588
Langmuir 1987, 3, 588-591
Solid Surface Tension: The Equation of State Approach and the Theory of Surface Tension Components. Theoretical and Conceptual Considerations J. K. Spelt and A. W. Neumann* Department of Mechanical Engineering, University of Toronto, Toronto, Ontario, Canada M5S 1A4 Received October 29, 1986 This paper considers the reasons for the apparent failure of the surface tension component theory as first proposed by Fowkes. This failure was illustrated in a previous paper, in which we showed that two liquids of the same surface tension always have the same contact angle on a given solid surface, irrespective of the intermolecular forces that are operative. It turns out that the key point is the thermodynamically proven existence of an equation of state for interfacial tensions which guarantees that ysL = f(ysv,yLv). With this restriction, there is no room in Young’s equation to accommodate a direct dependence of contact angles on dispersion, polar, or hydrogen-bonding forces. Thus, while surface tension components such as yd and yp might well exist, they cannot be determined from contact angles. It is shown that quantities which are at present claimed to be “dispersion”or “polar”components of surface tension are merely artifacts of the Rayleigh-Good equation and do not have the meaning commonly ascribed to them. Introduction In a previous publication’ we have shown that the contact angles of two liquids on a single solid surface are equal when the surface tensions of the liquids are the same, regardless of the relative magnitudes of the dispersion forces within each of the two liquids. These results cast into doubt both the legitimacy and the necessity of dividing surface tensions into components in order to predict solid and solid-liquid interfacial tensions.2 On the other hand, these same contact angle data clearly support the equation of state a p p r ~ a c hwhereby ,~ the solid-liquid interfacial tension is thought to be a function only of the total solid and liquid surface tensions, irrespective of the types and relative magnitudes of the intermolecular forces present within each phase. Thus, as far as the determination of solid surface tensions from contact angles is concerned, one might well discard the approach of surface tension components altogether in favor of an equation of state approach. Nevertheless, there are three points which merit further consideration. First, the preceding paper1 does not consider hydrogen-bonding liquids, and proponents of the Fowkes approach might argue that the contact angle of hydrogen-bonding liquids may not follow the same pattern. While it is possible to perform such measurements, the purpose of this paper is to consider theoretical aspects of the problem relevant to this question. Second, there is the question of whether the apparent breakdown of the Fowkes approach in the preceding paper1 could have been predicted and exactly where the fallacy in Fowkes’ arguments is. Finally, what is the correct interpretation of the quantities known as “dispersion components of the interfacial tension”? These questions are best approached from an historical perspective. Historical Perspective The calculation of solid surface tension, ysv, from the contact angle, 8, of a liquid of surface tension yLv,starts with Young’s equation YSL = Y s v - YLV cos 8 (1) (1) Spelt, J. K.; Absolom, D. R.; Neumann, A. W. Langmuir 1986,2, 620-625. (2) Fowkes, F. M. Znd. Eng. Chem. 1964, 56 (Dec.), 40-52. (3) Neumann, A. W.; Good, R. J.; Hope, C. J.; Sejpal, M. J. Colloid Interface Sci. 1974, 49, 291-304.
0743-746318712403-0588$01.50/0
T a b l e I. Solid S u r f a c e Tension (mJ/m2) of n -Hexatriacontane at 20 “C” liquid water glycerol thiodiglycol ethylene glycol hexadecane tetradecane dodecane decane nonane
Y1.V
72.8 63.4 54.0 47.7 27.6 26.7 25.4 23.9 22.9
0 104.6 95.4 86.3 79.2 46 41 38 28 25
YS
TSES
10.2 13.0 15.3 16.8 19.8 20.6 20.3 21.2 20.8
19.8 20.0 19.8 19.8 20.1 20.7 20.4 21.2 20.8
ys calculated by using eq 4 and ysEScalculated with the equation of state of Neumann et al.3 Contact angle data (deg) from ref 12.
where ysL is the solid-liquid interfacial tension. Of the four quantities in Young’s equation, yLvand 8 are readily measurable. Thus, in order to determine ysv, further information is necessary. Consequently, one obvious approach is to seek one more relation between the parameters in eq 1, such as an equation of state possibly of the form YSL
= f(YLV9YSV)
(2)
The simultaneous solution of eq 1and 2 would solve the problem. Note that if the commonly used assumption of negligible liquid vapor adsorption is applied, then eq 1 and 2 may be written in terms of yL and ys,rather than of yLv and YSV. An old equation of state for solid-liquid interfacial tensions is that due to Rayleigh and later Good et al. (see review in ref 4) YSL
= Ys + YL - 2(YsYL)1’2
(3)
where ys is the solid surface tension (equal to ysv if adsorption is neglected) and yL (or equivalently yLv) is the surface tension of the liquid. Combining eq 3 with 1gives ys =
1/4yL(1
+ cos 812
(4)
Early investigations by Good et al.4 showed that eq 4 yields consistent values of ys when ys and y L are both relatively small (for example, liquid alkanes on Teflon or paraffin wax). Contact angle data for liquids of higher surface tension, however, lead to values of solid surface (4) Good, R. J. J . Colloid Interface Sei. 1977, 59, 398-419.
