Entanglements parameters

Figure 11. Influence of the chain entanglement parameter, on the predicted viscosity.

Table 9.1 Entanglement parameters for flexible linear polymer melts...

Hess [13] neglected the hydrodynamic interactions among chain beads and treated the global motions of different chains as uncorrelated (this is to assume a small number of chain-chain contacts and thus to focus on the semi-dilute regime). He deduced that polymer self-diffusion consists of both lateral and longitudinal modes of chain motion until the entanglement parameter t/>(c, N) reaches unity, but it is dominated by the latter (i.e., chains move reptatively)... 

From the Marvin-Oser theory, the location of the maximum in J" on the frequency scale can also be related to entanglement parameters. However, both Me and Me are involved the frequency at the maximum is independent of M (if sufficiently high) but is inversely proportional to MeMc- The maximum in G" occurs at a frequency which is proportional to Mc/Me and hence is essentially independent of the entanglement spacing, though it is inversely proportional to A/. ... 

It is well-known that (or M ) will increase with temperature (melt or solution) for all flexible chains. However, the temperature dependence of o (din o/dT) can be negative, zero or positive. Thus, the assumption of a direct proportionality between the entanglement parameter and the unperturbed chain dimensions, as expressed by C , is incorrect. 

If the test chain is infinite, the free energy of the polymer solution is invariant by translation of the chain along itself, the effect of excluded volume is, therefore, very different for motions parallel to the chain or transverse to it this is the origin of a reptative motion. The reptation diffusion coefficient is obtained if the entanglement parameter ij/, proportional to the number of blobs per chain, is larger than a critical value at low values of this parameter the self-diffusion constant is given by a Rouse approximation. 

Fig. 11. Mode number dependence of the stretching-parameter /3p for the RRM (see Eq, 131) for different entanglement parameters The number of Kuhn segments was as-...

If the entanglement parameter obeys mode-number regime of the entangled motion exists for times t in the TRRM... 

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. 

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. 

The percolation parameters (p — Pc) associated with the disentanglement process are derived as follows p is the normalized entanglement density defined as... 

Coran and Patel [33] selected a series of TPEs based on different rubbers and thermoplastics. Three types of rubbers EPDM, ethylene vinyl acetate (EVA), and nitrile (NBR) were selected and the plastics include PP, PS, styrene acrylonitrile (SAN), and PA. It was shown that the ultimate mechanical properties such as stress at break, elongation, and the elastic recovery of these dynamically cured blends increased with the similarity of the rubber and plastic in respect to the critical surface tension for wetting and with the crystallinity of the plastic phase. Critical chain length of the rubber molecule, crystallinity of the hard phase (plastic), and the surface energy are a few of the parameters used in the analysis. Better results are obtained with a crystalline plastic material when the entanglement molecular length of the... 

The foregoing equations all express the multiphase viscosity as a function of the solids content, without any recourse to liquid parameters. A more realistic portrayal of the physical situation would include the fluid dynamic picture that compensates for entanglement and absorbed liquids carried along with the solid phase, thus effectively decreasing the liquid volume. An equation applicable to this case is [24] ... 

As a consequence of the irregular and rough structure of the xylan particles, entanglements between particles are promoted and this fact may explain the poor flow properties of this polymer (Kumar et al., 2002 Nunthanid et al., 2004). Additionally, rheological parameters of xylan powder have also been studied, such as bulk and tapped densities, Hausner ratio, Carr s index, and angle of repose values, and they are summarized in Table 1. 

In Equation 1, t is a thermal vibration frequency, U and P are, respectively activation energy and volume whereas c is a local stress. The physical significance and values for these parameters are discussed in Reference 1. Processes (a)-(c) are performed with the help of a Monte-Carlo procedure which, at regular short time intervals, also relaxes the entanglement network to its minimum energy configuration (for more details, see Reference 1). 

In addition to the Rouse model, the Hess theory contains two further parameters the critical monomer number Nc and the relative strength of the entanglement friction A (0)/ . Furthermore, the change in the monomeric friction coefficient with molecular mass has to be taken into account. Using results for (M) from viscosity data [47], Fig. 16 displays the results of the data fitting, varying only the two model parameters Nc and A (0)/ for the samples with the molecular masses Mw = 3600 and Mw = 6500 g/mol. 

According to Hess, the relative strength of the entanglement friction can be related to the more microscopic parameter q , describing the range of the true interchain interaction potential. A value of q 1 = 7 A, close to the average interchain distance of about 4.7 A, is obtained. 

Thus, with only two parameters, the values of which are both close to expectations, the Hess model allows a complete description of all experimental spectra. In the complex crossover regime from Rouse motion to entanglement controlled behavior, this very good agreement confirms the significant success of this theory. 

The cycle rank completely defines the connectivity of a network and is the only parameter that contributes to the elasticity of a network, as will be discussed further in the following section on elementary molecular theories. In several other studies, contributions from entanglements that are trapped during cross-linking are considered in addition to the chemical cross-links [23,24]. The trapped entanglement model is also discussed below. 

Equation (29) shows that the modulus is proportional to the cycle rank , and that no other structural parameters contribute to the modulus. The number of entanglements trapped in the network structure does not change the cycle rank. Possible contributions of these trapped entanglements to the modulus therefore cannot originate from the topology of the phantom network. 

The constrained-junction model was formulated in order to explain the decrease of the elastic moduli of networks upon stretching. It was first introduced by Ronca and Allegra [39], and Flory [40]. The model assumes that the fluctuations of junctions are diminished below those of the phantom network because of the presence of entanglements and that stretching increases the range of fluctuations back to those of the phantom network. As indicated by the second part of Equation (26), the fluctuations in a phantom network are substantial. For a tetrafunctional network, the mean-square fluctuations of junctions amount to as much as half of the mean-square end-to-end vector of the network chains. The strength of the constraints on these fluctuations is measured by a parameter k, defined as... 

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