Cross-linking and branching of primary polymer molecules are investigated using the Galton–Watson (GW) process. Starting with the probability generating function (pgf) of the primary molecular weight distribution (MWD), analytical expressions are derived for the bivariate pgfs g(nbr, s) of branched polymers which depend also on the number of branch points nbr. The bivariate MWDs n(nbr, i) (i: number of molecular units) are then derived as Taylor expansions in s. All three cases of random branching: X-shaped (cross-linking), T-shaped (only one end takes part in the branching process), and H-shaped (both ends can take part in the branching process) are treated. An extension of the formalism does not require the construction of the pgf and allows the direct use of the MWD of the primary chains. However, using pgfs allows to go past the gel point and to determine the MWD and content of the sol. Explicit expressions are given for special distributions: the mono modal, the most probable, the Schulz-Zimm, the Poisson, and the Catalan distribution for the cases of X-shaped and T-shaped branching.

Front Cover: The effects of the primary structure of reactive polymers on the structure and mechanical properties of gels are studied by a molecular dynamics simulation. The figure illustrates the improvement of structure uniformity by changing the arrangement of functional groups from a random one (top) to a periodic one (bottom), where red surfaces show low crosslinking density regions. This is reported by Tsutomu Furuya and Tsuyoshi Koga in article number 2200044.

Polydimethylsiloxane (PDMS) membrane in suitable-fluorinated level have excellent pervaporation performance as well as antibiological contamination performance. The pervaporation membranes with different PDMS/fluorosilane mass ratios, the adsorption and dissolution behaviors of acetone molecules on the membrane surface, as well as the diffusion and permeation behaviors in the membranes are studied by all-atom molecular dynamics simulation (AAMDS). The results show that when the mass ratio of PDMS/fluorosilane is 100/20, the surface solubility of acetone is 11.711 (J cm−3)0.5, and the interfacial interaction is −16897.0415 kcal mol−1, both of which are the highest. The results of wide-angle X-ray diffraction (WAXD) showed that there are amorphous regions in the membranes suitable for acetone penetration. The maximum chain spacing of the PDMS/fluorosilane(100/20)_membranes is 10.8482 Å, and the free volume fraction (FFV) is 3.03%, both of which are the largest. The change rate of long-term mean square displacement (MSD) in PDMS/fluorosilane(100/20)_Membrane with time is 0.45269. The Young's modulus E, shear modulus G, volume modulus K, and Poisson's ratio ν of PDMS/fluorosilane(100/20)_Membrane are 0.3249, 0.4061, 0.0492 GPa and -0.5999, respectively. The elasticity of the membrane enhances the diffusion behavior of acetone molecules, and the self-diffusion coefficient of acetone in the membrane is 0.07545 Å2 ps−1.

Front Cover: The dipoles of carboxylated acrylonitrile-butadiene rubber (XNBR) chains are oriented under the alternating electric field, illustrating the feasibility of a nonequilibrium molecular dynamics (NEMD) simulation method. To investigate the effect of XNBR molecular structure on dielectric properties, the nonequilibrium approach with an external temperature field and alternating electric field can be used. This is reported by Fei Cai, Sizhu Wu, and co-workers in article number 2200006.

A dynamic model is developed for gas-phase ethylene/1-hexene polymerization with a three-site hafnocene catalyst. The model accurately predicts molecular weight and comonomer composition distributions for 15 lab-scale copolymerization runs performed at different temperatures. The experimental runs used to fit this model are performed at temperatures between 60 and 85 °C. Gas-phase concentrations are measured every 2.7 min throughout each run. Predicted chain-length distributions are discretized to aid model development, keeping the number of ordinary differential equations manageable. Kinetic parameters at the reference temperature of 81 °C and activation energies are estimated. Using parameter subset selection techniques, it is determined that 53 of the 60 model parameters should be estimated using the product characterization and reactor data. An additional data set obtained at 85 °C is used for model validation, confirming the predictive power of the model. The proposed model and its parameter estimates can aid selection of operating conditions to achieve targeted polymer properties.

