Langmuir 2001, 17, 3997-4002
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Condensation Transport in Dynamic Wetting 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 December 6, 2000. In Final Form: April 17, 2001 Wetting and dewetting phenomena involve as yet incompletely understood kinetic behavior occurring near the solid/liquid/fluid triple line. Two major classes of theory exist to explain local motion: the molecular kinetics approach and the hydrodynamic model. We suggest another possibility here which is complementary to the former. Increased meniscus curvature near the triple line of an advancing wetting front leads to an effective slight supersaturation of the liquid vapor, and local condensation ensues. This mass transfer can be in addition to other effects, including hydrodynamic dissipation. Depending on wetting conditions, either effect may dominate, in general agreement with observations reported in the literature.
Introduction The motion of a solid/liquid/fluid triple line (TL) during the spreading of a drop of liquid is an everyday observation yet still presents barriers to total understanding, particularly of the very local phenomena occurring in the proximity of the three phase “line”. (The mathematical simplification of a line must break down at a molecular scale.) Two main classes of model have been proposed to explain the transfer mechanisms involved and their rate dependence on local conditions. The first is based on molecular kinetics theory, as initially proposed by Eyring et al.1 Following Cherry and Holmes’ suggestion of polymer spreading being controlled by activated viscous flow,2 Blake and Haynes3 and later Ruckenstein and Dunn4 proposed rate theories to explain TL motion, essentially by molecular “jumping”. These theories have attracted attention and been developed, particularly in more recent years.5-9 Although initially proposed as a theory to explain the transfer of individual molecules from the liquid to available sites ahead of the advancing TL, following an activated process biased by local wetting disequilibrium (unbalanced Young equation), it was pointed out that various other mechanisms could also be relevant,3 provided the masses of matter and energy involved are sufficiently small for Maxwell-Boltzmann statistics to apply.6 One such mechanism, that of adsorption, has similarities with condensation phenomena, the subject of this article. The second class of theories depends on hydrodynamic considerations. As instigated by Huh and Scriven,10 viscous dissipation during TL motion was considered to be the moderating effect dynamically balancing capillary disequilibrium. Later developments were made,11-13 but the (1) Glasstone, S.; Laidler, K. J.; Eyring, H.; The Theory of Rate Processes; McGraw Hill: New York, 1941. (2) Cherry, B. W.; Holmes, C. M. J. Colloid Interface Sci. 1969, 29, 174. (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? AIChE International Symposium on the Mechanics of Thin Film Coating, New Orleans, 1988; Paper 1a. (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.
general problem remained(s): when approaching the triple phase line, unless some slippage14 between liquid and solid is permitted, the shear stresses diverge to infinity! Notwithstanding, this class of model has shown considerable success in explaining observed wetting in various conditions, particularly at low values of contact angle, θ.15,16 Some attempts at compromise have been made, combining the two approaches, molecular kinetic and hydrodynamic, in the same model.17,18 More recent studies along these lines, using both theoretical and experimental approaches applied to sessile droplets, suggest the existence of as many as four domains of spreading, of which the middle two can be attributed, in chronological order, to molecular kinetics and to hydrodynamic behavior.19,20 Shikhmurzaev has developed a mathematical model describing motion of an interface between immiscible fluids along a solid surface21-23 and shows that there is a mutual interdependence between the dynamic contact angle and the local flow field. The mathematics of an activated process seems to explain well results obtained at high spreading speeds and large values of contact angle, θ, yet as indicated above, identification of the type of activated process involved seems rather delicate. Although activated mechanisms must surely play a significant role in various wetting phenomena (in particular in forced wetting, as in coating operations), the objective of this study is to suggest another approach, based on condensation. Even if the basic idea of liquid transport by evaporation is old,24 the development presented here is new, to our knowledge, and may explain certain experimental observations. It will be seen to be rather complementary to the molecular kinetics approach. (14) Thompson, P.; Robbins, M. O. Phys. World 1990, November, 35. (15) Cazabat, A. M.; Cohen Stuart, M. A. J. Phys. Chem. 1986, 90, 5845. (16) Shanahan, M. E. R.; Houzelle, M. C.; Carre´, A. Langmuir 1998, 14, 528. (17) Brochard-Wyart, F.; de Gennes, P. G. Adv. Colloid Interface Sci. 1992, 39, 1. (18) Petrov, P. G.; Petrov, J. G. Langmuir 1992, 8, 1762. (19) De Ruijter, M. J.; De Coninck, J.; Oshanin, G. Langmuir 1999, 15, 2209. (20) De Ruijter, M. J.; Charlot, M.; Voue´, M.; De Coninck, J. Langmuir 2000, 16, 2363. (21) Shikhmurzaev, Y. D. Int. J. Multiphase Flow 1993, 19, 589. (22) Shikhmurzaev, Y. D. AIChE J. 1996, 42, 601. (23) Shikhmurzaev, Y. D. J. Fluid Mech. 1997, 334, 211. (24) Hardy, W. Philos. Mag. 1919, 38, 49 (cited in ref 12).
