Introduction
The current best estimate of aerosol effective radiative forcing (relative to
the year 1750) is -0.9+0.8-1.0 W m-2,
in which the uncertainty range represents the 5–95 % confidence limits
(Alexander et al., 2013). Reducing the large uncertainty associated with
aerosol radiative forcing (RF) is crucial to improving the representation of
aerosol in climate models. Top-of-the-atmosphere radiative forcing
(RFTOA) is a common metric for assessing the contribution of
different aerosol particles to the warming or cooling of Earth's atmosphere
(Alexander et al., 2013; Erlick et al., 2011; Haywood and Boucher, 2000;
Ravishankara et al., 2015). RFTOA can be estimated for a uniform,
optically thin layer of aerosol in the lower troposphere using (Dinar et al.,
2008)
RFTOA=S0DTat21-Ac2Rs1-ω‾-β‾ω‾1-Rs2,
in which S0 is the solar constant (1370 W m-2), D is the
fractional day length, Tat is the solar atmospheric
transmittance, Ac is the fractional cloud cover and
Rs is the surface reflectance. The use of satellite, aeroplane
and ground-based observations allows quantification of the geographical
variation in D, Tat, Ac and Rs.
Importantly, Eq. (1) indicates the dependence of RFTOA on the
spectrally weighted single-scattering albedo, ω‾, and the
spectrally weighted backscattered fraction, β‾. Under the
assumptions that a particle is both spherical and homogeneous,
ω‾ and β‾ (in addition to other aerosol
optical properties) can be calculated using Mie theory with input values of
the aerosol particle size (a) and the complex refractive index (RI,
m=n+ki) dependent on wavelength (λ) (Bohren and Huffman, 1998;
Hess et al., 1998; Levoni et al., 1997). Although measurements of particle
size and size distribution are relatively straightforward using techniques
such as aerodynamic particle sizing and differential mobility analysis (DMA),
the measurement of m is more challenging, particularly as aqueous aerosol
droplets commonly exist at supersaturated solute concentrations that are not
accessible in bulk measurements. Zarzana et al. estimated that an uncertainty
in the real component of the RI, n, of 0.003 (0.2 %) leads to an
uncertainty in RF of 1 %, for non-absorbing (NH4)2SO4
particles with a 75 or 100 nm radius (Zarzana et al., 2014). Meanwhile,
Moise et al. reported that variation in n from 1.4 to 1.5 resulted in an
increase in the radiative forcing by 12 % (Moise et al., 2015).
Therefore, any strategy to reduce the uncertainty in RFTOA that
starts from characterisations of aerosol microphysical properties (a
bottom-up approach) requires the development of new in situ techniques for the accurate
characterisation of refractive indices dependent on wavelength and relative
humidity (RH).
Tang et al. levitated single, laboratory-generated aerosol particles in an
electrodynamic balance (EDB), measuring relative changes in the droplet mass and
recording the elastic light scattered at a fixed angle relative to the
propagation direction of a 633 nm laser beam. By comparing the measured
light scattering traces to calculations from Mie theory, n was
parameterised as a function of RH for a range of inorganic aqueous solution
droplets, key components of non-absorbing atmospheric aerosol. Tang's
parameterisations of n633 for aqueous inorganic solutes have become a
benchmarking standard for new techniques measuring refractive index
(Cotterell et al., 2015b; Hand and Kreidenweis, 2002; Mason et al., 2012,
2015) and are used as reference RI data for RF calculations of aqueous
inorganic aerosol (Erlick et al., 2011). However, Tang's measurements were
limited to λ=633 nm and knowledge of the optical dispersion is
required to calculate spectrally weighted values of the single-scattering
albedo and backscatter function.
Another method for determining the RI of a particle is to fit the measured
size dependence of optical cross sections to Mie theory (Moise et al., 2015).
In particular, the interaction of light with an aerosol particle is governed
by the particle extinction cross section, σext, (Bohren and
Huffman, 1998; Miles et al., 2011a) and measurements of σext
using cavity-enhanced methods are, in principle, highly precise. Ensemble
broadband cavity-enhanced spectroscopy involves σext
measurements on a cloud of aerosol particles over a broad range of
wavelengths. Zhao et al. (2013) have developed an ensemble broadband
cavity-enhanced spectrometer for measurements of σext over
the wavelength range 445–480 nm and reported measurements of
(NH4)2SO4 aerosol (Zhao et al., 2013). Rudich and co-workers
developed an ensemble broadband cavity-enhanced spectrometer for the
wavelength range 360–420 nm (Flores et al., 2014a, b; Washenfelder et al.,
2013). From measurements of Suwannee River fulvic acid aerosol (a weakly
absorbing species), m was measured to be 1.71 (±0.02) + 0.07
(±0.06)i at 360 nm and 1.66 (±0.02) + 0.06 (±0.04)i at
420 nm (Washenfelder et al., 2013). These uncertainties in the retrieved m
limit the accuracy of RF calculations.
Ensemble cavity ring-down spectroscopy (E-CRDS) is a related cavity-enhanced
spectroscopy technique used for σext measurements for
aerosol particles in both the laboratory (Dinar et al., 2008; Lang-Yona et
al., 2009; Mason et al., 2012) and in the field (Baynard et al., 2007;
Langridge et al., 2011). In E-CRDS, a flow of aerosol is introduced into an
optical cavity consisting of two highly reflective mirrors in which the
aerosol ensemble is probed using CRDS and an extinction coefficient, αext, is measured. In combination with measurements of particle
number concentration, N, using a condensation particle counter,
σext is calculated using
σext=αext/N. However, uncertainty in the
population distribution of aerosol within the cavity ring-down beam and
errors in the measured N can lead to imprecise measurements of
σext (Miles et al., 2011b). By size selecting aerosol using
DMA prior to admitting an ensemble into the optical cavity, the variation in
σext with particle size is measured and the particle RI can
be retrieved. However, the accuracy and precision in the retrieved RIs are
often too poor for reliable RF calculations owing to the combination of
imprecise σext measurement and the significant systematic
errors that can derive from the DMA size selection process. Mason et al.
reported a precision of ±0.02 (∼ 1.4 %) in the retrieved n
from E-CRDS measurements of NaNO3 particles at a range of RH values
(Mason et al., 2012), while Miles et al. 2011b found that errors in the
measured N can alone introduce a ∼ 2.5 % uncertainty in the
retrieved n, therefore limiting the precision of this technique. Under a
scenario incorporating best-case errors in variables governing E-CRDS
σext measurements, a theoretical study by Zarzana and
co-workers estimated the accuracy in the n retrieved from E-CRDS to be
0.6 % at best from simulations of σext measurements for
non-absorbing (NH4)2SO4 particles at 12 discrete particle sizes
(Zarzana et al., 2014).
