Expressions for the Evaporation and Condensation ... - ACS Publications

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Expressions for the Evaporation and Condensation Coefficients in the Hertz-Knudsen Relation Aaron H. Persad and Charles A. Ward* Department of Mechanical and Industrial Engineering, Thermodynamics and Kinetics Laboratory, University of Toronto, 5 King’s College Road, Toronto, Canada M5S 3G8 S Supporting Information *

ABSTRACT: Although the Hertz-Knudsen (HK) relation is often used to correlate evaporation data, the relation contains two empirical parameters (the evaporation and condensation coefficients) that have inexplicably been found to span 3 orders of magnitude. Explicit expressions for these coefficients have yet to be determined. This review will examine sources of error in the HK relation that have led to the coefficients’ scatter. Through an examination of theoretical, experimental, and molecular dynamics simulation studies of evaporation, this review will show that the HK relation is incomplete, since it is missing an important physical concept: the coupling between the vapor and liquid phases during evaporation. The review also examines a modified HK relation, obtained from the quantum-mechanically based statistical rate theory (SRT) expression for the evaporation flux and applying a limit to it in which the thermal energy is dominant. Explicit expressions for the evaporation and condensation coefficients are defined in this limit, with the surprising result that the coefficients are not bounded by unity. An examination is made with 127 reported evaporation experiments of water and of ethanol, leading to a new physical interpretation of the coefficients. The review concludes by showing how seemingly small simplifications, such as assuming thermal equilibrium across the liquid−vapor interface during evaporation, can lead to the erroneous predictions from the HK relation that have been reported in the literature.

CONTENTS 1. Introduction 1.1. Scope of Review 2. Problems with the Hertz-Knudsen Relation 2.1. Formulations of the Hertz-Knudsen Relation for the Evaporation Flux 2.2. Empirical Constants in the Hertz-Knudsen Relation 2.3. Inconsistent Coefficient Values in the HertzKnudsen Relation 2.4. Section Summary 3. Review of the Hertz-Knudsen Relation for Evaporation Flux 3.1. Modifications to the Hertz-Knudsen Relation 3.1.1. Simplifying Assumptions 3.2. Effect of Interfacial Curvature on Values of σe and σc 3.3. Effect of the Molecular Structure on Values of σe and σc 3.4. Capillary Waves and Energy Barriers 3.5. Effect of Impurities on Values of σe and σc 3.6. Interfacial Temperature Measurements 3.6.1. Thermocouples 3.6.2. Direction of Interfacial Temperature Discontinuities 3.7. Pressure Measurements 3.8. Molecular Dynamics Simulations

© 2016 American Chemical Society

3.9. Cloud Growth, Aerosols, and Multistep Models of Evaporation 3.9.1. Summary of Climatology Kinetic Studies 3.10. Section Summary 4. Review of the Statistical Rate Theory Expression for the Evaporation Flux 4.1. Expression for the Net Evaporation Flux 4.1.1. Expressing SRT in Terms of Known and Measurable Parameters 4.2. Examination of Predictions Made Using the SRT Evaporation Flux Expression 4.2.1. Purity of Evaporating Fluids during Experiments 4.2.2. Size of the Liquid−Vapor Interface 4.2.3. Steady-State Evaporation and Measurements of Evaporation Fluxes, Interfacial Temperatures and Vapor-Phase Pressures 4.2.4. Thermodynamically Consistent Saturation-Vapor Pressure Expressions Were Developed from the SRT Approach 4.2.5. Predictions of the Interfacial Temperature Discontinuities from the SRT Evaporation Flux Expression

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Received: August 31, 2015 Published: June 17, 2016 7727

DOI: 10.1021/acs.chemrev.5b00511 Chem. Rev. 2016, 116, 7727−7767

Chemical Reviews 4.2.6. SRT Evaporation Flux Expression Examined in Multicomponent Systems 4.3. Extension of SRT to Nonideal Vapors 4.4. Linearized Form of the SRT Evaporation Flux Expression 4.5. SRT Evaporation Flux Expression Neglecting Molecular Phonon Terms 4.6. Significance of Phonon-Dependent Terms in the SRT Evaporation Flux Expression 4.7. Section Summary 5. Thermal-Energy-Dominant (TED) Limit of the SRT Expression for the Evaporation Flux 5.1. Examination of the TED-SRT Evaporation Flux Expression 5.1.1. Determination of the Saturation-Vapor Pressure 5.1.2. Interfacial Temperature Discontinuity Predictions 5.2. Physical Meaning of the Evaporation and Condensation Coefficients 5.3. Effects of Common Assumptions on the TED-SRT Evaporation Flux Expression 5.3.1. Assuming Constant Values of the Evaporation and Condensation Coefficients 5.4. Section Summary 6. Concluding Remarks 7. Description of Experiment Acronyms Associated Content Supporting Information Author Information Corresponding Author Notes Biographies Acknowledgments References

Review

the coupling between the vapor and liquid phases during evaporation (i.e., the evaporation and condensation fluxes depend on both the conditions in the liquid and in the vapor phases). When the effects of both phases on the evaporation flux are taken into account in the HK relation, it is found to be as accurate as the quantum-mechanically based statistical rate theory expression for the evaporation flux.58,61,62 Without introducing empirical constants or other fitting parameters, we will show that the modified HK relation agrees with reported experiments (including experiments that had not agreed with the previous “traditional” HK relation63). The review concludes by showing how seemingly small simplifications, such as assuming thermal equilibrium across the liquid−vapor interface during evaporation, can lead to the erroneous predictions from the HK relation that have been reported in the literature.64

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1.1. Scope of Review

Numerous reviews have examined the application of classical kinetic theory to the evaporation problem and all have raised issue with the HK relation.1,56,59,64−66 One central issue identified in those reviews is the difficulty of determining the values of two fitting parameters found in the HK relation: the evaporation and condensation coefficients, σ e and σ c , respectively. It is generally accepted that the coefficients are no greater than unity, but inexplicably, they have been found to span 3 orders of magnitude.64 No satisfactory explanation for the scatter exists at present, since explicit expressions for these two coefficients have yet to be determined. In this review, we examine explicit expressions for the evaporation and condensation coefficients with the surprising result that their values are not bounded by unity. This review is organized as follows. In section 2, we describe problems with the HK relation that raise doubt about its validity. In section 3, we identify sources of errors in the HK relation by reviewing previous theoretical, experimental, and numerical studies. In section 4, we review another expression for the net evaporation flux, statistical rate theory (SRT) that is based on quantum and statistical mechanics, and highlight some challenges of applying that theory to evaporation. In section 5, we propose a new expression for the evaporation flux that is obtained from the SRT evaporation flux expression by taking a limit that removes the dependence of the evaporation flux on the values of the internal molecular vibration frequencies. The validity of taking this limit will be examined and shown to agree with experiments. We will show that this limit defines explicit expressions for both the evaporation and condensation coefficients in terms of measurable parameters. A summary of the review is provided in section 6. A list of symbols and nomenclature is provided in the Supporting Information. In section 7, a description of experiment acronyms is presented.

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1. INTRODUCTION Evaporation is a nonequilibrium process in which there is a net transport of liquid molecules across a liquid−vapor interface into the vapor phase. Evaporation is ubiquitous in nature and has numerous applications in many disciplines including physicochemical systems,1−5 biology,6−17 botany,18,19 climatology,1,20−22 agriculture,23−29 astronomy,30−32 and engineering,33−48 to name a few. Accurate evaporation models are needed to make evaporation processes efficient and costeffective.49,50 For the past 130 years, the Hertz-Knudsen (HK) relation has been used to model evaporation processes.1,51−56 However, the HK relation has led to inconsistent predictions that do no agree with experiments. For example, the vapor-phase temperature predicted from the HK relation is 3 orders of magnitude less than that observed experimentally.57,58 Surprisingly, as noted by Hołyst et al.,59 the HK relation continues to be used today by investigators who add correction terms and empirical constants to get better agreement with experiments. Investigators have acknowledged that inaccurate predictions result from the HK relation, yet they appear reluctant to leave it behind because of its simplicity.60 The objective of this review is to identify sources of error in the HK relation. We examine both theoretical and experimental sources of error. Our review will show that the HK relation is incomplete, since it is missing an important physical concept:

2. PROBLEMS WITH THE HERTZ-KNUDSEN RELATION 2.1. Formulations of the Hertz-Knudsen Relation for the Evaporation Flux

Past reviews of the HK relation indicate that many forms of the relation exist.1,56,59,64−66 For example, the HK relation is sometimes presented with both the liquid and vapor interfacial temperatures as parameters,63 while at other times thermal equilibrium is assumed across the liquid−vapor interface.64 Similarly, some have used the HK relation with both the evaporation and condensation coefficients present,63 while others have only considered one coefficient.64 Thus, it is 7728


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coefficient is also called the mass accommodation coefficient.78−80 The evaporation and condensation coefficients in eq 3 represent the fraction of molecules that strike the interface and change phases from their initial liquid or vapor states, respectively. If each molecule that collided with the interface were to change phase then the coefficients would have a value of unity. The coefficients represent the ratio of the “actual” unidirectional flux compared to the maximum flux predicted from classical kinetic theory (CKT).64,72,76 In accordance with CKT, the values of σe and σc must be between 0 and 1. Note that since eq 3 contains two unknowns, σe and σc, even if jLV, TLI , TVI , and PV were measured, the values of σe and σc cannot be determined simultaneously from eq 3. Even before being able to use eq 3, assumptions about the coefficients must be made to reduce the number of unknown parameters. Others have introduced additional equations for energy flux and momentum, in order to formulate a closed system of equations.63,72,76 Additional coefficients have been introduced to account for how vapor molecules interact with the interface. When vapor molecules collide with the interface, a fraction of the molecules will penetrate the interface (i.e., σc) and the remainder (i.e., 1 − σc) will reflect back into the vapor phase. There are two mechanisms by which the vapor molecules are reflected: specular and diffuse.63 In specular reflection, the vapor molecules do not thermally interact with the interface, whereas in diffuse reflection, the molecules are thermally accommodated with the interface. Thus, molecules in specular reflection conserve their energy, but those in diffuse reflection do not.63,72 The parameter αt is introduced into the HK relation to describe the fraction of molecules that are specularly reflected. This parameter has a value between 0 and 1 and is called the thermal accommodation coefficient. The fraction of diffusively reflected vapor molecules is 1 − αt. Many investigators have assumed55,59,64,66,81 σe = σc (4)

important for us to define the HK relation that is to be examined. One approach to developing the HK relation is to approximate the vapor as an ideal gas in equilibrium with the liquid−vapor interface and with the liquid phase. The velocity distribution of vapor molecules can then be approximated using the Maxwell-Boltzmann (M-B) velocity distribution function.67−69 The M-B distribution function leads to an expression for the collision frequency of vapor molecules with the interface per unit of area per second:67,68 jeV

⎛ m ⎞1/2 = Ps(Te)⎜ ⎟ ⎝ 2πkBTe ⎠

(1)

where jVe denotes molecular flux, P pressure, T temperature, m the mass of a molecule, and kB the Boltzmann constant. A superscript L or V denotes a property to the liquid or the vapor phase, and the subscript e indicates equilibrium. In a system where the liquid−vapor interface is flat, the equilibrium pressure is the saturation-vapor pressure, Ps(T) and depends on temperature.70 Under equilibrium conditions, there can be no net transfer of molecules across the interface. Therefore, the liquid molecular collision frequency with the interface, denoted by jLe , must be equal in magnitude to jVe . Under equilibrium, the magnitude of jVe represents the maximum theoretical collision frequency of vapor molecules with the interface.51,52,67,68 Note that a liquid− vapor system in equilibrium is dynamic, since there are two fluxes that are equal and in opposite directions: jVe is the unidirectional condensation flux in the direction from the vapor phase to the interface, and jLe is the unidirectional evaporation flux in the direction from the liquid phase to the interface. The M-B velocity distribution is only strictly correct under equilibrium, when the net evaporation flux is null. This does not make the M-B velocity distribution valid to the evaporation problem. However, investigators often assume that the M-B distribution is still applicable under near-equilibrium conditions.71−75 If the interfacial liquid and vapor temperatures are used instead of Te, and PV replaces the Ps(T) term from eq 1: j

LV

=

1/2 ⎛ m ⎞1/2 m ⎞ V ⎟ ⎟ −P ⎜ L V ⎝ 2πkBTI ⎠ ⎝ 2πkBTI ⎠

TIL = TIV

when using the HK relation, eq 3. However, in section 3, the above assumptions will be shown to result in errors, since the HK relation does not agree with evaporation experiments. Thus, a question that remains open is how to evaluate the HK relation, eq 3, without introducing simplifying assumptions.

⎛

Ps(TIL)⎜

(2)

where the subscript I indicates an interfacial property. Equation 2 is called the Hertz relation. jLV denotes the net evaporation flux and is the difference between the unidirectional evaporation flux and unidirectional condensation flux. Note that the unidirectional fluxes only depend on the properties of their respective phases. We will revisit this point later in this review.

2.3. Inconsistent Coefficient Values in the Hertz-Knudsen Relation

The HK relation, eq 3, has been used for the past 130 years to study evaporation experiments, but inconsistent results have been reported.54,64,81 One unresolved difficulty lies in defining the condensation and evaporation coefficients. Numerous numerical, theoretical, and experimental studies have been performed to test the HK relation, eq 3, but determinations of the coefficient values have produced inconsistencies; for water alone,54,64,66,81 the coefficient values varied from an order of 10−3 to 1.64 Attempts to explain such a spread in coefficient values have produced several classes of experiments. The list includes jets tensimeters,82 expansion cloud chambers,83 radioactive tracers,84 surfactant studies,85,86 molecular beam experiments,87 acoustic fields,88 sonoluminescence,89 shock tubes,90 droplet electrolevitation in light scattering cells,91,92 and droplet train flow reactors.93−95 A review of many of these techniques is provided by Davis54 and

2.2. Empirical Constants in the Hertz-Knudsen Relation

Equation 2 has been shown to disagree with numerous experimental studies.64,76 It was reformulated by Knudsen52 who introduced two parameters, the condensation and evaporation coefficients, σc and σe, respectively: j LV =

m 2πkB

⎛ L V ⎜σ Ps(T ) − σ P e c ⎜ TIL TIV ⎝

⎞ ⎟ ⎟ ⎠

(5)

(3)

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experimental σe values averaged 0.243, but he claims they were “no less than σe = 0.25 and the coefficient probably approximates unity”.105 It is typically claimed that values less than unity are due to several factors including problems with the HK relation itself, the configurations of the experiments, the presence of contaminants, and improper surface temperature and pressure measurements. Each of these factors will be discussed in what follows. Molecular dynamics (MD) studies are one way to mitigate the challenges of experimental investigations, though they present challenges of their own. Results from MD studies are also presented below.

by Davidovits et al.1 However, the actual values of the coefficients remain unresolved and there is a strong need, expressed in the literature, for accurate evaporation studies with precise interfacial measurements to correctly determine the coefficient values.56,63,66 Reviews of the experiments described above1,54,64,66 have brought to light several issues with the experiments performed to determine values of the evaporation and condensation coefficients: (1) the HK relation clearly indicates that two coefficients are present; however, in many of the experiments reviewed, it is assumed that σe and σc are equal and constant; (2) the HK relation has two interfacial temperatures, TVI and TLI , but thermal equilibrium across the interface is assumed in many studies, and it is not clear if the assumed temperature is TVI or TLI ; (3) the surface temperature reported in experiments such as cloud expansion chambers, electrolevitation, and shock tubes cannot be directly measured and must be calculated from heat and mass transfer equations. But each equation introduces new coefficients, such as thermal accommodation coefficients (αt), and these are also unknown.79,83,96−98 There is an added complexity that results from nonsteady conditions, since determination of the coefficient values must be from the interfacial conditions measured on the time scale of microseconds.74 The above issues are not restricted to experimental studies but appear in numerical and theoretical treatments of the problem as well.99−102 The validity of the HK relation has been questioned in previous reviews.64,66,81 Without explicit expressions for σe and σc, it is impossible to test the HK relation, eq 3, on its own. Thus, there is a need to examine the validity of other evaporation theories or to develop explicit expressions for σe and σc.

3.1. Modifications to the Hertz-Knudsen Relation

We have seen that, in a strict sense, the HK relation applies at equilibrium since the assumed M-B velocity distribution is for an ideal vapor in equilibrium with the liquid phase. Classical kinetic theory indicates that the maximum collision rate of molecules with the interface occurs at equilibrium. Thus, both σe and σc would be unity at equilibrium.72 In processes where there is a net evaporation rate, provided conditions are not too far from equilibrium, coefficient values of unity are still expected.105 However, if the coefficients were constant then they would always have a value of unity. The scatter in the coefficient values reported by Marek and Strauß clearly indicates that the coefficients cannot be assumed to be constant. Schrage raised concerns with the assumed velocity distribution in the HK relation.53 He claimed if there were a net evaporation rate, the velocity distribution of molecules leaving the interface would be different than those impinging on the interface; the evaporating or condensing molecules would have a Maxwellian velocity distribution shifted by a mean or drift velocity.73,106 He reformulated the HK relation into the Hertz-Knudsen-Schrage (HKS) equation given in eq 6.64,72

2.4. Section Summary

In this section, we introduced the Hertz relation for the evaporation flux, eq 2. It was formulated by assuming the Maxwell−Boltzmann distribution function is valid in the vapor phase. Two empirical coefficients, σc and σe, were introduced in the final form of the Hertz-Knudsen relation given in eq 3. However, several problems arise when the HK relation is used to examine evaporation experiments: (1) The M-B distribution function is only strictly valid at equilibrium, not during evaporation. (2) The coefficient values be cannot determined from the HK relation without introducing simplifying assumptions. (3) Conflicting values of the coefficients have been reported. Thus, we see that there are several problems with the HK relation. However, the sources of errors are still not known. In the next section, we will review studies of the HK relation and examine sources of errors.

LV jHKS

2 = 2 − σc

m 2πkB

⎛ L V ⎜σ Ps(TI ) − σ P e c ⎜ TIL TIV ⎝

⎞ ⎟ ⎟ ⎠

(6)

When the coefficients are equal to unity, the HKS relation gives an evaporation flux value double that of the HK relation. However, the HKS relation does not satisfy conservation of momentum or energy.72,76 Barrett and Clement proposed a new theory which does satisfy momentum and energy conservation but assumed σe was equal to σc in their derivation.72 Tsuruta and Nagayama took a different approach.102 They thought assuming uniform coefficients values and equality between σe and σc in eq 3 would result in “misunderstandings on the interface transport phenomena.” Instead, they applied transition-state theory to the HK relation to incorporate the effects of translational, rotational, and vibrational degrees of freedom for monatomic and polyatomic molecules.107−110 Tsuruta and Nagayama showed that the value of σc was directly affected by the internal degrees of freedom of the molecules.108,109 However, neither the rotational nor vibrational degrees of freedom influenced the condensation process.108 Rather, only the internal translational motions of the molecules were significant. They assumed that σc and σe were equal and that the interface was thermally equilibrated. They then proposed an explicit expression for the average condensation coefficient, σc , which only depended on the specific volumes of the liquid and vapor, vL and vV, and later102 updated their expression to include the particle velocity normal

3. REVIEW OF THE HERTZ-KNUDSEN RELATION FOR EVAPORATION FLUX This section describes the depth and breath of studies undertaken by others to examine sources of error in the HK relation, eq 3, and to explain variations in the coefficient values that have been reported. As indicated earlier, the values of σe and σc reported in the literature range between 10−3 and 1 for water alone.64 However, many researchers think that the correct value should be unity, despite the large number of experimental studies showing otherwise. For example, Cammenga et al. claim, “there is now no doubt” σe and σc are unity.103,104 In Hickman’s jet tensimeter work, the 7730


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strated that even if the evaporation and condensation coefficients were close to each other in value, assuming that they were equal “increases the error of evaporation− condensation determination strongly.” Even if the coefficients were in fact equal, the application of the HK relation requires a reliable expression for Ps(T) to be used in the calculations of σ. For example, at low temperatures (T ≪ critical temperature of the fluid) for both water and ethanol, questions have been raised regarding the accuracy of Ps(T) expressions reported in the literature.117,118 Furthermore, the latest study included in the Marek-Strauß compilation was reported in 1989. The standard assumption up to that time had been that the interfacial temperatures were equal during evaporation. However, measurements were reported which challenged this assumption.58 Thus, possible sources of the inconsistency in the σ values reported by Marek and Strauß are the assumptions indicated in eqs 4 and 5. The assumption in eq 4 was also made by Badam et al.119 They measured both TLI and TVI in a series of evaporation experiments (a description of the experiments is provided below in section 4.2) and used the HKS relation, eq 6, to determine the values of σ. They reported σ had to vary by at least an order of magnitude in their experiments in order to satisfy the HKS relation. This suggests that even when both TLI and TVI are taken into account, the HKS relation may still lead to poor predictions if σe and σc are assumed to be equal.

to the interface measured relative to the normal interface velocity, cz. Their expression for the condensation (and evaporation) coefficient was given as102 ⎡ ⎢ −2 − 3 σc = exp⎢ ⎢ 2 − 23 ⎣

vL vV vL vV

⎤ ⎡ mc 2 ⎤⎞ ⎥⎛ vL z ⎥⎟⎟ − 1⎥⎜⎜1 − 2 3 V exp⎢ − v ⎣ 2kBTIL ⎦⎠ ⎥⎝ ⎦ (7)

The velocity-dependent condensation coefficient proposed by Tsuruta and Nagayama102 was used by Bond and Struchtrup63 to investigate the evaporation experiments of Ward et al.58,111,112 A detailed discussion of the experiments by Ward et al. will be given later in section 4.2. Bond and Struchtrup did not use the explicit velocity-dependent condensation expression in eq 7 but instead kept the relation general by introducing two parameters.63 Their expression for σc did not assume thermal equilibrium across the interface, nor did they assume equality between σe and σc but instead expressed σe as a function of σc. Their expression for σc was ⎛ ⎡ −mc 2 ⎤⎞ z ⎟ ⎥ σc = ψ ⎜⎜1 − ω exp⎢ L ⎟ 2 k T ⎣ ⎝ B I ⎦⎠

(8)

where ψ and ω were constants with values between 0 and 1. In eq 8, ω scaled the temperature and velocity dependence of σc: the faster the particles collided with the interface, the more likely they were to condense; the colder the interface, the more likely impinging molecules would condense. Aside from the difficulty of still having two unknown constants, ψ and ω, in their expression for σc, they also used the mass accommodation coefficient, αm, in their mass and heat flux expressions. Their final expressions for σc and σe were complicated functions of ψ, ω, αm, TLI , and TVI and it was beyond the scope of their paper to examine the full expressions. Instead, the authors assigned values to the three constants and compared the results of their simulations to the evaporation experiments of Ward et al.58,111,112 Bond and Struchtrup were able to show their velocity-dependent coefficients led to a better agreement with the experiments than assuming constant coefficients, but they thought that a more realistic model, such as eq 7, accounting for the internal degrees of freedom of the molecules would yield even better results.63 The expressions for σc in eqs 7 and 8 indicate the coefficient values in eq 3 are expected to decrease with an increase in temperature, TLI . Others have noted a similar dependence of the coefficient values with temperature.81,97,113−115 Marek and Strauß also noted that values of σe and σc depended on the vapor-phase pressure, PV.64 While the experiments they analyzed showed σe and σc tended to decrease with an increase in PV, the numerical studies they investigated showed the opposite trend.64 3.1.1. Simplifying Assumptions. Marek and Strauß investigated a more restrictive assumption on the coefficients.64 They considered the consistency of the coefficients inferred from experimental data when it was assumed that both σe and σc were equal and constant and denoted it by σ. They also assumed that the interface was thermally equilibrated and compiled the values of σ reported in 29 different investigations. For water, they found σ varied between 10−3 and 1 for similar experimental conditions. Although this raises questions about the HK relation, eq 3, and its coefficients, it does not establish where the difficulty lies. Kruyukov and Levashov116 demon-

3.2. Effect of Interfacial Curvature on Values of σe and σc

The curvature of the interface in experiments is another reason given for coefficient values less than unity. Barrett and Clement noted that experiments with evaporating or condensing droplets resulted in coefficient values closer to unity than experiments with flat surfaces.72 They claimed that different limitations applied between the two geometries during evaporation or condensation. In planar surfaces, they thought kinetic processes were typically limited by the heat transfer or diffusion across the buildup of impurities on the surface. In contrast, they did not think that there would be any significant buildup of impurities on droplets, hence heat transfer would likely limit evaporation and condensation. They concluded that the accumulation of impurities in a gas layer above the planar surfaces resulted in lower values of σe and σc. Similarly, Varilly and Chandler also reported that the curvature of the liquid surface affected the molecular flux.120 They found that, “evaporation correlates with negative mean curvature,” indicating that the evaporation of a single molecule actually involved the concerted motion of a collection of neighboring molecules. 3.3. Effect of the Molecular Structure on Values of σe and σc

Others have investigated how the liquid−vapor surface roughness and the polarity of the molecules affect the coefficient values. Experimental studies of fluid−fluid interactions near the interface suggest the surface is not smooth.121 While the surface may appear smooth on a macroscopic scale, thermal fluctuations and capillary waves at the free liquid− vapor interface can cause the surface to become irregular at the microscopic scale.122,123 Such studies raise questions regarding the application of the HK relation, eq 3, to evaporation problems since interfaces may in reality be rough and not smooth at the microscopic scale. A rough interface would affect the applicability of the M-B velocity distribution function in determining the collision frequency of vapor molecules with the interface. 7731


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molecules at the liquid−vapor interface oriented themselves such that their hydrophobic methyl groups protruded into the vapor phase while their hydroxyl groups formed hydrogen bonds on the liquid side of the interface.129,130 These findings for polar molecules indicate that the liquid and vapor phases are coupled during evaporation and condensation, unlike the model presented in eq 3 that treats the phases as independent from each other.

