On the Spreading of Partially Miscible Liquids

Journal of Colloid and Interface Science 234, 375–383 (2001) doi:10.1006/jcis.2000.7287, available online at http://www.idealibrary.com on

On the Spreading of Partially Miscible Liquids Maria Santiago-Rosanne,∗, † Mich`ele Vignes-Adler,∗,1 and Manuel G. Velarde† ∗ Laboratoire de Physique des Mat´eriaux Divis´es et des Interfaces, Universit´e de Marne la Vall´ee, F-77454 Marne la Vall´ee Cedex 2, France; and †Instituto Pluridisciplinar, Universidad Complutense, Paseo Juan XXIII no. 1, 28040 Madrid, Spain E-mail: [email protected] Received May 18, 2000; accepted October 16, 2000

As time proceeds, partially miscible liquids spread as a cap surrounded by a primary film according to power laws, t n , for both the leading edge (front) and the central cap. The corresponding exponents depend on the thickness, H, of the liquid aqueous substrate and the deviation of concentration from its saturation value, 1C = C − C sat . As long as H is thick enough, here H ≥ 5 mm, the exponents are n = 1/2 and n = 1/3 for the front and the central cap, respectively. For thinner layers, H ∼ 2 mm, and moderate 1C, the spreading is drastically hindered and the measured values can go down to n = 0.1, due to the additional friction imposed by the confinement of the convective cells generated by dissolution below the primary film which anchor on the solid surface beneath the liquid substrate. °C 2001 Academic Press Key Words: miscible liquids; spreading; Marangoni effect.

1. INTRODUCTION

When a volume of liquid, e.g., a drop (denoted O), is deposited on the surface of an immiscible liquid (denoted w), gravitational forces cause spreading which is instantaneously opposed by inertia forces arising from the acceleration of the slick. As time proceeds, viscous drag with the substrate replaces inertia as the retarding force. Furthermore, as the spreading layer becomes thinner, gravity becomes less of an effective driving force and capillary forces take over. This is also the initial situation when the effective gravity is not significant. Whether “o” continues to spread or not depends on the sign of the spreading coefficient, ¡ ¢ S = σw − σo + σo/w ,

[1]

where σw and σo are the surface tensions of the phases w and o, and σo/w is their interfacial tension. The S is usually constant in most cases (1, 2). When a surface gradient exists the spreading driving force derives from the Marangoni shear stress (3) integrated over the length of the spreading film, Z S= 0 1

R(t)

∂σ d x. ∂x

To whom correspondence should be addressed.

For a spherical drop, axisymmetrically spreading on a liquid layer of thickness H , it has been established, both theoretically and experimentally, that when capillary forces dominate with viscous dissipation in the underlying liquid substrate, then the drop radius, R, grows according to a power law (4), R = k tn,

[2]

The prefactor k depends on the liquid physical properties. If dissipation mostly takes place in the viscous boundary layer of the underlying liquid (the case with a thick enough substrate) (5), n = 3/4

and

k ∝ S 0.5 /(µρ)0,25 ,

[3]

while if the underlying liquid is so thin that dissipation takes place over the whole bulk (as in the lubrication approximation) (7, 8), then n = 1/2

and

k ∝ (H S/µ)0,5 .

[4]

Here, ρ and µ are, respectively, the density and dynamic viscosity of the liquid in the underlying layer. This analysis ignores the drop finite size. For miscible liquids there is dissolution during the spreading, thus causing bulk and surface concentration gradients. If the drop is of denser material than the liquid substrate, like in our experiments, then buoyancy-driven convection may accompany the mixing process, thus altering the spreading process. Moreover, if due to the surface concentration gradient interfacial motions develop at the interface between the two liquids, these can give rise to interfacial instability also altering the spreading process. The mentioned convective phenomena occurs when evaporative processes accompany spreading. Dussaud and Troian (5) noted that the dissipation induced by buoyancy-driven convection lowers in their experiment the theoretically expected value of n = 3/4 down to n = 1/2. Although the spreading problem with miscible liquids is of great technological interest, it has received little attention with the exception of the work by Ruckenstein and collaborators

375

0021-9797/01 $35.00

C 2001 by Academic Press Copyright ° All rights of reproduction in any form reserved.

