Calorimetry at the Solid–Liquid Interface

Zajac, Jerzy Jozef

doi:10.1007/978-3-642-11954-5_6

Jerzy Jozef Zajac⁸

Part of the book series: Springer Series in Materials Science ((SSMATERIALS,volume 154))

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Abstract

Broad principles of Solid-Liquid calorimetry together with some illustrative examples of its use in the field of catalysis are presented here. The first use is related to the determination of surface properties of catalysts, adsorbents and solid materials in contact with liquids. In particular, it is shown how to evaluate the capacity of a given solid to establish different types of interaction with its liquid environment or to calculate its specific surface area accessible to liquids. The second use includes the measurement of the heat effects accompanying catalytic reactions and the related interfacial phenomena at Solid-Liquid and Liquid-Liquid interfaces. Examples of competitive ion adsorption from dilute aqueous solutions, as well as the formation of surfactant aggregates either in aqueous solution or at the Solid-Liquid interface are considered in view of potential applications in Environmental Remediation and Micellar Catalysis.

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Calorimetry Techniques

Adsorption Calorimetry on Well-Defined Surfaces

Determining Surface Energetics of Solid Surfaces

Keywords

1 Introduction

Microcalorimetry, also nanocalorimetry to follow the recent trends in thermal instrumentation and analysis, is a measuring technique that can be used to study interfacial phenomena occurring at the Solid-Liquid interface. Immersion of a solid in a pure liquid or a solution, wetting of a solid initially in contact with a gas or vapour by a liquid, adhesion between two condensed phases upon their “molecular” contact are examples of exothermic phenomena which are accompanied by significant heat evolvement. Competitive adsorption from solution is an important exception to the exothermicity of interfacial phenomena. This is because certain components of the solution may compete against each other to adsorb at the interface. Adsorption from solution is thus considered as an exchange process and formally split into several adsorption and desorption steps. If one of the components is to be preferentially accumulated at the interface, the transfer of its molecules to the interface must be accompanied by the transfer of an equivalent amount of molecules of another component in the reverse direction, i.e., from the interface to the interior of the solution. The “displacement” is a frequent term for this process. The overall effect of such a displacement may be endothermic in numerous systems, thereby giving rise to an entropy-driven phenomenon. The complexity of displacement process is the main reason why the van’t Hoff procedure for heat determination based on the measurement of the temperature dependence of adsorption isotherms frequently leads to unreliable values. In consequence, direct measurement of the thermal effect in these systems by calorimetry is strongly recommended.

Frequently the heat values for the displacement are relatively small, and thereby difficult to be detected. With the recent progress in ultra-sensitive heat flow measurements and the use of a wide variety of accessories to control the experimental conditions, new commercially available calorimeters offer maximum sensitivity, flexibility, and productivity. Nowadays it becomes possible to study the competitive adsorption phenomena with increased sensitivity and lower detection limits than previously possible.

In the field of catalysis, calorimetry may be used in two manners. The first use is related to the determination of surface properties of catalysts, adsorbents and solid materials in contact with liquids. In particular, it is possible to evaluate the capacity of a given solid to establish different types of interaction with its liquid environment or to calculate its specific surface area accessible to liquids. The second use of calorimetry includes the measurement of the heat effects accompanying catalytic reactions and the related interfacial phenomena at Solid-Liquid and Liquid-Liquid interfaces. In the present chapter, this group of calorimetry applications will be illustrated by following the examples of competitive ion adsorption from dilute aqueous solutions and the formation of surfactant aggregates either in aqueous solution or at the Solid-Liquid interface.

The present chapter does not pretend to be an exhaustive record of Solid-Liquid calorimetry applications in Surface Science and Technology. It should be rather regarded as an introductory course with some illustrative examples. It is important to realise that the individual author’s experience in the field has been the principal criterion for selection of specific instruments and their uses, without any intention of neglecting other contributions. The presentation of calorimetry methods will be restricted only to interfacial systems composed of a pure liquid or a dilute binary, at the most, solution in contact with a solid which does not dissolve in the liquid phase. This formalism may be still employed in the case of solutions which are not strictly binary but may be viewed as such (e.g., solutions containing ionizable solutes, background electrolytes or other additives that may be lumped together as constituting a mean solvent or a mean solute).

2 Thermodynamic Treatment of the Solid–Liquid Interface and the Related Interfacial Phenomena

This opening paragraph reviews some basic ideas and methods relating to interfacial phenomena at the Solid-Liquid interface. Usually, the subject is characterised by two main approaches: the presentation of these phenomena in terms of thermodynamics and their molecular interpretation. The detailed treatment of such general concepts and relationships can be found in numerous standard texts [1–5] and the interested reader should consult these texts. Here only a very brief review is provided on which to base the entire text, in particular the language that will be widely used in further discussion.

2.1 Surface Excess Functions and Surface Phase Model

Contrary to the bulk liquid phase which is homogeneous in three directions in space, has a characteristic composition, and is also autonomous (i.e., its extensive properties depend only on the intensive variables characterising this phase such as the temperature T, the pressure P, and the chemical potentials of the solvent $\mu _1 $ and the solute $\mu _2 )$, the formal thermodynamic description of a Solid-Liquid interface presents a serious difficulty. In the interfacial region, the density $\omega $ of any extensive quantity $\Omega $ changes continuously throughout the thickness (Fig. 6.1a).

For real solids, even the two-dimensional homogeneity of the interface is very difficult to attain, because the solid boundary is heterogeneous both in a physical and in a chemical sense (surface heterogeneity and roughness) [6, 7]. In order to overcome this difficulty, the only possible way is to introduce the so-called excess thermodynamic functions [1, 6, 8].

The widely used definitions of excess functions are based on the Gibbs model of the system, in which a flat interface is regarded as a mathematical dividing plane (the Gibbs dividing surface—GDS) [1, 6]. The two phases $\alpha $ and $\beta $ in contact are assumed to remain homogeneous up to the GDS (Fig. 6.1b depicts such a model). Since the Gibbs model provides a complete description of the heterogeneous system in physico-chemical equilibrium, the formal thermodynamic study of interfacial phenomena is commonly based on this approach. Compared to bulk phases, the thermodynamic expressions for the interface contain additional terms relating to interfacial tension and adsorption of chemical species. This model has the advantage of leading rapidly to the principal thermodynamic relationships between the interfacial quantities (e.g., the so-called Gibbs adsorption equation) [1, 6].

Nevertheless, the Gibbs formalism has some serious drawbacks. Firstly, the interfacial properties (i.e., the Gibbs excess functions) adopt different values depending on the position of the GDS and, consequently, have no direct experimental significance. For dilute solutions, this imperfection may be removed through introducing the so-called relative interfacial quantities. Secondly, the Gibbs surface has zero thickness and volume and this is at variance with the obvious physical picture of an interface. Furthermore, the physical meaning of the Gibbs excess functions is difficult to translate into molecular terms.

The alternative surface phase model popularized by Guggenheim is conceptually simpler since the interfacial region is approximated by a thin, homogeneous layer having an arbitrary thickness [6, 8]. This is illustrated in Fig. 6.1c where the two phases $\alpha $ and $\beta $ remain homogeneous up to the imaginary planes which constitute the boundaries of the surface phase. This representation may be directly used for the purpose of constructing molecular models of the interface. The related interfacial properties have a real physical significance and are compatible with the experimentally measured quantities. For systems containing a thermodynamically inert solid in contact with a pure liquid or a dilute solution, one of the separating planes is chosen to coincide with the surface of the solid. Therefore, the area A of the interface (i.e., its cross-sectional area) is identified with the surface area of the solid phase accessible to the liquid. In practice, the main challenge is always to evaluate correctly the thickness $\tau ^{s}$ of the interfacial region.

In the Guggenheim convention [8], the value of any extensive property of the surface phase, per unit area of the interface, may be expressed as

$$\begin{aligned} \Omega ^{s}=\tau ^{s}\cdot \varpi ^{s}=\frac{1}{A}\left[ {\Omega -\left( {\varpi ^{\alpha }V^{\alpha }+\varpi ^{\beta }V^{\beta }} \right) } \right] \end{aligned}$$

(6.1)

where $\Omega $ is the total extensive property of the whole system; $\varpi ^{\alpha }$, $\varpi ^{\beta }$ and $\varpi ^{s}$ are the densities of $\Omega $, respectively, in the two bulk phases and in the surface phase of thickness $\tau ^{s}$ and surface area A; V $^{\alpha }$ and $V^{\beta }$ represent the volume of the bulk phases $\alpha $ and $\beta $. The interfacial enthalpy $H^{s}$, interfacial Gibbs energy $G^{s}$, and interfacial entropy $S^{s}$ are defined in such a manner.

It is satisfactory to define the interfacial tension $\gamma $ as the work required to create isothermally and reversibly a unit area of an interface [1, 8]: $\gamma _{SG} $ (Solid-Gas interface), $\gamma _{SL} $(Solid-Liquid interface), and $\gamma _{LG} $(Liquid-Gas interface). Since $\gamma $ is referred to as an energy per unit area in this formulation, the privileged SI unit is J m$^{-2}$. Nevertheless, interfacial tensions reported in J m$^{-2}$ and N m$^{-1}$ have the same numerical value. Usually more convenient is the submultiple mJ m$^{-2}$ or mN m$^{-1}$ (numerically equivalent to the previously used c.g.s. units). Conceptually, the interfacial tension can be also seen as a new excess quantity that is attributed to the surface phase (or to the GDS) for the Guggenheim (or Gibbs) model to be thermodynamically equivalent to the real system [1, 8]. In heterogeneous systems where adsorption does not occur, like those containing the interface between one-component liquid and gas phases, interfacial tension is numerically equal to interfacial Gibbs energy $G^{s}$ (per unit area of the interface). Otherwise, adsorption takes place with a change in interfacial tension.

From a mechanical standpoint, the interface between a pure liquid and its own equilibrium vapour (or air, as adsorption is to be neglected here) behaves as a membrane of infinitesimal thickness stretched uniformly and isotropically by a force exerted tangential to it. Rapid relaxation towards equilibrium is the hallmark of liquid surfaces: when the viscosity of the liquid is not too high, the freshly formed area has enough time to relax completely and the equilibrium interfacial tension will attain the same value in all surface parts. It is important to realise that, owing to non-equilibrated cohesive forces operating in the interfacial region, the liquid squeezes itself together until it has the locally lowest surface area possible. Therefore, $\gamma _{LG} $ is regarded as a force (per unit length of surface edge) which opposes any attempt to increase the surface area. The terms surface and interfacial tensions are used interchangeably for $\gamma _{LG} $. The surface tension of most liquids (against equilibrium vapour or air) near room temperature ranges between 10 and 80 mJ m$^{-2}$, and decreases in a nearly linear fashion as the temperature rises. The description of the common experimental methods with comments on their suitability may be found in Refs. [4, 9].

In pure water, the collective action of intermolecular hydrogen bonds together with classical Van der Waals forces make water molecules stay close to one another (one water molecule is capable of forming four hydrogen bonds since it can accept two and donate two hydrogen atoms). When pure water is in contact with air, the great surface tension tends to minimize the area of hydrophobic-hydrophilic contact: the experimental value of $\gamma _{LG} $ at 298.15 K is equal to 71.99 $\pm $ 0.05 mJ $\mathrm{m}^{-2}$ [10]. In the temperature range 273.15–323.15 K, the effect of temperature on the surface tension of water against air is given by [11, 12]:

$$\begin{aligned}&\gamma _{LG} =75.716-0.1416\cdot \left( {T-273.15} \right) +\nonumber \\&\quad -0.25054\times 10^{-3}\cdot \left( {T-273.15} \right) ^{2} \end{aligned}$$

(6.2)

Figure 6.2 illustrates the temperature dependence of the surface tension and the numerical procedure leading to the estimate of the surface enthalpy $H_{LG}^s $ of water at room temperature.

The value of $\gamma _{LG} $, which is still equal to 58.9 mN $\mathrm{m}^{-1}$ at T $=$ 373.15 K, approaches zero in the vicinity of the critical temperature T $=$ 647.4 K, where there is no longer an interface between the liquid and the vapour.

In the case of solids, the term surface tension is used only to designate the interfacial tension $\gamma _{S0} $ operating at the boundary between the solid phase and the surrounding vacuum. The determination of $\gamma _{S0} $ for solid surfaces by stretching the surface area against the surface stress is not possible, since the latter is not equal to $\gamma _{S0} $. Solids do not deform reversibly and are capable of retaining their non-equilibrium shapes for a long time [4]. The determination of $\gamma _{S0} $ is sometimes possible from the calculation of reversible work of cleaving a crystal, i.e., by creating fresh surface having the same properties as the original. Several known examples are given in Table 6.1. These results are, however, subject to considerable uncertainty, because the cleavage technique is not entirely reversible [14]. Furthermore, the different crystallographic faces have somewhat different surface tensions owing to the differences in packing density of the atoms.

Table 6.1 Surface tensions of several solid crystals against vacuum, as obtained from the work of cleavage, $w_{cliv} =2\cdot \gamma _{S0} $ [14–17]

Full size table

2.2 Adhesion and Cohesion

Adhesion between a pure liquid and a solid may be described in terms of the interfacial and surface tensions [18]. Consider the reversible process of splitting a unit area of the Solid-Liquid interface in such a way as to create a unit area of the Solid-Vacuum interface and a unit area of the Liquid-Gas interface (as shown in Fig. 6.3a).

Adhesion between the two phases is defined as the reversed process and the Gibbs energy of adhesion is given, at fixed P and T, by the Dupré equation [4, 19–21]

$$\begin{aligned} \Delta _{adh} G_{SL} =\gamma _{SL} -\gamma _{S0} -\gamma _{LG} , \textit{per unit area of the interface} \end{aligned}$$

(6.3)

where $\gamma _{SL} $ is the interfacial tension between both phases; $\gamma _{S0} $ and $\gamma _{LG} $ are the individual surface tensions of the solid against vacuum and the liquid against its equilibrium vapour (or air). For a single solid or liquid phase an analogous procedure (as shown in Fig. 6.3b) yields the Gibbs energy of cohesion (per unit area of the interface)

$$\begin{aligned} \Delta _{coh} G_S =-2\gamma _{S0}\quad \mathrm{or}\quad \Delta _{coh} G_L =-2\gamma _{LG} \end{aligned}$$

(6.4)

Combination of Eqs. (6.3) and (6.4) yields

$$\begin{aligned} \gamma _{SL} =\Delta _{adh} G_{SL} -\frac{1}{2}\left( {\Delta _{coh} G_S +\Delta _{coh} G_L } \right) \end{aligned}$$

(6.5)

In a sense, the right-hand side of Eq. (6.5) may be considered as a generalised Gibbs energy of mixing. If the two phases mix spontaneously in all proportions, the Gibbs energy will decrease during such a process, thereby rendering $\gamma _{SL} $ negative. In thermodynamic terms this means that there is no stable interface. When the phases are immiscible, separation is spontaneous and the interfacial tension becomes positive. In that case, the interface is stable.

In broad outline, the cohesion of molecules (atoms, ions) to form the bulk phase of matter is due to long-range physical interactions (mainly of the van der Waals type) and short-range chemical forces (giving rise to covalent, ionic, metal, or hydrogen bonds). Amongst these various types of interactions encountered more or less frequently in interfacial phenomena involving liquid and solid phases, the London (dispersion) forces and the Lewis acid-base ones are really crucial in the construction of a thermodynamic treatment of interfaces [19, 21–23]. Solid and liquid substances may be classified according to their capacity of forming Lewis acid-base bonding. Materials that can be both Lewis acids (electron acceptors) and Lewis bases (electron donors) are termed bipolar. The monopolar acidic or monopolar basic substance can act exclusively as a Lewis acid or a Lewis base (the other property is negligible). Inert materials, capable of neither acid nor base interactions, are called apolar. It should be noted here that this nomenclature [22, 23] has nothing in common with polarity of molecules per se, as measured by their respective dipole moments.

Dispersion forces are universal because they attract all molecules together, regardless of their specific chemical nature. The potential energy of dispersion attraction between two isolated molecules decays with the sixth power of the separation distance. Based on the so-called Hamaker theory (i.e., the method of pair-wise summation of intermolecular forces) or the more modern Lifshitz macroscopic treatment of strictly additive London forces, it is possible to develop the so-called Lifshitz-Van der Waals expression for the macroscopic interactions between macroscopic-in-size objects (i.e., macrobodies) [19, 21]. Such an expression strongly depends on the shapes of the interacting macrobodies as well as on the separation distance (non-retarded or retarded interaction). For two portions of the same phase of infinite extent bounded by parallel flat surfaces, at a distance h apart, the potential energy of macroscopic attraction is:

$$\begin{aligned} U(h)=-\frac{A_{11} }{12\pi \cdot h^{2}} \end{aligned}$$

(6.6)

where $A_{11} $ is the so-called Hamaker constant which depends on the chemical nature of the molecules (atoms, ions) constituting the phase under consideration and the number of molecules per unit volume in two interacting bodies. The more gradual fall-off of the potential energy (6.6) with distance compared to the molecule-molecule interaction indicates that macroscopic attractions are of a more long-range type and they are expected to make a significant contribution to the total energy of attraction even at longer distances.

The values of $A_{11} $ for the various substances interacting across vacuum or across a medium can be found in Refs. [24, 25]. It should be noted that Hamaker constants for interaction across a medium are usually much lower in comparison with the related values under vacuum (e.g., in the case of two macroscopic bodies of quartz at short distances apart, $A_{11} $equals $41.3\times 10^{-20}$ J for interaction across vacuum and $1.3\times 10^{-20}$ J for interaction across water [25]).

The macroscopic effect of cohesion due to dispersion forces is usually calculated from the Lifshitz-Van der Waals expression (6.6), providing that the separation distance h is known. Israelachvili [21, 22] has proposed a universal value of 0.165 nm to describe the effective spacing h between molecular planes in all liquids with molecules interacting solely through dispersion forces. In this case, the Gibbs energy of cohesion may be evaluated as

$$\begin{aligned} \Delta _{coh} G_L \approx 9.74\times 10^{17}A_{11}\, in\, J m^{-2},\quad \textit{for apolar liquids} \end{aligned}$$

(6.7)

where $A_{11} $ is the Hamaker constant for the liquid substance. In general, considerable theoretical and experimental evidence is consistent with the postulate that the Keesom-Debye contribution to the Gibbs energy of cohesion is very small and has no significant importance between macroscopic bodies in the condensed systems [21, 26]. As a result, the Lifshitz-Van der Waals (LW) component $\Delta _{coh} G^{LW}$ of the Gibbs energy of cohesion for any liquid or solid material is commonly identified with the dispersion contribution.