This study presents a numerical investigation of the dimerization of polyglutamine homo-peptides of varying length. It employs the PRIME20 intermediate resolution protein model and studies it with a flat-histogram type Monte Carlo simulation that gives access to the thermodynamic equilibrium of this model over the complete control parameter range (for the simulations this is temperature). For densities comparable to typical in vitro experimental conditions, this study finds that the aggregation and folding of the polyglutamine chains occur concurrently. However, as a function of chain length the sequence of establishment of intra- and intermolecular hydrogen bonding contacts changes. Chains longer than about N = 24 polyglutamine repeat units fold first and then aggregate. This agrees well with the experimental finding that, beyond N = 24 the single polyglutamine chain is the critical nucleus for the aggregation of amyloid fibrils. A finite size scaling of the ordering temperatures reveals that for this chain length (and longer chains) folding occurs at physiological (respectively larger) temperatures, whereas shorter chains are disordered at physiological conditions.

Brownian Dynamics is used to investigate the scattering functions of ideal and excluded volume linear polymers in two to seven spatial dimensions. The scattering functions for ideal and excluded volume polymers examined are in agreement with theoretical predictions in all dimensions. As the dimension is increased, the scattering functions for the excluded volume chains converge toward the ideal results. These findings indicate that excluded volume chains behave more and more as ideal ones as the dimension gets larger.

Metropolis Monte–Carlo simulation is carried out for microphase-separated bulk state of AB diblock copolymers with various compositions. The distribution probability of end segments in long B-block chain are explored to determine the Wigner–Seitz(WS) cells as primitive cells for four known periodic structures, lamellar-, Gyroid-, cylindrical-, and spherical ones. The end segments are commonly turned to be localized at the several distinct far sites from the lattice points of WS cells for all morphologies investigated. Among them, when the fraction of A segments is 0.25, a hexagonal prism type column appears as a WS, while when the fraction is much lower at 0.1, body-centered cubic(BCC) lattice is formed and its end segments are found to be localized at hexagonal frames and also on the six square faces of truncated octahedron or Kelvin's Tetrakaidecahedron(KT), which has rarely been found in real soft material ever. This achievement is strongly pointing that each micelle formed by self-assembled diblock coplymers in bulk have essentially the framework of equivolume KT in real material systems.

The rational development of sustainable polymeric materials demands tunable properties using mixtures of polymers with chemical variations. At the same time, the sheer number of potential variations and combinations makes experimentally or numerically studying every new mixture impractical. A direct predictive tool quantifying how material properties change when molecular features change provides a less time- and resource-consuming route to optimization. Numerically solving Flory–Huggins theory provides such a tool for mono-disperse mixtures with a limited number of components, but for multi-component systems the large number of equations makes numerical computations challenging. Approximate solutions to Flory–Huggins theory relating miscibility and solubility to molecular features are presented. The set of approximate relations show a wider range of accuracy compared to existing approximations. The combination of the analytical, lower-order, and more accurate higher-order approximations together contribute to a broader applicability and extensibility of Flory–Huggins theory.

The effect of topology on the collapse transition and instantaneous shape of an energy polydisperse polymer (EPP), a model heteropolymer is studied by means of computer simulations. In particular, three different chain topologies, namely, linear (L), ring (R), and trefoil knot (T), are considered. The heteropolymer is modeled by assigning each monomer an interaction parameter, εi, drawn randomly from a Gaussian distribution. Through chain size scaling, the transition temperature, θ, is located and compared among the chains of different topologies. The influence of topology is reflected in the value of θ and observed that θ(L) > θ(R) > θ(T) in a similar fashion to that of the homopolymer counterpart. Also studied chain size distributions, and the shape changes across the transition temperature characterized through shape parameters based on the eigenvalues of the gyration tensor. It is observed that, for the model heteropolymer, in addition to chain topology, the θ-temperature also depends on energy polydispersity.

The effects of copolymerized monomer vinyl acetate (VAc) on processing properties and thermal stability of poly(vinyl chloride-co-vinyl acetate) (PVCA) are investigated via experiment and molecular dynamics simulation. Experimental results showed that PVCA with higher VAc content has larger loss tangent (tanδ), lower complex viscosity (η*), and glass transition temperature (Tg), which improved the processing properties of PVCA. A series of PVCA models are constructed to study the microstructure on the processing properties of PVCA, and the results showed the PVCA with higher VAc content exhibits larger molecular chain mobility and free volume fraction (FFV), smaller intermolecular interactions, and the mean square end-to-end distance (). Furthermore, the IR spectra of gas products indicated that thermal degradation of PVCA mainly generated hydrogen chloride (HCl), carboxylic acid, and aliphatic hydrocarbons between 200 and 500 °C, and the removal of HCl and carboxylic acid is almost simultaneous. The degradation models of PVCA chains demonstrated the CCl bond in vinyl chloride (VC) and CO bond in VAc have similar thermal stability, which corresponded to the experimental results. In a word, the work provides a promising technique to study the structure and property of PVCA at molecular dynamic level.

