Determining Number Concentrations and Diameters of PolystyreneParticles by Measuring the Effective Refractive Index of ColloidsUsing Surface Plasmon ResonanceJani Tuoriniemi,† Beatriz Moreira,† and Gulnara Safina*,†,‡

†Department of Chemistry and Molecular Biology, University of Gothenburg, Kemigarden 4, 412 96 Gothenburg, Sweden‡Division of Biological Physics, Department of Physics, Chalmers University of Technology, Kemigarden 1, 412 96 Gothenburg,Sweden

ABSTRACT: The capabilities of surface plasmon resonance (SPR) for character-ization of colloidal particles were evaluated for 100, 300, and 460 nm nominal diameterpolystyrene (PS) latexes. First the accuracy of measuring the effective refractive index(neff) of turbid colloids using SPR was quantified. It was concluded that forsubmicrometer sized PS particles the accuracy is limited by the reproducibility betweenreplicate injections of samples. An SPR method was developed for obtaining theparticle mean diameter (dpart) and the particle number concentration (cp) by fitting themeasured neff of polystyrene (PS) colloids diluted in series with theoretical valuescalculated using the coherent scattering theory (CST). The dpart and cp determined using SPR agreed with reference valuesobtained from size distributions measured by scanning electron microscopy (SEM), and the mass concentrations stated by themanufacturer. The 100 nm particles adsorbed on the sensing surface, which hampered the analysis. Once the adsorption problemhas been overcome, the developed SPR method has potential to become a versatile tool for characterization of colloidal particles.In particular, SPR could form the basis of rapid and accurate methods for measuring the cp of submicrometer particles indispersion.

■ INTRODUCTION

Measurement of the refractive index (n) of colloids is a usefulbut largely overlooked characterization principle. For inhomo-geneous materials such as particle dispersions the n is anef fective refractive index (neff) whose real part gives the phasevelocity (vp) of a light beam propagating through the material.The imaginary part takes into account losses due to bothabsorption and scattering. Zimm and Dandliker,1 and van derHulst2 developed models stating the neff of a colloid as afunction of the Mie scattering3 intensity in the forwarddirection and the volume or fill fraction of the particles ( f).These models are most accurate for low f and small particlediameters (dpart). The dpart and the n of the particles (npart) ofpolystyrene (PS) colloids can be determined accurately andsimultaneously using the Zimm−Dandliker model.4 Alimitation is that the specific turbidity as a function of theseparameters must be known. Sanchez-Perez et al. developed aparticle sizing method based on the van der Hulst model.5

However, its applicability is restricted because the numberconcentration (cp) must already be known, and the requirementto measure both the real and imaginary parts of the differencebetween the neff and the n of the dispersion media (nm).Marquez-Islas et al.6,7 developed another method fordetermining the npart, cp, and dpart based on the van der Hulstmodel. It requires the measurement of the neff of two samplesthat are diluted to the same concentrations with media havingdifferent nm. These parameters could be determined accurately.However, the dispersions must be dilute, and the dpart has to beconsiderably smaller than the wavelength.

The aim of this study is to develop a particle characterizationmethod based on the measurement of neff with surface plasmonresonance (SPR).8 The technique is sensitive and may resolvedifferences in n that are as small as 10−6 refractive index units inthe vicinity of surfaces. It has therefore found numerousapplications in biosensing, and practical instruments completewith liquid handling systems are widespread for this purpose.9

SPR has also been used for characterizing the optical propertiesof turbid industrial fluids,10,11 and the response to Au particlesadsorbed on surfaces has been quantified.12,13 However, athorough assessment of its capabilities to measure the neff ofcolloids is still lacking, and there are yet no practical methodsfor extracting the dpart and cp.In this study, these parameters are obtained by fitting

coherent scattering theory14−16 (CST) to the neff values ofcolloids measured using SPR. The CST contains the van derHulst model as its low concentration limit but is valid for higherconcentrations of particles. The CST has been experimentallyverified,16 and it is probably the most accurate theory that isavailable for predicting the neff of colloids. First the accuracy ofsuch characterization is assessed for PS colloids as a function oftheir concentration and sizes. The cp and dpart are subsequentlymeasured for 100, 300, and 460 nm nominal diameter PScolloids. These results are compared with reference valuesobtained by scanning electron microscopy (SEM). Finally,

Received: July 19, 2016Revised: September 23, 2016

Article

pubs.acs.org/Langmuir

© XXXX American Chemical Society A DOI: 10.1021/acs.langmuir.6b02684Langmuir XXXX, XXX, XXX−XXX

further developments are suggested to render SPR into an abletechnique for routine characterization of colloids.

