Thin-liquid-film evaporation at contact line

Abstract

When a liquid wets a solid wall, the extended meniscus near the contact line may be divided into three regions: a nonevaporating region, where the liquid is adsorbed on the wall; a transition region or thin-film region, where effects of long-range molecular forces (disjoining pressure) are felt; and an intrinsic meniscus region, where capillary forces dominate. The thin liquid film, with thickness from nanometers up to micrometers, covering the transition region and part of intrinsic meniscus, is gaining interest due to its high heat transfer rates. In this paper, a review was made of the researches on thin-liquid-film evaporation. The major characteristics of thin film, thin-film modeling based on continuum theory, simulations based on molecular dynamics, and thin-film profile and temperature measurements were summarized.

Keywords

meniscus / thin film / contact line / disjoining pressure / evaporation

Introduction

When liquid wets a solid wall, as illustrated in Fig. 1, the meniscus extends to the contact line on the solid surface. The extended meniscus can be divided into three regions based on the variations of forces [1-3]: a nonevaporating region, where the liquid is adsorbed on the wall due to strong intermolecular forces; a transition or thin-film region, where effects of long-range molecular forces (disjoining pressure) are felt; and an intrinsic meniscus region, where capillary forces dominate. The transition region is mostly below 1 μm because that thickness disjoining pressure is already very weak. In Refs. [4,5], the extended meniscus is divided into a “microregion” and a “macroregion” instead. It was described in Ref. [4] that the macro region has an almost constant interface curvature, while in the micro region, the curvature sharply turns and ends in nonevaporating region where the curvature is zero.

When the solid wall is heated, the heat is transferred from the wall surface to the liquid and then reaches, through the liquid film, the liquid-vapor interface, where evaporation occurs. Since the evaporation heat transfer is very efficient, in most cases, the conduction is the bottleneck of this heat transfer system. However, when the liquid film comes to be thin enough, the conduction through the film could be as efficient as the evaporation. For water in the atmosphere, for example, the conduction heat-transfer coefficient reaches about 50 W/(cm²•K) if the film thickness is 1 μm. Regardless of much debate on the accomomdation coefficient in evaporation calculation, this value is comparable to the evaporation heat-transfer coefficients of 100 W/(cm²•K) order provided in Ref. [6].

Therefore, “thin-film evaporation” means high heat-transfer efficiency, and “thin film” might be used to generally represent the liquid film near the contact line with small thickness and high heat-transfer coefficient. According to this definition, a thin film could consist of transition region and the beginning section of intrinsic meniscus.

Thin-film heat transfer can be found in various applications. Porous or grooved surfaces can help to form thin films for enhanced operation of heat transport devices such as heat pipes and capillary pumped loops. Fig. 2 (a) shows such a meniscus, which is formed in capillary. The meniscus extends to the wall surface, while the thin film is along the contact line. In boiling heat transfer, evaporation occurs mostly at the bubble bottom near the heated wall, as illustrated in Fig. 2 (b). The thin film is often referred to as “microlayer” in boiling literature. Its evaporation is intensive and causes a sharp temperature drop of nucleation site, which has been considered as one of the most important mechanisms in boiling [4,7-10].

A thorough understanding of thin-film characteristics is critical for the modeling of phase-change heat exchangers. On the other hand, the prompt developments of MEMS, microelectronics, and space technologies require thermal managements with high heat fluxes. For example, the flux of CPU chips has reached 100 W/cm². More efficient heat dissipation solution rather than traditional fan cooling is in urgent demand. Thin-film evaporation has great potential in this area.

disjoining pressure

At contact line, the interactions among liquid, vapor, and solid molecules greatly change the meniscus profile and the evaporation characteristics. Disjoining pressure is widely used in continuum models to represent the effect of these intermolecular forces. When liquid wets a solid wall in an ambient of air, the disjoining pressure has been defined as being equal to the difference between the pressure applied by the liquid film on the air and solid phase, as well as the pressure in the bulk of the liquid film in a state of isothermal and isobaric equilibrium [11]. The point is that when the liquid film is thin enough, the liquid-gas and liquid-solid interfaces interfere with each other, giving rise to disjoining pressure. Disjoining pressure consists of several components arising from various origins, including a molecular component dependent on the effect of molecular or dispersion forces; an ionic-electrostatic component, dependent on the overlapping of diffuse ionic atmospheres; an adsorption component, dependent on the overlapping of diffuse atmospheres of adsorbed molecules; a structural component; and an electronic component, dependent on the overlapping of near-surface layers of liquid metals (like mercury), in which the wave functions of electrons are different from the bulk [11].

