Curvature Dependence of the Mass Accommodation Coefficient

Apr 12, 2019 - ... but the applicability of planar accommodation coefficients to the high-curvature interfaces present in very small droplets is not c...
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Curvature Dependence of the Mass Accommodation Coefficient Paul L. Barclay* and Jennifer R. Lukes* Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States

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S Supporting Information *

ABSTRACT: Mass accommodation coefficient, a parameter that captures molecular transport phenomena at liquid−vapor interfaces, is essential for predicting the growth of liquid droplets during condensation processes but is difficult to obtain experimentally. Molecular simulations have been widely used to obtain accommodation coefficients for planar interfaces, but the applicability of planar accommodation coefficients to the high-curvature interfaces present in very small droplets is not clear. In this work, molecular dynamics simulations are used to compute equilibrium mass accommodation coefficients at different temperatures for small droplets of various fluids, including Lennard-Jones and Buckingham fluids, benzene, butane, methane, methanol, and water. For all fluids studied, the mass accommodation coefficient increases with droplet size to a constant limiting value and decreases with temperature. Furthermore, the accommodation coefficient curvature dependence collapses onto a universal curve when appropriately scaled.

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INTRODUCTION Condensation is of central importance in a broad range of areas in both nature1−9 and industry.10−20 Aerosol−cloud interactions, currently a significant open question in climate modeling, are driven by the condensation of atmospheric water vapor onto cloud condensation nuclei.1−4 Water harvesting, which occurs in living organisms, including cacti,6,9 beetles,7,8 and spiders,5 and in engineered systems for irrigation,11 supplementation of domestic water supply,11−13 and power plant cooling tower steam recovery,12 relies heavily on droplet condensation from the surrounding air or fog. Electronic cooling technologies such as vapor chambers14 and heat pipes21 employ condensation to achieve high cooling performance, and it has been found that adding nanoscale topographical21−25 or surface energy26−28 patterning to the condensing surfaces can greatly enhance the overall heat transfer coefficient. Additionally, steam power plants15−17 and thermal desalination plants17−20 are driven by the condensation of water vapor into liquid water. Theoretical models are often used to predict the rates of liquid water and latent heat production during condensation processes.21−24,29−31 A key component of these condensation models is the liquid−vapor interfacial thermal resistance. In kinetic-theory-based models,32 this resistance depends upon the mass accommodation coefficient, α, which represents the likelihood that a vapor molecule that impinges upon a liquid− gas interface condenses into the bulk liquid. This interfacial resistance increases as droplet radius decreases and dominates the condensation heat and mass transfer for droplets with radii below 1 μm.23,24 In these models, the value for α is often © XXXX American Chemical Society

assumed or estimated from computer simulations as it is difficult to obtain experimentally.21,22,29,30,33−36 Molecular dynamics (MD) simulations, which track condensation and evaporation processes at the molecular level, are well suited for mass accommodation studies and have been used to calculate α for a wide range of simple fluids and for more complex systems, including binary liquid mixtures, noncondensable gases, and surface monolayer coatings.37−51 All but one of the previous studies in the literature focused on planar liquid−gas interfaces. The applicability of planar accommodation coefficients to the high-curvature interfaces of very small droplets, for example, the submicron droplets produced on advanced condenser surfaces21,23−25 and the critical water nuclei responsible for cloud formation,45 is unclear. Curvature effects are known to be important for other interfacial properties such as surface tension52−58 and may also be relevant for mass accommodation. Julin et al.45 performed the only previous study of the effect of curvature on α. For water droplets with radii of approximately 2 and 4 nm at 273.15 K, they found that α slightly increased with radius but that it did not significantly deviate from its planar value. In the present work, a systematic MD study of the mass accommodation coefficient at equilibrium liquid−vapor interfaces is presented for a variety of pure fluids at a range of temperatures. In contrast to the previous work, this work shows that α is size-dependent, varying from the bulk value for Received: February 22, 2019 Revised: April 8, 2019 Published: April 12, 2019 A

