Cluster entanglement network

This chapter discusses the independent determination of parameters for a cluster entanglement network enabling the accurate description of the experimental data on the molecular orientation in polymethyl methacrylate. This confirms the validity of the earlier proposed structural model for the polymer amorphous state [1, 2]. 

The results of the present investigation have shown that independently determined parameters of cluster entanglement network enable rather accurate description of the experimental data on molecular PMMA orientation. The cluster model of structure is postulated in [1, 2] and this is the one that is used in this chapter. 

And at last, the relation between cp and cluster entanglements network density v, is given as follows [160] ... 

FIGURE 58 The dependence of cluster entanglements network density vcl on macromolecular coil fractal dimension for hnear polyaiylates. 

Figure 1.12 The dependences of cluster entanglement network density on the parameter for HDPE, corresponding to the Landau equation for second-...

The thermofluctuational origin of the cluster entanglement network allows theoretical estimation of its density temperature variation. For this purpose the authors [91] have used the model [92] in which the following expression for relaxation time... 

Xmax, clustering around the line X z = 0.6 Xn,a. In all cases however Xdz is less than X of crazes in the same polymer. This observation indicates that the chain scission/chain slippage necessary for fibrillation in a craze modifies the entanglement network (increases the effective X ,an of the polymer fibrils) and so increases X above... 

The cluster model assumes availability in the amorphous polymers structure of local order domains (clusters), consisting of several densely packed collinear segments of different macromolecules (amorphous analog of crystallite with stretched chains). These clusters are connected between themselves by tie chains, forming by virtue of this physical entanglements network, and are surrounded by loosely packed matrix, in which all fluctuation free volume is concentrated [107],... 

Lately the large attention is given to macromolecular entanglements fluctuation network influence on crazes formation process in polymers [20], Much less is known about entanglements network influence on shear deformation zones (ZD) formation processes, which by their essence are crazes, without microvoids. The authors of Ref [21] fulfilled quantitative estimation of such influence on deformation processes in ZD on the example of polyarylatesulfone (PASF) samples in the form of films with a sharp notch. In this analysis two models of macromolecular entanglements network were used of binary hooking s [20] and cluster ones [22], The PASF films were prepared with the aid of nine various solvents that allows their structure wide enough variation [21]. 

Based on the literary data for p and estimation of (Ap/p) (°o) at 7 temperatures ( Tj is the analogue of for semi-crystalline polymers connected with similarly to Equation 1.26) it has been shown that the mentioned value is approximately constant at these temperatures. This observation assumes that as some critical value of (Ap/p) is reached, formation of local order domains, i.e., thermofluctuational cluster networks of macromolecular entanglements, is impossible owing to high thermal mobility of macromolecules. On the contrary, below the critical (Ap/p) values local order domains (clusters) in the polymer melt are formed, which according to the present treatment are identified as nodes of the macromolecular entanglements network, i.e., a state is formed, which was defined by Boyer as a liquid with fixed structure [59]. 

To complete this section, let us consider one more principal question associated with entanglement cluster network formation in semi-crystalline polymers. Equation 1.24 is valid for polymers in which the entire volume is involved in the formation of a macromolecular entanglement network. In this case, if only the part of the polymer with the volume fraction is involved in this process. Equation 1.24 should be rewritten as follows [110] ... 

In Figure 1.33 the dependence of the front-factor A value on molecular weight is shown. As follows from the adduced plot, the cluster fluctuations constraint is systematically changed from 1.0 at = 0 to 0.5 at = 3600 g/mole. In other words, if the amorphous phase of a semi-crystalline polymer represents a single cluster (supercluster), fluctuations are completely suppressed, and its behaviour corresponds to the affine model [110]. The value = M = 3600 g/mole corresponds to polyethylene melt [20], where constraints imposed by crystallites are absent, and in this case entanglement network behaviour for polyethylenes corresponds to the phantom alternative [113]. 

Special attention is always paid to the questions of estimation of molecular mobility of polymer chains [54-56]. The reasons are obvious thermodynamically non-equilibrium solid-like media, particularly relaxation media, and their physical properties are defined by passing relaxation molecular processes in them, which in turn depend on features of the chemical constitution of the molecular chains and the structural organisation of the polymers [56]. As for parameters, there exist different points of view in describing these processes. So, for example, it is assumed that fast relaxations are defined by the mobility of free chains placed between densely packed domains, which are at the same time nodes of macromolecules physical entanglements network. Such treatment corresponds to the main postulates of the cluster model of the structure of polymers in the amorphous state [13], with the aid of which structure elements can be quantitatively described. 

An important role in the present model is played by the strongly non-linear elastic response of the rubber matrix that transmits the stress between the filler clusters. We refer here to an extended tube model of rubber elasticity, which is based on the following fundamental assumptions. The network chains in a highly entangled polymer network are heavily restricted in their fluctuations due to packing effects. This restriction is described by virtual tubes around the network chains that hinder the fluctuation. When the network elongates, these tubes deform non-affinely with a deformation exponent v=l/2. The tube radius in spatial direction p of the main axis system depends on the deformation ratio as follows ... 

Bond distributions can be analyzed for any molecular architecture. The notion of bonds can be taken in its widest meaning to comprise weakly bound solvation networks, molecular meshes, multiple entangled chains (including double-stranded DNA), and cross-linked polymers. Hydration clusters... 

The density of cluster network junctions V,] is a function of temperature, which decreases as temperature increases. This network decay, is complete at T = T. The increasing of V i (the density of the entanglement of the cluster network) caused by decreasing of temperature, is slowed down drastically for T < T = - 223 °C. 

One can determine the density of entanglement cluster network from the value of Poisson s ratio p by means of the following approximation [21] ... 

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