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Determining Number Concentrations and Diameters of Polystyrene Particles by Measuring the Effective Refractive Index of Colloids using Surface Plasmon Resonance Jani Tuoriniemi, Beatriz Moreira, and Gulnara Safina Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b02684 • Publication Date (Web): 23 Sep 2016 Downloaded from http://pubs.acs.org on September 25, 2016
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Determining Number Concentrations and Diameters of Polystyrene Particles by Measuring the Effective Refractive Index of Colloids using Surface Plasmon Resonance Jani Tuoriniemi1, Beatriz Moreira1, Gulnara Safina1,2* 1
2
Department of Chemistry and Molecular Biology, University of Gothenburg, Kemigården 4, 412 96 Gothenburg, Sweden
Division of Biological Physics, Department of Physics, Chalmers University of Technology, Kemigården 1, 412 96 Gothenburg, Sweden *Corresponding author. Email:
[email protected] Abstract The capabilities of surface plasmon resonance (SPR) for characterization of colloidal particles were evaluated for 100, 300, and 460 nm nominal diameter polystyrene (PS) latexes. First the accuracy of measuring the effective refractive index (neff) of turbid colloids using SPR was quantified. It was concluded that for submicron sized PS particles the accuracy is limited by the reproducibility between replicate injections of samples. An SPR method was developed for obtaining the particle mean diameter (dpart) and the particle number concentration (cp) by fitting the measured neff of polystyrene (PS) colloids diluted in series with theoretical values calculated using the coherent scattering theory (CST). The dpart and cp determined using SPR agreed with reference values obtained from size distributions measured by scanning electron microscopy (SEM), and the mass concentrations stated by the manufacturer. The 100 nm particles adsorbed on the sensing surface, which hampered the analysis. Once the adsorption problem has 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 submicron particles in dispersion. Introduction Measurement of the refractive index (n) of colloids is a useful but largely overlooked characterization principle. For inhomogeneous materials such as particle dispersions the n is an effective refractive index (neff) whose real part gives the phase velocity (vp) of a light beam propagating through the material. The imaginary part takes into account losses due to both absorption and scattering. Zimm and Dandliker,1 and van der Hulst2 developed models stating the neff of a colloid as a function of the Mie scattering3 intensity in the forward direction and the volume or fill fraction of the particles (f). These models are most accurate for low f and small particle diameters (dpart). The dpart, and the n of the particles (npart) of polystyrene (PS) colloids can be determined accurately and simultaneously using the Zimm-Dandliker model.4 A limitation is that the specific turbidity as a function of these parameters must be known. Sánchez-Pérez et al. developed a particle sizing method based on the van der Hulst model.5 However, its applicability is restricted because the number concentration (cp) must already be known, and the requirement to measure both the real and imaginary parts of the difference between the neff and the n of the dispersion media (nm). Marquez-Islas et al.6-7 developed another method for determining the npart, cp, and dpart based on the van der Hulst model. It requires the measurement of the neff of two samples that are diluted to the same concentrations with media having different nm. These parameters could be determined accurately. However, the dispersions must be dilute, and the dpart has to be considerably smaller than the wavelength.
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The aim of this study is to develop a particle characterization method based on the measurement of neff with surface plasmon resonance (SPR).8 The technique is sensitive and may resolve differences in n that are as small as 10-6 refractive index units in the vicinity of surfaces. It has therefore found numerous applications in biosensing, and practical instruments complete with liquid handling systems are widespread for this purpose.9 SPR has also been used for characterizing the optical properties of turbid industrial fluids,10-11 and the response to Au particles adsorbed on surfaces has been quantified.12-13 However, a thorough assessment of its capabilities to measure the neff of colloids is still lacking, and there are yet no practical methods for extracting the dpart and cp. In this study these parameters are obtained by fitting coherent scattering theory14-16 (CST) to the neff values of colloids measured using SPR. The CST contains the van der Hulst model as its low concentration limit but is valid for higher concentrations of particles. The CST has been experimentally verified,16 and it is probably the most accurate theory that is available for predicting the neff of colloids. First the accuracy of such characterization is assessed for PS colloids as a function of their concentration and sizes. The cp and dpart are subsequently measured for 100, 300, and 460 nm nominal diameter PS colloids. These results are compared with reference values obtained by scanning electron microscopy (SEM). Finally, further developments are suggested to render SPR into an able technique for routine characterization of colloids. Theory The reader is first provided with how to extract the neff from a measured SPR resonance angle. This is followed by a recapitulation of CST theory, and a description of the fitting procedure for determining colloid properties. The neff is calculated as a function of dpart and cp for a range of particle concentrations and sizes in order to delineate the method 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 of incidence equals the resonance angle (θr) the beam matches the resonance condition of surface plasmons, which are oscillations in electron density at the gold-sample interface. The θr is detected as a dip in the reflectance curve (Figure 1b). The θr is related to the relative permittivities (ε=n2 for non-magnetic materials) of the sample and Au by the following equation.8
2 2 = + =
−′′ +
(1)
Here the m and m’, and s and s’ are the real and imaginary parts of the ε of the Au and the sample respectively. The nprism is the refractive index of the prism (1.518). For colloidal samples the s’ takes into account attenuation of the beam due to scattering. When the turbidity is low enough for s’ to be neglected, it is possible to state the n of the sample (nsample) as a function of θr. "
! = % "
# " $
! #
" $
(2)
The nsample determined by SPR is thus only accurate for transparent liquids. For turbid samples the error increases with the imaginary part of the refractive index. The magnitude of such bias for the PS colloids measured here will be assessed below. The used SPR instrument allows measuring θr values up to ~78o. That corresponds to an n of ~1.38.
