Dynamics of Wetting

Journal of Colloid and Interface Science 229, 155–164 (2000) doi:10.1006/jcis.2000.6967, available online at http://www.idealibrary.com on

Dynamics of Wetting Rachid Chebbi Department of Chemical Engineering, University of Qatar, P.O. Box 2713, Doha, Qatar Received January 24, 2000; accepted May 8, 2000

We consider the dynamics of spreading of a small drop over a smooth solid surface. The analysis is concerned with complete wetting and accounts for capillary, viscous, and van der Waals effects, with two spreading geometries considered: cylindrical and axisymmetric. A complete description of the drop thickness, slope, and curvature profiles is provided. The model is not based on any fitting parameter, and the spreading laws are found to be in good agreement with published data obtained by different experimental techniques. The study evaluates the relative magnitudes of capillary, viscous, and disjoining-pressure effects and discusses the asymptotic form of the film profile and the existence of a singularity at the three-phase contact line. The model allows determination of the complete distribution of viscous dissipation in the whole drop, and the rate of viscous dissipation is related to the losses of the liquid drop free energy. °C 2000 Academic Press Key Words: liquid spreading; complete wetting; spreading laws; spreading dynamics; drop spreading; spontaneous spreading; dynamic contact angle; moving contact line; disjoining pressure; viscous dissipation; drop free energy.

1. INTRODUCTION

We study spreading of a small liquid drop over a horizontal smooth solid surface. This problem is reviewed in (1– 4). In particular, the existence of a precursor film preceding the advancement of a thicker film has been shown both experimentally (1) and theoretically (3, 5). Also, both experimental (6) and theoretical (7, 8) studies show that the free surface of the liquid drop presents an inflexion point. In the major part of the drop, the curvature of the surface is constant, and viscous forces are negligible. In the vicinity of the inflexion point, viscous forces come into play, and significant changes of the curvature occur. In the near vicinity of the contact line, van der Waals forces become important and cannot be ignored. Therefore, we distinguish two solutions: an outer solution, valid in the major part of the liquid drop, which takes into consideration both the capillary and the resisting viscous forces which oppose them, and an inner solution, valid for small thicknesses, which accounts in addition for van der Waals forces. A complete solution is obtained by matching the inner and outer solutions at the inflexion point. Asymptotic forms of the inner solution are given in (4, 5). The dynamics of wetting was obtained by Starov et al. (4) by matching the constant-curvature profile with the asymptotic form of 155

the inner solution profile. The model was found to compare favorably with the experimental data of Chen (4, 9). The model of Chebbi and Selim (8) takes into consideration the deviations between the outer solution and the constant-curvature profile near the inflexion point. The proposed outer solution (8) is valid in the major part of the drop and accounts for viscous effects, which become important in the vicinity of the inflexion point. Matching this outer solution (8), at the inflexion point, with the asymptotic form given by Starov et al. (4), allows determination of the spreading laws (8). The new changes proposed resulted in a closer agreement between theoretical results and experimental data. The model of Hervet and de Gennes (5) is restricted to the inner region and basically assumes the average velocity to be equal to the spreading velocity. This model is shown in our study to yield the same results as those obtained by the mathematical treatment of Starov et al. (4). A numerical solution (5) was obtained by integration, starting from the inflexion point, and using one arbitrary value of the dimensionless thickness calculated by the asymptotic form in (5), with values of the slope taken slightly larger than the one corresponding to the maximal profile (5). In the case of spreading of a liquid drop, we expect both the thickness and the slope to change with time at the inflexion point. To obtain the spreading laws and a complete determination of the drop profile and viscous dissipation in the whole drop, we need to match the inner solution with the outer one. To achieve this purpose, we study the effect of changing both the dimensionless thickness and the slope on the behavior of the inner solution. This study of the inner region allows, after matching with the outer solution, determination of the spreading kinetics without the necessity of resorting to the approximations made in (4, 8). This paper is concerned with a complete description of the dynamics of wetting. The governing equations are given in Section 2. In the next section, the model equations for the inner and outer regions are presented. The outer region equations include both capillary and viscous terms, whereas the inner region equations account in addition for van der Waals effects. The relative importance of capillary, viscous, and disjoining-pressure terms is determined in Section 3 also, followed by a study of the asymptotic behavior of the solution in the near vicinity of the three-phase contact line. A numerical study of the inner solution is shown to yield the deviations from the asymptotic approximation used in (4, 8) for the inner solution at the inflexion point. At the end of Section 3, the matching conditions between the inner 0021-9797/00 $35.00

C 2000 by Academic Press Copyright ° All rights of reproduction in any form reserved.

