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Langmuir 2007, 23, 1234-1241
Experimental Investigation of Evaporation and Condensation in the Contact Line Region of a Thin Liquid Film Experiencing Small Thermal Perturbations Rajendra Argade, Sombuddha Ghosh, Sirshendu De, and Sunando DasGupta* Department of Chemical Engineering, Indian Institute of Technology, Kharagpur 721302, India ReceiVed July 19, 2006 Image-analyzing interferometry technique is successfully used to study microscale transport processes related to a curved microfilm on a solid substrate. Digital image processing is used to analyze the images of interference fringes, leading to the evaluation of liquid (heptane) film thickness and curvature profiles at different inclinations on a high refractive index glass surface. The curvature profiles obtained at different inclinations clearly demonstrate that there is a maximum in curvature near the junction of the adsorbed film (of uniform thickness) and the curved film, and then it becomes constant in the thicker portions of the film. The adsorbed film thickness is measured for horizontal as well as inclined positions. Experimentally obtained values of the dispersion constants are compared to those predicted from the Dzyaloshinskii-Lifshitz-Pitaevskii (DLP) theory, and reasonable agreements were obtained. A parameter R is defined and experimentally evaluated to quantify the closeness of the system to equilibrium. The nonequilibrium behavior of this parameter R is also observed with certain heat input at a particular inclination. A small thermal perturbation is used to force the liquid meniscus to undergo a cycle of alternating condensation and evaporation. High-speed video-microscopy and subsequent image analysis are used for data analysis. The numerical solution of a model that takes into account the balance between the suction and the capillary force is compared with the data to elicit new insights into the evaporation/condensation phenomena and to estimate the interfacial temperature differences for near-equilibrium situations.
1. Introduction The intermolecular interactions between a thin film of liquid, its vapor, and a solid surface are crucial to many equilibrium and nonequilibrium processes such as adsorption, spreading, evaporation and condensation, wetting, and stability of thin films. The applications can be as varied as coating (photographic films, automobile exteriors, photoresistant deposition), surface cleaning, printing, spraying, and microscale transport processes. Furthermore, the ability to understand and control these interactions is growing in importance for the understanding and optimization of lab-on-a-chip processes. The reason for analysis of thin films as independent cases, rather than deducing it directly from the force balance of fluid mechanics as in macroscopic physics, is because free energy of a very thin film of liquid on a solid substrate is different from that of a bulk fluid. Derjaguin and co-workers pioneered the use of the disjoining pressure concept to model the variation in force with film thickness in the study of the mechanics and thermodynamics of very thin films.1-4 They measured the stability and equilibrium thickness of adsorbed ultrathin films as a function of an interfacial temperature jump. The presence of a liquid-vapor interfacial pressure jump has also been experimentally demonstrated. Blake5 used a vapor/air bubble pressed against a liquid film on a glass substrate. Potash and Wayner6 developed a Kelvin Clapeyron model to describe evaporation from an extended meniscus on a vertical flat plate. The model included the curved evaporating and the adjacent flat, equilibrium, adsorbed thin film regions. * Corresponding author. E-mail:
[email protected]. (1) Derjaguin, B. V.; Zorin, A. M. Proceedings of the 2nd International Congress on Surface ActiVity, London, Apr 8-13, 1957; Butterworths Scientific Publications Ltd.: London, England, 1957; p 145. (2) Derjaguin, B. V.; Shcherbakov, L. M. Colloid J. USSR 1961, 23, 33. (3) Derjaguin, B. V.; Churaev, N. V. Colloid J. USSR 1976, 38, 438. (4) Derjaguin, B. V.; Nerpin, S. V.; Churaev, N. V. Bull Rilem. 1965, 29, 93. (5) Blake, T. D. J. Chem. Soc., Faraday Trans. 1 1975, 71,192. (6) Potash, Jr. M.; Wayner, P. C., Jr. Int. J. Heat Mass Transfer 1972, 15, 1851.
