High-precision sizing of nanoparticles by laser transmission spectroscopy

Frank Li; Robert Schafer; Ching-Ting Hwang; Carol E. Tanner; Steven T. Ruggiero

doi:10.1364/AO.49.006602

1. Introduction

Many modern nanoparticle detection techniques rely on diffraction or dynamic light scattering (DLS), also known as photon correlation spectroscopy or quasi-elastic light scattering [1, 2, 3, 4]. These techniques use one or more fixed laser wavelengths and analyze light scattered at fixed angles. Other techniques have employed light scattering and/or transmission at various angles plus spectral inversion to obtain particle size distributions. These include turbidimetric techniques [5, 6, 7], nephelometry [8, 9], and spectral extinction using modified commercial spectrophotometers [10, 11]. General discussions of this area of particle sizing can be found in the literature [12, 13]. We note finally that ultrasonic waves (for particles in the $\sim 1 μm$ range) [14, 15] have also been applied to particle sizing.

2. Laser Transmission Spectroscopy

In this paper we describe the use of laser transmission spectroscopy (LTS) to obtain the size and density of nanoparticles in suspension based on wavelength-dependent light extinction. Our methodology relies on the measurement of the transmission (at zero angle with respect to the incoming beam) of laser light through a suspension of nanoparticles as a function of wavelength. The extinction so obtained is inverted, as discussed in Subsection 2D, to obtain the particle size distribution. The present methodology is applicable in the low-volume-fraction regime ( $\leq 0.5 %$ ).

Our technique employs an optical system and light detection scheme that allows us to take advantage of laser systems that are tunable over a wide wavelength range. Our implementation provides accurate transmittance spectra with high signal-to-noise ratios over the entire laser tuning range of 210 to $2300 nm$ . For spherical particles, Mie theory (the Mie solution to Maxwell’s equations) is used to obtain the total extinction cross section for electromagnetic radiation [16, 17] over this wavelength range. The technique thereby provides the distribution density for particles ranging from 5 to $3000 nm$ in diameter (including both the mean particle diameter and absolute particle count) in the interaction region. In this paper, we restrict our discussions to dilute ( $\leq 0.5 vol. %$ ) suspensions of spherical gold and polystyrene nanospheres. We measure the zero-angle extinction of a well-collimated laser beam as it pas ses through these low-density, low-volume-fraction, randomly distributed particle samples using a detector with a narrow field of view and small receiving aperture. Therefore, the effects of small-angle distortions due to multiple scattering are insignificant in our measurement regime [18, 19, 20, 21].

2A. Electromagnetic Model

Typical nanoparticle sizing devices measure the amount of single-wavelength light scattered into fixed angles, $(θ, ϕ)$ , then employ Mie theory to calculate the differential scattering cross section, $[d σ_{scat} (λ, r) / d Ω] (θ, ϕ)$ , in order to obtain the particle size distribution by mathematical inversion of the data. In our approach, we measure the zero- scattering-angle transmittance of light as a function of wavelength. We then employ Mie theory to calculate the extinction cross section as a function of wavelength and mathematically invert the data to obtain the absolute particle count as a function of size.

The Bouguer–Beer–Lambert law, Eq. (1), relates the transmittance, $T (λ)$ —the ratio of the transmitted (output) power, $P_{out}$ , to the incident (input) power, $P_{in}$ —of light of a particular wavelength through a sample of path length, z, to the extinction coefficient, $α (λ)$ :

T (λ) = \frac{P_{out} (λ)}{P_{in} (λ)} = \exp [- α (λ) z] .

The extinction coefficient due to particles of one particular radius,

r_{j}

, is described theoretically by the number of particles per unit volume,

n_{j}

, multiplied by an extinction cross section,

σ_{ext} (λ_{i}, r_{j})

, as given in Eq. (2). It is a combination of the absorption cross section,

σ_{abs} (λ_{i}, r_{j})

, and the total scattering cross section,

σ_{scat} (λ_{i}, r_{j})

. Both the scattering and absorption of light by nanoparticles depend on the size of the particle and the complex index of refraction at wavelength

λ_{i}

:

σ_{ext} (λ_{i}, r_{j}) = σ_{abs} (λ_{i}, r_{j}) + \oint \frac{d σ_{scat} (λ_{i}, r_{j})}{d Ω} d Ω = σ_{abs} (λ_{i}, r_{j}) + σ_{scat} (λ_{i}, r_{j}) .

