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ARTICLES Local Field Effects on Laser-Induced Heating of Metal Nanoparticles Samantha Bruzzone*,† and Marco Malvaldi‡ Dipartimento di Chimica e Chimica Industriale, UniVersita` di Pisa, Via Risorgimento 35, Pisa, Italy, and Dipartimento di Chimica Bioorganica e Biofarmacia, UniVersita` di Pisa, Via Bonanno 33, Pisa, Italy ReceiVed: January 13, 2009; ReVised Manuscript ReceiVed: June 22, 2009
In this paper, we study the local heating induced by a strong laser field on the surface of a nanoparticle resonant close to the laser field wavelength and on the embedding matrix. The ability of the metal particle to convert laser radiation into heat is related to the optical properties of the particle and can be studied with different models with growing degree of accuracy. We obtain a satisfactory agreement with experimental temperature range evidenced in the literature by phase transition phenomena (ice vaporization) on analogous systems. To explain the high temperatures obtained experimentally, surface field enhancement effects and the electronic smear out beyond the particle surface have to be considered. 1. Introduction The local heating induced by irradiated metal nanostructures is a promising tool for a wide range of applications, principally in the biomedical field.1-4 In this phenomenon, a metal nanoparticle perturbed by a laser field with frequency close to its plasmon resonance efficiently converts electromagnetic energy into thermal energy, with an overall rising of the temperature in the surroundings of the particle. Despite the increasing number of experimental works on the subject, a theoretical quantitative understanding of the phenomenon is still lacking. The motivation of this lack of knowledge is two-fold: first, the inherent difficulty in measuring a local microscopic temperature rise around a nanosized structure (giving uncertain experimental results) and, second, a noticeable degree of unmatching between theoretical and experimental conditions togheter with the absence of a systematic approach to the problem. In addition, few works about the theoretical modeling of phenomenon have appeared in the literature to date mostly focusing on a single sphere in an infinite medium. For what concerns the stationary state of this system, that is, the temperature profile at nanostructure-radiation thermal equilibrium the usual treatment is based on the foundamental solution of the heat conduction problem for a single heat-generating sphere embedded in a medium given by Goldenberg and Tranter.5 In this approach the heat-generation rate Q(r) has been expressed in two formally different ways6-8 which nevertheless can correspond exactly provided the sphere size is small compared with the incident laser light wavelength.9 In the first one6 the dissipation of radiation energy into heat is evaluated according to the Joule-Lenz relation,10 while in the second it is expressed through the nanoparticle absorption cross section. In both models, explicitly or implicitly, it is assumed that the electric field is locally homogeneous inside the metal nanoparticle. The results obtained by these models are not quantitatively * Towhomcorrespondenceshouldbeaddressed.E-mail:
[email protected]. Fax: +39-050-2219260. † Dipartimento di Chimica e Chimica Industriale. ‡ Dipartimento di Chimica Bioorganica e Biofarmacia.
