Determining the Effective Density and Stabilizer Layer Thickness of Sterically Stabilized Nanoparticles

A series of model sterically stabilized diblock copolymer nanoparticles has been designed to aid the development of analytical protocols in order to determine two key parameters: the effective particle density and the steric stabilizer layer thickness. The former parameter is essential for high resolution particle size analysis based on analytical (ultra)centrifugation techniques (e.g., disk centrifuge photosedimentometry, DCP), whereas the latter parameter is of fundamental importance in determining the effectiveness of steric stabilization as a colloid stability mechanism. The diblock copolymer nanoparticles were prepared via polymerization-induced self-assembly (PISA) using RAFT aqueous emulsion polymerization: this approach affords relatively narrow particle size distributions and enables the mean particle diameter and the stabilizer layer thickness to be adjusted independently via systematic variation of the mean degree of polymerization of the hydrophobic and hydrophilic blocks, respectively. The hydrophobic core-forming block was poly(2,2,2-trifluoroethyl methacrylate) [PTFEMA], which was selected for its relatively high density. The hydrophilic stabilizer block was poly(glycerol monomethacrylate) [PGMA], which is a well-known non-ionic polymer that remains water-soluble over a wide range of temperatures. Four series of PGMAx–PTFEMAy nanoparticles were prepared (x = 28, 43, 63, and 98, y = 100–1400) and characterized via transmission electron microscopy (TEM), dynamic light scattering (DLS), and small-angle X-ray scattering (SAXS). It was found that the degree of polymerization of both the PGMA stabilizer and core-forming PTFEMA had a strong influence on the mean particle diameter, which ranged from 20 to 250 nm. Furthermore, SAXS was used to determine radii of gyration of 1.46 to 2.69 nm for the solvated PGMA stabilizer blocks. Thus, the mean effective density of these sterically stabilized particles was calculated and determined to lie between 1.19 g cm–3 for the smaller particles and 1.41 g cm–3 for the larger particles; these values are significantly lower than the solid-state density of PTFEMA (1.47 g cm–3). Since analytical centrifugation requires the density difference between the particles and the aqueous phase, determining the effective particle density is clearly vital for obtaining reliable particle size distributions. Furthermore, selected DCP data were recalculated by taking into account the inherent density distribution superimposed on the particle size distribution. Consequently, the true particle size distributions were found to be somewhat narrower than those calculated using an erroneous single density value, with smaller particles being particularly sensitive to this artifact.

. DMF GPC traces for a PGMA 63 macro-CTA and PGMA 63 -PTFEMA y diblock copolymers analysed using (A) a RI detector (with an apparent low molecular weight shoulder) and (B) using UV detection at λ max 305 nm (with no low molecular weight shoulder). Figure S3. TEM images recorded for various PGMA x -PTFEMA y diblock copolymer nanoparticles prepared by RAFT aqueous emulsion polymerization of TFEMA using a PGMA x macro-CTA at 20 % w/w solids. Figure S4. Small-angle X-ray scattering patterns (red circles) obtained at 1.0 % w/w copolymer concentration and 20 °C for four series of PGMA x -PTFEMA y spherical nanoparticles with x = (A) 28, (B) 43, (C) 63 and (D) 98. Solid black lines represent fits to the data using a spherical micelle model (see section C). Figure S5. Weight-average particle size distributions determined by disk centrifuge photosedimentometry for; (A) G 28 -F y , (B) G 43 -F y , (C) G 63 -F y , (D) G 98 -F y nanoparticles. The single particle densities used to determine the size distributions are indicated on the graphs and were calculated as described in the text. Figure S6. Weight-average particle size distributions determined by disk centrifuge photosedimentometry for; G 63 -F 184 (blue traces), G 63 -F 430 (green traces), G 63 -F 615 (purple traces) and G 63 -F 1106 (red traces) particles. The blue dotted line shows a representative example of an erroneous size distribution obtained for G 63 -F 184 nanoparticles when an upper limit density of 1.47 g cm -3 (corresponding to the solid-state density of dry PTFEMA homopolymer) is used for analysis. Solid lines show particle size distributions obtained when a single effective particle density is used. Dashed traces indicate recalculated particle size distributions that account for an effective density distribution superimposed on the particle size distribution (see Table 2). Table S1. Parameters obtained for PGMA x polymer chains from theoretical calculations and Gaussian coil fits of scattering data (see Figure 6 and section C).  Table S2. Summary of all SAXS fitting parameters used for data presented in Figure 6; ϕ = volume fraction of the copolymer, R core = radius of the spherical core, σ Rcore = standard deviation of the core radius, R g = radius of gyration of the corona, V brush = calculated volume of the corona block chain, V chain = calculated volume of the core block chain, N agg = aggregation number. Data was fitted using a mean value for the solvation of the PTFEMA core (x sol ) of 0.05, with X-ray scattering length densities of ξ PTFEMAy = 12.76 x 10 10 cm -2 , ξ PGMAx = 11.94 ×10 10 cm -2 and ξ H2O = 9.42 × 10 10 cm -2 . where dΣ/dΩ(q) is the scattering cross-section per unit sample volume, ϕ is the volume fraction of polymer in solution, ∆ξ is the excess contrast scattering length density of the copolymer, and V mol is the total volume of the molecule. The form factor for a Gaussian polymer chain is:  Figure 6). The absolute SAXS intensity scale provides an opportunity for estimation of the degree of solvation. If the polymer chains are fully dissolved in water, then the volume of a scattering object should be equal to the volume of a molecule V mol = 5.7, 8.7, 12.8, 19.9 nm 3 for the four PGMA x chains studied here. According to equation (C1) the SAXS intensity at low q should be approximately 0.03, 0.04, 0.06 and 0.10 cm -1 for the four PGMA x macro-CTAs studied, (from the shortest to longest).