DOI: 10.1021/acs.langmuir.9b00537 Langmuir XXXX, XXX, XXX−XXX

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Figure 1. Continuum (left) and atomistic (right) representations of a Lennard-Jones droplet. Ninc, Nacc, and Nref refer to the number of incident, accommodated, and reflected molecules, respectively. The solid green line indicates the incident surface. The blue line indicates the liquid−vapor interface, defined as the location where the density is midway between the bulk liquid and vapor densities.

incident surface or was accommodated by the liquid. If the center of mass of the incident molecule remained inside the liquid region, defined as the interior of the droplet radius (inside the blue circle in Figure 1), for a time τacc, it was marked as accommodated. Following the method of Liang et al.,40 τacc was computed as the time for a molecule of mass m to travel 2δ if it were traveling at the thermal speed.32 Molecules that were not accommodated by the liquid as described above were marked as reflected. For the results reported in this paper, the first equality in eq 1 based on Nacc was used; no significant differences were found when using the second equality based on Nref. Further details regarding the MD simulations and interaction parameters are provided in the Supporting Information.

larger droplets down to zero for the smallest droplets, and that its bulk value is not generally equal to one. In addition, scaling reveals a universal curvature dependence for all fluids and temperatures considered.

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SIMULATION DETAILS To compute α, stable spherical equilibrium droplets of varying sizes were first prepared in MD using cubic domains with periodic boundary conditions. Lennard-Jones, Buckingham, benzene, n-butane, methane, methanol, and water droplets were simulated at multiple temperatures (Table 1). LennardJones fluids with repulsive forces proportional to r−k, where k = {12,9}, and Buckingham fluids with repulsive forces proportional to e6k(1−r), where k = {3,2}, were studied. Benzene, butane, methane, and methanol were modeled using transferable potentials for phase equilibria (TraPPE).59−62 Benzene and methane were modeled using the explicit hydrogen TraPPE potentials, while butane and methanol were modeled using the united atom TraPPE potentials. Water was modeled using the TIP4P/2005 potential.63 Figure 1 shows a continuum representation of the entire liquid−vapor simulation domain and a zoomed-in atomistic snapshot of a stable Lennard-Jones droplet with k = 12. The radius of each droplet, indicated by the blue line in Figure 1, is defined as the location where the fluid density is the average of the bulk liquid and vapor densities, ρ(R) = (ρL + ρV)/2. After establishing stable droplets, canonical ensemble MD simulations were then run to calculate α. Formally, α is the ratio of the number of accommodated molecules Nacc to the number of incident molecules Ninc N 1 − Nref α = acc = Ninc Ninc (1)

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RESULTS AND DISCUSSION To verify the validity of the above method to calculate the mass accommodation coefficient α, simulations on planar interfaces were run, and the results agreed well with the previous literature,37−40 as seen in Figure 2. The method was

Figure 2. Mass accommodation coefficient calculated in planar simulations versus temperature for the Lennard-Jones potential with k = 12.

where Nref is the number of reflected molecules. Here, Ninc was determined by calculating the number of molecules moving toward the droplet that crossed a hypothetical surface denoted the “incident surface” (the green line in Figure 1). As in previous accommodation coefficient studies,40−42 the incident surface was placed at a distance δ = rc beyond the liquid−vapor interface, where rc is the MD cutoff radius (Table 1). The cutoff radius was chosen for δ because within this distance, the vapor molecule begins to interact (or “collide”) with the liquid droplet. After the center of mass of a molecule crossed the incident surface, it was monitored until it recrossed the

then applied to calculate α for spherical droplets. Figure 3a shows α as a function of droplet radius for the Lennard-Jones k = 12 potential at T = 96.8 K. A clear decrease in α is observed as droplet radius decreases. This decreasing trend is due to several effects both physical and geometric in nature. Physically, the Kelvin equation predicts that vapor density at coexistence increases as droplet size decreases. This is evident from the coexistence droplet size expression64 B

