## Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

### 15. Some Special Topics: Surfaces of Liquids and Solutions

### 15.6. Characteristic Adsorption Behavior

Generally, three types of behavior of *γ* for the various types of solutes with concentration can be distinguished, as illustrated schematically in __Figure 15.6__. Type I behavior includes *hydrophilic* compounds which prefer to be in water and therefore deplete at the interface, resulting in a, generally small, increase in *γ*. The most common examples are salts such as NaCl. As an illustration we quote that, at room temperature, *γ* rises from ∼72.6 mJ m^{−2} for pure water to ∼76.0 mJ m^{−2} for a 2 M aqueous solution of NaCl. Type II behavior comprises the *hydrophobic* compounds; these do not prefer to be in the water and therefore segregate at the interface, resulting in a decrease in *γ*. Aliphatic alcohols with a somewhat longer chain such as hexanol provide an example. Type III behavior includes *amphiphilic* compounds or *surfactants* that generally contain strongly hydrophobic “tails” and strongly hydrophilic “heads.” One can distinguish between *anionic* surfactants generating negative molecules (and a positive counterion) and *cationic* surfactants generating positive molecules (and a negative counterion). Typical examples are sodium dodecylsulfate (SDS; C_{12}H_{25}OSO_{3}Na) and hexadecyl trimethylammonium bromide (CTAB; C_{12}H_{25}N(CH_{3})_{3}Br). Moreover, one has *nonionic* surfactants containing highly polar groups, such as polyethylene oxide, and *amphoteric*surfactants carrying a positive and negative charge while being on the whole neutral. The cationic and anionic compounds strongly enrich at the interface because their tails prefer to be out of the water while their heads prefer to be in the water. The polar groups of the nonionic surfactants prefer the water phase with a similar effect. Most surfactants are anionic, followed by nonionic surfactants. Cationic surfactants often pose environmental problems, while amphoteric surfactants are generally expensive and therefore are only used for special applications. Surfactants generally enrich strongly at the interface, so that *γ* is decreased considerably at small concentrations. Above a certain concentration they do not further enrich at the surface (the surface is “occupied”) but rather form *micelles*, clusters of typically 50 surfactant molecules in which the tails stick together in order to minimize the contact with water, and of which the structure is shown schematically in __Figure 15.6__. This concentration is called the critical micelle concentration (CMC). The solubility of surfactants is generally low but increases with temperature. The temperature at which the CMC equals the solubility is normally addressed as the *Kraft temperature*. Above this temperature, micelle formation is possible and surfactants are much more active above this temperature than below.

** Figure 15.6** (a) Schematic representation of the interfacial tension with concentration for hydrophilic (type I), hydrophobic (type II), and amphiphilic solutes (type III, surfactants). The CMC is indicated; (b) Schematic representation of the structure of micelles occurring at

*c*>

*c*

_{CMC}for surfactants, where the hydrophilic head is indicated by O and the hydrophobic tail by –––.

**15.6.1 Amphiphilic Solutes**

For amphiphilic solutes (surfactants), we may suppose that the concentration in the bulk is small and that segregation occurs essentially at the first surface layer (a higher concentration leads to micelles). Thermodynamically, this means that we assume that , so that and . We discuss a kinetic and a statistical–mechanical approach, both leading to the Langmuir isotherm.

For the kinetic approach we consider the surface as a plane with a certain density of surface sites. The process in the surface layer can be represented as a chemical reaction where the solute dissolved in the bulk (B) “reacts” with an empty surface site (S) to an occupied adsorbed site (A); that is, B + S ↔ A. Obviously, we assume that the molecules at the surface do not interact. Since the solubility is low, we approximate the activity *a* by concentration^{11)}*c* = *N*_{B}/*V*, where *N*_{B} is the number of dissolved molecules in the volume *V*. We denote the fraction occupied surface sites by *θ*, that is, *θ* = *N*_{A}/*N*_{max}, where *N*_{max} is the number of sites at monolayer coverage. Hence, *N*_{S}/*N*_{max} = 1 − *θ*. The equilibrium constant *K* for this adsorption “reaction” is then

__(15.86)__

usually called the *Langmuir adsorption isotherm*. The amount of segregated material *Γ*^{(σ)} is then, with *Γ*_{max} a proportionality constant representing monolayer coverage,

(15.87)

The latter form can be used for regression analysis of experimental data.