The combined effects of disjoining pressure, capillarity, and the temperature on the vapor pressure were demonstrated. The concept of an extended evaporating meniscus7 as a result of the long range Van der Waals forces is found to be extremely important in spreading and wetting. Sharma8 investigated the dependence of the equilibrium macroscopic contact angle of a drop on the free energy of a thin flat film coexisting with a drop. The models, which describe why a polar liquid is partially wetting, are based on the augmented Young-Laplace equation. The augmented Young-Laplace equation is used by many researchers to take into account the effect of both liquid-vapor and liquid-solid interfaces on the effective pressure jump at the interface. This pressure jump is a function of the liquid-vapor surface tension and interfacial radius of curvature. However, near the solid-liquid interface, additional changes in the stress field within the liquid occur because of changes in the intermolecular force field due to solid molecules replacing the liquid molecules. The augmented Young-Laplace equation relates the “pressure jump” at the evaporating liquid-vapor interface in the contact line region to the capillary pressure and the intermolecular force field present in the liquid meniscus. The works by Wayner,9 Troung and Wayner,10 and Renk et al.11-12 have resulted in the validation of the applicability of the augmented Young-Laplace equation to model the equilibrium shape of the isothermal thin film region as well as in nonequilibrium systems.13 Moosman and Homsy14 modeled the transport process in the contact line region of an evaporating wetting meniscus and (7) Wayner, P. C., Jr.; Kao, Y. K.; LaCroix, L. V. Int. J. Heat Mass Transfer 1976, 19, 487. (8) Sharma, A. Langmuir 1993, 9, 3580. (9) Wayner, P. C., Jr. Colloids Surf. 1991, 52, 71. (10) Troung, J. G.; Wayner, P. C., Jr. J. Chem. Phys. 1987, 87, 4180. (11) Renk, F. J.; Wayner, P. C., Jr. ASME J. Heat Transfer 1979a, 101, 55. (12) Renk, F. J.; Wayner, P. C., Jr. ASME J. Heat Transfer 1979b, 101, 59. (13) DasGupta, S.; Plawsky, J. L.; Wayner, P. C., Jr. AIChE J. 1995, 41, 2140. (14) Moosman, S.; Homsy, G. M. J. Colloid Interface. Sci. 1980, 73, 212.
10.1021/la062098m CCC: $37.00 © 2007 American Chemical Society Published on Web 12/02/2006
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demonstrated a high heat flux in the contact line region. Hockings studied the influence of intermolecular forces on thin films, developed an expression for the disjoining pressure in terms of the contact angle, and used the same to find the equation of the shape of a free surface governed by capillarity, gravitational forces, and Van der Waal’s forces.15 Evolution and instabilities of liquid-vapor interfaces were studied by various researchers. For example, de Gennes presented a unified model of dry spreading by considering a precursor film around a spreading drop16 and showed a scaling analysis of the upward flow of a wetting fluid.17 Dussan et al.18 and Marsh et al.19 experimentally validated a theoretical expression for the slope of a fluid interface by using an asymptotic form of the slope at the contact line as the boundary condition. Oron et al.20 presented a comprehensive review on the evolution of thin films. The effects of hydrodynamics and spreading on contact line motion were also studied in the past. For example, Goodwin and Homsy21 modeled the flow near the contact line for a finite contact angle system and concluded the presence of a recirculation region near the contact line. Lo´pez et al.22 developed a model based on the contact angle at the leading edge to study the contact line stability of a coating film. Zheng et al.23 have presented data on an unstable oscillating, evaporating thin film of pentane where moving velocities of the oscillating film were obtained. A force balance for the oscillating meniscus based on intermolecular and shape-governed forces was used to describe the oscillating velocities. However, the details of the region below a film thickness of 0.1 mm were not adequately addressed because of the relatively high velocity. Panchamgam et al.24 presented experimental data on the thickness, interfacial slope (a measure of contact angle), and curvature profiles of a moving, evaporating, thin liquid film of pentane on a quartz surface with emphasis on the region where the film thickness is below 0.1 mm. Herein, image-analyzing interferometry with an improved data analysis technique is used to study experimentally the liquidvapor interfacial profile, including the profile in the contact line region during condensation and evaporation in a specially designed experimental cell. The experimental system consists of a wetting liquid (heptane) on a high refractive index glass. The high refractive index glass substrate enhances the contrast of the interference fringes substantially. The thickness and curvature profiles of the fluid are measured at different inclinations to calculate the value of dispersion constants, in situ. The system is then thermally perturbed from its isothermal condition by the addition of a very small amount of heat to cause evaporation, thereby causing departure of the system from equilibrium. The heat source is manipulated such that the film undergoes an evaporation-condensation cycle. A large number of the images of the interference fringes of the liquid film undergoing phase change are captured, and the adsorbed film thickness and curvature profiles are evaluated. The solution of a model based on the augmented Young-Laplace equation provides valuable insight into the physics of the evaporation-condensation process in terms of the balance between capillary and suction forces. (15) Hockings, L. M. Phys. Fluids A 1993, 5, 794. (16) Joanny, J. F.; de Gennes, P. G. J. Phys. (Paris) 1986, 47, 121. (17) de Gennes, P. G. ReV. Mod. Phys. 1985, 57, 827. (18) Dussan, E. B. V.; Rame, E.; Garoff, S. J. Fluid Mech. 1991, 230, 97. (19) Marsh, J. A.; Garoff, S.; Dussan, E. B. V. Phys. ReV. Lett. 1993, 70 (18), 2778. (20) Oron, A.; Davis, S. H.; Bankoff, S. G. ReV. Mod. Phys. 1997, 69 (3), 931. (21) Goodwin, R. R.; Homsy, G. M. Phys. Fluids A 1991, 3, 515. (22) Lo´pez, P. G.; Miksis, M. J.; Bankoff, S. G. Phys. Fluids 1997, 9, 2177. (23) Zheng, L.; Plawsky, J. L.; Wayner, P. C., Jr.; DasGupta, S. ASME J. Heat Transfer 2004, 126, 169. (24) Panchamgam, S.; Gokhale, S. J.; Plawsky, J. L.; Wayner, P. C., Jr.; DasGupta, S. ASME J. Heat Transfer 2005, 127, 232.
