Oscillatory deformations

Experimentally a variety of quantities are used to characterise linear viscoelasticity (Ferry 1980). There is no need to consider all the characteristics of linear viscoelastic response of polymers which are measured under different regimes of deformation in linear region, they are connected with each other. The study of the reaction of the system in the simple case, when the velocity gradients are independent of the co-ordinates and vary in accordance with the law 

Since the velocity gradient is related to the displacement gradient by the expression v 2 = —iojA12, it follows that, instead of the dynamic viscosity, the use may be made of another characteristic - the dynamic modulus 

The components of the above complex quantities are linked by the relation 

Dynamic modulus is a convenient characteristic of viscoelasticity. To analyse the results, it is convenient also to consider the asymptotic behaviour of the dynamic modulus at high and low frequencies. In the latter case 

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Oscillatory deformations and stresses used experimentally for solids... 

The terms are arranged into sections dealing with basic definitions of stress and strain, deformations used experimentally, stresses observed experimentally, quantities relating stress and deformation, linear viscoelastic behaviour, and oscillatory deformations and stresses used experimentally for solids. The terms which have been selected are those met in the conventional mechanical characterization of polymeric materials. 

There are three modes of free and forced oscillatory deformations which are commonly used experimentally, torsional oscillations, uniaxial extensional oscUlations and flexural oscillations. 

The oscillatory deformations and stresses can be used for solids and liquids. However, the apparatuses employed to measure them are usually designed for solid materials. In principle, they can be modified for use with liquids. 

Oscillatory deformation of a material specimen with the motion generated without the continuous application of an external force. 

Note For any real sample of material the resulting oscillatory deformation is one of decaying amplitude. 

Garnot, P. and Olson, N. F. 1982. Use of oscillatory deformation technique to determine clotting times and rigidities of milk clotted with different concentrations of rennet. J. Food Sci. 47, 1912-1915. 

Kowalchyk, A. W. and Olson, N. F. 1978. Firmness of enzymatically-formed milk gels measured by resistance to oscillatory deformation. J. Dairy Sci. 10, 1375-1379. 

G. V. Vinogradov, A. I. Isayev, D. A. Mustafaev, and Y. Y. Podolsky, Polarization-optical investigation of polymers in fluid and high elastic states under oscillatory deformation, J. Appl. Polym. Sci., 22,665 (1978). 

The most common dynamic method is oscillatory testing, in which the sample is subjected to a sinusoidal oscillatory strain, and the resulting oscillatory stress measured. The more sophisticated rotational viscometers have the additional capability of dynamically testing liquid-like materials using small angle oscillatory shear. A parallel disc viscometer can be set up for testing solid-like materials (e.g., butter), in oscillatory shear. Some UTM-type solids rheometers, in which the moving crosshead can be made to reciprocate sinusoidally, can be used to test solid-like materials in oscillatory deformation in compression, tension or shear. 

D is the diffusion coefficient of the surfactant (0 is the angular frequency of the oscillatory deformation. 

Figure 3.8 Storage and loss moduli versus reduced frequency in small-amplitude oscillatory deformation for a cross-linked polyurethane rubber, Sorbothane 70, at 20°C. The imposed deformation was uniaxial extension, yielding tensile moduli E and E" which were converted to shear moduli using G = E /3 and G" = E"/3. The data at 16 different temperatures ranging from —81°C to 80°C were collapsed to a master curve using methods described in Section 3.5.2, (From Larson et al. 1996, reprinted with permission from Steinkopff Publishers.)...

The relaxation times, t, and corresponding moduli, G, constitute what is called the distribution or spectrum of relaxation times. The relaxation spectrum given in Eq. (3-40) is a distinctive feature of the Rouse model that can be tested experimentally. A simple type of rheological experiment from which this spectrum can be obtained is small-amplitude oscillatory deformation, discussed in Section 1.3.1.4. In this test, at low frequencies, (o < /x, the Rouse model predicts the usual terminal relaxation behavior G — Gco rf, and G" = Gcnxi. More significantly, at higher frequencies, where co is in the range 1/t, oj < 1/tat, the Rouse model predicts a power-law frequency dependence of G and G" ... 

Unusual stress waveforms are observed during oscillatory deformation on such materials when the strain amplitude is much greater than the yield strain. Watanabe and Kotaka (1984) plotted large-amplitude oscillatory shear data in the form of Lissajous figures— that is, plots of periodic stress versus periodic strain. In the nonlinear regime, above the yield strain, unusual rhombic Lissajous figures were obtained at low frequencies, and bent ellipses were obtained at higher frequencies. Qualitatively, the waveforms are similar to those depicted in Fig. 13-23. 

Above relation (1) between cr and y is exact in linear response, where nonlinear contributions in 7 are neglected in the stress. The linear response modulus (to be denoted as g (f)) itself is defined in the quiescent system and describes the small shear-stress fluctuations always present in thermal equilibrium [1, 3]. Often, oscillatory deformations at fixed frequency co are applied and the frequency dependent storage- (G (m)) and loss- (G"((u)) shear moduli are measured in or out of phase, respectively. The former captures elastic while the latter captures dissipative contributions. Both moduli result from Fourier-transformations of the linear response shear modulus g (f), and are thus connected via Kramers-Kronig relations. 