10.1021/la001714q CCC: $20.00 © 2001 American Chemical Society Published on Web 06/02/2001
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δr0 )
nVm 2φr0
(3)
where Vm is the molar volume of the condensed state. This corresponds to a reduction of liquid/vapor interface area of 2φδr0 and a consequent free energy gain pertaining to this interface of -2φγ δr0. Substituting for n in expression 2, from expression 3, and equating this to -2φγ δr0 leads to an energy balance. Reducing the resulting expression to that corresponding to the virtual transfer of one molecule of liquid leads to eq 1. Equation 1 must be modified in the presence of a solid. Consider the presence of a flat, solid surface at a distance h0 from the liquid/vapor interface (Figure 1). For any effects to be significant, h0 must be of the order of nanometers. We assume “simple”, long-range, dispersive interactions of the van der Waals type to exist.12 Thus, a long-range contribution, E(h0), to the overall energy must be taken into account: Figure 1. Schematic representation of a concave liquid/vapor meniscus of radius of curvature r0 near solid/liquid interface (liquid film thickness, h0).
A Modified Form of Kelvin’s Equation We start by considering the classic Kelvin equation25 relating the equilibrium partial pressure of vapor, pv0, of a liquid above a meniscus to the curvature of this meniscus. This equation is often discussed in the context of droplets of liquid (e.g., fog formation), for which the meniscus is convex with respect to the vapor phase, and two (identical) radii of curvature are presented (sphericity). For our development below, we are interested in a concave (with respect to the vapor phase), cylindrical (one effective radius of curvature, the other being infinite) meniscus, for which Kelvin’s equation can readily be shown to take the form
[ ]
pv0 ) p0 exp
-γv r0kT
(1)
where p0 is the equivalent partial pressure of vapor for a planar meniscus, γ is the liquid surface tension (liquid/ vapor interfacial tension), v is the effective volume occupied by a molecule of the liquid in the condensed state, r0 is the radius of curvature of the meniscus, and kT represents the usual product of Boltzmann’s constant and absolute temperature. This equation may be easily derived by consideration of a “virtual condensation” at equilibrium. Consider Figure 1 which represents the concave, cylindrical meniscus of radius of curvature r0 of a liquid film at distance h0 from a solid surface. Taking the vapor to obey the ideal gas equation, the virtual work required to condense (a small number), n, moles of vapor onto the meniscus is
( )
) nRT ln ∫pp p dV ) nRT ∫dV V 0
v0
pv0 p0
(2)
where p and V are variables, pressure and volume, and R is the gas constant. Assuming this condensation to take place evenly over an increment of liquid/vapor interface of radius of curvature r0, (small) angular extent 2φ in the plane (of Figure 1), and unit width perpendicular to the figure, there will be a reduction in radius of curvature, δr0 given by (25) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces, 6th ed.; Wiley: New York, 1997; p 53.
