Gold Nanorod Resonance Shift in
Interferometric Scattering
Microscopy
THESIS
submitted in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE
in PHYSICS
Author : Peter S. Neu
Student ID : 2093812
Supervisor : Michel Orrit
Martin D. Baaske
2ndcorrector : Daniela Kraft
Gold Nanorod Resonance Shift in
Interferometric Scattering
Microscopy
Peter S. Neu
Huygens-Kamerlingh Onnes Laboratory, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands
November 8, 2018
Abstract
Gold nanorods are used in various sensing applications. Through observation of their Surface Plasmon Resonance (SPR) at optical wavelengths, they offer a bridge to length scales below the diffraction
limit. In a confocal Interference Scattering Microscopy (iSCAT) setup fluctuations of the SPR may be detected fast enough to draw conclusions
about diffusion in the vicinity of the nanorod. We characterize such a setup and find its optimal working point.
Contents
1 Theory 7
1.1 Surface Plasmon Resonance in Gold Nanoparticles 7 1.1.1 Gold Nanorods in Quasistatic Approximation 7
1.1.2 Resonance Shift as Plasmonic Sensor 9
1.2 Confocal Microscopy 10
1.2.1 Interference Scattering (iScat) 10
1.3 Interference Signal Optimization for Detection of SPR Shift 14
1.4 Translational Diffusion 16
1.4.1 Diffusion Time Scales 17
1.4.2 Dynamic Light Scattering 17
2 Characterization of Nanorods in iScat Setup 19
2.1 Experimental Setup 19
2.2 Point Spread Function 21
2.3 Calibration of the Tunable Laser 21
2.4 Nanorod Spectrum in iScat 23
2.5 Determining the Orientation of Nanorods 27
3 Sensing Diffusing Nanoparticles 29
3.1 Polystyrene Sticks to Nanorod 29
3.2 Diffusion Signal 33
3.2.1 Polystyrene Beads 34
3.2.2 Gold Nanospheres 37
4 Conclusion and Outlook 39
Chapter
1
Theory
1.1
Surface Plasmon Resonance in Gold
Nanopar-ticles
The interaction of metallic nanoparticles (NPs) with light can be largely attributed to their shape, size and conductive properties. In 1908 Gustav Mie [1] calculated absorption and scattering of gold spheres from Maxwell’s theory, explaining earlier observations of Michael Faraday [2]. In recent years the spectra of a multitude of shapes have been investigated [3–5].
Especially the prolate gold nanorods (NRs) have a sharp resonance in the optical range, that is largely tunable through their aspect ratio (length/ diameter). As their spectrum also depends on the dielectric properties in the vicinity, they can be seen as sensors for local processes as done in biosensing [4].
1.1.1
Gold Nanorods in Quasistatic Approximation
In contrast to the spherical particles of arbitrary size considered by Mie, the gold nanorods used here are small compared to the wavelength λ. Mie theory shows that such small particles radiate as a dipole [6]. The scattering and absorption cross sections are
σscat = k 4 6πe20|α| 2 (1.1) σabs = k e0 Im{α}, (1.2)
respectively,where α denotes the polarizability parallel to the incident elec-tric field E and k the absolute value of the wave vector.
In this case the electric field at an instant of time is approximately con-stant over the nanorod (Quasistatic Approximation). The electric field dis-places the electrons collectively with respect to the heavier nuclei. The Coulomb force between electron cloud and protons acts as the restoring force in this driven harmonic oscillator. At resonance the scattered field follows the incident field with a phase difference of π/2 (see figure 1.1).
101 102 | |/V 0 aspect ratio 3:1 major axis minor axis 400 450 500 550 600 650 700 750 800 incident wavelength [nm] -- /2 0 re s
Figure 1.1:The polarizability along the major axis of an ellipsoidal nanorod (here 3:1) is much larger than along the minor axes (top). The scattered field is strongly
polarized along the major axis. The scattered electric field Escat is in phase with
the incident field at large wavelengths λ. For smaller wavelengths the scattered
field follows the incident field with a phase lag ϕres, which passes ϕ = −π/2 at
resonance.
The polarizability α in a given geometry is then calculated from elec-trostatic theory. For an ellipsoid with major axis length w and minor axis length d the formula from Gans theory [7, 8] is
αj =e0V
e1−em
1.1 Surface Plasmon Resonance in Gold Nanoparticles 9
with the volume V =wd2π/6, the geometrical factors Lwand Ld
Lw = 1 −e2 e2 1 2e ln 1+e 1−e −1 (1.4) Ld = (1−Lw)/2 (1.5)
and the eccentricity e
e2 =1− d
2
w2. (1.6)
One should note that also the relative permittivities e1 and em, of the nanorod and surrounding medium respectively, are functions of the wave-length. For simulations the values reported for water [9] and single-crystal gold [10] are used. The normalized polarizabilities parallel and perpendic-ular to the major axis of an ellipsoid are shown in figure 1.1.
