BME1450 Term Paper
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A Statistical Model for the Estimation of Gold
Nanorod Extinction Coefficients
Tanya Hauck

Abstract—Due to their tunable optical properties, gold
nanorods are being studied, characterized and modified for
biomedical applications. At present, rod concentration values are
estimations, based on Transmission Electron Microscope (TEM)
and Inductively Coupled Plasma (ICP) measurements, which are
prone to error. A model is presented to identify the main sources
of error in extinction coefficient calculations, and to relate
calculated values to literature values. Simulations show that the
presence of spherical byproducts, which usually form during the
synthesis of nanorods, present the greatest source of error when
calculating the extinction coefficient.
Index
Terms—gold nanorods,
concentration, statistical model
extinction
coefficient,
I. INTRODUCTION
N
materials such as fullerenes, quantum dots and
metallic nanoparticles have unique properties because of
their high surface area to volume ratio [1]. Gold nanospheres
and nanorods also have unique optical properties because of
the quantum size effect [2]. Gold nanorods are cylindrical
rods which range from less than ten to over forty nanometers
in width and up to several hundred nanometers in length.
These particles are typically characterized by their aspect ratio
(length divided by width) [3], [4].
The free electrons in gold nanoparticles oscillate at specific
wavelengths which relate to the nanoparticle’s size, and
produce an absorbance peak [1], [2], [5]. Nanorods have two
absorbance maxima, corresponding to their “length”
(longitudinal) and “width” (transverse) dimensions [6]. The
longitudinal absorbance bands are tunable and the peak
wavelength increases linearly with size [7], making nanorods
suited for a variety of biomedical applications.
Gold nanorods are considered excellent candidates for
biological sensing applications because the absorbance band
changes with the refractive index of local material [5],
allowing for extremely accurate sensing. In addition, nanorods
with near-infrared absorption peaks can be excited by a laser at
the absorbance band wavelength to produce heat, potentially
allowing for the selective thermal destruction of cancerous
tissues [6].
ANOSCALE
Manuscript submitted on November 7, 2005. Financial support from
NSERC is acknowledged.
T. Hauck is with the Institute of Biomaterials and Biomedical Engineering,
University of Toronto, Toronto, ON M5S 1A1, CANADA (e-mail:
Tanya.Hauck@utoronto.ca).
In order to study and exploit the unique properties of
nanorods, it is necessary to have a robust extinction coefficient
which can predict the concentration of a solution at a particular
absorbance. It is difficult to accurately obtain a measure of
nanoparticle (as opposed to metal atom) concentration in
moles per litre [6]. No spectroscopic device can provide
concentration data, and only approximations are currently
available. Biomedical applications of nanoparticles require
nanorods to be capped with biological molecules such as
antibodies. Concentration measurements that are off by orders
of magnitude can result in the waste of thousands of dollars of
antibodies and the irreversible aggregation of nanorod
samples, if excess or insufficient quantities of biological
capping agents are added.
The El-Sayed method of nanorod concentration
determination [7], [8] is currently the standard way of
measuring extinction coefficients (ε), and involves the
coupling of bulk gold concentration, Transmission Electron
Microscopy (TEM) size analysis and absorbance data.
Recently, Liao and Hafner calculated ε values of nanorods by
preparing films of immobilized nanorods [6]. Liao and Hafner
note that spherical byproducts lower the extinction coefficient
calculated by the El-Sayed method, but they do not discuss the
sensitivity of ε to this kind of error.
In this paper, a review of the El-Sayed method is presented
and the sensitivity of nanorod extinction coefficients to
changes in gold concentration and nanorod dimensions are
presented in a statistical model. It is shown that the presence
of spherical byproducts is the greatest source of error in
concentration estimates.
II. THEORY: EXTINCTION COEFFICIENT DETERMINATION
Extinction coefficients are used to relate concentration and
absorbance. From Beer’s Law [9]:
A  bc
(1)
A is absorbance (unitless), ε is the extinction coefficient
(M-1cm-1), b is pathlength (cm) and c is concentration (M). ε is
found by plotting A against c. A can be measured with a UVVisible Spectrophotometer and entered into (1) to obtain c.
The following measurements are used in calculations:
1) TEM images: Used to measure the length and width of
particles. From TEM data, average nanorod dimensions and
BME1450 Term Paper
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volume can be calculated. The volume measurement is
combined with a literature gold density value (atoms/nm3) to
obtain an average number of atoms of gold per nanorod.
