Langmuir 1987,3, 1009-1015 a maximum a t 420 nm. Colloids are formed by nucleation and grow via combination and Ostwald ripening processes. The structured absorption disappears when colloids undergo corrosion processes. The anodic corrosion of In2Ses results in the formation of Seowith very pronounced decomposition of particles. The anodic corrosion cannot be completely suppressed with triethanolamine a t a concen-
1009
tration of 1 X mol dm-3. Fewer changes in the absorption spectra of semiconductor are observed upon reduction of colloids, when Ino is probably formed. Registry No. PVA, 9002-89-5; In2Se3,12056-07-4; OH’, 3352-57-6; cop*-, 85540-96-1; (CH~)&OH,5131-95-3; H2Se, 7783-07-5; 1n(CiO4),, 13529-74-3; sodium metaphosphate, 10361-03-2.
Calculation of Solid Surface Tensions Eddy N. Dalal Xerox Corporation, Webster, New York 14580 Received April 23, 1986. I n Final Form: May 5,1987 The harmonic mean and geometric mean equations have been evaluated for calculating solid surface tensions ys from contact angle data. These equations have the advantage of semitheoretical bases and provision for a nondispersive component. In the application of these equations to many liquids to obtain an overdetermined set of equations, pairwise and simultaneous solution methods have been developed. Stability problems arise with the pairwise solution method, but can be overcome by rejecting ill-conditioned pairs of equations. These methods have been used to calculate -ye for 12 common polymers, using published contact angle data for a set of 6 testing liquids. The geometric mean equation fits the data better than the harmonic mean equation does, but in either case the resulting values of yshave low standard deviations and are in good agreement with each other and with ys values obtained by other independent methods. The commonly used critical surface tension yc is always lower than ys. 1. Introduction The ability of a liquid to wet or adhere to a solid is determined essentially by the surface tensions (i.e., the excess specific surface free energies) of the two materials. Consequently, the ability to measure surface tension is crucial to the development of products ranging from pesticides to detergents to adhesives. In xerography, the surface tensions of the fuser roll, toner, and release agent primarily determine the release performance of a roll fuser. The surface tension of a liquid can be measured directly, using tensiometry, drop profiles, capillary rise, etc. However, the surface tension of a solid cannot be measured directly because its surface cannot be reversibly deformed. Many indirect methods have been proposed, and of these, the critical surface tension method is by far the most popular. In this method, proposed by Fox and Zisman,l the contact angles Bi of sessile drops of liquids with different known surface tensions yi are measured. Extrapolation of a plot of cos Bi vs. yi to cos 19 = 1 defines the critical surface tension yc of the solid. The critical surface tension yc is a useful parameter, widely employed in characterizing the surface properties of polymers. However, while yc is considered by some to be equivalent to the true surface tension ys of the solid, Good2has shown that ycwill always be less than ya. Also, in practice, problems are often encountered with curvature or excessive scatter in the cos Bi vs. yi plot unless a homologous series of liquids is used. Even if this is done, the value of yc obtained varies significantly with the liquid series. These problems arise because the method is inadequate in that it ignores the effects of nondispersive interactions between the solid and the liquid. The inadequacies of the critical surface tension method have been discussed in greater detail in an excellent review by W U . ~ (1) Fox,H. W.; Zisman, W. A. J . Colloid Sci 1950,5, 514; 1952, 7,109; 1952, 7, 428.
(2)Good, R. J. J. Colloid Interface Sci. 1973, 44,63.
It is now quite generally accepted that there are at least two distinct components of the surface tension of a material, involving contributions from dispersive and nondispersive or “polar” forces: yi = y?
+ yip
(1)
In this context, the polarity Xi of the material is the fractional contribution of the polar component to its total surface tension Xi = yip/yi ’ (2) It should be noted that the polar component is not limited to dipole interactions but includes all of the nondispersive forces, such as hydrogen bonding. In fact, in condensed phases dipole interactions are small, and Fowkes4 has concluded that “polarity” as measured by dipole momenta is not a significant factor in intermolecular interactions in liquids and solids. The difficulties caused by neglecting polarity in the critical surface tension method are avoided in other methods, such as those proposed by W U , ~Kaelble,6 ,~ and Fo~kes.~~ These ’ methods are briefly discussed below. The work of adhesion W, between two bulk phases is the work required to reversibly separate them. Ignoring the equilibrium spreading pressure re,which is negligible for contact angles greater than about loo (e.g., see ref 3) w, = yi ys - yis = (1 + cos !9i)yi (3) where Bi is the contact angle for a testing liquid i on a solid surface s, yi and ysare the liquid and solid surface tensions, and yis is the interfacial tension. Practically useful ex-
+
(3) Wu, S. Polymer Interface and Adhesion; Marcel Dekker: New York, 1982. (4) Fowkes, F. M. In Physicochemical Aspects of Polymer Surfaces; Mittal, K.L., Ed.; Plenum: New York, 1983; Vol. 2, p 583. (5) Wu, S. J. Polym. Sci. 1971, C34, 19. (6) Kaelble, D.H. Physical Chemistry of Adhesion; Wiley-Interscience: New York, 1971. (7) Fowkes, F. M.In Adhesion and Adsorption of Polymers; Lee, L.-H., Ed.; Plenum: New York, 1980; p 43.