Disjoining pressure consists of complicated components, and much work has been devoted to clarify them [4, 11-16]. Basically, the forces between the solid wall and the liquid film can be attractive or repulsive. As long as range forces are of concern, they can be effective from distances of about 0.2 nm to 10 nm. They are quantum mechanical in origin and exist even if molecules are nonpolar [15].

Among the components, dispersion forces arising from Van der Waals interactions were intensively studied due to its importance. A model was developed to convert dispersion forces into a pressure in the liquid film. The disjoining pressure for a nonpolar liquid could be expressed as [1,15,17]

(1)p_{\mathrm{d}}=A/{\delta }^{\mathrm{3}},

where A is the dispersion constant and δ the film thickness. Hamaker [16] derived the Hamaker constant, which is equal to 6πA and which can be calculated from bulk properties such as dielectric constants and refractive indices, as described in Ref. [15]. The value of A is positive for completely wetting liquid, which means that the disjoining pressure is repelling the liquid–vapor interface. Note that the disjoining pressure was also expressed in Refs. [18-20] as

(2)p_{\mathrm{d}}=-A/{\delta }^{\mathrm{3}},

where there is a negative sign before A, and A has a negative value for completely wetting liquids; thus, Eq. (1) and Eq. (2) are the same indeed.

The expression of disjoining pressure for a nonpolar liquid is also expressed as

(3)p_{\mathrm{d}}=B/{\delta }^{\mathrm{4}}.

As specified in Ref. [15,21,22], Eq. (1) is for δ ≤ 20 nm, and Eq. (3) is for δ ≥ 40 nm. In most modeling literature, Eq. (1) was employed.

On the other hand, for polar liquids such as water, interactions besides dispersion forces are present and playing important roles. The positive charge of one molecule attracts the negative charge of another molecule, which leads to strong dipole-dipole interaction. These interactions could be stronger than dispersion forces. Potash and Wayner [23] analyzed a completely wetting film formed on a vertical flat plate and developed an expression for disjoining pressure, which was expressed in a logarithmic function of the film thickness and the thermal properties:

(4)p_{\mathrm{d}}=-{\rho }_{\mathrm{l}}R{T}_{\mathrm{l}\mathrm{v}}\ln {\displaystyle \frac{p_{\mathrm{v},\mathrm{l}\mathrm{v}}}{p_{\mathrm{v},\mathrm{l}\mathrm{v}}^{\mathrm{0}}}},

where R is the gas constant, T_lv is liquid-vapor interface temperature, p_v,lv is the reduced saturation pressure of the film, and p_{\mathrm{v},\mathrm{l}\mathrm{v}}^0 is the saturation pressure corresponding to T_lv. Holm and Goplen [24] further developed the expression as

(5)p_{\mathrm{d}}=-\rho R{T}_{\mathrm{l}\mathrm{v}}\ln (\alpha h{(x)}^{\beta }).

For water on quartz glass, for example, it was given in Ref. [24] that α = 1.49, β = 0.0243.

Continuum models

In Section 2, it can be seen that disjoining pressure was derived to represent the intermolecular forces in the thin film. Continuum models can be developed by assuming the thin film continuous and disjoining pressure as a force applied to the liquid-vapor interface.

Early studies [18,23,25] focused on the effects of the disjoining pressure on pressure field in the thin film and evaporation at liquid-vapor interface. Deryagin et al. [25] demonstrated liquid pressure reduction in the thin-film region due to disjoining pressure. Potash and Wayner [23] concluded that the variation of disjoining pressure along the meniscus provided the necessary pressure gradient for liquid supply into the thin-film region. Wayner et al. [18] discussed the effects of disjoining pressure on suppressing evaporation. Due to the lack of literature on the experimental data of disjoining pressure for polar liquids, extensive researches were done just for nonpolar liquids.

Governing equations

As illustrated in Fig. 1, the pressure difference between vapor and liquid at the liquid-vapor interface is due both to the capillary and disjoining pressures and is expressed using the augmented Young-Laplace equation:

(6)p_{\mathrm{v}}=p_{\mathrm{l}}+p_{\mathrm{c}}+p_{\mathrm{d}}.

The capillary pressure is the product of interfacial curvature K and surface tension coefficient σ\begin{array}{ll}{p}_{\mathrm{c}}=\sigma K,& K={\delta }^{"}{(\mathrm{1}+{\delta }^{\prime \mathrm{2}})}^{-\mathrm{1.5}},\end{array}where δ' and δ" are respectively the first and second derivatives of thickness with respect to length x.

