Polymer packing models

The parameters l0, Kb, 0o, K , K,p, n, 8, ay, by, qit qy and r belong to the fit parameters, which can be determined by fitting of Equation 1.1 to a sufficient set of data calculated by QM and/or determined experimentally (e.g., X-ray scattering, IR spectroscopy, heats of formation). From a numeric point of view the pair interaction terms (van der Waals and Coulomb) are most demanding. In this connection the typical size of polymer packing models is limited to typically 3000-10000 atoms (leading to lateral sizes of bulk models of a few nm), although in other connections now also models with up to 100000 atoms have been used. 

Equation 1.3 represents a system of usually several thousand coupled differential equations of second order. It can be solved only numerically in small time steps At via finite-difference methods [16]. There always the situation at t + At is calculated from the situation at t. Considering the very fast oscillations of covalent bonds, At must not be longer than about 1 fs to avoid numerical breakdown connected with problems with energy conservation. This condition imposes a limit of the typical maximum simulation time that for the above-mentioned system sizes is of the order of several ns. The limited possible size of atomistic polymer packing models (cf. above) together with this simulation time limitation also set certain limits for the structures and processes that can be reasonably simulated. Furthermore, the limited model size demands the application of periodic boundary conditions to avoid extreme surface effects. 

In the packing model [50,62,68] the entanglement distance is interpreted by the gradual build-up of geometrical restrictions due to the existence of other chains in the environment or, more precisely, the entanglement distance is determined by a volume which must contain a defined number of different chains. This approach is based on the observation that, for many polymer chains, the product of the density of the chain sections between entanglements is... 

The already mentioned limited lateral dimensions of packing models of just several nm makes it impossible to simulate complete membranes or other polymer-based samples. Therefore, on the one hand, bulk models are considered that are typically cubic volume elements of a few nanometers side length that represent a part cut out of the interior of a polymer membrane (cf. Figure 1.1). On the other hand interface models are utilized, for example, for the interface between a liquid feed mixture and a membrane surface or between a membrane surface and an inorganic filler (cf. Figure 1.2). 

Figure 1.1 Atomic representation of a typical 3-dimensional packing model (thickness about 3 A) starting with a single Hyflon AD60X polymer chain. Atom colors gray = carbon, red = oxygen, light blue = fluorine [15].

D. Gowanlock, R. Bailey, and F. F. Cantwell, Intra-particle sorption rate and liquid chromatographic bandbroadening in porous polymer packings I. Methodology and validation of the model, /. Chromatogr. A 726 (1996), 1-23. 

M. Heuchel, D. Fritsch, P. M. Budd, N. B. McKeown, D. Hofmann, Atomistic packing model and free volume distribution of a polymer with intrinsic microporosity (PIM-1), J. Membr. Sci., 318, 84-99 (2008). 

The construction of a correct model for the polymer structure is the prerequisite for obtaining accurate results and it is always the first step in the FV modelling. Nowadays several well established simulation methods exist for the preparation of atomistic packing models, both for high and for low free volume polymers [33]. In molecular modelling of amorphous polymers, usually a cubic characteristic volume element is filled with polymer... 

Figure 4.7 Representative slice of an atomistic packing model of ffyflon AD80X. The polymer chains are represented by the small sticks. The free volume elements are shown as the darker continuous regions, represented by a collection of neighbouring and overlapping spheres. Slice thickness ca. 5A...

The question about the conformations of main chains in many side-chain and main-chain LC polymers remains unsolved for various types of mesophases. Without a doubt, this problem is of prime importance for the consideration of an adequate model treating the packing of all structural elements of such LC polymers. This model is necessary for optimization of the molecular design and synthesis of LC polymers with desired optical and other physicochemical properties. 

Me represents the molecular weight that must be reached in order to have the zero shear viscosity follow the scaling of Figure 10.10. Rheological measurements combined with the packing model allow a comparison of various molecular parameters for PLA with other known polymer structures. 

Fig. 3.6 2D-GIXD patterns, AFM height images and cartoon illustration of the proposed packing models of a, d, and g PII2T, b, e, and h PCII2T and c, f, and 1 PCII2Se films. Films were prepared by spin-coating from the DCB solutions of the polymers (6 mg/mL) and annealed at 180 °C...

Finally - as a word of caution - it is to emphasize that the concept summarized by Fig. 3 is not meant to exclude other possible packing modes which may lead to the electronic interactions necessary to create an organic metal. It is nothing but a helpful concept to point out a packing mode which has often been encountered and to emphasize the possibilities and consequences of electronic interchain interactions in "doped polymers. The model does not say, that conductivity arises solely from interactions between segments of adjacent chains but it stresses the importance of contributions of such interactions to the total electric and electronic behaviour. 

We begin with the most rigorous version of self-consistent PRISM based on a Monte Carlo evaluation of the effective single-chain problem. Theoretical predictions of Grayce and co-workers" are compared with many-chain simulation results for the mean-square end-to-end distance of the hard-core chain model as a function of polymer packing fraction in... 

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