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Spreading of Water: Condensation Effects Martin E. R. Shanahan Ecole Nationale Supe´ rieure des Mines de Paris, Centre des Mate´ riaux P.M. Fourt, CNRS UMR 7633, B.P. 87, 91003 Evry Ce´ dex, France Received July 11, 2001. In Final Form: October 15, 2001 Although the macroscopic, bulk behavior of liquids wetting (or dewetting) can be adequately explained by viscous flow, the intricate details near the solid/liquid/fluid triple line (TL) are still not totally appreciated.These two aspects correspond to the bases of the hydrodynamic approach and of the molecular kinetics approach, respectively, to explain spreading. We have suggested a complementary model to that of molecular kinetics, based on the modified curvature of a dynamic, wetting meniscus near the TL, which leads to local condensation of vapor from the surrounding atmosphere, further contributing to motion. An earlier paper dealt with liquids whose thin film behavior (i.e., near the TL) is dominated by long-range van der Waals forces. Here, we turn our attention to the spreading of water. Basic equations are modified to allow for non-negligible meniscus slope and then applied to a model for water, consisting of two different, thickness dependent, disjoining pressure isotherms.
Introduction Dynamic wetting or dewetting involves complex behavior near the solid/liquid/fluid triple line (TL). Two main, basic schools of thought exist to explain motion in the vicinity of the TL, although several variants exist. Cherry and Holmes’1 model of activated viscous flow, based on the Eyring rate theory,2 led to the development of various molecular kinetics approaches by Blake and Haynes,3 Ruckenstein and Dunn,4 and others (e.g., refs 5-9). Although a popular interpretation of such dynamic processes was that of molecular “hopping”, or local mass transfer by adsorption, it was pointed out that other methods of transport could also be relevant, provided the scale involved is small enough for Maxwell-Boltzmann statistics to be pertinent.3,6 The second principal class of models derives from a dynamic balance between capillary forces and viscous dissipation, as suggested by Huh and Scriven10 and later developed by others. See, for example, refs 11-13. Various propositions for combining the two approaches have been made,14,15 and recent work indeed suggests a transition from molecular kinetics to hydrodynamic behavior, as the dominant process, as spreading evolves.16,17 (1) Cherry, B. W.; Holmes, C. M. J. Colloid Interface Sci. 1969, 29, 174. (2) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Processes; McGraw-Hill: New York, 1941. (3) Blake, T. D.; Haynes, J. M. J. Colloid Interface Sci. 1969, 30, 421. (4) Ruckenstein, E.; Dunn, C. S. J. Colloid Interface Sci. 1977, 59, 135. (5) Petrov, J. G.; Radoev, B. P. Colloid Polym. Sci. 1981, 259, 753. (6) Blake, T. D.; Wetting Kinetics-How Do Wetting Lines Move? AICh E International Symposium on the Mechanics of Thin Film Coating; New Orleans, 1988; Paper 1 a. (7) Petrov, J. G.; Petrov, P. G. Colloids Surf. 1992, 64, 143. (8) Hayes, R. A.; Ralston, J. J. Colloid Interface Sci. 1993, 159, 429. (9) Hayes, R. A.; Ralston, J. Langmuir 1994, 10, 340. (10) Huh, C.; Scriven, L. E. J. Colloid Interface Sci. 1971, 35, 85; (11) Voinov, O. V. Fluid Dyn. 1976, 11, 714. (12) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (13) Cox, R. G. J. Fluid Mech. 1986, 168, 169. (14) Petrov, P. G.; Petrov, J. G. Langmuir 1992, 8, 1762. (15) Brochard-Wyart, F.; de Gennes, P. G. Adv. Colloid Interface Sci. 1992, 39, 1. (16) De Ruijter, M. J.; De Coninck, J.; Oshanin, G. Langmuir 1989, 152, 2209. (17) De Ruijter, M. J.; Charlot, M.; Voue´, M.; De Coninck, J. Langmuir 2000, 10, 2363.