According to Eq. 6.4, the apolar component of the surface tension of a solid against vacuum or a liquid against its equilibrium vapour (or air) becomes:

$$\begin{aligned} \gamma _{S0}^{LW} =-\frac{1}{2}\Delta _{coh} G_S^{LW}\quad \mathrm{or}\quad \gamma _{LG}^{LW} =-\frac{1}{2}\Delta _{coh} G_L^{LW} \end{aligned}$$

(6.8)

Lewis acid-base interaction between molecules (atoms, ions) differs from a classical covalent bond in that only one of the partners supplies the pair of electrons [27]. Electron pair donors (EPD) are molecules which donate the lone pair of nonbonding electrons (n-EPD), the electron pair of a $\sigma $-bond ( $\sigma $-EPD), or the pair of $\pi $-electrons ( $\pi $-EPD). Electron pair acceptor (EPA) molecules may use a vacant valence orbital (n-EPA), a nonbonding $\sigma $-orbital ( $\sigma $-EPA), or a $\pi $-bond system with electron-withdrawing substituents ($\pi $-EPA). The combinations between all the above donor and acceptor types result in nine types of EPD-EPA complexes, with the bond strength ranging from high values for n-EPD/n-EPA associations to very week $\pi $-EPD/$\pi $-EPA interactions between neutral molecules. Formally, Lewis acid-base interaction includes a hydrogen bond which is usually situated at the lower end of the chemical bond range.

It has long been an operational premise that the Gibbs energy of cohesion for any non-metallic liquid or solid phase can be split into two contributions: an apolar one $\Delta _{coh} G^{LW}$ originating chiefly from dispersion forces and a polar one $\Delta _{coh} G^{AB}$ arising from the Lewis acid-base interaction between the constituent molecules (atoms, ions). Several semi-empirical methods have been proposed in the literature, e.g., [18, 26, 28–30], to determine both contributions. The results can be collated for different liquids and solids and subsequently used to predict the behaviour of new interfaces. Nevertheless, it is important to understand the approximate nature of this approach and to consider the resulting conclusions with caution.

In consequence, the total surface tension of a given non-metallic material against vacuum or its own equilibrium vapour is expressed by the sum of the Lifshitz-van der Waals (LW) and Lewis acid-base (AB) contributions [19, 22]

$$\begin{aligned} \gamma _{S0} =\gamma _{S0}^{LW} +\gamma _{S0}^{AB} =-\frac{1}{2}\left( {\Delta _{coh} G_S^{LW} +\Delta _{coh} G_S^{AB} } \right) \end{aligned}$$

(6.9a)

$$\begin{aligned} \gamma _{LG} =\gamma _{LG}^{LW} +\gamma _{LG}^{AB} =-\frac{1}{2}\left( {\Delta _{coh} G_L^{LW} +\Delta _{coh} G_L^{AB} } \right) \end{aligned}$$

(6.9b)

The change of Gibbs energy during adhesion between two phases is the macroscopic outcome of interactions between the microscopic constituents of the different phases. According to the empirical Berthelot principle [21], the London energy of attraction between two dissimilar macrobodies is a geometric mean of the mutual interactions between similar objects, so that

$$\begin{aligned} A_{12} \approx \sqrt{A_{11} A_{12} } \end{aligned}$$

(6.10)

where $A_{12} $ is the Hamaker constant referring to macroscopic interaction between two different phases.

To obtain the apolar (LW) contribution to the Gibbs energy of adhesion $\Delta _{adh} G_{SL}^{LW} $ between a solid and a liquid, it is assumed that expressions analogous to that given by Eq. 6.7 are still valid. The Berthelot principle Eq. 6.10 may be therefore used to evaluate$\Delta _{adh} G_{SL}^{LW} $. The combining rule for this component of $\Delta _{adh} G_{SL} $ is given by the Good-Girifalco-Fowkes relation [18, 31]:

$$\begin{aligned} \Delta _{adh} G_{SL}^{LW} =\sqrt{\Delta _{coh} G_S^{LW} \cdot \Delta _{coh} G_L^{LW} }=-2\sqrt{\gamma _{S0}^{LW} \cdot \gamma _{LG}^{LW} } \end{aligned}$$

(6.11)

If one of the condensed phases is apolar, dispersion forces are the only important type of interaction operating across the interface and $\Delta _{adh} G_{SL} =\Delta _{adh} G_{SL}^{LW} $. The apolar (LW) surface tension component of any solid can be thus determined by the measurement of the Gibbs energy of adhesion between this material and an apolar probe substance. The latter may be a liquid alkane, methylene iodide, or $\alpha $-bromonaphthalene, for which the surface tension $\gamma _{LG} =\gamma _{LG}^{LW} $ has already been measured.

The two-condensed-phase analog of Eq. 6.9a, 6.9b is:

$$\begin{aligned} \Delta _{adh} G_{SL} =\Delta _{adh} G_{SL}^{LW} +\Delta _{adh} G_{SL}^{AB} \end{aligned}$$

(6.12)

The polar (AB) contributions to the surface tension and Gibbs energy of adhesion are sometimes expressed in terms of Van Oss-Chaudhury-Good parameters [18, 19, 22, 32]

$$\begin{aligned}&\gamma ^{AB}=2\sqrt{\gamma ^{A}\cdot \gamma ^{B}} \end{aligned}$$

(6.13a)

$$\begin{aligned}&\Delta _{adh} G_{SL}^{AB} =-2\sqrt{\gamma _{S0}^A \cdot \gamma _{LG}^B }-2\sqrt{\gamma _{S0}^A \cdot \gamma _{LG}^B } \end{aligned}$$

(6.13b)

where $\gamma ^{A}$ and $\gamma ^{B}$ are the Lewis acid and Lewis base parameters of surface tension $\gamma $, respectively. This approximation leads to the following interesting conclusions: the polar (AB) surface tension component of a pure substance is equal to zero, if the substance is monopolar or apolar. There is no acid-base interaction across the interface, i.e., $\Delta _{adh} G_{SL}^{AB} =0$, if one of the components is apolar or if the two components are monopolar in the same sense, i.e., both being monofunctional acids or both monofunctional bases.

The Lewis acid and base contributions to the surface tension of solids can be derived from measurements of the Gibbs energy of adhesion between the material and a polar probe liquid. At least two different polar liquids (e.g., water and appropriate monofunctional liquid) must be used as probes, provided that their surface components $\gamma _{LG}^{LW} ,\gamma _{LG}^A ,\gamma _{LG}^B $ are known.

2.3 Wetting in Solid–Liquid Systems

Wetting includes the spreading of a pure liquid over the surface of a solid, displacing the gas (or vapour) initially in contact with that surface [18, 33]. Hence the phenomenon involves three interfaces, namely Solid-Gas, Solid-Liquid, and Liquid-Gas ones. The spreading coefficient $W_S $ is defined as [4, 33]

$$\begin{aligned} W_S =\gamma _{SG} -\left( {\gamma _{SL} +\gamma _{LG} } \right) \end{aligned}$$

(6.14)

where $\gamma _{SG} $, $\gamma _{SL} $ and $\gamma _{LG} $ are the appropriate interfacial tensions for the three interfaces at equilibrium. When a portion of the liquid is placed on a uniform, perfectly flat, and non-deformable solid surface and the two phases are allowed to come to equilibrium with the surrounding gas phase, one of the two events may happen:

1.
When $W_S $ is positive or zero, the liquid wets the solid material, i.e., spreads out spontaneously over its surface, providing there is enough liquid to eliminate a unit area of the Solid-Gas interface while exposing a corresponding amount of the Solid-Liquid and Liquid-Gas interfaces.
2.
When $W_S $ is negative, the liquid remains as a drop having, at equilibrium, a definite angle of contact with the solid surface (the liquid does not wet the solid). This case is illustrated in Fig. 6.4.

The equilibrium contact angle $\Theta $ between the liquid and the solid phases is determined by the following balance of interfacial tensions [4, 33, 34]:

$$\begin{aligned} \cos \Theta =\frac{\gamma _{SG} -\gamma _{SL} }{\gamma _{LG} } \end{aligned}$$

(6.15)

known as Young’s equation. This relation applies for contact angles $\Theta $ less than, equal to, or greater than 90$^{\circ }$. In the limiting case where $\Theta =0$, the liquid wets out the solid.

The classical form of Young’s equation, which describes the equilibrium balance of forces meeting at the three-phase contact line in the plane of the solid surface (see Fig. 6.4), is one of the most controversial expressions in Surface Science and there is a long list of objections to it (e.g., see Refs. [4, 33, 34] for details). In spite of that, it is still very widely encountered in the literature. To derive it, one must assume an ideal solid: chemically homogeneous, thermodynamically inert (e.g., it cannot swell under the action of the liquid neither can dissolve in the liquid), and flat at an atomic scale. In practice, appreciable hysteresis of the contact angle is observed in real systems (chemical heterogeneity and roughness of solid surfaces), depending on whether the liquid is advancing or receding across the solid surface [33]. Advancing contact angles are larger than receding angles, and the difference may be sometimes as much as 20$^{\circ }$–30$^{\circ }$. A very detailed critical discussion of the various methods for measuring contact angles can be found in Ref. [34]. The contact angles of powdered solids (e.g., clay minerals) are technically important but are difficult to measure. In the case of numerous fine-grained minerals, which do not occur as large, perfect single crystals with well-developed faces, the contact angles are determined indirectly by column and thin layer wicking [35, 36].

If the liquid is volatile, the gas phase will contain its vapour. In consequence, even though both the fluid phases are nominally pure components, there is in general finite adsorption at the Solid-Gas interface. The equilibrium value of $\gamma _{SG} $ for this interface will be, therefore, lower than its pure-component value $\gamma _{S0} $ by an experimentally determinable quantity, which is called the two-dimensional or surface pressure $\pi _{SG} $ [4, 5]. For the one-component gas phase under the conditions of sufficiently low pressures in contact with an inert solid adsorbent, the value of $\pi _{SG} $ can be evaluated using the adsorption isotherm for the vapour of the liquid on the solid surface:

$$\begin{aligned} \pi _{SG} =\gamma _{S0} -\gamma _{SG} \left( {p^{{*}}} \right) = RT\int \limits _0^{p*} {\Gamma ^{s}} d\ln p\quad \mathrm{{T, P = const}} \end{aligned}$$

(6.16)

where R is the molar gas constant; $\Gamma ^{s}$ denotes the number of moles of gas adsorbed per unit area of the solid adsorbent related to the equilibrium bulk pressure p in the bulk gas phase, at constant temperature T and pressure P; $p^{{*}}$ is the equilibrium pressure at which the actual adsorbed film has been formed on the solid surface: the integration of the adsorption isotherm $\Gamma ^{s}=\Gamma ^{s}\left( p \right) $ is carried out over a p-range from 0 to $p^{{*}}$. On applying Eq. 6.16 to the solid surface saturated with the vapour, i.e., when the latter forms an equilibrium, physically adsorbed film on the available adsorbent surface at p equal to the saturation vapour pressure $p_{sat} $, the surface concentration $\Gamma ^{s}$ and the surface pressure $\pi _{SG} $ are found to reach their (positive) maximum values, $\Gamma _m^s $ and $\pi _{SG}^m $, respectively.

In the case of solids having relatively small values of surface tension $\gamma _{S0} $ against vacuum (usually less than 100 mN $\mathrm{m}^{-1}$ [2]), the effect of gas adsorption is thought to be of little importance. Polymers and many other solid organic compounds are usually given as examples of this category of substrates. Solid materials for which $\pi _{SG}^m $ is small for any adsorbate are named the low-energy solids. For high-energy solids (e.g., mineral oxides, metal sulphides, inorganic salts), the decrease in surface tension due to adsorption is significant [30, 37, 38].

Equations 6.3, 6.15, and 6.16 may be combined to give the Gibbs energy of adhesion between the solid and liquid phases

$$\begin{aligned} \Delta _{adh} G_{SL} =-\gamma _{LG} \left( {1+\cos \Theta } \right) -\pi _{SG}^m \end{aligned}$$

(6.17)

This is a very useful relation, in which $\gamma _{LG} $, $\cos \Theta $, and $\pi _{SG}^m $ can be measured and calculated quite easily and accurately. Therefore, the operating procedures for the adhesion experiment are usually based on Eq. 6.17.

2.4 Hydrophobic and Hydrophilic Substances

Commonly the distinction between hydrophobic and hydrophilic substances is based on the analysis of interactions between their molecules and water as a solvent. A more precise classification of liquid and solid substances as hydrophobic and hydrophilic may be constructed basing on the apolar (LW) and polar (AB) components of their surface tensions. This three-parameter approach is of great importance for the understanding of surface behaviour [22, 32, 39, 40]. A non-metallic substance is hydrophobic if it interacts with water by exhibiting only LW character. It has very little (or none at all) Lewis acid or Lewis base character. Typical substances at the hydrophobic end have low $\gamma ^{LW}$ surface parameters and their $\gamma ^{A}$ and $\gamma ^{B}$ components are equal to zero. Hydrophilic substances have non-zero $\gamma ^{LW}$ components of the surface tension and at least one of their $\gamma ^{A}$ and $\gamma ^{B}$ parameters is significant.

The values of surface tension components that have been derived from the appropriate measurements of the Gibbs energy of adhesion for numerous liquids and solids are listed in Table 6.2. The determination of a set of $\gamma _{LG}^A $ and $\gamma _{LG}^B $ values for probe liquids is based on the choice of the first reference liquid. For this purpose, Van Oss et al. [39] assumed that $\gamma _{LG}^A =\gamma _{LG}^B $ for water. In consequence, all of acid-base parameters in Table 6.2 are relative to those of water.

Table 6.2 Surface tension parameters (in mJ $\mathrm{m}^{-2})$ of some liquids and polymers [23, 32, 39]

Full size table

3 Calorimetry Applied to Evaluate Surface Properties of Solids

The determination of the apolar and polar components of the surface tension of liquids and solids may be a powerful tool for classification of various substances with respect to their hydrophobic-hydrophilic character. In the case of solids, this macroscopic approach provides important information about the potential “global” behaviour of their surfaces against a given environment, however, giving no direct indication in regard with the heterogeneity of the solid surface. In consequence, it cannot replace methods based on the adsorption of probe molecules from the gas phase, but it does complement them by providing a different level of surface scanning. In practice, the experimental procedure is quite long and fastidious referring to the successive determinations of the Gibbs energy of adhesion $\Delta _{adh} G_{SL} $between a given solid and an apolar or polar liquid, which requires, in accordance with Eq. 6.17, measurement of the surface tension $\gamma _{LG} $, contact angle $\Theta $, and vapour adsorption isotherm (to calculate $\pi _{SG}^m )$ for each solid-liquid couple. To make matters worse, the very precise measurement of $\Theta $ is possible only for atomically smooth surfaces. Finally, the additive approximation expressed by Eqs. 6.9a, 6.9b and 6.12 is better suited to calculation of the enthalpy term than to that of the free energy, since the interactions may have both mechanical and entropic contributions [41].

The determination of the related surface enthalpy terms $H_{S0}^s $ and $H_{LG}^s $ from direct calorimetry measurements provides an alternative way to evaluate the hydrophobic-hydrophilic character of a solid surface [38, 42–44], since the following deconvolution procedures may be proposed for the surface enthalpy in analogy with those holding for the Gibbs energy (Eqs. 6.8–6.13a, 6.13b):

surface enthalpy of a non-metallic solid against vacuum:
$$\begin{aligned} H_{S0}^s =H_{S0}^{LW} +H_{S0}^{AB} =H_{S0}^{LW} +2\sqrt{H_{S0}^A \cdot H_{S0}^B } \end{aligned}$$
(6.18a)
surface enthalpy of a non-metallic liquid against its equilibrium vapour (or air):
$$\begin{aligned} H_{LG}^s =H_{LG}^{LW} +H_{LG}^{AB} =H_{LG}^{LW} +2\sqrt{H_{LG}^A \cdot H_{LG}^B } \end{aligned}$$
(6.18b)
Solid-Liquid interfacial enthalpy (Berthelot principle):
$$\begin{aligned} H_{SL}^s&=H_{SL}^{LW} +H_{SL}^{AB} =H_{S0}^s +H_{LG}^s +\nonumber \\&\quad -2\left( {\sqrt{H_{S0}^{LW} \cdot H_{LG}^{LW} }+\sqrt{H_{S0}^A \cdot H_{LG}^B }+\sqrt{H_{S0}^B \cdot H_{LG}^A }} \right) \end{aligned}$$
(6.18c)

where $H_{S0}^{LW}$, $H_{LG}^{LW}$ and $H_{S0}^{AB}$, $H_{LG}^{AB}$ are, respectively, the apolar (LW) and polar (AB) contributions to the appropriate surface enthalpy; $H_{S0}^A$, $H_{LG}^A$ denote the Lewis acid and $H_{S0}^A$, $H_{LG}^A$ the Lewis base components of the surface enthalpy ; the $H_{SL}^{LW}$, $H_{SL}^{AB}$ components refer to the Solid-Liquid interface.

It should be always remembered that the global treatment of surface hydrophobicity-hydrophilicity based on surface and interfacial enthalpies does not include the entropy effects.

For liquids and solids, specific orientation and conformation of unsymmetrical molecules (ions) in the interfacial regions result not only in the maximization of their interaction energy, but also yield entropy effects that cannot be neglected. For example, molecular dynamics calculations of the intermolecular potential function points to a predominant orientation of the water dipoles at the Liquid-Gas interface [45]. Other examples are an icelike structuring of water molecules in the vicinity of crystalline solid surfaces [46] and a specific orientation of the alcohol molecules in the interface between a liquid n-alkanol and water [47].

