Newtonian polymer liquids

The viscosity of a fluid arises from the internal friction of the fluid, and it manifests itself externally as the resistance of the fluid to flow. With respect to viscosity there are two broad classes of fluids Newtonian and non-Newtonian. Newtonian fluids have a constant viscosity regardless of strain rate. Low-molecular-weight pure liquids are examples of Newtonian fluids. Non-Newtonian fluids do not have a constant viscosity and will either thicken or thin when strain is applied. Polymers, colloidal suspensions, and emulsions are examples of non-Newtonian fluids [1]. To date, researchers have treated ionic liquids as Newtonian fluids, and no data indicating that there are non-Newtonian ionic liquids have so far been published. However, no research effort has yet been specifically directed towards investigation of potential non-Newtonian behavior in these systems. 

Though experimental data on suspensions of fibers in Newtonian dispersion media give more or less regular picture, a transition to non-Newtonian viscoelastic liquids, as Metzner noted [21], makes the whole picture far or less clear. Probably, the possibility to make somewhat general conclusions on a longitudinal flow of suspensions in polymer melts requires first of all establishing clear rules of behavior of pure melts at uniaxial extension this problem by itself has no solution as yet. 

The nature of the liquid in contact with a surface is also very important, with respect to boundary conditions. Although slip has long been observed for highly non-Newtonian, viscoelastic liquids such as polymer flows and extrusions, many recent studies have reported slippage of Newtonian liquids under a variety of experimental conditions. This clearly indicates that care must be taken when modeling any type of micro- or nanofluidic system, no matter which liquid is employed. 

The ease of rotation of chain segments has a great influence on the properties of a polymer structure. As previously discussed, this is a function of polymer structure and temperature. The glass transition temperature of a polymer is that temperature at which backbone segments begin to rotate. An ideal noncrystalline polymer is a glass below the transition temperature and a non-Newtonian viscous liquid at temperatures above Tg. Thus, normally, plastics have Tg values above the use temperature, while elastomers have Tg values below the use temperature. 

There are several aspects of rheological behavior exhibited by polymeric liquids that set these materials apart from Newtonian fluids. An excellent summary of the differences in fluid response between Newtonian liquids and non-Newtonian polymeric liquids under various scenarios has been given by Bird and Curtis [3]. Two very well-known atypical phenomena exhibited by polymeric liquids are the Weissenberg effect (a polymer melt or solution tends to climb a rotating rod) and extrudate swelling, which are illustrated in Figure 22.4. 

For both heat and mass transfer in laminar boimdaiy layers, it has been assumed that the momentum boimdary layer is everywhere thicker than the thermal and diffusion boundary layers. For Newtonian fluids (n = 1), it can readily be seen that s varies as Pr / and Sm oc Sc. Most Newtonian liquids (other than molten metals) have the values of Prandtl number > 1 and therefore the assumption of < 1 is justified. Likewise, one can justify this assumption for mass transfer provided Sc > 0.6. Most non-Newtonian polymer solutions used in heat and mass transfer studies to date seem to have large values of Prandtl and Schmidt numbers [Ghosh et al., 1994], and therefore the assumptions of 1 and m 1 are valid. 

For particle-liquid heat and mass transfer in non-Newtonian polymer solutions flowing over spheres fixed in tubes (0.25 < d/D, < 0.5), Ghosh et al. [1992, 1994] invoked the usual heat and mass transfer analogy, that is, Sh = Nu and... 

It is usual, in practice, to ignore the Rabinowitsch equation and to assume that the velocity profile for the polymer liquid is parabolic to within the required error. With this assumption the wall shear rate is taken to be given by the Newtonian value, so that, from eqns 7.13 and 7.17, the apparent viscosity is... 

In most studies dealing with heat and mass transfer, it has been generally assumed that the thermo-physical properties, such as thermal conductivity, specific heat, molecular diffusivity of non-Newtonian polymer solutions, are the same as that for water, except for their non-Newtonian viscosity. Intuitively, one would expect the surface tension to be an important variable by way of influence on bubble dynamics and shape, but only a few investigators have controlled/measured/included it in their results. The available correlations can be broadly classified into two types first, those which directly relate the volumetric mass transfer coefficient with the liquid viscosity and gas velocity. The works of Deckwer et al. [36], Godbole et al. [42] and Ballica and Ryu [60] illustrate the applicability of this approach. All of them have correlated their results in the following form ... 

Viscosity can also be determined from the rising rate of an air bubble through a liquid. This simple technique is widely used for routine viscosity measurements of Newtonian polymer solutions. A bubble tube vicometer consists of a glass tube of a certain size to which liquid is added until a small air space remains at the top. The tube is then capped. When it is inverted, the air bubble rises through the liquid. The rise time may be taken as a measure of viscosity or matched to that of a member of a series of standards (ASTM D1545). 

Robertson Model. The model developed by Robertson in 1966 (36) is also based on an activated process. He stated that the rigidity of a glass is a result of the intermolecular forces between adjacent chains, though for polymer glasses they suppose that the intramolecular forces are also important. Thus, to cause a glassy polymer to move into the liquid state it is necessary to reduce the effect of either the intramolecular or intermolecular forces. Robertson posited that a shear stress alone could achieve this and so induce Newtonian flow in the material. A shear stress fleld set up in the material can increase the number of flexed bonds (conformations) to a level above the preferred level of the equilibrium glass and may increase to the level that would be typically seen in a polymer liquid. 

