Macroscopic model, polymer

Bridging Scales from Microscopic Through Semi Macroscopic Models of Polymers... 

In this review, the state of the art of the bridging of the gap between quantum chemical, atomistic, coarse-grained (and almost macroscopic) models of polymers has been discussed. Simulations with coarse-grained models provide the promise of the equilibration of models of dense amorphous polymers, whereas such equilibration is extremely difficult if the models are expressed in fully atomistic detail. The review presents the status of this rapidly developing field as of the beginning of 1998. A few minor additions were incorporated in the page proof, early in 2000, in response to suggestions from the reviewer. 

It should be noted that the rigid band model and the tunnelling process discussed above are an idealization of the real device. It is unlikely that the barrier is exactly triangular as sketched in Fig. 5.2. The results presented in this book are aimed to give a further insight into the microscopic features of the metal-polymer interfaces and how these can be related to the macroscopic models such as the relations above. 

Proton diffusion can occur via two mechanisms, structural diffusion and vehicle diffusion [37]. It is the combination of these two diffusion mechanisms that confers protonic defects exceptional conductivity in liquid water. The conductivity of protons in aqueous systems of bulk water can be viewed as the limiting case for conductivity in PFSA membranes. When aqueous systems interact with the environment, such as in an acidic polymer membrane, the interaction reduces the conductivity of protons compared to that in bulk water [37]. In addition to the mechanisms described above, transport properties and conductivity of the aqueous phase of an acidic polymer membrane will also be effected by interactions with the sulfonate heads, and by restriction of the size of the aqueous phase that forms within acidic polymer membranes [32]. The effects of the introduction of the membrane can be considered on the molecular scale and on a longer-range scale, see Refs. [16, 32]. Of particular relevance to macroscopic models are the diffusion coefficients. As the amount of water sorbed by the membrane increases and the molecular scale effects are reduced, the properties approach those of bulk water on the molecular scale [32]. 

We have presented a review of experimental and macroscopic modelling aspects of transport phenomena in polymer electrolyte membranes. This included examination of the connection between the hydration scheme and the behaviour of the membrane, a discussion of the so-called Schroeder s paradox, and the influence of the membrane phase on transport mechanisms. We also provided a critical examination of various approaches to modelling transport phenomena in membranes, and established that binary friction model provide a correct and rational framework for modelling membrane transport. 

The Phantom Model. In this model polymer chains are allowed to move freely through one another and the network junctions fluctuate around their mean positions [3,91-93], The conformation of each chain depends only on the position of its ends and is independent of the conformations of the surrounding chains with which they share the same region of space. The junctions in the network are free to fluctuate around their mean positions and the magnitude of the fluctuations is strain invariant. The positions of the junctions and of the domains of fluctuations deform affinely with macroscopic strain. The result is that the deformation of the mean positions of the end-to-end vectors is not affine in the strain. This is because it is the convolution of the distribution of the mean positions (which is affine) with the distribution of the fluctuations (which is strain invariant, i.e., nonaffine). The elastic free energy of deformation is given by... 

Macroscopic Models of Diffusion in Porous Polymer Matrices... 

When macroscopic diffusion models are compared to experimental data, high values of tortuosity (corresponding to low values of Detr) are obtained (Tables I and II). For a random porous medium, tortuosity values due to windiness ofthe diffusional path should be between 1 and 3 (Pismen, 1974 Bhatia, 1986). Because the overall tortuosities predicted by the macroscopic models are much larger, other physical properties ofthe matrix must influence the rate of protein diffusion within the matrix. Microgeomet-ric models and percolation theory have been used to study the factors that might control protein diffusion within these polymer matrices. 

One of the main challenges of polymer dynamics is to map a multichain model (which presumably can describe the real chain as detailed as necessary) onto a single-chain model, which is simpler and can eventually be solved analytically or mapped to a tube model to produce constitutive equations for the macroscopic modeling (see Figure 36). 

A. Weber and J. Newman. Macroscopic modeling of polymer-electrolyte membranes. In T. S. Zaho, K.-D. Kreuer, and T. Van Nguyen, editors. Advances in Phiel Cells, volume 1, pages 47 118, Elsevier, Amsterdam, 2007. 

In section 14.2 we encountered numerous macroscopic models already in use to model the effect of fire damage on the structural integrity of polymer composite components in a variety of common applications. Such models belong to the fire-endurance category and normally consist of finite element computer programmes, which are primarily designed to deal with mechanical properties, and are normally decoupled from the thermal modeling we have discussed in section 14.3. 

Many simulations attempt to determine what motion of the polymer is possible. This can be done by modeling displacements of sections of the chain, Monte Carlo simulations, or reptation (a snakelike motion of the polymer chain as it threads past other chains). These motion studies ultimately attempt to determine a correlation between the molecular motion possible and the macroscopic flexibility, hardness, and so on. 

Molecular dynamics, in contrast to MC simulations, is a typical model in which hydrodynamic effects are incorporated in the behavior of polymer solutions and may be properly accounted for. In the so-called nonequilibrium molecular dynamics method [54], Newton s equations of a (classical) many-particle problem are iteratively solved whereby quantities of both macroscopic and microscopic interest are expressed in terms of the configurational quantities such as the space coordinates or velocities of all particles. In addition, shear flow may be imposed by the homogeneous shear flow algorithm of Evans [56]. 

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