Many years ago, Hardy18 suggested the importance of liquid transfer by evaporation and recondensation at the TL, in wetting processes. Following early work on the effects of disjoining pressure and capillarity (among other factors) in the contact line region of a stationary curved film,19 Wayner considered evaporation/condensation effects in the spreading of liquid films (e.g., refs 20-23). This approach was further developed recently by the present author.24-26 By allowing for the presence of longrange forces near the triple line region, and considering a nonequilibrium liquid meniscus in motion, it was shown how recondensation of ambient vapor of the drop liquid may contribute to spreading rate, provided the TL is already moving (consistent with ideas of mutual interaction between flow field and contact angle, as proposed by Shikhmurzaev27). Long-range forces were assumed to be of the van der Waals, dispersive type, leading to a positive value of Hamaker constant, A (using the sign convention of Adamson28), and thus a negative disjoining pressure, Π, tending to thin the local liquid film. The single most important liquid on this planet is water, and thus the main purpose of this paper is to extend the ideas of the previous work in order to consider the significance of condensation transport during the spreading of this ubiquitous liquid. Unfortunately, the detailed behavior of thin films of polar liquids, and in particular water, would still seem to be relatively poorly understood,29-31 but we shall proceed by assuming various, more (18) Hardy, W. Philos. Mag. 1919, 38, 49 (cited in ref 12). (19) Potash, M., Jr.; Wayner, P. C., Jr. Int. J. Heat Mass Transfer 1972, 15, 1851. (20) Wayner, P. C., Jr. Colloids Surf. 1991, 52, 71. (21) Schonberg, J.; Wayner, P. C., Jr. J. Colloid Interface Sci. 1992, 152, 507. (22) Wayner, P. C., Jr. Langmuir 1993, 9, 294. (23) Wayner, P. C., Jr. Colloids Surf., A 1994, 89, 89. (24) Shanahan, M. E. R. C. R. Acad. Sci., Paris 2001, 2 (IV), 157. (25) Shanahan, M. E. R. Langmuir 2001, 17, 3997. (26) The author was unaware of the significant contributions of Wayner to the problem when refs 24 and 25 were published. (27) Shikhmurzaev, Y. D. AIChE J. 1996, 42, 601. (28) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces, 6th ed.; Wiley: New York, 1997; p 232. (29) Israelachvili, J. N. Intermolecular and Surfaces Forces; Academic Press: New York, 1985. (30) Sharma, A. Langmuir 1993, 9, 3580. (31) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces, 6th ed.; Wiley: New York, 1987.
10.1021/la011065y CCC: $20.00 © 2001 American Chemical Society Published on Web 12/01/2001
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Figure 1. Liquid (L) film of thickness, h(x), and local radius of curvature, r(x), on solid (S), presenting convex meniscus toward vapor phase (V).
Figure 2. Continuity considerations in liquid flow at speed U toward the left. Jp and Jc represent respectively Poiseuille current and condensation current.
or less accepted, long-range force relations for watersolid interactions.
is the usual product of Boltzmann’s constant and absolute temperature. Since pvo ) po at equilibrium, the far-field environment being in (molecularly dynamic) equilibrium with the local vapor pressure above the TL region, the contents of the brackets of eq 2 are equal to zero and eq 1 is a natural consequence. In principle, eq 1 (or eq 2) could be used to elucidate the equilibrium meniscus profile near the TL, provided a suitable form for Π(h) is available (discussed below). However, our present concern is to consider a local meniscus perturbed from equilibrium due to TL motion. The perturbed, or dynamic, profile is denoted hd, rather than h. With rd as the local, dynamic radius of curvature of the meniscus, we may calculate from eq 2 the appropriate equilibrium partial pressure of vapor, p˜ o, were the drop profile to be at equilibrium with this geometry. Excess vapor pressure, ∆p, provoking local condensation from the atmosphere is then given by
Basic Equations In this section, we present essentially a summary of the basic theory given in ref 25 but modified to allow for profile slopes, h′, not necessarily small. However, we shall restrict our attention to systems of (macroscopic) equilibrium contact angle, θo, less than π/2. Dynamic Condensation Rate. The well-known Young equation adequately describes (macroscopic) equilibrium at the three-phase solid/liquid/fluid wetting TL, but assessment of the local equilibrium liquid profile is more delicate, due to the existence of long-range forces. By considering the liquid phase, L, to be thin (a film) of thickness, h, in the presence of solid, S, and vapor, V, as shown in Figure 1, local equilibrium is assured by a balance of the local pressures, viz., the ambient (atmospheric) pressure, PA, the hydrostatic pressure in the liquid, P, Laplace’s excess pressure due to meniscus curvature, γ/r, where γ is liquid surface tension and r is the local radius of curvature (N.B. a cylindrical meniscus is assumed, thus eliminating the second curvature term), and the disjoining pressure of Derjaguin,32,33 Π(h).
PA - P ) -
γ γ h′′ + Π(h) ) 0 + Π(h) ) r (1 + h′2)3/2
(1)
In eq 1, h′ and h′′ have their usual meanings of first and second derivatives of (equilibrium) liquid thickness with respect to distance x. Equation 1 is a form of the augmented Young-Laplace relation (AYL), as suggested several times in the past (e.g., refs 30, 34-36). Note that for convenience, we have assumed a convex meniscus with respect to the vapor phase for the definition of sign of r. Another method for deriving the AYL was recently described.25 By considering the classic Kelvin equation,37 but modified to take into account the presence of longrange interactions (and also the existence of a single meniscus radius of curvature near the TL), it was shown that
[kTv (γr - Π(h))]
pvo ) po exp
(2)
where pvo is the partial pressure of the vapor of the liquid compatible with meniscus curvature, r-1, and po is that for a flat meniscus, v is liquid, molecular volume, and kT (32) Derjaguin, B. V. Zh. Fiz. Khim 1940, 14, 137. (33) Derjaguin, B. V. Kolloidn. Zh. 1955, 17, 191. (34) White, L. R. J. Chem. Soc., Faraday Trans. 1 1977, 73, 390. (35) Wayner, P. C., Jr. J. Colloid Interface Sci. 1980, 77, 495. (36) Wayner, P. C., Jr. J. Colloid Interface Sci. 1982, 88, 294. (37) Reference 31; p 53.
