1. Introduction

It is now well accepted that the development of cloud properties and subsequent precipitation processes becomes a primary influence during the early stage of formation. When moist air lifts past the cloud base to continually expand, cool, and generate supersaturation, the cloud condensation nuclei (CCN) activate from the low supersaturation end of the activity spectrum. The nucleated and growing droplet begins to remove water vapor from and to return the generated heat of condensation to the air environment and reduce the rate of supersaturation increase. During increasing supersaturation, more and more CCN nucleate, and the formed droplets grow. The reduction rate of supersaturation eventually catches up to the generation rate to a balance, and at this supersaturation maximum, the CCN activation ceases. There, the initial number density of droplets is decided. The process is thus a nucleation–growth interaction in the thermodynamically changing air environment. A similar interaction takes place in the process of aerosol generation.

In the nucleation–growth interaction above the cloud base, the contribution of the CCN activity spectrum is well recognized (Twomey 1977). In the droplet growth that follows the nucleation, incoming vapor flow toward and outgoing flow of the generated condensation heat from the surface establish a balance. The two balanced counterflows connect the respective values of saturation at the surface. Nevertheless, the processes became known to depend on the efficiency of exchanging both properties at the droplet surface, for heat the thermal accommodation coefficient, α, and for the mass of vapor the condensation coefficient, β (sometimes called the sticking or mass accommodation coefficient). In the early days, such inefficiencies were not considered, and the droplet growth process is now referred to as the Maxwellian (Maxwell 1890), which assumes continuities of the two properties at the droplet surface. The inefficiency appears when the free molecular gas kinetic flows just above the surface come into play as a part of continuum diffusional flows outside. The growth process that involves the two serial flows is called the “diffusion-kinetic” (Smirnov 1971). Early diffusion-kinetic theories involved one of the two flows only; that is, heat or vapor mass flow. When the first analytic solution including diffusion-kinetic flows of both vapor and heat was obtained (Fukuta and Walter 1970), the significant contributions in cloud microphysical processes were realized, such as the nucleation in gaseous-phase homogeneous condensation in particular (Fukuta 2004). The droplet size spectrum spreads by slowing down growth more on the smaller side, and the number density increases to influence cloud lifetime, albedo, and precipitation (Zou and Fukuta 1999).

In the application of the diffusion-kinetic theory for measurement and analysis, the thermal accommodation coefficient used to be assumed as unity. Measurements were difficult and values of the condensation coefficient reported in the last half a century scatter over two orders of magnitude (Fukuta and Walter 1970; Pruppacher and Klett 1978, 1997; Mozurkewich 1986). The condensation coefficient seems to group about two values around unity and 0.04. The problem thus boils down to accurately measure both coefficients in a properly designed experiment.

We have found during the theoretical development that contributions of atmospheric pressure to the condensation and thermal accommodation coefficient terms are different. Two sets of conditions can be used to solve for the two independent accommodation coefficients simultaneously. Measurements were designed and carried out under conditions close to atmospheric cloud formation, selecting the droplet size range and the method of measurement sensitive to distinguish the diffusion-kinetic effect. In the analysis of measured data and comparison with the work of others, it was necessary to clarify the previously neglected effect arising from the shift of steady-state temperature and vapor fields, caused by the movement of growing droplet surface boundary. The attempt for the theoretical derivation was fruitful. It appeared to be the missing link between the two groups of reported coefficients dissolving the gap. The correction was obtained for original data of the present measurement. It even showed a new effect for broadening of droplet size spectrum. It is the purpose of this work to report the result.

2. Theory

For experimental determination of the condensation and thermal accommodation coefficients, the use of a growing or evaporating droplet system is advantageous since all the fluxes of heat and vapor are in the steady-state (or quasi steady state) balance in between the outer boundary of air environment and the inner boundary at the droplet surface.

a. Behavior of the diffusion-kinetic droplet growth equation

In the steady growth of droplets in air to determine α and β, the analytic solution exposes the relationships among all the variables and constants. Direct application of the primitive transport equations without obtaining the solution and adopting them in the curve fitting method tends to miss some hidden relationships. While linearized suitably for natural cloud conditions, the solution of Fukuta and Walter (1970, hereafter FW70) with a minor improvement for treatment of both accommodation coefficients (Fukuta and Xu 1996, hereafter FX96) gives the basis for analysis of droplet growth under supersaturated vapor of the environment.

