340
Langmuir 1994,10, 340-342
Notes The Molecular-Kinetic Theory of Wetting Robert A. Hayes and John Ralston’ School of Chemical Technology, University of South Australia, The Levels, SA 5095, Australia Received June 28, 1993. In Final Form: October 12, 1993
Introduction Wetting and dewetting processes are central to a vast number of everyday processes, both natural and industrial. It is often assumed that the understanding of these processes is commensurate with the degree of application. However,despite widespread experimental and theoretical investigation, generally applicable predictive models do not exist. Studies to date have inevitably focused on contact line (CL) movement and, in particular, the contact angle, 8, due to ita experimental accessibility. In the dynamic situation, where the CL is mobile due to either spontaneous or forced shear, one typically observes a change in 8 as a result of the tendency of the CL to resist movement. Advancing liquid movement (wetting) gives rise to an increase in 8 while receding liquid movement (dewetting) reduces 8. Theoretical models of dynamic wetting (and dewetting) processes fall into two broad categories. A molecular-kinetic (MK) model, which addresses CL movement from an activated rate perspective involving molecular scale “jumps,” has been developed by Blake.lI2 Hydrodynamic discount surface chemical events and instead aim to describe CL motion in terms of bulk viscous dissipation. The application of theoretical models to experimental data has inevitably involved separate analyses of wetting and dewetting CL movement and the use of an experimentally determined static contact angle (eo). It is the difference between the dynamic contact angle (8d) and the corresponding static angle which is presumed to provide the driving force for contact line movement. The molecular-kinetic theory describes CL movement as a stress modified molecular rate process involving adsorption of molecules of the advancing phase and concurrent desorption of molecules of the receding phase. When a shear is applied to the CL, by for example forced liquid movement (at velocity, VI, the out of balance force initiates CLmovement in the direction of the applied shear. If it is assumed that the shearing force is provided by the out-of-balance interfacial tension forces acting on the tpcl and, further, that there is no dissipation of this force other than in molecular displacement than one may write, after Eyring,6 the following equation relating 0 and V
V = 2K0 X sinh(yl,(cos 0 O - cos ed)/2nk7‘) (1) where KO is a quasi-equilibrium rate constant, X is the distance between adsorption sites, n is the surface density
* Author to whom correspondence should be addressed.
(1) Blake, T. D.; Haynes, J. M. J. Colloid Interfoce Sci. 1969,30,421. ( 2 ) Blake, T. D. Wetting Kinetics-How Do Wetting Lines Moue?; AICM International Symposiumon the Mechanics of Thin Film Coating, New Orleans, 1988; paper la. (3) Voinov, 0.V. Fluid Dyn 1976,11, 714. (4) Cox, R. G. J. Fluid Mech. 1986,168, 169. ( 5 ) Glaestone, S.; Laidler, K. J.; Eyring, H. J. The Theory of Rate Processes; McGraw-Hill: New York, 1941;
of these sites, and 8” and ed are the “static” and dynamic contact angles, respectively (TI”, k,and T have their usual significance). A positive velocity corresponds to advancing movement. If it is further assumed that the distribution of adsorption sites is isotropic, then X = n-1/2 and the equation is reduced to a two parameter, single variable equation (eq 2). This may be readily solved without further approximation by nonlinear least-squares (NLLS) analysis. Adherence of dynamic contact angle data to the molecularkinetic model requires that a good fit to eq 1 is obtained and further that the values of KO and X obtained are physically meaningful. While the generally adopted MK approach has been shown2to have considerable merit in terms of data fitting, one may question its phenomenological validity. Other than demonstrating the KO and X values are “physically significant”, a meaningful trend in KO and X values with a range of physicochemicalproperties of liquid and solid phases has yet to be established. For example if one were to consider liquid movement across a series of solids of varying and known “site density” (A), would there be a meaningful correlation with the parameter values obtained by conventional molecular-kinetic analysis? Indeed the link between X and solid surface properties has been inferred rather than proven and may depend additionally, or even entirely, on liquid phase properties. Application of this and other theories to date has ignored the regime of great and unresolved interest due to contact angle hysteresis. The exclusion of the difference between the experimental eao and Br0 from dynamic contact angle analyses has implied that these values represent nonencroachable boundaries. Recent work: involving the study of contact line relaxation, questions this verdict. For water on poly(ethy1eneterephthalate) (PET)tensiometric measurements demonstrated that the onset of mechanical equilibrium, as observed by for example optical means, did not correspond with microscopicequilibrium. Equilibration took hours, or even days, and was critically dependent on the direction of contact line movement prior to relaxation and also water vapor pressure. To explain these observations, an essentially evaporative mechanism of CL relaxation was proposed. Surface roughness and/or chemical heterogeneity trap the contact line in a metastable state of mechanical equilibrium with the contact line relaxing to thermodynamic equilibrium by evaporation. The finite sensitivity of the tensiometric technique precluded any definitive statement of precisely when true equilibrium was established. However what could be stated unequivocally was that contact angles corresponding to the onset of mechanical equilibrium were significantlydifferent from those measured at some later stage. These results have some important implications in terms of previous applications of theories of contact line movement. The additional aim of the current communication is to assess whether the molecular-kinetic theory has broader application than has previously been demonstrated.
