Micelle Structure Details and Stabilities of Cyclic Block Copolymer Amphiphile and Its Linear Analogues

In this study, we investigate structures and stabilities of the micelles of a cyclic amphiphile (c-PBA-b-PEO) composed of poly(n-butyl acrylate) (PBA) and poly(ethylene oxide) (PEO) blocks and its linear diblock and triblock analogues (l-PBA-b-PEO and l-PBA-b-PEO-b-PBA) by using synchrotron X-ray scattering and quantitative data analysis. The comprehensive scattering analysis gives details and insights to the micellar architecture through structural parameters. Furthermore, this analysis provides direct clues for structural stabilities in micelles, which can be used as a good guideline to design highly stable micelles. Interestingly, in water, all topological polymers are found to form ellipsoidal micelles rather than spherical micelles; more interestingly, the cyclic polymer and its linear triblock analog make oblate-ellipsoidal micelles while the linear diblock analog makes a prolate-ellipsoidal micelle. The analysis results collectively inform that the cyclic topology enables more compact micelle formation as well as provides a positive impact on the micellar structural integrity.

was stirred at 90 C for 3 h, cooled with liquid nitrogen, and allowed to warm to 20 C.
From the 1 H NMR spectrum, the number of repeat unit in the PBA block was determined from the integrations of the BA unit' methyl peak (-CO 2 CH 2 CH 2 CH 2 CH 3 ) and the linker unit' dimethyl peak (-C(CH 3 ) 2 -), while the number of repeat unit in the PEO block was estimated from the integrations of the EO unit' methylene peak (-OCH 2 CH 2 O-) and the linker unit' dimethyl peak (-C(CH 3 ) 2 -). This analysis gave the composition and molecular weight of the linear block copolymer product: [CH 3 O(EO) 126.2 CO][C(CH 3 ) 2 (BA) 8.8 CH 2 CH=CH 2 ] and n,NMR = [5620][1210]. The polymer product was further analyzed using a gel permeation chromatography (GPC) system (Shodex GPC-101), which was equipped with a refractive index detector (Shodex RI-71S) and two columns (Shodex GPC KF-804L). The GPC system was calibrated with polystyrene standards. Tetrahydrofuran (THF) was used as an eluent at a flow rate of 1.0 mL/min at 40 °C. The linear diblock copolymer product was determined to have a number-average molecular weight n,GPC of 5950 and a polydispersity index (PDI) of 1.07.
An amphiphilic linear triblock copolymer, poly(n-butyl acrylate)-b-poly(ethylene oxide)-b- An amphiphilic cyclic diblock copolymer, poly(n-butyl acrylate)-b-poly(ethylene oxide) (cyclic PBA-b-PEO) was prepared from some of the above linear triblock block copolymer product; here, the linear PBA-b-PEO-b-PBA having two allyl end groups was used as the precursor. The second-generation Hoveyda-Grubbs catalyst (87 mg, 0.14 mmol) was added to a toluene solution In general, the scattering intensity I(q) of particles in a solvent medium is expressed by the where K x is a constant, N p is the number of particles, P(q) is the form factor of the particle, and S(q) is the structure factor for the particles. q is the magnitude of the scattering vector defined as q = (4π/λ)sinθ in which 2θ is the scattering angle and λ is the wavelength of the X-ray beam used. The P(q) term contains the information of the shape and the size of the particle while the S(q) term describes the interparticle distance as the following: 2,3 where r l is the position of particle l. The first term of the structure factor equals 1 due to the perfect positional correlation stemming from the particle l itself. The second term is the interference function between particles.
For general solution scattering cases in dilute conditions, the interparticle distance (r l -r l' ) become random and the interference function reaches 0, Then, thereby allowing the approximation of the S(q) term as unity over the entire q range. This approximation allows the isolation of P(q) from the overall scattering intensity I(q). Therefore, equation 1 can be rewritten as: (5) Dilute conditions, however, present the problem of having considerably low intensity in the high q region as opposed to the low q region, which is detected in much higher intensity. This problem causes the intensity in the high q region to be obscured by the background noise, producing significant errors in analysis of the scattering data. As a way of overcoming such low intensity, semidilute conditions are often used with the assumption that the interparticle interaction is low enough where S(q) ≈ 1 over the entire q range. In this study, the X-ray scattering measurements were conducted with polymeric micelle solutions with concentration ranging from 1.0 to 5.0 mg/mL to obtain high quality scattering data with negligible S(q).