■ THEORYThe reader is first provided with how to extract the neff from ameasured SPR resonance angle. This is followed by a recapitulation ofCST theory, and a description of the fitting procedure for determiningcolloid properties. The neff is calculated as a function of dpart and cp fora range of particle concentrations and sizes in order to delineate themethod for their determination.Measurement of neff with SPR. The experimental setup for SPR

is outlined in Figure 1a. A laser beam is directed on a thin (∼30 nm)Au layer twixt between the prism and the sample. When the angle ofincidence equals the resonance angle (θr) the beam matches theresonance condition of surface plasmons, which are oscillations inelectron density at the gold-sample interface. The θr is detected as adip in the reflectance curve (Figure 1b). The θr is related to the relativepermittivities (ε = n2 for nonmagnetic materials) of the sample and Auby the following equation.8

θε ε

ε ε=

+= − ′ ′

+n

ms m sm s

sin ( )real( )

real( )prism2 2

rmetal sample

metal sample (1)

Here the m and m′, and s and s′ are the real and imaginary parts of theε of the Au and the sample, respectively. The nprism is the refractiveindex of the prism (1.518). For colloidal samples, the s′ takes intoaccount attenuation of the beam due to scattering. When the turbidityis low enough for s′ to be neglected, it is possible to state the n of thesample (nsample) as a function of θr.

θ

θ=

−n

n m

m n

sin ( )

sin ( )sampleprism

2 2r

prism2 2

r (2)

The nsample determined by SPR is thus only accurate for transparentliquids. For turbid samples, the error increases with the imaginary part

of the refractive index. The magnitude of such bias for the PS colloidsmeasured here will be assessed below. The used SPR instrumentallows measuring θr values up to ∼78°. That corresponds to an n of∼1.38.

Modeling of SPR Data with CST. According to CST, the neff thatdescribes the propagation of a light beam in a particle dispersion isgiven by16

π π

θ= + +

−

π θ−

⎡⎣⎢⎢

⎤⎦⎥⎥

n nic S

k

c

kS d n k

S d n k

14 (0) 4

cos ( )( ( , , )

( , , ))

eff mp

s3

2p

2

s6 2

m2

2part p s

02

part p s

1/2

m

(3)

Here i is the imaginary unit ( −1 ). The cp is stated in m−3. The ks(m−1) is the wave vector in the sample given by nm2πλ0

−1, with the λ0being the wavelength (m) in vacuum. The θm is the angle ofpropagation in the sample with respect to an axis perpendicular to theAu layer. The S0 and Sπ−2θm are the scattering matrix elementscalculated by Mie theory for p polarized light in the directions ofpropagation, and specular reflection from a thought plane of particles,respectively. Cartoons illustrating the system can be found inreferences.14−16 In the experiments here, the laser beam hits the Aulayer at a high enough angle for total internal reflection to occur. Onlyan evanescent field penetrates into the sample. For an evanescent fieldthe θm is of the form π/2 + αi, where α is the magnitude of theimaginary part of the angle. The θm can be calculated from the θr bysequentially applying Snells law on the prism−Au, and Au−sampleinterfaces. Only the solution with a negative value of α has a physicalmeaning in this situation.

To illustrate the measurement principle for PS colloids, the neff wascalculated by eq 3 for dpart ranging between 0 and 1000 nm and fbetween 0 and 25% (Figure 1c,d). The f range covers and extendsbeyond the concentration ranges of the samples measured here. The fwas calculated from the cp assuming that the particles are spherical.

Figure 1. Measuring the effective refractive index (neff) by SPR, and calculating it as a function of particle diameter (dpart) and fill fraction ( f) usingthe coherent scattering theory (CST). (a) Surface plasmons (curved arrow) are induced in the Au layer when the laser beam is shined on it at theresonance angle (θr). (b) The reflectance is plotted as a function of the angle of incidence. The θr is detected as a dip in the reflectance curve. Here itis shown how the θr shifts toward higher angles as the refractive index increases with the concentration of the 460 nm polystyrene (PS) colloid.(Dilution factors: 0.1, blue; 0.5, solid red; and 1, dashed red). (c) The refractive index (right scale) calculated theoretically using CST as a function ofthe f and dpart. The dashed vertical lines indicate sizes for which the neff is calculated as a function of f in (d). The neff as a function of PS concentrationfollows a unique curve for each dpart. It is therefore possible to determine simultaneously both the dpart and number concentration (cp) by fitting CSTto neff values measured by SPR for a dilution series of colloids.