Before we discuss the behavior and some consequences of the Langmuir isotherm, we first derive it via statistical mechanics. In this approach, we consider the solvent and the surface phase and equate the chemical potential of the solute in the bulk phase *μ*^{(α)} with that of the surface phase *μ*^{(σ)}. In view of the limited solubility, we assume ideal behavior so that the *N* solute molecules can move freely through the solvent with volume *V* with potential energy *ϕ*_{α}. Hence, we have for the partition function *z*^{(α)} of a single molecule (see __Chapter 5__)

(15.88)

where is the internal partition function and the external partition function, representing the translation motion of the molecule as a whole, with *Λ* ≡ (*h*^{2}/2π*mkT*)^{1/2} the thermal length. For the total partition function *Z*^{(α)}and Helmholtz energy *F*^{(α)} in the bulk we have, respectively,

(15.89)

so that

(15.90)

For the surface we suppose that we have *M* sites of which *S* are occupied. Hence, the fraction adsorbed molecules *θ* = *S*/*M* and the fraction of nonoccupied sites is 1 − *θ* = (*M *− *S*)/*M*. The partition function for a single molecule becomes

(15.91)

where similar labels are used as for the bulk. The total partition function *Z*^{(σ)} and Helmholtz energy *F*^{(σ)} for the surface then read, respectively,

(15.92)

where the factor *M*!/*S*!(*M *− *S*)! is introduced in view of the indistinguishability. Employing the Stirling approximation for the factorials, we have . The chemical potential *μ*^{(σ)} accordingly becomes

(15.93)

Equating *μ*^{(α)} with *μ*^{(σ)} we obtain, using for the bulk concentration *c* = *N*/*V*,

__(15.94)__

equivalent to __Eq. (15.86)__ if we take^{12)}*K* = *b*(*T*).

Let us now evaluate the Langmuir behavior. For small *c*, the increase in *θ* ≅ *Kc*, that is, the occupied surface site fraction is linear in *c*, while for large *c*, we have *θ* → 1, that is, the occupied surface site fraction saturates at 1. This behavior is displayed in __Figure 15.7__ for various values of *K*. Taking *K* = 10^{4}*c*_{0} as an example, we note that *θ* ≅ 1 at *c*/*c*_{0} ≅ 0.002, so that the CMC in this model will be about 0.002*c*_{0}. A few more remarks can be made. First, recall that the equilibrium constant for adsorption *K* is related to the Gibbs energy of adsorption *G*_{ads} by *K* = exp(−*G*_{ads}/*kT*), to be compared with __Eq. (15.94)__. Having determined *K* as a function of *T*, all other thermodynamic properties, such as the heat of adsorption, can be determined in the usual way. Second, substituting the Langmuir isotherm *Γ*^{(σ)} = *Γ*_{max}*θ* = *Γ*_{max}*Kc*/(1 + *Kc*) in the Gibbs adsorption expression *Γ*^{(σ)} = −(*RT*)^{−1}∂*γ*/∂ ln*c* and solving for *γ*, one obtains the *Szyszkowski equation* [35] reading

__(15.95)__

which appears to describe experimentally the amount of adsorbed material often rather well.

** Figure 15.7** (a) The Langmuir isotherm

*θ*(

*c*) =

*K*(

*c*/

*c*

_{0})/[1 +

*K*(

*c*/

*c*

_{0})] for

*K*= 10

^{2}

*c*

_{0}, 10

^{3}

*c*

_{0}, and 10

^{4}

*c*

_{0}; (b) The Fowler–Guggenheim adsorption isotherm for various values of

*α*=

*zw*/

*kT*. The critical value is

*α*= −2, while

*α*= 0 corresponds to the Langmuir isotherm.

Obviously, other assumptions lead to different adsorption isotherms. In fact, if we assume that the molecules at the surface do attract each other so that the energy changes by 2*zwθ*, where *z* denotes the surface coordination number and 2*w* the energy when a new pair of neighbors is formed, but that the surface entropy is not affected__ ^{13)}__, we have

(15.96)

This leads to the *Fowler–Guggenheim adsorption isotherm* reading

(15.97)

Plotting *θ* as a function of ln(*c*/*c*_{½}), where *c*_{½} is the concentration where *θ* = ½, we see that this isotherm produces a van der Waals shape with two loops for *α* = *zw*/*kT* < −2, characteristic of a phase transformation. The critical temperature is thus determined by , occurring at *θ*_{cri} = ½. __Figure 15.7__ shows the behavior for the various values of *α*. In view of the fact that the solution is obtained via the zeroth approximation, more sophisticated solutions lead to different values for . For example, the first-order approximation leads for a square lattice (*z* = 4) to while an exact solution yields (see Appendix 3).

