The composition of atmospheric aerosol particles has been found to influence
their micro-physical properties and their interaction with water vapour in
the atmosphere. Core–shell models have been used to investigate the
relationship between composition, viscosity and equilibration timescales.
These models have traditionally relied on the Fickian laws of diffusion with
no explicit account of non-ideal interactions. We introduce the
Maxwell–Stefan diffusion framework as an alternative method, which explicitly
accounts for non-ideal interactions through activity coefficients.

Aerosol particles are an uncertain component of the Earth's atmosphere,
interacting directly by scattering and absorbing radiation and indirectly by
acting as nuclei for the formation of cloud droplets and ice crystals

These core–shell models have relied upon Fickian laws of diffusion to
simulate the mixing of compounds through individual particles. Fickian
diffusion frameworks have played an important role in the investigation of
mixing in glassy aerosol particles, where viscosity has been used as a proxy
for phase state changes between a liquid and an ultra-viscous particle

Direct measurements of diffusion coefficients rely on single-particle
equilibration timescales and find the Fickian diffusion coefficient in
binary mixtures

Mixing rules provide mutual diffusion coefficients in mixtures and describe
the relationship between diffusion and concentration. Diffusion coefficients
of pure substances act as limiting values in these functions at the extremes
of concentration and are referred to as self-diffusion coefficients. Many
different mixing rules have been suggested, from a simple constant
relationship

The limited database of diffusion coefficients and viscosities has restricted
the testing of Fickian diffusion models. Although the Fickian framework has
been used to successfully model simple binary mixtures,

The Maxwell–Stefan diffusion equation differs from the Fickian case as mixing
is driven by a gradient in chemical potential. The Maxwell–Stefan equation is
given by

The aim of this study is to investigate the effect of solubility on diffusion
timescales, by comparing both Fickian and Maxwell–Stefan model simulations.
Binary mixtures of a representative organic compound and water are used to
investigate the sensitivities of these models to both self-diffusion
coefficient and solubility at room temperature. Self-diffusion coefficients
are investigated in the range of

The numerical diffusion frameworks are solved for the spherically symmetric
shell model shown in Fig.

The model has a moving boundary, which allows growth and shrinkage of the
particle depending on the ambient conditions by assuming that equilibrium
relative humidity equals the liquid water mole fraction in the outer aerosol
shell. By assuming that the outer aerosol shell is in equilibration with the
ambient relative humidity, the study focuses solely on the effect
non-ideality has on the rate of condensed phase diffusion and equilibration
timescales. We appreciate that the assumption that the surface layer is
always in equilibrium with the ambient relative humidity may not always be
valid in the case of viscous aerosol particles at low temperatures and
relative humidities. Laboratory evidence has suggested that amorphous
particles could efficiently absorb water into the particle bulk under certain
conditions, in contrast to glassy particles, where water vapour may be
limited to surface adsorption

To ensure that numerical effects did not accelerate the rate of diffusion, a maximum shell width is defined. When the radius of the particle grows or shrinks, the number of shells increases or decreases and the volume of the outer shell can vary depending upon the number of moles it contains. More details about the moving boundary can be found in the Appendix.

The aerosol shell model, where diffusion equations are solved
numerically to find concentration flux across the shell boundaries. The flux
across the outer shell is set to zero, denoted as

By assuming that concentration depends only on time and the radial component,
Eq. (

The Maxwell–Stefan law of diffusion from Eq. (

In our experiments the organic component is assumed to be non-volatile and
only water enters and leaves the aerosol particle. Water is added and removed
from the particle at the end of each time step, which allows us to assume
that the flux at the shell boundary is zero during the diffusion step. The
UNIFAC model is a semi-empirical system that uses both the interactions
between functional groups found in the molecules and binary interaction
coefficients to calculate the activity coefficients of each component within
a solution

The Fickian and Maxwell–Stefan diffusion coefficients are both functions of concentration, temperature, pressure and composition. For higher-order systems, the diffusion coefficient describing each component in a mixed solvent is required. This is mathematically expressed as a matrix of individual binary diffusion coefficients.

A mixing rule is used to estimate the relationship of a mutual diffusion
coefficient with mole fraction. These mixing rules are based on self-diffusion coefficients of pure substances, found at the limits of mole
fraction,

A variety of different mixing rules have been investigated

The Darken equation assumes a linear relationship between mole fraction and
diffusion coefficient

The Vignes equation assumes a logarithmic relationship between mole fraction
and diffusion coefficient

The Maxwell–Stefan framework separates the ideal and non-ideal effects of
diffusion, unlike the Fickian model. Therefore, solutions to the Fickian and
Maxwell–Stefan equations only coincide when the mixture is ideal and
solubility does not affect the rate of mixing. The two frameworks are related
through the so-called thermodynamic factor

The dependence of diffusion coefficient and thermodynamic
factor on water mole fraction. The solid and dashed black lines represent the
Darken and Vignes mixing rules from Eqs. (

In this study small self-diffusion coefficients of the non-volatile organic
compound have been used as a proxy for systems with a pronounced glassy
state. To investigate the model sensitivities to viscosity and solubility,
the self-diffusion coefficient of the non-volatile compound was varied
between

Figures

By first considering the Fickian simulations in Figs.

Now we move on to discuss the Maxwell–Stefan simulations of sucrose, butanoic
and hexanoic acids, which also take into account the non-ideal effects of
diffusion through the activity coefficients from the UNIFAC model, which is
assuming the liquid state. First notice that there is little difference
between any of the simulations when relative humidity was instantaneously
increased from 10 to 30 % in the first column of both Figs.