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RACHID CHEBBI

and the outer solutions are presented. In Section 4, the threephase contact line location is determined based on the value of the spreading coefficient for both axisymmetrical and cylindrical geometries. In Section 5, the rate of viscous dissipation is related to the losses of the free energy of the liquid drop. In addition the distribution of viscous dissipation is determined in the whole drop, followed by a discussion of the existence of a singularity at the contact line. In Section 6, the numerical procedure and results are presented. The validity of the matching technique is discussed based on the thickness, slope, and curvature profiles obtained from the inner and outer solutions. Finally, the theoretical results obtained by the proposed model are compared with the models and published experimental data in (4, 9) and in (7). 2. GOVERNING EQUATIONS

We consider the two cases of spreading corresponding to ribbon (ζ = 0) and axisymmetric (ζ = 1) geometries. The free surface profile h, which varies as function of time t and distance x, is shown schematically in Fig. 1. We consider the case of small slopes, which allows the use of the lubrication approximation (10). In addition, inertia and gravity effects are neglected, which shows, with nonretarded van der Waals forces, that the profile satisfies (4, 8) ½ µ ¶ ¸¾ · 1 ∂ 3A ∂h ∂ ∂ 2 h ζ ∂h ∂ ζ + (x h) = − x ζ h3 σ − , ∂t 3µ ∂ x ∂x ∂x2 x ∂x h4 ∂ x [1] where µ is the viscosity, σ represents the surface tension of the liquid, and A denotes Hamaker’s constant. For symmetric drops, the profile h satisfies ∂h ∂ 3h = 3 = 0. ∂x ∂x

[2]

3. SOLUTION

We seek a similarity solution of the form (7, 8) ¯ h = hϕ(η),

[4]

where h¯ denotes the apex thickness, and η = x/x¯ . A solution of this form imposes (8) ωη = ϕ 2

d dη

µ

¶ ζ dϕ d 2ϕ + , dη2 η dη

[5]

where ω is determined by matching with the inner solution as ¯ are indicated in Section 6. The spreading laws x¯ (t) and h(t) given respectively by (8) x¯ = x¯ 0 (1 + αT )1/(7+3ζ ) ,

[6]

I0 h¯ = h¯ 0 (1 + αT )−(1+ζ )/(7+3ζ ) , I

[7]

and

where T is a dimensionless time, t T = ; t¯

t¯ =

3µx¯ 04 , σ h¯ 30

[8]

subscript 0 denotes initial values, I is I = 2π

ζ

Z

1

ηζ ϕ dη,

[9]

0

and α is related to ω by

On the other hand, the equation V = 2π ζ

relates the liquid free surface profile h to the volume of the liquid V and the macroscopic limit of the drop x¯ .

Z

x¯

xζ h dx

µ [3]

0

α = ω(7 + 3ζ )

I0 I

¶3 .

[10]

In view of the minimum variation of the free-surface slope at the inflexion point, we define x¯ as the intersection of the tangent line at the inflexion point with the horizontal solid surface, which imposes at the inflexion point (8) ηm = 1 +

ϕm . ϕm0

[11]

¯ the dynamic contact angle θd is also govSimilarly to x¯ and h, erned by a power law (8) FIG. 1. Schematic figure of the right side of a symmetric drop spreading over a smooth solid surface.

tan θd = −

h¯ 0 I0 (1 + αT )−(2+ζ )/(7+3ζ ) ϕm0 . x¯ 0 I

[12]

157

DYNAMICS OF WETTING

To integrate [5], we need both ω and the value K 0 of K at the origin: µ K 0 = K |η=0 =

d 2ϕ ζ dϕ + 2 dη η dη

¶ |η=0

.