Another extreme for a liquid surface would be that the adsorbed molecules can move freely, much like a 2D-gas. This is suggested by the fact that the *surface pressure* *π* = *γ*_{0} − *γ*, where *γ*_{0} is the surface tension of the pure liquid and *γ* is the surface tension of the solution, for low concentration *c* is linearly related to *c*, that is, *π* = *λc*, with *λ* a constant. The surface pressure is usually measured with a *surface pressure balance*, often denoted as *Langmuir–Blodgett trough*, of which a schematic is shown in __Figure 15.8__. On one side of a movable barrier, surfactant is added to the surface while the other side has a pure water surface. By measuring the surface pressure as function of surface area *A* and using the Gibbs adsorption expression −d*γ* = d*π* = *kTΓ* dln*c* with in conjunction with the experimental relation *π* = *λc*, we obtain

(15.98)

** Figure 15.8** (a) Schematic representation of a Langmuir–Blodgett trough; (b) Schematic representation of the surface pressure

*π*=

*γ*

_{0}−

*γ*versus surface area

*A*and the associated structure, showing the regimes of gas-like (G), liquid-expanded (L

_{e}), liquid-condensed (L

_{c}), and solid (S).

This expression resembles that of the ideal gas law, and suggests similar improvements for the surface as has been used for bulk gases. By employing the van der Waals approximation, in analogy with the 3D expression (*P *+ *aρ*^{2})(1 − *bρ*) = *ρRT* (see __Chapter 4__), for 2D reading

(15.99)

with *a* and *b* parameters and where *θ* (or *σ* = 1/*θ*) in 2D corresponds to *ρ* (or *V*_{m} = 1/*ρ*) in 3D, the *van der Waals adsorption isotherm* becomes, with the constant *ε* = 2*a*/*b*,

__(15.100)__

This isotherm predicts a phase change, similarly as in the case of localized adsorption with attractive interaction. Characteristics are *θ*_{cri} = ⅓ and . This isotherm is not symmetric around *θ* = ½. While for *θ* > ½, the vdW isotherm resembles the Langmuir isotherm, for *θ* < ½ it decreases much more rapidly and resembles the Fowler–Guggenheim isotherm.

In fact, the surface pressure *π* = *γ*_{0} − *γ* of a film of amphiphilic molecules as a function of area *A* shows several features akin to gas compression. A schematic is provided in __Figure 15.8__, which shows the regimes labeled solid (S), liquid-condensed (L_{c}), intermediate (I), liquid-expanded (L_{e}), and gas (G). The structure changes during compression and this change is also schematically indicated. We have already mentioned the gas-like region in which the tails still make a considerable amount of contact with the liquid surface. In the liquid-expanded region the surface contains still a disordered, homogeneous distribution of molecules, but the tails lift off the surface. There is an intermediate region where gas-like and liquid-like areas coexist. The liquid-condensed phase is actually more solid-like because the tails are already aligned. Here, the molecules are relatively (surface) close-packed but are still rather mobile. In the solid-like region, the molecules are really close-packed and compressed to each other. Occasionally, a phase intermediate between the L_{c} and L_{e} phase is observed. The transition point between the S and L_{c}phase, the *Pockels point*, is a measure for the area occupied by the molecules. By knowing the amount of molecules on the surface, one can estimate the molecular area, since at that point they are just close-packed.

A pioneering experiment was that of Langmuir [36], who studied amphiphilic molecules with different chain lengths. For example, he used CH_{3}(CH_{2})* _{n}*COOH with

*n*= 14, 16 and 24, which resulted in 21, 22 and 25 Å

^{2}, respectively. From this, Langmuir concluded that the “head” area or COOH area is about 23 Å

^{2}, independent of the chain length. This field has undergone tremendous development since then and now contains many interesting structures and phenomena, detailed discussions of which are available elsewhere [37, 38].

It should be noted that, although the interpretation of the above transition between solid and liquid compressed phase of the adsorption isotherm is straightforward, and has generally been accepted as correct, debate still persists in the literature [39]^{14)}

.

**15.6.2 Hydrophobic Solutes**

Although, for hydrophobic solutes the change in *γ* is relatively small, even with fully soluble solutes segregation and orientation still occur. To illustrate such behavior we use the example of water and ethanol. Estimating the surface energy of hydroxyl groups by extrapolation from water as ∼190 mJ m^{−2}, and that of hydrocarbon groups as ∼50 mJ m^{−2}, one can easily reach the conclusion that the hydrocarbon tails should “stick out” of the solution. Supporting this suggestion is the fact that the actual surface tension of ethanol is 22 mJ m^{−2}, which is quite similar to the value for hydrocarbons. To illustrate such behavior further, we can use the data on water–ethanol solutions [40]. From measurements of the vapor pressure of water *P*_{1}, of ethanol *P*_{2}, and the surface tension *γ*, the quantity −∂*γ*/∂ln *P*_{2} can be calculated, and thus . __Table 15.4__ provides the relevant data. A useful intermediate is . Estimates for the individual values of *Γ*_{1} and *Γ*_{2} can be made only if extra assumptions are made. If we assume that the surface is covered with a monolayer, and that each molecule contributes a constant area *A _{i}* to the interface, we have

