Hygroscopic growth models are currently of interest as aids for targeting the deposition of inhaled drug particles in preferred areas of the lung that will maximize their pharmaceutical effect. Mathematical models derived to estimate hygroscopic growth over time have been previously developed but have not been thoroughly validated. For this study, model validation involved a comparison of modeled values to measured values when the growing droplet had reached equilibrium. A second validation process utilized a novel system to measure the growth of a droplet on a microscope coverslip relative to modeled values when the droplet is undergoing the initial rapid growth phase. Various methods currently used to estimate the water activity of the growing droplet, which influences the droplet growth rate, were also compared. Results indicated that a form of the hygroscopic growth model that utilizes coupled-differential equations to estimate droplet diameter and temperature over time was valid throughout droplet growth until it reached its equilibrium size. Accuracy was enhanced with the use of a polynomial expression to estimate water activity relative to the use of a simplified estimate of water activity based on Raoult’s Law. Model accuracy was also improved when constraining the film of salt solution surrounding the dissolving salt core at saturation.

Interest in the properties and behavior of hygroscopic particles in the atmosphere has been ongoing for decades. For example, the physics and chemistry associated with cloud droplet formation have been studied extensively with the text by

The travel time for an inhaled particle to reach the first respiratory bifurcation is approximately 0.2 s under normal breathing conditions (

Deliquescence and hygroscopic growth are considered separate phenomena but necessarily have overlapping characteristics (_{0}, below which this process cannot occur (_{0}, the atmosphere exhibits a water vapor pressure high enough to cause an imbalance in the thermodynamics of the salt-vapor system that initiates salt particle dissolution (_{2}O at 37°C (internal human body temperature). This equates to a mass percent of salt in the solution, _{p}, as:
_{s} is the density of NaCl (2165 kg/m^{3}) and _{w} is the density of water (997 kg/m^{3}). Solving _{p} for NaCl.

Deliquescence of a NaCl particle, therefore, results in a droplet with a completely dissolved core that is almost twice the size of the original dry particle. At _{0} the vapor pressure at the surface of the droplet solution, p_{s,d}, is assumed to be at saturation and in equilibrium with the vapor pressure exerted by the water vapor in the atmosphere, _{v,a}. A high-concentration salt solution exerts a lower vapor pressure than a more dilute solution (_{0} is the relative humidity that results in the most concentrated (at solution saturation) and smallest fully-dissolved salt solution droplet.

Prior research has determined _{0} for various water-soluble salts as well as the subsequent equilibrium size of the salt solution droplet as _{0} (_{0} = 75.7% for pure NaCl in water at 25 °C (_{0} for NaCl is not constant for all conditions, for example _{0} is lowered for multi-component NaCl mixtures (_{0} is raised for extremely small (<100 nm) NaCl particles (

Experimental methods applied to determine _{0} and subsequent growth beyond the minimum droplet size at _{0} frequently involved the use of a tandem differential mobility analyzer (TDMA) consisting of two DMA’s in series to both produce a monodisperse aerosol and measure the droplet size distribution exiting a humid atmosphere while _{0} (

Further growth of the saturated droplet when _{0} is the process that Martin (_{0}. This sudden immersion into a high humidity environment results in a two-stage process (_{s,d} = _{v,a} resulting in an equilibrium droplet diameter as conceptualized in

Numerous papers and texts have described the development of a mathematical model to estimate the growth of hygroscopic compounds over time (_{w} is the molecular weight of water, _{d} is droplet density, _{a} is ambient temperature, and _{d} is droplet temperature. The density of a growing droplet is the density of a diluting salt solution with a constant mass of salt that can be expressed as:
_{R}, can be incorporated into the model given its association with _{v,a} in _{s,w} is the saturation water vapor pressure over a flat surface for a given _{a}.

The computational form of the growth model must account for the effect of the curvature of the droplet, the droplet temperature, and its salt concentration on _{s,d}(_{d}) (_{d} relative to that at _{a} to obtain an equation that relates _{s,d}(_{d}) as a function of the known value of _{s,w}(_{a}) and water activity, _{w}, included to account for droplet salt concentration. In combined form these adjustments resulted in the following expression:
_{v} is the molecular diffusivity of water vapor at _{a}, _{c} is a fraction of the mean free path of water molecules in air, _{c} is the mass accommodation coefficient – the fraction of water vapor molecules hitting the droplet surface that are attached to that surface.

_{p,w} is the specific heat of water, and _{a} given by:
_{m} is the thermal conductivity of humid air (air-vapor mixture), _{t} is the relative thickness of the noncontinuum layer, _{t} is the thermal accommodation coefficient, _{m} is the density of the air-vapor mixture at _{a}, _{p,m} is the specific heat of the air-vapor mixture at _{a}, _{a} is the molecular weight of air.

