The average magnetic anisotropy of polystyrene in polymersomes self-assembled from poly(ethylene glycol)-b-polystyrene

Using the diamagnetic anisotropy of polymers for the characterization of polymers and polymer aggregates is a relatively new approach in the field of soft-matter and polymer research. So far, a good and thorough quantitative description of these diamagnetic properties has been lacking. Using a simple equation that links the magnetic properties of an average polymer repeating unit to those of the polymer vesicle of any shape, we measured, using magnetic birefringence, the average diamagnetic anisotropy of a polystyrene (PS) repeating unit, ΔχPS, inside a poly(ethylene glycol)–polystyrene (PEG–PS) polymersome membrane as a function of the PS-length and as a function of the preparation method. All obtained values of ΔχPS have a negative sign which results in polymers tending to align perpendicular to an applied magnetic field. Combined, the same order of magnitude of ΔχPS (10−12 m3 mol−1) for all polymersome shapes proves that the individual polymers are organized similarly regardless of the PS length and polymersome shape. Furthermore, the value found is only a fraction (∼1%) of what it can maximally be due to the random coiling of the polymers. We, therefore, predict that further ordering of the polymers within the membrane could lead to similar responses at much lower magnetic fields, possibly obtainable with permanent magnets, which would be highly advantageous for practical applications.


Supporting Information 1: Derivation of surface parameter
In order to calculate the magnetic properties of a polymersome vesicle, it is necessary to transform the magnetic susceptibilities of every single polymer in the membrane (both  ∥  and  ⊥  ) from the polymer axes to the axes in which the whole vesicle will be described (see Supporting Figure 1).This transformation gives: with θN defined as: and  the angle describing the orientation around the vesicle's symmetry axis.
Supporting Figure 1: Schematic representation of a polymersome disc and its internal organization.The magnetic anisotropy of the vesicle as a whole, the individual polymers are indicated.The angle θN is indicated.Only the PS chain is visualized as it dominates the magnetic anisotropy of the PEG-PS polymer.
In a previous publication, 1 we formulated the following parametrization in which any cylinderically symmetric vesicle shape can be described: With  from 0 to π and u from 0 to 2π.
The magnetic anisotropy of the whole vesicle can be calculated by integrating the parametrization over the surface of the vescile.In terms of this parametrization this can be expressed as: with N the number of polymers in the vescile, A the surface area of the vesicle and () the Jacobian of the parametrization used: 1 Equation (S6) can be shortened to: since terms within the integral have no dependency in u due to their cylindrical symmetry.In general, the relation between Δ ves and Δ  can be written as: Since the magnetic anisotropy of a PEG-PS block copolymer is dominated by the PS units, as was demonstrated before, 1 we can describe Δ ves in terms of a the magnetic anisotropy of a single PS repeating unit, Δ ves , by: With  the number of PS units in a single PS polymer.The number of polymers in a vesicle N, can be written as function of surface area A, membrane thickness t, polystyrene density ρ PS and weight of a PS repeating unit M PS by: Combining equations (S9), (S10) and (S11) gives: With SP given by:  Supporting Table 2: Fitting parameters and surface parameter (Sv) for discs, obtained from the fittings as shown in Supporting Figure 4.

Supporting Figure 5: Cryo-TEM images showing the cross sections of the PEG44-b-PS178
stomatocytes.Every cross section has been fitted using the parameterization stated in ref 1.The fitting parameters and the corresponding surface parameter (Sv) calculated are given in Supporting Table 3. Supporting Table 3: Fitting parameters and surface parameter (Sv) for the stomatocytes, obtained from the fittings as shown in Supporting Figure 5.  Supporting Table 4: Fitting parameters and surface parameter (Sv) for tubess, obtained from the fittings as shown in Supporting Figure 6.

3 cos 2 (Supporting Figure 2 :Supporting Figure 3 :Supporting Table 1 :Supporting Figure 4 :
TEM image of the PEG44-b-PS178 discs.The whole sample consists of discs although there is some spread in the size of the discs.Scale bar is 1 µm.TEM image of the PEG44-b-PS178 stomatocytes.The whole sample consists of open stomatocytes although there is some small spread in the size and shape.Scale bar is 1 µm.Accuracy of the Δχ PS of the magnetic birefringence curves for PEG44-b-PS178 and PEG44-b-PS195.Cryo-SEM images showing PEG44-b-PS178 disc cross sections.Every cross section has been fitted using the parameterization stated in ref 1.The fitting parameters and the corresponding surface parameter (Sv) calculated are given in Supporting Table2.

Figure 6 :
Cryo-SEM images showing PEG44-b-PS195 tube cross sections.Every cross section has been fitted using the parameterization stated in ref 1.The fitting parameters and the corresponding surface parameter (Sv) calculated are given in Supporting Table 4.

Figure 7 :
Cryo-SEM images showing PEG44-b-PS195 ellipsoid cross sections.Every cross section has been fitted using the parameterization stated in ref 1.The fitting parameters and the corresponding surface parameter (Sv) calculated are given in Supporting Table 5.