376

SANTIAGO-ROSANNE, VIGNES-ADLER, AND VELARDE

(9–11), who investigated the spreading of isobutanol steadily dissolving in water. Here we take up this line of study for the spreading of a nitroethane or an ethyl acetate drop carefully deposited on a water surface. Nitroethane is partially soluble in water, it is denser and less viscous, and it has a much lower surface tension than water. Ethyl acetate is also partially soluble in water but has a lower surface tension, and being lighter than water it yields in solution a stably stratified layer. Just after deposition, the drop spreads as a central cap surrounded by a primary film. As earlier reported (12), after 1 s, surface waves sustained by the solutal Marangoni stress appear in the primary film which evolve toward patterns like daisy flower petals. Subsequently, those patterns disappear and interfacial turbulence spreads over the surface. Top and side views of the motions generated by the drop spreading were visualized with a suitable Schlieren device sensitive to density gradients and surface deformations. In the present report we focus attention on the role of the drop dissolution during the initial stage of the spreading.

When necessary, we have used a superscript N or E to make a distinction between nitroethane and ethyl acetate. Unless explicitly mentioned, the aqueous solution is not totally saturated with the organic liquid, whereas the liquid of the spreading drops is used pure.

2. MATERIALS AND METHODS

2.1. Materials Nitroethane was obtained from Carlo Erba (purity 99%) and ethyl acetate from Aldrich (purity 99.5%). Their relevant physical properties are summarized in Table 1. The density and surface tension variations of the nitroethane and ethyl acetate aqueous solutions with solute concentration C are also reported in Table 1. Experiments were conducted with the initial solute concentration, C, ranging from 0 to the saturation concentration Csat . TABLE 1 Physical Properties of the Drop of Organic Liquid

Molecular weight (g/mol) Density (g/cm3 ) Saturation concentration in water at 20◦ C (mol/m3 ) Dynamic viscosity (cP) Surface tension (mN/m) Interfacial tension against water (mN/m) Diffusion coefficient (cm2 /s) Solubility in water at 20◦ C (wt%) Solubility of water in organic liquid (wt%) Refractive index at 20◦ C ∂ρ/∂C (kg · mol−1 ) ∂σ/∂C (mN · m−2 · mol−1 ) a

Nitroethane

Ethyl acetate

75 1.04a 626

88 0.90a 917

0.643a 31.9b 14.65

0.426a 23.75b 6.65

5.10−6 4.4 0.9

10.2.10−6 8.08 2.94

1.3919c 1.71.10−2d −0.04

1.3716c 1.91.10−2e concentration dependent

From “Handbook of Chemistry and Physics,” 77th ed., CRC Press. Measured with a de No¨uy type tensiometer. c Measured with an Abbe refractometer. d Calculated with the Lorentz–Lorenz formula (13). e From Ref. (14). b

FIG. 1. Experimental setup: Ca, capillary; Dg, organic liquid drop; W, water–organic liquid mixture; T, PTFE ring; H1 , H2 , upper and lower optical plates; S, source light; D1 , rectangular diaphragm; D2 , knife; M1 , M2 , concave mirrors; Mp1 , Mp2 , mp1 , mp2 , plane mirrors; CCD, camera.