1.1.2
Resonance Shift as Plasmonic Sensor
We see that the SPR strongly depends on the dielectric properties of both the nanorod and the surrounding medium (eq. 1.3). Also a small particle in the near field of the NR shifts the SPR. Simulations of the near field of the NR (Discrete Dipole Approximation) show strong field enhancement at the ends of the rod. The SPR shift is strongest in these sensitive volumes. The wavelength shift∆λ decreases exponentially with nanorod-particle separation s [4]
∆λ(s)
∆λ(0) =exp(−s/s0) (1.7)
1.2
Confocal Microscopy
In a confocal microscope an objective focuses an incoming excitation light beam on the sample and collects the light coming from the sample. Only light originating from a small focal volume then passes a pinhole and reaches the photo-detector. The small focal volume is achieved by the use of an objective lens with high numerical aperture (NA) and a pinhole that blocks light originating outside of the focal volume. In contrast to wide-field microscopy images are obtained by moving the sample through the focus (scanning). Figure 1.2 shows a schematic setup.
probe with piezo scanning objective lens probe laser beam splitter confocal pinhole tube lens photodetector polarizer analyzer
Figure 1.2: Sketch of a confocal microscope with polarization selection. The lin-early polarized probe laser is reflected at the beam splitter, collimated through the objective lens and illuminates the probe in the focal volume. Light emit-ted/scattered from the probe passes back through the objective lens and through the beam splitter. Another polarizer, the analyzer, can be rotated to (partially) block the reflection/switch between dark-field and bright-field mode. Another lens collimates the light rays at the confocal pinhole, where out-of focus rays are blocked. The last lens collimates the beam on the detector area.
1.2.1
Interference Scattering (iScat)
In a confocal microscope that holds a sample (in aqueous solution) on a glass slide the glass-water interface reflects light. The interference between the reflected light and scattered light from the probe located in the confocal volume in proximity to the water glass interface is detectable∗. For small particles bright-field microscopy has the advantage that the interference part of the detected intensity
Ipol =cnε0h|Eref(t) +Escat(t)|2i
= cnε0
2 ˆE 2
ref+2·EˆrefEˆscat·cos∆ϕ+Eˆ2scat
(1.8) ∗P. Kukura notes that most confocal microscopes already produce interference
1.2 Confocal Microscopy 11
4
2
0
2
4
r/
8
6
4
2
0
2
z/
wavefront reflection
wavefront scattering
(a)Light scattered from a gold nanorod in-terferes with the reflection from the glass-water interface (cut perpendicular to the NR’s long axis). The reflected Gaussian beam acquires a Gouy shift of π/2, com-pared to a spherical wave starting in phase at the NR. The dipole-like scattering of the nanorod depends on the frequency driving the oscillator, giving−π/2 at resonance.
- - /2 0 res 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 z/zR 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 co s
(b)Effect of axial displacement (z) and phase difference between incident and scattered field (ϕres) on the interference term, that is
proportional to cos∆ϕ. Total destructive in-terference, i. e. cos∆ϕ= −1, can be reached for ϕ∈ (−3π/4,−π/4)by shifting the focus
by up to one Rayleigh length. z is given in units of the the Rayleigh length zR=πw20/λ.
Figure 1.3: In interferometric scattering microscopy (iScat) the electric field scat-tered by the nanorod interferes with the reflection from the glass-water interface. The phase difference ϕ between the two fields determines the strength of the in-terference term, that is proportional to cos ϕ.
is proportional to the amplitude of the scattered field ˆEscat. The field scat-tered by a NP is in turn proportional to its Volume V. For small volumes the interference term scales more favorably than the scattering term (that is ∝ V2, see section 1.1.1). The interfering light waves and phase differences are illustrated in figure 1.3a. As a driven oscillator the nanorod causes a phase shift between incident field and the scattered field. The field of a Gaussian beam undergoes the Gouy phase shift when passing through the focus. When the focus is placed at distance z before the reflection, the Gouy phase between reflection and a plane wave reflected in phase is ϕGouy =π/2−arctan(z/z0), with the Rayleigh length zR. For a reflection in the center of the focus and resonant scattering (as illustrated in figure 1.3a), the phase difference between scattered and reflected light∆ϕ is π. The role of polarization If the probe laser is linearly polarized, then the light reflected from the glass-water interface is polarized in the same direc-tion. The detected intensity from reflection (without a NR present) under the analyzer angle θ is Iref = Iref(θ) ∝ cos θ ˆEref
2
. With the analyzer par-allel to the polarizer the detected intensity ˆIref = Iref(0)is maximized and will be used as the reference unit for further normalized values.
θ θ θ-θ θ-θNRNR θ θNRNR Analyzer Analyzer E Erefref E Escatscat
Figure 1.4: Orientation of reflected and scattered field (unit vectors) and
associ-ated angles. The reflection from the glass-water interface Erefconserves the
polar-ization of the incoming laser beam. The scattered field Escatis polarized along the
long axis of the NR, which encloses an angle θNRwith the incoming polarization.
Both fields are projected onto the analyzer axis, that encloses the angle θ with the polarization of the reflected field. The amplitude of the interference term is
determined by the phase difference∆ϕ between scattered and reflected field.
1.2 Confocal Microscopy 13
axis than along the minor axes (example in figure 1.1), the scattered field is (mainly) polarized along the nanorod. Angles are defined in figure 1.4. Consider that the scattered field polarization encloses an angle θNR with the incoming polarization and – by convention – 0 ≤ θNR < π. With a polarizer between beam splitter and tube lens set to enclose an angle θ with the reflected light, the intensity after passing the analyzer is Ipolwith
Ipol = Ipol(θ, θNR) ∝ cos2
θ ˆEref2 +cos θ cos(θ−θNR) ·2·EˆrefEˆscat·cos∆ϕ+cos2(θ−θNR)Eˆscat2 . (1.9) 700 720 740 760 780 800 incident wavelength [nm] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 = 105 = 135 = 165 = 195 = 225 = 240 Ipol /Iref ( )
Figure 1.5: Calculated bright-field spectrum for different analyzer angles θ, nor-malized to the corresponding detected intensity from reflection. Calculation for
ˆ
Escat/ ˆEref = 1/2 and NR oriented along θNR = 45◦ with resonance wavelength
λres = 750 nm (ϕGouy = π/2). Note that the intensity detected
perpendicu-lar to the NR (θ = 135◦) is only from reflection. The intensity goes to 0 at
θ = 240◦, where the projection on the analyzer of scattered and reflected field,
ˆ
Escatcos(θ−θNR) and ˆErefcos(θ), cancel. Refractive indices as reported in [10]
and [9].
