Dynamic, conformational and topological properties of ring–linear poly(ethylene oxide) blends from molecular dynamics simulations
Introduction
Our interest in the study of ring–linear polymer blends has been renewed in the last years after the landmark paper by Kapnistos et al. [1] on the stress relaxation of entangled ring polystyrene melts which demonstrated that small amounts of linear chains can have a drastic effect on their rheology. As reported by Kapnistos et al. [1], even present in such small concentrations as 50 times below the overlap concentration, linear chains can be bridged with rings to form a long-lived transient percolating network, which can dramatically affect the form of the stress relaxation modulus G(t) at intermediate times (instead of a power law behavior, an entanglement plateau appears). The observations of Kapnistos et al. [1] are consistent with an earlier study by McKenna et al. [2] who observed that re-fractionation of 180 kg/mol polystyrene rings raised the recoverable compliance with respect to the original sample in the intermediate time range; adding linear chains, on the other hand, was found to lower the plateau compliance in this time regime. Unlike their drastic effect on rheology, it turns out that small amounts of linear additives affect weakly the diffusion of rings. Kawaguchi et al. [3] reported diffusivity data for polystyrene rings purified by the LCCC method: compared to linear chains, ring diffusion coefficients were higher by a factor of about 2.
Computer simulations have also addressed issues related with chain conformation and dynamics in blends of ring–linear polymers. Halverson et al. [4] have studied structure, dynamics and rheology in ring–linear polymer blends of equal chain length through equilibrium and non-equilibrium MD simulations with a semi-flexible bead-spring model. The results were analyzed in terms of the chain length N and volume fraction φL of linear chains. Linear chains were found to be Gaussian for all combinations of N and φL. At small values of φL the rings were partially collapsed, but as the fraction of linear chains was increased, their size grew up due to threading by linear chains. At infinite dilution, the rings were found to be multiply threaded by linear chains and nearly Gaussian. Regarding diffusion coefficients, rings were found to diffuse slower as the volume fraction of linear chains was increased. At high enough fractions of linear chains, in particular, rings became multiply threaded and their diffusion was severely hindered. In contrast, the motion of linear chains was found to be largely independent of blend composition. Additional NEMD simulations allowed Halverson et al. [4] to compute the zero shear rate viscosity of blends containing long linear and ring chains, comparable to experimental systems. Linear contaminants were found to increase the zero-shear viscosity of the ring melt by about 10% around one-fifth of the corresponding overlap concentration. For equal concentrations of linear and ring polymers, the blend viscosity was about twice that of the pure linear melt.
Chain self-diffusion in binary blends of cyclic and linear polymers has also been studied by Subramanian and Shanbhag [5] with the help of the bond fluctuation lattice model; this, further, enabled them to construct a minimal constraint release model for the dynamics of a cyclic polymer infiltrated by neighboring linear chains. In a subsequent study [6], the authors used an adapted version of the annealing algorithm to identify primitive paths (PPs) in the blend and compute the PP length and the average number of entanglements of linear and ring components. For the molecular systems addressed in the simulations, the PP length and the average number of entanglements on a ring molecule increased approximately linearly with the fraction of linear chains, and for large chain lengths these approached values comparable to linear chains. Threading of ring molecules by linear chains and ring–ring interactions were observed only in the presence of linear chains. It was conjectured that for large chain lengths, these latter interactions facilitate the formation of a percolating entangled network, thereby resulting in a disproportionate retardation of the underlying dynamical processes.
It turns out that threading events in ring–linear polymer melts have also been addressed in an older paper by Helfer et al. [7] through a coarse-grained Monte Carlo algorithm on a diamond lattice. Helfer et al. [7] argued that, in general, cyclic molecules can be categorized in three groups: (A) the first contains small cyclic molecules with very simple conformations in two dimensions, such as a circle or an ellipse; (B) the second contains intermediate cyclic molecules in which the structure changes from two to three dimensions, but they can still be represented by a series of local planes in two dimensions; (C) the last category includes very large cyclic molecules with very complicated shapes in three dimensions. In their simulations, the size of the cyclic molecules fell into category B, in which the three-dimensional shape of molecules could be divided into a series of two dimensional planar structures. This assumption was verified by looking into the conformations of the cyclic molecules as evaluated by the principal moments of the radius-of-gyration tensor. And then they proposed an iterative procedure for determining the threading of a cyclic molecule in the melt by a linear chain by approximating the space spanned by this cyclic molecule with a series of local planes and checking if these local planes are crossed by a linear chain. As the size of the cyclic increased, Helfer et al. [7] observed the occurrence of multiple threading as well; these corresponded to two cases: either a linear chain is threading more than one cyclic molecule or a cyclic molecule is threaded by more than one linear chains.