Application of the Molecular-Kinetic Theory Our approach involves treating the static contact angle as an additional adjustable parameter in eq 2. In this way uncertainties in the experimentally determined value of (6) Hayes, R. A.; Ralston, J. Colloids Surf., in press.
0743-7463/94/2410-0340$04.50/0 0 1994 American Chemical Society
Notes
Langmuir, Vol. 10, No. 1, 1994 341 80
x u0 20
0
-0.2
0.0 0.2 Velocity (cm/s)
-
0.4
0
-0.10
0.00 0.10 Velocity (cm/s)
Figure 1. Tensiometric' contact angle vs contact line velocity for water (0, y = 72.8 mN/m, r) = 1.002 cP) and glycerol/water ( 0 ,y = 67.9mN/m,7)= 82.86 cP) on PET. Solid lines correspond to nonlinear least squares (NLLS)fitting of both advancing and receding experimentaldata to eq 2 simultaneously. Parameter values obtained for water were 8' = 68.2', X = 2.90 X lp7cm, KO = 8.12 X 106 a-1, with AG, = 94.6 kJ mol-', and (dO/dV),, = 1.73 X 1010 deg cm a-1, and for glycerol/water were eo = 55.0°, X = 1.64 X 10-7 cm, K O = 133 s-l, with AGw = 59.8 kJ mol-', and (dB/dV), = 7.10 X l(r deg cm s-l.
Figure 2. Optical8contactangle vs contactline velocity for water (0, y = 72.4 mN/m, r) = 0.8902 cP) and glycerol/water ( 0 , y = 51.9 mN/m, r) = 85.1 cP) on PET. Solid lines correspond to NLLS fitting of both advancing and receding experimentaldata to eq 2 simultaneously. Parameter values obtained for water were eo = 56.6O, X = 2.88 X le7cm, K O = 2.65 x lo-' 8-1, with AG, = 91.8 kJ mol-', and (dO/dv), = 6.06 X 109 deg cm 8-1 and for glyceroVwater were Bo = 44.5', X = 2.36 X 10-7 cm, K O = 3.7 s-l, with AG, = 68.5 kJ mol-', and ( dO/dV),, = 1.31 X 108 deg
Bo are avoided. In addition both advancing and receding dynamic contact angle data are fitted to eq 2 simultaneously.
Table 1. Molecular-Kinetic Parameters for PET/Water
V = 2K0A sinh[y,,(cos Bo - cos Bd)A2/2kTJ
(2) One may also write expressions for the maximum velocities of wetting (ed = 180O) and dewetting (ed = 0'): Vellme = 2K0A sinh{y,,(l
+ COS B0)A2/2k!I')
Ve+ = 2 K o A sinh(yl, (1- cos B0)A2/2kTj
kT = y l F A 3 sin 80
advancing
receding
82.1' 77.1 54.44 59.4
7.2 X 10-8 1.9 x 1v7 4.4 x 10-7 3.8 x 10-7
l.6X 1.1x 1.4 X 3.1 X
106
109 lV*
106 0 Experimental value measured 100 8 after cessation of forced CL movement.
(4)
for data obtained by both tensiometric and optical methods. The values of 8' obtained from the curve fitting procedure are in all cases slightly larger than the arithmetic means of the minimum velocity advancing and receding contact angle values. The values of A, the distance between "adsorption" sites, were found to be of the order of 3 nm for water on PET and 2 nm for the more viscous glycerol/ water mixture. These values are very similar in magnitude to those obtained by conventional molecular-kinetic analysis,' where advancing and receding data are treated separately and an experimental value of the static contact angle, eo, is used. In contrast the rate constant parameter, KO, was found to be much more sensitive to the choice of MK analysis. Values of KO obtained in the modified analysis were significantly smaller in magnitude than those obtained by conventional molecular-kinetic analysis. In Table 1 we compare parameter values obtained by conventional analysis using firstly the experimentally determined Bo and secondly a value of the contact angle, e*, 5 O less (greater in the receding case) than 8'. A value of 5 O was chosen simply because for PET/water this is a lower bound for the contact angle change that has been observed6 due to relaxation effects. One finds that only in the case of water advancing on PET using eo in eq 2 is a "physically reasonable" value of KO obtained. If B* is used in the advancing case or either contact angle value in the receding case, then the values of KO obtained appear to be unreasonably low. Furthermore the values obtained using B*, which is a better approximation to the equilibrium contact angle than eo, are 3 orders of magnitude smaller than when eo is used. In this context the low values of KO obtained by the modified MK analysis proposed here do not appear as extreme. The physical meaning of the rate
AG, = -NkT ln(hKo/kT) (5) In the present context it is also of interest to obtain the maximum value of the contact angle-velocity gradient which corresponds to V =0. This value, which may readily be obtained by differentiation of eq 2, assesses the sensitivity of the contact angle to contact line movement at low velocities.
dB
Obtained by Conventional Analysis
(3)
Physically these velocities correspond to vapor and liquid entrainment, respectively. The activation free energy of wetting, AGw, may be obtained from KO via eq 5,where N and h have their usual significance.
