Rheology and Viscoelasticity of Polymeric Materials

When an amorphous polymer possesses a certain amount of rotational freedom, it can be deformed by application of force. Application of force will cause the polymer to flow and the molecules will move past each other. The science of deformation and flow is called rheology. In the event that the force is applied quickly, and then withdrawn rapidly, the polymer molecules will tend to revert back to their previous undisturbed configuration. This is called relaxation. Thus, the amorphous polymers exhibit some elastic behavior due to disruption of thermodynamically favorable arrangements. If, however, the force is applied gradually and consistently, the molecules will f low irreversibly. Due to chain entanglement that increases with molecular weight and due to frictional effects, the viscosity of the flowing liquid will be high. Thus, molecular weight control is very important in polymer processing. In a way this is a paradoxical situation. Higher molecular weights usually yield better mechanical properties. On the other hand, higher molecular weight materials are harder to process. The molecular weight control, therefore, is quite critical. The combination of properties of polymeric liquids, elasticity, and viscous flow is referred to as viscoelasticity. It means reversible uncoiling of the polymeric chains. By the same terminology, viscous flow means irreversible slipping of the chains past each other. Thus, viscoelastic materials exhibit simultaneously a combination of elastic and viscous behavior. This type of behavior is particularly prominent in polymeric materials. The flow behavior of polymeric liquids is also influenced by the molecular weight distribution. At proper temperatures the mechanical properties of many amorphous polymers may approach the physical properties of three idealized materials individually. These are [20, 22, 24]: A Hookian or an ideal elastic solid, whose small reversible deformations are directly proportional to the applied force. 2. A Newtonian liquid that flows with a viscosity independent of the rate of shear. 3. An ideal elastomer that is capable of reversible extension of several hundred percent, with a much smaller stiffness than that of a Hookian solid. In the elastic response for a Hookian liquid the stress–strain relationship is: s(t) ¼ Gg(t). For the Newtonian liquid it is s(t) ¼ g0(t). In these equations, g(t) and g0(t) represent shear strain and shear rate at time t, while s(t) is the shear stress. Many forces can be applied to polymer deformation. The force that rheologists are particularly concerned with is tangential stress or shear. This is due to the fact that many polymers are extruded and forced to flow into dies for shaping and commercial use. If a shearing force f is applied to a cube of molten polymer per unit area it causes the top layer of the liquid to move a distance x from a fixed point at the bottom of the material with a velocity v. Shear causes the molecules to move past each other. This is illustrated in Fig. 2.4. The above assumes that the viscosity of the material is sufficiently small so that the cube is not very distorted during the process. The viscosity of the material, is defined as the ratio between the applied force and the velocity gradient, ∂v/∂x or as the rate of shear g, where  The above equation can be rearranged into another form, N=Fv/v² Fv is the energy used up per second on the cube. The shear stress, t is defined as

Fig. 2.4 Illustration of the movement of the upper layer due to applied force

where gis the shear rate. In this instance it should be equal to v. If the viscosity is independent of the shear rate, the liquid exhibits ideal flow and is a Newtonian liquid. To bolsky commented [12] that probably all Newtonian liquids, even those like water and benzene, that are very fluid, possess some elastic as well as viscous behavior. However, flow of most polymer liquids deviates strongly from an ideal behavior and either the viscosity decreases with the rate of shear, or no flow occurs until a certain minimum force is applied. The decrease of viscosity with the rate or shear is called shear thinning. It is believed to be a result from the tendency of the applied forces to disturb the long chains from their favored equilibrium conformations. In this case, the polymer is at an yield point or at an yield value. A liquid with an yield point is called a Bingham Newtonian fluid, provided that it exhibits Newtonian behavior once it starts flowing. It is defined by  where t_c is the critical shear stress, or threshold stress that is needed to initiate the flow. The Bingham fluid behavior might be attributed to structural arrangements of the molecules that give rise to conformational and secondary bonding forces. The non-Newtonian behavior occurs when shear stress does not increase in proportion to shear rate. In addition, there are thixotropic liquids that exhibit high viscosity or even resemble gelation at low shear rate but flow readily and exhibit low viscosity upon application of high shear. High shear rates can cause chain rupture and result in loss of molecular weight. In some cases, the shear rate may increase due to increase in molecular entanglement. In the case of flexible chain polymers, there is a critical molecular weight at which chain entanglement may show an increase. For most common polymers this may be in the molecular weight range of 4,000–15,000. Flow is also influenced by chain branching. The higher the degree of branching in a polymer, the lower will be the degree of entanglement at a given molecular weight. In general, the viscosity is higher with linear polymers than with branched ones at a given shear rate and molecular weight. Flow behavior is also influenced by molecular weight distribution (see Sect. 2.7.1). Usually, the broader the molecular weight distribution in polymers the lower is the shear rate that is needed to cause shear thinning .On the other hand, for polymers with narrow molecular weight distribution, shear thinning, once it starts, is more pronounced. This is due to absence of chain entanglement of the higher molecular weight polymeric chains. The viscosity of low molecular weight polymers is related to their temperature by an Arrhenius type relationship [19, 22, 25]: Mn=Ae ^-E/RT where E is the activation energy for viscous flow,andA is a constant related to molecular motion, and M isthe viscosity average molecular weight. For branched polymers, the larger or bulkier the side chains the greater is the activation energy, E. The same is true of polymers with large pendant groups. The activation energy of flow, E does not increase proportionately to the heat of vaporization for polymers but rather levels off to a value that is independent of molecular weight. This implies that for long chains the unit of flow is much smaller than the whole molecule. Instead, segments of the molecules, no larger than 50 carbon atoms move by successive jumps.