0743-7463/87/2403-lOO9$01.50/00 1987 American Chemical Society
1010 Langmuir, Vol. 3, No. 6, 1987 pressions for the work of adhesion have been derived by making certain simplifying approximations. If the ionization potentials of the two phases are assumed to be equal, and the polar component is assumed to be dominated by dipole-dipole interactions, one obtains the geometric mean (GM) equation6 On the other hand, if the polarizabilities of the two phases are assumed to be equal (and assuming that the polar component has the same form as the dispersive component), one obtains the harmonic mean (HM) equation3g5 c
Despite the theoretical bases claimed for the GM and HM equations, some of the assumptions involved in deriving them are not fully justified. The derivation of the polar component is especially weak. There is, however, a major argument in their favor: when applied to available contact angle data, they yield reasonable values for ys, which are compatible with those obtained by other independent methods. Consequently, the GM and HM equations are semiempirical in nature. Some researchers have reservations about attributing a phenomenological meaning to a ye value calculated on the basis of semiempirical equations. On the other hand, it is clear that these methods provide practically useful ways of characterizing solid surfaces. The surface tension yi of a testing liquid can be measured directly, and its dispersive component y? can be calculated from the contact angle on nonpolar solids of known surface tension, although the value of yid is slightly dependent on whether the HM or the GM equation is used in its calculation. Thus, in the GM method, all of the variables in eq 4 can be independently determined, except for the components 72 and ysP of the solid surface tension. Using two different testing liquids (i = 1,2) results in two equations that can be solved simultaneously for the two unknowns 7 2 and ysP. In the HM method, an identical procedure is applied to eq 5. Wu3has used a single liquid pair, water and methylene iodide, in his method using the HM equation. He has shown that surface tension values thus calculated for many polymers are in good agreement with values obtained by other methods, while yc values obtained by the Zisman method are always lower. However, since two unknowns are calculated from two simultaneous equations, there can be no measure of the reliability of the results obtained from a given experiment. Kaelble6has utilized the GM equation and has extended his method to include data on several testing liquids. This is a desirable feature, since the extent of compatibility between the data for the different liquids can be used to evaluate the experimental data as well as the theory. Kaelble analyzed the data by solving the equations pairwise, and averaging the results. He found that some liquid pairs gave widely scattered y$ and yapvalues but that these could be prevented from contaminating the mean values by rejecting those pairs of liquids for which the determinant of the coefficient matrix was less than some arbitrary minimum value. The standard deviation of the mean is a measure of the reliability of the results obtained. Kaelble has applied this method to data on several polymers and obtained mean and y a p values that have small standard deviations. However, since the standard deviations decrease as the minimum allowable value of the determinant is increased, an element of arbitrariness has
Dalal
been introduced in this method. Also, his standard deviation values appear to have been incorrectly calculated, being too low by a factor of n1I2,where n is the number of equation pairs used. SherrifP has pointed out that in Kaelble's method, the data for more than two liquids constitute a set of overdetermined equations, which cannot be fully satisfied by any one vector (y,d,y:). It is, however, possible to calculate a vector which minimizes the errors over the entire data set, for example, by minimizing the sum of the residuals in a least-squares fit. Sherriff tested this approach by reevaluating Kaelble's data for PTFE. He found that the root-mean-square error from this procedure was less than that from a paired equation analysis such as that used by Kaelble. Unfortunately, Sherriff did not report calculations for any other polymers. Also, his method is not directly applicable to other equations, such as the HM equation, which cannot be written in linear form. Fowkes (e.g., ref 4 and 7) has offered a different treatment of the nondispersive forces. He has suggested that acid-base interactions are primarily responsible for the polar forces and has therefore used the Drago constants for the acid (CA, E A ) and for the base (CB, EB) to represent the polar interactions. In one paper,' he has proposed a specific form for the work of adhesion: Wa = 2(yAdyBd)'/'
+ NAB(CACB +
-k
Wap
(6)
where NAB is the moles of acid-base pairs per unit area, f = 1is a constant to convert enthalpy per unit area into surface free energy, and the dipole-dipole contribution W,P is usually negligible. (Fowkes used a negative sign for the second term on the right-hand side of this equation.) While Fowkes' concept is intuitively reasonable and potentially very useful, little if any data are available for the quantitative application of eq 6 to polymers. In this work, the surface tensions of 12 polymers are calculated from published contact angle data for 6 different liquids, using both the GM and the HM equations. The Fowkes equation has not been utilized because of a lack of input data; however, the procedures described here are also applicable to his equation and to similar problems in general. The analysis in the following section is performed in two different ways. The first procedure involves pairwise solution of the C(6,2) = 15 pairs of equations resulting for each polymer, using the included angle @ (see section 2.2) as a measure of the ill-conditioning of any given pair of equations. The second procedure, which is preferred, involves obtaining a best-fit solution to all of the liquids simultaneously. The numerical methods used are described and explained in terms of graphical representations. These procedures are used to compare the ability of the GM and HM equations to fit the data. 2. Analytical Procedures 2.1. Graphical Representation. All of the results presented in this paper have been numerically computed. These procedures can be conveniently represented in terms of their graphical analogues, which will be developed in this section. The variables x and xL(i) defined here should not be confused with the polarity Xi defined in eq 2. Harmonic Mean Method. The harmonic mean equation (eq 5) may be rewritten as
(8) Sherriff, M. J . Adhes. 1976, 7, 257.
