Article pubs.acs.org/Langmuir
Nanowire Heating by Optical Electromagnetic Irradiation Paden B. Roder,† Peter J. Pauzauskie,*,† and E. James Davis‡ †
Department of Materials Science and Engineering and ‡Department of Chemical Engineering, University of Washington, Seattle, Washington 98195-2120, United States ABSTRACT: The dissipative absorption of electromagnetic energy by 1D nanoscale structures at optical frequencies is applicable to several important phenomena, including biomedical photothermal theranostics, nanoscale photovoltaic materials, atmospheric aerosols, and integrated photonic devices. Closed-form analytical calculations are presented for the temperature rise within infinite circular cylinders with nanometer-scale diameters (nanowires) that are irradiated at right angles by a continuous-wave laser source polarized along the nanowire’s axis. Solutions for the heat source are compared to both numerical finite-difference time domain (FDTD) simulations and well-known Mie scattering cross sections for infinite cylinders. The analysis predicts that the maximum temperature increase is affected not only by the cylinder’s composition and porosity but also by morphology-dependent resonances (MDRs) that lead to significant spikes in the local temperature at particular diameters. Furthermore, silicon nanowires with high thermal conductivities are observed to exhibit extremely uniform internal temperatures during electromagnetic heating to 1 part in 106, including cases where there are substantial fluctuations of the internal electric-field source term that generates the Joule heating. For a highly absorbing material such as carbon, much higher temperatures are predicted, the internal temperature distribution is nonuniform, and MDRs are not encountered.
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INTRODUCTION Understanding the absorption of electromagnetic energy by nanoscale materials at optical frequencies is important in a range of phenomena, including the modeling of the photophoretic dynamics of atmospheric aerosols1 that have been implicated in global climate cycles,2 the generation of electrical energy by nanoscale photovoltaic devices,3−5 the laser-thermal processing of semiconductor nanostructures,6,7 the heating profiles of metallic nanocrystals used in photothermal biomedical theranostics,8 the soft ionization of biological macromolecules in mass spectrometry,9 laser-assisted timed ionization events in atom-probe tomography,10 the characterization of nanowires with Raman spectroscopy,11−13 and more recently in the performance devices fabricated to achieve optomechanical laser refrigeration.14 Heat transfer from laserirradiated nanomaterials into a surrounding matrix is the most crucial determinant of the maximum temperature rise, but unlike the case of nanoscale spheres,15 the authors are unaware of quantitative heat-transfer calculations for a cylindrical nanostructure (nanowire) irradiated at normal incidence by a coherent electromagnetic plane wave. It is well known that the absorption of electromagnetic energy can be used to heat nanostructures and can also be applied to determine their optical properties and some physical/chemical properties. For example, Pluchino et al.16 and Allen et al.17 suspended single microspheres by means of electric fields to study electromagnetic energy absorption. Pluchino and co-workers measured the complex index of © 2012 American Chemical Society
refraction of carbon spheres by illuminating a levitated sphere with an argon ion laser beam and measuring the scattered light. The complex index of refraction was determined by comparing the angular light scattering data with Lorenz−Mie theory. Allen and co-workers used an electrodynamic balance to suspend single droplets also illuminated with an argon ion beam. As the beam intensity was increased, the evaporation rate of the droplet increased because of the absorption of electromagnetic energy. They determined the complex index of refraction by computing the evaporation rate based on calculations of the internal electric field and the internal temperature distribution, comparing the calculated evaporation rates with the measured values. Lorenz−Mie theory was used to compute the internal heat source and to determine the droplet size by comparison with measurements of the morphology-dependent resonances (MDRs). An electromagnetic beam can have two additional interesting effects on small particles.18 The first is the radiation pressure exerted by the beam on the particle, and the second is the nonuniform heating of the particle that leads to a photophoretic force. The theory of the scattering of electromagnetic radiation by small particles, including spheres and infinitely long circular cylinders, has been treated in detail by van de Hulst,19 Kerker,20 and Bohren and Huffman.21 Although the external electrical Received: August 10, 2012 Revised: September 29, 2012 Published: October 12, 2012 16177
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and magnetic fields are usually of interest, here we focus on the internal fields to establish the heat source using the notation and conventions of Bohren and Huffman. Although numerous studies of light scattering by microspheres have been reported as discussed by Davis and Schweiger,22 there have been fewer studies of light scattering and electromagnetic energy absorption by cylinders. Chang and co-workers (Owen et al.,23,24 Benincasa et al.25) explored the internal electric field distributions for cylinders with a particular interest in resonances, and Ruppin26 and Liu et al.1 applied the theoretical results for long cylinders to calculate the electromagnetic energy density and internal electric field distribution in an irradiated cylinder, respectively. Liu and co-workers were particularly interested in highly asymmetric internal fields that could lead to photophoresis. Additional computations of the internal electric fields for cylindrical microparticles were reported by Lednyeva and Astafieva,27 with particular interest in morphology-dependent resonances (MDRs) that produce high internal fields as well as strong light-scattering intensities. The purpose of this article is to analyze the heating of cylinders with nanoscale diameters by electromagnetic irradiation and to show that although asymmetric heating is necessary for photophoresis it is not sufficient. In particular, closed-form analytical calculations show that heat conduction within the cylinder can produce extremely uniform temperature distributions even when the internal electromagnetic heating source is extremely nonuniform. The effects of MDRs on the temperature of nanowires are explored further below.