When multiple types of particles are present in the same sample, the total transmittance, Eq. (3), at any one wavelength is the product of the transmittances of all the particles present where the extinction coefficients add in the exponent:

T (λ_{i}) = \prod_{j = 1}^{N} T (λ_{i}, r_{j}) = \exp [- z \sum_{j = 1}^{N} σ_{ext} (λ_{i}, r_{j}) n_{j}] .

Equation (3) is linearized by taking the natural logarithm of the transmittance and dividing by the path length, z. This gives the theoretical extinction coefficient,

α_{theory} (λ_{i})

, due to all particles present in the sample as in Eq. (4):

α_{theory} (λ_{i}) \equiv \sum_{j = 1}^{N} σ_{ext} (λ_{i}, r_{j}) n_{j} = \frac{- \ln T (λ_{i})}{z} |_{theory} .

The theoretical (Mie) values of the extinction cross section at each value of

λ_{i}

and

r_{j}

form the elements of a two-dimensional array,

σ_{ext} (λ_{i}, r_{j})

.

2B. Apparatus

The apparatus consists of a computer controlled light source and a balanced optical system, which provides accurate and precise measurements of sample transmittance over a wide range of wavelengths, from 210 to $2300 nm$ . A schematic diagram is shown in Fig. 1. The light source is a commercially constructed tunable laser (TL), which is comprised of a pulsed Nd:YAG laser with second- and third-harmonic generation, an optical parametric oscillator (OPO), and doubling crystals. The OPO, consisting of a resonant cavity surrounding two Type II beta-barium-borate (BBO) crystals, is capable of producing a signal beam from 420 to $709 nm$ and an idler beam from 710 to $2300 nm$ when pumped by the third harmonic of the pulsed Nd:YAG laser. Tunability of the OPO is achieved by rotating the BBO crystals within the resonant cavity. The OPO output is frequency doubled with two additional BBO crystals to produce light from 210 to $419 nm$ . The laser outputs beams through three separate ports, which correspond to the ranges: the near-infrared (NIR), 2300 to $710 nm$ ; the visible (VIS), 709 to $420 nm$ ; and the ultraviolet (UV), 419 to $210 nm$ . The result is tunable laser light, which ranges from 210 to $2300 nm$ with a pulse width of $\sim 5 ns$ , a pulse rise time of $1 ns$ , and a repetition rate of $10 pulses / s$ . The laser output power varies substantially from port to port, generally producing $1 - 3 mJ$ of power per pulse in an elliptically shaped output mode with major and minor axes of $\sim 10 mm \times 8 mm$ . The laser beam is well collimated, and a series of three $2 mm$ diameter apertures spaced several centimeters apart is used to select the central maximum of the roughly Gaussian intensity profile. Because of power variations with wavelength, filters (F) are used in front of each port to make the output power at the detectors over the entire wavelength range within the same order of magnitude ( $\sim 1 μJ / pulse$ ), eliminating the possibility of saturation over the entire range of wavelengths. The laser is stepped in $1 nm$ wavelength increments (at $5 s / step$ ).

For the NIR region, the balanced optical system consists of a $50 / 50$ metalized beam splitter (BS), a broadband mirror (BM), and a matched pair of germanium photodiodes (GPD) arranged as shown in Fig. 1. Light from two beams (A and B) of nearly equal intensity are directed into the two germanium photodiodes, where beam B reflects off of a NIR broadband mirror (BM). A computer controlled rotating sample holder (RSH) moves sample cuvettes (C) in and out of the beams as discussed further below. The transmitted and reflected beams of the $50 / 50$ beam splitter are well balanced in the NIR spectral region. However, this type of beam splitter is not suitable for the VIS and UV regions, so a different scheme is used for these wavelengths.

For the VIS to UV spectral regions, a second balanced optical system is employed, as shown in Fig. 1. A broadband mirror (BM) on a computer controlled translation stage (TS) alternatively directs the VIS and UV beams through the rest of the optical system. In this system, a single uncoated fused silica optical flat (OF) cut in two along its diameter is used to construct two identical beam splitters. Since the transmitted and reflected beams are substantially different, only the two reflections from the first half of the OF (beam A, front-external; and beam B, back-internal) are used to create two beams. These two beams are then reflected off of the second half of the OF (beam B, front-external; beam A, back-internal) resulting in two beams with nearly identical intensity and wavelength dependence. These two beams are steered with a pair of identical broadband mirrors (BM) onto a matched pair of silicon photodiodes (SPD). The computer controlled RSH moves the identical sample cuvettes (C) in and out of the selected pair of laser beams during a wavelength scan.