Langmuir, Vol. 23, No. 3, 2007 1235
Figure 1. A schematic diagram of the experimental setup.
2. Experimental Section A. Experimental Setup. A schematic diagram of the experimental heat transfer cell with the image processing setup with an enlarged view of the liquid meniscus is shown in Figure 1. The circular heat transfer cell encompasses the substrate, a high refractive index glass (LaFN21, R.I.1.788) having dimensions of 35 mm × 25 mm × 10 mm in a stainless steel cubical casing consisting of three main parts, namely a stainless steel base, middle circular disc with cubical casing, which is having slightly larger dimensions than the glass, and a top plate with provision of a glass cover through which the substrate can be viewed at all times. A stainless steel base plate and a cover plate isolate the glass substrate from the environment with the help of Teflon gaskets. The top cover plate with a matching viewing port is placed just above the glass substrate, and the whole setup is tightened with brass screws to prevent liquid leakage from the corners of the cell. Test liquid is added through a small orifice using a glass syringe, and then the orifice is closed with a screw. A very thin strip heater is also attached to the top viewing-glass plate that can provide very small amounts of heat during the perturbation (condensation/ evaporation) experiments. The whole setup is placed on the stage of a Leica DM-LM microscope, attached to the image-processing computer. Interference fringes can be viewed immediately near the corner menisci. A Canon Power shot S 70 digital camera with 6× optical zoom is used to capture the images at regular intervals of time. The camera is remotely controlled by Zoombrowzer EX software. Once the microscope is focused at the desired interference fringe pattern, remote shooting is used to capture the images and the images are transferred to the computer for further analysis. B. Experimental Procedure. The cleaning techniques are extremely important for this study, as the presence of even minute dust particles on the substrate surface significantly reduces the surface energy and changes the fringe patterns. They also act as nucleation sites during condensation and can result in a thick film region even in front of the adsorbed film region of the meniscus. The cleaning is carried out in a class 100 clean hood where laminar, filtered airflow is maintained. Initially the system is dipped in ethanol for 3-4 h and then further cleaned inside the clean hood with wet tissue paper using ethanol to get rid of any adsorbed impurities. The whole
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Figure 2. Typical interference pattern obtained for heptane on glass. setup is dipped again in ethanol for at least 0.5 h and rinsed repeatedly. Next the setup is dried in an oven at 120 °C for 1 h to remove remaining ethanol. The setup is transferred to the clean hood and rinsed thoroughly with the working liquid (Heptane, HPLC grade of 99.5+ % purity). The setup is assembled inside the clean hood. The working fluid is introduced through the small orifice with the help of a cleaned glass syringe. The screws are tightened properly to ensure no leakage of the working fluid from the corners and the orifice. The assembled setup is then placed under the microscope and examined carefully. Interferences fringes are readily formed and can be viewed clearly in the monochromatic light (of 546 nm wavelength). If dust particles are present, they will be visible immediately as lenses will form around them on the high refractive index glass surface. If the system is not properly cleaned, the whole cleaning process is repeated. In this way the cleanliness of the system is ensured for each experimental run. In order to calculate the gray value of the bare surface (Go), before rinsing, the assembled setup is placed under the microscope and Go is measured at multiple locations on the glass substrate. Images of the interference fringes are captured at different inclinations including the horizontal position. During the perturbation experiments to study evaporation and condensation of the liquid film, a strip heater, attached to the top glass plate, is operated. The images of the interference fringes are captured using a Canon Powershot S70 digital camera with 6× optical zoom. The camera can be operated in the remote shooting mode. Once the microscope is focused at the desired interference fringe pattern, remote shooting is used to capture the image. The reflectivity images are analyzed with Image-Pro Plus software (version 4.5). The images captured are digitized into 1024 (horizontal) × 768 (vertical) space pixels and assigned one of 256 possible gray values representing intensity from 0 (black) to 255 (white). The gray value at each pixel is a measure of the reflectivity. Thus, each microscopic pixel acts as an individual, simultaneous light sensor. For the magnification used in this study each pixel represented an area of 0.144 µm in diameter. C. Image Analysis Technique. From each image of the straight interference fringes a Line Profile Analysis of the gray values of the pixels are extracted using Image Pro Plus as shown in Figure 2. Thus an array of pixel gray value (G) versus pixel position (x) is obtained. To smoothen out local fluctuations, 3 × 3 Median Smoothing is used occasionally. Thus, a line profile plot of the gray value is obtained as shown in Figure 3. The details of the image analysis technique are described elsewhere.19,20 It is to be noted that when the film gets very thin (adsorbed film region), the index of refraction of the liquid may not remain constant, which makes interpretation more difficult. This necessitates the use of a relative gray value (G h ), as G h (x) )