Immersional and wetting calorimetry is an important method for studying interactions at the Solid-Liquid interface [48–50], especially in the case of finely divided and porous solids where the direct measurements of contact angle are hardly possible.

3.1 Enthalpy Changes in the Thermodynamic Cycle of Immersion-Adsorption–Wetting

Consider a simple experiment in which a clean solid surface (free of adsorbed liquid and vapour impurities) is immersed in an excess of pure liquid (Path 1 in Fig. 6.5). If thermal effects arising from absorption, solubility, and swelling of a solid may be eliminated, the whole enthalpy change on immersion is ascribed only to the interface. Sometimes the immersion of a solid in a liquid is accompanied by the formation of an electrical double layer. For mineral oxide-water systems [51, 52], the double-layer effects (i.e., generation of surface charge by protonation or deprotonation of some surface hydroxyl groups, and adsorption of counterions in the Stern or/and diffuse layers) are clearly secondary in comparison with the basic wetting (this contribution is 10–15 % of the total heat effect, at the most).

The total enthalpy change, at constant P and T, is then written as

$$\begin{aligned} \Delta _{imm} H=A_{SL} \cdot \Delta _{imm} H^{{*}}=A_{SL} \left( {H_{SL}^s -H_{S0}^s } \right) \end{aligned}$$

(6.19)

where $\Delta _{imm} H$ is called the enthalpy of immersion; $A_{SL} $ is the area of the Solid-Liquid interface (often identified with the surface area of a non-microporous solid); $H_{SL}^s $ and $H_{S0}^s $ are the interfacial enthalpies per unit area for the Solid-Liquid and Solid-Vacuum interfaces, respectively.

The experiment of immersion is sometimes performed under completely different conditions. The solid may be first put in equilibrium with the vapour of the immersional liquid at a given equilibrium pressure p (Paths 2 or 3 in Fig. 6.5). The adsorbed gas may be at submonolayer, monolayer or multilayer coverage, depending chiefly on the value of p, but also on the nature of liquid and solid. When the solid is subsequently immersed in the liquid, the measured enthalpy change, called the enthalpy of immersional wetting, $\Delta _W H$, will be different from $\Delta _{imm} H$. The various stages of the immersion-adsorption-wetting cycle are shown in Fig. 6.5.

In the case of ideal wetting (this means that the contact angle among the solid, immersional liquid, and vapour of this immersional liquid $\Theta =0)$, the final state is always the same, irrespective of the path followed. Therefore, one can write for Path 2 in Fig. 6.5:

$$\begin{aligned} \Delta _W H&=A_{SL} \cdot \Delta _W H^{{*}}=A_{SL} \left[ {H_{SL}^s -H_{SG}^s \left( p \right) } \right] \nonumber \\&=\Delta _{imm} H-\Delta _{ads} H\left( p \right) \end{aligned}$$

(6.20)

where $H_{SG}^s \left( p \right) $ is the interfacial enthalpy for the Solid-Gas interface at equilibrium pressure p. The area of the Solid-Gas interface is taken to be identical with $A_{SL} $ (the accessibility of the solid surface does not change when passing from the vapour to the liquid phase).

The enthalpy change

$$\begin{aligned} \Delta _{ads} H\left( p \right) =A_{SL} \left[ {H_{SG}^s \left( p \right) -H_{S0}^\sigma } \right] \end{aligned}$$

(6.21)

refers to the formation of an adsorbed film onto solid in equilibrium with the current gas phase and is therefore named the enthalpy of adsorption from vapour.

Since adsorption at the solid-gas interface is in general exothermic, $\Delta _W H$ increases (becomes less negative) monotonously with precoverage from the initial value $\Delta _{imm} H$ at p = 0 to a limiting steady value $\Delta _W H_m $ at p $=$ p $_{sat}$. Examples of such curves are presented in Fig. 6.6.

As far as the enthalpy is concerned, the interface between the saturated (multilayer) film on the solid surface and the equilibrium vapour phase at p $=$ p $_{sat}$ may be identified with the interface between the liquid and its own vapour [49, 54, 55]. Therefore,

$$\begin{aligned} \Delta _W H_m =A_{SL} \left[ {H_{SL}^s -H_{SG}^s \left( {p_{sat} } \right) } \right] =-A_{SL} H_{LG}^s \end{aligned}$$

(6.22)

where $H_{LG}^s $ is the surface enthalpy of the immersional liquid.

3.2 Immersional and Wetting Calorimetry Experiments

Measurements of the enthalpy changes accompanying the immersion and wetting phenomena may be performed with a Tian-Calvet type differential calorimeter [49, 54]. The experimental procedure includes several intermediate stages which may take much time, especially when the wetting enthalpy is measured as a function of the surface coverage by the vapour of the immersional liquid.

The first stage corresponds to the sample preparation during which a solid sample of a given mass is placed in a bulb made of high temperature glass and closed by a brittle tail (Fig. 6.7a). The subsequent sample evacuation and pre-coverage steps are performed with the solid enclosed in this bulb, as represented schematically in Fig. 6.7. The bulb fixed to the end of a glass tube is placed in the outgassing rig where the sample is evacuated at a high temperature to remove all adsorbed impurities from its surface. Then the tube is sealed off a few cm from the bulb end, fixed to the end of glass rod and transferred to the calorimeter. Prior to wetting experiment, the solid sample is outgassed in a glass bulb and then brought into contact with the vapour of the immersional liquid at a given pressure and at constant temperature $T_{M}$. The equilibrium pressure of the pre-coverage step is controlled by the temperature $T_{b}$ of the thermostated bath ($T_{b}$ must be lower than ambient temperature and $T_{M})$.

The Tian-Calvet type calorimeter system contains a massive calorimetric bloc which acts as a heat sink (its temperature is constant) and a removable calorimetric cell made of stainless steel and designed to fit in a cylindrical hole inside the calorimetric bloc. The instrumental signal is obtained by measuring the heat flux between the cell and the bloc. The temperature signal is derived from a sensor in the bloc: two symmetrical thermal flux meters, each constructed by a series of thermocouples surrounding a cylindrical hole for the measurement cell. The electric signal delivered by the difference in output voltage of the two flux meters is proportional to the temperature difference $\theta $ between the bloc and the cell. The electric signal is directly fed into a computer; the digitized signal is recorded on the computer hard disk and then processed using special software. To control the heat evolvement (or absorption) during each calorimetric run, a pen recorder may be used in analog recording of the signal; here, the vertical pen deflection $\Delta l$ perpendicular to the direction of feed of the recording chart is proportional to the temperature difference $\theta $. The scheme of the calorimetric cell is shown in Fig. 6.8, together with a trace showing a representative thermal profile for immersion.

According to the theoretical equation of Tian [56–59] for a conduction calorimeter working under ideal conditions (e.g., the thermal delay between the temperature change $\theta $ in the calorimetric cell and the response of the sensor is to be neglected), the total heat effect occurring in the calorimetric cell during the time of experiment $t_{exp}$ is given by the following expression:

$$\begin{aligned} Q_{\exp } =\int \limits _0^{t_{\exp } } {P(t)dt} = \frac{\lambda }{g}\int \limits _0^{t_{\exp } } {\Delta dt+\frac{\Lambda }{g}} \int _{\Delta _1 }^{\Delta _2 } {d\Delta } \end{aligned}$$

(6.23)

where P(t) is the heat generation (absorption) rate at time t, $\lambda $ is the thermal conductivity of conducting surface separating the calorimetric cell and the calorimetric bloc, $\Lambda $ is the heat capacity constant of the calorimetric cell with its content, $\Delta $ is the voltage signal produced by the sensor at time t, and g is the proportional constant in the relation $\Delta =g\cdot \theta ; \Delta _{1}$ and $\Delta _{2}$ denote the initial (t = 0) and final (t = $t_{exp})$ sensor indications, respectively. The first term on the right hand side of Eq. 6.23 represents the heat exchange between the bloc and the cell monitored by the sensor during the experiment, whereas the second term corresponds to the temperature rise (decrease) in the calorimetric cell. When the duration of the experiment $t_{exp}$ is chosen such that $\Delta _{1} = \Delta _{2}$ (i.e., the sensor signal returns to the baseline), this second term is always equal to zero and the total heat effect $Q_{exp}$ is determined by integrating the sensor record $\Delta =\Delta $(t) from $\Delta _{1}$ to $\Delta _{2}$:

$$\begin{aligned} Q_{\exp } =\frac{\lambda }{g}\int \limits _0^{t_{\exp } } {\Delta dt=K} \int \limits _0^{t_{\exp } } {\Delta dt} \end{aligned}$$

(6.24)

The calorimeter constant $K=\frac{\lambda }{g}$ is evaluated during the calibration run. Calibration of the area under each thermal peak in the thermogram $\Delta =\Delta (t)$ is carried out by dissipating a known amount of energy in the cell (heating through Joule effect). For this purpose, a special calibration resistor is placed in a glass bulb and introduced into the calorimetric cell under the conditions of real experiment. The operator decides both the power dissipated in the resistor $I^{2}\cdot R$(I—current flowing through the resistor and R—its resistance) and the duration of the calibration step $t_{cal}$. The integration of the resulting thermal peak allows the calibration constant K to be calculated as follows:

$$\begin{aligned} K=\frac{A_{cal} }{I^{2}\cdot R\cdot t_{cal} } \end{aligned}$$

(6.25)

where A $_{cal}$ is the area under the calibration peak.

The standard operating procedure for heat measurement is as follows [49, 54]. The glass bulb containing the solid sample after the sample evacuation (and pre-coverage) stage is introduced into the calorimetric cell, which has been previously filled with the immersional liquid (usually about 15cc). A large part of the glass rod remains outside when the cell is closed. Complete gas-tightnest of the cell is ensured by a special toric seal placed around the rod. At a given temperature, there is still some vapour of the immersional liquid occupying the dead volume inside the calorimetric cell. Then the calorimeter is left overnight to come to thermal equilibrium, giving a steady baseline on the recorder.

After attaining thermal equilibrium, the glass rod is pushed gently down and the glass tail of the bulb is broken against the bottom side of the cell. The immersional liquid penetrates into the bulb and comes into contact with the solid sample. Thermal effects accompanying the related exothermic and endothermic phenomena induce changes in the temperature inside the calorimetric cell and the concomitant heat flux between the bloc and the cell. The global thermal effect is recorded as a thermal peak A $_{exp}$ which can be integrated and transformed to the heat quantity Q $_{exp}$ (Fig. 6.8) making use of the calibration constant K:

$$\begin{aligned} Q_{\exp } =K\cdot A_{\exp } \end{aligned}$$

(6.26)

To extract the net enthalpy of immersion $\Delta _{imm} H$ or net enthalpy of wetting $\Delta _W H$, several correction terms have to be subtracted from Q $_{exp}$. These correction terms are related to (i) breaking the tail of the glass bulb inside the calorimetric cell, (ii) changes in the dead volume of the calorimetric cell and the bulb during the experiment, (iii) evaporation of the immersional liquid and condensation of its vapour, (iv) decrease in the adsorbent mass during the evacuation step. They may be determined in a blank test recording. Nevertheless, only two of them were found to give a noticeable contribution to the total heat effect [43, 54].

When a given volume of the immersional liquid enters the bulb, the liquid level in the cell is lowered and there is some evaporation of the liquid to equilibrate the vapour pressure in the dead volume. This phenomenon yields an endothermic effect which is proportional to the dead volume $V_0 $ of the bulb:

$$\begin{aligned} Q_{cor} =-V_0 \cdot \Delta _{vap} h \end{aligned}$$

(6.27)

where $\Delta _{vap} h$ is the evaporation enthalpy per unit volume measured in a blank run with an empty bulb (without a solid sample). The values of $\Delta _{vap} h$ obtained for some immersional liquids are given in Table 6.3.

Table 6.3 Enthalpies of evaporation per unit volume $\Delta _{vap} h$ for selected immersional liquids, as measured in a blank calorimety run with an empty bulb [60]

Full size table

The second correction term of great importance concerns the loss of the adsorbent mass during the evacuation step [43]. Prior to immersion and wetting measurements, the solid sample is subjected to thermal treatment in the glass bulb under vacuum. The total mass of the sample certainly decreases, depending on the amount of liquid and vapour impurities pre-adsorbed on its surface. The present experimental procedure does not allow the actual mass to be determined precisely because the bulb containing the sample cannot be weighed at the end of the outgassing process. In consequence, the measured enthalpy values may be underestimated, especially if the weight loss is significant (e.g., hydrophilic solids with a high specific surface area may contain much water vapour adsorbed on the surface). The most reliable way of quantifying the mass decrease is to carry out the outgassing procedure separately under the same conditions as those used in the calorimetry experiment (the outgassed sample may be weighed and the mass difference calculated).

Finally, the enthalpy change upon immersion of a solid in a given liquid per unit surface area of the solid is calculated as follows:

$$\begin{aligned} \Delta _{imm} H^{{*}}=\frac{\Delta _{imm} H}{A_{SL} }=-\frac{1}{m_S \cdot S}\left( {Q_{\exp } -V_0 \cdot \Delta _{vap} h} \right) , \mathrm{in}\, \mathrm{mJ\, m}^{-2 } \end{aligned}$$

(6.28)

where $m_{S}$ and S are the mass and the specific surface area of the “dry” solid sample, respectively.

3.3 Hydrophilic-Hydrophobic Series and Harkins-Jura Method

Using approximation Eq. 6.18c to express the interfacial enthalpy $H_{SL}^s $ in terms of the apolar (LW), Lewis acid (A) and Lewis base (B) components of the surface enthalpy for the solid and immersional liquid, the final explicit form for the enthalpy of immersion is as follows [42–44]:

$$\begin{aligned} \Delta _{imm} H=H_{LG}^s -2\cdot \left( {\sqrt{H_{S0}^{LW} \cdot H_{LG}^{LW} }+\sqrt{H_{S0}^A \cdot H_{LG}^B }+\sqrt{H_{S0}^B \cdot H_{LG}^A }} \right) \end{aligned}$$

(6.29)

Provided that the surface enthalpy components for the liquids, i.e., $H_{LG}^{LW} $, $H_{LG}^A $, $H_{LG}^B $ are known, the apolar ($H_{S0}^{LW} )$, Lewis acid ($H_{S0}^A )$, and Lewis base ($H_{S0}^B )$ contributions to the surface enthalpy of solids can be derived from measurements of the enthalpy of immersion. At least one apolar liquid and two polar liquids must be used. The values of $H_{LG}^{LW} $, $H_{LG}^A $, $H_{LG}^B $ for selected immersional liquids, i.e., apolar n-heptane, bifunctional water, and monofunctional basic formamide, are given in Table 6.4.

Table 6.4 Surface enthalpy components for n-heptane, water and formamide [43]

Full size table

The examples of the use of apolar n-heptane, bifunctional water, and monofunctional basic formamide to study the surface hydrophobic-hydrophilic character of several solids are shown in Figs. 6.9, 6.10, and 6.11 [38, 43, 44, 60]. n-heptane and formamide (Merck HPLC grade materials with purity exceeding 99 %), were additionally dried with 3A zeolite molecular sieves. This purification procedure to remove even traces of water from organic solvents is of great importance and strongly recommended for all immersion experiments. Water was deionised and purified with a Millipore Super Q System.

The immersion experiment was repeated three times for each solid sample and the average value taken. The amount of immersional liquid penetrating into the glass bulb upon immersion was determined by weighing the bulb after every run (to evaluate the correction term 6.27). Reproducibility of the calorimetry measurement in regard to the enthalpy of immersion expressed by weight of the solid was within 3 %. To obtain the enthalpy of immersion per unit surface area of the solid $\Delta _{imm} H^{{*}}$, the experimental enthalpy values were divided by the corresponding BET specific surface areas (the adsorption model of Brunauer, Emmet, and Teller applied to the experimental results of gaseous nitrogen adsorption at 77 K taking a cross sectional area of 0.162 $\mathrm{nm}^{2}$ per $\mathrm{N}_{2}$ molecule), i.e., $S=S_{BET} $. Finally, each solid was separately outgassed under the same conditions as those used in the immersion experiment with the purpose of quantifying the mass of dried sample. Then the immersion data were corrected for the mass loss during outgassing (following Eq. 6.28). In consequence of all these additional steps, the uncertainty in the enthalpy determination increased, but was, on average, better than 5 %.

Figure 6.9 shows the experimental values of $\Delta _{imm} H^{{*}}$ for one series of solids immersed in n-heptane. Here the interfacial phenomenon involves only van der Waals interactions between the immersional liquid and the solid, irrespective of the actual surface hydrophobic-hydrophilic balance (SHB) of the latter. Based on the enthalpy results obtained, it is possible to classify these solids with respect to hydrophobic character of their surface.

According to the criteria of hydrophobicity given in Sect. 6.2.4, the most hydrophobic substance has the lowest $H_{S0}^{LW} $ (more precisely $\gamma _{S0}^{LW} )$ surface parameter. Therefore, its enthalpy of immersion in n-heptane per unit area of the solid should have the smallest value. Among the solids presented in Fig. 6.9, only sulphur and polytetrafluoroethylene (PTFE) appear clearly at the hydrophobic end. Such mineral oxides as clay minerals with a lamellar structure (kaolinite, illite, talc, chlorite), precipitated amorphous $\mathrm{SiO}_{2}$ (silica), crystalline $\mathrm{SiO}_{2}$ (quartz), amorphous $\mathrm{Al}_{2}\mathrm{O}_{3}$ (alumina), or crystalline aluminosilicates (zeolite) are known to be more or less hydrophilic (commonly, the presence of numerous functional groups with a polar character in the surface of these materials is advanced as a typical argument). The comparison of the $\Delta _{imm} H^{{*}}$ values reported in Fig. 6.9 indicates that the overall intensity of Lifshitz-Van der Waals interactions between apolar n-heptane and a unit area of the solid surface is not the same for various solids. This means that all microscopically unsaturated structures which may be encountered in solid surfaces differ also in “apolar character” and, consequently, the apolar (LW) component to the surface enthalpy $H_{S0}^{LW} $ is a solid-dependent parameter. Surprisingly, non-porous graphitized carbon black (Graphon), regarded as a weakly hydrophilic solid, yield $\Delta _{imm} H^{{*}}$ greater than those of kaolinite, illite, or quartz which certainly possess more polar groups per unit surface area. Of course, it may be argued that graphitic basal planes in the surface of Graphon act as electron pair donors, thereby participating in $\pi $-EPD/ $\pi $-EPA or $\pi $-complexation interactions with the foreign molecules. Nevertheless, it is more reasonable to evaluate both the apolar (LW) and the polar (AB) components to the surface enthalpy $H_{S0}^s $ in order to compare the differences in the SHB among various solids.

