Measurement of second normal stress

E. F. Brown, W. R. Burghardt, H. Kahvand, and D. C. Venerus, Comparsion of Optical and Mechanical Measurements of Second Normal Stress Difference Relaxation Following Step-Strain Rheol. Acta 34, 221—234 (1995). 

Brown,E. F., Burghardt, W. R., Kahvand, H., Veneras, D. C.Comparison ofopticalandmechanical measurements of second normal stress difference relaxation following step strain. Rheol. Acta... 

Kuo Y, Tanner RI (1974) Use of open-channel flows to measure the second normal stress differences Rheol Acta 13 931... 

Normal stress measurements for some MLC nematics was reported to be consistent with that of a second-order fluid, that the low frequency limit of G /co equaled the low shear limit of N /(dy/dty [36]. Coleman and Markowitz demonstrated that for a second-order fluid in slow Couette flow, the viscoelastic contribution to the normal thrust must have a sign opposite to the inertial contribution on thermodynamic grounds [37]. A textbook by Walters stated that the measurements of first normal stress difference have invariably led to a positive quantity except for one case which was later found to be in error [38]. Adams and Lodge reported the possible observation of a negative value for Nj for solutions of poly isobutylene + decalin [39]. This result was obtained by a combination of obtained from radial... 

Here we describe the strain history with the Finger strain tensor C 1(t t ) as proposed by Lodge [55] in his rubber-like liquid theory. This equation was found to describe the stress in deforming polymer melts as long as the strains are small (second strain invariant below about 3 [56] ). The permanent contribution GcC 1 (r t0) has to be added for a linear viscoelastic solid only. C 1(t t0) is the strain between the stress free state t0 and the instantaneous state t. Other strain measures or a combination of strain tensors, as discussed in detail by Larson [57], might also be appropriate and will be considered in future studies. A combination of Finger C 1(t t ) and Cauchy C(t /. ) strain tensors is known to express the finite second normal stress difference in shear, for instance. 

Of major interest in this review are t](y) and (O) for which a large quantity of data has now been accumulated on well-characterized polymers. Some limited information is also available on the shear rate dependence of The second normal stress function has proved to be rather difficult to measure N2 appears to be negative and somewhat smaller in magnitude than N2 82). 

A number of experimental techniques have been developed for measuring the viscosity and the first (and second) normal stress coefficient. In Table 15.2 a survey of these methods has already been given. 

Prediction of the second normal stress difference in shear and thermodynamic consistency obviously requires the use of a different strain measure including of the Cauchy strain tensor in the form of the K-BKZ model. With the ratio of second to first normal stress difference as a new parameter, Wagner and Demarmels [32] have shown that this is also necessary for accurate prediction of other flow situations such as equibiaxial extension, for example. 

The Wagner equation finds its theoretical basis in the derivation of the more general K-BKZ equation. Unfortunately, it loses part of its original thermod3mamic consistency since, for simplification purposes, only the Finger strain measure is taken into account. Doing so, it is no more derivable from any potential function and additionally it does not predict second normal stress differences any more. 

Laun (1994) has measured the first and second normal stress differences and N2 of suspensions at steady state in the shear-thickened state. He found, remarkably, that Ni is negative and N2 is positive, which are opposite the usual signs for these quantities He also found the following relationship between A i, N2, and the shear stress [Pg.308]

Magda, J.J. Back, S.G. DeVries, K.L. Larson, R.G. Shear flows of liquid crystal polymers measurements of the second normal stress difference and the Doi molecular theory. Macromolecules 1991, 24, 4460-4468. 

Here t, 4, and 4 2 are three important material functions of a nonnewtonian fluid in steady shear flow. Experimentally, the apparent viscosity is the best known material function. There are numerous viscometers that can be used to measure the viscosity for almost all nonnewtonian fluids. Manipulating the measuring conditions allows the viscosity to be measured over the entire shear rate range. Instruments to measure the first normal stress coefficients are commercially available and provide accurate results for polymer melts and concentrated polymer solutions. The available experimental results on polymer melts show that , is positive and that it approaches zero as y approaches zero. Studies related to the second normal stress coefficient 4 reveal that it is much smaller than 4V and, furthermore, 4 2 is negative. For 2.5 percent polyacrylamide in a 50/50 mixture of water and glycerin, -4 2/4 i is reported to be in the range of 0.0001 to 0.1 [7]. 