Figure 3. Typical grey value profile with fitted minima and maxima at each pixel position. for most of the data, a third-order polynomial for the interpolatory envelope is sufficient and the evaluated film thickness profiles are found to be independent of the order of this polynomial. The values of G and thereby Gmin(x) and Gmax(x) are evaluated at every pixel (which, for the experimental system considered herein, represents an area of diameter equal to 0.144 µm). If a line parallel to the y-axis is drawn at each pixel location, its intersection with the interpolatory envelopes of Gmin(x) and Gmax(x) are the relevant values of these two quantities to be used subsequently for thickness evaluation. The thicknesses at the inflection points are explicitly known by standard formulas for constructive and destructive fringe formations.13 The values of Gmin(x) for positions ahead of the zeroth dark fringe are taken to be the value at the zeroth dark fringe. The value of Gmax(x) in the adsorbed flat film region is equal to the measured gray value (G0) of the bare glass surface. This gray value corresponds to a film thickness equal to zero and was measured using a dry cell, as explained previously with the same light intensity.13,24 A linear interpolation was used to connect the gray value of the first bright fringe and this value of G0 at the beginning of the adsorbed section. Since reflectivity is a function of the adsorbed thin film thickness, the change in reflectivity in the flat adsorbed film region, given by the difference between G0 and G(x), gives the thickness of the adsorbed thin film. Since the gray value at each pixel location is known from the experimental data, the reflectivity of the liquid film can be calculated at each pixel position using eq 2 and eq 3 RL(x) ) G h (x)[RLmax - RLmin] + RLmin RL )
Gmax (x) - Gmin(x)
,
(3)
where, θl )
2πnlδ ; λ
r1 )
nl - nv ; nl + nv
r2 )
ns - nl ; ns + nl
δ is the film thickness of the liquid at any pixel and R ) r12 + r22;
G(x) - Gmin(x)
R + β cos 2θl κ + β cos 2θl
(2)
β ) 2r1r2;
κ ) 1 + r12r22
(4)
(1)
where, Gmin(x)and Gmax(x) are the interpolatory envelopes to the various order minima and maxima (constructive and destructive fringes). The interpolatory envelopes [Gmin(x) and Gmax(x) ] are drawn by fitting the respective maxima and minima with polynomials of the order corresponding to an error of less than 1%. It is found that
Here nv, nl , ns is the refractive index of the vapor, liquid, and solid, respectively, and λ is the wavelength of light. The film thickness at each pixel location is related to the gray value at that pixel location according to eqs 2-4. The error associated with the film thickness measurement was estimated to be (0.01 µm in the transition and capillary region of the film as discussed in detail in a previous publication.24
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Langmuir, Vol. 23, No. 3, 2007 1237
Figure 5. Slope (dδ/dx) as a function of relative distance at an inclination of 11.32°.
Figure 4. The variation of thickness profile with respect to distance at horizontal position and at an angle of 11.32°. Table 1. Characteristics of the Heptane Meniscus (Zero Heat Input) angle of inclination (deg )
δ0 (nm)
K (m-1)
a
Bexpt (J·m)
0 5.21 11.32 12.64 13.94
20.1 18.3 15.1 10.5 10.93
213 403 430 408 486
0.832 0.881 0.941 0.821 0.861
-3.21 × 10-31 -5.24 × 10-31 -3.38 × 10-31 -0.434 × 10-31 -0.738 × 10-31
Once the film thickness is obtained at every pixel position, the slope (dδ/dx) of the film thickness profile (local tangent angle) and the curvature are obtained at every pixel number by fitting a thirdorder polynomial to the film thickness in the thicker portions of the meniscus (as the variation of curvature in this region is expected to be small). The starting point of this third degree polynomial is at a thickness of 0.2 µm. Since each pixel represents an area of diameter of 0.144 µm, a third degree polynomial for the film thickness at three consecutive pixel positions is found to be highly satisfactory. The curvature will change appreciably in the thinner region (below the first dark fringe, i.e., about 0.1 µm), especially if nonequilibrium effects are present, and hence a fifth degree polynomial is used for any points below 0.2 µm. The orders of the respective polynomials are decided when the values of the curvature are independent of the order of the polynomials. The curvature at each position is calculated using the following relation K)
d2δ/dx2 [1 + (dδ/dx)2]3/2
(5)
3. Results and Discussions A. Isothermal Studies. Image-analyzing interferometry is successfully used to measure the film thickness profiles of a curved microfilm. An estimate of the adsorbed thin film in front of the curved meniscus is also obtained. For each of the captured images, the film thickness is obtained experimentally at each pixel position. The number of fringes taken into consideration for calculation of film thickness depends on the magnification of the system and the cleanliness achieved prior to the experiments, as dust particles and/or adsorbed films present can significantly alter the nature and spacing of the fringes. Representative thickness profiles at isothermal condition are shown in Figure 4. The isothermal experiments are carried out at four different inclinations of 5.21°, 11.32°, 12.64°, and 13.94° with respect to the horizontal.