[13]

Equation [11] provides an implicit relation between ω and K 0 , which is however insufficient to resolve the problem completely, and it is necessary to match the outer solution with the inner one. In the near vicinity of the three-phase contact line, van der Waals effects become important and need to be included. The inner solution considered is of the form (4, 8) µ h=

3A x¯ 02 σ

¶1/4 Hˆ i (t)ψ(ξ ),

[14]

in which the stretched variable ξ is defined as x − x¯ . x¯ β(t)

ξ=

[15]

FIG. 2. Inner solutions obtained for ψm0 = −2.5 and the values Cf = 2, 2.01, 2.018, and 2.0184 (a) in the range ξ = 12 to 18, and (b) in a wider range for ξ starting from the inflexion point.

A solution of the form [14] requires (4, 8) dψ d = dξ dξ

µ

¶ 1 dψ ψ − ψ , dξ 3 ψ dξ 3d

3

[16]

below, matching the inner and outer solutions requires a relation between ψm and ψm0 at the inflexion point. The asymptotic forms in (4, 5) offer approximations for this relation. The one derived from Starov et al.’s work (4) is of the form

with (8) µ β=

¶1/2 3A x¯ 02 σ h¯ 40

µ

α 7 + 3ζ

¶−2/3

ψam = −ψm0 exp[(−ψm0 )3 /3]. (1 + αT )(3+2ζ )/(7+3ζ ) , [17]

and µ Hˆ i =

3A x¯ 02 σ h¯ 4

¶1/4 µ

0

α 7 + 3ζ

¶−1/3

(1 + αT )(2+ζ )/(7+3ζ ) . [18]

In terms of the capillary number Ca = µU/σ , in which the spreading speed U can be obtained directly from the expression for x¯ , the new variables ψ and ξ can be written in the following more compact forms ψ=

31/6

x − x¯ h , ξ = −1/6 −2/3 , −1/3 Ca a 3 Ca a

[19]

where a = (A/σ )1/2 . Integrating [16] and selecting the integration constant as zero (4) lead to 1 dψ d ψ , ψ = ψ3 3 − dξ ψ dξ

[21]

In order to relax this approximation, we integrate [20] numerically, starting from the inflexion point (5). For a given value of ψm0 , the asymptotic solution value ψam is corrected as follows ψm = Cf ψam .

[22]

As an example, numerical results are shown for the value ψm0 = −2.5 in Fig. 2. As shown in this figure, there is a narrow interval of values for Cf for which the profiles present precursor films of different extents, while having all very closely the same macroscopic profile. In addition, very close to a certain value Cfl (2.018 in Fig. 2), the profiles become identical except in the near vicinity of the contact lines. As mentioned in (5), the position of the contact line depends on the value of the spreading coefficient. This point is further extended and discussed in Section 4. The relative importance of capillary and disjoining-pressure terms appearing in Eq. [20] is compared to viscous terms using χc = −ψ 2 ψ 000 ; χd = −ψ 0 /ψ 2 .

[23]

3

[20]

a result consistent with the one obtained by Hervet and de Gennes (5) who assumed the average velocity of the liquid to be independent of x and equal to the spreading speed U . As indicated

Results, plotted in Fig. 3, show that, in the near vicinity of the three-phase contact line, capillary and disjoining-pressure terms become equal and predominant compared to viscous terms. Neglecting viscous terms (in comparison with the two other terms representing capillary and disjoining-pressure effects) in [20]

158

RACHID CHEBBI

FIG. 3. Relative magnitudes of viscous, capillary, and disjoining-pressure effects. Results obtained for the case ψm0 = −2.5 and Cf = 2.018 in the range (a) ψ = 10−1 to 102 , and (b) ψ = 0.2 to 1.

and integrating, we get 5s = −3ψ 3 ψ 00 → 1

as ξ → ξˆ ,

[24]

in which ξˆ denotes the contact-line value of ξ . Multiplying by ψ 0 ψ −3 and integrating yield 5d = −31/2 ψψ 0 → 1

as ξ → ξˆ ,

[25]

or −hh x /a → 1

as x → xˆ ,

FIG. 4. Asymptotic behavior of the solution in the vicinity of the three-phase contact line. Results obtained for the case ψm0 = −2.5 and Cf = 2.018.

