## Abstract

The extended meniscus and the intermolecular and capillary forces that govern its behavior and connection to change-of-phase heat transfer have been the subject of an increasing body of research over the past 50 years. We have been fortunate to be at the forefront of this effort starting from the development of a capillary feeder, in Earth's gravity, to stabilize film boiling to running a series of transparent heat pipe experiments aboard the International space station hoping to better understand the role of intermolecular forces in microgravity. The use of ellipsometry and interferometry to highlight the location and state of the vapor–liquid interface have been key to these studies and have helped to uncover many new, interesting, and sometimes unexpected phenomena associated with fluid flow and change-of-phase heat transfer.

## Introduction

The interest in the evaporating extended meniscus started with the evaluation of the effect of temperature and the intermolecular force of attraction on the controlled phase change in the capillary region of flow in the porous medium presented in Fig. 1. Essentially, the evaporation region of the porous medium acted like a portion of a bundle of heat pipes. In preliminary work, experimental equipment to study standard film boiling of liquid nitrogen on a porous heat source with flow through the heat source was being developed. In the Ph.D. thesis, this mode of heat transfer was unstable due to large-scale turbulence. However, stability was achieved by placing a porous flow control element to promote internal evaporation on the liquid side of the porous heater. The evaporating meniscus stabilized the process.

Fig. 1
Fig. 1
Close modal

A diagram of the experimental system is presented in Fig. 1, where (P) is the porous element with the liquid–vapor interface contained within; (H) is the porous heater with vapor flow out. The temperature of the outgoing vapor flow was a function of the movement of the plane of evaporation in the porous media with changes in heat input. In retrospect, the original device developed in the thesis was unstable because of the presence of gravity with no capillarity. Once capillarity within the porous media was established the liquid was stabilized in a gravitational field. Eventually, studies in microgravity on the International space station were used to minimize the effect of gravity and emphasize the expanded contact line region present in capillarity. The saying “What goes around, comes around” seems to apply. Additional results concerning the experiments are given in Ref. [1].

## Extended Evaporating Meniscus on a Vertical Flat Plate

Interfacial phenomena, in general, and more specifically, the phenomena associated with evaporation from a meniscus, play a major role in many current engineering applications, which require high local heat fluxes. For example, the extended evaporating meniscus based on the intermolecular force of attraction is one of the limiting parameters in heat pipes, grooved evaporators, vapor chambers, fuel cells, porous media, suction nucleate boiling devices, and sweat cooling devices. As described in Ref. [2], the study of the transport processes occurring in an evaporating, extended meniscus proved useful in the improvement of such applications. In Fig. 2, the following terms apply: extended meniscus—the combined intrinsic meniscus and the adsorbed thin film due to disjoining pressure extending above it (if present); intrinsic meniscus—that portion of the extended meniscus profile which is described by the conventional equation of capillarity while excluding the effects of disjoining pressure; extended thin films—that portion of the extended meniscus above the intrinsic meniscus, which is mainly described by the disjoining pressure, Pd, which is due to the force of attraction.

Fig. 2
Fig. 2
Close modal
The effect of disjoining pressure, Pd, on the vapor pressure of the film, Pvlv, is specified by an expanded form of the Kelvin equation in which the effects of disjoining pressure and curvature on the film vapor pressure may be combined, with the result given in Eq. (1). In this equation, K is the film curvature, σlv, is the interfacial tension, Mw is the molecular weight, ρl is the density of the fluid, and Tlv is the temperature at the vapor–liquid interface
$PvlvPvlvi=exp[(Pd−σlvK)MwρlRTlv]$
(1)

This form of the equation allows for a smooth transition from the curvature-controlled region to the disjoining pressure-controlled region. Pd is negative for a wetting fluid since the Hamaker constant will be negative.

A schematic diagram representing the change-of-phase process in the evaporating meniscus is given in Fig. 3. In the general case where we have the wall heated, there can be both evaporation and condensation occurring simultaneously in different parts of the meniscus. The degree of condensation depends upon the curvature of the meniscus and evaporation peaks at the contact line. The situation is far from equilibrium and heat fluxes, temperatures, and pressures in the liquid all vary with position. Due to surface tension gradients, temperature and concentration-driven Marangoni flows are possible. The figure also highlights how the adhesion force, meniscus curvature, heat flux, and thermal resistance are related to one another and how they vary throughout the different portions of the meniscus.