Figure 6. Curvatures profiles at two different inclinations.
The thickness profiles with respect to distance at horizontal position and at an angle of 11.32° are shown in Figure 4. The adsorbed film thicknesses (δ0) are 20.1 nm for the horizontal case and 15.1 nm for an inclination of 11.32°. From this figure it is clear that as the angle of inclination increases, the adsorbed film thickness decreases and the shape of the film becomes steeper. Results at other inclinations are presented in Table 1. The slopes, (dδ/dx), are calculated by analyzing the experimental data using the method described earlier, and an example of the slope profile is presented in Figure 5. The slope is always negative, as the film thickness decreases with increases in distance for the coordinate scheme chosen herein. In subsequent discussions only the absolute values of the slopes will be referred. The plot clearly demonstrates that the profile essentially starts from a flat adsorbed film region (where slope and curvature are zero), passes through a region where the slope (dδ/dx) increases rapidly, and then stabilizes to almost constant values. This implies the presence of a nearly constant curvature (K∞) region in the thicker portion of the liquid menisci. The liquid film thickness profiles and slope profiles are very sensitive to small changes in the environmental conditions and will be discussed in detail in the next section. The curvatures of the thickness profiles obtained using the experimental data and eq 5 at every pixel position at two different inclinations are plotted in Figure 6. From the figure it is clear that the curvature is zero in the adsorbed, flat film region. The
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curvature then increases and goes to a maximum in an intermediate region. The experimental results show that the liquid vapor interfacial profile of a thin liquid film has maximum curvature near the contact line region, and then it becomes constant in the thicker part of the film. A maxima in curvature near the contact line region was previously reported for an evaporating system.13,23,25 It was mentioned that the high curvature in the micro region leads to very high capillary forces and a strong flow toward the interface. DasGupta et al.25 experimentally obtained a curvature maximum in the contact line region of an evaporating thin film for a completely wetting case. The augmented Young-Laplace equation can be written for a point in the thicker portion of the meniscus, where the disjoining pressure effects are negligible. For the isothermal nonevaporating (no additional heat input, Q ) 0) case considered here, the liquid pressure will remain same, irrespective of the position at a constant gravitational level (different inclined positions). At equilibrium situation, where no evaporation or condensation is taking place, the augmented Young-Laplace equation can be written as
Pl - PV ) -σK -
∏
(6)
Figure 7. Comparison of experimental and theoretical slope profiles at an inclination of 11.32°. The experimental data at every 20th data point are only shown for clarity.
A dimensionless parameter, R, is defined next as
where
R4 ) dδ 2 -3/2 d2δ K(curvature) ) 2 1 + dx dx
[ ( )]
∏(disjoining pressure) )
-B δn
(7) (8)
In these equations, δ represents the film thickness, σ represents the surface tension, and B is a modified Hamaker constant (B < 0 for completely wetting systems). In the limit of very thin films of pure simple fluid (nonretarded region), n ) 3 and B ) A/6∏ ) A h in which A is the classical Hamaker constant, while in the thick film region (retarded region), n ) 4 and B is a dispersion constant. Equation 6 can be used to construct the slope profiles from the known values of A h , δ0, the curvature at the thicker end of the meniscus (K∞), and the physical properties of the fluid. Based on the methodology proposed by DasGupta et al.,13 the following equation can be written for two points, one in the adsorbed region and the other in the transition region.
σK -
B ) σK∞ δ4
Q)0
(9)
The curvature at the thicker portion of the meniscus (K∞) is nearly constant in the region.17 Using the simplified form of curvature, eq 9 can be modified as
σ
d2δ B - ) σK∞ dx2 δ4
(10)
The following non-dimensional variables are defined and introduced in eq 10 to obtain eq 12.