Langmuir, Vol. 3, No. 6,1987 1011
Calculation of Solid Surface Tensions
--
) .
a m
i\
h
U 4
'
1 0
n
OISPERSIVE COMPONENT 7:
OISPERSIVE COMPONENT
(x)
Figure 1. Schematic plot relating the polar and dispersive Components of surface tension of a solid for two polar (1,Z) and two nonpolar (3, 4) testing liquids.
where x = :y and y = y e p are the dispersive and polar components, respectively, of the surface tension of the solid. aL(i),xL(i),and yL(i)are constants for a particular liquid-solid combination and are given by
Note that all three constants, aL(i),xL(i),and yL(l'),contain only the liquid surface tension components yid and yip (which can be independently determined as described earlier) and the contact angle Oi (which can be measured). Thus, on a given solid, each testing liquid can be represented by a known curve of y vs. x . In the ideal case, all of the curves would intersect at a single point, which would define the surface tension components of the solid. However, even if the theory is correct, experimental errors will cause the curves to be distorted and, hence, not intersect at a single point. Such a situation is schematically illustrated in Figure 1. Geometric Mean Method. The geometric mean equation (eq 4) may be rewritten as x = [(I + cos ei)yi - 2(n~y)1/2~2/(4nd) (11) where x =:y and y = ysP as before. This form, with x on the left-hand side, is preferred, in order to avoid computational problems with nonpolar liquids. As in the previous case, on a given solid, each testing liquid can be represented by a curve of y vs. x . This situation also can be schematically represented by Figure 1. 2.2. Pairwise Solution. The intersection of each pair of curves in Figure 1represents the solid surface tension given by that pair of testing liquids. The results obtained by every possible pairwise combination of liquids can be averaged to give a mean value of the solid surface tension. When this is done with experimental data, the resulting values typically have much scatter and, consequently, high standard deviations. The angle included between any two curves a t their intersection is a function of the difference in their polarity (Xi= y,P/yi). Thus,liquids with similar polarities intersect a t low angles; i.e., they result in ill-conditioned pairs of simultaneous equations, and small errors in the input data lead to relatively large errors in the result. On the other hand, liquids with widely differing polarity result in well-conditioned pairs of equations. Note that the two testing liquids recommended by W U ,water ~ and methylene
d a 7:
(XI
Figure 2. Error representation in the simultaneous best-fit
solution.
iodide, have very different polarities. This effect is evident in Figure 1,in which curves 1and 2 refer to polar testing liquids and curves 3 and 4 refer to liquids with low polarity. The four pairs of liquids with differing polarity intersect very close together (points a, b, c, d in Figure l),while those pairs having similar polarity are widely scattered (points e, f). Moreover, the similar liquids 1and 2, for example, are quite close together at the dissimilar liquids intersection point b, but the dissimilar liquids 1 and 3 are very far apart a t the similar liquids intersection point e. For the two reasons mentioned above, the results should be more reliable if only dissimilar liquid pairs are chosen. When the experimental data are analyzed in this way, the results have much smaller standard deviations. The degree of ill-conditioning of a given pair of simultaneous equations may be quantified in terms of the angle 4 included between the tangents to the corresponding pair of curves at their intersection. In Figure 1, curves 1 and 2 intersect at low angles while curves 1 and 3 intersect a t much larger angles. The included angle can be calculated from eq 7 (harmonic mean) or eq 11 (geometric mean). By defining a (arbitrary) minimum acceptable included angle, amin, it is possible to reject ill-conditioned equation pairs in machine computations. 2.3. Simultaneous Best-Fit Solution. A preferable method would be to obtain a best-fit solution that simultaneously satisfies the equations corresponding to all of the testing liquids. Consider the point (xo, yo)in Figure 2, and let the curve i represent a particular testing liquid. Let the point (xi, yi) on the curve be that which is closest to (xo, yo);Le., the error ri is a minimum. The total RMS error for all n testing liquids is R: n
n
,=I
i=l
R2 = C r i 2 = C[(xi- xo)2 + Cy, -
(12)
If (xo, yo) is chosen such that R is minimized, then this point will define the surface tension components of the solid that best fit the data for all of the liquids simultaneously. A computer program has been written for determining (xo, yo);the source code is available from the author on request. It is written in Fortran 77 and uses standard IMSL library subroutines for function minimization and polynomial solution. This program also finds the pairwise solutions, with rejection of ill-conditioned pairs. 3. Results This section presents the data obtained by applying the harmonic mean and geometric mean equations to experimental data, using both of the methods (pairwise solution and simultaneous best-fit solution) described in the previous section. The data used in this analysis are taken from compilations by W U . ~ Virtually all of these data are traceable to Zisman and his co-workers.
