Photothermal Heating of Nanowires - The Journal of Physical

Article pubs.acs.org/JPCC

Photothermal Heating of Nanowires Paden B. Roder,† Bennett E. Smith,‡ E. James Davis,§ and Peter J. Pauzauskie*,†,⊥ †

Materials Science & Engineering Department, ‡Chemistry Department, and §Chemical Engineering Department, University of Washington, Seattle, Washington 98195, United States ⊥ Fundamental & Computational Sciences Directorate, Pacific Northwest National Laboratory, Richland, Washington 94720, United States S Supporting Information *

ABSTRACT: A theoretical model is developed here in tandem with singlebeam laser trapping experiments to elucidate the effects of the numerous thermal, optical, and geometric parameters that affect internal temperature distributions within finite nanowires (NWs) during laser irradiation. Analytical solutions to the heat-transfer equation are presented to predict internal temperature distributions within individual nanowires based on numerical calculations of the internal electromagnetic heat source. Single-beam lasertrapping experiments are performed to measure photothermal heating of silicon NWs. Silicon has not been considered to date for photothermal heating applications due to its indirect band gap and low absorption coefficient in the near-infrared tissue-transparency window. We also show here that ion implantation may be used to increase the optical absorption of silicon nanowires (SiNWs), leading to significant heating to temperatures greater than 42 °C in an aqueous environment at an irradiance of 3 MW/cm2. Experimental observations of photothermal heating agree well with theoretical predictions. Calculations for comparison with amorphous carbon NWs reveal significantly greater heating effects, as well as internal radial gradients not observed for SiNWs.

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INTRODUCTION Photothermal heating of nanoscale materials is currently under investigation in many laboratories for the treatment of cancerous tissues. Recently, nanomaterials including noble metals,1 semiconductors,2 metal oxides,3 and sp2 carbon allotropes4 have been considered for clinical use as absorbers for photothermal therapy applications.5 Gold nanocrystals have been shown to exhibit a large photothermal heating response when irradiated at near-infrared (NIR) wavelengths within the spectral transparency window of biological tissues (800 nm < λ < 1 μm) due to the presence of surface plasmons that produce large dipole absorption moments, typically >100 D.6 Recently, experimental extinction coefficients from plasmon absorption in gold nanocrystals have been reported to be greater than 109 M−1 cm−1.7 Gold is also known to have a high surface diffusion coefficient >1016 cm2/s,8 which results in significant morphological changes after exposure to high doses of laser radiation9,10 (i.e., rod to sphere). These topological changes have a pronounced impact on the spectral character of surface plasmon resonances. Furthermore, gold is a precious metal, and the price of feed-stocks motivates the search for complementary earth-abundant elements that may also exhibit appreciable laser heating in the NIR spectral window. Silicon nanomaterials have received a tremendous amount of attention in recent years for applications in biological sensing11 and in vivo therapeutic applications12,13 due to their low toxicity and cost. In particular, elemental silicon is known to biodegrade into silicic acid (Si(OH)4) at physiological ionic © 2013 American Chemical Society

strength and pH, which has been reported to have a half-life of less than 11 h in a human subject.14 At room temperature, crystalline silicon has an indirect band gap of 1.11 eV (λ = ∼1116 nm) which enables optical absorption within the electromagnetic transparency window of biological tissue. The absorption coefficient of silicon is small in this spectral region (∼100/cm)15 when compared with gold due to the need for momentum transfer to assist indirect optical transitions. However, point defects are known to increase free carrier absorption as well as produce recombination centers in the silicon crystal lattice that increase the relative rates of optical transitions well above those for a perfect crystal.16 Here, we show that the introduction of point defects within silicon nanowires (SiNWs) via ion implantation increases photothermal heating of SiNWs in a single-beam laser trap. Additionally, we present a numerical model of heat transport from NWs in close agreement with experimental observations that can be used to predict the heating of other photothermal materials illustrated here by an example of amorphous carbon. These results open up novel prospects for using silicon nanomaterials as a low-cost, earth-abundant alternative to gold in photothermal therapy applications. Received: August 6, 2013 Revised: December 12, 2013 Published: December 13, 2013 1407

dx.doi.org/10.1021/jp4078745 | J. Phys. Chem. C 2014, 118, 1407−1416

The Journal of Physical Chemistry C

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Figure 1. Synthesis and characterization of ion-implanted SiNWs. (a) MACE of a SiNW array from a ⟨111⟩ p-type 10 Ω·cm wafer. (b) Crosssectional scanning electron microscope image of the SiNW array following silver etching (inset: optical micrograph of the wafer; scale bar = 1 cm). (c) Transmission electron microscope image of a SiNW following MACE etching (scale bar = 150 nm; inset: select area electron diffraction). (d) Schematic of phosphorus ion implantation used to create point defects in the SiNW array. (e) Numerical simulation of the stopping depth of phosphorus ions in silicon using an implantation energy of 75 keV (inset: optical micrograph of the wafer after implantation; scale bar = 1 cm). (f) Transmission electron microscope image of a SiNW following phosphorus implantation (scale bar = 225 nm; inset: select area electron diffraction).