2.2. Experimental Setup (Fig. 1) The experimental setup used to observe surface motions consists of a parallelepipedic cell (350 × 350 × 70 mm3 ) with the upper and the lower plates (H1 and H2 ) made of optical glass, smoothly polished in their central part, and placed in frames made of stainless steel. Two of the opposite lateral walls have Pyrex windows. A stainless steel tube (Ca) passes across another vertical wall: it provides a tightly closed connection with the outside. A PTFE ring (T) (φ = 300 mm) is placed inside the cell and it is pressed against the lower plate. PET o-rings guarantee watertightness. The inner region of the PTFE ring acts as a Petri dish with a bottom of optical quality. The cell is placed in the field of a transmitted light Schlieren device which is sensitive to surface deformations and density gradients. A rectangular light source (D1 ), placed at the focus of a concave mirror (M1 ), provides a parallel light beam (φ = 200 mm). This parallel light beam crosses the experimental cell. It is then collected by a second concave mirror (M2 ) which concentrates the light in its image focus (itself conjugated with the light source) where a knife edge (D2 ) is placed. Finally, the parallel light beam converges on a CCD camera located at the conjugate focus of the cell. The position of the knife edge is adjusted in such a way that the rays in half of the rectangular source image do not go through. The uniform illumination I in the CCD camera is then reduced by half. The principle of the Schlieren device is that the rays are deviated by a small angle ε when the surface is deformed and, accordingly, the rectangular source image on the knife edge is shifted. Our system was developed to visualize and measure angular deviations down to 10−4 rad, which corresponds to surface

377

SPREADING OF PARTIALLY MISCIBLE LIQUIDS

elevations of the order of 1 µm (12). The whole apparatus is placed on a vibration-free table. A side view of bulk motions in the underlying liquid was precluded by the opaque PTFE ring. Visualization was done in another transparent square cell (123 × 123 × 55 mm3 ) made of window glass, much simpler in its design, using the Schlieren device as a shadowgraph by removing the knife. The images are less sharp in this cell because of the small difference between the water (1.33) and organic liquid refractive indices (Table 1). Nevertheless, the shadowgraph is sensitive to density gradients, and composition gradients in the solutions are well observed. Drop motions and deformations are visualized and recorded by means of a Lhesa CCD camera connected to a videotape recorder. The video images are digitized and processed using Synetic and Visilog softwares from No¨esis. Since the motions are so fast that the moderate recording frequency of the tape recorder (25 frames/s) makes the pictures slightly blurred, the two frames of each image are split; hence the actual processing frequency is 50 frames/s. 2.3. Experimental Procedure All parts of the cells in contact with the liquids were very carefully cleaned. Glassware was degreased with acetone and alcohol and passed through fresh sulfochromic acid. PTFE pieces were soaked in boiling aquaregalia, then rinsed profusely with Milli-Q water. The optical glasses were rinsed in ethyl alcohol, ether, and boiling pure water. The outer cell was tightly closed, and it was saturated with organic vapors by letting the organic liquids evaporate for 24 h from two beakers placed in the annulus around the inner cell. Just before the experiment was begun, the upper plate was slightly open for the time necessary to fill the inner cell with a solution layer. The layer thickness ranged from 2 to 16 mm. A small drop with a volume V of 11 mm3 was formed at the tip of a syringe fixed on a pushing device that delivered calibrated volumes of pure organic liquid, and the drop was carefully deposited on the aqueous surface. Two parameters were varied, the thickness H of the solution layer and the solute concentration C in the aqueous solution.

3. RESULTS

3.1. Surface Activity Results are reported in Fig. 2. There are two main differences in the surface activity of the two organic liquids: (i) the surface tension of the ethyl acetate solution is always lower than that of nitroethane; (ii) the nitroethane solution behaves ideally and its surface tension varies linearly with C N in the limit of miscibility, µ σ WN = σ0 +

N¶

C N Csat

¢ ¡ · σWN sat − σ0 ,

[5]

FIG. 2. Surface tensions of the solutions against air as a function of the organic liquid concentration, C: (s) nitroethane (∗) ethyl acetate.

where σ0 is the surface tension of the pure water and σWN sat = 47 mN/m. The surface tension of ethyl acetate solution is fit by a Langmuir–Szyskowski equation again in the limit of miscibility, σ

WE

¶ µ CE , = σ0 − RT 0∞ · ln 1 + a

[6]

where R and T are the perfect gas law constant and the temperature, respectively; 0∞ = 6.3 · 10−6 mol/m2 is the saturation adsorption and a = 79 mol/m3 is the Szyskowski concentration. 3.2. Spreading Laws Typically, as soon as the organic liquid drop is deposited on the aqueous surface, it spreads with the formation of a central cap surrounded by a primary film exhibiting a relatively thicker rim (Fig. 3a). Besides, two bumps could be observed in the rim (Fig. 3b). The rim width is about 20 mm. As already discussed by Fraaije and Cazabat (7), two domains ought to be considered in the spreading drop, the central cap and the primary film with the leading edge (Fig. 3a). The drop radius R(t) and the central cap radius Rcap (t) have therefore been systematically measured, and their corresponding dependencies on C and H have been investigated. Results are plotted in Figs. 4 –7. All the data can be fit by power laws represented by solid lines in the figures, R = k · tn

