Comment pubs.acs.org/Langmuir
Misconceptions of the Relation between Surface Energy and Surface Tension on a Solid
I
n a recent paper,1 a relationship among the surface tension, surface energy, and surface chemical potential on a solid is derived. We point out inconsistencies in this derivation and outline that no physical interpretation exists for the origin of surface tension as defined in the article. Hui and Jagota1 presented an analysis that is claimed to provide a relationship among the surface energy, surface tension, and surface chemical potential. They begin by referencing the Shuttleworth equation that was recently proven2 to hold no implications for the relationship between surface tension and surface energy as defined by Hui and Jagota.1 Putting that aside, the derivation by Hui and Jagota1 is discussed in the following text. Their starting point is an equation for the change in the Helmholtz free energy for a solid surface, Fs, upon stretching: dFs = σ dA + μs dNs
mathematical structure of thermodynamics.3−6 The correct definition of γ at a constant temperature reads
γ=
∂Fs ∂A
(1)
(2)
Ns
Here, A is the surface area, Ns is the number of particles at the surface, and μs is the surface chemical potential. The temperature T of the system is assumed to be constant. In a solid, molecules cannot be lifted onto the surface by lateral stretching in the same manner as in a liquid. Rather, the distance between molecules increases, which causes an elastic stress in the entire object. This is not at all analogous to capillarity in liquids where the surface tension arises from the work spent in lifting molecules onto the surface. Indeed, eqs 1 and 2 define σ at a fixed Ns. Hence, σ is defined by Hui and Jagota1 for the case where no molecules are lifted onto the surface against the surface potential and no work related to that process is therefore involved. Elastic excess energy is not considered by Hui and Jagota1 either, as later in their paper they adopt a constant surface molecular density assumption. What then is the physical origin of σ and the related mechanical work in eq 1? Adding an excess surface term to a fundamental thermodynamic relation would preferably require some understanding of the physical process that it attempts to describe. Next, Hui and Jagota1 present their eq 7, which in terms of the surface energy γ reads
γ=
Fs A
dFs = γ dA
(5)
Because γ is defined to include all work in creating a new surface irrespectively of how it is done, eq 5 is applicable to a solid as well. Hence, eq 5 is the fundamental thermodynamic relationship for a surface at a constant temperature. When the constant surface density approach by Hui and Jagota1 is used, eq 1 becomes dFs = σ dA + μsρs dA = (σ + μsρs ) dA
(3)
(6)
which also points to
They state that eq 3 is the definition of surface energy. The erroneous nature of eq 3 as a thermodynamic definition has been thoroughly discussed,3−6 emphasizing that in thermodynamics an intensive property must be defined as a derivative of a state function with respect to an extensive property. Accordingly, the use of the definition in eq 3 violates the © 2014 American Chemical Society
(4)
The surface energy γ in eq 4 relates to the work spent in creating a new surface whereas the surface tension σ in eq 1 relates to the work spent in stretching a solid surface. These terms are presented in eqs 9 in 10 by Hui and Jagota1 and then equated. The gravity of the error made by equating the work in two different processes has been pointed out by many authors,6−9 including Gibbs himself, who specifically warned against this error on p 315 of his thesis.8 Hui and Jagota1 then derive the Shuttleworth equation by combining eqs 2 and 3. However, the definition in eq 3 equals the correct definition in eq 4 only when ∂γ/∂A = 0. When A is increased by stretching, ∂γ/∂A = 0 is valid only in the case of a liquid. It is inconvincible to stretch a solid, keeping ρs constant. The error in equating the work done in the stretching of two different systems, liquid and solid, has been made before and has been warned against repeatedly.5,6,9 The problems in the analysis by Hui and Jagota1 can now be summarized. First, no specific physical interpretation is given for the origin of variable σ in their starting eq 1. Second, the work spent in creating a new surface is replaced by the work spent in stretching, although these processes are fundamentally different in the case of a solid. Third, the thermodynamic theory is applied, but the surface energy is defined in eq 3 in a way that is incompatible with the mathematical structure of thermodynamics. Finally, combining eqs 2 and 3 in order to obtain an equation for a solid includes an equation that, upon stretching, is valid for a liquid only. Any of these four errors alone is sufficient to render the results of Hui and Jagota1 invalid. The total work of creating a new surface at a fixed temperature can be expressed in terms of the thermodynamic surface energy γ, which includes all excess surface energy:
The definition of variable σ, called the surface tension, follows from eq 1 as σ=
∂Fs ∂A
γ = σ + μsρs
(7)
Received: December 30, 2013 Published: February 11, 2014 2580
dx.doi.org/10.1021/la404921t | Langmuir 2014, 30, 2580−2581
Langmuir
Comment
(8) Gibbs, J. W. The Scientific Papers of J. Willard Gibbs: In Two Volumes; Ox Bow Press: Woodbridge, CT, 1993; Vol. 1. (9) Gutman, E. M. Comments on the article entitled “Additional remarks related with the discussion inaugurated by the article Incompatibility of the Shuttleworth equation with Hermann’s mathematical structure of thermodynamics. Surf. Sci. 2011, 605, 644−645. (10) Gurtin, M. E.; Murdoch, A. I. Surface stress in solids. Int. J. Solids Struct. 1978, 14, 431−440. (11) Wolfer, W. G. Elastic properties of surfaces on nanoparticles. Acta Mater. 2011, 59, 7736−7743.