Extra, unused beams are captured with beam dumps (BD) to eliminate the possibility of their causing scattered light. Additional baffles not shown are also used to prevent scattered laser light from entering the detectors. The apertures of our detectors are $\sim 2 mm$ in radius and the apertures are $80 mm$ from the particle sample. This detection geometry corresponds to a collection solid angle for scattered light of $\sim 1.4 \times 10^{- 4}$ while we collect 100% of the transmitted laser beams, which pass through the apertures on their way to the detectors. Our detection geometry has a half-field-of-view of $0.025 rad$ or $1.4 °$ , and the optical depths of our particle samples are of order 1.4 or less.

During data taking, the laboratory is darkened, a blackout curtain surrounds the laser table, and a light-tight box encloses the entire optical system to eliminate external background light. A shutter (not shown) is used to temporarily block the laser beam path prior to each wavelength scan, and the detector amplifiers are triggered to zero their outputs using an analog offset prior to data taking. Such signal offsets represent a fraction of a percent of the total signal, and, in tests conducted without this zeroing procedure, there was no discernible effect either in the raw extinction data taken or in any resulting particle density versus size results. The photodiode signals are amplified and digitized with a matched pair of sensor amplifiers (SA), which are also computer controlled. Because of time variations in laser power, the amplifier modules for a given pair of photodetectors are simultaneously triggered when acquiring data at a given wavelength. A computer controls the data acquisition using a LabVIEW program that utilizes USB-GPB and Ethernet interfaces.

2C. Measurement Method and Data Acquisition

A typical sample consists of nanoparticles suspended in a fluid contained in a quartz cuvette (C). To obtain the properties of the nanoparticles alone, one must also measure the transmission properties of the suspension fluid. The suspension fluid is placed in a second separate identical quartz cuvette. With one cuvette in each of the closely matched pair of laser beams A and B, the wavelength of the laser is scanned using $1 nm$ steps. The computer controls the RSH and selects which pair of photodiode (GPD or SPD) outputs to record, depending on the wavelength range to be measured. After the total desired wavelength range is scanned, the sample cells are interchanged using the RSH, and the process is repeated. The following quantities are thus recorded as the raw data at each wavelength in the scan:

D_{A P} (λ_{i}) = signal through particles and fluid in beam A, D_{B F} (λ_{i}) = signal through fluid alone in beam B, D_{A F} (λ_{i}) = signal through fluid alone in beam A, D_{B P} (λ_{i}) = signal through particles and fluid in beam B .

At any given wavelength, the detector signal can be considered to be a product of several quantities:

P_{A}

and

P_{B}

are the powers in the laser beams A and B,

T_{P}

is the transmittance of the particles,

T_{F}

is the transmittance of the fluid, and

ε_{A}

and

ε_{B}

are the gain-efficiency products of the detectors in beams A and B. For each case in Eq. (5), the corresponding signals are given in Eq. (6):

D_{A P} (λ_{i}) = ε_{A} (λ_{i}) T_{P} (λ_{i}) T_{F} (λ_{i}) P_{A} (λ_{i}), D_{B F} (λ_{i}) = ε_{B} (λ_{i}) T_{F} (λ_{i}) P_{B} (λ_{i}), D_{A F} (λ_{i}) = ε_{A} (λ_{i}) T_{F} (λ_{i}) P_{A} (λ_{i}), D_{B P} (λ_{i}) = ε_{B} (λ_{i}) T_{P} (λ_{i}) T_{F} (λ_{i}) P_{B} (λ_{i}) .

D_{A P}

and

D_{B F}

are recorded simultaneously in a first wavelength scan. The samples are interchanged and

D_{A F}

and

D_{B P}

are recorded simultaneously in the second wavelength scan. While pulse-to-pulse power fluctuations can be significant, the powers

P_{A} (λ_{i})

and

P_{B} (λ_{i})

are measured simultaneously with the same trigger pulse using matched detectors. Thus, for a given wavelength,

λ_{i}

, the ratio

P_{A} (λ_{i}) / P_{B} (λ_{i})

is determined by the optical system and does not vary significantly from pulse to pulse. By computing the ratio-of-ratios, R, we obtain the square of the transmittance due to the particles versus wavelength. As seen in Eq. (7), the ratios