For a droplet with a radius, a, and sufficiently dilute solution with a negligible Kelvin effect (for notations, see appendix A), FW70 and FX96 give

View Expanded

Equation (1) is now found to be valid in the entire range of a and integrable under the assumed condition. The integral form, with a = 0 at t = 0, takes the form

View Expanded

or

View Expanded

where C₁ and C₂ are constants for the air environment in question; α and β are thermal accommodation and condensation coefficients defined by Knudsen (1934) and included in continuum heat and vapor fluxes at the droplet surface (Chapman and Cowling 1970; FX96); α′ and β′ are representative values of α and β that appear in the free molecular flux density of Hertz–Knudsen equation as well as FW70’s.

In Eqs. (1)–(3), it may be seen that only when a → ∞ (and fictitious condition of α′ = β′ = ∞ or α = β = 2), f_α = f_β = 1, and Eq. (1) approaches to the limit; that is, the Maxwellian with C₂ = 0 in Eq. (3). In Eq. (3), C₂ ≠ 0 describes the diffusion-kinetic. This is to say that both equations take the same parabolic Maxwellian-functional style and they would be indistinguishable if an arbitrary parallel shift were applied in a curve fit analysis. The parallel shift for the diffusion-kinetic is restricted with t = 0, a = 0, which is accompanied by another unreal condition, t = 0, a < 0 (see Fig. 1).

The slowdown effect of the diffusion-kinetic droplet growth rate from the Maxwellian is a function of the droplet size, a, and it is desirable to plan the measurement in the a range where the ratio between Maxwellian and diffusion-kinetic is sufficiently large. Figure 2 shows r and dm/dt ratios between Maxwellian and diffusion-kinetic growths for a case of α′ = 1, β′ = 0.03, at 0°C, 1 atm, and (S − 1) = 0.003. Under these assumed conditions, the ratios are expected to be larger than 2 when a_MW < 4 μm.

The fall velocity of droplet thus growing may be expressed by the Stokes–Cunningham equation

View Expanded

where C₃ is a constant near unity. The equation provides a basis to assess the range of fall distance of the growing droplet to stay within the width of the laser beam of the measurement.

It should be pointed out that the diffusion-kinetic equation used above does not include the Kelvin (or the curvature) and solute effects of the nucleus particle, which produce a single vapor pressure maximum, reduce the vapor pressure difference or the effective (S − 1) between the environment and lower the droplet growth rate. The maximum leads to saturation as the droplet grows. The maximum becomes smaller as the dry mass of the nucleus increases. Through selection of the nucleus mass to keep the maximum sufficiently low and use of (S − 1) in the growth environment sufficiently high, one can find a combination to make the effects essentially negligible on the condition used in the above diffusion-kinetic growth equation. The details of the combination actually applied are given in section 2d.

b. Droplet growth environment

In clouds, a droplet grows in supersaturated air caused by the adiabatic expansion of lifting. In the laboratory, the adiabatic expansion method is apt to cause a damping oscillation in pressure. In addition, supersaturation deviations from the aimed value due to wall disturbances may occur. The thermal diffusion chamber method offers a steady profile of supersaturation, both in static and flow types, if and when both warmer top and cooler bottom plates are securely saturated with respect to water. The temperature field in the chamber establishes linearly with respect to the height. The saturation ratio, S, and supersaturation, (S − 1), are a function of the vapor pressure ratio between the real and saturation values or S = e/e_s. The e_s may be easily determined from the temperature since e_s = f (T). The e profile varies depending on the property assumed in the linear steady-state profile, such as e (Katz and Mirabel 1975) or ρ (Chodes et al. 1974). It is, however, most reasonable to take the linear vapor flux density profile with height or a constant rate of molecular transport in the existing temperature field (Fukuta and Gramada 2003). Between the parallel top and bottom plates of the chamber under the steady state,

View Expanded

and w² ∝ T and n ∝ e/T in the gas kinetics lead to

View Expanded

or

View Expanded

The e value at the midpoint of the chamber estimated based on relationship (7) between the top and bottom plates gives (S − 1) = 0.0032 at 4°C with a temperature difference of 2.5°C. Under the conditions, assumption of linear profiles with height for e and ρ leads to an overestimation of (S − 1) by 6% and an underestimation by 8%, respectively.

c. Method for simultaneous determination of thermal accommodation and condensation coefficients