Figure 1. Configuration of the electromagnetic plane wave and the infinite nanoscale circular cylinder (nanowire).
a symmetry condition
∂T (r , 0, t ) = 0 (4) ∂θ and a convective heat transfer boundary condition at the surface −κ1
THEORY The problem of heat transfer from electromagnetically heated spheres for steady-state heat transfer was solved by Allen et al.17 using a Green’s function approach to determine the temperature distribution within the sphere, and Foss and Davis28 extended that approach to the transient heating of microspheres. In both cases, Mie theory was applied to obtain the source term. More recently, Baffou and Rigneault29 used numerical methods to explore femtosecond-pulsed heating of gold nanospheres. Their formulation of the problem is essentially that of Foss and Davis. The solution methodology used here for cylinders is analogous to the solution of Allen et al.17 for spheres. If a long cylinder is suddenly exposed to an electromagnetic source (a laser beam), then the temperature distribution in the cylinder is described by the energy equation ⎡ 1 ∂ ⎛ ∂T ⎞ ∂T 1 ∂ 2T ⎤ ⎜r ⎟ + = κ1⎢ ⎥ + S‴(r , θ , t ) ∂t ⎣ r ∂r ⎝ ∂r ⎠ r 2 ∂θ 2 ⎦
Nu =
T (0, θ , t ) = bounded
2ah κ2
(6)
in which κ2 is the thermal conductivity of the fluid. A representative correlation for the Nusselt number for free convection to and from horizontal cylinders is that of Churchill and Chu.30 In the limit of no fluid flow over the surface, the result of the Nusselt number averaged over the surface of the cylinder is Nu = 0.36. McAdams31 correlated a large amount of data for heating and cooling for air normal to cylinders. Their correlation for low Reynolds numbers is Nu = 0.32 + 0.43Re 0.52
where the Reynolds number is defined by 2aρf v∞ Re = μf
(7)
(8)
in which ρf is the density of the fluid, μf is its viscosity, and v∞ is the fluid’s velocity. An alternate approximation is obtained by extrapolating data for low Reynolds number flow around the cylinder presented by Davies and Fisher.32 This yields Nu = 0.30. On the basis of the data published by McAdams, we shall use the McAdams correlation, eq 7, here. It is convenient to write the energy equation in nondimensional form by introducing the nondimensional variables given by
(1)
in which ρ1 is the density of the cylinder, C1 is its heat capacity, κ1 is its thermal conductivity, T is its temperature, and S‴(r, θ, t) is the volumetric heat source per unit time due to electromagnetic heating. Although the heat source can be a function of time, we shall consider it to be a function of only the radial position, r, and the angle, θ; that is, we shall write the volumetric heat source as S‴(r, θ). The coordinates of the system are shown in Figure 1. The energy equation must be solved subject to auxiliary conditions, which are the initial conditions T (r , θ , 0) = T∞
(5)
in which h is a heat transfer coefficient that depends on the convective transport of heat to the surrounding medium. Theory and empirical correlations for the heat transfer coefficient are usually written in terms of the Nusselt number defined by
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ρ1C1
∂T (a , θ , t ) = h[T (a , θ , t ) − T∞] ∂r
ϕ=
(2)
αt (T − T∞) r , x = , τ = 12 , T∞ a a
S(x , θ ) =
(3) 16178
a2 S‴(r , θ ) κ1T∞
(9)
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where a is the cylinder radius, T∞ is the temperature of the surrounding medium, which is also the initial temperature of the cylinder, and α1 = κ1/ρ1C1 is the thermal diffusivity of the cylinder. Using these nondimensional variables, we transform the energy equation to ⎡ 1 ∂ ⎛ ∂ϕ ⎞ ∂ϕ 1 ∂ 2ϕ ⎤ ⎜x ⎟ + =⎢ ⎥ + S(x , θ ) ∂τ ⎣ x ∂x ⎝ ∂x ⎠ x 2 ∂θ 2 ⎦