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Figure 1d shows that if a colloid is diluted to a concentration series,then the neff as a function of the dilution factor traces a curve that isunique for each dpart and cp of the undiluted stock. The dpartcharacterized here covers the range where the neff( f) changes froman approximately linear, to a more rapidly increasing function (Figure1d).The dpart and cp are determined by fitting eq 3 to the neff measured

by SPR. The procedure exploits that the cp are known fractions of thatin the stock dispersion. The θm is a function of θr. It depends thereforeon the neff that increases with cp. In order to simplify the calculations, itwas assumed that the θm was for all samples the angle that is producedwhen the θr is 70° and the neff 1.3334 (θm = π/2 − 0.36327i). Themagnitudes of the deviations from accurate CST theory brought bythis approximation are assessed below.

■ EXPERIMENTAL SECTIONChemicals. The particles were the 100, 300, and 460 nm nominal

diameter aqueous PS colloids obtained from Sigma-Aldrich (SaintLouis, MO). Concentration series of the colloids were diluted inultrapure water (Milli-Q, resistivity 18 MΩ·cm, Millipore, BillericaMA) and sonicated prior to measurements.Dilutions of the Colloids. The samples with concentrations

ranging from 10% to 80% of that in the undiluted colloid were madeby pipeting from the stock dispersion. The concentrations between 1%and 8% were made by 10 times dilution of the previously madesamples. Further dilutions were made analogously to the schemedescribed above.SPR Instrument. The measurements were conducted using a SPR

Navi 220 instrument (Bionavis, Ylojarvi, Finland). The Au coatedsubstrates were provided by the instrument manufacturer. The usedwavelength was 670 nm.Measurements. The gold substrates were cleaned by boiling them

during 10 min in a 1:1:5 mixture of 25% NH3, 30% H2O2, and Milli-Qwater prior to measurements. The substrates were rinsed in ultrapurewater and mounted into the instrument. After measuring a blank ofultrapure water, the colloids were injected, measured, and removed byrinsing with ultrapure water. The injections started from the lowest,and proceeded in order of increasing concentrations.Calculations. The θr was extracted using the instrument software.

All subsequent computations were performed in the Matlab R2015asoftware. The Mie scattering calculations were done using the Matlabcode suite by Maztler.17

Materials Properties. The n of gold (0.1457 + 3.7614i) was takenfrom a fit to the data of Johnsson and Christy.18 The npart of the PScolloids (1.5811) was taken from Ma et al.19 The density of theparticles was 1.053 g cm−3.20 The concentration of these colloids wasstated as 10 wt % by the manufacturer. This would imply an f of 9.5%.Scanning Electron Microscopy. The colloids were diluted in

ultrapure water and sonicated for 30 min prior to sample preparation.Droplets of 1 μL, 5 μL, and 10 μL of each suspension were placed onthe centers of 15 nm nominal cut off Whatman NucleporePolycarbonate filters obtained from Sigma-Aldrich (Saint Louis,MO). The filters were placed in clean sterilized Petri dishes thatwere sealed after the droplet evaporated. Secondary electron imageswere acquired using a FEI Quanta 200 variable pressure SEM with alarge field detector. The accelerating voltage was kept at 20 kV, and thechamber pressure was 0.5−0.6 Torr in the low vacuum mode. Fiji/ImageJ software was used for measuring the particle size distributionfrom the micrographs. The particle diameters were obtained bymanually measuring the cross sections using the line tool.

■ RESULTS AND DISCUSSION

This section is divided into two parts. In the first, calculationsare carried out to ensure that the SPR method and CST areapplicable for the measured colloids. In the second, the dpart andcp are measured with SPR and compared with reference valuesobtained by SEM.

Determining the Validity Range of SPR Based neffMeasurement and CST. Before proceeding to the character-ization of the test colloids it is necessary to ascertain thatneither the SPR method, nor the CST calculations are afflictedwith any significant bias in the investigated concentration andsize ranges. The first error to be assessed is that brought by theturbidity.