Fig. 2.5 Illustration of a physical crosslink in molten polymers. This is accomplished with some degree of coordination, but results in the whole chain shifting. Based on experimental evidence, the viscosity can be defined as,  where C is a constant. Mw is weight average molecular weight (see Sect. 2.7) Chain length, Z, or the molecular weight of polymers is an important variable that influences flow properties of polymers. The relationship of a Newtonian viscosity of an amorphous polymer to the chain length when shear stress is low can be expressed as [19, 22, 25],  where the constant k is temperature-dependent. By the same token, based on experimental evidence, the relationship of viscosity to temperature and to chain length at low shear rates, for many polymers can be expressed as follows:  where constant k0 is specific for each polymer and Tg is the glass transition temperature (Tg is discussed in Sect. 2.2.3). Although linear molten polymers exhibit well-defined viscosities, they also exhibit elastic effects. These effects are present even in the absence of any crosslinks or a rubber network. It is referred to as creep. This creep is attributed to entanglement of polymeric chain to form temporary physical crosslinks: This is illustrated in Fig. 2.5. Deviations from Newtonian flow can occur when shear stress does not increase in direct proportion to shear rate. Such deviation may be in the direction of thickening (called dilatent flow) and in the direction of thinning (called pseudo plastic). Related to non-Newtonian flow is the behavior of thixotropic liquids when subjected to shear, as explained above. Flow behavior can be represented by the following equation:  where A is a constant and B is the index of flow. For Newtonian liquids A ¼ and B ¼ 1. All polymers tend to exhibit shear thinning at high shear rates. This is commonly explained by molecular entanglement, as mentioned above. Certainly, in the amorphous state there is considerable entangle ment of the polymeric chains. Low shear rates may disrupt this to a certain degree, but the chains will still remain entangled. As the shear rates increases, disruption can occur at a faster rate than the chain

Fig. 2.6 Illustration of a plot of modulus of elasticity against time canreentangle.Thedecreasedamountofentanglementresultsinlowerviscosityoftheliquid,allowing the molecules to flow with less resistance. Actually, two factors can contribute to chain entanglement. These are high length of the chains for very large molecules and/or bulky substituents. Stress applied to a Newtonianliquid, outside of an initial spike, is zero. Stress, however applied to a viscoelastic fluid starts at some initial value. This value decreases with time until it reaches an equilibrium value due to the viscoelastic property of the material. Figure 2.6illustrates what aplot of the modulus of elasticity G(t), which depends on the temperature, when plotted against time, looks like: The equation for shear–stress relaxation modulus that varies with temperature can be written as follows:  With constant stress, s(t) ¼ Gg0, where creep strain g(t) is constant [g(t) ¼ s0/G] for a Hookean solid. It would be directly proportional to time for a Newtonian liquid [(g(t) ¼ s0/ )t]. Here t is the initial time at which recovery of the viscoelastic material begins. For a viscoelastic fluid, when stress is applied, there is a period of creep that is followed by a period of recovery. For such liquids, strains return back toward zero and finally reach an equilibrium at some smaller total strain. For the viscoelastic liquid in the creep phase, the strain starts at some small value, but builds up rapidly at a decreasing rate until a steady state is reached. After that the strain simply increases linearly with time. During this linear range, the ratio of shear strain to shear stress is a function of time alone. This is shear creep compliance, J(t) The equation of shear creep compliance can be written as follows:  The stress relaxation modulus and the creep compliance are both manifestations of the same dynamic process at the molecular level and are closely related. This relationship, however, is not a simple reciprocal relation that would be expressed as G(t) ¼ 1/J(t), but rather in an integral equation that is derived from the Boltzman superposition principle. It relates recoverable compliance, J0 s to 0, zero shear viscosity [22].