particular, it has been shown that the cylinder’s size parameter at resonance increases as the angle of incidence of the plane wave increases. The effects of MDRs on rod heating are explored below. The solutions of the light-scattering problem reviewed by van de Hulst, Kerker, and Bohren and Huffman are analytical solutions of the governing Maxwell equations. An alternate approach to the determination of the electric fields is the application of the well-known finite-difference time-domain method5,38 that is freely available through the MIT Electromagnetic Equation Propagation (MEEP) package.39 We have applied the classical approach of van de Hulst and Bohren and Huffman as well as the finite-difference time-domain approach to compute the internal electric field needed to calculate the internal heat source. Results for the two methods are compared here. Ruppin26 calculated the electromagnetic energy density in an irradiated cylinder, and Liu et al.1 presented plots of the electric field intensity for microcylinders for various complex refractive indices suspended in an absorbing gaseous medium. The motivation for the latter paper was to explore the effects of energy absorption on the photophoretic force exerted on the rod by the electromagnetic irradiation. However, they did not compute the temperature field, which could be uniform even if the internal energy source is nonuniform. That would lead to no photophoretic force. The volumetric rate of heat generation in the cylinder is given by Allen et al.17 as
(10)
The auxiliary conditions become ϕ(x , θ , 0) =
∂ϕ (x , 0, τ ) = 0, ϕ(0, θ , τ )=bounded ∂θ (11)
and ∂ϕ (1, θ , τ ) = −Biϕ(1, θ , τ ) ∂x
(12)
where Bi is the Biot number, which is related to the Nusselt number by κ Nu Bi = 2 κ1 2 (13) The Biot number is a measure of the ratio of the internal resistance to heat transfer to the external resistance to heat transfer.33 For Bi ≪ 1, it can be anticipated that the internal temperature distribution is uniform or nearly uniform.
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ELECTROMAGNETIC HEAT SOURCE The electromagnetic fields outside and inside the cylinder depend on the polarization state of the laser beam and the orientation of the beam with respect to the axis of the cylinder as well as other parameters considered below.34−36 In applications such as particulate matter in the atmosphere and in particle suspensions in aqueous systems, the particle orientation with respect to the light source is random. In experimental studies using laser tweezers or electrodynamic levitation to trap single particles, a laser light source can be aligned to be perpendicular to the axis of a small rod. In general, any orientation of the rod with respect to the light beam can be analyzed by the theory developed here, but we shall confine our attention to a single orientation. Two special cases, initially outlined by van de Hulst,19 were also presented by Kerker20 and Bohren and Huffman.21 In these cases, it is assumed that the direction of propagation of a plane wave is normal to the axis of the cylinder (the Z axis). For case I, the incident electric field is parallel to the YZ plane, and for case II, the incident electric field is perpendicular to the YZ plane. For the purposes of this article, we shall consider case I, but the theory can be extended to other orientations by using the internal electric field equations presented by van der Hulst, Kerker, and Bohren and Huffman to determine the heat source function. Although Mie theory has most frequently been applied to light scattering by small particles, Roll and Schweiger37 used geometrical optics to explore light scattering by obliquely illuminated dielectric cylinders with particular emphasis on morphology-dependent resonances (MDRs). Roll et al. used such a method to analyze spherical resonances as well.37 Lock34 also explored the MDRs of an infinitely long circular cylinder illuminated by a diagonally incident plane wave or a focused Gaussian beam using the S matrix approach for scattering. In
1 σ(E1·E1*) 2
S‴(r , θ ) =
(14)
where E1 is the internal electric-field amplitude vector, E1* is its complex conjugate, and σ is the electrical conductivity of the cylinder, which at optical frequencies is given by σ=