Probing a single particle, instead of an aerosol ensemble, resolves many of
the problems inherent in E-CRDS that degrade the RI retrieval precision. We
have previously reported the application of two single-particle cavity
ring-down spectroscopy (SP-CRDS) instruments to measure σext at wavelengths of λ=405 and 532 nm for single particles confined within a Bessel laser beam
(BB) optical trap (Cotterell et al., 2015a, b; Mason et al., 2015; Walker et
al., 2013). No measurement of particle number density is required and
measurements of σext are made with continuous variation in
the particle size as the particle evolves with time, either through an RH
change or through particle–gas partitioning of semi-volatile components. In
our measurements, the particle size is precisely determined from fitting the
angularly resolved elastic light-scattering distributions to Mie theory. For
single-component evaporation measurements, we measured the precision in
retrievals of n to be ±0.0007 for the 532 nm SP-CRDS instrument (Mason
et al., 2015) and ±0.0012 using the 405 nm SP-CRDS instrument (Cotterell
et al., 2015b). Also, we demonstrated the retrieval of RI from hygroscopic
response measurements for aqueous droplets containing inorganic solutes
(Cotterell et al., 2015a, b). By simulating σext data using
the parameters of the 532 nm SP-CRDS instrument, we demonstrated the
expected retrieval accuracy in n to be 0.0002 (0.014 %) for
single-component particles evaporating over the radius range
1–2 µm. Meanwhile, we showed that the retrieval accuracy is
< 0.001 for coarse-mode particles when n varies with particle
size, such as in a hygroscopic response measurement (Cotterell et al., 2016).
In this paper, we report the application of the 405 and 532 nm SP-CRDS
instruments for the measurement of n at the four wavelengths of 405, 473,
532 and 633 nm (n405, n473, n532 and n633,
respectively). In particular, we report comprehensive RI measurements for the
hygroscopic response of aqueous droplets containing atmospherically relevant
inorganic solutes and quantify the RI retrieval precision. We compare the
differences in the RI retrieval precision and accuracy when sizing particles
using either (i) light elastically scattered from the BB optical trap or (ii)
light elastically scattered from a Gaussian probe beam. Measurements of
n633 as a function of RH from phase functions, Raman spectroscopy (from
aerosol optical tweezer (AOT) measurements) and from the Tang et al. (Tang
et al., 1997; Tang and Munkelwitz, 1994) parameterisation are compared.
Finally, our multi-wavelength RI retrievals, combined with n650
measurements from aerosol optical
tweezers, n589 measurements of bulk solutions and the n633
descriptions provided by Tang et al. (Tang et al., 1997; Tang and Munkelwitz,
1994), are parameterised using an optical dispersion model. The following
section describes the 405 and 532 nm SP-CRDS instruments and the aerosol
optical tweezer instrument, while Sect. 3 reports the retrieved RI variation
with RH for aqueous aerosol particles containing the inorganic solutes NaCl,
NaNO3, (NH4)2SO4, NH4HSO4, or Na2SO4
and compares different methods of retrieving nλ. Indeed, these
solutes represent the most abundant species found in inorganic atmospheric
aerosol. Finally, Sect. 4 presents a Cauchy dispersion model for
parameterising the variation in the RI with both wavelength and RH.
Experimental and numerical methods
Aerosol optical tweezers
Measurements of RI at 650 nm were accomplished using
commercial AOTs (AOT-100, Biral).
The experimental set-up has been described in detail previously (Davies and
Wilson, 2016; Haddrell et al., 2017). Briefly, a single aqueous droplet from
a plume produced by a medical nebuliser (NE-U22, Omron) is captured by a
gradient force optical trap formed from focussing a 532 nm laser (Opus 2W,
Laser Quantum) through a high numerical aperture microscope objective
(Olympus PLFLN 100×). Inelastically backscattered (Raman) light is
imaged onto the entrance slit of a 0.5 m focal length spectrograph
(Princeton Instruments, Action Spectra SP-2500i), dispersed by a 1200 line
pairs per millimetre grating onto a cooled charge-coupled device (CCD)
camera. The Raman spectrum of a spherical droplet consists of a broad
underlying Stokes band with superimposed resonant structure at wavelengths
commensurate with whispering gallery modes (WGMs), from which the radius, RI,
and dispersion can be determined with accuracies better than 2 nm, 0.0005
and 3 × 10-8 cm respectively (Preston and Reid, 2013). RH in
the trapping chamber is controlled by adjusting the relative flows of dry and
humidified nitrogen and is measured to ±2 % using a capacitance probe
(Honeywell HIH-4602C). A typical experiment involves trapping an aqueous
droplet containing one of the studied solutes, decreasing the RH in discrete
steps over several hours, and monitoring the RH-dependent changes to droplet
radius, refractive index and dispersion. Note that because dispersion is also
determined, it is possible to compare refractive indices measured using the
optical tweezers with other approaches at different wavelengths (e.g. 633 nm
by Tang et al.).