satisfying giving temperatures that are too low, even of an order of magnitude, with respect to the corresponding experimental data.11,12 In this paper, our principal intent is to show that such understimation is due to the neglect the surface effects given by the coupling of giant electromagnetic fields13 on the surface with spill out electron density14 and to the assumption that the local field inside the particle is homogeneous, losing in this way the peculiar importance of surface plasmon modes. In the following section we clarify the problem from the theoretical point of view; then, in the next section, numerical calculations performed with different models are presented. Finally, a discussion of the results obtained and of their implications is reported. 2. Theoretical Outline In the absence of phase transformation, the temperature of a system consisting of a homogeneous sphere of radius R embedded in an infinite medium can be found solving the heattransfer system of equations:
F1(r)c1(r)
(
)
∂T1(r, t) K1 ∂ 2 ∂T1(r, t) r + Q(r, t), ) 2 ∂t ∂r r ∂r 0 e r e R (1)
F2(r)c2(r)
(
)
∂T2(r, t) K2 ∂ 2 ∂T2(r, t) r , ) 2 ∂t ∂r r ∂r
r>R
(2)
with the boundary condition T1(r,0) ) T2(r,0), T1(R,t) ) T2(R,t) and K1(∂T1(r,t))/(∂r) ) K2(∂T2(r,t))/(∂r).5 Here T1 and T2 are the temperatures of the sphere and medium, respectively, and F, c, and K are the mass density, heat capacity, and thermal conductivity, respectively (where the subscript 1 indicates the sphere and 2 the medium), while Q is the heat-generation rate per unit volume and unit time. The involved solution of this system, given by Goldenberg and Tranter,5 becomes simpler in the limit t f ∞ assuming Q is homogeneous on the whole sphere
10.1021/jp9003517 CCC: $40.75 2009 American Chemical Society Published on Web 08/13/2009
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volume. In particular, the variation of temperature distribution outside the nanoparticle is given by
∆T2(r) )
VNPQ 4πK2r
(3)
with VNP the nanoparticle volume. As already indicated in the Introduction, two different ways to calculate Q have been proposed in the past years. The first one6 is based on the Joule-Lenz law and, assuming that the field inside the particle is given by10
E(r) )
3εm E 2εm + εp 0
(4)
gives the resultant heat generation rate per volume unit:
[
]
εp - 1 1 Q(r, t) ) 〈j(r, t) · E(r, t)〉t ) - Re iω |E(r)| 2 2 4π
(5)
homogeneous inside the whole particle, with the local electric field E(r) given by eq 4. Here, εm and εp are the dielectric function of the matrix and nanoparticle, respectively, while j(r,t) is the current density. The Joule-Lenz law meaning can be understood from the energy conservation law of electromagnetic fields,
∂u ) -∇ · S + E · j ∂t
(6)
where the rate of variation of energy density of the field, u, in a given volume of space is divided in two contributions: • ∇ · S, the energy transported (inside or outside the volume) in the form of electromagnetic waves (incoming or scattered fields); • E · j, the energy lost by the electromagnetic wave by its interaction with the matter contained in the volume. In the second one,7,8 Q is obtained according to
Q)
CabsI VNP
(7)
where Cabs is the absorption cross section of the particle and I is the laser flux intensity. For a spherical particle, in a general point of view, the absorption cross section can be calculated by the Mie theory as the difference between the extinction cross section and the scattering cross section: ∞
Cext )
∑
2π (2L + 1)Re(aL + bL) |k| 2 L)1
(8)
∞
Csca )
∑
2π (2L + 1)(|aL | 2 + |bL | 2) 2 |k| L)1 Cabs ) Cext - Csca
(9)
(10)
where aL and bL are the expansion coefficients of the scattered field. It is straightforward to show that these two approach are identical, recalling that I ) (cE02εm)/(8π), provided that the scattering cross section is negligible with respect to the absorption cross section and the electrostatic approximation holds. Indeed, in this hypothesis, the absorption cross section can be expressed as9
Cabs )
(
8π2R3√εp εp - εm Im λ 2εm + εp
)
(11)
that, substituted in eq 7, gives, through simple operations, eq 5. Our description starts by observing that the local electromagnetic field inside the sphere is not homogeneous: in particular, two separated aspects have to be considered. First, the variation of the radial component of the electric field inside the nanoparticle for its n-normal mode is given by the relation
Er(r) ∝ rn-1
(12)