Enough experimental data are now available that a clear picture of polymer behavior in extension is emerging. For Newtonian liquids, it is found, by both experiment and theory, that the extensional viscosity equals exactly three times the shear viscosity. For non-Newtonian polymer melts, though, the extensional viscosity can exceed three time the zero-shear viscosity by more than a factor of ten. The extensional viscosity, however, is a function of the stretch rate and is frequently a decreasing function of the stretch rate, especially at large values of the stretch rate (see Figure 5). 

For simple liquids, such as water, the coefficient of viscosity is defined as the ratio of shear stress to shear rate. When the viscosity is independent of both shear stress and shear rate the liquid is said to be Newtonian. Polymer melts are almost invariably non-Newtonian in that their viscosity decreases with increasing shear and the term apparent viscosity is used instead of the coefficient of viscosity. As a general rule the broader the molecular weight distribution the more non-Newtonian the polymer becomes, a phenomenon observed with both rubbery and non-rubbery polymers. 

When surface tension effects are small, as will be the case for large bubbles at high Reynolds number, Eqs. (12) and (13) and virtually the same. The experimental results of Astarita and Apuzzo (1965). Calderbank et al. (1970), Acharya et al. (1977), and Haque et al. (1988) on bubble velocities in non-Newtonian polymer solutions are well represented by Eq. (12) and (13) at high Reynolds number, thereby lending support to the notion that the liquid rheology plays little role in high Reynolds number bubble motion. 

Viscosity is defined as the ratio of shear stress to shear rate. The viscosity of a Newtonian fluid is a material constant that depends on temperature and pressure but is independent of the rate of shear that is, the shear stress is directly proportional to the shear rate at fixed temperature and pressure. Low molar-mass liquids and aU gases are Newtonian. Complex liquids, such as polymers and suspensions, tend to be non-Newtonian in that the shear stress is a nonlinear function of the shear rate. Some typical melt viscosities are shown in Figure 1.7. The viscosity approaches a constant value at low shear rates, known as the zero-shear viscosity and denoted... 

It is worth giving especial consideration to a particular polymer solution which has been extensively researched around the world and has now become a commercially-available, non-Newtonian standard liquid. It is a 2% w/ v solution of poly(isobutylene) (molecular weight 4.3. lO daltons) in a mixture of cis- and trans-decalin. It is marketed by the Pure and Applied Chemistry department. University of Strathclyde, Scotland, as Standard Fluid SUAl. 

Beyond short periods of time, a polymer cannot be regarded as a purely elastic material for the ratio of the stress applied to the strain undergone does not remain constant with time. Polymers indeed behave simultaneously like Hookean objects and like Newtonian (or non-Newtonian) viscous liquids. The matter in the latter case is not elastic and does not strain but flows under mechanical forces ... 

We begin by looking at twelve experiments (see Figure 1) in which the behavior of Newtonian fluids is contrasted with that of polymeric fluids. These experiments, and many more, show that the differences between Newtonian flow and polymer-liquid flow are not just small quantitative deviations, but rather major qualitative differences. 

The ratio of the fluid relaxation time to the timescale for flow If defines a dimensionless group termed the Deborah number, De = /tf. This group has been used in the literature to characterize deviations from Newtonian flow behavior in polymers [6]. Specifically, in flows such as simple steady shear flow where a single flow time tf = y can be defined, it has been observed that for De 1, a Newtonian fluid behavior is observed, whereas for De 1, a non-Newtonian fluid response is observed. However, in flows where multiple timescales can be identified, for example, shear flow between eccentric cylinders, the Deborah number is clearly not unique. In this case, it is generally more useful to discuss the effect of flow on polymer liquids in terms of the relative rates of deformation of material lines and material relaxation. In a steady flow, this effect can be captured by a second dimensionless group termed the Weissenberg number, Wi = k A, where k is a characteristic deformation rate and A is a characteristic fluid relaxation time. For polymer liquids, A is typically taken to be the longest relaxation time Ap, and for steady shear flow, k = y, which leads to Wi = y Ap. 

However, a number of liquids (polymer liquids and suspensions [18,19]) show a non-Newtonian behavior. The aim of this section is to extend the similarity solution method used in Chapter 3 to the case of spreading of non-Newtonian liquids (Ostwald-de Waele liquids) over solid surfaces and to deduce the corresponding spreading laws for both gravitational and capillary regimes of spreading. 

One of the most prominent rheological features of entangled polymer liquids is the nonlinear (non-Newtonian) behavior... 

Polymers owe much of their attractiveness to their ease of processing. In many important teclmiques, such as injection moulding, fibre spinning and film fonnation, polymers are processed in the melt, so that their flow behaviour is of paramount importance. Because of the viscoelastic properties of polymers, their flow behaviour is much more complex than that of Newtonian liquids for which the viscosity is the only essential parameter. In polymer melts, the recoverable shear compliance, which relates to the elastic forces, is used in addition to the viscosity in the description of flow [48]. 

Flow behaviour of polymer melts is still difficult to predict in detail. Here, we only mention two aspects. The viscosity of a polymer melt decreases with increasing shear rate. This phenomenon is called shear thinning [48]. Another particularity of the flow of non-Newtonian liquids is the appearance of stress nonnal to the shear direction [48]. This type of stress is responsible for the expansion of a polymer melt at the exit of a tube that it was forced tlirough. Shear thinning and nonnal stress are both due to the change of the chain confonnation under large shear. On the one hand, the compressed coil cross section leads to a smaller viscosity. On the other hand, when the stress is released, as for example at the exit of a tube, the coils fold back to their isotropic confonnation and, thus, give rise to the lateral expansion of the melt. 

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