Enhanced growth rates of nanodroplets in the free molecular regime: The role of long-range interactions

ABSTRACT

Recently, Pathak et al. (2013) conducted a series of non-isothermal D₂O nanodroplet growth studies in the free molecular regime. They found that under highly non-equilibrium conditions, the condensation (q_c) and evaporation coefficients (q_e) can differ from each other and from the expected value of 1. Here, we confirm these observations by analyzing comparable experiments using n-propanol. We show that the best agreement with the non-isothermal Hertz–Knudsen growth law corresponds to setting (q_c, q_e) = (1, 0.6) or (q_c, q_e) = (1.3, 1). The approach of retarded evaporation yields values close to those observed by Pathak et al. for D₂O, but is difficult to justify theoretically. Enhancing the condensation coefficient is consistent with long-range attractive interactions between the vapor molecules and droplets in the nanometer size range.

© 2016 American Association for Aerosol Research

EDITOR:

• Yannis Drossinos

1. Introduction

Almost all vapor-to-liquid phase transitions initiated by homogenous nucleation involve droplet growth in the free molecular regime. Understanding this phenomenon is, therefore, critical to a broad range of natural and industrial applications. In the atmosphere, for example, new particle formation yields clusters with diameters on order of 1–1.5 nm, but most instruments can only detect particles after they have grown to be 3–10 nm in size. Obtaining accurate nucleation rates, based on the appearance of the larger particles can be quite sensitive to assumptions made regarding the free molecular growth of the initial nuclei to detectable size (Korhonen et al. 2014). In an industrial setting, improving the efficiency of steam turbines requires droplet growth laws that are accurate from the free molecular to continuum regime under conditions that are far from equilibrium (Hughes et al. 2015).

Recently, Pathak et al. (2013) investigated the growth rates of nonane and D₂O nanodroplets formed via homogenous nucleation in a supersonic nozzle. In their experiments, the Knudsen number was always greater than 10, where λ is the mean free path of the carrier gas and is the average droplet radius. Thus, droplet growth was always in the free molecular regime and the rate of droplet growth by condensation should be governed by the Hertz–Knudsen equation,[1] where is the molecular volume of the condensate, is the mass of the condensate monomer, is the Boltzmann constant, is the partial pressure of the vapor, T is the gas temperature, and is the droplet temperature. The vapor pressure above a droplet of radius and temperature is denoted as and is given by[2] where is the equilibrium vapor pressure above a flat surface, is the surface tension of the liquid, and is determined by solving the coupled heat and mass fluxes to the growing droplets as summarized in Section 3 of this article and described in more detail in Pathak et al. (2013). Finally, the condensation and evaporation coefficients q_c and q_e, are most often assumed to both equal 1, but there are arguments, based on experimental results, that these values do not need to be the same when the system is not at equilibrium (Young 1982; Marek and Straub 2001; Pathak et al. 2013). Furthermore, recent molecular dynamic (MD) simulations also concluded that the condensation and the evaporation coefficients can differ from each other and from 1 under non-equilibrium condensation or evaporation (Meland et al. 2004).

In the nonane nanodroplet growth experiments, Pathak et al. (2013) found that coagulation was negligible on the timescale of the experiment because the aerosol number density was low enough. Furthermore, the condensational growth rates predicted by the Hertz–Knudsen equation (Equation (1)[1] ) agreed very well with the experimentally determined values when q_c = q_e = 1. In these experiments, the high degree of supersaturation () of the alkane ensured that the condensational growth rates were essentially independent of the temperature of the growing droplets. Growth was only limited by the rate of arrival of monomer to the droplet surface and not by the re-evaporation of molecules from the rapidly growing droplet, and, thus, it was not possible to test whether q_e was <1.

