Gibbs Adsorption Equation

Several models have been developed to predict the equilibrium-adsorbed amount of some substance at some interface as a function of the amount of nonadsorbed substance pres-ent. The Gibbs adsorption equation is the thermodynamic expression which relates the adsorbed amount (or surface excess concentration) of a species to γ and the bulk activity or fugacity of that species. It is widely applied to study of adsorption phenomena, in and outside of food science and technology, especially at the air–water interface. The Gibbs adsorption equation is written as

−^d =

∑

i

γ Γ µ (2.3)

where Γi (mol/m²) is the excess surface concentration of component i, and μ_i (J/mol) is its chemical potential [1]. A more useful form of Equation 2.3 can be developed as follows.

Consider adsorption of a component i, dissolved in a liquid, α, at the liquid–vapor inter-face. For this system, Equation 2.3 becomes −dγ = Γα dμ_α + Γi dμ_i. If the three-dimensional interfacial region is defined such that the excess concentration of liquid α contained within it is zero [1–4], then −dγ = Γi dμ_i. The chemical potential, μ_i, is equal to µi = µi⁰ +Rtln( ), ai

where mi0 is the standard chemical potential of component i in solution, R is the gas con-stant, t is temperature, and a_i is the activity of component i in solution. Thus, dμ_i = Rtd ln (a_i), and Equation 2.3 becomes

Γi

Rt i

d d a

= −  1  γ

ln( ) (2.4)

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For the majority of systems characterized by strongly adsorbed monomolecular films, the surface excess concentration and the total surface concentration of the adsorbate are more or less identical. Adsorption involves a profound reduction in γ, even for dilute solu-tions, and in that case a_i can be approximated by c_i, the concentration of species i in solu-tion. Equation 2.4 can then be written in its simplest and most used form,

Γi

Thus adsorption reduces γ, and a plot of this surface tension reduction vs. the equi-librium solution concentration of a given adsorbate allows for the determination of the adsorption isotherm: an expression for Γ as a function of solution concentration.

However, proper application of the Gibbs adsorption equation requires an under-standing of its origin. For example, equilibrium “spreading pressure” data have been recorded as a function of solution concentration for a wide variety of proteins, surfactants, and other molecules used as stabilizers in food foams and emulsions. The spreading pressure, denoted by Π (J/m²), is simply a positive measure of surface energy reduction at a given solution concentration; that is, Π is the difference between γ evaluated when Γ = 0, and γ evaluated at a selected solution concentration (i.e., Π = γ_ci − γ_ci=0). The concentra-tion dependence of Π is shown in Table 2.1 for each of two milk proteins at the air–water interface: α-lactalbumin and bovine serum albumin. These data were recorded at 25°C in each case, from 0.010 M sodium phosphate buffer, pH 7.0 [5,6]. One could use these data to derive an expression for the isotherm, Γ = f(ci), according to Equation 2.5, where −dγ = dΠ, and compare the isotherm to independent experimental determinations of adsorbed amounts for each of these proteins. Table 2.2 lists the results of such an exercise, provid-ing a comparison of adsorbed amounts of α-lactalbumin and BSA recorded by direct

Table 2.1 Concentration Dependence of Equilibrium Spreading Pressure for α-Lactalbumin and Bovine Serum Albumin at the Air–Water Interface

Equilibrium Spreading Pressure, Π (mJ/m²)

Protein Concentration, C (mg/mL) α-Lac BSA

0.05 23.9 11.8

surface spectroscopy at a hydrophobic, silanized silica–water interface (fundamentally similar to the model hydrophobic air–water interface), and calculated with Table 2.1 and Equation 2.5 as a function of equilibrium concentration. The comparison reveals impor-tant issues surrounding the utility of the Gibbs adsorption equation in this context. In particular, although it did provide estimates within an order of magnitude of actual val-ues, the Gibbs adsorption equation clearly overpredicted the actual adsorbed amounts in each case. Also, the plateau value of adsorbed mass for α-lactalbumin is about twice that of BSA, consistent with the respective plateau values of Π, but not consistent with experi-mental values of adsorbed mass, where Γplateau,α−lac is several times greater than Γplateau,BsA. The Gibbs adsorption equation treats every incremental change in Π as an increase in Γ.

This feature serves to limit its application in analysis of protein adsorption, where one protein molecule can bind to the surface by multiple noncovalent contacts [7,8]. In this way, through unfolding and formation of new contacts with the apolar interface, γ may continue to decrease even after adsorption has ended and Γ is constant. However, the Gibbs adsorption equation can be a useful starting point for analysis of small-molecule surfactant adsorption.

The study of adsorption from solution onto solids is important and wide-ranging, and in many cases the most relevant to practical applications. The solid–liquid interface is cer-tainly the least understood of the four major interfaces, however, due to its complexity, and this contributes to it being the most actively studied interface today. Solid surfaces have different electrical and optical properties than found in the bulk, and can be characterized by atomic- or molecular-level textures and roughnesses. They are generally energetically heterogeneous. For example, although a solid surface may be assigned a particular “wetta-bility,” it would most likely be the result of a distribution of heterogeneous surface regions of varying wettability [9].

Measurement of γ at liquid–liquid and liquid–gas interfaces is straightforward, and is briefly summarized in Section 2.3. Evaluation of solid surface energetics is considerably less straightforward. In the following section, we discuss evaluation of properties relevant to

Table 2.2 Adsorbed Amounts of α-Lactalbumin and BSA, Determined Experimentally at a Model Hydrophobic Interface and Calculated with Application of the Gibbs Adsorption Equation

Adsorbed Amount, Γ (nmol/m²) Protein Concentration,

C (mg/mL)

Experimental Determination

Gibbs Adsorption Equation

α-Lac BSA α-Lac BSA

0.10 114 35 569 305

0.30 115 60 604 325

0.60 221 81 628 338

1.00 205 75 646 348

1.50 236 78 660 357

2.00 218 74 671 363

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γ for solid surfaces, through analysis of a very widely used macroscopic thermodynamic approach.