Rcap = kcap · t ncap ,

[7]

where k, kcap are material constants, and n and n cap are spreading powers. The values are given in Table 2. Generally n cap 6= n. The linear correlation coefficients for the fit of the two parameters are also reported in Table 2.

378

SANTIAGO-ROSANNE, VIGNES-ADLER, AND VELARDE

FIG. 3. Nitroethane drop spreading on the surface of a layer (H = 13.6 mm) of a nitroethane solution at 1C N = 602 mol/m3 : (a) Top views at different times and schematic drop spreading; t = 0 is the drop deposition time. (b) Profile of the primary film rim of the drop (Dg = 2.75 mm) measured by Schlieren.

3.2.1. Influence of the organic liquid concentration. First of all, it was checked that the cell diameter was large enough to rule out the influence of its outer boundary upon the spreading process. It was also checked that the initial concentration of water in the drop has a negligible influence on the process due to the lower water solubility in nitroethane (0.9 wt%) and in ethyl acetate (2.94 wt%) with respect to the opposite situation. Accordingly, in all the experiments, the organic liquids were used in pure form. Results are provided in terms of 1C = Csat − C, which is the initial deviation of C from the saturation concentration of the organic solute in water. For each organic liquid,

the saturation degree was introduced as s = so − C ·

M ρ

with so = Csat ·

M , ρ

[8]

where M and ρ are the molecular weight and the density, respectively; s is directly related to solute solubility so in water. Nitroethane drops always spread following the steps already described. However, if C is too small, i.e., 1C is too large, the ethyl acetate drops do not spread but there is rather an immediate turbulent mixing of the drop liquid with water. There is a critical

379

SPREADING OF PARTIALLY MISCIBLE LIQUIDS

3.2.2. Influence of the aqueous layer thickness. Experiments were performed for various values of H ranging from 2.3 to 16 mm (Figs. 5 and 6). Typically, for the same initial drop volume, larger values of both R and Rcap are obtained when H increases, whatever the organic liquid. However, there exist two critical values H ∗ ∼ = 7 mm and H ∗∗ ∼ = 3.5 mm of the water layer thickness such that (Table 2) if H > H ∗

kN ∼ = 70 mm/s n

n ∼ 0.5

kE ∼ = 53 mm/s n for all values of H, if H ∗ > H > H ∗∗

n = 0.5

k increases when H increases for both organic liquids, if H ∗∗ > H

n < 0.4

k depends on the organic liquid (k N > k E ) and on 1C.

FIG. 4. Time dependence of the radius R of a drop spreading on aqueous substrate layer of thickness H = 5 mm at different solute concentrations: (a) nitroethane; (b) ethyl acetate.

value s ∗ = 5.9% above which interfacial turbulence instantaneously occurs. When C increases, i.e., s decreases below s ∗ , the onset of the interfacial turbulence is significantly delayed, and then the spreading experiments can be made. Experiments were systematically performed for several values of the solute concentration in water with H = 5 mm (Figs. 4a and 4b). For both organic liquids, n ∼ 0.5 whereas k depends on the organic liquid used; it is independent of the ethyl acetate concentration and it varies slightly with the nitroethane concentration. The lower the value of s, the more extended is the central cap and the narrower is the primary film. When the water and the organic liquid are initially mutually saturated, i.e., s = 0%, the primary film is no longer visible, and the drop spreads as a disk, although there is still formation of a thicker rim with a bump at the periphery. In this case n ∼ 0.56 for nitroethane and n ∼ 0.4 for ethyl acetate. Note that pattern and interfacial turbulence do not occur during the subsequent spreading stages (12).