2) Total quantity of gold: In order to obtain a value of
nanorods/volume, the total gold concentration in the sample is
needed. There are several ways of obtaining this data.
One option is to assume that every atom of gold added to
the sample becomes a nanoparticle.
This assumption
overestimates the amount of gold, as some gold does not form
particles and remains in solution as ions. This method
theoretically underestimates the extinction coefficient.
Inductively coupled plasma (ICP) can be used to measure
the total concentration of gold atoms. ICP measures trace
amounts of metals in samples [10]. ICP data also contain error
– to measure the gold content of a sample it is necessary to
calibrate the ICP device with gold standards, which are diluted
in the lab and thus inherently erroneous.
3) Absorbance Measurements: Finally, absorbance
measurements must be obtained to produce a Beer’s law plot
(absorbance against concentration).
TABLE II
PREPARATION OF GOLD SEED
Reagents
Quantity (mL)
0.01M HAuCl4•3H2O
0.1 M CTAB (cetyltrimethylammonium bromide)
0.01 M AgNO3
0.1 M Ascorbic Acid
Seed solution from table I
4.25
100
0.625
0.675
1.05
B. Characterization and Measurements for Extinction
Coefficient Calculations
A solution of unknown concentration “c” was diluted to
produce eight different solutions (refer to fig. 1 (A)).
Absorbance measurements were taken at each concentration,
yielding a set of concentration/absorbance data.
The
longitudinal absorbance peak was used for the Beer’s Law
plots in the statistical model.
longitudinal
A
transverse
III. EXPERIMENTAL RESULTS USED IN STATISTICAL MODEL
In order to create a statistical model, sample data were
required for baseline measurements.
The following
experimental results are used later as starting values in the
statistical model. Normally distributed noise is added to the
data values during statistical modeling to examine the effect of
different types of errors.
A. Synthesis
Synthesis of gold nanorods has recently undergone dramatic
improvements. It is possible to produce high yields of nearly
monodispersed short gold nanorods [4], [7], [11]. The rods
synthesized for this paper were synthesized using Murphy’s
method [11]. First, a “seed” solution of spherical gold
nanoparticles was prepared by adding the following:
TABLE I
PREPARATION OF GOLD SEED
Reagents
0.01M HAuCl4•3H2O
0.1 M CTAB (cetyltrimethylammonium bromide)
0.01 M NaBH4
Quantity (mL)
0.250
7.5
0.6
NaBH4 reduces the gold salt to form nanoparticles, and
CTAB is a surfactant which stabilizes the seeds to prevent
aggregation. To make nanorods, the reagents in Table II are
added in order from top to bottom. Gold rods are synthesized
with a small amount of silver to control rod size and make
short rods [7], [11]. CTAB is a directing surfactant; without it,
only spheres would form. CTAB forms a rod shaped template
that is filled with gold atoms as they are reduced by ascorbic
acid. Ascorbic acid is a weaker reducing agent than NaBH4,
but in the presence of seeds and a CTAB template, it reduces
gold ions at the seed surface [4].
B
Fig. 1. (A) Dilution of nanorod solution. From left to right, starting with
unknown concentration “c”, 0.75c, 0.5c, 0.375c, 0.25c, 0.125c, 0.1c, 0.05c.
(B) Absorbance of diluted samples. Absorbance at the longitudinal peak is
used to calculate the Beer’s law plot.
C. Transmission Electron Microscopy Data
TEM images were taken with a Philips Tecnai 20 TEM.
Sample images in Fig. 2 show the high degree of
polydispersity. Fig. 2 (A) and (B) are from the same sample.
Even within a 1 μm radius on a TEM grid, the distribution of
nanoparticle shape ranges from regions of entirely rods to
regions where there are large numbers of spherical byproducts.
TEM grids were prepared by placing a drop of concentrated
rod solution onto a plastic-coated TEM grid.
A
B
Fig. 2. (A) and (B). TEM Images showing different size distributions
within the same sample. Scale bars are 100 nm.