finite at h = 0 is not a serious limitation, at least for thicknesses larger than 1 nm. For a macroscopic description of the profile, it is necessary to determine the dependency of Cf , which we approximate as Cfl , on ψm0 . As indicated in Section 1, the numerical solution in (5) considers only one arbitrary value of ψm , whereas we expect this value to change during spreading. The procedure explained above to obtain Cfl for the case ψm0 = −2.5 is repeated using different values for ψm0 . The relation we obtain between Cfl and ψm0 is

[26]

Cfl =

4 X

c j (−ψm0 ) j ,

[29]

0

and −1/4

ψ →3

q 2(ξˆ − ξ )

as ξ → ξˆ ,

[27]

or h→

p

2a(xˆ − x)

as x → xˆ ,

[28]

which proves the claimed asymptotic profile (5) in the near vicinity of the three-phase contact line. Numerical results, plotted in Fig. 4, are also consistent with the asymptotic behavior of the solution near the three-phase contact line: as h becomes small (ξ tends to ξˆ ), both 5d and 5s tend to 1. We also note that for thicknesses as small as 1 nm, the slope remains less than 0.23 for the typical values (4) A = 10−14 erg, and σ = 20 dyn/cm, which still satisfies reasonably the lubrication approximation requirement. Therefore, since the drop thickness cannot reach zero strictly speaking, the fact that the slope is not

where the constants c j are obtained using least-squares polynomial fitting. Results are given in Table 1, and a graphical representation of this relation is shown in Fig. 5. Matching the inner and outer solutions is made at the inflexion point by imposing equality for the values of the thickness and the slope (8), ψm ϕm =β 0 , ϕm0 ψm

[30]

TABLE 1 Quartic Fit Coefficients in Eq. [29] c0

c1

c2

c3

c4

1.3716

0.67079

−0.25564

0.042754

−2.5679 × 10−3

159

DYNAMICS OF WETTING

tion (x = xˆ ). In order to calculate the resultant of the viscous forces at the solid wall Fv , we need to evaluate the shear stress at the wall, which requires an expression for the velocity profile. This is given in (8) by u=

1 ∂P 2 (y − 2hy), 2µ ∂ x

[33]

in which µ

∂ 2h ζ ∂h + P = pg − σ ∂x2 x ∂x

¶

A , h3

−

[34]

pg is the gas pressure, and y is the vertical coordinate (Fig. 1). From [33], we get the wall shear stress u¯ τw = 3µ , h FIG. 5. Variation of Cfl with respect to ψm0 .

in terms of the average velocity u¯

and

u¯ = − ψm0 = ω−1/3 ϕm0 ,

·

¸ ϕm ln − 0 = ln βϕm

0

¡

c j −ω−1/3 ϕm0

¢j

1 ∂P 2 h . 3µ ∂ x

[36]

[31]

which gives in view of [21], [22], and [29]–[31], while approximating Cf by Cfl " 4 X

[35]

#

For the axisymmetric case, we obtain by applying the momentum balance Z

xˆ

Fv = 1 − ω−1 (ϕm0 )3 . [32] 3

The dimensional expression for the total curvature is k = ∂ 2 h/∂ x 2 + (ζ /x)∂h/∂ x. By equating the values of the slope at the inflexion point (where ∂ 2 h/∂ x 2 = 0), it is clear that we also ensure equality between the curvatures calculated from the inner and outer solutions. The validity of the method presented above in providing good matching between the inner and the outer solutions is shown in Section 6. 4. CONTACT LINE LOCATION

As previously stated in Section 3, for a given value of the slope ψm0 at the inflexion point, there is a small range of values for ψm for which the profile presents a precursor film (5). These profiles are essentially the same, with the basic difference being the contact-line location and the extent of the precursor film. As indicated in (5), the profile corresponding to a given situation is determined by the value of the spreading coefficient S = σSG − σSL − σ (with σSG , and σSL denoting the solid-gas and solid-liquid interfacial tensions, respectively). The treatment in (5) is extended here to include the case of axisymmetric spreading. In both cases of spreading, we start by applying the momentum equation to the right side (for instance) of the drop located between the inflexion point (x = xm ) and the contact line posi-