Error Due to Turbidity. Equation 3 was used to calculatethe complex neff for PS colloids with dpart of 100, 300, 460, and1000 nm and f reaching up to 25%. The largest colloid wasincluded in order to investigate the factors limiting theapplicability range of the SPR method. The computed valueswere used for calculating the θr by eq 1. The next step was toinsert the recently calculated θr into eq 2 to obtain the apparentneff that would be measured by SPR. If the imaginary part of theneff calculated by eq 3 is negligible, then the apparent neff, i.e.,the neff as it seems from the measurements, will equal the valueof the real part of the actual neff. Otherwise the apparent neffbecomes biased. The relative bias, defined as (real(neff) −neff‑apparent)/real(neff), is presented in Figure 2a. Negative valuesmean that the neff measured by SPR is overestimated, while forpositive bias the measurements provide smaller than the actualvalues.

The magnitude of the bias due to turbidity should becompared with the uncertainty inflicted by the variation amongreplicate measurements. The relative standard deviation in neffamong repeated injections of particle dispersions was found tobe ∼0.056%, which means that the neff of a single measurementis surrounded by a 95% confidence interval (Δ) of ∼± 0.11%due to random error. Figure 2a shows that the bias brought bylight scattering tends to increase with f and dpart as the colloidsbecome more turbid. However, it is not likely to become the

Figure 2. (a) Relative bias in effective refractive index (neff) measuredby SPR due to turbidity. (b) Relative bias in neff calculated by CST dueto using a fixed value of the propagation angle in the sample (θm).

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accuracy limiting factor for the colloids characterized herehaving f ≤ 9.5%.Error Due to Using a Fixed Value of θm. Simplified

calculations using a fixed value of θm lead to deviations fromaccurate CST. The relative bias ((neff‑exact − neff‑approximate)/neff‑exact) introduced into calculated neff values was assessed bythe following procedure. First the neff was calculated as afunction of f for a range of fixed θm (68°, 69°, 70°, 71°, 72°,73°, 74°, 75°, 76°, and 77°). For each tested θm, the CSTprovides the exact neff for that f which produces thecorresponding θr. These accurate values were compared withthose calculated for the same f using the fixed θm that waschosen for fitting the experimental data. The relative bias isshown for a range of dpart and f in Figure 2b. The approximationis valid for the dispersions characterized here in the sense thatthe bias is unlikely to exceed the uncertainty due toreproducibility. However, applying it when fitting the neff oflarger colloids (e.g., the 1000 nm polystyrene beads) for whichthe angle dependency of scattering is more prominent couldproduce spurious results.Validity Range of Mie Theory. Mie theory assumes that

the light scattering is not influenced by neighboring particles.Equation 3 is therefore only valid for dispersions dilute enoughfor the cp to be in the independent scattering regime where theparticles do not influence their neighbors. For moreconcentrated dispersions in the dependent scattering regime,the amplitude of scattered light decreases because ofinterference with light scattered by other particles in thevicinity. The risk that photons already scattered in somedirection undergo a second scattering event also increases.21−24

Hottel et al.23 stated based on empirical observations that thedependent scattering regime is limited by c/λ0 < 0.3 and lpart/rpart < 2.8. Here c is the interparticle clearance, which is theaverage distance between the surfaces of two neighboringparticles. The lpart is the mean distance between particle centers,and rpart is the particle radius. It has been noted that thisempirical criterion coincides with the c where the near fields ofeach scattering particle starts to overlap with that of theirneighbors.25 The criterion thus conforms to physical theory. Ifthe criteria of Hottel et al. is combined with the approximateexpression for the lpart as a function of f given by eq 6.12 inTorquato et al.,26 then the range of f for which the computed Sare not reliable is given by the following set of inequalities:

λ−−

<d f

f f

(1 )

12 (2 )0.3part

3

0

+−

−< → >

ff f

f2(1 )

6 (2 )2.8 0.0835

3

(4)

The onset of dependent scattering calculated by inequalities 4 isshown in Figure 3. The undiluted PS samples having an f of9.5% are predicted to be in the dependent scattering regime.On the other hand, an evanescent field only probes a thin sliceof the sample and interferences between particles are likely tobe less important.27 It is therefore possible that the range ofindependent scattering is extended for the SPR setup.Validation of SPR Based Particle Characterization. In

previous section it was concluded that SPR is likely toaccurately measure the neff for the test colloids, and that thevalues can be fitted by CST. Next the capabilities of SPR basedparticle characterization are evaluated.

