Rheological functions

Fig. 12. The rheological functions G ((o) and G"(co) for an H-shaped PI of arm molecular weigh 20 kg mol and backbone 110 kg mol" [46]. The high-frequency arm-retraction modes can be seen as the shoulder from co 10 to co 10 together with a low-frequency peak due to the cross-bar dynamics at co 10. The smooth curves are the predictions of a model which takes Eq. (33) as the basis for the arm-retraction times and a Doi-Edwards reptation spectrum with fluctuations for the backbone. The reptation time is correctly predicted, as is the spectrum from the arm modes, though the low frequency form is more polydisperse than the simple theory predicts...

Wagner et al. (63-66) have recently developed another family of reptation-based molecular theory constitutive equations, named molecular stress function (MSF) models, which are quite successful in closely accounting for all the start-up rheological functions in both shear and extensional flows (see Fig. 3.7). It is noteworthy that the latest MSF model (66) is capable of very good predictions for monodispersed, polydispersed and branched polymers. In their model, the reptation tube diameter is allowed not only to stretch, but also to reduce from its original value. The molecular stress function/(f), which is the ratio of the reduction to the original diameter and the MSF constitutive equation, is related to the Doi-Edwards reptation model integral-form equation as follows ... 

Hwang, J., and Kokini, J. (1991). Structure and rheological function of side branches of carbohydrate polymers. /. Texture Studies 22 123—167. 

The structure factor (P(M)) describes the topological relaxation of the macromolecular chains this is the function which will be described by molecular models, P(M) being the distribution of moleciilar weights. Here lies a very important point if one wishes to "isolate" the topological effects in order to test molecular models, one has to use rheological functions defined at the same segmental mobility, and hence the same value of the mobility factor as far as viscosity is concerned, the reduced function (T) will be used instead of the viscosity itself. 

Part 2 presents a summary of the theoretical considerations and basic assumptions that lead to the model equations. Part 3 discusses some experimental aspects and focuses on the measmements in various shear and uniaxial elongational flow situations. Part 4 and 5 are devoted to the comparisons between experimental and predicted rheological functions. Problems encountered in the choice of correct sets of parameters or related to the use of each type of equation will be discussed in view of discrepancies between model and data. 

In the last decade of the nineteenth century, Maurice Couette invented the concentric cylinder viscometer. This instrument was probably the first rotating device used to measure viscosities. Besides the coaxial cylinders (Couette geometry), other rotating viscometers with cone-plate and plate-plate geometries are used. Most of the viscometers used nowadays to determine apparent viscosities and other important rheological functions as a function of the shear rate are rotating devices. 

This book is concerned mainly with the study of the viscoelastic response of isotropic macromolecular systems to mechanical force fields. Owing to diverse influences on the viscoelastic behavior in multiphase systems (e.g., changes in morphology and interfaces by action of the force fields, interactions between phases, etc.), it is difficult to relate the measured rheological functions to the intrinsic physical properties of the systems and, as a result, the viscoelastic behavior of polymer blends and liquid crystals is not addressed in this book. 

From the morphological point of view, the existence of these critical concentrations, has a more profound influence on some rheological functions than on others, ie., strain recovery, Sj. Thornton et al. (44) reported step-increases in vs. w. function at concentrations and 0. The steps were not visibile in n vs. w plots. 

LDPE, and with polypropylene, PP, was studied In steady state shear, dynamic shear and uniaxial extenslonal fields. Interrelations between diverse rheological functions are discussed In terms of the linear viscoelastic behavior and Its modification by phase separation Into complex morphology. One of the more Important observations Is the difference In elongational flow behavior of LLDPE/PP blends from that of the other blends the strain hardening (Important for e.g. fllm blowing and wire coating) occurs In the latter ones but not In the former. 

Since dissolution of UHMWPE strongly affects the molecular weight distribution, MWD, of the matrix polymer, any rheological function sensitive to MWD can be used to follow the dissolution process. The dynamic viscoelastic data provided a simple and easy tool (10, 11). As will be discussed later, to accomplish this the dynamic data were examined using either (1) Zelchner-Patel... 

T, is a complex function dependent on sample polydlsperslty which can not be readily correlated with a single molecular or rheological function ( ). However, the correlation can be obtained through Hq as Intermediary. 

Equation 4-3b may be regarded as the basic equation of capillary viscometry. The properties of the liquid enter the equation by way of the rheological function which expresses the relation between rate of shear and shear stress. For Newtonian flow from Eqn 4-2 we get... 

Two types of rheological phenomena can be used for the detection of blend s miscibility (1) influence of polydispersity on the rheological functions, and (2) the inherent nature of the two-phase flow. The first type draws conclusions about miscibility from, e.g., coordinates of the relaxation spectmm maximum cross-point coordinates (G, CO ) [Zeichner and Patel, 1981] free volume gradient of viscosity a = d(lnT]) / df the initial slope of the stress growth function S = d(lnr +g)/dlnt the power-law exponent n = d(lnOj2)/dlny = S, etc. The second type involves evaluation of the extrudate swell parameter, B = D/D, strain (or form) recovery, apparent yield stress, etc. 

In multiphase systems, there are many possible configurations of the interacting phases. Following the concepts of statistical mechanics, the rheological functions must be volume-averaged [Hashin, 1964]. The averaged quantities are sometimes known as bulk quantities. For example, the bulk rate of strain tensor, (y ), is expressed as ... 

Thus, for miscible polymer blends, the relaxation spectrum is a linear function of the relaxation spectra of the components and their weight fractions, Wj, hence one may use rheological functions to detect miscibility/immiscibility of polymer blends. An example is presented in Figure 7.14 [Utracki and Schlund, 1987]. 

Over the years, there has been an effort to describe the rheological functions of liquid mix-mres from those of neat ingredients and their content. The theoretical treatment has been discussed, to some extent, in Part 7.3.2.3. A summary of these efforts is given in Table 7.11. 

Rheological function Simple Fluid Polymer Blend... 

Direct measurements of and indicate a parallel dependence of both these functions plotted vs. ( ), even when these have a sigmoidal form. Considering the steady shear flow of a two-phase system, it is generally accepted that the rate of deformation may be discontinuous at the interface, and it is more appropriate to consider variation of the rheological functions at constant stress than at constant rate, i.e., = Nj(Oj2). 

Eros. I. Thaleb. A. Rheological studies of creams. 1. Rheological functions and Structure of creams. Aoa Pharm, Hung,. 64(3) 101-103. 1994,... 

K. Minagawa, K. Koyama, Electro- and magneto-rheological materials, stim-uli-induced rheological functions, Curr. Org. Chem., 2005,9,1643. 

---