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Langmuir 1996, 12, 3728-3732
Contact Angle and Vapor Adsorption Malcolm E. Schrader† Department of Inorganic and Analytical Chemistry, The Hebrew University of Jerusalem, Jerusalem, Israel Received December 5, 1994. In Final Form: May 2, 1996X It is shown that the original Good-Girifalco derivation of an equation for cos θ (where θ is the contact angle of a liquid on a solid) based on intermolecular force (IMF) theory contains an inconsistency which leads to dependence of cos θ on the term -πe/γL (where πe is the reduction in free energy on equilibraton of the solid surface with the vapor of the liquid at its saturation pressure). It is pointed out, furthermore, that the inconsistency has been carried over into the succeeding IMF approaches, such as, for example, that of Fowkes, that of Owens and Wendt, and that of Wu. Reported high experimental values for πe on low-energy surfaces and the possible effect on contact angle prediction are discussed in light of the above and of other factors.
Introduction 1
In 1957 Girifalco and Good proposed an equation for the prediction of interfacial surface tensions, essentially liquid-liquid, from the values of the individual surface tensions
γ12 ) γ1 + γ2 - 2φ(γ1γ2)
1/2
(1)
where γ12 is the interfacial tension between two condensed phases, γ1 is the surface tension of one condensed phase, γ2 that of the other, and φ is a system constant. In 1960, Good and Girifalco2 used their method for the study of the liquid-solid interface. The methods for measuring interfacial tension of liquid-liquid interfaces are not applicable to the solid-liquid interface. However, useful practical and fundamental information can be gained by the measurement of contact angles, whose equilibrium energetics are described by3
γL cos θ ) γSV - γSL
(2)
where θ is the contact angle, γSV the free energy per unit area of the solid surface in equilibrium with the vapor of the liquid, and γSL the free energy per unit area of the solid-liquid interface. Good and Girifalco let the subscript 1 in (1) be the solid, designated by S, and the subscript 2 be the liquid, designated by L, so that (1) becomes
γSL ) γS + γL - 2φ(γSγL)1/2
(3)
Substituting (3) in (2), they wrote
γL cos θ ) γSV - [γS + γL - 2φ(γSγL)1/2]
(4)
surface tension, liquid or solid, could be resolved into various components so that for example
γ ) γd + γh
(6)
where γ is the total surface tension, γd is the dispersion component of the surface tension and γh its hydrogenbonding component. The solid-liquid interfacial free energy was then written
γSL ) γS + γL - 2(γSdγLd)1/2
(7)
and, substituting into (2), there is
γL cos θ ) γSV - [γS + γL - 2(γSdγLd)1/2]
(8)
Since γSV ) γS - πe, the equation seems to reduce to
cos θ ) -πe/γL - 1 + 2(γSdγLd)1/2/γL
(9)
Once again, cos θ is equal to minus 1, plus an interaction term, minus πe/γL. In 1969 Owens and Wendt5 suggested the use of the following highly empirical formula for interfacial tension, obtained in the following manner:
γ ) γd + γp
(10)
where γp is a “polar” component of the surface free energy. The solid-liquid interfacial energy is then given as
γSL ) γS + γL - 2(γSdγLd)1/2 - 2(γSpγLp)1/2
(11)
Substituting into (2), there is Since γSV ) γS - γe, the above equation seemed to reduce to
cos θ ) -πe/γL - 1 + 2φ(γSγL)1/2/γL
γL cos θ ) γSV - [γS + γL - 2(γSdγLd)1/2 - 2(γSpγLp)1/2] (12)
(5)
In 1964 Fowkes4 treated the hypothetical case where all interaction between solid and liquid could be attributed to London “dispersion” forces. He assumed that any
Since γSV ) γS - πe, the equation seems to reduce to
cos θ ) -πe/γL - 1 + 2(γSdγLd)1/2/γL + 2(γSpγLp)1/2/γL (13)
†
E-mail:
[email protected]. FAX: 972-2-6585319. X Abstract published in Advance ACS Abstracts, July 1, 1996. (1) Girifalco, L. A.; Good, R. J. J. Phys. Chem. 1957, 61, 904. (2) Good, R. J.; Girifalco, L. A. J. Phys. Chem. 1960, 64, 561. (3) Young, T. Philos. Trans. R. Soc. London 1805, 95, 65. (4) Fowkes, F. M. Ind. Eng. Chem. 1964, 56 (12), 40.