The greater the order of the normal mode, the more the field is localized near the surface of the sphere. Second, for a metal particle the electron density is smoothly going to zero outside the particle surface: this phenomenon is usually indicated as spill out.14,15 In a schematic representation of a metal cluster it is assumed that every atom loses the valence electrons. These electrons are delocalized on the whole volume of the particle. The remaining positive ions constitute a background, the ion core, whose radius typically is considered equal to the radius of the particle. Such outlying charge density can couple to the giant electromagnetic fields originating on the external nanoparticle surface when the same particle is submitted to light with wavelength near to its plasmon resonance. The importance of surface effects on the optical properties of metal particles has been pointed out, between of others, in a series of works by Apell et al.16-19 The authors criticize the classical picture of boundary scattering and emphasize the increased ability of the electrons to respond to the external field if the electron density profile is not terminated at the cluster surface. In particular, on the surface of the nanoparticle the radial component of the local electric field is absolutely not negligible, giving rise to the well-known nearfield enhancement effect.13,20 The local field on the external surface is given by the sum of the incident and scattered vector fields on the same surface. In this framework, we have calculated the heat generation rate Q from the Joule-Lenz law, eq 5, in which the local electric field has been obtained following the Mie theory for a single spherical particle.9,21 According to this theory, the internal Ei, incident E0 and scattered Es fields can be expressed by a vector spherical harmonics expansion.9 The coefficients of such expansions can be found by solving the boundary conditions on the sphere surface
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(E0 + Es - Ei) × eˆr ) (H0 + Hs - Hi) × eˆr ) 0
(13)
and depend on the sphere radius and particle and embedding medium dielectric functions at the used wavelength. From eq 13 its is evident that only the tangential components of external field (incident plus scattered fields) and internal one must be equal. The radial component of scattered field close to the surface is responsible for the huge enhancement of near electromagnetical field. A complete treatment of this problem can be found in fundamental literature.9,13 Following several former works in the literature, the dielectric function employed in our calculations has been obtained in a semiempirical fashion, starting from the experimental dielectric data for bulk metal. We considered the dielectric function as given by the sum of two contributions: an interband contribution (not dependent on the sphere size) and a free electron contribution (which depends on the sphere radius).22 The second one is calculated according to the venerable Drude model, while the first has been obtained by subtracting the bulk Drude contribution to the experimental data:23
bulk R bulk εp(ω) ) εbulk exp (ω) - εDrude(ω) + εDrude(ω) ) εexp (ω) + ωp ωp (14) ω + iΓbulkω ω + iΓRω
-1 -1 where, as usual, Γbulk ) τFermi and ΓR ) τFermi + VFermi/R.24 The values of physical parameters referred to Au bulk (τFermi, VFermi) employed in eq 14 are reported by the seminal text on solid state by Ashcroft and Mermin.25 Explicitly, τFermi ) 3 × 10-14 s and VFermi ) 1.4 × 108 cm/s. ωp is the plasmon frequency, defined as ω2p ) (ne2)/(ε0me), where n is the bulk number electron density, e and me are the electron charge and mass, respectively. The number electron density is calculated according to its definition, n ) (3)/(4πrs3), where rs is the Wigner-Seitz radius of bulk Au (rs ) 3.01 au).25 As the electronic density in the spill out region is lower than the one inside the particle, even the dielectric function should be a function of the radial position. However, at this first stage, we have preferred to consider the dielectric function as nonlocal to limit the complexity of the mathematical treatment. For what concerns the dielectric constant of the medium, its value acts on the local fields and on the absorption cross section of eq 7.9 In this work, we limit ourselves to consider a single nanoparticle embedded in ice; thus, we took the dielectric constant of the medium to be equal to the one of ice in the reported range of frequencies; at these conditions, it can be assumed that εm ) εice is independent upon the temperature. Following Govorov6 the value of εice = εwater is 1.8.