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Figure 1. Measuring the effective refractive index (neff) by SPR, and calculating it as a function of particle diameter (dpart) and fill fraction (f) using the coherent scattering theory (CST). a) Surface plasmons (curved arrow) are induced in the Au layer when the laser beam is shined on it at the resonance 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 it is shown how the θr shifts towards 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 of the 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 concentration follows a unique curve for each dpart. It is therefore possible to determine simultaneously both the dpart and number concentration (cp) by fitting CST to neff values measured by SPR for a dilution series of colloids.
Modeling of SPR data with CST. According to CST the neff that describes the propagation of a light beam in a particle dispersion is given by:16 neff= &1 +
()#* +,
-.
+
() " * " / - *0" $!
2 1)%2$ 3456 , , 8 9 − 1,2 3456 , , 8 9
!
(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 nm2π;%< , , with the λ0 being the wavelength (m) in vacuum. The θm is the angle of propagation in the sample with respect to an axis perpendicular to the Au layer. The S0 and Sπ-2θm ACS Paragon Plus Environment
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are the scattering matrix elements calculated by Mie theory for p polarized light in the directions of propagation, and specular reflection from a thought plane of particles respectively. Cartoons illustrating the system can be found in references.14-16 In the experiments here, the laser beam hits the Au layer at a high enough angle for total internal reflection to occur. Only an evanescent field penetrates into the sample. For an evanescent field the θm is of the form π/2+αi, where α is the magnitude of the imaginary part of the angle. The θm can be calculated from the θr by sequentially applying Snells law on the prism–Au, and Au–sample interfaces. Only the solution with a negative value of α has a physical meaning in this situation. To illustrate the measurement principle for PS colloids, the neff was calculated by Eq. 3 for dpart ranging between 0-1000 nm and f between 0-25% (Figure 1c-d). The f range covers, and extends beyond the concentration ranges of the samples measured here. The f was calculated from the cp assuming that the particles are spherical. 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 is unique for each dpart and cp of the undiluted stock. The dpart characterized here covers the range where the neff(f) changes from an approximately linear, to a more rapidly increasing function (Figure 1d). 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 that in the stock dispersion. The θm is a function of θr. It depends therefore on the neff that increases with cp. In order to simplify the calculations, it was assumed that the θm was for all samples the angle that is produced when the θr is 70o and the neff 1.3334 (θm =π/20.36327×i). The magnitudes of the deviations from accurate CST theory brought by this approximation are assessed below.
Experimental Chemicals. The particles were the 100, 300, and 460 nm nominal diameter aqueous PS colloids obtained from Sigma Aldrich (Saint Louis MO, USA). Concentration series of the colloids were diluted in ultrapure water (Milli-Q, resistivity 18 MΩ·cm, Millipore, Billerica MA, USA) 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 made by pipetting from the stock dispersion. The concentrations between 1 % to 8 % were made by ten times dilution of the previously made samples. Further dilutions were made analogously to the scheme described above. SPR instrument. The measurements were conducted using a SPR Navi 220 instrument (Bionavis, Ylöjärvi, Finland). The Au coated substrates were provided by the instrument manufacturer. The used wavelength 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-Q water prior to measurements. The substrates were rinsed in ultrapure water and mounted into the instrument. After measuring a blank of ultrapure water, the colloids were injected, measured, and removed by rinsing 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 R2015a software. The Mie scattering calculations were done using the Matlab code suite by Mäztler.17 Materials properties. The n of gold (0.1457 + 3.7614i) was taken from a fit to the data of Johnsson and Christy.18 The npart of the PS colloids (1.5811) was taken from Ma et al.19 The density of the particles was 1.053 g cm-3.20 The concentration of these colloids was stated 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 minutes prior to sample preparation. Droplets of 1 µL, 5 µL, and 10 µL of each suspension were ACS Paragon Plus Environment
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placed on the centers of 15 nm nominal cut off Whatman Nuclepore Polycarbonate filters obtained from Sigma Aldrich (Saint Louis MO, USA). The filters were placed in clean sterilized Petri dishes that were sealed after the droplet evaporated. Secondary electron images were acquired using a FEI Quanta 200 variable pressure SEM with a large field detector. The accelerating voltage was kept at 20 kV, and the chamber pressure was 0.5-0.6 torr in the low vacuum mode. Fiji/ImageJ software was used for measuring the particle size distribution from the micrographs. The particle diameters were obtained by manually measuring the cross sections using the line tool.
