Tensor Spaces and Numerical Tensor Calculus

Xu G., Mattice W.L.

Monte Carlo (MC) simulations of structure formation for short polyethylene chains at low temperature are performed based on a recent developed method that uses coarse-grained chains on a high coordination lattice. Local short-range interactions based on rotational isomeric state (RIS) model and long-range interactions obtained from Lennard–Jones (LJ) potential are introduced during the simulation. Properties evaluated from the simulations are the mean square dimensions, anisotropy of the radius of gyration tensor, local conformation determined by the occupancy of trans state and orientation correlation functions, energy of the system, and chain packing reflected by the pair correlation functions and structure factors. All of these parameters reveal an ordering process that produces an approximation to a hexagonal crystal phase. The hexagonal structure is imposed by the presence of a diamond lattice underlying the high coordination lattice on which the simulation is performed. Folding of the chains in the crystal is mandatory, because they have fully extended lengths in excess of the dimension of the simulated periodic box. Nevertheless, the simulations demonstrate that a high degree of crystallinity can be achieved in reasonable computer time. The simulation technique should be applicable to other choices of periodic boundary conditions that do not affect the results as strongly as in the present case.

Michalke W., Kreitmeier S., Lang M., Buchner A., Göritz D.

The article presents the results of Monte Carlo simulations of bimodal networks performed with the Bond-Fluctuation-Algorithm. First the sol-fractions of networks with different ratios of short chains were studied and found to be always less than 2%. Concerning clustering behaviour, we saw that while random networks always form a main cluster containing more than 95% of all chains, simulated networks with less than 80% short chains do not form a main cluster. The density profiles during the swelling process show that clustering is reflected in a lower swelling degree and a sharper transition zone between the inner part and the boundary regions of the network. Finally, comparing the density distributions of crosslinkers of unimodal and bimodal networks, we found that all unimodal networks have a more ordered structure in their interior than in the melt. On the other hand, bimodal networks, where the ratio between long and short chains leads to equal masses of the fractions, show a superposition of two separate density distribution peaks, leading to a broader distribution than the Gaussian distribution found for a melt.

Ding J., Carver T.J., Windle A.H.

A lattice Monte Carlo (MC) simulation was applied to the study of block copolymers in selective solvent or amphiphilic surfactant solution on the segment level, hydrodynamic interactions being neglected. The code was found to be very efficient, employing a partial reptation mode as the elementary movement of the self-avoiding lattice chains. Typical self-assembled structures of block copolymers such as micelle, lamellae, hexagonal cylinder and bicontinuous networks have been successfully reproduced without any priori specification of structure. Order–disorder and order–order transitions of diblock copolymers are systematically studied by adjusting the temperature, the concentration or the block length ratio in a series computer simulations. The structural differences between micelles composed of ABA and BAB triblock copolymers are also explicitly revealed by direct visualisation of the underlying chain configurations. The simulation results are consistent with the experimental observations in the literature. This simulation approach is thus a very useful tool in the extensive investigation of self-assembled structures. It has the advantage that both micro-domains and chain configurations can be studied with only a comparatively modest call on computational resources.

Nakazawa H., Fujinami S., Motoyama M., Ohta T., Araki T., Tanaka H., Fujisawa T., Nakada H., Hayashi M., Aizawa M.

Polymerization-induced phase separation in polymer-dispersed liquid crystal is studied by computer simulations in two dimensions. The domain morphology resulting from phase separation is investigated by solving the coupled set of equations for the local volume fraction and the nematic order parameter, taking into account the viscoelastic effects and gelation due to polymerization. Comparing the morphology of phase separation by temperature quench, it is shown that the viscoelastic effects and gelation enable the polymer-rich phase to form a stable interconnected domain even when the polymer component is minority. The experimental evidence consistent with this characteristic feature is also given.

Ramos J.I.

Asymptotic methods based on the slenderness ratio are used to obtain the leading-order equations that govern the fluid dynamics of axisymmetric, isothermal, Newtonian, annular liquid jets such as those employed in the manufacture of textile fibres, annular membranes, composite fibres and optical fibres, at low Reynolds numbers. It is shown that the leading-order equations are one-dimensional, and analytical solutions are obtained for steady flows at zero Reynolds numbers, zero gravitational pull, and inertialess jets. A linear stability analysis of the viscous flow regime indicates that the stability of annular jets is governed by the same eigenvalue equation as that for the spinning of round fibres. Numerical studies of the time-dependent equations subject to axial velocity perturbations at the nozzle exit and/or the take-up point indicate that the annular jet dynamics evolves from periodic to chaotic motions as the extension or draw ratio is increased. The power spectrum of the annular jet's radius at the take-up point broadens and the phase diagrams exhibit holes at large draw ratios. The number of holes increases as the draw ratio is increased, thus indicating the presence of strange attractors and chaotic motions.