The thermal accommodation and condensation coefficients in Eqs. (1) and (2) appear combined with terms of different pressure dependence. Measurements of droplet growth under two different pressures, therefore, yield two sets of growth equations to be simultaneously solved for the coefficients. By exposing the pressure dependence, the solutions may be obtained as

View Expanded

and

View Expanded

where

View Expanded

α′ and β′ are representative values used in FW70 and others and are related to the true values, α and β, as shown in Eq. (8). When they are small, both values coincide. The method thus requires experimental determination of both da/dt and a or alternatively t values under two pressures, p₁ and p₂, at constant temperature.

d. The initial method for droplet growth rate estimation

There are three methods commonly used for droplet size determination: the fall velocity measurement in the Stokes regime with analysis by Eq. (4), the electrodynamic levitation method, and the Mie peak identification method of monochromatic light scattering. The fall velocity measurement offers a simultaneous determination of r in Eq. (4), but when droplets are small, the measurement will receive the updraft/downdraft effect from the heat of condensation/evaporation as well as the diffusio- and thermophoretic effect if used in a horizontal thermal diffusion chamber. The electrodynamic levitation method compensates the fall of a droplet to a standstill, but for the large droplet size that is usable in levitation, the diffusion-kinetic effect diminishes. We selected the Mie-peak detection method of light scattering utilizing a He–Ne laser (λ = 0.6328 μm) in a flow-type horizontal thermal diffusion chamber.

Figure 3 shows the computed intensity of scattered light for a He–Ne laser at 90° plotted as a function of droplet radius. Upon an inspection of the figure, it is clear that the interval between two adjacent scattering peaks, Δa(= 0.162 μm), is constant so that by measuring the time elapsed between the two adjacent flashes of light or peaks, Δt, the radial growth rate,

View Expanded

may thus be easily measured. The second value needed in Eqs. (8) and (9), a, may be determined by measuring t, the time elapsed until the moment of (da/dt) measurement, and applying it to Eq. (1).

The conditions used in the present measurement are typically (S − 1) = 3.2 × 10⁻³ and T = 277 K with two naturally occurring CCN compounds, NaCl and (NH₄)₂SO₄, prepared by drying nebulized a 0.1% aqueous solution with an average particle mass at about 10⁻¹⁴ g. Equations used here are without Kelvin effect for the increase of the droplet vapor pressure and the solute effect for depression of the pressure, which tend to overestimate the effective supersaturation to the droplet. With the NaCl nucleus of 10⁻¹⁴ g, the maximum supersaturation of the solution droplet exists at about (S − 1) = 3.9 × 10⁻⁴ and a = 2 μm (Mason 1957). Taking an average value for the entire growth process at 75% of the maximum or effective (S − 1) = 2.9 × 10⁻⁴ with Δa = 0.162 μm, the growing droplet experiences the environmental value less this amount or 3.2 × 10⁻³ − 2.9 × 10⁻⁴ = 2.91 × 10⁻³, to alter β by a factor of 1/0.87. It was found that an increase of Δa to 0.17 μm from 0.162 μm by holding the effective (S − 1) = 2.91 × 10⁻³ would change β by a factor of 0.88. This is to say that [(S − 1), Δa (μm)] = (2.91 × 10⁻³, 0.162), the realistic condition, leads to about the same β with (3.2 × 10⁻³, 0.17). The data analysis was carried out based on the latter. This correction, however, was smaller than the error of measurement as will be shown later.

e. Effect of transitional steady state of a moving boundary in droplet growth kinetics and the correction

During the data analysis of measurements based on the above method and comparison with others, it was realized that an important effect was missing in the contemporary theoretical structure of small droplet growth. The effect is necessary for the steady-state vapor and temperature fields in air surrounding the droplet to adjust to new levels owing to the growth-based movement of the boundary at the droplet surface by transferring the field difference existing between them before and after the growth. Although a complete formulation of the theory is beyond the scope of this paper and will be done elsewhere, the summary derivation, and a corresponding correction factor γ for both α′ and β′, due to the transitional steady state of the moving boundary were obtained (see appendix B) as

View Expanded

where

View Expanded

and

View Expanded

The factor, f_MB, becomes very large as a reduces to and below about the 1-μm level. When the data were obtained by the method described in section 2c, the vapor flux or da/dt becomes a known constant. Since f_MB affects the outer continuum vapor field only, the factor and β in the free molecular process at the droplet surface, both affect the measured vapor flux or da/dt. Based on this effect, the data of the initial measurement were reinterpreted for β and similarly for α.