E(h0) )
-γa2 -A ≈ 12πh02 2h02
(4)
where A is an overall Hamaker constant for the given combination solid/liquid/vapor and a is of the order of a molecular diameter. (N.B. A may be positive or negative, but we consider it positive here.) In the above derivation of eq 1, we note an incremental change from r0 to r0 - δr0 and consequentially h0 becomes h0 + δr0 and E, from eq 4, becomes greater (its absolute value gets smaller). It can be seen after simple algebra that in the limit of small δro
E(h0 + δr0) - E(h0) )
A δr0
(5)
6πh03
and thus the overall free energy gain after virtual condensation of n moles of liquid now becomes 2φ δr0[-γ + Ar0/(6πh03)]. Equating this energy gain, modified by the proximity of the solid, to eq 2, using expression 3, and reducing to transport of one molecule leads to the modified Kelvin equation:
[ (
pv0 ) p0 exp
)]
A v -γ + kT r0 6πh03
)
[ (
p0 exp
)]
v -γ - Π(h0) kT r0
(6)
where Π(h0) is the disjoining pressure defined by Π(h0) ) -dE/dh0. Note that under conditions of equilibrium, as will be seen, (-γ/r0 - Π(h0)) ) 0, explaining why no (net) condensation occurs, despite meniscus curvature. Also, we have derived eq 6 on the understanding that the Hamaker constant, A, is positive. Had we assumed a convex meniscus with respect to the vapor phase, we should simply have found an equation compatible with a negative Hamaker constant, viz.,
[ (
pv0 ) p0 exp
)]
v γ |A| kT r0 6πh 3 0
(7)
in which A < 0, and the term in brackets equals zero at equilibrium. Static Profile Near the Triple Line Before proceeding with dynamic wetting, let us briefly consider the static profile adopted by the liquid meniscus
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Figure 2. Schematic representation of microscopic zone near the static triple line of a liquid of negative spreading coefficient on a solid (a) when the Hamaker constant is positive and (b) when it is negative.
near the triple line when long-range forces of the van der Waals type come into play. We assume the spreading coefficient to be negative, to ensure a finite yet small value of macroscopic equilibrium contact angle, θ0. At equilibrium, pv0 ) p0 and thus, in eq 6, [-γ/r0 + A/(6πh03)] ) 0, for A positive. Setting x as the abscissa, parallel to the solid surface, and h0 as the local film thickness (see Figure 2), we note that r0-1 ≈ d2h0/dx2 provided the gradient of the liquid profile is relatively small. Using the fact that A ≈ 6πa2γ (see eq 4), we write
d2h0
)
a2 h03
(8)
Figure 3. The advancing triple line compared to the static case. The local radius of curvature is smaller for the former [r ≈ (d2h/dx2)-1] than for the latter [r0 ≈ (d2h0/dx2)-1].
Equation 8 can be integrated exactly. Using the assumptions that the gradient, dh0/dx, is constant at macroscopic distances (x g a/θ02) and equal to θ0, the equilibrium contact angle, as boundary conditions, we find a hyperbolic profile:
Solomentsev and White.29 This suggests that for a positive Hamaker constant, even with a finite value of macroscopic contact angle, θ0, the mesoscopic value is zero (before we get to the last few liquid molecules, which cannot be considered here). Similarly, for A negative, the mesoscopic contact angle is expected to be π/2.
dx2
h02 ) θ02x2 +
a2 θ02
(9)
If we consider a negative Hamaker constant, eq 7 leads to
d2h0 dx2
)-
a2 h03
(10)
and consequently
h02 ) θ02x2 -
a2 θ02
(11)
again hyperbolic. The profiles corresponding to eqs 9 and 11 are shown schematically in Figure 2. Expressions 9 and 11 were previously derived by Brochard-Wyart et al. using a rather different approach26 (apart from slight confusion concerning the definition of sign of a Hamaker constant; we have used Adamson’s definition27). Thus, the modified Kelvin equation (eq 6 or eq 7) can be employed to obtain the equilibrium profile near the triple line. This, of course, is on a mesoscopic scale; the molecular tip cannot be treated using such continuum methods.26,28 We note from eq 9 that dh0/dx (x ) 0) is zero, in agreement with (26) Brochard-Wyart, F.; di Meglio, J. M.; Que´re´, D.; de Gennes, P. G. Langmuir 1991, 7, 335. (27) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces, 6th ed.; Wiley: New York, 1997; p 232. (28) de Gennes, P. G.; Hua, X.; Levinson, P. J. Fluid Mech. 1990, 212, 55.
