The role of entanglements

On the other hand, microhardness studies of short paraffin crystals reveal the influence of the molecular packing and hence of the chain ends on b (Ania et al, 1986). 

The crystal thickness c and the surface free energy of the PEO crystals may be changed simultaneously by changing the molecular weight of the samples. 

It should be noted that with increasing molecular weight there is a transition from extended-chain to folded-chain crystals. This transition must dramatically affect the surface free energy of the crystals. As pointed out by Buckley Kovacs (1976) and Balta Calleja et al. (1969), when the crystals have extended chains there is a high proportion of hydrogen bond formation between the -OH groups at adjacent chain ends. 

Let us next compare the variation of the b and b parameters with molecular weight and analyse the relationship, b/b, between them. The initial decrease in b is confirmed by a similar decrease of b with increasing molecular weight. 

It is noteworthy that the ratio b/b (Table 4.2) does not remain constant as in the case of the samples crystallized at different temperatures it diminishes with increasing molecular weight. The reason for this is that b diminishes faster with M than does b. Since Og is the same for both b and b, the difference in the variation of these parameters with M must originate in the different variations of Ah and Ah/ with M . As Ahf should have a constant value which is independent of M the observed variation in b/b must be due to changes in Ah. One possible explanation for this effect may lie in the occurrence of tie molecules between adjacent crystals. Tie molecules may be expected to appear with increasing molecular weight, thus providing a reinforcement of the material which could contribute to an increase in 

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We have speculated that when the melt is kept above the melting temperature, ve gradually increases with an increase of At and approaches the equilibrium ve (ve = 1). It was shown that the melt relaxation process takes a long time, i.e., it takes several hours or a few days depending on the annealing temperature (Tmax) and Mn. However, this experimental fact was indirect evidence of the role of entanglement in the nucleation of polymers. 

These considerations, while far from complete, suggest that the role of entanglements in the equilibrium force is fairly small. Chains do not become strongly entangled with one another simply because they tend not to reverse their directions frequently. 

F. J. Balta -Calleja, C. Santa Cruz, R. K. Bayer, and H. G. Kilian, Microhardness and Surface Free Energy in Linear Polyethylene The Role of Entanglements, Colloid Polymer Sci., 268,440 (1990). 

The findings described above form a basis for our current work that examines the influence of the interphase on the structure and ultimately on the properties of linear polymers. In this chapter we will first show that, depending on the crystallization conditions, the amount of loops or entanglements in the interphase, and thus the deformation behavior of polymers, can be varied. We will initially consider the example of ultrahigh molecular weight polyethylene (UHMW-PE) and the role of entanglements upon its drawability. 

As we have outlined above, it is important to consider the role of entanglements within the interphase. Entanglements as such are ill-defined topological constraints, which are usually visualized in textbooks as four strands leading away from a mutual contact, see Fig. 2. 

What is the role of entangled states inside the cavity Terms, for example, such as those in 3) and 4) appear as linear superpositions (omitting common factors) ... 

To conclude, we may say that the role of entanglement in quantum information processing is not yet well understood. 

Ever since Einstein, Podolsky, and Rosen in their seminal paper from 1935 [Einstein 1935] introduced the possibility of entangling two quantum system, entanglement has been viewed as one the most curious and spectacular phenomena in quantum mechanics. In the past few years the role of entanglement in quantum mechanics has shifted dramatically from being a fundamental test of the foundation of the entire quantum mechanical theory to being a techni-... 

Let us now consider an interface A/ B between two molten polymers, with 1 > X K X is large, there will be no entanglements between A and B, and slippage may occur. This problem was first considered by Furukawa ( 8) -but without a full appreciation of the role of entanglements versus Rouse friction. A slightly improved (qualitative) discussion is given in ref. (29) and ill be summarized here. 

The mechanical properties of polymer/polymer interfaces are clearly very sensitive to the detailed structure of the interface. We have seen here two major examples of this correlation a) the role of chain ends, and of their spatial distribution in A/A healing b) the role of entanglements in A/B fracture or in A/B slippage. 

When M < Me, the role of entanglements is no longer present and Eqns. 2 and 3 cannot be used since Gie = 0 at Me. However, the Nail solution applies for weak interfaces (see Eqn. 3 in Polymer-polymer adhesion models) and the chain segments simply pullout at fracture such that (Gic-Gq) M, where Gq is the surface energy term, and we obtain. 

A second model has similar static properties (such as d(s), eqn ]44]) to the KG MD model, but softer bonded and nonbonded potentials, allowing easy chain crossing. We will use this model mainly in the next section to elucidate the role of entanglements. The potentials are given by... 

Also, the theory does not directly address the role of entanglements, but treats them in the manner of Flory and Erman (12-13) as imposing restrictions on crosslink mobility. We regard it as an open question as to whether this approach is adequate for accounting for the effect of entanglements and its local variation on influencing local swelling. 

Chains in Networks. One of the first studies of chains in networks is by Gao and Weiner (235) where they performed extended simulations of short chains with fixed (affinely moving) end-to-end vectors. The first extensive molecular dynamics simulations of realistic networks were performed by Kremer and collaborators (236). These calculations were based on a molecular dynamics method that has been applied to study entanglement effects in polymer melts (237). The networks obtained by cross-linking the melts were then used to study the effect of entanglements on the motion of the cross-links and the moduU of the networks. The moduli calculated without any adjustable parameters were close to the phantom network model for short chains, and supported the Edwards tube model for long ones. Similar molecular dynamics analyses were used to understand the role of entanglements in deformed networks in subsequent studies (238-240). 

Balth Calleja F J, Santa Cruz C, Bayer R K and Kilian H G (1990) Microhardness and surface free energy in linear polyethylene the role of entanglements, Colloid Polym Sci 268 440-446. 

Using trouser tear tests, Gent and Tobias [21] showed that the Lake-Thomas model held reasonably well for a number of elastomers. In their analysis, they took the molecular mass of a strand as that extracted from the elastic modulus and thereby incorporated the effect of entanglements (Section 9.8.2). More recent results on poly(dimethyl siloxane) (PDMS) elastomers using simple shear tests [22,23] over a wide range of structure (extent of cross-links and trapped entanglements) indicate that the role of entanglements in elastomer fracture needs to be elucidated further. 

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