S0743-7463(94)00962-5 CCC: $12.00
Once again -πe/γL appears as a thermodynamic term in the equation for cos θ. (5) Owens, D. K.; Wendt, R. D. J. Appl. Polym. Sci. 1969, 13, 741.
© 1996 American Chemical Society
Contact Angle and Vapor Adsorption
Langmuir, Vol. 12, No. 15, 1996 3729
Still another approach is to use the framework of (11) with a different type of interaction in place of the root mean square, such as6
γ1γ2/(γ1 + γ2) in which case we have
γSL ) γS + γL - 2(γSdγLd)/(γSd + γLd) p
p
substitution to be valid, γSL must have the same meaning in both equations. Since it is the solid-liquid interfacial free energy existing in an atmosphere of liquid vapor in (2), it must be the interfacial free energy remaining after the solid and liquid are combined in an atmosphere of liquid-vapor, in eq 3. For that to be the case, the correct rendition of (3) should be
γSL ) γSV + γL - 2φ(γSVγL)1/2 p
(3′)
p
2(γS γL )/(γS + γL ) (14) Since γSV ) γS - γe, the equation seems to reduce to
cos θ ) -πe/γL - 1 + 2(γSdγLd)/(γSd + γLd)γL + 2(γSpγLp)/(γSp + γLp)γL (15) Once again, the thermodynamic term -πe/γL appears. All the above have in common an interaction term, which is a proposed work of adhesion based on independently known characteristics of the liquid and solid, and a πe/γL term, which comes from the manner of fitting this work of adhesion into the Young equation. Specifically, the latter involves the use of γS as the free energy of the separated solid surface in eqs 7, 11, and 14. In their 1960 paper, Good and Girifalco stated that for systems with reasonably high contact angles, πe would be near 0. Good subsequently7 calculated low values for series of liquids on low-energy surfaces. He dropped πe from his equations on this basis. Fowkes also stated πe should be negligible for above-zero contact angle systems, ultimately claimed that it was zero, and also dropped it from his equations. Adamson and co-workers8,9 were skeptical about the alleged ubiquitous near-zero value for πe. They conducted a series of experiments measuring adsorption isotherms on low-energy surfaces, using ellipsometry to follow the thickness of the adsorbed layer. Their results yield substantial, often high, values of πe on substances where it “should” be zero. This has been confirmed by other investigations,10 where tens of mJ/m2 can sometimes be added to the surface energy. The presence and amount of πe consequently is of substantial importance. In this paper it is shown that the term -πe/γL appears in all the intermolecular force (IMF) theories listed above, as well as others, as a result of an inconsistency in the original Good-Girifalco derivation. The consequences are discussed. Results and Discussion Origin of the Inconsistency. In their 1960 publication Good and Girifalco correctly used the Bangham modificaton11 of the Young equation to represent the balance of free energies yielding a stable reproducible contact angle for a liquid on a smooth solid surface. The Young-Bangham relation, eq 2 in this paper, represents the equilibrium contact angle in a system where the sample is completely equilibrated with the vapor of the liquid. For correct reproducible measurements of thermodynamic significance, this vapor equilibration is a minimum requirement. The authors then substitute the value of γSL from their own eq 3 into (2). Clearly, for this (6) Wu, S. Polymer Interface and Adhesion, 1st ed.; Dekker: New York, 1982. (7) Good, R. J. J. Colloid Interface Sci. 1975, 52, 308. (8) Tadros, M. E.; Hu, P.; Adamson, A. W. J. Colloid Interface Sci. 1974, 49, 184. (9) Adamson, A. W.; Hu, P. J. Colloid Interface Sci. 1977, 59, 605. (10) Busscher, H. J.; Kip, G. A. M.; van Silfhout, A.; Arends , J. J. Colloid Interface Sci. 1986, 114, 303. (11) Bangham, D. H.; Razouk, R. I. Trans. Faraday Soc. 1937, 33, 1459.