In early studies, p_c is often neglected to simplify the problem by assuming that disjoining pressure dominates in the thin film. Schonberg and Wayner [19] developed simplified models for thin-film profile and an analytical solution for thin-film heat transfer that was applicable only to insulating fluids. However, p_c was later found critical for the film profile even in the transition region. It is not appropriate to neglect p_c in the profile calculation. Avoiding the profile calculation, Wang et al. [26] made an attempt to develop an analytical solution to estimate the total heat transfer only in the transition region. Based on the developed expression, the heat transfer in the transition region was to increase in disjoining pressure and to decrease in liquid viscosity.

To include p_c in the model, further studies such as Hallinan et al. [2] and Dasgupta et al. [27] developed fourth-order ordinary differential equations to solve Eq. (6) and obtained the thickness profile of the extended meniscus. Similar approach could be found in Refs. [1,3,17,28,29]. Combining Eqs. (1), (6), and (7) assuming uniform p_v along the meniscus, and differentiating with respect to x, the following differential equation is obtained for the thin-film profile δ (x)

(8){\delta }^{"\prime }-{\displaystyle \frac{\mathrm{3}{\delta }^{\prime }{\delta }^{"\mathrm{2}}}{\mathrm{1}+{\delta }^{\prime \mathrm{2}}}}+{\displaystyle \frac{\mathrm{1}}{\sigma }}\left({\displaystyle \frac{\mathrm{d}{p}_{\mathrm{l}}}{\mathrm{d}x}}-{\displaystyle \frac{\mathrm{3}A}{{\delta }^{\mathrm{4}}}}{\delta }^{\prime }\right){(\mathrm{1}+{\delta }^{\prime \mathrm{2}})}^{\mathrm{1.5}}=\mathrm{0.}

In view of the very low Reynolds number and the large length-to-height ratio of the thin film, lubrication theory is employed to obtain the pressure gradient dp_l/dx in Eq. (8) A no-slip boundary condition at the wall and a no-shear boundary condition at the liquid-vapor interface are imposed. Under these assumptions, the liquid pressure gradient dp_l/dx may be related to the mass flow rate m'(x). At steady state, the mass flow rate m'(x) at a position x is equal to the integral of the net evaporative mass flux m''(x) from the beginning of the film to the local position. The liquid pressure gradient may then be obtained as

(9){\displaystyle \frac{\mathrm{d}{p}_{\mathrm{l}}}{\mathrm{d}x}}={\displaystyle \frac{\mathrm{3}\nu }{{\delta }^{\mathrm{3}}}}{\displaystyle {\int }_{-\infty }^x{m}^{"}\mathrm{d}x}.

Substituting the pressure gradient into Eq. (8) and further differentiating with respect to x, a fourth-order ordinary differential equation is obtained for the thin-film profile:

(10){\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}\left\{\left[{\displaystyle \frac{\sigma {\delta }^{"\prime }}{{(\mathrm{1}+{\delta }^{\prime \mathrm{2}})}^{\mathrm{1.5}}}}-{\displaystyle \frac{\mathrm{3}\sigma {\delta }^{\prime }{\delta }^{"\mathrm{2}}}{{(\mathrm{1}+{\delta }^{\prime \mathrm{2}})}^{\mathrm{2.5}}}}-{\displaystyle \frac{\mathrm{3}A{\delta }^{\prime }}{{\delta }^{\mathrm{4}}}}\right]{\displaystyle \frac{{\delta }^{\mathrm{3}}}{\mathrm{3}\nu }}\right\}=-m^{"}.

Evaporation mass flux

The evaporation mass flux m” at local liquid-vapor interface is determined by the interfacial mass transport. Wang et al. [3] calculated m^” using the kinetic theory-based expression developed by Schrage [30]. Assuming that the evaporation coefficient and the condensation coefficient are equal to the accommodation coefficient \widehat{\sigma } for simplicity as suggested in the reviews in Refs. [6,31], the net mass flux may finally be written as

(11)m^{"}={\displaystyle \frac{\mathrm{2}\widehat{\sigma }}{\mathrm{2}-\widehat{\sigma }}}{\left({\displaystyle \frac{\overline{M}}{\mathrm{2}\mathrm{\pi }\overline{R}}}\right)}^{\mathrm{1}/\mathrm{2}}\left({\displaystyle \frac{p_{\mathrm{v}\_\mathrm{e}\mathrm{q}\mathrm{u}}(T_{\mathrm{l}\mathrm{v}})}{T_{\mathrm{l}\mathrm{v}}^{\mathrm{1}/\mathrm{2}}}}-{\displaystyle \frac{p_{\mathrm{v}}}{T_{\mathrm{v}}^{\mathrm{1}/\mathrm{2}}}}\right).