Fig. 3
Fig. 3
Close modal

The curvature of the vapor–liquid interface plays a major role in defining the Gibbs energy and so measuring that curvature was important to verify the theory. Experimental measurements of the extended meniscus thickness profile in the contact line region of a completely wetting film of ethanol on a flat plate were first measured using a microcomputer-enhanced interferometer and the results were presented in Ref. [3]. A picture of the recorded fringe pattern is presented in Fig. 4. The extreme sensitivity of the system to the nonequilibrium effects associated with volatile liquids was demonstrated in these studies and evaluated using the reflectivity profile. The inferred equilibrium profile agreed with theoretical predictions for the width of the transition region, the curvature, and the thickness of the adsorbed film. The procedures allowed the interfacial properties of the system to be evaluated in situ, at the start of dynamic studies, and then to be used to describe the transport processes associated with evaporation and condensation. The effect of transport processes on the thickness profile agreed with previous theoretical models. The convenience of microcomputer-enhanced video microscopy naturally leads to a better understanding of the transport processes in the contact line region. The results demonstrated that the near-equilibrium processes of change-of-phase heat transfer and fluid flow in thin films are intrinsically connected because of their common dependence on the intermolecular force field and gravity.

Fig. 4
Fig. 4
Close modal

The availability of enhanced data from interferometry helped advance theory. In Ref. [4], the method of matched asymptotic expansions was used to show the details of the horizontal meniscus, which is influenced by both capillarity and the adsorbed thin film. These results would be compared with and validated by images such as in Fig. 4. Interferometry has limitations in the range of thicknesses that can be measured. This is especially important in the contact line and adsorbed film regions where the film thickness can be below the resolution of interferometry. Ellipsometry can measure film thicknesses down to a monolayer. In Ref. [5], the use of a scanning microphotometer with ellipsometry was first used to determine the evaporative heat transfer characteristics of the extended contact line region. A cross section of the combined system is presented in Fig. 5 (Table 1).

Fig. 5
Fig. 5
Close modal
Table 1

Legend for Fig. 5

AAnalyzerVCR IPMImage processing computer
CCompensatorLHelium-neon laser
PPolarizerMULMonochromatic, unpolarized light
MMicroscopeCPLCircularly polarized light
DDetectorPPLPlane polarized light
FFilterPMPhotomultiplier
FCGFrame code generatorVAVariable aperture
HRVImage resolutionSSScanning stage
VCVideo camera
AAnalyzerVCR IPMImage processing computer
CCompensatorLHelium-neon laser
PPolarizerMULMonochromatic, unpolarized light
MMicroscopeCPLCircularly polarized light
DDetectorPPLPlane polarized light
FFilterPMPhotomultiplier
FCGFrame code generatorVAVariable aperture
HRVImage resolutionSSScanning stage
VCVideo camera

A picture of the interference fringes and the results of the profile analysis are presented in Fig. 6. This figure was one of the first showing that the effect of evaporation could be easily seen in a plot of the square root of film thickness, δ1/2 versus position. Deviations from a straight line signified the presence of phase change.

Fig. 6
Fig. 6
Close modal

The contact line dynamics of surfactant-laden microdrops were studied in Ref. [6]. An optical technique based on the reflectivity measurements of a thin film was developed to extend interferometry and eliminate the need for an ellipsometer to estimate the adsorbed film thickness. While the ellipsometer was accurate, its spot size was large, and the extent of the contact line region is much smaller than the spot size. The new technique was used to experimentally study the spreading, evaporation, contact line motion, and thin film characteristics of drops consisting of a water-surfactant (polyalkyleneoxide-modified heptamethyltrisiloxane, called a superspreader) solution on a fused silica surface. Based on the experimental observations, we concluded that the surfactant adsorbs primarily at the solid liquid and liquid–vapor interfaces near the contact line region. At equilibrium, the completely wetting corner meniscus was associated with a flat adsorbed film having a thickness of 31 nm. The calculated Hamaker constant, A= –4.47 × 1020 J, shows that this thin film was stable under equilibrium conditions. During a subsequent evaporation/condensation phase-change process, the thin film of the surfactant solution was unstable, and it broke into microdrops having a finite contact angle. The surfactant pinned the contact line of the drops so that evaporation and condensation repeatedly occurred over the same area. A picture of the observations is presented in Fig. 7.

Fig. 7
Fig. 7
Close modal

## Constrained Vapor Bubble Experimental System

Some experimental observations using a relatively simple, sealed system consisting only of a liquid in contact with its vapor give direction to developing a heat pipe and obtaining information on the general force of attraction. Using a transparent system allows one to map the entire vapor–liquid interface in the device. The details of the following brief description of the constrained vapor bubble (CVB), experimental system are given in Refs. [713]. In Fig. 8, the not-to-scale cross-sectional profile of the evaporating pentane meniscus in a square quartz cuvette is presented [13]. Some results from the first isothermal and heated menisci measured are given in Fig. 9.