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METHODS AND RESULTS SiNW Synthesis and Ion Implantation. SiNWs shown in Figure 1a−c were fabricated with a range of diameters using metal-assisted chemical etching (MACE)17 of single-crystal Si ⟨111⟩ wafers in hydrofluoric acid. Following MACE synthesis, point defects were introduced selectively within SiNWs cavities via implantation (Figure 1d,f) of positive phosphorus ions at an energy of 75 keV with a flux of ∼1015 ions/cm2 at an incidence angle of 7° off of normal. Calculations were made using freely available Stopping Range of Ions in Matter (SRIM) software18 in order to select for implantation depths of approximately 100 nm (Figure 1e) in order to ensure that implanted ions remained within the NWs synthesized from the MACE process. Transmission electron microscopy images (Figure 1c,f) reveal that SiNWs are crystalline after implantation, although there is apparent crystallographic damage to the top and side surface of the NWs following ion implantation in Figure 1f. Optical Trapping Experiment and Data Analysis. Single-beam laser tweezers have been used extensively in recent years both to assemble NWs into complex structures19 as well as to characterize their composition and spectroscopic properties at the single-NW level.20 If incident laser radiation is absorbed by the particle, it can lead to substantial thermal heating. In an extensive review of optical trapping, Neuman and Block21 reported that heating of a particle can be explored from measurement of the power spectrum of a particle undergoing Brownian motion in a laser trap. A detailed analysis of the power spectrum of particles trapped using optical tweezers was reported by Berg-Sorensen and Flyvbjereg.22 Photothermal heating alters the particle’s thermal kinetic energy, kBT,

associated with the power spectrum. Here, kB is the Boltzmann constant, and T is the absolute temperature. Peterman et al.23 analyzed the thermal motion of polystyrene and silica beads trapped in water and glycerol and showed that light absorption by the fluid has a small but measurable effect on the temperature change for these low-absorption particles. Photothermal heating has also been suggested to affect fluid drag on a trapped NW.20 In an extreme recent example, optically trapped gold nanocrytals have been reported to superheat water to temperatures above 400 °C due to large Young−Laplace interfacial surface pressures that arise at nanometer length scales.24 The nonequilibrium Brownian motion of hot particles has recently been analyzed with a Markovian description that accounts for the fact that the fluid viscosity can no longer be treated as being a constant isotropic parameter in the vicinity of the heated particle.25,26 In this work, the photothermal heating for both native and implanted SiNWs was investigated through the use of a customized single-beam laser trapping instrument shown in Figure 2a. The NWs were suspended in an aqueous bath via sonication and underwent Brownian motion in a coordinate frame shown in Figure 2c. Light scattered off of optically trapped NWs in the forward direction and created a dynamic interference pattern on a quadrant photodiode (QPD) in the back focal plane of the trapping instrument. Fluctuating photovoltage signals from the detector could then be used to measure the three-dimensional center-of-mass as well as the temperature of an optically trapped particle.21 Brownian motion was detected through analysis of the normalized timedependent photovoltage from the QPD shown in the figure, 1408



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Figure 2. Optical trapping instrument and coordinate system. (a) Schematic of the optical trapping instrument used to measure the temperature of individual SiNWs. (b) Illustration of coordinates and boundary conditions used for solving the energy-transfer equation. (c) Labels for Cartesian coordinates and angles relative to a NW with arbitrary orientation used to analyze QPD data.

trapping stiffness. Combining both Cartesian x- and y-channels creates a scatter plot position distribution map shown in Figure 3e. To determine the particle temperature from the normalized QPD voltage, it remains to establish the deconvoluted trap stiffness, which can be extracted from the autocorrelation of Vx(t). The autocorrelation for motion in the x-direction is defined by

which recorded x and y positions of the particle due to a combination of translational and rotational fluctuations. Following the analysis of Marago et al.,27 the displacement was obtained from the normalized Vx and Vy components of the voltage output of the QPD, which were influenced by the dynamic motion of the NW. These voltages are related to the Cartesian displacements Xi and the angular displacements Θi by βi Vi = δi = (Xi + L Θi)

i = x, y

(1)