The equilibrium vapor pressure p_{v_equ} is the pressure at which the vapor is in equilibrium with the liquid. For a flat interface on a bulk liquid (no capillary or disjoining pressure), p_{v_equ} is equal to the saturation pressure p_sat at T_lv. In a thin film, however, the disjoining pressure and capillary pressure affect the equilibrium state, and p_{v_equ} is not equal to p_sat but smaller:

(12)p_{\mathrm{v}\_\mathrm{e}\mathrm{q}\mathrm{u}}(T_{\mathrm{l}\mathrm{v}})=p_{\mathrm{s}\mathrm{a}\mathrm{t}}(T_{\mathrm{l}\mathrm{v}})\exp \left[{\displaystyle \frac{p_{\mathrm{v}\_\mathrm{e}\mathrm{q}\mathrm{u}}(T_{\mathrm{l}\mathrm{v}})-p_{\mathrm{s}\mathrm{a}\mathrm{t}}(T_{\mathrm{l}\mathrm{v}})-(p_{\mathrm{d}}+p_{\mathrm{c}})}{{\rho }_{\mathrm{l}}{T}_{\mathrm{l}\mathrm{v}}\overline{R}/\overline{M}}}\right].

It is seen that the role of disjoining and capillary pressures is to reduce p_{v_equ} (T_lv) and, thus, to suppress evaporation.

1. Note that there is much debate about the value of the accommodation coefficient in Eq. (11), as reviewed by Paul [32], Carey [6] and Marek [31] , especially for polar liquids. Nonpolar liquids including carbon tetrachloride, benzene, and hexadecane have been experimentally found to have an accommodation coefficient of close to unity.

This equation is seldom directly employed in literature. Wayner et al. [18] proposed a simplified evaporation expression:

(13)m^{"}=a(T_{\mathrm{l}\mathrm{v}}-T_{\mathrm{v}})-b(p_{\mathrm{d}}+p_{\mathrm{c}}),

where

\begin{array}{ll}a=C{\left({\displaystyle \frac{\overline{M}}{\mathrm{2}\mathrm{\pi }\overline{R}{T}_{\mathrm{l}\mathrm{v}}}}\right)}^{\mathrm{1}/\mathrm{2}}{\displaystyle \frac{p_{\mathrm{v}}\overline{M}{h}_{\mathrm{f}\mathrm{g}}}{\overline{R}{T}_{\mathrm{v}}{T}_{\mathrm{l}\mathrm{v}}}},& b=C{\left({\displaystyle \frac{\overline{M}}{\mathrm{2}\mathrm{\pi }\overline{R}{T}_{\mathrm{l}\mathrm{v}}}}\right)}^{\mathrm{1}/\mathrm{2}}{\displaystyle \frac{V_{\mathrm{l}}{p}_{\mathrm{v}}}{\overline{R}{T}_{\mathrm{l}\mathrm{v}}}}.\end{array}

The expression was simplified from Schrage’s original expression by using an extended Clapeyron equation [18] and the approximations T_lv ≈ T_v and p_{v_equ} ≈ p_v. The second group on the right side of Eq. (13) represents the suppression of evaporation by disjoining and capillary pressure. A comparison of the evaporation mass flux calculated from the original expression, i.e., Eq. (11), with that calculated from the simplified Eq. (13) was conducted in Ref. [3] for different superheats. It is seen that the two equations yield nearly identical results for superheat under 5 K.

In the development so far, T_lv is unknown. Note that the evaporation heat flux on the interface is equal to the conduction heat flux through the thin film:

(14)m^{"}{h}_{\mathrm{f}\mathrm{g}}={\displaystyle \frac{k_{\mathrm{l}}(T_{\mathrm{w}}-T_{\mathrm{l}\mathrm{v}})}{\delta }}.

T_lv and m^” are obtained by solving Eqs. (13) and (14).

Solutions

The equation (10), the forth-order governing equation, may be treated as an initial-value problem by specifying the conditions at the beginning of the thin-film region, x = 0, where the nonevaporating film ends and the evaporating thin film starts.