With very few exceptions, most of the above-mentioned simulation studies have been performed with coarse-grained (non-atomistic) models. In these models (e.g., in the bead-spring model of Ref. [4], in the bond fluctuation lattice model of Refs. [5], [6], and in the diamond lattice model of Ref. [7]), chemical and/or chain architectural details are ignored; instead, the emphasis is put on capturing the universal behavior of different polymers, typically in the form of scaling laws. A very important point about coarse-grained models is that dynamics is significantly accelerated at the coarse-grained level (and this can often lead to incorrect results for quantities such as diffusivity and viscosity), unless one indirectly accounts for the entropy loss (i.e., the extra dissipation) associated with the neglected degrees of freedom [8].
Full atomistic simulations, on the other hand, can be significantly accelerated today through the use of massively parallel algorithms and/or graphics processors, and the development of multiple-time step techniques. For relatively short polymers, such as the model ring molecules synthesized today, they constitute ideal simulation tools for studying their properties and elucidating many of the mechanisms and interactions at the atomistic level, and topological constraints at the level of entire chains or molecules that shape these properties [9]. In a recent study, e.g., full atomistic MD simulations with a short ring PEO melt [10] were found to yield results fully consistent with a neutron spin echo spectroscopy (NSE), confirming the faster diffusion of the centers-of-mass of these molecules in comparison to the linear polymer. This agrees perfectly with the earlier experimental findings of Kawaguchi et al. [3]. In some sense, the work here extends that of Ref. [10] by addressing ring dynamics in a binary blend with linear chains.
The rest of the paper is organized as follows: In Section 2, we provide details of the simulation method and of the PEO atomistic model employed in the present study. A thorough comparison with other simulation results and experimental data for relatively short PEO melts (pure linear or pure cyclic) that served as validation tests for our work is presented in Section 3. Then, Section 4 follows with a presentation and a detailed discussion of the simulation predictions for the selected blends. And the paper concludes with Section 5 where we summarize important findings and briefly discuss future plans.
Section snippets
Simulation details
The ring poly(ethylene oxide) chains considered in our simulations are represented by the formula –CH2–O–(CH2–CH2–O)n–CH2– while the linear analogues by the formula CH3–O–(CH2–CH2–O)n–CH3, with n in both cases denoting the number of monomers per chain (the same for both species). In the following discussion, the ring and linear components of the polymer blend will be denoted as R and L, respectively. All simulations were executed with R and L species in each system having the same chain length.
Validation tests
Before employing the molecular model described in Section 2 for production runs with the selected PEO systems, we thoroughly tested it by comparing its predictions for several properties of pure linear and pure ring PEO systems against other simulation results already reported in the literature and/or experimental data. In Table 2, for example, we present results for the density of several short linear PEO systems at various temperatures, along with experimental data and previous simulation
Thermodynamic properties
In Table 4, we report the MD simulation predictions for the density ρ of the ring–linear PEO blends (at T = 413 K and P = 1 atm), and how they depend on the size of the PEO molecules and the mass fraction wL of linear chains. The data indicate a negligible effect of blend composition on ρ. Unfortunately, it was not possible to find experimental data in the literature for the densities of the simulated pure PEO-1800 and PEO-5000 melts (R and L) with which we could directly compare our simulation
Conclusions
We have carried out a detailed study of the conformational, dynamic and topological properties of ring–linear PEO blends by carefully analyzing long-time atomistic trajectories accumulated in the course of detailed MD simulations. Ring contamination by linear chains seems to have a minor (if any) effect on the structural and conformational properties of the system; however, it strongly affects chain dynamics. In the presence of linear chains, rings relax and diffuse dramatically slower. In
Acknowledgments
We are grateful to PRACE for the allocation of significant CPU time through their JUROPA center in Germany, Jülich. We feel deeply indebted to Prof. Dimitris Vlassopoulos for many fruitful discussions, his warm support of this work, and his continued interest in our simulation findings.
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