Equilibrium Vapor Research Articles

A Simple Method for Triple Stable Isotope Analysis of Cellulose, Sugar, and Bulk Organic Matter-Advances and Limitations.

Determining several isotope ratios in one analysis multiplies the information that can be retrieved from a sample in a cost-efficient way. The stable isotope ratios of hydrogen (δ2H), carbon (δ13C), and oxygen (δ18O) in organic compounds are highly relevant due to their complimentary hydroclimatic and physiological signals. Different types of organic material reflect different processes and integration times, like short term in leaf sugars and long term in tree ring cellulose, but currently, no simple method exists for their triple isotope analysis. Here, we present a method that enables the isotopic analyses of the three elements H, C, and O in one run and is applicable to different types of carbohydrates and bulk organic matter. We discuss all steps required from water vapor equilibration necessary for obtaining reliable δ2H values of carbon-bound H to high-temperature conversion (HTC) of the sample to CO and H2 and to the mass-spectrometric isotope-ratio analysis. We show that reliable triple isotope analysis is possible for a large range of samples, although it results in some reduction of precision compared to individual isotope analysis. Important considerations are the equilibration procedure, the type of autosampler, selection of HTC reactor, the influence of nitrogen in the sample, the verification of δ13C values obtained by HTC versus combustion, and the selection of reference materials. By presenting a relatively simple triple-isotope method, we promote the use of multi-isotope studies in environmental sciences, which helps in addressing many important climate and ecological research challenges that we face today.

Study on the effect of pore structure changes induced by freeze-thaw-wetting-drying cycles on water retention characteristics of compacted loess

Deterioration of earthworks (including compacted soil collapse and loss of suction) as a result of climate change has had a significant negative impact on engineering. To explore the water retention characteristics and microstructural evolution of loess after freeze-thaw-wetting-drying (FTWD) cycles. In this paper, the soil water characteristic curve (SWCC) of unsaturated loess in the full suction range is obtained by using the filter paper method (FPM) and vapor equilibrium technique (VET). Scanning electron microscopy (SEM) and mercury intrusion porosimetry (MIP) were used to test the two-dimensional images and pore size distribution (PSD) curves of the specimens. The results of the study showed that the SWCC of compacted loess before and after FTWD cycles showed an unimodal pattern. Compacted loess has inter-aggregate pores and intra-aggregate pores, and the PSD curves show a bimodal pattern. The FTWD cyclic action reduces the water retention capacity of loess in the low to medium suction range (0kPa-288kPa). Inter-aggregate pore porosity decreases with an increasing number of FTWD cycles. Finally, to easily predict the SWCC of loess under the condition of pore variation, a prediction model was established in the double-porosity media framework, considering the changes of inter-aggregate pores and intra-aggregate pores. The rationality and applicability of the present model were verified by experimental data. The present model can reasonably predict the water retention properties of compacted clays with unimodal or bimodal type SWCC under microscopic pore changes, which is suitable for the engineering application of compacted clays.

Strength and dilatancy of an unsaturated expansive soil at high suction levels

A series of suction-controlled triaxial tests was conducted on Nanyang expansive clay to investigate the effects of dry density and suction on dilatancy and strength. The suction of the soil samples was controlled using a vapour equilibrium technique, with four suction levels ranging from 3.29 MPa to 198.14 MPa, where water retention is dominated by adsorption. The experimental results show that the tested soil exhibits a brittle failure mode under high suction, significantly distinguishing the hydro-mechanical behaviour of the soil at high suction from that observed at low suction. This brittle failure mode significantly increases the contribution of suction to peak strength compared to residual strength, causes the soil to fail before reaching the critical state, a phenomenon not observed in soils under high suction, and results in dilatancy caused by damage to the soil particle aggregates rather than particle rearrangement. The dilatancy data obtained from the triaxial tests reveal that significant soil dilatancy occurs during shear after reaching peak strength, with the maximum dilatancy angle increasing with suction and decreasing with confining pressure. However, the initial dry density has a negligible impact on the soil's dilatancy under high suction levels. This observation further supports that, for unsaturated soils under high suction levels, dilatancy is attributed to damage to soil particle aggregates rather than the rearrangement of soil particles.