In contrast, in D₂O nanodroplet growth experiments, rapid condensational growth gave way to coagulation-driven growth as the condensable was depleted from the vapor phase. More importantly, the non-isothermal droplet growth rates calculated using Equation (1)[1] and q_c = q_e = 1, underestimated the experimentally measured values even though the droplet temperatures predicted by theory were lower than those estimated directly from the experimental data. The differences between the experimental and theoretical growth rates and droplet temperatures were greatly reduced by setting q_e = ∼0.5 while maintaining q_c = 1. Pathak et al.'s (2013) approach was motivated by the work of Young (1982) who found that models of low pressure steam condensation in supersonic nozzles only matched experimental results when the ratio q_e/q_c was less than 1. Evaluating Young's Equation (31), using the parameters suggested by author (q_c = 1.0, α = 9) under Pathak et al.'s conditions gives q_e/q_c ∼ 0.65. The value q_e = 0.5 determined by Pathak et al. was also comparable to the value (q_e = 0.57 ± 0.06) reported by Drisdell et al. (2008) who examined the evaporation of liquid D₂O jets into vacuum.

One concern with Pathak et al.'s (2013) D₂O experiments is that the droplets were highly supercooled. With droplet temperatures in the range of 220 K < T_d < 260 K, uncertainty in the analysis could arise from the physical properties for D₂O being extrapolated by as much as 50 K. To examine whether evaporation coefficients that differ from 1 are reasonable or not requires commensurate experiments using a material whose growth also occurs under moderate supersaturation but where the liquid is not in a supercooled state. In addition, comparable number densities and droplet sizes would ensure equivalent influence from coagulation on the experimental results.

The first goal of this work is, therefore, to analyze nanodroplet growth experiments conducted with n-propanol to confirm the enhanced growth rate effect. Alcohols are a good choice because they condense in the supersonic nozzle at reasonable supersaturations and at temperatures well above their melting points (Ghosh et al. 2008; Mullick et al. 2015). Mullick et al. (2015) recently published a set of position resolved measurements for n-propanol droplet growth that used the same set of matched nozzles (Nozzle C3/C2) as Pathak et al.'s (2013) D₂O experiments. Furthermore, the aerosol droplet sizes (D₂O: ∼5.2 nm, n-propanol: ∼6.2 nm) and number densities (D₂O: ∼3.2 × 10¹⁹ kg⁻¹, n-propanol: ∼6.8 × 10¹⁹ kg⁻¹) matched quite closely, and Kn was always greater than 100.

A second goal of this work is to better understand the origin of the enhanced growth rates. Pathak et al. (2013) only considered the option of reducing q_e to ∼0.5 while maintaining q_c = 1. Here, we explore the alternate choice, i.e., allowing q_c > 1 while maintaining q_e = 1. Although the condensation rate expression generally only considers the velocity and density of vapor molecules and the surface area of a droplet, Vasil'ev and Reiss (1996a,b) pointed out that the attractive potential of the liquid droplet should also play a role. In particular, the presence of such a potential could increase the number of vapor molecules impinging onto the droplet surface and, as a result, increase the condensation rate. Vasil'ev and Reiss defined the condensation enhancement factor as[3] where β and β^(c) are the condensation rates of vapor molecules onto a droplet in the presence and absence of an attractive potential. For small enough droplets under dilute vapor conditions, they found that by considering van der Waals interactions η could be as high as 2 or 3. They also showed that when the droplet size is large enough, or the carrier gas is dense enough, η again approaches 1 (Vasil'ev and Reiss 1996a).

Similar considerations also govern enhanced coagulation of nanoparticles in the free molecular regime. Here, experimental data (Fuchs and Sutugin 1965; Graham and Homer 1972; Okuyama et al. 1984; Mullick et al. 2015) and simulations (Marlow 1980, 1982; Kennedy 1987; Harris and Kennedy 1988; Gopalakrishnan and Hogan 2011; Ouyang et al. 2012) both show enhanced coagulation rates that depend on the strength of the long-range van der Waals interactions between particles and are a strong function of particle size and gas pressure. In a sense, condensation can be thought of as coagulation between a vapor molecule and the growing droplet. Indeed, the MD simulations by Gopalakrishnan and Hogan (2011) showed that the dimensionless collision rates for both condensation and coagulation are essentially identical over the whole range of Kn. Thus, it would seem reasonable that if long-range interactions are important in coagulation, they should also be observable in nanodroplet growth studies.