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IV. STATISTICAL MODEL AND SIMULATION
A. Overview of the Statistical Model
To examine the relationship between nanorod extinction
coefficients, nanorod size distributions and measured gold
concentrations, the measured variables (size and gold
concentration) were varied so that their effect on the extinction
coefficient could be observed. Using estimates based on the
experimental data presented above, normally distributed data
was randomly generated to simulate different size distributions
and ICP error. From the literature [7], it is assumed that
nanoparticle sizes are normally distributed. It is also assumed
that ICP measurement error is normally distributed, since it is
random human dilution error.
The error in absorbance data is considered statistically
insignificant relative to TEM and ICP error since the line fitted
to the absorbance measurements alone had an R2 value of
0.9985. Furthermore, the purpose of this paper is to focus on
novel errors that arise from nanoparticle experiments. It has
been noted that when the absorbance reading is repeated for
the same set of solutions, the slope has very little error. There
is usually an offset error that occurs as the machine is
calibrated before each use. This value is subtracted from the
data to maintain the slope while setting the y intercept to zero.
B. Simulation of Error in Particle Dimension Distribution
The volume of nanorods can be calculated using the
following model and equation:
Length = L
R
Width = W
W = 2R
Radius = R
Fig. 3. Calculation of rod volume.
Volume  43 R 3  R 2 L  2 R 
(2)
From the TEM images, the average length of the rods was
approximately 30 nm and the average width was
approximately 8 nm, giving an aspect ratio of 3.75. These
values were used as estimates for the size calculations.
C. Effect of Polydispersity and Spherical Byproducts
To model changes in polydispersity, rod volume was
changed while the total amount of gold was held constant. The
standard deviation of the length and width were varied:
TABLE III
POLYDISPERSITY MODEL
Length (nm)
Width (nm)
30 ± 1
8±1
30 ± 5
8±3
30 ± 15
8±5
While simulating the effect of volume changes, the gold
concentration value was held constant, and it was assumed that
every atom of gold became a nanoparticle when the original
solution of concentration “c” was synthesized. This solution
had a total gold concentration of 402 μM. The concentrations
of the other solutions were calculated by dilution (e.g. 0.75 x
402 μM = 301.5 μM). The nanorod dimensions and standard
deviations in Table III were used with (2) to simulate a sample
of 1000 normally distributed random nanorods.
This
simulation of 1000 nanorods was repeated three times to
obtain an average volume and standard deviation values (for
the error bars in Fig. 4). The average volume was used to
calculate the atoms of gold per gold rod (from [7], gold
density is 59 atoms/nm3). The total concentration of gold is
divided by the gold per rod to obtain an estimate of rod
concentration for each data point. Finally, rod concentration is
plotted against absorbance and the slope and standard error are
calculated.
Spherical byproducts are formed during gold nanorod
synthesis because they are thermodynamically favoured [1].
To see the effect of sphere contamination on ε, the calculation
was performed with rods of 30 ± 5 x 8 ±3 nm, at varying
concentrations of 20 ± 5 nm diameter spheres (5%, 10%, 20%,
and 40% spheres), using the equation for the volume of a
sphere. Total gold concentration is kept constant, while three
simulations of 1000 nanoparticles are averaged.
D. Simulation of Error in Total Gold Content
To model errors in ICP data, the volume values were kept
constant while the gold concentration values were simulated.
1000 random, normally distributed rods (30±5 nm by 8±3 nm)
were generated, and the average volume of these rods was
used for all calculations. To observe the effect of errors in
ICP data, a baseline estimate for gold was taken as the
maximum possible gold content, assuming all gold ions
formed nanoparticles (402 μM).
For each absorbance
measurement a total gold concentration value was randomly
generated (with a mean of 402 μM and a standard deviation of
5, 25 and 100 μM), and scaled to the appropriate dilution
factor (multiplying by 0.75, 0.5, etc.). This randomization of
gold concentration was also repeated three times to obtain
error bars.
V. RESULTS
A. Errors from Polydispersity and Colloidal Byproducts
Changes in nanorod dimensions have the greatest impact on
extinction coefficients. Fig. 4 shows the change in the Beer’s
law plot with increasing polydispersity. As variation in the rod
dimensions increases, the extinction coefficient increases.
However, these increases are not excessive – from 3.85 ± 0.07
to 4.90 ± 0.08 x 10-8 M is not significant, and errors in
concentration resulting from this magnitude of change in ε
would not be enough to cause aggregation or substantial
miscalculations of antibodies. However, the effect of spherical
byproducts is more pronounced, and the presence of spheres
can alter the extinction coefficient enough to cause concern.
