Shear modulus, temperature dependence

Shear modulus MPa Depends on compounding ingredients, temperature. 0.3-20 (14, 26)... 

Experimental phase diagrams for amorphous block copolymers were explored by Khandpur and co-workers (29). First, low-frequency isochronal shear modulus-temperature curves were developed on a series of polyiso-prene-h/ocA -polystyrene polymers to guide the selection of temperatures for the transmission electron microscopy and SAXS experiments to follow see Figure 13.14 (29). Both order-order (OOT) and ODT transitions were iden-tihed. The OOT are marked by open arrows, while the ODT are shown by hlled arrows. Since the ODT occurs as the temperature is raised, an upper critical solution temperature is indicated, much more frequent with block copolymers than with polymer blends. The regions marked A, B, C, and D denote lamellar, bi-continuous, cylindrical, and perforated layered microstructures, respectively. The changes in morphology are driven by the temperature dependence of Xn,... 

These parameters were also shown to depend on the system variables such as composition and temperature. The elastic modulus at high frequency is equivalent in these systems to the shear modulus and depends on the interfacial tension, droplet radius and volume fraction in the way predicted by eqn. (11.1) within... 

The solidity of gel electrolytes results from chain entanglements. At high temperatures they flow like liquids, but on cooling they show a small increase in the shear modulus at temperatures well above T. This is the liquid-to-rubber transition. The values of shear modulus and viscosity for rubbery solids are considerably lower than those for glass forming liquids at an equivalent structural relaxation time. The local or microscopic viscosity relaxation time of the rubbery material, which is reflected in the 7], obeys a VTF equation with a pre-exponential factor equivalent to that for small-molecule liquids. Above the liquid-to-rubber transition, the VTF equation is also obeyed but the pre-exponential term for viscosity is much larger than is typical for small-molecule liquids and is dependent on the polymer molecular weight. 

Estimates of the ultimate shear strength r0 can be obtained from molecular mechanics calculations that are applied to perfect polymer crystals, employing accurate force fields for the secondary bonds between the chains. When the crystal structure of the polymer is known, the increase in the energy can be calculated as a function of the shear displacement of a chain. The derivative of this function is the attracting force between the chains. Its maximum value represents the breaking force, and the corresponding displacement allows the calculation of the maximum allowable shear strain. In Sect. 4 we will present a model for the dependence of the strength on time and temperature. In this model a constant shear modulus g is used, thus r0=gyb. 

The temperature dependences of the isothermal elastic moduli of aluminium are given in Figure 5.2 [10]. Here the dashed lines represent extrapolations for T> 7fus. Tallon and Wolfenden found that the shear modulus of A1 would vanish at T = 1.677fus and interpreted this as the upper limit for the onset of instability of metastable superheated aluminium [10]. Experimental observations of the extent of superheating typically give 1.1 Tfus as the maximum temperature where a crystalline metallic element can be retained as a metastable state [11], This is considerably lower than the instability limits predicted from the thermodynamic arguments above. 

Most polymers are applied either as elastomers or as solids. Here, their mechanical properties are the predominant characteristics quantities like the elasticity modulus (Young modulus) E, the shear modulus G, and the temperature-and frequency dependences thereof are of special interest when a material is selected for an application. The mechanical properties of polymers sometimes follow rules which are quite different from those of non-polymeric materials. For example, most polymers do not follow a sudden mechanical load immediately but rather yield slowly, i.e., the deformation increases with time ( retardation ). If the shape of a polymeric item is changed suddenly, the initially high internal stress decreases slowly ( relaxation ). Finally, when an external force (an enforced deformation) is applied to a polymeric material which changes over time with constant (sinus-like) frequency, a phase shift is observed between the force (deformation) and the deformation (internal stress). Therefore, mechanic modules of polymers have to be expressed as complex quantities (see Sect. 2.3.5). 

Free Volume Versus Configurational Entropy Descriptions of Glass Formation Isothermal Compressibility, Specific Volume, Shear Modulus, and Jamming Influence of Side Group Size on Glass Formation Temperature Dependence of Structural Relaxation Times Influence of Pressure on Glass Formation... 