Schematic diagrams of the two SP-CRDS instruments used in this
work. OI is an optical isolator, λ/2 is a half-wave plate, PBS is a
polarising beam splitter cube, AOM is an acousto-optic modulator, PZT
represents a piezo ring actuator, PD is a photodiode, and M1 and M2 are highly
reflective mirrors of the optical cavity. The 633 nm laser in (b)
is used in selected measurements indicated in the text.
Single-particle cavity ring-down spectroscopy instrument
Figure 1 summarises the experimental arrangement of the 405 and 532 nm
SP-CRDS instruments. The 532 nm SP-CRDS instrument has been described previously (Mason
et al., 2015), while Cotterell et al. (2015a) describes modifications to
the instrument to improve particle size retrieval. The 405 nm SP-CRDS instrument is
described elsewhere (Cotterell et al., 2015b), and recent modifications to
this instrument to improve size determination are described below. The
following section provides a general description of σext
and elastic light-scattering measurements using SP-CRDS.
Overview of single-particle cavity ring-down spectroscopy
The continuous-wave cavity ring-down spectrometer
The single-particle CRDS measurements were performed on two separate
instruments summarised in Fig. 1, with each instrument performing CRDS at
either 405 or 532 nm. The beam from a continuous-wave single-mode laser
(with a spectral bandwidth < 5 MHz) passes through an acousto-optic
modulator (AOM). The first-order diffraction spot is injected into an optical
cavity, while the zero and higher-order spots are attenuated with beam
blocks. The optical cavity consists of two highly reflective mirrors with
radii of curvature of 1 m and reflectivities > 99.98 % at
the CRDS wavelength and has a free spectral range of ∼ 300 MHz. The
cavity is aligned such that the TEM00 mode is preferentially excited. A
piezo ring actuator, affixed to the rear cavity mirror, continuously varies
the cavity length across several free spectral ranges. When the cavity length
is such that a longitudinal cavity mode is excited, a photodiode monitors the
build-up of light inside the cavity and measures the intensity escaping from
the rear mirror as an output voltage, V, which is sent to both a Compuscope
digitizer and a digital delay generator. The digital delay generator outputs
a 5 V transistor–transistor logic (TTL) pulse to the AOM when the leading
edge of the photodiode signal reaches a 1 V threshold value, rapidly
switching off the first-order AOM diffraction beam and initiating a ring-down
decay. The subsequent time variation in V obeys a single exponential decay
and is fitted to V=V0exp-t/τ+b,
with τ the characteristic ring-down time (RDT) and b a baseline
offset. RDTs are measured at a rate of 5–10 Hz. To reduce the contributions
of airborne dust particles to light extinction and to prevent the mirrors
getting dirty, nitrogen gas flows are directed across the faces of the cavity
mirrors and through flow tubes that extend to a trapping cell at the centre
of the cavity.
The Bessel beam optical trap
Both the 405 and 532 nm SP-CRDS instruments use a 532 nm laser beam to
generate a BB optical trap. A Gaussian 532 nm laser beam is passed through a
2∘ axicon to produce a BB, which has a circularly symmetric intensity
profile consisting of a central core and multiple rings. A pair of lenses
reduces the BB core diameter to 3–5 µm. A 45∘ mirror
propagates this beam vertically into the trapping cell. This mirror and the
trapping cell are mounted on the same translation stage, allowing the
position of the BB optical trap to be translated in the horizontal transverse
direction to the CRDS optical axis. In the trapping cell, the radiation
pressure exerted by the BB on a trapped particle is balanced by a humidified
nitrogen gas flow of 100–200 sccm, allowing control over RH. This gas flow
is also used to purge the cell of excess aerosol. The RH inside the trapping
cell is monitored using a capacitance probe. A plume of aerosol particles is
introduced into the cell using a medical nebuliser. Typically, a single
aerosol particle is isolated from this plume by the BB optical trap.
Occasionally multiple particles are trapped, in which case the trapping cell
is evacuated and aerosol particles are again nebulised into the cell. This
process is repeated until only a single particle is trapped.
Size and RI retrieval using phase function images
A camera coupled to a high numerical aperture objective, positioned at
90∘ with respect to the BB propagation direction, records images
(referred to as phase function, PF, images) that describe the
angular variation in elastically scattered light at a selected polarisation.
For 532 nm SP-CRDS measurements, we record PFs of the light elastically
scattered from a 473 nm probe laser beam. The probe beam is aligned
collinearly to the BB propagation direction using a polarisation beam
splitter (PBS) cube, the alignment of which is shown in Fig. 1a. The PBS
merges the two beams prior to the final lens (f=50 mm) and, therefore,
the probe beam is weakly focussed into the trapping cell. In the trapping
cell, the probe beam has a Gaussian transverse intensity profile and a beam
diameter ∼ 8 times larger than the BB core diameter. A 532 nm
laser line filter ensures that the camera does not collect elastically scattered
light from the BB. For 405 nm SP-CRDS measurements, we predominantly collect
PFs of light elastically scattered from the beam forming the BB optical trap
or occasionally from a 633 nm probe laser beam (LHP073, Melles Griot). The
633 nm probe beam is aligned collinearly to the BB propagation direction and
is merged with the BB using a PBS cube as shown in Fig. 1b. In the trapping
cell, the probe beam has a Gaussian transverse intensity profile and a beam
diameter that is ∼ 3 times larger than the BB core diameter. When
recording 633 nm PFs, a 532 nm laser line filter ensures that the imaging camera
does not collect elastically scattered light from the BB, while this filter
is removed when collecting 532 nm PFs corresponding to BB illumination.
The variation in particle radius and nλ is determined by fitting
the complete measured PF data set to Mie theory in a self-consistent step. We
previously reported the algorithms used to perform this fitting (Cotterell et
al., 2015a, b; Preston and Reid, 2015). For all hygroscopic response
measurements reported in this work, the RI varies with the particle size.