Cxx(t ′) = Vx(t )Vx(t + t ′)

in which δi represents a convolution of the position (Xi) and angular (Θi) displacements, βi are calibration constants, L is the length of the NW, Θy = θ·sin(ϕ), and Θx = θ·cos(ϕ) in Figure 2c. Here, θ is the angle between the z-direction and the axis of the NW, and ϕ is the azimuthal angle in the x−y plane, as shown in Figure 2c. Note that the voltages, Vi , depend on both the center-of-mass displacement as well as angular fluctuations of the NW in the trap. In the absence of the laser trap, NWs undergo Brownian motion (Figure 3a), but once confined in the optical trap, the NWs point along the optical axis of the trapping laser (Figure 3b). The optical potential can be approximated to be Hookean22 and acts to restore the particle to its stationary position after small displacements from the center of the trap. An example of the time-dependent position of the particle as detected by the QPD is shown as an inset in Figure 3c for a time period of 3 s. A histogram of the time-dependent voltage signal is presented in Figure 3d for an irradiance value of I = 3 MW/cm2. The symbol Vx(t) represents the voltage from the x channel processed from the QPD that has been normalized to the total sum voltage. The normalized mean-square voltage time signal is equal to the variance, σ2V,x, of the normalized voltage histogram shown in Figure 3d. As the irradiance increases, the variance decreases due to an increase of the

(2)

where t′ is the lag time between the signal and itself. Figure 3c shows a representative sample of the autocorrelation for a small silicon NW. A similar result applies to the Cyy autocorrelation. For the x normalized voltage signal, Vx(t), it is possible to deconvolve the center-of-mass motion from the angular fluctuations by fitting Cxx(t′) to a double exponential, that is Cxx(t ′) = Co, x·exp( −fx |t ′|) + Co, θ ·exp( −fθ |t ′|)

(3)

where Co,x and Co,θ are the amplitudes for x and θ degrees of freedom, respectively. By means of a least-squares fit of the autocorrelation data, the characteristic relaxation frequency in the x-direction, f x, is obtained.27 The stiffness is then obtained from kx = fx ·γNW

(4)

where γNW is the drag coefficient of the NW. The drag coefficient depends on the geometry of the NW as well as the temperature-dependent viscosity, η, of the surrounding fluid, that is20 γNW = 1409

4πL · η(T ) ln(L /2R ) + 0.84

(5)



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Figure 3. Extraction of temperature from analysis of Brownian motion. (a) Bright field optical micrograph of an isolated SiNW in Brownian motion (scale bar = 2 μm). (b) Bright field optical micrograph of an individual SiNW confined within a single-beam laser trap (scale bar = 1 μm). (c) Autocorrelation function from a normalized voltage time trace for the X-channel output from the QPD. Inset: Normalized voltage signal measured over a 3 second time interval at 100 kHz. (d) Histogram of the normalized voltage time trace from the X-channel output from the QPD. (e) Scatter plot of the center-of-mass coordinates of a single SiNW over the course of 3 seconds. (f) Average experimental temperatures measured from both native SiNWs (left) and ion-implanted SiNWs (right) at two separate laser irradiances. Error bars are standard deviations from the mean.

where L is the NW’s length, R is the NW’s radius, and the factor 0.84 is an approximation for large aspect ratio cylinders and arises from end-facet effects from the NW’s finite length.28 The translational relaxation frequency determined from the data in Figure 3c is determined to be f x ≈ 1000 Hz. Using these results, the voltage variance is related to the fitted parameters by σV2 , x =

simulations that are used in subsequent theoretical temperature calculations discussed below. At the highest trapping irradiance of 3 MW/cm2, the SiNWs without ion implantation are observed to heat, on average, to approximately 27 °C, starting with a surrounding water temperature of 22 °C. In comparison, SiNWs that have been implanted with phosphorus ions are observed to heat to temperatures greater than 42 °C, presumably due to the formation of optical recombination centers that have been shown to increase the optical absorption coefficient of silicon by several orders of magnitude.16 Future efforts to increase the defect density or to dope with strong absorbers may reduce the necessary heating irradiance to levels compatible with photothermal therapy (∼1 W/cm2).

kBTβx2 γNW ·fx

(6)

Determining the variances for experiments at two different irradiances, the ratio of these variances is σV2 , x ,2 σV2 , x ,1

=

fx ,1 T η(T ) · 2 1 fx ,2 T1η(T2)