Note that Ipol is π-periodic in θ, as the analyzer defines an axis of pro-jection (not a direction). At θ =π/2 the reflected beam is suppressed.
The effect of axial displacement∆z from the focus on the phase differ-ence∆ϕ was considered in figure 1.3b. The bright-field spectra in figure
700 720 740 760 780 800 incident wavelength [nm] 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Ipol /Iref ( ) z/zR= 1 z/zR= 0.5 z/zR= 0 z/zR= 0.5
Figure 1.6: Calculated bright-field spectrum for different axial displacements z
from the focus in units of the Rayleigh length zR. Calculation for ˆEscat/ ˆEref =1/2
and NR oriented along θNR=45◦with resonance wavelength λres =750 nm. The
analyzer angle is set to θ = 240◦ where the scattered field on resonance and the
reflected field have same amplitude.
1.6 are calculated at an analyzer angle where the sum of scattered and re-flected intensity is comparable to the interference term. At ∆z = 0 (as in figure 1.5) the interference term is symmetric around the resonance wave-length, as it is proportional to cos∆ϕ and ∆ϕ = π at resonance. In the other cases the bright-field spectrum is asymmetric around λres. The min-imum of Ipolis red-shifted for rising∆z. The calculation does not take into account that the incident intensity changes with the displacement of the focus, and therefore also the ratio Escat/Erefchanges.
1.3
Interference Signal Optimization for
Detec-tion of SPR Shift
The SPR shift ∆λres is determined by the rod, the optical properties of the analyte particle and their mutual distance. The difference in detected intensity caused by a shift of SPR is then∆λres·∂Ipol/∂λ. The shot noise scales asp Ipol. For a shot noise limited detector the configuration of the setup should maximize the signal to noise ratio (SNR)|∂Ipol/∂λ|/p Ipol.
1.3 Interference Signal Optimization for Detection of SPR Shift 15
The intensity change with SPR shift in relation to shot-noise at differ-ent analyzer angles θ and wavelengths λ is shown in figure 1.7. At the resonance wavelength λres the SNR is zero (as ∂Ipol/∂λ = 0), except for one analyzer angle where Ipol →0.
700 720 740 760 780 800 wavelength [nm] 0 20 40 60 80 100 120 140 160 180 [ ] 0.000 0.002 0.004 0.006 0.008 0.010 SN R/ Iref (0 ) [1/ nm ]
(a) Under ideal conditions Ipol → 0 at
θ=θminand λ=λres.
700 720 740 760 780 800 wavelength [nm] 0 20 40 60 80 100 120 140 160 180 [ ] 0.000 0.002 0.004 0.006 0.008 0.010 SN R/ Iref (0 ) [ 1/ nm ]
(b)With a background of Iref(θ)/20 the SNR
is zero at λres. The band with an SNR larger
than half its maximum is contoured in red.
Figure 1.7: Intensity change with SPR shift in relation to shot noise at different
analyzer angles θ and wavelengths λ normalized to p Iref(θ=0). Numerical
calculations for ˆEscat/ ˆEref = 1/10 and NR oriented along θNR = 45◦ with
reso-nance wavelength λres =750 nm. At θmin ≈ 85◦the scattered and reflected field
have the same amplitude when projected on the analyzer, ˆEscatcos(θmin−θNR) =
ˆ
Erefcos(θmin).
In the shot noise limited case the signal to noise ratio diverges for Ipol →0. In practice that means that the scattered and reflected field can-cel and thus the incident power can be increased without saturating or damaging the APD. Still too high power incident on the NR will cause it to heat and reshape, limiting the power one can apply. Then any slight perturbation of scattering phase will give a detectable intensity change. In fact, at the resonance wavelength, i. e. at∆ϕ=π, there is always an angle
θminthat satisfies
Ipol =0 (1.10)
⇐⇒ 0=cos2θmin−2·cos θmincos(θmin−θNR) · ˆ Escat ˆ Eref + cos(θmin−θNR) ˆ Escat ˆ Eref 2 (1.11)
⇐⇒ Eˆscatcos(θmin−θNR) =Eˆrefcos(θmin), (1.12) that is when the projection of the scattered field amplitude is the same as the projection of the reflected field amplitude on the analyzer.
In figure 1.7b the more realistic case that Ipol > 0 is considered. The calculation assumes that there is a background intensity of Iref(θ)/20, de-pendent on the analyzer angle θ. That means at θ = θmin the destructive interference spectrum reaches 1/20 of the intensity on the reflection, as we find in the spectrum in figure 3.3b. In this case the SNR at the resonance wavelength is zero, because ∂Ipol/∂λ = 0 (it is a minimum/maximum). The maximum SNR is reached at λres±5 nm. At θ =θminthe SNR is larger than its maximum in a rather broad band from 2 nm< |λ−λres| <12 nm, marked in red in figure 1.7b.
1.4
Translational Diffusion
Particles inside a fluid show Brownian motion, i.e. a random walk driven by collisions with the thermally moving molecules of the fluid [12]. From a macroscopic view entropy drives the particles to a uniform concentration c0. The characteristic parameter of such a concentration relaxation is the diffusion coefficient D in Fick’s second law
∂c0(x, t)
∂t =D∆c(x, t). (1.13)
For spherical particles of radius R the diffusion coefficient is the Stokes-Einstein coefficient D0, given by the formula
D0 = kBT
6πηR (1.14)
where T is the temperature, kBthe Boltzmann constant and η the viscosity of the medium.
