Entanglements, diffusion

Parallel studies on PAMAM dendrimers, the Frechet type polyether den-drons, and other dendrimer families have generated an extensive list of unique properties driven by the dendritic state/ Figure 1.18 compares several significant physical property differences between the linear and dendritic topologies related to conformations, crystallinity, solubilities, intrinsic viscosities, entanglement, diffusion/mobility and electronic conductivity. 

Figure 7.25. Comparative chromatograms illustrating the effect of pore size on protein separation. Two larger proteins (peaks 5 and 6) show substantial peak broadening due to entangled diffusion within the smaller pores of the second column. Reprinted with permission from reference 35.

Superficial reducte mobile phase veM [Eq. (1-3) see Appendix 1] Entangled diffusion paiameter for small molmlet (Eq. (41)] ReetrictetMiffuskm paiameter, equal to (P /DJ (Eq. (68)]... 

In dilute solutions, tire dependence of tire diffusion coefficient on tire molecular weight is different from tliat found in melts, eitlier entangled or not. This difference is due to tire presence of hydrodynamic interactions among tire solvent molecules. Such interactions arise from tire necessity to transfer solvent molecules from tire front to tire back of a moving particle. The motion of tire solvent gives rise to a flow field which couples all molecules over a... 

The tube is a construct which we might continue to sketch around an emerging chain as it diffuses out of the original sleeve. Instead, it is convenient to start with the tube initially in place and consider how long it takes for the molecule to escape. The initial entanglements which determine the contours of the tube comprise a set of constraints from which the molecule is relaxing, even if only to diffuse into another similar set. Accordingly, we identify this reptation time as a relaxation time r for the molecule. 

The alcohol swells the poly (ethyl methacrylate) beads, rapidly promoting diffusion of the plasticizer into the polymer. As a result of the polymer-chain entanglement, a gel is formed. The conditioner is applied to the denture and provides a cushioning effect alcohol and plasticizer are slowly leached out, and the material becomes rigid. To ensure resiliency, the conditioner must be replaced after a few days. Some materials exhibit high flow over a short period compared with others with low initial flow the latter remain active longer. 

In Section 4.2.2 the central role of atomic diffusion in many aspects of materials science was underlined. This is equally true for polymers, but the nature of diffusion is quite different in these materials, because polymer chains get mutually entangled and one chain cannot cross another. An important aspect of viscoelastic behavior of polymer melts is memory such a material can be deformed by hundreds of per cent and still recover its original shape almost completely if the stress is removed after a short time (Ferry 1980). This underlies the use of shrink-fit cling-film in supermarkets. On the other hand, because of diffusion, if the original stress is maintained for a long time, the memory of the original shape fades. 

The interdiffusion of polymer chains occurs by two basic processes. When the joint is first made chain loops between entanglements cross the interface but this motion is restricted by the entanglements and independent of molecular weight. Whole chains also start to cross the interface by reptation, but this is a rather slower process and requires that the diffusion of the chain across the interface is led by a chain end. The initial rate of this process is thus strongly influenced by the distribution of the chain ends close to the interface. Although these diffusion processes are fairly well understood, it is clear from the discussion above on immiscible polymers that the relationships between the failure stress of the interface and the interface structure are less understood. The most common assumptions used have been that the interface can bear a stress that is either proportional to the length of chain that has reptated across the interface or proportional to some measure of the density of cross interface entanglements or loops. Each of these criteria can be used with the micro-mechanical models but it is unclear which, if either, assumption is correct. 

Adhesion development depends on diffusion of the CPO component of the primer through the crystalline boundary layers followed by swelling and entanglement with the rubber rich layer [75]. 

The mechanism by which the primers are thought to work is relatively straightforward. The primer first diffuses into the polyolefin surface, and subsequently becomes entangled in the polyolefin. The primer molecule can then act as an anchor in the substrate surface for the adhesive polymer, which forms after the primer initiates polymerization of the alkyl cyanoacrylate monomer [37]. 

Appropriately, this was called the Folded Chain Theory and is illustrated in Fig. A.ll. There are several proposals to account for the co-existence of crystalline and amorphous regions in the latter theory. In one case, the structure is considered to be a totally crystalline phase with defects. These defects which include such features as dislocations, loose chain ends, imperfect folds, chain entanglements etc, are regarded as the diffuse (amorphous) regions viewed in X-ray diffraction studies. As an alternative it has been suggested that crystalline... 

Fig. 7 gives an example of such a comparison between a number of different polymer simulations and an experiment. The data contain a variety of Monte Carlo simulations employing different models, molecular dynamics simulations, as well as experimental results for polyethylene. Within the error bars this universal analysis of the diffusion constant is independent of the chemical species, be they simple computer models or real chemical materials. Thus, on this level, the simplified models are the most suitable models for investigating polymer materials. (For polymers with side branches or more complicated monomers, the situation is not that clear cut.) It also shows that the so-called entanglement length or entanglement molecular mass Mg is the universal scaling variable which allows one to compare different polymeric melts in order to interpret their viscoelastic behavior. 

N. Nemoto, M. Kishine, T. Inoue, T. Osaki. Tracer diffusion of linear polystyrene in entanglement networks. Macromolecules 22 659-664, 1990. 

If no entanglements are present, the diffusion coefficient of polymer molecule is, according to Beuche (7 ), given as... 

The diffusion coefficients of entangled polymers in solution will most certainly depend on the viscosity of the medium and vice versa. It is reasonable therefore to expect that the diffusion coefficient would correlate well with the weight average molecular weight of the polymer. M is therefore used with equation (lO) giving... 

The remaining problem in the model development is to estimate the decrease in kp as a function of conversion. As the reaction proceeds beyond the point of chain entanglement, a critical conversion is reached where the propagation reaction becomes diffusion controlled and kp begins to fall with further increase in polymer concentration. At the critical conversion, one may write... 

Figure 6. Thermogravimetric analysis (TGA) of free 55 K PVP and 7.1 nm Pt-PVP nanoparticles in oxygen. Oxidative decomposition of free PVP begins at 573K, while significant weight loss due to the catalyzed oxidation of PVP on PVP-protected Pt nanoparticles occurs at 473 K. It appears that PVP layer is not a complete monolayer or the entanglement of PVP chains causes a porous polymer layer enabling oxygen diffusion to the nanoparticle surface [17]. (Reprinted from Ref [17], 2006, with permission from Springer.)...

Although this athermal bond fluctuation model is clearly not yet a model for any specific polymeric material, it is nevertheless a useful starting point from which a more detailed chemical description can be built. This fact already becomes apparent, when we study suitably rescaled quantities, such that, on this level, a comparison with experiment is already possible. As an example, we can consider the crossover of the self-diffusion constant from Rouse-like behavior for short chains to entangled behavior for longer chains. 

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