ACPAtmospheric Chemistry and PhysicsACPAtmos. Chem. Phys.1680-7324Copernicus PublicationsGöttingen, Germany10.5194/acp-18-14939-2018A thermodynamic description for the hygroscopic growth of atmospheric aerosol particlesRefined Köhler theoryCastarèdeDimitrihttps://orcid.org/0000-0002-2812-6401ThomsonErik S.erik.thomson@chem.gu.sehttps://orcid.org/0000-0003-2428-7539Department of Chemistry and Molecular Biology, Atmospheric Science, University of Gothenburg, Gothenburg, Swedenformerly at: Observatoire Midi-Pyrenees, University of Toulouse (Paul Sabatier, Toulouse III), FranceErik S. Thomson (erik.thomson@chem.gu.se)17October2018182014939149488May201827June201826September20188October2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://acp.copernicus.org/articles/18/14939/2018/acp-18-14939-2018.htmlThe full text article is available as a PDF file from https://acp.copernicus.org/articles/18/14939/2018/acp-18-14939-2018.pdf
The phase state of atmospheric particulate is important to atmospheric
processes, and aerosol radiative forcing remains a large uncertainty in
climate predictions. That said, precise atmospheric phase behavior is
difficult to quantify and observations have shown that “precondensation” of
water below predicted saturation values can occur. We propose a revised
approach to understanding the transition from solid soluble particles to
liquid droplets, typically described as cloud condensation nucleation – a
process that is traditionally captured by Köhler theory, which describes
a modified equilibrium saturation vapor pressure due to (i) mixing entropy
(Raoult's law) and (ii) droplet geometry (Kelvin effect). Given that
observations of precondensation are not predicted by Köhler theory, we
devise a more complete model that includes interfacial forces giving rise to
predeliquescence, i.e., the formation of a brine layer wetting a salt
particle at relative humidities well below the deliquescence point.
Introduction
The role of aerosols in the radiative budget of the planet is a source of
large uncertainty in climate modeling and prediction . One
significant source of uncertainty comes from inadequate understanding of
aerosol phase state in the atmosphere . The phase
behavior of atmospheric particles depends on both the environmental
conditions (pressure, temperature, humidity, etc.) and the particle
properties (Fig. ).
Soluble particles are ubiquitous in the atmosphere, with primary
sources including biomass burning and the sea surface illustrated here. Solid
salt particles, which can be directly emitted or form when solution droplets
evaporate and effloresce, provide hygroscopic surfaces for cloud condensation
nucleation.
The phase state influences surface as well as bulk phase chemistry, cloud
forming potential, particle deposition, and other aspects of the global water
and bio- and geochemical cycles. Thus fundamental knowledge of aerosol particle
phase behavior is a key aspect to understanding and modeling atmospheric
processes. For example, mixed phase clouds in the Arctic demonstrate a
surprising persistence that is not predicted by the current understanding of
ice-water saturation vapor pressure gradients and the resulting competition
for H2O (Wegener–Bergeron–Findeisen process)
.
For soluble atmospheric particulate, Köhler theory is generally used to
quantify and parameterize atmospheric phase state .
Köhler theory describes the equilibrium size of solution droplets in the
atmosphere as determined by the saturation vapor pressure, which depends upon
component mixing (Raoult's law) and the curvature of
the interface (Kelvin or Gibbs–Thomson effect) .
For a fully soluble particle, implicit in the theory is a sudden transition
from a dry (solid) particle to a saturated droplet. The relative humidity at
the transition point is referred to as the deliquescence relative humidity
(DRH). The Köhler model is useful because it provides a simple physical
and mathematical description of nucleated condensation and can be modified to
include compounds of limited solubility . As such it has
remained the tool of choice in atmospheric models
, although many measurements suggest that a
precondensation of water even on pure soluble surfaces may occur below the
DRH
.
For pure compounds, such observations are not predicted and thus hint that
Köhler theory can be refined.
Here we suggest a theoretical refinement by considering the stability of a
salt particle that is gradually engulfed and dissolved by a brine layer
(Fig. ). The model includes the understood bulk phase
equilibria established by Köhler theory but invokes a transition region
where metastable liquid layers exist on particle surfaces. The general
shortcoming of Köhler theory for single component systems is the
activated transition from dry particle to liquid droplet and the exclusion of
an interfacial system that reduces the global free energy. The theory we
propose herein evolves from previous explorations of wetting of soluble
surfaces , where previous work has used bulk
thermodynamics and ascribed disjoining pressures (also modeled using bulk
properties or fitting parameters) to explore the stability of thin films on
solvating particles
. Other studies
have used similar theoretical models to treat particle interfaces in the
presence of substrate surfaces , or for terrestrial
systems like brine crusts where geometric effects can be excluded
. In this treatment we include terms to explicitly
account for the intermolecular origins of the interfacial forces in a manner
akin to surface melting .
