1. Introduction

In recent years, it has been found that metallic nanoparticles (NPs), apart from their intense use for enhanced-spectroscopy applications [1], also exhibit a great potential in optical heating applications such as drug delivery [2], imaging [3–5], and photothermal cancer therapy [6–10]. Heating metallic nanoparticles, previously allocated into cancerous cells, with an impinging laser beam, results in the stop of the cellular activity, thus leading to either shrinking the tumour size, or slowing down its spread. Obviously, the larger the NPs temperature, the more effectively the cell activity is stopped. That is the reason why most of the effort in this novel field is focused on exploiting the huge electromagnetic fields produced on the nanoparticles through localized surface plasmon resonances (LSPR). Indeed, it has been shown that Au NPs can be heated up to temperatures five orders of magnitude larger than those reached with dyes originally used in early demonstrations of photothermal tumour therapy [9]. This fact leads to a reduction of the irradiation energy, considerably shrinking the probability of damaging non-cancerous cells.

Along with these promising experimental results, on the other hand, it has been recently shown that metallic nanostars (NSs) notably enhance local electromagnetic fields at LSPRs, making them good candidates for surface-enhanced Raman spectroscopy (SERS) substrates without any induced aggregation [11–15]. Additionally recent thermometric measurements of metallic NPs [16–18] have shown the need of deep theoretical studies of the thermal properties of metallic NPs. In this Letter, we show that metallic nanostars are not only suitable for sensing application, but can also play the role of heat sources considerably more efficient than the commonly studied nanospheres, as a result of the large absorption cross section at the LSPR. Making use of the 3D Green’s Theorem (surface integral equations) method (3DGTm) for flexible-shape NPs that we have recently introduced [19], we calculate rigorously the local electromagnetic field distribution on the surface of the nanostars at the LSPRs, from which the absorption cross section is worked out, as needed to determine the steady-state temperature of a metallic NP.

2. LSPRs of gold nanostars

Let us consider a NS whose geometrical shape is described as a deformable parametric surface called Supershape [20], which depends on certain parameters that basically modify the sphere formula in spherical coordinates. Particularly, we fix a configuration of parameters that allow us to change the number of tips in a star-like volume, preserving its shape. In order not to introduce too much complexity in the system under study, we only vary the number of tips in the plane Π (see Fig. 1(a)) from n = 1 to n = 6. In the perpendicular plane Σ, the symmetry of the number of star tips is kept constant in such a way that two opposing tips, one pointing upwards and the other one downwards. We will refer to the nanostar with n tips in the plane Π as S_n. Such star-like shapes indeed closely resemble those of fabricated colloidal NSs [13].

 

Fig. 1 (a) Schematic representation of a 4-fold nanostar, with two relevant symmetry planes: the plane Π where the number of tips is varied and the polarization plane Σ. (b) Absorption cross sections for Au nanostars with different number of tips from S₁ to S₆, along with that for the equivalent Au nanosphere.

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In Fig. 1(b) the absorption cross section is plotted for NSs made of gold described by the dielectric constant reported in reference [21]. The incident field is a plane wave impinging from the bottom, its plane of incidence being the plane Σ, Fig. 1(a). NSs have arms 40 nm long, and increasing number of tips. The LSPRs for Au-NSs are red-shifted when the number of tips in the plane Π increases from S₁ to S₆: λ_LSP (nm)=521, 548, 560, 587, 628, 688, respectively. In Fig. 2(a–f) we have plotted the norm of the surface electric field (SF) of the NSs in logarithmic scale. As a general result, the SF is accumulated on the vertex of the NSs, thereby providing large field enhancements, about |E|² ∼ 10⁴. Incidentally, these values make nanostars suitable for SERS applications [11, 13]. Essentially, such SF patterns results from the dipolar character on the plane Π of the corresponding lowest-energy LSPR. In addition, SF enhancements at the LSPR grow for increasing number of tips from S₁ to S₆, in agreement also with the absorption cross section in Fig. 1(b), that shows the same trend from S₁ to S₆; except for S₂, which has an absorption cross section and SF slightly larger than expected, presumably due to its 2-tip symmetry along the polarization axis, specially suited to match the dipolar LSPR pattern. In this regard, the NS asymmetry on the plane Σ along the polarization axis reduces its quality as dipole-like resonant cavity.

 

Fig. 2 Distribution of electric field amplitudes in logarithmic scale on the surface of the Au nanostars at their corresponding LSPR wavelengths for an incident electric field contained into the plane Σ (see Fig. 1(a)).

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3. Temperature profile of optically heated gold nanostars

From the point of view of the applicability for thermal therapy, such large absorption cross sections make Au NSs good candidates to be used as heat sources.

In fact, in recent works [22–24], G. Baffou and co-workers have developed a useful and simple expression that relates the absorption cross section σ_abs with the steady-state temperature T of a NP under the effect of a laser beam, namely:

Table 1. LSPR wavelengths λ_p and relative temperature factors τ_E for Au-NSs with given geometrical parameters: n, number of tips in the plane Π; s, the parameter that controls NS tip sharpness; and r_eff, effective radius.