Panczyk simulated the deposition of argon on either smooth or rough surfaces. For both cases, he calculated the collision frequency between the gas and surface.124 He compared his calculated collision frequency with that of the HK relation under equilibrium conditions for a flat and smooth surface. He found that the collision frequency decreased with surface roughness (specifically the fractal dimension), although his results for an ordered grainy surface indicated collision frequencies greater than those obtained from the HK equation. The surface roughness could change (either increase or decrease) the collision frequency calculated from the HK relation by 24 to 48% and would directly affect values of either σe or σc. He noted his simulations were simplified to consider only ideal gases and did not account for fluid−fluid interactions. Other studies have investigated how the values of σe and σc depend on the polarity of the molecules. Some have claimed that a molecule’s dipole moment does not affect the coefficient values. For example, Ishiyama et al.125 performed molecular dynamic simulations of water, methanol, and argon evaporation. They assumed thermal equilibrium at the interface and that the coefficients σe and σc were equal. They found that the coefficients of water, of methanol, and of argon approached unity with a decrease in temperature. They did not report any dependence of the coefficient values on the polarity or size of the molecules. A later MD study provided additional support of their findings; Ishiyama et al.126 wrote, “The condensation coefficient is found to be constant and equal to the evaporation coefficient determined by the liquid temperature only.” However, results from other studies have suggested that the coefficients do depend on the polarity of the molecules. Some have reported that polar liquids can have values of σe and σc near unity104,105 or much less than unity.127 The conflicting conclusions reached in the studies resulted from the different assumptions investigators have applied to eq 3 and from differences in their understanding of evaporation. Wyllie assumed that the evaporation and condensation coefficients were equal and the interface was thermally equilibrated. He noted that liquids with large dipole moments had σc values much less than unity.127 He found that polar liquids that were able to form hydrogen bonds had condensation coefficients approximately equal to their free angle ratio (i.e., the ratio of the rotational partition functions of molecules in the liquid phase to those in the vapor phase assuming evaporating molecules share the same vibrational partition function as those in the liquid phase).127 He thought that polar molecules resulted in an ordered surface and impinging molecules had to be properly oriented before they could condense, resulting in lower values of σc. Wyllie’s work indicated that the liquid and vapor unidirectional rates were not independent but coupled through noncovalent intermolecular bonding. For water, the idea that the phases are coupled was further supported by the work of Stiopkin et al.121 They investigated the thickness of the interface layer for an air−water system by using water molecules with one or two deuterium atoms.121,128 They reported that the interface was only a molecular diameter thick on average and had properties markedly distinct from both the bulk liquid and vapor phases. In particular, they found that water molecules in the surface layer were hydrogen bonded to each other, but some molecules had unbonded hydrogens protruding into the vapor phase.121 These “dangling bonds” were then free to hydrogen bond to other water molecules in the vapor phase. Similarly, for methanol or ethanol, the

3.4. Capillary Waves and Energy Barriers

In his examination of evaporation by thermally excited capillary waves, Phillips131−133 considered the liquid surface as a membrane with the thickness of a molecular diameter, clearly distinct from the bulk liquid, and capable of supporting normal vibrational modes in three dimensions. He claimed131 that thermally excited capillary waves of water at 300 K would produce displacements on the surface with a root-mean-square on the order of 0.4 nm, indicating that the liquid surface would be “highly planar on the scale of the mean free path.” He examined two extreme cases of evaporation: one in which the evaporating molecule was emitted normal to the planar surface and another in which the evaporating molecule was emitted from the tip of a sharp caret-shaped wavelet. In either case, the escaping (evaporating) molecule required a sufficiently high velocity (kinetic energy) to overcome the energy barrier to evaporation (the barrier being on the order of the latent heat per molecule). Phillips claimed that the energy barrier was lowest for a planar surface and increased as the wavelet became more pointed. However, he also noted,131 “evaporation is a dynamic process for which the probability of escape is not necessarily equal to the probability of possessing sufficient energy to surmount the activation barrier.” He calculated that the probability, Pw, of emission at an angle θw to the normal for an individual molecule with a fixed amount of energy, εw, in the normal coordinate to the surface having a temperature Tw was given by ⎡ −ε tan 2 θ ⎤ w ⎥sin εw Pw[θw , εw] = exp⎢ w kBTw ⎦ ⎣

(9)

and indicated “the most probable direction of emission is not far from the normal to the surface.” Phillips concluded that the newly evaporated molecules would have nonrandom directions of velocity and high kinetic energy in comparison with those in the gas phase at the same nominal temperature (i.e., a temperature discontinuity would be expected to exist across the liquid−gas interface). He explicitly writes, “the surface might, under some conditions, cause the temperature of the gas adjacent to the surface to be higher than the surface temperature.” In fact, Phillips describes the interfacial temperature discontinuities as paradoxes that “do appear to have some basis in reality.” Note that interfacial temperature discontinuities were not considered in his expression for the probability of the emission of a molecule from the surface in eq 9. In a subsequent paper, Phillips used a capillary-wave model of the evaporation process to investigate the “paradox” of the interfacial temperature discontinuities.132 He raised doubt about using macroscopic fluid properties in his capillary-wave model because of the molecular dimensions involved. He remarked that, on the distance scale of an evaporating molecule, there was no local surface of well-defined slope. Instead, Phillips considered a distribution of slopes at the surface. He visualized evaporation132 “as occurring when a molecule at the top of a fast-moving column of liquid continues to travel away from the 131

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mean surface after the rest of the column has ceased moving upwards.” He applied his capillary-wave model to vapor leaving a water droplet 1 mm in diameter at 298 K but found that the shape of the calculated angular distribution was sensitive to the value of the heat of vaporization and very sensitive to the effective surface area of a molecule. He concluded, “there was no experimental evidence to suggest that the velocity distribution functions of molecules leaving the surface of a liquid are as complicated and variable as this would imply, so the model must be wrong.” However, he notes that for molecular evaporation times of 0.1 ps or less, “the surface vibrations are effectively frozen...Hence the concept of a local slope is a valid one at high kinetic energies, and this should apply to an escaping molecule as well as to an incoming one.” Thus, Phillips identified conditions under which his capillarywave model would not be applicable. Phillips furthered his analysis of local-mode capillary-wave theory in studies to calculate evaporation rates for 12 different liquids.133 He attempted to bridge the gap between “small-scale motions of individual molecules that are treated by molecular dynamics calculations and the larger-scale fluctuations, termed capillary waves, that can be regarded as Brownian motion of the membrane created by surface tension,” by extending capillarywaves models to very small distance scales. The test of his localmode capillary-wave evaporation model was to predict the saturation-vapor pressure. He assumed that the evaporation and condensation coefficients were unity and found that “the calculations are very sensitive to the value assumed for the collision diameter, which implies that treating the molecules as non-polar hard spheres must introduce significant error.” He concludes, “Clearly the calculations do not capture the whole temperature dependence,” indicating that the capillary-wave models he had investigated did not adequately capture the physical process of evaporation. Varilly and Chandler used a capillary-wave model, transition path sampling methods, and energy barrier studies to investigate the evaporation of water.120 They treated the uptake coefficient (condensation coefficient) as equal to the evaporation coefficient. They wrote that “the chief advantages of our approach are that we do not need to introduce the approximation that the velocities and angular momenta of the evaporated water molecule are Boltzmann-distributed, with a temperature equal to that of the liquid, and that we are able to generate a large number of evaporation trajectories (about 5000), which we can characterize statistically instead of anecdotally,” thus overcoming some of the challenges encountered by Phillips. Varilly and Chandler used the extended simple point charge (SPC/E) model of water because it “adequately captured a broad swatch of liquid water’s properties.” However, many of the fluid properties at 298 K contained significant errors: their calculated surface tension had an error of 10% compared to experimental values, and their saturation-vapor pressure values were “within a factor of 2” of those reported by Errington and Panagiotopoulos.120 However, Errington and Panagiotopoulos remarked,134 “The SPC/E model vapor pressure curve is consistently a factor of 2 lower than the experimental value, an unacceptably large deviation for many applications.” Furthermore, the experimental saturationvapor pressure values that Errington and Panagiotopoulos used were obtained from the National Institute of Standards and Technology (NIST), and doubts have been raised regarding its accuracy as a source of saturation-vapor pressure expressions of fluids.117,118

Nonetheless, using their evaporation model, Varilly and Chandler120 found that all water molecules with any initial trajectory toward the water slab always condensed, implying a condensation coefficient (and by their assumptions, an equal evaporation coefficient) of unity. They claimed that their coefficient value of unity was consistent with other simulation studies but “are in apparent contradiction with the most recent experimental results.” They also remarked that, “the general lack of consensus between experiments makes it unclear whether our result of apparent unit evaporation coefficient agrees with reality or if it is an artifact of our simulations.” Varilly and Chandler attempted to explain the discrepancy between their results and those from experiments (see for example the experiments by Li et al.93 described in the next section); they suggested an error in the experimental calibration of temperature sensors: “A systematic error of 2% in absolute temperature in the experiments (equal to ∼10% in the temperature change during the course of the measurements) would be sufficient to account for the discrepancy between the experiments and our calculations.” Varilly and Chandler acknowledge,120 “Of course, it is impossible to obtain arbitrarily precise quantitative agreement with experiments using SPC/E or any other classical model of water. However, the consistency of this model with general measures of liquid-vapor coexistence, interfacial energetics, and molecular fluctuation amplitudes and time scales gives us confidence in its qualitative predictions about the molecular dynamics of water evaporation.” Bellucci and Trout, whose work support the results of Varilly and Chandler, provided a similar disclaimer,135 “our approach to the calculation of the evaporation coefficient is only meant for qualitative, not quantitative purposes.” Their coefficients values ranged between 0.83 and 3.08, which they described as “unphysical since evaporation coefficient should be bound by 1.” Nagata et al.136 extended the work of Varilly and Chandler to examine “how a water molecule can gain sufficient kinetic energy to move from the interface to the vacuum; the free energy of 10−12kBT (25−30 kJ/mol) is required for a water molecule to move from the interface to the vacuum at room temperature.” They used molecular dynamics simulations and found that the evaporation of a molecule across the water−air interface required a particular molecular pathway where the forming and breaking of hydrogen bonds between at least three surface water molecules in a specific sequence caused the recoil of one molecule and allowed it to escape (evaporate). Nagata et al. took into account a “temperature dependence of the experimentally measured evaporation coefficients” in their proposed mechanism, but they did not account for the effect of interfacial temperature discontinuities across the water−air interface. In summary, capillary-wave models have been used to study evaporation and to determine values of the condensation and evaporation coefficients. However, some capillary-wave models have regimes in which they are not valid.131−133 While studies using the capillary-wave and energy barrier approaches agree with each other,120,135,137 they do not show agreement with experimental studies.120 Thus, the source of the scatter of the evaporation and condensation coefficient values remains unresolved by those models. Some have suggested that the source of the disagreement lies in the correct determination of the interfacial liquid temperature.120 Moreover, interfacial temperature discontinuities have been described as “paradoxes” by investigators using the capillary-wave and energy barrier 7733


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that in order to measure the evaporation flux to an accuracy of ±2% (this was the precision in the measured values of σc claimed by Li et al.), the vapor pressure would have to be measured to within ±10−3 Pa. Moreover, Polyakov et al. have noted that the isotopic self-exchange is reversible:146

methods,131 but none (to our knowledge) have considered the effects of the interfacial temperature discontinuities on the values of the evaporation and condensation coefficients. 3.5. Effect of Impurities on Values of σe and σc

As mentioned earlier, small values of σe and σc have been attributed to contamination of the interface. Experiments to determine the coefficient values are typically divided into two classes: aging and renewing surfaces. In experiments where the interface ages, coefficient values much less than unity have been reported, whereas in experiments with renewing surfaces, coefficient values near unity were reported.64,138 The general explanation is that in aging surfaces, contaminants accumulate on the surface and impede the evaporation process, leading to coefficient values orders of magnitude less than unity.72,81,85,102,139−141 In contrast, renewing surfaces prevent the accumulation of contaminants allowing the evaporation flux to proceed unimpeded, resulting in higher coefficient values.105 The difficulty with these explanations is that even undetectable or trace impurities are hypothesized to reduce the coefficient values,139 and it is difficult to know when impurities are present in the system. Shi et al.142 investigated the uptake of ammonia gas into water droplets using a droplet train flow reactor. They considered the measured uptake of a gas into a liquid droplet as consisting of three combined effects: gas diffusion to the droplet surface, mass and thermal accommodation of the gas molecules at the interface, and the chemical reaction of the solvated gas in the droplet. Li et al.93 used a similar approach to study the σc value of water. They only considered the unidirectional condensation flux and assumed thermal equilibrium of the liquid−vapor interface with the bulk liquid. They used a droplet train flow reactor to measure the uptake of H17 2 O and D2O into H2O droplets. In the experiments, the water droplets were kept in equilibrium with the H2O vapor but were exposed to trace amounts of H17 2 O and D2O in the vapor phase. Only the trace elements were not in equilibrium and Li et al. were able to determine values of σc by monitoring the disappearance of the trace isotopes from the vapor. They found values of σc between 0.17 and 0.32 over a range of conditions. Others have questioned those values and suggested Li et al. did not account for the effect of the carrier gas and that the values of σc from their experiments should have been between 0.2 and 1.120,143,144 However, Davidovits et al.55 noted different experimental techniques and physical models of interfacial mass transfer could lead to different values of σc that are valid, depending on the interpretation of the results. It is interesting to note that in their determinations of the σc values, Li et al. made questionable assumptions:93 (1) they did not account for the adsorption of methane onto the water droplets during their 50 ms transit in the reference gas used to monitor the H17 2 O and D2O species; (2) the water isotopes did not change the total vapor pressure by more than a factor of 10−3 (or ±0.19 Pa) and therefore had a negligible effect on the H2O liquid−vapor equilibrium; and (3) evaporation of the D2O species from the liquid droplet was negligible. However, as noted earlier, trace amounts of impurities can have a significant effect on lowering the coefficient value:139 Cammenga145 noted, “most phase transitions, especially evaporation and condensation at liquid surfaces, are extremely sensitive to minute contaminants which are very difficult to exclude.” In addition, the results of Ward and Fang61 indicated

D2 O(v) + H 2O(l) ⇌ D2 O(l) + H 2O(v)

(10)

since the H2O(l) and H2O(v) were in equilibrium, the evaporation rate of D2O would not be negligible compared to its condensation rate. Hence, the assumptions made by Li et al.93 are open to question, and their values of σc need further verification. 3.6. Interfacial Temperature Measurements

Accurate measurements of surface temperatures and pressures during evaporation or condensation are important to correctly determine values of σe and σc. Erroneous measurements of these properties are considered to be one of the reasons why coefficient values between 10−3 and 1 have been reported. However, as noted earlier, the interface is approximately only a few molecular diameters thick and only one molecular diameter thick for water,105,121 for ethanol, and for methanol.129,130 Thus, the methods of determining the interfacial conditions must make accurate surface measurements on each side of the interface. Several techniques have been employed in the literature to measure the surface temperature including thermocouples (some with bead diameters as small as 25 μm58,112,147,148), thermistor probes,149 filament resistance probes, infrared thermography,150−152 interferometric techniques,87,153,154 and using the dependence of surface tension on temperature to infer the surface temperature of a droplet (having a radius smaller than the capillary length so that gravitational forces could be assumed negligible) from measuring the curvature of the liquid−vapor interface.64,103 A more detailed discussion of thermocouple measurements will be provided below in section 3.6.1. Cammenga et al. provide numerous examples of studies using these techniques as well as their advantages and shortcomings when applied to measure the surface temperature.103 For example, in measurement techniques that physically probe the surface, questions have been raised regarding the disturbance of the surface by the probe, the introduction of spatially averaged errors from the size of the probe, and heat conduction along the probe that may introduce reading errors.103 Conversely, noninvasive techniques measure the temperature indirectly and are less accurate since the temperature is averaged over a large volume and accurate fluid properties are required to convert the measurements into temperature values.155 Even for accurate experiments, inaccurate thermophysical fluid properties, such as Ps(T), can introduce errors in the calculated surface temperature.117,118 Additional examples of measuring interfacial temperatures are provided by Derkachov et al.156 They used the results from Hołyst et al.157 who had developed an expression, based on molecular dynamics simulations, for the temporal evolution of the radius of curvature of an evaporating droplet. Derkachov et al. developed a method to determine the temperature in the evaporating droplets (pure or multicomponent) to within fractions of mK using the initial composition of the droplet and the ambient temperature as inputs into the temporal evolution expression proposed by Hołyst et al. 7734


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Barnes noted that, for both invasive and noninvasive techniques, the temperature cannot be directly measured at the interface. It must be inferred and extrapolated from measurements and assumptions of temperature gradients in the bulk phases and from fluid properties.81 3.6.1. Thermocouples. Thermocouples have played an important role in understanding evaporation processes. Their bead diameters can be made as small as 25 μm and brought to within a mean-free-path of the interface58,117 to accurately measure the interfacial vapor (and liquid) temperatures. They can be also be calibrated to within ±0.04 K118,147,158 and do not have to be recalibrated for different fluids. They were instrumental in measuring interfacial temperature discontinuities (see the discussion that follows in section 3.6.2). Moreover, they can be used with ice point cells and formed into U-shapes to help mitigate effects of Thompson electromotive force and unwanted thermal conduction along the thermocouple wires.58,118,147,159 The small size of thermocouples also makes their intrusive effects negligible. For example, a study of evaporating sessile multicomponent droplets compared the measurement of temperature pulsations in the droplets made using infrared thermography (noninvasive) with those made using thermocouples submerged in the evaporating droplets.148 The study indicated that the thermocouples gave the same quantitative measurements as the noninvasive method. Thus, any intrusive effects of the thermocouples were negligible. 3.6.2. Direction of Interfacial Temperature Discontinuities. Another important consideration is the existence of an interfacial temperature discontinuity across the liquid−vapor interface. The idea of an interfacial temperature discontinuity is not new,160−165 but the direction of the discontinuity has been contentious; results from classical kinetic theory have indicated the that vapor side of the interface would be colder than the liquid side, whereas some experiments indicate the opposite is true. Only in the past 20 years have the first experimental measurements indicating the existence and direction of the interfacial temperature jump during evaporation and condensation been reported.58,112,147 One test of kinetic theories is to predict both the direction and magnitude of the interfacial temperature discontinuities during evaporation or condensation. Pao160−162 and Koffman et al.73 applied CKT to a simple conceptual evaporation system: two parallel plates (semi-infinite) were supposed separated by a small distance but greater than the thickness of the Knudsen layer, as indicated in Figure 1. The top plate was supposed to be maintained at a constant temperature T0 + 0.5ΔT and the bottom plate at a temperature of T0 − 0.5ΔT. A film of liquid water covered both plates. Water would evaporate from the hot plate and condense onto the cold plate. If the cold plate were kept at 372 K and the hot plate at 373 K, the CKT approach predicted the interfacial water-vapor temperature above the cold plate would be 373 + 0.38 K and the water-vapor temperature below the hot plate would be 372−0.38 K. In other words, the vapor temperature near the hot plate would be colder than the temperature of the cold plate; similarly, the vapor temperature near the cold plate would be hotter than the hot plate’s temperature. These predictions are nonphysical and Koffman et al. wrote,73 “This clearly does not make sense.” Pao acknowledged experiments would be required to determine if his predictions were correct; he wrote,161 “it would be interesting to see some experimental measurements capable

Figure 1. Water theoretically evaporates between two semi-infinite parallel plates. The top plate is hotter than the cold plate at the bottom. The temperature difference between them is ΔT. Thus, water evaporates from the top plate and condenses onto the cold one. When Pao’s CKT approach was applied to the system, the vapor temperature near the cold plate was predicted to be hotter than the hot plate’s temperature. Likewise, the vapor temperature at hot plate was predicted to be colder than cold plate’s temperature. The inverted temperature profile is shown. Adapted with permission from ref 161. Copyright 1971 AIP Publishing. Adapted with permission from ref 73. Copyright 1984 AIP Publishing.

of determining whether such a predicted negative temperature gradient does occur.” Similar results were found by many other investigators.166,167 Regarding CKT, Koffman et al.73 wrote, “A result of the theory is that the temperature profile in the vapor for the continuum problem is inverted from what would seem physically reasonable. This paradox is significant in that it casts a shadow of doubt on the fundamental theory.” Aoki and Cercignani168 investigated the conditions leading to the inverted temperature jump, but they wrote about the need for experimental investigations: “Perhaps, experimental checks to prove or disprove the present results are possible.” In a later paper, Cercignani et al.169 noted the paradox was still unresolved and “still waiting for experimental solution.” Thus, CKT predicted a nonphysical inverted-temperature profile at the liquid−vapor interface of an evaporating system, and the CKT approach was still questioned in 1985. There was a clear need expressed for fundamental experiments to investigate the conditions at the liquid−vapor interface during evaporation. Results from CKT had led many to assume that the liquid and vapor interfacial temperatures were equal171 since they were only separated by distances on the order of a molecular diameter.172 The first experiments to investigate the question of the interfacial temperature jumps were reported by Shankar and Deshpande.170,173 They evaporated water, mercury, and Freon 113 into air and used type K thermocouples with bead diameters ∼600 μm to measure the temperature to within ±0.05 K. They reported measured temperature discontinuities of approximately 1 K for water, see Figure 2. Similarly, in their Freon 113 and mercury evaporation studies, they reported interfacial temperature discontinuities of 0.5 and 10 K, respectively. As indicated in Figure 2, the interfacial vapor temperature at the evaporating surface was colder than that of the liquid. However, it is not clear if the measurements reported by Shankar and Deshpande are of the interfacial liquid and vapor temperatures. For example, their temperature measurements were not close to the interface but were made approximately 1 mm away from the interface. In addition, their 7735


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was continuous between the two phases. Thus, they did not consider the temperature jump as real. Instead, they claimed that near the surface, the thermocouple bead was only partially wetted and was affected by the evaporation of water from the bead. Using a type K, 25 μm-diameter bead microthermocouple, experiments were performed by Fang and Ward.58 They evaporated pure water steadily into its own vapor and were able to measure the interfacial vapor temperature to within onemean-free path (MFP) of the water-vapor interface where an evaporating molecule was more likely to collide with the microthermocouple bead than with another water molecule. As indicated in Figure 4, interfacial temperature discontinuities of Figure 2. Water heated from the bottom evaporates into air. Temperature measurements of the liquid (■) and air (□) show that the interfacial liquid temperature is warmer than that of the vapor. Adapted with permission from ref 170. Copyright 1990 AIP Publishing.

experiments were not in the steady state and were exposed to contamination. They thought that in a “pure” system, the temperature jumps would have been even greater but in the opposite direction. Thus, the direction and magnitude of the interfacial temperature jump they reported are open to question. Nonetheless, they raised an important objection to CKT: “Continuum calculations which assume that the temperature jumps are absent, such as those of Plesset [M. S. Plesset, J. Chem. Phys. 20, 790 (1952)] are certainly in error.” In other experiments by Hisatake et al.,174,175 water evaporated into atmospheres with different relative humidities and air flow velocities. They used a type K thermocouple with a bead diameter of 127 μm to measure the temperatures above and below the air−water interface, as indicated in Figure 3. They measured the temperatures on either side of the interface and within 100 μm of it. They observed temperature discontinuities of about 1 K, and the temperature on the vapor side of the interface was greater than that on the liquid side. However, Hisatake et al. assumed the temperature profile