Dynamic Considerations and Liquid Transfer In the following, we shall develop uniquely the case of the Hamaker constant, A, being positive as defined in eq 4. Similar considerations, with appropriate sign changes, apply equally well to negative A but will not be treated here, for brevity. Conditions of flow will be taken to be steady state. Low spreading speeds, U, and small contact angles are assumed. We must initially consider the profile of the meniscus near the TL under dynamic conditions. This is not an easy task, and so some plausible approximations will be employed. Using the arguments of de Gennes,12 we first consider the horizontal Poiseuille current, Jp, parallel to the solid surface and toward the left in Figure 3. In the lubrication approximation, this is given by12
Jp )
h3 ∂P 3η ∂x
(
)
(12)
where h ) h(x) is the dynamic film thickness, η is liquid viscosity, and P is the pressure within the liquid. Abscissa x is counted in the reference frame of the moving TL. We henceforth assume the existence of a condensation current from the surrounding vapor, taken to be saturated at a partial vapor pressure corresponding to a flat liquid meniscus, p0. This is reasonable because in the far field, the curvature of the drop surface will be weak; it is only locally, near the TL, that such curvature effects need to be treated in detail. With an (anticipated) excess local (29) Solomentsev, Y.; White, L. R. J. Colloid Interface Sci. 1999, 218, 122.
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curvature (see Figure 3), eq 6 predicts condensation. For reasons of continuity, the condensation current, Jc (volume per unit surface and per unit time), and the Poiseuille current are related by
∂Jp + Jc ) 0 ∂x
(13)
Taking r to be the local dynamic radius of curvature of the meniscus, we may calculate, from eq 6, the appropriate equilibrium value of the partial pressure of vapor, p˜ 0, were the drop profile to be static with this geometry. The excess vapor pressure, ∆p, leading to condensation may then be expressed as
{
∆p ) p0 - p˜ 0 ) p0 1 - exp
[ (
)]}
v -γ A ≈ + kT r 6πh3 d2h A f γ 2(14) dx 6πh3
[
]
where f is the molar fraction of the vapor of the liquid contained in the surrounding atmosphere (f ) p0/PA where PA is the total atmospheric pressure surrounding the drop). Thus, the pressure in the liquid, P, can be expressed as
d2h P ) PA - γ 2 - Π(h) ≈ PΑ - ∆p/f dx
(15)
[
]
(16)
where K ) v/(2πmkT)1/2, m being the molecular mass of the liquid,30 and thus, from eqs 12, 13, 15, and 16, we obtain
( )
(17)
Realizing that the current in the frame of reference of the solid is -UTh, where UT is the overall rate of advance, we obtain
∫Xx(P - PA) dx ≈
∂P 3Kηf + 3 ∂x h
3ηUT h2
(21)
allowing us to rewrite eq 18 as
z′′′ +
[
∫1y
3Kηf 3 z′ ≈ z4 aθ0z3
-z′′ +
]
1 dy - 2 z3 z
(18)
where X is an upper cutoff corresponding to the transition between the microscopic and macroscopic zones, the liquid surface becoming virtually flat (X ≈ a/θ02). As did de Gennes et al.28 and Joanny31 in somewhat related problems, we proceed using a perturbation approach. It is useful to introduce dimensionless variables,
y)
θ02x a
(19)
z)
θ0h a
(20)
and a perturbation parameter, (30) Hudson, J. B. Surface Science; Wiley: New York, 1998; p 297. (31) Joanny, J. F. Le Mouillage. Ph.D. Thesis, Universite´ de Paris 6, 1985.