Z

xm

Z

Z

2σSG xˆ sin λ dλ

0

π/2

Z

π/2

2σ xm cos θd sin λ dλ −

0

2σSL xm sin λ dλ

0 π/2

−

π/2

2τw x sin λ dλ d x =

0

− Z

π/2

2(Pm − pg )h m xm sin λ dλ,

[37]

0

in which λ is the cylindrical angular coordinate, and h m is the thickness at the inflexion point. Integrating and simplifying [37], while using the approximation x m ≈ xˆ ≈ x¯ , we obtain Z

xˆ

τw d x = S + σ

xm

θd2 + (Pm − pg )h m . 2

[38]

The last term in [38] is equal to (Pm − pg )h m = σ tan θd

hm , xm

[39]

and can be neglected. Therefore, extending the result of Hervet and de Gennes to add the axisymmetric case, we obtain Fv = (2x¯ )ζ

Z

xˆ xm

τw d x = S + σ

θd2 , 2

[40]

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RACHID CHEBBI

in which the lower integration limit is now well defined, and Fv represents the viscous force per unit length in the cylindrical geometry case. In a dimensionless form, the shear stress force, Fv = σ (3 Ca)2/3 (2x¯ )ζ

F˜ v =

Z

ξˆ

1 dξ, ψ

[41]

¶ θ2 S + d . σ 2

[42]

ξm

Considering now the rate of loss of free energy and using Leibniz’s rule, we write [43] as

ξˆ ξm

1 dξ = (3 Ca)−2/3 ψ

µ

It is clear from Eq. [42] that different profiles correspond to different contact-line locations with different values for ξˆ , and therefore, correspond to different values of the spreading coefficient S. For instance, for the four values, Cf = 2, 2.01, 2.018, and 2.0184 previously considered in Fig. 2, we obtain respectively by numerical integration the values 6.91, 7.21, 8.20, and 8.74 for the left-side term appearing in Eq. [42], which clearly shows that the values of S are different and increase with the size of the precursor film.

F 2π ζ

¶

¶ ½Z k xˆ µ d xˆ A ∂h ∂h ∂ 2 h ζ = −S xˆ x σ + lim − 3 dx k→1 dt ∂ x ∂ x∂t h ∂t 0 µ ¾ ¶¯ A ¯¯ 2 ζ d xˆ 1 2 σ h x + 2 ¯ k xˆ . [48] + 2 h dt k xˆ ζ

is related to the spreading coefficient by Z

µ

d dt

Manipulating the term below, which appears in [48], we obtain after integrating by parts Z

k xˆ 0

∂h ∂ 2 h dx = − x σ ∂ x ∂ x∂t ζ

k xˆ

xζ σ

0

0

"

σ 2 −S + 2

µ

∂h ∂x

¶2

) # A ζ + 2 (π x) d x . [43] 2h

This expression extends the one given by de Gennes (3) to include the case of axisymmetric spread. The loss of free energy is accompanied by viscous dissipation at the rate of Z Ev = µ 0

xˆ

"Z

hµ 0

∂u ∂y

¶2

dy 2(π x)ζ d x.

[44]

Substituting for u from [33] and integrating with respect to y, while using [36], we get Ev = 3µ 2π ζ

Z 0

xˆ

u¯ 2 ζ x d x. h

σ

0

A σ ∂h ∂ 2h + 3 +ζ ∂x2 x ∂x h

¶

∂h dx ∂t [49]

Z

k xˆ µ

σ

F 2π ζ

¶ Z xˆ µ

A σ ∂h ∂ 2h + 3 = −S xˆ U − σ 2 +ζ ∂ x x ∂ x h 0 µ ¶ A σ + xˆ ζ U lim − h 2x + 2 . x→xˆ 2 2h

¶

∂ ζ (x h) d x ∂t [51]

Finally, substituting Eqs. [A5] and [A6] from the Appendix into [47] and [51], we get Z xˆ µ