Figure 4. Water evaporates into its own vapor, and temperature measurements in the liquid and vapor phases show that the interfacial liquid temperature is colder than that of the vapor by up to 7.8 K. The interfacial vapor temperature, measured to within 1 MFP of the interface, was greater than the interfacial liquid temperature. Adapted with permission from ref 58. Copyright 1999 American Physical Society.

up to 7.8 K were reported, with the temperature in the vapor higher than that in the water,58,117 in contradiction to the predictions of Pao160−162 and of Koffman et al.73 Other studies have supported the findings of Fang and Ward.58,147,158,176 Investigators have found that the magnitude of the interfacial temperature discontinuity depends on the evaporation flux, but the direction has always been positive (TVI > TLI ).58,118,147,176 The possible factors affecting the measurement of the interfacial temperature discontinuity, including radiation, thermocouple bead diameter, and evaporative cooling of the thermocouple bead, were investigated and concluded to be negligible.58,177 Badam et al. measured even larger interfacial temperature discontinuities when they heated the water vapor phase with an electrical heater. They reported interfacial temperature discontinuities of more than 15 K.119,159 Duan and He178 placed three microthermocouples near the interface of steadily evaporating water and measured the temperatures simultaneously in the vapor and in the water. They reported temperature discontinuities of up to 6.5 K. David et al.179 measured interfacial temperature discontinuities of up to 0.45 K of water evaporating into room air using a thermocouple with bead diameter of 250 μm. Fang and Ward also reported temperature discontinuity values of ∼5 K during the steady evaporation of methylcyclohexane and of octane.111

Figure 3. Water evaporates into a humid atmosphere and temperature measurements of the liquid (○) and air (●) show that the interfacial liquid temperature is colder than that of the vapor. Adapted with permission from ref 174. Copyright 1993 AIP Publishing. 7736


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study, Hołyst and Litniewski183 used simulations consisting of ∼106 particles and proposed that the evaporation of a nanosized droplet was limited by the transfer of heat from gas molecules to the liquid surface. In their model, the nanodroplet was initially heated from “the condensation of hot vapor at a colder surface of the droplet; thus, paradoxically, evaporation starts with initial condensation of the vapor,” and the nanodroplet initially increased in size and temperature. This transient period lasted a few nanoseconds before transitioning to a quasistationary regime, where mechanical equilibrium was established very quickly in comparison to thermal equilibrium. Thus, Hołyst and Litniewski claimed that the evaporation process was limited by the heat transfer between gas molecules and the liquid surface, not from mass diffusion inside the vapor. In the quasistationary regime, they observed that the interfacial temperature discontinuity increased with the ratio between the densities of the liquid and of the vapor. When the density ratio was less than 10, no interfacial temperature discontinuities were observed, but when the ratio was above 30, the interfacial vapor temperature was observed to be “as large as nearly 30% of liquid temperature at the interface.” They claimed that their MD simulation studies, “show a clear need for the reformulation of the current view on the dynamics of the evaporation.” Thus, the measurement of interfacial temperature discontinuities is not at all unusual. Hołyst et al. remarked,59 “the temperature discontinuity at the liquid-vapor interface discovered by Fang and Ward (1999 Phys. Rev. E 59 417−28) is a rule rather than an exception.” However, predictions of temperature discontinuities made from CKT are smaller and in the opposite direction to the measured experimental values and indicate that CKT does not give a proper description of the evaporation process. Therefore, there is a need for another, more accurate theory of evaporation than that provided by the HK relation, eq 3.

Moreover, Ward and Stanga reported interfacial temperature discontinuities in water evaporation and condensation experiments.112 In their condensation experiments, the values of TVI were greater than those of TLI . Similar observations have been reported in other water condensation studies.180 Zhu et al.181 used a 50 μm-diameter bead type K microthermocouple and measured negative interfacial temperature discontinues during the evaporation of silicone oil, as shown in Figure 5. They claimed the interfacial temperatures

Figure 5. Silicone oil evaporates into the room atmosphere, and temperature measurements of the liquid and vapor phases show that the interfacial liquid temperature is warmer than that of the vapor. (Temperature discontinuity of 0.65 cSt and 1.5 cSt silicone oil close to the interface when the temperature difference is 8 °C). Adapted with permission from ref 181. Copyright 2009 Springer.

were measured 50 μm apart from each other, but this would mean that the microthermocouple bead was partially wetted during the interfacial measurement. Evaporative cooling of a partially wetted bead could explain the colder interfacial vapor temperature they measured. However, Zhu et al.181 provided an alternate explanation: “Note that our results are different from Ward’s results in temperature scales and temperature gradients (Ward and Duan 2004). It is mainly because of differences in experimental conditions and the low evaporating rates in our experiments.” However, there are other problems with their measurements: in Figure 5, the temperature profile in the liquid phase shows a kink, indicating a localized heat source; however, no explanation of the heat source was provided making their measurements suspect. In a subsequent study, Zhu and Liu182 described in more detail the evaporation of 2 mm thick layers silicone oil exposed to horizontal temperature gradients under room conditions. They indicated that a kink in the measured vertical temperature profile probably resulted from strong convective flows generated by a combination of evaporative and thermocapillary effects. They claimed to have measured both negative and positive interfacial temperature discontinuities. However, it was still not clear from their recent work if their 50 μm-diameter bead type T microthermocouple was partially wetted during the interfacial temperature measurements; the time interval between measurements was only 15 s and may not have been long enough for the bead to have completely dried after crossing from liquid phase into vapor phase. The direction of the interfacial temperature discontinuity has also been examined with molecular dynamics simulations (details of MD studies are provided in section 3.8). In one

3.7. Pressure Measurements

Another important quantity to measure in eq 3 is the vaporphase pressure, PV. The vapor-phase pressure cannot be directly measured at the interface. Early evaporation studies used mercury manometers to measure pressure.51,184 The pressure was typically read using a cathetometer to within ±0.01 mmHg or ±1.33 Pa.185 Chang and Davis used a McLeod gauge in addition to a manometer in their apparatus.91 They assumed σe was unity and found that the measured pressures agreed reasonably well with the values predicted from applying classical kinetic theory, thus they concluded the coefficient value should be unity. Mass spectrometry, ion gauges, and optical interference methods were used by Brown et al.87 to measure the vaporphase pressure during water vapor condensation on ice. They calculated the value of PV by assuming the sticking coefficient, αs, was unity and that TVI was equal to TLI . They reported “excellent agreement” with the measured values over the temperature range from 100 to 273 K and a pressure range between 10−8 and 103 Pa. Their values of σc (note they treated αs and σc as distinct) were dependent on the surface temperature and the incident beam flux, indicating that σc in eq 3 was not a constant.87 Others have used manometers of the test liquid instead of mercury and kept the closed end of the manometer at a known temperature103 while positioning the open end near the interface where the surface temperature was measured. The 7737


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interfacial pressure was then determined by assuming PV was equal to Ps(TLI ) and using saturation-vapor expressions to convert measured temperatures into pressures.81 Bertrand et al. built an evaporator-condenser and measured the vapor-phase pressure using a similar idea.186 When the PV is calculated in this manner, small errors in measuring the surface temperature propagate and amplify.187 There is the added complexity that different Ps(T) expressions differ considerably over the same temperature range.103 Only a few studies have considered the impact of uncertainties in PV on the values of σe and σc,81,104 but simplifying assumptions [σe equal to σc, PV equal to Ps(TLI )] used in those studies leave their findings open to question. Cammenga et al.104 demonstrated the importance of accurate pressure measurements when determining values of σe for glycerol. They used an effusion cell to measure the vapor-phase pressure of glycerol to within ±0.6%. By changing the size of the orifice of the cell and measuring the weight loss by effusion, they were able to perform experiments over a range of evaporation conditions. They found that assigning σe a value of unity provided the best fit to their experiments and suggested others who had reported σe values of 0.05 and lower did not correctly measure the PV of glycerol.104 Thus, the coefficient values in the HK relation were indicated to be sensitive to measurements of PV.

Table 1. Water and Ethanol Parameters for the LJ-Potential in Eq 11.195,196 δLJ (nm)

ϵ/kB

0.4530 0.2641

362.6 809.1

predicted from an approximate HK relation where the condensation flux was neglected, and the evaporation coefficient was set to unity. Depending on the value of the temperature assumed to exist at the interface, they found their approximate HK relation either over or under predicted the evaporation rate obtained from their simulations.198 Despite the gross simplifications they made to eq 3, Hołyst and Litniewski were able to show that the evaporation coefficient was not constant but had a temperature dependence. Cheng et al.193 carried out simulations similar to those of Hołyst and Litniewski with the objective of investigating the effects of molecular composition on evaporation. Their motivation was to address one draw back of the LJ model: “Namely it has a very high vapor pressure. This results in a very large vapor density which does not match the properties of most liquids.” They proposed a method to make the LJ model more realistic, which was to model the liquid as short chains of LJ atoms. Thus, they considered liquids composed of monomers, dimers, or trimers. They did not take the interfacial temperature discontinuity into account and assumed the evaporation and condensation coefficients were equal. They used the expression for the condensation coefficient proposed by Tsuruta and Nagayama102 (see eq 7). Their calculations of the coefficient values agreed closely with those of Hołyst and Litniewski, even for the dimers or trimers, namely that the coefficient value decreases with increasing temperature. Molecular dynamics simulations were used by Hołyst et al.157 to develop an analytical equation for the radius of an evaporating, mutlicomponent, LJ fluid as a function of time. They used their analytical equation for the radius to examine the evaporation of (1) water microdroplets into air and (2) glycerol, diethylene glycol, and triethylene glycol microdroplets into nitrogen gas. The analysis by Hołyst et al.157 indicated that the condensation coefficient in their MD simulations had a value of 1.003, or approximately unity. This result was in close agreement with their water experiments, where the coefficient had a value of 1.08. In the remaining experiments (evaporation of glycerol, diethylene glycol, or triethylene glycol microdroplets into the nitrogen gas), the coefficient value was between 0.74 and 0.9. The error in their calculated coefficient values was ±20%. In order to compare their MD simulations to the HK relation, eq 3, Hołyst et al.157,199 assumed thermal equilibrium of the interface with the liquid phase. They assigned values to the evaporation coefficient that would lead to agreement with their MD simulation and found,157 “Finally, for LJ liquid evaporating directly into vacuum at constant temperature the mass flux was given by the Hertz-Knudsen (HK) equation multiplied by the evaporation coefficient of value 2 instead of 1.” Note that the evaporation coefficient value of 2 is inconsistent with the CKT definition since it is greater than unity. Physically, a σe value of 2 would mean that for every molecule that crossed the interface from the liquid side, two molecules would evaporate, which is clearly nonphysical.53,63 The difficulty in their approach highlighted a problem with

3.8. Molecular Dynamics Simulations

One way to mitigate the experimental challenges described above is to use molecular dynamics (MD) simulations. These simulations offer several advantages over experimental investigations: no explicit thermodynamic properties or expressions for the kinetic behavior of the system are required, impurities are no longer a concern, and systems can be studied in time scales which are difficult to study experimentally.188,189 Instead of specifying the behavior of a system, MD simulations determine the positions and velocities of atoms as a function of time from the Newtonian equations of motions and a prescribed model for atomic interactions. 189 However, investigators using MD simulations still face the problem of specifying boundary conditions that reflect the physical behavior of evaporation systems. Atoms are typically modeled as spheres, and one interatomic model commonly used in MD simulations is the Lennard-Jones (LJ) potential: ⎡⎛ δ ⎞12 ⎛ δ ⎞6 ⎤ LJ LJ VLJ = 4ϵ⎢⎜ ⎟ − ⎜ ⎟ ⎥ ⎢⎣⎝ Γ ⎠ ⎝ Γ ⎠ ⎥⎦

liquid ethanol water

(11)

where VLJ is the potential energy between two molecules, Γ is the distance between molecules, δLJ is a molecule’s diameter or the distance at which VLJ = 0, and ϵ is the depth of the potential well and defines the unit of energy.190 The LJ potential has been used in many MD simulations to investigate evaporation and to determine values of σe and σc.116,191−194 The paramenters of the LJ potentials of water and ethanol are given by Assael et al. and Cussler195−197 and listed in Table 1. When calculations based on the HK relation are compared with MD simulations, the sensitivity of the HK relation to the assumed interfacial temperature has been illustrated: a recent review of the MD literature193 highlighted the work of Hołyst and Litniewski.198 Hołyst and Litniewski studied the evaporation of a monatomic LJ fluid into a vacuum and calculated the evaporation flux. They compared their calculated flux with that 7738


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using the HK relation, eq 3, and they wrote:157 “Before an appropriate HK equation is formulated, two problems have to be resolved: what is the temperature that we should use in the estimation of the saturation pressure in the Hertz-Knudsen equation? How should we account for the change of partial pressure of the evaporating substance near the interface?” In other words, Hołyst et al. still had to make the assumption of thermal equilibrium, eq 5, but it was unclear to them if they should have used TLI or TVI . This question will be addressed later in section 5.3. Furthermore, Hołyst et al.199 reported a substantial difference between the evaporation flux predicted from the HK relation and that determined from their MD simulations: “The flux determined in the simulations was 3.6 times larger than that computed from the HK equation...this observation suggests that the HK equation may not properly grasp the physical mechanism of evaporation.” Nagayama et al.200 used MD simulations to determine the effects of the molecule’s size and orientation relative to the interface on the values of the evaporation and condensation coefficients. They investigated simple molecules such as water and argon and long chain molecules such as octane and dodecane. Following a similar approach as Cheng et al.,193 Nagayama et al. used an expression for condensation coefficient proposed by Tsuruta and Nagayama102 and generally found that the evaporation and condensation coefficients decreased with increasing temperature of the interface. The surface was modeled to be 1 or 2 molecules thick, but they assumed a thermally equilibrated surface (i.e., they did not consider the temperature difference across the liquid−vapor interface). Nagayama et al.200 noted: “A condensation/evaporation coefficient of unity means a zero energetic barrier; i.e., all molecules approaching the interface completely condense into the liquid phase without reflection and those departing from the interface completely evaporate to the vapor phase.” They thought that at equilibrium, any energetic barrier to phase change would be zero, but under nonequilibrium conditions, an energetic barrier to phase change would be present and result in coefficient values less than unity. They report, “the mechanism of the energetic barrier has been attributed to the restricted translational motion of molecules...the molecular structure or shape is irrelevant to the condensation/evaporation coefficient.” Their results were consistent with other MD simulation studies that did not find any importance of the molecule’s structure on the values of the coefficients.114,129,201−203 Thus, the MD simulations studies give results that are mostly consistent with one another but are inconsistent with the HK relation.198 The studies have indicated that the underlying CKT formulation of the HK relation does not properly grasp the physical mechanism of evaporation. However, the MD simulations studies have not yet been shown to agree with independent experimental data.

Cloud formation can be initiated by aerosol particles that act as cloud condensation nuclei (CCN). Voigtländer et al.205 investigated the mass accommodation coefficient of water condensing onto or evaporating from sodium chloride aerosol particles under “realistic lower atmospheric conditions” using both experimental and theoretical studies. They treated the mass accommodation coefficient as equivalent to the condensation coefficient in the HK relation and as equal to the evaporation coefficient. They also considered the thermal accommodation coefficient, αt, in their evaporation model, but noted they were not able to determine both the mass and thermal coefficients simultaneously. Instead, they considered two values of αt: 0.85 and 1. In their experiments, aerosol NaCl particles with diameters of either 54 or 108 nm were saturated with water vapor and forced through a 1 m long laminar tube with a sheath of air. By controlling the wall temperature of the tube, Voigtländer et al. were able to vary the saturation ratio inside the tube between approximately 1.00 and 1.02. Depending on the experimental conditions, as the fluids were forced through the tube, the water vapor would first condense onto the NaCl particles to form droplets that reached a maximum size. Then, as the droplets were carried further into the tube by the air sheath flow, the droplets’ diameters would shrink (evaporate). The droplets finally exited the tube after a transit time of about 1.5 s. Voigtländer et al. plotted the mean droplet diameter as a function of the tube wall temperature and fitted the value of σe that gave the best agreement with their data. Due to experimental uncertainties, they concluded that the coefficient value was anywhere between 0.3 and 1 but could not comment on the effect of temperature on σe. Smith et al.95 took a careful experimental approach to determine the evaporation coefficient of water in the absence of molecular condensation fluxes. They evaporated microdroplet trains of water between 12 and 40 μm in diameter in a controlled vacuum chamber and measured the bulk water temperature using Raman spectroscopy to within ±2 K. The water microdroplets’ temperatures were between 245 and 295 K, and the measured organic content of the water was between 3 and 4 parts per billion. They considered the effect of evaporative cooling on the surface temperatures of the water droplets (but they ignored the vapor phase and thus any interfacial temperature discontinuity) in their HK evaporation flux expression and found that the value of σe in their experiments was 0.62 ± 0.09. They assumed σe and σc were equal and only found weak effects of temperature or droplet surface curvature on the values of the coefficients. They concluded that, since their coefficient values were less than unity, there existed a small entropic barrier to water evaporation. Their findings were inconsistent with others reported in the above sections, but Smith et al. stated that the disagreement could have resulted from the different constraints (droplet size and temperature) of their evaporation model. In a later study, Cappa et al.206 examined the apparent weak temperature dependence of σe reported by Smith et al.95 Cappa et al. used a transition state theory model to examine the evaporation of isotope mixtures of water, also in the absence of molecular condensation fluxes. They modeled the liquid surface as consisting of molecules that were more ordered and directionally oriented than molecules in the bulk phases. They found that the evaporation flux was sensitive to the molecular orientation at the surface (specifically to the surface molecules’ librational and hindered translational vibration modes) and was strongly sensitive to the isotope mixture.

3.9. Cloud Growth, Aerosols, and Multistep Models of Evaporation

It was briefly mentioned in the introduction that the HK relation was used in climatology studies. In this section, we expand on the application of the HK relation to cloud growth rates. Aerosol composition has been known to affect cloud growth rates through molecular mechanisms; in particular, organic surfactants in atmospheric aerosols are suspected to suppress growth rates, but the mechanisms involved remain poorly understood.204 7739


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Their Arrhenius-type evaporation model related σe to an experimentally determined activation energy for evaporation, Ea, and an effective area of the transition state, Atst. Interestingly, both Ea and Atst could not be determined from their methods, but they were able to input a desired temperature-dependence of σe and then calculate values for Ea and Atst. When they used the weak temperature dependence reported by Smith et al., they found that Ea was 41.7 kJ/mol and was in agreement with the evaporative latent heat of water (Atst was 2.45 × 10−21 m2). When Cappa et al. investigated the drastically different temperature-dependence of σe reported by Li et al.,93 they found that Ea was 24.7 kJ/mol and Atst was 4.7 × 10−25 m2. Thus, Cappa et al. obtained two very different results depending on their inputs, but the reason for the discrepancy was unknown; they wrote,206 “Without explicit knowledge of [Ea] and [Atst], it is not possible to establish, through use of this model, the validity of the temperature dependencies as deduced from the two experiments.” Cappa et al. additionally reported calculated σe values greater than unity for isotopic mixtures of water but did not comment further on whether or not σe could in reality be greater than unity. In another study by Drisdell et al.,207 the evaporation of ammonium sulfate solution was investigated because it is a prevalent atmospheric inorganic aerosol found in the troposphere where clouds are formed. In an aqueous solution, ammonium sulfate dissociates into ammonium and sulfate ions, but only the ammonium ion is thought to interact with water molecules at the liquid surface. However, the interaction was reported to be weak since only about 11% of the water molecules were within the interaction range of the ion. Drisdell et al. concluded that while 3 M solutions of ammonium sulfate decreased the equilibrium vapor pressure of the solution by 13% compared to the saturation-vapor pressure of pure water, the solution surprisingly was not found to change the value of σe (0.58 ± 0.05) for pure water. This result appears to be in contradiction with the earlier work by Cappa et al.206 who found that the value of σe was sensitive to the molecular orientation and vibrations at the surface. Indeed, Drisdell et al.207 expected that the σe value for pure water would decrease in the presence of surface impurities, but their experimental results from the evaporation of ammonium sulfate solution suggested that this was not always the case. The work reported by Drisdell et al.207 thus indicated that surface impurities in general do not necessarily result in lower values of σe. Similar results were obtained by Duffey et al.204 who found that the value of σe was 0.53 ± 0.12 for 2 M concentrations of acetic acid (which is ubiquitous in atmospheric aerosol) and was “indistinguishable from that of pure water.” To investigate which surface impurities would be expected to lower the σe value of pure water, Drisdell et al.208 repeated their evaporation studies using a 4 M sodium perchlorate solution. Four molar was the maximum concentration that could be used in their droplet train experimental apparatus, but atmospherically relevant concentrations were expected to be at least twice as high. Perchlorate was selected since its ion was predicted to strongly interact with the air−water surface. They found that the σe value for the sodium perchlorate solution was 0.47 ± 0.02, or 25% less than that of pure water. Drisdell et al.208 claimed that approximately 27% of the interfacial water molecules were within the interaction range of the ion and that the ion possibly hindered the librational modes of the water molecules resulting in lower evaporation rates and hence lower values of σe. Thus, Drisdell et al. suggested that surface-

active ions of inorganic solutes can significantly hinder the evaporation rate of water. Miles et al.209 were able to heat or cool aqueous aerosol droplets of water held in optical tweezers. Using a heating laser, they were able to control (change) the temperature of the droplet to within ±0.001 K and could measure the droplet’s change in size as it re-equilibrated with its surroundings. Miles et al. assumed σe and σc were equal and that the liquid−vapor surface was thermally equilibrated. They found that their experiments were consistent with assuming that σe and αt were both unity. They claimed it would be invaluable, “to understand the dynamic coupling between the liquid and gas phases at a liquid surface.” Their statement stands in contrast to the approach taken by Drisdell et al.208 who neglected the coupling between the liquid and gas phases during evaporation: “In nonequilibrium conditions, the condensation rate will depend on the gas pressure, but the evaporation rate will not, as the activity of the liquid is constant.” The investigations reported thus far in this section have treated cloud growth as a one-step process: a vapor is accommodated into the bulk liquid phase of an aerosol droplet. Julin et al.80 examined a multistep process of gas uptake using MD simulations of water evaporation (condensation) onto (from) nanosized droplets and planar surfaces at different temperatures. In the multistep process, water vapor molecules that collided with the liquid surface were first adsorbed onto the surface and then absorbed from the surface into the bulk liquid. Whereas the one-step process led to a single mass accommodation coefficient or σc, the multistep process led to a distinct coefficient for each step: a surface mass accommodation coefficient and a bulk mass accommodation coefficient. In their MD simulation studies, Julin et al. found that the surface and bulk mass accommodation coefficients were nearly exact and reduced to the one-step model. The value of σc in their studies was assumed to be unity (and equal to σe), even though the evaporation rates determined from their model and from experiments differed by a large factor of ∼2. In fact, σc values greater than unity would have yielded better agreement, but this was not a possibility Julin et al. were willing to consider. Instead, they suggested that the simulated saturation-vapor pressure could have been underestimated by a few hundred pascals, thus indicating the importance of having accurate thermophysical properties in the determination of the coefficient values. They also investigated the dependence of σc with temperature and the curvature of their nanosized droplets. They found a slight dependence on each, but no more than a 0.5% deviation from unity for the conditions they considered. Miles et al.210 further highlighted the need to have accurate thermophysical properties when measuring values of the evaporation and condensation coefficients of water. In their manuscript, they assessed and compared measurements of the coefficient values made using five different techniques (expansion chambers, continuous cloud condensation nuclei chambers, laminar flow tubes, aerosol optical tweezers, and electrodynamic balances) and attempted to reduce the uncertainty in the coefficient values from 3 orders of magnitude to just one. They investigated the sensitivities of each experimental technique to five thermophysical quantities (diffusion constants, thermal conductivities, saturation pressure of water, latent heat, and solution density) and to experimental parameters (pressure and temperature). In their approach, they assumed that σe and σc were equal to the mass accommodation 7740