{
[ (
)]}
v γ - Π(hd) kT rd
∆p ) po - p˜ o ) po 1 - exp
f
[
γhd′′
(1 + hd′2)3/2
≈
]
+ Π(hd) (3)
where f is the molar fraction of the vapor of the liquid contained in the atmosphere (f ) po/PA). Defining a (net) rate of condensation, Jc(x), at abscissa x, taken parallel to the solid surface (see Figure 2), we expect this to be proportional to the vapor pressure excess, ∆p
Jc ) K∆p ≈
[
]
γhd′′ fv + Π(hd) (2πmkT)1/2 (1 + hd′2)3/2
(4)
where K ) v/(2πmkT)1/2, with m representing the molecular mass of the liquid in question.38 Continuity. To make our task more tractable, we shall assume steady-state flow conditions at relatively low spreading speeds, U, although we do not assume small contact angles, θ, as we did in the earlier paper,25 for reasons which will become clear below. To ensure continuity, or mass conservation, during TL motion involving both a horizontal, hydrodynamic, Poiseuille current, Jp(x), and a condensation current, Jc(x), we have from Figure 239
( ) ( )
∂hd dhd ) dt ∂t
x
+U
∂hd )0 ∂x t
(5)
(38) Hudson, J. B. Surface Science; Wiley: New York, 1998; p 297. (39) Teletzke, G. F.; Davis, H. T.; Scriven, L. E. Chem. Eng. Commun. 1987, 55, 41.
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Langmuir, Vol. 17, No. 26, 2001 8231
where t represents time, leading to
Jc ∂Jp ∂Jp + ) + Jc(1 + hd′2)1/2 ) 0 ∂x cos R ∂x
(6)
R represents the local slope of the liquid meniscus (see Figure 2). If we assume that the lubrication approximation12 is valid, we may write
Jp )
hd3 ∂P 3η ∂x
(
)
(7)
where η is liquid viscosity. (This expression may, however, be somewhat open to doubt for large contact angles.) Using eqs 1, 3, 4, 6, and 7, we obtain
[
]
3 ∂ hd ∂ (P - P) + Kf(1 + hd′2)1/2(PA - P) ) 0 ∂x 3η ∂x A
(8)
Equation 8 is our key expression, but for it to be exploited we must use a perturbation approach. Perturbation Solution. We start by integrating eq 8 once to give
∫xx
3ηKf ∂ (P - PA) + ∂x h3 d
(1 + hd′2)1/2(PA - P) dx ≈
max
3ηUT hd2
(9)
where xmax is an upper cutoff value for x at which the condensation effect may be considered negligible and UT is the total speed of TL motion with both hydrodynamic (U) and condensation contributions. In a first-order perturbation approach, we adopt hd ) h for the meniscus profile and realize that the integral of eq 9 is then zero, to this approximation. We can then write
P - PA ≈ 3ηU
+ constant ∫dx h2
dh + constant ∫h′h 2
∫xx
(11)
Jc(1 + h′2)1/2 dx ) |Jp| +
max
min
∫hh
max
min
Jc
Π(h) )
A1 ; h
h1 e h e h2
(13)
Π(h) )
A2 ; h2
h ˜ 2 e h e h3
(14)
Regime 2:
To estimate overall spreading speed, UT, we note that
|Jp| +
Regime 1:
(10)
and thus we obtain a first-order approximation for the condensation current by using eqs 1, 3, and 4
Jc ) K∆p ≈ -3ηKfU
the negative disjoining pressure leads to film thinning and in the region of most interest, near the TL, the slope of the meniscus is small, allowing several simplifications to be made in the calculation of spreading/condensation behavior. In treatment of the case of water, several complications arise. The first is that, whatever the actual functional form of Π(h), it is to be expected, at least under most circumstances, that Π(h) will be positive, tending to thicken the film and leading to a convex meniscus with respect to the vapor phase. Following the arguments of White,34,40 independently of the macroscopic contact angle, if Π(h) is negative, a microscopic contact angle of zero may be expected (just before the final molecular region, defining the extent of the liquid phase). Conversely, if Π(h) is positive, a microscopic contact angle of π/2 is appropriate.25,41 It is for these reasons that we have retained more exact formulas in the previous section, in which terms of the form (1 + h′2)n/2, with n ) 1 or 3, have not been simplified to unity. A second, major problem when considering the behavior of water is that there is still, to the author’s knowledge at least, significant doubt as to the thickness dependence of