Note that _{m} is essentially equivalent to _{a}, the thermal conductivity of dry air, and therefore utilizes an equation similar to _{a} substituting for _{m} and _{p,a}, the specific heat of dry air, substituting for _{p,m}. However, both _{m} and _{p,m} are functions of

The _{a}.

The only property of NaCl applied to the hygroscopic growth model is true (absolute) density (_{w}, must also be applied to the growth model (_{s} is the saturation vapor pressure over a salt solution in an open container, _{w} is the mole fraction of water in the solution, _{w} is the number of moles of water in the solution, and _{s} is the number of moles of salt in the solution. Comparing _{w} is the equilibrium relative humidity established over a flat salt solution and is dependent on solution temperature and salt concentration (

However, the relationship between _{s} and _{s,w} is not ideal, and various methods have been employed to account for the discrepancy. _{w}:
_{w} used by other researchers (_{o} (_{s} is the molecular weight of the salt. A reduced form of _{o} is also seen in the literature in which the term, _{o}/_{s}, is eliminated under the assumption that its value is very low relative to the volume of the droplet when the droplet is dilute as _{R} approaches unity (

Raoult’s Law is mathematically simplistic and, therefore, often used in computational fluid dynamic (CFD) models developed to estimate the deposition of hygroscopic particles in the human lung to minimize computation time (_{w} for high salt solution concentrations that occur when a droplet is first forming (

In addition to the use of Raoult’s Law, _{w} has also been calculated for varying droplet salt concentrations as a function of the molal osmotic coefficient, Φ, (_{NaCl} = 2), and _{w}.

A third method for determining _{w}, for the case of a NaCl solution, is to apply a polynomial fit directly to measurements of _{w}. A number of studies report theoretically-derived and measured values of _{w} relative to salt solution molality, or relative to _{w} values published by Robinson and Stokes (_{w} values regressed on ^{2} = 1.000), and therefore was suitable for accurately representing NaCl _{w} over the range of pure water to a salt solution near saturation (

During the core dissolution phase, the dissolving salt core is surrounded by a film of salt solution. The vapor pressure in proximity to the film surface is, therefore, interacting with the atmospheric vapor pressure to induce growth by providing a difference between _{v,a} and _{s,d}(_{d}). Given an available source of solid salt and rapid transfer of dissolved salt from the dissolving core to film surface, it is reasonable to assume that the film consists of a near saturated salt solution. Therefore, during the growth period when _{p} the hygroscopic model should be configured to constrain

The primary objective of this study was to determine the relationship between measured and modeled values during both the transient growth phase of a hygroscopically growing salt particle and its size at equilibrium. A secondary objective was to evaluate the effect of the method used to calculate _{w} on model results. The comparison was conducted with the use of MATLAB and while incorporating either the R&S cubic expression for _{w} (_{w} (

The coupled differential _{w} with either _{w} at its saturation value while _{p}. This level represents a hypothetical upper limit on

For any initial particle size, the model was run over a time period until there was a < 0.1% change in

A novel system was developed to measure hygroscopic growth after a salt particle was instantaneously enveloped by a high humidity atmosphere. Initially, salt was aerosolized by pushing filtered, compressed air at 10 L min^{−1} through a single-jet Collison nebulizer containing a 10% m/v salt solution. Water in the exhausted droplets evaporated as they traveled through a heated brass tube. Excess water vapor was then removed with a condenser. A glass coverslip was then passed through the exiting air stream several times to dust the coverslip with dry aerosolized salt particles.

To measure the hygroscopic growth of particles on a slide, a second system was developed to inject an atmosphere with known temperature and relative humidity onto the surface of the slide while viewing growth with the use of an inverted microscope. As shown in ^{−}1 was directed either through a needle valve or into a glass bottle containing heated water. The air in the bottle passed through a fritter to create bubbles that maximized the transfer of water vapor into the air stream. Relative humidity was adjusted by blending the output from the bottle with air passing through the needle valve. To collect any condensed water vapor before entering the rest of the system, an impinger was placed in-line as a water trap. The combined flow was directed through a 4-way tube fitting. Relative humidity and temperature were measured with a sensor (Model HM70, Vaisala, Helsinki, Finland) every 0.5 s for the duration of the experiment. The probe of the sensor was wrapped with Teflon® tape to create a tight seal and extended down through the upper hole of the fitting and into the center area. The probe was previously calibrated using a two-point calibration procedure in which the probe was tightly sealed in a container containing a saturated KNO_{3} solution and then a saturated NaCl solution, which produces equilibrium RH levels of 75.41% and 94.53%, respectively.