Recent studies indicate the importance of using bead-rod models for polymer chains, resolving to a single Kuhn step, especially in strong flows. Earlier, researchers suggested the Fene-Fraenkel spring as a computationally efficient approximation to the rod for Brownian dynamics (BD) simulations. This analysis reveals that the Fraenkel spring is an even better alternative. The predictions from BD simulations with stiff Fraenkel springs are nearly identical to those using Fene-Fraenkel springs and rigid rods. Significantly, such excellent agreement is obtained when the spring lengths at every time step are updated once, without any further iterations for refinement. The computational speeds for single-step update of the spring lengths, using Fraenkel springs, is an order of magnitude faster than the usual procedure of iterating until convergence. Even when multiple iterations are allowed within each time step, the computational time remains about 20–25% lower for longer chains with Fraenkel springs than Fene-Fraenkel springs. The merits of using highly resolved chains, using Fraenkel springs, in preaveraged models are also highlighted. The predictions of such preaveraged models agree reasonably well with BD simulations. This is particularly advantageous since the predictions for shear flows are poor, especially at higher shear rates, for well-known models like FENE-P.

The renormalized bond lifetime model (RBLM) is a popular scaling theory for the effective lifetime of reversible bonds in transient networks. It recognizes that stickers connected by a reversible bond undergo many (J) cycles of dissociation and reassociation. After finally separating, one of these stickers finds a new open partner in time τopen via a subdiffusive process whose mean-squared displacement is proportional to tα, where t is the time elapsed, and α is the subdiffusion exponent. The RBLM makes convenient mathematical approximations to obtain analytical expressions for J and τopen. The consequences of relaxing these approximations is investigated by performing fractional Brownian motion (FBM) simulations. It is found that the scaling relations developed in the RBLM hold surprisingly well. However, RBLM overestimates both τopen and J, especially at lower values of α. For α = 0.5, corresponding to the Rouse limit, it is found that τopen is overestimated by a factor of approximately 4x, while the approximation for J is nearly exact. The degree of overestimation worsens as α decreases, and increases to 1–2 orders of magnitude at α = 0.25, corresponding to the reptation limit. This has important ramifications for experimental studies that use RBLM to interpret rheology and dielectric spectroscopy observations.

Linear alkanes (n-alkanes) are chemically the most simple linear hydrophobic molecules in nature. Studying the incorporation of n-alkanes into lipid membranes is therefore a good starting point toward understanding the behavior of hydrophobic molecules in lipid membranes and to assess how accurately molecular dynamics models describe such systems. Here, the miscibility and structure of different n-alkanes—n-decane (C10), n-eicosane (C20), and n-triacontane (C30)—in dipalmitoylphosphatidylcholine membranes are investigated using two of the most used force fields for lipid membrane molecular dynamics simulations (CHARMM36 and Slipids). The n-alkanes are miscible in the membrane up to a critical volume fraction, ϕc, that depends on the force field interaction parameters used. ϕc is dependent on alkane chain length only for the model with more disordered chains (Slipids). Below ϕc, a comparison with 2H nuclear magnetic resonance (NMR) spectra indicates that a more realistic structure of the longer alkane molecules (C20 and C30) is obtained using the Slipids force field. On the other hand, for the shorter alkane (C10), Slipids simulations underestimate molecular order and CHARMM36 simulations enable a precise prediction of its experimental spectrum. The predicted 2H NMR spectra are highly sensitive to 1–4 electrostatic interactions, and suggest that a reduction of the partial charges of the longer alkanes and acyl chains in CHARMM36 results in a better performance. The results presented indicate that lipid membranes with incorporated alkanes are highly valuable systems for the validation of force fields designed to perform lipid membrane simulations.

