Oscillatory shear, response

Oscillatory shear response of two PDMS networks at 30 °C. Filled symbols are the storage modulus G and open symbols are the loss modulus G". The circles are... 

Diblock copolymers with roughly equal block lengths can microphase— separate into a lamellar phase, with alternating layers of mostly A monomer and mostly B monomer. When quenched into the lamellar phase from the isotropic phase, these layers form roughly parallel to each other locally. A polydomain texture is created from this quenching, with a typical grain size of order 0.1 pm. The oscillatory shear response of such a quenched sample is observed to have at the lowest measurable frequencies. Can this observed response be the real terminal response of the sample Is this sample a viscoelastic solid or a viscoelastic liquid ... 

Figure 1. Linear dynamic oscillatory shear response of the 50K PBA based Si02 hybrid sample. The data collected at temperatures between 30 and 80 °C were reduced to a single master curve using the principle of time-temperature superpositioning. The horizontal frequency shift factors (af were similar to that for the pure PBA homopolymer.

Dyna.mic Viscometer. A dynamic viscometer is a special type of rotational viscometer used for characterising viscoelastic fluids. It measures elastic as weU as viscous behavior by determining the response to both steady-state and oscillatory shear. The geometry may be cone—plate, parallel plates, or concentric cylinders parallel plates have several advantages, as noted above. 

Rheometric Scientific markets several devices designed for characterizing viscoelastic fluids. These instmments measure the response of a Hquid to sinusoidal oscillatory motion to determine dynamic viscosity as well as storage and loss moduH. The Rheometric Scientific line includes a fluids spectrometer (RFS-II), a dynamic spectrometer (RDS-7700 series II), and a mechanical spectrometer (RMS-800). The fluids spectrometer is designed for fairly low viscosity materials. The dynamic spectrometer can be used to test soHds, melts, and Hquids at frequencies from 10 to 500 rad/s and as a function of strain ampHtude and temperature. It is a stripped down version of the extremely versatile mechanical spectrometer, which is both a dynamic viscometer and a dynamic mechanical testing device. The RMS-800 can carry out measurements under rotational shear, oscillatory shear, torsional motion, and tension compression, as well as normal stress measurements. Step strain, creep, and creep recovery modes are also available. It is used on a wide range of materials, including adhesives, pastes, mbber, and plastics. 

Oscillatory shear experiments are the preferred method to study the rheological behavior due to particle interactions because they directly probe these interactions without the influence of the external flow field as encountered in steady shear experiments. However, phenomena that arise due to the external flow, such as shear thickening, can only be investigated in steady shear experiments. Additionally, the analysis is complicated by the different response of the material to shear and extensional flow. For example, very strong deviations from Trouton s ratio (extensional viscosity is three times the shear viscosity) were found for suspensions [113]. 

Typical for the spectroscopic character of the measurement is the rapid development of a quasi-steady state stress. In the actual experiment, the sample is at rest (equilibrated) until, at t = 0, oscillatory shear flow is started. The shear stress response may be calculated with the general equation of linear viscoelasticity [10] (introducing Eqs. 4-3 and 4-9 into Eq. 3-2)... 

Due to the viscoelastic nature of the material the stress response, after the application of the oscillatory shear strain, is also a sinusoidal but out of phase relative to the strain what can be represented by equation (2.7) as... 

The viscoelastic response of a liquid in oscillatory shear is markedly different. The terminal response (at low frequency) of any liquid is dominated by the loss modulus because the stress is very nearly in-phase with the shear rate. The viscoelastic liquid has G" G at low frequencies. G" is... 

An example of the linear viscoelastic response in oscillatory shear for a nearly monodisperse linear polybutadiene melt is shown in Fig. 1.2%. Extrapolation of the limiting power laws oiG uP and G" u (the dashed lines in Fig. 7.28) to the point where they cross has special significance. The intersection of the power laws G = J qrj uP and G = t]uj using the above two equations allows us to solve for the frequency where they cross uj= l/(/)/eq), which is the reciprocal of the relaxation time r [Eq. (7.132)]. The modulus level where the two extrapolations cross, obtained by setting a = 1/r = 1 /(/ /eq) iti either equation, is simply the reciprocal of the steady... 

While in the linear response regime, modulus and density correlator are measurable quantities outside the linear regime, both quantities serve as tools in the ITT approach only. The transient correlator and shear modulus provide a route to the stationary averages because they describe the decay of equilibrium fluctuations under external shear and their time integral provides an approximation for the stationary distribution function. Determination of tlie frequency dependent moduli under large amplitude oscillatory shear has only recently become possible [58], and requires an extension of the present approach to time dependent shear rates in (3) [59],... 

The loss and storage moduli of small amplitude oscillatory shear measurements [1, 3] follow from (5b) in the linear response case at y = 0 ... 

For the mechanical response one subjects the material to an oscillatory shearing motion with the imposed velocity... 

These differences are shown in the following examples where measurements of the dynamic moduli, G and G" are used to monitor the structure of gel networks. Measurements are performed by imposing an oscillatory shear field on the material and measuring the oscillatory stress response. The stress is decomposed into a component in phase with the displacement (which defines the storage modulus G ) and a component 90 out of phase (which defines the loss modulus G"). The value of G indicates the elastic and network structure in the system (15, 17, 18) and can be interpreted by using polymer kinetic theories. 

In a dynamic experiment, a small-amplitude oscillatory shear is imposed to a molten polymer confined in the rheometer. The shear stress response of the polymeric system can be expressed as in Equation 22.14. In this equation, G and G" are dynamic moduli related to the elastic storage energy and dissipated energy of the system, respectively. For a viscoelastic fluid, two independent normal stress differences, namely, first and second normal stress differences can be defined. These quantities are calculated in terms of the differences of the components of the stress tensor, as indicated in Equation 22.15a and 22.15b, and can be obtained, for instance, from the radial pressure distribution in a cone-and-plate rheometer [5]. Some other experiments used in the determination of the normal stress differences can be found elsewhere [9, 22] ... 

The rheological behavior of these materials is still far from being fully understood but relationships between their rheology and the degree of exfoliation of the nanoparticles have been reported [73]. An increase in the steady shear flow viscosity with the clay content has been reported for most systems [62, 74], while in some cases, viscosity decreases with low clay loading [46, 75]. Another important characteristic of exfoliated nanocomposites is the loss of the complex viscosity Newtonian plateau in oscillatory shear flow [76-80]. Transient experiments have also been used to study the rheological response of polymer nanocomposites. The degree of exfoliation is associated with the amplitude of stress overshoots in start-up experiment [81]. Two main modes of relaxation have been observed in the stress relaxation (step shear) test, namely, a fast mode associated with the polymer matrix and a slow mode associated with the polymer-clay network [60]. The presence of a clay-polymer network has also been evidenced by Cole-Cole plots [82]. 

Because of the complications caused by the stress-induced orientation of clay platelets resulting in different rheological responses, the studies of CPNC flow focus on smaU-amplitude oscillatory shear flow (SAGS). As the discussion on the steady-state flow indicates, there is a great diversity of structures within the CPNC family. Whereas some nanocomposites form strong three-dimensional structures, others do not thus while nonlinear viscoelastic behavior is observed for most CPNCs, some systems can be smdied within the linear regime. 

In the case of oscillatory shear experiments, for example, the strain amplitude must usually be low. For large and more rapid deformations, the linear theory has not been validated. The response to an imposed deformation depends on (1) the size of the deformation, (2) the rate of deformation, and (3) the kinematics of the deformation. 

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