Short Cyclic Structures in Polymer Model Networks: A Test of Mean Field Approximation by Monte Carlo Simulations

On the Swelling of Polymer Network Strands

Large scale computer simulations are employed to analyze the conformations of network strands in polymer networks at preparation conditions (characterized by a polymer volume fraction of ϕ0) and when swollen to equilibrium (characterized by a polymer volume fraction ϕ < ϕ0). Network strands in end-linked model networks are weakly stretched and partially swollen at preparation conditions as compared to linear polymers in the same solvent at ϕ0. Equilibrium swelling causes non-ideal chain conformations characterized by an effective scaling exponent approaching 7/10 on intermediate length scales for increasing overlap of the chains. The chain size in a network consists of a fluctuating and a time average "elastic" contribution. The elastic contribution swells essentially affinely ∝(ϕ0/ϕ)2/3, whereas the swelling of the fluctuating part lies between the expected swelling of the entanglement constraints and the swelling of non-cross-linked chains in a comparable semi-dilute solution. The total swelling of chain size results from the changes of both fluctuating and non-fluctuating contributions.

Structural Homogeneity of Macromolecular Networks by End-to-End Click Chemistry between Discrete Tetrahedral Star Macromolecules

Cross-linking via the end-to-end click chemistry of multiarm star polymers creates polymer networks with minimal inhomogeneities. Although it has been suggested that the mechanical and swelling properties of such networks depend on the absence of defects, the structural details of homogeneous networks created by this method have not yet been studied at the molecular level. Here, we report the synthesis of discrete tetrahedral star macromolecules (dTSMs) composed of polylactide (PLA) arms with discrete molecular weight and sequence. Polymer networks prepared by 4 × 4 cross-linking by Cu-free strain-promoted cyclooctyne-azide click chemistry (SPAAC) reaction exhibited a high degree of swelling (>40 fold by weight) in solvents without sacrificing mechanical robustness (elastic modulus >4 kPa). The structural details of the networks were investigated by network disassembly spectrometry (NDS) using MALDI-TOF mass spectrometry. By implementing a cleavable repeating unit in the discrete PLA arms of dTSM in a sequence-specific manner, the networks could be disassembled into fragments having discrete molecular weights precisely representing their connectivity in the network. This NDS analysis confirmed that end-to-end click reactions of dTSM networks resulted in the formation of a homogeneous network above the critical concentration (∼10 w/v%) of building blocks in the solution.

Characterization of Monte Carlo Dynamic/Kinetic Properties of Local Structure in Bond Fluctuation Model of Polymer System

We report the results of the characterization of local Monte Carlo (MC) dynamics of an equilibrium bond fluctuation model polymer matrix (BFM), in time interval typical for MC simulations of non-linear optical phenomena in host-guest systems. The study contributes to the physical picture of the dynamical aspects of quasi-binary mosaic states characterized previously in the static regime. The polymer dynamics was studied at three temperatures (below, above and close to the glass transition), using time-dependent generalization of the static parameters which characterize local free volume and local mobility of the matrix. Those parameters play the central role in the kinetic MC model of host-guest systems. The analysis was done in terms of the probability distributions of instantaneous and time-averaged local parameters. The main result is the characterization of time scales characteristic of various local structural processes. Slowing down effects close to the glass transition are clearly marked. The approach yields an elegant geometric criterion for the glass transition temperature. A simplified quantitative physical picture of the dynamics of guest molecules dispersed in BFM matrix at low temperatures offers a starting point for stochastic modeling of host-guest systems.

Modeling of Nonlinear Optical Phenomena in Host-Guest Systems Using Bond Fluctuation Monte Carlo Model: A Review

We review the results of Monte Carlo studies of chosen nonlinear optical effects in host-guest systems, using methods based on the bond-fluctuation model (BFM) for a polymer matrix. In particular, we simulate the inscription of various types of diffraction gratings in degenerate two wave mixing (DTWM) experiments (surface relief gratings (SRG), gratings in polymers doped with azo-dye molecules and gratings in biopolymers), poling effects (electric field poling of dipolar molecules and all-optical poling) and photomechanical effect. All these processes are characterized in terms of parameters measured in experiments, such as diffraction efficiency, nonlinear susceptibilities, density profiles or loading parameters. Local free volume in the BFM matrix, characterized by probabilistic distributions and correlation functions, displays a complex mosaic-like structure of scale-free clusters, which are thought to be responsible for heterogeneous dynamics of nonlinear optical processes. The photoinduced dynamics of single azopolymer chains, studied in two and three dimensions, displays complex sub-diffusive, diffusive and super-diffusive dynamical regimes. A directly related mathematical model of SRG inscription, based on the continuous time random walk (CTRW) formalism, is formulated and studied. Theoretical part of the review is devoted to the justification of the a priori assumptions made in the BFM modeling of photoinduced motion of the azo-polymer chains.

Direct characterization of a polymer network through its retainable units

A partially decrosslinkable network provides a general protocol for full, direct and quantitative characterization of polymer networks through its retainable units.

Elasticity of Phantom Model Networks with Cyclic Defects

The impact of finite cycles on the phantom modulus in an otherwise perfect network is computed exactly. It is shown that pending cycles reduce the phantom modulus of the network by kT/V independent of junction functionality. The correction for nonpending cycles is larger than estimated previously within this particular approximation of the surrounding network structure. It is discussed that loop formation inevitably leads to streched chain conformations, if the loops are built step by step as part of the network structure. All network loops tend to contract simultaneously to optimize conformations, which leads to an increasing stretch of chains in larger loops that can be observed in computer simulations. Possible other corrections to the phantom modulus that were left aside in previous work are discussed briefly.