Since the electric fields are inhomogeneous, to obtain an heat transfer rate per volume unit we average the Joule-Lenz results over the volume:
Q(r, t) ) 〈j(r, t) · E(r, t)〉t )
(15)
ω Im(εP(ω)) 8VNPπ ω Im(εP(ω)){ 8VNPπ
∫ |E(r)|2 dV )
(16)
∫0R |E(r)|2r2 dr dΩ + ∫R(R+R') |E(r)|2r2 dr dΩ}
(17)
where the argument of the integral inside the particle is the square modulus of the electric internal field obtained from Mie theory, while the argument of the integral in the spill-out shell (from R to R + R′) is the square modulus of the local field obtained as the sum of the incident and scattered fields (the latter giving the main contribution due to the closeness to particle surface). Since we want to understand the relative importance of the two effects (inhomogeneous fields inside the particle and spill out in the giant field region close to the surface), we find it useful to separate the heat transfer rate of the irradiated particle in two contributions: the inner contribution, which takes into account the field inhomogeneity, and the external or surface one. The value of R′, spill out parameter defined from the approximative decay of the electron density beyond the ion core with radius R, is set equal to a few atomic units (0.2 nm), according to several works,14,26 and has been validated by comparison with electron density decay functions obtained by DFT calculations for spherical jellium particles. In these calculations the smear out of electron density beyond the particle surface has been always found to deviate from the R′ value (0.2 nm) of no more than 10%. The spill out extension actually depends on the particle dimensions, being a function of the number of free electrons.15 Since our external field rapidly fades away as far as we move of about 1 Å from the particle surface, the leading contribution to the heat generation is the one arising directly on the surface or in its immediate closeness. The correct spill out calculation would be not negligible, in principle, in the accurate calculation of resonance frequency; nevertheless, in this work, we do not consider such aspect, and we have considered the spill out parameter independent of the particle size. 3. Results and Discussion We begin our analysis by calculating the absorption cross section for single Au particles with different radius as a function of incident wavelength, according to Mie theory, as well as electrostatic approximation. From inspection of Figure 1, it is evident that up to R ) 30 nm the electrostatic approximation gives satisfactory results; beyond this radius, however, the deviation from the Mie calculations begin to be considerable in both intensity and position of the plasmon resonance. As a consequence, we expect that for particles with radius R > 30 nm the calculation of heat transfer rate Q according to eq 5, Qel.ap., or 7, QMie, will differ appreciably. In Figure 2, we report the actual comparison between the heat transfer rates per unit volume Qel.ap. and QMie divided by the laser field intensity, I. The reported values are obtained with a laser flux intensity of 1.8 × 105 W/cm2, that is, the same (strong) intensity used in ref 11. As expected, the Qel.ap. is overestimated heavily with respect to QMie when the particle dimension becomes such that electrostatic approximation begins to fail. In the same figure, we report the total contribution (as calculated from eq 17) to the heat transfer rate per volume unit, which takes into account the local (non homogeneous) electric field vector where the
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Figure 1. Comparison between absorption cross section obtained by Mie theory and in the framework of electrostatic approximation.
Figure 2. Comparison between heat rate transfer per volume unit divided by laser intensity Q/I following eqs 7 (green), 5 (red), and 17 (blue).
electron density is nonzero. The same calculation limited to the first term of eq 17, i.e., the integral from zero to R, (equivalent to consider only the internal particle volume) is practically superimposed to the QMie curve, confirming the reliability of our calculated fields. In Figure 3, we display the internal (Figure 3a) and surface contribution (Figure 3b) to the heat transfer rate as separate quantities, together with their sum (Figure 3c). The first consideration is about the relative importance of the two contributions: the surface term is almost twice the volume one, thus revealing itself as the most important contribution. As a second aspect we note that, as the particle volume is increased, the heat rate transfer per unit volume decreases at all wavelengths. It is well known that the absorption cross section calculated according to Mie theory scales with a thirth order law versus the radius of the particle: this has been confirmed experimentally.27 We tested the behavior of the total (volume plus surface contribution) Q/I calculated by us together with the volume contribution (equivalent to the Mie theory result) only as a function of radius: the result is reported in Figure 4. Both values apparently follow a third-order law with respect to
the particle radius; thus, this aspect of Mie theory is retained even when the surface contribution is taken into account. In order to compare our results with experimental findings, we calculated the temperature enhancement for an Au nanoparticle of radius R ) 25 nm, embedded in ice (K2 ) 1.6, T ) -20 °C11) irradiated with a laser light at 530 nm and flux intensity 1.8 × 105 W/cm2, to reproduce the experimental condition given by Govorov.11,28 In the cited work, it is reported that at this light intensity, water around the nanoparticle goes into vapor phase. This indicates that the temperature around nanoparticle is superior to 100 °C (and not 0 °C as supposed by the authors11). This requires a temperature enhancement of, at least, 120 °C. In Table 1 the temperature enhancement for all the three models described in the text is reported. It appears immediately that neglecting the field inhomogeneity and surface effect results in a strong underestimation of the experimental temperature enhancement. The value of local temperature around the nanoparticle can not be measured directly and thus does not allow for a strict comparison between theory and experiments. To support further our hypothesis we note that, for the field
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Figure 3. Internal and surface contribution to heat rate transfer per volume unit togheter with the total calculated according to the local field description using the Joule-Lenz relation.