Allington R.D., Attwood D., Hamerton I., Hay J.N., Howlin B.J.

Cyanate ester resins are an emerging family of high performance polymers that are being studied for a variety of technological applications. The structure of the aromatic sym-triazine ring formed during cure of these polymers is investigated here by analysis of X-ray crystallographic data from a number of model compounds. The data show a preferred conformation of the ring structure with alternating internal bond angles of ca. 112° at nitrogen (C–N–C) and 128° at carbon (N–C–N). The C–N bond lengths are also shorter than those found in pyridine or pyrimidine, leading to a non-planar ring conformation. Force constants for the bond stretch, bend and torsional motions of the sym-triazine ring have also been calculated, using mopac , the semi-empirical quantum mechanics package.

von Lockette P.R., Arruda E.M.

A conjugate gradient Monte Carlo algorithm was used to simulate the annealing of two and three dimensional end-linked unimodal and bimodal polydimethylsiloxane networks. Equilibrium is satisfied at every crosslink during network energy minimization resulting in distinct differences in network characteristics from classical assumptions. Annealed unimodal networks were found to retain the uniformly dispersed arrangement of crosslinks generated during the crosslinking algorithm. Radial distribution functions of chain vector lengths for various unimodal systems show a shift in the mean chain length from the rms length prior to annealing to shorter lengths upon annealing. Short chains in bimodal networks cluster during the annealing process in agreement with experimental investigations of short chain agglomeration in the literature. This work provides the first predictions of bimodal chain network clustering via simulated network formation and demonstrates the critical role of network annealing in determining the initial configurations of deformable elastomeric networks. This information is extremely useful in the development of accurate constitutive models of bimodal networks.

Şen M., Güven O.

Prediction of swelling behaviour of hydrogels containing cationic and anionic moieties, sensitive to pH and ionic strength changes of the swelling medium was investigated. The equations derived for the prediction of the theoretical swelling curves are based on the phantom network theory and the approaches of Peppas et al. For all predictions, a number of polymer based parameters, solution property parameters and polymer–solvent combination type parameters were evaluated typical of amphiphilic copolymers. The advantages of the derived equations for the determinations of average molecular weight between the cross-links, and also polymer–solvent interaction parameter have been exemplified.

Tashiro K.

The molecular dynamics (MD) technique was used to calculate the temperature dependence of the structure, molecular motion, and mechanical property of the orthorhombic polyethylene (PE) crystal. The potential functional parameters reported by Karasawa et al. (J Phys Chem, 95 (1991) 2260) were refined further so that the vibrational frequencies of infrared and Raman bands, measured by us at ultra-low temperatures for the normal and fully deuterated PE, could be reproduced well. The flip-flop motion around the chain axis and the torsional motion of the skeletal chains were found to start above ca. 350 K and increase the amplitude of these motions progressively. Coupling these two types of chain motion resulted in a steep increase of the thermal vibration parameters or the mean-square-displacements of carbon and hydrogen atoms, corresponding well with the X-ray data. The lattice constants and the related linear thermal expansion coefficients were also found to be in good agreement with the observed data. The calculated Young's modulus along the chain axis decreased gradually with the increasing temperature: 330 GPa at 0 K to 280 GPa at room temperature. The latter was in good agreement with the value of 280–305 GPa evaluated from the Raman measurement of the longitudinal acoustic mode. Young's modulus was found to relate intimately with the chain contraction caused by the skeletal torsional motion. Only 0.3% contraction of the chain resulted in the reduction of the modulus by ca. 35%. A similar behavior was also seen in the trigonal polyoxymethylene and nylon 6 α forms.

Goddard W.A., Cagin T., Blanco M., Vaidehi N., Dasgupta S., Floriano W., Belmares M., Kua J., Zamanakos G., Kashihara S., Iotov M., Gao G.

Advances in theory and methods are making it practical to consider fully first principles (de novo) predictions of structures, properties and processes for organic materials. However, despite the progress there remains an enormous challenge in bridging the vast range of distances and time scales between de novo atomistic simulations and the quantitative continuum models for the macroscopic systems essential in industrial design and operations. Recent advances relevant to such developments include: quantum chemistry including continuum solvation and force field embedding, de novo force fields to describe phase transitions, molecular dynamics (MD) including continuum solvent, non equilibrium MD for rheology and thermal conductivity and mesoscale simulations. To provide some flavor for the opportunities we will illustrate some of the progress and challenges by summarizing some recent developments in methods and their applications to polymers and biopolymers. Four different topics will be covered: (1) hierarchical modeling approach applied to modeling olfactory receptors, (2) stabilization of leucine zipper coils by introduction of trifluoroleucine, (3) modeling response of polymers sensors for electronic nose, and (4) diffusion of gases in amorphous polymers.