P_{A} / P_{B}

, the fluid transmittance

T_{F}

, and the detector factors

ε_{A}

and

ε_{B}

all drop out of the expression at each wavelength value:

R (λ_{i}) = \frac{\frac{D_{A P} (λ_{i})}{D_{B F} (λ_{i})}}{\frac{D_{A F} (λ_{i})}{D_{B P} (λ_{i})}} = \frac{\frac{ε_{A} (λ_{i}) T_{P} (λ_{i}) T_{F} (λ_{i}) P_{A} (λ_{i})}{ε_{B} (λ_{i}) T_{F} (λ_{i}) P_{B} (λ_{i})}}{\frac{ε_{A} (λ_{i}) T_{F} (λ_{i}) P_{A} (λ_{i})}{ε_{B} (λ_{i}) T_{P} (λ_{i}) T_{F} (λ_{i}) P_{B} (λ_{i})}} = [T_{P} (λ_{i})]^{2} .

The square root of Eq. (7) gives the experimental values for the nanoparticle transmittance (due to both scattering and absorption) versus wavelength:

T_{P} (λ_{i}) |_{meas} = \sqrt{R (λ_{i})} = \sqrt{\frac{\frac{D_{A P} (λ_{i})}{D_{B F} (λ_{i})}}{\frac{D_{A F} (λ_{i})}{D_{B P} (λ_{i})}}} .

Dividing the transmittance, Eq. (8), by the known sample path length z, an experimental value for the extinction coefficient versus wavelength is obtained as shown in Eq. (9):

α_{meas} (λ_{i}) = \frac{- \ln T_{P} (λ_{i})}{z} |_{meas} .

Before sample data is taken, the cuvettes are tested to see if their transmittances are equal. If the two cuvettes show dissimilar transmittances, then a set of data similar to that described above is taken with both cuvettes empty to obtain the ratio of the cuvette transmittances at each wavelength. This ratio can then be divided out of the sample transmittance ratio at each wavelength.

2D. Data Inversion

A set of linear equations [Eq. (10)] with unknowns, $n_{j}$ , is obtained by setting the measured extinction coefficients, Eq. (9), at each wavelength equal to the theoretical extinction coefficients as given in Eq. (4):

α_{meas} (λ_{i}) \equiv \sum_{j = 1}^{N} σ_{ext} (λ_{i}, r_{j}) n_{j} .

A particle size distribution is obtained by solving this set of linear equations for the particle densities

n_{j}

. With Eq. (10) written in matrix form, Eq. (11), it is apparent that finding the particle size distribution is essentially a matrix inversion problem:

[\begin{matrix} \dots \\ α_{i} \\ \dots \end{matrix}] = [\begin{matrix} \dots & \dots & \dots \\ \dots & σ_{i j} & \dots \\ \dots & \dots & \dots \end{matrix}] \cdot [\begin{matrix} \dots \\ n_{j} \\ \dots \end{matrix}] or more compactly α = σ \cdot n .

In our samples, the number of particle types and their sizes are unknown. This is referred to as an ill-posed problem, for which inversion algorithms typically employ least-squares minimization. However, this approach alone may lead to multiple solutions, most of which are non-physical. A regularization term can be introduced to filter out the non-physical solutions. The most common method is called Tikhonov regularization [22], where a regularization parameter, a, and matrix, Γ, are used to impose restrictions that give preference to non-oscillatory solutions. Solutions are found by minimizing the sum of the squares of the deviations along with the regularization term as shown in Eq. (12):

‖ σ \cdot n - α ‖^{2} + ‖ a Γ \cdot n ‖^{2} .

To perform the inversion, we use the publicly available FORTRAN program FTIKREG [23], which calculates the inversion of the Fredholm integral of the first kind using Tikhonov regularization. FTIKREG takes two inputs, and we impose constraints on the inversion to restrict the output densities to positive values lying in the range of 1 to

3000 nm

. The first input is the kernel, which is the matrix of theoretical extinction cross sections for each wavelength and particle size in the desired range with the desired resolution (

1 nm

for samples containing one type and size of particle and

3 nm

for mixtures of particles). The kernel is generated by the publicly available FORTRAN program MIEV0 [24]. MIEV0 [25] and FTIKREG [26, 27, 28] have been used to solve inversion problems similar to the one described here, and this combination works efficiently for our application. The kernel is stored in memory so that it does not have to be calculated while the inversion program is running. MIEV0 calculates the extinction cross section of spherical particles from known values for the real and imaginary parts of the index of refraction over the range of 210 to