1.4 Translational Diffusion 17
1.4.1
Diffusion Time Scales
The average time it take particles to travel a distance L, in this case the 1/e decay length of the nanorods near field LNF, is
τNF= L2NF/D0. (1.15)
As this relation is quadratic in L, diffusion in the near-field is two to four orders of magnitude faster than diffusion in the confocal volume.
At an instant of time the probability of finding a particle in the near field is c0VNF. It will diffuse out of the near field after τNFon average.
The rate of particles passing the near field (NF) at concentration c0 is estimated as
fNF =c0VNF/τNF∝ c0LNFD0 (1.16) neglecting particle-particle interaction†.
1.4.2
Dynamic Light Scattering
In Dynamic Light Scattering (DLS) the diffusion of particles through the focus is observed. The scattering signal from the particles is detected. Sample volumes are in the milliliter range with very low concentrations, as to allow for a detectable number fluctuation in the focal volume and minimize multiple scattering [13]. The diffusion time, and thus the parti-cles’ (hydrodynamic) radius, can be extracted from the intensity correla-tion funccorrela-tion
g(2)(τ) = hI(t)I(t+τ)i
hI(t)i2 . (1.17)
The use of a confocal microscope with two polarizers is described in [14]. They report an apparent diffusion coefficient DA in opaque samples, that is lower than the actual one and lies between D0/2 and D0.
†Which is valid for solutions up to∼100 µM concentration, where c
Chapter
2
Characterization of Nanorods in
iScat Setup
2.1
Experimental Setup
The setup is mounted on a floating optical table. The laser source is a tunable Ti-sapphire laser. After a row of adjustable attenuators the beam is coupled into a polarization maintaining single-mode fiber. After the fiber the beam is collimated and enters the setup as sketched in 1.2 and shown in figure 2.1. The sample is mounted horizontally on the piezo stage. The top of the glass slide is covered with ultrapure water (Milli-Q). After the analyzing polarization filter the light is collected with an Avalanche Photo Detector (APD). The dead time of the APD limits the sampling rate to 1 MHz. The diffusion in the NR’s near field is expected to take place in few microseconds, thus be detectable. In general a higher sampling rate, as provided by analog photo detectors, is desirable.
In practice the available oscilloscope was just able to store a 10 ms mea-surement every second. Sufficient data for e. g. calculation of an autocor-relation function cannot be obtained in reasonable times. Also permanent changes of the NR that hinder the further measurement, like diffusors cov-ering the NR, would likely be missed.
beam splitter sample holder piezo stage (mirror underneath) polarizer tube lens confocal pinhole analyzer (on rotational stepper motor) incoming beam to detector
Figure 2.1:Setup in practice. Incoming beam on the top, confocal pinhole and an-alyzer on the right. The sample is mounted horizontally with a mirror underneath the objective. A box (usually closed) shields the detector path from background light.
Figure 2.2: Scans of the sample in dark-field configuration (left) and bright-field
configuration (right). For the bight-field scan, the analyzer is rotated by 10◦ from
the crossed polarization setting (to θ=100◦) and the scattered and reflected
elec-tric fields interfere. Depending on the NR orientation θNR, the NR will appear
2.2 Point Spread Function 21
2.2
Point Spread Function
The Abbe-Rayleigh criterion [15] states the minimum resolvable distance to be
dmin =1.22× λ
2N A ≥480 nm (2.1)
for the used air objective with NA = 0.9 and wavelengths λ > 720 nm. As this exceeds the nanorod dimensions in any direction, a close-up scan of the nanorod images the point spread function (PSF) rather than the nanorod itself (figure 2.3).
In a bright-field setting the glass slide reflection is visible, convoluted with the axial PSF. Above and below the slide plane the Airy-rings are imaged as cones of constructive and destructive interference (fig. 2.3f).
Gaussian fits to the dark-field and bright field images along the the lat-eral axes are performed in figure 2.3c and 2.3d, respectively. The dark-field fits exhibit beam sizes in the order of dminwith a FWHM of 648 nm along the x-axis and 727 nm along the y-axis. Both values are significantly larger than the expected FWHM size of√2 ln 2 λ/2NA. That indicated that the numerical aperture of the objective is lower at λ =750 nm than indicated for the visible spectrum. Also the incoming beam is not overfilling the objective, which leads to larger focus sizes than expected.
The widths extracted from the fits to the bright-field image are larger. This is expected, as in bright field configuration the additional interfer-ence term is proportional to the electrical field amplitude of the Gaussian beam. The field amplitude of a Gaussian beam decreases as EGauss(r) ∝ exp −r2/σ2 with radial distance r from the center. The intensity of a Gaussian beam decreases as IGauss(r) ∝ E2Gauss(r) ∝ exp −2r2/σ2, which is a factor√2 narrower than the decrease of the interference term.
2.3
Calibration of the Tunable Laser
The probe laser is a tunable titanium-sapphire laser. Laser modes are lected with a stepper motor that turns a set of birefringent filters that se-lect the mode of the cavity. The wavelength is read on a spectrometer with 1 nm resolution.
The recorded wavelengths and corresponding motor positions are shown in figure 2.4. To check reproducibility the screw is turned from low to high and high to low. We find that reliable tunability is given from 720 nm to 770 nm. The linear fit in figure 2.4 is used to convert between motor posi-tion reading and wavelength further on.