Therefore, we parameterise nλ in terms of particle radius using
the expression
nλ=nλ,0+nλ,1a3+nλ,2a6,
in which a is the particle radius, nλ,1 and nλ,2
are fitting parameters, and nλ,0 is the real RI of pure water at
wavelength λ. This latter term is known precisely from bulk
measurements and we use the data of Daimon and Masumura for water at T= 24 ∘C (Daimon and Masumura, 2007). Specifically, we use
n405,0= 1.343, n473,0=1.338, n532,0=1.335, n633,0= 1.332 and n650,0=1.331.
Measuring σext for a single particle
Once a single aerosol particle is optically trapped, the particle position
is optimised to obtain a minimum in the measured RDT, corresponding to the
particle being located at the centre of the ring-down beam. The position is
varied in both transverse directions to the CRD beam, in the vertical
direction by changing the laser power and in the horizontal direction by
scrolling the translation stage on which the trapping cell is mounted. When
the particle is centred in the CRD beam, a computer-controlled laser
feedback is initiated to maintain a constant particle height over the
duration of the measurement and values for τ are collected. After
measurements have been made on the particle, the empty-cavity RDT, τ0, is recorded over several minutes. From knowledge of τ and
τ0, σext is calculated using
σext=Lπw22c1τ-1τ0,
in which L is the separation distance between the two cavity mirrors, c
is the speed of light and w is the focal beam waist of the intra-cavity
ring-down beam, which is either calculated using Gaussian optics (Kogelnik
and Li, 1966) or is treated as a variable parameter when fitting the
σext data to the prediction of a light-scattering model. We
use the latter method for determining w and have previously demonstrated
that the fitted w value agrees with the predictions of Gaussian optics (Cotterell
et al., 2015b; Mason et al., 2014, 2015)
Computational analysis of σext data
Mie theory assumes that a travelling plane wave illuminates a homogeneous
spherical particle. However, the field inside the optical cavity is a
standing wave and not a travelling wave, with the extinction of light by
aerosol varying for different phases of the standing wave (Mason et al.,
2014; Miller and Orr-Ewing, 2007). The two cases of a particle centred at a
standing wave node or antinode provide limiting values for
σext. A confined particle undergoes Brownian motion within
the BB core over distances of micrometres, allowing the particle to sample
standing-wave nodes, antinodes and intermediary phases. Therefore, the
standing wave leads to a broadening in the measured σext,
with the limits in the recorded data corresponding to the particle located at
either a node or antinode. To fit our measured σext vs.
radius data, we use the cavity standing wave generalised Lorenz–Mie theory
(CSW-GLMT) equations that we have derived previously to calculate
σext for the limiting cases of a node- and antinode-centred
particle (Cotterell et al., 2016). These calculations provide a CSW-GLMT
envelope, within which all the measured data are expected to lie (in the
absence of experimental noise) for the best-fit simulation. A residual
function, R, is defined by Eq. (4), in which the σext,j
points included in the summation are only those with measured values that
reside outside the simulated σsim envelope and
σsim,j is the closer of the node- or antinode-centred
simulation values at the given radius.
R=1J∑j=1Jσext,j-σsim,jρj
The density of the measured σext data points within a 1 nm
range of the particle radius, ρj, is used as a weighting factor and
prevents biasing of the fit to regions where the measured number of data
points is high in the radius domain. The RI is varied to fit CSW-GLMT to the
σext data, with the RI parameterised by Eq. (2) in terms of particle
radius. The value of R was minimised by varying the parameters
nλ,1 and nλ,2, in addition to varying the beam waist,
w. The nλ,1 and nλ,2 values that correspond to the
minimum in R define the best-fit refractive index. For the
1–2 µm radius particles studied using either a 405 or 532 nm
SP-CRDS, nλ,1 was typically varied over a range of 0 to
1 × 108 nm3 in steps of 1 × 106 nm3,
while nλ,2 was varied over a range of -1× 1018 to
0 nm6 in steps of 1 × 1016. A grid search was used to
vary the three parameters nλ,1, nλ,2 and w such that all
points in the three-dimensional search space were sampled.
SP-CRDS measurements for hygroscopic inorganic aerosol
particles
Single aqueous aerosol droplets containing one of the inorganic solutes of
interest (NaCl, NaNO3, (NH4)2SO4, NH4HSO4 or
Na2SO4) were optically trapped using the SP-CRDS instruments at
high (∼ 80–85 %) RH. Following nebulisation, the ambient RH was
unsteady and thus no action was taken during a conditioning period of
∼ 10 min until the RH had stabilised. The RH was subsequently lowered
over time at a near-constant rate of 0.4–0.5 % per minute, until the
particle effloresced or the particle size was such that it was unstable and
was ejected from the optical trap. While lowering the RH, τ and PFs
were recorded at a rate of ∼ 10 and 1 s-1, respectively. Repeat
measurements (∼ 5 droplets) were performed for all inorganic solutes of
interest using both of the SP-CRDS instruments and, in the case of 405 nm
SP-CRDS, using the 532 nm BB for PF measurements. This work considers
measurements of particles with mean particle radii
> 1 µm only, as we have previously found that
nλ retrievals for hygroscopic particles in this size range are
accurate to < 0.001, while this accuracy degrades significantly for
smaller particles (Cotterell et al., 2016).
(a) The fitted radius, n473, and correlation
coefficient variation with frame number for an aqueous NaNO3 droplet.
Red points correspond to initial n473 parameter values, green points to
further optimised values and purple points to the best-fit values.
(b) The mean correlation coefficient, c‾n473, and the parameters n473,1 and n473,2 were
varied.