Experimental Investigation of Post-Dryout Heat Transfer with R-134a at High Pressures

An experimental study of post-dryout (PDO) heat transfer in the coolant R-134a was performed in a vertical round tube with upward flow. Experiments were conducted at high pressures from 28.4 bars up to 39.8 bars, corresponding to a reduced pressure of 0.7 to 0.98, respectively. Mass flux was varied in the interval of 300 to 1500 kg/m2∙s, and overall equilibrium vapor quality was between −1.88 and 4.89. Depending on the settings of the experimental parameters, the heat flux was varied from around 11 to 100 kW/m2. The uniformly heated tube had an inside diameter of 10 mm and a heated length of 3000 mm. In total, more than 10 000 PDO data points were obtained. In the PDO region, the wall temperature distributions had similar behavior across the pressure range. At the occurrence of dryout, the wall temperature suddenly increased until it reached a peak. For higher mass flux, the wall temperature decreased after reaching the peak, followed by a second temperature increase with a lower slope. For lower mass flux, the wall temperature kept increasing after the dryout point. The temperature peak after the dryout point was smaller at higher pressure, while this effect was even stronger near the critical pressure. Likewise, the vapor quality corresponding to the first peak shifted to even lower values with increasing pressure. Furthermore, it was found that at increasing pressure and at increasing mass flux, the dryout location and the total temperature distribution shifted toward lower vapor qualities. In addition, several PDO correlations were assessed, and a new correlation for high-pressure conditions was developed and compared with the results of the existing ones.

Ice Formation, Exsolution, and Multiphase Equilibria in the Methane–Ethane–Nitrogen System at Titan Surface Conditions

Titan is unique among the icy satellites in that it has a thick atmosphere, stable surficial bodies of liquid, and a precipitation system that promotes interactions between the two. Although Titan’s surface conditions are typically assumed to be above the freezing point temperatures of the major constituent species of the climate system (methane, ethane, and nitrogen), conditions may be sufficiently cool across parts of Titan to allow for ice formation alongside known liquid-vapor phases. In this study, we used Raman spectroscopy, visual inspection, and the CRYOCHEM 2.0 equation of state to map the appearance of first ice and to quantify the amount of nitrogen dissolution into liquid in the methane–ethane–nitrogen system along a 1.5 bar isobaric cooling path in the temperature range 80–95 K. This was with the intent of (1) determining the effects nitrogen has on the phase behaviors of the methane–ethane binary system, and (2) establishing the temperatures and ternary mixing ratios needed for ice formation on Titan’s surface. We found that ethane-rich mixtures enter a three-phase solid–liquid–vapor equilibrium and are characterized by nitrogen-rich exsolution upon freezing and ice that form at the bottom of the sample. With sufficient methane content, the mixtures cross a univariant four-phase solid–liquid–liquid–vapor boundary, which contributes to a distinct isothermal freezing point profile and ice that forms starting at the liquid–liquid interface. Our results generally agree with findings from previous studies of the methane–ethane–nitrogen system and are intended to add to our current understanding of Titan’s geochemical processes.

Drying model and drying characteristic analysis of multiphase porous medium for flake materials

The main purpose of this paper is to optimize the drying process and gain a fundamental understanding of the movement of different phases during the drying process of cigarette flakes. Therefore, the drying model of a multiphase porous medium for cigarette flakes has been established in this paper. Binary diffusion of water vapor and gas pressure transport, as well as capillary diffusion of liquid water and gas pressure transport, are considered in this model. The non-equilibrium equations are used to calculate the evaporation rate, and the fluxes of water vapor and liquid water are also obtained. The results indicated that the stochastic stacking model is suitable for studying cigarette flake drying. Because water evaporation absorbs more heat than hot air, the temperature of the cigarette flakes decreases during the drying process. The water activity, vapor density, and vapor mole fraction of the medium layer of cigarette flakes were significantly higher than those of the upper and lower layers. The highest water vapor and liquid water fluxes occur in the medium of the drying process. Additionally, it was observed that the saturated vapor pressure was higher than the equilibrium vapor pressure and steam pressure, non-equilibrium evaporation occurs in the medium of drying. The correlation between the evaporation rate and the drying temperature and humidity of the cigarette flakes was found to be positive.

