Soft Character of Star-Like Polymer Melts: From Linear-Like Chains to Impenetrable Nanoparticles

The importance of microscopic details in the description of the behavior of polymeric nanostructured systems, such as hairy nanoparticles, has been lately discussed via experimental and theoretical approaches. Here we focus on star polymers, which represent well-defined soft nano-objects. By means of atomistic molecular dynamics simulations, we provide a quantitative study about the effect of chemistry on the penetrability of star polymers in a melt, which cannot be considered by generic coarse-grained models. The “effective softness” estimated for two dissimilar polymers is confronted with available literature data. A consistent picture about the star penetrability can be drawn when the star internal packing is taken into consideration besides the number and the length of the star arms. These findings, together with the recently introduced two-layer model, represent a step forward in providing a fundamental understanding of the soft character of stars and guiding their design toward advanced applications, such as in all-polymer nanocomposites.


I. PREPARATION AND EQUILIBRATION OF THE MODEL STAR POLYMERS IN MELT
The preparation of the small model PS and PEO star systems in melt, which serve as building units for the systems studied here, was described in detail elsewhere. [1] Here we only briefly describe the methodology depicted in Fig.1(a,b,c) of the main manuscript. The star-shaped topology was prepared by attaching fully stretched chains of 40 monomers consisting of either styrene or ethylene oxide monomers into a central dendritic structure. The central dendritic structure, also called kernel, is built of C, CH and CH 2 units and it is made of 1 to 3 generations depending on the functionality (number of arms, f ) of the prepared star. Overlaps and local tension, that can occur by forming such a topology, were eliminated by a serie of short runs consisting of energy minimization and molecular dynamics with a low time step of 0.1 fs. Then, the stars were randomly placed in a simulation box. The equilibration was performed by combining NPT runs and runs with stochastic dynamics at higher temperatures, as described in detail in ref. [1]. The production runs of 100 ns were performed at the temperature of 600K for (PS) f and 450K for (PEO) f , employing Nosé-Hoover thermostat and Parrinello-Rahman barostat at a constant pressure of 1 atm. The same procedure was used for the DS2a systems and for the systems consisting of 15 and 30 stars in DS2c data set (see Tab. S1 below).
The last configurations from the production runs of the systems described above and in ref. [1] were taken and used as initial configurations for the preparation of the bigger systems (i.e., for the DS1, DS2b and (PS c ) 8 listed in Tab. S1). The coordinates of the initial configuration were unwrapped to retain the information about the connectivity. We replicated this initial configuration 8 times to create a new, bigger cubic simulation box as depicted in Fig.1(d) of the main manuscript. The number of stars in so-prepared system is listed in Tab. S1. In order to avoid overlaps of unwrapped molecules, an extra gap was kept between the replicated small boxes. To eliminate this gap and to obtain homogeneous systems, we run an equilibration run in the NPT ensemble. The NPT simulation of PEO systems lasted 100 ns at 550 K with the pressure equal to 1 atm. The Langevin dynamics was used for the NPT run of PS systems for 50 ns, with the temperature of 700K and the pressure of 1 atm to speed up the equilibration. The box size was adjusted within the first 4 nanoseconds and then the box size fluctuated around an average value, indicating a successful elimination of the gaps. In the end of these equilibration runs, the density profiles along the sides of the simulation box were calculated to confirm the homogenous character of the systems. Before the production run, we set a pre-production run of 100 ns with the same settings as the production run. The production runs were 500 ns long and were performed at 600 K in the case of PS systems and at 450 K in the case of PEO stars. The temperature was maintained by Nosé-Hoover thermostat and the pressure was kept at 1 atm by Parrinello-Rahman barostat. All the simulations were performed by Gromacs simulation package [2] with a timestep of 1 fs.

II. COMPARISON TO THE THEORETICAL SCALING PREDICTIONS
A. Comparison to the standard blob model The standard blob model of an isolated regular homopolymer star applies geometrical arguments to predict three regions of different scaling of the density (concentration) profile, as a function of distance from the center of a star molecule. [3][4][5] The framework was developed for stars in solutions with ideal arms following random walk statistics and therefore, it is not expected to provide quantitative predictions for star melts studied here. However, as it has been used previously for the interpretation of the experimental data in star melts [6], we report our data together with the scaling predictions in Fig. S1.