It is worth noting that any arrangement of solid surfaces by hydrophobic character, as well as any ordering or ranking of solids in regard to their SHB may be used only for comparative purposes and do not give the “absolute ranking position” of a given solid within the class of hydrophobic and hydrophilic substances.

Figure 6.10 shows examples of several hydrophobic and hydrophilic solids, together with their $H_{S0}^{LW} $ and $H_{S0}^{AB} $ parameters obtained on the basis of Eq. 6.29. SiC14, SiAl32C14, and SiAl8C14 are mesoporous silica-based materials of the MCM-41 type prepared by a surfactant-assisted synthesis and doping with aluminium (e.g., abbreviation “SiAl32” refers to a molar Si-to-Al ratio of 32). Only PTFE is a clearly hydrophobic substance since it has low $H_{S0}^{LW} $and zero $H_{S0}^{AB} $ component. Amorphous fumed $\mathrm{SiO}_{2}$ (Cab-O-Sil) and mesoporous MCM-41 silica or aluminoslicates can be reckoned among low-energy solids possessing surfaces with a weakly hydrophilic character. For high-energy solids with surface enthalpy $H_{S0}^s $ greater than 500 mJ $\mathrm{m}^{-2}$, the ratio between $H_{S0}^{AB} $ and $H_{S0}^{LW} $ varies from 7 (illite) to 0.2 (SiC); alumina is characterized by an intermediate SHB value of 0.8.

Figure 6.11 illustrates the Lewis acid-base character of the solid surfaces presented in Fig. 6.10. The Lewis acid $H_{S0}^A $ and Lewis base $H_{S0}^B $ components are related to the polar (AB) contribution to the surface enthalpy of each solid: $H_{S0}^{AB} =2\sqrt{H_{S0}^A \cdot H_{S0}^B }$. Crystalline $\mathrm{SiO}_{2}$ (quartz) and, to a smaller extent, amorphous precipitated $\mathrm{SiO}_{2}$ (silica) provide strongly acidic surfaces, whereas the surfaces of MCM-41 mesoporous silicas and Cab-O-Sil have a predominantly basic character.

The Harkins-Jura method for estimating the specific surface area $S_{HJ} $ of a solid available to a given immersional liquid is based on Eq. 6.22 [20, 54, 55]. In practice, water is by far the most frequently used liquid because of the small size of its molecules (the van der Waals diameter of a water molecule is about 0.28 nm). The specific surface area of a hydrophilic solid is thus calculated as follows:

$$\begin{aligned} S_{HJ} =-\frac{\Delta _W H_m }{m_S \cdot H_{LG}^s }=-\frac{\Delta _W h_m }{H_{LG}^s }=-\frac{\Delta _W h_m }{0.118} \end{aligned}$$

(6.30)

where $\Delta _W h_m $, expressed in J g$^{-1}$, denotes the minimum value of the specific enthalpy of immersional wetting, i.e., wetting enthalpy per unit mass of the solid sample $\Delta _W h_m =\frac{\Delta _W H_m }{m_S }$. This enthalpy value is determined from the plot of $\Delta _W h$ against the relative pressure $\frac{p}{p_{sat} }$ at which surface pre-coverage with water vapour has been carried out (Fig. 6.12).

The application of the above procedure is limited only to non-microporous materials, for which the solid surface area available to water is equal to the area of the interface between the adsorbed water film on the surface and the equilibrium water vapour phase. It is always necessary to check the condition of the ideal wetting. Water does not spread on hydrophobic surfaces, like the surface of the Graphon, and the value of $\Delta _W H$ decreases (becomes more negative) as the surface adsorbs more water vapour [20]. This proves that the empty surface has less affinity for water than has the surface of the adsorbed water. Equations 6.22 and 6.30 cannot be applied to such a system.

4 Enthalpy Changes Accompanying Competitive Adsorption from Dilute Solution

The competitive aspect of the phenomenon, nowadays commonly accepted, shows up most clearly in adsorption from concentrated solutions (e.g., mixtures of completely miscible liquids) [6]. In dilute solution, only a very detailed analysis of the adsorption system may reveal that the phenomenon is indeed competitive. In this context, calorimetry has proven very useful in studying the competitivity of adsorption from dilute solution onto solids.

The driving force of competitive adsorption from binary solution onto solids is a macroscopic outcome of intermolecular forces belonging to the following categories: (i) interactions of the solute and the solvent with the solid surface, (ii) interactions among the solution components in the interfacial region, and (iii) solute—solvent, solvent—solvent, and solute—solute interactions in the bulk solution. In numerous adsorption systems, a considerable enhancement of the amount adsorbed is often the result of co-operative effects involving various interaction types. The most spectacular examples are related to the adsorption of amphiphilic substances (e.g., surfactants and polymers) that exhibit reduced water solubility [61, 62]. The general behaviour of amphiphilic molecules in aqueous solution reflects the opposing tendencies of the hydrophobic portion of the molecule to escape from the aqueous environment while the hydrophilic moiety tends to remain immersed in the water. Depending on the surface hydrophilic-hydrophobic character of the solid particules immersed in the solution, such molecules can be compelled to accumulate in an oriented fashion at the Solid-Solution interface or to associate between themselves within the solution (i.e., micellization). Aggregation of surfactant units both at the Solid-Solution interface and in the bulk solution is mainly governed by hydrophobic bonding, which refers to the apparent attraction between hydrophobic moieties being much stronger in water than it is in non-aqueous media [26, 61, 63]. Hydrophobic bonding is a typical example of the so-called thermodynamic interactions having both mechanical and entropic contributions.

Adsorption experiment may be carried out in two different ways: as (1) immersion of a dry adsorbent in a binary solution of a given composition or (2) displacement of the solvent from the solid-solvent interface by the adsorbing solute supplied in a stock solution [6, 64, 65]. Additionally, the effects of displacement are determined either in a batch displacement or in a flow displacement experiment. Depending on the experimental procedure applied, the accompanying changes of macroscopic properties differ to a great extent. The amount adsorbed and enthalpy change on adsorption can be precisely defined in operational terms by considering models of the immersion and displacement experiments.

4.1 Thermal Properties of Dilute Solutions

A dilute solution typically means a solution containing no more than about $10^{-2}$ mol $\mathrm{l}^{-1}$ of solute. For the solvent, a convenient reference state is its own pure liquid state at 1 bar. However, a more useful reference state for the solute, including aqueous solutions of electrolytes and non-electrolytes, is that of infinite dilution in the solvent [66, 67].

The general form of the chemical potential of a real solute in a binary solution of non-electrolytes is therefore [66, 67]

$$\begin{aligned}&\mu _2 =\mu _2^{*} +RT\ln \left( {\frac{m_2 }{1}\cdot f_2 } \right) , f_{2} \, \rightarrow 1~\mathrm{as}~m_{2} \, \rightarrow 0 \end{aligned}$$

(6.31a)

$$\begin{aligned}&\mu _2 =\mu _2^{*} +RT\ln \left( {\frac{c_2 }{1}\cdot f_2 } \right) , f_{2} \, \rightarrow 1~\mathrm{as}~c_{2} \, \rightarrow 0 \end{aligned}$$

(6.31b)

where $\mu _2^{*} $ is the standard chemical potential of the solute; $m_2 $ is the molality of this component and $c_2 $ the corresponding molar concentration. The unity on the right-hand side is written to remind the reader that the molality (the molar concentration) is expressed in mol $\mathrm{kg}^{-1}$ (mol $\mathrm{L}^{-1})$, whereas the activity is dimensionless. The employment of concentration units is often discouraged because the concentration of a given solution varies with temperature, while the molality is independent of temperature. The latter also allows the composition determination to be achieved with greater precision, since any solution may be prepared by weighing the solute and solvent, or the stock solution and solvent (dilution of the stock solution).

The interactions between ions contained in an ionic solution are so strong that the solution approaches ideality in the sense of obeying Henry’s law only at very low values of total ion concentration, usually less than $10^{-3}$ mol $\mathrm{kg}^{-1}$ [68, 69]. In precise considerations, ion activities must be used. In the case of strong electrolytes in aqueous solution, the total Gibbs energy (and any other extensive property) of the solute is the sum of the partial molar Gibbs energies for the individual ions produced by the solute. Suppose that a strong electrolyte $\mathrm{M}_{p}\mathrm{X}_{q}$ dissociates into ions according to the following reaction

$$\begin{aligned} \mathrm{M}_p \mathrm{X}_q =\mathrm{p}\cdot \mathrm{M}^{z+}+\mathrm{q}\cdot \mathrm{X}^{z-}\quad , \quad p\cdot z_+ =q\cdot z_- \end{aligned}$$

(6.32)

where p and q are the number of ions of M and X type, respectively; $z_+ $ and $z_- $ represent the valencies of the respective ions (positive number for cations and negative number for anions). The chemical potential of the solute j in a real solution may be written as [68, 69]

$$\begin{aligned} \mu _j =\mu _j^{*} +(p+q)RT\ln m_\pm +(p+q)RT\ln f_\pm \end{aligned}$$

(6.33a)

where the mean ionic molality $m_\pm $ and the mean ionic activity coefficient $f_\pm $ are defined as follows:

$$\begin{aligned} m_\pm =\left[ {\left( {m_+ } \right) ^{p}\left( {m_- } \right) ^{q}} \right] ^{\frac{1}{p+q}}\quad \mathrm{and}\quad f_\pm =\left[ {\left( {f_+ } \right) ^{p}\left( {f_- } \right) ^{q}} \right] ^{\frac{1}{p+q}} \end{aligned}$$

(6.33b)

Now both types of ion share equal responsibility for the non-ideality.

Strong and long-range Coulombic forces acting between ions are primarily responsible for the departures from ideality (the activity coefficients are lowered) and dominate all other contributions. The effect has been evaluated in the Debye-Hückel theory and there exist several equations, which are useful in estimating the mean activity coefficient [68, 69]. The latter is related to the ionic strength of the solution:

$$\begin{aligned} I=\frac{1}{2}\sum _j {z_j^2 \cdot m_j } \end{aligned}$$

(6.34)

where the sum extends over all ions present in the solution; $m_j $ is the molality of the jth ion. For example, the value of $f_\pm $ can be calculated from the Debye-Hückel limiting law

$$\begin{aligned} \log f_\pm =-A\left| {z_+ \cdot z_- } \right| \cdot \sqrt{\frac{I}{1}} \end{aligned}$$

(6.35a)

where A is the dimensionless constant characteristic of the solvent at the specified temperature and pressure (A = 0.5085 for an aqueous solution at 298.15 K [68, 69]).

The approximation (6.35a) is in good agreement with the experiment at very low molalities: less than 0.01–0.001 mol kg$^{-1}$, depending on charge type (the range of validity becomes narrower when divalent or multivalent ions are present in the solution). Nevertheless, when the departures from the experimental results are large, the activity coefficient may be estimated from the extended Debye-Hückel law

$$\begin{aligned} \log f_\pm =-\frac{A\left| {z_+ \cdot z_- } \right| \cdot \sqrt{\frac{I}{1}}}{1+B\cdot \sqrt{\frac{I}{1}}} \end{aligned}$$

(6.35b)

where B is an adjustable empirical parameter (which involves numerical factors, the dielectric constant, the temperature and the mean distance of nearest approach of the ions [68, 69]).

In thermodynamics of solutions, the partial molal quantity and the apparent molal quantity for a component are usually defined [66]. In the properties of enthalpy, one deals with quantities which cannot be measured in an absolute sense. It is thus necessary to use a reference state from which to make the evaluations. Since the state of infinite dilution of the solute in the solvent is taken as reference, the specification of the composition of the solution by means of the molality $m_2 $ of the solute is consequently most convenient for the calculation of enthalpy.

If the enthalpy H of a two-component solution is expressed as a function of the composition, at constant pressure and temperature, the partial molal enthalpy $\overline{h} _2 $ for the solute is defined as the rate of change of the enthalpy H with change in the number of moles of this component, with the number of moles of the solvent being held constant; that is

$$\begin{aligned} \overline{h} _2 =\left( {\frac{\partial H}{\partial n_2 }} \right) _{T,P,n_1 } \end{aligned}$$

(6.36)

The basic partial molal equation for the total enthalpy of the binary solution is

$$\begin{aligned} H=\sum _{j=1}^2 {n_j \cdot \overline{h} _j } \quad ,\mathrm{{T, P = const}} \end{aligned}$$

(6.37)

where $n_j $ is the number of moles of the jth component in the solution.

The partial molal enthalpy of the solute in a solution may not always be conveniently measured experimentally with the required precision. Usually, the apparent molal property for the solute is most used in connection with dilute solutions. For a solution of two components, the apparent molal enthalpy of the solute $\Phi _H $ is

$$\begin{aligned} \Phi _H =\frac{\Delta H}{n_2 }\quad ,\; \mathrm{{T, P = const}} \end{aligned}$$

(6.38)

where$\Delta H$ is the enthalpy change produced when the solution is formed by adding $n_2 $ moles of solute to $n_1 $ moles of pure solvent.

The total enthalpy of the solution may be conveniently expressed in terms of the apparent molal enthalpy of the solute as

$$\begin{aligned} H=\frac{M_1 }{10^{3}}m_2 \Phi _H +M_1 h_1^{*} \end{aligned}$$

(6.39)

where $M_1 $ is the total mass of the solvent in the solution and $h_1^{*} $ is the specific enthalpy (per unit mass) of the pure solvent.

In the infinitely dilute solution, the value of the molal enthalpy for the solute is identical with the corresponding value of the partial molal enthalpy, namely

$$\begin{aligned} \Phi _H \left( \infty \right) =\overline{h} _2 \left( \infty \right) \quad \textit{at infinite dilution} \end{aligned}$$

(6.40)

For any other solution, the partial molal enthalpy of the solute referred to the state of infinite dilution may be calculated if the apparent molal enthalpy is known as a function of the composition. In such a case,

$$\begin{aligned} \overline{h} _2 -\overline{h} _2 \left( \infty \right) =\Phi _H -\Phi _H \left( \infty \right) +m_2 \frac{d\left[ {\Phi _H -\Phi _H \left( \infty \right) } \right] }{dm_2 } \end{aligned}$$

(6.41)

It frequently happens that the apparent molal enthalpy is expressible as an appropriate function of the composition in order to obtain a curve which does not depart greatly from a simple function and which has not too great a curvature over the given range. For aqueous solutions of strong electrolytes, the value of $\Phi _H $ is sometimes linear with the square root of the molality, over a given range of molality [68]. Converting Eq. 6.41 to the slope of $\Phi _H $ against $\sqrt{m_2 }$, one obtains

$$\begin{aligned} \overline{h} _2 -\overline{h} _2 \left( \infty \right) =\Phi _H -\Phi _H \left( \infty \right) +\frac{1}{2}\sqrt{m_2 }\frac{d\left[ {\Phi _H -\Phi _H \left( \infty \right) } \right] }{d\left( {\sqrt{m_2 }} \right) } \end{aligned}$$

(6.42)

The difference $\Phi _H -\Phi _H \left( \infty \right) $, called the relative apparent molal enthalpy, is the quantity most readily evaluated from calorimetric measurements of enthalpies of dilution. For the following reaction of dilution:

$$\begin{aligned}&\left( {\frac{\mathrm{{M}}_1^0 }{10^{3}}\mathrm{m}_2^0 \times \mathrm{solute} + \mathrm{M}_1^0 \times \mathrm{solvent}} \right) (\mathrm{liq})+\left( {\infty -\mathrm{M}_1^0 } \right) \times \mathrm{solvent(liq)} \nonumber \\&\rightarrow \left( {\frac{\mathrm{M}_1^0 }{10^{3}}\mathrm{m}_2^0 \times \mathrm{solute}+\infty \times \mathrm{solvent}} \right) (\mathrm{liq}) \end{aligned}$$

(6.43)

the change in the enthalpy is equal to

$$\begin{aligned} \Delta _{dil} H\left( {m_2^0 \rightarrow \infty } \right) =-\frac{M_1^0 }{10^{3}}m_2^0 \left[ {\Phi _H -\Phi _H \left( \infty \right) } \right] \end{aligned}$$

(6.44)

where $M_1^0 $ denotes the initial mass of the solvent and $m_2^0 $ the initial molality of the solute. Values of the relative apparent molal enthalpy may be obtained by measuring enthalpy changes $\Delta _{dil} H$ for the dilution of a given stock solution of molality $m_2^0 $ to a series of different molalities $m_2 $,

$$\begin{aligned} \Delta _{dil} H\left( {m_2^0 \rightarrow m_2 } \right) =-\frac{M_1^0 }{10^{3}}m_2^0 \left[ {\Phi _H \left( {m_2^0 } \right) -\Phi _H \left( {m_2 } \right) } \right] \end{aligned}$$

(6.45)

and suitably extrapolating the curve $\Delta _{dil} H=\Delta _{dil} H\left( {m_2 } \right) $ to infinite dilution.

4.2 Macroscopic Description of Competitive Adsorption

Compared to interfacial phenomena occurring at the boundary between a solid and a pure liquid, Coulombic forces constitute an additional type of interaction that should be taken into account when analysing the adsorption phenomenon at an electrified interface from solution containing free charges (ions and electrons) and associated charges (dipolar molecules and polarised atoms). In such systems, there is a variation in the charge density across the interfacial region and an electric double layer forms [70–72] (the Gouy-Chapman-Stern-Grahame model of the EDL is shown in Fig. 6.13).

Transfer of the individual ionic species between the interface and the solution leads to a thermodynamic equilibrium at which the interface is electrically neutral as a whole (and so is the equilibrium bulk solution).