3. Experimental apparatus and measurement

A double-decked, low-height, flow-type thermal diffusion chamber was constructed for the measurement (see Fig. 4). The measuring device consisted of a saturation cell followed by a diffusion cell. The walls of the cells were constructed of 0.67-cm (1/4″) copper plates for strength under low pressure operation and uniform temperature distribution. The saturation cell was 0.5 cm high, 7.6 cm wide, and 40.6 cm long. The bottom plate of the cell was covered with fiberglass filter paper, which was adhered by Dow Corning RTV Sealant silicone glue and treated with heat for degassing. It was wetted with water during the tests. The glass fiber filter paper and silicone glue combination was chosen because of the inertness of glass and to secure adhesion between them and copper plates under steam cleaning, heat treatment for degassing, and prolonged wet conditions. The saturation cell was held at the same temperature as the bottom plate of the diffusion cell. The air entered into the diffusion cell and established the steady-state profile in about 1 s. The airflow rate was typically 300 mL s⁻¹ with the airspeed 1.2 cm s⁻¹ in the middle of the cell.

The diffusion cell was the same dimensions as the saturation cell except that the effective height was 0.8 cm. Both the top and bottom plates were covered with glass fiber filter paper that held the water. While the rest of the system was constructed of copper, the sidewalls of the diffusion cell were fabricated from polycarbonate to provide visibility in the cell and insulation between the plates comprising the bottom and top of the cell. A 0.3-cm-diameter port covered with optical glass was placed in the end of the cell to allow a laser beam (Oriel Helium-Neon Laser 79245) to traverse the length of the cell. A second port 3.8 cm long and 0.3 cm high with a glass cover was cut in the middle of the sidewall centered at 30.5 cm from the end of the cell to obtain droplet observation in the cell. Silicone rubber “O” ring seals were used to make the cells accessible. Ports were also provided to allow the introduction of water to the top and bottom of the diffusion cell and the bottom of the saturation cell, with outlet ports to remove excess water. The system was tilted at an angle of approximately 1° to ensure water would flow down the fiberglass filter paper, keeping it wet. The water supply and removal ports were sealed during the tests at reduced pressure. The top plate temperature was controlled by a Neslab Endocal Model ULT-80 low temperature bath circulator. The bottom plate and the saturation cell temperature control utilized a Forma Scientific Model 2160 bath and circulator. The temperature of the cell was measured with copper-constantan thermocouples in ports drilled in the middle of upper and lower plates. These were located 25 cm from the end of the cell. The thermocouple readings were taken using a Yokogawa Model LR421OE recorder. Prior to use, the entire system was steam cleaned for approximately 1 h, followed by heating to ∼80°C on a hotplate over a weekend with a slow flow of air passing through the system.

The introduction of nucleus samples was made through an 18-gauge (0.125 cm O.D., approximately 0.076 cm I.D.) hypodermic needle that had a hole drilled in the side and the end plugged. The needle port was located at 20.3 cm from the inlet of the cell. The needle was positioned so that the hole was in the center of the cell and the laser beam. With extreme care, sample introduction provided a thin stream of nuclei in low concentration, and droplets were not influencing the growth of adjacent ones to allow the identification of individual droplets. The introduction was accomplished in two ways: Nuclei were produced using a small solution nebulizer blowing into a 250 mL Erlenmyer flask that held clean, dry air. For the room pressure tests, samples of the flask contents were taken with a 10-mL syringe and very slowly introduced into the system through the needle. An alternative of this method was to let the nucleus sample be sucked into the chamber air under slight negative pressure. The sample air was estimated to reach the supersaturation of the surrounding air in about 2 × 10⁻² s. For the low pressure tests, the flask containing the suspended nuclei was connected to the introduction needle by a small silicon rubber tube that had a variable restrictor in the line as well as an open or closed valve. The pressure in the flask was reduced somewhat and the restrictor adjusted to allow a very small leakage into the needle. The use of the valve enabled a very small amount of sample to be injected by the difference in pressure between the system and the sample flask by quickly opening and closing the valve, usually much less than one second during this process. The flashing points of red laser light by the droplet growth were recorded using a Sony Model No. DC-TRV-18 digital camera recorder and analyzed using Pinnacle Studio DV, Version 7 software.