Fig. 8
Fig. 8
Close modal
Fig. 9
Fig. 9
Close modal
Fig. 10
Fig. 10
Close modal

The upper extended menisci in the device are fed by capillary flow from the corner grooves. These grooves are connected to the liquid reservoir at the bottom. At equilibrium, the accuracy of the optical measurements can be checked using statics, pitting the force of gravity against the capillary and disjoining pressure forces. At any height, using the liquid curvature in the corner, and a measurement of the adsorbed film thickness toward the center, allows one to calculate an in situ estimate of the Hamaker constant.

To use the CVB as a form of linear heat pipe, a heater was attached to the top of the cuvette, a condenser of various forms, at the bottom. The outside surface axial temperature profile was measured. Since one end of the cuvette was open to enable filling, a pressure transducer was attached that allowed one to measure the internal vapor pressure and predict where condensation would start on the walls of the device. Due to the interference of the reflected light from the liquid–solid interface with that from the liquid–vapor interface, a naturally occurring interference pattern can be easily observed using a microscope with an attached video camera (Figs. 8 and 9). It is important to emphasize that, since the reflectivity is a continuous function of film thickness, the continuous film thickness profile can be obtained as a function of the heat input. Extremely small thicknesses below that of the first destructive interference fringe can be measured using the reflectivity and suitable calibration.

A CCD camera with a maximum frame rate of 30 fps was used to capture reflectivity images of the extended meniscus. The captured images were digitized to set the grayscale reflectivity spectrum. Based on the microscope's large magnification, each pixel covers an area with a diameter of 0.177 microns. The size of the cuvette was partially dictated by the planned comparison of our earth-based experimental results with those obtained using a similar cell on the International Space Station. The final dimensions of the CVB were 3 × 3 × 40 ml and ensured that terrestrial gravitational effects would be large (based on the Bond number) and that operation in microgravity would be significantly different from on Earth. A sketch of this version of the CVB is shown in Fig. 10 from Ref. [14].

Although the performance of this device as a “heat pipe” can be relatively poor due to the glass walls and especially in the Earth's environment where the heater is at the top to preserve symmetry, the square corner, transparent design is well suited for basic research on interfacial phenomena. In addition, internal improvements for higher heat fluxes are feasible. We also note that, in the not measurable limit, K → ∞ in a square corner, and that concentration gradients and a slip velocity would help performance. We have used both a mixture of air/vapor at one atmosphere and a pure vapor. When using a pure vapor, we found that the measured pressure in the vapor space correlated with the thermodynamic vapor pressure of the liquid surface temperature at the junction where condensation was evident. Therefore, the pressure drop for vapor flow is minimal for a pure vapor. However, in the condensation region of an air/vapor system, diffusion controls the flow of vapor. Vapor flow is pressure driven near the evaporating meniscus in both cases.

To understand how the device might operate in microgravity and on Earth, a very simplified model was developed to predict how the heat input would affect the curvature of the meniscus as one moved from the condenser end to the evaporator end. This model did not include the effects of intermolecular force interactions. The differential equation, Eq. (2), was solved for the two different cases assuming that Q was the net heat that made it into the device and that we would not lose heat to the surroundings. In these equations, K is the curvature, kf is the thermal conductivity of the liquid, σl is the interfacial tension, ν is the kinematic viscosity of the liquid, hfg is the heat of vaporization, and ρl is the density of the liquid. Later, a consideration of the effects of intermolecular forces would be added. Equation (3) and Fig. 11 show the results of that simplified model assuming microgravity conditions and the estimate is the curvature varies with $x−1/3$ power. Gravity severely restricts the return flow of condensate and so the behavior of the actual CVB in 1-g approximated the behavior of the model. In microgravity, things were decidedly different.

Fig. 11
Fig. 11
Close modal
What these simple models failed to take into account was the poor thermal conductivity of the glass, the resistance to removing heat from the device by radiation, and the Marangoni stresses that would arise due to the steep temperature gradients. This led to several interesting phenomena that can be seen in Figs. 12 and 13. The interferometry technique developed to study the meniscus provided vital clues to what was occurring in the system and also provided high contrast images of what the liquid was doing that could not have been obtained in any other way.
$dKdx=−kfνCl3σlQK4−ρlgσl$
(2)
$1K3=1Kcond3−3νkfQCL3σhfg(xcond−x)$
(3)
Fig. 12
Fig. 12
Close modal
Fig. 13
Fig. 13
Close modal

The first phenomenon in Fig. 12 was heater-end flooding in what we eventually called the interfacial region of the CVB. As more heat was input, the interfacial region grew in extent. Flooding was a result of Marangoni flows originating from the heater end and traveling toward the condenser end opposing the capillary return flow from the condenser. The junction of the two flows led to spillover of the liquid in the corner to the flat faces of the cuvette and formed a central drop. The central drop itself was composed of two counter-rotating vortices as illustrated in Fig. 12. The origin of some of the flow directed from the heater to the condenser was condensation at the heater end. The liquid film on the flat surfaces near the heater was inherently unstable, and thickness oscillations in that film led to curvature extremes that promoted capillary condensation. The condensate then fed the liquid in the corners.