The measured neff of the PS colloids fitted by CST are shownin Figure 4. The highest measured concentration was for eachsize the undiluted colloid. During calculations the values of dpart,and the cp in the nondiluted sample, were first adjusted

Figure 3. Theoretically calculated regimes of dependent andindependent scattering. The line demarcating the dependent andindependent scattering is dependent on the refractive index of neitherthe particle nor the dispersion media, and therefore is valid generallyfor Mie scattering at a wavelength of 670 nm.

Figure 4. Coherent scattering theory (CST) (black solid line) fitted tothe experimental effective refractive index (neff) values obtained bySPR (blue squares) for the 100, 300, and 460 nm nominal diameterpolystyrene (PS) colloids.

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manually until the neff as a function of dilution factor roughlyfitted to the results. The fits were then refined using theLevenberg−Marquardt algorithm. The n of water is a relativelystrong function of temperature.28 It was therefore most reliableto allow the nm to vary as a fit parameter during the CSTcalculations. The dpart measured by SPR is here called the neffweighted average diameter (dpart,n). The determined values arelisted in Table 1 that summarizes the information on dpart andcp.Electron micrographs of the PS particles are shown in Figure

5. The particle size distributions (PSD) extracted from theseimages are found in Figure 6. The particles are spherical. It istherefore possible to compare the measured dpart,n withreference values calculated from the PSDs.Calculated Reference Values for dpart,n. The first step in

obtaining the reference values was to calculate for a given cp theneff produced by the SEM based PSDs. The next step was tofind the dpart,n that reproduces the previously calculated neff formonodisperse colloids having the same cp. The neff for the PSDswere computed by inserting the sums of the S containing termscalculated for each size fraction into eq 3 (For the diameter dranging from a to b: Σd=a

d=bCpdS0d, and Σd=ad=bCpd

2(Sπ−2θmd2 − S0d

2)).Because the neff as a function of cp follow different curves foreach size fraction, the calculated dpart,n could depend on theassumed cp. However, such variation was found to be too smallto be significant within the size and concentration ranges of themeasured colloids.The particle count in SEM was statistically significant with

the PSDs containing enough particles to reduce the relateduncertainty in the number weighted dpart (dpart‑num) to <1%.29

However, possible error in magnification calibration andimprecision in locating the edges of the particles limit theaccuracy of SEM to ±7%.29 Therefore, the 95% Δ in thereference values for dpart,n were calculated by shifting thenumber based PSDs ± 7%.

The calculated reference values are both listed in Table 1,and shown together with the measured dpart,n in Figure 6. The

Table 1. Properties of Polystyrene (PS) Colloidsa

100 300 460

SEMdpart‑num, nm

b 79 ± 6 287 ± 20 418 ± 29cpc (3.5 ± 0.8) × 1020 (7.3 ± 1.7) × 1018 (2.0 ± 0.5) × 1018

calculated SPRdpart,n, nm

d 83 ± 6 292−18+24e 420 ± 30

measured SPRdpart,n, nm 86f 236 ± 3 430 ± 21cp 3.5 × 1020f (7.2 ± 0.08) × 1018 (2.0 ± 0.1) × 1018

aThe uncertainty values are 95% confidence intervals. cp = number concentration. dpart = particle diameter. bParticle number-averaged diametermeasured by SEM. cCalculated from the concentration provided by the manufacturer and the SEM based particle size distribution (PSD).dCalculated from the SEM based PSD. eThe uncertainty interval is asymmetric. fUncertainties not possible to calculate because error is not randomdue to adsorption of colloids.

Figure 5. SEM micrographs of the polystyrene (PS) particles. Scale bar indicates 2 μm.

Figure 6. Size distributions (PSD) for polystyrene (PS) colloidsmeasured by SEM. Solid vertical lines indicate the refractive indexweighted diameter (dpart,n) determined using SPR. Dashed linesindicate the calculated reference values.