The air supply to the system was prepared by simply passing the room air through an electrostatic precipitator to remove the existing particles after treatment of the carrier room air by a liquid nitrogen trap, bubbling through an oxidizer bath followed by electrostatic particle removal or use of nitrogen gas from the liquid reservoir. No recognizable difference in condensation coefficient was observed from that measured in air treated through the electrostatic particle precipitator alone. In a separate estimation, we found that there did not exist a sufficient amount of adsorbable organic molecules to form a monolayer on the surface of a formed droplet for growth impediment (Fuentes et al. 2000). Airflow was obtained using a dry vane vacuum pump and controlled by needle valves placed after the system for the atmospheric pressure runs and before the system for the reduced pressure tests.

Deionized water was used in the tests for keeping the filter paper wet and for making solutions. The water production system had a charcoal trap followed by a reverse osmosis system, which, in turn, was followed by mixed ion exchange resin beds. All tubing used in the system was Teflon or silicon, which gave no volatiles and contained no plasticizers.

Analysis of the data was done by observing the position, distance, and time elapsed between two adjacent Mie scattering peaks or flashes for a given particle. The time between the two flashes gave Δa = 0.162 μm. The airflow rate was given by the distance to the middle of two flashes and the time between them. The distance to the middle of the two flash positions gave the average time from nuclei injection [cf. Eq. (2)] using the flow speed to convert distance to time. The average droplet radius when measured was ∼2 μm. The temperature at the center of the cell was approximately 277 K, with a temperature difference across the cell of ∼2.5°C. Particles introduced at the top of the beam would fall a maximum of ∼2 mm during their traverse from the injection point to the end of the observation. They were kept within the fall distance limit by controlling the flow speed.

4. Results of measurement and discussion

In the course of the preliminary test, under some conditions that we never pursued nor identified, the scattered beams from the He–Ne laser were observed colored other than the original red. The reason for this was not known to us. It appeared that possibly harmonics of the laser light occurred in the scattering. The original red laser beam did scatter at the position of the measurement.

b. The moving boundary effect of droplet growth and data comparison

The theoretical development of the moving boundary effect of droplet growth happened to be, as mentioned earlier, posterior to the present measurement. It appears to close the last large remaining gap in the growth kinetics of droplet and to provide the missing link among the existing results of measurements. There are two typical experimental conditions: One has the moving boundary effect attached to the main diffusion-kinetic process expressed in Eq. (B7). This condition represents the growth process with no expansion or continuous expansion like the growing cloud droplets in the updraft. The continuous expansion process holds steady-state growth by continuous generation of supersaturation within the cell, the space claimed by a droplet, until the supersaturation generation rate becomes overcome by the increased removal rate with larger droplets and the steady-state field collapses. The other is the growth under instantaneous expansion in which the established boundary of the cell permits neither mass nor heat flux to cross, and the process also does not allow the continuous generation of supersaturation. Failing to establish diffusion-kinetic growth in the instantaneous expansion process, (1 + f_MB) of the no-expansion process in Eq. (B7) goes to f_MB.

Figure 5 summarizes the moving boundary effect in two typical cases for no expansion and instantaneous expansion in the ratio

View Expanded

In the figure, the moving boundary effect increases R_MW for (S − 1) from about a = 10⁻⁵ to 10⁻² cm. For both no expansion and instantaneous expansion cases, when (S − 1) ≥ 0.3, the ratio approaches and exceeds the Maxwellian growth rate ratio, R_MW = 1 before falling back. This effect is clearly visible in the unsuspected expansion experiment of Vietti and Schuster (1973) in comparison with FW70 under very high supersaturations. It is important to point out that the moving boundary effect on the diffusion-kinetic process, due to the growth rate increase, thus tends to give a deceptive impression as if the growth behavior were truly Maxwellian. For (S − 1) = 0.0032, the value taken in the present measurement, the R_MW increase from the diffusion-kinetic process is sufficiently small. However, for the upper limit of natural cloud processes, say (S − 1) ≈ 0.032, the deviation becomes quite recognizable. This is likely to enhance the broadening effect of the droplet size spectrum by increasing the growth rate on the larger side, in addition to another broadening effect due to the growth rate reduction of the diffusion-kinetic theory on the smaller end, the latter being the original contention of FW70 as mentioned in Hagen et al. (1989). It may enhance early formation of large droplets to let the collision–coalescence process proceed toward raindrop generation or graupel and hail processes, especially in the convective clouds of high updraft.