DOI: 10.1021/acs.langmuir.9b00537 Langmuir XXXX, XXX, XXX−XXX

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where α∞ and R0 are fitting parameters. This functional form was observed to fit the MD data well for all fluids and temperatures modeled in this study. These data, normalized for each fluid and temperature, collapse onto a master curve (Figure 4). The normalizing values for α∞ and R0 can be found

Figure 3. Radial dependence of (a) mass accommodation coefficient α, (b) inverse of normalized vapor density ρV*, and (c) normalized droplet attraction energy U* versus droplet radius for the LennardJones potential with k = 12 and T = 96.8 K. Dashed horizontal lines are limiting values for a planar interface.

yz ρ∞ 2σ ij R = ∞ ∞ jjjj ln(ρV*) − V∞ (ρV* − 1)zzzz ρL kBT k ρL {

Figure 4. Normalized mass accommodation coefficient versus normalized radius. Data are normalized by α∞ and R0 (Table 1).

in Table 1. The physical meaning of α∞ is simple to understand as it is the accommodation coefficient for an infinitely large (zero-curvature) droplet, which was confirmed by comparing the fitted α∞ values to mass accommodation coefficient values directly computed in planar simulations (see Figure 2 and the Supporting Information). Similar to the trend observed in Figure 2, a decreasing trend with temperature is evident for α∞ (Table 1), which is consistent with observations in the literature.37−40,47−49 The physical meaning of R0 requires some discussion. From eq 3, R0 is the radius below which no molecules can accommodate onto the droplet. The question arises: why does accommodation not occur below this size? Clearly, no molecules can condense onto a droplet that is below the minimum thermodynamically stable size. While the standard Kelvin equation cannot explain a nonzero minimum stable droplet size, the extended Kelvin equation can, in the limit of very high vapor density (Figure 3b). Mathematically, this minimum size arises from accounting for a nonzero vapor/ liquid density ratio in the second term in eq 2; the standard Kelvin equation neglects this term. Since the minimum size predicted by eq 2 was found to be significantly smaller than R0 for all MD cases, some other factor must contribute to the magnitude of R0. Figure 5 shows that R0 correlates fairly well with the liquid−vapor interfacial width Lint. Since minima in the free energy landscape occur at the bulk liquid and vapor densities, a possible reason for this correlation is that the

−1

σ∞, ρ∞ L,

(2)

ρ∞ V

where and are the surface tension, liquid density, and vapor density for a planar interface, respectively, and ρV* = ρV/ρ∞ V is equivalent to the supersaturation. Equation 2 is an extended version of the Kelvin equation that explicitly includes the ratio of vapor and liquid densities in the second term on the right-hand side. MD-computed results for the reciprocal of normalized vapor density ρV* agree well with eq 2 at temperatures for which the coexisting phases are well described as incompressible liquid and ideal gas (Figure 3b). For temperatures close to the critical temperature, deviations from the Kelvin equation were observed, but the general trend of decreasing 1/ρV with decreasing droplet size still held. Since Ninc ∼ ρV,37,40,45 the increase in vapor density contributes to a decrease in α, as is evident from eq 1. Furthermore, the number of accommodated molecules Nacc is expected to scale with the energetic attraction between an incident vapor molecule and the liquid droplet. This attraction, U, was estimated by averaging the effective molecule−droplet interaction energy from Yasuoka et al.65 in the region between R and R + δ (Figure 1). Additional details regarding the calculation of interaction energy are provided in the Supporting Information. Figure 3c shows that the normalized attraction, U* = U/U∞, where U∞ represents the average attraction of a molecule to a planar liquid slab, decreases with the droplet size. Smaller droplets exert a reduced pull because their high curvature dictates a smaller number of pairwise attractive interactions, within the cutoff radius, between the incident molecule and molecules in the droplet. This reduced interaction contributes to a reduction in Nacc and, accordingly, a reduction in α with decreasing droplet radius. The trend of the MD data in Figure 3a is consistent with the curvature dependence manifested in both the reciprocal of vapor density (Figure 3b) and the vapor molecule−droplet interactions (Figure 3c). This trend suggests a functional form for α as follows R y i α(R ) = α∞jjj1 − 0 zzz R{ k

