Loops, and entanglements

Free chain ends (unreacted functionalities) reduce the number of active network chains in a network compared with the same network without free ends. Disregarding possible presence of loops and entanglements, C — 1 C crosslinks are necessary according to Flory (55) to connect C chains into one giant macromolecule. Additional crosslinks will be elastically effective. Their number is given by... 

Increase in number of loopings and entanglements between two or more molecular chains (physical cross-linking)... 

Fig. 4.34 Schematic diagram of the molecules in an elastomer showing normal crosslinking, chain ends, intramolecular loops and entanglements.

In addition to the chemical nature of its constituting monomeric units, a polymer network can be defined through two essential parameters, namely, the number (v) of its elastic chains and the number (p.) of its junction points or cross-links. According to their definition, elastic chains are connected by their two ends to junction points from which emanate at least two other elastic chains. However, a network not only is a collection of v chains connecting p junction points, but also contains also a fraction of loose ends, or dangling chains, loops, and entanglements. A model network is defined as a defect-free one where all cross-links have the same functionality and all elastic chains are of the same size (identical number of monomeric units). 

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. 

Networks obtained by anionic end-linking processes are not necessarily free of defects 106). There are always some dangling chains — which do not contribute to the elasticity of the network — and the formation of loops and of double connections cannot be excluded either. The probability of occurrence, of such defects decreases as the concentration of the reaction medium increases. Conversely, when the concentration is very high the network may contain entrapped entanglements which act as additional crosslinks. It remains that, upon reaction, the linear precursor chains (which are characterized independently) become elastically effective network chains, even though their number may be slightly lower than expected because of the defects. 

Here the 2 in the denominator is to avoid double counting because it takes two sites to form one link. However closed loops and chain entanglements are both possibilities and Equation (2.57) must be modified for these effects. We can write the network modulus as... 

In 1944, Flory (3) noted that the moduli of cross-linked butyl rubbers generally differ somewhat from values calculated from the crosslink density according to the kinetic theory of rubber elasticity. In many cases, the modulus also depends on the primary (uncross-linked) molecular weight distribution of the polymer. He attributed both observations to three kinds of network defects chain ends, loops, and chain entanglements. The latter are latent in the system prior to cross-linking and become permanent features of the network when cross-links are added. 

Other imperfections are developed in the process of crosslinking network defects (unreacted functionalities, intramolecular loops, chain entanglements), inhomogeneity in crosslink distribution, or heterogeneity of the network due to phase separation. These four types of network imperfections are interdependent and a sharp borderline between them does not exist. 

Even if completely homogeneous and disordered in the relaxed state, a real network differs from the ideal network, defined in Chapter I. Three types of network defects are commonly considered to be present in polymer networks unreacted functionalities, closed loops, and permanent chain entanglements. Within each group there are several possibilities dependent on the arrangement of chains the effect of defects on the elastic properties of the network is thus by no means simple, as has been stressed e.g. by Case (28). Several possible arrangements are shown in Fig. 1, where only nearest neighbour defect structures have been drawn. 

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. 

Fig. 1 Chemical and physical crosslinks associated with covalently bonded polymer gels. (A) Bi- and trifunctional chemical crosslinks (B) simple and trapped physical entanglements and (C) ineffective chemically bonded loop and dangling ends.

Brown has proposed a simple model to calculate the amount of coupling by entangled loops and found that, in the PMMA/random copolymer system, the model predicted a higher toughness than that observed for narrow interfaces. He suggested that this deviation could be caused by a decrease in entanglement density close to narrow interfaces. 

A similar study was undertaken by Beck Tan et al. on the adhesion between poly(styrene-r-sulfonated styrene) and poly(2-vinylpyridine). In this case, however, the variable was the mole fraction of sulfonated styrene in the random copolymer [95]. The results of the maximum fracture toughness Qc vs. mole fraction of functional groups are plotted in Fig. 51. The reinforcement shows a very sharp maximum with degree of functionalization consistent with the multiple stitching giving rise to short loops poorly entangled with the homopolymer however, in this case as well, the bulk properties of PS are modified by the presence of the styrenesulfonic acid and this could contribute to the decrease in Qc at high levels of functionalization. 

In addition to terminal chains and entanglements, there are other types of network imperfections. Figure 6-2 shows that if a short chain were crosslinked only once, the crosslink is a wasted one because the chain cannot support elastic stress. Also, if a crosslink forms an intrachain loop, it is again an ineffective crosslink. Unfortunately, owing to its very complexity, it is at present impossible to completely characterize the network structure of an elastomer. 

The main difference between physical entanglements (loops) and chemical crosslinks is that the former allow creeping of chains. The density of both types of link are determined using the same rubber elasticity equation, a factor of 0.8 being frequently added to take creeping into account [75]. 

The number average molar mass of a chain section between two junction points in the network is an important factor controlling elastomeric behavior when is small, the network is rigid and exhibits limited swelling, but when is large, the network is more elastic and swells rapidly when in contact with a compatible liquid. Values of can be estimated from the extent of swelling of a network, which is considered to be ideal but rarely is, and interpretation of the data is complicated by the presence of network imperfections. A real elastomer is never composed of chains linked solely at tetrafunctional junction points but will inevitably contain defects such as (1) loose chain ends, (2) intramolecular chain loops, and (3) entangled chain loops. 

FIGURE 14.8 Diagram showing defects in an elastomeric network. (A) Loose chain ends, (B) intramolecular chain loops, and (C) entangled chain loops. 

As the seed crystal meets the gel layer on the rotor surface, the adsorbed polymer molecules make loops extending into the solution and other molecules become entangled with those loops and with each other. Hms, a network with physical crosslinks adheres to the rotor sxuface. As shearing motion is encoimtered, the network of adsorbed molecules is extended and small embryonic crystallites will... 

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