4141
Langmuir 1996,11, 4141-4142
An Algorithm for Estimating Contact Angles Sujit Banerjee*>? Institute of Paper Science and Technology, 500 Tenth Street NW, Atlanta, Georgia 30318
Frank M. Etzler" Boehringer Ingelheim Pharmaceuticals Inc., P.O. Box 368, Ridgefield, Connecticut 06877 Received February 9, 1995. I n Final Form: July 14, 1995@ Contact angles of a variety of polar and nonpolar liquids on polymer surfaces are correlated with surface tension and the interfacial free energy between the solid and the liquid. It is assumed that intimate molecular contact is made between the liquid and the polymer surface. The interfacial free energy is represented by the residual infinite dilution activity coefficient of the liquid monomer as computed by the UNIFAC equation. Estimated contact angles are within an average of 7" of measured values. Zisman and co-workers1 found a nearly linear dependence of the cosine of the contact angle with the liquid surface tension when a homologous series of liquids was placed on a given polymer surface. The intercept of the linear plot with cos 8 = 1 has been referred to as the Zisman critical surface tension a,. Zisman noted that the value of the critical surface tension was dependent on the specific surface chemistry of the solid surface. van Oss et aL2recently separated surface tension into components based on London dispersion forces and Lewis acid-base interactions. While valuable for gaining insight into the origins of contact angle, the approach does not provide for general contact angle estimation. The relationship between contact angle and basic physicochemical properties can be understood in terms of surface thermodynamics. Recalling the Young equation as, - os1= a,, COS
e
(1)
and noting that, at cos 8 = 1, the liquid surface tension aivis just the Zisman critical surface tension, uC,it follows that 0,= os,
- 081
(2)
Recalling further that the work of adhesion W, is related to interfacial tensions via the following definition wa
= as, + 01" - as1
(3)
w,= a, + a,,
(4)
thus, The empirical Zisman equation may be written as COS
e = 1 - pc~,,- 0,)
(5)
Recalling the above relationship between W, and a,, the Zisman equation can be rearranged to cos
e = 1 + pw, - zpa,,
(6)
Equation 6 expresses the contact angle in terms of two fundamental parameters: the liquid vapor surface tension and the adhesion energy. In order to evaluate eq 6, it is necessary to estimate W,. E-mail:
[email protected]. Abstract published in Advance A C S Abstracts, September 15, 1995. (1)Zisman, W. A. In Contact Angle Wettability and Adhesion, 1964; ACS Advances in Chemistry Series; Gould, R. F., Ed.; American Chemical Society: Washington, DC, p 1. (2) van Oss,C. J.;Chaudhury, M. IC;Good,R. J.Adu. Colloid Interface Sci. 1987,28,35-64.
4 4
00
04
08
12
Cos theta (measured)
Figure 1. Comparison of contact angles estimated by eq 8 to measured values.
We have chosen the logarithm of the UNIFAC residual activity coefficient for the solute (in this case, the liquid) at infinite dilution in the monomer of the solid polymeric material as a measure of the interaction between the liquid and the polymeric surface. UNIFAC is a group contribution method used principally in the chemical engineering ~ o m m u n i t y .It~ represents the activity coefficient as the product of a combinatorial component that reflects differences in size and shape between the components in the mixture and a residual contribution that measures heat effects. The combinatorial part is obtained from van der Waals volume and area parameters. The residual part is obtained from experimentally obtained interaction parameters of every group with every other group. For the calculation, all the components in the system are factored into their functional groups, and the activity coefficient of each group is calculated. The activity coefficient of the parent molecules is then assembled from the group activity coefficients. In our case, the surface is approximated by a monomer, and the combinatorial component is unnecessary. The residual component of the activity coefficient was calculated using the interaction parameters listed by Hansen et al.4
+
@
(3) Fredenslund, A,; Jones, R. L.; Prausnitz, J . M. AIChE J. 1975, 21, 1086.
(4) Hansen, H. K.; Rasmussen, P.; Fredenslund, A,; Schiller, M.; Gmehling, J.Ind. Eng. Chem. Res. 1991, 30, 2352-2355. (5) Fox, H. W.; Zisman, W. A. J . Colloid and Interface Sci. 1950, 514-531.