Vaporisation thermodynamic studies over and biphasic regions of U-Sn system using high temperature mass spectrometry

High-temperature vaporisation thermodynamic studies over <U3Sn7(cr) + USn2(cr)> and <USn2(cr) + USn(cr)> two-phase regions were carried out by employing Knudsen Effusion Mass Spectrometry (KEMS) in the temperature range 1148–1465 and 1222–1471 K, respectively. Sn(g) was the only species observed in the mass spectra of the equilibrium vapour phase over both the biphasic regions. The partial pressure-temperature relations of Sn(g) were deduced aslog(pSn/Pa) = (−15,737 ± 73)/(T/K)) + (9.52 ± 0.06) (1148–1465 K) <U3Sn7(cr) + USn2(cr)> andlog(pSn/Pa) = (−16,151 ± 67)/(T/K)) + (9.70 ± 0.05) (1222–1471 K) <USn2(cr) + USn(cr)>Using p-T relations, the enthalpies of the following heterogeneous reaction equilibria were evaluated by the second law method: U3Sn7(cr) = 3USn2(cr) + Sn(g) and USn2(cr) = USn(cr) + Sn(g). Subsequently, the Gibbs energies of the formation of U3Sn7(cr) and USn2(cr) were derived. Knudsen effusion mass spectrometric studies over these two biphasic regions are being reported for the first time.

Molecular kinetic modelling of non-equilibrium evaporative flows

Recent years have seen the emergence of new technologies that exploit nanoscale evaporation, ranging from nanoporous membranes for distillation to evaporative cooling in electronics. Despite the increasing depth of fundamental knowledge, there is still a lack of simulation tools capable of capturing the underlying non-equilibrium liquid–vapour phase changes that are critical to these and other such technologies. This work presents a molecular kinetic theory model capable of describing the entire flow field, i.e. the liquid and vapour phases and their interface, while striking a balance between accuracy and computational efficiency. In particular, unlike previous kinetic models based on the isothermal assumption, the proposed model can capture the temperature variations that occur during the evaporation process, yet does not require the computational resources of more complicated mean-field kinetic approaches. We assess the present kinetic model in three test cases: liquid–vapour equilibrium, evaporation into near-vacuum condition, and evaporation into vapour. The results agree well with benchmark solutions, while reducing the simulation time by almost two orders of magnitude on average in the cases studied. The results therefore suggest that this work is a stepping stone towards the development of an accurate and efficient computational approach to optimising the next generation of nanotechnologies based on nanoscale evaporation.

Equilibrium vapor pressure of aqueous sodium acetate and potassium acetate solutions used as working fluids in frost-free air–source heat pumps at 263–328 K

Aqueous solutions of sodium acetate (CH3COONa) and potassium acetate (CH3COOK) are considered effective alternative heat exchange media for frost-free air–source heat pumps (FFASHPs) owing to the low corrosiveness, low viscosity, and low cost. Equilibrium vapor pressure is the most crucial property that characterizes freezing point and latent heat transfer. However, studies on this property, especially in subzero temperature ranges are scarce. This study was conducted to experimentally investigate the equilibrium vapor pressure of CH3COONa and CH3COOK solutions. The developed apparatus and procedures were based on the static method, and validated by evaluating the vapor pressure of distilled water, n-heptane, and calcium chloride (CaCl2) aqueous solution. Their average absolute deviations were within 1.94%. As the temperature increased from 263 to 328 K and solute concentration increased from 9.27 to 33.81 wt%, 124 data points of the vapor pressure of the CH3COONa and CH3COOK solutions were obtained, ranging from 0.2759 to 13.2608 kPa. Modified Antoine equation and ion interaction (Pitzer) model were established for correlation of the experimental data. The average absolute deviations of the CH3COONa and CH3COOK solutions produced by Antoine equation were 2.08 and 2.48%, respectively. Those produced by Pitzer model further decreased to 1.36 and 1.45% due to the import of osmotic coefficient and the accuracy improvement. Furthermore, according to the experimental and calculation results, the vapor pressure of the CH3COONa solution was lower than that of the CH3COOK solution. Therefore, under the same antifreezing conditions, the CH3COONa solution facilitates latent heat absorption from ambient air, while the CH3COOK solution is conducive to achieve regeneration. The results of this study provide foundational data for the vapor pressure of the two solutions and can promote their application in FFASHPs.