4πRe{N1} Im{N1} λ i μc
(15)
in which N1 = n1 + ik1 is the complex refractive index of the cylinder, n1 = Re{N1}, k1 = Im{N1}, λi is the wavelength of the incident beam, μ is the magnetic permeability, and c is the velocity of light in vacuum. For case I, the transverse magnetic (TM) wave and the incident, internal, and scattered electric fields are given by Bohren and Huffman21 as follows ∞
E i = E0
∑
( −i)n einθJn (ρ2 )ez
n =−∞
(16)
∞
∑
E1 = E0
( −i)n einθdnJn (ρ1)ez
n =−∞
(17)
and ∞
Es = −E0
∑
( −i)n einθbnHn(ρ1)ez
n =−∞
(18)
where i = −11/2, ez is the unit vector in the z direction, and ρ1 and ρ2 are defined by 2π 2π N1r , ρ2 = k 2*r = N2r ρ1 = k1*r = λi λi
(19)
Here, k1* and k2* are wavenumbers in the cylinder and in the surrounding medium, respectively, and N2 is the complex index 16179
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of refraction of the surrounding medium. Functions Jn(r) and Hn(r) are Bessel and Hankel functions of the first kind, respectively. The amplitude Eo of the electric vector is related to the irradiance Iinc of the incident beam by Eo
2
which follows from eq 12. The steady-state solution is obtained by taking the limit as τ → ∞. The result is ϕ(x , θ ) =
2cε = Iinc N2
(20)
(21)
1 [Jn (ρ2 ) Hn′(ρ2 ) − Jn′(ρ2 ) Hn(ρ2 )] m [Jn (ρ1) Hn′(ρ2 ) − mJn′(ρ1) Hn(ρ2 )]
(22)
∞ ⎧ ⎫ ⎪ d0J0 (ρ1) − 2 ∑ ( −1) j − 1d 2jJ2j (ρ1) cos(2jθ )⎪ ⎪ ⎪ j=1 ⎪ ⎪ ∞ ⎬ez E1 = Eo⎨ ⎪ ⎪ − i 2 ∑ ( −1)k − 1d 2k − 1J2k − 1(ρ1) ⎪ ⎪ k=1 ⎪ ⎪ cos[(2 1) )] − θ k ⎩ ⎭
(23)
Finally, the source function S‴(r, θ) is obtained by using eq 23 and the complex conjugate of E1 in eq 14.
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SOLUTION OF THE ENERGY EQUATION Assuming a product solution of the form (24)
the solution becomes ∞
ϕ(x , θ , τ ) =
∞
∑ ∑ Ajk (τ)Jk (λj ,kx) cos(kθ) (25)
j=1 k=0
where Ajk (τ ) =
2 2 [1 − exp(−λj , k τ )] π λj , k 2 || Njk ||2
π
∫0 ∫0
1
x′Jk (λj , k x′)
× cos(kθ′) S(x′, θ′) dx′ dθ′
(26)
in which the prime indicates a dummy variable of integration and the norm squared ∥Njk∥2 of the eigenfunctions Jk(λj,kx) is defined by || Njk ||2 =
∫0
1
x′[Jk (λj , k x′)]2 dx′
(27)
The eigenvalues λj,k satisfy the boundary condition d J (λj , k x)|x = 1 = −BiJk (λj , k ) dx k
π
1
x′Jk (λj , k x′) cos(kθ′)
j=1 k=0
(29)
REPRESENTATIVE SOURCE FUNCTIONS AND TEMPERATURE DISTRIBUTIONS Recently, several groups have shown that it is possible to trap semiconductor nanowires with multibeam40 and singlebeam41−51 laser tweezers in the near-infrared (NIR) spectral region. One common laser wavelength for NIR laser trapping is 980 nm, and at this wavelength the complex refractive index of silicon is given by N1 = 3.6014 + 0.0005259i from Palik.52 Calculations were limited to infinite cylinders with diameters ranging from 10 to 1000 nm. The thermal conductivities of these were estimated wherever possible from experimental data that have been measured to be in the range of 7 to 55 W·m−1·K−1.53 These are smaller than the thermal conductivity of bulk silicon (κ1 = 149 W·m−1·K−1) because of the size dependence of the Umklapp boundary scattering of phonons. Although in laser tweezer experiments the beam axis is parallel to the cylinder axis, it is informative to consider the configuration of perpendicular illumination given that absorption is expected to be maximized when the electric field vector of the incident radiation is parallel to the cylinder’s axis. To compare source functions for various systems and parameters, it is convenient to plot the dimensionless source function (E1·E*1 )/Eo2. Sample source functions calculated using the analytical solution and the finite-difference time-domain method (MEEP)39 for silicon nanowires with diameters of 500 nm are presented in Figure 2a−d. The FDTD results are shown at a time that is sufficient for the system to reach its steady state because the FDTD approach solves the