0743-7463/95/2411-4141$09.00/00 1995 American Chemical Society
4142 Langmuir, Vol. 11, No. 10, 1995
Banerjee and Etzler
Table 1. Contact Angles and Residual Activity Coefficients of Some Liquids In yr diiodomethane dibromomethane 1,2-dibromoethane pyridine
2.48 2.17 1.93 1.18
water bromoform diiodomethane 1,1,2,2-tetrabromoethane 1,2,3-tribromopropane iodobenzene 1-bromonaphthalene 1-chloronaphthalene
7.48 3.65 2.48 4.77 3.09 1.48 1.71 0.67
water bromoform diiodomethane 1,1,2,2-tetrabromoethane 1,2,3-tribromopropane
2.16 0.29 1.14 0.35 0.14
hexadecane tetradecane n-dodecane undecane decane nonane
4.56 4.03 3.49 3.23 2.97 2.70
water propanol
7.37 3.05
water
7.37
water
5.45
water propanol
2.18 1.02
contact angle (deg)
In Yr
Polyethylene (20 " C ) (Ref 6) 52 N,N-dimethylformamide 27.5 dimethyl sulfoxide 47 water 32 Polyethylene (20 "C) (Ref 7) 96.1 bromobenzene 36.7 chlorobenzene 52.8 2-nitrotoluene 50.9 nitromethane 43.1 nitrobenzene 33.1 3-nitrotoluene 41.1 carbon disulfide 36.3 Poly(ethy1eneTerephthalate) (20 " C ) (Ref 7) 76.5 iodobenzene 21.4 1-bromonaphthalene 40.8 1-chloronaphthalene 37.8 bromobenzene 26.1 nitromethane
Poly(tetrafluoroethy1ene) (Ref 5) 46 n-octane 44 n-heptane 42 n-hexane 39 n-pentane 35 benzene 32 tert-butylnaphthalene Paraffin (25 "C) (Ref 8) 108 a-bromonaphthalene 36 Polyethylene (25 " C ) (Ref 8) 109 a-bromonaphthalene Polystyrene (25 " C ) (Ref 8) 84 a-bromonaphthalene Poly(methy1methacrylate) (25 " C ) (Ref 8) 79 a-bromonaphthalene 13
Contact angle and surface tension data were taken from the literature; the former are for sessile drops and are listed in Table 1. The surfaces were generally hydrophobic (ranging from paraffin to polytetrafluoroethylene)), and the liquids covered a wide range of hydrophobicity from water to hexadecane. UNIFAC yr values were obtained with a commercial package (PC-UNIFAC 5.0, bri, PO Box 7834, Atlanta, GA 30357). Regression of contact angle against In yr and surface tension (0) led to the equation cos 8 = 1.42 - 0.102 In yr - 0.0115 (n = 54, r = 0.95) (7) The relationship is illustrated in Figure 1. The sign of the In yr coefficient in eq 7 is correct, since In y increases (6)Fowkes, F. M. J.Adhesion Sci. Technol. 1987,I , 7-27. (7) Bronislaw, J.;Bialopiotrowicz, T. J.Colloid Interface Sci. 1989, 127, 189-204. (8) Busscher, H. J.; van Pelt, A. W. J.; de Jong, H. P.; Arends, J. J. Colloid Interface Sci. 1983,95, 23-27.
4.15 4.75 7.48
contact angle (deg) 41 60 107
1.55 0.51 2.46 4.00 2.20 2.46 0.54
21.8 10 39.2 55 47.3 40.8 0
0.46 0.21 0.10 0.16 0.14
19.2 22.7 18.5 14.8 30
2.43 2.17 1.90 1.33 1.33 2.78
26 21 12 0 46 65
1.68
42
1.68
15
0.21
10
0
14
with increasing solute-surface incompatibility, which, in turn, results in larger contact angles. Not surprisingly, In yr and o are weakly correlated a t r = 0.43. While eqs 6 and 7 are similar in form, the coefficients in eq 7 cannot be quantitatively assigned to the variables in eq 6. For example, our intercept of 1.42in eq 7 is higher than the expected value of one, probably because of intercept error. If the intercept in eq 7 is forced to 1, r decreases to 0.89. Also, since In yr and u are not orthogonal but weakly correlated, any change in one coefficient is partially compensated by a change in the other. In summary, eq 7 appears to be able to handle a variety of liquids and surfaces, and is the most universal method available for estimating contact angle. Estimates are within 7" of measured values, which compares quite well with the precision reported for measurements made across several laboratories. LA9501010