σ

0

¶ A ∂ ζ σ ∂h ∂ 2h + (x h) d x, + ζ ∂x2 x ∂ x h 3 ∂t

[46] meaning that the loss of free energy is converted into viscous dissipation. In order to study the distribution of viscous dissipation in the drop, we use the following dimensionless form

∂ ζ (x h) d x ∂t

+ xˆ ζ U lim (σ hh x x + A/ h 2 )xˆ . x→xˆ

µ

dF = −E v = −2π ζ dt

along with Eqs. [34] and [36], we write [45] as Z xˆ µ

¶

[52]

∂ ζ ∂ ¯ (x h) = − (x ζ uh), ∂t ∂x

Ev = 2π ζ

d dt

[45]

Integrating by parts and using the continuity equation

∂h x σ ∂x ζ

and then, substituting into [48], we obtain

ζ

#

µ

¶ σ ∂h ∂ ζ ∂ 2h + ζ (x h) d x ∂x2 x ∂ x ∂t 0 ¢¯ ¡ [50] − σ U x ζ h 2x ¯k xˆ ,

∂h ∂ 2 h dx = − ∂ x ∂ x∂t

As the drop spreads, there is a loss of its free energy equal to xˆ

0

∂ ∂x

Making use of the continuity equation [46], we get Z

(Z

k xˆ

+ (x ζ σ h x h t )|k xˆ .

5. ENERGETIC CONSIDERATIONS

d dF = dt dt

Z

[47]

E˜ v =

Ev 2 ζ 2σ (π x¯ ) 32/3

Ca5/3 /µ

.

[53]

161

DYNAMICS OF WETTING

increase takes place in the near vicinity of the contact line. Also, we can note that the differences between the curves relative to Cf = 2 and 2.018 are limited to the near vicinity of the contact lines, with an increase in viscous dissipation due to a larger extent of the precursor film in the case Cf = 2.018. It should be pointed out here that, although the shear stress is infinite at the three-phase contact line, this does not yield an infinite force since thepimproper integral in [41] converges due to the fact that limξ →ξˆ ξˆ − ξ /ψ is finite (see Eq. [27]). Using this same argument, and referring to Eq. [54], it is also clear that the rate of viscous dissipation is also finite. 6. NUMERICAL PROCEDURE AND RESULTS

Solving Eq. [5] requires determination of ω and K 0 . For different values of β, the problem is solved by trial and error so as to satisfy [11] and [32]. The numerical procedure is similar to the ones detailed in (8, 11). Final results are expressed as FIG. 6. Viscous dissipation distribution. Results obtained using ψm0 = −2.5 and the two values Cf = 2 and 2.018 in the range (a) η = 0 to 1.2, and (b) ψ = 10−4 to 1.

By making use of Eqs. [5], [19], [34], [36], and [45] we get Z

ηm

E˜ v = ω1/3 0

η2+ζ dη + ϕ

Z

ξˆ ξm

1 dξ. ψ

[54]

Defining

Iv =

 Z  1/3  ω    1/3  ω

Z

η 0

0

η2+ζ dη ϕ

ηm

η2+ζ dη + ϕ

for η ≤ ηm Z

ξ ξm

1 dξ ψ

[55] for η > ηm ,

we can see how viscous dissipation is distributed in the drop. This requires determination of both the inner and the outer solutions which can be accomplished as indicated in Section 6. Considering the axisymmetric case and ψm0 = −2.5, with the values Cf = 2 and 2.018 previously considered in Fig. 2, we can see from Fig. 6 that, comparatively with the viscous dissipation in the major part of the drop which is insignificant, a noticeable change occurs near the inflexion point, and a very significant

f =

4 X

b j log j β,

[56]