In the cases where relative humidity has been instantaneously increased from
10 to 80 %, the variation in growth rates and aerosol composition is
greater than when relative humidity is increased to 30 %. Variation in mixing
timescales can be explained by the diverging thermodynamic factors at high
water mole fractions in Fig.

As expected, in Figs.

Butanoic and hexanoic acid, which are the immiscible examples in Figs.

The simulations in Figs.

The change in water mole fraction as a function of radius
through time after an initial increase in relative humidity from 10 to
30, 80 and 99 %, at

The change in water mole fraction as a function of radius
through time after an initial increase in relative humidity from 10 to
30, 80 and 99 %, at

Figure

Through investigating both high and low water mole fractions by increasing
the relative humidity from 10 to 30 and 80 % in Fig.

Figure

The

The main aim of this study has been to introduce the Maxwell–Stefan law of
diffusion to describe the changing composition of atmospheric aerosol
particles with time. The Maxwell–Stefan equation could act as an alternative
framework to the widely used Fickian framework, which has limitations as it
does not inherently account for solubility effects. From comparing the
sensitivities of these models we found the following:

Observed aerosol partitioning in laboratory studies cannot be replicated using a Fickian framework, which is driven by a gradient in concentration without modifying the Fickian diffusion coefficient to account for the non-ideal effects.

Inclusion of the solubility effects arising from intermolecular interactions is essential to model sustained component separations within aerosol particles. The Maxwell–Stefan framework accounts for these through activity coefficients, calculated using the UNIFAC model.

At low water mole fractions, viscosity was shown to be the most influential factor on equilibration times within aerosol particles.

At high water mole fractions, the variation in equilibration timescales is due to solubility effects, which is especially significant in the atmosphere where there is an abundance of water.

Through simple binary systems of water and a non-volatile secondary organic aerosol, we have shown that there is a complicated relationship between the viscous and soluble effects of mixing. Atmospheric particles are far more complicated systems, with a far greater number of components, all with differing properties. Therefore it is essential to ensure the most suitable framework is used to model this system.

This area of research aims to understand the key micro-physical processes that underpin cloud development and therefore produce models that better predict these processes. We found that at low water mole fractions, equilibration times were most sensitive to changes in viscosity. However at high water mole fractions solubility became a more important factor to consider. This could have significant implications for atmospheric processes – especially for the activation of cloud condensation nuclei and ice nuclei, which occur at high water mole fractions.

This study highlights one key question that needs to be addressed before we continue to investigate the impact of partitioning within aerosol particles on atmospheric models, which is whether current frameworks used to model aerosol composition are suitable to apply to highly complex atmospheric systems. The Fickian model works well for simple two-component systems, where diffusion coefficients can be measured directly. However, for complex systems with multiple components, the Maxwell–Stefan framework offers an alternative which allows mixing against the concentration gradient and phase separations to form.

The Fickian approach has been preferred, as direct measurements of mixing
based on equilibration timescales gives a mutual Fickian diffusion
coefficient. However, the application of these binary diffusion coefficients
to complex, multicomponent atmospheric systems is questionable. On the other
hand, the Maxwell–Stefan laws are inherently multicomponent. The difficulty
we have is selecting the most appropriate mixing rule, which relates a mutual
diffusion coefficient to mole fraction as a function of the self-diffusion
coefficients at infinite dilution. Predictive models for Maxwell–Stefan
diffusivities have been found experimentally from self-diffusion coefficients

To ultimately answer the question of whether current frameworks describing aerosol composition successfully model atmospheric processes, more laboratory studies are required to test model predictions. Through these investigations, we will then be able to better understand the competition between viscosity (or phase state) and solubility to dominate partitioning within atmospheric aerosol particles.

The code has been made open access at

Fick's second law predicts how concentration changes with time,

To solve this equation it was assumed that there was no external source of
material diffusing through the drop. We specify Neumann boundary
conditions, where the flux through the shell boundary is set to zero. The
flux conditions are given as

We begin with the Maxwell–Stefan equation

The numerical model is solved using matrix algebra and written in that form
gives

Each time step is separated into a diffusion step and a moving boundary step, which allows the amount of water in the particle to change. This was not used during the model runs in this paper. However, the development of a moving boundary is vital for the inclusion of a diffusion step into a parcel model and to investigate the effect of aerosol composition on cloud micro-physical properties.

If the change in volume is positive and water is deposited onto the surface
of the aerosol particle, then the steps involved to change the outer boundary
of the particle are as follows:

The new radius,

The change in volume is then used to calculate the molar concentration in the outer shell as follows:

Shell boundaries are then initiated and filled with the total moles in each layer. If the final shell is filled the molar
concentration is distributed over to a new shell; see Fig.

Shell boundaries are initially fixed distances apart, however in the final step, the outer shell radius,

If the change in volume is negative and water is being removed from the
particle surface, then the steps involved to change the outer boundary of the
particle are as follows:

The total volume of water is calculated as sum of the volume of each individual shell

Only water is removed from the aerosol particle, therefore if

After the water has been removed, a new outer shell radius is found,

This study has assumed that the mole fraction of water in the condensed phase
in the outer shell is in equilibration with the ambient saturation relative
humidity. This assumes ideality of the accommodation coefficient, which
enables the study to focus on the sensitivity of diffusion timescales to the
framework used. The equation for water mole fraction,

To find the

The authors declare that they have no conflict of interest.

This work was funded by the Natural Environment Research Council (NERC) through the PhD studentship of Kathryn Fowler under the grant reference number NE/L002469/1. Paul J. Connolly acknowledges funding from the European Union's Seventh Framework Programme (FP7/2007-2013) under grant agreement number 603445 EU FP7. Edited by: Yafang Cheng Reviewed by: two anonymous referees