Exhaust air traveled through one of two solenoid valves controlled by a software routine created in LabVIEW (National Instruments, Austin, Texas). The routine was also coded to continuously monitor the RH level measured with the sensor (temperature was noted at intervals by the operator). The solenoid valves were operated simultaneously so that only one was open at any time. During start-up, the open solenoid valve allowed air through the system to exhaust into the atmosphere of the laboratory until the desired RH was consistently measured. The on/off pattern of the solenoids was then switched to direct the high relative humidity air onto the microscope slide via a short tube connected to a custom-made glass tube with a downward curving exit hole. The glass tube exit hole was first pointed away from the slide to fill the tube with the humidified air. The entire switching process was then repeated after pointing the exit hole onto the slide containing the particles.

The slide was placed on an inverted microscope (CKX31, Olympus Corp., Center Valley, PA) set to 400x magnification. A cell phone (SM-G965U1, Samsung Corp., Seoul, South Korea) was mounted via an eyepiece attachment so that its camera could video record particles on a slide. The digital magnification of the cell phone was maximized and the slide was positioned to have a single particle in the video stream at the best clarity. Particle growth was recorded for at least 30 s. After a trial, a video recording was also briefly made of a stage micrometer with scaling every 0.1 mm. Both video recordings were then processed by a video-to-JPG conversion software (DVDVideoSoft, Digital Wave Ltd) to extract and save individual frames of the video. The cell phone was set to video record at the standard capture rate of 30 photos s^{−1} or once every 0.033 s. Micrograph analysis software (Image J) was used to measure the diameter of a particle in photos spanning a 30 s trial. Only the diameters in successive micrographs that revealed a noticeable increase from a previously measured micrograph were recorded in a spreadsheet. The image resolution was 0.1 μm/pixel. Of the particles available to be viewed in a micrograph, one with a clear outline and distant from other particles was measured. The spreadsheet was also used to convert pixel distance to actual distance from their ratio obtained from the extracted micrograph of the stage micrometer. The start of the growth process was the micrograph just before the first frame that revealed any change in the shape or coloration of the measured particle.

An example of micrographs of a growing droplet is given in _{EV}, diameter of a sphere with equal volume) was computed from each measurement by dividing that measurement by the cube root of 2.

At least six trials were conducted by two of the authors independently to ensure the repeatability of the measurement process. Three trials were chosen from each set based on the criteria that the RH during a trial remained within ± 0.3% of the average RH value over the 30-s trial. The six trials were conducted at temperatures between 23–25 °C and RH between 98–100%. As shown in Figure 4 of the

Trial results were then compared to results from the growth model implemented in MATLAB given the trial average RH and temperature, and Φ_{EV} of the dry particle measured. Given that the three-dimensional structure of the dry particle was not measurable, and therefore the initial mass of salt was unknown, the starting mass of the salt was adjusted until the root mean squared error (RMSE) of measured-to-modeled results over the 30 s trial period were minimized. This adjustment, therefore, resulted in a comparison of the growth patterns developed by the measured droplet on the coverslip to that of the modeled values for the same RH and temperature over the entire growth curve, but, in particular, a comparison of the relationship between measured-modeled pairs during the initial fast-growth period.

_{p} and _{w} was fixed at its saturation value. Although essentially linear over this short time period, growth during this time follows a steeply upward curving trajectory of growth established by maximizing _{w} at its minimum value. Droplet temperature during the initial 0.1 s of growth is shown in _{p} after which the IF statement released the constraint on the droplet solution concentration at saturation. That plot also suggests that the addition of

_{w}. Models of KCl water activity were developed in a manner similar to those developed for NaCl (_{w} values than NaCl for any _{s,d}) and therefore a lower computed vapor pressure differential between ambient, _{v,a}, and droplet surface, p_{s,d}, vapor pressures (_{w}. For both salts, the use of Raoult’s Law produces a larger equilibrium diameter than the R&S equation when modelling at high humidity (99.5% RH). As shown in _{w} value than the R&S equation as _{w} is high. A lower _{w} value represents a lower vapor pressure exerted at the droplet surface and, therefore, a greater difference between _{v,a} and p_{s,d}, which results in more growth as it approaches equilibrium. This effect is more pronounced for KCl than for NaCl because Raoult’s Law for KCl is greater than the R&S equation for KCl over a wider range of _{w} is also shown in

_{w} values than does Raoult’s Law for the same

Modeled diameter change (growth ratio), _{0}, using the R&S equation closely compares to a best-fit curve established by _{0} at low RH and overestimate _{0} at high RH (> 95 %). This comparison demonstrates that the relationship between Raoult’s equation and the R&S equation for _{w} discussed in the preceding section is only valid for high RH conditions. As shown in _{0} to an increasingly greater extent as RH approaches _{o}. This result can also be explained by the relationship in model output resulting from the use of the two equations shown in _{w} values relative to _{w} values are low, which are the case for equilibrium at low RH. These higher _{w} values result in the lower predicted growth ratios shown in

A comparison of _{EV} obtained with the use of the video capture method with droplet diameters obtained from the hygroscopic growth model while using the R&S model (

The results provided in _{w} and with a constraint on its saturation value during the deliquescence growth phase is valid when modeling NaCl hygroscopic growth. _{p}. Over the entire growth period and during the initial growth phase, the use of the R&S values to produce _{p} produced the best fit to the measured values (lowest RMSE and highest R^{2}).