Based on isothermal reversible work done in transferring one mole of ions of the solute 2 from infinity (in vacuum) to a given part in the interfacial region which has a non-zero average charge and where the electrostatic potential is$\Psi $, it is possible to define the so-called electrochemical potential of component 2 [70]

$$\begin{aligned} \mu _2 =\mu _2 +z_2 \cdot \mathsf{{F}}\cdot \Psi \quad ,\mathrm{{T, P = const}} \end{aligned}$$

(6.46)

where $\mu _2 $ is the chemical potential of the solute, $z_2 $ is the valency of an ion of the solute, and F is the Faraday constant (F is a product of the Avogadro number L and the elementary charge e). If the transferred species carries no net charge, i.e., $z_2 =0$, the electrochemical potential $\overline{\mu }_2 $ becomes simply the chemical potential $\mu _2 $. The new potential $\overline{\mu }_2 $ is now the quantity which must have the same value everywhere in the system for thermodynamic equilibrium to be established at constant temperature and pressure [70].

According to the surface phase model (cf., Sect. 6.2.1), when the concentrations of the components of the binary solution $\beta $ in the solid phase $\alpha $ are to be neglected, i.e., $c_j^\alpha =0$ (j =1,2), the interfacial concentration $\Gamma _j^s $ of component j is given by [6, 8]

$$\begin{aligned} \Gamma _j^s \equiv \frac{n_j^s }{A}=\tau ^{s}\cdot c_j^s =\frac{1}{A}\left( {n_j -c_j^\beta V^{\beta }} \right) ,\quad (\mathrm{j} = 1,2) \end{aligned}$$

(6.47)

where $c_j^\beta $ and $c_j^s $ are the concentrations (in moles per unit volume of solution) of the jth component in the bulk solution $\beta $ and in the surface phase of thickness $\tau ^{s}$ and surface area A; $V^{\beta }$ represents the volume of the bulk phase $\beta $ in the surface phase model. The subscripts 1 and 2 denote the solvent and the solute, respectively; this notation will be used throughout the present chapter.

The case of $\Gamma _j^s >0$ defines the preferential adsorption of component j at the Solid-Liquid interface. In a given adsorption system, the interfacial concentration $\Gamma _j^s $ is a function of both the actual composition of the bulk solution (i.e., $c_1^\beta $ and $c_2^\beta )$ and the relative affinities of the components for the interface (chiefly, for the solid surface). Contrary to the case of adsorption at the Solid-Gas interface, the interfacial region is always completely filled with molecules of solvent and solute. Further consideration will be based on the preferential adsorption of solute against solvent, taking the interface between the solid and the pure solvent as a starting point for thermodynamic treatment of competitive adsorption. In such a case, the adsorption equilibrium can be represented schematically as follows:

$$\begin{aligned} \mathrm{r}\cdot \mathrm{n}_2^\beta +\mathrm{n}_1^\mathrm{s} \,\displaystyle {\rightleftarrows }\, \mathrm{r}\cdot \mathrm{n}_2^\mathrm{s} +\mathrm{n}_1^{\beta } \end{aligned}$$

(6.48)

where r is the molar ratio of displacement introduced to account for unequal molecular sizes of both the components. It is worth noting that the same formalism may be applied in the treatment of adsorption at the interface between a binary liquid solution and a gas phase.

When the pure liquid phase is replaced by a binary liquid solution $\beta $, the interfacial tensions $\gamma _{SL} $ and $\gamma _{LG} $ will be lowered owing to the preferential adsorption of the solute at the Solid-Liquid and Liquid-Gas interface. The relationship between the extent of adsorption and the resulting interfacial (surface) tension change (T, P $=$ const) is given by the following equation developed by Guggenheim [6, 8]

$$\begin{aligned} d\gamma&=-\left[ {\Gamma _2^s -\Gamma _1^s \left( {\frac{c_2^\beta }{c_1^\beta }} \right) } \right] d\mu _2 =-\left[ {\Gamma _2^s -\Gamma _1^s \left( {\frac{n_2^\beta }{n_1^\beta }} \right) } \right] d\mu _2 \nonumber \\&=-\left[ {\Gamma _2^s -\left( {\frac{M_1^s }{10^{3}A}} \right) \cdot m_j^\beta } \right] d\mu _2 \end{aligned}$$

(6.49)

where $\mu _2 $ is the chemical potential of solute in the bulk solution $\beta $: at equilibrium, $\mu _2 $ is the same throughout the system; when expressing the solution composition in terms of the molality $m_2^\beta $ (in moles per kilogram of solvent), $M_1^s $ is the mass of solvent in the surface phase. It is worth noting that Eq. 6.49 is formally analogous to the Gibbs adsorption isotherm for a binary solution [6, 8].

4.3 Competitive Adsorption Measurements

The amount of solute molecules preferentially adsorbed from dilute solution onto a given solid can be measured in a separate adsorption experiment, independently of the calorimetry measurement. In the case of the titration calorimetry procedure, this is even the only possibility to determine the amount adsorbed after each injection step and subsequently calculate the differential molar enthalpy of adsorption. The main difficulty here, contributing to a significant uncertainty of the experimental result, is related to the necessity of reproducing strictly the same experimental conditions in both types of experiment (i.e., the same solid surface-to-solution volume ratio, evolution of the pH and ionic strength in the equilibrium bulk solution, charging behaviour of the solid surface, etc.).

The quantity of adsorption is usually measured by means of the solution depletion method [6] in glass stoppered tubes or flasks (Fig. 6.14). A known mass of the solid sample and a certain amount of dilute binary solution of given composition are put into each tube or flask, and the mixture is shaken at a constant temperature (e.g., in a thermostated box) for a period of time necessary to attain the adsorption equilibrium. Then the solid particles are separated from the supernatant by centrifugation or filtration (special attention should be paid during fitration to avoid the retention of the solute on the filtration membrane). After centrifugation when the solid particles have been precipitated at the bottom of the tube or flask, a sample of the supernatant may be collected by using a special syringe with a long needle. The equilibrium composition of the supernatant is determined by referring to the appropriate analytical technique (UV spectroscopy, refractometry, total carbon analysis, etc).

The amount $n_2^s $ of solute adsorbed at the Solid-Liquid interface is calculated by means of the following formulas:

$$\begin{aligned} \frac{n_2^s }{m_S }=\frac{V^{0}}{m_S }\left( {C_2^0 -C_2^\beta } \right) \quad \mathrm{or}\quad \frac{n_2^s }{m_S }=\frac{M_1^0 }{10^{3}\cdot m_S }\left( {m_2^0 -m_2^\beta } \right) \end{aligned}$$

(6.50)

where $m_S $ is the mass of the solid sample in a given tube or flask; $C_2^0 $ and $m_2^0 $ are respectively the molarity and the molality of the initial solution (before adsorption); $C_2^\beta $ and $m_2^\beta $ are respectively the molarity and the molality of the supernatant solution $\beta $ (after the attainment of adsorption equilibrium); $V^{0}$ denotes the volume of the solution put initially into the tube or flask and $M_1^0 $ is the total mass of the solvent in the heterogeneous system. When the specific surface area of the solid $S_{BET}$ or $S_{HJ}$ is known, it is even possible to calculate the interfacial concentration $\Gamma _2^s $ of the solute. It should be realised that expressions 6.50 hold only for dilute binary solutions containing a solute that is preferentially adsorbed at the Solid-Liquid interface.

For a given adsorption system, the amount of solute adsorbed at equilibrium depends on the temperature T, the pressure P, and the composition of the equilibrium solution phase $\beta $. The experimental results of adsorption measurements are usually reported in the form of individual adsorption isotherms showing the quantity $\frac{n_2^s }{m_S }$ or $\Gamma _2^s $ as a function of the solution composition:

$$\begin{aligned}&\frac{n_2^s }{m_S }=\frac{n_2^s }{m_S }\left( {C_2^\beta } \right) \quad \mathrm{or}\quad \frac{n_2^s }{m_S }=\frac{n_2^s }{m_S }\left( {m_2^\beta } \right) , \mathrm{{T, P = const}}\end{aligned}$$

(6.51a)

$$\begin{aligned}&\Gamma _2^s =\Gamma _2^s \left( {C_2^\beta } \right) \quad \mathrm{or}\quad \Gamma _2^s =\Gamma _2^s \left( {m_2^\beta } \right) , \mathrm{{T, P = const}} \end{aligned}$$

(6.51b)

Figure 6.15 shows three selected shapes of adsorption isotherms at the Solid-Liquid interface for various charged and uncharged solutes exhibiting a limited solubility in the solvent employed. In general, typical adsorption curves present the amount adsorbed as a smooth, monotonically increasing function of the solute content in the equilibrium bulk solution. Many adsorption isotherms end up in an upper composition range at a constant amount adsorbed (a plateau of the isotherm).

The appearance of an adsorption plateau region at equilibrium concentrations (molalities) approaching the solubility limit indicates that the phenomenon involves only single solute species that are individually dissolved in the solvent. When the adsorption plateau is observed at lower concentrations (molalities), it is usually argued that surface sites of a given type have been saturated by the adsorbing solute species.

The experimental adsorption isotherms may be plotted on different scales, thereby allowing a more detailed analysis of the subsequent adsorption stages to be made. When the adsorption of homologous substances onto the same solid is due only to their different solubilities in the solvent, the use of a reduced concentration (molality) scale to represent the corresponding isotherms results in a single adsorption curve. Any changes in the adsorption mechanism with increasing bulk concentration (molality) show up most clearly in double logarithmic (log-log) plots. The transitions between such adsorption regions are obviously more gradual in reality than they seem to be on a log-log scale. The plateau adsorption region markedly manifests itself only on a double linear (lin-lin) scale. In the linear-logarithmic (lin-log) plot, it is difficult to observe the effects of surface heterogeneity and co-operative adsorption at low bulk concentrations (molalities) but the top parts of the isotherm become clearer.

By analogy to Eq. 6.16, the surface pressure per unit surface area of the Solid–Liquid interface may be determined as follows:

$$\begin{aligned} \pi _{SL} \left( {C_2^{*} } \right) =\gamma _{SL1} -\gamma _{SL} \left( {C_2^{*} } \right) =\frac{RT}{m_S \cdot S}\int \limits _{C_2 =0}^{C_2^{*} } {n_2^s d\ln C_2 } \end{aligned}$$

(6.52)

where $\gamma _{SL1} $ is the interfacial tension of the solid in equilibrium with the pure solvent; S is the specific surface area of the solid (i.e., $S = S_{BET}$ or $S = S_{HJ})$; $C_2^{*} $ denotes the equilibrium bulk concentration taken into consideration.

4.4 Immersion in Dilute Solutions

The solution depletion procedure can be generalised to construct a simplified model of immersion experiment [64, 65]. This model is very useful to define the measurable enthalpy quantities, which does not necessarily mean that they can be easily obtained experimentally. The immersion process is schematically represented in Fig. 6.16.

Before immersion the system separately contains an outgassed solid sample of a given mass $m_S $ and a given surface area S in vacuum and a binary solution of molality $m_2^0 $ consisting $M_1^0 $ grams of the solvent, all at constant temperature T and pressure P. Then the solid is immersed in the solution under conditions of constant T, P. The phenomenon of adsorption induces an uneven partition of solvent and solute molecules between the solid-solution interface and the bulk solution. After attaining the adsorption equilibrium, the molality of the solute in the bulk of the solution (sufficiently far from the solid-liquid interface) becomes equal to $m_2^\beta $. Since absorption, dissolution and swelling effects are not taken into consideration, the adsorbent is considered as thermodynamically inert, i.e., its mass, specific surface area and bulk phase properties do not change during adsorption. Furthermore, the adsorption of the solute at the Solution-Gas interface and the relative surface enthalpy for this interface are omitted for convenience: changes of both effects during immersion in an excess of dilute solution are negligible small.

The balance of enthalpy in the initial (init) and final (fin) states of the model system can be written as follows (cf. Sect. 6.4.1):

$$\begin{aligned}&H^{init}=m_S \cdot S\cdot H_{S0}^s +m_S \cdot h_S +M_1^0 \cdot h_1^{*} +\frac{M_1^0 }{10^{3}}m_2^0 \cdot \Phi _H \left( {m_2^0 } \right) \end{aligned}$$

(6.53a)

$$\begin{aligned}&H^{fin}=m_S \cdot S\cdot H_{SL}^s +m_S \cdot h_S +\frac{M_1^0 }{10^{3}}m_2^\beta \cdot \Phi _H \left( {m_2^\beta } \right) +M_1^0 \cdot h_1^{*} \end{aligned}$$

(6.53b)

where $H_{S0}^s $ and $H_{SL}^s $ are the interfacial enthalpies for solid in vacuum and for Solid-Liquid interface in equilibrium with a bulk solution of molality $m_2^\beta $, respectively; $h_S $ is the specific (per unit mass) enthalpy of the bulk of the solid phase; $\Phi _H \left( {m_2 } \right) $ is the value of the apparent molal enthalpy of the solute in a binary solution corresponding to the molality $m_2 $; $h_1^{*} $ is the specific enthalpy (per unit mass) of the pure liquid solvent. Equation (6.53b) has been developed based on the hypothesis that changes in the mass of the solvent can be neglected during immersion in excess solution.

The total thermal effect of the immersion experiment is:

$$\begin{aligned} \Delta H_{\exp }&=H^{fin}-H^{init}=m_S \cdot S\cdot \left( {H_{SL}^s -H_{S0}^s } \right) \nonumber \\&\quad +\frac{M_1^0 }{10^{3}}\left[ {m_2^\beta \cdot \Phi _H \left( {m_2^\beta } \right) -m_2^0 \cdot \Phi _H \left( {m_2^0 } \right) } \right] \end{aligned}$$

(6.54a)

Taking into account the reference state for the solute defined in Sect. 6.4.1 and using Eqs. (6.40), (6.47), and (6.50), this expression may be transformed as follows:

$$\begin{aligned}&\Delta H_{\exp } =m_S \cdot S\cdot \left( {H_{SL}^s -H_{S0}^s } \right) -m_S \cdot S\cdot \Gamma _2^s \cdot \overline{h} _2 \left( \infty \right) + \nonumber \\&-m_S \cdot S\cdot \Gamma _2^s \left[ {\Phi _H \left( {m_2^\beta } \right) -\Phi _H \left( \infty \right) } \right] -\frac{M_1^0 }{10^{3}}m_2^0 \left[ {\Phi _H \left( {m_2^0 } \right) -\Phi _H \left( {m_2^\beta } \right) } \right] \end{aligned}$$

(6.54b)

Since the adsorption of the solvent is here considered negligible (i.e.,$M_1 =M_1^0 )$, the first two terms on the right hand side of Eq. (6.54b) represents the enthalpy of immersion in a binary solution (1+2) with respect to the bulk solution at infinite dilution taken as the reference state [64]; the third term corresponds to the enthalpy of dilution of $m_S \cdot S\cdot \Gamma _2^s $ moles of the solute from molality $m_2^\beta $ to infinite dilution in the solvent (cf., Eq. 6.44) and it is often incorporated into the previous enthalpy contribution to give the enthalpy of immersion $\Delta _{imm} H_{12} \left( {m_2^\beta } \right) $ of the solid in excess solution of molality $m_2^\beta $ [48, 76, 77]. The last term represents the dilution effects in the bulk solution: $\frac{M_1^0 }{10^{3}}m_2^0 $ moles of the solute are diluted from molality $m_1^0 $ to molality $m_2^\beta $. Finally,

$$\begin{aligned} \Delta H_{\exp } =\Delta _{imm} H_{12} \left( {m_2^\beta } \right) +\Delta _{dil} H\left( {m_2^0 \rightarrow m_2^\beta } \right) \end{aligned}$$

(6.55)

When the dilution term is evaluated independently in appropriate dilution experiment, the enthalpy of immersion $\Delta _{imm} H_{12} \left( {m_2^\beta } \right) $ can be determined experimentally by means of the same calorimetry equipment as that used to measure the enthalpy of immersion in a pure liquid (Sect. 6.3.2). However, great difficulties may be encountered when evaluating the usual correction terms in case of solutes which are volatile or surface-active (the composition of the vapour occupying the dead volume of the bulb is unknown). This method is also tedious. For systems containting electrified interfaces, the effects of EDL formation additionally contribute to the complexity of the adsorption phenomenon. Here the displacement experiment yields the enthalpy data easier to interpret.

4.5 Model of Flow Calorimetry Experiment

Liquid-flow microcalorimetry is a reliable method to measure simultaneously the enthalpy changes and amounts of adsorption under dynamic conditions. Calorimetry experiments may be carried out in two different ways by following a pulse or saturation operating mode [64, 78–83]. In the pulse mode, small aliquots of a stock solution at a known concentration are injected into the carrier liquid (pure solvent) flowing through the adsorbent bed placed inside the calorimetric cell. In this case, the calorimetric system contains an additional loop injection facility (a manual injection valve with appropriate injection loops). The interpretation of the enthalpy data obtained is straightforward only when the whole amount of the solute injected is irreversibly adsorbed on the solid surface.

The saturation mode (continuous-flow method) is more frequently used. Here changes in enthalpy and amount adsorbed of the solute correspond to the formation of a Solid-Liquid interface being in thermal and material equilibrium with the percolating stock solution of a given composition. Repeated adsorption and desorption cycles with the liquid phase in contact with the solid surface for a time required to reach equilibrium can be used to assess reversibility of the phenomenon, and quantify the reversible and irreversible adsorption components [79, 80]. In addition, the same equipment allows probing for some active sites in the solid surface.

The physical meaning of thermodynamic quantities measured in the flow calorimetry experiment, may be discussed based on a simplified model of the system (Fig. 6.17). The model system is composed of three parts: (i) a reservoir R containing a given volume of the stock solution of molality $m_2^0 $, (ii) a cell C with the solid sample of mass $m_S $, in contact with the solvent or the stock solution, (iii) a trap T for the effluent [65, 78].