1012 Langmuir, Vol. 3, No. 6, 1987
Dalal
Table I. Surface Tensions (dunlcm) of the Testing Liauidsa harmonic mean liquid WT, water MI, methylene iodide FA, formamide BN, a-bromonaphthalene GL, glycerol TP, tricresyl phosphate
" Calculated by Wu3 from contact
Tz
Y,d
72.8 50.8 58.2 44.6 63.4 40.9
22.6 49.0 36.0 44.6 40.6 39.8
Y&P 50.2 1.8 22.2 0.0 22.8 1.1
geometric mean
x, 0.69 0.04 0.38 0.00 0.36 0.03
Yid
22.5 48.5 39.5 44.6 37.0 39.2
Y? 50.3 2.3 18.7 0.0 26.4 1.7
x, 0.69 0.05 0.32 0.0 0.42 0.04
angle data.
Table 11. Contact Angles (deg) of the Testing Liquids of Table I on Various Solids" solid PE, polyethylene PTFE, poly(tetrafluoroethy1ene) P3FE, poly(trifluoroethy1ene) PVDF, poly(viny1idene fluoride) PVF, poly(viny1 fluoride) PVDC, poly(viny1idene chloride) PVC, poly(viny1 chloride) PS, polystyrene PMMA, poly(methy1 methacrylate) PCTFE, poly(chlorotrifluoroethy1ene) PA66, poly(hexamethy1ene adipamide) PET, poly(ethy1ene terephthalate)
WT 102 108 92 82 80 80 87 91 80 90 70 81
MI 53 77 71 63 49 29 36 35 41 64 41 38
FA 77 92 76 59 54 61 66 74 64 82 50 61
BN 35 73 61 42 33 9 11
15 16 48 16 15
GL 79 100 82 75 66 61 67 80 69 82 60 65
TP 34 75 49 28 28 10 14
19 44
"Data compiled by W U . ~
The six testing liquids used are listed in Table I, along with their surface tensions. Their dispersive components, ,:y were calculated by Wu3 from contact angle data on nonpolar substrates (low-density polyethylene and triacontane). Calculations from interfacial tension data would probably be more reliable, but these are not available for all of these liquids, so the contact angle calculations have been used throughout for consistency. The values of yid presented here for the geometric mean method are virtually identical with those obtained by Kaelble.6 Note that three of the liquids in Table I are polar (WT, FA, GL), while the other three are essentially nonpolar (MI, BN, TP). In the case of BN, the calculated values of yid are 47.7 and 47.0 dyn/cm, but they have been set equal to yi = 44.6 dynlcm because yid cannot be greater than yi. The difference is not major and might be attributable to experimental error or to inadequacies in the theory. The 12 solids used in this analysis are listed in Table 11, along with contact angles made by the 6 testing liquids of Table I. The variability of the pairwise solutions is illustrated in Table 111, using harmonic mean calculations for poly(methyl methacrylate) as an example. Such results for the other polymers are not shown to conserve space but are available from the author on request. Table I11 shows that the mean values of :y and y s P have high standard deviations if all possible pairs are used but that the standard deviations can be significantly improved if ill-conditioned liquid pairs are rejected. This is true for all the polymers analyzed. The (arbitrary) minimum acceptable included angle amin (see section 2.2) has been set at 0.5 rad for all of the calculations. At this level, the results are fairly insensitive to reasonable variations in particularly in the case of the geometric mean. The variation in ys is often smaller than that in either component :y or ysP, because all of the curves have negative slopes in the area of interest (see Figure 1)and therefore errors in one component are partially offset by errors in the other. Consequently, ys is virtually independent of over a wide range. Since ill-conditioned equation pairs result when liquids of similar polarity are combined, an alternative method
Table 111. Harmonic Mean Pairwise Solutions of the Data for Poly(methy1 methacrylate) Along with Their Means, Standard Deviations, and Included Angle @ (rad)" liquid pairs typeb Y. 7.d 7.P x. 0 WT/MI WT/BN WT/TP FA/MI FA/BN FA/TP GL/MI GL/BN GL/TP WT/FA WT/GL FA/GL MI/BN MI/TP BN/TP mean of all pairs mean of well-conditioned pairs = 0.5)
D
D D
D D D D D D
S S
S S S
S
44.9 36.1 8.8 0.195 50.4 42.9 7.5 0.150 44.8 36.0 8.8 0.196 40.2 37.3 3.0 0.074 44.3 42.9 1.4 0.032 39.8 36.6 3.2 0.080 39.9 37.7 2.2 0.055 43.6 42.9 0.7 0.016 39.3 36.8 2.5 0.062 32.8 18.4 14.4 0.440 32.9 18.5 14.4 0.438 32.7 19.3 13.4 0.409 40.9 40.9 0.0 0.000 46.6 36.0 10.6 0.228 39.8 39.8 0.0 0.000 6.1 40.9 34.8 0.158 f5.2 f8.7 f5.4 f0.158 43.0 38.8 4.2 0.095 f3.6 f3.1 f3.2 *0.068
1.28 1.41 1.32 0.85 1.34 1.02 0.69 1.33 0.90 0.41 0.43 0.03 0.29 0.03 0.25
"Input data are from Tables I and 11. y values in dyn/cm. Similar (S) or dissimilar (D)polarity of liquid pair.