Thermal effects on the strain rate-dependent behavior of highly compacted GMZ01 bentonite

Investigation of thermal effects on the strain rate-dependent properties of compacted bentonite is crucial for the long-term safety assessment of deep geological repository for disposal of high-level radioactive waste. In the present work, cylindrical GMZ01 bentonite specimens were compacted with suction-controlled by the vapor equilibrium technique. Then, a series of temperature- and suction-controlled stepwise constant rate strain (CRS) tests was performed and the rate-dependent compressibility behavior of the highly compacted GMZ01 bentonite was investigated. The plastic compressibility parameter λ, the elastic compressibility parameter κ, the yield stress p0, as well as the viscous parameter α were determined. Results indicate that λ, κ and α decrease and p0 increases as suction increases. Upon heating, parameters λ, α and p0 decrease. It is also found that p0 increases linearly with increasing CRS in a double-logarithm coordinate. Based on the experimental results, a viscosity parameter α(s, T) was fitted to capture the effects of suction s and temperature T on the relationship between yield stress and strain rate. Then, an elastic-thermo-viscoplastic model for unsaturated soils was developed to describe the thermal effects on the rate-dependent behavior of highly compacted GMZ01 bentonite. Validation showed that the calculated results agreed well to the measured ones.

Impact of solvent dry down, phase change, vehicle pH and slowly reversible keratin binding on skin penetration of cosmetic relevant compounds: II. Solids

We extended a mechanistic, physics-based framework of the dry down process, previously developed for liquids and electrolytes, to solids and coded it into the latest UB/UC/P&G skin permeation model, herein renamed DigiSkin. The framework accounts for the phase change of the permeant from dissolved in a solvent (liquid) to precipitated on the skin surface (solid). The evaporation rate for the solid is reduced due to lower vapor pressure for the solid state versus subcooled liquid. These vapor pressures may differ by two orders of magnitude. The solid may gradually redissolve and penetrate the skin. The framework was tested by simulating the in vitro human skin permeation of the 38 cosmetically relevant solid compounds reported by Hewitt et al., J. Appl. Toxicol. 2019, 1–13. The more detailed handling of the evaporation process greatly improved DigiSkin evaporation predictions (r2 = 0.89). Further, we developed a model reliability prediction score classification using diverse protein reactivity data and identified that 15 of 38 compounds are out of model scope. Dermal delivery predictions for the remaining chemicals have excellent agreement with experimental data. The analysis highlighted the sensitivity of water solubility and equilibrium vapor pressure values on the DigiSkin predictions outcomes influencing agreement with the experimental observations.

Effect of calcium sources on enzyme-induced carbonate precipitation to solidify desert aeolian sand

Enzyme-induced carbonate precipitation (EICP) is a promising technique for soil reinforcement. To select a suitable calcium source and a suitable solution amount for aeolian sand stabilization using EICP, specimens treated with different solution amounts (1.5, 2, 2.5, 3, and 3.5 L/m2). Surface strength, crust thickness, calcium carbonate content (CCC) and water vapor adsorption tests were performed to evaluate the effect of two calcium sources (calcium acetate and calcium chloride) on aeolian sand solidification. The plant suitability of solidified sand was investigated by the sea buckthorn growth test. The suitable calcium source was then used for the laboratory wind tunnel test and the field test to examine the erosion resistance of solidified sand. The results demonstrated that Ca(CH3COO)2-treated specimens exhibited higher strength than CaCl2-treated specimens at the same EICP solution amount, and the water vapor equilibrium adsorption mass of Ca(CH3COO)2-treated specimens was less, indicating that Ca(CH3COO)2-solidified sand was more effective and had better long-term stability. In addition, plants grown in Ca(CH3COO)2-treated sand had greater seedling emergence percentage and higher average height, which indicated that calcium acetate is a more suitable calcium source for EICP treatment. Furthermore, the surface strength and crust thickness of solidified sand increased with increasing the solution amount. For sand treated with 3 L/m2 of solution, the excessive strength and thickness of the crust made plants growth difficult, and the performance of sand treated with more than 2 L/m2 of solution significantly improved. Thus, the solution amount of 2–3 L/m2 is suggested for engineering applications. The sand solidified using EICP in the field could effectively mitigate wind erosion and facilitate the growth of native plants. Therefore, EICP can be combined with vegetative method to achieve long-term wind erosion control in the future.

