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J. Phys. Chem. B 2006, 110, 3448
Reply to “Comments on the Thermodynamic Justification of an Equation of State for Monolayers” by Anatoly I. Rusanov V. B. Fainerman† and D. Vollhardt*,‡ Medical Physicochemical Centre, Donetsk Medical UniVersity 16 Ilych AVenue, Donetsk 83003, Ukraine, and Max Planck Institute of Colloids and Interfaces, D-14424 Potsdam/Golm, Germany ReceiVed: October 19, 2005 In the past decade, the fundamental progress in the experimental determination of the main characteristics of Langmuir monolayers in microscopic and molecular scale required improved thermodynamic possibilities to describe the surface pressure-area (π-A) isotherms, as the classical equations of state allowed only the characterization of the fluid (gaseous, liquid-expanded) state. To filling this gap, we developed thermodynamically based equations of state that consider also the aggregation of the monolayer material to the condensed phase. Thus, for the first time (as far as we are aware) those parts of the π-A isotherm can be characterized that represent the transition region to the condensed phase and are decisive for the features of the monolayer. We have shown in numerous papers that the application of these equations of state are in good (very often in excellent) agreement with the experimental results obtained for monolayers of very different amphiphiles. In a recent paper, Rusanov gave another derivation for an equation for the fluid state of Langmuir monolayers.1 In his Comment5 Rusanov claims that our basic differential equation of state (eq 5) and thus, all other equations derived on its basis, are incorrect. We disagree absolutely with Rusanov’s conclusion. In the following reasoning we refer to refs 1-3 and the formula presented in Rusanov’s Comment.5 In our publication1 the equation of state for the surface layer was derived by the simultaneous solution of the differential equation for the surface potential eq 2 and the Gibbs’ adsorption equation (eq 1). In ref 1, eq 5 derived in this way was then used to obtain known (Frumkin’s, van der Waals’, and Volmer’s) equations of state. This fact testifies, at least implicitly, to the correctness of the method used. Also in ref 1, the generalized Volmer’s equation (for the multicomponent insoluble monolayer) was derived, which was then extended to include the liquid-expanded and liquid-condensed coexistence region of the monolayer. In our subsequent publications these equations were further generalized to take into account the dissociation/ association processes in Langmuir monolayers, a further phase transition between two condensed phases, and were successfully used to describe the experimental π-A isotherms for various insoluble amphiphilic compounds. In Rusanov’s paper2 the so-called “excluded area” (aex) of the molecule in the insoluble monolayer was taken into account, * Corresponding author. † Donetsk Medical University. ‡ Max Planck Institute of Colloids and Interfaces.
and therefore the solution of the equation proposed by Rusanov for the chemical potential simultaneously with the Gibbs’ equation resulted in eq 6, which involves the excluded area, aex, instead of the partial molar area of a molecule, ω. Assuming this difference to be insignificant, we cited his study2 in our publication,3 and indicated the similarity between eqs 5 and 6. This conclusion was obtained on the basis of the fact that the values for the area per one molecule in the extremely saturated monolayer, as calculated from Rusanov’s equation of state (which assumes the excluded area factor), are very close to the values calculated from our equation, see ref 3. We agree, however, that the differential equation for the chemical potential, derived by Rusanov with the excluded area taken into account, differs essentially from Butler’s equation in the differential form (eq 2) which was used by us. Now in the Comment5 the attempt is made to prove that eq 5 (and, therefore, the equations derived from this eq 5 in ref 1 and our subsequent publications) is erroneous. However, this proof is deficient. The error made by Rusanov is that eq 7 was used incorrectly. Both in our publication1 and in Rusanov’s study,2 Gibbs’ equation (eq 1) was used for a certain choice of the dividing surface, which implies zero adsorption of the subphase (the solvent). For example, in the case that one insoluble component is present, eq 1 becomes: dγ ) -Γdµ (or dΠ ) Γdµ). Introducing the relation dµ ) dΠ/Γ into eq 2 and taking into account eqs 3 and 4, one obtains eq 5. It follows from eq 9 used by Rusanov that, for this (one-component) insoluble monolayer, Γω ) 1. This is, however, incorrect. The correct relation is Γω ) θ, where θ is the fraction of the surface occupied by insoluble molecules whose molar area ω is either fixed or relatively slowly varied with γ and T, see eq 10. In this case, the fraction of the surface equal to 1 - θ is the fraction of the unoccupied places (vacancies) in the insoluble monolayer. Therefore, the relation Γω ) 1 is valid only for extremely saturated monolayers. Only in this case, the surface pressure becomes infinitely large. If one assumes (following Rusanov) that Γω ) 1 for any state of the one-component monolayer, an absurd result follows: for a close-to-zero adsorption (zero surface excess) value the molar area of the insoluble component is infinitely large. The same applies to eq 7 for the mixture of several insoluble components: if the presence of vacancies is taken into account, then ∑ωiΓi e 1. Rusanov himself shows in a recent paper4 the fact that the excluded area is not the defining factor. In ref 4 he has taken the factor of the excluded area equal to unity as an approximation. This results precisely in eq 5. Summarizing we believe that the comments are clearly erratic. References and Notes (1) Fainerman, V. B.; Vollhardt, D. J. Phys. Chem. 1999, 103, 145. (2) Rusanov, A. I. J. Chem Phys. 2004, 120, 10736. (3) Fainerman, V. B.; Vollhardt, D., Liu, F. J. Phys. Chem. B 2005, 109, 11706. (4) Rusanov, A. I. Colloids Surf. A 2004, 239, 105. (5) Rusanov, A. I. J. Phys. Chem. B 2006, 110, xxxxx.
10.1021/jp0582742 CCC: $33.50 © 2006 American Chemical Society Published on Web 01/31/2006