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E

Table

2.Com

parisonof

Selected

Metho

dsCom

mon

lyUsedforCharacterizingParticles

inDispersiona

measuredphysicalpa-

rameter

method

PSD/d

part

c pn p

art

properties

comments

dynamiclight

scat-

terin

g(D

LS)38

bc

chydrodynam

icdiam

etersobtained

from

diffusioncoeffi

cients;size

range1nm

to10

μm;

concentrationrangefrom

∼0.00001to

0.01

wt%

towhere

errordueto

multip

lescatterin

gand

interactions

betweenparticlesbecomes

toosevere

at1−

40wt%

themostaccurate

determ

inationof

d partrequiresmeasurin

gitforaseriesof

concentrations

and/or

scatterin

gangles

andextrapolatingforzero

angleand

concentration;

rapidanalysis;allscatterin

gtechniques

areineffi

cientin

opaque

media

andsuffer

from

interferencesfrom

,e.g.,dustandairbubbles

staticlight

scatter-

ing(SLS

)38and

laser

diffraction3

8,39

bc

cdiam

etersderived

from

angledependence

oflight

scatterin

g;size

rangefrom

∼10

nmto

∼1mm

dependingon

experim

entalsetup;concentrationrangereaching

from

0.00001to

0.01

wt%

tothe

onsetof

multip

lescatterin

gevents;givesshapeinform

ation

mostdata

evaluatio

nschemes

relyon

assumptionof

sphericalshape

ultrasonicattenua-

tion

spectroscopy

40

bb

cpropertiesextractedfrom

thefrequencydependence

ofultrasonicattenuationcoeffi

cient;size

range

10nm

-1000μm

;works

forconcentrated

samples

upto

60wt%

works

forturbidandabsorbingsamples;d

ataevaluatio

nrequiresseldom

availabledataon

thermophysicalp

roperties;interferencesfrom

airbubbles;lowresolutio

nin

d part;rapid

analysis

nanoparticletrack-

inganalysis

(NTA)41

bb

bcountin

gmethod;

diffusioncoeffi

cientsmeasuredusinglaserbasedultram

icroscopy;

size

range

dependingon

materialfrom

15−50

to1000

nm;concentrationrange10

12−10

16particlesm

−3 ;a

methodforestim

atingn p

artfrom

thescatteredlight

intensity

hasbeen

developed4

2

bias

inc pandPS

Dpossibleifopticalpropertiesof

samplediffer

from

c pcalibratio

nstandard;results

depend

onsubjectiveimageprocessing

parameters

spICP-MS4

3b

bc

countin

gmethod;

individualparticlesdetected

assignalspikes

inICP-MS;

tracelevelconcentratio

nsof

108 −10

10particlesm

−3or

downto

100ppt;sizesas

smallas

∼10

nmforAu;

higher

forother

elem

entsandmaterials;chem

icalselectivity

suitableforparticlesof

metalsandoxides

forelem

entsforwhich

theICP-MSissensitive;

unsuitablefororganiccolloids

opticalparticle

counter44

bb

ccountin

gmethod;

particlespassingthroughalight

beam

aredetected

asflashesof

scatteredlight;the

PSD

canbe

inferred

from

theirintensities;size

range0.05−2000

μmcalibratio

nwith

know

nc pstandard

required;

sensitivity

might

depend

onparticlesoptical

properties,andthey

should

thereforematch

thoseof

thesample;

rapidanalysis

smallangleX-ray

scatterin

g(SAXS)

45

bb

cpropertiesextractedfrom

X-ray

scatterin

gtheory;sizerange0.1−

1000

nm;concentratio

nrange0.1−

10wt%;obtainingshapeinform

ationispossible

lowsignalforelem

entswith

lowatom

icnumber;expensivebulkyequipm

entor

accessto

synchrotronfacilitiesrequired

scanning

mobility

particleanalyzer

(SMPS

)46

bb

ccountin

gmethod;

size

classificatio

nbasedon

mobility

inan

electricfield;

subsequent

countin

gwith

condensatio

nnucleuscounter;size

range0.001−

1μm

;concentrationrange109−10

13mL−

1calibratio

nwith

know

nc pstandard

requiredforconcentrationmeasurement;fastand

accurate

sizing

turbimetry36

bb

bbasedon

accessingtheMiescatterin

gfunctio

nviameasuredscatterin

gcrosssections;size

range

0.035−

50μm

;concentrations

∼<0.2wt%

tediousdata

evaluatio

n;requiring

onlyaspectrophotometer

SPR

bd

bbe

diam

etersandc paredeterm

ined

viaMiescatterin

gmatrix

elem

entsextractedfrom

theconcentration

dependence

ofn e

ff;forpolystyrened p

art∼<1μm

;polystyrenecolloid

concentrations

>1wt%

relieson

assumptionof

sphericalshape;

thecapabilitieshave

yetnotbeen

completely

assessed

ac p=numberconcentration,d p

art=particlediam

eter,n

part=refractiveindexofparticle,and

PSD=particlesize

distrib

ution.bIndicatesthatparametercanbe

measured.c Indicates

thatparametercannotb

emeasured.

dAverage

d partonly.eIn

principlepossible,h

owever

notyettested

experim

entally.