Critical state behaviour of an unsaturated kaolin mixture

In this paper, a series of suction-controlled triaxial tests were conducted to investigate the hydromechanical behaviour of an unsaturated kaolin mixture. The studied variables included the mean net stress, suction, and initial dry density. The suction was controlled over a wide range – in the span of 0–390 MPa using the axis translation method and the vapour equilibrium technique, aiming to holistically study the mechanical response of unsaturated soils. The discussion focused on the critical state behaviour and dilatancy characteristics of the tested soil. The experimental results indicate that the ductility of the soil decreases and shifts towards brittleness with increasing suctions. The critical state stress ratio (Ma) of the soil exhibits a significant increase with suction, and this increase becomes more pronounced with higher initial dry density. Conversely, the impact of suction on the critical state line in the specific volume-mean net stress plane (v-lnpnet plane) diminishes with higher initial dry density. Furthermore, suction notably influences the stress-dilatancy relationship of kaolin mixtures, while the effect of confining pressure appears to be marginal. The influence of suction on the stress-dilatancy relationship primarily stems from its impact on the critical stress ratio (Ma), which can be normalized by dividing the stress ratio by the critical state stress ratio. Lastly, the stress-dilatancy equation of the modified Cam-clay model is extended to capture the effect of suction on the dilatancy characteristics of unsaturated soils.

Cubic-two-state equation of state for modeling liquid-liquid and vapor-liquid equilibria of nitriles + hydrocarbons or water mixtures

The cubic-two-state (CTS) equation of state (EoS) was employed for modeling the liquid-liquid and vapor-liquid equilibria of nitriles + hydrocarbons (alkanes and aromatics) or water mixtures. For nitriles, the five pure compound parameters were determined by fitting the model simultaneously to experimental saturation pressure, density of saturated liquid, enthalpy of vaporization and liquid-liquid equilibrium data when mixed with n-dodecane. For alkanes, the pure parameters were those determined from conventional corresponding states correlations for Soave-Redlich-Kwong EoS. The van der Waals mixing and combining rules were applied for dispersive parameter and covolume, with a temperature-dependent binary parameter. In the case of nitrile mixtures with aromatics and water, cross-association were considered with the previously published CTS combining rules, and no association binary parameters were required. The correlations were very good in all cases. The model was also able to predict selective extraction of sulfur compounds (benzothyophene and benzothyophene 1,1-oxide) from alkanes. Because of its simplicity, therefore, the CTS EoS is a very attractive option for process simulation involving this class of mixtures.

Challenges in determining individual chemical potentials of system elements by Gibbs energy minimization

Accounting for the elusive gas phase in thermodynamic equilibrium calculations is demonstrated to be important for achieving accurate and unique determination of constituent element chemical potentials, when Gibbs energy minimization (GEM) encounter indeterminacy due to rank deficiency in mass-balance equations. The molar GEM of the equilibrium vapor phase offers a robust solution for unambiguously and accurately determining element chemical potentials. To enable this, a novel optimization method for mass-balance equation is presented, paving the way for reliable thermodynamic calculations.

Novel working fluid pair of methanol/betaine-urea for absorption refrigeration system driven by low-temperature heat sources

An absorption refrigeration system (ARS) designed for low-temperature heat sources like industrial waste heat, solar-heated water, or geothermal heat can offer substantial energy-saving benefits. This study explores a novel working fluid pair comprising methanol and betaine-urea (BE-UR), forming a deep eutectic solvent (DES) system. This design capitalizes on methanol's high enthalpy of vaporization, the low heat capacity of the working pair, and the appropriate intensity and strength of the hydrogen bonding network within the solution. The proposed fluid pair shows full solubility of BE-UR in methanol at a mole ratio of 1.81 to 1 (methanol to BE-UR), with an equilibrium vapor pressure dropping to 8.59 kPa at 303.15 K. Molecular dynamics simulations unveil a microscale solution structure, showcasing a hydrogen bond network and unbonded methanol in the concentrated working fluid pair. This configuration enhances the absorption process of methanol vapor in the ARS absorption unit. Aspen Plus simulations predict a coefficient of performance (COP) exceeding 0.85, comparable to H2O/LiBr in single effect-ARS and outperforming NH3/H2O in SE-ARS by approximately 30 %, while maintaining a lower cycle ratio of about 2.

