Model flexible polymer

Fig. 8.8 The bond fluctuation model. In this example three bcmds in the polymer arc incorporated into a singk effecti bond between effective moncmers . (Figure adapted from Baschnagel J, K Binder, W Paul, M Laso, U Sutcr, I Batouli [N ]ilge and T Burger 1991. On the Construction of Coarse-Grained Models for Linear Flexible Polymer-Chains -Distribution-Functions for Groups of Consecutive Monomers. Journal of Chemical Physics 93 6014-6025.)...

It is somewhat difficult conceptually to explain the recoverable high elasticity of these materials in terms of flexible polymer chains cross-linked into an open network structure as commonly envisaged for conventionally vulcanised rubbers. It is probably better to consider the deformation behaviour on a macro, rather than molecular, scale. One such model would envisage a three-dimensional mesh of polypropylene with elastomeric domains embedded within. On application of a stress both the open network of the hard phase and the elastomeric domains will be capable of deformation. On release of the stress, the cross-linked rubbery domains will try to recover their original shape and hence result in recovery from deformation of the blended object. 

These models are designed to reproduce the random movement of flexible polymer chains in a solvent or melt in a more or less realistic way. Simulational results which reproduce in simple cases the so-called Rouse [49] or Zimm [50] dynamics, depending on whether hydrodynamic interactions in the system are neglected or not, appear appropriate for studying diffusion, relaxation, and transport properties in general. In all dynamic models the monomers perform small displacements per unit time while the connectivity of the chains is preserved during the simulation. 

G. Guillot, L. Leger, F. Rondelez. Diffusion of large flexible polymer chains through model porous membranes. Macromolecules 5 2531-2537, 1985. 

Equation (23) predicts a dependence of xR on M2. Experimentally, it was found that the relaxation time for flexible polymer chains in dilute solutions obeys a different scaling law, i.e. t M3/2. The Rouse model does not consider excluded volume effects or polymer-solvent interactions, it assumes a Gaussian behavior for the chain conformation even when distorted by the flow. Its domain of validity is therefore limited to modest deformations under 0-conditions. The weakest point, however, was neglecting hydrodynamic interaction which will now be discussed. 

The non-free draining character of flexible polymer chains was considered in the Zimm model [48], In this model, the effect of hydrodynamic interaction at the location of bead i is taken into account by an additional fluid velocity term vj ... 

The Zimm model predicts correctly the experimental scaling exponent xx ss M3/2 determined in dilute solutions under 0-conditions. In concentrated solution and melts, the hydrodynamic interaction between the polymer segments of the same chain is screened by the host molecules (Eq. 28) and a flexible polymer coil behaves much like a free-draining chain with a Rouse spectrum in the relaxation times. 

Biomimetic materials are materials that are modeled after naturally occurring materials. Gels or flexible polymers (Chapter 19) modeled after natural membranes and tissues are biomimetic materials with remarkable properties. Some can be made to crawl on their own like tiny nanometer worms, others pulsate to an internally generated rhythm, and still others respond in seemingly lifelike ways to stimuli. 

The examples outlined above are intended to show the utility of a generalized computer model for polymer processing problems. Such a model is able to adapt itself to a wide variety of situations simply by adjustments to the input dataset, rather than requiring alterations to the code itself. This flexibility makes the code somewhat more difficult to learn initially, but this might be minimized by embedding the finite-element code itself in a more "user-friendly" graphics-oriented shell. 

Diffusion of flexible macromolecules in solutions and gel media has also been studied extensively [35,97]. The Zimm model for diffusion of flexible chains in polymer melts predicts that the diffusion coefficient of a flexible polymer in solution depends on polymer length to the 1/2 power, D N. This theoretical result has also been confirmed by experimental data [97,122]. The reptation theory for diffusion of flexible polymers in highly restricted environments predicts a dependence D [97,122,127]. Results of various... 

Mapping Atomistically Detailed Models of Flexible Polymer Chains in Melts to Coarse-Grained Lattice Descriptions ... 

We will then examine other flexible polymer crystallization instances which may be interpreted, at least qualitatively, in terms of the bundle model. We will concentrate on crystallization occurring through metastable mesophases which develop by quenching polymers like isotactic polypropylene, syndiotactic polypropylene etc. In principle also hexagonal crystallization of highly defective polymers, and order developing in some microphase-separated copolymer systems could be discussed in a similar perspective but these two areas will be treated in future work. A comparison between the bundle approach and pertinent results of selected molecular simulation approaches follows. 

