Transient normal stresses

In fact, this relationship had been previously found to describe experimental data for LDPE for 7 up to 30 [37], a polystyrene solution for yup to at least 8 [38] and polybutadiene solution for yup to at least 3.3 [39]. It should be noted that the precise measurement of transient normal stress differences requires great care, as it is very difficult to avoid errors due to instrument compliance and temperature variations associated with the operation of the heating/cooling system [34]. 

The measurement of normal stress differences in transient deformations is extremely sensitive to small variations in gap spacing, which can arise from instrument compliance or minute temperature variations. Venerus and Kahvand [43] have shown how to evaluate the effect of instrument compliance by measuring the response using several sets of cone-plate fixtures. If a Force Rebalance Transducer is used for a transient normal stress measurement to compensate continuously for compliance in order to keep the gap constant, the response time of the transducer may affect the data. Also, the thermal expansion that results from the power dissipated in the transducer can affect the gap spacing and is of particular concern when normal stresses are being measured [104]. 

In Section 10.8.1 it was noted that molecular orientation results in flow birefringence in a polarizable polymer, and if the melt is transparent, optical techniques can be used to determine the three components of the stress tensor in uniform, shear flows [91-93]. To determine the transient normal stress differences, the phase-modulated polarization technique was developed by Frattini and Fuller [ 126]. Kalogrianitis and van Egmond [ 127] used this technique to determine the shear stress and both the normal stress differences as functions of time in start-up of steady simple shear. Optical techniques are particularly attractive for measurements of normal stress differences, since such methods do not require the use of a mechanical transducer, whose compliance plagues measurements of normal stress differences by mechanical rheometry. 

Elasticity is another manifestation of non-Newtonian behavior. Elastic Hquids resist stress and deform reversibly provided that the strain is not too large. The elastic modulus is the ratio of the stress to the strain. Elasticity can be characterized usiag transient measurements such as recoil when a spinning bob stops rotating, or by steady-state measurements such as normal stress ia rotating plates. 

The slower rise and decay of normal stress transients compared to shear stress arises quite simply and directly from the polydispersity of relaxation times (78), and probably has no direct bearing on entanglement mechanisms per se. Likewise, the depression of the superimposed moduli at low frequencies follows from rather non-specific continuum models, the loss moduli by Eqs.(8.49) and (8.50) from the simple fluid model, and the storage moduli from the following properties of the more specific but still quite general BKZ model (366,372) ... 

The trouble is that, under transient conditions, the shear recovery vs. preceding shear deformation can be much more sensitive to deviations from the strict behaviour of a second order fluid than the shear viscosity or the normal stress difference. A few entanglements between extraordinarily long chain molecules may be responsible for a maximum in the shear recovery. If this is the case, a shear recovery higher than the one... 

For systems where the stress-optical rule applies, birefringence measurements offer several advantages compared with mechanical methods. For example, transient measurements of the first normal stress difference can be readily obtained optically, whereas this can be problematic using direct mechanical techniques. Osaki and coworkers [26], using a procedure described in section 8.2.1 performed transient measurements of birefringence and the extinction angle on concentrated polystyrene solutions, from which the shear stress and first normal stress difference were calculated. Interestingly, N j was observed to... 

Transient birefringence measurements were used by Larson et al. [112] to test the validity of the Lodge-Meissner relationship for entangled polymer solutions. This relationship states that the ratio of the first normal stress difference to the shear stress following a step strain is simply Nx/%xy - y, where y is the strain. Those authors found the relationship was valid, except for ultrahigh molecular weight materials. 

Optical measurements often have a greater sensitivity compared with mechanical measurements. Semidilute polymers, for example, may not be sufficiently viscous to permit reliable transient stress measurements or steady state normal stress measurements. Chow and coworkers [113] used two-color flow birefringence to study semidilute solutions of the semirigid biopolymer, collagen, and used the results to test the Doi and Edwards model discussed in section 7.1.6.4. That work concluded that the model could successfully account for the observed birefringence and orientation angles if modifications to the model proposed by Marrucci and Grizzuti [114] that account for polydispersity, were used. 

From Eqs. (15.61) and (15.62) it follows that the shear stress and the first normal stress difference gradually increase from 0 to the steady state value. In this respect sometimes the following definitions are suggested for the transient values of viscosity and first normal stress coefficient... 