B. Comparison to the Gaussian approximation
In Fig. S2 we fit the ρ intra (d) data to the Gaussian function, similarly to ref. [7] where this approach was used to estimate the extension of the impenetrable region. Our profiles exhibit a Gaussian shape only in a certain range of d, therefore, we do not further discuss this method here.  Similar procedure was followed in ref. [7], where the position of the Gaussian maximum dGauss was deemed to represent the range of the impenetrable core, h dry . However, such a quantitative estimation was avoided in this work because of the following reason: the range of d where the profiles can be described by a Gaussian decay varies for both chemistries, particularly at large d. Therefore, as the position of the maximum strongly depends on the selected range of d for the fitting procedure, the estimated h dry has a huge error and its value is unreliable. Note also, that the density profile in ref. [7] is measured from the center of mass of the molecule, while here we use the central carbon atom.  S3. Percentage of the monomers contributing to the intramolecular density profile before the intersection point as a function of the functionality, f . Note that the limit of 100% of the monomers would correspond to a sphere with a so-called corona radius Rc [8], which is of the order of the longest arm end-to-end vector in the system and it is more than factor of 2 larger than the average Rg (see Tab. S1).

V. OVERVIEW OF THE TWO-LAYER MODEL
The two-layer model assumes a spherical shape of the polymer layer around the central particle of radius R [9]. The total volume of the particle is expressed as: where R tot is an expected radius of the whole object, N k is the number of Kuhn segments per grafted chain (arm) and ρ k is the number density of the Kuhn segments. The overall radius R tot is a sum of the radius of the impenetrable layer, h dry , the radius R and half of the interpentrable region, h inter : The model solves equation for an unknown parameter n inter , which denotes the number of Kuhn segments in the interpenetration layer (for more details see the Supplementary Information in ref. [9]). The main parameter of the solution, the so-called overcrowding parameter x was formulated as: where l k is the Kuhn length. In the derivation, the prefactor 4/3 which appears in eq. S1 was omitted. Moreover, in order to achieve the final expression of eq. S3, one must assume a Gaussian conformation of the chains in the interpenetration layer. Alternatively, the eq. S3 can be reformulated as: where ν 0 is the volume of the Kuhn monomer. [10] Without doing the above-mentioned assumptions, we can rely on the physical meaning of the overcrowding parameter. The quantity x can be defined as a ratio of the actual number of monomers in a single star, within a distance R ′ , N intra,mers (R ′ ), to the number of monomers that would occupy the same volume in an unperturbed melt, N ideal,mers (R ′ ). By choosing the radius of gyration Rg as a characteritic radius of the spherical volume for the definition of x, i.e. R ′ = Rg, we obtain: where N arm is the number of monomers per arm and ρ N the monomer number density. Note that N k and ρ k , instead of N arm and ρ N , can be also used in the above definition without a significant change in the value of x parameter. We opt for the monomer-based quantities as these can be easily calculated from the atomistic simulations and also they can be found in the literature. The overcrowding parameter defined by using eq. S3 and eq. S5 is plotted in Fig. S5 and listed in Tab. S1. 16 32 x f (PS) f , eq.S3 (PEO) f , eq.S3 (PS) f , eq.S5 (PEO) f , eq.S5 The product of intra-and inter-molecular density profile used for the estimation of h = hinter/2 + h dry . The dashed line represents a Gaussian fit and the arrows, starting at d = h, indicate the length equal to hinter/2 for each system with the corresponding color scheme. hinter/2 is defined as one half of the full width at half-maximum of the function ρintra.ρinter. (b) Thickness of the interpenetration layer hinter as a function of the overcrowding parameter, x. The dashed lines represent the theoretical prediction of Kapnistos et al. [12]. The upper line corresponds to β = 0.34 and the bottom line to β = 0.4, using the values of l k and N k corresponding to the polystyrene stars.