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gave the same Ps(T) values. However, they argued that some parameter combinations, though mathematically valid, may not be physically valid. They concluded that the σe value of water was between 0.5 and 1 (consistent with the values reported by Miles et al.210) but were not able to assess the temperature dependence of the coefficient. Their Ps(T) data agreed well with other reported values, especially those of Murphy and Koop.212 Davies et al.211 did not refer to another similar study by Duan et al.117 who also used evaporation experiments to determine Ps(T) values of water down to 250 K. We note that there is good agreement between the saturation-vapor pressure data reported by Davies et al.211 and by Duan et al.,117 and both studies are consistent with a σe value near unity. The work by Duan et al.117 is discussed in more detail in section 4. Finally, Julin et al.213 examined the mass accommodation coefficient of atmospheric molecules other than water. They considered four organic molecules: adipic acid, succinic acid, naphthalene, and nonane. They assumed σe and σc were equal to the mass accommodation coefficient and performed MD simulation and cloud expansion studies to determine σc. They considered bulk and surface condensation coefficients and were also able to investigate the effects of molecular structure on the coefficients. While the number of molecules simulated was small (they only considered 100 molecules, whereas other MD studies discussed in this section considered 10000 molecules), Julin et al.213 found that the bulk condensation coefficient value was sensitive to how the location of the interfacial region was defined. They found that a 5% change in the interface position (or about 0.3 Å) could result the coefficient changing by a factor of 2. They concluded that for all four molecules, the surface and bulk condensation coefficients were consistent with a value of unity and appeared independent of the molecular structure, surface curvature, or phase state (solid-like or glassylike properties) of the condensed phase. 3.9.1. Summary of Climatology Kinetic Studies. In summary, the formation and growth kinetics of clouds changes the dynamics of radiative effects in the atmosphere. Understanding the mechanisms of cloud growth rates from aerosol composition is thus important in understanding climate change. All of the works reviewed in this section investigated the evaporation dynamics of droplets (or planar surfaces) of water under atmospherically relevant conditions or of aerosol organic molecules. We note common assumptions made by the authors of these works: (1) the unidirectional evaporation flux was assumed to be independent of the vapor or gas phases and (2) σe and σc were assumed to be equal to each other and were often treated as constants. The effects of these assumptions on evaporation models will be addressed later in this review. Common conclusions that we noted were as follows. (1) There is a need to correctly account for the sensitivities of σe and σc to thermophysical properties. (2) The value of σe and σc for water is most likely constrained between 0.5 and 1. (3) The complete buildup of organic films on aerosols results in values of σe and σc orders of magnitude less than unity. (4) Although different methods were used to determine the values of σe and σc, the results are not consistent since the “most likely” mass accommodation coefficients of pure water (0.62 or 1) do not overlap within the reported uncertainty of ±0.1; the question of why the coefficient values span an order of magnitude remains unanswered.

coefficient and that the thermal accommodation coefficient could be assigned a value of unity. They also accounted for the effect of evaporative cooling on the liquid surface temperature in their evaporation model but assumed thermal equilibrium across the liquid−vapor interface. Miles et al. were careful to point out that their results were not universally valid but only under a certain range of temperatures and pressures. They claimed that for all five experimental techniques, the coefficient values were most sensitive to the ratio of the vapor-phase pressure and the saturation-vapor pressure. Thus, they emphasized that it was important to have accurate knowledge of the partial pressure of water, the system temperature, the droplet curvature, and the saturation pressure of water. They questioned the experimental techniques that performed single particle measurements, such as optical tweezers and electrodynamic balances, since those methods were very sensitive to the thermophysical parameters. Regarding values of the evaporation and condensation coefficients of water, they concluded, “In reconciling the values reported by these techniques, we consider that the value can be safely assumed to be larger than 0.5 for water adsorbing to a water surface, independently verifying assessments made by previous authors.” Thus, with regard to atmospheric conditions relevant in cloud kinetics, Miles et al. constrained the coefficient value of pure water to be between 0.5 and 1. Davies et al.140 investigated surface active organic films on single aqueous aerosol droplets between 5 and 20 μm in radius and their effect on reducing the evaporation coefficient of water. They assumed αt was unity, σe and σc were equal to the mass accommodation coefficient, and found that σe decreased from 2.4 × 10−3 to 8.5 × 10−6 as the number of carbon atoms, n, in long-chain alcohols (CnH(2n+1)OH) increased from 12 to 18. Interestingly, for aerosols coated in condensed long-chain alcohol films, Davies et al. found that, as water condensed onto the organic film, the long-chain alcohol molecules strongly cohered to each other and led to “broad areas of uncoated surface” onto which the water could more rapidly condense. They concluded that, under atmospheric conditions, it was unlikely that aerosols would be completely coated in organic films. Thus, it was unlikely that σe would be less than 0.1. Davies et al.211 took a different approach to determine the evaporation coefficient water. They assumed σe and σc (or the mass accommodation coefficient) were at least equal but considered the possibility that σc could be greater than σe. They performed experiments in which droplets of either pure water or solutions of sodium chloride were trapped in an electrodynamic balance and their evaporation rates were measured under different conditions of relative humidities (RH) and temperatures. The temperature range of their experiments was between 248 and 298 K and the RH values were between 80 and 90%. Using their experiments and a model of evaporation that took into account evaporative cooling of the surface temperatures (but not the interfacial temperature discontinuity), they were able to determine values for the saturationvapor pressure of water well below its freezing point. Aware of the sensitivity analysis reported by Miles et al.,210 Davies et al.211 took care to account for sources of error on their determination of σe. Their method to determine Ps(T) values from their evaporation experiments reduced to identifying the values of three key parameters for a given temperature: the gas flow velocity, an effective gas phase diffusion coefficient, and σe. They found that various combinations of parameters could give the same Ps(T) value; for example, choosing σe to be 0.1 or 1 7741


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water,238 and ethanol.118 This review will also examine the application of the SRT evaporation flux expression to multicomponent systems of water−ethanol and water− methanol.148 The full nonlinear SRT expression 61,215,216 for the evaporation flux requires material properties such as the specific volume of the liquid, liquid−vapor surface tension, and saturation-vapor pressure. It also uses molecular properties, namely the internal molecular vibration frequencies, ωS . Thus, the full SRT evaporation flux expression can be used when the molecular and fluid properties are known. Many of the fluid properties are well-documented for numerous substances.239 However, not all ωS values have been been reported.118 As we will see, this has been a barrier to investigations using the SRT approach.240 One of our objectives in this review is to examine the importance of retaining the internal molecular vibration frequencies in the SRT expression for the evaporation flux. We do this by examining evaporation flux expressions with and without the ωS terms. One approximation we consider to remove the terms is by taking the limit in which ℏωS /kBT goes to zero in the full evaporation flux expression, where ℏ is the reduced Planck constant. We define this limit in section 5 as the “thermal-energy-dominant (TED) limit” and examine its validity experimentally. We will show that the TED limit leads to a HK-type relation, similar to eq 3, but with explicit expressions for σe and σc that fundamentally change the interpretation of the limiting values of these coefficients. We also consider the classical limit (ℏ going to zero) in section 5.


In order to test the HK relation given in eq 3, an accurate expression of Ps(T) is required. In addition, accurate measurements of the interfacial conditions are needed: TLI , TVI , PV, and jLV. Even when such measurements are available, assumptions must be made to eq 3 to even apply it. We have reviewed theoretical, experimental, and MD simulation studies of the HK relation for the evaporation flux. Several sources of error in studies examining the HK relation have been identified in our review: (1) The M-B distribution function is only correct under equilibrium conditions, but the HK relation is formulated by assuming the M-B function is still valid in nonequilibrium systems that are “not too far” from equilibrium. This has led to predictions of the interfacial temperature discontinuities that are smaller in magnitude and in the reverse direction of those observed in evaporation experiments. (2) The evaporation and condensation coefficients in the HK relation are often treated as equal and constants. This has led to coefficient values that span 3 orders of magnitude.64 (3) Thermal equilibrium across the interface continue to be assumed, even though interfacial temperature discontinuities as high as 7.8 K have been reported in experiments since 1999.58 Discontinuities of 15 K were reported in 2009 when the vapor phase was heated with a heating coil.159 (4) Experimental investigations of the HK relation, eq 3, have made inaccurate interfacial measurements of temperature and pressure. (5) Experiments performed in the open atmosphere are prone to effects from impurities that can accumulate on the liquid−vapor interface. (6) Molecular dynamics simulations are limited to systems that contain hundreds or thousands of particles often based on LJ fluids and have not yet been shown to agree with experiments. (7) Thermophysical fluid properties, such as Ps(T), used in investigations are assumed to be correct, but they are not always supported by experiments.117,118 Therefore, studies investigating the validity of the HK relation, eq 3, remain open to question. Even in the most recent studies considered in this review (by Nagayama et al.200 and by Hołyst et al.199 in 2015) many of the questionable assumptions listed above, such as assuming thermal equilibrium between the interface and the liquid, are still made. Thus, our review in this section indicates that the HK relation in eq 3 does not provide a physical description of evaporation that is experimentally verified.

4.1. Expression for the Net Evaporation Flux

The complete derivation of the SRT evaporation flux expression is provided elsewhere.61,62 The Gibbs dividing surface approximation70 was used in the SRT derivation, but as described in section 3.3, the liquid−vapor interfaces of some substances have been reported to have finite thicknesses on the order of a molecular diameter. This raises questions about the SRT model of the interface. However, assigning a thickness to the interface was not found to affect the final SRT result.241 In developing the SRT expression for the evaporation flux, Ward and Fang performed a first-order perturbation analysis of their evaporation system and approximated the exchange rate of molecules across the liquid−vapor surface, KLV, as a constant: KLV = C

4. REVIEW OF THE STATISTICAL RATE THEORY EXPRESSION FOR THE EVAPORATION FLUX Statistical rate theory (SRT) provides an alternate expression for the evaporation flux.61,62 It was developed in 1977 to describe the rate of particle transport across a phase boundary using the concept of transition probabilities between quantum states.214−216 The SRT approach has been examined in numerous experiments, including gas sorption,214,217−230 thermal desorption,231,232 chemical reactions,215 ion transport,233−235 condensation,112 solidification, and melting studies.236,237 The SRT approach was applied to the evaporation problem58,61,111 in 1999. In this review, we focus on the application of SRT to evaporation. As will be discussed, the SRT expression for the evaporation flux has the advantage of being complete in the sense that it does not contain any fitting parameters. We will show that the SRT evaporation flux expression agrees with measurements from evaporation studies of water,58,117,147 heavy

2π |Vve|ζ ℏ

(12)

where C was a proportionality constant, ζ was the energy density of the available quantum states, and |Vve| was the matrix element corresponding to transitions of quantum states between system configurations.61 They found that the numerator of eq 12 was too complex to allow for computation, so they made the approximation that KLV could be interpreted as the equilibrium exchange rate of molecules across the liquid−vapor interface. Their final expression for the net evaporation flux, jLV, was61,62,220 ⎡ −Δs LV ⎤ ⎡ Δs LV ⎤ j LV = Ke exp⎢ ⎥ ⎥ − Ke exp⎢ ⎣ kB ⎦ ⎣ kB ⎦

(13)

In eq 13, Δs is the change in entropy when one molecule evaporates, and Ke is the unidirectional molecular flux across the liquid−vapor interface under equilibrium conditions in an isolated liquid−vapor system. The superscript LV indicates the LV

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net evaporation from the liquid to the vapor. The first term on the right side of eq 13 is the unidirectional evaporation flux, and the second term is the unidirectional condensation flux. The expression for Ke was obtained from classical kinetic theory and the equilibrium Maxwell−Boltzmann velocity distribution function, which were discussed earlier in section 2.1: ⎛ m ⎞ ⎟ Ke = Ps(T L)⎜ L ⎝ 2πkBTI ⎠

θS =

and

(15)

(20)

must satisfy

− γ LV(T )(C1 + C2) = ηPs(TIL)

(21)

4.2. Examination of Predictions Made Using the SRT Evaporation Flux Expression

When applied to evaporation, the SRT evaporation flux expression, eqs 16 to 21, can be used to predict a number of parameters when the values of TLI , TVI , PV, and jLV are measured. One such parameter is the saturation-vapor pressure, Ps. It is a thermophysical property that depends on temperature70 and has been independently measured and reported in many investigations.239 Using SRT, Ps values were determined from evaporation experiments of water and of ethanol, as reported by Duan et al.117 and by Persad and Ward,118 respectively. They examined the SRT approach by comparing their predictions of several thermophysical properties from the theory with independently reported values. The experiments reported by Duan et al. and by Persad and Ward were designed and performed to mitigate many of the sources of errors identified earlier in section 3 regarding the experimental studies that had been performed to test the validity of the HK relation. We review the experimental studies to test the SRT expression for the evaporation flux and provide a general description of the experimental setup. 4.2.1. Purity of Evaporating Fluids during Experiments. The experiments to examine the SRT expression for the evaporation flux were performed in evaporation chambers sealed from the room atmosphere58,111,118,147,158,159,176,238 (see Figure 6). This reduced the chance of impurities affecting the evaporation flux. Before the start of an experiment, the evaporation chamber was further cleaned by connecting it to a turbo molecular pump that reduced the pressure to less than

(16)

(17)

⎡⎛ V ⎞ 4 ⎤ ⎛ P (T L ) T TV ⎞ Δs LV = ln⎢⎜ IL ⎟ s VI ⎥ + 4⎜1 − IL ⎟+ ⎢⎣⎝ TI ⎠ P ⎥⎦ kB TI ⎠ ⎝ kBTIL

e−θ1/2T 1 − e−θ1/ T

The parameters are defined as follows: vf is the specific volume of the liquid at saturation, γLV is the surface tension at the liquid−vapor interface, and PLe is the liquid pressure that would exist at equilibrium when the interface has a curvature. If the liquid−vapor interface were spherical then the sum of principal curvatures of the interface, C1 + C2, would be replaced with 2C0, where C0 would be the curvature of the spherical interface. The term “DOF” indicates the vibrational frequency degrees of freedom: Hill242 reports that the DOF values for nonlinear and linear molecules are 3n − 6 and 3n − 5, respectively, where n is the number of atoms in the molecule. The term denoted by θS is the characteristic temperature corresponding to ωS , and qvib is called the vibration partition function. Note that qvib, η, and (ΔsLV/kB) are all dimensionless quantities. Also note ΔsLV in eq 18 is a function of PV, TLI , TVI , and known material properties. Thus, the unidirectional condensation and evaporation rates in eq 13 both depend on the properties of the liquid and vapor phases. Whenever this review refers to the “full” or “complete” SRT expression for the evaporation flux, it is referring to the system of equations, eqs 16 to 21. Again, note the full SRT expression for the evaporation flux has no fitting parameters; all parameters may be evaluated from fluid properties and experimental measurements.

(14)

where

vf (TIL)m

PLe

PeL

where hV is the specific enthalpy of the vapor phase and μ is a chemical potential of the phase indicated by the superscript. The entropy change when a molecule spontaneously transitions from the liquid phase to the vapor phase is denoted by ΔsLV. The entropy change when a molecule transitions in the reverse direction, from the vapor phase to the liquid phase, is −ΔsLV as indicated in eq 13. 4.1.1. Expressing SRT in Terms of Known and Measurable Parameters. In order to apply SRT to evaporation experiments, expressions for the specific enthalpy and chemical potential are needed in eq 15. Such expressions are provided by Hill242 and have been used by Ward and Fang61 to produce the final form of the SRT evaporation flux expression. Assuming the liquid phase is incompressible and the vapor phase behaves as an ideal gas, the SRT expression for jLV is given by61,118,148,220,243,244

⎡ v (T L ) ⎤ f I L L ⎥ η = exp⎢ − ( P P ( T )) e s I ⎢⎣ kBTIL ⎥⎦

∏ S=1

Note Ke is an equilibrium property and SRT (unlike CKT) correctly applies the M-B distribution function only under equilibrium conditions. Bond and Struchtrup63 have raised an issue with eq 14 claiming it is “just the first term of the HK mass flux...which assumes that all vapor particles that hit the interface condense. Thus, the same assumption [of σc being unity] is present in the SRT expression.” However, no such assumption was made in eq 14, since according to CKT, under equilibrium conditions, σc must be unity.52,72 The expression for ΔsLV in eq 13 is61,62

⎛ Δs LV ⎞ j LV = 2ηKe sinh⎜ ⎟ ⎝ kB ⎠

(19) DOF

qvib(T ) =

1/2

⎛ 1 ⎛ μL μV ⎞ 1 ⎞ Δs LV = ⎜ L − V ⎟ + hV ⎜ V − L ⎟ TI ⎠ TI ⎠ ⎝ TI ⎝ TI

ℏωS kB

(PV + γ LV(C1 + C2) − Ps(TIL))+

DOF ⎡ ⎡ q (T V ) ⎤ ⎛ ⎤ θ θS 1 1 ⎞ I ln⎢ vib L ⎥ + ⎜ V − L ⎟ ∑ ⎢ S + ⎥ V ⎢⎣ qvib(TI ) ⎥⎦ ⎝ TI TI ⎠ S = 1 ⎣ 2 e θS / TI − 1 ⎦

(18) 7743


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chamber was monitored with a residual gas analyzer. Air was the only impurity identified, and the partial pressure of nitrogen gas was at least 2 orders of magnitude less than the partial pressure of the test fluid. 4.2.2. Size of the Liquid−Vapor Interface. Inside of the chamber, a temperature-controlled column supported either funnels or substrates. Several different funnels were used. These included funnels with circular mouth openings that were made of glass,58,111 stainless steel,118,147,238 or poly(methyl methacrylate) (PMMA).117 Additional funnels included those with rectangular mouth openings that were made of stainless steel245 or polyvinyl chloride (PVC).119,159 Some studies used substrates instead of funnels. The substrates were cylindrical and made of silica,241 copper,176 or gold.158 The test liquid was either forced through the funnel throat, or a small center hole at the top surface of the cylindrical substrate, to form the liquid phase on top. Three liquid−vapor surface geometries would form depending on whether a funnel or cylindrical substrate was used: spherical caps formed on the funnel substrates with circular mouth openings; cylindrical caps formed on the funnel substrates with a rectangular mouth opening; and ellipsoidal caps formed on the cylindrical substrates. The liquid phase in each experiment was typically between 7 and 18 mm in diameter and between 0.5 and 5 mm in height. 4.2.3. Steady-State Evaporation and Measurements of Evaporation Fluxes, Interfacial Temperatures and Vapor-Phase Pressures. The liquids were brought into steady-state evaporation by replenishing them at their base such that their heights did not change by more than ±10 μm during the experiments. The liquid replenishment rates were controlled with syringe micropumps to within a pumping accuracy of ±0.5% of the set rates.58,111,118,147,158,176,238 Under steady state, the net evaporation flux, jLV, could be determined from the pumping rate of the syringe micropump. In the

Figure 6. Schematic of the experimental apparatus used to conduct steady-state evaporation studies. Adapted from ref 118. Copyright 2010 American Chemical Society.

10−5 Pa. All components in contact with the fluid were cleaned by soaking in acetone for 24 h then in detergent for 24 h, followed by rigorous rinsing with water. The water was deionized, distilled, nanofiltered, and had a resistivity greater than 18.2 MΩ cm, which indicated its high purity. The test liquids were pure and single-component (water,58,147,158,159,176 heavy-water,238 anhydrous ethanol,118 octane, and methylcyclohexane111). They were degassed and afterward never exposed to the room atmosphere. Before, during, and after each experiment, the gas composition of the

Table 2. Experimental Conditions Measured during the Steady-State Evaporation of Watera experiment

PVm (Pa)

TVI (K)

TLI (K)

ΔTLV I (K)

EVCu1 EVCu2 EVCu3 EVCu4 EVCu5 EVPD1 EVPD2 EVPD3 EVPD4 EVPD5 EVAu1 EVAu2 EVAu3 EVAu4 EVAu5 EVAucl1 EVAucl2 EVAucl3 EVAucl5

665 722 737 756 777 517 464 396 383 317 686 651 621 601 576 689.4 661.6 627.8 584.1

275.18 276.23 276.62 276.79 277.30 272.48 271.24 269.30 268.53 266.53 275.85 275.02 274.42 273.93 273.42 276.57 276.05 275.47 274.49

274.33 275.49 275.78 276.14 276.53 270.89 269.44 267.33 266.91 264.44 274.44 274.05 273.38 272.93 272.35 274.86 274.29 273.66 272.58

0.85 0.74 0.84 0.65 0.77 1.59 1.80 1.97 1.62 2.09 1.41 0.97 1.04 1.00 1.07 1.71 1.76 1.81 1.91

jLV 0

mg

( ) m2s

134 82 66 46 32 34 66 77 93 152 49 57 61 77 84 3,068 3,332 2,004 3,383

Ps* (Pa)

P†s (Pa)

Ps(T) (Pa)

σc*

σe*

664.66 721.59 736.58 755.55 776.54 516.74 463.80 395.85 382.84 316.92 685.64 650.64 620.66 600.67 575.70 690.42 662.75 628.38 585.32

664.66 721.60 736.59 755.56 776.54 516.74 463.80 395.85 382.84 316.93 685.65 650.64 21.01 600.68 575.70 − − − −

665.11 722.67 737.73 756.81 777.98 517.54 464.60 396.15 383.65 316.98 670.38 651.84 621.03 601.07 576.18 690.88 663.20 633.74 585.94

1.00158 1.00137 1.00155 1.00120 1.00142 1.00305 1.00349 1.00387 1.00316 1.00416 1.00266 1.00181 1.00195 1.00188 1.00202 1.00324 1.00335 1.00345 1.00367

1.00073 1.00069 1.00067 1.00066 1.00065 1.00068 1.00071 1.00074 1.00076 1.00087 1.00067 1.00068 1.00068 1.00070 1.00071 1.00262 1.00288 1.00206 1.00322

Err(σc*) 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.2 1.1 1.2 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1

× × × × × × × × × × × × × × × × × × ×

10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04

Err(σe*) 2.4 1.3 9.9 6.6 4.3 1.0 2.4 3.9 5.0 1.2 8.5 1.1 1.3 1.7 2.0 5.2 6.1 4.1 7.9

× × × × × × × × × × × × × × × × × × ×

10−06 10−06 10−07 10−07 10−07 10−06 10−06 10−06 10−06 10−05 10−07 10−06 10−06 10−06 10−06 10−05 10−05 10−05 10−05

a

The measurements are made on the centerline of the interface. The experiment acronyms are described in section 7. The values of the saturationvapor pressure determined from the full SRT evaporation flux expression (eqs 16 to 21), P†s , the TED-SRT evaporation flux expression (eqs 38 to 40), Ps*, and the water Ps(T) expression (eq 22) are also listed. The values of σe* ± Err(σe*) and σc* ± Err(σc*) are also given. PVm is the measured vapor-phase pressure. Note that, as described in the text, the number of decimal places does not represent the accuracy of the calculations. Continued in Table 3. 7744


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Table 3. Continued from Table 2a experiment

PVm (Pa)

TVI (K)

TLI (K)

ΔTLV I (K)

EVPM6 EVPM7 EVPM8 EVSS1 EVSS2 EVSS3 EVSS4 EVSS5 EVSS6 EVSS7 EVSS8 EVSS9 EVSS10 EVSS11 EVSS12 EVSS13 EVSS14 EVSS15 EVSS16 EVSS17 EVSS18 EVSS19

188.3 168.3 127.3 797.2 795.7 786.8 791.5 787.9 786.6 783.9 777.3 770.1 765.3 745.3 665.3 591.9 505.3 398.6 301.3 285.3 264.0 258.6

260.60 259.51 257.74 278.10 278.03 277.77 277.80 277.83 277.77 277.71 277.61 277.50 277.35 277.88 276.52 274.99 272.95 270.50 267.49 265.96 265.84 265.56

257.55 256.23 253.89 276.85 276.91 276.67 276.72 276.71 276.71 276.62 276.50 276.41 276.26 275.88 274.26 272.62 270.33 267.11 263.48 261.54 261.29 261.09

3.05 3.28 3.85 1.25 1.12 1.10 1.08 1.12 1.06 1.09 1.11 1.09 1.09 2.00 2.26 2.37 2.62 3.39 4.01 4.42 4.55 4.47

jLV 0

mg

( ) m2s

487 548 494 27 34 43 50 56 63 66 69 75 76 139 235 346 508 733 931 1,167 1,142 1,205

Ps* (Pa)

P†s (Pa)

Ps(T) (Pa)

σc*

σe*

188.47 168.51 127.52 796.76 795.25 786.36 791.06 787.47 786.17 783.47 776.88 769.68 764.89 745.02 665.13 591.82 505.36 398.88 301.74 285.89 264.59 259.21

188.47 168.52 127.53 796.76 795.26 786.37 791.06 787.47 786.17 783.48 776.88 769.69 764.89 745.03 665.14 591.83 505.36 398.88 301.75 285.90 264.60 259.21

182.10 163.13 133.80 795.73 799.10 785.70 788.48 787.92 787.92 782.93 776.33 771.41 763.27 742.99 661.77 587.65 496.49 389.56 294.00 252.03 247.03 243.09

1.00639 1.00695 1.00836 1.00233 1.00208 1.00204 1.00200 1.00208 1.00196 1.00202 1.00206 1.00202 1.00203 1.00380 1.00435 1.00460 1.00516 1.00689 1.00839 1.00941 1.00973 1.00955

1.00180 1.00209 1.00235 1.00065 1.00065 1.00066 1.00066 1.00066 1.00067 1.00067 1.00067 1.00068 1.00068 1.00072 1.00080 1.00090 1.00109 1.00146 1.00201 1.00245 1.00255 1.00270

Err(σc*) 1.3 1.3 1.4 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.2 1.2 1.2 1.2 1.3 1.3 1.3 1.3

× × × × × × × × × × × × × × × × × × × × × ×

10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04

Err(σe*) 1.0 1.4 2.2 3.6 4.4 5.7 6.5 7.4 8.3 8.8 9.3 1.0 1.1 2.0 4.3 7.9 1.6 3.6 7.9 1.1 1.2 1.4

× × × × × × × × × × × × × × × × × × × × × ×

10−04 10−04 10−04 10−07 10−07 10−07 10−07 10−07 10−07 10−07 10−07 10−06 10−06 10−06 10−06 10−06 10−05 10−05 10−05 10−04 10−04 10−04

a

Experimental conditions measured during the steady-state evaporation of water. The measurements are made on the centerline of the interface. The experiment acronyms are described in section 7. The values of the saturation-vapor pressure determined from the full SRT evaporation flux expression (eqs 16 to 21), P†s , the TED-SRT evaporation flux expression (eqs 38 to 40), Ps*, and the water Ps(T) expression (eq 22) are also listed. The values of σ*e ± Err(σ*e ) and σ*c ± Err(σ*c ) are also given. PVm is the measured vapor-phase pressure. Note that, as described in the text, the number of decimal places does not represent the accuracy of the calculations. Continued in Table 4.


