Entanglement

How do we describe the entanglements and how do we calculate the relaxation times of a chain and other dynamical properties of the entangled system The statistical mechanics of such a disordered system with the uncrossability constraints is very difficult to formulate. Nevertheless there have been three kinds of approaches. 

If there are two or more entanglement points per molecule, a type of network is set up. In order to flow, a molecule needs to pull neighboring molecules with which it is entangled. These molecules, in turn, may be coupled with others and so on throughout the system. When a stress is imposed on such a system, there is a resistance to flow so that the system resembles a cross-linked polymer. However, it is different from a cross-linked system in that, if the stress is maintained, slippage of the chains 

Non-cross-linked chain molecules, as well as chemically weakly cross-linked materials, show entropy-elastic behavior under given stresses. It is also conceivable that the chains of long, non-cross-linked, flexible macromolecules 

The Barus effect can be studied with the aid of the Bagley plot. Solving 

Equation (7-32) for the pressure p and replacing the shear stress oi for non-Newtonian liquids by oi = i7app1,  

In the Bagley plot, the pressure is plotted against the nozzle geometry Lj.R at a constant shear rate y. With Newtonian liquids, according to Equation (7-32), for L / = 0,/ will also be 0 the slope of the straight lines is 

Usually/ o is identified with the loss of pressure caused by the elastically accumulated energy of the flowing liquid and by the formation of a stationary flow profile at both ends of the capillary. Since the loss in pressure disappears at large LjR values, it may be assumed that the elastic deformation can equilibrate in time in very long capillaries (disentangling). This is also 

As we move away from dilute solutions to more concentrated systems, we can no longer look at isolated molecules, and a significant theoretical problem is How does one model intermolecular entanglement Before attempting to answer this question, let us look at some experimental evidence indicative of entanglement. 

Stress relaxation modulus for methyl acrylate-methyl methacrylate copolymers. Dashed line illustrates a typical Rouse theory prediction. Adapted from Fujino et al. (1961). 

Storage modulus versus frequency for narrow distri-totion polystyrene melts of increasing molecular weight, reduced to I60°C by temperature-frequency superposition. Molecular weight ranges from - 8900 (L9) to s 581,000 (L18). From Onogi et al. 1970). 

Entanglements have become part of the folklore of polymer science the idea they represent is a particular type of intermolecular interaction, to be distinguished from the coil overlap type of interaction mentioned earlier. Howev their exact topological charactm is quite difficult to define. 

Viscosity versus the product cM for polystyrenes at concentrations between 23 and 100%. Data at the various concentrations have been shifted vertically to avoid overlap. From Graessley (1974). 

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Kremer K and Grest G S 1990 Dynamics of entangled linear polymer melts a molecular-dynamics simulation J Chem. Phys. 92 5057... 

Figure C2.1.13. (a) Schematic representation of an entangled polymer melt, (b) Restriction of tire lateral motion of a particular chain by tire otlier chains. The entanglement points tliat restrict tire motion of a chain define a temporary tube along which tire chain reptates.

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... 

Under compression or shear most polymers show qualitatively similar behaviour. However, under the application of tensile stress, two different defonnation processes after the yield point are known. Ductile polymers elongate in an irreversible process similar to flow, while brittle systems whiten due the fonnation of microvoids. These voids rapidly grow and lead to sample failure [50, 51]- The reason for these conspicuously different defonnation mechanisms are thought to be related to the local dynamics of the polymer chains and to the entanglement network density. 

Solvents exert their influence on organic reactions through a complicated mixture of all possible types of noncovalent interactions. Chemists have tried to unravel this entanglement and, ideally, want to assess the relative importance of all interactions separately. In a typical approach, a property of a reaction (e.g. its rate or selectivity) is measured in a laige number of different solvents. All these solvents have unique characteristics, quantified by their physical properties (i.e. refractive index, dielectric constant) or empirical parameters (e.g. ET(30)-value, AN). Linear correlations between a reaction property and one or more of these solvent properties (Linear Free Energy Relationships - LFER) reveal which noncovalent interactions are of major importance. The major drawback of this approach lies in the fact that the solvent parameters are often not independent. Alternatively, theoretical models and computer simulations can provide valuable information. Both methods have been applied successfully in studies of the solvent effects on Diels-Alder reactions. 

As is suggested frequently , this term might well result from the restriction of the hydrogen bonding possibilities experienced by the water molecules in the first hydration shell. For each individual water molecule this is probably a relatively small effect, but due to the small size of the water molecules, a large number of them are entangled in the first hydration shell, so that the overall effect is appreciable. This theory is in perfect agreement with the observation that the entropy of hydration of a nonpolar molecule depends linearly on the number of water molecules in the first hydration shell ". ... 

Function 4 provides van der Waals entanglement via long carbon chains. 

There are a number of important concepts which emerge in our discussion of viscosity. Most of these will come up again in subsequent chapters as we discuss other mechanical states of polymers. The important concepts include free volume, relaxation time, spectrum of relaxation times, entanglement, the friction factor, and reptation. Special attention should be paid to these terms as they are introduced. 

Sec. 1.8, where polydispersity in ordinary samples was emphasized. Polydis-persity clearly complicates things, especially in the neighborhood of n, where a significant number of molecules are too short to show entanglement effects while an equally significant fraction are entangled. We simply note that any study conducted with the intention of a molecular interpretation should be conducted on a sample with as sharp a distribution as possible. 

We further note that p = SjPj and below the threshold for entanglements = 2jT j. Substituting these into a combination of Eqs. (2.37) and (2.38)... 

The segmental friction factor introduced in the derivation of the Debye viscosity equation is an important quantity. It will continue to play a role in the discussion of entanglement effects in the theory of viscoelasticity in the next chapter, and again in Chap. 9 in connection with solution viscosity. Now that we have an idea of the magnitude of this parameter, let us examine the range of values it takes on. 

In a similar fashion, a fraction of the velocity of the molecules with first-order coupling is transmitted to other molecules entangled with the latter. This is called second-order coupling (subscript 2). Still higher orders of effect radiate from the original molecule in the manner suggested by Fig. 2.13. Because of the... 

Figure 2.13 Model of several orders of coupling through entanglements according to Bueche theory.

Since the slippage factor is a fraction, Eq. (2.59) states in mathematical terms something we realize must be the case, namely, that the effects of entanglements on the neighbors of the original molecule must diminish as we move away from that molecule to prevent the coupling from producing an infinite viscosity. 

If this approach is to have any success, the weighting factors Cj must also decrease with increasing i to avoid a catastrophic increase in viscosity due to the proposed web of entanglements. We shall not detail the entire derivation of these C values as developed by Bueche but shall only note the following points ... 

Since each molecule has M/M entanglements, and each could entrain a different molecule, an upper limit for the number of couplings of order i is given by (M/Mg). ... 

This last factor overcounts the number of couplings, since the random placement of chain segments makes it improbable that each entanglement will involve a new molecule. Thus an entanglement may be redundant the chain might already be coupled to the original molecule. 

Equation (2.61) predicts a 3.5-power dependence of viscosity on molecular weight, amazingly close to the observed 3.4-power dependence. In this respect the model is a success. Unfortunately, there are other mechanical properties of highly entangled molecules in which the agreement between the Bueche theory and experiment are less satisfactory. Since we have not established the basis for these other criteria, we shall not go into specific details. It is informative to recognize that Eq. (2.61) contains many of the same factors as Eq. (2.56), the Debye expression for viscosity, which we symbolize t . If we factor the Bueche expression so as to separate the Debye terms, we obtain... 

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