(a)dark-field, lateral-lateral plane (b)bright-field, lateral-lateral plane
(c)dark-field, Gaussian fit (d)bright-field, Gaussian fit
(e)dark-field, lateral-axial plane (f)bright-field, lateral-axial plane
Figure 2.3:Nanorod scans to image the point spread function in the lateral-lateral
and lateral-axial plane. Taken with λ = 750 nm. In the axial scan in bright-field
2.4 Nanorod Spectrum in iScat 23
12 13 14 15 16 17 18
Motor gauge reading [mm]
710 720 730 740 750 760 770 laser wavelength [nm] linear fit series 1 series 2
Figure 2.4: Wavelength of the laser, tuned with a motorized micrometer screw that turn a birefringent filter . Data points are consistent over multiple runs, in-dependent of direction of travel. Tunability is linear from 715 nm to 770 nm, with
slope−9.6 nm nm−1and intercept 890.0 nm.
The laser output power depends on the selected wavelength, as can be seen in the spectrum in figure 2.5a. The intensity measured on the re-flection of the slide Ireffluctuates when the laser wavelength is tuned. The fluctuations are mostly due to changes of laser output power. A method to compensate for these fluctuations is introduced in the following chapter.
2.4
Nanorod Spectrum in iScat
Nanorod scattering spectra show a distinctive peak at the resonance wave-length λres. In an iScat setup these peaks may also show up as valleys if the interference term is negative, according to equation 1.8.
We use the tunable laser to record iScat spectra, that is Ipol dependent on the wavelength, at fixed analyzer angle θ. As the laser intensity fluctu-ates with wavelength and slowly with time, we need a reference intensity. This reference is taken on the glass slide reflection (as it is a bright-field spectrum) immediately after the measurement on the NR for each wave-length. The spectrum is the intensity on the rod, divided by the intensity on the reflection for the corresponding wavelength. This reference to the reflection is possible because the piezo stage travels much faster than the
timescale of laser intensity drifts.
The expected spectra (see fig. 1.5) follow from the scattered field (eq. 1.3) and the interference formula (eq. 1.9). The expected width of a peak is 40 nm (FWHM).
(a)nanorod, θ=85◦ (b)nanorod, θ =95◦
Figure 2.5: Spectra taken in bright-field configuration,±5◦ from the crossed po-larizer angle. The blue and orange line are the count rates measured on the NR and on the slide reflection, respectively (right scale bar). The black dots form the
normalized iScat spectrum, obtained from division Ipol/Iref at each wavelength.
The interference term changes from destructive to constructive as the projection of the reflected electric field on the analyzer changes sign.
The angle θ between polarizers is determined by turning the analyzer such that the transmitted reflection from the glass slide is minimized. That angle is denoted as θext in experimental data and signifies the crossed-polarizer state θ= ±90◦. It can be determined with±0.1◦ accuracy.
The nanorods used have a nominal resonance at λres = 750 nm (d = 40 nm), lying in the laser range. Recorded spectra of one nanorod are shown in 2.5, together with the intensity count rates Ipol and Iref on and off the NR. Some spectra of nanorods have no maximum or minimum in the scanning range. They may be multiple clustered rods that couple to each other or unresolvable rods of different orientations with their scat-tered fields interfering.
The spectra of the nanorod are shown in figures 2.5a and 2.5b.
The shape of the spectrum (notably the resonance wavelength) is inde-pendent of displacement between NR and beam focus up to 200 nm each direction (see Appendix A). That means that the piezo stage can be po-sitioned accurately enough (reproducibility < 100 nm) to record reliable spectra.
2.4 Nanorod Spectrum in iScat 25
Changing the Refractive Index of the Surrounding Medium
The SPR depends on the refractive index of the medium (1.3). Adding glycerol to water increases the refractive index of the solution, thus red-shifting the nanorod. Furthermore the reflectivity of the glass-solution in-terface is decreased, thus decreasing the count rate.
(a) n=1.33 (b) n=1.35
Figure 2.6:Spectra of one NR, recorded at two different refractive indices n of the surrounding medium. Other parameters are held constant. Note the lower count
rate at n=1.35 as the reflectivity decreases.
Spectra are recorded in the center of the PSF of each nanorod. Then the water (n=1.33) is mixed with glycerol (volume fraction of glycerol:water is 1:6), giving a solution with refractive index n = 1.35 [16]. Then the spectra of the NRs are measured again.
An exemplary nanorod is shown in 2.6. The SPR shift of ≈ 11 nm matches the calculation from Gans theory∗. As the NR is partly covered by the glass interface of constant refractive index we would actually expect a smaller change of SPR.
Note on Experimental Considerations
Reflected and scattered light have different polarizations until they reach the analyzer. Initial experiments show that a dielectric broadband mir-ror (mounted underneath the objective) caused a wavelength-dependent phase-lag between the two polarizations. This interference effect is visi-ble in the spectra in figure 2.7. The peak at ∼ 755 nm is too narrow for
∗The Gans theory calculation for this aspect ratio gives dλ
a gold nanorod resonance. The fact that it changes sign when the ana-lyzer angle passes 90◦ shows that the mirror changes the outgoing beam. The nanorod SPR shift with change of the medium’s refractive index is not measurable under these circumstances. The spectrum looks similar for non-resonant nanorods. Other dielectric mirrors produce peaks at dif-ferent wavelengths; We have solved the problem by use of a silver mirror. Alternatively we could use a white-light source and a spectrometer to record the spectrum without scanning the laser wavelength. On the other hand the signal with SPR shift is better estimated from the iScat spectrum.