SAXS model for PGMA x -PTFEMA y spherical micelles
In general, the X-ray intensity scattered by a dispersion of spherical micelles [usually represented by the scattering cross-section per unit sample volume, dΣ/dΩ(q)] can be expressed as a product of their form factor, F s-mic (q,r) and the structure factor, S s-mic (q): It is also assumed in this expression that the micelle core radius, r, is polydisperse, where Ψሺ‫ݎ‬ሻ is the distribution function and N is the number density per unit volume. The spherical micelle form factor used in equation (C3) is given by: where the core block and the corona (stabilizer) block X-ray scattering length contrast is given by β c = V chain (ξ c -ξ sol ) and β s = V brush (ξ s -ξ sol ) respectively. Here ξ c , ξ s , and ξ sol are the X-ray scattering length densities of the core block, ξ PTFEMA = 12.76 x 10 10 cm -2 , the corona block, ξ PGMA = 11.94 ×10 10 cm -2 , and the solvent, ξ H 2 O = 9.42 × 10 10 cm -2 , respectively. V chain and V brush are volumes of the core and the corona block. The volumes were obtained from ܸ = ெ ౭ ே ఽ ఘ , using homopolymer densities determined by helium pycnometry (ρ PTFEMA = 1.47 g cm -3 and ρ PGMA = 1.31 g cm -3 ). The sphere form factor amplitude is used for the core self-term, The self-correlation term of the corona block is given by the Debeye function The contribution of the coronal blocks to the scattering signal is comparable to that from the micelle core for the micelles formed by copolymers with small DP [e.g., for PGMA 28 -PTFEMA 100 , ሺߚ ୱ / ߚ ୡ ሻ ଶ ≈ 0.05]. Thus, a rigorous corona scattering length density radial profile was used for SAXS data fitting: The radial profile, µ s (l), is expressed via a linear combination of two cubic b splines, with two fitting parameters s and a corresponding to the width of the profile and the weight coefficient, respectively. This information can be found elsewhere, 2, 3 as can the approximate integrated form of equation (C7). A sharp interface (i.e. no sigmoidal interface) between the two blocks was assumed for the micelle form factor [equation (C4)]. In addition, it is assumed that there is no penetration of the coronal stabilizer chains within the micelle cores. A Gaussian distribution is used for equation (C3) to describe the polydispersity for the micelle core radius: where R core is the average micelle core radius and σ Rcore is the standard deviation for R core . The aggregation number, N agg , for the spherical micelles is given by The number density per unit volume is expressed as: where φ is the total volume fraction of copolymer in the spherical micelles and V(r) is the total volume of copolymer in a spherical micelle ]. An effective structure factor expression previously proposed for interacting spherical micelles 4 is used in order to describe possible aggregation of the spherical micelles: Herein the form factor of the average radial scattering length density distribution of micelles is ‫ܣ‬ ୫୧ୡ ሺ‫,ݍ‬ ‫ݎ‬ሻ = ܰ ୟ [ߚ ୡ ‫ܣ‬ ୡ ሺ‫,ݍ‬ ‫ݎ‬ሻ + ߚ ୱ ‫ܣ‬ ୱ ሺ‫,ݍ‬ ‫ݎ‬ሻ] and S PY (q, R PY , f PY ) is a hard-sphere interaction structure factor based on the Percus-Yevick approximation, 5 where R PY is the interaction radius (R PY = r +∆R PY , ∆R PY is a fitting parameter) and f PY is the hard-sphere volume fraction. A numerical integration was used for equations (C3), (C7) and (C9) during the fitting.