of selection would be to include only those pairs that contain one polar liquid (WT, FA, GL) and one nonpolar liquid (MI, BN, TP). When this is done, the results are very similar, in most cases, to the included-angle method described above with @min of the order of 0.5 rad. In the data in Table I11 the correspondence is exact, with dissimilar liquid pairs having included angles @ 1 0.69 rad and similar liquids having @ I 0.43 rad. Formamide and glycerol have very similar polarities (see Table I), and, consequently, the FA/GL pair have an included angle of only 0.03 rad (Table 111). That liquid pairs of similar polarity form an ill-conditioned system is illustrated in Table IV. Here, the effect of experimental errors on the solution is investigated by changing the contact angle of formamide (OF*) on poly(methyl methacrylate) from its reported value of 64" to a value of 69"; Le., an error of 5" in the contact angle has been simulated. The output error hy,d is the difference
Langmuir, Vol. 3, No. 6, 1987 1013
Calculation of Solid Surface Tensions
Table IV. Data from Harmonic Mean Calculations on Poly(methy1 methacrylate) for Ill-Conditioned and Well-Conditioned Pairsn error &A = 69' &A = 64' liquid pair typeb AX: adrad Ysd YE? Ysd YSP AYsd AYaP +42.5 -16.0 -2.6 61.8 13.4 19.3 S 0.02 0.03 FA/GL S D D D
FA/WT FA/MI FA/TP FA/BN 0 7values in dyn/cm. angle (at 64').
0.31 0.35 0.35 0.38
0.41 0.85 1.02 1.34
18.4 37.3 36.6 42.9
14.4 3.0 3.2 1.4
14.2 38.5 37.3 42.9
16.9 1.2 1.6 0.2
-4.2 +1.2 +0.7 0.0
Similar (S) or dissimilar (D) polarity of liquid pair. AX, = difference in polarity (see Table I).
f2.5 -1.8 -1.6 -1.2 0 = included
Table V. Solid Surface Tensions y, and Components 7.d and y,P Calculated by the Harmonic Mean (HM)Equation Using the Data in Tables I and 11' pairwise solutions all pairs
WT/MI solid PE
YI
Y,d
33.9
32.9
1.0
PTFE
22.7
20.9
1.8
P3FE
29.7
22.4
7.2
PVDF
36.3
25.9
10.5
PVF
42.0
32.4
9.6
PVDC
48.9
41.1
7.8
PVC
44.1
38.7
5.4
PS
43.3
39.6
3.8
PMMA
44.9
36.1
8.8
PCTFE
32.7
25.6
7.1
PA66
49.2
35.8
13.4
PET
45.5
37.5
8.1
YSP
well-conditioned pairs
simultaneous best-fit solution
Ys
Ysd
Yap
Ys
Ysd
YSP
Ys
Ysd
YSP
34.6 f1.5 19.9 f3.0 35.6 f24.9 36.4 f4.2 40.2 f3.2 44.8 f8.3 41.3 f6.4 37.0 f8.0 40.9 f5.2 31.7 f5.5 45.1 f5.1 41.8 f5.0
34.1 f1.9 18.5 f3.3 24.0 f4.0 29.9 f6.1 33.7 f4.7 40.7 f10.5 39.1 f7.3 33.2 f11.4 34.8 f8.7 28.2 f9.4 35.7 f8.6 36.2 f8.1
0.5 f0.6 1.4 f2.3 11.6 f24.3 6.5 f4.6 6.5 f4.0 4.1 f4.8 2.2 f4.4 3.8 14.3 6.1 f5.4 3.5 f5.3 9.4 f6.5 5.6 f4.2
35.3 f2.0 20.4 f3.8 29.7 fl.9 37.8 f3.6 41.7 f2.6 44.6 A5.1 43.3 f2.6 44.3 f1.8 43.0 f3.6 33.2 f3.2 47.5 f4.4 44.9 f3.1
34.5 f2.3 19.3 f2.4 25.4 f2.1 31.6 f4.2 34.8 12.4 41.0 13.2 40.4 f2.8 41.9 f2.1 38.8 13.1 29.0 f2.9 39.5 f3.7 40.6 f2.8
0.8 f0.4 1.0 f2.2 4.3 fl.9 6.2 13.4 6.8 f2.3 3.6 f3.6 2.9 f1.9 2.4 fl.9 4.2 13.2 4.2 f2.9 7.9 f4.0 4.3 f2.6
34.8 f0.3 22.6 f0.9 29.5 f2.0 37.4 13.9 41.5 f2.5 45.7 f2.9 41.4 f2.0 42.0 f2.3 40.5 13.1 30.8 f2.5 41.2 f4.3 44.7 f3.0
34.5 fO.0 22.7 f0.3 25.3 f1.6 31.5 f3.3 34.7 f2.0 41.1 f2.2 39.0 f0.6 41.7 f0.4 37.1 f0.6 29.1 f0.7 39.2 f2.9 40.8 f1.9
0.3 f0.3
The prefix f refers to the standard deviation.