0

in which f denotes ω, ϕm0 , I , or K 0 /2. The coefficients b j in Table 2 are given for the case of axisymmetric spreading. For a given value of ψm0 (−2.5 in the case of Fig. 2), the values of β, ω, K 0 , ψm , ϕm , and ϕm0 can be calculated by trial and error using Eqs. [30]–[32] and [56]. Variations of the thickness, the slope, and the curvature profiles are plotted for Cf = 2.018 in Figs. 7–9. We note excellent matching between the inner and the outer solutions. In addition, significant changes are seen to occur for the curvature, starting from the inflexion point and increasing significantly in the near vicinity of the three-phase contact line. The slope variation is minimum at the inflexion point, but becomes drastic in the near vicinity of the contact line. For a given value of time t, the values of β, α, ω, K 0 , ϕm0 , ¯ and θd are calculated by making use of [6], [7], [10], ¯ h, I, x, [12], and [17], in addition to the relations of the form [56] as in (8, 11). In particular, it is clear that α, which is related to ω and I by [10], is time dependent since both ω and I depend on β (see Eq. [26] and Table 2), which is itself time dependent (see Eq. [17]). The model is applied to compare the results obtained by the present method, with the experimental data in (4), and with the previous models in (4, 8). Results are shown in Fig. 10. The

TABLE 2 Quartic Fit Coefficients in Eq. [56]

b0 b1 b2 b3 b4

ω

0 ϕm

I

K 0 /2

2.4039397122 1.2205741563 0.2777901655 0.0299676085 1.2482910139 × 10−3

−1.3794273284 0.2294804165 0.0460653587 4.6383212485 × 10−3 1.8492196301 × 10−4

1.4296598083 −0.0685842823 −0.0152695692 −1.6252942173 × 10−3 −6.7090347286 × 10−5

−2.3548766497 −0.1758435124 −0.0395129032 −4.2286249065 × 10−3 −1.7517563114 × 10−4

162

RACHID CHEBBI

TABLE 3 Deviations from the Experimental Data of Diez et al. (7) for x¯ Volume, mm3

0.12

0.19

0.42

0.20

Average % deviation Maximum % deviation

1.2 3.2

2.1 3.4

1.5 3.7

3.1 5.6

FIG. 9. Matching between the outer and the inner curvature profiles. Variations of K for the case ψm0 = −2.5 and Cf = 2.018 (a) in the range η = 0.94 to 1.02, and (b) in the whole liquid drop.

FIG. 7. Matching between the outer and the inner thickness profiles. Variations of the thickness for the case ψm0 = −2.5 and Cf = 2.018 (a) in the range η = 0.94 to 1.02, and (b) in the whole liquid drop.

exponents in Starov et al.’s power laws (4) for the drop radius, the apex height, and the dynamic contact angle are the same as those listed by Tanner (6), who qualified these exponents as approximate. Tanner, who did not provide the actual values for the exponents, expected them to be slightly higher for the radius and slightly lower for the apex height and the dynamic contact angle in comparison with those he listed in (6). As in (8), we also find that α increases with time, and the effect of this variation is to increase slightly the slope for the radius as seen in Fig. 10a, and to lower slightly the slope for the apex height (see Fig. 10b), and also for the dynamic contact angle (see Fig. 10c), in consistency with what Tanner anticipated. On the other hand, the present model is also compared with the theoretical treatment and the experimental data in (7). The corresponding results are shown in Fig. 11 and Tables 3–5. For comparison with the data of Diez et al., Hamaker’s constant is estimated to be nearly equal to the one reported in (4), since experiments were made for the same liquid-solid system, silicone oil-PDMS, while using two different experimental techniques. As shown in Fig. 10, the approach used allows relaxation of the approximations made in (4, 8), and this leads to a better match with the experimental data in (4), in agreement with the expectations reported in (11). Also, it can be seen from Fig. 11 and Tables 3–5 that the model agrees well with the experimental data reported by Diez et al. (7). TABLE 4 ¯ Deviations from the Experimental Data of Diez et al. (7) for h

FIG. 8. Matching between the outer and the inner slope profiles. Variations of the slope for the case ψm0 = −2.5 and Cf = 2.018 (a) in the range η = 0.94 to 1.02, and (b) in the whole liquid drop.

Volume, mm3

0.12

0.19

0.42

0.20

Average % deviation Maximum % deviation

0.3 1.1

3.8 7.3

7.4 12.7

1.2 2.0

DYNAMICS OF WETTING

FIG. 10. Comparison with Chen’s run7 data for (a) radius, (b) apex height, and (c) dynamic contact angle.

