Synthesis and electrokinetics of cationic spherical nanoparticles in salt-free non-polar media

The electrokinetics of cationic sterically-stabilized diblock copolymer nanoparticles prepared in salt-free non-polar media depend on whether the charge is located in the stabilizer shell or in the nanoparticle core.


SAXS model
The scattering cross section per unit volume from a dispersion of nanoparticles (dΣ/dΩ(Q)), equivalently referred to as the intensity (I(Q)), can be given by the following expression, where P (Q) is the form factor, S(Q) is the structure factor, n is the number density, φ is the volume fraction, and V t is the volume of the object.
is related to the geometry of the scattering object. For composite objects, such as polymer micelles, P (Q) depends on the scattering length density (SLD, ρ) difference between parts of the system as well as their volume, so is a function of both. S(Q) accounts for deviations from random scattering, which only become appreciable for either concentrated dispersions or for strongly interacting species. The SAXS data reported here are for dilute dispersions, and in this case, S(Q) = 1.
The spherical nanoparticles studied here are diblock copolymer micelles, and the form factors of these have previously been reported in the literature. 1,2 The form factor (P m ) of a spherical diblock copolymer micelle consists of four terms: two self-terms (for the spherical core, P s , and the chains on the surface, P c ) and two cross-terms (between the core and the chains, S sc , and between different chains on the surface, S cc ). The self-term for the chains in the corona given by the Debye function, assuming that they are Gaussian chains with a radius of gyration R g . 5 To mimic non-penetration of the Gaussian chains, they are set as starting a distance dR g away from the surface of the core, where d ≈ 1. The cross-term between core and chains is given by the following expression.
The functions Φ(x) and ψ(x) are given below.
The interference term between chains in the corona is given by the following expression.
Two modifications have been to this standard model for spherical diblock copolymer micelles. First, A sigmoidal interface was assumed to account for a varying scattering length density at the micellar interface. The interface width σ was set to 2.5. This modified the interface by the term exp (−Q 2 σ 2 /2). Next, a radial profile was used to define scattering in the micelle corona using a linear combination of two cubic splines with fitting parameters cor-responding to the width and weight coefficient. Further information on these modifications can be found elsewhere. [6][7][8] The core radius of the micelles was fit with a Gaussian distribution with standard deviation σ G , given in the expression below.
SAXS data fitting SAXS data were fit to models as explained in the previous sections. The core radius was allowed to vary, the stabilizer R g was fixed, and the volumes and SLDs of the two blocks were fixed from the known mass densities of the materials.
The radii of gyration for the two PSMA stabilizers were calculated using an approach described by Derry et al. 8 The contour length of a PSMA or PMOTMA monomer is set to 2.55 Å (two C-C bonds in all-trans conformation), and the total contour length is the product of this length and the DP. The Kuhn length is considered to be equal to that of The volume of the polymer blocks are calculated from the mass density and the molar mass of the monomer unit, which is then multiplied by the DP of the unit to give a total block volume. The mass densities have been taking from the literature. For PSMA and PBzMA, these are available experimentally. 8,9 For PMOTMA + , only the monomer density is available, and the polymer density was by scaling this by the ratio of the density of PMMA to MMA. 10 For TFPhB − , the molecular volume was calculated by Hartree-Fock theory. 11 For n-dodecane, the mass density is available in the literature. 12 For the ionic species, the volumes are calculated assuming complete ion binding, as the dissociation constant for ions in non-polar solvents is extremely low.  To calculate the scattering length densities, both the coherent scattering length (b i ) and the molar volumes must be known.
For X-rays, as scattering arises from the interaction between X-rays and the atomic electron cloud, b i is related to the atomic number. At X-ray energies away from absorption edges, the atomic scattering factor f 1 is well approximated by the atomic number. In this case, b i is equal to the product of the atomic number and the classical electron radius (r e ). 13,14  To fit the SAXS data, the molecular volume of the two blocks (Table S10), the scattering length densities (Table S11), and the PSMA R g were fixed initially. The only modification to the these calculated values was the SLD of the stabilizer for the charged shell nanoparticles, which was allowed to vary and fit to a value slightly below the calculated value. This is likely due to the small fraction of anions that must be dissociated from the particles to make them electrophoretic. The best fit values are shown in Table S12 below along with the fit scattering curves in Figure S1.   Figure S1: SAXS data for uncharged, charged shell, and charged core diblock copolymer nanoparticles. Experimental data has been fit to a diblock copolymer micelle model as explained in the text.