Fig. 4 (B) shows that the extinction coefficient nearly
doubles as percentage of spherical by products is increased
from 5% to 40%. Spherical byproducts are an ongoing
challenge in nanorods synthesis, and it is important to consider
how they can change the extinction coefficient.
BME1450 Term Paper
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Error from Changes in Polydispersity
2
8
-1
C. Comparison with Literature Values
The range of extinction coefficients found in this model are
quite different from the values calculated by [4] and [6], which
were on the order of 109 M. However, the rods used in these
calculations, while of approximately the same aspect ratio, had
dimensions of 50 x 15 nm instead of 30 x 8 nm. The same
aspect ratio implies the same peak wavelength, but can result
in a different rod volume (and thus different concentration
measurements).
Small changes in aspect ratio can
substantially change absorbance peak location [7]. Literature
values must be used cautiously in nanorod studies.
-1
ε = 4.18 ± 0.07 x 10 M cm
1.8
1.6
Absorbance
1.4
8
-1
-1
ε = 4.90 ± 0.08 x 10 M cm
1.2
ε = 3.85 ± 0.07 x 108 M-1cm-1
1
0.8
0.6
0.4
Increasing σ
0.2
0
0.00E+00
1.00E-09
2.00E-09
3.00E-09
4.00E-09
5.00E-09
6.00E-09
Nanorod Concentration (M)
30±1 nm by 8±1 nm
30±5 nm by 8±3 nm
30±15 nm by 8±5 nm
VI. CONCLUSIONS
Error from Spherical Byproducts
B2
8
-1
-1
8
ε = 6.13 ± 0.11 x 10 M cm
1.8
-1
-1
ε = 5.11 ± 0.09 x 10 M cm
ε = 8.01 ± 0.14 x 108 M-1cm-1
1.6
Absorbance
1.4
1.2
1
0.8
ε = 4.64 ± 0.08 x 108 M-1cm-1
0.6
0.4
Increasing % colloids
0.2
0
0
5E-10
1E-09
1.5E-09
2E-09
2.5E-09
3E-09
3.5E-09
4E-09
4.5E-09
Nanorod Concentration (M)
5% Spheres
10% Spheres
20% Spheres
40% Spheres
Fig. 4. Effect of error as (A) polydispersity of particles change and (B)
number of spherical byproducts increases. In both cases, total gold
concentration was held constant. Error bars are very narrow.
B. Errors in ICP
Fig. 5 shows that errors in ICP data have a low impact on
the calculated ε value. When taking ICP data, a calibration
curve of several gold standards is used to calibrate the machine
– even if one of these solutions is not accurate, the overall
calibration curve will be fairly accurate. If ICP data is not
available, it is reasonable to assume that all gold atoms
become incorporated in nanoparticles, without significantly
miscalculating ε. The large error bars in this model result from
the small simulation size – only eight absorbance values were
simulated (in triplicate), as opposed to 1000 rods.
Effect of ICP Error
2
1.8
1.6
Absorbance
1.4
σ = 5: ε = 4.49 ± 0.05 x 108 M-1cm-1
8
-1
-1
σ = 25: ε = 4.38 ± 0.08 x 10 M cm
8
-1
σ = 100: ε = 4.36 ± 0.38 x 10 M cm-1
1.2
1
0.8
0.6
0.4
0.2
0
0.00E+0 5.00E0
10
1.00E09
1.50E09
2.00E09
2.50E09
3.00E09
3.50E09
4.00E09
4.50E09
5.00E09
Nanorod Concentration
400 ± 5 μM
400 ± 25 μM
400 ± 100 μM
Fig. 5. Effect of error in ICP values. Nanorod volume was held constant as
the variation in the total concentration was manipulated.
Randomly generated, normally distributed values were used
to model error in the estimated extinction coefficients of gold
nanorods. Errors resulting from spherical byproducts resulted
in the greatest changes in the calculated value of ε. Errors in
the total gold concentration had a low impact on ε. For this
sample, ε values on the order of 4 x 108 M-1cm-1 were
obtained. These deviations from literature [6] underscore the
importance of performing a new extinction coefficient
calculation for each sample, as rods with nearly identical
aspect ratios may have very different volumes. To use
nanorods in biomedical applications, it is advised that samples
are produced in bulk and ε is calculated for each batch.
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