Results and discussion This section is divided into two parts. In the first calculations are carried out to assure that the SPR method and CST are applicable for the measured colloids. In the second the dpart and cp are measured with SPR and compared with reference values obtained by SEM. Determining the validity range of SPR based neff measurement and CST Before proceeding to the characterization of the test colloids it is necessary to ascertain that neither the SPR method, nor the CST calculations are afflicted with any significant bias in the investigated concentration and size ranges. The first error to be assessed is that brought by the turbidity. Error due to turbidity. Eq. 3 was used to calculate the complex neff for PS colloids with dpart of 100, 300, 460, and 1 000 nm and f reaching up to 25 %. The largest colloid was included in order to investigate the factors limiting the applicability range of the SPR method. The computed values were used for calculating the θr by Eq. 1. The next step was to insert the recently calculated θr into Eq. 2 to obtain the apparent neff that would be measured by SPR. If the imaginary part of the neff calculated by Eq. 3 is negligible, then the apparent neff, i.e. the neff as it seems from the measurements, will equal the value of the real part of the actual neff. Otherwise the apparent neff becomes biased. The relative bias, defined as (Real(neff) - neff apparent) /Real(neff), is presented in Figure 2a. Negative values mean that the neff measured by SPR is overestimated, while for positive bias the measurements provide smaller than the actual values. The magnitude of the bias due to turbidity should be compared with the uncertainty inflicted by the variation among replicate measurements. The relative standard deviation in neff among repeated injections of particle dispersions was found to be ~ 0.056 %, which means that the neff of a single measurement is surrounded by a 95 % confidence interval (∆) of ~ ±0.11 % due to random error. Figure 2a shows that the bias brought by light scattering tends to increase with f and dpart as the colloids become more turbid. However, it is not likely to become the accuracy limiting factor for the colloids characterized here having f ≤9.5%. Error due to using a fixed value of θm. Simplified calculations using a fixed value of θm lead to deviations from accurate CST. The relative bias ((neff exact - neff approximate )/neff exact) introduced into calculated neff values was assessed by the following procedure. First the neff was calculated as a function of f for a range of fixed θm (68o, 69o, 70o, 71o, 72o, 73o, 74o, 75o, 76o, and 77o). For each tested θm the CST provides the exact neff for that f which produces the corresponding θr. These accurate values were compared with those calculated for the same f using the fixed θm that was chosen for fitting the experimental data. The relative bias is shown for a range of dpart and f in Figure 2b. The approximation is valid for the dispersions characterized here in the sense that the bias is unlikely to exceed the uncertainty due to reproducibility. However, applying it when fitting the neff of larger colloids (e.g. the 1000 nm polystyrene beads) for which the angle dependency of scattering is more prominent could produce spurious results.
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Figure 2. a) The relative bias in effective refractive index (neff) measured by SPR due to turbidity. b) The relative bias in neff calculated by coherent scattering theory (CST) due to using a fixed value of the propagation angle in the sample (θm). Validity range of Mie theory. Mie theory assumes that the light scattering is not influenced by neighboring particles. Eq. 3 is therefore only valid for dispersions dilute enough for the cp to be in the independent scattering regime where the particles do not influence their neighbors. For more concentrated dispersions in the dependent scattering regime, the amplitude of scattered light decreases because of interference with light scattered by other particles in the vicinity. The risk that photons already scattered in some direction undergo a second scattering event also increases.21-24 Hottel et al.23 stated based on empirical observations that the dependent scattering regime is limited by c/λ0 < 0.3 and lpart/rpart < 2.8. Here c is the interparticle clearance, which is the average distance between the surfaces of two neighboring particles. The lpart is the mean distance between particle centers, and rpart is the particle radius. It has been noted that this empirical criterion coincides with the c where the near fields of each scattering particle starts to overlap with that of their neighbors.25 The criterion thus conforms to physical theory. If the criteria of Hottel et al. is combined with the approximate expression for the lpart as a function of f given by equation 6.12 in Torquato et al.,26 then the range of f for which the computed S are not reliable is given by the following set of inequalities: ACS Paragon Plus Environment
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