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CST predicts the dpart,n to be rather close to the modal values ofthe PSDs.Calculated Reference Values for the cp. Reference values

for the cp were calculated from the SEM based PSDs and massconcentrations stated by the manufacturer. This was done byfinding the cp that produces a mass concentration of 10% whensumming the masses contained in each size fraction. The 7%uncertainty in dpart is propagated cubed to the cp, and its rootsum of squares with the ±5% uncertainty in the reported massconcentration result in a 95% Δ of ±23%. The values are listedin Table 1.Accuracy of dpart and cp Measured by SPR. For the 100

nm particles the measured neff were for low concentrationshigher than what the CST predicts (Figure 4). A likely reason isadsorption of particles on the Au surface. Particles of smallersize have higher diffusion coefficients and surface-to-volumeratios.30 This increases the rate of encounters and probability tostick on surfaces. The adsorption rates (m−2 s−1) have beenconsequently reported to increase with decreasing size.31

Therefore, no adsorption was detected here for the 300 and460 nm PS colloids. However, Meeten and North encounteredspurious adsorption when measuring the neff of 500 nm PScolloids with a critical angle refractometer.32 Adsorption couldthus hamper the characterization of colloids in this size range aswell. The adsorption was reversible because the neff returned tothe value of pure water when the measurement cell was rinsedbetween the samples. Note that the sedimentation of the PScolloids is too slow to play any role within the time scale of themeasurements. The CST agrees better with experimental datafor the three highest concentrations measured. This could bebecause the particles in dispersion outnumbered the adsorbedones. These samples govern the fitted values of the dpart,n and cp(Figure 6, Table 1). They are within the margin of uncertaintyof the reference values.For the 300 and 460 nm particles, the fitted CST agrees with

the measured neff in the whole concentration range (Figure 4).The differences between the values predicted by CST and theneff measured by SPR do not for any sample of the 300 nmcolloid exceed 0.06%, while the deviations amount up to ∼3%for some samples of the 460 nm colloid.For these two larger colloids, it might be valid to assume that

the measured neff values are normally distributed around thefitted CST. In this case, the 95% Δ in the fitted dpart and cp aregiven by33

Δ = ± −J J1.96ssqdf

( )t 1

(5)

Here ssq is the sum of squares of the residuals. The df is thedegrees of freedom, which is the difference between thenumber of samples measured and the fitted parameters. The Jand Jt respectively are the vectors containing the derivatives ofeq 3 with respect to dpart or cp evaluated at the measuredconcentrations, and its transponate. The 1.96 is the z-score ofthe normal distribution for calculating 95% of the area under aGaussian curve.The Δ of dpart,n are ±1.1% and ±4.9% for the 300 and 460

nm particles respectively, while those of the cp are ±1.2% and±6.9%, respectively (Table 1). The random error limitedaccuracy of refractive index based particle sizing can thus becompared with the 0.8%29 for ∼35 nm silica, 0.3−1.8%34 for∼10−200 nm Au and polystyrene, and 1−8.3%35 for 20−100nm polystyrene colloids reported for DLS. Turbidimetry that isalso based on scattering theory reaches an accuracy of ∼2%.36

The dpart,n measured for the 460 nm particles is only 2.3% largerthan its reference value, which is not a statistically significantdifference. However, despite of the better CST fit, the dpart,nmeasured for the 300 nm colloid is significantly (∼19%)smaller than the calculated value. This indicates that we do notyet have full understanding of all aspects of the measurementprocess.All of the cp determined by SPR are within the confidence

intervals of the reference values (Table 1). Measuring the neffcould therefore be an alternative to turbidimetry that alsoprovides accurate values for the cp.

37

No deviations from CST attributable to dependent scatteringeffects were observed for the undiluted samples whose cp arepredicted to be above the onset of dependent scattering. Thissupports the arguments for that evanescent fields can probemore concentrated dispersions than conventional setups relyingon light beams transmitted through the sample. However,dedicated studies are required for confirming this.