where γSV is the surface tension of the solid in equilibrium with the vapor of the liquid. Substituting (3′) in (2), we have
γL cos θ ) γSV - [γSV + γL - 2φ(γSVγL)1/2]
(4′)
Now, the γSV terms, representing the surface free energy of the solid in an atmosphere of liquid-vapor, cancel out, and we have, instead of (5)
cos θ ) -1 + 2φ(γSVγSL)1/2/γL
(5′)
We now note that the term -πe/γL is no longer present. Furthermore, the eq 5′ cannot be written in any manner which will reintroduce it. For example, if the γSV present under the square root sign is written as (γS - γe), and separated out, it will not reappear as -πe/γL. The canceling out of γSV during substitution into (2) is a fundamental phenomenon with no relation to the form or type of the hypothesized “interaction term”. This will be discussed further in the next section. Work of Adhesion and Interfacial Free Energy. The original Dupre equation12,13 may be written
WAB ) γA + γB - γAB
(16)
where γA is the surface free energy of the condensed phase A, γB is that of the condensed phase B, and WAB is the work of adhesion of A to B. Where A is a solid and B a liquid, S is substituted for A, and L for B. We then have
WSL ) γS + γL - γSL
(17)
We now write the Young equation
γL cos θ ) γS - γSL
(18)
Assuming that the liquid, as well as the solid, has no vapor pressure, then no vapor adsorbs to the solid, and γS in eqs 17 and 18 refers to the surface free energy of the clean solid. Combining (17) and (18)
WSL ) γS + γL + γL cos θ - γS
(19)
WSL ) γL(1 + cos θ)
(20)
the well-known Young-Dupre equation.14,15 Now, consider the more usual case of a liquid which does have a vapor pressure. The vapor may now adsorb to the solid. At equilibrium, we have
WSL ) γSV + γL - γSL
(21)
(12) Dupre, A. Theorie Mechanique de la Chaleur; Gauthier-Villars: Paris, 1869; p 369. (13) Adam, N. K. The Physics and Chemistry of Surfaces, 3rd ed.; Oxford Press: London, 1941; p 8. (14) Dupre, A. Theorie Mechanique de la Chaleur; Gauthier-Villars: Paris, 1869; p 393. (15) Adam, N. K. The Physics and Chemistry of Surfaces, 3rd ed.; Oxford Press: London, 1941; p 179.
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Langmuir, Vol. 12, No. 15, 1996
Schrader
of adhesion. Rearranging
and
γL cos θ ) γSV - γSL
(2)
γL(1 + cos θ) ) 2φ(γSγL)1/2
(24)
where γSV in eqs 21 and 2 is the surface free energy of the solid surface in equilibrium with the vapor. Combining 21 and 2
Then, since the interaction term 2φ(γSγL)1/2 equals the work of adhesion WSL, there is
WSL ) γSV + γL + γL cos θ - γSV
γL(1 + cos θ) ) WSL
(22)
therefore
WSL ) γL(1 + cos θ)
(20)
and eq 20 appears once again. Equation 20 therefore is valid for all cases of solid-liquid-vapor equilibrium, from liquids of near-zero saturation vapor pressure up to high values. The solid surface free energy used in eq 17 must of course be the same as the solid surface free energy used in eq 18 if the two equations are to be combined. It is clear then that the solid surface free energy of the Dupre equation (17) always cancels the solid surface free energy of the Young equation (16), regardless of whether we refer to the clean surface or vapor adsorbed surface. It must be kept in mind, then, that (20) refers to the general case where there may be, and probably is, vapor pressure from the liquid, so that WSL refers to the separation of the liquid mass (retaining its shape16 ) from the solid in an atmosphere of the liquid vapor. This is what Bangham called the work of adhesion as generally defined.11 Let us now examine the Good-Girifalco equation of 1960. It is set up as follows:
γSL ) γS + γL - 2φ(γSγL)
1/2
(3)
We now rewrite (17), the Dupre equation, in the form
γSL ) γS + γL - WSL
(17)
It can be seen that the Girifalco-Good interaction term is exactly equal to the Dupre work of adhesion. When Good and Girifalco combine their eq 3 with the Young equation, the result, if correctly done, must be the same as the Dupre combination of (17) with (18) to obtain the Young-Dupre equation (20). Of specific interest is the term for the solid surface free energy, which occurs in both the Dupre equation (17) and the Young equation (18). If both equations are to be combined, obviously the symbol must have the same meaning in both. The only correct meaning it can have, in fact, is that of a system that is completely equilibrated. If there is a vapor pressure, then the solid of eq 21 has the free energy γSV, where the V indicates equilibration with the vapor, and the solid of eq 2 also has the free energy γSV, where the V indicates equilibrium with the vapor. When this is correctly done, a solid surface free energy term (γSV in this case) from the Dupre equation (21) will always cancel a solid surface free energy term from the Young equation (2). Now, we have shown that the GG interaction term is a work of adhesion. It can then be seen that the final equation for cos θ of Good-Girifalco, obtained after they assume the approximation πe ) 0, which is
