Mechanistic investigation of radiolysis-induced gold nanoparticle formation for radiation dose prediction

Gold(III) chloride trihydrate and L-ascorbic acid were purchased from Sigma-Aldrich. Cetyltrimethylammonium bromide was purchased from MP chemicals. All chemicals were used as received without any additional purification.

HAuCl₄.3H₂O (30 μl of 10 mM) was mixed with 600 μl of 50 mM C₁₆TAB. Upon addition of 900 μl (5.88 or ∼5.9 mM) of 10 mM l-ascorbic acid, the solution turned colorless. The pH of the final solution was ∼5. All samples were prepared in Nanopure water (18.2 MΩcm) at Banner-MD Anderson Cancer Center, Gilbert, AZ, immediately prior to radiation treatments.

A Varian TrueBeam linear accelerator was used to irradiate samples with x-rays at a dose rate of 15.6 Gy min⁻¹ at room temperature. Energy of the beam was 6 MeV and the field size was 30 cm by 30 cm [21]. The samples were irradiated at doses of 0 (control), 5, 10, 15, 25 and 35 Gy, respectively. All post-irradiation analyses were performed at Arizona State University in Tempe, AZ (one-way travel time of approximately 20–30 min).

UV–visible absorbance spectra of the irradiated and the control samples were measured using a BioTek Synergy 2 plate reader. Absorbance spectra, from 300 nm–900 nm with a step size of 10 nm, were determined using 150 μl sample in a 96-well plate. Nanopure water (18.2 MΩcm) was used as a blank in all cases. The absorbance value corresponding to the peak wavelength (between 500 and 700 nm) was used as an indicator for gold nanoparticle formation. The final absorbance values used in the mathematical model were offset by subtracting absorbance at 900 nm (A₉₀₀).

The hydrodynamic size of the nanoparticles was measured using a Zetasizer Nano instrument and 50 μl volume of the sample. Thereafter, the average diameter values were recorded based on the software readout.

Samples of the irradiated and the control (no irradiation) with a volume of 1.5 ml were centrifuged for 10 min at 10 000 rpm. After centrifuging the samples, the clear supernatant was removed carefully without disturbing the nanoparticle pellet at the bottom of the Eppendorf tube. This nanoparticle pellet was dissolved in 0.32 M HNO₃ in order to oxidize the metallic (i.e. gold) nanoparticles to metal ions to facilitate detection using a High Resolution Multicollector ICP-MS. Gold ions were detected using an internal calibration curve performed independently. This was performed to quantify the conversion of precursor (gold ion) to gold nanoparticles in the absence and presence of radiation.

Because of the extremely short time scale of the water radiolysis and gold ion reduction reactions [22, 23], the reduction reactions are not included explicitly in either of the mathematical models described below, and, instead, the initial concentration of zero-valent gold atoms is specified from experimental measurements at different radiation doses. Two different mathematical models are examined here that predict the formation of GNPs at low radiation doses such as those needed for practical radiation dose measurement.

The first kinetic model describing the kinetics of gold nanoparticle synthesis begins with a reaction describing the formation of stable gold nanoparticle nuclei:

$$\begin{eqnarray*}&&A\mathop{\longrightarrow }\limits^{{k}_{1,fw}}B\end{eqnarray*}$$

where A is the concentration of gold atoms, B represents the total concentration of gold atoms contained in relatively stable gold nanoparticles and ${k}_{1,fw}$ is the rate of nanoparticle nuclei formation. The stable nuclei formation step is often the slow step in nanoparticle synthesis. The second reaction in the Finke-Watzky model describes the growth of stable nuclei through the incorporation of additional gold atoms:

$$\begin{eqnarray*}&&A+B\mathop{\longrightarrow }\limits^{{k}_{2,fw}}2B\end{eqnarray*}$$

The Finke-Watzky model ultimately predicts the rate at which insoluble gold atoms, A, are assimilated into gold nanoparticles, B. However, a significant limitation of the Finke-Watzky model is that it does not predict either nanoparticle size or the concentration of particles.