The forth-order ordinary differential equation (ODE) needs four initial conditions: δ(0), δ^’(0), δ^"(0), and δ^’’’(0). The initial thickness δ(0) should be equal to the nonevaporating thickness δ₀, which is obtained by setting m^” as zero in Eq. (13). For a completely wetting liquid, δ^’(0), δ^" (0), and δ^’’’(0) should be zero. However, with these four initial conditions, the forth-order ODE can yield only a film with constant thickness. In order to avoid the trivial solutions, a small perturbation has to be applied to the initial thickness and the slope [2,3,17]. With perturbation of initial thickness, Wee [17] used the sixth-order Runge-Kutta-Fehlberg method to solve the equation. An iterative technique was employed to guess the initial slope δ^’(0) such that the solution converged to the appropriate curvature in the bulk meniscus region. It was found that the solution was extremely sensitive to the initial second derivative. Wang et al. [3] found that since the evaporation at the very beginning is weak due to the suppression by disjoining pressure, a small perturbation of initial thickness has an insignificant influence on the overall thin-film heat transfer. The initial slope δ^’(0) = 1×10^-11 was used, and it was found that further reduction of the value to zero did not significantly change the resulted film profile. The second derivative δ^" (0) was chosen such that the solution converged to the appropriate curvature in the bulk meniscus, which could be considered as a far field boundary condition. It was found that the imposition of the far-field condition is critical. Small changes in δ(0) or δ^’(0) made negligible difference to the thin film profile and heat transfer characteristics as long as the far-field condition was the same.

After solving the forth-order ODE and getting the film profile, the thin-film characteristics such as heat transfer, evaporation, and pressure field can be easily derived. The preferable heat-transfer coefficient of thin film evaporation has been concluded in literature. The micro region, which was defined as extending from the nonevaporating region to a location where the film is 1 mm thick, was found to account for 50% of the total heat transfer of the whole meniscus in a microchannel [3]. In Stephan and Busse’s study of an ammonia meniscus in a groove of 1 mm height and 1 mm width [5], 45% of the overall heat transfer was found to be dissipated from the micro region.

Note that although the beginning of the film is the thinnest, it does not have the strongest local evaporation, because the disjoining pressure is great, and it suppresses evaporation, as seen in Eq. (12) or (13).

Figure 3 is an example of the variation of the different pressures along the film length presented in Ref. [3]. The variation of disjoining pressure (p_d), capillary pressure (p_c), and liquid pressure change (Δp_l = p_l - p_l (x=0)) along the length of the liquid film are plotted. The sum (p_sum= p_d + p_c+ Δp_l) is always a constant, as expected from the Young-Laplace equation. The evaporated liquid is replenished by the liquid pressure gradient, i.e., by the Δp_l dropping from the bulk to the contact line. It was seen that the gradient in liquid pressure is solely supported by the reduction in disjoining pressure for the first 40 nm, while beyond this length, the capillary pressure also decreases and begins to contribute. How to accurately distinguish “transition region” and “intrinsic meniscus” is still vague. In Fig. 3, the transition region is identified as ending at a location when the disjoining pressure drops to about 1/2000th of p_d0, which is the disjoining pressure in the nonevaporating region. Based on this division, in this case the transition region ends at approximately x = 150 nm, with a thickness of about 20 nm. The microregion in general, including the transition region, is much more important to the overall heat transfer than the transition region only.

To-be-clarified details

Continuum models have been developed for several decades and gained great success. However, there are still unsolved problems left. The complexity may lie with the following aspects: first, the solid-liquid-vapor interactions at contact line are still not very clear, especially for complicated molecules, and disjoining pressure can hardly represent all types of interactions accurately; second, in the nonevaporating region and the beginning of the transition region, the thin film is at nanoscale and the continuum assumption is doubtable and so is the kinetic theory of evaporation; finally, the film thickness varies from macroscale to nanoscale, which is basically a multiscale problem. It is difficult to satisfactory solve the whole problem with just one continuum model, and it is also difficult to deal with the transitions between the scales.

Due to the above reasons, a lot of work is still waiting to be completed in the thin-film modeling. For example, in Section 3.3, the forth-order ODE equation needs four initial conditions. However, the definition about the initial point, i.e., the transition or junction between the nonevaporating region and the transition region, is not clear.

For another example, the no-evaporating region is often considered dominated by disjoining pressure, and liquid evaporation is suppressed, as indicated by Eq. (12) or (13). The nonevaporating thickness is then obtained by setting m^” as zero in Eq. (12) or (13). By setting m^” as zero means that the disjoining pressure has to be great to suppress the evaporation to zero when the wall superheat is at high level. At this time, the disjoining pressure is even greater than the vapor pressure. Based onp_{\mathrm{v}}=p_1+p_{\mathrm{c}}+p_{\mathrm{d}}, however, vapor pressure should always be greater than disjoining pressure (p_l is always positive, and p_c is zero in nonevaporating region). This contradiction means that either the augmented Young-Laplace equation is not valid at film beginning, or the suppression mechanism in nonevaporating region is not quite clear yet.