long-range interactions, or what amounts to disjoining pressure isotherms. Exponential dependence on thickness of the disjoining pressure at asymptotic distances was predicted long ago,12,29,31,42,43 based on electric double-layer concepts. At smaller thicknesses, it may be expected that Π(h) ∼ h-2, as given by the Langmuir equation.29,44 Finally, Pashley,45 studying water on glass or silica, suggested a scaling dependence of the form Π(h) ∼ h-1 for thin water layers (up to ca. 40 nm, or more). There is clearly still some doubt as to the behavior of water in thin films on solid surfaces (the type of solid may also play a nonnegligible role), but for the following, we shall adopt the suggestions of Pashley, as used by Teletzke et al.,39 in a different calculation:
(1 + h′2)1/2 dh ≈ UThmin (12) h′
where xmin, xmax, hmin, and hmax are suitable lower and upper cutoff limits for condensation-assisted spreading. The next problem is to obtain suitable functional forms concerning long-range interactions, to exploit the above equations. Form of Disjoining Pressure In the earlier work,25 we assumed that the liquid was apolar. Apolar interactions are notably simpler to treat, as discussed in some detail by Sharma,30 for example. We adopted a typical van der Waals dispersive form for the disjoining pressure, viz., Π(h) ) -A/(6πh3), in which A is the Hamaker constant. With A positive, as defined above,
Despite the apparent existence of a poorly understood ˜ 2,39 we shall assume continuity, such region, h2 e h e h ˜ 2. The two regimes are shown schematically that h2 ≡ h in Figure 3. Typical values of A1, A2, h1, h2, and h3 will be discussed below. (Consideration of the fundamental ingredients contained in A1 and A2 constitutes a totally different topic and is outside the scope of the present work.) Calculation of Condensation Current and Spreading Rate Estimation of condensation current, Jc, is possible if the integral of eq 11 (or eq 10) may be evaluated. It is convenient at this stage to introduce the variables R and β, where R represents the angle of the liquid meniscus (40) Solomentsev, Y.; White, L. R. J. Colloid Interface Sci. 1999, 218, 122. (41) Brochard-Wyart, F.; di Meglio, J. M.; Que´re´, D.; de Gennes, P. G. Langmuir 1991, 7, 335. (42) Langmuir, I. J. Chem. Phys. 1938, 6, 873. (43) Frumkin, A. J. Phys. Chem. USSR 1938, 12, 337. (44) Langmuir, I. Science 1938, 88, 430. (45) Pashley, R. M. J. Colloid Interface Sci. 1980, 78, 246.
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where φ ) A2/(γh3) and L ) ln[(B - β2)/φ]. In the final expression of eq 18, it is assumed that φ is considerably smaller than B. Finally, use of eq 18 in eq 12 leads to
{
UT2 ≈ U 1 +
Figure 3. Two assumed regions of disjoining pressure, Π, dependence on film thickness, for water, and illustration of various heights (film thicknesses) and angles used in the text.
A1 γh′′ )0 + 2 3/2 h (1 + h′ ) h 1 e h e h2
(15) cos R )
h 2 e h e h3 which, upon a first integration, yields
(
)
A2 1 1 1 ) cos θo 2 1/2 γ h h3 (1 + h′ )
(16)
where the integration constant has been evaluated by assuming that R is equal to the macroscopic contact angle, θo, at the upper limit of regime 2, h ) h3. Here, the disjoining pressure is no longer significant. (The assumption of an abrupt cutoff is, of course, a physical absurdity but constitutes a reasonable mathematical simplification.) Using R as the independent variable and thus transforming the integral of eq 11 by employing relation 16 and noting that h′ ) tan R, we obtain for the regime 2 condensation current
3ηKfγU 3ηKfγU 2 (sin R - sin θo) ≈ (B - β2) A2 2A2 (17)