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From equations 1 and 2, it is clear that the steady-state temperature of two metallic NPs with the same r_eff = r = r′, and different frequencies ω̃, ω follow:

In Fig. 3(a) we study the dependence of Q_abs and Q_sca on the tip sharpness for the Au NS S₄. In order to isolate the effect of the change of the sharpness in the optical response of the NSs, their volume is kept constant, and equal to the volume of the effective sphere. Two main effects are addressed: i) LSPR shifts to the red for increasing sharpness; ii) The value of Q_abs at the LSPR becomes larger. Interestingly, the imaginary part of the dielectric constant of gold, Im(ɛ_Au), has a minimum in the near IR at a wavelength (700nm) slightly larger than the onset of interband transitions, beyond which Im(ɛ_Au) monotonically increases following a Drude metal behavior. Accordingly, a reduction of Q_abs in the range of wavelengths (λ ≳ 600 nm) relevant to medical applications might be expected. Q_abs of the equivalent sphere is plotted in order to illustrate this fact. The LSPR wavelength of the sphere is λ = 510 nm; for larger wavelengths the imaginary part of the dielectric constant decreases leading to a decrease of Q_abs as well. On the contrary, Q_abs of the NSs increases when the LSPR is shifted to the red, even if the imaginary part of the dielectric constant decreases. However, the ratio Q_abs/Q_sca at the LSPR decreases as expected. In Table (1) we show the values of τ_E at the LSPR for varying tip sharpness in the case of S₄ NS [19]. It clearly reveals that, the sharper the NS tips are, the larger the absorption cross section is at the LSPR. Moreover, the temperature relative factor for the sharper S₄ is about thirty times larger than τ_E for the effective sphere. Thus, sharp Au NSs provide a huge heating efficiency compared to that of the effective Au nanosphere. In Fig. 3(b) the temperature distribution on the plane Π is shown for the sharper Au-NS (S₄ with s = 0.125 in Fig. 3(a). The temperature distribution clearly follows the NS shape, the effective heating range being almost restricted to the touching regime. Thus, cells will only be affected by the NSs when both are in contact. In Fig. 3(c), the corresponding SF is shown for the sake of completeness, revealing strong SF enhancements with dipolar character at the LSPR.

 

Fig. 3 (a) Absorption Q_abs (solid curves) and scattering Q_sca (dashed curves) cross sections for S₄ (Au) NSs with constant volume and increasing sharpness for the vertexes, along with those for the equivalent Au nanosphere. (b) Temperature relative factor distribution map near to the sharpest Au nanostar S₄, labeled as s = 0.125 in Fig. 3 (cyan), illuminated at its LSPR, λ = 860.2 nm. (c) Corresponding SF map at the LSPR.

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There are four results that allow us to state that Au NSs are good candidates as heat sources. Firstly, every single star heats more efficiently than its equivalent sphere, since for all the stars τ_E > 1. Moreover, S₁, S₂, S₃, S₄ are able to heat up from two to seven times more efficiently, and S₅, S₆ more than an order of magnitude. Secondly, τ_E increases when the number and/or the sharpness of the NS tips increases. Sharp (realistic) Au NS with S₄ symmetry yield a temperature enhancement of τ_E = 32. Bear in mind that the fabrication of metallic NSs by chemical shaping of spheroidal nanoparticles in a colloidal suspension [13, 25–28] does not allow full control over the number of tips or the sharpness of the NPs. Therefore it is extremely important that any nanostar in a colloid with r_eff ≳ 40 nm be a more efficient heater that the equivalent nanosphere, thus making unnecessary the use of any technique to isolate NSs with specific number of tips, dimensions, or tip sharpness. In addition, NSs obtained by chemical techniques usually have a large number of tips; we have shown that this fact improves the heating efficiency of the nanoparticles either. Moreover, it has been shown that a larger number of tips red-shifts the LSPR of the nanostar. The availability of LSPR in the IR reduces the energy of the impinging beam needed to heat NPs up, which makes more unlikely damaging non-cancerous cells. Thirdly, the sharpness of the NS vertexes can be used to tune the ratio Q_abs/Q_sca. It has been reported [6] that metal NPs with large values of Q_abs along with Q_sca have applications in photothermal ablation of tumours, with the advantage that the increased near-infrared scattering results also in an enhancement of the optical contrast for optical coherence tomography imaging. Fourthly, note that the reported Q_abs are of the same order of that of Au [24]; then ${\tau }_E^{\mathit{NS}}\gtrsim {\tau }_E^{\mathit{dim}}$, since $r_{\mathit{eff}}^{\mathit{NS}}<r_{\mathit{eff}}^{\mathit{dim}}$. Thus metallic NSs not only have comparable performance to dimers in SERS, but also in optical heating.

Finally we have to point out that the discussion of the whole article is focused on the parameter τ_E, which is normalized to that of the equivalent sphere, so that the thermal conductivity of the surrounding medium is irrelevant. In practice, to determine absolute values of temperature, water should be considered as the most appropriate surrounding medium.

4. Conclusions

In conclusion, we have described in this work the properties of Au nanostars as thermal heaters based on their large absorption cross sections at the LSPR, and their suitability for cancer thermal therapy. We have shown that their heating properties, resulting from the NS symmetry and geometrical dimensions, are excellent for a variety of optical heating applications; indeed, a ∼ 30-fold increase in the steady-state temperature is found for realistic NSs. Additionally, the red-shift induced on the LSPR for increasing number and/or sharpness of NS tips shows that a wide range of frequencies in the visible and near-IR can be covered with various NS shapes. Moreover, the NS tip sharpness rules also the ratio Q_abs/Q_sca; tuning this ratio, in such a way that Q_sca is also large, is a key feature for increasing the optical contrast, thus improving the quality of the optical coherence tomography imaging. Our results confirm that typical Au NSs available with current nanofabrication techniques should no doubt be suitable for photothermal cancer applications.