Initially, the outgassed solid sample is immersed in $M_{1C}^0 $ grams of pure solvent and the total quantity of stock solution in the reservoir is given by the mass of solvent $M_{1R}^0 $; the trap is empty. Then the flow of stock solution from the reservoir is directed to the cell under constant liquid-flow conditions. The temperature and pressure are assumed to be uniform throughout the system and there is neither loss of energy nor loss of matter between the reservoir, the cell and the trap. As the stock solution flows through the cell with the adsorbent, the boundary between the solution and the solvent remains sharp, and the effects of adsorption and desorption change its position according to the ideal chromatographic behaviour. In practice, this means that the change of liquid composition should be sufficiently small for heat of mixing contribution to be neglected. The final state is achieved just before the front of the stock solution reaches the trap. At this point, the trap is filled with $M_{1T} $ grams of pure solvent and the equilibrium molality of the solute within the cell (far from the solid surface) is equal to $m_2^0 $. The composition of the stock solution in the reservoir does not change but its amount decreases; the final mass of the solvent is $M_{1R} $.

For the initial state, the amount of the solute $n_2^{init} $ in the model system and the total enthalpy of the system $H^{init}$ are given by (cf. Sect. 6.4.1):

$$\begin{aligned} n_2^{init}&=\frac{M_{1R}^0 }{10^{3}}m_2^0 \end{aligned}$$

(6.56a)

$$\begin{aligned} H^{init}&=m_S \cdot S\cdot H_{SL1}^s +m_S \cdot h_S +M_{1C}^0 \cdot h_1^{*} \nonumber \\&\qquad +\frac{M_{1R}^0 }{10^{3}}m_2^0 \cdot \Phi _H \left( {m_2^0 } \right) +M_{1R}^0 \cdot h_1^{*} \end{aligned}$$

(6.56b)

where $H_{SL1}^s $ is the interfacial enthalpy for the solid-solvent interface; other quantities have been introduced previously (cf., Eqs. 6.53a, 6.53b).

Applying the law of mass conservation to the solvent in the system, the corresponding quantities in the final state may be written as follows:

$$\begin{aligned} n_2^{fin}&=m_S \cdot S\cdot \Gamma _2^s +\frac{M_{1R} }{10^{3}}m_2^0 +\frac{M_{1R}^0 +M_{1C}^0 -M_{1R} -M_{1T} }{10^{3}}m_2^0\end{aligned}$$

(6.57a)

$$\begin{aligned} H^{fin}&=m_S \cdot S\cdot H_{SL}^s +m_S \cdot h_S +\frac{M_{1R} }{10^{3}}m_2^0 \cdot \Phi _H \left( {m_2^0 } \right) +M_{1R} \cdot h_1^{*} \nonumber \\&\quad +\frac{M_{1R}^0 +M_{1C}^0 -M_{1R} -M_{1T} }{10^{3}}m_2^0 \cdot \Phi _H \left( {m_2^0 } \right) \nonumber \\&\quad +\left( {M_{1R}^0 +M_{1C}^0 -M_{1R} -M_{1T} } \right) \cdot h_1^{*} \end{aligned}$$

(6.57b)

where $H_{SL}^s $ refers to the interfacial enthalpy for the solid-solution interface in equilibrium with a bulk solution of molality $m_2^0 $. The comparison of expressions (6.56b) and (6.57b) indicates that the adsorbent is considered as thermodynamically inert, i.e., its parameters $m_S ,S,$ and $h_S $ do not change during adsorption.

The law of mass conservation applied to the solute gives rise to the following relation:

$$\begin{aligned} \Gamma _2^s =\frac{\left( {M_{1T} -M_{1C}^0 } \right) }{10^{3}\cdot m_S \cdot S}m_2^0 \end{aligned}$$

(6.58a)

This relation shows that the quantity of adsorption for the solute may be directly measured in the flow calorimetry experiment if the values of $M_{1C}^0 $ and $M_{1T} $ are known. When the solid bed in the cell contains a non-adsorbing solid, the difference $M_{1T} -M_{1C}^0 $ is equal to zero; the value of $M_{1C}^0 $ can thus be measured in the appropriate “blank” run. Otherwise, this difference depends on how much longer the solute is retained by the adsorbent. In the model flow experiment, the concentration curves obtained with a non-adsorbing and adsorbing solid represent obviously square profiles (Fig. 6.18a).

If the feed solution flow rate $d_{pump} $ is constant and the solvent migrates at the same velocity throughout the whole system, the mass of solvent entering the trap is directly proportional to the retention time $\tau _A $:

$$\begin{aligned} M_{1T} =\frac{10^{3}\cdot d_{pump} }{10^{3}+m_2^0 \cdot M_{solute} }\cdot \tau _A \end{aligned}$$

(6.58b)

where $M_{solute} $ is the molar mass of the solute. The amount of solute adsorbed onto solid sample can be thus calculated from the corrected retention time, i.e., $\tau _A -\tau _{NA} $ (Fig. 6.18a). In a real calorimetry run, the shape of the $m_2 $ vs. time plot depends on the underlying equilibrium isotherm of solute adsorption, as well as diffusion and mass-transfer kinetics. This results in a breakthrough curve, as shown in Fig. 6.18b. The quantity of solute adsorption is calculated from the difference between the areas over the breakthrough curves obtained with a non-adsorbing and adsorbing solid [79, 80].

When using a pen recorder, the retention measurements are made in terms of chart distances and the area difference is determined by graphical integration. In modern systems, this theoretically correct analysis is also easy to perform when the digitized signal is recorded on the computer hard disk. In practice, the calculation of the areas always includes signal noise and it is very sensitive to the integration limits (the mass transfer is often slow and the plateau concentration $m_2^0 $ is reached slowly, thereby resulting in significant systematic errors). Therefore, it is easier and better to handle the retention times (chart distances) defined by characteristic points method (e.g., the retention parameters derived from the inflection point or the half-height [84]).

The thermal balance in the flow calorimetry experiment can now be expressed in the following way:

$$\begin{aligned} \Delta H_{\exp } =H^{fin}-H^{init}&=m_S \cdot S\cdot \left[ {\left( {H_{SL}^s -H_{S0}^s } \right) -\Gamma _2^s \cdot \Phi _H \left( {m_2^0 } \right) } \right] +\nonumber \\&\quad -m_S \cdot S\cdot \left( {H_{SL1}^s -H_{S0}^s } \right) \end{aligned}$$

(6.59a)

Taking into account the discussion in Sect. 6.4.4, the first term on the right hand side represents the enthalpy of immersion $\Delta _{imm} H_{12} \left( {m_2^0 } \right) $ of the solid in excess solution of molality $m_2^0 $ and one obtains:

$$\begin{aligned} \Delta H_{\exp } =\Delta _{imm} H_{12} \left( {m_2^0 } \right) -\Delta _{imm} H_1 =\Delta _{dpl} H \end{aligned}$$

(6.59b)

where $\Delta _{dpl} H$ denotes the integral enthalpy of displacement [64], which is the main enthalpy effect measured in the liquid-flow calorimetry experiment.

The molar integral enthalpy of displacement is calculated as follows:

$$\begin{aligned} \Delta _{dpl} h=\frac{\Delta H_{\exp } }{m_S \cdot S\cdot \Gamma _2^s } \end{aligned}$$

(6.60)

This quantity provides information about the excess of component-adsorbent interactions averaged over all surface domains from which the solvent has been displaced by the adsorbing solute species. In consequence, it is not easy to monitor subtle changes in the adsorption mechanism based on usually small variations of the $\Delta _{dpl} h$ values with increasing quantity of adsorption. Compared to the differential molar enthalpy of displacement, the enthalpy $\Delta _{dpl} h$ is less sensitive to the energetic heterogeneity of the solid surface.

4.6 Model of Batch Calorimetry Experiment

Liquid titration calorimetry is a microcalorimetric batch technique most often used to study the mechanism of solute adsorption onto solids from binary dilute solution [83, 85, 86]. It differs from the flow variant in that a stock solution is injected into the calorimetric cell where the solid sample is kept in homogeneous suspension in a liquid (solvent or solution). The possibility of measuring the pseudo-differential molar enthalpy effects is the most important advantage of this technique. The differential enthalpy is very sensitive to various partial processes of adsorption occurring at the Solid-Liquid interface, thus allowing changes in the interfacial properties with surface coverage to be continuously monitored. From this standpoint, calorimetric data are very useful for theoretical consideration and modelling. Additionally, titration calorimetry may be an important analytical tool for determining the enthalpy effects accompanying the dilution of solutions with given compositions.

A typical operational procedure involves injection of a stock solution by small steps into the calorimetric cell containing either dilute solution (dilution experiment) or suspension of powder solid sample in a dilute solution (adsorption experiment). Serious drawbacks to the use of this calorimetry technique in studying the adsorption phenomena are due to the difficulty of direct evaluation of the related amounts adsorbed and to insufficient control of the environment of the liquid phase (e.g., the pH and ionic strength cannot be maintained constant through the whole run) [73, 87]. The progress in the adsorption quantity during successive injections of the adsorbate to the cell is quantified from the adsorption isotherm measured separately under exactly the same experimental conditions. For the adsorption of ionic solutes at electrified interfaces, it is impossible to well reproduce the charging behaviour of the adsorbent at a constant pH since this parameter cannot be re-adjusted during injections.

The model of dilution experiment is depicted in Fig. 6.19. Prior to injection sequence, there is $M_1^0$ grams of pure solvent in the calorimetric cell. Then, small aliquots of the stock solution of molality $m_2^0 $ are injected by the syringe pump operating at a constant flow rate $d_{pump}$. When the time of injection $t_{inj} $ is maintained constant, the amount of the solute $n_2^{inj} $ introduced into the calorimetric cell is always the same. As a result, the equilibrium solution in the cell becomes more and more concentrated, i.e., $m_2^i \rightarrow m_2^0 $. The dilution data may be further processed in two different ways, by calculating either differential molar or cumulative molar enthalpy changes.

The total enthalpy change during the ith injection per mole of the solute is expressed as follows [73, 74, 88]:

$$\begin{aligned} \frac{\Delta _{inj} H_i }{n_2^{inj} }&=\Phi _H \left( {m_2^i } \right) +(i-1)\cdot n_2^{inj} \cdot \frac{\left[ {\Phi _H \left( {m_2^i } \right) -\Phi _H \left( {m_2^{i-1} } \right) } \right] }{n_2^{inj} }+\nonumber \\&\quad -\Phi _H \left( {m_2^0 } \right) \end{aligned}$$

(6.61a)

where $n_2^{inj} $ depends on $d_{pump} $, $t_{inj} $, and $m_2^0 $, according to the formula given in Fig. 6.19.

The equilibrium molality of the solution $m_2^i $ in the cell after this injection is calculated by means of the following expression:

$$\begin{aligned} m_2^i =\frac{10^{3}\cdot i\cdot n_2^{inj} }{M_1^0 +\frac{10^{3}\cdot i\cdot n_2^{inj} }{m_2^0 }} \end{aligned}$$

(6.61b)

The determination of all $\frac{\Delta _{inj} H_i }{n_2^{inj} }$ terms in function of $m_2^{i-1} $ and $m_2^i $ yields the results in the form of a histogram, as shown in Fig. 6.20. Referring to the general definition of the partial molal enthalpy $\overline{h} _2 $ and apparent molal enthalpy $\Phi _H $ for any particular solute in a solution of two components (Eqs. 6.36 and 6.39), it is to be noted that, for the sufficiently small quantities $n_2^{inj} $ of the solute injected during each injection, Eq. (6.61a) can be transformed to give:

$$\begin{aligned} \frac{\Delta _{inj} H_i }{n_2^{inj} }&\approx \mathop {\lim }\limits _{n_2^{inj} \rightarrow 0} \frac{H\left( {m_2^i } \right) -H\left( {m_2^{i-1} } \right) }{n_2^{inj} }-const=\overline{h} _2 \left( {m_2 } \right) -const,\nonumber \\&\qquad m_2^{i-1} <m_2 <m_2^i \end{aligned}$$

(6.62)

where H denotes the enthalpy of a binary solution in the calorimetric cell.

Thus, values of $\overline{h} _2 $ may be obtained experimentally by measuring both enthalpy changes $\Delta _{inj} H_i $ and very small quantities $n_2^{inj} $ of the solute injected during successive dilution of the stock solution. It is clear from Eq. (6.62) that the above relation does not hold for the first injection: the first points in the calorimetric curves of dilution should not be taken into consideration.

The thermal effects of successive injections can be also summed up to obtain the molar cumulative enthalpy of dilution $\Delta _{dil}{h}_{cum}$. After k injections, one obtains:

$$\begin{aligned} \Delta _{dil} h_{cum}&=\sum _{i=1}^k {\frac{\Delta _{inj} H_i }{n_2^{inj} }} =k\cdot \left[ {\Phi _H \left( {m_2^k } \right) -\Phi _H \left( {m_2^0 } \right) } \right] ,\nonumber \\&\quad \quad \mathrm{{with}}\, m_2 =m_2^k \end{aligned}$$

(6.63)

The experimentally measured values of $\Delta _{dil} h_{cum} $ are further plotted against the equilibrium solute molality $m_2 $ and only such a representation may have clear physical meaning from a thermodynamic standpoint. Nevertheless, it is sometimes more useful to present the enthalpy of dilution curve in terms of $\Delta _{dil} h_{cum} $ as a function of the injection number k, especially when the dependence of $\Delta _{dil} h_{cum} $ vs $m_2 $ is linear or contains several linear portions (cf., Fig. 6.21).

In the adsorption calorimetry experiment, a small amount $n_2^{inj} $ of the stock solution injected during a given injection is diluted in the supernatant liquid inside the cell and some of the resulting species subsequently adsorb onto solid particles. They displace a certain amount of solvent molecules and can exchange with some pre-adsorbed molecules or ions, because of the limited extent of the adsorption space. The effects of desolvation and re-solvation of various compounds taking part in the displacement process contribute to the competitive character of adsorption at the solid-solution interface. The flow chart of the batch displacement experiment is shown in Fig. 6.22. Since the enthalpy effects accompanying dilution of the stock solution inside the cell should be known, both the dilution and adsorption experiments are carried out under the same conditions (cf., Fig. 6.19).

When the adsorbent is initially immersed in $M_1^0 $ grams of pure solvent and the final state corresponds to the formation of a Solid-Liquid interface in equilibrium with a bulk solution of molality $m_2^k $, the mass balance inside the cell leads to the following expression for the amount of solute adsorbed at the Solid-Liquid interface:

$$\begin{aligned} n_{2,k}^s =n_2^s \left( {0\rightarrow m_2^k } \right) =\sum _{i=1}^k {n_2^{inj} -\frac{m_2^k }{10^{3}}} \left( {M_1^0 +\sum _{i=1}^k {M_1^{inj} } } \right) \end{aligned}$$

(6.64)

where $M_1^{inj} $ is the mass of solvent introduced into the calorimetric cell during one injection of the stock solution (see the appropriate equation in Fig. 6.22).

Equation 6.64 represents the so-called calorimetric line, since the amount adsorbed $n_{2,k}^s $ varies linearly with $m_2^k $ between two characteristic points: (P1) the whole amount of the solute injected is retained by the adsorbent surface and $m_2^k =0$; (P2) the whole amount of the solute injected remains in the bulk solution and $n_{2,k}^s =0$. In practice, it is impossible to measure directly the values of $n_2^s $ for all injections (it would not be reasonable to interrupt the calorimetry run after each injection and remove the calorimetric cell in order to analyse its content). Nevertheless, the partition of the solute between the adsorbed and bulk phases is strictly determined by the adsorption equilibrium at a given temperature and does not depend on the path by which the adsorption system passes from its initial state to the equilibrium. Hence, the values of $m_2^k $ and $n_{2,k}^s $ are evaluated from the intersection between the calorimetric line (6.64) and the experimental isotherm $\Gamma _2^s =\Gamma _2^s \left( {m_2^\beta } \right) $, obtained under exactly the same experimental conditions (Fig. 6.23).

The total change in enthalpy during the ith injection in the adsorption experiment may be written as follows:

$$\begin{aligned}&\Delta _{inj} H_i =m_S \cdot S\cdot \left[ {H_{SL}^s \left( {m_2^i } \right) -H_{SL}^s \left( {m_2^{i-1} } \right) } \right] -n_{2,i}^s \cdot \Phi _H \left( {m_2^i } \right) +\nonumber \\&\qquad \qquad +n_{2,i-1}^s \cdot \Phi _H \left( {m{ }_2^{i-1} } \right) +\nonumber \\&+n_2^{inj} \cdot \left\{ {\Phi _H \left( {m_2^i } \right) +(i-1)\cdot n_2^{inj} \cdot \frac{\left[ {\Phi _H \left( {m_2^i } \right) -\Phi _H \left( {m_2^{i-1} } \right) } \right] }{n_2^{inj} }-\Phi _H \left( {m_2^0 } \right) } \right\} \end{aligned}$$

(6.65a)

where S is the specific surface area of the adsorbent; $H_{SL}^s \left( {m_2 } \right) $ is the interfacial enthalpy for a Solid-Liquid interface in equilibrium with a bulk solution of molality $m_2 $. The above equation has been developed by neglecting any changes in the enthalpy of the bulk of the adsorbent $m_S \cdot h_S $. The comparison with Eqs. (6.54a, 6.54b) and (6.59a, 6.59b) in Sects. 6.4.4 and 6.4.5 indicates that the first three terms on the right hand side of Eq.(6.65a) represent the difference in the enthalpy of displacement $\Delta _{dpl} H\left( {m_2^i } \right) -\Delta _{dpl} H\left( {m_2^{i-1} } \right) $ when passing from $m_2^{i-1} $ to $m_2^i $. The forth term has the same form as the overall enthalpy change recorded in the dilution experiment (cf., Eq. 6.61a). For small $n_2^{inj} $ values accompanied by small increments in the amount adsorbed $n_2^s $, the batch adsorption experiment allows measuring the differential enthalpy of displacement per mole of the solute adsorbed at the Solid-Liquid interface, since:

$$\begin{aligned} \Delta _{dpl} h_{diff}&=\mathop {\lim }\limits _{\Delta n_2^s \rightarrow 0} \frac{\Delta _{dpl} H\left( {m_2^i } \right) -\Delta _{dpl} H\left( {m_2^{i-1} } \right) }{n_{2,i}^s -n_{2,i-1}^s }\nonumber \\&\approx \frac{\Delta _{dpl} H\left( {m_2^i } \right) -\Delta _{dpl} H\left( {m_2^{i-1} } \right) }{n_{2,i}^s -n_{2,i-1}^s }=\nonumber \\&=\frac{\Delta _{inj} H_i -n_2^{inj} \cdot \left[ {\overline{h} _2 \left( {m_2^\beta } \right) -const} \right] }{n_{2,i}^s -n_{2,i-1}^s } \end{aligned}$$

(6.65b)

where the equilibrium molality $m_2^\beta $ is located in the middle of interval $\left[ {m_2^{i-1} ,m_2^i } \right] $.

