Filled polymers model

Some of these questions have strict and unambiguous answers, in a mathematical model, to other answers are derived from extensive empirical material. The present paper will discuss the problems formulated above, but concerning only rheological properties of filled polymer melts, leaving out the discussion of specific hydrodynamic effects occurring during their flow in channels of different geometrical form. 

Protein structures are so diverse that it is sometimes difficult to assign them unambiguously to particular structural classes. Such borderline cases are, in fact, useful in that they mandate precise definition of the structural classes. In the present context, several proteins have been called //-helical although, in a strict sense, they do not fit the definitions of //-helices or //-solenoids. For example, Perutz et al. (2002) proposed a water-filled nanotube model for amyloid fibrils formed as polymers of the Asp2Glni5Lys2 peptide. This model has been called //-helical (Kishimoto et al., 2004 Merlino et al., 2006), but it differs from known //-helices in that (i) it has circular coils formed by uniform deformation of the peptide //-conformation with no turns or linear //-strands, as are usually observed in //-solenoids and (ii) it envisages a tubular structure with a water-filled axial lumen instead of the water-excluding core with tightly packed side chains that is characteristic of //-solenoids. 

Figure 2.1 Exploded views showing the nonporous membrane size-exclusion phenomenon in the uptake and loss of organic compounds. Middle illustration shows the movement of contaminant molecules through transient pores in the membrane and retention (membrane exclusion) of much larger lipid molecules. Upper illustration shows similarly scaled space-filled molecular models of some organic contaminants and triolein, along with the hypothetical polymer pore (transient) size. Reprinted with permission from the American Petroleum Institute (Huckinset al., 2002).

The interaction of two substrates, the bond strength of adhesives are frequently measured by the peel test [76]. The results can often be related to the reversible work of adhesion. Due to its physical nature such a measurement is impossible to carry out for particulate filled polymers. Even interfacial shear strength widely applied for the characterization of matrix/fiber adhesion cannot be used in particulate filled polymers. Interfacial adhesion of the components is usually deduced indirectly from the mechanical properties of composites with the help of models describing composition dependence. Such models must also take into account interfacial interactions. 

Application of the dual mode sorption and diffusion models to homogeneous polymer blend-gas systems 26,65) and filled polymers 66) has also been reported. 

The CPK space-filling molecular model of the threo structure represents a helical loop in which all oxygen atoms point toward the center (Fig. 1) whereas that of the erythro structure tends to form an extended rigid chain in which oxygen atoms tend to alternate along the chain because of steric crowding of the methine hydrogens (Fig. 2). As to the threo polymer, it is reasonable to assume by... 

Both the Carreau and the Cross models can be modified to include a term due to yield stress. For example, the Carreau model with a yield term given in Equation (2.16) was employed in the study of the rheological behavior of glass-filled polymers (Poslinski et al., 1988) ... 

Figure 7.10. Schematic model of morphological transformations in filled polymers. A - silica content less than 10 wt% (d>d ), B - silica content 10 wt% (d=d ), C - silica content 20 wt% (d

The measurement of yield stress at low shear rates may be necessary for highly filled resins. Doraiswamy et al. (1991) developed the modified Cox-Merz rule and a viscosity model for concentrated suspensions and other materials that exhibit yield stresses. Barnes and Camali (1990) measured yield stress in a Carboxymethylcellulose (CMC) solution and a clay suspension via the use of a vane rheometer, which is treated as a cylindrical bob to monitor steady-shear stress as a function of shear rate. The effects of yield stresses on the rheology of filled polymer systems have been discussed in detail by Metzner (1985) and Malkin and Kulichikin (1991). The appearance of yield stresses in filled thermosets has not been studied extensively. A summary of yield-stress measurements is included in Table 4.6. 

Doraiswamy et al. (1991) developed a non-linear rheological model combining elastic, viscous and yielding phenomena for filled polymers. The model predicts a modified Cox Merz relationship for filled melts ... 

The critical indices estimated from these relations fall into the admissible ranges of variation P = 0.39-0.40, V = 0.8-0.9, and t = 1.6-1.8, determined in terms of the percolation model for three-dimensional systems. The researchers [7] noted that not only numerical values but also the meanings of these values coincide. Thus the index P characterises the chain structure of a percolation cluster. The 1/p value, which serves as the index of the first subset of the fractal percolation cluster in the model considered [7], also determines the chain structure of the cluster. The index v is related to the cellular texture of the percolation cluster. The 2/df index of the second subset of the fractal percolation cluster is also associated with the cellular structure. By analogy, the index t defines the large-cellular skeleton of the fractal percolation cluster. The relationship between the critical percolation indices and the fractal dimension of the percolation cluster for three-dimensional systems and examples of determination of these values for filled polymers are considered in more detail in the book cited [7]. Thus, these critical indices are universal and significant for analysis of complex systems, the behaviour of which can be interpreted in terms of the percolation theory. 

Samples of unfilled polyetherimide plaques or films were pretreated and metallized with copper. Peel strengths of -170 g/mm were achieved. Both sets of samples failed cohesively within the polymer layer. This failure mode has been discussed previously for filled-polymer resinsS S.H. Fracture patterns on the polymer side of the peel were found to be similar for both materials when viewed at high magnification (20,000 X), Figure 5. The ductile failure model 8 observed for the polyetherimide film was not apparent at low magnifications (300X). 

Figure 12.5. Models for filled polymers. (Nielsen, 1966.) (a) Perfect adhesion (b) no...

Figure 12.24. Model for the permeability of a liquid through a filled polymer. Reprinted from Nielsen 1967h, p. 933, by courtesy of Marcel Dekker, Inc.

In modeling filled polymer composites, Schrager (30) has recently proposed an equation useful for predictittg the break strength in the case where the filler is not treated to improve adhesion. The form of the Schrager equation is given as... 

We now consider the case in which a layer of grafted polymer chains is in contact not with a solvent, but with a dense polymer melt. This situation is of intrinsic interest as a much simplified model for the interphase between a polymer matrix and the reinforcing particles or fibres of a filled polymer or polymer composite it is also closely related to the situation encountered when a block copolymer is used to modify an interface between immiscible polymers. 

Theoretical models for other systems, such as star, branched, and ring polymers, random and alternating copolymers, graft and block copolymers are discussed in the book by Mattice and Suter [1]. Block copolymers are discussed in Chap. 32 of this Handbook [2]. Theories of branched and ring polymers are presented in the book by Yamakawa [3]. Liquid-crystalline polymers are discussed in the book by Grosberg and Khokhlov [4], and liquid crystalline elastomers in the recent book of Warner and Terentjev [5]. Bimodal networks are discussed by Mark and Erman [6,7]. Molecular theories of filled polymer networks are presented by Kloczkowski, Sharaf and Mark [8] and recently by Sharaf and Mark [9]. 

The model describes the compositirai dependence of tensile yield stress or tensile strength of particulate filled polymers. The expressirai for yield stress takes the form ... 

In this chapter, the fracture of WPCs as particle-filled polymer composites was elaborated. The characterization of particulate polymer composites fracture behavior and the influencing factors such as particle size as well as orientation, temperature, and loading were discussed. The fracture observation using special setup was described and the diverse numerical methods to analyze the fracture of such composites were reviewed. Finally the finite element simulation of the fracture for WPG specimen with real geometrical model was conducted and the agreement of results compared to the experimental ones was demonstrated. 

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