The integration constant has been inferred assuming zero condensation current in the macroscopic part of the drop, h > h3, where meniscus curvature is low and the contact angle is θo. B is the complementary angle to θo, and both here and in the following it is assumed that R is everywhere sufficiently large for the approximations sin β ≈ β and cos β ≈ 1 - β2/2 to be valid. We assume a range of, say, π/4 e θo , π/2. We may now employ eq 12 in order to estimate the overall spreading rate, UT2, noting that, in regime 2, hmin and hmax are represented by h2 and h3, with corresponding meniscus slope angles of R2 and R3 (≡θo) and complements β2 and β3 (≡B). The Poiseuille current, |Jp|, becomes Uh2. The integral of eq 12 may be transformed, with the above approximations of β small, to give
(1 + h′2)1/2 dh 3ηKfU ≈ J c2 h2 h′ 2
∫
h3
2 B (B
- β2) dβ
∫β (B - β + φ)2 ≈ 2
(
(20)
A first integration yields
A2 γh′′ + 2)0 2 3/2 (1 + h′ ) h
Jc2 ≈
(19)
Equation 19 thus estimates the overall spreading rate in regime 2, UT2, taking into account both intrinsic, hydrodynamic, and condensation contributions. Regime 1. Let us now consider regime 1. Combination of expressions 1 and 13 leads to
with respect to the solid surface and β is its complement (π/2 - R) (see Figure 3). We shall commence with regime 2, for a reason that will become clear below. Regime 2. Insertion of expression 14 into eq 1 leads to:
cos R )
}
3ηKfB(L + β2/(2B) - 3/2) h2
3ηKfBU L +
)
β2 3 (18) 2B 2
A1 1 ln h + constant ) 2 1/2 γ (1 + h′ )
(21)
The contact angle, defined from a continuum point of view, neglecting the very first, thin layer of liquid molecules in direct contact with the solid surface, will be π/2 (positive disjoining pressure).25,34,40,41 We may thus evaluate the integration constant to obtain
()
A1 h 1 ln ) 2 1/2 γ h1 (1 + h′ )
(22)
h 1 e h e h2 h1 representing the thickness of the thin contact layer. Before proceeding, let us briefly consider eq 22, which demonstrates a point of interest. The left-hand member corresponds to cos R and is thus constrained to the range 0 e cos R e 1 (angles >π/2 are not treated here). Thus, eq 22 implies that
(
h2 ) h1 exp
)
()
γ cos R2 γ e h1 exp A1 A1
(23)
where R2 is the value of R at h2. This effectively limits the range of validity of expression 13. Depending on the ratio γ/A1, the range of h1 to h2 may be significantly reduced. As for regime 2, explicit evaluation of h, from eq 22, is unnecessary. Using β, the complement of R, as independent variable and combining eqs 11 and 12 leads to the condensation current in regime 1, Jc1
Jc1 ≈
-3ηKfγU h1A1
[
]
β ∫ sin β exp -γAsin 1
dβ + constant (24)
Using the small angle approximation, sin β ≈ β, the integral of eq 24 is readily evaluated. Denoting by ζ the pressure difference, PA - P, at h ) h2, we obtain
Spreading of Water
Jc1 ≈
{(
Langmuir, Vol. 17, No. 26, 2001 8233
) ( )
A1 3ηKfU -γβ β+ exp h1 γ A1 A1 -γβ2 β2 + exp + ζKf ≈ γ A1 3ηKfγU 2 (β - β2) + ζKf (25) 2h1A1 2
(
) ( )}
to leading order of β. The reason for considering regime 2 prior to regime 1 now becomes clear. We require to evaluate ζKf, which is effectively the condensation current at h ) h2. With the assumption of continuity between regimes 1 and 2 (stated after eqs 13 and 14), and to be discussed later, we can evaluate ζKf from eq 17
ζKf ≈
3ηKfγU 2 (B - β22) 2A2
(26)
Combination of eqs 25 and 26 leads to
Jc1 ≈
[
]
2 2 2 2 3ηKfγU (β2 - β ) (B - β2 ) + 2 h1A1 A2
(27)
We may now return to eq 12 in order to estimate the overall spreading rate in regime 1, UT1, containing both intrinsic, hydrodynamic, and condensation contributions. Although eq 27 is of the same mathematical form as eq 17, in the limit of small angles β, the integral of eq 12 is somewhat different because of its dependence on the relationship between h and R (or β), as demonstrated by eq 22. With integration limits of β1 ()0) and β2, to leading order in β we obtain
(1 + h′2)1/2 dh ≈ h′ 3(B2 - β22) ηKfγ2β2h1U β22 + (28) A1 h1A1 2A2
∫hh Jc1 2
1
[
]
Since |Jp| ≈ Uh1, we finally obtain for the overall spreading rate in regime 1
{
UT1 ≈ U 1 +
[
]}
ηKfγ2β2 β22 3(B2 - β22) + A1 h1A1 2A2
(29)
Solution Matching We assumed at the outset that the water is spreading under steady-state conditions. This implies that UT2 and UT1, from eqs 19 and 29, respectively, must be equal. Put mathematically, we must have