Schematic of the system considered herein – an idealized solvating
atmospheric particle. The use of an idealized spherical geometry is justified
by previous findings that NaCl crystal corner and step sites dissolve
preferentially to surface sites, in a manner that quickly roughens and rounds
faceted crystals .
Thus we extend previous work to combine and define entropic,
geometric, and explicit intermolecular interactions that result in modified equilibrium
vapor pressures above particle surfaces. We do so in a manner wherein
achievable analytic solutions help to illuminate the underlying
physicochemical processes that control the system and may be used to benefit
modeling at a range of scales. The implications of this work are far-reaching
given that within the current state of understanding the existence of
atmospheric aqueous surfaces is limited to deliquesced materials and/or
coated insoluble particles. The model we propose herein suggests that aqueous
surface layers may exist in some range of relative humidity (RH) below the
DRH and thus resemble liquid droplet behavior for some processes and
detection techniques.
Refinement of Köhler theory
Köhler theory considers and balances the effects on droplet stability due
to the interfacial curvature and soluble components
. Thus the equilibrium vapor
pressure pr at temperature T above a brine droplet of radius r is
described by its departure from the analogous vapor pressure at the flat,
pure water interface pb∞ at the same temperature,
prpb∞=psolpb∞⋅prpsol,
where psol represents the equilibrium vapor pressure over the flat brine
solution surface, also at temperature T. In detail the geometric term in
Eq. () is given by the Kelvin equation,
prpsol=exp2σvsolRgTr,
where the surface free energy of the liquid–gas interface σ≡σ(ρsol) is a function of the bulk composition of the solution
droplet as expressed by the solution density ρsol, Rg is the
ideal gas constant, and vsol is the molar volume of the solution.
Likewise the modified equilibrium due to the addition of soluble components
is given by the water activity,
psolpb∞=aw,
where a theoretical model (e.g., Van't Hoff – ; E-AIM
– ) or empirical parameterization can be
chosen to best match the system of interest.
Thus from the combination of these effects the DRH is implicitly given when r=RDRH at the radius of the deliquesced droplet,
DRH100=awexp2σvsolRgTRDRH.
This traditional Köhler formulation () for a fully soluble
particle predicts the deliquescence point and, when r is allowed to evolve
above RDRH, the equilibrium relative humidity above the
evolving brine droplet. However, in this formulation the sudden transition
from a solid to liquid particle remains implicit.
Although the transition from solid to liquid solution may proceed quickly, it
is important to consider whether or not there exists an intermediate state(s)
of importance. Thus we consider whether a particle may be wetted by a thin
film and what, if any, stability conditions may (de-)stabilize such a system.
Such thin films are a common phenomenon and are generally treated within the
rubric of adsorption or wetting , which are incorporated into applications in many fields from biology to surfactant
physics .
Reformulating atmospheric particle dissolution as an interfacial problem
leads to the system postulated in Fig. , where a charged
soluble particle is engulfed and dissolved by a brine layer. However, in
order for the interfacial system to endure it must yield some energetic
benefit that can be captured by minimizing the global free energy of the
system. Thus in addition to the bulk free energies tacit in Köhler theory
(Eqs. –) an interfacial contribution ζ(l)
must be included. For thin films, Derjaguin–Landau–Verwey–Overbeek (DLVO)
theory, which assesses the balance between short- and long-range
intermolecular interactions, has been used to great success
.
In the case of an ionic electrolyte a surface charge results from the
differential solubility of ions , while the solvated
ions affect both the mixing entropy and the electric potential of the film.
The ions in the brine layer are organized to offset the surface charge in a
manner described by linearized Poisson–Boltzmann theory, where the
characteristic falloff of the electric field is a Debye length κ-1=ϵϵokbT/e2NAρsol1/2, where ϵ0 is the vacuum
permittivity, ϵ is the relative permittivity of the brine, kb
is the Boltzmann constant, T is the absolute temperature, e is the
elementary charge, and NA is Avogadro's constant. Thus the contribution to
the free energy is
Felec(d)=2qs2κϵϵ0e-κl,
where qs is the surface charge and l is the thickness of the liquid
layer. The colligative effect of the ions remains unchanged as in
Eq. ().