()
(11)
-B 1 d2η + ) 1. 2 dZ σK∞δ04 η4
(12)
η)
δ δ0
Z)x
(
)
K∞ δ0
1/2
(25) DasGupta, S.; Kim, I. Y.; Wayner, P. C., Jr. ASME J. Heat Transfer 1994, 116, 1007.
-B σK∞δ04
(13)
The factor R is a measure of the deviation of a specific meniscus from the equilibrium situation. For the equilibrium case, Q ) 0 and R )1. Equation 12 can be integrated to obtain the following expression for the slope of the meniscus.
(dZdη) ) 2η + 32 Rη + C 4
2
3
1
(14)
where C1 is the constant of integration, and using the boundary condition that at η ) R, dη/dZ ) 0, the slope of the meniscus can be expressed as
x
dδ ) -(K∞δ0)1/2 dx
2η +
2 R4 8 - R 3 η3 3
(15)
Note that this can be viewed as both an artificial boundary condition for a nonequilibrium system for which R > 1 and the actual boundary condition for an equilibrium system for which R ) 1. Hence, if the curvature at the thicker end of the meniscus, K∞ , along with B, δ0, and σ are known, the slope of the meniscus can be directly calculated as a function of the film thickness, using only the augmented Young-Laplace equation. The minus sign in eq 15 is indicative of the fact that for the reference frame selected, the meniscus slope should always be negative (film thickness decreases as distance increases). A slight shift from the equilibrium situation, during the experiments, can have a drastic effect on the slope profiles. Figure 7 shows one example of the close match between the experimental slope and the slope obtained by the solution of the augmented Young-Laplace equation, with R being the adjustable parameter. Herein we compare the slope of the meniscus obtained by numerical analysis of the experimental data for a system with unknown values of the Hamaker constant (or dispersion constant depending upon the adsorbed film thickness) and the slope predicted by the augmented Young-Laplace equation eq 15. The slope, a function of R and the R corresponding to the least rms error between the theoretical and experimental values of slope at each pixel position, is chosen, and those profiles are presented herein. It can be observed that for most of the cases,
InVestigation of EVaporation and Condensation
R values corresponding to the minimum rms error lie in the range 0.8-1.0. The R values evaluated in this way are subsequently used to determine the value of the dispersion constant. The values of the dispersion constants obtained from analysis of experimental data are tabulated in Table 1. However, a direct comparison with the values predicted from DLP theory cannot be made, as the wavelength-dependent optical properties required for calculation are not available for the specific high refractive glass used in this study. An order of magnitude calculation using the refractive index data for quartz glass has however been made. These approximate calculations show that the values of the dispersion constant so obtained are approximately 50-80% larger than those evaluated from the experiments. It should be emphasized here that apart from the optical properties of the substrate, the values of the dispersion constant are extremely sensitive to the cleanliness of the system.10 This underscores the importance of “in situ” evaluation of these constants, as is done in this study, for any further use. As expected, specifying an experimentally obtained curvature results in a relatively closer match at the thicker end of the meniscus, but the deviation (as a percentage from the actual slopes) increases in the thinner region (where disjoining pressures effects are present). The values of the dispersion constants obtained from the values of R are also tabulated in Table 1. However, a direct comparison with the values predicted from DLP theory cannot be made, as the wavelength-dependent optical properties required for calculation are not available for the specific high refractive glass used in this study. It should be emphasized here that the values of the dispersion constant are extremely sensitive to the cleanliness of the system,10 underscoring the importance of “in situ” evaluation of these constants for any further use. At isothermal conditions the value of R should be equal to 1. However, it must be emphasized that a very small interfacial temperature change due to uncontrolled surrounding conditions (even of the order of 10-5 K) is sufficient to disturb the equilibrium and may cause evaporation or condensation.26 This will result in departure in the values of R from its expected value of unity. Table 1 shows the variations of adsorbed film thickness δ0, curvature in the thicker portion K∞, R, and dispersion constants with respect to angle of inclination. From Table 1 we clearly see that as the angle of inclination increases, the adsorbed film thickness decreases. This is due to the body force (gravity) acting in the opposite direction that drains the film and results in a thinner adsorbed film. Also with increase in the angle of inclination, the curvature in the thicker part of the film increases as the film adjusts its shape to a higher value of the opposing body force. As the values of R suggest, the systems are not at equilibrium; in fact, in all the cases, slight evaporation is taking place (R < 1). B. Non-isothermal Studies. Perturbation Experiments. In order to study the effect of thermal perturbations on the shape and behavior of the menisci, a small strip heater is added to the upper end of the viewing glass. The idea is to perturb the system in such a way so as to force the film to undergo an evaporationcondensation cycle. A large number of images are captured during this process and analyzed to study the behavior of the system during condensation as well. The values of the adsorbed film thickness, slope, and curvature for the system during this imposed departure from equilibrium are calculated. Next the values of R are evaluated by comparing the numerical solution with the (26) Gokhale, S. J.; Plawsky, J. L.; Wayner, P. C., Jr. J. Colloid Interface Sci. 2003, 259, 354.