ener,48 as presented by Pashley,45 for water on hydrophobic silica and quartz, the value of A1 may be expected to be of the order of 10-2 N m-1. On a hydrophilic surface such as glass, a value of ca. 0.3 N m-1 would seem appropriate.45,49 Since we treat relatively large values of macroscopic contact angle, θo, in this study, the former value would seem more suitable. Taking β2 ≈ 0.2 and γ ) 73 mN m-1 (at ambient temperature), eq 23 sets an upper ratio of h2/h1 for regime 1 of ca. 4. With an upper limit for regime 1 of h2 ≈ 40 nm, this gives h1 ≈ 10 nm, which is plausible.45 However, too much importance should not be attached to this last finding: although it is an inevitable consequence of eq 23, the isotherm given by eq 13 would seem to be an empirical finding with, to our knowledge, no deep theoretical understanding, as yet.12 Notwithstanding, we may adopt, in addition, plausible values for the other parameters of eq 30, viz., A2 ≈ 3 × 10-11 N,33 h3 ≈ 150 nm,45 and taking β ≈ 0.4, admittedly rather arbitrarily, since this will very much depend on the properties of the solid (via Young’s equation), we obtain values of ca. 7 × 108 m-1 for the left-hand member of expression 30 and ca. 108 m-1 for the right-hand member. Clearly these are significantly different, but of the same order of magnitude, corroborating the compatibility of eqs 19 and 29. (We are not prepared to “force” an equality, in the light of the poor knowledge concerning certain quantities!) Nevertheless a point should be remembered: we have assumed an abrupt transition from regime 1 to regime 2 at film thickness h2 (i.e., h2 ≡ h ˜ 2). Notwithstanding, this is physically unrealistic and, as hinted at above, there is a vague region between ca. 50 nm and ca. 100 nm where, presumably, some sort of gradual transition occurs between different disjoining pressure film thickness dependences.39 From this we infer that the angle B in the left-hand member of eq 30 is not, strictly speaking, the same B as that in the right-hand member (the former would be less than the latter, tending to improve matching). Spreading and Dissipation We shall consider dissipation phenomena occurring near the TL during spreading, as in the manner of the earlier article.25 Since the mathematics are rather less tractable in the case of disjoining isotherms, as defined by relations 13 and 14, than for van der Waals liquids, it is instructive to consider the basic physics initially, using simple scaling concepts (which should be quite general). Scaling Argument. Hydrodynamic, or macroscopic, M, dissipation, in the lubrication approximation,10-12 may be expressed as
) 3ηU2I ∫dx h
TS˙ M ≈ 3ηU2
(31)
In addition, considering eq 23, the ratio h2/h1 is restricted. Unfortunately, to our knowledge at least, data available concerning the various parameters in expression 30 are at best approximate and, anyway, will depend on the solid substrate to some extent. Judging from the results of Derjaguin and Zorin,46 Hall,47 and Pashley and Kitch-
where the integral I will depend on the meniscus profile, h, function of distance from the TL, x. The exact form of I is immaterial for present purposes but, at least very close to the TL, it clearly depends on the nature of longrange forces in play. The integration domain is usually taken to extend from a microscopic cutoff near the TL to some macroscopic distance, such as drop contact radius. Admittedly, the validity of the lubrication approximation is open to doubt for large values of contact angle, θo, but we shall take it to hold for our purposes (the essential, scaling argument will remain valid).
(46) Derjaguin, B. V.; Zorin, Z. M. Proceedings, 2nd International Congress of Surface Activity; 1957; Vol. 2, p 145. (47) Hall, A. C. J. Phys. Chem. 1970, 74, 2742.
(48) Pashley, R. M.; Kitchener, J. A. J. Colloid Interface Sci. 1979, 71, 491. (49) Garbatski, U.; Folman, M. J. Phys. Chem. 1956, 60, 793.