The long-range dipole fluctuations in the system are volume–volume
interactions and can be depicted to first order as the nonretarded
dispersion or van der Waals forces. Assuming that retardation is not
important the van der Waals contribution to the free energy can be expressed
as
Fdisp(d)=-Ah12πl2,
where the Hamaker constant Ah is determined for a given layered system
. This formulation assumes a planar geometry, which given
the relative scales of the layering versus the system size is taken as
accurate to first order . For small nanoparticles other
considerations including the entire particle volume may become important, as
we discuss later.
The combined effects of the short- and long-range interactions captured by Felec(l) and Fdisp(l) yield a total interaction
potential,
ζ(l)=Fdisp(l)+Felec(l),
whose derivative with respect to l, ζ′(l) represents the interfacial
contribution to the free energy, and whose sign and strength will depend upon
the material properties and geometry of the specific system.
To re-express Köhler theory including the effect of these intermolecular
interactions we define an equilibrium vapor pressure over the curved brine
surface pζ that includes the interfacial term, and thus in analogy
to Eq. (),
pζpb∞=psolpb∞⋅prpsol⋅pζpr.
Assuming that the thermodynamic equilibrium condition at the solution–vapor interface is set by the free exchange of water molecules
∂G∂Nv=∂G∂Nl and
that near deliquescence temperature is constant, an expression for
pζ can be calculated by considering the difference in chemical
potentials between the solvated and layered states,
pζ=prexpvsolζ′(l)RgT,
where the details of the derivation as presented in Hansen-Goos et
al. (2014, hereafter, HG14) are included in the
Supplement for completeness. Thus Eq. () can be
rewritten as
pζpb∞=RH100=awexp2σvsolRgTr′expvsolζ′(l)RgT,
where
r′<RDRH→r′=Rs+l-lβ,r′≥RDRH→r′=r,
Thus at r′<RDRH the system size evolves like
Rs+l-lβ, which captures the change in radius due to
condensation and dissolution, where a linear solvation is assumed. The exact
solution considering the volume–volume equivalence of dissolution requires
numerically solving a third-degree polynomial and yields a
negligible correction factor. The growth factor at DRH is β=RDRH/Rs, where the dry particle radius is
Rs. At r′≥RDRH the entire particle is dissolved
and thus r is the solution droplet's radius and the theory re-converges to
classical Köhler theory as ζ′(l) vanishes.
Equation () is a general result describing the equilibrium vapor
pressure over a dissolving salt particle, from the dry particle state to the
totally dissolved state. Although herein we treat an idealized monovalent
electrolyte system using modified DLVO theory to constrain the functional
behavior of ζ′(l), natural systems may require more complex treatments
that would likely yield a host of interesting behavior, and simultaneously
strain the ability to achieve analytical and/or computational solutions.
Applying refined Köhler theory
It is instructive to use the refined Köhler formulation (Eq. ) to
model a NaCl particle, as this might represent an idealized marine aerosol.
Although considerable information concerning bulk salt solutions is available
it is difficult to assess the applicability of these values to thin brines.
For example, the assumptions that the interfacial brine layer is a saturated
solution whose thickness is controlled by electrostatic interactions may not
be self-consistent. The ion availability within a saturated NaCl brine will
allow efficient charge screening, and thus a very short Debye length should
result. That said, the uncertain theoretical parameters (qs, Ah,
κ-1, and brine concentration C) can also be used as fitting
parameters in order to illuminate the range of possible physical behavior.
Here for an idealized case we choose to apply Eq. () to a sodium
chloride particle of a representative atmospheric diameter ≃0.8µm
in the accumulation mode . The growth
factor is assumed such that the particle will lead to a solvated brine
droplet of radius 1.36 µm, and for consistency with previous work we
choose a temperature of 20 ∘C and a saturated concentration of
[NaCl]sat=5.4molL-1, which is also used to
calculate the Debye length. The value for surface charge qs=0.12Cm-2
is taken from , while the HG14 value Ah=-1.5×10-20 is used, and the expression for the water activity of a NaCl
solution is taken from . The result is illustrated in
Fig. , where both the classical Köhler behavior and the
refined model of predeliquescent hygroscopic growth are captured. With the
refined interfacial model the DRH is captured (≈75 % RH) as reported
in many previous studies , but
near the deliquescence transition (Fig. ) a wetted
interface is predicted below DRH when considering the balance of the
intermolecular interactions.
Model of the growth of the solvating surface of a 0.8 µm
accumulation mode NaCl particle as outlined in the text. Below deliquescence
(≲75 % RH and and in the inset, zoomed plot) the metastable brine layer behavior is
governed as in Eq. () – red curve. Above deliquescence the
layer thickness l becomes the radius r of the growing solution droplet as
is captured by classical Köhler theory
(Eqs. – and Eq. for r′=r,
where the interfacial term has vanished – blue curve).