From the above viewpoint of the moving boundary effect riding on the diffusion-kinetic process, Fig. 5 also suggests the experimental condition most suitable for α and β determination; the moving boundary effect is sufficiently suppressed while a well-established diffusion-kinetic process prevails. This is to say that measurement in the no-expansion condition like a thermal diffusion chamber with small (S − 1), the moving boundary effect and droplet size for enhanced diffusion-kinetic effect, still being outside the Kelvin and solute effects, should be advantageous over the others to avoid a large correction.

In comparison of the present result with others, the common demographic approach in the values of α and β does not solve the problem and is even misleading. Reported data are influenced by the method of measurement as well as the theory applied in the analysis. Their choice is crucial to the comparison. As far as the measurement methods are concerned, the one that employs droplet growth has a distinct advantage with measurable macroscopic environmental properties, which are continuously connected to the droplet surface, the place where α and β effects appear. For this reason and the process being a direct reproduction of what happens in the clouds, here data of droplet growth-based measurements are selected for comparison. For the theory of data analysis, almost all of the past measurements applied the diffusion-kinetic theory of FW70 and FX96, obviously without consideration of the moving boundary effect. The moving boundary effect is hence the key point for the comparison, and the latter in turn may test the effect. The selected data of measurements are categorized into three groups based on the type of chambers or the method used as discussed below.

1) Expansion chamber

The chamber was used in two different modes—instantaneous expansion and continuous expansion. In the instantaneous expansion mode of operation applying the Mie scattering method of the He–Ne laser for droplet size determination under S = 1.49–3.45, Vietti and Schuster (1973) obtained β′ = 0.0065 with the α′ = 1 assumption.² A very large positive deviation of droplet radius from that of the diffusion-kinetic theory was observed, which later switched into a negative. The reported β′ value is therefore questionable but is consistent and explainable with the behavior of the moving boundary effect under high (S − 1) in Fig. 5.

The instantaneous expansion mode was also used in the recent work of Winkler et al. (2004), under the condition of T = 250–290 K, S = 1.3–1.5 with droplet size determination of the Mie scattering of laser and data analysis based on a curve-fitting method for simultaneous determination of α′ and β′ under pressure modulation. The Maxwellian behavior of α′ = β′ = 1 was reported. Whereas, the measured data were showing da/dt ≠ 0 at t = 0, the diffusion-kinetic behavior, as shown in Fig. 1. The high (S − 1) of the measurement most likely placed the condition in the zone close to R_MW = 1 for moving boundary effect in the instantaneous expansion mode (see Fig. 5).

The results of extensive expansion chamber measurement in the continuous expansion mode were reported by Hagen et al. (1989) using the Mie scattering method with an argon continuous wave (CW) laser for the droplet size detection with the starting T = 16°C, cooling rate of 10°C min⁻¹ corresponding to 17 m s⁻¹ cloud-base updraft, which would reach nominal (S − 1) = 0.3 in 30 s in our estimate. The reported maximum supersaturation was around 5% or 6%, and droplets were measured to a = 12–15 μm. Droplet growth was found to be more rapid during the first test, possibly indicating impurity accumulation in the following test. The main result is that, under the α′ = 1 assumption, β′ was found to decrease in the neighborhood of 0.01 from a value near unity at the outset of the test. Nevertheless, with a careful examination of the reported data, one can find a clear increase of β′ with a size in the range below a = 2–5 μm. Furthermore, for the size range, data points may be seen crammed at the β′ = 1 level, an indication of growth exceeding the Maxwellian behavior. These behaviors are unexplainable with the purity or lack of accumulation of contamination in the early stage of the growth. They are, however, compatible with the moving boundary effect of the diffusion-kinetic theory.

In Fig. 5, the upward deviation of droplet growth by the moving boundary effect from the diffusion-kinetic line, with α′ and β′ of our present measurement, is shown by iso-(S − 1) lines and other diffusion-kinetic lines with larger α′ and β′ (not shown) lying above. The maxima of the moving boundary effect for no expansion that correspond to a = 2–5 μm may be found with the (S − 1) = 32% line and can be seen in a portion of the line exceeding R_MW = 1 or the Maxwellian. The zone of (S − 1) = 5%–6% should exist close to this. Under continuous expansion and increasing (S − 1) before the maximum, a droplet grows with the moving boundary effect and passes through the iso-(S − 1) lines to approach the Maxwellian and give a β′ increase if interpreted by the diffusion-kinetic theory alone and vice versa as it grows past the maximum. This tendency always exists with growth along (S − 1) = const.