163

FIG. 11. Comparison with Diez et al.’s data (V = 0.12 mm3 ) for (a) radius, (b) apex height, and (c) dynamic contact angle.

164

RACHID CHEBBI

TABLE 5 Deviations from the Experimental Data of Diez et al. (7) for tan θd Volume, mm3

0.12

0.19

0.42

Average % deviation Maximum % deviation

1.6 3.6

2.5 6.1

7.4 17.3

Then, integrating by parts provides µ ¶ µ ¶ A A σ 2 lim − h x + 2 + lim σ hh x x + 2 = S. [A2] x→xˆ x→xˆ 2 2h h In a dimensional form, Eq. [20] can be written as ∂ 3h 3A ∂h 3µU . =σ 3 − 4 2 h ∂x h ∂x

7. CONCLUSION

The models of Starov et al. (4) and Hervet and de Gennes (5) for the inner region are shown to agree. In order to obtain the spreading laws, a complete description of both the inner region and the outer region solutions is needed and matching is required. This is achieved in this study, and complete numerical treatments are provided. The solution is shown not to violate the lubrication theory for thicknesses as small as 1 nm. By equating the thicknesses, and the slopes given by the inner and outer solutions at the inflexion point, we can determine the complete drop profile. Extending the inner and outer solutions beyond both sides of the inflexion point and comparing the two solutions show the validity of the procedure in providing excellent matching among the thickness, the slope, and the curvature profiles obtained by using the outer and inner solutions. Results show drastic changes of the slope in the near vicinity of the contact line, and minimum slope variations at the inflexion point. The analysis provides a complete description of the viscous dissipation distribution in the whole drop. Changes in the curvature and in the viscous dissipation distribution are found to become significant in the vicinity of the inflexion point, and to increase very significantly in the near vicinity of the three-phase contact line. The losses in the drop free energy are shown to be converted into viscous dissipation, and both shear stress force and viscous dissipation are shown to remain finite, although the model uses the no-slip condition at the contact line. The model is compared with the theoretical results and experimental spreading data in (4, 9) and in (7). Results show good agreement with the data for the radius, apex height, and dynamic contact angle reported in these references. APPENDIX

Substituting for τw from [35] into [40], while using [34] and [36], yields µ 2 ¶ Z xˆ ∂ A ∂ h θ2 h σ 2 + 3 dx = S + σ d . [A1] ∂x h 2 xm ∂ x

[A3]

In order to determine the second limit in Eq. [A2], we integrate Eq. [A3] to obtain σ hh x x

A + 2 = 3µU h h

Z

x xm

1 d x. h2

[A4]

Finally, √ using the asymptotic form of the solution [28] and limx→xˆ xˆ − x ln (xˆ − x) = 0 gives µ ¶ A lim σ hh x x + 2 = 0, x→xˆ h

[A5]

which, upon substitution into Eq. [A2] yields µ ¶ A σ lim − h 2x + 2 = S, x→xˆ 2 2h

[A6]

and this proves the corresponding claimed statement in (5). REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Dussan V., E. B., Annu. Rev. Fluid Mech. 11, 371 (1979). Marmur, A., Adv. Colloid Interface Sci. 19, 75 (1983). de Gennes, P. G., Rev. Modern Phys. 57, 827 (1985). Starov, V. M., Kalinin, V. V., and Chen, J. D., Adv. Colloid. Interface. Sci. 50, 187 (1994). Hervet, H., and de Gennes, P. G., C.R. Acad. Sci. II 299, 499 (1984). Tanner, L. H., J. Phys. D 12, 1473 (1979). Diez, J. A., Gratton, R., Thomas, L. P., and Marino, B., Phys. Fluids 6, 24 (1994). Chebbi, R., and Selim, M. S., J. Colloid Interface Sci. 195, 66 (1997). Chen, J. D., J. Colloid Interface Sci. 122, 60 (1988). Lopez, J., Miller, C. A., and Ruckenstein, E., J. Colloid Interface Sci. 56, 460 (1976). Chebbi, R., J. Colloid Interface Sci. 211, 230 (1999).