TABLE 1: Heat-Generation Rate and Temperature Enhancement for an Au Nanoparticle with R ) 25 nm Irradiated with Laser Light at 530 nm and Flux Intensity of 1.8 × 105 W/cm2
Q (1016 W/cm3) ∆T (K)
Figure 4. Size dependence of Q/I calculated according to eq 17 (black circles) and QMie/I (red squares) at 532 nm. Solid lines are fitting with a third-order law.
intensity used and particle radius considered, local enhancement on the surface of gold nanoparticles of the order of hundreds of degrees are expected from SERS experiments29 and finite elements simulations.30 4. Conclusions In this work, we studied theoretically the heat generation by irradiated metal particles with different classical models. We evidenced that for particles with radius of some tens of nanometers (a size range widely used in experiments) the
electrostatic approximation
Mie theory
field inhomogeneity
12.9 16.8
16.5 21.4
112.2 146.1
approach based on the electrostatic approximation and the one using the correct Mie absorption cross section, even if apparently corresponding, do not lead to equivalent results. We stress that, in calculating the local heating induced by radiation, the intertwined effects of local field inhomogeneity inside the particle and the coupling between electronic spill out and giant radial field on particle external surface can not be neglected. Taking into account these two effects, we obtained a temperature enhancement for a single particle of the correct order of magnitude with respect to experimental extimations, without the need to invoke collective effects, that actually seem to be absent in experimental situations to which we refer. We have to stress that collective effects are known to enhance the electromagnetic absorption and the photothermal efficiency of metal nanoclusters; this has been demonstrated both theoretically7 and experimentally. When more clusters are close together, such effects cannot be neglected. The actual unease of measuring a local temperature on a nanometer scale, joined with the difficulty to perform single-particle experiments, could in our view have generated a situation in which experimental results are not straightforward to be interpreted. More accurate experiments, separing these two aspects, should be performed
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in order to estabilizh which of the two effects (surface or collective effects) plays the leading role. The hypothesis that a detailed description of the surface is necessary to understand the photothermal properties of nanoparticles is supported by previous experimental results, even for slightly different systems.31 However, our calculations suffer from several drawbacks: as a first point, the mathematical description of heat flow diffusion eq 2 is valid only if the mean free path of the heat carrier is smaller than the characteristic particle size. For gold metal the electron mean free path l is about 20 nm32 and in a metal nanoparticle it is widely accepted that the effective electron mean free path leff is given by (1)/(leff) ) (1)/(R) + (1)/(l).33 These observations lead to admit the solution proposed by5 to be absolutely reliable only for large particles, measuring at least some tens of nanometers. In addition, the solution presented in eq 3 has been obtained under the assumption that Q is constant over all the volume in the particle. This situation is not realized in real particles. Last but not least, we describe the electronic response of the particle with a homogeneous (non local) dielectric function. Actually, the response of electrons in the spill out region and on the external surface (according to some papers even close to the surface on internal side34) should be correctly described by a local dielectric function. Nevertheless, the utilization of a local dielectric function would complicate consistently the calculation of electromagnetic response fields; in this preliminary work this is beyond our purposes, since the main aim of the present work was to stress the importance of local fields in the calculation of heat generation by laser irradiated nanoparticles. Acknowledgment. Prof D. Leporini, Prof M. Ausloos, and Prof A. Lucas are gently acknowledged for helpful discussions. References and Notes (1) Jain, P. K.; Lee, K. S.; El-Sayed, I. H.; El-Sayed, M. A. J. Phys. Chem. B 2006, 110, 7238–7248. (2) Huang, X.; Jain, P. K.; El-Sayed, I. H.; El-Sayed, M. A. Nanomedicine 2007, 2, 681–693. (3) Huang, X.; El-Sayed, I. H.; Qian, W.; El-Sayed, M. A. J. Am. Chem. Soc. 2006, 128, 2115–2120. (4) Sperling, R. A.; Gil, P.; Zhang, F.; Zanella, M.; Parak, W. J. Chem. Soc. ReV. 2008, 37, 1896–1908. (5) Goldenberg, H.; Tranter, C. Brit. J. Appl. Phys. 1952, 3, 296–298.