PVm (Pa)

TVI (K)

TLI (K)

ΔTLV I (K)

EVrSS2 EVrSS3 EVrSS4 EVrSS5 EVG1 EVG2 EVG3 EVG4 EVG5 EVG6 EVG7 EVG8 EVG9 EVG10 EVG11 EVG12 EVG13 EVG14 EVG15

551.9 469.3 317.3 256.0 596.0 493.3 426.6 413.3 310.6 342.6 333.3 269.3 277.3 264.0 269.3 245.3 233.3 213.3 194.7

273.09 271.29 266.14 263.37 276.41 273.81 272.49 272.09 269.29 270.39 271.49 268.49 268.79 268.19 268.99 267.09 267.89 266.89 266.29

271.70 269.74 264.52 261.50 272.79 270.19 268.29 267.89 264.19 265.39 265.29 262.39 262.79 262.09 262.49 261.19 260.69 259.59 258.49

1.39 1.55 1.62 1.87 3.62 3.62 4.20 4.20 5.10 5.00 6.20 6.10 6.00 6.10 6.50 5.90 7.20 7.30 7.80

jLV 0

mg

( ) m2s

431 671 1,060 1,272 280 254 305 417 370 348 397 408 435 410 486 417 494 509 539

Ps* (Pa)

P†s (Pa)

Ps(T) (Pa)

σc*

σe*

551.78 469.34 317.59 256.43 596.10 493.39 426.81 413.56 310.95 342.94 333.87 269.79 277.80 264.49 269.89 245.74 233.96 213.95 195.40

551.79 469.35 317.60 256.43 596.11 493.40 426.82 413.57 310.96 342.95 333.88 269.80 277.80 264.50 269.90 245.75 233.97 213.96 195.41

549.38 475.14 318.97 251.22 594.98 491.35 426.09 413.38 310.85 341.29 338.66 269.71 278.41 263.35 271.86 245.06 235.39 215.32 196.79

1.00265 1.00298 1.00319 1.00375 1.00723 1.00730 1.00865 1.00867 1.01091 1.01062 1.01352 1.01344 1.01317 1.01346 1.01445 1.01301 1.01638 1.01672 1.01815

1.00099 1.00128 1.00213 1.00283 1.00085 1.00088 1.00097 1.00110 1.00118 1.00111 1.00118 1.00133 1.00135 1.00134 1.00145 1.00141 1.00159 1.00171 1.00187

Err(σc*) 1.1 1.1 1.2 1.2 1.2 1.2 1.3 1.3 1.4 1.3 1.4 1.4 1.4 1.4 1.5 1.4 1.5 1.5 1.6

× × × × × × × × × × × × × × × × × × ×

10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04 10−04

Err(σe*) 1.1 2.4 8.1 1.5 6.3 8.3 1.3 1.9 3.0 2.3 2.8 4.3 4.3 4.5 5.1 5.3 6.8 8.4 1.1

× × × × × × × × × × × × × × × × × × ×

10−05 10−05 10−05 10−04 10−06 10−06 10−05 10−05 10−05 10−05 10−05 10−05 10−05 10−05 10−05 10−05 10−05 10−05 10−04

a

Experimental conditions measured during the steady-state evaporation of water. The measurements are made on the centerline of the interface. The experiment acronyms are described in section 7. The values of the saturation-vapor pressure determined from the full SRT evaporation flux expression (eqs 16 to 21), P†s , the TED-SRT evaporation flux expression (eqs 38 to 40), P*s , and the water Ps(T) expression (eq 22) are also listed. The values of σe* ± Err(σe*) and σc* ± Err(σc*) are also given. PVm is the measured vapor-phase pressure. Note that, as described in the text, the number of decimal places does not represent the accuracy of the calculations.

experiments, the liquid−vapor surface was renewed from the replenishment by the syringe micropump and from thermocapillary flow.147,158,176,238,245

Under steady conditions, temperature measurements were made with Alumel−Chromel microthermocouples with bead diameters of either ∼25,58,111,158,176 50,118,147,238 or 8058,111 μm 7745


Chemical Reviews

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Table 5. Experimental Conditions Measured during the Steady-State Evaporation of Ethanola experiment

PVm (Pa)

TVI (K)

TLI (K)

ΔTLV I (K)

EthEVSS1 EthEVSS2 EthEVSS3 EthEVSS4 EthEVSS5 EthEVSS6 EthEVSS7 EthEVSS8 EthEVSS9 EthEVSS10 EthEVSS11 EthEVSS12 EthEVSS13 EthEVSS14 EthEVSS15 EthEVSS16 EthEVSS17 EthEVSS18 EthEVSS19 EthEVSS20 EthEVSS21 EthEVSS22

127.3 168.3 188.3 661.3 551.9 469.3 317.3 256.0 797.2 795.7 786.8 791.5 787.9 786.6 783.9 777.3 770.1 765.3 745.3 665.3 591.9 505.3

259.82 261.65 268.39 266.99 264.88 253.37 253.92 254.64 254.59 258.82 260.58 261.62 264.71 263.82 263.21 262.36 256.08 256.37 256.52 256.65 259.78 260.33

256.30 258.88 266.02 264.56 262.13 249.06 250.03 250.60 250.61 255.60 257.38 258.87 262.29 261.62 260.81 259.89 252.03 252.36 252.62 252.53 256.37 256.88

3.52 2.77 2.37 2.43 2.75 4.31 3.89 4.04 3.98 3.22 3.20 2.75 2.42 2.20 2.40 2.47 4.05 4.01 3.90 4.12 3.41 3.45

jLV 0

mg

( ) m2s

264.785 104.271 140.944 115.209 123.366 224.346 24.359 102.461 157.475 191.121 141.335 79.983 96.155 114.794 110.835 79.983 172.975 172.365 171.671 234.597 166.083 210.547

Ps* (Pa)

P†s (Pa)

Ps(T) (Pa)

σc*

σe*

Err(σc*)

Err(σe*)

399.03 496.53 865.06 777.01 636.39 232.02 243.01 256.57 254.10 374.21 432.26 501.86 668.88 620.47 576.87 536.49 282.43 289.84 294.76 289.72 385.46 396.56

399.34 497.23 866.49 778.27 637.29 231.74 243.09 256.62 254.16 374.60 432.71 502.57 669.97 621.56 577.82 537.35 282.18 289.91 294.86 289.75 385.79 396.89

403.31 497.07 865.71 774.98 642.58 218.61 237.85 249.85 250.06 380.77 440.44 496.67 650.63 617.49 579.45 538.77 282.38 290.41 296.88 294.62 405.63 422.89

1.00920 1.00677 1.00544 1.00564 1.00661 1.01235 1.01077 1.01127 1.01106 1.00826 1.00813 1.00671 1.00567 1.00508 1.00565 1.00587 1.01123 1.01107 1.01067 1.01145 1.00884 1.00894

1.00262 1.00247 1.00240 1.00241 1.00244 1.00277 1.00253 1.00260 1.00266 1.00259 1.00252 1.00246 1.00243 1.00244 1.00245 1.00245 1.00265 1.00264 1.00263 1.00269 1.00256 1.00258

2.E-04 2.E-04 2.E-04 2.E-03 2.E-04 2.E-04 2.E-04 2.E-04 2.E-04 2.E-04 2.E-04 2.E-04 2.E-04 2.E-04 2.E-04 2.E-04 2.E-04 2.E-04 2.E-04 2.E-04 2.E-04 2.E-04

2.E-06 6.E-07 4.E-07 4.E-06 5.E-07 4.E-06 6.E-07 2.E-06 3.E-06 1.E-06 9.E-07 5.E-07 4.E-07 5.E-07 5.E-07 5.E-07 2.E-06 2.E-06 2.E-06 3.E-06 1.E-06 1.E-06

a

The measurements are made on the centerline of the interface. The experiment acronyms are described in section 7. The values of the saturationvapor pressure determined from the full SRT evaporation flux expression (eqs 16 to 21), P†s , the TED-SRT evaporation flux expression (eqs 38 to 40), Ps*, and the water Ps(T) expression (eq 23) are also listed. The values of σe* ± Err(σe*) and σc* ± Err(σc*) are also given. PVm is the measured vapor-phase pressure. Note that, as described in the text, the number of decimal places does not represent the accuracy of the calculations.

from SRT at the measured value of TLI . For water and ethanol, a series of 50 and 22 data points, respectively, were used in a curve fitting procedure to develop the temperature-dependent expressions given in eqs 22 and 23.

located on a positioning-micrometer. The micrometer had a positioning accuracy of ±10 μm in three perpendicular directions. The vertical temperature profiles in the liquid and vapor phases were measured above and below the apex of the surface. In some experiments, the center of the microthermocouple bead was brought to within ∼24 μm of the interface from the liquid and vapor sides and was never partially exposed during a measurement. The measurements indicated that the interfacial vapor temperature, TVI , was always greater than the interfacial liquid temperature, TLI , which, as discussed in section 3.6.2, contradicts predictions that were made from classical kinetic theory. The vapor phase pressure, PV, was measured with either a mercury manometer to within ±13.3 Pa or with capacitance gauge transducers to within ±2 Pa. The pressure was typically measured ∼25 cm above the apex. Some experiments had multiple pressure transducers at different heights in the chamber, but no measurable differences between the pressure readings were reported. Thus, it was assumed in the experiments that the vapor-phase pressure was nearly uniform to within ±2 Pa in the chamber. This assumption was supported by numerical studies and MD simulations.198,199,246 4.2.4. Thermodynamically Consistent SaturationVapor Pressure Expressions Were Developed from the SRT Approach. In the experiments described in the preceding discussion, measurements were made of TLI , TVI , PV, and jLV 0 (the subscript 0 indicates the local evaporation flux). Their values are listed in Tables 2, 3, 4, and 5. From those measurements, Ps values were determined from eqs 16 to 21 using the “FindRoot” function of Mathematica. Details of the analysis are provided by Duan et al.117 and by Persad and Ward118 for water and for ethanol, respectively. The main result from the analysis was that each experiment provided a data point: the Ps value derived

water: Ps(T ) = 611.2 exp[1045.8511577 − 21394.6662629 T −1 + 1.0969044T − 1.3003741 × 10−3T 2 + 7.7472984 × 10−7T 3 − 2.1649005 × 10−12T 4 − 211.3896559 ln T ]

(22)

ethanol: Ps(T ) = exp[52.165 − 6445T −1 − 3.75424ln T − 3.0053 × 10−6T 2 + 3.0053 × 10−6T ] (23)

The number of decimal places in the expressions were not indications of precision but were necessary to fit the SRT data to within ±1%, see Figures 7 and 8. Equation 22 provides a Ps(T) expression for water below the triple point temperature, which could only be developed from evaporation studies because of the propensity of water to freeze below that temperature.211,247 Persad and Ward determined the Ps(T) expression of ethanol from experimental data between 246 and 267 K, but they found that eq 23 gave good agreement with independent measurements of Ps(T) even when extrapolated up to 380 K, Figure 8. Note that nonequilibrium experiments and SRT were used to predict an equilibrium fluid property, Ps(T), for two different liquids. The accuracy of the predicted Ps(T) expressions, eqs 22 and 23, were further examined by using them to predict the evaporative latent, hfg(T), and constant-pressure specific heat of the liquid, cLp (T): 7746


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Figure 7. The saturation-vapor pressure of water at temperatures below the triple point are shown. The Ps(T) values reported by Duan et al. are shown as open circles, and their SRT-derived Ps(T) expression is shown as a dashed line. Adapted from ref 117. Copyright 2008 American Chemical Society.

Figure 9. The cLp (T) curve calculated from the SRT-derived Ps(T) expression in eq 22 is shown to agree well with independent cLp measurements of water reported by Angell et al.248 and by Archer and Carter.249 Adapted from ref 117. Copyright 2008 American Chemical Society.

Figure 8. The Ps(T) expression of ethanol (dashed) derived from the full SRT expression and the experimental data of Persad and Ward (○) is shown to agree with the independent values reported by the Design Institute for Physical Properties Research (DIPPR)239 (●), even in an extrapolated temperature range. Adapted from ref 118. Copyright 2010 American Chemical Society.

(25)

Figure 10. Comparison of the predicted values of cLp (T) obtained from the SRT-derived Ps(T) expression in eq 23 with the independently measured values available in the literature in the temperature range considered. The measurements of the liquid constant-pressure specific heat at low temperatures reported in the literature are shown as △ (data from Touloukian and Ho250), ▽ (data from Kelley251), □ (data from Texas A&M252), ○ (data from Parks253), ◇ (data from Wilhoit et al.254), and ◆ (data from Green255). Note the good agreement, even in an extrapolated temperature range. Adapted from ref 118. Copyright 2010 American Chemical Society.

where R is the universal gas constant, is the constantpressure specific heat capacity of the vapor phase, and the fluid properties were evaluated at T equal TLI . See Figures 9 and 10 for the predictions of cLp (T). The hfg values calculated by Duan et al.117 from eqs 24 and 22 were reported to agree with those independently measured by Kell et al.256 and by Wagner and Pruß.257 Similar agreement was found by Persad and Ward118 when they compared the hfg values calculated from eqs 24 and 23 to those independently reported by the Design Institute for Physical Properties Research (DIPPR). 239,258 Thus, the Ps (T) expressions determined from SRT, eqs 22 and 23, led to correct predictions of hfg that agreed with independent measurements. Moreover, the Ps(T) expressions in eqs 22 and 23 were used in eq 25 to determine cLp values of water117 and of ethanol,118 respectively. As indicated in Figure 9, the cLp (T) values of water determined from the SRT approach agreed with independent measurements. Similarly for ethanol, SRT-derived cLp (T) values agreed with independent measurements, even in an extrapolated temperature range of 40 K (see Figure 10).

Thus, the Ps(T) expressions derived from the SRT approach, eqs 22 and 23, were shown to be thermodynamically consistent since they led to predictions of the evaporative latent heat and of the constant-pressure specific heat of the liquid that agreed with independently measured values.117,118 This experimental support indicates that the SRT evaporation flux expression, eqs 16 to 21, provides a physically accurate model of evaporation. 4.2.5. Predictions of the Interfacial Temperature Discontinuities from the SRT Evaporation Flux Expression. Additional examinations of the SRT expression for the net evaporation flux were provided by Duan et al.159 They investigated interfacial temperature discontinuities in the water evaporation experiments reported by Badam et al.119 Duan et al. found that the SRT expression for the evaporation flux led to predictions of the magnitude and direction of the interfacial temperature discontinuities that were in complete agreement with the evaporation experiments, even when the vapor phase was artificially heated to increase the interfacial temperature discontinuity (see Figure 11). Statistical rate theory was able to

hfg (T ) = T (v V (T ) − vf (T ))

cpL(T ) = cpV(T ) −

dPs(T ) dT

(24)

d ⎛ RT 2 dPs(T ) ⎞ ⎜ ⎟ dT ⎝ Ps(T ) dT ⎠

cVp

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temperature pulsations of water evaporating into a methanol atmosphere were the inputs into statistical rate theory, the pressure pulsations in the water vapor as a function of time could be predicted. The predictions were found to agree with the measurements. Similarly, the bulk liquid temperature pulsations were correctly predicted from the measured watervapor-partial-pressure pulsations (see Figure 12). As seen there,

Figure 11. Schematic of the experimental apparatus with an electrical heating element in the vapor phase. The funnel was constructed of polyvinyl chloride. The test liquid was pure water. Reprinted with permission from ref 159. Copyright 2008 American Physical Society. Figure 12. A sessile water droplet evaporates into a controlled watermethanol atmosphere. The water vapor mole fraction in the atmosphere is 0.143. The measured droplet temperature, TL(t), as a function of time is shown as a thin line. The thick line is the SRT calculation of droplet temperature, ; L . The agreement between TL(t) and ; L provides experimental support of the SRT evaporation flux expression. Adapted from ref 148. Copyright 2013 American Chemical Society.

correctly predict interfacial temperature discontinuity values of up to 15 K. At higher values, they found that the SRT predictions did not agree with the measurements. After they investigated the source of the error, they discovered that a pressure transducer had failed during the experiment because of the high temperature of the vapor phase (above 370 K).159 Thus, the SRT evaporation flux expression gave such a good description of the evaporation process that it was actually used to verify the integrity of the experimental measurements. 4.2.6. SRT Evaporation Flux Expression Examined in Multicomponent Systems. Statistical rate theory has been used to model multicomponent evaporation and condensation systems. Kapoor and Elliott investigated multicomponent systems of water and ethanol.259 They took a theoretical approach and used SRT to determine how water and ethanol in the multicomponent droplets evaporated and changed the concentration in the droplet with time. However, their results were not examined experimentally. Persad et al.148 examined the SRT evaporation flux expression with experimental studies of multicomponent droplets of water, methanol, and ethanol. Previous studies of such droplets with IR thermography and shadowgraphs had indicated temperature pulsations during evaporation, but similar studies with pure fluids had not indicated such pulsations. Using microthermocouples, Persad and Ward found that the liquid and vapor phase temperatures pulsated when sessile droplets of water evaporated into two-component atmospheres of water-methanol or water-ethanol. Additionally, they observed vapor-phase pressure pulsations that were coupled to the temperature pulsations. However, when a droplet evaporated into its own vapor, they did not observe temperature or pressure pulsations.148 Their observations led them to propose a new mechanism for the temperature and pressure pulsations: the adsorption-absorption of one component from the atmosphere into the droplet consisting of another component produced a pulse in energy (because of the heat of solution) that caused a local temperature pulsation. Consequently, the local evaporation flux and vapor-phase pressure also pulsed. Persad et al. used the SRT evaporation flux expression to examine their proposed mechanism and showed that when the measured droplet

the SRT-predicted temperature pulsations, ; L(t ), mostly lie on top of the measured liquid temperature pulsations, T L(t ). Thus, the SRT evaporation flux expression was demonstrated to agree with unsteady, multicomponent evaporation experiments. 4.3. Extension of SRT to Nonideal Vapors

In all of the evaporation experiments described above to examine the SRT expression for the evaporation flux, the compressibility factor, Z, was within 1% of unity. However, nonideal behavior of the vapor phase becomes important near the critical temperature, Tc.260 Kapoor and Elliott243 extended the SRT expression for jLV to include nonideal effects. Their expression for Ke became Ke =

Ps(TIL) ⎛ m ⎜⎜ Z ⎝ 2πkBTIL

⎞ ⎟⎟ ⎠

(26)

and their expression for Δs also depended on the parameter Z. They then compared their “Peng-Robinson equation-of-state based SRT model” with the “ideal SRT model.” They found both models gave the same result when used to predict the vapor-phase pressure of octane steadily evaporating into its own vapor. The agreement between the two models were best at “low temperatures (below room temperature) and pressures (below 1 kPa)”. However, they found disagreement “at sufficiently higher temperatures and pressures (near the critical temperature). The Peng-Robinson based SRT model predicted higher vapor pressures than the ideal SRT model irrespective of the temperature jump across the interface.” They also noted that more experiments were needed to test their nonideal SRT expression. They wrote, “Experiments should be done in the LV

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experiments of Ward and Stanga. 112 They concluded: “Altogether we can state that the SRT model...gives a good description of the [Ward and Stanga] evaporation experiments.” Thus, Bond and Struchtrup claimed to have validated the linearized form of the SRT evaporation flux expression, and they questioned the necessity of using the full SRT evaporation flux expression.

near-critical region (temperatures higher than 0.8Tc) to investigate the predictions that have been made.” Thus, given the agreement between the “ideal” and “nonideal” SRT expressions, the vapor phase in the SRT evaporation experiments described in section 4.2 may be approximated as ideal. 4.4. Linearized Form of the SRT Evaporation Flux Expression

4.5. SRT Evaporation Flux Expression Neglecting Molecular Phonon Terms

We have seen that the SRT expression for the evaporation flux has been examined with numerous evaporation and condensation experiments, and they strongly support it (refs 58, 111, 117, 118, 147, 148, 159, 244, 245, and 261). However, the SRT evaporation flux expression is much more complex than the HK relation, eq 3, and some have found the SRT evaporation flux expression, eqs 16 to 21, difficult to use in its nonlinear form. Badam et al.119 considered a linearized form of the SRT expression. Their values of ΔsLV were at least 3 orders of magnitude smaller than unity. Furthermore, their measurements indicated that the ratios TVI /TLI and Ps(TLI )/PV were approximately unity. Thus, they introduced the following approximations to the full SRT evaporation flux expression: sinh[Δs LV /kB] ≈ Δs LV /kB

(27)

ln[(TIV /TIL)4 ] ≈ 4(TIV /TIL − 1)

(28)

ln[(Ps(TIL)/PV )] ≈ (Ps(TIL)/PV − 1)

(29)

While the full nonlinear SRT expression61,215,216 for evaporation flux does not contain any fitting constants, it does contain the internal-molecular-vibration frequencies of the molecule, ωS . The full SRT expression can be used for a substance like water since all the ωS values are known and, as described above, gives accurate descriptions of the experiments. However, for most substances not all molecular frequencies can be measured.118,262−264 This was the case in the evaporation experiments of octane and methylcyclohexane by Fang and Ward.58,111 Since they wanted to use the SRT expression for the evaporation flux to examine the evaporation of these hydrocarbons but did not know all of the values of ωS (octane has 72 and methylcyclohexane has 57 values of ωS ), they simply neglected all the terms with ωS . In the SRT evaporation flux expression neglecting molecular phonon terms, only the entropy change term, eq 18, is modified: ⎡⎛ V ⎞ 4 ⎤ ⎛ P (T L ) T TV ⎞ Δs LV ≈ ln⎢⎜ IL ⎟ s VI ⎥ + 4⎜1 − IL ⎟ ⎢⎣⎝ TI ⎠ P ⎥⎦ kB TI ⎠ ⎝

Moreover, they reported that all the terms containing ωS were small compared to the other terms and were thus ignored. They also found that the experimental values of η in eq 17 were approximately unity, thus the curvature terms were also neglected compared to the value of Ps(TLI ). Using eqs 27 to 29, the SRT evaporation flux expression was linearized: ⎛ m jev ≈ 2⎜⎜ L 2 k π ⎝ BTI

