�P�T�� 199�

where �r�0 � �CSL. The major difference between differential capacity and thecapacity of an electric condenser is that it depends on the potential across thedouble layer, whereas that of the electric condenser does not. For the purpose ofintegration the potential difference is referred to the electrocapillary maximum(UECM), which for pH scales occurs at pHPZC (see Fig. 1.24).

1.7.2External Illumination

Let Vh� be the electrical potential generated by photonic energy. Then the poten-tial difference in relation to an arbitrary reference potential ��Vh� � Vh� � V0�is [85]

d�SL�d d�SL�dVh� �200�

We have defined the work of immersion at constant P, T and � (Eq. 6d), whichis also termed the adhesion tension:

ISL � �SV � �SL � �LV cos�SL �201�

The Young equation is utilized to define the surface charge density differencebetween the SV and the SL interfaces [85]:

�� d�ISL��dVh� � d��SV � �SL��dVh� �202 a�

�� LV�d�cos�SL��dVh�� LV sin�SL�d�SL�dVh�� 202 b�

The latter expression is symmetrical with Eq. (144 c) and shows that the changein the contact angle on illumination is proportional to the charge density differ-ence.

Table 1.13 illustrates the dependence of the contact angle on the illumination.As shown, the contact angle decreases slowly with time, indicating that the solidmay store the optical energy.

As shown in Fig. 1.32, the hydrophilization (reduction in contact angle) is re-lated to the degradation time of stearic acid reflected as a reduced absorbancewhich is characteristic for hydrocarbon groups [87]. It is assumed that the hy-drophilization is due to the (radical?) formation of hydroxyl groups at the sur-face [86]. This activation is obviously also related to the degradation mechanism.When energy-rich UV light was used, the time needed to reduce the contact


angle was shortened. The UV irradiance was measured [87] as 1.1 mW cm–2 =11 J m–2 s–1, which when multiplied by the irradiation time gives the illumina-tion energy at the surface �Es

h�� reported in Table 1.13.By differentiation, we obtain the differential capacity difference between the

SV and SL interfaces [85]:

�CSL � d2�ISL��d�Vh��2 � d��dVh� �203�


Table 1.13 Dependence of the contact angle of water on mesoporoustitania (anatase) surfaces on the UV and fluorescent visible illuminationas a function of time (from [86, 87], with permission).

Time (min)

0 15 30 60 180 360 540

Fluorescent visible light

Contact angle, �SW (�) 33 30 24 18 17

Ultraviolet light

Contact angle, �S (�) 49 32 26 9Es

h� (kJ m–2) 0 10 20 40

Fig. 1.32 Reduction in the characteristic infrared peaks for stearic acid asa function of time due to catalytic degradation by titania. Pilkington glass,atomic layer deposition and (sol–gel) surfaces under illumination(from [87], with permission).

where ISL is denoted adhesion tension. Recalling that the process is carried outat constant P, T and �, we replace �� and CSL by �� and �CSL (Eqs. 199 and203), respectively, to give [85]

�0 � �Vh� � ��CSL�T�P� � �204�

A plot equivalent to the electrocapillary curve should be obtained on plotting ISL

against �0 or �Vh� (Fig. 1.33).

The left-hand side from the maximum refers to n-type semiconductor solidsand the right-hand side to p-type. The maximum corresponds to Vh� �V0 � UECM, i.e. the point of zero charge (cf. pHPZC) or the flat band potential.It is assumed that at PZC both the space charge layer and the electrical doublelayer vanish.

The energy gap of intrinsic semiconductors, �Eg�, approximately equals twicethe equivalent heat of formation ��Hf

eq � ��Gfeq� for several binary inorganic

substances. For defect-free (intrinsic) semiconductors, we find that [88]

1�2�Eg� ��Hfeq � EF � �Eb� � k �205�

where �Eb� is the average bond energy and k = 2.7 eV for a large number of sol-ids. Using this equation, Vijh [88] found an energy gap of 4.7 eV for titania,which is not far from the values reported below.

The band gap can also be determined spectroscopically through the transmit-tance (Fig. 1.34). First the refractive index (n) and extinction coefficient (k) aredetermined. The following relationship has been defined for the photonic en-ergy [89]:

Eh� � h� � �e�kn �206�

where �e is the conductivity of the solid. The absorption coefficient () is thendetermined as [90]

h� � l�h�� Eg��m �207�


Fig. 1.33 Schematic illustration of thedependence of the work of immersion �ISL�on the photonic potential (Vh�), resemblingan electrocapillary curve.

where l is a model-dependent constant relating to the transition considered andis determined by �Eg� and m. The exponent m equals 1/2 for allowed direct, 2for allowed indirect, 3/2 for forbidden direct and 3 for forbidden indirect transi-tions. Extrapolating the linear part of �h��m against h� (both in eV) to the ab-scissa gives m as the slope and �Eg� at �h��m � 0 [87].

The bulk value for rutile is 3.0 eV and for anatase 3.2 eV, which correspondfavorably with the values found (3.3–3.4 eV) in Fig. 1.34.


Fig. 1.34 Band gap �Eg� determinations of TiO_2 films for an alloweddirect transition (m = 1/2) (from [87], with permission).

Since the Hamaker constant is also described in terms of refractive indicesand dielectric constants, including a characteristic frequency in the ultravioletregion (� 8 � 1015 s–1), the relationship between the photonic energy parame-ters is obvious [16, 31]. Converting the band gap energies to thermodynamic en-ergy units, we obtain for the 3.0–3.4 eV range (48.1–54.5) � 10–20 J, which is notfar from published Hamaker constant values for titania in air (vacuum) [90]:(15.5–16.8) � 10–20 J, representing the cohesive energy of the solid (Eq. 12 b).

1.8Conclusions

It has been demonstrated that nearly all phenomena occurring at solid surfacescan be related to and investigated by the use of different work of wetting. Thisholds true for the adsorption isotherms, which can be considered as represent-ing the surface pressure in liquids competing for surface sites.

Despite their sensitivity to salient molecular interactions, higher order ther-modynamic parameters, such as heat capacity, expansivity and compressibility,are not determined. In contrast, first-order parameters such as enthalpy are (un-successfully) related to the Gibbs free energy.

The relationships between wetting models currently considered have beenpresented and evaluated. Although the absolute values vary between the models,the overall trends remain the same, frequently being dependent on the probeliquids used. Since the choice is to a large extent a matter of convenience, sim-plification of most current models is suggested.

The equations presented are restricted to equilibrium conditions excludingconsiderations on dynamic wetting, hysteresis effects and differences in advanc-ing and retreating (receding) contact angles. Moreover, only the interfaces be-tween pure solids, liquids and probe molecules are evaluated. The properties ofaqueous electrolyte solutions at a distance from the solid surfaces is discussedonly to a limited extent.

The influence of chemical and structural heterogeneity of the surface hasbeen considered with a few examples. Moreover, it has been shown that externalstimuli influence the surface energy and contact angle. All models can, how-ever, be rationalized in a straightforward way.

Acknowledgments

This chapter is written in support for the PINTA-ShinePro and NETCOAT-MOL-PRINT projects financed by the National Technology Agency of Finland. Theauthor is grateful to Dr. Folke Eriksson for figure drawings and to Dr. MarkkuLeskelä, Dr. Mikko Ritala, Dr. Sami Areva, Björn Granqvist and Mikael Järn forpermission to use unpublished data.

1.8 Conclusions 81


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