Film under complex conditions

The menisci in complex geometries have also been studied. Stephan and Busse [5] studied the heat transfer in the microregion within open grooves and then combined the solution with the macroscopic meniscus. Xu and Carey [33] conducted a combined analytical and experimental investigation on liquid flow in V grooves and emphasized the importance of disjoining pressure on overall heat transfer. Ma and Peterson [29] proposed a mathematical model for the heat-transfer coefficient and temperature variation along the axial direction of a groove, which led to a better understanding of the axial heat transfer coefficient and temperature distribution on grooved surfaces. Morris [34] suggested a universal relationship between heat flow, contact angle, interface curvature, superheat, and material properties, which can be extended to different geometries.

Evaporation under complex conditions and other interfacial effects were also considered. In Ref. [28], the gas domain was assumed to consist of a mixture of air and vapor, and the thin film was maintained throughout at a temperature below the saturation temperature corresponding to the imposed pressure. The vapor diffusion in the gas domain was calculated to obtain evaporation flux. Park et al. [1] proposed a mathematical model that included the vapor region and a slip boundary condition. It was concluded that the pressure gradient in the vapor region significantly affected the thin-film profile. Wee [17] discussed the effects of liquid polarity, slip boundary, and thermocapillary effects on thin-film profile. The polarity effect was found to elongate the transition region but suppresses evaporation. Recently, binary liquids [35] have been found to induce a distillation-driven capillary stress to counteract the thermocapillary stress, leading to an elongation of thin-film length.

Various factors cause instabilities or even breakdown of liquid films. Many researches have been conducted for different films under various conditions [36-49]. Person [36] and Sternling and Scriven [37] addressed the importance of thermocapillary stress in the instabilities of an evaporating liquid film. Smith and Davis [38,39] theoretically studied the instability of a planar liquid layer with a temperature gradient along with it. Miladinova et al. [41] considered the liquid layer falling down an inclined plate with nonuniform heating and studied the coupling of thermocapillary instability and surface-wave instability. Kalliadasis et al. [47] developed a stability theory by using the integral-boundary-layer approximation of the Navier-Stokes equations and free-surface boundary conditions. Oron et al. [48] concluded a unified mathematical theory based on an asymptotic procedure and simplified the original complex free-boundary problem. Oron and Bankoff [49] later proposed a model for the evolution of thin nonpolar liquid film by taking account of the effects of attractive and repulsive intermolecular forces.

The instability of contact line was also considered. Münch and Wagner [50] studied the linear stability of dewetting thin polymer films driven by intermolecular forces using a lubrication model. Lyushnin et al. [51] investigated the fingering instability of growing dry patches in an evaporating film of a polar liquid placed on a solid substrate. It was found that the instability manifests itself as fingering of mobile fronts between growing “dry” (thin) and shrinking “wet” (thick) regions of the film. The influence of the evaporation rate, polar intermolecular forces, and chemical heterogeneity of the substrate were investigated.

Related to the instabilities, the critical heat flux of liquid film evaporation, beyond which the film dries out and the heat transfer deteriorates greatly, was also considered. Fujita and Ueda [52] studied the film breakdown for subcooled water films flowing downwards on a tube outside. The film breakdown was found in a thin region. Hoke and Chen [53] developed an analysis to predict film breakdown with a force-balance model. Pautsch and Shedd [54] experimentally investigated the thin liquid film formed in spray cooling. The film thickness profiles were measured, and the critical heat flux was discussed.

Molecule-based models

In the above descriptions, disjoining pressure was used in continuum models to take account of intermolecular forces. To get direct insight, molecule-based methods have to be employed. MD (molecular dynamics) simulation that has been widely used in the nanoscale heat-transfer studies, which is recently used in thin-film studies.

The very fundamental process of evaporation is the molecular transport at liquid-vapor interface. Although much work has been done, an accurate description or modeling of evaporation or condensation is still not available [6,31,32]. Using the MD method, Yang and Pan [55] investigated the evaporation of a thin water layer into vacuum. Microscale physical phenomena near the liquid-vapor interface such as evaporation, condensation, recoil, and molecular exchanges were observed. The simulations showed that the hydrogen bond had a significant effect on the molecular behavior near the interface and might reduce the evaporation coefficient. This study also demonstrated that the combination of molecular dynamics simulations, and the classical Schrage model [30] might provide an elegant methodology to determine the evaporation coefficient of water at different temperatures or corresponding saturation pressures. Kikugawa et al. [56] focused on the microscopic structure of the liquid-vapor interface and the effect of impurities on interfacial properties, such as surfactants or electrolytes in the aqueous solution. They performed molecular dynamics simulations of a simple planar interface system and a bubble system in which a nanometer-sized void region was maintained. As a result, the number density profile of the water was found distorted in the concentrated electrolyte solution, and it had the peak at the liquid-side edge of the interface. In the surfactant solutions, it was found that the effect of slowing the dynamics was strong mainly at the interface region.