unsteady-state system of equations. There is mutually consistent agreement between the analytical and numerical approaches, and for the results presented below, the analytical solution was used. All calculations presented here were performed with an incident irradiance of 103 W·cm−2 unless specified otherwise. Source functions for silicon nanowires are shown in Figure 3a−c for rod diameters of 10, 536, and 1000 nm, and the corresponding dimensionless temperature distributions are presented in Figure 3d−f. As expected, the internal electric field exhibits much more complex structure as the diameter increases as a result of the intricate wave interference and reenforcement within the nanowire. However, for the case of silicon nanowires the internal temperature is very nearly uniform on the order of 1 part in 106 for all of the diameters considered. For a silicon nanowire with a diameter of 10 nm in water, eq 29 yields a very nearly uniform temperature distribution with ϕ(x, θ) ≈ 2.5 × 10−8. Thus, the nanowire temperature is that of the surrounding fluid. A uniform temperature is to be expected in this case because the Biot number is 0.0292. When the diameter is increased to 536 nm, eq 29 again yields a very nearly uniform temperature, as shown
in which m = N1/N2. Computations show that d−n = dn. When these results are used in eq 17, the internal electric vector becomes
ϕ(x , θ , τ ) = U (x) V (θ ) W (τ )
⎡
■
and dn =
∞
For the particle and surrounding medium properties of interest here, the time required to reach steady state is on the order of 10−9 to 10−6 s.
[Jn (ρ1) Jn′(ρ2 ) − mJn′(ρ1) Jn (ρ2 )] [Jn (ρ1) Hn′(ρ2 ) − mJn′(ρ1) Hn(ρ2 )]
∞
∑ ∑ ⎢⎣∫ ∫ 0 0
⎤ J (λj , k x) cos(kθ ) × S(x′, θ′) dx′ dθ′⎥ k ⎦ λj , k 2 || Njk ||2
Liu et al. showed that the internal field intensities for a transverse electric (TE) wave are qualitatively and, to some extent, quantitatively similar to those for a TM wave. Consequently, we limit our analysis to the TM wave. Coefficients bn and dn are obtained by applying appropriate boundary conditions on the electric and magnetic vectors at r = a as discussed by Bohren and Huffman. The results are 1
bn =
2 π
(28) 16180
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sequently, even though the heat source is highly nonuniform for this larger nanowire the temperature distribution is very uniform with ϕ(x, θ) ≈ 1.6 × 10−4. The temperature increases as the size of the nanowire increases, but for the low Biot numbers associated with the silicon−water system, the internal temperature is nearly completely uniform. Table 1 illustrates Table 1. Results for (Tmax − Tmin)/Tavg for Silicon Nanowires in Air and Water air diameter (nm) 10 500 649.9a 1000
water
(Tmax− Tmin)/Tavg 1.05 3.63 2.40 1.79
× × × ×
−9
10 10−8 10−6 10−7
diameter (nm) 10 500 949a 1000
(Tmax− Tmin)/Tavg 1.38 5.69 2.02 2.00
× × × ×
10−9 10−8 10−6 10−7
a
Denotes the diameter corresponding to a resonance condition leading to the highest calculated temperature.
the uniformity of the internal temperature for silicon nanowires of various diameters in air and water. For both systems, the ratio (Tmax − Tmin)/Tavg is very small. Consequently, even though the heat source may be highly asymmetric, these calculations lead to a counterintuitive prediction that heat conduction within the nanowire can lead to an extremely uniform internal temperature. The results for the intermediate size (536 nm) in Figure 3b correspond to a resonance (whispering gallery mode) discussed further below. In contrast to silicon, which has a weakly absorbing indirect band gap, the calculated source functions are substantially
Figure 2. Comparison of the electric field distributions based on the closed-form analytical solution (a and c) and MEEP (b and d) for side views (a and b) and top-down views (c and d) for an infinitely long silicon cylinder with a diameter of 500 nm irradiated in water with a free-space wavelength of λ = 980 nm.
