Size and shape fluctuations of ultrasoft colloids

An ultrasoft colloidal particle fluctuates due to its flexibility. Such fluctuation is essential for colloidal structure and dynamics, but is challenging to quantify experimentally. We use dendrimers as a model system to study the fluctuation of ultrasoft colloids. By considering the dynamic polydispersity in the small-angle neutron scattering (SANS) model and introducing the fluctuation of invasive water into the contrast in SANS, we reveal the fluctuating amplitudes of the size and shape of the dendrimer of generation 6 at finite concentrations. The size fluctuation is suppressed while the shape fluctuation increases as the weight fraction of dendrimers passes 11%. With neutron spin echo data, we suggest that such a crossover originates from the competition between the inter- and intraparticle dynamics. Further investigation on lower-generation samples shows a contrary result, which suggests a structural basis for these dynamic phenomena.

Figure 1

Illustration of the molecular structure of the G4-OH PAMAM dendrimer [42].

Figure 2

SANS analyses of the G6 PAMAM dendrimers dissolved in ${\mathrm{D}}_2\mathrm{O}$ at $c=1$, 5, 10, 12.5, 15, and 20 wt.%. Symbols in (a) and (b) denote the measured spectra (they are vertically shifted for better visibility). (a) Fits with the size fluctuation model and the model with both the size and shape fluctuations. (b) Fits with the monodisperse fuzzy-ball model and the model only considering the shape fluctuation. These two models, without explicitly considering the size fluctuation, underestimate the low-$Q$ intensity at $c\ge 10$ wt.%. (c) Size fluctuation ${\sigma }_R$ of the G6 dendrimer as a function of $c$. The results are obtained with the size fluctuation model and the model with both the size and shape fluctuations. (d) Shape fluctuation ${\sigma }_{\epsilon }$ of the G6 dendrimer as a function of $c$. More details of the SANS fitting can be found in the Supplemental Material [54].

Figure 3

Scattering contrast $A$ as a function of the average scattering length of solvent molecule $b_{\mathrm{w}}$ at (a) $c\le 10$ wt.% and (b) $c>10$ wt.%. Symbols denote the experimental results. Solid and dashed lines denote the fits using Eq. (6) (with size fluctuation) and Eq. (7) (without size fluctuation), respectively. The data are shifted vertically for better visibility. More details of the calculation can be found in the Supplemental Material [54].

Figure 4

(a) Fluctuations of volume (${\sigma }_V/V_{\mathrm{p}}$) and invasive water ($\alpha /N\propto {\sigma }_N/N$) of a G6 dendrimer as a function of $c$. (b) Radius of the G6 dendrimer as a function of $c$. Here $R_{\mathrm{fit}}$ is obtained by fitting with the size fluctuation model and $R_{\mathrm{eff}}$ is the effective radius defined by the condition of compressibility equivalence between the soft particle and the hard sphere [61]. (c) Interparticle collisional time ${\tau }_{\mathrm{inter}}$ and intraparticle relaxation time ${\tau }_{\mathrm{intra}}$ of G6 dendrimers as a function of $c$ [62]. The dashed line marks the crossover concentration $c^{*}$.

Figure 5

(a) SANS analysis of G4 PAMAM dendrimers dissolved in ${\mathrm{D}}_2\mathrm{O}$ at $c=1$, 5, 10, 12.5, 15, and 20 wt.%. Symbols denote the measured spectra. Solid and dashed lines denote the fits using the monodisperse fuzzy-ball model and the size fluctuation model, respectively. (b) Scattering contrast of G4 dendrimers as a function of $b_{\mathrm{w}}$. Symbols denote the experimental results. Solid and dashed lines denote the fits using Eqs. (6) and (7), respectively. Data are vertically shifted for better visibility.