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THEORY Governing Heat-Transfer Relations. The temperature established within an optically trapped SiNW depends on two key factors. First, the trapping laser causes photothermal heating within SiNWs related directly to the magnitude of the internal electromagnetic field.29 Second, interfacial heat transport occurs from the SiNW to the surrounding bath and depends on the interfacial resistance to heat flow from the solid to the fluid phase.31 In order to simulate photothermal heating, one can approximate the incident Gaussian beam (TEM00 mode) as a plane wave given that the NW’s cross section is significantly smaller than the diameter of the diffraction-limited laser beam. For a cylinder with the alignment shown in Figure 2b, this is a reasonable approximation if the NW’s diameter, D, is small (∼100 nm) compared with the waist radius, wo (∼1 μm). The heat-transfer analysis presented below is general and may also be used with more complex incident laser polarization profiles. In the case of a finite NW considered here, the dimensionless energy equation presented by Roder et al.29 must be extended

(7)

We assume based on recent calculations29 that if the incident irradiance is sufficiently low, then the laser heating of the NW will be small, and the temperature of the NW will be that of the surrounding fluid, that is, TNW = T∞. Consequently, eq 7 can be used to determine temperature T2 at trapping conditions with a higher irradiance where laser heating is not negligible. We also assume that that the calibration constants are the same at different powers when using normalized voltage signals from the QPD detector.30 A series of seven native and four ion-implanted wires were trapped and analyzed thermally with the single-beam laser trapping instrument at various irradiances (∼0.7, ∼1.8, and ∼3 MW/cm2) shown in Figure 3f at a distance of ∼75 μm from the lower coverslip (half of the chamber thickness). Each NW was then affixed to the top coverslip of the trapping chamber (Figure 2b) to facilitate atomic force microscopy (AFM) analysis, as shown in Figure 4a. Measuring NW geometries via AFM is critical for designing computational electromagnetic 1410



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above to heating by pulsed lasers using a time-dependent source function. Chamber boundary conditions are shown in Figure 2b, where the temperatures of the upper and lower coverslips are assumed to be at the temperature of the fluid (T∞ = 22 °C) far from the NW. The length SA is the distance between the lower surface of the NW and the lower coverslip, while SB is the distance between the upper surface of the NW and the top coverslip of the chamber. The dimensionless initial and boundary conditions for the NW are given above in Table 1.

to take into account heat transfer from the upper and lower surfaces of the NW by including axial conduction. The dimensionless energy equation governing the NW temperature becomes ⎛ R ⎞2 ∂ 2Θ 1 ∂ ⎛ ∂Θ ⎞ 1 ∂ 2Θ ∂Θ = + S*(ξ , ζ , τ ) ⎜ξ ⎟ + 2 2 + ⎜ ⎟ ⎝ L ⎠ ∂ζ 2 ∂τ ξ ∂ξ ⎝ ∂ξ ⎠ ξ ∂ϕ (8)

in which the dimensionless variables are defined by TNW − T∞ r ξ= T∞ R κNW t τ= ρNW Cp,NW R

Θ=

ζ=

z L

Table 1. Boundary Conditions (9)

Here, TNW is the NW’s temperature, T∞ is the fluid temperature far from the NW, κNW is the thermal conductivity, ρNW is the density, L is the length, R is the radius, and Cp,NW is the specific heat capacity at constant pressure per unit mass. The dimensionless source function is S*(ξ , ϕ , ζ , τ ) =

R2 S(ξ , ϕ , ζ , τ ) κNWT∞

(10)

which represents the intrinsic electromagnetic heating of the NW. Using a continuum approximation for internal electromagnetic heating, the time-averaged point source function is given by32 1 S = σ E·E* 2

4π Re{NNW } Im{NNW } λ incμc

2 Iinc NBc ϵ0

axial conduction from NW to the bottom chamber surface:

∂Θ (ξ , ϕ , 0, τ ) = + Bi1Θ(ξ , ϕ , 0, τ ) ∂ζ

radial heat transfer to the stagnant fluid:

∂Θ (1, ϕ , ζ , τ ) = − Bi 2 Θ(1, ϕ , ζ , τ ) ∂ξ

axial conduction from NW to the top chamber surface:

∂Θ (ξ , ϕ , 1, τ ) = − Bi3Θ(ξ , ϕ , 1, τ ) ∂ζ

no heat flow in the angular direction:

∂Θ ∂Θ (ξ , 0, ζ , τ ) = (ξ , π , ζ , τ ) = 0 ∂ϕ ∂ϕ Bi1 = Bi 2 =

(11)