in Figure 3e, with ϕ(x, θ) ≈ 3.75 × 10−4. Because Bi = 0.0012, a uniform temperature is again to be expected. That is also the case for a silicon nanowire diameter of 1000 nm. If the bulk thermal conductivity is used, then Bi = 0.000658. Con-
Figure 3. Plots of the calculated normalized cross-sectional electromagnetic heating source term (E1·E1*)/Eo2 (a−c) and corresponding reduced cross-sectional temperature ϕ = (T − T∞)/T∞ (d−f) for silicon nanowires in water with outer diameters of 10 nm (a and d), 536 nm (b and e), and 1 μm (c and f) irradiated at Iinc = 103 W·cm−2 with a free-space wavelength of λ = 980 nm. 16181
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Figure 4. Plots of the calculated normalized cross-sectional electromagnetic heating source term (E1·E1*)/Eo2 (a and b) and the corresponding reduced cross-sectional temperature ϕ = (T − T∞)/T∞ (c and d) for carbon rods in water (a and c) and air (b and d) with an outer diameter of 1 μm irradiated at room temperature with an irradiance of Iinc = 103 W·cm−2 at a free-space wavelength of λ = 488 nm. Unlike silicon, an appreciable variation in the reduced temperature is observed for the carbon material.
different for strongly absorbing materials such as amorphous carbon. Pluchino et al.16 measured the complex refractive index of micrometer-sized spheres by levitating them in air using an electric field and illuminating them with a laser beam. The measurement of the scattered light was used to extract the refractive index by comparison with Mie theory. They reported the refractive index of amorphous carbon spheres of about 4 μm in radius to be approximately given by N1 = 1.70 + 0.80i for a laser wavelength of 488 nm. They did not report either the laser beam intensities used or some of the relevant properties of the carbon. The thermal conductivity of carbon at 298 K varies from 0. 0159 W·m−1·K−1 for amorphous carbon to 138 W·m−1·K−1 for some graphite to 2200 W·m−1·K−1 for diamond. Zdrojek et al.54 recently measured the Raman backscattering from individual multiwalled carbon nanotubes dispersed on an SiO2 substrate. The tubes were illuminated normal to the tube axis, and the Raman shift was used to measure a temperature rise of 860 °C for irradiances on the order of 105 W·cm−2. Figure 4a shows the dimensionless source for an amorphous carbon rod with a diameter of 1 μm in water, and Figure 4b shows the source for a carbon rod in air. The source functions are similar for the highly absorbing carbon rod with little penetration of the electromagnetic wave into the core of the rod. The lack of penetrating electromagnetic waves into the carbon rod prevents internal wave interference, and thus there are no resonances expected for the carbon case. The corresponding dimensionless temperature distributions are presented in Figure 4c,d. In this case, the surrounding fluid was at 298 K, and λι = 488 nm. The temperature rise for the carbon rod in air is an order magnitude greater than in water. In contrast to silicon, a nonuniform temperature is predicted for both cases of carbon, which could lead to photophoresis.
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Figure 5. Comparison of calculated values for the maximum reduced temperatures of laser-heated silicon nanowires in water scaled by the incident irradiance with the calculated diameter-dependent lightscattering efficiency. (a) Diameter-dependent maximum reduced temperature scaled by the incident irradiance of a single-crystalline silicon nanowire irradiated in water at a free-space wavelength of λ = 980 nm. (b) Calculated scattering cross section for a single-crystalline silicon nanowire in water irradiated at a free-space wavelength of λ = 980 nm as a function of nanowire diameter.