The model prediction is the total moles of gold atoms cumulatively contained in all the particles as a function of time, and this quantity can be converted to an estimate of absorbance using experimental measurements of average particle size when available. A prediction of absorbance is desired so that it can be compared with the experimental measurements taken using UV–vis spectrometry. Beer's Law was used to convert the model prediction of particle concentration to absorbance [24]:

$$\begin{eqnarray}&&A=\varepsilon bC\end{eqnarray} \tag{ 1 }$$

where, b is path length (i.e., the distance of the path that the light travels through the sample), ε is the extinction coefficient, and C is the particle concentration. The particle concentration was calculated by dividing the total concentration of gold atoms in particle form by the mean particle volume:

$$\begin{eqnarray}&&C=\displaystyle \frac{B}{\tfrac{4}{3}\pi {r}^{3}{N}_{A}{M}_{Au}}\end{eqnarray} \tag{ 2 }$$

where r is the mean particle radius, N_A is the Avogadro's number and M_Au is the molar density of gold, 98 $\tfrac{{\rm{mol}}}{{\rm{L}}}.$ The diameters of the nanoparticles synthesized under different radiation doses were measured experimentally, and for the Finke-Watzky model, all synthesized particles were assumed to be the mean particle size. The extinction coefficient is also dependent on the particle size and is calculated using:

$$\begin{eqnarray}&&\mathrm{ln}\,\varepsilon =k* \,\mathrm{ln}\,D+a\end{eqnarray} \tag{ 3 }$$

where, D is the mean particle diameter and, k and a are constants that were previously determined to be 3.321 11 and 10.805 05 for gold nanoparticles [24].

Population balance models explicitly recognize particles of different sizes and allow prediction of growth based on size. The governing population balance equation is:

$$\begin{eqnarray}&&\displaystyle \frac{\partial C(r)}{\partial t}+\displaystyle \frac{\partial }{\partial r}[\mu (r)C(r)]=N\cdot \delta (r-{r}_{c})\end{eqnarray} \tag{ 4 }$$

where C(r) is the concentration of particles of size r, μ(r) is the growth rate for particles of size r, and N is the nucleation rate of particles of critical radius size, r_c. The numerical approximation of particle balance models is frequently achieved by discretizing particles of different radii into size groups with each group representing a specific radius range. As a result, nanoparticle growth is split into numerous reactions that each describe the growth of particles in radius range i and, after sufficient time, some of these particles grow into radius range i + 1. In other words, when the particles reach a certain size, they move into a grouping of larger sized particles. After discretizing in size groups, the following reactions for nucleation and growth apply:

$$\begin{eqnarray*}&&A\mathop{\longrightarrow }\limits^{{k}_{1}}{C}_{0}\,\,{\rm{Nucleation}}\end{eqnarray*}$$

$$\begin{eqnarray*}&&A+{C}_{i}\mathop{\longrightarrow }\limits^{{k}_{2,i}}{C}_{i+1}\,\,{\rm{Growth}}\end{eqnarray*}$$

where, C₀ is the concentration of stable gold nanoparticle nuclei and C_i is the concentration of nanoparticles in size range i. The smallest stable nuclei (C₀) are assumed to consist of approximately 20 atoms, which is used to calculate the minimum particle radius [17]. The maximum particle radius is estimated from long-time span simulations and verified using experimental measurements, and the range of possible particle radii is divided into N discrete groups (N = 50 was used for results here after confirming the model predictions did not change significant when using additional groups that each represent a smaller range of particle sizes). The growth reaction leads to particles of larger size, and particles continue to grow at different rates (given below).

Material balances on the nucleation and growth reactions results in a model consisting of the following system of differential equations:

$$\begin{eqnarray}&&\displaystyle \frac{dA}{dt}=-{k}_{1}\cdot {A}^{n}-\displaystyle \sum _{i=0}^{N-1}{k}_{2,i}\cdot A\cdot {B}_{i}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\displaystyle \frac{d{C}_{0}}{dt}={k}_{1}\cdot {A}^{n}-{k}_{\mathrm{2,0}}* A\cdot {C}_{0}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{d{C}_{i}}{dt}={k}_{2,i-1}\cdot A\cdot {C}_{i-1}-{k}_{2,i}\cdot A\cdot {C}_{i}\\ \,{\rm{for}}\,i=0,\,1,\,\ldots ,\,N\end{array}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\displaystyle \frac{d{C}_{N}}{dt}={k}_{2,N-1}\cdot A\cdot {C}_{N-1}\end{eqnarray} \tag{ 8 }$$

The exponent, n, is an unknown parameter that will be determined using the experimental data. Mathematical models of similar systems have used n = 1 to 4 [25]. Therefore, only values over that range are considered here. These equations were non-dimensionalized before obtaining a numerical approximation of the solution.

