Model-free description of polymer-coated gold nanoparticle dynamics in aqueous solutions obtained by Bayesian analysis of neutron spin echo data

We present a neutron spin echo study of the nanosecond dynamics of polyethylene glycol (PEG) functionalized nanosized gold particles dissolved in ${\mathrm{D}}_2\mathrm{O}$ at two temperatures and two different PEG molecular weights (400D and 2000D). The analysis of the neutron spin echo data was performed by applying a Bayesian approach to the description of time correlation function decays in terms of exponential terms, recently proved to be theoretically rigorous. This approach, which addresses in a direct way the fundamental issue of model choice in any dynamical analysis, provides here a guide to the most statistically supported way to follow the decay of the intermediate scattering functions $I(Q,t)$ by basing on statistical grounds the choice of the number of terms required for the description of the nanosecond dynamics of the studied systems. Then, the presented analysis avoids from the start resorting to a preselected framework and can be considered as model free. By comparing the results of PEG-coated nanoparticles with those obtained in PEG2000 solutions, we were able to disentangle the translational diffusion of the nanoparticles from the internal dynamics of the polymer grafted to them, and to show that the polymer corona relaxation follows a pure exponential decay in agreement with the behavior predicted by coarse grained molecular dynamics simulations and theoretical models. This methodology has one further advantage: in the presence of a complex dynamical scenario, $I(Q,t)$ is often described in terms of the Kohlrausch-Williams-Watts function that can implicitly represent a distribution of relaxation times. By choosing to describe the $I(Q,t)$ as a sum of exponential functions and with the support of the Bayesian approach, we can explicitly determine when a finer-structure analysis of the dynamical complexity of the system exists according to the available data without the risk of overparametrization. The approach presented here is an effective tool that can be used in general to provide an unbiased interpretation of neutron spin echo data or whenever spectroscopy techniques yield time relaxation data curves.

Figure 1

Sketch of the PEG400 and PEG2000 functionalized NPs. An indication of the characteristic sizes of the Au core ($R_{\text{Au}}$) and of the hydrodynamic radius $R_h$ are also shown.

Figure 2

(a) A representative series of NSE curves representing $I(Q,t)/I(Q,0)$ of the PEG2000 AuNPs dispersed in ${\mathrm{D}}_2\mathrm{O}$ at neutron wavelength $\lambda =10\AA$ and a temperature of 280 K (symbols). The $Q$ values increase from above to bottom. The lines represent the best fits to the data. (b) $Q$ dependence of the $A_1$ parameter obtained by the RJ-MCMC analysis and of the $I\left(Q\right)$ measured for a solution of PEG2000 AuNPs in ${\mathrm{D}}_2\mathrm{O}$ at the 1 wt.% concentration; the red diamond symbols correspond to a SANS measurement performed on the sample. (c) The two relaxation times ${\tau }_D$ and ${\tau }_{\mathrm{pol}}$ obtained by the fitting analysis at different temperatures. The full and empty symbols correspond to the values obtained at 10 and 16 Å. The red solid lines indicate the relaxation time obtained from the translational diffusion coefficient as derived by DLS at the respective temperatures.

Figure 3

(a) An example of the posterior distribution for the parameter $D$ for the PEG2000 AuNPs at 318 K, $\lambda =10\AA$, $Q=0.72{\AA }^{-1}$. The solid red curve is a Gaussian fit to the simulated posterior distribution. (b) Posterior distribution for the number of relaxation components for the same example in (a). (c) Trace plot for the relaxation time ${\tau }_D$ (bottom curve) and ${\tau }_{\mathrm{pol}}$ (upper curve). (d) Trace plot for the $A_1$ parameter.

Figure 4

Joint posterior distribution at four selected $Q$ values, of the two relaxation times, conditional on $k=2$ and marginalized with respect to the other model parameters for the PEG2000 AuNP sample at $T=318\mathrm{K}$ and $\lambda =10\AA$.

Figure 5

(a) Representative set of NSE curves measured on the PEG400 AuNPs at $\lambda =8\AA$ and $T=318$ K; the lines represent the best fits to the data. (b) Diffusion relaxation time as derived by the RJ-MCMC algorithm for the PEG400 AuNPs (circles) and for the PEG400 polymers in 5% solution in ${\mathrm{D}}_2\mathrm{O}$ (squares). The empty and solid symbols refer to $\lambda =8$ and 16 Å, respectively. The continuous red lines correspond to the relaxation times obtained from the translational diffusion coefficients as measured by DLS measurements for the PEG400 AuNPs. The dashed lines indicate the relaxation times obtained from the translational diffusion coefficient of PEG400 solution as obtained from literature [47]. (c) An example of the simulated posterior distribution for the diffusion coefficient obtained at $T=318$ K and $Q=0.048{\AA }^{-1}$.

Figure 6

(a) Representative set of NSE curves measured on the PEG2000 at $\lambda =8\AA$ and $T=280$ K. The lines represent the best fits to the data. (b) The two relaxation times ${\tau }_1$ and ${\tau }_2$ obtained by the fitting analysis at different temperatures.

Figure 7

NSE curves representing $I(Q,t)/I(Q,0)$ measured on the PEG2000 homopolymer 10% concentration, at 280 K (top panels) and 318 K (bottom panels) at three selected $Q$ values. For single relaxation ($Q=0.078$ ${\AA }^{-1}$, $Q=0.065$ ${\AA }^{-1}$) the lines are the best fit to the data. For multiple relaxations ($Q=0.013$ ${\AA }^{-1}$, $Q=0.019$ ${\AA }^{-1}$) the black lines fully overlapping to the data points are the best fit. We show also the relaxation components (red curves).

Figure 8

We provide as example the posterior distribution obtained for PEG2000 homopolymer at 280 K and at $Q=0.19{\AA }^{-1}$ for (a) the number $k$ of relaxation components (b) ${\tau }_1$ and (c) ${\tau }_2$.

Figure 9

NSE curves for the data collected with IN11 (blue filled symbols) and IN15 (green empty symbols) joined together on a PEG2000 homopolymer solution in ${\mathrm{D}}_2\mathrm{O}$ at 10% concentration, at four selected $Q$ values and $T=280$ K.

Figure 10

Characteristic relaxation times ${\tau }_1$ and ${\tau }_2$ for the homopolymer PEG2000 solution at 10% concentration and relaxation time of the polymer in the AuNP, ${\tau }_{\mathrm{pol}}$, at (a) $T=280$ K and (b) $T=318$ K.

Figure 11

The values of ${\mathrm{\Gamma }}_D=1/{\tau }_D$ and ${\mathrm{\Gamma }}_{\mathrm{pol},\mathrm{cor}}$ as a function of $Q^2$ obtained for PEG2000 AuNP dispersed in ${\mathrm{D}}_2\mathrm{O}$ at the temperatures $T=280$ K (a), $T=318$ K (b).

Figure 12

Ratio between the two relaxation times estimated by the RJ-MCMC algorithm for the PEG2000 solution, at the two investigated temperatures.