We conclude that, while some contamination could have existed in this work to reduce the β′ value in the later stage of measurement, β′ near unity at the beginning of the growth and the steady decrease thereafter toward 0.01 are the deceptive results caused by data reduction with diffusion-kinetic theory alone without the moving boundary effect. The decrease may also be attributable to the loss of the supersaturated vapor to the wall during the rather prolonged measurement.

2) Thermal diffusion chamber

Three operational modes exist with the thermal diffusion chamber; the static mode and the vertical, as well as the horizontal, flow mode. Chodes et al. (1974) made use of the thermal diffusion chamber in the static mode at T ≈ 20°C, (S − 1) = 0.0040–0.0075, with the droplet size directly measured by the fall velocity in the chopped light beam, to report β′ = 0.033 on average. The released latent heat of condensation during growth of the droplet could have slowed down the fall velocity to give a slight smaller β′. Phoretic effects were not considered but are expected to be small. The moving boundary effect to increase the growth rate and therefore β′ is also expected to be small due to small (S − 1). The reported data should thus be regarded close to the true value.

Sinnarwalla et al. (1975), on the other hand, applied the chamber vertically with flow at (S − 1) = 0.005–0.01, T = 20°C, and photographic determination of the fall velocity and the size, to obtain β′ = 0.026 under the α′ = 1 assumption. Constraints here are basically the same as those of Chodes et al. (1974), and the reported β′ can be considered sufficiently close to reality. The present work is an example of α′ and β′ measurements that make use of a horizontal flow thermal diffusion chamber.

3) Electrodynamic levitation

The electrodynamic levitation method uses a charged particle, few tens of micrometers in diameter, which happened to be in the zone of low sensitivity for the diffusion-kinetic effect (cf. Fig. 2) and also low moving boundary effect (cf. Fig. 5), suspended in the quadrupole, and through the fall behavior with balancing voltage, to measure growth or evaporation. Sageev et al. (1986) measured the growth rate of an aqueous (NH₄)₂SO₄ solution drop of diameter between 10 and 20 μm to report α′ ≈ 1. At T = −34.9°C, Shaw and Lamb (1999) could determine the possible range for α′ = 0.1–1 and β′ = 0.04–0.1 applying the pressure modulation method for evaporation of a = 20 μm drop. While the accuracy of these measurements is low, the results are generally in line with those of the thermal diffusion chamber method.

5. Conclusions

A new droplet growth theory of moving boundary effect with second-order (S − 1) dependence was derived and found to play an important role as a “missing link” between the two controversial groups of data for α, β measurements. The theory explained and closed the gap. In turn, the data confirmed validity for the moving boundary effect of the theory. The high (S − 1) of expansion chamber methods, not the rapid renewal of water surface, was found to enhance the moving boundary effect for faster growth and lead to deceptive interpretation toward larger α, β. The theory provided the basis for bettering the measurement, correction for data analyzed without the moving boundary effect. It suggests a new broadening effect in the droplet size spectrum by accelerating the growth on the larger end together with the diffusion-kinetic effect of growth slowdown on the smaller, toward onset of the collision–coalescence process and precipitation development.

Measurements employing a flow-type horizontal thermal diffusion chamber and the Mie scattering method using a He–Ne laser at T = 277 K, (S − 1) = 0.32%, a ≈ 2 μm under two pressures of modulation, p = 100 and 860 hPa, with correction of the moving boundary effect for nuclei of NaCl and (NH₄)₂S0₄, resulted in the simultaneous determination of α = 0.81 and β = 0.043 on average. For paper smoke, α = 0.68 and β = 0.022.

Comparison with other data of low moving boundary effect suggests a slight β increase with temperature lowering. Based on the result of the present measurement and other theoretical and experimental works, the mechanism of β being much smaller than α is suggested mostly as a kinetic process controlled by the removal rate of the released latent heat at the moment of molecular impact at the surface for β and without the process for α. An identical situation applies to evaporation.