■ CONCLUSION AND OUTLOOKIt has been shown that both dpart and cp can be determined byfitting CST to the neff measured by SPR for a dilution series.The determined values agreed with reference values determinedby SEM, except for the dpart of the 300 nm colloid that was foran unknown reason smaller than expected. This indicates thatthe description of the measurement process and error sourcesgiven here is not complete. Further studies are thereforeneeded to deepen the understanding of how SPR signals fromcolloids arise, and to investigate the feasibility of determiningalso the npart and nm by including them among the fitparameters. It was observed that the 100 nm particles adsorbedon the sensing surface. Although it had only little influence onthe results here, the risk of adsorption must be reduced in orderto render the SPR method reliable.For the PS colloids investigated here, the f must exceed ∼1%

to produce detectable shifts in neff. Although this study providedonly limited experimental data on the applicability range ofSPR, some guidelines can be drawn for future studies. Thesmallest detectable concentration is likely to decrease withincreasing dpart or npart as the particles scatter more light. On theother hand, at the same time the highest concentration, forwhich the error due to turbidity stays within acceptableboundaries, also decreases. Furthermore, the angular depend-ency of light scattering increases with dpart, and the error due toassuming a fixed value of θm could therefore impede theanalysis of micron sized and larger particles. The concentrationmust also at any time be within the range of independentscattering. The results presented here indicate that this rangemight be extended for an evanescent field setup, though furtherstudies are required to confirm this.In Table 2, the properties of SPR based particle character-

ization are compared with more established techniques.36,38−46

Determining the dpart by SPR is limited to relativelyconcentrated dispersions, because it requires f > ∼1% toobtain sufficient changes in neff. SPR is probably moreinteresting as a tool for measuring the cp, considering that theaccuracy of most particle counting methods rely on calibrationwith optically similar cp standards. Measurements with singleparticle inductively coupled plasma mass spectrometry (spICP-MS) are both absolute and accurate. However, this techniqueonly works for trace level concentrations. The dilution step hasbeen found to cause discrepancies with reference cp obtained atorders of magnitude higher concentrations.47,48 SPR and other

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techniques based on scattering theory are therefore attractivefor developing improved metrology for the cp.The main hurdle that needs to be overcome before SPR

based particle characterization can be applied to real analyticaltasks is the propensity of particles to adsorb on surfaces.Because gold is easy to functionalize with different monolayers,a possible direction could be to develop surface coatings thatreduce particle adsorption. Because the evanescent field reachesdeeper into the sample for longer wavelengths, infrared SPR49

could be interesting because the relative importance of theparticles in contact with the surface would diminish. A biggerleap in this direction would be to directly measure the angle ofpropagation of a laser beam through the sample. Such a devicehas been constructed where the beam passes through a hollowprism filled with colloid.50 The need to prepare several manualdilutions of the sample in series is time-consuming. However,one could envision continuous online dilution of the sample ina flow injection system that would rapidly generate a largenumber data points for the curve fitting. It might be possible toeliminate the need to dilute the sample altogether if the neff wasmeasured for different wavelengths with, e.g., a tunable laser.Because there are indications that dependent scattering effectsare suppressed for evanescent fields, such a SPR system couldbe used for characterizing more concentrated colloids thanwhat is possible with for instance using DLS.Modeling SPR data with CST might find biological

applications in monitoring processes that involve submicrom-eter cell compartments in living cells grown on the sensor slide.SPR is a very versatile technique in this niche because it is inprinciple possible to follow any kind of biological processresulting in morphological or chemical changes within the reachof the evanescent field. The applications include, e.g.,monitoring of the cholesterol content of cell membranes,49

and swelling of cells upon exposure to drugs.51 So far, theinterpretation of the shifts in neff have been based on theassumption that the cells can be considered to be composed oflayers of homogeneous material with different n parallel to thesensing surface. In such a case, the angle shifts that result fromthe modeled process can be calculated using the Fresnelequations.51 However, the scope of this approach is limitedbecause, for instance, vesicles, large protein compartments, andbulges in the cell membrane are more pertinent to beconsidered as particle-like scatterers of light. The authors arecurrently investigating the potential of SPR to measure the rateof neurotransmitter exocytosis by quantifying the vesicles in thevicinity of the cell membrane.

■ AUTHOR INFORMATION

Corresponding Author*E-mail: gulnara.safina@chem.gu.se.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

The work has been supported by the Swedish Research Council(Young Investigator Grant 621-2011-4395), The Olle EngkvistByggmastare Foundation (Grant 2012/428), and The RoyalAcademy of Sciences (Grant FOA12 V-111). B.M. is grateful tothe Erasmus+ students exchange programme for financialsupport.

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