Cagin T., Wang G., Martin R., Zamanakos G., Vaidehi N., Mainz D.T., Goddard W.A.

Dendrimers and hyperbranched polymers represent a novel class of structurally controlled macromolecules derived from a branches-upon-branches structural motif. The synthetic procedures developed for dendrimer preparation permit nearly complete control over the critical molecular design parameters, such as size, shape, surface/interior chemistry, flexibility, and topology. Dendrimers are well defined, highly branched macromolecules that radiate from a central core and are synthesized through a stepwise, repetitive reaction sequence that guarantees complete shells for each generation, leading to polymers that are mono-disperse. This property of dendrimers makes it particularly natural to coarsen interactions in order to simulate dynamic processes occurring at larger length and longer time scales. In this paper, we describe methods to construct 3-dimensional molecular structures of dendrimers (Continuous Configuration Boltzmann Biased direct Monte Carlo, CCBB MC) and methods towards coarse graining dendrimer interactions (NEIMO and hierarchical NEIMO methods) and representation of solvent dendrimer interactions through continuum solvation theories, Poisson–Boltzmann (PB) and Surface Generalized Born (SGB) methods. We will describe applications to PAMAM, stimuli response hybrid star-dendrimer polymers, and supra molecular assemblies crystallizing to A15 colloidal structure or Pm6m liquid crystals.

Denniston C., Orlandini E., Yeomans J.M.

Lattice Boltzmann simulations are used to explore the behavior of liquid crystals subject to Poiseuille flow. In the nematic regime at low shear rates we find two possible steady state configurations of the director field. The selected state depends on both the shear rate and the history of the sample. For both director configurations there is clear evidence of shear-thinning, a decrease in the viscosity with increasing shear rate. Moreover, at very high shear rates or when the order parameter is large, the system transforms to a log-rolling state with boundary layers that may exhibit oscillatory behavior.

Wang Z., Lupo J.A., Patnaik S., Pachter R.

Molecular dynamics simulations for 4- n -pentyl-4′-cyanobiphenyl (5CB) with as many as 944 molecules are reported. The order- N fast multipole method (FMM) is used to treat the long-range interactions. For a droplet of 944 molecules, the simulation shows a correlation between the droplet shape and the nematic order and a strong surface effect; little nematic order is found in a 118 molecule droplet. Simulations of the bulk system result in similar order parameters for both the 118 and 944 molecular ensembles. Although the nematic–isotropic transition was not observed at temperatures as high as 400 K using the CHARMM force field, a modification of the force field using ab initio determined partial atomic charges lowers the order parameters.

Katti D.R., Katti K.S., Sopp J.M., Sarikaya M.

Nacre (mother-of-pearl), the inner layer of seashells is a ceramic laminated biocomposite with exceptional mechanical properties of fracture toughness and strength. The organic layers in the composite play a significant role in the mechanical response of nacre to stresses. In this work, three dimensional finite element models of nacre (constructed in our previous work to design ‘brick and mortar’ micro-architecture of nacre) were used to study influence of nonlinear response of organic component. In this work, nonlinear elasto-plastic models for organic component are applied to model the mechanical response of nacre. Nanoscale material parameters (elastic modulus and hardness) were obtained using nanoindentation experiments. The yield stress of the organic was maintained at 40×10−6, 50×10−6, 60×10−6, 80×10−6, 120×10−6, 240×10−6, 320×10−6, and (40–400 MPa). The choice of initial value of yield stress of organic phase is the onset of nonlinearity in nacre response at that value. Tensile tests were simulated for each of these values of yield stress of organic phase under identical loading conditions of in increments of For each value of organic phase yield stress stress–strain response of nacre is plotted. The resulting yield stress of nacre was compared to experimentally obtained value. This indicates that a much higher yield stress of organic is necessary to obtain the experimentally obtained yield stress of nacre. Microstructural implications of this result are suggested.

Fried J.R., Li B.

Molecular dynamics has been used to determine the glass transition temperature of the amorphous phase of five di-substituted polysilanes from plots of specific volume versus temperature. In each case, good agreement was obtained between the simulation values and the reported DSC results. The effect of amorphous cell dimensions and equilibration time on Tg has been investigated. The use of larger cells provides better agreement with experimental Tg and probably more accurate densities as suggested by earlier studies. The effect of pressure on the Tg of two different polysilanes was also investigated. Although experimental data for comparison is unavailable, values obtained for dTg/dp are consistent with those reported for other polymers. Vectorial autocorrelation analysis was used to explore the mobility of the polysilane main chains and side groups relative to polyalkanes, polyphosphazenes, and polysiloxanes.