Langmuir, Vol. 23, No. 3, 2007 1239
Figure 8. Curvature in the thicker portion, K∞, versus adsorbed film thickness, δ0, at an inclination of 5.21° during condensation.
experimentally obtained values of the slope during this condensation and evaporation cycle. It is anticipated that with a large number of images taken at very small time intervals it will be possible to capture a case that will be extremely close to equilibrium. The analysis of these images should provide valuable insights into the interplay of the different forces present in a thin liquid film. Once the liquid film stabilizes after being introduced into the experimental cell, a very small heater is switched on (supplying approximately 0.03 W to the top viewing plate), and the curved microfilm starts receding (evaporation). After a while (approximately 10 min), the heat input to the system is switched off and condensation starts immediately, resulting in the gradual accession of the liquid film. The cycle of backward movement (evaporation) and forward movement (condensation for about 15 min) of the film can clearly be seen during the experiments through the microscope. At the start of heat supply cycle, the bottom glass substrate (of high refractive index) is slightly cooler than the top viewing plate so immediate condensation takes place on the bottom glass (substrate of interest). A forward movement of the liquid meniscus is immediately noticed along with an apparent increase in the adsorbed film thickness (reduced reflectivity of the surface) and larger separation between the fringes (less curved film). During this entire cycle, the softwaretriggered camera captures snaps of the interference fringes formed. Of the large number of images captured, the ones with good resolution are chosen, keeping in mind that they should be nearly equally spaced as far as time is concerned to capture a representative picture of the cyclic process. Thus, the results presented are not averaged over time; rather they are snapshots of the evaporation-condensation cycle at different instants of time. The images are analyzed using the methods already described. Figure 8 shows the variation of curvature in the thicker portion, K∞ versus the adsorbed film thickness, δ0, at an inclination of 5.21° during this perturbation studies. The figure clearly shows that the curvature in the thicker portion decreases with an increase in the adsorbed film thickness, i.e., during condensation. Thus, the meniscus becomes less curved and spreads forward during condensation, with an associated increase in adsorbed film thickness. The presence of a curvature gradient implies fluid flow toward or away from the contact line region. As all of these situations are concerned with either evaporation or condensation (in effect flow near the contact line), this figure clearly demonstrates the importance of a curvature gradient in contact line motion. The parameter R signifies the closeness of the system to
1240 Langmuir, Vol. 23, No. 3, 2007
Argade et al. Table 2. Interfacial Temperature Difference during Experiments with No Heat Inputs and at Near-Equilibrium Conditions
Figure 9. Plot of variation of R with adsorbed film thickness (δ0) during condensation and evaporation at an inclination of 13.94°.
Figure 10. Plot of Suction potential (B/δ04) against capillary force (σ K∞).
equilibrium. Ideally, the value of R should be 1 if the liquid film is at equilibrium. However, as presented earlier, a very small disturbance caused by uncontrolled conditions in the surroundings can cause a shift of the system from equilibrium, resulting in either evaporation or condensation. Figure 9 shows the variation of R with adsorbed film thickness δ0 during condensation and evaporation experiments at an inclination of 13.94°. Since at equilibrium R should be equal to 1, it can be seen from the figure that the specific experiment corresponding to an adsorbed film thickness of 3.39 × 10-8 m is closest to equilibrium. It is clear that a value of R less than 1 results in a lower adsorbed film thickness, thereby signifying evaporation. The opposite trend is seen for the three data points to the right of the point corresponding to the film thickness of 3.39 × 10-8 m, thereby signifying condensation (R > 1). The results are entirely consistent with visual observation of the behavior (receding during evaporation, advancing during condensation) of the film under the microscope. Thus, the values of R obtained by the analysis of experimental data points are definite indicators of the closeness of the system to equilibrium. The results presented in Figures 9 and 10 are evaluated from the same set of experiments. The points in Figures 9 and 10 are first selected from the evaporation (heat on) and condensation (heat switched off) cycle and then analyzed for various quantities
adsorbed film thickness, δ0 (m)
interfacial temp difference, ∆Tlv (K)
2.01 × 10-8 1.83 × 10-8 1.51 × 10-8 1.05 × 10-8 1.09 × 10-8 3.39 × 10-8 (perturbation experiment)
4.40 × 10-5 6.30 × 10-5 1.29 × 10-4 4.78 × 10-4 4.10 × 10-4 5.82 × 10-6
