Techniques in oscillatory shear rheology - Indian Institute of ...

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8 Abhijit P. Deshpandemers, colloids, gels, liquid crystals, micelles etc, are recognized based on microscopic/molecularcooperative organization [17]. The viscoelastic behaviour in thelinear and nonlinear regimes is dependent on how the organization responds to deformation.1.4 Linear and nonlinear responseThe response is termed linear if scaled change in input leads to change in the outputwith the same scaling. Describing linear response in other words, the output of acombination of inputs is the same as the combination of outputs of individual inputs.This is also referred to as superposition principle. In the context of SAOS, if thestrain amplitude is changed by a factor, the stress amplitude of the sinusoid alsochanges by the same factor. Therefore, material functions such as G ′ and G” arenot functions of strain amplitude. The linear viscoelastic limit (or the maximumstrain amplitude at which linear viscoelastic behaviour is observed) can be found bymeasuring the material functions as functions of strain amplitude.The onset of nonlinear response is therefore, expected at larger amplitudes ofstrain/stress as discussed in Sect. 2. Nonlinear response, in the context of rheologicalresponse, can be classified as due to large deformations, structural changes andphase transitions [18]. Analysis of nonlinear response is complicated, because interplayof these factors is difficult to resolve and their mathematical representationsare difficult to propose and solve. However, in recent decades, lot of progress intheoretical and experimental tools has led to significant understanding. The use ofoscillatory shear in the nonlinear regime is an example of such endevour.As in the case of linear viscoelastic behaviour, several methods can be used toexamine the nonlinear response of the materials. These include creep and recovery,stress relaxation, oscillatory shear etc. These different methods serve to highlightparticular structure-property relations or they are more relevant for an application.Along with experimental characterization, development and use of comprehensivemodels that explain behaviour of different materials in various methods of probingare continuously being undertaken.1.5 SusceptibilityLinear and nonlinear response of materials to excitation is of interest in variousfields such as mechanical, electrical, thermal and their combinations. The materialproperty relating the response and excitation is referred to as susceptibility. At lowlevels of excitation, linear response is observed. In other words, the implication is (incase of oscillatory excitation) that the susceptibility is independent of the amplitudeof excitation.
Techniques in oscillatory shear rheology 9Linear response is usually very important in understanding basic mechanismsresponsible for material behaviour. Nonlinear response, on the other hand, is morerelevant for applications and is also more difficult to characterize. Measuring thenonlinear response of a material to an excitation is a way to examine properties,which cannot be characterized by examining the linear response. Understandingnonlinear effects has led to breakthroughs in different materials such as elastic,plastic, viscoelastic, optical materials, ferroelectric, freezing, or dipolar glass transitions,isotropic-liquid crystal transition or binary mixtures, superconductivity, fieldor heating effects in electrical transport, heating due to electric field excitation ofsupercooled liquids [24].Similar techniques are used for the analysis of nonlinear response in these diverseareas. As an example, the response in dielectric spectroscopy and mechanical spectroscopycan be understood in relation to each other. For small amplitude of electricfield, similar to Eq. 7, we can write the relation between electric displacement (D)and electric field (E) as,D = ε ∗ E , (16)where ε ∗ is permittivity of material. The modes of material response, as discussedin earlier sections, can also be identified by examining the dielectric response as afunction of frequency. In case of rheology, load and displacement are measured andanalysis is carried out for stress, strain or strain rate. In case of dielectric response,we measure current and voltage and carry out the analysis with electric field, polarisationor electric displacement.When material is subjected to large amplitude of electric field, the above equationis no longer valid. However, material behaviour can be described by writing higherorder terms in electric field [3],D = ε ∗ 1 E + ε ∗ 2 E 2 + ε ∗ 3 E 3 +... , (17)where ε n is the permittivity of n th order. It can be shown that only the odd poweredterms of the above equation are non-zero. In addition to such general descriptions,variety of theoretical and experimental tools are common in investigations ofnonlinear response of materials. In the next few sections, various features of nonlinearresponse in oscillatory shear will be described.2 Oscillatory testing at large amplitudeThe classes of overall oscillatory rheological response of materials can be understoodfrom the diagram shown in Fig. 4. The diagram (referred to as Pipkin diagram)can be recast in the form of dimensionless numbers, Weissenberg number(We = ˙γλ) and Deborah number (De = ωλ). At low frequencies, and at lowstrain amplitudes, the material response is purely viscous and Newtonian. As mentionedearlier, this is at timescales larger than the largest relaxation time in the ma-
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