The effect of temperature relative to the glass transition, Tg, is illustrated further with the shear modulus, G. The variation of shear modulus with temperature for three common engineering polymers is shown in Figure 5.71. In general, the temperature dependence is characterized by a shallow decline of log G with temperature with... 

Figure 5.71 Dependence of shear modulus on temperature for three common engineering polymers crosslinked natural rubber, amorphous polyvinyl chloride (PVC), and crystalline Nylon 6. The typical use temperatures are indicated by dotted lines. Reprinted, by permission, from N. G. McCrum, C. P. Buckley, and C. B. Bucknall, Principles of Polymer Engineering, 2nd ed., p. 154. Copyright 1997 by Oxford University Press.

In addition to knowing the temperature shift factors, it is also necessary to know the actual value of ( t ) at some temperature. Dielectric relaxation studies often have the advantage that a frequency of maximum loss can be determined for both the primary and secondary process at the same temperature because e" can be measured over at least 10 decades. For PEMA there is not enough dielectric relaxation strength associated with the a process and the fi process has a maximum too near in frequency to accurately resolve both processes. Only a very broad peak is observed near Tg. Studies of the frequency dependence of the shear modulus in the rubbery state could be carried out, but there... 

Fig. Z4 (a) Temperature ramp at a frequency a> = lOrads (strain amplitude A = 2%) for a nearly symmetric PEP-PEE diblock with Mn = 8.1 X 104gmol l, heating from the lamellar phase into the disordered phase. The order-disorder transition occurs at 291 1 °C, the grey band indicates the experimental uncertainty on the ODT (Rosedale and Bates 1990). (b) Dynamic elastic shear modulus as a function of reduced frequency (here aT is the time-temperature superposition shift factor) for a nearly symmetric PEP-PEE diblock with Mn = 5.0 X 1O g mol A Shift factors were determined by concurrently superimposing G and G"for w > and w > " respectively. The filled and open symbols correspond to the ordered and disordered states respectively. The temperature dependence of G (m < oi c) for 96 < T/°C 135 derives from the effects of composition fluctuations in the disordered state (Rosedale and Bates 1990). (c) G vs. G"for a PS-PI diblock with /PS = 0.83 (forming a BCC phase) (O) 110°C (A) 115°C ( ) 120°C (V) 125°C ( ) 130°C (A) 135°C ( ) 140°C ( ) 145°C. The ODT occurs at about 130°C (Han et at. 1995).

The increase in fiber-matrix interfacial shear strength can be predicted from a purely mechanistic viewpoint. Rosen [20], Cox [21], and Whitney and Drzal [22] have shown that the square root of the shear modulus of the matrix appears explicitly in any model of the interfacial shear strength. It has been demonstrated experimentally [23, 24] that the fiber-matrix interfacial shear strength has a dependence on both the product of the strain-to-failure of the matrix times the square root of the shear modulus and on the difference between the test temperature and Tg when the interfacial chemistry is held constant. 

Fig. 4. Temperature dependence of the shear loss modulus of poly(methyl methacrylate) (1), poly(n-propyl methacrylate) (2), poly(2-hydroxyethyl methacrylate) (3), poly(5-hydroxy-3-oxapen-tyl methacrylate) (4), and poly(8-hydroxy-3,6-dioxaoctyl methacrylate) (5)...

Fig. 5. Temperature dependence of the shear loss modulus of poly(pivaloyl-2-oxyethyl methacrylate) atactic (—), isotactic (---)...

Fig. 12. Effect of the volume fraction of methacrylic acid on the temperature dependence of the shear loss modulus of copolymers with 2-hydroxyethyl methacrylate 1 = 0.00 2 = 0.14 3 = 0.39 4 = 0.72 5 = 1.00...

Figure 5.1 Temperature dependence of log (shear) modulus in a polymer system showing molecular mechanism of the deformations taking place at different point...

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