Simulation methods have been proved to be useful in the study of many different molecular systems, in particular in the case of flexible polymers chains [ 14]. According to the variety of structures and the theoretical difficulties inherent to branched structures, simulation work is a very powerful tool in the study of this type of polymer, and can be applied to the general problems outHned above. Sometimes, this utility is manifested even for behaviors which can be explained with simple theoretical treatments in the case of linear chains. Thus, the description of the theta state of a star chain cannot be performed through the use of the simple Gaussian model. The adequate simulation model and method depend strongly on the particular problem investigated. Some cases require a realistic representation of the atoms in the molecular models [10]. Other cases, however, only require simplified coarse-grained models, where some real mon-... 

A basic theoretical model for flexible polymers is the Gaussian chain which assumes N ideal beads with intramolecular distance between them following a Gaussian distribution, so that the mean quadratic distance between two beads separated by n-1 ideal and not correlated bonds is given by [ 15,20]... 

In an early attempt to model the dynamics of the chromatin fiber, Ehrlich and Langowski [96] assumed a chain geometry similar to the one used later by Katritch et al. [89] nucleosomes were approximated as spherical beads and the linker DNA as a segmented flexible polymer with Debye-Huckel electrostatics. The interaction between nucleosomes was a steep repulsive Lennard-Jones type potential attractive interactions were not included. 

Recently, monoclonal antibodies were attached to gas-filled microbubbles using this spacer coupHng technology. Testing in vitro, in a cell culture system, demonstrated selective accumulation of decafluorobutane-based lipid shell MP1950 microbubbles with covalently attached anti-lCAM-1 antibodies, onto the surface of activated endotheUum [8]. Anti-P-selectin antibodies attached to such microbubbles via an avidin-biotin scheme showed selective accumulation in the areas of inflammation and ischemia/reperfusion injury in an animal model [93]. In the latter example, biotin was also connected to the microbubble surface via a flexible polymer spacer arm. 

Free-draining models were among the first to be considered [14-18]. For flexible polymer chains of sufficient length, [77] behaves as if the polymer coil occupied a spherical volume through which the solvent cannot flow. Under these conditions,... 

The distinct properties of liquid-crystalline polymer solutions arise mainly from extended conformations of the polymers. Thus it is reasonable to start theoretical considerations of liquid-crystalline polymers from those of straight rods. Long ago, Onsager [2] and Flory [3] worked out statistical thermodynamic theories for rodlike polymer solutions, which aimed at explaining the isotropic-liquid crystal phase behavior of liquid-crystalline polymer solutions. Dynamical properties of these systems have often been discussed by using the tube model theory for rodlike polymer solutions due originally to Doi and Edwards [4], This theory, the counterpart of Doi and Edward s tube model theory for flexible polymers, can intuitively explain the dynamic difference between rodlike and flexible polymers in concentrated systems [4]. 

With increasing polymer concentration, we may expect that the polymer global motion changes from the fuzzy cylinder model mechanism to the repta-tion model mechanism. The onset of the crossover should depend on the degree in which the lateral motion of a polymer chain is suppressed by entanglement with its surrounding chains, but it is difficult to estimate this degree. There are some disputes over it in the case of flexible polymers [20]. 

The equilibrium structure is considered of flexible polymer chains within the RIS model. This model is solved by an irreducible tensor method which is somewhat different from, and simpler than, the approach of Flory and others. The results are used to compute the light scattering intensities from dilute solutions of flexible polymer chains, and the angle dependence is found... 

An approximate analytical solution is presented for the RIS model of flexible polymer configurations. The method is applied to calculate and . 

In an earlier paper (S 044], anomalies in the angular moments of the monomer-monomer distribution function for flexible polymers are established. It is shown here how these anomalies arise from the tetrahedral symmetry of the three-state RIS model and how they disappear in the continuum limit of torsional conformations. It is concluded that the eighth and higher radial moments contain spurious contributions when calculated within the usual three-state model. 

The conformational entropies of copolymer chains are calculated through utilization of semiempirical potential energy functions and adoption of the RIS model of polymers. It is assumed that the glass transition temperature, Tg, is inversely related to the intramolecular, equilibrium flexibility of a copolymer chain as manifested by its conformational entropy. This approach is applied to the vinyl copolymers of vinyl chloride and vinylidene chloride with methyl acrylate, where the stereoregularity of each copolymer is explicitly considered, and correctly predicts the observed deviations from the Fox relation when they occur. It therefore appears that the sequence distribution - Tg effects observed in many copolymers may have an intramolecular origin in the form of specific molecular interactions between adjacent monomer units, which can be characterized by estimating the resultant conformational entropy. 

Accordingly, given the necessity from equilibrium coil dimensions that bt> 1, the shear rate and frequency departures predicted by FENE dumbbells are displaced from each other. Moreover, the displacement increases with chain length. This is a clearly inconsistent with experimental behavior at all levels of concentration, including infinite dilution. Thus, finite extensibility must fail as a general model for the onset of nonlinear viscoelastic behavior in flexible polymer systems. It could, of course, become important in some situations, such as in elongational and shear flows at very high rates of deformation. 

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