0 1987 American Chemical Society
Langmuir, Vol. 3, No. 4, 1987 589
Solid Surface Tension tension which become progressively smaller as the liquid surface tension increases. The fourth column of Table I illustrates this for the contact angles of a wide range of liquids on solid hexatriaconbe. On the assumption that the surface tension of the hexatricontane should be approximately constant for all of these liquids, it is evident that eq 3 is inadequate as an equation of state. G o d et al.4 proposed that eq 3 could be modified so that it yields constant values of ys for all yL by incorporating an adjustable parameter, 9, the Good interaction parameter. Equation 3 is then written as YSL
= Ys + YL - 2@(YsYL)1’2
liquid surface tension, yL, its dispersive component, yLd, and the contact angle, 6. The approach of Neumann et al.3 begins with the question of the existence of an equation of state of the form of eq 2. For a three-phase two-component system consisting of a pure liquid and an inert solid which does not dissolve in the liquid and has zero vapor pressure, the three Gibbs-Duhem equations for the solid-vapor, the solidliquid, and the liquid-vapor interfaces can be written as5 dYsv = -ssv d T +
(5)
and eq 4 becomes 1 YL ys = - -(1 492
+ cos %)2
with the understanding (from experimental observation) that 9 approaches 1whenever ys and yL are both relatively small. It should be noted, however, that Good et ala4believe that it is not the magnitudes of the solid and liquid surface tensions which govern the condition 9 = 1 but rather the similarity in the types of intermolecular forces in the solid and liquid. These two points of view are often coincidental, since as surface tension decreases below, say, 30 mJ/m2, it is generally found the London dispersion forces are predominant in both solids and liquids. At this point three options present themselves for the further development of eq 3 or 5. (1)Attempts can be made to calculate 9 in terms of statistical mechanics, which is the very complex path followed by Good et al.4 (2) Fowkes’ arguments start with the conviction that the total surface tension can be decomposed into surface tension components: y = yd
+ y h + ...
(7)
where yd and yh are considered as unique physical properties, called the dispersion and hydrogen-bonding components of surface tension. In the context of eq 3 this approach interprets the decrease in ys with increasing yL (cf. Table I) as a reflection of the decrease in the relative importance of dispersion forces in the higher surface tension liquids found in Table I. Equation 3 is thus thought to be deficient because it does not take into account the nature of the intermolecular forces present in these various liquids. This reasoning led to the Fowkes equation YSL
d
d 112
= Ys + YL - 2(Ys YL )
(8)
where -ysd and Y~~ are, respectively, the dispersion components of the solid and liquid phases. Equation 8 is supposed to be valid only for cases where at least one phase is completely “dispersive”. With respect to Table I this procedure would, e.g., in the case of water, have the following consequence: Since Y~~ C yL, the value for ysL would be larger than the value of ysL obtained from eq 3; hence, in view of Young’s equation, ysv would also be larger, potentially equal to the values obtained with low surface tension liquids, which, coincidentally, are also the liquids having dispersion forces only. Combining eq 8 with Young’s equation, eq 1, yields 1
(104
dYsL = -SSL d T + rsL dpL
(lob)
dYLv = -SLV d T +
(10c)
rLv
dpL
where s denotes the specific surface entropy, T the temperature, the specific interfacial masses, and pL the chemical potential of the liquid. Since the three equations (10a)-(lOc) contain, on their right-hand sides, only the two independent properties T and pL, it is apparent that the left-hand sides of the equations are not independent, i.e., that a relation of the form of eq 2 exists. An explicit equation of state was formalized3empirically from contact angle data, utilizing the experimental fact that 9 = 9(ysL). It should be noted that the final product of this approach is not an analytical equation but rather a computer program3s6 or, alternatively, a set of tables’ which permit the determination of ys and ysL from contact angles. It may also be noted that the use of 9 in the development of the equation of state was solely a matter of convenience. Since the approach utilizes curve-fitting procedures, the selection of 9 as a correlating variable, while convenient, was essentially arbitrary.