*A*

_{1}

*Γ*

_{1}+

*A*

_{2}

*Γ*

_{2}= 1. Estimating the molecular areas for H

_{2}O and CH

_{3}CH

_{2}OH as

*A*

_{1}= 0.4 × 10

^{10}cm

^{2}mol

^{−1}(

*A*

_{1}/

*N*

_{A}≅ 7 Å

^{2}) and

*A*

_{2}= 1.2 × 10

^{10}cm

^{2}mol

^{−1}(

*A*

_{2}/

*N*

_{A}≅ 20 Å

^{2}), one can estimate

*Γ*

_{1}and

*Γ*

_{2}from the above two equations, as given in

__Table 15.4__. Further calculating

*y*=

*Γ*

_{2}/(

*Γ*

_{1}+

*Γ*

_{2}), which can be considered as the mole fraction in the surface, we see that

*y*increases rapidly with the bulk mole fraction

*x*and reaches already

*y*≅ 0.6 at

*x*= 0.1. Later analysis supports this image, including the sticking out of the surface of the hydrocarbon tail [41] (suggesting an average angle of ∼40 ° with respect to the surface with a distribution width of about 30 °). Even for this relatively simple system the structure of the surface layer appears to be not completely known.

** Table 15.4** Mixture of water and ethanol at 25 °C.

^{a)}

The molecular structure of liquid solution surfaces is still intensively investigated. Today, the literature is extended and many – often opposing – views have been expressed. Important experimental methods, sum-frequency spectroscopy [42] and neutron reflection [43], as well as theoretical methods [44], have been reviewed.

**15.6.3 Hydrophilic Solutes**

Contrary to hydrophobic and amphiphilic solutes, hydrophilic solutes show depletion at the interface, leading to an increase in surface tension, albeit of a much smaller magnitude as compared to the decrease in surface tension for the other two types of solute. Theories fall into three groups: (i) direct calculation of the work of separation for a surface originally in the interior of the solution; (ii) calculation of the depletion, followed by the use of the Gibbs adsorption equation; and (iii) calculation by statistical mechanics using the pair correlation function using an extended Kirkwood–Buff approach. Here, we briefly limit the discussion to theories of the second type, since those of the first type are used only to a limited degree, while those of the third type are limited to more or less spherical molecules.

Within the theories of type two, two basic pictures have been used. In the oldest case, the increase in *γ* has been modeled by Wagner, Onsager, and Samaras [45], invoking the image charges that repel the ions from the surface. In this picture, polarizability is entirely neglected, but Stairs [46] has modified the model so as to include this effect. The second picture uses the fact that the hydration shell of ions cannot stick out of the surface. Disruption of the hydration shell would involve a large amount of energy as the hydration energies are large; hence, to keep the shell intact ions cannot approach the surface too closely [47]. Again, the field is wide, but has been reviewed [48].

It should be mentioned that an alternative picture for the surface structure of salt solutions has been proposed [49] on the basis of MD simulation in which the polarizability of the ions plays a significant role. Whilst for the small and hardly polarizable ions, such as Na^{+} and F^{−}, depletion does indeed occur, for the larger and more polarizable ions an enrichment was actually calculated. This was attributed to an anisotropic solvation, leading to a substantial dipole on the ion. The positive ions were indeed depleted from the surface, whereas the negative ions were enriched. The solvated-ion-dipole/water-dipole interactions were shown to compensate for the loss of pure-ion-charge/water-dipole interactions. This picture was consistent with, for example, surface potential measurements.

Problem 15.11: Langmuir isotherm

Show that, taking *θ* = *n*/*n*_{max} = *Kc*/(1 + *Kc*), with *n* the number of moles adsorbed and *n*_{max} the number of moles adsorbed at full monolayer coverage, the Langmuir isotherm can also be linearized as *c*/*n* = (1/*n*_{max}*K*) + (*c*/*n*_{max}).

Problem 15.12

Derive __Eq. (15.95)__.

Problem 15.13: van der Waals isotherm

Using the 2D vdW equation (*π *+ *aσ*^{−2})(*σ *− *b*) = *kT* and the Gibbs adsorption expression *Γ* = −(*RT*)^{−1}∂*γ*/∂ ln*c*, derive the vdW adsorption expression __Eq. (15.100)__. Hint: write d ln*c* as a function of d*π* and integrate the resulting expression term by term.