A bunch of studies focus on vapor nucleation in liquid. Wolde and Frenkel [57] simulated homogeneous gas-liquid nucleation in a Lennard-Jones system. They computed the free energy of a cluster as a function of its size and determined the height of the nucleation barrier as a function of supersaturation. It was found that the relationship between the critical nucleus size and the degree of supersaturation was in excellent agreement with the nucleation theorem. However, it was found that the nucleation barrier was differing from the prediction of classical nucleation theory. Wang et al. [58] conducted a simulation in a larger domain, up to 68 nm × 68 nm for argon, to investigate the liquid-vapor nucleation using the two-dimensional molecular dynamics method. They found that if the simulation domain is not large enough, the simulated liquid-vapor nucleation process would be confined to the cavity growth stage, and in classical nucleation theory sense, no bubble could be formed. The simulation in large domain obtained the liquid-vapor nucleation with three stages, i.e., cavity growth, cavity coalescence, and bubble formation. Maruyama and Kimura [59] simulated a heterogeneous nucleation of a vapor bubble on a solid surface, and they have successfully demonstrated the nucleation of a three-dimensional vapor bubble on the solid surface using the molecular dynamics method. Dynamic behaviors of the low-density patches leading to the bubble nucleation were visualized for several wettable conditions.

MD simulations of thin-film evaporation have been conducted in recent years. In 2002, Yi et al. [60] studied an ultrathin liquid argon layer on a platinum surface. Different vaporization behaviors were observed depending on the magnitude of the wall temperature. The simulations were also used to investigate condensation. A similar work was done by Freund in 2005 [61], in which the atomistic simulations of a simple Lennard-Jones fluid were conducted to investigate the very near-wall dynamics and thermodynamics of evaporating menisci. A two-dimensional liquid drop was centered at a cool spot of the solid wall, and the meniscus extended into the hot spot of the solid wall to start evaporation. It was found that the continuum model with disjoining pressure was accurate for predicting the interface shape and mass flux of evaporating menisci even down to the atomic dimensions of the wall-bound film. In making comparisons, the continuum model was solved with boundary conditions consistent with the atomistic simulation results. In the strong wetting cases, it was found that the fluid atoms right on the wall was effectively immobile. The thermal resistance at the solid-liquid interface was found significant in very thin films. A Kapitza resistance was employed to obtain the solid-liquid interface temperature.

experiments

Thin-film profile

Interferogram is one of the most employed technologies in film thickness measurement. In early studies, it was often used to measure the profile of microlayer at bubble bottom in nucleate boiling [62,63]. Later in Chen and Wada’s work [64], the spreading dynamics of a drop edge was studied using a laser interference microscopy method. Based on the movie of the interference fringes, the spreading speed at the drop edge was measured, and the edge profile was reconstructed.

Wayner et al. [21,65-70] conducted a series of measurements on thin liquid film profiles using interferogram technique. In Ref. [65], isothermal profiles of an extended meniscus in a quartz cuvette were measured using an image-analyzing interferometer that was based on computer-enhanced video microscopy of the naturally occurring interference fringes. In Ref. [67], an improved image analyzing technique was developed to obtain simultaneously the curvature and the apparent contact angle of a thin film. Green light was used, and the zeroth dark fringe corresponded to 100 nm film thickness. The profile measured in the thicker portion of the meniscus fitted well with theoretic analysis.

Further, in Ref. [21], a relative reflectivity concept was introduced so that the film thickness was determined based on the gray value of the interference image, which enhanced the measurement resolution. The critical region, where the thickness was below 100 nm, was emphasized. It was found that the nonevaporating region, the thickness, and interfacial slope were consistent with continuum models if the dispersion constant for disjoining pressure was evaluated using experimental data. Isothermal equilibrium conditions were used to verify the accuracy of the procedures. Later, Panchamgam et al. [69] studied the mixture of pentane-octane with the similar experimental setup and the same analysis method. The presence of large Marangoni interfacial shear stresses with the mixture was demonstrated by comparing with pure pentane. A control volume model was developed to evaluate the differences between the two systems.