cos θ ) 2φ(γS/γL)1/2 - 1
(23)
is simply a Young-Dupre equation with the GirifalcoGood geometric-mean approximation proposed as the work (16) Schrader, M. E. Langmuir 1995, 11, 3585.
(20)
which is the Young-Dupre equation. Note that whereas Good and Girifalco derive it only after neglecting γe, it was derived here, (20), in a natural rigorous manner by using a consistent definition of the solid-surface free energy. It is emphasized that this issue has nothing to do with the form of the proposed interaction term (ie, geometric mean with φ, geometric mean of dispersion, γp, acid-base, etc.) but rather with the manner in which the interaction term is fitted into the Young equation. This, in turn, is a fundamental characteristic of the YoungDupre formulation. The Inconsistency as a Correction Procedure. An attempt at justifying the Good and Girifalco derivation (which results in the term πe being included in their original equation for cos θ and retained whenever its value is not negligible) could be presented as follows: One starts with the interaction term, inside the Dupre relation
γSL ) γS + γL - 2φ(γSγL)1/2
(3)
where, from the practical point of view, if one cannot use γS, the free energy of the clean solid surface, the entire approach will be of little use. For self-consistency, all solid surface free energies, including that in the Young equation itself, should then be that of the clean solid surface. In that case, of course, πe does not appear anywhere in the equation for cos θ. However, since the equilibrium contact angle will ultimately be measured at saturation vapor pressure, the Young equation is written
γL cos θ ) γSV - γSL
(2)
where the free energy at the solid-gas interface is given by γSV, that of the solid in equilibrium with the vapor of the liquid. The interaction term therefore contains a deliberate error, since it uses γS instead of γSV. According to the prevailing view, γS > γSV, by the amount πe, so the calculated interaction will be too great. However, since the Young equation is written (where γS - πe ) γSV)
γL cos θ ) γS - πe - γSL
(25)
when the proposed value for γSL is combined with it we have
cos θ ) -πe/γL - 1 + 2φ(γSγL)1/2/γL
(5)
According to this approach, therefore, the value of cos θ was originally too high due to an interaction term which is too great. So, it is corrected for by the term -πe/γL which lowers the calculated cos θ. It is clear however, that this procedure does not work. First, while γS is allegedly greater than γSV by πe, the calculated interaction is not greater by the amount πe, since it is under the square root sign. Second, the free energy of a surface with a physically adsorbed adlayer cannot necessarily be used under the square root sign, so there is no baseline for a
Contact Angle and Vapor Adsorption
calculated excess. As Good and van Oss17 point out, “The Good-Girifalco-Fowkes equations ... make it an essential premise of the model that the composition of each phase is homogeneous right up to the physical surfaces that separate the phases. If the surface region consists of a layer of close-packed, oriented molecules, such that the outer surface is “autonomous”, that is effectively uninfluenced by the substrate, then the conditions for validity of the Good-Girifalco theory are fulfilled. But if the outer surface is not close packed; for example if it is a partial monolayer or liquidlike adsorbed film, then the conditions will not be fulfilled. In such cases there may be no grounds for believing a priori that the geometric mean relations will hold for the LW (dispersion) component of surface energy; and this denial will also hold with respect to the acid-base combining rules....” Furthermore, it will be shown in a forthcoming publication that (γS - γe), as presently measured and calculated, does not, in the general case, yield a correct value for the free energy of the solid surface. Recommended Nomenclature Adjustments. Bangham originally emphasized that in the Young equation for contact angles, the free energy of the solid surface is that existing when the solid