Consequences for the Fowkes Approach of the Existence of an Equation of State The existence of an equation of state, generically written as eq 2, which is firmly based on thermodynamics, has far-reaching consequences. Most important, an equation such as eq 8 would also have to satisfy eq 2; but this is apparently impossible, since the dispersion components of the surface tensions would have to be functions of the total surface tensions. This would preclude the possibility of yd being a unique function of intermolecular forces, as stipulated by the Fowkes approach. This impossibility carries over into contact angles. Combining eq 1 and 2 yields cos 6 =
f(YS,YL) - Ys YL
so that cos 6 = AYS,YL)
(12)
indicating that the contact angle is completely determined by the total surface tensions ys and yL. This implies that surface tension components cannot be determined from contact angles. Clearly, any arbitrary solid-liquid-vapor system with the same total surface tensions ys and yL must have the same contact angle, 6. This is true regardless of the relative magnitudes and types of intermolecular forces, including hydrogen bonding. The results of the previous publication’ constitute experimental confirmation of this thermodynamic result.
r:,
= - -(1 + cos 612 4 YLd Thus, within the context of the present surface tension components approach, ysd can be determined from the Ysd
d@L
r*Y
(5) Ward, C. A.; Neumann, A. W. J. Colloid Interface Sci. 1974, 49, 286-290. (6) Taylor, C. P. S. J . Colloid Interface Sci. 1984, 100, 589-594. (7) Neumann, A. W.; Absolom, D. R.; Francis, D. W.; van Oss, C. J. Sep. Purif. Methods 1980, 9,69-163.
590 Langmuir, Vol. 3, No. 4, 1987
Spelt and Neumann
Table 11. Comparison of y~~ Calculated from Fowkes’ Approach and @yLCalculated from the Equation of State of Neumann et al.*O liquid water glycerol thiodiglycol ethylene glycol
YT.~
@YT.
36.2 40.2 40.3 39.1
37.8 41.6 42.3 40.9
YT.
72.8 63.4 54.0 47.7
A% 4.4 3.5 5.0 4.6
YrdlYr.
0.50 0.63 0.75 0.82
a A% is the percentage difference between these two results, the basis of contact angle data on hexatriacontane as per Table I. ys = 20.6 mJ/m2 at 20 “C for both calculations.
Overall, it is apparent that while eq 7 might be correct, eq 8 and 9 are in conflict with the existence of an equation of state, i.e., with eq 2. Thus any attempt to develop a theory of surface tension components would have to come to terms with eq 2. Particularly, there is no apparent way of using contact angles to measure surface tension components.
Status of the Parameter yd While it is apparent from the above that yd parameters as obtained from the Fowkes approach cannot be dispersion components of surface tension, the question remains as to whether they may not carry some meaning nevertheless. For a liquid which, in Fowkes’s sense, is capable only of interaction by means of dispersion forces, eq 8 becomes YSL
= Ys + YL - 2(ysdyL)1’z
(8a)
Comparing eq 8a with eq 5 yields Ysd = @%s
(13)
Considering eq 5 as the definition of CP and using eq 2, we see that Ysd = r(:(ys,YL)
(14)
i.e., ysd would have to be some function t of ys and yL. Similarly, for a solid which, in Fowkes’ sense, is capable of interaction only by means of dispersion forces, one obtains YLd
= 9%
(13a)
and YLd
=
dYL,YS)
(144
Not only is there, within eq 14, no room for a direct dependence of ysd on intermolecular forces; there is also a dependence of ysdon yL which destroys Fowkes’ stipulation that ysd is a property of the solid phase alone. It becomes apparent from eq 13 and 13a that yd is a correction factor playing virtually the same role as @. Since the equation of state approach of Neumann et al.3 utilizes an empirical relation between @ and ysL, it is possible to calculate @ and hence ysd or yLdvalues from that approach; this is illustrated in Table 11, which presents a comparison of yLd calculated from Fowkes’ methodology and the calculated from mathematically equivalent quantity a2yL the equation of state approach, using some of the hexatriacontane contact angle data listed in Table I. In other words, we use the equation of state of Neumann et aL3to evaluate ysL in eq 5 and then substitute @ into eq 13 or 13a. For both the a2yL(equation of state) and yLd (Fowkes) calculations, ys was taken to be 20.6 mJ/m2. This value is an average of the five liquid alkane contact angle measurements which yields ys = 20.54 mJ/m2 (0.53 standard deviation) by eq 9 and ys = 20.64 mJ/m2 (0.42 standard deviation) using the equation of state of Neu-
Table 111. Comparison of yLdand aZyLCalculated from the ysEs Values of Table I for Each Particular Liquid Instead of the Constant ys = 20.6 mJ/mZa liquid YT. Yr 0% water 72.8 37.4 37.4 glycerol 63.4 41.2 41.2 thiodiglycol 54.0 41.7 41.7 ethylene glycol 47.7 40.4 40.4 “Contact angle data on hexatricontane as per Tables I and 11.