3000 nm

. The wavelength-dependent dielectric constant has been catalogued for many materials, including polystyrene, a common reference material [24, 29, 30], and many metals, semiconductors, oxides, molecules, water, [29, 30, 31, 32, 33, 34] polymers [35, 36], and numerous organic compounds via the Reaxys database [37]. The kernel can be modified to include multiple particle types; however, these types of mixtures are not discussed in this paper. In Section 4 we discuss mixtures of polystyrene particles of different diameters. The second input to FTIKREG is our measured extinction coefficient as a function of wavelength.

Historically, various methods have been employed in the context of spectral inversion for particle size analysis. These include analytical [38, 39, 40, 41], numerical [42], Monte Carlo [43], and iterative [44] methods. Because of its compatibility with modern computational techniques, we use FTIKREG with double precision on a Windows laptop computer. A typical inversion involves extinction inputs at $\sim 400$ wavelength values and produces density outputs at $\sim 1000$ possible diameters and takes $\sim 15 s$ . The output of the inversion program is the particle density distribution versus particle size (diameter). FTIKREG also outputs an uncertainty for each value in the resulting density distribution. For all cases presented here, the inversion uncertainty is under 2% indicating the high reliability of the results. As discussed in Section 3, this general approach reliably produces results that accurately represent the known diameter and density of test particles for homogeneous (single particle size), and non-homogeneous (more than one particle size) samples under conditions we have carefully delineated.

3. Results

3A. Polystyrene Spheres

Polystyrene spheres were used as standard test particles because commercial samples of uniform diameter, known concentration, and accurate sphericity can be obtained in a variety of sizes, and the wavelength-dependent dielectric constant for this material is known [29]. Commercial samples can be accurately diluted with measured amounts of deionized water. Transmission data on such samples were collected for polystyrene spheres with various diameters. In this section we show experimental results for a number of diameters as a function of dilution, and compare to theoretical results. We also show results for samples containing a mixture of particles of different sizes compared with theory.

Typical data are shown in Fig. 2a, where we plot experimental extinction, defined as $E = - \ln T_{P} (λ)$ , for $1025 nm$ diameter polystyrene spheres at three densities as measured by quantitative dilution of the manufacturer’s samples. Quantitative dilution resulted in the particle samples numbered 1, 2, and 3 with densities of $2.00 \times 10^{8}$ , $6.00 \times 10^{8}$ , and $1.20 \times 10^{9} particles / ml$ (ratios $1 ∶ 3 ∶ 6$ ), respectively. Also shown is a theoretically constructed version of the extinction calculated using Mie theory for $1025 nm$ diameter polystyrene spheres at density 3. In all cases, the sample path length is $z = 1.00 \pm 0.001 cm$ . All theoretical and measured extinction spectra are then inverted using the Mie-based algorithm discussed in Subsection 2D to obtain the particle density dis tribution versus diameter, as shown in Fig. 2b. Inverting the three measured extinction spectra (for densities 1, 2, and 3) gives the particle density distributions shown. Inverting the theoretically generated (Mie model) extinction spectrum returns the correct density and size values that were input to the Mie model. For purposes of presentation, the density distribution peak produced by inversion of the Mie model extinction has been linearly scaled by a factor of 0.0741 in Fig. 2b. Without scaling, the area under the Mie model peak is equal to the area under the broader experimental peak for density 3. Finally, the total number of particles associated with a given peak is obtained by integrating the density distribution for all sizes associated with a given peak. For the curves shown, this gives total experimental particle densities of $2.15 \times 10^{8}$ , $6.75 \times 10^{8}$ and $1.29 \times 10^{9} particles / ml$ (ratios $1.00 ∶ 3.14 ∶ 6.00$ ) for densities 1, 2, and 3, respectively. Note that the total densities and ratios of the total densities—obtained by integration of the distributions in Fig. 2b—are in good agreement with the expected values. Also note that the maximum extinction values found in Fig. 2a are not sufficient to provide an accurate determination of the total density or the ratio of densities using the Bouguer–Beer–Lambert law alone, because, as expected, the commercial particle samples contain a distribution of particle sizes. A detailed discussion of the uncertainty and accuracy of our technique is presented in Subsections 3B, 3C.