2. Experiments

A detailed description of the flow system and the experimental methods used to conduct the n-propanol experiments are included in earlier papers (Ghosh et al. 2008; Mullick et al. 2015). To summarize briefly, expansions started from a stagnation temperature T₀ of 49.80°C and stagnation pressure p₀ of 30.18 ± 0.03 kPa. The carrier gas was N₂ and the partial pressure of n-propanol was 0.43 kPa. In the Pressure Trace Measurements (PTM), the static pressure was measured as a function of position along the centerline of the flow with (wet trace) and without (dry trace) the condensate. Small Angle X-ray Scattering (SAXS) measurements were conducted using the 12-ID_C beam line at the Advanced Photon Source, Argonne National Labs, Argonne, IL to determine the mean radius of the droplets , the width of the size distribution σ, the aerosol number density N, and, thus, the mass fraction of condensate, g. The other properties of the flow, i.e., temperature T, density ρ, and velocity u, are then derived in by solving the equations that describe supersonic flow in the nozzle using p and g as the measured inputs. Further details of this integrated data analysis approach are available in Pathak et al. (2013). The physical properties of n-propanol and N₂ used in the calculations are those presented by Ghosh et al. (2008).

3. Growth models

The two main processes that affect droplet growth in supersonic nozzle experiments are condensation and coagulation. We assume that these growth mechanisms are independent and, thus, the total growth rate is given by[4]

The expression for condensational growth is given in Equation (1)[1] , where the droplet temperature is determined by balancing the mass and heat fluxes from the droplets to the surrounding gas, i.e.,[5] and is the latent heat of condensation. Equation (5)[5] neglects the small contribution from that corresponds to the change in temperature of condensable vapor from T to T_d, where c_p,v is the specific isobaric heat capacity of the condensable vapor. In the free molecular regime, is given by[6] and by[7] where is the specific isobaric heat capacity of nitrogen. To calculate the theoretical droplet temperature, T_d,exact, we solve Equations (5)–(7) using the experimental values for , T, , and . To calculate the experimental droplet temperature, T_d,exp, we equate in Equation (7)[7] with the experimentally derived heat flux (Tanimura et al. 2010)[8] where is the number of particles per kg of flow and is the experimentally determined mass flux.

For coagulation, we use the equation derived by Pathak et al. (2013) based on the experimental values for and :[9]

4. Condensation enhancement factor

In a series of two papers, Vasil'ev and Reiss explored the effect long-range interactions could have on the growth of small clusters (Vasil'ev and Reiss 1996a,b). Their motivation was to examine the effect of increased capture cross sections on vapor-liquid nucleation rates. Although the droplets in our growth studies are well past the critical cluster size characteristic of supersonic nozzle nucleation experiments, they are reasonably close in size to critical clusters at lower saturations (Manka et al. 2012; Viisanen and Strey 1994). Thus, Vasil'ev and Reiss' reasoning should apply equally well to our situation.

To evaluate the condensation enhancement factor η, Vasil'ev and Reiss considered both the simple potential (Vasil'ev and Reiss 1996a) and the more realistic potential (Vasil'ev and Reiss 1996b)[10] derived by integrating the simple potential over a sphere of radius R₀ where , is the droplet radius, and a is the dimension of the hard repulsive molecular core. The values of and are related to the characteristic Lennard–Jones energy and length by and . Finally, is the distance between the particles and is the molecular number density in the condensed phase.

The increase in over is found by integrating[11] where is the molecular velocity, ρ is the density of vapor, and corresponds to the largest initial velocity a molecule with impact parameter R can have and still be captured by the drop. For the given potential, they found that the enhancement factor η could be written in terms of the two independent parameters μ[12] and ν[13] as[14] where[15]

Here, we use the approach in the second paper of Vasil'ev and Reiss (1996b) to calculate η for n-propanol, D₂O, and nonane where .

Vasil'ev and Reiss (1996a) also noted that if the carrier gas density is high enough or the particle size is large enough because the attractive potential from other molecules also affects the condensing vapor molecules, and as a result, cancels out the enhancement. The screening effect becomes important when the intermolecular distance of carrier gas is much smaller than droplet size, i.e., , where[16] and the subscript 1 indicates the properties of the carrier gas.