Figure 5. Normalizing droplet radius versus interface width. Solid line is a linear fit to all of the data with slope and intercept of 0.95 ± 0.14 and 0.30 ± 0.13 nm, respectively.

(3) C

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Langmuir Table 1. Parameters for Curved Droplet Simulationsa fluid Lennard-Jones k = Lennard-Jones k = Lennard-Jones k = Lennard-Jones k = Lennard-Jones k = Lennard-Jones k = Lennard-Jones k = Buckingham k = 3 Buckingham k = 3 Buckingham k = 2 Buckingham k = 2 benzene benzene butane butane butane methane methane methane methanol methanol water water water water

12 12 12 12 12 9 9

T (K)

δ (nm)

τacc (ps)

α∞

R0 (nm)

R* (nm)

72.6 84.7 96.8 108.9 121.0 108.9 145.1 84.7 96.8 108.9 133.1 365 415 295 325 360 120 135 150 375 425 500 525 550 575

1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.40 1.40 1.50 1.50 1.50 1.50 1.50 1.50 1.40 1.40 1.00 1.00 1.00 1.00

41.67 38.58 36.09 34.02 32.28 34.02 29.46 38.58 36.09 34.02 30.77 35.61 33.39 36.61 34.88 33.14 30.15 28.43 26.97 22.50 21.13 10.44 10.18 9.95 9.73

1.02(2) 0.919(3) 0.782(3) 0.595(2) 0.418(4) 0.848(5) 0.395(5) 0.801(3) 0.593(7) 0.783(3) 0.453(8) 0.76(2) 0.43(4) 0.40(2) 0.33(2) 0.15(1) 0.62(2) 0.37(3) 0.17(1) 0.63(1) 0.41(1) 0.28(1) 0.228(7) 0.169(9) 0.101(5)

0.82(3) 0.843(6) 1.045(5) 1.242(6) 1.62(2) 0.977(8) 1.68(1) 0.98(5) 1.18(1) 1.012(4) 1.42(1) 1.49(2) 2.15(7) 1.82(4) 2.2(6) 2.79(7) 1.53(2) 1.63(5) 2.29(4) 0.81(2) 1.15(3) 1.04(3) 1.04(2) 1.24(3) 1.28(3)

16.5(5) 16.9(1) 20.9(1) 24.8(1) 32.4(3) 19.5(2) 33.6(3) 19.6(1) 23.6(2) 20.24(9) 28.4(3) 29.7(5) 43(1) 36.3(7) 44(1) 56(1) 30.7(5) 33(1) 45.7(9) 16.3(4) 23.0(5) 20.7(7) 20.8(5) 24.7(7) 25.6(6)

Value in (.) indicates uncertainty in the preceding digit(s), i.e., 1.23(4) = 1.23 ± 0.04.