(22)
where z′ ) dz/dy, and so forth. In eq 22, we have made use of the fact that
P - PA )
(
)
γθ03 1 -z′′ + 3 a z
(23)
In dimensionless variables, the static solution (eq 9) is
z02 ) y2 + 1
(24)
which, when inserted into eq 23, shows that P ) PA, as expected, and thus no condensation occurs when the TL is at rest [cf. eq 15]. Thus, we must “perturb” the static profile with speed U, or more specifically parameter , to obtain any condensation. We wish to calculate the pressure difference, P - PA, to first order in . For this, we may use eqs 22 and 23, together with the static profile given by eq 24:
(
fv d2h A γ 1/2 2 (2πmkT) dx 6πh3
∂ 3∂P h + 3Kηf(P - PA) ) 0 ∂x ∂x
3ηU 3ηUT ≈ γθ03 γθ03
)
γθ03 γθ03 3 d (P - PA) ) -z′′′ - 4z′ ≈ dy a z az02
where Π(h) is the local disjoining pressure. The rate of condensation is given by
Jc(x) ) K∆p ≈
)
(25)
leading to
γθ03 γθ03 -1 tan y - const ≈ (y - 1) (26) (P - PA) ) a a Equation 26 gives us the necessary pressure difference, to first order, to be used to calculate the condensation current, Jc. Using eqs 15, 16, and 26, we obtain
Jc ≈
3ηfa2U(1 - y)
(27)
(2πmkT)1/2
where we have taken v ) a3. Spreading and Dissipation Let us consider motion of the TL, where the basic speed term, U, as used in our perturbation calculation above, is associated with hydrodynamic dissipation, TS˙ M, and M refers to macroscopic (viscous) effects. In the hydrodynamic picture,12,28 we have
TS˙ M ≈
3ηU2 θ0
∫0y
max
dy 3ηU2I ) z θ0
(28)
Using the static approximation, z ≈ z0 (eq 24), we find that
(
I ≈ ln(2ymax) ) ln
)
2θ02xmax a
(29)
where ymax and xmax refer to some large macroscopic limit (e.g., the contact radius for a small sessile drop). We thus obtain a natural cutoff for the logarithmic flow singularity, as was done previously28 (although, in the present case,
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the sign of the Hamaker constant has been reversed; we have not allowed for dissipation in the molecular tip). The microscopic dissipation term, TS˙ m, concerning local condensation, essentially in close proximity to the TL, may be written
TS˙ m ≈
∫0XJc ∆p dx ) K1 ∫0XJc2 dx
(30)
which, on using eq 27, returning to dimensional abscissa, x, leads to
TS˙ m ≈
3η2f 2a2U2 (2πmkT)1/2θ02
(31)
Equation 31 thus gives the condensation contribution to dissipation, but it is possibly more familiar in a different form. The hydrodynamic energy balance during spreading may be written as12
TS˙ M ≈ Uγ(cos θ0 - cos θ)
(32)
where θ is the dynamic contact angle (θ > θ0 for spreading). Thus, isolating U from eqs 28 and 32 and substituting in eq 31, we find
TS˙ m ≈
f 2a2γ2(cos θ0- cos θ)2 3(2πmkT)1/2I2
(33)
and finally, the total dissipation during spreading is given by
TS˙ ) TS˙ M + TS˙ m ≈ 2 2 2 2 3ηU2I f a γ (cos θ0 - cos θ) + (34) θ0 3(2πmkT)1/2I2
which is our main result. The actual spreading speed will now be somewhat higher than U, because of the condensation perturbation. We note that
|Jp| +
∫0XJc dx ≈
UTa θ0
(35)
where UT is total spreading speed, evaluated at x ) 0, containing both hydrodynamic and condensation contributions. By use of eqs 19 and 27, it is readily shown that
{
UT ≈ U 1 +
}
3ηfa2 2θ0(2πmkT)1/2
(36)
where the first term on the right-hand side is the hydrodynamic part and the second term is the condensation contribution to overall spreading speed. Discussion Putting aside hydrodynamic considerations for the moment and considering the wetting “motor” to be γ(cos θ0 - cos θ), corresponding to natural wetting (cf. eq 32), eq 33 suggests that the mass transfer process by condensation near the TL will lead to a scaling law of the form