Injectable synthetic hydrogel for bone regeneration: Physicochemical characterisation of a high and a low pH gelling system

Hybrid poly(ethylene glycol)-co-peptide hydrogels are a versatile platform for bone regeneration. For the use as injectable scaffolds, a good understanding of reaction kinetics and physical properties is vital. However, these factors have not yet been comprehensively illuminated. We show that gelation time can be effectively controlled by pH without affecting the elasticity of the formed hydrogels. Maleimide functionalised PEG gels at lower pH and produces more densely cross-linked networks than vinylsulfone functionalised PEG. Both form non-ideal networks. The elastic moduli on the order of a few kPa are in good agreement with the structural characterisation. Primary human osteoblasts cultured in proximity to bulk gels were not adversely affected in vitro. The results demonstrate that hybrid PEG-peptide hydrogels can be tailored to the requirements of in situ gelation. Attributed to their increased structural properties and a higher tolerance towards low pH, maleimide functionalised hydrogels might provide a better alternative for injectable applications.

Counting Secondary Loops Is Required for Accurate Prediction of End-Linked Polymer Network Elasticity

To predict and understand the properties of polymer networks, it is necessary to quantify network defects. Of the various possible network defects, loops are perhaps the most pervasive and yet difficult to directly measure. Network disassembly spectrometry (NDS) has previously enabled counting of the simplest loops-primary loops-but higher-order loops, e.g., secondary loops, have remained elusive. Here, we report that the introduction of a nondegradable tracer within the NDS framework enables the simultaneous measurement of primary and secondary loops in end-linked polymer networks for the first time. With this new "NDS2.0" method, the concentration dependences of the primary and secondary loop fractions are measured; the results agree well with a purely topological theory for network formation from phantom chains. In addition, semibatch monomer addition is shown to decrease both primary and secondary loops, though the latter to a much smaller extent. Finally, using the measured primary and secondary loop fractions, we were able to predict the shear storage modulus of end-linked polymer gels via real elastic network theory (RENT).

Kinetic Monte Carlo Simulation for Quantification of the Gel Point of Polymer Networks

Accurate prediction of the gel point for real polymer networks is a long-standing challenge in polymer chemistry and physics that is extremely important for applications of gels and elastomers. Here, kinetic Monte Carlo simulation is applied to simultaneously describe network topology and growth kinetics. By accounting for topological defects in the polymer networks, the simulation can quantitatively predict experimental gel point measurements without any fitting parameters. Gel point suppression becomes more severe as the primary loop fraction in the networks increases. A topological homomorphism theory mapping defects onto effective junctions is developed to qualitatively explain the origins of this effect, which accurately captures the gel point suppression in the low loop limit where cooperative effects between topological defects are small.

Theoretical Study of Network Formation and Mechanical Properties of Physical Gels with a Well-Defined Junction Structure

A statistical-mechanical theory of thermoreversible gelation considering loops for the system consisting of bifunctional polymer units carrying A functional groups and trifunctional units carrying B functional groups at their ends is constructed. We obtain the sol-gel transition line and the properties of the post-gel region as functions of the polymer concentration, temperature, association constant, and loop parameter using the present theory. In this article, we calculate the number concentration of elastically effective chains in the gel region and obtain the shear modulus by an application of the phantom network theory. The shear modulus obtained by this theory is lower than that obtained by conventional theory because of loop formation. We find that these theoretical results are in good agreement with the experimental data.

Multiscale approach for the construction of equilibrated all-atom models of a poly(ethylene glycol)-based hydrogel

A multiscale modeling approach is presented for the efficient construction of an equilibrated all-atom model of a cross-linked poly(ethylene glycol) (PEG)-based hydrogel using the all-atom polymer consistent force field (PCFF). The final equilibrated all-atom model was built with a systematic simulation toolset consisting of three consecutive parts: (1) building a global cross-linked PEG-chain network at experimentally determined cross-link density using an on-lattice Monte Carlo method based on the bond fluctuation model, (2) recovering the local molecular structure of the network by transitioning from the lattice model to an off-lattice coarse-grained (CG) model parameterized from PCFF, followed by equilibration using high performance molecular dynamics methods, and (3) recovering the atomistic structure of the network by reverse mapping from the equilibrated CG structure, hydrating the structure with explicitly represented water, followed by final equilibration using PCFF parameterization. The developed three-stage modeling approach has application to a wide range of other complex macromolecular hydrogel systems, including the integration of peptide, protein, and/or drug molecules as side-chains within the hydrogel network for the incorporation of bioactivity for tissue engineering, regenerative medicine, and drug delivery applications.

Counting primary loops in polymer gels

Much of our fundamental knowledge related to polymer networks is built on an assumption of ideal end-linked network structure. Real networks invariably possess topological imperfections that negatively affect mechanical properties; modifications of classical network theories have been developed to account for these defects. Despite decades of effort, there are no known experimental protocols for precise quantification of even the simplest topological network imperfections: primary loops. Here we present a simple conceptual framework that enables primary loop quantification in polymeric materials. We apply this framework to measure the fraction of primary loop junctions in trifunctional PEG-based hydrogels. We anticipate that the concepts described here will open new avenues of theoretical and experimental research related to polymer network structure.