(a) θ=80◦ (b) θ=100◦
Figure 2.7: iScat spectra with dielectric mirror in detection path. The dielec-tric mirror induces a wavelength dependent phase lag between the reflected and scattered light. The recorded spectra are independent of the nanorod. All other experiments are conducted with a silver mirror instead which does not cause a wavelength dependent shift.
2.5 Determining the Orientation of Nanorods 27
2.5
Determining the Orientation of Nanorods
As the nanorods are spin-coated on the glass slide, their long axis is paral-lel to the surface. The orientation in the plane of the glass-water interface is random. A scan of analyzer angles θ shows the interference minimum. It also reveals the NR orientation θNRand the interference pre-factor cos∆ϕ.
Recalling equation 1.9, the interference term
cos θ cos(θ−θNR) ·2· |Eref||Escat| ·cos∆ϕ (2.2) vanishes perpendicular to the incoming polarization (cos θ = 0) and per-pendicular to the NR (cos(θ−θNR) = 0). There exists one analyzer angle where Ipol = Iref for all wavelengths, that is θ = θNR±90◦. In the experi-ment (figure 2.8, top) this angle can be read most accurately if the slope of the interference term|Eref||Escat| ·cos∆ϕ is large, i.e. close to the resonance wavelength.
Figure 2.8: Difference between intensity measured on the NR and off the NR as function of analyzer angle θ. The minimum of intensity depends on wavelength. With the analyzer perpendicular to the NR the intensity on and off the NR are the
same, highlighted in green at θ = −35±2◦and θ = 145±2◦. This angle is read
Chapter
3
Sensing Diffusing Nanoparticles
With the iScat setup we can observe changes of the nanorods and their surrounding. A shift of the SPR, as induced by particles in the nanorod near field, will lead to a change in scattered intensity. After choosing a suited nanorod and setup configuration, we record time-resolved intensity traces on this NR. As diffusors spherical polystyrene beads (d = 100 nm) and gold nanospheres (d=20 nm) are used.
3.1
Polystyrene Sticks to Nanorod
In these experiments the NRs have a nominal resonance wavelength of 750 nm, diameter d =40 nm and the diffusing particles are spherical poly-styrene beads of 100 nm diameter.
The measurements are performed with an Avalanche Photo Detector (APD) that allows for very low laser powers reaching the nanorod (be-low 1 µW). At higher laser powers one can expect additional effects, like heating or thermal reshaping of the nanorod. For computation of the au-tocorrelation function the photons in each microsecond are counted. For visualization in time traces the count rate is computed from the number of photons in a millisecond window.
To begin with, a nanorod that shows destructive interference at λ =
750 nm and θ = 110◦ is selected. The SPR shift under the addition of glycerol (10%, like sec. 2.4), is checked to be sure that the NR’s spectrum is sensitive to changes of the surrounding. Then the analyzer is set to the interference minimum, that is at θ =115◦for the selected nanorod. There we expect the largest signal to noise ratio (see section 1.3).
modula-tions of intensity, especially scattering directly from the diffusing particles (DLS, section 1.4.2). This added intensity is distinguished from nanorod SPR shifts by selecting a laser wavelength above the resonance wavelength (λ = 749 nm > λres ≈ 742 nm). The expected red-shift of the SPR upon a particle entering the nanorod near field now causes a drop in detected intensity. The NR spectrum is shown in figure 3.1a. Also the possible vi-brations and drifts of the sample holder would cause a higher detected intensity, as the beam is focused on the center of the NR’s PSF, that is the darkest spot.
A background measurement without diffusors is performed. In this configuration spherical (d = 100 nm) polystyrene particles are added to the water/glycerol solution, giving a particle concentration of 10 nM. A time trace is taken during the injection of particles and afterwards. How-ever the autocorrelation function matches the one obtained in a particle-free solution.
We gradually add sodium chloride (NaCl) in order to shield the surface charges of the polystyrene particles. No change in autocorrelation func-tion, compared to the background, is observed at 0 mM, 1.1 mM, 2.2 mM or 3.2 mM NaCl concentration. At 6.2 mM NaCl concentration the count rate fluctuates and then stays low, shown in figure 3.1. The spectra 3.1a and 3.1b show the nanorod spectrum immediately before and after the time trace was recorded. They show a shift of the interference minimum from 743 nm to 746 nm.
The experiment is repeated with a new sample. The analyzer is set to θminagain. The minimum of the spectrum is located at λ∼765 nm, which is near the end of the scanning range.
For this NR the derivative of intensity with respect to wavelength is large between 740 nm and 760 nm. So this time the time trace is recorded at λ <λres, where the SPR shift causes an increase in detected intensity.
A permanent change of intensity occurs at 3.9 mM NaCl concentration. The part of the time trace with a step in intensity is shown in figure 3.2. The shift of SPR is unclear in the destructive interference spectra, as the minimum is at the end of the laser scanning range.
In both measurements it takes tens of seconds until a change in inten-sity is visible. At the given diffusor concentration a diffusor should enter the near field every two seconds (see table 3.1). So it is likely that the shift of SPR is caused by agglomerated diffusors, that stuck together in the so-lution.
The first execution of the experiment, where the resonance wavelength could be traced, shows that a sticking particle causes a detectable SPR shift of 3±1 nm.
3.1 Polystyrene Sticks to Nanorod 31
(a)Spectrum before sticking. (b)Spectrum after sticking.
40 60 80 100 120 time [s] 20000 40000 60000 80000 100000 counts/s
Figure 3.1:Spectra before and after polystyrene sticks to the NR. The laser wave-length is indicated by the vertical line in each spectrum. A red-shift of the SPR causes a lower count rate in the time trace. It is probable that the polystyrene spheres agglomerated to form larger particles.