⎞ ⎟⎟(PIV − Ps(TIL)) ⎠

+

vf (TIL)m kBTIL

(PV + γ LV(C1 + C2) − Ps(TIL)) (31)

Mathematically, this was justified since the neglected terms were smaller than the other terms. In addition, since this was the only simplification made, they were able to use their simplified SRT evaporation flux expression to examine the conditions at the liquid−vapor interface58,111 because it still contained the parameters PV, TLI and TVI . Similarly, Kapoor and Elliott neglected the molecular phonon terms when they applied SRT to evaporation studies of ethane, butane, and octane. They wrote,243 “Since the characteristic temperatures of vibration which appear in the vibrational partition function are not available for hydrocarbons, both the ideal SRT and Peng-Robinson based SRT expression are evaluated neglecting the vibrational terms in the final rate expressions.” In another study by Kapoor and Elliott, on the evaporation and condensation of multicomponent systems, they again neglected the molecular phonon terms because,259 “[the characteristic temperatures of vibration are] not readily available for most substances.” They went on to say,259 “the need for this data arises only when the liquid-phase temperature is appreciably different from the vapor-phase temperature...However, it was shown that the terms involving the vibrational partition functions and characteristic temperatures of vibration can be neglected for such small temperature jumps without altering the qualitative behavior for evaporation systems.” In summary, SRT offers the advantage of giving an accurate description of the evaporating interface without introducing fitting parameters. However, in studies where the values of ωS are not known for the test liquid, investigators who wished to

(30)

However, the linearized form of the full SRT expression becomes an expression for the condensation flux! Mathematically speaking the approximations in eqs 27 to 29 are justified from the experimental measurements, but the linearized form in eq 30 has lost physical meaning. Badam et al. introduced a minus sign in front of eq 30 to make it an expression for evaporation. They then noted their linearized expression for the evaporation flux did not depend on the interfacial vapor temperature, TVI , and pointed out that their linearized SRT expression could not be used to predict the temperature discontinuity.119 They wrote, “Furthermore, the temperature jumps near the liquid-vapor interface in the phase change process are not explained by the SRT since there is no expression for the temperature jump.” However, this was only because the approximations they had made to the SRT expression had removed all TVI terms! A subsequent study by Duan et al., in which the full SRT expression was used and the nonlinear terms kept, examined the data of Badam et al. Duan et al. found that the full SRT expression for the evaporation flux led to accurate predictions of the experimentally measured interfacial temperature discontinuities.159 The linearized evaporation flux expression was also used by Bond and Struchtrup63 to examine the steady water evaporation 7749


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Table 6. Entropy Change Terms from Eqs 32 to 37 in Section 4.6 Are Given for the 22 Steady-State Ethanol Evaporation Experiments Listed in Table 5a experiment

PVcalc (Pa)

EthEVSS1 EthEVSS2 EthEVSS3 EthEVSS4 EthEVSS5 EthEVSS6 EthEVSS7 EthEVSS8 EthEVSS9 EthEVSS10 EthEVSS11 EthEVSS12 EthEVSS13 EthEVSS14 EthEVSS15 EthEVSS16 EthEVSS17 EthEVSS18 EthEVSS19 EthEVSS20 EthEVSS21 EthEVSS22

402.95 496.83 865.41 774.70 642.27 218.30 237.64 249.58 249.78 380.48 440.15 496.44 650.39 617.29 579.22 538.57 282.06 290.09 296.56 294.26 405.31 422.54

j0LV 2K e

× 106

246.73 79.84 68.37 62.793 79.26 388.87 43.74 166.16 246.36 197.45 129.72 66.67 61.15 85.06 78.78 55.84 237.66 232.03 224.19 306.75 162.94 198.17

Δs LV kB

× 106

246.72 79.89 68.37 62.79 79.26 388.86 43.74 166.16 246.36 197.45 129.72 66.67 61.15 85.06 78.78 55.84 237.66 232.03 224.19 306.74 162.94 198.17

ΔsctLV kB

× 106

911.99 486.65 355.68 367.02 473.58 1,424.70 883.44 1,068.96 1,122.54 756.34 676.59 467.61 366.35 338.20 381.22 377.68 1,138.00 1,113.15 1,056.56 1,235.93 787.10 835.34

LV Δs ph

kB

× 106

−665.27 −406.76 −287.31 −304.23 −394.32 −1035.84 −839.70 −902.80 −876.19 −558.89 −546.88 −400.94 −305.20 −253.13 −302.44 −321.84 −900.34 −881.12 −832.37 −929.19 −624.17 −637.18

LV ΔsΔ T kB

× 106

−373.82 −227.36 −157.81 −167.71 −218.59 −592.11 −479.15 −514.27 −499.15 −314.77 −306.62 −224.11 −169.21 −140.64 −168.33 −179.52 −510.99 −499.70 −471.83 −526.63 −350.73 −357.55

ΔsωLVS /kB × 106 −291.44 −179.41 −129.50 −136.53 −175.73 −443.73 −360.55 −388.53 −377.04 −244.12 −240.26 −176.82 −135.98 −112.49 −134.12 −142.32 −389.34 −381.42 −360.54 −402.56 −273.44 −279.63

a The experiment acronyms are described in section 7. The PVcalc values determined from the full SRT expression for the evaporation flux, eqs 16 to V 21, are given as well as the ratio jLV 0 /2Ke. Note that the number of decimal places in Pcalc is not a reflection of the accuracy of the calculation but are necessary to reduce rounding errors; the values reported here are slightly different than those reported by Persad and Ward118 because of rounding errors.

use the SRT evaporation flux expression have either neglected the molecular phonon terms or used linearized forms. Simplifications made to the SRT expression may appear mathematically justifiable and may even retain some qualitative features of the full SRT expression. However, the physical meaning of the equations become questionable, certainly if all the terms containing TVI are neglected, since the interfacial temperature discontinuities cannot then be predicted.

In eq 33, the ln[PV/Ps(TLI )] term dominates, and it is therefore important to correctly determine this ratio. To eliminate any errors introduced from the Ps(T) expression or measurement errors of PV, the vapor-phase pressure of each V experiment is predicted, Pcalc , by applying the full SRT evaporation flux expression to the experimental conditions. Then, PVcalc was used instead of the measured PV values to determine values of Δsct/kB. Note that eq 33 does not involve TVI or any molecular phonon terms, and the only continuum terms in eq 34 are the interfacial temperatures. In none of the 65 water evaporation experiments considered by Duan et al. was the value of ΔsLV ph /kB 159 negligible compared to the value of ΔsLV /k . However, it was ct B still not clear if this meant the ωS values could be neglected compared to the interfacial temperature terms in ΔsLV ph /kB. To examine this question further, Persad and Ward118 LV segmented Δsph /kB into terms involving the interfacial temperature discontinuities or the internal vibration frequen-

4.6. Significance of Phonon-Dependent Terms in the SRT Evaporation Flux Expression

Duan et al. examined the importance of retaining the internalmolecular-vibration frequencies in the SRT expression.159 They divided the ΔsLV/kB term from eq 18 into what they called LV continuum and phonon terms, Δs ctLV/kB and Δs ph /kB , respectively: LV Δsph ΔsctLV Δs LV = + kB kB kB

(32)

⎡ P (T L ) ⎤ vf (TIL)m V ΔsctLV = ln⎢ s VI ⎥ + (P + γ LV(C1 + C2) kB kBTIL ⎣ P ⎦ − LV Δsph

kB

Ps(TIL))

LV cies, ΔsLV ΔT/kB or ΔsωS / kB , respectively:

(33)

LV Δsph

⎡⎛ V ⎞ 4 ⎤ ⎡ q (T V ) ⎤ ⎛ T 1 1 ⎞ I = ln⎢⎜ IL ⎟ ⎥ + ln⎢ vib L ⎥ + ⎜ V − L ⎟ ⎢⎣⎝ TI ⎠ ⎥⎦ ⎢⎣ qvib(TI ) ⎥⎦ ⎝ TI TI ⎠ 3n − 6

∑ l=1

⎛ ⎤ ⎡ θl θl TV ⎞ ⎥ + 4⎜1 − IL ⎟ ⎢ + V ⎣2 TI ⎠ ⎝ e θl / TI − 1 ⎦

kB

ΔsωLVS ΔsΔLVT = + kB kB

⎡⎛ V ⎞ 4 ⎤ ⎛ ΔsΔLVT T TV ⎞ = ln⎢⎜ IL ⎟ ⎥ + 4⎜1 − IL ⎟ ⎢⎣⎝ TI ⎠ ⎥⎦ kB TI ⎠ ⎝

(34) 7750

(35)

(36) DOI: 10.1021/acs.chemrev.5b00511 Chem. Rev. 2016, 116, 7727−7767

Chemical Reviews ΔsωLVS kB

Review

⎡ q (T V ) ⎤ ⎛ 1 1 ⎞ I = ln⎢ vib L ⎥ + ⎜ V − L ⎟ ⎢⎣ qvib(TI ) ⎥⎦ ⎝ TI TI ⎠ 3n − 6

∑ l=1

⎡ θl θl ⎢ + V / T θ ⎣2 el I −

⎤ ⎥ 1⎦

small changes in PV. Thus, if the entropy change terms in eqs 36 and 37 were neglected from the full SRT evaporation flux expression, PV would need to change by no more than 0.3 Pa or 0.2% in order to account for the missing terms. This is the reason why the mathematical results of the simplified forms of the SRT expression agree very closely with those from the full SRT expression for the evaporation flux. However, the simplified forms of SRT lose physical meaning. LV For instance, when the ΔsLV ΔT/kB and ΔsωS / kB terms are neglected from the full SRT expression, then the simplified expression cannot be used to explain, or to predict, the interfacial temperature discontinuities. Thus, there are serious consequences in physically understanding the evaporation process when the simplified forms of SRT are used. However, if the values of ωS are not known, the question arises of how to simplify the SRT evaporation flux expression without it losing physical meaning. This is the problem that will be addressed in section 5.

(37)

The entropy change values in eqs 32 to 37 from 22 steady ethanol evaporation experiments are given in Table 6. Note how none of the entropy change terms in Table 6 are negligible when compared to each other, and in fact, many of the terms have the same order of magnitude. Also, all of the ΔsLV/kB values are positive, which is expected since the ethanol in all experiments was evaporating. LV However, the values of ΔsLV ΔT/kB and ΔsωS / kB are negative. This indicates that the interfacial temperature discontinuities and the internal molecular vibration frequencies are important mechanisms in the evaporation process; they represent energy barriers that molecules must overcome in order to evaporate. As a result, it is the vapor phase that gets the high-energy molecules and thus has a higher interfacial temperature than the liquid phase, as observed in many experiments described in section 4.2. The role of the internal molecular phonons on the energy flux across the liquid−vapor interface during evaporation was examined by Persad using statistical rate theory.241 He reported that the quantum of energy transported across the water interface corresponded to frequency shifts of the interfacial water molecules as they broke their hydrogen bonding in order to evaporate. According to Table 6, the values of ΔsLV ct /kB are the only positive contribution to ΔsLV/kB. This suggests that molecules get their energy to evaporate from the liquid interface; the liquid interface thus loses energy and has a colder temperature than the interfacial vapor interface. Other investigators have expressed a similar opinion. Bond and Struchtrup63 wrote that, “the heat of evaporation [is] absorbed from (in the condensation case) or provided by (for evaporation) the liquid. This is plausible, since the heat conductivity of the liquid is substantially larger than the heat conductivity of the vapor”. Furthermore, Duan and Ward,147 Persad and Ward,118 and Ghasemi and Ward158,176 reported that the energy required for evaporation is transported to the liquid interface by thermal conduction from the liquid and vapor phases and by thermocapillary flow along the liquid surface of an evaporating sessile droplet. As mentioned earlier, simplified forms of SRT gave approximately the same mathematical results as the full SRT expression (i.e., the predicted PV values were in agreement with the measured values within the measurement error). However, Table 6 clearly shows that the ΔsLV/kB values can change by up to a factor of 2 when the ωS terms are omitted from the full SRT evaporation flux expression. The question then arises of how ΔsLV/kB can change so dramatically without resulting in similarly large changes in the calculated values of PV. Fang and Ward reported that the full SRT evaporation flux expression is most sensitive to the parameter PV.58 They found that an uncertainty of 10−4 Pa in the PV value would would result in a 0.5% error in jLV. Moreover, an uncertainty of 13.3 Pa in the PV value (i.e., their measurement error) could result in negative values of jLV being calculated from the full SRT evaporation flux expression. This sensitivity of the full SRT expression to PV means that large errors in ΔsLV/kB can be compensated by


The full SRT expression for the evaporation flux, eqs 16 to 21, was reviewed. It was used with measurements from evaporation experiments of water and of ethanol to determine expressions for the saturation-vapor pressure for each liquid. These expressions led to predictions of the evaporative latent heat and of the liquid-phase constant-pressure specific heat. The predictions agreed with values of these properties that were independently measured, even in an extrapolated temperature range. Interfacial temperature discontinuities were measured in the experiments. The value of TVI was in each case greater than that of TLI (see section 4.2). Given measured values of TVI , the SRT evaporation flux expression was used to predict the values of TLI and of the interfacial temperature discontinuities. The good agreement between the predictions and the measurements indicated that the measured interfacial temperatures made with the microthermocouple were sufficiently accurate to examine the SRT expression for the evaporation flux. The role of the molecular phonon vibrations during evaporation was examined using the full SRT expression for the evaporation flux. The phonon terms took energy away from evaporation, since their contribution to the entropy change was negative. This meant that some energy went into exciting the internal molecular phonons in order for the molecules to break their intermolecular bonding and evaporate from the liquid. The contribution of the phonon terms to evaporation could only be examined using the full SRT expression when all internal molecular phonon vibration frequencies of the molecule were known. In cases where not all of the internal molecular phonon vibration frequencies of the molecule were known, investigators have simplified the SRT evaporation flux expression by linearization or by neglecting the molecular phonon vibration terms. Those simplifications may have appeared mathematically justifiable, but they resulted in the SRT expression losing physical meaning, such as losing the ability to predict the interfacial temperature discontinuities. Therefore, there is a need to correctly extend the SRT approach to molecules for which not all molecular phonon frequencies, ωS , are known. 7751


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Note that σ*e and σ*c in eqs 38 and 39 are not constants, as is often assumed in studies of the HK relation. Also note that the unidirectional evaporation and condensation fluxes in eq 40 depend on the conditions in both the liquid and the vapor phases. This is in contrast to the HK treatment where the underlying physical model indicated the unidirectional fluxes only depended on the properties of their respective phases, eq 3. This indicates that σe* and σc* have different physical meanings than those of σe and σc. We will discuss this issue later in section 5.2. Although the validity of the HK relation, eq 3, has been questioned, it had not been possible to examine it experimentally without explicit expressions for σe and σc. As was discussed earlier in section 3, only simplified forms of the HK relation have been investigated since assumptions such as equality of the coefficients and a thermally equilibrated interface were commonly made.64 However, the TED-SRT evaporation flux expression, eq 40, can be examined by (1) using it with reported water and ethanol experimental measurements to determine Ps(T) for both liquids. The validity of the Ps(T) expressions can then be assessed by comparing them with those obtained from the full SRT expression that were discussed earlier in section 4.2.4 and (2) by using the TED-SRT evaporation flux expression to predict the interfacial temperature discontinuities reported in the independent evaporation experiments previously discussed in section 4.2.5.

5. THERMAL-ENERGY-DOMINANT (TED) LIMIT OF THE SRT EXPRESSION FOR THE EVAPORATION FLUX In this section, we develop a new expression for the evaporation flux that defines explicit expressions for σc and σe but has no dependence on ωS . The new expression is derived by taking the thermal-energy-dominant (TED) limit of the full SRT evaporation flux expression given in eq 16 for jLV. We define the TED limit as one in which ℏωS /kBT goes to zero in the jLV expression. The TED limit allows us to derive an HK-like relation from SRT (i.e., to derive an evaporation flux expression that has no dependence on ωS ). We will explore the validity of TED limit experimentally. When taking the TED limit, ℏ can still have nonzero values and does not eliminate quantum effects from SRT, since the resulting expression for the evaporation flux will be shown to retain nonlinear terms that are derived from the transition probability concept of quantum mechanics. Thus, the TED limit is conceptually different from the classical limit (ℏ goes to zero) that was described by Schiff.265 In fact, the quantum-mechanical basis of SRT requires ℏ to be nonzero;61 if the classical limit were applied to SRT then the denominator in eq 12 would be zero, and the molecular exchange rate would be predicted to be infinite, unrealistically requiring that any evaporation system instantaneously reached equilibrium. We also assume that the values of η, see eq 17, are approximately unity and that the vapor phase can be approximated as an ideal gas. These assumptions are supported by the experiments we will discuss. We introduce the following definitions to obtain a similar form to the HK relation, eq 3: σe* ≡

Ps(TIL) PV

5.1. Examination of the TED-SRT Evaporation Flux Expression

The TED-SRT expression for the evaporation flux given in eq 40 was examined using the experiments described in section 4. The experimental procedures were described earlier in section 4.2 and in detail elsewhere.58,111,117,118,147,158,241,266−268 In the discussion that follows, we initially focus on water and ethanol evaporation studies because the availability of accurate Ps(T) expressions for each liquid helps to reduce sources of error in the analysis. Later in our discussion, we focus solely on the water results since there are much more experiments available for it than for ethanol. 5.1.1. Determination of the Saturation-Vapor Pressure. As was discussed earlier in section 4.2.4, determination of Ps values provides a rigorous test of an evaporation theory. Using the measurements of TVI , TLI , PVI , and jLV 0 listed in Tables 2 to 5, values of Ps were determined from eqs 38 to 40 and are denoted by P*s in the tables. For comparison, the Ps values determined from the full SRT expression, eqs 16 to 21, are denoted by P†s . The errors in the values of Ps* and P†s were ±13.3 Pa for the water results and ±2 Pa for the ethanol results. Two decimal places are shown in the tables to illustrate the very good agreement between the values of Ps* and P†s . The average error between Ps* and the Ps(T) expressions in eqs 22 and 23 can be determined as follows. Let P*s (TLn ) denote the value of P*s at the liquid interfacial temperature TLn and let there be r total data points. Let Ps(TLn ) be the saturation-vapor pressure expression evaluated at the temperature TLn . Then the average error, denoted Err(Ps), is given by117,187

DOF + 4 ⎡ ⎛ T V ⎞⎤⎛ T V ⎞ exp⎢(DOF + 4)⎜1 − IL ⎟⎥⎜ IL ⎟ ⎢⎣ TI ⎠⎥⎦⎝ TI ⎠ ⎝

(38)

σc* ≡

TIV TIL

DOF + 4 ⎡ ⎛ T V ⎞⎤⎛ T L ⎞ exp⎢ −(DOF + 4)⎜1 − IL ⎟⎥⎜ IV ⎟ ⎢⎣ TI ⎠⎥⎦⎝ TI ⎠ ⎝

(39)

Using the above assumptions and definitions, we find that the TED limit of the SRT evaporation flux expression simplifies to * = jev

m 2πkB

⎛ L V ⎞ ⎜σ * Ps(TI ) − σ * P ⎟ e c ⎜ ⎟ TIL TIV ⎠ ⎝

(40)

where the net local evaporation flux is denoted by jev * . We call eq 40 the TED limit of the SRT evaporation flux expression, or the TED-SRT evaporation flux expression, and have used the superscript * to indicate the parameters associated with it. In eq 40, jev * is the difference between the unidirectional evaporation and condensation fluxes. It satisfies the requirements63 that j*ev is zero and that both σ*e and σ*c are unity under equilibrium conditions. There are no fitting parameters in eqs 38 to 40, thus the system of equations is closed. By contrast as described earlier in section 3, the HK relation is undefined since it contains two unknowns but only one equation. Eq 40 also has no dependence on ωS values, making it easier to apply than the full SRT evaporation flux expression described in section 4.

Err(Ps) =

1 r

r

∑ n=1

[Ps(TnL) − Ps*(TnL)]2 Ps*(TnL)

(41)

For both water and ethanol, Err(Ps) was less than 1%. The good agreement between Ps determined from the full SRT evaporation flux expression and from its TED limit is shown 7752


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The values of σ*c and σ*e in Tables 2 to 5 were found to vary with PV, TLI , and TVI . General trends can be identified: both σ*c and σe* increased with decreasing PV, TLI ,and TVI values. These trends follow those described earlier in section 3 where the coefficient values were noted to decrease with an increase in temperature (TLI )1,74,81,97,113−115,193,271 or an increase in the vapor-phase pressure (PV).64 Thus, the explicit coefficient expressions agree with several evaporation studies. 5.1.2. Interfacial Temperature Discontinuity Predictions. In section 4.6, the ωS terms in the full SRT evaporation flux expression were important in providing an explanation for the direction of the interfacial temperature discontinuities. While the TED-SRT evaporation flux expression does not use any ωS values as inputs, the expression does contain both TLI and TVI as parameters. Thus, the accuracy of the TED-SRT evaporation flux expression can examined by predicting the V L interfacial temperature discontinuities, ΔTLV I ≡ TI − TI . The 117,118 procedure requires using the Ps(T) expressions in eqs 22 and 23. However, those Ps(T) expressions were formulated using the experiments in Tables 2 to 5. To avoid circular calculations in the prediction of the interfacial temperature discontinuities, we require independent experiments. Badam et al.119 reported 45 water evaporation experiments that are independent and they are listed in Tables 7 and 8. However, there are only 5 independent ethanol experiments, as reported by Persad.241 Since there are more independent experiments available for water than for ethanol, we focus on the water studies in what follows. An iterative procedure was used to calculate ΔTLV I * and the superscript * is added to identify it as a quantity calculated from eqs 38 to 40. In the procedure, an initial value of TLI is assumed and used with the measured values of TVI and jLV 0 to calculate a value of PV. If the calculated value of PV were different than the measured value, PVm, a new value of TLI (with an increment of 0.01 K) would be assumed and the procedure would continue until there were no longer any differences between the predicted and measured PV values. The predicted values of TLI were then used to determine ΔTLV I *. The values of σe* and σc* were also obtained in the procedure and are listed in Tables 7 and 8. The values of ΔTLV I * determined from the above procedure were found to agree with the measured ΔTLV I values to within the measurement error, as indicated in Figure 14. Errors in the calculations are listed in Tables 7 and 8. They were determined from eq 42 and account for the uncertainties in the measurements and in the Ps(T) expressions. Thus, the TEDSRT evaporation flux expression, eq 40, is further validated since it leads to accurate predictions of the interfacial temperature discontinuities. The TED-SRT evaporation flux expression does not contain any ωS values, but the expression still leads to correct determinations of the interfacial temperature discontinuity. This result should not be interpreted as indicating that the ωS terms from the full SRT evaporation flux expression are not important in evaporation. To the contrary, it is the ωS terms that give the TED-SRT evaporation flux expression its final form in the TED limit. They are thus inherently a part of the TED-SRT evaporation flux expression.

in Figure 13. Thus, the TED-SRT evaporation flux expression is as accurate as the full SRT expression, indicating that the Ps(T)

Figure 13. Ps(T) expressions for water and ethanol, eqs 22 and 23, derived from the full SRT expression (see section 4.2.4) are shown as dashed lines. The values of P*s calculated from the TED-SRT evaporation flux expression, eq 40, are shown as ○ and ◊ for water and ethanol, respectively. They are seen to agree with their respective Ps(T) expressions in eqs 22 and 23 to within 1%. Errors bars are contained within data points. The P*s values are listed in Tables 2 to 5.

expressions in eqs 22 and 23 do not depend significantly on the ωS values. The values of σ* e and σ* c (see eqs 38 and 39, respectively), were obtained in the procedure to calculate P*s . Their values are listed in Tables 2 to 5, and as seen there, both σc* and σe* are slightly larger than unity. Note that their values are not bounded by unity since they represent a different physical meaning than the coefficients in the HK relation, eq 3. The physical interpretation of the coefficients is discussed further in section 5.2. Others have reported coefficient values greater than unity79,269,270 but have claimed these resulted from experimental error or large standard deviations in the measured values of temperature or pressure. The effect of measurement errors on the values of σe* and σc* can be determined as follows. If we denote the error of a quantity z by Err(z), and suppose that x is a function of the independent variables i, j, and k, then187 Err(x) = {[x(i + Err(i), j , k) − x(i , j , k)]2 + [x(i , j + Err(j), k) − x(i , j , k)]2 + [x(i , j , k + Err(k)) − x(i , j , k)]2 }0.5

(42)

The errors in the calculated σe* and σc* values, denoted by Err(σc*) and Err(σe*) respectively, are given in Tables 2 to 5 and account for the uncertainties in the measurements of PV, TLI , and TVI . As seen there, the errors are small, indicating that values of σc* and σe* greater than 1 were not due to measurement errors.