5. Result and discussion

The n-propanol static pressure (p/p₀) and SAXS () measurements we will analyze are presented in Figure 1 along with the derived quantities (T, g, S and scaled nucleation rate J/J_max). All the data are shown with respect to the flow time where the latter is calculated from the position and flow velocity in the nozzle via . The time origin, t = 0, corresponds to the point in the flow where the supersaturation S reaches its maximum value.

Figure 1. The n-propanol data set comprises both PTM and SAXS experiments. Directly measured quantities (pressure ratios,) are shown in black. The remaining quantities, derived from the integrated analysis, are shown in gray (blue). (a) The static pressure ratio and estimated temperatures. (b) The mean droplet size and the spread in the size distribution . (c) The mass fraction condensate g and the n-propanol supersaturation S. The upper dashed-dot line corresponds to the mass fraction of n-propanol entering the nozzle, ginf. The lower long gray dashed line indicates whereas the short dashed line (red) indicates , i.e., the supersaturation corresponding to that of the droplets at T_d,exp. (d) The aerosol specific number, , and normalized nucleation rate (J/J_MAX). The number density is normalized by the gas density in order to account for the continued expansion of the flowing mixture. In the absence of coagulation is constant. Size and number density data were previously published in Mullick et al. (2015). The error bars indicate the ± 5% uncertainties in the absolute calibration for the SAXS measurements.

As illustrated in Figure 1a, both the temperature and pressure profiles of the condensing flow follow those of the non-condensing flow until the curves deviate as heat is released to the flow due to droplet formation and growth. The onset of sudden particle growth is also reflected in the values of , Figure 1b and g, Figure 1c. In the first ∼30 μs after onset, particles grow from ∼2 nm to ∼4.8 nm and g increases to ∼80% of the incoming value. Particle growth slows considerably after this point, and by t ∼ 80 μs, particle size reaches ∼6.2 nm and ∼93% of the available n-propanol has condensed. The width of the size distribution, σ also increases gradually as the aerosol evolves. Figure 1c shows the supersaturation S peaks sharply near the onset point before approaching the supersaturation consistent with the droplet size and temperature when t ∼30 μs. The behavior of S and g suggests that condensational droplet growth should be less important for t > 30 μs and that the continued increase of is largely due to a competing mechanism. The rapid decrease of in Figure 1d suggests that droplet coagulation is the dominant mechanism for continued droplet growth when t > 30 μs. The normalized nucleation rate J/J_MAX in Figure 1d, calculated using the Becker–Döring version of Classical Nucleation Theory and the values of S and T derived from the experiments (Becker and Döring 1935), suggests that particle formation is essentially complete when we are first able to detect particles.

Figure 2. (a) The n-propanol droplet temperatures and the gas temperature, and (b) the temperature difference between droplets and gas. (c) The experimental growth rates (circles) are calculated from a fit to the measured values of . The lines correspond to the growth rate predicted using the Hertz Knudsen growth model, Equation (1)[1] , using q_c = q_e = 1 and the indicated droplet temperatures.

Figures 2a and b show the droplet temperatures and the difference in temperature between the droplets and the gas mixture. The experimental droplet temperature (T_d,exp) was determined by setting the heat flux, arising from the collisions of the nitrogen gas with the growing n-propanol droplets (Equation (7))[7] , equal to the heat released from the condensation of n-propanol (Equation (8)[8] ). The theoretical droplet temperature (T_d,exact) was determined by the implicit solution of Equations (5)–(7) assuming that q_c = q_e = 1. During the early stages of condensation, droplet temperatures are 10–15 K higher than the gas temperature but this temperature difference (ΔT) approaches zero by t ∼30 μs. After this time point, ΔT is almost zero because the condensation rate has decreased and the heat release due to coagulation is negligible. Since we use the raw dg/dt values rather than fitting and smoothing the data, ΔT_exp fluctuates even when we expect it to be zero. Since ΔT_exp is always less than 1.3 K in this region, we consider 1.3 K to be a measure of the experimental uncertainty in this quantity. Thus, at earlier times the difference between T_d,exp and T_d,exact is too large to be explained by experimental uncertainty alone.

