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**Introduction to Viscoelasticity**

All viscous liquids deform continuously under the influence of an applied stress – They exhibit viscous behavior. Solids deform under an applied stress, but soon reach a position of equilibrium, in which further deformation ceases. If the stress is removed they recover their original shape – They exhibit elastic behavior. Viscoelastic fluids can exhibit both viscosity and elasticity, depending on the conditions. Viscous fluid Viscoelastic fluid Elastic solid Polymers display VISCOELASTIC properties Viscoelasticity CHEE 390

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**Introduction to Viscoelasticity**

The response of polymeric liquids, such as melts and solutions, to an imposed stress may resemble the behavior of a solid or a liquid, depending on the situation. Viscoelasticity CHEE 390

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**Linear Viscoelasticity**

Viscoelasticity is observed in polymer melts and solutions irrespective of the magnitude and rate of the applied deformation. LINEAR viscoelastic behaviour is restricted to mild deformations such that polymer chains are disturbed from their equilibrium configuration and entanglements to a negligible extent. Under these conditions, stress is LINEARLY proportional to strain, or s(t) = G(t) go where s is the stress, G is the modulus and g is the strain. Since polymer processing operations very often involve severe deformations, studies of linear viscoelasticity have little practical utility. The value of data acquired under linear viscoelastic conditions lies in its amenability to fundamental analysis, which leads to useful inferences on polymer structure (molecular weight distribution, branching) and valuable insight into the characteristics of polymer flow (relaxation phenomena, elastic recovery). Viscoelasticity CHEE 390

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**Dynamic (Oscillatory) Rheometry**

Linear viscoelasticity in polymer melts is examined by dynamic measurements Examine the dynamic elasticity as a function of temperature and/or frequency. Impose a small, sinusoidal shear or tensile strain (linear t-e region) and measure the resulting stress (or vice versa) Stress Strain Viscoelasticity CHEE 390

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**Dynamic (Oscillatory) Rheometry**

A. The ideal elastic solid A rigid solid incapable of viscous dissipation of energy follows Hooke’s Law, wherein stress and strain are proportional (s=Ee). Therefore, the imposed strain function: e(w)=eo sin(wt) generates the stress response t(w)=Geosin(wt) = to sin(wt) and the phase angle, d, equals zero. B. The ideal viscous liquid A viscous liquid is incapable of storing inputted energy, the result being that the stress is 90 degrees out of phase with the strain. An input of: t(w)=to sin(wt+p/2) and the phase angle, d, p/2. Viscoelasticity CHEE 390

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**Dynamic (Oscillatory) Rheometry**

Being viscoelastic materials, the dynamic behaviour of polymers is intermediate between purely elastic and viscous materials. We can resolve the response of our material into a component that is in-phase with the applied strain, and a component which is 90° out-of-phase with the applied strain, as shown below: Viscoelasticity CHEE 390

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**Dynamic (Oscillatory) Rheometry**

The dynamic analysis of viscoelastic polymers the static Young’s modulus is replaced by the complex dynamic modulus: G* = G’ + i G” The storage (in-phase) modulus, G’, reflects the elastic component of the polymer’s response to the applied strain. Reflects the portion of the material’s stress-strain response that is elastic (stored). The loss (out-of-phase) modulus, G”, reflects the viscous component of the response. Reflects the proportion of the material’s stress-strain response that is viscous (dissipated as heat). The ratio of the two quantities is the loss tangent, tan d = G”/G’, which is a function of temperature, frequency and polymer structure. Viscoelasticity CHEE 390

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**Dynamic (Oscillatory) Rheometry**

Logarithmic plots of G’ and G” against angular frequency for uncrosslinked poly(n-octyl methacrylate) at 100°C (above Tg), molecular weight 3.6x106. Zones relevant to polymer melt processing Viscoelasticity CHEE 390

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**Dynamic (Oscillatory) Rheometry: HDPE**

Viscoelasticity CHEE 390

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**Simple models of Viscoelasticity**

Liquid-like behavior can be described by a Newtonian model, which can be represented by using a “dashpot” mechanical analog: The simplest elastic solid model is the Hookean model, which can be represented by a “spring” mechanical analog. Viscoelasticity CHEE 390

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Maxwell Model A simple model of a viscoelastic fluid requires at least two components, one to describe the elastic component and the other viscous behavior. One such model is the Maxwell model: which responds with a stress, t, when deformed by a strain, g: Viscoelasticity CHEE 390

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**Maxwell Model l = h/G (s) is called the relaxation time .**

The deformation rate of the Maxwell model is equal to the sum of the individual deformation rates: l = h/G (s) is called the relaxation time If the mechanical model is suddenly extended to a position and held there (g=const., g=0): . Exponential decay in stress – Stress Relaxation Viscoelasticity CHEE 390

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**Viscoelasticity and Stress Relaxation**

Stress relaxation can be measured by shearing the polymer melt in a viscometer (for example cone-and-plate or parallel plate). If the rotation is suddenly stopped, ie. g=0, the measured stress will not fall to zero instantaneously, but will decay in an exponential manner. . Relaxation is slower for Polymer B than for Polymer A, as a result of greater elasticity. These differences may arise from polymer microstructure (molecular weight, branching). Viscoelasticity CHEE 390

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**Viscoelasticity and Stress Relaxation**

The Maxwell model is conceptually reasonable, but it does not fit real data very well. l1 l2 l3 ln Instead, we can use the generalized Maxwell model Viscoelasticity CHEE 390

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**Viscoelasticity and Stress Relaxation**

The relaxation of every element is: l3 l1 l2 l4 ln The relaxation of the generalized model is: where Gi is a weighting constant or “relaxation strength” and li the “relaxation time” Viscoelasticity CHEE 390

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**Viscoelasticity and Dynamic Rheology**

The Generalized Maxwell model can also be used to analyze dynamic oscillatory measurements, by fitting G’ and G” with an appropriate number of elements, each having a unique relaxation strength (Gi) and relaxation time (li): Viscoelasticity CHEE 390

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**Viscoelasticity and Dynamic Rheology**

This example illustrates the storage and loss modulus of uncrosslinked polybutadiene, plotted as a function of oscillation frequency An adequate representation of G’ and G” as a function of frequency required four elements, whose Gi and li are tabulated. li(s) 8.04x10-3 5.93x10-2 1.46x10-1 7.61x10-1 Gi(Pa) 3.00x105 4.83x105 2.98x104 1.04x102 Viscoelasticity CHEE 390

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**Dynamic and Stress Relaxation Testing**

Recall stress relaxation data from page and dynamic rheology from 23.17 G(t) vs t G’(w) vs w A is monodisperse with M<Mc; B is monodisperse with M>>Mc and C is polydisperse The information contained in a stress relaxation plot is complementary to that acquired in a dynamic measurement. Stress relaxation measurements are used when very low frequencies are needed to characterize terminal flow behaviour. Viscoelasticity CHEE 390

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Stress Analysis in Viscoelastic Materials

Stress Analysis in Viscoelastic Materials

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