Admittedly, the use of Raoult’s law to calculate _{w} (_{w} we purposely simplified the growing droplet solution to that of a binary mixture containing a salt with well-established properties from which the relationship between _{w} and _{w} models without sacrificing computation time (Petters and Kreidenweis 2007;

The application of a constraint on _{p} produced much better associations between measured and modeled results than when left unconstrained. Use of the unconstrained R&S equation produced a low slope relative to unity and high intercept (_{p}, the slopes resulting when the R&S constraint was imposed (0.930) relative to when it was not imposed (1.061) are nearly identical. This suggests that the constraint may be relaxed to a more realistic level near, but not equal to, saturation.

The assumption that the film layer remains near saturation while _{p} may not be true for other salts as it will only occur when the rate of dissolution of the salt core is high relative to the water uptake rate of the growing droplet. Increasing the accuracy of the hygroscopic model by incorporating the kinetics associated with the dissolving salt core is a research topic of interest (

Furthermore, the equations given by Broday and Georgopoulos (2002) followed here to model hygroscopic growth do not consider the thermodynamics associated with a dissolving salt core. Salt dissolution absorbs heat that is otherwise released during condensation (_{0} (

The results presented here are particularly important when using the hygroscopic growth model to determine the growth of particles entering the lungs. Given the fast transit time of a particle from mouth to lung bifurcations, the accuracy of the growth model during the initial growth phase is critical for accurately estimating the eventual deposition of the particle. For particles with hygroscopic properties similar to NaCl, model results indicate that particles larger than 140 nm entirely within a 99.5% RH environment will not reach equilibrium within the 0.2 s transit time to the first bifurcation (

The hygroscopic growth model, as formulated by

This work was supported by the FDA under Grant U01-FD005837; the NIEHS under Grant P30 ES005605; CDC/NIOSH under Grant T42OH008491; and the NIH under Grant U01-HL114494.

Stages of hygroscopic growth of a dry salt particle. The ambient relative humidity, _{w}, and the vapor pressure exerted by droplet solution, _{s,d}, are given in relationship to the minimum _{0}, to achieve deliquescence and the vapor pressure exerted by water vapor in the atmosphere, _{v,a}. Adapted from

(A) Changes in water activity relative to mass percent of NaCl and KCl solutions. Values displayed are those reported by Robinson and Stokes (

Schematic diagram of the hygroscopic growth and video-capture system.

Modeled growth curve (A) of a 1000 nm NaCl particle in 37°C (310.15 K), 99.5% RH atmosphere, and modeled droplet temperature (B). The inset for plot (A) expands the time scale of the larger plot; the open circle indicates the point on the curve related to the time at which the particle has fully dissolved before which its solution concentration was constrained to its saturation value.

Modeled KCl and NaCl growth curves using either Raoult’s equation or Equation (19) based on Robinson and Stokes values (R&S) to calculate water activity. Modeling was performed with a temperature of 37 °C and 99.5% RH. (A) Comparison of entire growth curves developed for both salt types and both water activity equations. (B) Initial growth period when modeling NaCl using both equations and when no constraint is applied to the saturation of the growing droplet.

Comparison of modeled values to measurements made by Tang et al. during deliquescence experiments using 400 nm NaCl particles at 25 °C and the RH noted on the x-axis.

Comparison of modeled and measured values when using the Robinson and Stokes equation to compute water activity. (A) Example of comparison between measured values and model curve – results from one trial. (B) Correlation of all measured-to-modeled pairs for the six trials - dashed line indicates a 1:1 relationship.

Summary of Comparisons Between Modeled and Measured Results for Three Variations of the Water Activity Calculation in the Hygroscopic Growth Model.

RMSE | Slope | Intercept | R^{2} | |
---|---|---|---|---|

Measurements over 30 s | ||||

R&S Constrained | 1.007 | 0.979 | 0.008 | 0.950 |

Raoult’s Law Constrained | 1.392 | 0.977 | −0.374 | 0.923 |

R&S Unconstrained | 1.649 | 0.740 | 3.602 | 0.887 |

Measurements when φ < 1.905 φ_{p} | ||||

R&S Constrained | 0.607 | 0.930 | 0.596 | 0.892 |

Raoult’s Law Constrained | 0.865 | 0.816 | 1.000 | 0.814 |

R&S Unconstrained | 2.247 | 1.061 | 1.438 | 0.718 |