[ (
)
]
3B[L + β2/(2B) - 3/2] γ2β2 2 1 3 3B2 β2 + ) A1 A1h1 2A2 2A2 h2 (30)
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In the context of spreading being assisted by a condensation current, Jc, we also have a microscopic dissipation term
TS˙ m ≈
∫
Jc ∆p
(1 + h′2)1/2 dh ) h′ K
∫∆p2
(1 + h′2)1/2 dh (32) h′
(cf. eq 11). Since ∆p ∼ U, to a first-order approximation, we have
TS˙ m ∼ U2
TS˙ M ∼ Uγ(cos θo - cos θ)
(34)
where θ is the dynamic contact angle. Separating U by using relations 31 and 34 and combining with the scaling relation 33, we obtain
TS˙ m ∼ (cos θo - cos θ)2
U ∼ γ(cos θo - cos θ)
(36)
which is the scaling law expected for spreading, at low speeds, from the molecular kinetics approach, in which the following, more precise, relation is often employed
(37)
K ˜ represents a molecular oscillation frequency, and λ is a “hopping” distance for molecules between adsorption sites. The crux of the above simple argument is that the effective pressure, causing condensation, scales with hydrodynamic spreading rate, to a first approximation. This should be independent of the nature of the longrange forces involved, provided a disjoining pressure results. Specific Case of Water. Having established the basic physics of condensation transport, we shall present (approximate) expressions for the case of water. Two forms of the condensation current, viz., expressions 17 and 27 have been derived. We therefore require two dissipation terms, corresponding to expression 32. For regime 1, we may write, for the condensation contribution to dissipation
∫0
2
[ ]
β2
2
(β2 - β ) h1A1
[
(B2 - β22) + A2
2
2
4A2
A dβ
∫βB(B2 - β2)2 Kγ (B +2 φ - β)2 2
3η2f 2γKU2 [8B3 - (B + β2)3] 4A2
TS˙ m ≈
[
()
4β22(B2 - β22) 9η2f 2γ3h1Kβ2U2 8β24 + + 4A1 3h1A1A2 15h12A12
]
(B2 - β22)2 A22
(38)
]
2 9η2f 2γKB3U2 h1γ β2B 7 + 4A2 A1A2 3
(40)
Exploiting the scaling argument presented above, we may write finally for the condensation contribution to dissipation during spreading
TS˙ m ≈
[
]
f 2γ3KB3(cos θo - cos θ)2 h1γ2β2B 7 + A1A2 3 4A I2
(41)
where we recall that B ) (π/2 - θo) and K ) v/(2πmkT)1/2. Discussion One of the basic findings of this study is that, by changing from the “simpler”, van der Waals long-range forces governing disjoining pressure25 to two regimes, apparently reasonably representative of water, the basic physics of condensation assisted spreading remains similar. We may write an approximate form for the total dissipation during spreading, TS˙ , dynamically balancing the power supplied by the capillary motor, viz., Uγ(cos θo - cos θ). Using the “wedge” approximation for the liquid profile in the macroscopic domain,12 the integral, I, of eq 31 becomes l/θ, where l is the logarithm of the ratio of a macroscopic distance (say, drop contact radius) to a microscopic cutoff. We may thus write
3ηU2l + θ f 2γ3KB3(cos θo - cos θ)2θ2 h1γ2β2B 7 + (42) A1A2 3 4A l2
TS˙ ) TS˙ M + TS˙ m ≈
2
h1γ γβ exp dβ ≈ KA1 A1
(39)
Thus, the total microscopic dissipation term (per unit length of TL and per second) is given by TS˙ m ) TS˙ m1 + TS˙ m2 (assuming, as before, that the blurred transition from regime 1 to regime 2 may be acceptably approximated by a sharp transition occurring at film thickness h2). Simplifying eqs 38 and 39 to take into account only the dominant contributions, assuming β2 small, we obtain
2
2K ˜ λ3 γ(cos θo - cos θ) kT
(3ηKfγU)2 4
≈
(3ηKfγU)2
(35)
Since the wetting “motor”, for natural wetting, may be expressed as γ(cos θo - cos θ), from relation 35 it may be construed that
TS˙ m1 ≈
TS˙ m2 ≈
(33)
Now, the hydrodynamic energy balance during spreading may be written as12
U≈
where use has been made of eqs 22 and 27, and the usual small angle approximations, and evaluation has been truncated at the leading value of β. (The last term in square brackets, of the last expression, may be expected to dominate.) The detailed mathematics of regime 2 are also cumbersome, but using similar levels of approximation and eqs 16 and 17, we ultimately obtain (for φ small)
[
]
Two distinct, dissipation mechanisms are thus active. Although θ is a dynamic contact angle, we shall consider it to change only relatively slightly during (slow) spreading near equilibrium. Thus the difference of cosines is a more sensitive variable. For small values of θ, we may expect the hydrodynamic term, TS˙ M, to be dominant, with its dependence on the inverse of contact angle. However, for large contact angles, the condensation term may be more
Spreading of Water
Langmuir, Vol. 17, No. 26, 2001 8235
significant. This is often observed in practice.15,50,51 Indeed, during spreading, one may expect a transition from the TS˙ m term being important, to hydrodynamically dominated spreading, as the contact angle decreases.16 There is nevertheless interplay between the two modes since, as pointed out by Shikhmurzaev,27 there is mutual interdependence of the dynamic contact angle and flow field. In our case, without local distortion of the meniscus near the TL, due to hydrodynamic flow, there will be no condensation. A note of precaution is however justified at this point. Not all of the parameters in eq 42 (and 43) are completely independent. With further understanding of disjoining pressure isotherms for water, some simplification may be possible. Let us consider the compatibility of the theory portrayed above with the molecular kinetics interpretation. The two relations to be compared are eq 37, corresponding to a commonly used version of the molecular kinetics theory,3 and the condensation transport counterpart, viz.