5.2. Physical Meaning of the Evaporation and Condensation Coefficients

As described above, the TED-SRT evaporation flux expression was analyzed with a total of 105 water and 22 ethanol 7753


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Table 7. Forty Five Experiments that Were Not Used to Formulate the Water Ps(T) Expression (eq 22) Were Used in the TEDSRT Expression for the Evaporation Flux (eqs 38 to 40) to Calculate the Interfacial Temperature Discontinuitya experiment

PVm (Pa)

Err(PV) (Pa)

TVI (K)

TLI (K)

ΔTLV I (K)

jLV 0

EVPVC1 EVPVC2 EVPVC3 EVPVC4 EVPVC5 EVPVC6 EVPVC7 EVPVC8 EVPVC9 EVPVC10 EVPVC11 EVPVC12 EVPVC13 EVPVC14 EVPVC15 EVPVC16 EVPVC17 EVPVC18 EVPVC19 EVPVC20 EVPVC21 EVPVC22 EVPVC23 EVPVC24

561.0 490.0 389.1 336.5 292.4 245.3 736.0 569.5 483.3 391.2 295.2 240.3 736.0 567.0 485.0 392.3 288.5 236.6 847.9 743.0 572.4 391.4 288.5 236.0

13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3

273.93 272.27 269.44 267.91 266.37 264.78 279.84 276.13 274.24 271.94 269.19 267.11 281.17 278.02 276.22 273.77 270.98 269.20 284.11 282.70 279.57 275.97 273.08 271.23

272.05 270.19 267.13 265.25 263.54 261.48 275.80 272.24 269.97 267.13 263.64 261.30 275.80 272.18 270.03 267.20 263.35 260.97 277.81 275.94 272.23 267.12 263.33 260.94

1.88 2.08 2.31 2.66 2.83 3.30 4.04 3.89 4.27 4.81 5.55 5.81 5.37 5.84 6.19 6.57 7.63 8.23 6.30 6.76 7.34 8.85 9.75 10.29

268.567 278.542 315.990 335.066 348.442 342.124 343.883 367.853 378.046 417.884 435.331 505.540 411.754 426.590 463.680 465.089 526.514 539.808 514.209 500.350 500.712 549.937 572.702 595.937

mg

( ) 2

m s

ΔTLV I * (K)

Err(ΔTLV I *) (K)

σ*c

σe*

Err(σc*)

1.90 2.08 2.31 2.66 2.92 3.54 4.04 3.89 4.22 4.73 5.60 6.11 5.37 5.83 6.15 6.51 7.67 8.37 6.29 6.75 7.24 8.72 9.75 10.41

0.32 0.36 0.44 0.50 0.57 0.66 0.25 0.31 0.36 0.44 0.55 0.67 0.25 0.31 0.37 0.44 0.57 0.67 0.23 0.25 0.31 0.44 0.57 0.67

1.00366 1.00405 1.00458 1.00535 1.00596 1.00740 1.00805 1.00783 1.00864 1.00991 1.01214 1.01355 1.01101 1.01226 1.01313 1.01418 1.01742 1.01951 1.01305 1.01425 1.01567 1.01990 1.02313 1.02532

1.00043 1.00040 1.00030 1.00022 1.00014 1.00006 1.00043 1.00036 1.00030 1.00018 1.00002 0.99976 1.00038 1.00031 1.00023 1.00013 0.99987 0.99968 1.00036 1.00033 1.00025 1.00004 0.99980 0.99958

0.00065 0.00074 0.00092 0.00107 0.00124 0.00149 0.00056 0.00069 0.00082 0.00103 0.00136 0.00170 0.00059 0.00075 0.00091 0.00112 0.00154 0.00187 0.00054 0.00060 0.00080 0.00122 0.00167 0.00203

Err(σe*) 4.0 5.6 1.0 1.5 2.1 2.9 3.0 5.5 8.1 1.4 2.5 4.5 3.7 6.6 1.0 1.6 3.3 4.9 3.5 4.4 7.7 1.9 3.6 5.4

× × × × × × × × × × × × × × × × × × × × × × × ×

10−06 10−06 10−05 10−05 10−05 10−05 10−06 10−06 10−06 10−05 10−05 10−05 10−06 10−06 10−05 10−05 10−05 10−05 10−06 10−06 10−06 10−05 10−05 10−05

LV LV The calculated values are labelled ΔTLV I * ± Err(ΔTI *) and measured values are labelled ΔTI . The values of σe* ± Err(σe*) and σc* ± Err(σc*) are also listed. The experiment acronyms are described in section 7. Note that, as described in the text, the number of decimal places does not represent the accuracy of the calculations. Continued in Table 8.

a


PVm (Pa)

Err(PV) (Pa)

TVI (K)

TLI (K)

ΔTLV I (K)

jLV 0

EVPVC25 EVPVC26 EVPVC27 EVPVC28 EVPVC29 EVPVC30 EVPVC31 EVPVC32 EVPVC33 EVPVC34 EVPVC35 EVPVC36 EVPVC37 EVPVC38 EVPVC39 EVPVC40 EVPVC41 EVPVC42 EVPVC43 EVPVC44 EVPVC45

866.0 743.9 569.2 386.3 291.7 235.5 966.8 850.5 747.0 573.1 389.2 290.7 215.6 1076.8 946.3 855.1 744.5 569.2 388.7 288.1 213.0

13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3 13.3

286.10 284.27 281.16 277.03 274.53 272.66 283.92 286.57 285.58 282.80 278.76 276.43 274.83 291.02 289.15 288.10 286.88 284.04 280.97 277.89 275.47

278.19 275.95 272.23 267.18 263.74 261.11 279.77 277.90 276.01 272.28 267.21 263.57 260.16 281.26 279.58 277.86 276.07 272.38 267.19 263.39 259.73

7.91 8.32 8.93 9.85 10.79 11.55 4.15 8.67 9.57 10.52 11.55 12.86 14.67 9.76 9.57 10.24 10.81 11.66 13.78 14.50 15.74

675.290 627.476 622.211 658.403 708.858 719.253 766.398 726.691 723.583 682.442 711.041 746.497 759.722 779.699 847.751 794.422 816.438 779.172 789.032 751.058 764.331

mg

( ) 2

m s

ΔTLV I * (K)

Err(ΔTLV I *) (K)

σc*

σe*

Err(σc*)

7.97 8.29 8.89 9.94 11.05 11.85 4.23 8.69 9.53 10.42 11.55 12.96 15.06 9.70 9.72 10.12 10.86 11.74 13.74 14.51 15.84

0.23 0.25 0.31 0.44 0.56 0.67 0.21 0.23 0.25 0.31 0.44 0.56 0.72 0.19 0.21 0.22 0.25 0.31 0.43 0.56 0.74

1.01709 1.01806 1.01992 1.02327 1.02689 1.02967 1.00833 1.01893 1.02128 1.02406 1.02790 1.03274 1.04031 1.02124 1.02147 1.02267 1.02489 1.02782 1.03461 1.03777 1.04304

1.00028 1.00026 1.00016 0.99991 0.99961 0.99935 1.00027 1.00025 1.00020 1.00012 0.99986 0.99955 0.99916 1.00030 1.00022 1.00021 1.00015 1.00004 0.99977 0.99954 0.99913

0.00057 0.00064 0.00085 0.00128 0.00172 0.00214 0.00045 0.00059 0.00067 0.00090 0.00135 0.00184 0.00258 0.00050 0.00056 0.00060 0.00070 0.00094 0.00143 0.00194 0.00272

Err(σe*) 4.7 5.6 9.8 2.3 4.3 6.6 4.3 5.2 6.5 1.1 2.5 4.5 8.2 3.5 4.9 5.4 7.5 1.2 2.7 4.6 8.6

× × × × × × × × × × × × × × × × × × × × ×

10−06 10−06 10−06 10−05 10−05 10−05 10−06 10−06 10−06 10−05 10−05 10−05 10−05 10−06 10−06 10−06 10−06 10−05 10−05 10−05 10−05

Forty five experiments that were not used to formulate the water Ps(T) expression (eq 22) were used in the TED-SRT expression for the LV evaporation flux (eqs 38 to 40) to calculate the interfacial temperature discontinuity. The calculated values are labelled ΔTLV I * ± Err(ΔTI *) and measured values are labelled ΔTLV I . The values of σe* ± Err(σe*) and σc* ± Err(σc*) are also listed. The experiment acronyms are described in section 7. Note that, as described in the text, the number of decimal places does not represent the accuracy of the calculations. a

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Hence, the HK relation, as written in eq 3, is missing an important physical concept: the coupling of the liquid and vapor phases during evaporation. This idea of coupling between phases is empirically supported by other studies. The works of Stiopkin et al.,121 Vinaykin and Benderskii,272 Zhang et al.,273 and Hsieh et al.274 have shown that water molecules at the water−air interface act as physical “bridges” between the bulk phases and interact through vibrational energy transfer. Recall that the TED-SRT evaporation flux expression, eq 40, does not explicitly contain ωS terms. This does not mean that the internal molecular vibration frequencies are not important during evaporation. Rather, it is the ωS terms that give eq 40 its final form and was shown to agree with experiments. However, the full SRT expression for the evaporation flux must be used to investigate precisely how the ωS values affect evaporation. As was discussed in section 4.6, the full SRT evaporation flux expression indicated that the ωS terms are important in understanding the direction of the interfacial temperature discontinuities measured during evaporation. The idea that the evaporation and condensation coefficients represent the coupling of the liquid and vapor phases during evaporation is further supported by the experiments by Ward et al.58,61,111,112,118,147,158,176,238,261 All of their experiments indicate that the evaporation flux is controlled by energy transfer from both the liquid and vapor phases. Thus, the evaporation and condensation coefficients in the HK relation ought to be interpreted as terms that account for the coupling between the liquid and vapor phases during phase change, where their explicit expressions are given by eqs 38 and 39, respectively.

Figure 14. Comparison between measured interfacial temperature discontinuities and those predicted from the TED-SRT evaporation flux expression, eq 40, for 45 steady water evaporation experiments reported in Tables 7 and 8.

experiments. The analysis indicated that the TED-SRT evaporation flux expression is as accurate as the full SRT expression in predictions of the saturation-vapor pressure and of the interfacial temperature discontinuities. While the TEDSRT evaporation flux expression may have the same form as the HK relation or even appear identical to it (compare eqs 3 and 40), the reader is reminded they represent two distinct physical ideas. The difference in their physical meaning is captured in the definitions of σ*e and σ*c , eqs 38 and 39. The coefficients in the TED-SRT evaporation flux expression indicate a coupling between the liquid and vapor phases during evaporation because each coefficient depends on both the liquid and phases. In contrast, the coefficients in the HK relation represent the number of molecules that change phase compared to a maximum theoretical amount and each coefficient only depends on the properties of one phase. Furthermore, the implicit assumptions used to formulate either the HK or the TED-SRT evaporation flux expressions are different. For example, the HK relation assumes that the M-B velocity distribution function is valid and applies it outside of equilibrium, see the earlier discussion in section 2. Conversely, the TED-SRT evaporation flux expression applies the M-B velocity distribution only when it is valid under equilibrium conditions. The difference between the two approaches is further illustrated by considering the following example. Suppose a liquid were to evaporate into its vapor where PV was only slightly lower than Ps(TLI ). Then, suppose that the vapor-phase pressure were suddenly decreased. The net evaporation flux would naturally be expected to increase. If the HK relation, eq 3, were applied to this system, it would predict that the unidirectional condensation flux would decrease, thereby increasing the net evaporation flux. However, the unidirectional evaporation flux would remain unchanged. In contrast, if the TED-SRT evaporation flux expression, eq 40, were applied to the system, it would predict that the unidirectional evaporation flux would increase, the unidirectional condensation flux would decrease, and the net evaporation flux would increase. Thus, while both approaches would give the same final result, the net evaporation flux would increase, the HK relation should not be viewed as equivalent to the TED-SRT evaporation flux expression in terms of physical meaning.

5.3. Effects of Common Assumptions on the TED-SRT Evaporation Flux Expression

The definitions of σ*e and σ*c in eqs 38 and 39 can be used to examine the effects that seemingly simple approximations can have on the TED-SRT evaporation flux expression. One simplifying assumption often made in evaporation studies is to neglect the interfacial temperature discontinuity (see our previous discussion in section 4.2.5) and to assume the interface is thermal equilibrated (i.e., has one temperature TI). The effect of this assumption was examined using the experiments reported by Badam et al.119 and listed in Tables 7 and 8 to predict values of PV. As indicated in Figure 15, the PV values were calculated from TED-SRT evaporation flux expression, eqs 38−40, for three conditions: (1) the measured conditions, (2) TI equal TLI , and (3) TI equal TVI . The PV values obtained from applying the above three conditions to the water experiments are tubulated in Tables 9 and 10. They indicate that if thermal equilibrium were assumed at the interface, then using TLI would give a more accurate prediction of PV than using TVI . This is because using TVI instead of TLI increases the value of the saturation-vapor pressure [i.e., Ps(TVI ) > Ps(TLI )] which causes the predicted value of PV to also increase. Another frequently used assumption in the HK relation, eq 3, is that the evaporation and condensation coefficients are equal. When this assumption is applied to the TED-SRT evaporation flux expression (i.e., σe* and σc* are equal to σ*), the system of eqs (eqs 38 to 40) becomes overdefined. For example, the assumption of equal coefficients leads to three equations to determine the vapor-phase pressure, denoted by PVi (i = 1, 2 or 3), one of which does not even require the evaporation flux to be known: 7755


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⎛

P2V

=

L ⎜ P (T ) TIV ⎜ s I ⎜ TIL ⎝

⎞ j0LV ⎟ m ⎟ ⎟ 2kBπ ⎠

−

3n − 2 ⎞ ⎛ ⎡ ⎛ TIV TIV ⎞⎤⎛ TIL ⎞ ⎟ ⎜ exp⎢ −(3n − 2)⎜1 − L ⎟⎥⎜ V ⎟ ×⎜ L ⎟ ⎥ ⎢ T T T ⎠ ⎝ ⎠ ⎝ ⎦ ⎣ I I ⎠ ⎝ I

(44)

P3V

=

⎛ V ⎞2(3n − 2) L TI Ps(TI )⎜ L ⎟ ⎝ TI ⎠

TIL TIV

⎞ j0LV ⎟ m ⎟ ⎟ 2kBπ ⎠

⎫0.5 ⎛ V ⎞ ⎤ ⎛ V ⎞ 3n − 2 ⎞ ⎪ ⎡ ⎛ T T ⎟⎬ × ⎜⎜Ps(TIL) exp⎢(3n − 2)⎜1 − IL ⎟⎥⎜ IL ⎟ ⎟⎪ ⎥ ⎢ T T ⎠ ⎝ ⎠ ⎝ ⎦ ⎣ I I ⎠⎭ ⎝

Figure 15. Predicted values of PV obtained from the TED-SRT evaporation flux expression are compared with the measured values of PV reported in 45 evaporation experiments by Badam et al.119 and listed in Tables 7 and 8. Three different approximations on the interfacial temperatures are considered: they are assumed to be the measured values (×), assumed to be thermally equilibrated with TLI (□), and assumed to be thermally equilibrated with TVI (▽).

P1V

⎧ ⎛ ⎪ V ⎜ Ps(TIL) = ⎨ TI ⎜ − ⎜ TIL ⎪ ⎝ ⎩

(45)

PVi

The calculated values of (and the corresponding values of σ*i ) from eqs 43 to 45 are listed in Tables 11 and 12 for the water experiments. As seen there, the calculated values agree with the measured values, PVm, to within the measurement error. In addition, the values of the coefficients are within 5% of each other. Thus, the assumption that the evaporation and condensation coefficients are equal does not lead to notable errors when applied to the TED-SRT evaporation flux expression. Note that the assumption of equal coefficients does not imply that σ* is constant. The assumption of constant coefficients is investigated below in section 5.3.1.

⎡ ⎛ T V ⎞⎤ exp⎢2(3n − 2)⎜1 − IL ⎟⎥ ⎢⎣ TI ⎠⎥⎦ ⎝ (43)

Table 9. Simplifying Assumptions Are Made to the TED-SRT Expression (eqs 38 to 40) and Are Applied to the Water Experiments Listed in Tables 7 and 8a assumptions

TI = TLI and σc* = 1

none V

experiment

P

EVPVC1 EVPVC2 EVPVC3 EVPVC4 EVPVC5 EVPVC6 EVPVC7 EVPVC8 EVPVC9 EVPVC10 EVPVC11 EVPVC12 EVPVC13 EVPVC14 EVPVC15 EVPVC16 EVPVC17 EVPVC18 EVPVC19 EVPVC20 EVPVC21 EVPVC22 EVPVC23

566.67 492.90 391.25 338.64 298.26 256.22 741.12 573.26 482.90 388.40 299.38 253.38 740.23 569.71 484.55 390.69 291.78 243.19 852.08 746.78 567.39 385.47 289.78

σ*e

σ*c

P

0.99584 0.99796 0.99826 0.99793 0.99130 0.97970 0.99738 0.99764 1.00141 1.00473 0.99430 0.97539 0.99802 0.99860 1.00154 1.00324 0.99581 0.98799 0.99845 0.99840 1.00544 1.00894 0.99931

1.00352 1.00395 1.00447 1.00525 1.00565 1.00674 1.00794 1.00772 1.00864 1.00998 1.01190 1.01266 1.01090 1.01216 1.01311 1.01420 1.01718 1.01898 1.01295 1.01415 1.01580 1.02012 1.02298

566.85 493.09 391.44 338.87 298.49 256.50 742.20 574.05 483.72 389.25 300.28 254.23 742.13 571.49 486.28 392.29 293.45 244.84 855.06 749.83 570.19 388.35 292.48

V

TI = TVI and σc* = 1

σ*e

P

σ*e

0.99568 0.99776 0.99801 0.99759 0.99091 0.97917 0.99666 0.99695 1.00056 1.00362 0.99281 0.97376 0.99674 0.99704 0.99976 1.00119 0.99299 0.98467 0.99671 0.99637 1.00297 1.00520 0.99469

739.66 665.29 550.93 505.74 459.96 428.61 1,296.37 997.90 897.57 793.62 699.05 626.38 1,554.91 1,307.39 1,186.47 1,032.93 933.61 872.87 2,004.07 1,894.10 1,605.47 1,418.44 1,270.01

0.87163 0.85898 0.84124 0.81659 0.79825 0.75748 0.75412 0.75614 0.73452 0.70287 0.65068 0.62036 0.68860 0.65919 0.64003 0.61699 0.55670 0.52150 0.65103 0.62689 0.59771 0.52596 0.47734

V

a The effect of the assumptions described in Section 5.3 on the prediction of PV is examined and plotted in Figure 15. The experiment acronyms are described in section 7. Note that, as described in the text, the number of decimal places does not represent the accuracy of the calculations. Continued in Table 10.

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Table 10. Continued from Table 9a assumptions

TI = TLI and σc* = 1

none V

experiment

P

EVPVC24 EVPVC25 EVPVC26 EVPVC27 EVPVC28 EVPVC29 EVPVC30 EVPVC31 EVPVC32 EVPVC33 EVPVC34 EVPVC35 EVPVC36 EVPVC37 EVPVC38 EVPVC39 EVPVC40 EVPVC41 EVPVC42 EVPVC43 EVPVC44 EVPVC45 mean SD

241.68 877.81 745.26 569.12 393.36 304.97 248.12 982.43 857.36 746.95 567.76 390.84 296.51 230.54 1,074.26 971.43 845.75 754.10 578.66 388.04 289.57 216.90

TI = TVI and σc* = 1

σ*e

σ*c

P

σ*e

PV

σ*e

0.98993 0.99422 1.00010 1.00119 0.99238 0.97968 0.97617 0.99299 0.99700 1.00110 1.00587 0.99935 0.99191 0.96918 1.00213 0.98800 1.00656 0.99473 0.99303 1.00240 0.99925 0.99316 0.99551 0.00851

1.02480 1.01681 1.01801 1.01989 1.02287 1.02594 1.02855 1.00805 1.01875 1.02125 1.02420 1.02775 1.03225 1.03872 1.02127 1.02093 1.02286 1.02460 1.02742 1.03458 1.03756 1.04249 1.01823 0.00978

244.23 882.64 749.87 573.29 396.99 308.44 251.41 983.89 863.04 753.05 573.52 395.80 301.30 235.49 1,083.06 979.17 853.55 761.95 585.86 395.03 295.51 222.29

0.98475 0.99150 0.99702 0.99755 0.98783 0.97417 0.96977 0.99225 0.99372 0.99704 1.00081 0.99308 0.98401 0.95894 0.99805 0.98409 1.00195 0.98959 0.98692 0.99350 0.98916 0.98107 0.99197 0.00956

1,183.51 2,550.47 2,332.80 2,006.86 1,668.95 1,548.17 1,462.16 1,713.45 2,761.12 2,761.47 2,487.67 2,109.19 2,036.27 2,184.59 3,856.42 3,461.51 3,348.03 3,283.40 2,955.63 2,861.31 2,522.54 2,422.85

0.44734 0.58327 0.56527 0.53316 0.48178 0.43482 0.40213 0.75189 0.55556 0.52065 0.48053 0.43019 0.37851 0.31485 0.52891 0.52339 0.50590 0.47671 0.43938 0.36914 0.33856 0.29717 0.57964 0.15113

V

a

Simplifying assumptions are made to the TED-SRT expression (eqs 38 to (40) and are applied to the water experiments listed in Tables 7 and 8. The effect of the assumptions described in section 5.3 on the prediction of PV is examined and plotted in Figure 15. The experiment acronyms are described in section 7. SD is the standard deviation. Note that, as described in the text, the number of decimal places does not represent the accuracy of the calculations.

was determined from eq 3 by inputting the measured values of TLI , TVI , PV, and jLV 0 :

However, if both assumptions, that the evaporation and condensation coefficients are equal and that interface is thermally equilibrated, are applied to eqs 38 to 40, we find that σe* and σc* are unity and that PV is equal Ps(TI) (i.e., the system would be in equilibrium). These assumptions are exactly those made by many who have studied evaporation, as discussed in Section 3. It is no wonder then that the reported values of the condensation and evaporation coefficients from the HK relation, eq 3, to date vary by orders of magnitude despite careful experimental investigations; making both assumptions simultaneously, as seemingly small as they may have appeared, are not valid for evaporation. 5.3.1. Assuming Constant Values of the Evaporation and Condensation Coefficients. In the above considerations, the values of σe* and σc* were not forced to be constants but were functions of TLI , TVI , and PV. If the restriction of being constants were imposed then the TED-SRT evaporation flux expression would reduce to the HK relation and the physical meaning of the TED-SRT expression would be lost (i.e., the predicted coupling of the vapor and liquid phases during evaporation would be lost). We showed earlier in sections 5.1.1 and 5.1.2 that the water evaporation experiments reported by Badam et al.119 and by Duan et al.159 supported the TED-SRT evaporation flux expression. In what follows, we use the same water evaporation experiments to investigate the HK relation, eq 3. When the HK relation is applied to the water experiments listed in Tables 7 and 8, neither σe nor σc can be determined simultaneously. We thus must assume σe and σc are equal and are denoted by a single constant coefficient, σ. The value of σ

j0LV

σ= m 2kBπ

⎛ Ps(TIL) ⎜ L − ⎝ TI

PV TIV

⎞ ⎟ ⎠

(46)

The calculated values of σ from eq 46 are listed in Tables 11 and 12 and, as seen there, are much less than unity. The scatter in their values of 2 orders of magnitude is clearly indicated in Figure 16. Instead of using the HK relation, Badam et al.119 examined their experiments using the HKS relation, eq 6. They also assumed σe and σc were equal to σ. The values of σ obtained from the HKS relation are shown in Figure 16. Note there is little difference in the σ values between the HK and HKS approaches; both indicate that a variation in the σ values of 2 orders of magnitude is required to get agreement with the experiments. Similar results were reported by Ward and Fang61 (0.0586 < σ < 0.1247) when they examined the experiments labeled “EVG” in Table 4 using the HKS relation. Earlier in section 3, we described Marek and Strauß’s64 examination of the values of σ from the HK relation, eq 3, that were reported when it was also assumed that the liquid−vapor interface was thermally equilibrated. The assumption of a thermally equilibrated interface raises the question of which temperature should be assumed as the interfacial temperature.59 In Figure 17, the calculated values of σ are shown as squares when the interfacial temperature was chosen to be TVI and σ was calculated from the experiments in Tables 7 and 8 using eq 7757