FIG. 5. Time dependence of the radius R of a drop spreading on aqueous substrate layer of different thicknesses: (a) nitroethane 1C N = 602 mol/m3 ; (b) ethyl acetate 1C E = 517 mol/m3 . Solid lines are fitting curves to the data according to power laws [7].

380

SANTIAGO-ROSANNE, VIGNES-ADLER, AND VELARDE

(i) At t < 0.30 s, no dissolution occurs either below the central cap or below the primary film except under the outer rim, where a propagating axisymetric vortex close to the interface is visible. Its length, l, is about 4–5 mm and its thickness, L, about 1.5–2.5 mm. The propagation of the axisymetric vortex follows, approximately, the same power law and prefactor as the drop spreading. (ii) At 0.30 s < t < 0.60 s, there is also formation of small eddies close to the interface in the inner region of the propagating axisymetric vortex. They are smaller and they appear less bright than the above-mentioned main vortex, and their length (l ≈ 4 –5 mm) is three or four times smaller than the rim width 3 (Fig. 3b). There is a third step, which has been discussed in (12). When t > 0.60 s, the eddies penetrate into the bulk, and they spread over the layer adjacent to the interface until it is no longer possible to distinguish the vortex from the smaller eddies. Then turbulent motions invade the entire organic liquid layer. We have explored the influence of the thickness of the aqueous layer, H , on the convective motions generated by the dissolution

FIG. 6. Time dependence of the central cap radius R of a drop spreading on aqueous substrate layer of different thicknesses: (a) nitroethane 1C N = 602 mol/m3 ; (b) ethyl acetate 1C E = 517 mol/m3 .

For nitroethane at H = 2.3 mm, the value of n drastically decreases. We found n = 0.31 if 1C N = 602 mol/m3 , i.e., at low saturation, and ∼0.1 if 1C N = 314 mol/m3 . Note that if H is still lower, the drop does not spread at all but comes into contact with the lower glass plate of the cell, and it instantaneously dewets it. We could not observe such low n values with ethyl acetate. For the spreading of the central cap (Figs. 6a and 6b), the spreading power law follows the same trend, H > H ∗∗

n cap = 0.3–0.4

E N > kcap , for both organic liquids but kcap

and n cap drastically decreases when H < H ∗∗ . 3.3. Dissolution Effects Side views (Fig. 7) clearly show that the dissolution of the nitroethane drop occurs in two steps:

FIG. 7. Side views of the spreading drop showing the convective cells: (i) t ≈ 0.30 s; (ii) t ≈ 0.50 s. The solid arrows indicate the convective roll below the rim propagating with the leading edge. The dotted arrows indicate the small Marangoni cells.

381

SPREADING OF PARTIALLY MISCIBLE LIQUIDS

TABLE 2 Spreading Values for the Leading Edge, R(t), and the Central Cap, Rcap (t), Measured in mm

Nitroethane

Ethylacetate

H (mm)

1C (mol m−3 )

S (mN/m) from [9]

n

k

k from [4]

r2

n cap

kcap

r2

2.3 3.4 5 13.6 16 2.33 5 5 10.2

602 602 602 602 602 312 312 69 0

39.5 39.5 39.5 39.5 39.5 27.7 27.7 17.9 0.45a

0.316 0.428 0.475 0.535 0.491 0.082 0.509 0.503 0.564

48 59 64 73 69 27.6 59.8 48.9 39.5

302 367 445 733 795 254 372 299 67

0.993 0.999 0.997 0.998 0.991 0.918 0.999 0.999 0.999

0.069 0.102

6.3 6.4

0.899 0.9137

0.36 0.317 0.13

11.32 11.57 9.4

0.994 0.981 0.994

17 23

0.9937 0.9806

2.33 3.4 5 6.8 10.2 13.6 16 5 5 5

517 517 517 517 517 517 517 317 117 0

21 21 21 21 21 21 21 16 10 1.8a

0.44 0.44 0.48 0.54 0.54 0.56 0.52 0.487 0.501 0.407

47 49 57 64 65 67 65 55 54.7 32.7

221 269 326 380 466 538 583 285 226 94

0.998 0.999 0.999 0.999 0.997 0.999 0.998 0.995 0.999 0.995

0.3 no cap observed 0.156 0.277

4.9 8

0.8262 0.9841

0.329

11.25

0.9948

0.395 0.353

12.9 12.74

0.9843 0.9959

0.124

11

0.8796

Note. H, thickness of the underlying layer; C, initial organic liquid concentration in water; S = σWsol − σsol except for mutually saturated liquids; n and n cap , spreading exponents; k and kcap , spreading prefactors; r 2 = correlation coefficient. a From [1].