mann et aL3 The close agreement between the two types of results is typical in all cases when the solid and liquid surface tensions are both relatively small, say less than 30 mJ/m2. The equation of state “prediction” of yLd,namely, @yL,was thus computed by using only ys = 20.6 mJ/m2 and the total surface tension, yL,of the particular liquid. The contact angle did not enter the calculation at all. Note that for water yLd and CPzyLequal respectively 36.2 and 37.8 mJ/m2, in disagreement with the commonly quoted value of 22.0 mJ/m2 derived from liquid-liquid interfacial tension measurements.E This points to a discrepancy between the behavior of the Fowkes equation in liquid-liquid and in solid-liquid systems. From the perspective of the Fowkes theory, it is amazing that the agreement between a2yL and yLdis within 5% as one would have to ask how these two completely unrelated quantities, ys and yL, combine to yield a remarkably accurate prediction of the liquid property yLd. The answer must be that yLdis, in fact, not a property of the liquid. The “dispersion component of liquid surface tension” is an adjustable parameter defined by yL and ys of the particular system in question. From the vantage of the equation of state approach, the agreement between a2yLand yLd was, of course, to be expected. The only question is why these two parameters are not identically equal. The answer is to be found in the errors inherent in most contact angle measurements. Even though the contact angles of Table I are perhaps the most accurate available (other than the alkanes, the data were obtained by the highly accurate technique of the capillary rise at a vertical plateg on a hexatriacontane substrate which was made essentially hysteresis-free by vapor deposition), the contact angles still possess some error which is made evident by the small scatter in the ysES(i.e., the solid surface tension as calculated by the equation of state) values of Table I. Each liquid surface tension and contact angle pair corresponds, through the equation of state, to an apparent solid surface tension. In general, this apparent value of ys may be in some error due to surface roughness, heterogeneity, and vapor adsorption. Therefore, the (O,yL) pairs do not necessarily correspond to the mean value of the ys determined with the liquid alkanes (20.6 mJ/m2). These small inconsistencies manifest themselves as errors in the value of yLdas calculated by eq 9. This may be corrected by using for each liquid of surface tension yLnot the mean alkane value of ys but rather the equation of state value (or the apparent value) corresponding to the (O,yL)pair. In other words, for each liquid we may use the ysESof Table I in the calculation of both yLdand @yL. If this is done, the parameters aZyL and yLdbecome exactly equal, as illustrated by Table 111. More details are given in ref 10. Given that ysd and yLdare not physical properties but rather artifacts of the Rayleigh-Good equation, it is obvious that the considerable number of extensions and F. M. J. Phys. Chem. 1980, 84, 510-512. (9) Neumann,A. W. 2.Phys. Chern. (Frankfurt)1964, 41, 339-352. (10) Spelt, J. K. Ph.D. Thesis, University of Toronto, Toronto, 1985.
(8) Fowkes,
Langmuir, Vol. 3, No. 4,1987 591
Solid Surface Tension
generalizations of the Fowkes approach are also deficient. As an example, we shall consider the approach of Owens and Wendt.’l We shall calculate, from the total surface tensions yL and ys,the supposed “dispersion” and “polar” components of surface tension, yd and yh of the Owens and Wendt approach. Clearly, while it should of course be possible to calculate the total surface tensions from the components, the opposite is, conceptually, impossible. Our ability to calculate “yd”and “yh”from y proves that “yd” and “yh”are not dispersion and polar components of interfacial tension. The approach of Owens and Wendtl’ involves the measurement, on a given solid surface, of the contact angles of water and methylene iodide, from which are determined the “dispersive” and “nondispersive components of solid surface tension”. These parameters, ysd and ySh, respectively, are obtained by the simultaneous solution of the following two equations for water (w) and methylene iodide (4:
where yw = 72.8, ywd= 21.8, ywh= 51, ym = 50.8, ymd= 49.5, and ymh= 1.3 mJ/m2. Let eq 2 be written as (17) = Ys + YL - 2KYL1i2 where K is some undetermined function of ys and yL. Combining eq 17 with Young’s equation, eq 1, yields YSL
1 + cos 8 = K2/yL1I2
(18)
where K =
Ys + YL - YSL 2YL1/2
The empirical equation of state of Neumann et al.3 may be used to evaluate K, employing as input the contact angle and the known surface tension yL. It is important to realize that such a calculation makes no reference at all to the particular types of intermolecular forces present in the system. The premise of the equation of state approach is that ysL is completely determined by ys and yL. The right-hand sides of eq 15 and 16 represent the Owens and Wendt interaction terms between the solid of interest and the surface tension components of water and methylene iodide, respectively. The left-hand sides of these same equations, can, however, be replaced by 2 K , / ~ , l / ~and 2K,/ym’/2 by using eq 18 with water and methylene iodide contact angles, respectively. Making these substitutions and solving for the Owens and Wendt variables ysd and Y~~ in terms of Kwand K, result in the equations (20) (ysd)‘I2 = -0*217Kw(ys,~W) + 1*13Km(~s,~m,) ( ~ s ~ ) ’= / ’ 1 . 3 4 K w ( ~ s , ~- W 0.741Km(~s,ym) ) (21) Equations 20 and 21 make it possible to use the equation of state of Neumann et al.3 to calculate values of ysd and -ysh for a given solid using the contact angle data of Owens and Wendt.8 The values of Kw and K , are determined from eq 19 by using the equation of state to calculate ysL and ys from the water and methylene iodide contact angles (11) Owens, D. K.; Wendt, R. C. J. A p p l . Polym. Sic. 1969, 13, 1741-1747. (12) Neumann, A. W. Adv. Colloid Interface Sci. 1974, 4, 105-191.