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Table 11. Simplifying Assumptions, As Described In Section 5.3, Are Made to the TED-SRT Evaporation Flux Expression (eqs 38 to 40 and Are Applied to the Water Experiments Listed in Tables 7 and 8a σe* = σc*

assumptions experiment

PV1

EVPVC1 EVPVC2 EVPVC3 EVPVC4 EVPVC5 EVPVC6 EVPVC7 EVPVC8 EVPVC9 EVPVC10 EVPVC11 EVPVC12 EVPVC13 EVPVC14 EVPVC15 EVPVC16 EVPVC17 EVPVC18 EVPVC19 EVPVC20 EVPVC21 EVPVC22 EVPVC23

561.61 489.32 388.32 335.73 293.62 249.01 732.43 566.80 478.82 385.87 293.78 243.73 729.87 561.35 478.39 385.96 285.27 235.48 838.82 734.25 560.88 380.75 282.70

σ1

PV2

1.00352 1.00395 1.00447 1.00525 1.00565 1.00674 1.00794 1.00772 1.00864 1.00998 1.01190 1.01266 1.01090 1.01216 1.01311 1.01420 1.01718 1.01898 1.01295 1.01415 1.01580 1.02012 1.02298

565.42 493.04 391.62 339.08 296.75 252.21 744.33 575.65 487.20 393.68 300.88 249.91 746.46 575.60 491.48 397.40 295.53 244.85 861.72 756.26 579.72 397.21 296.71

HK

HK with TI = TVI

HK with TI = TLI

σ2

PV3

σ3

σ

σ

σ

0.99804 0.99766 0.99732 0.99663 0.99635 0.99529 0.99308 0.99350 0.99257 0.99126 0.98934 0.98893 0.98970 0.98837 0.98741 0.98629 0.98318 0.98130 0.98727 0.98588 0.98405 0.97913 0.97599

563.87 491.49 390.22 337.63 295.38 250.77 738.82 571.57 483.30 390.01 297.50 246.97 738.59 568.80 485.21 391.89 290.55 240.28 850.73 745.65 570.59 389.15 289.81

1.00078 1.00080 1.00089 1.00093 1.00099 1.00100 1.00048 1.00058 1.00057 1.00058 1.00055 1.00072 1.00024 1.00020 1.00018 1.00015 1.00003 0.99996 1.00003 0.99992 0.99980 0.99942 0.99921

0.05244 0.07761 0.10346 0.10686 0.06744 0.04238 0.03825 0.05411 0.08814 0.15375 0.06857 0.04618 0.03769 0.04739 0.06898 0.08819 0.07011 0.06036 0.03651 0.03723 0.07106 0.10198 0.06940

0.08957 0.18372 0.26516 0.26668 0.10251 0.05431 0.11159 0.15923 5.33944 −0.35479 0.15261 0.06424 0.13393 0.18561 0.78440 −7.33207 0.18667 0.11467 0.14294 0.14358 −0.36079 −0.31873 0.24958

0.08927 0.18304 0.26405 0.26538 0.10197 0.05398 0.11079 0.15812 5.29819 −0.35167 0.15104 0.06354 0.13266 0.18366 0.77563 −7.24422 0.18404 0.11291 0.14136 0.14187 −0.35606 −0.31361 0.24510

The effect of the assumptions on the predictions of PV using eqs 43 to 45 are listed. Simplifying assumptions are also made to calculate σ from the HK relation, eq 46, and are plotted in Figures 16 and 17. The experiment acronyms are described in section 7. Note that, as described in the text, the number of decimal places does not represent the accuracy of the calculations. Continued in Table 12.

a

seen in Figures 16 and 17, when the coefficients are assumed equal, or when the interfacial temperature discontinuity is ignored, the values of σ are as much as 3 orders of magnitude less than unity and scattered over three orders. In other words, the seemingly small assumptions applied to evaporation studies (equality of the evaporation and condensation coefficients, constant coefficients, and the interface is thermally equilibrated) have a significant impact on the results. This suggests the variation in the σ values reported in the literature for the past 130 years are likely the result of repeatedly making incorrect assumptions regarding evaporation flux expressions.

46. By contrast, when same experiments are again considered, but the interfacial temperature was chosen to be TLI , the values of σ obtained are shown as circles in Figure 17. The σ values shown in Figure 17 are listed in Tables 11 and 12. The scatter in the values of σ in Figure 17 span 3 orders of magnitude, a range similar to that reported by Marek and Strauß.64 In the data considered by Marek and Strauß, the interfacial temperature was sometimes chosen to be TLI and sometimes TVI . Thus, the scatter in the coefficients that they reported could have resulted from the improper assumptions that were made. Some of the σ values shown in Figure 17 are greater than unity (or negative, Tables 11 and 12), which is in complete violation of their definitions in the HK and HKS relations. Others have also reported data indicating negative values of σ and values greater than unity.199,269,275−277 Davis et al.269 used the HK relation, eq 3, and evaporation experiments of water and isopropanol to determine values of σ and found the values were greater than unity, as indicated in Figure 18. They claimed errors in the experiments were likely the cause of such large values. However, from the above analysis, it is not clear that the “violating” σ values resulted from experimental error, and the values may in fact be valid, as has been discussed. Moreover, in MD simulation studies of argon evaporation, Hołyst et al.199 reported that the evaporation coefficient was significantly above unity, ∼3. Others have investigated both the HK and HKS relations but assumed that σe and σc were unity,63,72,76,103,104,160−162,198 as discussed earlier in section 3.1.1. This assumption may have appeared reasonable; however, the expressions in eqs 38 and 39 indicate that assuming the coefficients are unity would put the evaporating system in equilibrium, which is contradictory. As


The thermal-energy-dominant limit of the SRT evaporation flux expression leads to an expression for this flux that does not depend on internal molecular vibration frequencies, ωS , and has no fitting parameters. We denoted it the TED-SRT evaporation flux expression. While it does not explicitly have ωS values as input parameters, this does not mean that the ωS terms are insignificant to evaporation. To the contrary, the ωS terms were necessary to give the TED-SRT evaporation flux expression its final form, eqs 38 to 40. It has an HK-like form but with the condensation and evaporation coefficients explicitly defined. The TED-SRT evaporation flux expression was examined with 105 water and 22 ethanol evaporation experiments and was found to be as accurate as the full SRT expression, since it accurately predicted the measured interfacial temperature discontinuities and values of the saturation-vapor pressures. Thus, the TED-SRT expression for the evaporation flux was shown to be complete, functional, and physically accurate. 7758


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Table 12. Continued from Table 11a σe* = σc*

assumptions experiment

PV1

EVPVC24 EVPVC25 EVPVC26 EVPVC27 EVPVC28 EVPVC29 EVPVC30 EVPVC31 EVPVC32 EVPVC33 EVPVC34 EVPVC35 EVPVC36 EVPVC37 EVPVC38 EVPVC39 EVPVC40 EVPVC41 EVPVC42 EVPVC43 EVPVC44 EVPVC45 Mean SD

233.14 857.23 731.22 557.97 381.13 290.84 235.17 966.54 838.00 731.29 556.88 379.54 284.55 214.81 1,052.82 938.92 831.23 731.19 558.57 375.47 278.51 206.36

σ1

PV2

1.02480 1.01681 1.01801 1.01989 1.02287 1.02594 1.02855 1.00805 1.01875 1.02125 1.02420 1.02775 1.03225 1.03872 1.02127 1.02093 1.02286 1.02460 1.02742 1.03458 1.03756 1.04249 1.01823 0.00978

245.61 888.11 759.56 581.95 400.01 307.22 249.79 982.20 871.93 765.14 586.49 402.79 304.96 233.57 1,101.85 981.74 872.94 770.84 592.57 404.78 302.20 226.35

σ2

PV3

0.97409 0.98269 0.98127 0.97914 0.97588 0.97252 0.96964 0.99322 0.98035 0.97730 0.97374 0.96970 0.96444 0.95660 0.97704 0.97762 0.97521 0.97312 0.96972 0.96094 0.95748 0.95172 0.98115 0.01173

239.45 873.08 745.73 570.20 390.71 299.11 242.53 974.95 855.34 748.50 571.86 391.25 294.77 224.14 1,077.72 960.69 852.37 751.23 575.69 390.11 290.31 216.27

HK

HK with TI = TVI

HK with TI = TLI

σ3

σ

σ

σ

0.99913 0.99961 0.99947 0.99931 0.99910 0.99887 0.99866 1.00061 0.99936 0.99903 0.99865 0.99830 0.99777 0.99682 0.99891 0.99904 0.99875 0.99853 0.99815 0.99708 0.99671 0.99607 0.99946 0.00123

0.05976 0.03032 0.04102 0.05022 0.04736 0.04388 0.04814 0.04527 0.03448 0.04197 0.05590 0.05465 0.05350 0.04196 0.03367 0.02402 0.04987 0.03224 0.03463 0.05402 0.05754 0.06065 0.05740 0.02458

0.12647 0.07658 0.20457 0.28821 0.11044 0.07559 0.08011 0.08435 0.10982 0.22949 4.34046 0.19131 0.12412 0.06839 0.25549 0.04830 −0.76044 0.08695 0.08577 0.21960 0.17656 0.14279 0.14953 1.52704

0.12406 0.07552 0.20157 0.28362 0.10846 0.07409 0.07840 0.08374 0.10815 0.22563 4.25934 0.18732 0.12121 0.06654 0.25119 0.04749 −0.74687 0.08531 0.08400 0.21417 0.17191 0.13867 0.14744 1.50863

Simplifying assumptions, as described in section 5.3, are made to the TED-SRT evaporation flux expression (eqs 38 to 40) and are applied to the water experiments listed in Tables 7 and 8. The effect of the assumptions on the predictions of PV using eqs 43 to 45 are listed. Simplifying assumptions are also made to calculate σ from the HK relation, eq 46, and are plotted in Figures 16 and 17. The experiment acronyms are described in section 7. SD is the standard deviation. Note that, as described in the text, the number of decimal places does not represent the accuracy of the calculations. a

Figure 17. It is assumed that σe and σc equal σ and that the interface is thermally equilibrated with a temperature of TI. The values of σ were determined using eq 46 and the experiments reported by Badam et al.119 (listed in Tables 7 and 8). Absolute values of σ are shown. With TI chosen to be TVI , the resulting σ values are less than 0.01. When TI is chosen to be TLI , the magnitude and scatter of σ values is increased. The overall scatter in the σ values ranges over 3 orders of magnitude.

Figure 16. Water experiments reported by Badam et al.119 and listed in Tables 7 and 8 are used to calculate the values of σ from the HK (eq 3) and HKS (eq 6) relations, shown as red ★ and as blue ●, respectively. It is assumed that σe and σc are equal to σ. The σ values found from using the HK relation are listed in Tables 11 and 12.

Assuming that the interface was thermally equilibrated with TLI also appeared to be a valid assumption. However, making both assumptions simultaneously resulted in errors, such as incorrect predictions of PV (see Figure 15). Imposing the additional assumption that the coefficients were constant was equivalent to assuming that the system was in equilibrium and resulted in coefficient values spanning 3 orders of magnitude. Thus, we found that seemingly small assumptions made to the HK relation were improper and are the likely source of the scatter in the coefficient values reported by Marek and Strauß.64 While the TED-SRT evaporation flux expression has the same form as the HK relation, they represent different physical

The explicit expressions for the coefficients in eqs 38 and 39 indicated that the coefficient values increased with decreasing values of TLI for given values of TVI and PV. Thus, the explicit coefficient expressions agreed with trends previously reported in independent experimental studies and MD simulations (see section 5.1.1). The explicit coefficient expressions were used to examine the effects that seemingly small assumptions have on the TED-SRT evaporation flux expression. Assuming that the coefficients were equal was reasonable and led to predictions of the vapor-phase pressure that agreed with the experimentally measured values. 7759


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predicted the measured interfacial liquid temperature and the interfacial temperature discontinuities. The role of the internal molecular phonons, ωS , in the evaporation process was examined in section 4.6 using the full SRT expression for the evaporation flux. The ωS terms were indicated to play a role in establishing the direction of the interfacial temperature discontinuity.118,159 Not all ωS values have been measured for all liquids, making it difficult to apply the SRT evaporation flux expression in general. In order to remove the dependence on ωS , we have simplified the SRT evaporation flux expression by taking the thermal-energy-dominant (TED) limit: the limit in which the phonon energy of a molecule (ℏωS ) is negligible compared to its thermal energy (kBT) (i.e., ℏωS /kBT goes to zero). This limit removed the ωS terms, without compromising the physical relevance of the expression. As discussed in section 5, the resulting expression, called the TED-SRT evaporation flux expression, was similar in form to the HK relation but did not introduce any empirical constants. Instead, explicit expressions for the evaporation and condensation coefficients were defined from the TED-SRT evaporation flux expression in eqs 38 and 39. The explicit coefficient expressions revealed that they represent the coupling between the liquid and vapor phases during evaporation. This coupling results from the quantum and statical mechanics formulation of the SRT expression. As a result, the coefficients are not limited to values less than unity. Thus, they were conceptually different than those from the HK relation. The TED-SRT evaporation flux expression was examined with 105 water and 22 ethanol evaporation experiments. It was found to give as accurate predictions of several parameters as the full SRT expression, including the saturation-vapor pressure, the interfacial liquid temperature, and the interfacial temperature discontinuities. Once the TED-SRT evaporation flux expression was known to have experimental support, it was used to examine simplifying assumptions that had been made in many other theoretical, numerical, and experimental evaporation studies. One commonly made assumption was that the interfacial liquid and vapor temperatures were equal. However, the results in Figure 15 clearly show that making this assumption introduced errors when predicting the conditions at the liquid vapor interface, especially if the interface were assumed to have the vapor-phase temperature, TVI . Assuming that the coefficients were equal and that the interface was thermally equilibrated was equivalent to assuming that the evaporating system was in equilibrium. This contradiction resulted in coefficient values spanning 3 orders of magnitude. Thus, we found that seemingly small assumptions made to the HK relation were improper and were the source of scatter in the coefficient values reported by Marek and Strauß.64 Such an analysis of the simplifying assumptions would not have been possible without taking the thermal-energy-dominant limit of the full, quantum mechanical SRT expression for the evaporation flux, since it provided explicit expressions for the coefficients.

Figure 18. Davis et al.269 used the HK relation to determine values of σ from water (○) and isopropanol (■) evaporation experiments. They claimed σ values greater than unity were the result of experimental error. Adapted from ref 269. Copyright 1975 American Chemical Society.

concepts. In the HK approach, the evaporation coefficients cannot be greater than unity; however, there is no such restriction in the TED-SRT evaporation flux expression. The evaporation and condensation coefficients in the TED-SRT evaporation flux expression are the terms that couple the liquid and vapor phases during evaporation. This phase coupling concept is missing from the HK relation.

6. CONCLUDING REMARKS We have reviewed two theories of evaporation: the HertzKnudsen (HK) relation and the statistical rate theory (SRT) expression for the evaporation flux. Sections 2 and 3 described the HK relation, eq 3, and its examination over the past 130 years with numerous experimental, numerical, and molecular dynamics simulation studies. We examined the evaporation and condensation coefficients in the HK relation and described the simplifying assumptions that previous investigators had made to determine their values. The most commonly made assumptions were that the coefficients were equal and that the liquid−vapor interface was thermally equilibrated.1,56,59,64−66 However, those assumptions were found to be questionable. As was described in section 3.6.2, assuming a thermally equilibrated interface was not consistent with numerous evaporation experiments that indicated the interfacial vapor temperature was larger than the interfacial liquid temperature. Therefore, studies investigating the validity of the HK relation remained open to question. In section 4, we reviewed an expression for the evaporation flux developed from statistical rate theory that is based on quantum and statistical mechanics.61,62 The full SRT expression for the evaporation flux, eqs 16 to 21, has no fitting parameters and has been verified in several evaporation experiments conducted with both single and multicomponent fluids. As discussed in section 4.2, the SRT evaporation flux expression was used to predict expressions for the saturation-vapor pressure, Ps(T), of water and of ethanol. The Ps(T) expressions were shown to lead to predictions of the evaporative latent heat and of the liquid-phase constant-pressure specific heat that were consistent with independently measured values of these properties, even in an extrapolated temperature range.117,118 Furthermore, the SRT evaporation flux expression accurately

7. DESCRIPTION OF EXPERIMENT ACRONYMS Each of the water and ethanol experiments examined in section 5 and listed in Tables 2 to 12 are labeled by an acronym, which are defined as follows: (1) EVCu, sessile water droplets evaporated from the flat surface of a copper substrate (see the experiments labeled EVD by Ghasemi and Ward176,278); (2) 7760


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EVPD, sessile water droplets evaporated from the flat surface of a polydimethylsiloxane (PDMS) substrate;241,278 (3) EVAu, sessile water droplets evaporated from the flat surface of a gold substrate (see the experiments labeled EVA by Ghasemi and Ward158,278); (4) EVAucl, measurements are made 250 μm from the droplet contact line in the EVAu experiments, where the evaporation flux is very high (see the experiments labeled EVA by Ghasemi and Ward158,278); (5) EVPM, water evaporated from the circular mouth of a poly(methyl methacrylate) (PMMA) funnel (see the experiments labeled ET by Duan et al.117); (6) EVSS, water evaporated from the circular mouth of a stainless-steel funnel (see the experiments labeled EV by Duan and Ward147,261); (7) EVrSS, water evaporated from the rectangular mouth of a stainless-steel funnel (see the experiments labeled EVC by Duan et al.117); (8) EthEVSS, ethanol evaporated from the circular mouth of a stainless-steel funnel;118 and (9) EVPVC, water evaporated from the rectangular mouth of a polyvinyl chloride (PVC) funnel experiments reported by Badam and Duan et al.,119,245 where the temperature of the vapor phase could be imposed with an electrical heating element, as illustrated in Figure 11.

Tsai (University of Alberta), Hadi Ghasemi (University of Houston), Fei Duan (Nanyang Technological University), Ian Thompson, Gang Fang, and Dennis Stanga for providing data of their evaporation experiments.

REFERENCES (1) Davidovits, P.; Kolb, C. E.; Williams, L. R.; Jayne, J. T.; Worsnop, D. R. Update 1 of: Mass Accommodation and Chemical Reactions at Gas-Liquid Interfaces. Chem. Rev. 2011, 111, PR76−PR109. (2) Corsetti, S.; Miles, R. E. H.; McDonald, C.; Belotti, Y.; Reid, J. P.; Kiefer, J.; McGloin, D. Probing the Evaporation Dynamics of Ethanol/ Gasoline Biofuel Blends Using Single Droplet Manipulation Techniques. J. Phys. Chem. A 2015, 119, 12797−12804. (3) Kim, D.-H.; Lim, S.; Shim, J.; Song, J. E.; Chang, J. S.; Jin, K. S.; Cho, E. C. A Simple Evaporation Method for Large-Scale Production of Liquid Crystalline Lipid Nanoparticles with Various Internal Structures. ACS Appl. Mater. Interfaces 2015, 7, 20438−20446. (4) Lancaster, D. K.; Johnson, A. M.; Kappes, K.; Nathanson, G. M. Probing Gas-Liquid Interfacial Dynamics by Helium Evaporation from Hydrocarbon Liquids and Jet Fuels. J. Phys. Chem. C 2015, 119, 14613−14623. (5) Galloway, M. M.; Powelson, M. H.; Sedehi, N.; Wood, S. E.; Millage, K. D.; Kononenko, J. A.; Rynaski, A. D.; De Haan, D. O. Secondary Organic Aerosol Formation during Evaporation of Droplets Containing Atmospheric Aldehydes, Amines, and Ammonium Sulfate. Environ. Sci. Technol. 2014, 48, 14417−14425. (6) Fu, M.; Yu, T.; Zhang, H.; Arens, E.; Weng, W.; Yuan, H. A Model of Heat and Moisture Transfer through Clothing Integrated with the UC Berkeley Comfort Model. Build. Environ. 2014, 80, 96− 104. (7) Kakitsuba, N. Dynamic Changes in Sweat Rates and Evaporation Rates through Clothing during Hot Exposure. J. Therm. Biol. 2004, 29, 739−742. (8) Pérez-González, A.; Urtiaga, A. M.; Ibáñez, R.; Ortiz, I. State of the Art and Review on the Treatment Technologies of Water Reverse Osmosis Concentrates. Water Res. 2012, 46, 267−283. (9) Allan, R. P. The Role of Water Vapour in Earth’s Energy Flows. Surv. Geophys. 2012, 33, 557−564. (10) Lozano, A. L.; Cherblanc, F.; Cousin, B.; Bénet, J. C. Experimental Study and Modelling of the Water Phase Change Kinetics in Soils. Eur. J. Soil Sci. 2008, 59, 939−949. (11) des Rieux, A.; Fievez, V.; Garinot, M.; Schneider, Y.-J.; Préat, V. Nanoparticles as Potential Oral Delivery Systems of Proteins and Vaccines: A Mechanistic Approach. J. Controlled Release 2006, 116, 1− 27. (12) Sefiane, K.; Shanahan, M. E.; Antoni, M. Wetting and Phase Change: Opportunities and Challenges. Curr. Opin. Colloid Interface Sci. 2011, 16, 317−325. (13) Sharma, S.; Debenedetti, P. G. Evaporation Rate of Water in Hydrophobic Confinement. Proc. Natl. Acad. Sci. U. S. A. 2012, 109, 4365−4370. (14) Sefiane, K. On the Formation of Regular Patterns from Drying Droplets and Their Potential Use for Bio-Medical Applications. J. Bionic Eng. 2010, 7, S82−S93. (15) Goto, E.; Matsumoto, Y.; Kamoi, M.; Endo, K.; Ishida, R.; Dogru, M.; Kaido, M.; Kojima, T.; Tsubota, K. Tear Evaporation Rates in Sjögren Syndrome and non-Sjögren Dry Eye Patients. Am. J. Ophthalmol. 2007, 144, 81−85. (16) Bou Zeid, W.; Vicente, J.; Brutin, D. Influence of Evaporation Rate on Cracks’ Formation of a Drying Drop of Whole Blood. Colloids Surf., A 2013, 432, 139−146. (17) Droplet Wetting and Evaporation: From Pure to Complex Fluids, 1st ed.; Brutin, D., Ed.; Academic Press: San Diego, CA, 2015. (18) Schlesinger, W.; Jasechko, S. Transpiration in the Global Water Cycle. Agr. Forest Meteorol. 2014, 189−190, 115−117. (19) Jasechko, S.; Sharp, Z. D.; Gibson, J. J.; Birks, S. J.; Yi, Y.; Fawcett, P. J. Terrestrial Water Fluxes Dominated by Transpiration. Nature 2013, 496, 347−350.

ASSOCIATED CONTENT S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.chemrev.5b00511. List of symbols (PDF)

AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. Biographies Aaron H. Persad is a Post-Doctoral Fellow in the Mechanical and Industrial Engineering (MIE) Department at the University of Toronto, Canada. His work in the Thermodynamics and Kinetics Lab investigates several topics: liquid−vapor phase change processes, surface and wetting phenomena, bubble nucleation, adsorption onto porous and nonporous media, energy transport and conversion, and the stability of confined fluids in near-freefall environments. He was named top teaching assistant by the Faculty of Engineering in 2013. His doctoral thesis investigating energy transport during evaporation was nominated by the MIE department for the 2015 John LeyerleCIFAR Prize for Interdisciplinary Research and for the Governor General’s Gold Medal. Charles A. Ward is professor emeritus of Mechanical and Industrial Engineering at the University of Toronto. His research interest is in thermodynamics and kinetics. In 2009, he was awarded the Jules Stachiewicz Medal of the Canadian Society of Mechanical Engineering in recognition of his contributions to Heat Transfer. He has published over 150 research papers and supervised over 25 doctoral students.

ACKNOWLEDGMENTS We gratefully acknowledge the support of the Canadian Space Agency, the Natural Sciences and Engineering Research Council of Canada, and the European Space Agency. We thank Chunmei Wu (Chongqing University), Peichun Amy 7761


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