Among the best-known building blocks in nanotechnology applications are
nanoparticles of noble metals. For example, colloidal suspensions of silver or
gold nanoparticles (10-200 nm) exhibit vivid colors because of intense optical
absorption in the visible spectrum, making them useful in colorimetric
sensors. In the illustration shown below, a suspension of gold nanoparticles
of a fairly uniform size in water exhibits peak absorption near a wavelength
of \(525 \mathrm{nm}\) (near the blue region of the visible spectrum of light).
When one views the solution in ambient (white) light, the solution appears
wine-red because the blue part of the spectrum is largely absorbed. When the
nanoparticles aggregate to form large particles, an optical absorption peak
near \(600-700 \mathrm{nm}\) (near the red region of the visible spectrum) is
observed. The breadth of the peak reflects a fairly broad particle size
distribution. The solution appears bluish because the unabsorbed light
reaching the eye is dominated by the short (blue-violet) wavelength region of
the spectrum.
since the optical properties of metallic nanoparticles are a strong function
of their size, achieving a narrow particle size distribution is an important
step in the development of nanoparticle applications. A promising way to do so
is laser photolysis, in which a suspension of particles of several different
sizes is irradiated with a high-intensity laser pulse. By carefully selecting
the wavelength and energy of the pulse to match an absorption peak of one of
the particle sizes (e.g., irradiating the red solution in the diagram with a
\(525 \mathrm{nm}\) laser pulse), particles of or near that size can be
selectively vaporized.
(a) A spherical silver nanoparticle of diameter \(D\) at \(25^{\circ} \mathrm{C}\)
is to be heated to its normal boiling point and vaporized with a pulsed laser.
Considering the particle a closed system at constant pressure, write the
energy balance for this process, look up the physical properties of silver
that are required in the energy balance, and perform all the required
substitutions and integrations to derive an expression for the energy
\(Q_{\text {abs }}(\mathrm{J})\) that must be absorbed by the particle as a
function of \(D(\mathrm{nm})\)
(b) The total energy absorbed by a single particle \(\left(Q_{\text {abs
}}\right)\) can also be calculated from the following relation:
$$
Q_{\mathrm{abs}}=F A_{\mathrm{p}} \sigma_{\mathrm{abs}}
$$
where \(F\left(\mathrm{J} / \mathrm{m}^{2}\right)\) is the energy in a single
laser pulse per unit spot area (area of the laser beam) and
\(A_{\mathrm{p}}\left(\mathrm{m}^{2}\right)\) is the total surface area of the
nanoparticle. The effectiveness factor, \(\sigma_{\mathrm{ahs}},\) accounts for
the efficiency of absorption by the nanoparticle at the wavelength of the
laser pulse and is dependent on the particle size, shape, and material. For a
spherical silver nanoparticle irradiated by a laser pulse with a peak
wavelength of \(532 \mathrm{nm}\) and spot diameter of \(7 \mathrm{mm}\) with \(D\)
ranging from 40 to \(200 \mathrm{nm}\), the following empirical equation can be
used for \(\sigma_{\mathrm{abs}}\)
$$
\sigma_{\mathrm{abs}}=\frac{1}{4}\left[0.05045+2.2876 \exp
\left(-\left(\frac{D-137.6}{41.675}\right)^{2}\right)\right]
$$
where \(\sigma_{\text {abs }}\) and the leading \(\frac{1}{4}\) are dimensionless
and \(D\) has units of nm. Use the results of Part (a) to determine the minimum
values of F required for complete vaporization of single nanoparticles with
diameters of \(40.0 \mathrm{nm}, 80.0 \mathrm{nm},\) and \(120.0 \mathrm{nm}\). If
the pulse frequency of the laser is \(10 \mathrm{Hz}\) (i.e., 10 pulses per
second), what is the minimum laser power \(P(\mathrm{W})\) required for each of
those values of D? (Hint: Set up a dimensional equation relating \(P\) to \(F\).)
(c) Suppose you have a suspension of a mixture of \(D=40 \mathrm{nm}\) and
\(D=120 \mathrm{nm}\) spherical silver nanoparticles and a \(10 \mathrm{Hz} / 532
\mathrm{nm}\) pulsed laser source with a \(7 \mathrm{nm}\) diameter spot and
adjustable power. Describe how you would use the laser to produce a suspension
of particles of only a single size and state what that size would be.
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