Critical Viscoelastic Constants

The suppression of director fluctuations near the nematic-smectic A (N-A) transition because of divergence in the twist and bend elastic constants and the twist viscosity 7e [6.6] are now examined. Above the phase transition at T/vaj there are cybotactic smectic A clusters in the nematic phase, whose dimension is measured by a coherence length In fact, it is the coupling between the nematic director and the smectic order parameter that causes the viscoelastic constants to approach infinity at TnA Hence, A22J A33 oc while 7e oc Suppose there is interest in a frequency far below the high-frequency cutoffs such that A 1. In this limit, 

The analogy to the lambda transition in helium [6.28] is used for the N-A transition to give oc (T and 

Therefore, (lj) does not diverge at the N-A transition. The effects of translational diflhision were included by Brochard [6.6, 6.7] in treating spin relaxation above the N-A transition. Putting j and jDj, the components of translational diffusion tensor parallel and perpendicular to no, into Eq. (6.40), 

Note that this expression is identical to Eq. (6.43) when the high-frequency cutoffs are neglected. 

Again jpo ( ) goes to zero at Tna faster than (o ), but their dependences on are altered by the inclusion of translational diffusion of the molecule. It is noted that measurements of 1/Ti may lead to the determination of critical exponents of 22 33, and 7e. 

---

The viscoelastic creep modulus may be determined at a given temperature by dividing the constant applied stress by the total strain prevailing at a particular time. Since the creep strain increases with time, the viscoelastic creep modulus must decrease with time (Fig. 2-23). Below its critical stress for linear viscoelasticity, the viscoelastic creep modulus versus time curve for a material is independent of the applied stress. In other words, the family of strain versus time curves for a material at a given temperature and several levels of applied stress may be collapsed to a single viscoelastic creep-modulus-time-curve if the highest applied stress is less than the critical value. 

Viscoelastic creep data can be presented by plotting the creep modulus (constant applied stress divided by total strain at a particular time) as a function of time [23-26], Below its critical stress, the viscoelastic creep modulus is independent of stress applied. A family of curves describing strain versus time response to various applied stress may be represented by a single viscoelastic creep modulus versus time curve if the applied stresses are below the material s critical stress value. 

While the dynamic experiments described above are to be conducted in the linear viscoelastic range, another experiment can be conducted in which the results obtained in the non-linear range are useful. With a controlled-stress rheometer, one can conduct an experiment in which the stress is increased continuously at a constant oscillatory frequency, say 1 Hz. Results obtained in such an experiment are shown schematically in Figure 3-40. As the stress is increased continuously, initially, G and G" remain relatively constant until at a critical value of stress, Oc, the magnitude of G decreases sharply and that of G" also decreases not as sharply after a slight inerease. One may also use the value of the applied stress at which the curves of G and G" intersect... 

Differences between solid-like and liquid-bke complex fluids show up in all three of the shearing measurements discussed thus far the shear start-up viscosity t), the steady-state viscosity rj(y), and the linear viscoelastic moduli G co) and G (o). The start-up stresses a = y/ +()>, t) of prototypical liquid-like and solid-like complex fluids are depicted in Fig. 1-6. For the liquid-like fluid the viscosity instantaneously reaches a steady-state value after inception of shear, while for the solid-like fluid the stress grows linearly with strain up to a critical shear strain, above which the material yields, or flows, at constant shear stress. 

The frequency is fixed say at 1 H z (or 6.28 rad s ) and G, G and G" are measured as a function of strain amplitude this is illustrated in Figure 20.12, where G, G and G" are seen to remain constant up to a critical strain. This is the linear viscoelastic region where the moduli are independent of the applied strain. Above however, G and G start to decrease whereas G" starts to increase with further increase in y this is the nonlinear region. 

In this case, the oscillation is fixed (e.g., at 1 Hz) and the viscoelastic parameters are measured as a function of strain ampHtude. G, G and G" remain virtually constant up to a critical strain value, (this region is the linear viscoelastic region), but above G and G starts to fall, whereas G" starts to increase (this is the nonhnear region). The value of may be identified with the minimum strain above which the structure of the dispersion starts to break down (e.g., the breakdown of floes into smaller units and/or the breakdown of a structuring agent). 

Measurement of C requires more sophisticated and expensive rheometers and more involved experimental procedures. It must be remembered that experiments have to he carried out below the critical strain value (see Sec II), or in [he region of linear viscoelastic behavior. This region is determined by measuring the complex modulus G as a function of the applied strain at a constant oscillation frequency (usually 1 Hz). Up to 7, G does not vary with the strain above Yr, G tends to drop. The evaluation of oscillatory parameters is more often restricted to product formulation studies and research. However, a controlled-fall penetrometer may be used to compare the degree of elasticity between different samples. Creep compliance and creep relaxation experiments may be obtained by means of this type of device. In fact, a penetrometer may be the only way to assess viscoeIa.sticity when the sample does not adhere to solid surfaces, or adheres too well, or cures to become a solid or semisolid. This is the case of many dental products such as fillings, impression putties, sealants, and cements. 

The critical crack tip opening displacement 5c (CTOD) can be used as a fracture criterion. A constant 5c is equivalent to constant if ic and Go as used in LEFM. If 5c remains constant outside the LEFM range, it can be used to make predictions for viscoelastic and large-scale plasticity behavior. 

In dynamic (oscillatory) measurements, one applies a sinusoidal strain or stress (with amplitudes yo or < o and frequency co in rad s ) and the stress or strain is measured simultaneously. For a viscoelastic system, the stress oscillates with the same frequency as the strain, but out of phase. From the time shift of stress and strain, one can calculate the phase angle shift <5. This allows one to obtain the various viscoelastic parameters G (the complex modulus), G (the storage modulus, i.e. the elastic component of the complex modulus) and G" (the loss modulus or the viscous component of the complex modulus). These viscoelastic parameters are measured as a function of strain amplitude (at constant frequency) to obtain the linear viscoelastic region, whereby G, G and G" are independent of the applied strain until a critical strain above which G and G begin to decrease with further increase of strain, whereas G" shows an increase. Below y the structure of the system is not broken down, whereas above y the structure begins to break. From G and one can obtain the cohesive energy density of the structure... 

---