of the organic liquid. This is particularly significant in the nitroethane case. Due to limitations in the setup this was only possible for H > 5 mm. Note that for lower values of H the size of the eddies is about the depth of the solution layer and hence takes in the whole solution layer like in B´enard convection (15). If H > 7 mm the eddies deform and go deep into the aqueous phase such that for much higher values of H , like H = 37 mm, an array of streamers appears in the aqueous solution. With ethyl acetate solutions, although small eddies instantaneously form below the whole spreading drop including the central cap, there is, however, no formation of a propagating axisymetric vortex at the leading edge and, whatever the value of H , only an array of streamers is finally observed in the bulk layer. 4. DISCUSSION

The drop spreading process is driven first by the gravitational forces and subsequently, as the drop becomes thinner, by capillary forces. It is slowed down by the inertia forces and the viscous drag occurring in the primary film of the spreading drop and/or in the aqueous substrate. The dynamic balance of all these forces yields the earlier given spreading power laws. As recalled in the introduction, n = 0.5 (with prefactor ∼ (H S/µ)0.5 ) is obtained when viscous dissipation occurs in the whole bulk of the aqueous substrate, while n = 0.75 (with prefactor ∼S 0.5 /(µρ)0,25 ) if dissipation affects only a boundary layer in a relatively thick liquid substrate.

In the experiments performed with mutually saturated nitroethane and water (they can therefore be considered as practically immiscible) on a rather thin substrate layer (H = 10.2 mm), the spreading coefficient is S = 0.45 mN/m, and the measured spreading law is for R(t) expressed in mm, R(t) = 40 t 0.56 , which agrees well with the theoretical prediction, n = 0.5. When dissolution occurs, the measured exponents are also close to n = 0.5, as long as H is between 13 and 3.5 mm (equal to H ∗∗ ). We can calculate a spreading theoretical prefactor, k th , from Eq. [4] assuming that the spreading coefficient is given by S = σ WN − σ N

[9]

instead of [1] as we have an interface between two miscible liquids. The corresponding theoretical prefactors are much higher than the measured ones (Table 2). For a given 1C, we have plotted the ratio k th /k as a function of the sublayer thickness (Fig. 8). Ratio k th /k varies between 6 and 11 for the nitroethane and between 5 and 8 for the ethyl acetate, and it roughly increases with H . The expression [4] was derived for immiscible liquids. Obviously, the dissolution of the organic liquid, because of mass loss, decreases the prefactor and the corresponding speed of the leading edge, but not the power law. However, for lower thicknesses, H < H ∗∗ , organic liquid dissolution hinders the drop spreading. Furthermore, for a given value of the aqueous layer thickness, it can be also drastically slowed down