The response of complex materials, e.g., block copolymers, may not even be periodic, as the oscillatory deformation can lead to transient changes in the properties of the material. Eventually, of course, the response should become strictly periodic as the material transforms to its new structure, although some have reported chaotic behavior. 

This was done by Rothwell et al. [58] and Brown et al. [59-61]. The materials under investigation were mainly polystyrene, polycarbonate, and polymethyl methacrylate. Step-wise or oscillatory deformations were performed and the influence of thermal treatment or chemical additives, such as solvents and plasticizers was studied. 

Dynamic methods involve oscillatory deformation and it is therefore necessary to consider the strain generated by a sinusoidal applied stress. 

Additional information can be obtained from dynamic experiments, based on the application of oscillatory deformations of samples ... 

In processing there is little extrudate swelling, or none at all. This in spite of the feet that high elasticity is observed in small-amplitude oscillatory deformations. 

The actuation study of IPMC under a constant current is shown in Fig. 3.19. The figure shows that the oscillatory potential results in oscillatory deformation of IPMC. During the oxidation of formaldehyde, the intermediate (CO) of the reaction strongly binds to the platinum surface of the IPMC and blocks active sites. In this process, the resistance of platinum is increased, which leads to weaker field strength between electrodes of IPMC. Platinum also adsorbs OH which then oxidizes the CO on adjacent platinum sites to CO2. Due to this reaction, conductivity of platinum improves and results in a stronger field strength between the electrodes. This is believed to be the cause for self-oscillatory actuation of IPMC. 

A reversible crosslinked system forms a transient network that when placed under a macrodeformation shear rate exhibited shear flow. The properties depended on disruption and recombination of the reversible network. The linear response to oscillatory deformations has been determined for the reversible network with uniform chains with reversible crosslinking at end groups. Where molar mass (M) was less than the critical M for entanglements the dynamic moduli were related to temperature, M and crosslink bond... 

A linear viscoelastic constitutive model of dilute emulsion viscoelastic properties was proposed by Oldroyd [111, 112]. The model considered low deformation of monodispersed drops of one Newtonian liquid in another, with an interphase. Choi and Schowalter [113] extended their cell model to dilute emulsions with Newtonian matrix and viscoelastic drops under infinitesimally small oscillatory deformation. Oldroyd s model was modified by Palierne [126, 127] for dilute viscoelastic hquids emulsions with polydispersed spherical drops (thus, subject to small deformations) with constant interfacial tension coefficient, Vu, at concentrations below that where the drop-drop interactions start complicating the flow field, that is, < 0.1 ... 

While the Choi and Schowalter [113] theory is fundamental in understanding the rheological behavior of Newtonian emulsions under steady-state flow, the Palierne equation [126], Eq. (2.23), and its numerous modifleations is the preferred model for the dynamic behavior of viscoelastic liquids under small oscillatory deformation. Thus, the linear viscoelastic behavior of such blends as PS with PMMA, PDMS with PEG, and PS with PEMA (poly(ethyl methacrylate))at <0.15 followed Palierne s equation [129]. From the single model parameter, R = R/vu, the extracted interfacial tension coefficient was in good agreement with the value measured directly. However, the theory (developed for dilute emulsions) fails at concentrations above the percolation limit, 0 > (p rc 0.19 0.09. 

Fig. 1 Four types of SMPs (dual-shape effect) depicted as a function of their dynamic thermomechanical behavior. Plotted is the tensile storage modulus vs temperature as measured using a smtdl oscillatory deformation at 1 Hz for (a) Cat. A-I, chemically crosslinked amorphous polymer network (7, = Tg) (b) Cat. A-II, chemically crosslinked semicrystalline polymer networks (Ttrans = 7m) (c) Cat. B-I, physically crosslinked thermoplastic with r,ra,K = 7g and (d) Cat. B-II, physically crosslinked thermoplastic (Tlrans = Tm). Taken from ref [5], Copyright 2007. Reproduced by permission of the Roytd Society of Chemistry, http //dx.doi.org/10.1039/b615954k...

This chapter is devoted to the molecular rheology of transient networks made up of associating polymers in which the network junctions break and recombine. After an introduction to theoretical description of the model networks, the linear response of the network to oscillatory deformations is studied in detail. The analysis is then developed to the nonlinear regime. Stationary nonhnear viscosity, and first and second normal stresses, are calculated and compared with the experiments. The criterion for thickening and thinning of the flows is presented in terms of the molecular parameters. Transient flows such as nonhnear relaxation, start-up flow, etc., are studied within the same theoretical framework. Macroscopic properties such as strain hardening and stress overshoot are related to the tension-elongation curve of the constituent network polymers. 

Several rheometers do not subject the polymer to a steady rate of deformation but to an oscillatory deformation, usually sinusoidal simple shear. If the angular frequency is o, and the shear strain amplitude Yo. the shear strain y can be written as a function of time ... 

Dynamic mechanical methods (typically oscillatory parallel plate rheometry) are commonly used to measure the dynamic mechanical properties from the liquid state to the solid state. By using small-amplitude oscillatory deformations (linear viscoelastic regime), the dynamic storage and loss moduli can be obtained. From these quantities, the viscosity and modulus can be calculated (71) (see Dynamic Mechanical Analysis). 

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