κB L κNW SA

κB Nu κ = 0.16 B κNW 2 κNW Bi3 =

κB L κNW SB

The interfacial resistance for heat transport is known to depend critically on the atomistic interfacial structure at the junction between the solid and fluid phases and can produce abrupt interfacial temperature discontinuities.31 As discussed previously,29 the Nusselt number, Nu, is an empirical/ phenomenological dimensionless heat-transfer coefficient that accounts for all heat transfer processes between the solid and fluid phases. In the low Reynolds number (Re) limit, experimental measurements have shown Nu = 0.32 for cylinders in a stagnant fluid. Calculations suggest that this approximation is valid for the experiments considered here because Re < 10−6. To obtain the boundary conditions expressed in Table 1, we assume that the heat loss from the lower and upper surfaces of the NW is due to conduction in the fluid between the NW and the bounding surfaces of the laser trapping chamber. This approximation neglects effects of motion of the fluid relative to the NW that might actually exist experimentally due either to Brownian motion or to convective laminar flow in the event that a substantial steadystate temperature gradient is established. Solution of the Photothermal Heat Equation. Using the classical method of a product solution, the energy equation is first solved in the absence of a source. The solution to the homogeneous energy equation is assumed to have the form

(12)

in which NNW is the complex index of refraction of the NW, λinc is the wavelength of the incident wave, μ is the magnetic permeability of the NW, and c is the velocity of light in vacuum. The electric field amplitude of the incident plane wave, Einc, is related to the laser irradiance by 2 E inc =

Θ(ξ,ϕ,ζ,0) = 0 ∂Θ (0, ϕ , ζ , τ ) = 0 ∂ξ

Biot numbers at (1) the bottom, (2) the side, and (3) the top NW surfaces:

where E is the complex internal electric field and the variable σ is the conductivity of the NW at optical frequencies given by σ=

TNW = T∞ at time = 0: no heat conduction across the center line (radial symmetry):

(13)

where NB is the complex index of refraction of the bath (water), ε0 is the permittivity of free space, and Iinc is the laser irradiance in the fluid. Clearly, this source function S(ξ,ϕ,ζ,τ) for electromagnetic heating depends on the internal electromagnetic field. In general, the internal electromagnetic heating source is a function of radial coordinate ξ, angle ϕ, axial coordinate ζ, and time τ as well as the optical properties of the material and the characteristics of the laser light source. For the high frequencies associated with CW laser irradiation, we can use a timeaveraged source. Consequently, the dimensionless heat source reduces to the constant three-dimensional source S*(ξ,ϕ,ζ). If the internal electromagnetic field does not vary across the NW’s diameter, the heat source reduces further to a function of axial position only, that is, to S*(ζ). Homogeneous internal electromagnetic fields recently have been shown to be a good approximation for SiNWs with diameters on the order of 10s of nanometers.29 It is possible to extend the analysis presented

ΘH(ξ , ϕ , ζ , r ) = u(ξ)v(ϕ)w(ζ )χ (τ )

(14)

where u(ξ), v(ϕ), and w(ζ) are orthonormal eigenfunctions that satisfy the boundary conditions that are presented in Table 1. The function χ(τ) does not need to be considered at this stage because the time dependence of the inhomogeneous 1411



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here, it is necessary to use numerical methods to determine the internal electromagnetic field profile. Freely available discrete dipole approximation (DDA)33,34 and finite difference time domain (FDTD)35 computational electromagnetics software can be applied to compute the source function for the case of plane wave illumination along a NW’s axis. We assume that if the diameter of the NW is significantly less than the diffraction limit of the incident laser radiation, it is possible to justify approximating the incident electromagnetic field as a coherent plane wave. In laser trapping experiments, it is common21 to overfill the back aperture of the high numerical aperture lens (shown in Figure 2a). In this way, the relatively constant central intensity of the Gaussian beam profile fills the objective aperture and creates an optical trap that may be approximated as a focused coherent plane wave. To obtain the geometrical properties needed for numerical electromagnetic field simulations, NWs were fixed to the top coverslip of the chamber after laser trapping experiments were performed. AFM was used to measure the length and diameter of individual NWs. For example, the AFM image in Figure 4a was generated from the NW whose optical bright field image is shown in Figure 3a. The AFM data yielded a diameter (D) of 60 nm and a length (L) of approximately 5 μm for this particular NW. These dimensions were used in conjunction with DDA simulations to calculate the internal electromagnetic field.

problem is substantially different than that of the homogeneous problem. The solution of the inhomogeneous energy equation can be written in terms of these orthonormal functions as ∞