Table IV. Comparison of ysd and ygh (mJ/m2) Obtained by Owens and Wendt” with the Values Calculated by Eq 20 and 21 and by the Equation of State of Neumann et a1.3 Owens and equation of state Wendt ‘‘yshx solid Ysd YSh 32.0, 33.2 1.1, 0.0 31.9, 33.1 1.1, 0.0 polyethylene 40.0 1.5 39.9 1.5 poly(viny1 chloride) 7.1 23.0 7.2 poly(viny1idene fluoride) 23.2 12.5, 18.6 1.5, 0.5 12.4, 18.4 1.5, 0.5 Teflon TFE 33.9, 40.6 9.2, 6.3 34.1, 40.8 9.1, 6.2 Nylon 6-6 41.1 0.7 41.4 0.6 polystyrene 31.1 5.5 31.3 5.4 poly(viny fluoride) 3.0 41.7 3.1 poly(viny1idene chloride) 42.0 19.7 4.1 19.9 4.0 poly(trifluoroethy1ene) 35.7 4.3 35.9 4.3 poly (methyl methacrylate) eysd93
listed in ref 8. Therefore, from the perspective of the equation of state approach, the implication of eq 20 and 21 is that (ysd)’/2and (ysh)’I2are merely linear combinations of the K factors required in eq 17 for a solid of surface tension ys and liquids of surface tension 72.8 and 50.8 mJ/m2. In other words, the parameters 7sd and ysh are functions only of total surface tensions and are not directly determined by the relative magnitudes of dispersion and hydrogen-bonding forces in a given solid. Table IV presents a comparison of the ysd and ySh values of Owens and Wendt8 with those obtained from eq 20 and 21 using the water and methylene iodide contact angle data from ref 8 to evaluate K , and K,, respectively, by means of eq 19 and the equation of state of Neumann et al. I t is clear that the equation of state “predictions” of the “dispersive and nondispersive components of surface tension” closely match the values determined in ref 8. The parameters ysd and ysh are thus shown to be functions of ys (for the fixed water and methylene iodide surface tensions) and should not be viewed as material properties which depend directly on the types and relative magnitudes of the intermolecular forces. This conclusion is similar to that reached with regard to the dispersion components of surface tension in the Fowkes theory of interfacial behavior. Overall, we conclude that surface tension components which are reported in the literature as extensions of the Fowkes approach are incorrect. Such components cannot be regarded as physically realistic quantities. Moreover, since ysL is a known function of the total solid and liquid surface tensions, in many instances there is little incentive to attempt an evaluation and treatment of surface tension components. Total solid surface tensions can be determined from a single contact angle measurement from the equation of state a p p r ~ a c h . ~ Conclusions The existence of an equation state, which has been demonstrated thermodynamically, precludes the use of contact angles as a means of determining surface tension components. The quantities yLdand ySd,derived from the Fowkes equation, are not “dispersion components of surface tension” but rather are adjustable parameters which are functions of the total surface tensions ys and yL. The magnitudes and functional characteristics of the variables Y~~ and 7sd are an artifact of the behavior of the simple Rayleigh-Good equation of state which breaks down and requires corrective terms as the total surface tensions of the solid and liquid increase. Acknowledgment. This investigation was supported by NSERC Grant A8278.