U ∼ γ (cos θ0 - cos θ)
U≈
2K ˜ λ3 γ(cos θ0 - cos θ) kT
(38)
where K ˜ corresponds to a molecular oscillation frequency near the TL and λ is an effective “hopping” distance for molecules (between adsorption sites). We thus arrive at a complementary theory of mass transfer which may be involved in spreading, at least in certain cases but certainly not in all. (Blake and Haynes3 studied the system benzene/ water; with two liquids rather than a liquid and a gas, it would seem very difficult, if not impossible, to apply the theory developed here.) Considering eq 34, we see that two possible dissipation mechanisms are present. The condensation mechanism may be dominant under conditions of large dynamic contact angle, and conversely, for small dynamic contact angles one may expect hydrodynamic effects to be overriding. As indicated in the Introduction, this often seems to be the case in practice.15-17 De Ruijter et al.19 have discussed the presence of four regimes in droplet spreading, of which the middle two are of direct relevance in the present context. After a rapid initial stage of linear spreading with time, a period follows in which the molecular kinetics theory satisfactorily explains behavior. Subsequent behavior fits in better with the hydrodynamic explanation, and finally, there is exponential relaxation to equilibrium. The switch from molecular kinetics to hydrodynamics accompanies decreasing contact angle and is thus in agreement with eq 34. Also, if the microscopic contact angle is close to zero, as discussed in the above section on static profiles, then the higher values of macroscopic, dynamic angle in the initial stages of wetting favor larger departure from equilibrium curvature, thus exacerbating condensation effects. The development leading to eq 34, although based on a perturbation approach, shows a fundamental physical feature of the condensation-aided spreading. Without motion imposed on the TL leading to a nonequilibrium shape, more specifically, excess curvature, condensation cannot occur. This is in agreement with the conclusions of Shikhmurzaev.21-23 Using fluid mechanical arguments, he arrived at the important conclusion that there is mutual interdependence of the dynamic contact angle and the flow field, a feature lacking in earlier models. This argument is clearly substantiated by the development presented here; TL motion can lead to an added condensation component, and inversely, condensation will modify the liquid profile and therefore the flow field. It is generally accepted that silicone oils follow the hydrodynamic description of spreading.15,16 This could possibly be directly related to their very low values of partial vapor pressure at ambient temperatures. If f f 0, it can be seen, from eq 34 for example, that condensation effects will tend to zero. Similarly, dewetting seems to fit in better with the hydrodynamic approach.32 This could be because dewetting tends to occur at contact angles close to their equilibrium values, in which case the modified Kelvin equation derived above would tend to preclude vapor exchange. Although no “database” for spreading results would seem to exist at present, it is instructive to compare the prefactor of expression 37,that is, (fa)2/[3I2(2πmkT)1/2], with that of eq 38, that is, 2K ˜ λ3/(kT), as obtained by experiment. Taking a ≈ 3 × 10-10 m, f ≈ 2 × 10-2 (although the value
(37)
which is precisely the form expected, at low speeds, from the molecular kinetics approach, in which we may write3
(32) Redon, C.; Brochard-Wyart, F.; Rondelez, F. Phys. Rev. Lett. 1991, 36, 715.