The internal stratification of a polyelectrolyte complex (zipper brush) formed by mixing of an anionic charged polyelectrolyte brush (PEB) with cationic-neutral diblock copolymers bearing a very long neutral block is studied by means of molecular dynamics simulations. The authors find that the fraction of the neutralized PEB units in the mixture increases as the fraction of the PEB charged units (a) increase for high grafting densities (d). Due to the charge neutralization condition, from the initial PEB through the complexation with the appropriate choice of a cationic-neutral copolymer, neutral brush having grafting density lower, equal, or much higher than that of the PEB are obtained. The addition of monovalent salt in the mixture with concentrations 0.1 and 1 m leads to a reduction in complexed diblock copolymer chains of up to 91% and practically the initial PEB is recovered. The findings are in full agreement with existing experimental predictions and provide new insights into the structure and the shape of the coacervate. The latter progressively changes from a dense film to perforated film, to lamella, to pinned micelles, to stacks as a, d, and the molecular weights of the PEB and diblock copolymer blocks are altered.

A three-site metallocene catalyst is used in a gas-phase semi-batch reactor to produce ethylene/hexene copolymers. At the end of each batch, polyethylene (PE) is collected and analyzed to determine the carbon-13 nuclear magnetic resonance (13C-NMR) triad sequence distribution. Joint molecular weight (MW) and composition distribution data are obtained using gel permeation chromatography with an infrared detector (GPC-IR). Data from ten experimental runs are used for kinetic parameter estimation. Using a mean-squared error (MSE) selection methodology, 23 of the 36 model parameters are selected for estimation using the available polymerization rate and PE characterization data. The remaining parameters are held at initial guesses to avoid overfitting. Addition of the triad data to the parameter estimation problem allows for one additional parameter to be estimated and results in improved parameter estimates. Standard deviations of all but one of the estimated parameters decreased due to inclusion of triad data. The updated parameter estimates result in good fits for the triad data and for joint MW and composition data. The model accurately predicts four validation data sets not used for parameter estimation. The new model and its updated parameter estimates will be valuable for scaling up new polymer grades from laboratory-scale to commercial-scale.

The structure of polymer chains at interfaces still is not fully understood. A coarse-grained lattice model is used to study the structure of macromolecules strongly adsorbed on a flat surface. As a result, the polymers are strictly two-dimensional. Chains are flexible and athermal. A dynamic Monte Carlo algorithm consisting of local and non-local modification of chains’ conformations is employed. The scaling behavior of chains’ size is found to be similar to monodisperse systems. It is also shown that the introduction of polydispersity increases values of the percolation threshold, especially for longer chains. The influence of the type of polydispersity on the percolation threshold in two-dimensional polymer films is found to be significant.

Mean-square radius of gyration Rg2 and the graph diameter D of the random crosslinked network polymers are investigated to find a linear relationship, Rg2 = a D. The proportionality coefficient, a is dominated by the cycle (circuit) rank, or the number of intramolecular crosslinks kc, and a convenient equation is proposed for the relationship between a and kc. This relationship makes it possible to estimate Rg2 based on D and kc, which can reduce the required computational time to determine the Rg2-values greatly. This new method is applied to find that the contraction factor g decreases with kc, and the differences in the primary chain length distribution that constitute the network polymers vanish for large kc-values.

Front Cover: Phase behavior of symmetric ABCBD pentablock quarterpolymers has been investigated by Monte-Carlo simulation. Two alternating helices of red(A) and blue(B)) domains are trapped in {100} large homochiral holes of the level surface for the Schoen's Gyroid surface indicated in green. The red and blue helices are naturally packed tetragonally with the same helical sense, i.e., they keep homochirality. This is reported by Jiro Suzuki, Atsushi Takano, and Yushu Matsushita in article number 2200015.

Directing reaction-diffusion (RD) phenomena, through the use of external stimuli has been one of the widely used approaches for designing multifunctional soft materials. Using modeling and simulation, it is demonstrated that the nonuniform electric field can be harnessed to create intricate ordered patterns in polymer ionic liquid (PIL) blends. The investigation begins with the establishment of the equilibrium phase diagrams of electroresponsive PIL blends and subsequently, use the Poisson–Nernst–Planck equations to model the kinetics of pattern formation. The simulations reveal that in the presence of nonuniform electric field the ionic liquid (IL) rich domains self-aggregate in high electric field regions. Thus, the ordering of the electric field regions effectively dictates the ordering of the IL-rich phase in the PIL blends. It is also demonstrated that the mechanism of spatiotemporal pattern formation is quite robust and can be dynamically controlled by varying the distribution of electric field. It is believed that the methodology provides a simplistic mechanism for creating ordered patterns in soft materials through RD phenomena that can be exploited for designing other similar stimuli-responsive systems.