-0.1 f0.9 4.2 11.2 5.9 12.1 6.8 11.5 4.6 fl.9 2.3 f1.9 0.3 f2.2 3.5 f3.1 1.6 12.4 8.0 f3.2 3.9 f2.3
values in dynjcm.
in the resulting values of ysd for such a variatiw in OF*; the error AyZ is similarly defined. Dissimilar liquid pairs, having large differences in polarity, also have large included angles @ a t their intersection (e.g., FA/BN). Table IV clearly shows that such pairs of equations have solutions that are quite stable to relatively large input errors. In contrast, very similar liquid pairs, having very small included angles, produce ill-conditioned solutions that are extremely sensitive to input error (e.g., FA/GL). Table V presents the harmonic mean calculation results for all of the polymers listed in Table 11. The first set of data (labeled WT/MI) contains values of ys, ,:y and ysP obtained from the water/methylene iodide pair alone; this corresponds to the original Wu method. The numbers are slightly different from those reported by Wu3 because in thiswork the liquid polarities calculated from contact angle data (Table I) have been used, to maintain a consistent data set, whereas Wu used liquid polarities calculated from interfacial tension data. The next two sets of data contain the means of all liquid pairs and of well-conditioned liquid pairs (amin = 0.5 rad), respectively, together with their standard deviations. The final set of data refers to the best-fit solution to all of the testing liquids simultaneously. This set consistently has the lowest standard deviations. The corresponding geometric mean calculation results are presented in Table VI. When all liquid pairs are considered, the scatter is enormous, as evidenced by the very large standard deviations. Rejection of ill-conditioned
pairs is obviously essential to obtaining meaningful results. The harmonic mean calculations (Table V) did not produce these large standard deviations; this is because very illconditioned pairs (such as FA/GL) usually result in complex roots and are thus automatically eliminated. Solid surface tension values calculated by the harmonic mean and geometric mean equations are compared with those obtained by other independent methods in Table VII. The data for the alternative methods are taken from WU.~ The polymer melt method involves the extrapolation of melt surface tension (which can be measured directly, using methods suitable for liquids) to room temperature. The liquid homologue method similarly involves extrapolation of the surface tension of low molecular weight liquid homologues of the polymer to high molecular weight. Neither of these methods requires contact angle data. The equation of state method and the critical surface tension method require contact angle data but do not involve use of the harmonic mean or geometric mean equations. The "equation of state" method, for which results are reported here, is that proposed by W U ,which ~ is different from the one proposed by N e u " et al.l0 All of the methods used (9) Wu, S. J. Colloid Interface Sci. 1979, 71, 605; 1980, 73, 590. (10) Neumann,A. W.;Good, R. J.; Hope, C. J.; Sejpal, M. J. Colloid Interface Sci. 1974, 49, 291. (11) Good, R.J. J. Colloid Interface Sci. 1977, 59, 398.
1014 Langmuir, Vol. 3, No. 6,1987
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Table VI. Solid Surface Tensions y, and Components y t and A,p Calculated by the Geometric Mean (GM) Equation Using the Data in Tables I and 11" pairwise solutions all pairs
WT/MI
well-conditioned pairs
solid PE
Ys
Yad
YBP
Ys
Yad
Ysp
33.7
33.7
0.0
PTFE
19.1
18.7
0.4
P3FE
23.4
19.3
4.1
50.6 f61.0 111.4 f360.0 288.2
PVDF
29.5
22.2
7.2
604.4
13.8 f49.4 93.8 f361.1 245.2 f939.7 532.8
PVF
36.0
30.6
5.4
PVDC
44.7
41.8
2.9
PVC
41.6
40.3
1.3
PS
42.5
42.0
0.5
50.2 f43.7 132.3 f347.0 58.1 f66.9 38.9
PMMA
39.6
35.5
4.2
PCTFE
27.0
23.4
3.6
PA66
42.2
32.9
9.3
PET
40.8
37.5
3.4
36.8 f13.4 17.5 f3.9 43.0 f77.6 71.5 f149.5 37.8 f11.8 35.9 f13.4 36.3 f11.7 37.3 111.4 35.5 f8.0 36.0 f42.9 36.7 f7.9 36.6 f8.l
fb fb
hlO.1
39.8 16.0 139.7 f430.7 42.7 f4.0 39.9 14.7
f*
12.4 f33.7 96.4 f356.6 21.8 f74.7 1.6 f1.7 4.2 f5.9 103.7 f388.7 6.0 f4.6 3.3 f3.6
"The prefix f refers to the standard deviation. y values in dyn/cm.