(a)Spectrum before sticking. (b)Spectrum after sticking. 20 22 24 26 28 30 time [s] 100000 125000 150000 175000 200000 225000 250000 275000 counts/s
Figure 3.2:Spectra before and after polystyrene sticks to the NR. The laser wave-length is indicated by the vertical line in each spectrum. A red-shift of the SPR causes a higher count rate in the time trace.
3.2 Diffusion Signal 33
3.2
Diffusion Signal
As diffusing particles only cause a slight shift of SPR when they quickly (∼ µs timescale) pass the near field, the intensity change caused by the particle may not be clearly distinguishable from the noise floor. Nonethe-less the SPR perturbations from multiple particles passing the NF in sev-eral minutes may be sufficient to be visible on an autocorrelation function. On that autocorrelation function also the direct scattering of the diffusors in the confocal volume is present. However we expect that the time scale of the direct scattering is orders of magnitude longer than the characteris-tic SPR shift time, and therefore the two contributions to the autocorrela-tion funcautocorrela-tion should be easily distinguishable. We compare measurements on the NR, that is in the center of the PSF, and off the NR, that is with the focus at a clear spot of the glass slide and 2 µm into the solution.
An overview of available diffusors and their associated diffusion times is given in table 3.1. The length of the near field of a NR is estimated to be LNF = 5 nm. A NF length of 6 nm is given in [17] for a NR with diameter d =15 nm. The average time a diffusor spends in the NF is
τNF= L2NF/D0. (3.1)
If diffusors were to pass the NF in less than 1 µs on average, that is faster than the APD sampling rate, they would hardly be detectable in the au-tocorrelation function. Therefore we choose to rather underestimate LNF. The rate of diffusors passing the NF is estimated as fNF =c0VNF/τNF(eq. 1.15).
All diffusors used have negative surface charges at pH = 7 according to the manufacturer. Also the NR’s and glass slides surfaces are nega-tively charged, thus repel the diffusors. Surface charges are shielded with sodium chloride (NaCl). The characteristic length, after which the electric potential is decreased by 1/e is the Debye length
LD = s
εrε0kBT 2·NAq2ec
(3.2) where ε0is the permittivity of free space, εr is the dielectric constant, qe is the charge of the electron and c is the molar concentration of electrolytes per unit volume [18]. For NaCl in water εr =78 at room temperature.
Measurements are performed at different concentrations of NaCl such that the Debye length is around the expected near field length, allowing particles to enter the near field. That is between λD(c = 1 mM) = 9.6 nm and λD(c =10 mM) = 3.0 nm.
Table 3.1: Overview of diffusion times in the near field and confocal volume for different spherical diffusors. The calculations are based on the concentration of
the undiluted stock solution. The near field passing rate fNF (eq. 1.16) is linear
in the degree of dilution. The average time spent in the confocal volume and the
near field are calculated (eq. 1.15) for Lcf = 200 nm and LNF = 5 nm. Diffusors
faster than the detection bandwidth are marked (7).
Diffusors Molarity Time confocal τcf Time NF τNF Passingrate fNF Gold nanospheres d=5 nm 72.8 nM 456.0 µs 7 0.28 µs 80 /s d=10 nm 9.1 nM 912.0 µs 7 0.57 µs 5 /s d=20 nm 1.0 nM 1824.0 µs 1.1 µs 0.3 /s Polystyrene beads d=20 nm 7551.0 nM 1824.0 µs 1.1 µs 2089 /s d=50 nm 2428.3 nM 4560.0 µs 2.8 µs 268 /s d=100 nm 303.5 nM 9120.0 µs 5.7 µs 16 /s
3.2.1
Polystyrene Beads
The experiment is conducted with 100 nm polystyrene beads as diffusors, diluted to 30 nM concentration, in a 10% glycerol in water solution. The NRs have a nominal diameter of d = 40 nm and a resonance wavelength at 750 nm. The NaCl concentration is 1 mM.
One nanorod that allows for a low detected intensity in destructive interference is selected. At θ = 73◦ and λ = 751 nm the iScat spectrum (see fig. 3.3a) shows an intensity that is a factor 10 lower than on the glass slide’s reflection.
The autocorrelation function (figure 3.4) recorded off the NR shows an increase at times below 1 ms. An increase with lower amplitude is visible on the autocorrelation functions computed from the time trace on the NR at the same time scale.
The decay times of all measurements are the same, as rescaling g(2)(τ) to the interval[1, 2]in figure 3.5 shows. Each computed function is linearly rescaled so that the first 5 µs average 2 and the slow part from τ=0.1 s to τ =1 s averages 1.
3.2 Diffusion Signal 35
(a) iScat spectrum of the NR (d = 40 nm) as used in the experiment for sensing diffusion of 100 nm polystyrene beads (section 3.2.1). The vertical lines mark the wavelengths at which autocorrelation functions (figure 3.4) were extracted.
(b)iScat spectrum of the NR (d = 25 nm) as used in the experiment for sensing diffusion of 20 nm gold nanospheres (section 3.2.2).
Figure 3.3:With use of smaller NRs (d=25 nm, right) a lower relative count rate
Ipol/Iref, thus a better signal to noise ratio, is reached.
(700±5)µs. The diffusion coefficient follows from DLS [19] as
DDLS =
λ2 8π2n2
m
/τDLS =5.5 µm2/s , (3.3)
with the refractive index of the medium nm = 1.35 for the glycerol/water solution. The value calculated from Stokes-Einstein theory for a 100 nm diameter sphere and the viscosity of 10% glycerol (see [20]) is Dtheo. = 3.19 µm2/s. The fact that the diffusion measured is faster than expected may be due to finite volume effects of the focus and multiple-scattering effects, as the solution is rather opaque.