For some systems, the growth rate, k_2,i, is controlled by the association rate of new gold atoms (i.e., reaction rate limited systems) while in other systems, the growth rate is controlled by the diffusion of gold atoms to the nanoparticle surface (i.e., diffusion limited systems) [18, 19]. If I is defined as the molar flux of gold to the surface of the nanoparticle and V_M is the molar volume, then the change in particle volume and radius is:

$$\begin{eqnarray}&&I=\displaystyle \frac{1}{{V}_{M}}\displaystyle \frac{dV}{dt}=\displaystyle \frac{4\pi {r}^{2}}{{V}_{M}}\displaystyle \frac{dr}{dt}.\end{eqnarray} \tag{ 9 }$$

The total flux of molecules to the surface is proportional to particle radius (i.e., the characteristic diffusion length scale) for the diffusion limited case (i.e., $I=4\pi r{\mathscr{D}}({C}_{b}-{C}_{surface}),$ and the total flux is proportional to particle surface area for the reaction limited case (i.e., $I=4\pi {r}^{2}k{C}_{surface}$). For the diffusion limited case, the rate at which particles are predicted to move from one size group, C_i, to the next group, C_i+1, is inversely related to the particle radius because $\tfrac{dr}{dt}=\tfrac{{V}_{M}}{r}\cdot {\mathscr{D}}({C}_{b}-{C}_{surface}).$ For the reaction rate limited case, $\tfrac{dr}{dt}={V}_{M}k{C}_{surface}.$ Experimental data from the system of interest here suggested that the growth rate of the nanoparticles was dependent on particle size (i.e., the system was diffusion limited) so the particle grow rate, k_2,i, was set inversely proportional the particle radius increases:

$$\begin{eqnarray}&&{k}_{2,i}=\displaystyle \frac{{k}_{2}^{o}}{{r}_{i}}\end{eqnarray} \tag{ 10 }$$

where ${k}_{2}^{o}$ is an unknown growth rate constant. The population growth model predicts both the rate of nanoparticle formation and the size distribution of the final population of nanoparticles.

The primary prediction of the mathematical model, the concentration of particles in each size group, C(r), is converted to estimate of absorbance for comparison with the experimental measurements of absorbance obtained using UV-vis spectrometry. Equations (1) and (3) were used to convert concentration to absorbance. For equation (7), particles in the same size group were assumed to have the same diameter.

Algorithms to numerical solve the system of ordinary differential equations associated with each model were developed using Python and the Scipy library. The initial value equations (equations (5)–(8) were solved using ODEPACK from Lawrence Livermore National Laboratory [26]. The initial conditions were set by assuming that no nanoparticles were present at the start of the simulation, and the initial concentration of insoluble gold atoms was set based on experimental measurements at various radiation doses. Finally, for both models, the unknown reaction rates (k₁ and k₂ for Finke-Wazky and k₁ and ${k}_{2}^{o}$ for the population balance model) were determined by minimizing the L₂-norm of the difference between experimental measurements and model prediction using a Nelder-Mead simplex algorithm.

The Finke-Watzky model prediction for the conversion of gold atoms into gold nanoparticles using optimal values for the reaction rates (${k}_{1,fw}$ and ${k}_{2,fw}$) so that the model predictions match the experimental data [11] are shown in figure 1. The experimental data are from measurements of absorbance after exposure to a radiation dose of 15 Gy. The initial gold salt concentration is 0.0109 mM and DLS measurements provided the particle diameter of 88.46 nm. Under these conditions, the optimal values for the kinetic parameters, ${k}_{1,fw},$ and ${k}_{2,fw},$ are 1 min⁻¹ and 3 × 10⁵ min⁻¹M⁻¹, respectively.