382

SANTIAGO-ROSANNE, VIGNES-ADLER, AND VELARDE

by the particular value of the initial deviation of the concentraN . For instance, tion from its saturation value, 1C N = C N − Csat we have measured n ∼ 0.1 when H = 2.3 mm and 1C N = 312 mol/m3 . Dussaud and Troian (5) have reported a reduction in the spreading exponent from n = 0.75 to n = 0.5 in the spreading of volatile liquids on thick layers. They argued, and provided evidence, that the additional source of dissipation was due to a propagating buoyancy-driven (Rayleigh–B´enard) convective cell beneath the leading edge. We have observed a similar convective roll for nitroethane, but we have also observed small convective cells in the whole sublayer (Fig. 7). However, with ethyl acetate, only the small rolls were observed. As already mentionned, nitroethane is denser than water while ethyl acetate is lighter. Stabilizing density gradients (as expected with ethyl acetate) confine interfacial convection to narrow zones adjacent to the interface while destabilizing gradients (as expected with nitroethane) generate eddies streaming away from the interface and penetrating deeply into the bulk phases, as long ago recognized by Berg and Morig (16). In our experiments, density gradients are caused by the drop dissolution. Actually, nitroethane ideally mixes with water since ρ and σ depend linearly on C in the whole range of their mutual miscibility. As their interfaces behave as free surfaces, since there is no surfactant accumulation, a straightforward analogy exists between the present concentration-induced cells and the temperature-induced (Rayleigh–B´enard) cells in thin liquid layers. Hence, as the aqueous substrate layer thickness decreases, the convective cells become confined reaching the solid bottom. Accordingly, viscous dissipation drastically increases, due to the additional friction imposed by the lower sticky boundary. This justifies a lowering of n from the predicted value 0.5 down to the measured value n = 0.1, obtained when the sublayer thickness is 2.33 mm and 1C N has a moderate value (∼312 mol/m3 ). The thickness 2.33 mm is, approximately, the size of the main convective cell below the leading edge. If, however, 1C N is high enough, 1C N ∼ 602 mol/m3 , the measured value is n = 0.3. It is noticeable that, for the thinner layers and moderate values of 1C N , the power law, n = 0.1, is the same as that found for capillary-driven spreading on a solid surface when viscous dissipation occurs in the drop (17). On the other hand, since aqueous solutions of ethyl acetate are stably stratified with surface tension gradients leading to small rolls, such a cell confinement would have necessitated aqueous layer thicknesses below their limit of mechanical stability. The Marangoni and Rayleigh numbers are defined as Ma = ((∂σ/∂C)1Ch)/(µD) and Ra = ((∂ρ/∂C)1Cgh 3 )/(µD), where µ and D are the dynamic viscosity and the diffusion coefficient of the organic solute in water, respectively. The quantity h denotes the vertical penetration in the liquid substrate of the dissolved amount of drop material, and ∂σ/∂C is obtained from Eqs. [3] and [4], while ∂ρ/∂C is obtained from refractive index measurements and the application of the Lorentz–Lorenz formula (13) (Table 1). Taking, to a first

FIG. 8. Comparison between the experimental prefactor and the theoretical value [4] obtained for immiscible liquids as a function of the sublayer thickness. ( ) nitroethane, ( ) ethylacetate).

approximation, as critical values for B´enard–Marangoni and Rayleigh–B´enard instabilities (18) Mac ∼ = 1000 and Rac ∼ = 100, respectively, we have found that, for h = 2.33 mm, they correspond to 1CMa ∼ = 0.54 10−3 mol/m3 and 1CRa ∼ = 0.24 mol/m3 , respectively, which are values compatible with our experiments. Let us discuss further the influence of the dissolution on the spreading prefactor which is larger at thicker solution sublayers (Fig. 8) and which is larger for nitroethane than for ethyl acetate. As H increases, the convective rolls penetrate deeper into the sublayer, hence enhancing dissolution and decreasing the amount of material to spread, Ethyl acetate (and not nitroethane) solutions are stably stratified. The convective cells are confined to a narrower region near the interface than with nitroethane. The convective transfer is therefore reduced with ethyl acetate relative to nitroethane. Moreover, with nitroethane, there exists a large convective roll below the rim (Figs. 3 and 7), which does not exist with ethyl acetate, and which also enhances dissolution. For the spreading of the central cap, two regimes exist depending on the value taken by H . For thin aqueous layers, n cap ∼ 0.1 while n cap ∼ 1/3 for thicker layers as long as t < 0.50 s. These data are rather difficult to interpret, and we shall not comment much. However, for immiscible liquids, Fraaije and Cazabat (7) deduced thoretically that, when the dissipation occurs in the whole sublayer, n cap is equal to 1/8 if the spreading is capillarydriven, and n cap = 1/6 if it is gravity-driven. Assuming that dissipation only occurs in the viscous boundary layer, we find that n cap = 3/16 if the spreading is capillary-driven and n cap = 1/4 if it is gravity-driven. Actually, no convective cell was observed below the central cap during the very first stage of the spreading (t < 0.30 s). In the cap there is a huge excess of liquid, there is no surface tension gradient, and spreading occurs as if the liquids were immiscible. On very thin sublayers, our data compare fairly well with these theoretical predictions for spreading. On thicker, although not too thick, films, the measured spreading exponents are higher than the predicted values.