The experimental procedure and data processing leading to a smooth curve of $\Delta _{dpl} h_{diff}$ as a function of the amount adsorbed $n_2^s $ include the following stages: the quantity of the solute injected $n_2^{inj} $ and the successive enthalpy changes $\Delta _{inj} H_i $ are measured in the adsorption calorimetry experiment; the limit molality values $m_2^{i-1} $ and $m_2^i $ for each injection class and the related increments in the amount adsorbed $n_{2,i-1}^s $ and $n_{2,i}^s $ are evaluated with the aid of the adsorption isotherm; based on the “differential” enthalpy curve obtained in the dilution experiment, the correction terms for dilution, $n_2^{inj} \cdot \left[ {\overline{h} _2 \left( {m_2^\beta } \right) -const} \right] $, are determined for the values of $m_2^\beta $ taken as the middle of appropriate intervals $\left[ {m_2^{i-1} ,m_2^i } \right] $; with small increments $\Delta n_2^s =n_{2,i}^s -n_{2,i-1}^s $, the differential displacement terms $\Delta _{dpl} h_{diff} $ are finally calculated and ascribed to the $n_2^s $ values in relation with the equilibrium molalities $m_2^\beta $. It should be noted again that the enthalpy value corresponding to the first injection cannot be considered as the differential molar enthalpy of displacement and has to be removed from the experimental curve.

5 Calorimetry Applied to Study Competitive Adsorption from Dilute Solution

The integral and differential enthalpies of displacement can be measured directly using calorimeters of the isothermal type. Such instruments either are produced and marketed by some manufacturers of high-performance calorimetric systems for different applications (e.g., SETARAM Instrumentation, LKB-ThermoMetric, TA Instruments, Microscal Ltd), or are home-mode prototypes based on original work carried out in university laboratories and therefore documented in considerable detail in scientific publications. This section focuses on the properties of two calorimetry systems used by the author to study the competitive adsorption from dilute solution.

5.1 Flow Calorimetry System

A liquid-flow calorimetry system (commercialised by Microscal Ltd) is represented schematically in Fig. 6.24 [79, 80]. Here the calorimetric cell is simply a cylindrical cavity inside the calorimetric bloc and its volume is limited by two removable tubes. The outlet tube is also used as a holder for a powdered solid sample. Prior to each calorimetry run, the inlet and outlet tubes are taken away from the calorimeter and the calorimetric cell is cleaned.

The measuring termistors detect the temperature of the cell content via a PTFE cell wall membrane which protects them from chemical attack. A separate temperature sensor is installed in the calorimetric block to permit the digital display of block temperature. The thermistor sensors are part of a Wheatstone bridge, which has the advantage that only temperature differences between the calorimetric cell and the block will put the bridge out of balance. When the solution containing a preferentially adsorbing solute or the pure solvent reaches the adsorbent bed there is an exothermic or endothermic displacement with resultant evolution or absorption of heat. This causes a small resistance change to the measuring thermistors and the effect is registered in the form of a thermal peak, the area of which is proportional to the heat measured.

It is necessary to have an evenly packed adsorbent bed in the calorimetric cell and a smooth, steady flow of liquid to pass through the bed so as to obtain reproducible results. Since the volume of the cell is limited (about 0.17 mL), a suitable volume of powder or granular solid is weighed out in a special measuring tube provided by the manufacturer. Then the outlet tube is fitted in the calorimeter and the powder is poured into the sample holding space by means of an extended stainless steel funnel. Contrary to appearances, this operation is crucial for further measurements. Sometimes, the sample should be gently crushed to decrease the dead volume of the adsorbent bed. The use of monodispersed particles can be beneficial. When the particles are agglomerates of much finer material, they should not disintegrate in the liquid flow through the adsorbent. For solids having significant differences in their dry and wet packing densities (e.g., solids swelling in carrier liquids), the sample volume placed in the cell must take account of the volume change occurring on sample evacuation and immersion with the liquid. It is still possible to check whether the quantity of solid sample introduced and its packing in the cell are correct by fitting the inlet tube carefully. Any adsorbed species can be subsequently removed from the sample surface by evacuation, making use of a vacuum pump connected to the outlet tube. The temperature may be raised on evacuation by heating the calorimetric block with the aid of a stabilised D.C. power supply. The temperature limit depends on the thermal resistance of the materials of construction used. The efficiency of the evacuation step can be enhanced by flushing the adsorbent bed with the carrier liquid.

Further operating procedure for adsorption studies includes establishing a steady flow of the solvent used as a carrier liquid, awaiting thermal equilibrium, and setting the sensitivity controls. Different flow rates should be usually tested to optimise the operation for peak height, shape, sensitivity and duration. It should be remembered that the pump flow rate is limited by the syringe volume (normally 10 or 20 mL), compressibility of the solvent utilised, size of the solid particles, and their packing density in the sample holding space: the most commonly applicable flow rates are included in the range 0.05–0.1 mL per minute. Moreover, all occluded air should be carefully removed from the syringe and the whole line purged. As the equilibration proceeds, the thermal signal in function of time approaches a straight line asymptotically and this state is the criterion by which to judge the equilibration and to set the sensitivity of the calorimeter.

After attaining thermal equilibrium in the system, the flow of carrier liquid solvent is replaced with the identical flow rate of the solution of adsorbate at defined concentration by means of a changeover valve. The solution is fed by a syringe pump other than the one directing the solvent via the valve to the calorimeter. The progress of solute adsorption is monitored by the evolution or absorption of heat measured by thermistors sensing temperature changes in the calorimetric cell and simultaneously adsorbate transfer from the bulk phase to the Solid-Liquid interface is monitored by measuring composition changes in the effluent leaving the adsorbent and passing through the downstream detector. The principle of detection (UV, referactometry, thermal conductivity) and thus the choice of the detector depend on the chemical nature of both the solute and the solvent, the flow rates to be used, as well as the technical specification of the detector (e.g., operating parameters, sensitivity of detection, linearity of the signal with concentration, baseline stability, ease of operation). The adsorbent bed and the solution are allowed to remain in contact until heat evolution or absorption ceases (the signal returns to the thermal baseline) and no further change in the effluent composition is detected (the recorder trace shows a straight line), as illustrated in Fig. 6.24.

Since the calorimetric system is particularly well adapted to the study of the thermodynamic reversibility of the adsorption phenomenon, it is always worth carrying out the first desorption stage under exactly the same experimental conditions, just by returning to the flow of the carrier liquid through the adsorbent bed. The reader should be reminded that the heat effects of adsorption and desorption are opposite in sign. When the thermal and detector signals return to those of pure solvent, the adsorption-desorption cycle can be repeated to test the attainment of adsorption reversibility (e.g., the heat effects of adsorption and desorption are equal and repeatable). For some porous materials, the desorption process may last longer than the corresponding adsorption due to the slower mass transfer kinetics and the peak tailings effects are observed, thereby reducing the measurement accuracy and repeatability.

Calibration of the instrument is required at some stage for the particular operating conditions employed. Calibration of the areas under the thermal peaks is carried out by dissipating a known amount of energy in the adsorbed bed with the aid of a calibration probe incorporated into the outlet tube and encapsulated in PTFE. The related “exothermic” peak may be integrated making use of appropriate software facilities to process the digitized signal recorded on the computer hard disk. The data processing is the same as that described in Sect. 6.3.2 for immersional and wetting calorimetry. The downstream detector provides a plot of the effluent composition, the profile of this curve being influenced by the amount of solute molecules retained on the adsorbent surface during adsorption or released from the surface during desorption. To determine the related amount of solute adsorbed or desorbed, this profile is to be compared with composition changes obtained in a “blank” experiment with the use of a “non-adsorbing” solid through which pure solvent and the solution are passed at the same flow rate and temperature as those for the adsorbing sample. Glass or PTFE balls of low specific surface areas are usually used as non-adsorbing adsorbents, but the main difficulty here is to reproduce the same flow conditions of liquids through the adsorbent bed in the blank run (e.g., packing density of the solid, hydrophobic-hydrophilic character of its surface, pressure drop over the adsorbent bed). The two types of composition profile can be matched and presented in the form of net mass-transfer for adsorption and desorption. The areas of the segments, resulting from the subtraction of the peaks obtained on adsorbing and non-adsorbing solids, provide a direct measure of the quantity of the solute which either fails to emerge in the effluent solution due to its adsorption by the adsorbent bed in the calorimetric cell or is released to the carrier stream by the flow of pure solvent. The detector calibration factor is determined from the injection of a given volume of the solution into the stream of the solvent percolating through the detector (calibrated injection loop facility).

5.2 Measurements of Integral Enthalpy of Displacement

Despite the limited physical meaning of the integral enthalpy of displacement, measuring $\Delta _{dpl} H$ or $\Delta _{dpl} h$ may be very useful in several cases.

The first case certainly corresponds to the study of the thermodynamic reversibility of adsorption onto solids from binary solutions. Liquid-flow calorimetry measurements usually provide clear, unambiguous arguments for irreversible character of the phenomenon in numerous systems. An example of such systems is illustrated in Fig. 6.25. With non-porous Graphon possessing a very small number of surface polar sites, the adsorption of C$_{60}$ fullerene from toluene is completely reversible. In the case of porous active carbons, the phenomenon is only partially reversible, the degree of reversibility being evaluated from the difference between the values of $\Delta _{dpl} h$ measured for the adsorption and desorption stage.

The irreversible enthalpy component may be calculated from the following formula:

$$\begin{aligned} \Delta _{dpl} h_{IR} =\frac{\Delta _{ads} n\cdot \Delta _{ads} h+\Delta _{des} n\cdot \Delta _{des} h}{\Delta _{ads} n-\Delta _{des} n} \end{aligned}$$

(6.66)

where $\Delta _{ads} n$ and $\Delta _{des} n$ are positive changes in the number of moles of solute measured during the adsorption and desorption run, respectively; $\Delta _{ads} h$ and $\Delta _{des} h$ are the corresponding molar enthalpies of displacement observed in both stages.

Low sensitivity of the $\Delta _{dpl} H$ or $\Delta _{dpl} h$ values to the surface heterogeneity effects makes the integral enthalpy of displacement useful for probing specific sites on the surface of solid materials. The principle of this method lies in measuring the enthalpy of displacement per unit area of the adsorbent $\Delta _{dpl} H^{{*}}$ during adsorption of specific probe species (solute) capable of displacing non-specific solvent molecules from the targeted surface sites. For example, the polar contribution to the interfacial enthalpy $H_{SL}^s $ may be thus approximated by determining the integral enthalpy of displacement of an apolar solvent by a polar solute. Prior to calorimetry measurements, the solution composition should be carefully optimised to ensure monomolecular adsorption of the solute on the solid surface [90, 91].

Figure 6.26 shows the effect of heteroatom incorporation into the framework of ordered mesoporous silica, as inferred from the adsorption of 1-butanol from n-heptane.

When diluted in an apolar solvent, the molecules of 1-butanol (BuOH) may be regarded as monomer species, potentially acting as both hydrogen-bond donors and acceptors. Since the doping procedure aims at isomorphic substitutions of silicon by such heteroatoms as Al, Ti or Zr in the tetrahedral structures, the hydrophilic surface of the four mineral oxides should be dominated by surface hydroxyl groups. Butanol molecules can form hydrogen bonds with these silanols (Si-OH), aluminols (Al-OH), titanols (Ti-OH), or zirconols (Zr-OH). The monolayer adsorption of BuOH is accompanied by simultaneous desorption of apolar heptane molecules. In consequence, the integral enthalpy of displacement per unit area of the adsorbent surface $\Delta _{dpl} H^{{*}}$ is a function of the surface density of hydroxyl groups. In the case of materials doped with Ti(IV) and Zr(IV), there is no reason for a marked change in the surface density of hydroxyl groups.

This hypothesis is well illustrated by quite similar (within the experimental error) $\Delta _{dpl} H^{{*}}$ values obtained for SBA-15, Zr-SBA-15, and Ti-SBA-15, irrespective of differences in the surface area and porous structure among the samples. The partial replacement of Si(IV) by Al(III) results in additional surface hydroxyls related to the ‘bridging’ Si(OH)Al hydroxyl structures, thereby enhancing the value of $\Delta _{dpl} H^{{*}}$ (Fig. 6.26).

In the liquid-flow calorimetry experiment, the purified adsorbent bed remains in contact with a stock solution of constant composition. It is clear that the environment of the liquid phase does not change during the measurement. This is an important advantage of the flow calorimetry, especially in the case of solid-solution systems containing electrified interfaces. The study of ions adsorption from aqueous solutions onto mineral oxides bearing a pH-dependent surface charge always requires constant pH and ionic strength [71, 93].

Figure 6.27 presents the thermogram resulting from adsorption of a heavy metal cation from aqueous solution on the negatively charged surface of Spherosil registered during liquid-flow calorimetry measurements.

For a given concentration of the stock solution $m_2^0 $, continuous percolation of the solution through the calorimetric cell containing the solid sample leads to saturation of the adsorbent with the solute giving rise to a negative heat effect (i.e., adsorption is endothermic) in the form of a peak in which the beginning and the end depend on the solution concentration, flow rate of solution throughout the adsorbent bed, and the kinetics of adsorption. Then the solute is removed from the adsorbent by exchanging the flow of the solution for that of pure solvent. The solute desorption is exothermic and complete in each cycle since the adsorption and desorption enthalpy effects have the same absolute value. Therefore, the successive saturation-desorption cycles can be performed without changing the solid sample in the cell. Although the thermogram in Fig. 6.27 shows that the resulting values of $\Delta _{dpl} H$ increases with increasing concentration, the molar enthalpy $\Delta _{dpl} h$ is proven to be a monotonously decreasing function of $m_2^0 $ [94].

The comparison among thermal displacement effects accompanying the individual adsorption of an alkaline earth metal from aqueous solution is given in Fig. 6.28. The pH and ionic strength of the aqueous phase were identical in the four systems studied.

According to the appropriate speciation diagrams, each metal forms divalent cationic species in aqueous solution under the experimental conditions applied. In spite of the same electric charge of the four cations, the positive enthalpy values are very different, indicating that electrostatic attraction is not the only driving force of adsorption. Since there are $\mathrm{{Na}}^{+}$ ions in the heterogeneous system, the total displacement effect should also include ion exchange between sodium and a given divalent cation. Modelling attempts to reproduce the positive displacement effects lead to the conclusion that metal cations may form multidentate complexes with oxygen atoms of ionised silanol groups and changes in the hydration layers of the adsorbing and desorbing cations are the main reason for the endothermic character of the overall process [96].

5.3 Titration Calorimetry System

Liquid titration calorimeters contain a stirring device, which ensures the homogeneity of the liquid solution or solid suspension in the calorimetric cell, and an injection system permitting the controlled introduction of the reagents from outside the calorimetric cell. Contrary to the flow system, nothing flows out of the calorimeter to the surroundings since the reagents from a stock solution fed to the injection device are collected within the calorimetric cell. The heat detection is usually based on the principles of the isothermal batch or flow microcalorimeters. An example of home-made microcalorimeter [86] designed for study of the enthalpies of mixing of liquid and adsorption from dilute solution onto divided solids is represented schematically in Fig. 6.29.

The variations of temperature in the calorimetric cell induced by dilution or adsorption phenomena, as well as by the electrical calibration are recorded by thermistors arranged as a Wheatstone bridge. The two measuring thermistors, calibration coil and the inlet end of the injection tube are immersed in the solution or suspension. A precision syringe pump injects a stock solution into the cell at a constant rate ranging between 0.01 and 0.2 g $\mathrm{{min}}^{-1}$ without introducing any significant thermal perturbation. The injected solution flows through a heat exchanger tube where it is heated up to attain the temperature of the calorimetric block. The measurement of the enthalpy changes may be carried out also at higher temperatures (this microcalorimeter has been proven to give satisfactory results at temperatures ranging from 20 and $50^{\circ }$C [86]). For this purpose, an additional power unit provides a stabilised D.C. power supply for heating the calorimetric block to the desired temperature. To reduce the heat loss or temperature variations during injection, it is sometimes necessary to thermostate both the stock solution in the syringe and the injection tube outside the calorimetric block. A horizontal agitator with a variable speed of rotation is driven by a stepper motor fitted through a magnetic transmission.

The dissipation of a known amount of electrical energy inside the calorimetric cell by means of a calibration coil (i.e., the Joule effect) is used to relate the area of the thermal peaks recorded to the enthalpy effects which this represents. The difficulty with this type of calibration in the titration calorimetry systems is related to the fact that the mass of solution in the calorimetric cell is constantly increased by successive injections, thereby changing the calorific capacity of the cell. Therefore, thermal calibration should be regularly repeated after each series of injections in the same calorimetric run.

The calorimetric cell, together with the measuring termistors, stirring device and injection system, can be removed from the calorimetric block to facilitate cleaning and refilling the cell. After the instrument set up and attainment of thermal equilibrium, all steps of the run (injections of the stock solution containing the adsorbate, calibration processing, and recording of the heat effects) are carried out by the appropriate computer system. It is a best practice to weigh the calorimetric cell at the end of each run in order to calculate the real (mean) pump rate. One of the original features of the construction of this calorimeter is also the possibility of easily changing the thickness of the insulating barrier between the calorimetric cell and the metal block. This operation permits the modification of the calorimeter sensitivity and the time of the return of the thermal signal to the baseline.

The titration calorimetry technique presents some limitations due to the necessity of a strict correlation between the dilution and adsorption measurements. A special care must be taken to avoid experimental artefacts and erroneous interpretations in the study of solute adsorption onto solid supports which dissolve to a great extent in solutions or when the quantity of foreign substances released from the solid surface to the bulk solution is significant. In such cases, the composition of the equilibrium supernatant does not correspond to the pure solvent and it may even change constantly with increasing adsorption of the preferentially adsorbed solute [87, 93, 97]. Firstly, this evolution of the supernatant liquid should be monitored thoroughly during adsorption. Then, the evaluation of the correction term for dilution may pose serious problems since the stock solution has to be prepared by dissolving solute in the actual supernatant liquid and not in the pure solvent.