The result suggests that observations of precondensation in the existing
literature may also be explained by a
metastable interfacial equilibrium. In Fig. the theoretical
model is compared to measurement data from and
. Each of the solid lines in Fig. correspond
to nonlinear least-squares solutions to fit the data, where the identified
fitting parameters are presented in Table .
Fitting parameters yielding curves in Fig. , where superscripted
letters correspond to figure panels.
For physical consistency with the theoretical framework the best-fit
solutions have been calculated excluding data points that preclude a full
monolayer of water (l≤0.3nm) and excluding the data points that
represent the fully solvated particles (growth factor ≥1.2 in
Fig. ). Although sub-monolayer water adsorption is observed
, implicit in this theory is a bulk solution layer of
uniform thickness. In all cases the data are well represented within a narrow
window of the fitting parameters and the best-fit solutions also serve to
demonstrate the impact of the parameters on the shapes of the curves.
Furthermore, the best-fit solutions agree well with values extracted from the
literature, listed in the first row of Table and used
previously to construct Fig. , where their sources are
referenced.
Measurement data points of NaCl particle hygroscopic growth under
incrementally increasing RH. Colors correspond to different initial dry
particle radii indicated in the legend and the growth factor (GF) is the
ratio of the measured radius to the dry radius (GF-1 scaling is used to
suitably present data on logarithmic scales). (a) Lines represent theoretical
fits to data presented by . (b) Theoretical fits to
data presented by . In all cases fitting parameters are
presented in Table . Given the experimentally prescribed
increasing humidities, particles that grow beyond the activation barrier set
by the strong Kelvin effect (cf. Fig. ) are observed at
RH exceeding the equilibrium value.
For small particles (Rs≲5nm) the Kelvin term is strong
enough to compete with the intermolecular forces and thus the system retains
an activation barrier until DRH or above , as shown in
Figs. –. In practice a small dry
particle would be subjected to reversible uptake of water due to
intermolecular attraction until it suddenly dissolves into a brine droplet
when the deliquescence activation barrier is overcome. However, at short
length scales the veracity of the bulk approximations and several other
simplifying assumptions of this model must be questioned. For example, the
high brine layer concentrations predicted for small particles may be
indicative of the model limitations. Although the model represents the data
remarkably well even where it might be expected to fail, in those cases it
might be better to approach the activation problem in terms of adsorption
theory or using molecular dynamics simulations
.
The evolution of the individual terms in Eq. () are
shown for a solvating 5 nm NaCl particle. For such small particles the Kelvin
term dominates, yielding an activation barrier, as illustrated by the
inflection in the layer growth (magenta) curve. One also observes that the
Raoult term (yellow curve) changes only after the solid particle dissolves
(RDRH) and that the intermolecular interactions are very short
range (black curve).
The refinement of Köhler theory we have proposed yields a smooth
metastable transition from solid- to aqueous-phase atmospheric particles. It
captures observed behavior for specific compounds, yet remains general such
that its application to more complex systems may yield deeper understandings
of aerosol phase state and particle behavior.
Discussion
Real particles in the atmosphere tend to be more complicated than idealized
theories of solvation can easily capture. Atmospheric particulates that
assist nucleation are often internally mixed and include varying quantities
of soluble/insoluble, organic/inorganic materials etc.
. Furthermore, theoretical adaptations of
Köhler theory have been used to capture particle mixing state; for
example “modified Köhler theory” and
“κ Köhler theory” . However, these remain
limited to predicting critical supersaturations and droplet evolution. Our
contribution is general in that it predicts the complete evolution from the
dry particle through a metastable equilibrium characterized by a growing thin film, to the fully solvated droplet. Such films are not only consistent with droplet deliquescence but
also with previous observations of water absorption and ionic mobility at RH
far below the DRH (35%≤RH≤DRH)
. The implication is that given the
correct intermolecular force balance, the surface of any
soluble-material-containing atmospheric particle may “predeliquesce” and
thereby contribute to an as yet unquantified aqueous reservoir.