In relaxation back to equilibrium, the polymer assumes a new conformation. At first, the response is glassy. The modulus for such an organic glass is large, Gg ~10⁹ Pa. This modulus decreases with

Fig. 2.7 Illustration of a cone and plate rheometer time as the polymer begins to relax and continues along the whole length of the chain. For short chains this relaxation to zero takes place at a fairly constant rate. For very long chains, however, the relaxation rate tends to be in three stages. It starts at a certain rate at stage one, but after a while, noticeably slow down at some point, and at stage two the modulus remains relatively constant over some period of time. After that, at stage three, the relaxation is resumed again at a rapid rate until full equilibrium is reached. At stage two there is a period of relatively constant modulus that is not affected by the chain architecture and the material resembles a rubber network. The length of the third stage, however, is profoundly affected by the molecular weight, by the molecular weight distribution and by long-chain branching of the polymer. Koga and Tanaka [23] studied the behavior of normal stresses in associated networks composed of telechelic polymers under steady shear flow. They showed numerically that the first and second normal stress coefficients reveal thickening as a function of shear rate and that the sign of the second normal stress coefficient changes depending on the nonlinearity in the chain tension, the dissociation rate of the associative groups from junctions and the shear rate by analytic calculation they showed that in the limit of small shear-rate, the sign inversion occurs by the competition between the nonlinear stretching and dissociation of associative groups. Thus, the molecular mechanism of the sign inversion is shown to be similar to that of thickening of the shear viscosity. In the behavior of polymeric liquids two quantities are important. These are steady-state recover able shear compliance, J⁰s (as shown above) and steady-state viscosity at zero shear rate, 0. These quantities are related:

where y ss is the shear rate and yr is the total recoil strain. Both shear compliance and shear viscosity can be obtained from creep studies. The product of the two, zero shear compliance and zero shear viscosity is the characteristic relaxation time of the polymer:  where M is the torque in dynes per centimeter, and R is the cone radius in centimeters (or meters). The shear rate can be obtained from the following equation

Fig. 2.8 Illustration of a capillary viscometer

where a is the cone angle in radians or in degrees, and O is the angular velocity in radians per second or in degrees per second. The viscosity is obtained from the following relationship [15, 16]:

k is a constant, specific for the viscometer used. It can be obtained from the relationship [7]:

The cone and plate rheometers are useful at relatively low shear rates. For higher shear rates capillary rheometers are employed. They are usually constructed from metals. The molten polymer is forced through the capillary at a constant displacement rate. Also, they may be constructed to suit various specific shear stresses encountered in commercial operation. Their big disadvantage is that shear stress in the capillary tubes varies from maximum at the walls to zero at the center. On the other hand, stable operation at much higher shear rates is possible. Determination, however, of Z0 is usually not possible due to limitations of the instruments. At low shear rates. one can determine the steady state viscosity from measurements of the volumetric flow rates, Q and the pressure drop: 

where, P0 is the ambient pressure. A capillary viscometer is illustrated in Fig. 2.8, where the diameter of the capillary can be designated as D. For Newtonian liquids the viscosity can be determined from the following equation:

 where L represents the length of the capillary. The shear stress at the capillary wall can be calculated from the pressure drop:  Also, the shear rate at the walls of the capillary can be calculated from the flow rate [22]:  Wang et al. studied the homogeneous shear, wall slip, and shear banding of entangled polymeric liquids in simple-shear rheometry, such as in capillary viscometry, shown above [20]. They observed that recent particle-tracking velocimetric observations revealed that well-entangled polymer solutions and melts tend to either exhibit wall slip or assume an inhomogeneous state of deformation and flow during nonlinear rheological measurements in simple-shear rheometric setups. It is important to control the viscoelastic properties of confined polymers for many applications. These applications are in both, microelectronics and in optics. The rheological properties of such f ilms, however, are hard to measure. Recently, Chan et al. [27] reported that the viscoelasticity can be measured through thermal wrinkling. Thermally induced instability develops when polymer films are compressed between rigid, stiffer layers. This is due to differences in coefficients of thermal expansion between the polymer and the inorganic layers. A net compressive stress develops at the polymer–substrate interface when the composite layers are heated to temperatures that promote mobility of the polymer layer. Such wrinkling substrate surface is characterized by an isotropic morphology that can be approximated as a sinusoidal profile. Chan et al. utilized the thermal wrinkling to measure the rubbery modulus and shear viscosity of polystyrene thin films as a function of temperature. They used surface laser-light scattering to characterize the wrinkled surface in real time in order to monitor the changes in morphology as a function of annealing time at fixed annealing temperatures. The results were compared with a theoretical model, from which the viscoelastic properties of the PS thin film are extracted.

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مواضيع ذات صلة

The Crystalline State

Thermodynamics of Elasticity

Elasticity

The Glass Transition and the Glassy State

Induction Forces in Polymers

Effects of Dipole Interactions

Structure and Property Relationship in Organic Polymers

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الحجم والشكل

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