3B. Performance Tests with Theoretically Generated Extinction Values

To quantify numerical performance, we tested the inversion algorithm that is used to obtain the particle density versus diameter from extinction versus wavelength as described in Subsection 2D. To do this, we first generated theoretical (input) values of the extinction versus wavelength for spheres of a given size. As shown in Fig. 3a, we have plotted the normalized deviation of the resultant inverted results as a function of diameter. In this case, we see no measurable deviation of the output inverted size ( $D_{inverted}$ ) from the theoretical extinction input size ( $D_{input}$ ). The error bars in this case represent the $1 nm$ step size that is used in the inversion algorithm. This step size was chosen because we empirically found that, for $5 nm$ diameter particles (the smallest measured), the particle size distribution (obtained by inversion) is insensitive to step sizes smaller that $1 nm$ .

Similarly, in Fig. 3b, we show results for the output inversion density ( $n_{inverted}$ ) of theoretically generated extinction results for fixed diameter ( $771 nm$ ) particles of varying input density ( $n_{input}$ ). The error bars in this case represent the normalized statistical uncertainty of the particle count, as output by the inversion algorithm. We see that the normalized uncertainty in this case is $\sim 10^{- 7}$ , with a very small systematic offset, $\sim 10^{- 8}$ , which appears to be due to rounding errors. We minimize this and related effects by inverting extinction data in $100 nm$ wavelength intervals to obtain high resolution in the inversion, and combining the separate inversions to form a complete density distribution versus diam eter plot.

3C. Performance Tests with Measured Extinction Values

We also performed comparable tests with actual spheres. Polystyrene spheres were used as standard test particles for the reasons noted in Subsection 3A. Shown in Fig. 3c are results for three sizes of spheres, specified by the manufacturer to have diam eters of 404, 771, and $1025 nm$ , for which the standard deviation (the square root of the mean of the squared deviations) is represented by the horizontal error bars. We collected nine sets each of extinction versus wavelength data for these three different sizes, represented by the three sets of red data points in Fig. 3c. The fractional deviation of the measured diameter ( $D_{measured}$ ) relative to the manufacturer’s stated diameter ( $D_{reference}$ ) showed a standard deviation (the square root of the mean of the squared deviations), represented by the vertical error bar, of $σ \leq 0.5 %$ for each size, and the deviation of the mean diameter from the manufacturer’s reference value for all particle sizes was $\leq 1.5 σ$ .

Finally, in Fig. 3d, we show results for measurements of $771 nm$ polystyrene spheres with different densities. For each of three different densities, we collected nine sets of extinction versus wavelength data, shown as the red points. In Fig. 3d, we see that the fractional deviation of the measured density ( $n_{meas.}$ ) from the reference density ( $n_{reference}$ based on measured dilutions of the manufacturer’s samples) showed a standard deviation (vertical error bar) of $σ \leq 4 %$ for each density, and that deviations of the mean measured densities from the reference value are $\leq 1.5 σ$ . Each horizontal error bar is a combination of the manufacturer’s uncertainty in the density and our dilution uncertainty. The manufacturer determines the density of their samples using microscopic techniques that may account for the apparent systematic difference between the LTS results and those expected from the manufacturer’s stated density.

4. Resolution

4A. Two-Diameter Mixtures

To determine the size resolution of the system, we generated values of the extinction versus wavelength using the Mie model for a variety of pairs of particle diameters, $D_{1}$ and $D_{2}$ . We then summed the extinctions for the pair and added Gaussian noise to simulate the random noise present with typical data acquisitions, as exemplified by the data in Fig. 2a, and finally inverted this simulated data. Figure 4a is a plot of this measure of resolution or (size discrimination) as a function of $D_{1}$ and $D_{2}$ , which is symmetric about the line $D_{1} = D_{2}$ . The green regions represent mixtures of particles with two diam eters, where the resulting particle density distribution peaks are resolved. The yellow regions indicate where the resulting particle density distribution peaks are resolved, but show some line pulling for the smaller particle. The green diagonal line represents homogenous samples where the particles are the same size, and the inversion outputs the cor rect density and diameter. The white region represents mixtures of two diameters, where the peaks are unresolved.