Ys 37.2 f0.5 18.6 fl.1 25.2 f2.0 34.5 f3.8 38.0 321.9 44.1 f2.0 42.8 f2.1 43.5 10.3 42.1 f2.5 30.5 f2.6 43.7 f3.1 42.9 f1.9
simultaneous best-fit solution
Yad
YSP
Ya
Y,d
Yap
36.9 fO.0 18.2 f1.2 22.8 f2.8 31.0 f5.0 34.3 f3.4 42.0 f3.0 41.9 f2.7 43.1 10.0 40.3 *3.6 29.0 f3.2 39.1 f4.8 41.4 f2.6
0.3 f0.5 0.5 f0.4 2.4 f1.3 3.6 h1.8 3.7 f2.4 2.0 f1.4 0.9 f0.6 0.4 f0.3 1.8 f1.7 1.5 f1.3 4.6 f3.1 1.5 f1.3
36.8 10.4 18.8 10.3 27.1 f0.9 36.2 11.3 41.2 11.4 45.5 10.9 44.2 10.4 43.4 f0.3 44.1 11.0 31.4 f0.9 47.0 12.1 44.3
36.6 f0.2 18.5 f0.2 25.5 f0.6 33.3 f0.5 39.3 fO.9 44.3 f0.2 43.8 fO.1 42.9 fO.1 43.2 f0.2 30.6 f0.4 44.0 fl.0 43.5 f0.3
0.2 f0.3 0.3 f0.3 1.6 10.6 2.9
fl.O
fl.1
1.9 fl.0 1.2 fO.9 0.4 f0.4 0.4 f0.3 0.9 fl.0 0.8 f0.8 3.0 fl.9 0.9 fl.O
Standard deviation > f999.9.
Table VII. Comparison of Solid Surface Tensions y. (dyn/cm) Obtained by Several Independent Methods solid PE PTFE P3FE PVDF PVF PVDC PVC
PS PMMA PCTFE PA66 PET
harmonic mean method" 34.8 22.6 29.5 37.4 41.5 45.7 41.4 42.0 40.5 30.8 47.2 44.7
geometric mean methodn 36.8 18.8 27.1 36.2 41.2 45.5 44.2 43.4 44.1 31.4 47.0 44.3
polymer melt method3 35.7
40.7 41.1 30.9 46.5 44.6
liquid homologue method3 34.7 23.9
equation of state method3 35.9 22.6 29.5 36.5 37.5 45.2 43.8 43.0 42.5 32.1 43.8 44.0
critical surface tension y 2 31 18 22 25 28 40 39 33b 39 31 46 43
"By the simultaneous best-fit solution (Tables V and VI). b A "Good-Girifalco" plot of the same data yields yc = 43 dyn/cm for polystyrene."
in Table VI1 are described in greater detail by W U . ~ Note that the ysvalues calculated from the HM and GM methods are in good agreement. Both methods fit the data quite well, with acceptably low standard deviations, but the GM method provides a consistently better fit. This is in contrast with the conclusions reached by W U ,who ~ found the GM method to be inadequate for polymers. Wu's findings might be due to inclusion of data for liquid pairs corresponding to ill-conditioned pairs of equations. Not only are solid surface tension values calculated by the HM and GM methods in agreement with each other, but they also agree very well with those obtained independently by the other methods listed in Table VII. On the other hand, the widely used critical surface tension yo included in Table VII for comparison, is consistently lower than ys,as has been pointed out by several authors. Although the HM and GM methods agree well in the ys values obtained, they differ on the polarity of the polymers. The GM method consistently provides lower values of the polar component ysP. Moreover, when all of the liquid data are considered, pairwise or simultaneously, y s P values are generally obtained that are lower than those obtmned when the WT/MI liquid pair alone is considered,
as Wu does. This is true in both the HM and the GM methods. 4. Conclusions In the application of the geometric mean (GM) and harmonic mean (HM) equations to contact angle data, using many liquids in order to obtain an overdetermined set of equations, the data may be solved pairwise or simultaneously. It has been shown that the simultaneous solution procedure is well suited to the task, whereas the pairwise solution is ill-conditioned and produces intrinsically poor results unless the liquid pairs are suitably selected. Values of the solid surface tension ys have been calculated for 12 polymers, using the GM and HM methods with published experimental contact angle data for a set of 6 testing liquids. It is found that ysvalues obtained by the two methods are generally quite close, and neither of the two conceptually different equations (which cannot both be theoretically correct) is clearly incompatible with the available experimental data. As a semiempirical means of analyzing contact angle data on polymers, either method would be satisfactory. However, the more widely used GM
Langmuir 1987,3, 1015-1025 method is preferable because it consistently fits the data better. The values of Y~obtained by either method are in good agreement with those obtained by other independent methods, whereas the critical surface tension yCis always lower.