383

SPREADING OF PARTIALLY MISCIBLE LIQUIDS

5. CONCLUSION

The spreading of partially miscible liquids follows power laws, like that of immiscible liquids. However, the value of the spreading parameters, the power and the prefactor, are largely influenced by dissolution. Dissolution generates convective cells just below the interface between the two miscible liquids. It decreases the amount of materials to spread at a rate which depends on the penetration of the convective cells in the aqueous sublayer, decreasing therefore the prefactor. When the spreading liquid is less dense than the underlying liquid, the sublayer becomes stably stratified and the Benard–Marangoni convective cells are confined to a narrow layer. When the sublayer is unstably stratified, the convective cells penetrate deeply in the sublayer, which decreases still more the spreading prefactor. As long as the sublayer is thick enough that there is no hydrodynamic interaction between the convective cells and the bottom wall, only the spreading prefactor is modified. When the sublayer thickness becomes comparable to the characteristic dimension of the convective cell, there is an additional source of viscous dissipation which drastically decreases the spreading parameter.

ACKNOWLEDGMENTS This research has been supported by a predoctoral fellowship of the Spanish Ministry of Education and Science to M.S.-R., by DGICYT (Spain) Grant PB 96-599, and by the European Union Grant INCOPAC (Network). Partial financial support of the CNES (Centre National des Etudes Spatiales) is also acknowledged. One of us (M.G.V.) gratefully acknowledges the support of the

University of Marne-la-Vall´ee for a visiting professorship, and he wishes to express his gratitude to the LPMDI for the hospitality received.

REFERENCES 1. Harkins, W. D., “The Physical Chemistry of Surface Films.” Reinhold, New York, 1952. 2. Davies, J. T., and Rideal, E. K., “Interfacial Phenomena.” Academic Press, San Diego, 1963. 3. Edwards, D. A., Brenner, H., and Wasan, D. T., “Interfacial Transport Processes and Rheology.” Butterworth–Heinemann, Boston, 1991. 4. Hoult, D. P., Annu. Rev. Fluid Mech. 4, 341–368 (1972). 5. Dussaud, A., and Troian, S., Phys. Fluids 10, 23–38 (1998). 6. Gaver, D. P., and Grotberg, J. B., J. Fluid Mech. 235, 399–414 (1972). 7. Fraaije, J. G. E. M., and Cazabat, A. M., J. Colloid Interface Sci. 133, 452–460 (1989). 8. Starov, V. M., De Ryck, A., and Velarde, M. G., J. Colloid Interface Sci. 190, 104–113 (1997). 9. Suciu, D. G., Smigelschi, O., and Ruckenstein, E., AIChE J. 13, 1120–1124 (1967). 10. Suciu, D. G., Smigelschi, O., and Ruckenstein, E., AIChE J. 15, 686–689 (1969). 11. Ruckenstein, E., Smigelschi, O., and Suciu, D. G., Chem. Eng. Sci. 25, 1249–1254 (1970). 12. Santiago-Rosanne, M., Vignes-Adler, M., and Velarde, M. G., J. Colloid Interface Sci. 191, 65–80 (1997). 13. Brun, E. A., and Martinot-Lagarde, A., “M´ecanique des Fluides.” Dunod, Paris, 1960. 14. Mendes-Tatsis, A., private communication. 15. Koschmieder, E. L., “B´enard Cells and Taylor Vortices.” University Press, Cambridge, 1993. 16. Berg, J. C., and Morig, C. R., Chem. Eng. Sci. 24, 937–946 (1969). 17. Tanner, L. H., J. Phys. D 12, 473 (1979). 18. Normand, C., Pomeau, Y., and Velarde, M. G., Rev. Mod. Phys. 49, 581–624 (1977).