Bruzzone and Malvaldi (6) Govorov, A. O.; Zhang, W.; Skeini, T.; Richardson, H.; Lee, J.; Kotov, N. A. Nanoscale Res. Lett. 2006, 1, 84–90. (7) Keblinski, P.; Cahill, D. G.; Bodapati, A.; Sullivan, C. R.; Taton, T. A. J. App. Phys. 2006, 100, 54305. (8) Pustovalov, V. K. Chem. Phys. 2005, 308, 103–108. (9) Bohren, C. F.; Huffman, D. R. Absorption and Scattering of Light by Small Particles; Wiley: New York, 1983. (10) Landau, L. D.; Lifshitz, E. M. Electrodynamics of Continous Media; Pergamon: New York, 1960. (11) Richardson, H. H.; Hickman, Z. N.; Govorov, A. O.; Thomas, A. C.; Zhang, W.; Kordesch, M. E. Nano Lett. 2006, 6, 783–788. (12) Hamad-Schifferli, K.; Schwartz, J. J.; Santos, A. T.; Zhang, S.; Jacobson, J. M. Nature (London) 2002, 415, 152–155. (13) Quinten, M. Z. Phys. D 1995, 35, 217–224. (14) Lang, N. D.; Khon, W. Phys. ReV. B 1970, 1, 4555–4568. (15) Kreibig, U.; Vollmer, M. Optical Properties of Metal Cluster; Springer: New York, 1995. (16) Apell, P.; Penn, D. R. Phys. ReV. Lett. 1983, 50, 1316–1319. (17) Monreal, R.; Apell, P. Phys. ReV. B 1990, 41, 7852–7855. (18) Apell, P. Phys. Scr. 1983, 29, 146–149. (19) Apell, P.; Ljungbert, A.; Lundqvist, S. Phys. Scr. 1984, 30, 367– 383. (20) Messinger, B. J.; von Raben, K. U.; Chang, R. K.; Barber, P. W. Phys. ReV. B 1981, 24, 649–657. (21) Kerker, M. The Scattering of Light and Other Electromagnetic Radiation; Academic: New York, 1969. (22) Hummel, R. E. Electronic Properties of Materials; Springer: New York, 2001. (23) Palik, E. D. Handbook of Optical Constants of Solids; Academic: Orlando, 1985. (24) Kreibig, U.; Fragstein, C. W. Z. Phys. 1969, 224, 307. (25) Ashcroft, N. W.; Mermin, N. D. Solid State Physics; Thomson Brooks/Cole: New York, 1983. (26) Lica, G. C.; Zelakiewicz, B. S.; Costantinescu, M.; Tong, Y. J. Phys. Chem. C 2004, 108, 19896–19900. (27) Berciaud, S.; Cognet, L.; Blab, G. A.; Lounis, B. Phys. ReV. Lett. 2004, 93, 257402. (28) Govorov, A. O.; Richardson, H. H. Nanotoday 2007, 2, 30–38. (29) King, M. D.; Khadka, S.; Craig, G. A.; Mason, M. D. J. Phys. Chem. C 2008, 112, 11751–11757. (30) Downes, A.; Salter, D.; Elfick, A. Opt. Express 2006, 14, 5216– 5222. (31) Stietz, F.; Stuke, M.; Viereck, J.; Wenzel, T.; Trager, F. Surf. Sci. 1997, 389, L1153–L1158. (32) Haberland, H. Cluster of Atoms and Molecules; Springer-Verlag: Berlin, 1994. (33) Mulvaney, P. Langmuir 1996, 12, 788–800. (34) Lerme´, J.; Palpant, B.; Pre´vel, B.; Cottancin, E.; Pellarin, M.; Treilleux, M.; Vialle, J. L.; Perez, O.; Broyer, M. Eur. Phys. J. D 1998, 4, 95–108.
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