In recent years, Argade et al. [71] used image-analyzing interferometry to study microscale transport processes. The liquid profile from the images of interference fringes showed a maximum in curvature near the junction of the transition region and intrinsic meniscus. The film behavior under different thermal perturbation was studied. Deng et al. [72] studied the stability and oscillations of an evaporating curved wetting film of pentane. The film thickness was measured as a function of heat input, time, and axial position. Plawsky et al. [68] studied the profile details of an oscillating meniscus in a right angled corner as a function of time and axial position. Both the advancing and receding films were profiled. Gokhale et al. [70] measured the thickness profiles of partially wetting condensing drops of 2-proranol on a quartz surface. They found the curvature value of the drop near the thicker part is negative, while the drop was concave in the thinner region where the liquid merged with an adsorbed film. The pressure profiles inside the drop were calculated from the augmented Young-Laplace equation. Internal flow was found towards the point of maximum positive curvature. Churaev et al. [73] studied the spreading of surfactant solutions with interference method.

Besides interferogram, ellipsometric method is also attractive in thin-film measurement. Liu et al. [74,75] developed image scanning ellipsometry based on null ellipsometry to study nonuniform film thickness profiles. The thickness at every point on the meniscus can be measured with high spatial resolution and thickness sensitivity. The thickness from several nanometers up to tens of microns was measured with image scanning ellipsometry. In Ref. [75], a vertically draining film of a completely wetting fluid was measured. Four draining regions were identified as follows: the interfacial region, the transition region, the hydrodynamic region, and the capillary region. By comparing the experimental profiles with the theoretical results, the authors indicated that a purely hydrodynamic draining solution was valid for films with thickness at least 20 nm. Liu et al. [76] further studied the drainage of the partially wetting film dodecane on a vertical silicon substrate using image scanning ellipsometry.

Thin-film temperature

The information of temperature at thin film is critical for heat transfer calculation. The interface temperature directly determines the evaporation mass flux, as seen in Eq. (13), and the local solid wall temperature is an important indicator of the cooling effect of thin-film evaporation.

However, compared with the measurement on thin-film profile, the temperature measurement is even more challenging due to the very small scale. Conventional thermocouples are not appropriate for thin-film temperature measurement since their tips are mostly larger than hundreds of micrometers, and their invasion will alter the meniscus shape. Thermocouples can be sued to measure the solid temperature at thin film, but they should not be too close to the solid surface; for example, in Ref. [33], the thermocouples were embedded 2.5-15.2 mm below the surface.

Höhmann and Stephan [77] used a type of liquid crystal whose color was sensitive to temperature (thermochromic liquid crystal, TLC ) to measure the wall temperature right below the thin film . A capillary slot was created by two parallel flat plates. The setup of each plate is illustrated in Fig. 4. The color image of the TLC was recorded with a CCD camera, so that the two-dimensional temperature distribution was obtained. A theoretical spatial resolution of less than 1 μm and the temperature uncertain of 7.3% were reached. A temperature drop of about 0.2 K over a region of 35 μm length was detected near the contact line under relatively low heat flux. Buffone and Sefiane [78] used TLC to measure the wall temperature affected by the evaporation of a volatile liquid in a tube. Despite the thermal diffusion through the tube’s thickness, the technique was concluded in Ref. [78] for temperature measurement at a microscale.

Using infrared (IR) camera, Sefiane and his coworkers [79-83] conducted a series of measurements on the wall temperature and the meniscus temperature. In Ref. [79], an evaporating meniscus was located in a capillary tube. The temperature field on the tube outside surface was mapped using a high-resolution IR camera. Particle image velocimetry (PIV) technique was employed simultaneously to characterize the convection inside the meniscus. It was found that if there was no heat applied, the self-induced temperature gradient due to evaporative cooling effect would generate the convection from the center of the meniscus towards the edge. In Ref. [80], the evaporation of volatile liquids in capillary tubes ranging from 600 to 1630 μm was studied. The evaporative cooling effect at the meniscus contact line was captured by the IR camera with a spatial resolution of 30 μm and a sensitivity of 20 mK at 30 °C. Temperature profiles with different liquids and tube sizes were obtained. In Ref. [81], the meniscus interfacial temperature profile was partially measured and was demonstrated as a key for the onset of Marangoni or thermocapillary convection.

Concluding remarks

The thin liquid film at contact line, with thickness varying from nanometers to micrometers, is gaining increasing interest due to its high heat transfer rates and its great role in phase-change heat transfer. Continuum models and molecule-based simulations have provided great details in thin-film physics. There are still many unveiled issues about the solid-liquid-vapor interactions in thin film, especially for complicated molecules. A comprehensive multiscale modeling from macroscale to nanoscale is also of interest. With the development of microscale and nanoscale techniques, more detailed and reliable information about thin film is expected.

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