W"m
cm a-1.
(6)
The conventional application of the MK theory to the wetting of polyethylene terephthalate (PET) by polar liquids have previously been r e p ~ r t e d . ~Both ?~*~ tensiometric' and optical8 experimental data were used to test the alternative method of application of the MK theory described here. NLLS analyses for water and glycerol/ water on PET are summarized in Figures 1and 2.
Discussion It is clear from Figures 1 and 2 that the general form of an entire set (advancing and receding) of contact angle/ velocity data can be described by eq 2. This is the case (7) Hayes, R. A.; Ralaton, J. Colloid Interface Sci. 1993, 159,429. ( 8 ) Petrov, J. G.; Petrov, P. G. Colloids Surf. 1992, 64, 143.
342 Langmuir, Vol. 10, No. 1, 1994
constant parameter obtained by either standard analysis, using a more appropriate value of the equilibrium contact angle, or the modified MK analysis proposed here is not clear and remains to be addressed in future work. While the values of K O seem particularly low when compared to corresponding values of bulk viscous flow2 of 10" s-', it should be remembered that according to the original definition2 they are a quasi-equilibrium rate constant for wetting and correspond to the natural frequency of wetting under no applied shear. The activation free energies of wetting, AG,, calculated from eq 5 using values of KO obtained by modified MK analysis were 90 kJ mol-' for water and 60 kJ mol-' for glycerol/water. While one may readily calculate the maximum velocities of wetting and dewetting from eq 3 and eq 4, comparison between experiment and theory can only be made for the receding movement of the more viscous glycerol/water mixture on PET. In this case V,, is 0.33 cm s-l whiIe the corresponding experimental value is in the range 0.080.10 cm s-l. The lack of agreement between theory and experiment at the extremes of velocity and viscosity highlight deficiencies in the MK theory (as described in eq 1) in describing wetting kinetics in domains where hydrodynamic factors may be expected to prevail. A combined MK and hydrodynamic theory has recently been reported.9 The maximum values of contact angle/velocity gradient obtained from tensiometric data are 1.7 X 1O1O deg cm-1 s for water and 7.1 X lo4deg cm-l s for glycerol/ water. In the latter case then a CL velocity of 1pm s-' willgive rise to 7.1° contact angle hysteresis while for water on PET the contact angle is even more sensitive to CL movement. As Blake2 has already noted the magnitude of contact angle hysteresis can be explained simply by incidental CL movement beyond the range of observation prevailing in a typical contact angle measurement. In Figure 3 the effect of variation of the molecular parameters, X and KO,on the wetting of PET by glycerol/ water is explored. The values of alternatively X or KO were varied with the other (and Bo) being held constant. The contact angle/velocity dependency is clearly more sensitive to variation in X than KO. Variations of less than 1order of magnitude in these parameters have a significant impact upon both the magnitude of contact angle hysteresis and the maximum velocities of wetting and dewetting. (9) Petrov, J. G.;Petrov, P.G.Langmuir, 1992,8, 1762.
Notes l
o
00
o
-
w
______----.....................
.......................
-
h
i(" ........................................................................ :
"0P
P
....... ...: .,:
...................
r
Q
M 40-
s
V w
-
i:
...../ 20
-
i
.
"
-
!
-0.4
-0.2
r
.
0.0 0.2 Velocity (cm/s)
0.4
0.6
Figure 3. Effect of variation of molecular parameters in eq 2 on wetting kinetics. The solid line represents the NLLS fit to glycerol/water tensiometric data with eo = 55.0°, X = 1.64 X 10-7 cm, and KO = 133 s-l. The other Iines were obtained by single 3X, (- - -) 0.8X, (- - -) 1OKo, and parameter variation of (-a)
(-
*
-)
0.5K0.
Decreases in X and K O give rise to an increase in contact angle hysteresis and a decrease in the maximum velocities of wetting/dewetting.
Conclusions The molecular-kinetic theory of wetting kinetics developed by Blake2is capable of describing both advancing and receding contact angle/velocity data including contact angle hysteresis. The general form of the slip condition proposed by the molecular-kinetic theory V = A sinh(Bu) where Q is the shear stress, has been reinforced in the present work for both advancing and importantly receding liquid movement simultaneously. The physicochemical basis of the constant, A, remains uncertain. At extreme values of velocity and viscositythe theory does not describe experimental data as well and one presumes that an additional term reflecting hydrodynamic considerations is required.
Acknowledgment. The financial support of the Australian Research Council (ARC) is appreciated.
