Equation polymer melt

Many industrially important fluids cannot be described in simple terms. Viscoelastic fluids are prominent offenders. These fluids exhibit memory, flowing when subjected to a stress, but recovering part of their deformation when the stress is removed. Polymer melts and flour dough are typical examples. Both the shear stresses and the normal stresses depend on the history of the fluid. Even the simplest constitutive equations are complex, as exemplified by the Oldroyd expression for shear stress at low shear rates ... 

The Arrhenius equation holds for many solutions and for polymer melts well above their glass-transition temperatures. For polymers closer to their T and for concentrated polymer and oligomer solutions, the WiUiams-Landel-Ferry (WLF) equation (24) works better (25,26). With a proper choice of reference temperature T, the ratio of the viscosity to the viscosity at the reference temperature can be expressed as a single universal equation (eq. 8) ... 

Capillary viscometers are useful for measuring precise viscosities of a large number of fluids, ranging from dilute polymer solutions to polymer melts. Shear rates vary widely and depend on the instmments and the Hquid being studied. The shear rate at the capillary wall for a Newtonian fluid may be calculated from equation 18, where Q is the volumetric flow rate and r the radius of the capillary the shear stress at the wall is = r Ap/2L. 

Polymer melts are frequendy non-Newtonian. In this case the earlier expression given for the shear rate at the capillary wall does not hold. A correction factor (3n + 1)/4n, called the Rabinowitsch correction, must be appHed in such a way that equation 21 appHes, where 7 is the tme shear rate at the wall and nis 2l power law factor (eq. 22) determined from the slope of a log—log plot of the tme shear stress at the wad, T, vs 7. For a Newtonian hquid, n = 1. A tme apparent viscosity, Tj, can be calculated from equation 23. 

However, for the high strain rates appropriate for the analysis of typical extrusion and injection moulding situations it is often found that the simple Power Law is perfectly adequate. Thus equations (5.22), (5.23) and (5.27) are important for most design situations relating to polymer melt flow. 

R. G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworths, Boston (1988). 

Rheological methods of measuring the interphase thickness have become very popular in science [50, 62-71]. Usually they use the viscosity versus concentration relationships in the form proposed by Einstein for the purpose [62-66], The factor K0 in Einstein s equation typical of particles of a given shape is evaluated from measurements of dispersion of the filler in question in a low-molecular liquid [61, 62], e.g., in transformer oil [61], Then the viscosity of a suspension of the same filler in a polymer melt or solution is determined, the value of Keff is obtained, and the adsorbed layer thickness is calculated by this formula [61,63,64] ... 

Since non-Newtonian flow is typical for polymer melts, the discussion of a filler s role must explicitly take into account this fundamental fact. Here, spoken above, the total flow curve includes the field of yield stress (the field of creeping flow at x < Y may not be taken into account in the majority of applications). Therefore the total equation for the dependence of efficient viscosity on concentration must take into account the indicated effects. 

Though the accuracy of description of flow curves of real polymer melts, attained by means of Eq. (10), is not always sufficient, but doubtless the equation of such a structure based on the idea of relaxation mechanism of non-Newtonian polymer flow, correctly reflects the main peculiarities of viscous properties. Therefore while discussing the effect a filler has on the viscosity properties of polymer melts, besides the dependences Y(filler modifies the characteristic time of relaxation. According to [19], a possible form of the X versus

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The above considerations illustrate the difficulties of trying to formulate equations descriptive of rheological behavior of polymer melts with gas bubbles. An optimistic approach to the solution of this task is contained in [60, 61]. The content of these works is revealed by their titles On the Use of the Theory of Viscoelasticity for Describing of the Behaviour of Porous Material and for the Calculation of construction... 

Equation (22) was obtained, essentially, with examination of the energy balance equation with respect to flows of gas-containing polymer melts. The key moment of this analysis is, in our view, comprehension of the fact that the energy of gas dissolved in the polymer is transformed into the energy of movement of the two-phase medium. 

Larson RG (1988) Constitutive equations for polymer melts and solutions, Butter-worths, Boston, p 256... 

Dodd, L. R. and Theodorou, D. N. Atomistic Monte Carlo Simulation and Continuum Mean Field Theory of the Structure and Equation of State Properties of Alkane and Polymer Melts. Vol 116,pp, 249-282,... 

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. 

Heine, D. R., Grest, G. S. and Curro,J. G. Structure of Polymer Melts and Blends Comparison of Integral Equation theory and Computer Sumulation. Vol. 173, pp. 209-249. 

The integral equation theory is a simple means of studying the density profiles of dense polymer melts at surfaces where the structure is dominated by... 

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