(R, B, ∆T, suction potential, capillary force, etc.). The film thickness profiles of selected experiments during the evaporationcondensation cycle are measured, and using the theoretical approach presented before, various parameters, e.g., R, B, suction potential, and capillary force are evaluated. Figure 9 represents a plot of R, a measure of the deviation from equilibrium versus adsorbed film thickness. Figure 10 represents the delicate balance between the suction potential at the contact line and the capillary force at the relatively thicker portion of the meniscus. The results (deviation of R from unity, Figure 9; balance between the forces, Figure 10) demonstrate the coincidence of physical observation and data analysis. The suction potential of an evaporating thin film can be represented by the term B/δ0,4 which is a measure of the excess potential of a thin film (disjoining pressure). If for a system B/δ04 (in the adsorbed film region) is more than the capillary force (represented by σK∞) in the thicker portion of the film, the system will try to revert back to equilibrium (as B/δ04 is equal to σK∞ at equilibrium). This is possible if δ0 increases, thereby resulting in a decrease of B/δ0.4 This leads to condensation and advancing menisci under the action of a favorable pressure gradient. The opposite happens during evaporation. At equilibrium, this suction potential will be exactly balanced by the capillary force (σK∞) acting on the thicker part of the meniscus. Thus, there will be no flow across the contact line in either direction (evaporation or condensation) for a film at equilibrium. Figure 10 is a plot of the suction potential against capillary force for all the experiments. The points above the diagonal represent condensation during experiments (suction being larger than capillary resulting in forward movement of the meniscus and increasing adsorbed film thickness), and the points below the diagonal line signify evaporation (capillary force being larger than suction resulting in movement of the meniscus toward the capillary region and decreasing adsorbed film thickness). Thus, it is clear that all these experiments reported herein agree well with the basic physics of the process. It can be clearly observed that the experiment corresponding to an adsorbed film thickness of 3.39 × 10-8 m is extremely close to equilibrium, as it is very close to the diagonal, signifying the near equality of the two forces. Previous researchers26 have shown that a very small interfacial temperature difference is sufficient to cause evaporation/ condensation. The value of the interfacial temperature difference, ∆Tlv, can be estimated from the following equation:
δ0 )
(
)
BTi ∆h∆Tlv
1/4
(16)
where B is the retarded dispersion constant, Ti the interfacial temperature, δ0 the adsorbed film thickness, ∆h the latent heat of vaporization of the liquid per unit volume, and ∆Tlv the interfacial temperature difference. Table 2 lists the values of interfacial temperature differences and adsorbed film thickness at conditions with no heat input but at different angles of
InVestigation of EVaporation and Condensation
inclinations. Only one data point from the perturbation experiments is included in the last row of the table. From Table 2 it is clear that the value of the interfacial temperature difference ∆Tlv for the case corresponding to an adsorbed film thickness of 3.39 × 10 -8 m is 2 orders of magnitude smaller than the other cases with no heat inputs. Thus, though it is extremely difficult to maintain equilibrium in such systems, a carefully planned set of experiments have allowed us to capture a near-equilibrium situation. These results emphasize the critical role played by very small interfacial temperature differences for all the important transport processes near the contact line region of a curved microfilm.
4. Conclusions A heat transfer cell is designed and fabricated where a stable liquid meniscus with an adsorbed thin film was formed on a high refractive index glass surface (to enhance the contrast of the images). Image analyzing interferometry technique is used for accurate measurement of the liquid film thickness profile, including an estimate of the adsorbed film thickness based on the relative reflectivity of the surface. The profiles are obtained for menisci in equilibrium as well as in near equilibrium cases.
Langmuir, Vol. 23, No. 3, 2007 1241
The film thickness profile characterizes the pressure field, and important information is obtained by analyzing the slope, curvature, etc., of the profile. A model based on the augmented Young-Laplace equation is used to study the change in the effective pressure at the liquidvapor interface. The parameter R is calculated to check the sensitivity of the system to small perturbations in the environment. A small thermal perturbation is used to force the liquid meniscus to undergo a cycle of alternating condensation and evaporation. High-speed video-microscopy and subsequent image analysis are used to analyze the data, including a near-equilibrium condition. The modeling accurately captures the physics of the process, including the trend in the values of the adsorbed film thickness R and the balance between the suction and the capillary force, etc. The interfacial temperature difference for a large number of zero heat input cases at different inclination angles are also calculated. However, the temperature difference is at least 2 orders of magnitude smaller for the near-equilibrium situation captured during the perturbation experiments compared to the zero heat input cases showing the sensitivity of the system. LA062098M