being the work conjugate of the surface area in the system considered by Hui and Jagota,1 not σ as proposed.1 The result of Hui and Jagota1 is actually trivial when incorrectly using the constant surface density assumption for a solid because eq 7 then follows directly from eqs 1 and 5. Such a formal triviality does not signify any physical phenomenon, of course. Thus, the title of the paper1 “Surface tension, surface energy, and chemical potential due to their difference” is misleading because it implies a causality that does not exist. The issue here is simply that the same equilibrium can be presented by variables that are defined differently. In particular, one should note that γ = μs(∂Ns/∂A), so that eq 7 reduces to σ = 0. To give eq 1 a meaningful physical interpretation in the case of a soft solid material, σ should be defined as an elastic stress and dA should be defined as the area change related to the elastic deformation only. As for the elastic excess stress, the molecular structure at a surface is different from that in the bulk, and this has been connected to the concept of mechanical surface tension.10 However, it is not easy to apply such an approach to the Gibbsian analysis or to identify a representative surface layer for mechanical considerations.11 A force involves a counterforce, the implication of which is the well-known rule in engineering mechanics that prestressing layers of solids with respect to each other has no effect on the work done in stretching the entire system. Thus, even if it were possible to stretch a thin surface layer independently, one finds difficulties in identifying the origin of the excess stress and, consequently, in asserting how a physically relevant surface tension on a solid should be understood.
Lasse Makkonen
■ ■ ■
VTT Technical Research Centre of Finland, Espoo 02044 VTT, Finland
AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS This work was supported by the Academy of Finland and Norden. REFERENCES
(1) Hui, C.-Y.; Jagota, A. Surface tension, surface energy, and chemical potential due to their difference. Langmuir 2013, 29, 11310− 11316. (2) Makkonen, L. Misinterpretation of the Shuttleworth equation. Scr. Mater. 2012, 66, 627−629. (3) Bottomley, D. J.; Makkonen, L.; Kolari, K. Incompatibility of the Shuttleworth equation with Hermann’s mathematical structure of thermodynamics. Surf. Sci. 2009, 603, 97−101. (4) Bottomley, D. J.; Makkonen, L.; Kolari, K. Reply to: “Comment by H. Ibach on: Incompatibility of the Shuttleworth equation with Hermann’s mathematical structure of thermodynamics”. Surf. Sci. 2009, 603, 2356−2357. (5) Bottomley, D. J.; Makkonen, L.; Kolari, K. Reply to: Additional remarks related with the discussion inaugurated by the article “Incompatibility of the Shuttleworth equation with Hermann’s mathematical structure of thermodynamics”. Surf. Sci. 2010, 604, 2066−2068. (6) Láng, G.; Heusler, K. E. Can the integral energy function of solid interfaces be of a non-homogenous nature? J. Electroanal. Chem. 1999, 472, 168−173. (7) Makkonen, L. The Gibbs−Thomson equation and the solid− liquid interface. Langmuir 2002, 18, 1445−1448. 2581
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