Θ(ξ , ϕ , ζ , τ ) =

∞

∞

∑ ∑ ∑ ASmn(τ)uSn(ξ)vn(ϕ)wm(ζ ) S=1 m=1 n=0

(15)

where the eigenfunctions are uSn =

J (γ ξ ) XSn(ξ) = n Sn XSn XSn

vn =

Yn(ϕ) cos(nϕ) = Yn Yn

wm =

cos(δmζ ) + (Bi1/δm) sin(δmζ ) Zm(ζ ) = Zm Zm

(16)

Here, Jn (γSnξ) is an nth-order Bessel function, ∥XSn∥, ∥Yn∥, and ∥Zm∥ are the norms of the eigenfunctions, and γSn, n, and δm are their respective eigenvalues. Substituting eq 15 into eq 8 and applying the principle of orthogonality for each of the eigenfunctions, the time-dependent coefficients ASmn(τ ) are found to be π

1

1

∫0 ∫0 ∫0 ∫0

ASmn(τ ) =

τ

S*(ξ′, ϕ′, ζ′, τ′)

Table 2. Trapping Data

exp[−λS2mn(τ − τ′)]ξ′uSn(ξ′)vn(ϕ′)wm(ζ′) dξ′ dϕ′ dζ′ dτ′

(17)

where the primes indicate dummy variables of integration and λS2mn = γS2n +

⎛ R ⎞2 2 ⎜ ⎟ δ ⎝L⎠ m

(18)

If the source function is not a function of time, that is, if the time-averaged electromagnetic source function is used, eq 17 can be integrated over time to yield ASmn(τ ) =

1 − exp[− λS2mnτ ]

π

1

∫0 ∫0 ∫0

λS2mn

1

S*(ξ′, ϕ′, ζ′)ξ′uSn(ξ′)

vn(ϕ′)wm(ζ′) dξ′ dϕ′ dζ′

ASmn =

1 λS2mn

π

∫0 ∫0 ∫0

(19)

S*(ξ′, ϕ′, ζ′)ξ′uSn(ξ′)vn(ϕ′) (20)

and the steady-state temperature distribution reduces to ∞

Θ(ξ , ϕ , ζ ) =

∞

∞

∑∑∑ S=1 m=1 n=0 π 1 1

∫0 ∫0 ∫0

dξ′ dϕ′ dζ′

predicted (°C)

ΔT (°C)

diameter (nm)

length (μm)

native native native native native native native implanted implanted implanted implanted

26.5 24.9 25.2 28.8 24.5 27.6 23.5 45.8 35.3 37.0 45.9

26.4 22.1 23.2 22.4 22.4 33.6 31.6 44.2 28.6 45.6 45.9a

0.1 2.8 2.0 6.4 2.1 −6.0 −8.1 1.6 6.7 −8.6 0

92 31 62 41 40 112 107 97 76 112 37

10.5 4.5 4.9 8.2 7.9 9.4 12.9 16.2 16.7 13.7 13.9

For an unpolarized laser, which is the configuration used here, the electric field is very nearly uniform in the radial and angular directions. Consequently, we used the spatial average value of the electric vector at each axial position to determine the source function and the temperature distribution. In this case, the electric field is only a function of the axial position, as demonstrated in the left-hand plots of Figure 4b and c for a weakly absorbing SiNW and a strongly absorbing carbon NW, respectively. In both cases, the electric field is seen to decrease sharply near the bottom of the wire and then change more gradually in the axial direction. When the source function is independent of r and ϕ, eq 21 can be simplified by setting νn(ϕ) = ν0(ϕ) = 1 and carrying out the integration over ξ to give

1

wm(ζ′) dξ′ dϕ′ dζ′

observed (°C)

a The imaginary index of refraction of k = 0:025 was determined from this data point.

At steady state (as τ → ∞), this result reduces to constants given by 1

type

uSn(ξ)vn(ϕ)wm(ζ ) λS2mn S*(ξ′, ϕ′, ζ′)ξ′uSn(ξ′)vn(ϕ′)wm(ζ′) (21)

Lorenz−Mie theory can be used to calculate the source function that generates photothermal heating for particles with high symmetry such as spheres or infinite cylinders.29 However, for the more complex case of a finite cylindrical NW considered

∞

Θ(ξ , ζ ) =

∞

∑∑

XS0(ξ)Zm(ζ )J1(γS0)

γ λ2 S = 1 m = 1 S0 Sm0

XS0

2

Zm

2

∫0

1

⟨S*(ζ′)⟩Zm(ζ′) dζ′ (22)