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Figure 2 shows how the same kinetic parameters that provided the best fit for the median dosage, 15 Gy, fit the model to experimental data at other radiation dosages. Unfortunately, this model does not predict the size distribution of the nanoparticles so comparisons with nanoparticle diameter measurements are not possible. Instead, experimental measurements of particle diameter were used to predict absorbance. The model predictions are least accurate for the small radiation dose cases where GNP formation is predicted by the model to occur more rapidly than is observed experimentally.

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The next set of results are based on the population balance model. Figure 3 shows the model prediction of absorbance versus time for a range of different radiation doses from 5 to 35 Gy. The kinetic parameters, k₁, and ${k}_{2}^{o},$ are 8 × 10⁻¹⁰ min⁻¹ and 3 × 10¹³ m⁴min⁻¹μM⁻¹, respectively, and n is set to 4. The experimental measurements of absorbance as a function of time are also shown for comparison to the model predictions.

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Unfortunately, the earliest experimental measurements of absorbance are 60 min after the radiation dose was delivered so the initial rate of change in absorbance cannot be compared between the model and the experimental data. The agreement between the model predictions and the experimental measurements is the strongest at radiation doses of 15 Gy or more, and the agreement is weaker at the 10 Gy and 5 Gy radiation doses. This discrepancy could be due to error in the measurement of the initial, reduced gold atom concentration, i.e., the initial concentration of Au⁰. The initial concentration is determined by measuring the gold salt concentration (i.e., the Au³⁺ concentration) before and after radiation exposure (i.e., measuring the Au³⁺ that was not converted to Au⁰ by radiation exposure). At the higher radiation doses (25 and 35 Gy), most of the gold salt was converted to Au⁰ and only a small quantity remained as soluble gold salt. Hence, the experimental error in measuring the initial Au⁰ concentration is most likely lower for the higher radiation doses because a large concentration change can be measured more accurately.

Figure 4 shows the model prediction for the final particle diameter distribution at different radiation doses. The vertical, dashed lines represent the experimental measurements of median particle diameter at the various radiation doses. The model predictions agree with the experimental observation that the mean particle size decreases as the radiation dose is increased. For higher radiation doses, there is a higher initial concentration of Au⁰ atoms and the formation of the smallest stable nanoparticle (i.e., nucleation) is more rapid. The numerous, small nanoparticles quickly grow, but all the Au⁰ in the solution is consumed before the particles are very large (i.e., when they are still only 35–40 nm). Conversely, at lower radiation doses, the smaller Au⁰ concentration results in less nucleation and fewer nanoparticles, overall. Even though the Au⁰ concentration is lower, the small number of nanoparticles allows the particles to grow much larger before the Au⁰ concentration decreases to zero.

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The agreement between the model and the experimental measurements is best at the higher radiation doses (15 and 25 Gy), but in these cases the model prediction is for larger particles than what was measured experimentally. One possible explanation for this is that the model assumes a maximum particle size that cannot be exceeded. If a few experimental particles exceed this maximum particle size (an event that almost certainly occurred), then less Au⁰ was available for particle growth in the experiment than in the corresponding model prediction, and, as a result, the mean experimental particle size would be slightly smaller than would otherwise be expected.

Another possible explanation for the larger particles predicted by the model is that the model assumed diffusion was limiting the particle growth rate so that the rate constant, ${k}_{i,2},$ was inversely related to the particle radius (equation (10)). This assumption was based on a 1-dimensional approximation of the diffusion limitation effects, and the true reduction in the particle growth rate as the particle radius increases could be greater that this assumption implies, thus leading to greater slowing of the growth of the largest particles and reducing the mean particle size predicted by the model.

One more explanation for the smaller particles observed experimentally is that the nucleation rate is faster than modeled, especially for the lowest radiation doses. The nucleation rate is proportional to A⁴, or the reduced gold atom concentration ${[A{u}^{0}]}^{4}.$ This implies that a halving of the initial Au⁰ concentration results in a nucleation rate that is 16-times lower. Using a non-integer exponent [34] could result in slightly better agreement between the model and the experimental data.