Radius and refractive index retrieval from recorded phase
functions
Prior to analysis of the σext data, the particle radius and
nλ at the PF acquisition wavelength were determined from fitting
the PFs to Mie theory (see Sect. 2.1). We refer the reader to Fig. 1 in
Cotterell et al. (2015a) and Fig. 4 in Carruthers et al. (2012) for
examples of raw PF images recorded using our instrumentation. As
described previously (Cotterell et al., 2015a, b), a mean Pearson correlation
coefficient c‾nλ is used to quantify the
level of agreement between the measured PF data set and the set of best-fit
Mie theory simulations, with c‾nλ=1
corresponding to perfect agreement and lower values of c‾nλ indicative of a poorer fit. The entire PF data set is
made up of typically 5000 PFs, each with a corresponding Pearson correlation
coefficient value, c(nλ), which each contribute to the overall
mean Pearson correlation coefficient value, c‾nλ. Figure 2a shows the variation in the fitted Pearson correlation
coefficient cn473, radius and n473 with time
(labelled as frame number, with PF frame acquisition rate of 1 s-1)
when fitting 473 nm PFs from a measurement on an aqueous NaNO3
particle. Data are shown for the initial n473 fitting parameter values
(n473,1= 5.6 × 107 nm3, n473,2= 0.0 nm6), for further optimised values (n473,1= 7.3 × 107 nm3, n473,2= 0.0 nm6) and for
the best-fit parameter values (n473,1= 1.221 × 108 nm3, n473,2= -7.585 × 1015 nm6). Figure 2b shows how the mean
correlation coefficient c‾n473 varies with
n473,1 and n473,2 on their initial search cycles. Further data
points that correspond to refined grid-search cycles can be seen. There are
clear maxima in c‾n473 as n473,1 and
n473,2 are varied, with the maximum having a value of c‾n473=0.9951.
For the SP-CRDS measurements reported in this work, each hygroscopic response
measurement has a c‾nλ value associated with
the fitting of the PFs. When using 473 (or 633) nm probe laser beam
illumination, values of c‾n473 (and
c‾n633) averaged over all hygroscopic response
measurements performed at the respective wavelengths were 0.993 ± 0.003
(and 0.99646 ± 0.00004), in which the errors represent 1 standard
deviation in the c‾nλ. We also collected
PFs of the elastically scattered light from the BB illumination, which
resulted in lower values of c‾nλ
compared to illumination from a Gaussian probe beam; the corresponding
c‾n532 value from BB illumination was
0.985 ± 0.008. This lower c‾nλ
value indicates that the use of PFs from illumination with the BB light (core
of diameter ∼ 5.5 µm) is detrimental to radius and n532
determination, with more accurate determinations arising when analysing the
PFs from Gaussian illumination at 473 and 633 nm. The component wave vectors
constituting the BB make large angles with respect to the optical propagation
axis, while Mie theory assumes that a plane wave illuminates a spherical
particle. The 473 and 633 nm probe beams have Gaussian profiles with spot
diameters estimated to be ∼ 30 and ∼ 15 µm,
respectively, at the particle trapping location and are more representative
of plane wave illumination compared to that provided by the trapping BBs. Section 3.4 further analyses the consequential errors in the
derived particle size and n532 due to BB illumination.
Comparison of refractive index retrieval methods
The 633 nm probe beam was used for PF illumination to provide a comparison
of the refractive index retrieval accuracy from PF imaging with AOT
measurements (using Eq. 5) and measurements by Tang et al. (Tang et al.,
1997; Tang and Munkelwitz, 1994) for the aqueous NaCl system only. For the
final set of data presented in Sect. 4, the parameterisation by Tang et al.
is used to represent n633 for all compounds. In addition, for the
retrieval of RIs from the 405 nm SP-CRDS system, all droplet radii were
retrieved from PFs using BB illumination at 532 nm rather than at 633 nm
(Tang et al., 1997; Tang and Munkelwitz, 1994).
The refractive index from AOT measurements is retrieved from Raman
spectroscopy measured at 650 nm. The fitting of the Raman spectra yields not
only n650 but also dispersion terms, m1 and m2, which allow
the refractive index to be calculated at alternative wavelengths:
n=n0+m1ν-ν0+m2ν-ν02.
Here, n is the refractive index at the desired wavelength, n0 is the
refractive index at the measured wavelength (n650 in this case), ν
is the wave number of the desired wavelength and ν0 is the
wave number at the measurement illumination wavelength.
Comparison of the RH-dependant parameterisation of n633
reported by Tang et al. (1997) and Tang and Munkelwitz (1994) for aqueous
NaCl to n633 measured from 633 nm Gaussian-illuminated PFs and Raman
spectroscopy from aerosol optical tweezer measurements. The error bars
correspond to uncertainties associated with RH measurements.
Representative SP-CRDS measurements of σext (red
points) and best-fit CSW Mie envelopes (black lines) for aqueous aerosol
particles containing the inorganic solutes NaCl, NaNO3, and
(NH4)2SO4. The measured σext values were
collected using either 405 nm (left column) or 532 nm (right column)
SP-CRDS.
Figure 3 shows n633 for an aqueous sodium chloride particle retrieved
from AOT measurements and the average of three measurements from PFs measured
on the 405 nm SP-CRDS instrument, with good agreement between the
techniques. Furthermore, n633 values from both techniques are consistent
with the parameterisation of n633 from Tang et al., within the RH
uncertainty indicated corresponding to ±2 % for AOT and PF
measurements. Although the RH ranges of the new measurements do not perfectly
match for the limited number of measurements compared here, the consistency
between the new measurements and the previous parameterisation provided by
Tang et al. is sufficient for validating the measurement approaches.
Hargreaves et al. compare EDB and optical tweezer
measurements of aqueous NaCl to the Tang parameterisation, reporting good
agreement to Tang et al. at high RH (> 70 ± 0.2 %)
with increasing divergence at low RH (45 + 5 %), at which the error
in RH represents the equivalent offset in RH required to bring the different
measurements into agreement (Hargreaves et al., 2010). For
(NH4)2SO4, Tang et al. (Tang and Munkelwitz, 1994) acknowledged
that although their measurements agreed with those of Richardson and Spann
(Richardson and Spann, 1984) at high RH, there was divergence from
measurements by Cohen and co-workers by an RH equivalent of 4–5 % (Cohen
et al., 1987). Despite this, the n633 measurements from three different
techniques show good agreement within the uncertainty in RH, reinforcing the
premise that the different techniques offer an accurate means of retrieving
nλ as a function of RH and are compatible for combining in the RI
parameterisation presented in Sect. 4. This comparison also demonstrates that
PF measurements can be used to accurately retrieve particle radius and RI. It
should also be noted that the AOT measurements are centred at a wavelength of
650 nm and the values reported in the figure are the corrected values for
633 nm. Over this wavelength range, the RI change from 650 to 633 nm is
∼ -0.0017 at a typical RH of 80 %.