Figure 2c compares the droplet growth rates determined from the experiments to those calculated assuming condensation is the only growth mechanism. As expected, the experimental growth rates decrease rapidly until ∼30 μs, and more gradually thereafter. The growth rates calculated using Equation (1)[1] and T_d,exp or T_d,exact both lie below the measured droplet growth rate especially when t is less than ∼30 μs. Comparing Figures 2b and c leads to an interesting dilemma when we try to explain the discrepancy between the theoretical and experimental growth rates given the error in T_d,exact. Figure 2b suggests T_d,exact is too low, but the low growth rates in Figure 2c suggest that T_d,exact is still too high.

The results for n-propanol illustrated in Figure 2, i.e., that ΔT_d,exp > ΔT_d,exact at the same time that (dr/dt)_exp > (dr/dt)_exact, mirror the results of Pathak et al. (2013) for D₂O nanodroplet growth. Thus, the current experiments—experiments in which the liquid condensate is never supercooled—confirm and strengthen Pathak et al.'s observations.

To improve agreement between theory and experiments, we first adopt the approach used by Pathak et al. (2013), to match the theoretical and experimental growth rate of D₂O nanodroplets. As illustrated in Figure 3a, if we reduce q_e to 0.6 while maintaining q_c = 1, the agreement between ΔT_d,exp and ΔT_d,exact for the current n-propanol experiments improves significantly. Since the improvement under our experimental conditions reflects the change in the ratio q_e/q_c, rather than the particular value of either parameter, one can equally set q_e = 1 and increase q_c to a value larger than 1. Figure 3a shows that setting q_c = 1.3 and q_e = 1 also leads to good agreement between ΔT_d,exp and ΔT_d,exact. Furthermore, the growth rates at the shortest times are also predicted more accurately. Although not shown here, Pathak et al.'s D₂O data can also be well fit using q_c = 1.3 and q_e = 1.

Figure 3. (a) The experimental temperature differences between the droplets and surrounding gas are compared to theoretical droplet temperatures determined using the following two assumptions: i: (q_c, q_e) = (1, 0.6), and ii: (q_c, q_e) = (1.3, 1.0). (b) The theoretical growth rates calculated with T_d,exact for case ii agree quite well with the values measured during rapid particle growth. Including the effect of coagulation improves the agreement when t > ∼30 μs. The growth rates for case i lie almost on top of those for case ii and are not shown to maintain clarity.

While the refined values of q_e and q_c provide improved droplet temperatures and growth rate, the theoretical growth rates, Figure 3b, again starts to diverge from the experimental growth rate for t > ∼30 μs. As discussed earlier with respect to Figure 1, for t > ∼30 μs droplet growth is dominated by coagulation. Thus, as illustrated in Figure 3b, adding the coagulation growth rate to the condensational growth rate removes this difference. Once again, the results for n-propanol are very similar to those observed by Pathak et al. (2013) for D₂O.

Although it is not a standard approach in the droplet growth literature, the possibility of q_c > 1 was suggested by Vasil'ev and Reiss (1996a,b). Their interest was in exploring the effect a droplet's attractive potential could have on enhancing the droplet's capture cross section for vapor molecules and thereby change the nucleation rate. Using the equations in Section 4 of this article, we calculated the condensation enhancement factors η for n-propanol, D₂O, and nonane as a function of position in the nozzle and averaged the values. Table 1 summarizes the Lennard–Jones parameters used in the calculations and the average values of η.

Table 1. The Lennard–Jones parameters for n-propanol, D₂O, and nonane (Poling et al. 2001), and the average values for η calculated using Equations (10)–(13).

Condensable	(Å)	(K)	Average η
n-Propanol	4.549	576.7	1.47 ± 0.06
D2O	2.641	809.1	1.46 ± 0.06
Nonane	6.648	472.2	1.26 ± 0.07

In all cases, the values of η are greater than 1. Although not shown here, the values are not very sensitive to the particular set of Lennard–Jones parameters chosen for the calculation. For n-propanol and D₂O, the enhancement factors are in line with the experimentally determined values of q_c required to achieve good agreement with the experimental growth rates. The value for η for nonane initially seems to be inconsistent with the result of Pathak et al.'s (2013) experiments that showed good agreement between experimental and theoretical growth rates when q_c = q_e = 1. This discrepancy can, however, be explained by considering the screening effect proposed by Vasil'ev and Reiss (1996a), i.e., that η ≈ 1 when where the ∼ is the average carrier gas intermolecular distance and is the droplet size.