surface is in equilibrium with the vapor of the liquid of the sessile drop. To aid in remembering this, and in differentiating that surface from one which exists in vacuum, he proposed the notation γSV for the vapor-equilibrated surface and γS for that existing in vacuum. These notations are in common use to this day. Bangham also pointed out that the Young-Dupre work of adhesion could be interpreted in two ways. One, which he called the work of adhesion as commonly defined, is that required to separate a drop from the surface in an atmosphere of the vapor of the liquid. The other, which he designated as the work of adhesion as commonly understood, is that required to separate a drop from a surface while leaving behind a clean surface, such as exists in vacuum. Often the exact same symbols are used to describe these two different quantities, with the difference either being ignored or relying on an accompanying verbal description. As indicated previously in this paper, The Young-Dupre equation (20) is obtained for the normal case of a liquid with significant vapor pressure where the liquid and solid are separated in an atmosphere of the vapor, or for the less usual case of a liquid with negligible vapor pressure. There are investigators, however, as we have seen, who may choose to deal with a hypothetical situation where the sessile drop on a solid surface, in an atmosphere of the vapor of the liquid, is removed from the surface in such a manner as to take the adlayer of adsorbed vapor on the solid surface along with it. In a sense this may be regarded as a particular case of the general problem of the work of adhesion of a composite interface, where, say, the interfacial composition varies along the axis perpendicular to the interfacial bond. It is then important to specify where the bond is broken (or joined), by indicating the composition of the outer surface layer(s) after separation (or before joining). The Bangham notation of (21) where the solid-gas interface is designated γSV, is then not quite an adequate description of the process yielding a work of adhesion in view of the options available. Rewriting the equation so that the right hand side represents the free energies of the separated surfaces, while the left hand side represents the free energies associated with the bonded solid-liquid (17) Good, R. J.; van Oss, C. J. In Modern Approaches to Wettability: Theory and Applications; Schrader, M. E., Loeb, G. I., Eds.; Plenum Press: New York, 1992; p 22.
Langmuir, Vol. 12, No. 15, 1996 3731
interface
WSL + γSL ) γSV + γL
(21)
it should be, but is not generally, understood that the bond is to be broken in a manner that will leave an adsorbed adlayer on the solid surface. Furthermore, when eq 20 is obtained
WSL ) γL(1 + cos θ)
(20)
the free energy of the solid surface is now canceled out and there is no longer any explicit indication of where the bond is broken during dissociation. Also, in the Young equation
γL cos θ ) γSV - γSL
(2)
which is combined with (21) to yield (20), it is unfortunately not clear to many where the interfacial bond is broken, or joined, in obtaining (20). Furthermore, as we have shown, this can even lead to two different interpretations of where the interfacial bond is to be broken, with one used in (21) and the other used in (2). This of course is not a matter of semantics, since one interpretation will yield a separated solid surface with energy γSV, while the other will yield a separated solid surface with energy γS. It is consequently recommended that both the symbol representing the work of adhesion (energy of interaction at the interface), and the symbol for its complementary quantity the residual free energy at the interface (see (21) above), be written with subscripts to clearly indicate the two surfaces to which this interface is being separated on describing a work of adhesion and interfacial tension. For the case where there is a vapor pressure, and the separation is performed at equilibrium in an atmosphere of the liquid-vapor, (21) would then be written
WS(V)L ) γSV + γL - γS(V)L
(25)
and (2) would be written
γL cos θ ) γSV - γS(V)L
(26)