Extinction cross section measurements
Once the time evolution in particle radius was determined from the PFs, the
measured σext data were compared to CSW-GLMT calculations
using the procedure described in Sect. 2.2. Figure 4 shows example
σext data sets measured using either 405 or 532 nm SP-CRDS
and the best-fit CSW Mie envelopes. For the example data presented in this
figure, the particle radii for measurements using 405 nm SP-CRDS are from
fitting the 532 nm PFs (i.e. using BB illumination) and for 532 nm
SP-CRDS the radii are from fitting 473 nm PFs (from the probe beam
illumination). The σext data sets measured using 405 nm
SP-CRDS have a higher number of resonance features compared to the data sets
measured using 532 nm SP-CRDS, even though the particles evaporated over
similar radius ranges, owing to the 1/λ dependence of the particle
size parameter. The higher number of resonance features in the σext data at lower wavelengths is expected to facilitate higher
precision determinations of nλ.
Multi-wavelength determinations of the refractive index
The precision in nλ retrievals from σext and PF
data was tested by repeat SP-CRDS measurements on different aqueous droplets
containing the inorganic solutes of interest. The retrieved nλ
values were represented as a function of the RH, as measured by a calibrated
capacitance probe located ∼ 1 cm from the droplet trapping location.
The nλ data were binned in 2 % RH intervals because the
capacitance probe RH measurements have a standard error of ±2 %.
Repeat measurements of nλ were then averaged for each RH bin and
a standard deviation, sλ(RH), was calculated. The values of
n405 and n473 were retrieved from fitting the σext(λ= 405 nm) and PF(λ= 473 nm) data,
respectively. This section presents the average n532 calculated from
σext(λ= 532 nm) data only and neglects the
n532 value retrieved from PF analysis, assuming that the RI retrieval using
σext data is more precise than using PFs from BB scattering.
Section 3.5 presents a thorough analysis of the impact of
BB illumination on n532 precision. Table 1 reports the number
of particles studied for each inorganic species using either 405 or 532 nm
SP-CRDS. Data are shown for 405 nm SP-CRDS measurements using the BB probe
beam for PF collection.
Information relating to the precision of nλ measured
from SP-CRDS for aqueous NaCl, NaNO3, (NH4)2SO4,
NH4HSO4 and Na2SO4 droplets. N is the number of data
sets available for phase function and extinction cross section (σext) analysis from either 405 nm or 532 nm SP-CRDS. The mean
standard deviation in the retrieved nλ, snλ‾ is reported for both extinction cross section and
phase function (PF) measurements.
405 nm SP-CRDS
532 nm SP-CRDS
(using 532 nm PFs)
N
sn532‾
sn405‾
N
sn473‾
sn532‾
(PFs)
(σext)
(PFs)
(σext)
NaCl
5
0.0088
0.0044
7
0.0038
0.0030
NaNO3
7
0.0053
0.0034
5
0.0028
0.0015
(NH4)2SO4
4
0.0086
0.0041
9
0.0036
0.0027
NH4HSO4
3
0.0130
0.0095
4
0.0054
0.0025
Na2SO4
–
–
–
9
0.0072
0.0062
The average nλ variations with RH for aqueous droplets
containing the inorganic solutes of interest. The uncertainties in
nλ have not been included here for clarity; they are listed in
Table 1.
Figure 5 summarises the average retrieved RIs (n405, n473 and
n532) binned in 2 % RH intervals for each of the inorganic solutes
of interest. Also shown are the n633 variations reported by Tang et al.
(Tang et al., 1997; Tang and Munkelwitz, 1994) and values for n589
reported in the CRC Handbook of Chemistry and Physics (Haynes, 2015). The CRC
handbook reports measured values of n589 in terms of mass fraction of
solute. Therefore, the E-AIM model was used to relate solute mass fraction to
water activity (AIM, 2017; Clegg et al., 1998). Furthermore, the CRC handbook
does not report RI values for NH4HSO4 and the n589 values
plotted in Figs. 5d and 8d were taken from bulk solution measurements using a
refractometer (Misco, Palm Abbe).
In general, Fig. 5 shows that nλ increases towards shorter
wavelengths, as expected given the chromatic dispersion behaviour of typical
materials. Furthermore, the optical dispersion increases as the RH decreases
and the solute becomes more concentrated within the aqueous droplets. The
increasing separation between the measured RH-dependent nλ values
for progressively shorter wavelengths indicates the increasing influence of
optical dispersion. At higher RH values (> 80 %), the
measured values of n405, n473, and n532 approach one another
and, in the case of NaNO3, the values of n405 become lower than the
measured n473 and n532 values. The RI values are expected to
converge as the RH tends to 100 % (the RI variation with wavelength in
the visible spectrum is low for pure water) (Daimon and Masumura, 2007), but
this crossing is likely to derive from calibration errors in the RH probes,
noting that the set of {n405, n532} and {n473, n532}
measurements are made using different SP-CRDS instruments.
The n589 bulk measurements are limited by the solubility of each solute,
which constrains measurements to high water activity (RH) values. As
expected, in all cases, the bulk n589 literature data lie close to, or
match, the n633 values. Small deviations of n589 from n633
(> 0.003) for NH4HSO4 and NaNO3 can be observed
in Fig. 5. While the RH values for the n589 data are calculated from
measured solute mass fraction data (which are expected to be accurate to
< 1 %) using the E-AIM model (AIM, 2017; Clegg et al., 1998),
the RH measurements associated with the n633 parameterisation are
expected to be more uncertain, up to ±5 % at RH < 45 %, as
discussed in Sect. 3.2. Therefore, it is reasonable to see deviations in the
order of the data points at high RH in line with the uncertainty associated
with the RH measurements.