The central drop formation led to yet another odd occurrence. Interferometry showed a jet of liquid emanating from the drop and confirmed that the jet was moving in a direction from the drop toward the heater end. The pressure drop driving this flow rivaled the capillary pressure drop returning liquid from the condenser end. It took a while, but some simple simulations showed that the jet we were seeing was related to rip current phenomena in the ocean [16]. The pair of counter-rotating vortices in the central drop, shown in Fig. 13, swept liquid from the corners and flat faces of the cuvette and ejected that liquid toward the heater. Since the edges of the jet evaporate, this provided additional contact line area to enhance heat transfer. The linear velocities required in the vortices were of the order of one meter per second, much faster than anything we imagined could occur within the CVB.

We had mentioned that the thin film in the CVB near the heater end was unstable. In fact, oscillating menisci have been observed in nearly all our investigations and become more common as the heat input is increased. The frequency and amplitude of oscillation were also seen to increase with increasing heat input. Figure 14 shows one way of representing this time-dependent phenomenon. The figure is a strip chart of oscillations taken from interference images. The x-axis represents time and the y-axis is one line of pixels cut through the meniscus image and so represents a kind of amplitude of oscillation. One can see that there are several frequencies and amplitude swings that occur with occasional bursts where the meniscus suddenly floods the surface. Mathematical analyses by Bankoff and others [17] have shown that these films are unstable when the Marangoni and gravitational stresses exceed the disjoining pressure but give no clue about whether this behavior enhances or detracts from heat transfer. We developed a simple, transient 1D model based on their work and applied an oscillating temperature to the base of the film [18]. In sweeping the frequency of oscillation, it appeared that oscillation enhanced overall evaporative heat transfer but only if there was a period or region of condensation involved that relieved the disjoining pressure resistance to evaporation. Of course, this model was quite crude, and more work is needed to force the film to oscillate spontaneously. Once spontaneous oscillation is achieved, condensation will likely only occur in regions of highest curvature at the contact line and so the current model likely overestimates the role condensation plays. Still, it hints that there are resonant frequencies involved and that condensation on heated surfaces, like we observed in the CVB plays a role in enhancing heat transfer and destabilizing the film.

Fig. 14
Fig. 14
Close modal

## Conclusions

Over the past 50 years, the importance of the intermolecular force of attraction and capillary forces responsible for governing the interaction between vapor, liquid, and solid phases in change-of-phase heat transfer has become clear. Interest in microscale thermal phenomena is becoming increasingly important as we attempt to design enhanced surfaces and systems to promote phase change. Despite the advances that have been made, much more remains to be discovered, and visualization coupled with modeling and simulation will be key to optimizing phase change systems. In the areas discussed in this paper, there are two obvious directions for future research: (1) enhancing flow in a system like the CVB or (2) studying intermolecular forces at nanoscale thicknesses using enhanced optical methods and molecular dynamics simulations.

## Acknowledgment

The authors would like to thank NASA, NSF, DoE, and the Air Force for funding over the years.

## Funding Data

• Directorate for Engineering, NSF (Grant No: CBET-1603318; Funder ID: 10.13039/100000084).

## Nomenclature

• A =

Hamaker constant (J)

•
• B =

retarded Hamaker constant (J)

•
• Cl =

friction coefficient

•
• g =

gravitational acceleration (m/s2)

•
• K =

film curvature (m−1)

•
• Kcond =

curvature at condenser (m−1)

•
• kf =

thermal conductivity of liquid (W/mK)

•
• Mw =

molecular weight (kg/mol)

•
• Pd =

disjoining pressure (Pa)

•
• Pvlv =

liquid film vapor pressure (Pa)

•
• Pvlvi =

flat liquid film vapor pressure (Pa)

•
• Q =

heat flow (W)

•
• R =

gas constant (J/K/mol)

•
• Tlv =

temperature of vapor–liquid interface (K)

•
• δ =

liquid film thickness (m)

•
• ν =

kinematic viscosity

•
• ρl =

density (kg/m3)

•
• σ =

interfacial tension (J/m2)

•
• σlv =

interfacial tension (J/m2)

•
• σΔc =

interfacial tension concentration dependence (J/m2)

•
• σΔT =

interfacial tension temperature dependence (J/m2)

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