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Figure 4. The internal electromagnetic heating profiles for silicon and carbon NWs. (a) Atomic force microscope image of the optically trapped NW shown in Figure 3a/b. (b) Calculated internal axial electric field magnitude and the corresponding axial temperature profile for the SiNW shown in (a). (c) Calculated internal axial electric field magnitude and the corresponding axial temperature profile for an amorphous carbon NW with the same geometry as that used in (a) and (b). (d) Comparison of experimentally measured and numerically calculated temperatures for several optically trapped native and ion-implanted SiNWs. The arrow represents calculations for a single NW presented in Figure 3a and panel (a) here. Diameters of the NWs can be found in Table 2. For ion-implanted NWs, an imaginary index of refraction of k = 0.025 was used, determined from the data point marked with (*). (e) Diameter dependence of experimentally measured and numerically calculated temperatures for native SiNWs at 3 MW/cm2 irradiance. The dashed line represents a seventh-order polynomial fit. All error bars represent 95% confidence intervals.

where ⟨S*(ζ′)⟩ is based on the spatial average of the electric field at each axial position.

For low SiNW radii, it is likely that photothermal heating of the ambient water bath becomes increasingly important,22 even though it is neglected in the model presented here. For large SiNW radii, it is likely that heating occurs according to the predicted nonlinear dependence that leads to an inhomogeneous, nonlinear temperature field around the NW. Motion of cooler fluid near the NW acts to reduce the observed temperature relative to what is predicted with the model developed above. Inhomogeneous temperature fields have been predicted theoretically25 and reported for optical trapping experiments with gold nanocrystals.24,26 The laser trapping experiment here can only detect the local fluid temperature through the hot Brownian motion of the optically trapped NW. In this way, the ambient fluid temperature can be expected to be lower than the internal temperature of an optically trapped NW. Furthermore, the NWs move within the laser trap relative to the fluid due to Brownian motion, and this will increase the effective heat-transfer coefficient used here. These effects are not included in the heat-transport model developed in this work. However, the observed temperature increases scatter closely about theoretical predictions, which reflects the utility of

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DISCUSSION Comparison of Predicted Temperatures with Experiment. Identical experimental measurements of temperature using a combination of laser trapping and AFM were made for seven unique NWs at various irradiances. Numerical electric field source function simulations and corresponding temperature distribution calculations were made for each of these wires, and the agreement between experiments and theory are plotted in Figure 4d/e. The arrow in Figure 4d represents measurements made for the NW shown in Figures 3a and 4a at an irradiance of 3 MW/cm2. The theory developed here predicts steady-state NW temperatures that increase nonlinearly with increasing NW radii (Figure 4e). As shown in Figure 4e, this model is observed to underpredict the observed experimental temperatures for wires with diameters below ∼90 nm and overpredict temperatures for wires above this threshold. 1413



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window, opening up the prospect of using silicon nanomaterials in future photothermal therapy applications.

the model developed here. For instance, the average internal electromagnetic fields from DDA simulations can be used to estimate the total power dissipated within an individual NW. The total heat dissipation was calculated to be 18 μW for the SiNW of Figure 4b under and irradiance of 3 MW/cm2. The power associated with the incident beam over an equivalent cross-sectional diameter of the NW was found to be 81 μW, indicating that over 22% of the incident electromagnetic energy is dissipated as heat by the NW cavity. Furthermore, it is also possible to use this model to make quantitative comparisons of the internal electromagnetic field and temperature profiles for strongly absorbing NW materials such as amorphous carbon36,37 with those for crystalline SiNWs. Example results are presented in Figure 4c for an amorphous carbon NW (a-CNW) that has an identical cylindrical morphology as the SiNW in Figure 4b. A visible wavelength of λinc = 488 nm and irradiance of 1000 W/cm2 are used for these calculations given the availability of experimental absorption coefficients for amorphous carbon in this spectral region.36 The real and imaginary indices of refraction as well as the thermal conductivity are appreciably different for amorphous carbon, stemming from the complex mixture of sp2 and sp3 bonding.36 In the visible spectral region, the large imaginary index of refraction, k = 0.8, and low thermal conductivity, κNW = 0.0159 W/m·K, of amorphous carbon lead to several marked differences between weakly absorbing silicon and strongly absorbing a-CNW cavities. First, the internal electric field profile is much more complex than that for crystalline silicon, owing to the large imaginary index of refraction for amorphous carbon, and there is a sizable field along the length of the NW, leading to heating along the entire length of the nanoscale cavity. This would not be expected for planar amorphous carbon due to the large absorption coefficient at this wavelength. Second, there is an appreciable radial temperature gradient across the carbon NW that is not observed in the case of crystalline silicon. Third, the total temperature gradient along the length of the NW (∼20 °C over 5 μm) is significantly larger than that for crystalline silicon (∼1 °C over 5 μm), even though the incident irradiance is over 3 orders of magnitude lower. This observation implies that photothermal heating and subsequent effects of photophoresis are much more relevant to consider for amorphous carbon or amorphous silicon materials than for crystalline silicon. Finally, it is possible to use the model to infer the unknown absorption coefficient for the ion-implanted SiNWs for comparison with bulk silicon. Temperature measurements showed significantly increased heating compared with native SiNWs. The imaginary component of the refractive index for the implanted SiNW (k = Im{NNW}) was varied using DDA computations, and the resulting temperatures were calculated to match the observed temperature. For an ion-implanted SiNW with D = 37 nm and L = 13.9 μm (denoted with an asterisk in Figure 4d), we estimate k = 0.025, or equivalently, an absorption coefficient of αNW > 3000 cm−1. This is roughly 2 orders of magnitude larger than that for bulk intrinsic silicon (k = 5.6 × 10−4, αSi = 72 cm−1)38 and agrees well with prior absorption coefficient measurements for ion-implanted silicon wafers.16 Using this extracted value for k, predicted temperatures for the other ion-implanted wires were calculated and are shown in Figure 4d. Future efforts toward introducing point defects in silicon nanomaterials could lead to further increases of their absorption coefficient in the NIR tissue-transparency