heavily on the nanowire’s diameter. From eq 29 and the definitions of the source function (eqs 14 and 20), the dimensionless temperature rise, ϕ(x, θ), is proportional to the incident irradiance, Iinc, assuming that the irradiance is in the regime of linear optics. For large Iinc, the temperature rise can be so large that the physical properties of the rod are not those at the surrounding temperature. Consequently, the computations of ϕ(x, θ) need to be modified when large temperature rises are predicted. Furthermore, for carbon rods in water the rod temperature can be so high that boiling occurs, and the Biot number becomes large. Calculations were performed on a home-built PC with dual quad-core processors and 72 GB of RAM using the MATLAB programming language. Iterative calculations were performed for which an initial temperature was calculated on the basis of available room-temperature physical property data. The physical property data were updated on the basis of the new
MOPHOLOGY-DEPENDENT RESONANCES
As with the case of spherical particles, it is well known that large internal electric resonance conditions occur for particular nanowire diameters. Figure 5a shows how the reduced temperature normalized to the incident irradiance depends 16182
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temperature, and new calculations were performed until the newly calculated temperature changed by less than 0.1 °C. Table 2 presents some of the results of the iteration procedure Table 2. Convergence of the Maximum Temperature of a Silicon Nanowire for Resonance and Off-Resonance Conditions diameter, nm
thermal conductivity, W m−1 K−1
426 426 426
60.489 49.002 49.162
420 420 420 420
60.151 52.890 52.217 52.128
index of refraction
Resonance (3.6014 (3.6152 (3.6149 Off-Resonance 3.6014 3.6094 3.6103 3.6104
+ 5.763) × 10−4i + 6.359) × 10−4i + 6.348) × 10−4i + + + +
5.763 6.102 6.140 6.145
× × × ×
10−4i 10−4i 10−4i 10−4i
Tmax, °C 67.301 66.584 66.599
Figure 6. Comparison of calculated values for the maximum reduced temperatures of silicon nanowires laser heated in air and scaled by the incident irradiance with the calculated diameter-dependent lightscattering efficiency. (a) Diameter-dependent maximum reduced temperature scaled by the incident irradiance of a single-crystalline silicon nanowire irradiated in air at a free-space wavelength of λ = 980 nm. (b) Calculated scattering cross section for a single-crystalline silicon nanowire in air irradiated at a free-space wavelength of λ = 980 nm as a function of nanowire diameter.
49.574 52.266 52.625 52.675
for pure silicon nanowires having diameters of 426 and 420 nm with an incident irradiance of 106 W·cm−2 that we consider to be within the regime of linear optics where optical electric field amplitudes are well below 108 V/m. The former diameter corresponds to a resonance, which leads to a greater temperature rise than in the off-resonance case. In both examples, the value of the maximum temperature within the nanowire was reached within three iterations. The thermal conductivities and refractive indices involved are also given in the table, and the temperature of the surrounding water was 298 K. One clearly visible feature of these diameter-dependent calculations is the existence of morphology-dependent resonances in the temperature rise of laser-irradiated cylinders in air and water when nanowires are larger than approximately 200 nm. Figure 5b presents calculated results for the scattered light intensity, Qsca, and the maximum reduced temperature, ϕmax/Iinc, for silicon nanowires in water with diameters of between 10 and 1000 nm. The scattering efficiency is given by14 Q sca =
λ2 πa
this motivation to explore the effects of porosity on the heating of microporous silicon nanowires in water as shown in Figure 7.
∞
∑ n =−∞
|bn|2
Figure 7. Diameter-dependent maximum temperature change, scaled by irradiance, of a porous single-crystalline silicon nanowire in water irradiated at a free-space wavelength of λ = 980 nm with T∞ = 298 K. (Inset) Expansion of the region between 10 and 500 nm. Plots a−d are offset and correspond to the (a) single-crystalline, (b) 25% porous, (c) 50% porous, and (d) 75% porous cases.
(30)
It can be seen that the nanowires have several well-defined resonances at 427, 536, 642, 747, 849, and 949 nm, which appear in Qsca, as well as the temperature rise. It is also interesting that there is an absence of MDRs and a significant reduction in thermal heating observed for nanowires with diameters of less than 100 nm. Figure 6 presents similar temperature and scattering calculations for the silicon−air system. The calculated temperatures are higher in this case because of the lower thermal conductivity of the surrounding fluid. However, because the Biot numbers are smaller than for silicon−water a more uniform temperature distribution within the cylinder is predicted. Recent synthesis advances have shown that it is possible to prepare highly porous nanowire materials for metal-assisted chemical etching (MACE) of single-crystal wafers.55 One potential application of nanowires is in biosensing given their electrical conductivity and high intrinsic surface areas.56 In this case, the nanowire’s thermal conductivity and the complex index of refraction depend on the porosity and pore structure, so they can be modified by altering the porosity. We have used
Light scattering by voids within the porous silicon is an important consideration when calculating cavity resonances and is known to depend critically on the relative size between pores and the wavelength of incident radiation. Synchrotron-based grazing-incidence small-angle X-ray scattering methods have been used to measure typical pore sizes (D) of