The formulation of continuous dry particle dissolution and droplet growth as
represented by Eq. () and presented in the figures has several
advantages over previous treatments of such systems. First we treat the
intermolecular interactions explicitly in order to minimize the use of bulk
parameters to model the interfacial system. The interfacial free energy
minimization is then carried out and incorporated into Köhler theory as a
simple additional term that continues to allow for analytical solutions. The
approach is in contrast to other treatments of analogous systems that utilize
ascribed or phenomenological descriptions of short-range interactions
and/or do not account for the full particle geometry
and thus the atmospheric context. The model has the additional benefit of
highlighting why, in practice, deliquescence is often observed to be an
abrupt transition. The competition between the Kelvin term and the
intermolecular forces results in an activation barrier (seen as the
inflection points in Figs. and ),
which when exceeded leaves a solvating particle in a highly supersaturated
environment. As a result the particle grows suddenly until it reaches the new
completely solvated equilibrium. Thus, there may be implications for
nonequilibrium particle growth in the atmosphere.
Unfortunately, approximations remain inherent in the model, for example the
fitted concentrations of the brine layers increase with decreasing particle
size. These predicted equilibrium concentrations may seem physically
unrealistic for very small particles, which could be due to the geometric
limitations of our model – the dispersion forces are derived assuming
interactions between flat, parallel interfaces. But it is also possible that
the model is compensating for other physical effects not taken into account
in this study (such as surface depletion), effects that could lower the
chemical potential of the liquid phase for very small sizes. Applying the
model to more complex systems will also yield hurdles and likely make further
approximations necessary but, as previously stated, may also lead to deeper
insight. This model may allow some assessment of the relative importance of
the short- versus long-range interactions and which quantities limit surface
phase behavior. However, for mixtures and other materials each term of
Eq. () would need to be re-evaluated. If bulk parameters that
feed into the Köhler behavior (e.g., surface energy, water activity) are
poorly constrained, strict physical interpretations will remain challenging.
Given the importance of aerosol phase state there may also be significant
implications to even a limited RH range where stable aqueous interfacial
films exist. Most obvious is the significance for contributions to aqueous
phase chemistry . However, there are also potential cloud-
and climate-scale impacts that deserve some investigation. The radiative
absorption cross section of predeliquescing particles may significantly
change their optical properties, as has been shown for aerosol–particle
mixtures , and especially for particles that include soot
. There are also implications for understanding mixed
phase cloud stability given that the equilibrium vapor pressure above a
predeliquescence layer can be much lower than the analogous vapor pressure
above a liquid droplet (Fig. ).
Schematic of the equilibrium vapor pressures over droplets,
predeliquesced particles, and ice particles, where gradients lead to
Wegener–Bergeron–Findeisen processing. The range of equilibrium vapor
pressures above predeliquesced thin films straddle the equilibrium vapor
pressure over ice pice, resulting in a more stable coexistence between
ice and predeliquesced particles, relative to liquid droplets.
Mixed phase clouds are inherently unstable given that air saturated with
respect to water is supersaturated with respect to ice, and in fact most
precipitation globally originates from mixed phase processes
. However, in the Arctic and sub-Arctic
regions particularly, the unexplained persistence of mixed phase clouds has consequential
climate impacts . Simply put, the equilibrium vapor
gradient innate between supercooled liquid droplets and ice crystals is
greatly diminished if droplets are replaced by predeliquesced particles
(Fig. ), while to some observational techniques the two
morphologies may be indistinguishable. A potential result is slower-growing
ice crystals and thus a longer lifetime for mixed phase clouds. Although a
complete understanding of the mixed phase must also involve particle dynamics
, predeliquescing particles may play a contributing role
and recent studies suggest that there exist more sources for dry soluble
particles than previously thought .
Conclusions
This study has introduced a refinement of Köhler theory that tracks a
soluble atmospheric particle from its dry state to the solvated droplet
equilibrium. The model presumes that from molecular scale adsorption to the
growth of thin liquid films, the interface can be stabilized by the
intermolecular interactions in a system. Although the details of real
atmospheric systems would be subject to strict bookkeeping, even the highly
simplified model proposed here captures many important parameters, like
equilibrium vapor pressure and liquid layer thickness, that could contribute
to better parameterizations for aerosol–cloud interaction modeling efforts.
All data utilized herein were extracted from the cited references.
The supplement related to this article is available online at: https://doi.org/10.5194/acp-18-14939-2018-supplement.
The authors have contributed equally to this work.
The authors declare that they have no conflict of interest.
Acknowledgements
This research was supported by the Swedish Research Council VR, the Swedish
Research Council FORMAS, the Nordic Top-Level Research Initiative CRAICC, and
the European Commission for ERASMUS mobility. Hendrik Hansen-Goos, Markus Petters,
Sarah Petters, Fabian Mahrt, and Robert McGraw are thanked for
helpful discussions. Special thanks to Merete Bilde whose contributions
significantly helped to improve the paper.
Edited by: Ashu Dastoor
Reviewed by: two anonymous referees
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