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CONCLUSIONS A heat transport model was developed to predict temperature distributions for photothermal heating of finite cylindrical NWs trapped by a continuous-wave single-beam optical trap. In contrast to prior theoretical work focused on laser heating of infinite NWs,29 this work takes into consideration heat transfer from both the curved surface as well as the ends of the NW. The theoretical analysis developed in this study elucidates the effects of the numerous thermal, optical, and geometric parameters on the temperature distribution within NWs. For weakly absorbing materials such as silicon, relatively small temperature increases are encountered. For SiNWs with a diameter of 100 nm, a maximum temperature increase is predicted to be ∼5 °C for a laser wavelength of 975 nm at an irradiance on the order of 1 MW/cm2. Furthermore, the end-on illumination used to trap NWs can involve significant axial temperature gradients on the order of 1 °C/μm, whereas temperature gradients are not predicted for the case of infinite NWs. The high irradiance (∼1 MW/cm2) used for laser trapping particles can lead to significant heating of a NW, especially if it is composed of a strongly absorbing material such as amorphous carbon. Temperature increases are predicted to be significantly higher for strongly absorbing materials such as amorphous carbon. For carbonaceous particles having a relatively low thermal conductivity, significant axial and radial temperature gradients are predicted even when the heat source function is fairly uniform radially. For silicon, which has a much higher thermal conductivity, the predicted temperature is quite uniform radially. It is to be expected that much higher temperatures arise for particles trapped in air compared to those in water because the thermal conductivity of air is substantially lower than that of water. The optical properties of silicon can be modified by ion implantation. Single-beam laser trapping experiments are used to test the model using both native silicon NWs as well as SiNWs implanted with phosphorus ions. The observed temperature increases for native SiNWs scatter closely about theoretical predictions. During experiments, forward-scattered light from the laser trap is projected onto a QPD to provide interferometric measurements of the Brownian motion of individual trapped particles. Temperatures measured from Brownian motion agree well with the predicted values. For phosphorus-implanted silicon NWs, the complex refractive index was estimated by an iterative procedure, matching experimental and theoretical predictions for a single NW, and that complex refractive index was used for comparison with measurements of other implanted particles. Implantation is observed to increase the imaginary index of the SiNWs to a value of approximately k = 0.025 from k = 0.00059 for the native silicon at a wavelength of 975 nm, leading to local temperatures greater than 42 °C in laser trapping experiments. These results suggest the potential use of employing implanted silicon as a low-cost material in future photothermal heating applications.

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ASSOCIATED CONTENT

S Supporting Information *

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beam laser trapping experiments, experimental extraction of temperature, temperature extraction accuracy analysis, numerical electromagnetic source term calculations, as well as numerical heat-transfer simulations is available. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by an Air Force Office of Scientific Research Young Investigator Award (Contract #FA95501210400) as well as start-up funding from the University of Washington. P.B.R. thanks the NSF for a Graduate Research Fellowship under Grant Number DGE1256082.

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Photothermal Heating of Nanowires - The Journal of Physical

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