Here, Ttee is the pressure, which may be measured by flush-mounted pressure transducers located on the plate, and F is the total force applied on the plate to keep the tip of the cone on the surface of the plate. However, evaluation of both the first and second normal stress coefficients requires the pressure distribution on the plate. Only a few instruments have the capacity to measure both F] and P2. 

Whorlow [1992] notes that, of the many methods which have been proposed for the measurement of various combinations of the first and second normal stress differences, N and N2 respectively, few can give reliable estimates of N2- Combined pressure gradient and total force measurements in the cone-and-plate geometry, or combined cone-and-plate and plate-plate force measurements, appear to give reliable values [Walters, 1975] and satisfactory results may also be obtained from techniques based on the measurement of the elevation of the surface of a liquid as it fiows down an inclined open duct [Kuo and Tanner, 1974],... 

Cone-and-plate rheology represents one of the most ideal means of measuring viscosity of molten plastics. The melt is placed between a plate and a cone that are located a precise distance apart. The torque required to impose a constant rate of angular rotation is measured. Cone angles of 3° or less are used, which result in a geometry where the shear rate is constant with radius so that no assumptions about the flow kinematics are needed. The main drawback of the technique is that it is limited to very low shear rates. It remains an ideal method for the determination of the zero shear viscosity, a number that is found to correlate well to the molecular weight of the plastic. The first and second normal stresses can also be measured using this apparatus. 

Knowledge of the behavior of apparent viscosity and the normal stress coefficients is inversely proportional to the case of the experimental measurement needed. The most is known about the apparent viscosity and the least about the second normal stress coefficient (i /2). 

Data is provided on the complete rheological characterisation, relative to steady-state shear stress and first and second normal stress differences as a function of shear rate, as well as the complementary oscillatory data, see figxires 13 and 14. The measured extensional properties are also available, see figure 15 [7]. 

Still another method for obtaining information about the second normal stress difference involves measuring the pressure difference between the walls of two coaxial cylinders between which annular flow is occurring. The reduction of the data is complicated, however. Also, flow through a tube can be employed. ... 

In the case of an x-y shear flow, it is also possible to direct the light beam along either the x or the y axis, thus enabling determination of A 23 or Artis, respectively. According to the SOR, optical measurements of T22 - T33 or n 1 - T33 are then possible. > en either quantity is combined with the first normal stress difference obtained from the more common measurement of Arti2 and X, the second normal stress difference can be determined. Measurement of Art 13 in the x-y plane has been achieved by at least two geometries. 

C.-S. Lee, J. J. Magda, K. L. DeVries, and J. W. Mays. Measurements of the second normal stress difference for star polymers with highly entangled branches. Macromolecules, 25 (1992), 4744 750. 

The second normal stress difference is the most difficult to measure of the three viscometric functions. Several techniques have been proposed for the determination of the second normal stress difference. The most common method involves the use of a cone-plate rheometer, where... 

Turning to the behavior of typical melts, it is found that the damping function is not nearly as sensitive to molecular structure as are the linear viscoelastic properties, e.g. the storage and loss moduli. The rubberlike liquid, as well as the tube model, predict that the ratio of the first normal stress difference to the shear stress in step shear should be equal to the strain at all strains, and this is in fact observed. The other quantity measured in simple shear experiments is the second normal stress difference, but this is difficult to measure and few data are available. Of the shear histories other than step strain than have been used to study nonlinear viscoelasticity, start-up of steady simple shear has been the most used. If the shear rate is sufficiently large, some degree of chain stretch can be generated in the early stages. 

This flow is shown in Figure 2(a) where the velocity distribution is given by Vx = yy,Vy = 0,V2 = 0 and y = dv /dy is a constant. For this flow it is possible to measure a shear stress a first normal stress difference x — and a second normal stress difference These three quantities are in general strong functions of the shear rate y — dVx/dyl It is conventional to define three viscometric functions , namely the (non-Newtonian) viscosity rj (equation 1), the first normal stress coefficient Pi (equation 2) and the second normal stress coefficient 2 (equation 3), as follows... 

The normal stress response of suspensions has received rather less attention than the effective viscosity. However, since the first measurement of normal stress differences in suspensions, with the work of Gadala-Maria (1979), the normal stresses and their relation to the microstructure of the material on one hand (see Section 11.4) and to the bulk flow response on the other (see Section 11.5) have garnered increasing attention. Restricting attention to a simple shear flow, we first define the flow, gradient, and vorticity directions as the 1, 2, and 3, directions, respectively (1 = x, 2 = y, 3 = z in the shear flow Ux = Yy). The first and second normal stress differences are given, respectively, by... 

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