Fig. 5.4 (a, b and c) Initial and final (steady-state) strain rate of a hypothetical composite as a function of normalized stress. The dashed lines represent the creep rate of the constituents, (b) and (c), which detail the two framed regions in Fig. 5.4(a), show the transient paths of the stress and strain rate for the composite and its constituents for the two corresponding stress regimes cr and a<. The dashed lines in (b) and (c) show the creep rate of the constituents (excluding the elastic components), which follow the monolithic creep behavior of each phase the total strain rate of the composite and the constituents must remain equal. 

Figures 7 and 8 show the predictions of the Wagner model compared to experimental data for transient shear viscosity and first normal stress coefficient of LD. These have been calculated according to ...

The slip parameter can be easily determined from various experiments in shear situations by some fit of the steady state shear viscosity and primary normal stress coefficient. Analytic expressions are easily derived in steady state and transient flows in the form ... 

Alternatively, in transient flows the slip parameter can also be determined using the time position of the maximum of the experimental fimctions (tT for tangential stress and tN for normal stress). However, this can only be performed accurately if the stress overshoot is large enough to avoid uncertainties in these values and this can only be achieved at high shear rates. In this case, according to equations (50c) and (50d) ... 

Figure 1.10 Transient shear stress a and first normal stress difference A i after start-up of steady shearing for a low-density polyethylene melt, Melt I, at a shear rate y = 1 sec . (From Laun 1978, reprinted with permission from Steinkopff Publishers.)...

Figure 3 Predicted steady-state shear stress as a function of shear rate for values of the CCR rate paramenter Cy = 0,0.01,0.1,1. The inset displays the shear stress and normal stress transients in the limit of high shear rate (within the zero-stretch regime). The results displayed are extrapolations from numerical calculations on finite chain lengths n to the limit of large n ...

Transient Effects In system where the structure changes with time upon imposition of stress, the transient effects are important. For example, semi-concentrated fiber suspensions in shear and extension show large transient peaks in the first and the second normal stress dilference [Dinh and Armstrong, 1984 Bibbo et al., 1985]. It is interesting that the peaks appear at dilferent times, first for N, then for Np and finally for... 

The theory makes it possible to compute the drop aspect ratio, p = a /a, a parameter that can be directly measured in either transient or steady-state flows. Following the derivation by Hinch and Acrivos [1980] the flow-induced changes to the drop aspect ratio were assumed to be proportional to the first normal stress difference coefficient of the matrix fluid. The coalescence was assumed to follow the Silberberg and Kuhn [1954] mechanism. These assumptions substituted into Eq 7.110 gave a simple dependence for the aspect ratio ... 

The shear stress growth on the inception of shear flow may reflect the orientation of the liquid crystalline domains. Orientation seems to occur within less than 2 strain units in shear flow. This primary normal stress difference can exhibit different phenomena from the shear stress response. In particular for the 60 mole % PHB/PET system, values of N are positive and rise gradually to the equilibrium values whereas the 80 mole % PHB/PET system can exhibit negative values of N. Ericksen s transversely isotropic fluid theory can qualitatively handle some of the observed phenomena. Further studies which couple the transient flow behavior to the orientation and morphology need to be carried out. 

Equation (10) cannot be applied until A, the equivalent relaxation time for the fluid, is known. However, A is defined by the linear Maxwell model, and actual polymer solutions exhibit marked nonlinear viscoelastic properties [5,6,7]. For both fresh and shear degraded solutions of Separan AP 30 polyacrylamide, which exhibit pronounced drag reduction in turbulent flow, Chang and Darby [8] have measured the nonlinear viscosity and first normal stress functions, and Tsai and Darby [6] have reported transient elastic properties of similar solutions, A nonlinear hereditary integral function containing six parameters has been proposed to represent the measured properties [8], The apparent viscosity function predicted by this model is ... 

This represents the solid lines in Fig. 1, which shows data for three concentrations (500, 250 and 100 wppm) each of the fresh and shear degraded solutions. The parameters that govern the viscosity function in (11) also strongly influence other predicted response functions such as normal stress behavior, elongational viscosity, transient response, etc. 

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. 

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