1015
Acknowledgment. I acknowledge valuable discussions with Dr. I. D. Morrison. Registry No. PE, 9002-88-4; PTFE, 9002-84-0; P3FE, 24980-67-4;PVDF, 24937-79-9;PVF, 24981-14-4; PVDC, 900285-1;PVC, 9002-86-2;PS, 9003-53-6;PMMA, 9011-14-7;PCTFE, 9002-83-9; PA66, 32131-17-2; PET, 25038-59-9.
Adsorption of Boron on Molybdenum(100) and Its Effect on Chemisorption of Carbon Monoxide, Ethene, Propene, and 3,3,3-Trifluoropropene T. B. Fryberger,? J. L. Grant,i and P. C. Stair*l The Ipatieff Laboratory and Department of Chemistry, Northwestern University, Evanston, Illinois 60201 Received February 10,1987. In Final Form: May 11, 1987 The interaction of boron with Mo(100) and its effect on surface reactivity have been investigated for the first time. Boron-covered surfaces can be prepared by adsorption and decomposition of diborane (BZH6) at 300 K. The B(1s) binding energy (BE) measured by XPS increases approximately linearly with boron coverage from 186.9 eV at 0.2 monolayer to 187.6 eV at 1.1monolayers (saturation coverage),indicative of coverage-dependentB-B interactions. The absence of multiple B(1s) peaks or peak broadening suggests the overlayer grows by uniformly filling the available surface sites. After annealing to 1073 K and above, two distinct phases are formed: a low-coveragephase (-0.2 monolayer) with B(ls) BE = 186.8 eV and a three-dimensional, B-rich, surface "boride" phase (MOB,)with B(1s) BE = 188.2 eV. The assignment of the latter phase is established by comparison of the B(ls) BE with the literature value for MoBz. The B(1s) signal disappears upon heating above 1700 K, indicating loss of surface boron. Adsorption of carbon monoxide, ethene, propene, and 3,3,34rifluoropropene on B-covered surfaces was studied. Comparison of the results with data for adsorption of the same molecules on carbon- and oxygen-modified Mo(100) suggests that the B adatoms are not located in 4-fold hollow sites on the surface. 1. Introduction There is considerable interest in understanding the chemical modification of transition-metal surfaces by foreign adsorbate atoms. Surface modifiers receiving attention have primarily been either highly electronegative (e.g., C, 0, S ) or electropositive (alkali metals) elements compared to transition metals. Boron, however, has a Pauling electronegativity (2.0) which is comparable to that of transition metals. In addition, boron displays an unusual tendency toward forming B-B bonds. For example, in transition-metal borides, one-, two-, and three-dimensional boron networks are progressively formed with increasing boron content.' These properties suggest that the interaction of boron with a transition-metal surface may be quite distinct from the other chemical modifiers alluded to above and that the chemical interaction of the resulting surface with adsorbate molecules may be unusual. The present paper presents the results of X-ray and ultraviolet photoelectron spectroscopy (XPS and UPS) measurements of the interaction of boron with a Mo(100) surface and of the adsorption of CO and olefins on Bmodified Mo(100). Thermal desorption spectroscopy (TDS) measurements were also performed to confirm some of the conclusions from the electron spectroscopy data. Section 3.1 describes the preparation of B-modified Mot Present address: National Bureau of Standards, Gaithersburg, Maryland 20899. t Present address: 3M Corporation, St. Paul, Minnesota 551441000. Aldred P. Sloan Foundation Fellow 1984-1988.
0743-7463/87/2403-1015$01.50/0
(100) surfaces by adsorption and decomposition of diborane @&I6). From XPS, UPS, and TDS data on diborane adsorption a t low temperatures, it is shown that B2H6completely decomposes on the surface below room temperature, and Hzdesorbs in the temperature range 280-400 K. In section 3.2, the behavior of adsorbed boron as a function of coverage at room temperature is discussed. Measured B(ls) and B(2p) binding energy shifts suggest that interactions between boron adatoms become important with increasing coverage up to a saturation value of 1.1monolayers (1.0 monolayer is taken to be 1.0 X 1015/ cm2,which is equal to the Mo surface atom density). The effects of stepwise heating of B/Mo(100) surfaces are described in section 3.3. At boron coverages above 0.5 monolayer a phase transition occurs upon annealing to 1073K. At this temperature, the coexistence of two phases is observed: a three-dimensional "boride" phase with an approximate stoichiometry MoBz and the low-coverage (0.2 monolayer) chemisorbed phase. The two phases persist for annealing temperatures up to about 1600 K, where the boride phase is no longer detectable by XPS. Finally, after annealing above 2000 K no boron remains on the surface. Results are presented in section 4 for the adsorption of carbon monoxide, ethene, propene, and 3,3,3-trifluoropropene on the boron-modified Mo(100) surface. These molecules were chosen primarily because they have been studied on C- and 0-modified M O ( ~ O Oand, ) ~ ~ hence, ~ (1)Kiessling, R. Acta Chem. Scand. 1950, 4, 209. (2) Deffeyes, J. E.; Horlacher Smith, A.; Stair, P. C. Appl. Surf. Sci. 1986, 26, 517.
0 1987 American Chemical Society