where WS(V)L replaces WSL to indicate that the work of adhesion refers to an interface formed from a solid which had liquid-vapor adsorbed to it. Likewise, γS(V)L replaces γSL to indicate that when this interfacial energy is used in a Dupre equation for work of adhesion, the interface must be split in such a manner as to leave behind an adlayer of vapor molecules on the solid surface. The parentheses on the V in the subscript indicate that this is not a three-phase line but rather an instruction on how the interface is formed, i.e., what its components are when disjoined. Experimental Findings of πe on Low-Energy Surfaces. It is mentioned earlier in this paper that groups working with the ellipsometer obtained results that appeared to contradict the expectations of many investigators, such as, for example, Zisman,18 Fowkes,19 and Good2 that πe for low-energy high-contact angle systems should generally be very low. One striking result among these is obtained by both Adamson and Busscher during ellipsometric measurements of πe of water on PTFE. Expectations for this system, where the solid is from a type of material yielding the lowest solid surface energies presently obtainable and the liquid has the highest surface energy known among “ordinary” liquids (Hg is excluded), (18) Fox, H. W.; Zisman, W. A. J. Colloid Sci. 1952, 7, 428. (19) Fowkes, F. M.; Sawyer, W. M. J. Chem. Phys. 1952, 20, 1650.
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are for a truly negligible value of both the amount adsorbed and the reduction in free energy, πe. Both Adamson and, subsequently, Busscher obtained the far from negligible value of 8.8 to 9.0 mJ/m2. Of course, the results of our paper indicate that those ellipsometric results do not formally affect the values of cos θ predicted by GoodGirifalco or other IMF equations in the manner generally expected, since the equations, when correctly derived, do not contain the term πe/γL. Nevertheless, a PTFE surface containing that much bound water can obviously not be regarded as a molecularly clean polymer of bulk composition for use in the root-mean-square interaction term. At the same time, the ellipsometric results do seem to contradict the rather soundly based expectation of Zisman, Fowkes, and Good, that the adsorption here should be negligible. The answer has been aptly stated by Wu6 who suggested that the anomaly is caused by the presence of porosity or hydrophilic sites introduced during surface preparation. Certainly, experience with clean surfaces and their ubiquitous possible contaminants obtained through present-day use of ultrahigh vacuum analytical techniques such as , for example, Auger electron spectroscopy or X-ray photoelectron spectroscopy, would cause one to strongly doubt the feasibility of obtaining clean surfaces from the preparation techniques used for these ellipsometric adsorption isotherms, without subsequent ion bombardment or other in situ cleaning in an ultraclean vacuum system. Evidence that these surfaces are indeed contaminated arises from the fact that the contact angles observed are substantially lower than the established values for these materials. Adamson’s group utilized boiling ethanol, boiling water, contact with glass slides, more boiling water, and degassing in a “conventional”
Schrader
vacuum system at 10-5 Torr. Busscher’s group desribes the surface preparation as “grinding and polishing the plate materials”. Adamson himself points out that his contact angle of 98° of water on PTFE is “10 to 15 degrees lower than frequently reported values” and then states “We do find the more usual value of 109° for the unpolished PTFE and it may be either that there is a roughness effect or that our flow-smoothing method introduced polar sites to the surface.” In summation, there is no compelling evidence that πe values on low-energy surfaces are other than negligible or very low for those systems which yield high contact angles. If a system with substantial adsorption of water, or other liquid vapor, is found, it will probably have a low equilibrium contact angle (equilibrium contact angles only, or approximations thereof, should be used). In any event, the term -πe/γL should not be inserted into the GoodGirifalco or other IMF equation. However, the interaction term should be carefully re-evaluated in view of this information, which may clearly indicate an altered surface. The possibility should then be considered that the IMF method is unsuitable for treatment of that system. Conclusions 1. Correct derivation of the Good-Girifalco intermolecular force equation for liquid-solid contact angles yields a value for cos θ that does not depend on the term -πe/γL. This applies to all the subsequent IMF approaches as well. 2. Efforts to use the term -πe/γL as compensation for ignoring adsorption of liquid vapor to the clean solid surface are not correct. LA940962H