In conclusion an intensity correlation from the expected SPR shift, that should be in the µs range, is not visible. We suppose that the diffusors do not reach the near field of NR or pass it too quickly due to the repelling surface charges. Decreasing the Debye screening length cannot be done indefinitely, as then the diffusors agglomerate.
Another cause could be that the polystyrene beads are so large, that they are geometrically blocked from the NF. That could be the case if con-taminants or residue of CTAB settled on the NR and next to it on the glass slide.
Figure 3.4: Autocorrelation of diffusing 100 nm polystyrene beads recorded on the NR at the indicated wavelengths and off the NR for comparison. The char-acteristic correlation time scale is around 1 ms for all measurements. The ampli-tudes of the autocorrelation functions recorded on the NR are similar, the one recorded in the fluid has highest amplitude.
Figure 3.5: Autocorrelation of diffusing 100 nm polystyrene beads re-scaled to the [1,2] range. All autocorrelation functions have the same time constant within the noise. On the autocorrelation functions recorded on the NR an oscillation is visible above 1 ms, which is probably caused by vibrations transmitted to the sample holder. The dash-dotted line is the best mono-exponential fit to the auto-correlation function off the NR, used to calculate a diffusion constant from DLS
3.2 Diffusion Signal 37
3.2.2
Gold Nanospheres
For further measurements we switch to NRs with nominal diameter of d = 25 nm, resonance wavelength still at 750 nm. They scatter less light than the d = 40 nm nanorods (Escat ∝ V) and the minimum of detected intensity Ipol is reached at θmin=4◦ from extinction, thus suppressing the DLS. Also with these nanorods a lower intensity in destructive interfer-ence is reached, with Ipol/Iref ∼ 0.05 compared to Ipol/Iref ∼ 0.1 for the NRs previously used. A spectrum of the NR selected for the experiment is shown in figure 3.3b .
The diffusors in this experiment are gold nanospheres with diame-ter d = 20 nm. Their resonance at about 540 nm is far from the laser wavelength, so they have a weak scattering signal. 100 µl of the stock solution are diluted in 400 µl water, giving a concentration of 0.2 nM. At that concentration one nanosphere in the nanorod near field is expected in 15 s. Gold nanospheres have a larger polarizability than polystyrene beads, thus cause a larger SPR shift when passing the NR’s NF. The asso-ciated intensity changes could be strong enough to be detected as single events although they are rare.
The APD count rate is recorded for 180 s, both on and off the NR. Nei-ther shows an apparent signal.
After addition of NaCl (1 mM) individual spikes show, both in on and off the NR (see figure 3.6). The spikes are on the 10 ms timescale (inset to time traces) which is longer than expected even for the average time in the confocal volume.
The fact that their rate and length is similar with the focus positioned on and off the NR and that it is slow suggests that the gold spheres clumped together. If they were single spheres all the time, the intensity spikes would always be visible in the off NR configuration, regardless of NaCl content, as this is measured 2 µm from the glass slide.
(a)On NR.
(b)Off NR.
Figure 3.6: Time traces recorded with the focus on and off the NR, with gold
nanospheres (d=20 nm) diffusing and 1 mM NaCl added. In both configurations
single spike-like increases of intensity are visible. The effect cannot be attributed to the NF of the NR. No such intensity fluctuations were visible without added NaCl (neither on or off the NR), which suggests that the light is scattered from larger aggregates of gold particles.
Chapter
4
Conclusion and Outlook
In summary, the static characterization of the iScat setup works as ex-pected, whereas the time-resolved measurements need improvement to allow the targeted sensing of diffusing particles in the nanorod near field. In the iScat setup the nanorods can be characterized with regard to their orientation and spectra. The signal to noise ratio is favorable on the slopes of the spectrum of the nanorod. The calculation shows that the in-tensity incident on the detector can be increased by transmitting more of the reflected light, while maintaining close to optimal signal to noise ratio. In such a configuration measurements with detectors that offer nanosec-ond resolution should be possible.
Different diffusors were tested in different sizes and concentrations. After establishing that a sticking particle causes a distinct shift of SPR, it appears that the timescale of the signal is not resolvable with an Avalanche Photo Detector. If the diffusors are repelled from the nanorod, firstly, the time spent in the near field is shortened and, secondly, the possible SPR shift, proportional to the signal amplitude, is lowered exponentially with distance. Tuning the Debye screening length with NaCl in order to over-come surface charges lead to aggregation of the diffusing particles. For purposes of understanding diffusion near the nanorod one may switch to other diffusors that offer tunability of their ζ potential, e. g. micelles. The Dynamic Light Scattering (DLS) signal, that is visible also in iScat, matched the expected diffusion time for the diffusors used. Scattering from the diffusors could be circumvented in a total internal reflection illu-mination of the nanorod. Anyway, we expect that the time for passing the near field of a nanorod is orders of magnitude shorter than the character-istic times in DLS. Then the two effects are easily distinguishable and may act as a reference for each other under changes of the environment.
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Appendix
A
(a)centered, λfitres=743.8 nm (b)1 µm axial displacement, λfitres=744.7 nm
(c) 0.2 µm lateral displacement, λfit res =
742.8 nm
(d) 0.4 µm lateral displacement, λfit res =
741.4 nm
Figure A.1: Nanorod spectra when the focus is displaced from the PSF center. Deviations of the fitted resonance wavelength are within 1 nm for 0.2 µm lateral and 1.0 µm axial displacement.