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may be difficult to estimate in practice33), and m ≈ 3 × 10-26 kg (for water), with I ≈ 13, the prefactor of expression 37 is ca. 2 × 10-3 m2 N-1 s-1 at ambient temperature. This is, indeed, similar to the value given by Blake and Haynes3, although as stated above, condensation cannot be the relevant mechanism in this case, because the diffusion of molecules through a liquid is not comparable to gaseous transport. Petrov and Petrov34 present a table of values of K ˜ and λ for several systems, all containing water and corresponding to receding contact angles. The prefactor 2K ˜ λ3/(kT) is given in the range from 2.4 × 10-3 to 1.7 m2 N-1 s-1, most values being toward the lower end. Our estimate would then seem to be quite consistent with these results. Gribanova35 gives data, again for aqueous systems, corresponding to 2K ˜ λ3/(kT) of the order of 10-2 m2 N-1 s-1, similar in value to our prefactor. Despite the good agreement with experimental data, we should be cautious. Our reasoning is based on long-range interactions uniquely of the van der Waals type (see eq 4). Nevertheless, aqueous systems are much more complex (double-layer forces12). Semal et al. have studied the spreading of squalane, a nonpolar hydrocarbon, on surface-treated glass and thus a system more in keeping with the assumptions of the present study.36 They have considered a coefficient of friction, ζ, which corresponds to kT/(2K ˜ λ3) in our eq 38. Values of ζ-1 obtained, depending on experimental conditions, were of the order of 10-2-10-1 m2 N-1 s-1. Thus overall, we see comparable figures, thus possibly emphasising the conceptual similarity between adsorption/ desorption and condensation/evaporation phenomena. However, various features may differ between systems, potentially modifying numerical prefactors. Values may vary enormously between wetting and dewetting9 (although there is no clear consensus), and possibly geometry (capillary, sessile drop, etc.) may play a role. An interesting point is that the prefactor of eq 37 contains T-1/2, whereas eq 38 involves T-1. In principle, it may be possible to distinguish between the two spreading mechanisms by considering temperature dependence. However, other effects, such as variability of the value of γ with T, may obscure observed behavior. (33) Bourge`s, C.; Shanahan, M. E. R. C. R. Acad. Sci., Ser. II 1993, 316, 311. (34) Petrov, P. G.; Petrov, J. G. Langmuir 1995, 11, 3261. (35) Gribanova, E. V. Adv. Colloid Interface Sci. 1992, 39, 235. (36) Semal, S.; Voue´, M.; De Coninck, J. Langmuir 1999, 15, 7848.
Shanahan
The development above has taken the Hamaker constant as positive, but in principle the same treatment could be applied to systems with a negative Hamaker constant (convex micromeniscus near the TL, with respect to the vapor phase). Although small contact angles have been assumed, the basic physics should remain unaltered for larger angles. (In passing, eq 14 resembles a hyperbolic sine before any approximations are made and thus shows similarity with the general expression suggested by Blake and Haynes3). Finally, although the idea of liquid tranfer by condensation is not new,24,35 this contribution suggests a line to follow and could explain, in certain cases, a mechanism contributing to spreading and complementing the molecular kinetics theory. In this case, it is (a modified version of) Kelvin’s equation rather than Eyring’s approach which is at the heart of things, although the final mathematical formulation is rather similar. Conclusions We have considered the potential effects of condensation during the advance of a wetting triple line. Our main conclusions are the following. A modified form of Kelvin’s equation has been deduced which must be obeyed near the TL in order to preclude evaporation/condensation in the static case. This equation has been used as the basis of an alternative method to derive the equilibrium profile of the TL when van der Waals long-range forces are taken into consideration. Under dynamic conditions of an advancing TL, the added curvature of the liquid surface leads to condensation, which contributes to the overall spreading. Dissipation effects due both to hydrodynamic and to condensation currents have been considered. The condensation effect is likely to be greater at higher contact angle (e.g., in forced wetting), and the overall features are somewhat similar to those expected with the molecular kinetics approach. However, the basic equation is that of Kelvin (modified to take into account long-range forces) rather than that of Eyring. Our findings may be considered as complementary to the molecular kinetics approach. Acknowledgment. The author thanks P. G. de Gennes for useful discussions. LA001714Q