An example of the adsorption system investigated by means of titration calorimetry technique is given in Fig. 6.30 [74]. The experimental adsorption isotherm (Fig. 6.30b) has been determined separately based on the solution depletion method. The solute (dodecyldimethylammonio) butanoate (C12N3C) is a zwitterionic surfactant possessing a dipolar head-group and a linear aliphatic tail. In aqueous solution, the solute molecules self-assemble into aggregates called micelles. The concentration of the surrounding aqueous phase at which the surfactant monomers begin to form micelles is known as the critical micelle concentration (CMC).

When a micellar stock solution (i.e., its concentration is 10 times the CMC) is injected into a more dilute solution in the calorimetric cell, the constant value of partial molal enthalpy $\overline{h} _2 $ in the premicellar region is due to destruction of micelles and dilution of unmicellized species (Fig. 6.30a); the constant $\overline{h} _2 $ value in the postmicellar region is ascribed to dilution of micelles.

The molar change in partial molal enthalpy of the surfactant when monomers associate into a micelle at the cmc represents the standard enthalpy of micellisation $\Delta _{mic} h^{o}$ per mole of surfactant monomers [98]. In accordance with the variations of $\overline{h} _2 $ as a function of $m_2^\beta $ in Fig. 6.30a, the enthalpy of micellisation for C12N3C at 298 K is positive, indicating that the micellisation process is endothermic.

The experimental curves describing the adsorption of C12N3C onto Spherosil XOB015 from aqueous solution at 298 K (i.e., Figs. 6.30b and 6.30c) suggest that the phenomenon generally occurs in two stages. At very small quantities of adsorption $n_2^s $, the values of $\Delta _{dpl} h_{diff} $ are negative so that the phenomenon is exothermic. The negative enthalpies of displacement are usually attributed to individual adsorption of surfactant molecules on an empty surface, where there may be only a few adsorbed molecules, and therefore lateral adsorbate-adsorbate interactions can be neglected [74]. The subsequent adsorption stage is dominated by adsorbate-adsorbate interactions. At moderate and great adsorption amounts, the driving force of adsorption derives from the hydrophobic effect, i.e., lateral chain-chain attractions and the tendency of hydrophobic tails to escape from an aqueous environment. This mode of adsorption is characterised by a constant, positive enthalpy of displacement, showing much similarity to micelle formation in the bulk solution [74]. This surface aggregation is likely controlled by a pseudo nucleation step, i.e., individual adsorption: the first adsorbed monomers act as nucleation centres for future surface-bound surfactant aggregates formed through chain-chain association.

5.4 Scanning of Surfactant Aggregation by Titration Calorimetry

Surface-active molecules or ions with an amphiphilic structure are known to have low solubility in water and self-assemble into large aggregates [61]. According to the enthalpy curves presented in Fig. 6.30, surfactant aggregation may occur not only in aqueous solution but also at the Solid-Liquid interface. Since the topic of surfactant aggregation is a well-developed field of research [62, 71, 73, 93, 99, 100], only a brief review of the broad principles is proposed in the present paragraph to better illustrate the contribution of titration calorimetry.

In aqueous solution, the formation of surfactant aggregates is driven by the tendency of the surfactant units to densely pack their tails. The hydrophobic tails remain inside the liquid-like micellar core due to unfavourable interactions with water molecules, whereas the polar head-groups, due to favourable interactions with the solvent, form a hydrophilic outer layer protecting the hydrophobic core. In the case of ionic surfactants, the Coulombic repulsion among the ionised head-groups is moderated by the specific adsorption of some counter-ions close to them within the Stern part of a curved ionic double layer surrounding the micelle. The micelle morphology and size depend primarily on the nature and relative sizes of the hydrophilic and hydrophobic moieties, as well as on the composition and environment of the aqueous phase [61, 62]. Titration calorimetry may be very useful when studying surfactant micellisation under different experimental conditions. The fundamental thermodynamic parameters, namely the CMC and the standard molar enthalpy of micellisation $\Delta _{mic}h^{\circ }$, can be easily inferred from calorimetric measurements of successive dilutions of a micellar stock solution injected by small steps into aqueous solution having a given composition [73, 74]. Several examples of cationic and zwitterionic surfactants are given in Table 6.5.

It is important to note that the formation of micelles in pure water may be an exothermic, endothermic or even athermic phenomenon, depending on the detailed molecular structure of the surfactant. The CMC value is related to the standard Gibbs energy of micellization, $\Delta _{mic} G^{\circ }$, which always takes negative values. This negative energy results rather from a large increase in entropy, which is ascribed either to structural changes in the solvent, associated with loss of hydration of the hydrophobic tail when the surfactant enters the micelle [61] or to increased freedom of the hydrophobic chain in the interior of the micelle compared to the bulk aqueous medium [101]. Besides the volume and length of the hydrophobic tail, the area per head-group at a curved interface between the micelle core and the aqueous solution $\sigma _{mic} $ is a critical packing factor having an impact on the ultimate micelle structure [102]. For conventional ionic and zwitterionic surfactants with a single hydrocarbon chain, globular micelles are formed above but near the CMC. A significant increase in the overall surfactant content in aqueous solution may induce a change from spherical micelles to cylindrical (prolate) or disc-like (oblate) aggregates. This change is parralled by a decrease in $\sigma _{mic} $ which allows a larger number of monomers to be inserted into each aggregate. The maximum cohesion is attained in large lamellar sheets (flat bilayers) two molecules thick, though surfactant tails never attain such a close-packed arrangement in dilute solutions.

Table 6.5 Critical micelle concentrations, CMC, and standard enthalpies of micellisation,$\Delta _{mic}h^{\circ }$, per mole of surfactant monomers for selected quaternary ammonium surfactants in pure water at 298 K [73, 74, 88, 97, 103, 104]

Full size table

Dimeric or Gemini surfactants (composed of two surfactant units connected by a hydrophobic or hydrophilic chain—spacer) usually show a much stronger tendency for micellar growth and self-assemble into larger aggregates with a lower degree of curvature (e.g., linear thread-like and tree-like micelles, or spheroids) [105, 106].

The addition of solid particles (porous or non porous) into aqueous solution induces a decrease in the chemical (or electrochemical) potential of the surfactant solute and, in consequence, the adsorbing surfactant units form, at the solid surface, some periodic adsorbate self-assemblies closely related to the micellar structures encountered in the bulk solution at higher monomer concentrations. Such surface-bound aggregates are spoken of as interfacial aggregates or solloids [107]. In the general case, the shape and size of solloids are considered to be a compromise between the free curvature, as defined by the energetic, geometrical and packing factors arising from the molecular structure of the surfactant in a given environment, and some influences and constraints imposed, on the one hand, by direct solute-surface interactions and, on the other hand, by the porosity of the adsorbent. For example, the images of extended aggregate structures showing a closer registry with the underlying surface have been obtained by atomic force microscopy (AFM) on atomically smooth crystalline surfaces [99]. Such solloids have cylindrical and hemi-cylindrical morphologies, depending on the hydrophilic-hydrophobic character of the solid surface. In the case of powders or porous solids, where such microscopy techniques as AFM or ellipsometry have very limited applicability, the titration calorimetry measurements of the differential molar enthalpy of displacement as a function of the surfactant adsorption may provide important information about the self-assembled surfactant structures when compared with the thermal effects of surfactant micellisation in aqueous solution under the same experimental conditions. Nevertheless, calorimetry alone cannot be used to scan for the detailed solloid morphology and appropriate modelling of the adsorption system is necessary.

As far as the adsorption of ionic surfactants on the oppositely charged (and macroscopically flat) surfaces of mineral oxides is concerned, the following three types of solloid are frequently used in the empirical explanation or modelling of the experimental data:

(1)
monolayered hemimicelles, composed of surfactant units oriented ‘head-on’ towards the surface, with the surfactant tails forming a hydrophobic film in contact with the equilibrium aqueous solution [108],
(2)
bilayered admicelles, containing two adsorbed layers of the surfactant monomers directed ‘head-on’ and ‘head-out’ with respect to the surface [109],
(3)
small surface micelles, i.e., spherical isolated aggregates anchored to certain surface sites with aggregation numbers markedly smaller than in bulk micelles [110]. Each of these solloid morphologies is claimed to have a micelle-like character, although the contact area between the water molecules and the hydrophobic surfactant moieties is not always reduced to the minimum.

Typical plots of the differential molar enthalpy of displacement against the amount of the surfactant adsorbed reveal significant variations in the enthalpy value when the adsorption progresses. An example of the enthalpy curve is given in Fig. 6.31. Such trends in $\Delta _{dpl} h_{diff} $ with increasing $n_2^s $ suggest almost continuous evolution of the solloid morphology and size: the aggregates self-assembled from the adsorbing surfactant monomers at equilibrium concentrations in the bulk phase lower than the CMC may grow in the direction parallel and perpendicular to the solid surface. The most successful theoretical attempts to mimic complex shapes of both the experimental adsorption isotherms and enthalpy of displacement curves have been based on the assumption that the adsorbed phase at a given surface coverage can be seen as a mixture of mutually interacting surface-bound monomers, monolayered hemimicelles, and bilayered admicelles varying in size and number [65, 111]. The proportion between the various types of adsorbate species is shown to undergo significant changes with increasing surfactant adsorption, first monomers and monolayered aggregates and then bilayered admicelles dominating on the surface.

When surfactants are adsorbed onto fine-pore solids, the growth of solloids in the direction perpendicular to the pore walls should be limited by the pore volume [93]. For mesoporous ordered mineral oxides, it may be even that the surfactant monomers adsorb only head-on with respect to the hydrophilic surface and the hydrophobic tails of the surfactant units adsorbed on the opposite walls interpenetrate themselves in such a way as to produce “internal” aggregates, which fill the pore space. To check whether the head-out adsorption of surfactant monomers is to be excluded in such systems, one can refer to the micellar solubilisation of water-insoluble, hydrophobic materials as well as polar substances, which dissolve in water only to a limited extent [62, 69, 112, 113]. For low contents in aqueous solution, some small molecules may behave as molecular probes, occupying specific sites in surfactant aggregates without greatly disturbing their morphology and size. The exact locus of a given probe molecule in a micelle reflects the type of forces operating between the aggregate and the solubilised material. This justifies the use of titration calorimetry to study micellar solubilisation. One of the possible methodologies is to lump water and the additive together into the mean solvent and investigate the micellisation of the surfactant in this new medium. The molar enthalpies of micellisation for several cationic surfactants in the absence and the presence of phenol at various additive contents are compared schematically in Fig. 6.32.

On the addition of phenol to the aqueous phase, the enthalpy of micellisation per mole of the surfactant becomes more negative (and the CMC value is decreased), thereby indicating more favourable phenol-surfactant interactions after the transfer of the additive to the micellar phase. Therefore, phenol molecules are preferentially located in the outer portions of cationic micelles close to the surfactant head-groups, without involving much rearrangement of the micelle structure. When the phenol concentration increases (e.g., PhOH-HTAB systems), the existence of an endothermic contribution to $\Delta _{mic} h$ may be deduced from the evolution of the enthalpy value. To better understand this positive enthalpy component, one may refer to the detailed analysis of $^{1}\mathrm{H}$ NMR spectra recorded with the various PhOH-HTAB solutions: additional phenol units penetrate deeper into the micelle core producing unfavorable interactions with cationic micelles of HTAB [88]. Consequently, if phenol is to be used as a molecular probe for detecting the presence of the head-out adsorbed surfactants at the Solid-Liquid interface, the overall additive content in the system should remain low.

Based on the assumption that surfactant aggregation on the solid surface is a pre-requisite for the uptake of phenol and the aromatic molecules can be located only close to the “free” head-groups of the surfactant units within the interfacial aggregates, it may be helpful to compare the curves presenting the differential enthalpy of displacement as a function of the surface coverage by the surfactant adsorbate in the absence and in the presence of the additive. Figure 6.33 illustrates such a comparison for a cationic surfactant adsorbed onto ordered mesoporous aluminosilicate of the MCM-41 type.

The difference between the two curves is clearly pronounced at higher surface coverage ratios where surface aggregation is considered to be the predominant sorption mode. With a small amount of phenol added to the aquous phase, the displacement phenomenon is more exothermic in this region: the related portion of the enthalpy curve can be viewed as shifted towards more negative values by a constant value in comparison with the curve corresponding to the system without phenol. Similar enthalpy behaviour of this surfactant has been observed during its micellisation in aqueous solution (cf., Fig. 6.32). This analysis provides strong indication for phenol incorporation in the interfacial aggregates having their head-groups oriented outwards. Therefore, the image of all surfactant units interacting directly with the negatively charged surface and their hydrophobic tails filling the whole pore space available is rather to be excluded.

The coexistence of hydrophilic and hydrophobic nano-domains separated in space, with a local order and fluidity typical of liquids, confer to supramolecular surfactant structures remarkable properties, which are advantageous in applications involving molecular confinement within nanoscopic regions and reactivity in micro-heterogeneous media. Micelle-mediated reactions constitute the basis of the so-called micellar catalysis [62, 116], admicellar catalysis [117] or admicellar polymerisation [118] in which reaction mechanisms may be controlled at a molecular level to save energy and raw materials, as well as to avoid lengthy post-reaction purification and analytical steps.

6 Concluding Remarks

The intention of the present chapter was to present mostly the prospective advantages, but also some limitations, of the use of isothermal calorimetry at the Solid-Liquid interface as a powerful tool in the study of interactions between solid surfaces and the surrounding liquid phase. The operating principles and examples of applications were described for three calorimetry techniques frequently used to date in Surface Science and Technology: (i) wetting and immersional calorimetry, (ii) liquid flow calorimetry, (iii) titration batch calorimetry. The interested reader is encouraged to search for other outstanding examples of commercial or home-made instruments and their specific uses which have not been included here.

Wetting and immersional calorimetry may be employed to determine surface properties of catalysts, adsorbents and other solid materials in contact with liquids. Based on the calorimtery measurements of the various contributions to the total surface enthalpy of a solid, it is possible to evaluate the hydrophobic-hydrophilic character of its surface. The Harkins-Jura method for the evaluation of the specific surface area of a solid based on enthalpy changes in the so-called immersion-adsorption-wetting cycle gives the surface area of contact between this solid and a pure liquid or a solution, thereby sheding light on the availability of the solid surface under real experimental conditions.

Liquid flow or batch titration calorimetry techniques offer an opportunity for studying the macroscopic outcome of the various interactions involved in interfacial phenomena occurring at the Solid-Liquid Interface. The enthalpy changes appear very sensitive to the partial mechanisms through which a given phenomenon can occur. In particular, the effects of heterogeneity of a solid surface (i.e., surface sites with different adsorption energies, “confinement effects” due to adsorbent porosity) show up more clearly in heat quantities than in adsorption isotherms, thereby allowing easier interpretation of the phenomenon studied. The assessment of the thermodynamic reversibility of competitive adsorption from multicomponent solutions is one of the different possibilities of isothermal calorimetry. Self-assembled surfactant structures, defined by the regular assembly of small molecular entities into larger supra-molecular structures either in aquous solution or at the Solid-Liquid interface, may be thermodynamically described based on the results of titration calorimetry measurements. Here the next step would be to use high sensitivity isothermal calorimeters to determine the thermal effects of micelle-mediated reactions in micellar or admicellar catalysis.

In spite of many advantages, Solid-Liquid calorimetry alone is not capable of solving satisfactorily many detailed problems concerning the resulting interfacial mechanisms. It certainly cannot provide much information on entropy changes. In consequence, calorimetric measurements have to be always supplemented by other experimental studies reported on the system so as to obtain a more complete description of the phenomenon

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Institut Charles Gerhardt Montpellier, UMR-5253 CNRS-UM2 Equipe Agrégats, Interface, et Matériaux pour l’Energie (AIME), Université Montpellier 2, C.C. 1502 Place Eugène Bataillon, 34095, Montpellier Cedex 5, France
Jerzy Jozef Zajac

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Lyon (IRCELYON), Institut de Recherches sur la Catalyse et l’Environnement de, Villeurbanne CX, France
Aline Auroux

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Zajac, J.J. (2013). Calorimetry at the Solid–Liquid Interface. In: Auroux, A. (eds) Calorimetry and Thermal Methods in Catalysis. Springer Series in Materials Science, vol 154. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11954-5_6

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DOI: https://doi.org/10.1007/978-3-642-11954-5_6
Published: 18 September 2013
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Calorimetry at the Solid–Liquid Interface

Abstract

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Calorimetry Techniques

Adsorption Calorimetry on Well-Defined Surfaces

Determining Surface Energetics of Solid Surfaces

Keywords

1 Introduction

2 Thermodynamic Treatment of the Solid–Liquid Interface and the Related Interfacial Phenomena

2.1 Surface Excess Functions and Surface Phase Model

2.2 Adhesion and Cohesion

2.3 Wetting in Solid–Liquid Systems

2.4 Hydrophobic and Hydrophilic Substances

3 Calorimetry Applied to Evaluate Surface Properties of Solids

3.1 Enthalpy Changes in the Thermodynamic Cycle of Immersion-Adsorption–Wetting

3.2 Immersional and Wetting Calorimetry Experiments

3.3 Hydrophilic-Hydrophobic Series and Harkins-Jura Method

4 Enthalpy Changes Accompanying Competitive Adsorption from Dilute Solution

4.1 Thermal Properties of Dilute Solutions

4.2 Macroscopic Description of Competitive Adsorption

4.3 Competitive Adsorption Measurements

4.4 Immersion in Dilute Solutions

4.5 Model of Flow Calorimetry Experiment

4.6 Model of Batch Calorimetry Experiment

5 Calorimetry Applied to Study Competitive Adsorption from Dilute Solution

5.1 Flow Calorimetry System

5.2 Measurements of Integral Enthalpy of Displacement

5.3 Titration Calorimetry System

5.4 Scanning of Surfactant Aggregation by Titration Calorimetry

6 Concluding Remarks

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