Figure 5 summarizes the effect of changing the nucleation rate, k₁. Halving the nucleation rate to 4 × 10⁻¹⁰ min⁻¹ decreases the initial slope of the absorbance predictions while increasing the particle sizes. As expected, slower nucleation leads to fewer initial nuclei being formed, leaving more precursor for growth. This results in a slower increase in absorbance and decreased particle concentrations for all radiation doses. The lower nucleation rate allows time for the particles to grow, creating fewer, larger particles. Doubling the nucleation rate to 1.6 × 10⁻⁹ min⁻¹ has the opposite effect. Faster nucleation leads to more nuclei, which results in a more rapid increase in absorbance and increased particle concentrations for all radiation doses. The production of extra nuclei consumes precursor and limits the growth of the particles, creating more but smaller particles.

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Figure 6 shows the effects of changing the growth rate, ${k}_{2}^{o}.$ Halving the growth rate to 1.5 × 10¹³ m⁴min⁻¹μM⁻¹ decreases the initial slope of the absorbance predictions while also decreasing the particle sizes. As expected, slower growth allows more initial nuclei to form, leaving less precursor for growth. This results in a slightly slower increase in absorbance and an increased particle concentrations for all radiation doses. The production of extra nuclei consumes precursor and limits the growth of the particles, creating more, smaller particles. Doubling the growth rate to 6 × 10¹³ m⁴min⁻¹μM⁻¹, has the opposite effects. Faster growth leads to fewer nuclei having a chance to form before the precursor is consumed. The result is a more rapid increase in absorbance, decreased particle concentrations, and larger particles for all radiation doses.

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The particle size distributions are a result of the nucleation and growth reactions competing for the precursor. The exponent on the precursor concentration in the nucleation reaction, n, increases the rate at which nucleation occurs, and more rapid nucleation generates more competition with the growth reaction for the limited supply of precursor. Figure 7, shows the effect of changing the precursor consumption power on the GNP size distribution.

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The model prediction with n = 1 always result in larger particles when the radiation dose is higher because the higher dose creates more precursor and the additional precursor eventually results in larger GNPs. This result directly contradicts what is observed experimentally. For the n = 2 and n = 3 cases, nucleation is more rapid, especially at the higher radiation doses, so we start to see the highest doses generating smaller particles because of precursor depletion, but the particle sizes at different radiation doses are still less variable than what was observed experimentally. The optimal agreement with the experimental measurements is n = 4, as shown in figure 4.

The original motivation for the development of the model presented here was to help identify key mechanisms controlling the formation of GNPs. It was hoped that this understanding could guide the development of better radiation dosage measurements systems. Comparing a lower dose case, e.g., 10 Gy, to a higher dose case, e.g., 25 Gy, we can observe that the GNPs resulting from a 25 Gy dose absorbed 4 times the amount of light relative to 10 Gy, but the number of particles at 25 Gy was 6 times the number at 10 Gy. Clearly, fewer, larger particles at 10 Gy provide a greater color change relative to the number of particles than at 25 Gy. Any change that slows down nucleation slightly so that particles can grow larger has the potential to improve system performance. Exploring alternative templating molecules and adjusting gold salt concentrations both have the potential to improve system performance at low radiation doses. Seeding the solution with stable nuclei so that more of the gold is captured in particle growth also has the potential to improve low dose performance.

Of the two different models that were developed, the Finke-Watzky model and the population balance model, only the later model had the ability to predict the final particle size distribution so it is the more useful model for understanding some of the key relationships between the various physical phenomena that control GNP synthesis. The accuracy of both models is limited, and neither model can predict absorbance over a large range of radiation doses. If the model parameters are fit based on a dosage of 15 Gy, for example, that model is only reasonable accurate from 10 Gy to 45 Gy. The model parameters can be obtained based on data at lower radiation doses, but then the model would be limited to a range of doses around that lower level. The same limitation applies to the experimental system. Depending on the precursor gold concentration, the system can measure radiation doses only over a limited range before absorbance changes are not measurable. Another limitation of the mathematical model is that the parameters are only appropriate for the radiation dosage rate (i.e., 15.6 Gy min⁻¹) and temperature of the experimental system. A small change in the radiation dosage rate is unlikely to have a large impact on the kinetics of color change because the nanoparticle growth rate is much slower (i.e., 10's of minutes) that the radiation time (i.e., a few seconds to deliver 1 or 2 Gy). Increasing the temperature will increase both the reaction kinetics and diffusion rate, but large temperature changes are unlikely as the experimental system is relatively small and will be maintained mostly at core body temperature.