The equilibrium vapor pressures of ammonia and oxygen ices at outer solar system temperatures

Few laboratory studies have investigated the vapor pressures of the volatiles that may be present as ices in the outer solar system; even fewer studies have investigated these species at the temperatures and pressures suitable to the surfaces of icy bodies in the Saturnian and Uranian systems (<100 K, <10−9 bar). This study adds to the work of Grundy et al. (2024) in extending the known equilibrium vapor pressures of outer solar system ices through laboratory investigations at very low temperatures. Our experiments with ammonia and oxygen ices provide new thermodynamic models for these species’ respective enthalpies of sublimation. We find that ammonia ice, and to a lesser degree oxygen ice, are stable at higher temperatures than extrapolations in previous literature have predicted. Our results show that these ices should be retained over longer periods of time than previous extrapolations would predict, and a greater amount of these solids is required to support observation in exospheres of airless bodies in the outer solar system.

Vapor equilibrium models of accreting rocky planets demonstrate direct core growth by pebble accretion

The gaseous envelope of an accreting rocky planet becomes hot enough to sublimate silicates and other refractory minerals. For this work, we studied the effect of the resulting envelope enrichment with a heavy vapor species on the composition and temperature of the envelope. For simplification, we used the gas-phase molecule SiO to represent the sublimation of silicate material. We solved the equilibrium structure equations in 1D for planets in the mass range of 0.1 to 3 M⊕. The convective stability criterion was extended to take the stabilizing effect of the condensation of SiO clouds into account. We assumed that the envelope is both in hydrostatic equilibrium and in vapor equilibrium with the underlying magma ocean. This means that pebbles do not undergo sublimation in the envelope and therefore survive until they plunge into the magma ocean. We find that the emergence of an inner radiative region, where SiO condensation suppresses convection, increases the pressure and temperature in the inner envelope compared to pure H2/He envelopes once Mpl ≳ 0.3 M⊕. For Mpl &gt; 0.75 M⊕, the temperature and pressure close to the surface reach the supercritical point of SiO. The amount of SiO stored in the envelope is lower than the total planet mass for low mass planets. However, for Mpl &gt; 2.0 M⊕, all accreted pebble material must contribute to maintain the vapor equilibrium in the envelope. Therefore, the non-vapor mass of the planet ceases to increase beyond this threshold. Overall, our vapor equilibrium model of the planetary envelope allows for direct core growth by pebble accretion up to much higher masses than previously thought.

Soil warming increases the number of growing bacterial taxa but not their growth rates.

Soil microorganisms control the fate of soil organic carbon. Warming may accelerate their activities putting large carbon stocks at risk of decomposition. Existing knowledge about microbial responses to warming is based on community-level measurements, leaving the underlying mechanisms unexplored and hindering predictions. In a long-term soil warming experiment in a Subarctic grassland, we investigated how active populations of bacteria and archaea responded to elevated soil temperatures (+6°C) and the influence of plant roots, by measuring taxon-specific growth rates using quantitative stable isotope probing and 18O water vapor equilibration. Contrary to prior assumptions, increased community growth was associated with a greater number of active bacterial taxa rather than generally faster-growing populations. We also found that root presence enhanced bacterial growth at ambient temperatures but not at elevated temperatures, indicating a shift in plant-microbe interactions. Our results, thus, reveal a mechanism of how soil bacteria respond to warming that cannot be inferred from community-level measurements.

An Approximate Criterion for Morphological Transformations in Small Vapor Grown Ice Crystals

Abstract Observations and measurements show that crystals remain relatively compact at low ice supersaturations, but become increasingly hollowed and complex as the ice supersaturation rises. Prior measurements at temperatures &gt;−25°C indicate that the transition from compact, solid ice to morphologically complex crystals occurs when the excess vapor density exceeds a threshold value of about 0.05 g m−3. A comparable threshold is not available at low temperatures. A temperature-dependent criterion for the excess vapor density threshold (Δρthr) that defines morphological transformations to complex ice is derived from laboratory measurements of vapor grown ice at temperatures below −40°C. This criterion depends on the difference between the equilibrium vapor density of liquid () and ice (ρei) multiplied by a measurement-determined constant, . The new criterion is consistent with prior laboratory measurements, theoretical estimates, and it reproduces the classical result of about 0.05 g m−3 above −25°C. Since Δρthr defines the excess vapor density above which crystals transition to a morphologically complex (lower density) growth mode, we can estimate the critical supersaturation (scrit) for step nucleation during vapor growth. The derived values of scrit are consistent with previous measurements at temperatures above −20°C. No direct measurements of scrit are available for temperatures below −40°C; however, our derived values suggest some measurement-based estimates may be too high while estimates from molecular dynamics simulations may be too low.