[
]
2 f 2γKB3θ2 h1γ β2B 7 U≈ + γ(cos θo - cos θ) A1A2 3 4A2l2
(43)
It is the coefficient of γ(cos θo - cos θ) in eq 43 which is to be compared with 2K ˜ λ3/(kT) of eq 37. Whereas in the study of van der Waals liquids25 the equivalent coefficient was independent of contact angles, in the present case, there is dependence on B, θ (≈π/2 - B), and β2, and these will depend on the substrate. Notwithstanding, we shall adopt the values discussed above, viz., B ) 0.4 and β2 ) 0.2, to give some idea of orders of magnitude, at least. Also, we adopt the previously mentioned values of γ ≈ 73 mN m-1, A1 ≈ 10-2 N m-1, A2 ≈ 3 × 10-11 N, and h1 ≈ 10 nm; together with m ≈ 3 × 10-26 kg and v ≈ 3 × 10-29 m3, to calculate K (≈10-6 m3 N-1 s-1). Taking f ≈ 2 × 10-2 (for water in air at T ≈ 293 K, the assumed temperature) and l ≈ 13, we obtain a value of the prefactor in eq 43 of ca. 2 × 10-3 (Pa‚s)-1. It is interesting to note that the value is similar to that obtained for the van der Waals disjoining pressure isotherm in our earlier article.25 This compares favorably with the range of values of 2K ˜ λ3/(kT) indicated for aqueous systems by Petrov and Petrov,52 corresponding to 2.4 × 10-3 to 1.7 (Pa‚s)-1, most values being toward the lower end. Similarly the results of Gribanova53 suggest values of order 10-2(Pa‚s)-1, not too far removed from our estimation. We can, at present, do little more than show the compatibility of the present theory with reported experimental results: a complete study would be of interest. Such factors as wetting hysteresis9 and wetting geometry are not taken into account, although they may have considerable impact. The theory developed above will clearly only be of use under restricted conditions, due to the simplifying math(50) Cazabat, A. M.; Cohen Stuart, M. A. J. Phys. Chem. 1986, 90, 5845. (51) Shanahan, M. E. R.; Houzelle, M. C.; Carre´, A. Langmuir 1998, 14, 528. (52) Petrov, P. G.; Petrov, J. G. Langmuir 1995, 11, 3261.
ematical assumptions made in the development. Condensation effects are treated as perturbations, rather than primary phenomena. To overcome this shortcoming could well pose non-negligible mathematical hurdles, but let us nevertheless briefly consider one aspect, with no pretence at rigor. Take eq 29 describing overall spreading rate with both hydrodynamic and condensation contributions. (We could also use eq 19.) With the viscosity of water, η, of ca. 10-3 Pa‚s and other values as above, we find, in order of magnitude, that UT1 ∼ 1.01U, indeed corresponding to a small perturbation due to condensation. However, were the local atmosphere to be composed of saturated steam, we would have f f 1: this would result in UT1 ∼ 2U! The present development is not capable of handling such a case rigorously, because of its mathematical limitations, but the basic physics could point at least to a partial understanding of the recent results of Chaudhury’s group,54 in which exceedingly rapid drop motion resulted during condensation from saturated steam onto a substrate with a surface free energy gradient. Drop motion in the absence of such an atmosphere (ambient conditions) was slower by ca. 2 orders of magnitude. We only double the speed by saturation in this very approximate approach, but a full analysis may reveal interesting results! Conclusions We have extended earlier work25 concerning potential condensation transport during the spreading of liquids. In the region near the triple line, TL, at equilibrium, the meniscus is stationary due to a combination of balancing Laplace and disjoining pressures. Under TL motion, modified meniscus curvature leads to an unequilibrated (modified) Kelvin equation in the vicinity, causing condensation from the atmospheric vapor, adding to the already existent TL motion. The earlier study considered disjoining pressure due to van der Waals interactions. The local film was considered to be virtually flat and the macroscopic contact angle, θo, small. In this article, we remove these restrictions and attempt to employ the resulting equations to explain potential condensation transport in the case of water, when θo f π/2. A review of existing literature suggests plausible disjoining pressure dependence in two domains: in (film thickness)-1 near the solid surface and in (film thickness)-2 further removed. With these assumed disjoining pressure isotherms, we calculate condensation currents and spreading effects. Consideration of dissipation effects during spreading leads to the conclusion that even in a more general case than that in which the disjoining pressure is govered uniquely by long-range van der Waals forces, local condensation may contribute to wetting behavior, the essential phenomenon being controlled by the unbalanced Kelvin equation during TL motion. The findings are complementary to the molecular kinetics approach. LA011065Y (53) Gribanova, E. V. Adv. Colloid Interface Sci. 1992, 39, 235. (54) Daniel, S.; Chaudhury, M. K.; Chen, J. C. Science 2001, 291, 633.