Section D. Derivation of cubic equation for recalculating DCP particle size distributions

Derivation of cubic equation for density distribution correction
We have previously shown a method to correct disk centrifuge photosedimentometry particle size distributions for polymeric core/particulate shell nanocomposite particles which have a density distribution superimposed on the particle size distribution. 2 A similar approach can be used to derive a correction to the density profile for particles with a core-shell morphology in which the core is of variable diameter and the shell is of fixed thickness.
The situation which the model is designed to treat is represented schematically as: Here the shell material is less dense than the core, thus the particle density increases as a function of core diameter.
Let the core radius be R core and the shell thickness T shell . The densities of core, shell and spin fluid are ρ core , ρ shell and ρ fluid , with density differences ∆ c = ρ core −ρ fluid and ∆ s = ρ shell −ρ fluid , respectively. For particles of uniform density ρ moving according to Stokes' Law through a spin fluid of density ρ fluid , the relationship between the time of detection (t) and the apparent diameter (D t ) of the particles detected at that time is given by where ∆ 0 is the density difference (ρ −ρ fluid ), and C is a constant determined by the viscosity of the fluid, the spin speed and the cell geometry. In the usual mode of operation, this constant is determined by comparison with a reference sample of known diameter, then a suitable value for the particle density ρ is selected, and the instrument software computes D t as a function of time.
If the particles do not all have the same density, an average particle density can be used in (D1), but then the predicted diameter D t will in general differ from the true particle diameter, D p . For particles with a core-shell morphology, such as the sterically-stabilized particles studied here, the true diameter for particles detected at time t is given by It is useful to refer all distances to the fixed length T shell , and so a dimensionless radius variable r is defined as With these definitions, the density of the core-shell particle is 3 3 3 shell and, if we define ∆ p = ρ particle −ρ fluid , the relation between detection time and true diameter is The expanded form of (D4) in which (D3) is used for particle ρ yields a cubic equation in the dimensionless variable: 0 ) ( where the coefficients are: This is a cubic equation with real coefficients, and hence has either one or three real roots. , the cubic has either three negative real roots, or one negative real and two complex roots, but in either case there is no physically acceptable solution for r. However, physical solutions are available whenever d is negative, i.e., when ).
If d is negative, then there is one positive real root and either two negative real roots or a pair of complex roots. Hence (D5) has a unique physical solution (r is both real and positive) for , regardless of the sign of c.
The uniqueness condition is fulfilled for example by the thin-shelled polystyrene/silica core/shell particles of Retsch et al. 7 where the reported core diameters are in the range 300 -650 nm, shell thickness is 14 nm and densities of core and shell are 1.05 and 1.7 g cm -3 , respectively. Furthermore, for sterically-stabilized PGMA x -PTFEMA y particles with core diameters between 30 -150 nm, shell thicknesses of 2.7 -4.0 nm, ρ core = 1.47 g cm -3 and ρ fluid 1.02 g cm -3 , numerical investigation shows that particles with these parameters also correspond to physically realistic solutions of (D5).
Cubic equations of course have analytical solutions in terms of radicals, and Cardano's Formula 6 gives formal expressions for these, which yield all the real and complex solutions. For flexibility in practical computations, it may be easier to use a Newton-Raphson procedure with a range of positive starting guesses, than to pick out the desired real solution from formal expressions that may require complex arithmetic for their evaluation. This brute-force approach, which uses trivial amounts of computer time, was adopted here.
Once a value of D p has been obtained from each D t , the curve for the scattering cross-section as a function of diameter, Q net (D), can be used to convert the measured absorption vs. time profile into a corrected particle size distribution. The scattering cross section, Q net (D), is determined by the optical properties of the particles. In the present case, the function used by the CPS instrument software for PTFEMA latex particles was taken to apply also to the sterically-stabilized particles, and was found to be well represented by a cubic polynomial fitting function. Another approach would be to use the function for PGMA provided by the instrument software. In fact, this choice makes no discernible difference to the calculated size distribution, at least in this particular case.
As in our previous work, 8 a simple FORTRAN77 programme was written based on the method described above in order to recalculate the true weight-average particle size distributions for a given set of PGMA x -PTFEMA y data obtained by DCP, see below.