When we checked the importance of screening in the experiments, we found that for n-propanol and D₂O, is 3∼4 times larger than during the early stages of condensation and only becomes comparable to in the region where coagulation dominates growth. In contrast, for nonane the condition holds for all but the first few droplet measurements. Thus, even though we calculate an average value of η equal to 1.26, the actual value of η should be very close to 1 because of the dense carrier gas “screening effect.” This is consistent with the experimental growth rate of nonane being in good agreement with the theoretical growth rate with q_c = 1. Although Pathak et al. (2013) assumed q_e = 1, the experiments cannot constrain q_e because the theoretical results are not sensitive to this parameter.

Fundamentally, the (q_c, q_e) values determined here for n-propanol (or D₂O) are not unique. Rather they represent appropriate values corresponding to the assumptions of q_c = 1 or q_e = 1, and other choices are possible. For n-propanol (D₂O), however, our experiments support the observation of others that q_e/q_c < 1 (Young 1982). Furthermore, the analysis of Vasil'ev and Reiss (1996a,b) and results from the coagulation literature (Fuchs and Sutugin 1965; Graham and Homer 1972; Marlow 1980, 1982; Okuyama et al. 1984; Kennedy 1987; Harris and Kennedy 1988; Gopalakrishnan and Hogan 2011; Ouyang et al. 2012; Mullick et al. 2015) suggest that q_c > 1 is reasonable. Clearly, under equilibrium conditions above a flat surface, q_c must equal q_e, but under non-equilibrium conditions this is not a given.

In the Hertz–Knudsen equation (Equation (1)[1] ), the evaporation rate of condensable species is derived based on the fact that the evaporation and condensation rates are equal under metastable equilibrium conditions, i.e., , and T_d = T. The evaporation coefficient q_e in Equation (1)[1] should be unity when the following conditions are satisfied; (1) the condensation rate is equal to the collision rate on a hard sphere of radius in the absence of an attractive force at S_drop = 1, and (2) the evaporation rate at any S_drop is equal to that at S_drop = 1. Thus, the coefficient q_e indicates deviations from these two conditions and is more complicated to analyze than the coefficient q_c. We therefore avoid any further discussion regarding q_e in this article.

6. Summary and conclusion

We analyzed the growth rates of n-propanol nanodroplets produced in a supersonic nozzle using the results of PTM and SAXS experiments. Working with alcohols has the advantage, relative to experiments with H₂O or D₂O, that liquid droplets are not supercooled with respect to the solid phase. The key result of our analysis of the n-propanol growth rates is very much in line with the D₂O results of Pathak et al. (2013). In particular, the theoretical droplet temperatures and growth rates are inconsistent with the experimentally derived values if we assume q_c = q_e = 1. Good agreement is possible if we assumed either that (i) (q_c, q_e) = (1, 0.6) or (ii) (q_c, q_e) = (1.3, 1.0). Approach (i) is consistent with the observations of Young (1982) but is more difficult to justify from a theoretical standpoint. Approach (ii) can be justified by considering the attractive potential between the condensing vapor molecules and the growing droplets. Since long-range interactions are known to affect the coagulation rate of nanoparticles (Fuchs and Sutugin 1965; Graham and Homer 1972; Marlow 1980, 1982; Okuyama et al. 1984; Kennedy 1987; Harris and Kennedy 1988; Gopalakrishnan and Hogan 2011; Ouyang et al. 2012; Mullick et al. 2015), and condensation is, in a sense, coagulation between a vapor molecule and the droplet, it should not be surprising that long-range interactions can play a role in nanodroplet growth.

Acknowledgments

The authors would like to thank S. Seifert and R. Winans for help with the SAXS experiments.

Funding

Support was provided by NSF grants CHE-1213959 and CHE-1464924. This research used resources of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC02–06CH11357.