In most cases the n650 data lie below the data measured at shorter
wavelengths, which follows the expected trend of chromatic dispersion. In the
cases of NaNO3, (NH4)2SO4 and NH4HSO4, the
n650 values cross the n633 values, which is attributed to
uncertainties in RH measurements from both the AOT and Tang parameterisation
of ±2 and ∼ ±5 %, respectively (Tang and Munkelwitz, 1994).
Comparison of n532 values determined from either PFs recorded
using BB illumination or σext data using 532 nm SP-CRDS.
The shaded envelopes represent 1 standard deviation in the measurements.
Precision in refractive index determined using SP-CRDS
The precision in the RI values determined using SP-CRDS can be quantified by the
standard deviation in nλ within a 2 % RH interval. For ease
of reading, the plots in Fig. 5 do not show error bars representing this
standard deviation. Instead, the means of the standard deviation values over
all the RH bins are calculated and denoted, snλ‾. Table 1 presents the snλ‾
values for each inorganic solute of interest as measured on each of the
SP-CRDS instruments. In all cases, the retrieved RI from fitting
σext data is more precise than from the corresponding PF
analysis.
For measurements performed using the 532 nm SP-CRDS instrument, the
precision of n473 and n532 measurements (from PFs and CRDS,
respectively) was generally high, indicated by the majority of
sn473‾ values <0.004 and sn532‾ values <0.003 in Table 1. The only exception was
Na2SO4 for which the standard deviations for the measurements are
larger, sn473‾∼0.007 and
sn532‾∼0.006. Na2SO4
effloresces at relatively high RH (∼ 60 %), thereby limiting the
radius range accessed during drying. This reduces the extent of the resonant
structure observed, which provides the greatest constraint on RI retrievals
and limits the precision of the nλ determination. The
measurements from the 405 nm SP-CRDS instrument are not as precise as those
performed on the 532 nm SP-CRDS instrument owing to the collection of PFs
from BB illumination, in which sn532‾ and
sn473‾ values are in the range of 0.005–0.013
and 0.003–0.007, respectively. The sn532‾
values from (BB-illuminated) PFs are significantly higher than
sn405‾ retrieved from CRDS; the majority of
solutes (except NH4HSO4) have sn405‾<0.0045. The NH4HSO4 sn532‾
values are particularly high, which influences the retrieved
sn405‾ since information relating to the
geometric size of the particle from PF data is required in the
σext data fitting procedure. The NH4HSO4 values of
sn532‾= 0.013 and sn405‾∼0.01 emphasise the recommendation to use a probe Gaussian
beam for PF illumination to improve precision in both nλ from PFs
and nλ from CRDS in the long term.
Contour plots representing the parameterisation of RI as a function
of both wavelength and RH for aqueous aerosol particles containing the
inorganic solutes of interest.
Summary of the best-fit parameters to describe the RI variation with
both wavelength and RH for the inorganic solutes of interest. The parameters
and function forms are defined in Eqs. (7)–(9). N is the total number of
measured and literature data points (shown in Fig. 5) to which the
parameterisation was fitted. Δn‾ is the
mean absolute difference between the best-fit Cauchy model and the measured
and literature data.
NaCl
NaNO3
(NH4)2SO4
NH4HSO4
Na2SO4
n0,0
1.3495
1.4767
1.3764
1.4649
2.3810
n0,1 / 10-2
0.8770
-0.0761
0.8552
0.0690
-3.2494
n0,2 / 10-4
-2.6190
-0.1451
-2.7632
-0.4877
3.3485
n0,3 / 10-6
2.8864
0.2941
3.3769
0.6234
-0.5428
n0,4 / 10-8
-1.1586
-0.2147
-1.5096
-0.3336
-0.6018
n1,0 / 10-2
7.0981
3.1219
-5.7022
3.8202
34.8882
n1,1 / 10-4
-21.7961
-1.8033
46.6467
4.1621
-61.1350
n1,2 / 10-6
31.1422
-1.4789
-82.4706
-6.4225
27.6052
n1,3 / 10-8
-15.5951
1.3676
42.4736
-0.5106
0.0000
N
121
190
130
164
86
Δn‾
0.0020
0.0016
0.0018
0.0044
0.0027
We now provide further evidence that reliable RI retrievals from fitting Mie
theory to PFs cannot be performed when the illuminating light field is a
focussed BB. Figure 6 compares n532 values retrieved from CRDS and
BB-illuminated PFs. Measurements using 405 nm SP-CRDS were performed for
aqueous Na2SO4 but the non-linear dependence of nλ on
RH and possible impurities in the sample prevented fitting the 532 nm PFs to
Mie theory (and therefore 405 nm σext measurements to
CSW-GLMT). Therefore, Na2SO4 n405 measurements are not
discussed further. For each inorganic solute studied, the precision in the
n532 values retrieved from the PFs is poorer than retrievals from
σext data, indicated by PF sn532‾ values being approximately twice the equivalent sn532‾σext data, as indicated in Table 1.
Moreover, the slopes of the RI vs. RH plots in Fig. 6a–d are steeper for the
PF-retrieved, compared to the σext-retrieved, n532
data. The PF-retrieved n532 for NH4HSO4 (Fig. 6d) has both a
significantly larger standard deviation and divergence from n532σext data. The propagation of this uncertainty in
PF-retrieved RI (and particle size) into the
σext-retrieved n405 values is evident, with
sn405‾= 0.0095. Although the precisions of
n532 and radius retrievals are compromised when fitting PFs using BB
illumination, the corresponding sn405‾ from
fitting σext data is, on the whole, close to σext-retrieved sn532‾ from probe beam
PF illumination (with the exception of NH4HSO4), with values in the
range of 0.003–0.005 (Table 1). Furthermore, the sn405‾ values indicate a precision better than twice that of the
corresponding sn532‾ from fitting PFs.