a

minimum droplet radius must be at least Lint to form a bulk liquid phase and gain the associated energetic benefit. Additionally, R0 increases with temperature (Table 1), leading to a corresponding decrease in α (eq 3). This reduction in α is accompanied by an observed increase in the amplitude of interfacial ripples, which represent instantaneous local deviations from spherical droplet curvature. Evaporation rates are enhanced in the vicinity of local protrusions of liquid−vapor interfaces into the vapor phase; 66 such protrusions experience reduced condensation as compared to lower-curvature spherical interfaces. Accordingly, it is conjectured that the increase in R0 and decrease in α with temperature occur because the incident vapor molecules interact first with these “spiky” protrusions and are thus hindered in their ability to condense. Finally, it is noted that although eq 3 has a similar form to surface tension size dependence based upon the Tolman length, R0 was not found to correlate with the Tolman length. Details of the calculations of interface width and Tolman length are shown in the Supporting Information. From eq 3, the accommodation coefficient for large droplets approaches the planar value α∞. Defining R* as the droplet radius, where the accommodation coefficient is 95% of its planar value allows conclusions to be drawn about when the curvature dependence of α can be safely neglected. Values of R* ranging from 16.3 nm for methanol at 375 K to 56 nm for butane at 360 K (Table 1) were obtained from eq 3. This indicates that, for the fluids and temperatures studied in this work, the curvature effect on α is negligible for droplets with radii larger than a few tens of nanometers. For other fluids and temperatures, different R* values may be observed. The present observations are consistent with the results of Julin et al.,45 who reported a negligibly small increase in the α of water

at 273.15 K as droplet radius increased from 2 to 4 nm to infinity (planar interface). A linear fit to the temperaturedependent R0 data for water (Figure 6) results in R0 = 0.095

Figure 6. Normalizing droplet radius versus temperature for water. Solid line is a linear fit with slope and intercept of 0.0039 ± 0.0011 nm/K and −0.9787 ± 0.6046 nm, respectively.

nm at 273.15 K, which corresponds to R* = 1.908 nm. The water droplets simulated by Julin et al.45 were equal to or larger than this value of R*; therefore, the lack of significant curvature dependence is expected. To compare predictions of condensation heat and mass transport from kinetic-theory-based models,32 which rely on α, to predictions from nonequilibrium thermodynamics,31,67−71 which do not, it is important to clarify how the droplet radius is defined in both approaches. In the present work, the radius R for each droplet has been chosen for computational convenience as the point whose density is midway between the bulk liquid and bulk vapor densities. In previous work,31,67−72 the Gibbs equimolar radius Re has been used to define the droplet radius. Calculations of Re based on the fitted density profiles (see the Supporting Information eqs S19 and S20 and Figures S9−S13) show that Re is larger than R for all cases in this work. For the smallest droplets considered, R and D

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Re differ by less than 8%, and for the largest droplets, the difference decreases to less than 1.5%. The size dependence of α thus changes only slightly if the radius is defined as Re rather than R.

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CONCLUSIONS To summarize, molecular dynamics simulations have shown, for a variety of atomic and molecular fluids, that the condensation mass accommodation coefficient depends on the droplet size. This dependence collapses to a universal curve when the droplet size is scaled by the appropriate normalizing radius R0 and planar mass accommodation coefficient α∞. The decrease in α as droplet radius decreases is consistent with the decreased attraction between an incident vapor molecule and the liquid droplet and also with the increased vapor density predicted from the Kelvin equation as droplet radius decreases. Future studies to examine whether this curvature dependence holds in the presence of noncondensable gases or surfactants, for fluid mixtures, or for actively condensing droplets are recommended.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.9b00537. Interatomic potential parameters and verification of equilibrium droplet size through the extended Kelvin equation; energetic model for vapor interaction with a liquid droplet; calculations of the planar mass accommodation coefficient, interface width, surface tension, and Tolman length; calculations of the equimolar radius (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (P.L.B.). *E-mail: [email protected] (J.R.L.). ORCID

Jennifer R. Lukes: 0000-0002-7763-6141 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Support from the Department of Education GAANN grant number P200A160282 and the University of Pennsylvania Research Foundation is greatly appreciated. This work, under project TG-CTS170007, was performed using the Extreme Science and Engineering Discovery Environment, which is supported by the National Science Foundation grant number ACI-1548562. Specifically, this work was performed on the Bridges supercomputer at the Pittsburgh Supercomputing Center, which is supported by NSF award number ACI1445606.

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DOI: 10.1021/acs.langmuir.9b00537 Langmuir XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.langmuir.9b00537 Langmuir XXXX, XXX, XXX−XXX