In Fig. 4b we show experimental results for two sizes, in the “resolved” regime of Fig. 4a, of polystyrene spheres physically mixed together. The blue and red curves are the individually measured particle density distributions and the solid black curve is the density distribution of the physical mixture of the same particles. Before mixing, we measured diameters of $458 \pm 2$ and $772 \pm 4 nm$ , at densities of $(2.00 \pm 0.88) \times 10^{8}$ and $(1.95 \pm 0.08) \times 10^{8} particles / ml$ , respectively. After mixing, diameters of $462 \pm 2$ and $772 \pm 4 nm$ and densities of $(1.94 \pm 0.08) \times 10^{8}$ and $(1.88 \pm 0.08) \times 10^{8} particles / ml$ , respectively, were measured. Thus, although the peak widths increased with mixing, the position and area under the each peak remained constant to within the uncertainties noted.

4B. Multi-Diameter Mixtures and Continuous Distributions

We also undertook theoretical investigations to determine the inversion resolution for multi-diameter particle systems. In Fig. 5a we show the results for a theoretical comb distribution of particles 50 to $3000 nm$ in diameter, with a delta-function spike at each $50 nm$ . At diameters where the associated extinction versus wavelength generated by the Mie model for a given spike contains a peak, as in Fig. 2a, the resultant inversion peak (in red) generally has the same amplitude as a spike. This is true for diameters of $750 nm$ and greater. For diameters of 200 to $700 nm$ , the resultant inversion peak (in red) is reduced in amplitude and broadened compared to the input spike (in black), but still has an area of unity (as does each spike) and is centered about the correct diameter. The amplitude suppression occurs at the particle diameters for which there is no distinct maximum present in their extinction spectra within the wavelength range of 250 to $1000 nm$ . Finally for the 50, 100, and $150 nm$ spikes, the inversion produces a single broad peak with a total area 3 times that of a single spike. These diam eters lay below the lower wavelength bound of the theoretical extinction spectrum. These results are consistent with those presented in Fig. 4a. An experimental investigation of a two-particle- diameter mixture in the small-particle range is also discussed below in conjunction with Fig. 6b, where smaller ( $46 nm$ ) particles are resolved.

In Fig. 5b, we show the results for the inversion of a Gaussian distribution of sizes with a full width at half-maximum of $708 nm$ and unit integrated density. In this case, the fidelity of the reproduction is high except, again, for deviations at small particle diameter.

5. Sensitivity and Comparison to Dynamic Light Scattering

To quantify the sensitivity of LTS, we measured the minimum density required to acquire a value of density with 1% accuracy. To do this, we systematically diluted particle suspensions of spherical gold particles ranging from 5 to $40 nm$ and spherical polystyrene particles ranging from 46 to $3000 nm$ in diameter. For gold particles, plasmon resonances imply that particle geometry can have a strong effect on the extinction spectrum [45]. Therefore, the gold particles used in these measurements were imaged with transmission electron microscopy, which showed a maximum deviation from sphericity of at most 5% (based on measurements of 200 random particles for a given diameter).

We used tabulated values for the complex dielectric constants of gold [2, 24] and polystyrene in the inversion kernel. In Fig. 6a, we show our results compared to the published results of a high sensitivity commercial DLS instrument, for measurements of globular samples including proteins and latexes [46]. We observe a higher sensitivity with LTS throughout the size range of 5 to $3000 nm$ and a sensitivity $\sim 10^{6}$ times greater than DLS for particle sizes above $50 nm$ , where, for both methods, dielectric materials were studied. The uncertainty in the LTS results shown in Fig. 6 is less than or equal to the size of the plotting symbols, and the uncertainty in the DLS results is assumed comparable for the purpose of exposition. A detailed analysis of LTS uncertainties is presented in the discussion of Fig. 3 in Subsections 3B, 3C. Our highest sensitivity at present is $3580 particles / ml$ for $1025 nm$ polystyrene particles. For this measurement, the laser beam apertures were expanded to produce a roughly uniform beam over a diameter of $\sim 4.0 mm$ at the $1.00 \pm 0.01 cm$ long sample cell. These dimensions imply an exposure volume of $0.126 ml$ for a total of $\sim 451$ particles in the measurement volume.

We also performed an experimental resolution comparison between LTS and DLS, and, at the same time, investigated the resolution of LTS for samples containing mixtures of nearly equal volume percent samples with two particle diameters. In Fig. 6b, we show LTS results for individual homogeneous samples of 46 and $206 nm$ particles at 0.00204 and $0.00203 vol . %$ , respectively, as the red and blue curves. We also show results for the physical mixture of these same samples measured by LTS and DLS [47], as the solid black and green curves, respectively. In all cases. individual data points are also plotted. This combination of particle sizes is at the LTS limit for small- and larger-diameter mixtures for this particular smaller diameter. The first and second black peaks have integrated volume percents of 0.00201 and 0.00198, respectively, and peak maxima at 50 and $204 nm$ , respectively. Note that DLS does not detect the presence of the smaller particles, detects the presence of the larger-diameter particles with lower resolution, and provides only a relative measurement of particle density.