All snλ‾ values encompass contributions
from both variation in nλ and, more significantly, from
uncertainties in measuring the RH. Indeed, we have recently reported that the
nλ retrieval accuracy from fitting 532 nm SP-CRDS σext data is < ± 0.001 for the size range of
hygroscopic response measurements performed in this paper
(> 1 µm), although our previous analysis neglected the
influence of sample impurities on the retrieved nλ (Cotterell et
al., 2016). As an example, sn473‾ and
sn532‾ take respective values of 0.0028 and
0.0015 for the measurements of NaNO3. The uncertainty of ±2 % in
the recorded RH probe value is calculated to contribute uncertainties of
0.0023 and 0.0021 to sn473‾ and
sn532‾, respectively. Therefore, any errors in
particle sizing from 473 nm PFs and noise in the recorded τ data make
only a small contribution to the nλ precision in the RH domain.
Parameterising the refractive index with variation in wavelength and
RH
The plots in Fig. 5 contain all the information to characterise the RI in
terms of both wavelength and RH. Here, we develop an RI model
that accounts for variations in both wavelength and RH, and this model is
fitted to the data in Fig. 5 to parameterise n(λ, RH). The Cauchy
equation is an empirical relation describing the wavelength dependence in the
RI and can be written as (David et al., 2016)
n=n0+∑i=1Nniλ0λ2i-1,
in which n0 is the RI at reference wavelength λ0 and ni is the dispersion coefficient. In our model, we find that
expansion of the summation in Eq. (6) to i= 1 is required only when
n=n0+n1λ0λ2-1.
To incorporate the RH dependence of the RI into Eq. (7), we
note that n0 and n1 are expected to be smooth functions of RH and
thus we parameterise n0 and n1 as polynomial functions of the water
activity. In our Cauchy model, we use λ0= 525 nm since this
wavelength is at the centre of the wavelength range (405–650 nm) over which
we have RI data available. The values for n0 and n1 are described
by the following quartic and cubic polynomial equations, respectively, in terms
of aw :
n0=n0,0+n0,1100aw+n0,2100aw2+n0,3100aw3+n0,4100aw4n1=n1,0+n1,1100aw+n1,2100aw2+n1,3100aw3.
The parameters (n0,x, …) and (n1,x, …) are fitted
concurrently by performing a least squares fit to Eq. (7) by minimising the
residual between the measured and literature data (n405, n473,
n532, n650, n589, n633) and the nλ generated
by the Cauchy model. The Microsoft Excel generalised reduced gradient non-linear engine is used
to accomplish this fitting, constraining the modelled nλ to the
value of pure water at RH = 100 % (Daimon and Masumura, 2007).
Figure 7 presents the results as contour plots for each inorganic solute of
interest. These contour plots represent the most comprehensive description of
RI for atmospherically relevant inorganic aerosol, fully characterising the
RI variation with both visible wavelength and RH. The best-fit parameters for
Eqs. (8) and (9) that describe these contour plots are summarised in Table 2.
The best-fit Cauchy dispersion curves (solid lines) for measured and
literature RI data (points) shown at 10 % RH intervals (labels). The
parameterisations were fitted to data at 2 % RH intervals, and the total
numbers of data points included in the parameterisation are specified in
Table 2. The curves are found by a global fit of the Cauchy dispersion model
(Eqs. 7–9) to the measured and literature data in Fig. 5 using the procedure
described in the text. The error bars represent the uncertainty in the RH
measurements, indicated in the text, in the RI domain.
Figure 8 compares the measured data points with the Cauchy model curves from
the aforementioned global fitting procedure (i.e. using the relevant
parameters in Table 2). This plot only shows data at 10 % RH intervals
for simplicity. The total number of literature and measured data points, N,
used in the fitting procedure are indicated for each solute in Table 2. The
agreement between the Cauchy model and data at RH = 100 % highlights
that the model is constrained to the RI of pure water. The error bars
represent the ±2 % uncertainty associated with RH measurements in the
nλ domain. Within these error bars, there is generally good
agreement between the measured data and the Cauchy model. In all cases
(except NH4HSO4, as previously discussed) the Cauchy model
describes the measured n405, n473 and n532 data well. There is
good agreement between n633 data and the Cauchy model at high RH.
However, the literature n633 values are lower than the calculated
n633 for NaCl and (NH2)4SO4 at low RH. These discrepancies
are attributed to the reduced number of literature and measured data points
for λ > 600 nm at the low RHs. The bulk n589 data
points are described well by the Cauchy model; two of the NH4HSO4
measured n589 values are lower than the modelled n589. The
n650 values from AOT measurements lie marginally lower than is expected
from the Cauchy model. This is also evident in Fig. 3 in which AOT measurements
(calculated at n633) were compared to those parameterised by Tang et
al. and implies that the RHs reported for each of the n650 AOT measurements
are systematically low by a value close to the 2 % uncertainty associated
with the RH measurements.
The overall agreement between the global Cauchy fit (nfit) and
the measured and literature values (nj) is quantified by evaluating the
mean RI difference:
Δn‾=1J∑j=1Jnfit-nj,
in which the summation is over all J measured–literature values. Table 2
summarises the values of Δn‾ for the
inorganic solutes studied here. The solutes NaCl, NaNO3 and
(NH4)2SO4 have values of Δn‾ ≤ 0.002, which indicates that the measured and literature data points lie very
close to the Cauchy parameterisation. The solutes Na2SO4 and
NH4HSO4 give Δn‾≈ 0.003 and
Δn‾≈ 0.0044, respectively. The
limited number of measured and literature data for Na2SO4 is a
consequence of the high efflorescence RH and results in an RI description that is not as well constrained, which is reflected in the
Δn‾ value.