Guinier analysis. The Guinier analysis is a model independent method based on the law of
Guinier developed by André Guinier, which is expressed as the following equation where I o (q) is the incident beam intensity. According to the law of Guinier, the radius of gyration R g,G of particles in solution can be approximated from the low q region of the scattering data. This approximation, however, must satisfy the two boundaries conditions: the particles must be globular and the maximum qR g,G must be less than 1.33. Within these boundary conditions, the obtained scattering data can be plotted into a log I(q) vs. q 2 curve, in which the resulting slope of the curve m is used to yield the approximate R g,G of particles (i.e., micelles in this study) by the following expression: Indirect Fourier transformation analysis. The indirect Fourier transformation (IFT) analysis is another model independent method that could provide more structural information than the Guinier analysis. Parameters such as radius of gyration R g can be obtained through the IFT method, 4,5 which transforms the scattering data to its real space analogue, the pair distance distribution function p(r).
The pair distance distribution function describes the probability of finding two scatterers separated by a distance r inside the particle (i.e., micelle). This data transformation bypasses the utilization of a specific parameterized model and directly derives the structural information from the scattering data. The scattering intensity can be expressed by the Fourier transformation of p(r): 4,5 r d qr in which an estimate of R g,IFT can be derived based on p(r) as follows: In addition to the radius of gyration, the IFT method can offer the radial electron density distribution function ρ(r), which is related to the pair distance distribution function in the following: 4,5 where r is the distance from the center of the micelle.
Spherical core-fuzzy shell model. The spherical core-fuzzy shell (CFS) model is based on a spherical particle composed of two phases, namely a dense core and a fuzzy shell. 4,6 For the spherical CFS particle, the P(q) term in equation 5 can be expressed as a sum of two terms: 4,6 dr qr where the first term, P shape (q), is the form factor of the overall spherical particle composed of a dense core and a fuzzy shell, and the second term (i.e., P blob (q)) is the contribution from the density fluctuations that occur within the fuzzy shell. Here, the second term describes the Fourier transform of the correlation function γ(r) of density fluctuations on length scales (r) smaller than the blob radius,  is the amplitude of the blob scattering contribution, and  is the average correlation length. γ(r) is equal to zero for r >  , but not equal to zero for r ≤  . For r ≤ , γ(r) can be expressed by where  is the Flory-Huggins parameter, which equals 3/5 for the good solvent condition, 1/2 for the Θ solvent condition, and 2/3 in the case that the molecules are stretched. 4,6 In light of fuzzy-sphere, 6 P shape (q) for the core-fuzzy shell can be replaced by the following equation: where A CFS is the scattering amplitude of the core-shell structure with fuzziness, r c,CFS is the radius of the dense spherical core, r CFS is the outer radius of core-fuzzy shell (i.e., total radius of the micelle), which is related to the shell thickness by t f,CFS = r CFS -r c,CFS . 2σ f,CFS is a measure of the width of the soft region in the shell part, and A CFS is the scattering amplitude of the core-shell structure with sharp interface and uniform density distribution in core and shell, respectively. Further, the fuzziness of the shell part is defined by the ratio of σ f,CFS and t f,CFS , because only shell region is fuzzy.
The scattering amplitude of the core-shell is given by the following equation :  where P core is the self-correlation term of the core, A(q,r c,SCM ) is the scattering amplitude of spherical core, and r c,SCM is the radius of core.
Two phase ellipsoid model. In this study, we have developed a new model dependent method as follows. This proposed structural model is ellipsoidal in shape and is composed of two phases: (i) a dense core and (ii) a corona. For this two layer ellipsoid particle, the P(q) in equation 5 can be expressed by the sum of two terms in the following equation: P shape (q) describes an ellipsoidal particle consisting of a core and a corona. The second term P blob (q) describes the density fluctuations smaller than the blob radius within the corona.
In P shape (q), the form factor describes a spheroid shape with a pair of equal semi-axes (R, R) and a distinct third semi-axis (εR) (i.e., an ellipsoid of gyration). The ellipsoidicity ratio ε in this model is defined as the ratio between the polar axis (R p ) and the two equatorial axes (R e ): For a single spheroid with the equatorial radius of R (i.e., R e ), the form factor is described where F[q,r(R e ,ε,)] is the scattering amplitude and  is the modular angle of spheroid. The scattering amplitude is given as a function of the scattering vector q, and r(R e ,ε,): In the case of two layer ellipsoid model, the overall form factor can be expressed as thusly: Additionally, each scattering amplitude term include an interface term that functions to describe the gradual decay of polymeric micelle's density along the radial direction. This is especially prevalent in the corona due to the solvation but, because polymeric micelles are soft particles, the core-corona interface also possesses a sizeable region with decaying density. As a result, the regions of fuzzy, nonideal interface could be described with scattering as the following equation: where 2t f is the width (i.e., thickness) of the fuzzy interface in the core-corona interface and the corona-solvent interface.
The second term of the overall form factor, P blob (q), is expressed as the following:  10 This structural model is also ellipsoidal in shape and is composed of three phases: (i) a dense core, (ii) a dense corona, and (iii) a solvated corona. For this three layer ellipsoid particle, the P(q) in equation 5 could expressed as identical as equation 18. In this case, however, the first term P shape (q) describes an ellipsoidal particle consisting of three layer phases, namely a core, a dense corona, and a solvated corona. The core is unpenetrated by solvent molecules and the corona is divided into two regions of different densities depending on the level of solvent penetration. The second term P blob (q) describes the density fluctuations smaller than the blob radius within the corona regions.
In P shape (q), the form factor takes a spheroid shape with a pair of equal semi-axes (R, R) and a distinct third semi-axis (εR). The ellipsoidicity ratio ε (the ratio between the polar axis (R p ) and the two equatorial axes (R e )) in this model is defined in equation 19. The mathematical description of a spheroid with the equatorial radius of R (i.e., R e ) is defined in equations 20-22.
In the case of three layer ellipsoid model, the overall form factor can be expressed as thusly: Additionally, each scattering amplitude term include an interface term that functions to describe the gradual decay of polymeric micelle's density along the radial direction. The soft material characteristic of polymer micelles give rise to a fuzzy, nonideal interface rather than a sharp interface between layers. This is in effect at the core-dense corona interface as well as at the dense corona-solvated corona interface and the solvated corona-solvent interface. As a result, the scattering amplitude term for each phase can be rewritten in the same manner as equation 24 where 2t f is the width (i.e., thickness) of the corresponding fuzzy interface.
Moreover, the core radius, dense corona thickness, and solvated corona thickness within P shape (q) are assumed to possess Gaussian distribution n(A): where A corresponds to the core radius or the dense corona thickness or the solvated corona thickness, A is the mean value of the structural parameter A, and σ A is the standard deviation of A from A . Therefore, equation 24 can be rewritten as: The second term of the overall form factor, P blob (q), is expressed in equation 27, where r ≤ , γ(r) can be defined by equation 12; the  in γ(r) is defined by equation 13.