Radiolabelling of nanoparticles by proton irradiation: temperature control in nanoparticulate powder targets

Abstract

Radiolabelled nanoparticles are useful tools for biodistribution or cellular uptake studies related to the risk assessment of nanomaterials. Such studies are ideally carried out with industrially manufactured nanoparticles. Irradiation of small quantities of such nanoparticles, in the form of dry powders, with neutrons or light ions allows radiolabelling while preserving their biologically relevant properties. However, nanoparticle powders exhibit poor thermal conductivity and may overheat under irradiation. Their effective thermal conductivity is not known and conventional temperature measurement methods are difficult to apply. Reasonably accurate temperature data could be derived from post-irradiation X-ray diffraction studies on anatase ST-01 TiO₂-nanoparticles, with a primary particle size of 7 nm, subjected to proton beams of different intensities. The anatase-to-rutile phase transition starting at about 750 °C was identified by observing rutile peaks in X-ray diffraction patterns. The onset of growth of single diffracting TiO₂-domains at around 200 °C was revealed by shape analysis of the diffraction peaks. Identifying these reference temperatures allowed a calibration of the calculated temperature profile. The effective thermal conductivity in the TiO₂ powder target was found to be close to that of air trapped in interstices of the nanoparticulate powder. This suggests that the contribution of the nanoparticles to the heat removal from the target is negligible, thus necessitating the use of thin nanoparticle layers in the target in order to facilitate cooling and prevent thermally induced alterations of the nanoparticles.

Appendix: Stopping power correction

The energy dissipation rate η in Eq. (5) can only be considered as homogeneous in the whole target volume if the energy degradation dE _p/dx is constant over the spatial coordinate x. A simulation using SRIM (Ziegler et al. 2008) with a proton beam of E _p = 19 MeV passing 300 μm Al, 2350 μm water, 300 μm Al and a distance x of TiO₂-NP powder with an effective density of 0.955 g/cm³ gives the energy E _p(x) of those protons having passed a distance x (in μm) of TiO₂-NP powder and the fraction f(x) of the protons that have already been stopped within a distance x (in μm) in the TiO₂-NP powder. About 1/3 of the particles are stopped in the last 100 μm of the material and deposit their whole residual energy in the material. Taking this into account the effective energy loss in the powder material is higher than the difference E _p(x = 0) − E _p(x) and can be estimated according to

$$ \Updelta E_{\rm p,eff}(x) = (1 - f(x)) (E_{\rm p}(x=0) - E_{\rm p}(x)) + f(x)E_{\rm p}(x=0) $$

(11)

where x is the spatial coordinate in the powder layer ranging from 0 at the front and 400 μm at the rear side of the NP powder. Performing the SRIM simulation through aluminium windows and water layer implicitly takes into account the straggling of the proton beam and the fact that the maximum distance a proton travels in the NP powder may be larger that the thickness d of the capsule. E _p(x) and f(x) are shown in Fig. 8a and the degradation of E _p,eff(x) due to the energy loss ΔE _p,eff is depicted in Fig. 8b. E _p,eff(x) can be fitted quite well with a second order polynomial E _p,eff(x) = a + b ₁ x + b ₂ x ² with the coefficients a = (4.31642 ± 0.00012) MeV, b ₁ = (5.12 ± 0.12) × 10⁻³ MeV/μm and b ₂ = (8.20152 ± 0.29977) × 10⁻⁶ MeV/μm². The effective stopping power dE _p,eff(x)/dx = b ₁ + b ₂ x is the first derivative of this function and is hence linear. This energy loss per unit length is needed to deal with spatially varying energy dissipation rate η = η(x). It increases by a factor of 2.3 towards the rear side of the target capsule. In the case of irradiations at a primary E _p = 23.5 MeV the proton energy is degraded from 13.4 to 12.4 MeV with dE _p/dx increasing only by 12 % and no protons are stopped in the NP volume. Therefore, for the NP activation experiments η = const is a good approximation.

Fig. 8

Degradation of the proton energy in a TiO₂-NP layer of thickness x simulated by SRIM (Ziegler et al. 2008). E _p(x) denotes the energy of protons with a primary energy of 19 MeV after having passed 300 μm Al, 2350 μm water, 300 μm Al and x μm of TiO₂-NP powder with an effective density of 0.955 g/cm³. The right ordinate in (a) gives the fraction of protons that are stopped in the TiO₂-NP powder. Considering both contributions, slowing down and complete absorption, gives an effective proton energy E _p,eff(x), which is shown as squares and fitted second order polynomial in (b). Its first derivative dE _p,eff(x)/dx is the quantity required to treat the spatial dependence of the energy dissipation rate η = η(x)

The local variation of η(x) in Eq. (5) can be treated as

$$ \frac{{\hbox{d}}^2 T}{{\hbox{d}}x^2} = -\frac{I_{\rm p}}{\pi \lambda r_{\rm p}^2} \int\limits_{x}^{x+dx} \frac{{\hbox{d}}E_{\rm p,eff}(x)}{{\hbox{d}}x} \hbox{d}y . $$

(12)

Twice integrating this equation and determining the integration constants from the constraints T(x = 0) = T(x = d) = T ₀ yields

$$ T(x) = T_0 + \frac{ I_{\rm p} }{ \pi r_{\rm p}^2 \lambda } \left( \left[ \frac{ b_1 d }{ 2 } + \frac{ b_2 d^2 }{ 3 } \right] x - \frac{ b_1 }{ 2 } x^2 - \frac{ b_2 }{ 3 } x^3 \right) $$

(13)

whose temperature maximum can be determined solving the quadratic equation derived from d(T − T ₀)/dx = 0. In the present case the positive solution is x ₁ ≈ 213 μm. Thus, the temperature maximum is shifted by 13 μm towards the rear side of the NP powder capsule. For x ₁ = 213 μm one can calculate the maximum temperature difference

$$ T(x) - T_0 = - \frac{ I_{\rm p} }{ \pi r_{\rm p}^2 \lambda} \left( \frac{ b_2 }{ 3 } {x_1}^3 + \frac{ b_1 }{ 2 } {x_1}^2 - \left[ \frac{b_1 d}{2} + \frac{b_2 d^2}{3} \right] x_1 \right) $$

(14)

which differs from the value derived from Eq. (7) only by a numerical factor; the maximum temperature is <1 % higher in the asymmetric case.