6. Conclusions

We have shown that LTS can provide the diameter and density (and, thus, absolute particle count) for gold and polystyrene particles over the size ranges of 5 to $40 nm$ and 46 to $3000 nm$ , respectively. For polystyrene particles in the size range from 404 to $1025 nm$ in diameter, the particle diameter and density can be measured with accuracies of $\pm 0.5 %$ and $\pm 4 %$ , respectively, for samples with low volume fraction ( $5 \times 10^{- 8}$ to $0.5 vol. %$ ). To date, the greatest sensitivity achieved by LTS with $1025 nm$ polystyrene particles is $3580 particles / ml$ , or 451 total particles in the interaction volume, which is in excess of $10^{6}$ times the sensitivity of dynamic light scattering for dielectric materials. The sensitivity of LTS is due to a balanced optical technique whereby the incident beam intensity of a tunable laser is compared with its transmitted intensity after interaction with a suspension of particles as a function of wavelength.

We acknowledge support from the College of Science and Department of Physics of the University of Notre Dame, and theoretical support from Walter Johnson, who provided critical assistance on understanding the inversion problem.

Fig. 1 Schematic of the laser transmission spectrometer.

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Fig. 2 (a) Theoretical (Mie model) and experimental extinction versus wavelength for samples containing homogeneous polystyrene spheres at three densities. (b) Corresponding plots of the density distribution versus diameter after inversion The Mie model result has been linearly scaled for ease of presentation as noted in the text. The distribution given by the manufacturer is also shown.

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Fig. 3 Accuracy tests. (a) Inversion results for theoretically generated (input) extinction data for particles of a given diameter. (b) Inversion results for theoretically generated (input) extinction data for particles of a given density (n). (c) Multiple results for measurements of particle diameter compared with manufacturer’s reference values. (d) Multiple results for measurements of particle density compared with manufacturer’s reference values.

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Fig. 4 Evaluation of the resolution capabilities of LTS for mixtures of two diameters ( $D_{1}$ and $D_{2}$ ) of polystyrene spheres of near equal densities. (a) Outcome of the inversion for simulated mixtures of two particle diameters. Green areas indicate resolved peaks, yellow areas indicate conditional resolution (see text), white areas indicate lack of resolution. (b) Separate experimental results for two samples, each with a different particle diameter (red and blue curves and data points), and a sample comprising a mixture of particles with these same two diameters at near equal densities (solid black curve and data points). The peak positions and the integral under each peak are indicated on the plot.

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Fig. 5 Numerical inversion results for theoretically generated comb and Gaussian distribution functions. (a) Input comb distribution in black (simulating a sample with the same distribution) and resultant inversion in red. (b) Results for a theoretically generated Gaussian distribution (black) and inversion (red).

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Fig. 6 (a) Sensitivity comparison between DLS (green circles) and LTS (orange squares for gold and blue squares for polystyrene) as a function of particle diameter. (b) Resolution comparison of DLS and LTS. The blue and red data points and curves are LTS results for homogeneous equal-volume-percent samples of 46 and $206 nm$ particles, respectively. The lower black curve (square data points) with the two peaks is the LTS measurement of an equal-volume-percent mixture of these particles. The LTS peaks for the larger-diameter particles have been magnified for clarity. The green points and curve is the DLS result for the same mixed sample as measured by LTS (black curve and data points).

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High-precision sizing of nanoparticles by laser transmission spectroscopy

Abstract

1. Introduction

2. Laser Transmission Spectroscopy

2A. Electromagnetic Model

2B. Apparatus

2C. Measurement Method and Data Acquisition

2D. Data Inversion

3. Results

3A. Polystyrene Spheres

3B. Performance Tests with Theoretically Generated Extinction Values

3C. Performance Tests with Measured Extinction Values

4. Resolution

4A. Two-Diameter Mixtures

4B. Multi-Diameter Mixtures and Continuous Distributions

5. Sensitivity and Comparison to Dynamic Light Scattering

6. Conclusions

Cited By

Figures (6)

Equations (12)

Applied Optics