Complex viscosity dilute solutions

With appropriate caUbration the complex characteristic impedance at each resonance frequency can be calculated and related to the complex shear modulus, G, of the solution. Extrapolations to 2ero concentration yield the intrinsic storage and loss moduH [G ] and [G"], respectively, which are molecular properties. In the viscosity range of 0.5-50 mPa-s, the instmment provides valuable experimental data on dilute solutions of random coil (291), branched (292), and rod-like (293) polymers. The upper limit for shearing frequency for the MLR is 800 H2. High frequency (20 to 500 K H2) viscoelastic properties can be measured with another instmment, the high frequency torsional rod apparatus (HFTRA) (294). 

Chain-growth polymerizations are diffusion controlled in bulk polymerizations. This is expected to occur rapidly, even prior to network development in step-growth mechanisms. Traditionally, rate constants are expressed in terms of viscosity. In dilute solutions, viscosity is proportional to molecular weight to a power that lies between 0.6 and 0.8 (22). Melt viscosity is more complex (23) Below a critical value for the number of atoms per chain, viscosity correlates to the 1.75 power. Above this critical value, the power is nearly 3 4 for a number of thermoplastics at low shear rates. In thermosets, as the extent of conversion reaches gellation, the viscosity asymptotically increases. However, if network formation is restricted to tightly crosslinked, localized regions, viscosity may not be appreciably affected. In the current study, an exponential function of degree of polymerization was selected as a first estimate of the rate dependency on viscosity. 

The important conclusion drawn from the above studies on PS(OH)/PMMA in solution and bulk is that complexes formed in dilute solutions can be preserved during the process of film casting. In particular, when we use an inert solvent whose Ejp is close to zero, the critical hydroxyl contents in proton-donating polymers for complexation estimated by viscosity or LLS are comparable to that for the miscibility-to-complex transition in bulk, which can be easily detected by DSC or TEM. Therefore, by combining the results from both solution and bulk, it should be possible to construct a map for a given blend system visualizing how the immiscibihty, miscibihty and complexation of the blend depend on the content of interacting sites. 

The presence of a second type of repeat unit causes the dilute solution behavior to be more complex than that of homopolymers [1], Copyolymer composition and sequence distribution directly effect the intrinsic viscosity. Interactions between unlike chain segments and preferential interaction of solvent molecules with one of the comonomers are also of considerable importance. 

The dilution of solutions33 containing equimolar ratios of monomer units of the complex components results in the dissociation of the complexes of PMAA with low-molecular weight PEG. The reduced viscosity of solutions rapidly increases, which indicates the existence of the equilibrium PMAA + PEG complex. In the case of a relatively high-molecular weight PEG, the PMAA macromolecules are firmly connected with PEG and at the dilution of aqueous solution, an increase of the reduced viscosity typical of polyelectrolytes does not occur, i.e. the complex does not dissociate. The absence of temperature dependence of the relative viscosity in the temperature range 15-40 °C is indicative of the stability of this complex (Fig. 4). 

The relationship between viscosity and concentration of polymer solution is very complex. Several empirical equations are necessary to describe the viscosity behavior of a polymer solution s dependence upon concentration. As an example, the following equation can be used for dilute solutions [7] ... 

It can be shown that for most dilute solutions there exists a simple correlation between dynamic and steady state flow characteristics (16). For most detergent solutions the magnitude of the complex viscosity 1 n I at a certain angular frequency CO coincides with the steady state viscosity n, at the corresponding shear rate "f (12, 17). 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

In Figures 3 d, 3 e, and 3 f are shown the data on the dynamic properties of several dilute solutions. The quantities plotted there are the real and imaginary parts of the intrinsic complex viscosity ... 

A step-growth polycondensation route has been succesfully devised to prepare novel nickel polymers 207 with arene spacer groups. The procedure involved the polycondensation of the fluorinated dilithiated species 206 and an Ni(ii) complex (Equation (75))." " The rod-like structure of these polymers was established by dilute-solution viscosity measurements, and the results were similar to those reported for the related platinum polyyne polymers 166 (M = Pt(P Bu3)2 x = 2) (Section 12.06.5.2.3). 

For polymeric fiuids, early Idnetic-theory workers (40) attempted to calculate the zero-shear-rate viscosity of dilute solutions by modeling the polymer molecules as elastic dumbbells. Later the constants in the Rivlin-Ericksen (17) expansion were obtained for dumbbells (41, 42) and other more complex models and only recently have the kernel functions in the memory integral expansions been obtained (43), This rapidly expanding field has been summarized recently in a monograph (44) here, too, molecular dynamics simulation may prove fhiitful (45),... 

Dilute Solution Properties. The rheology of dilute polymer solutions has been used extensively to gain insight into the structure and conformation of polymers in solution (11). The intrinsic viscosity provides a measure of the molecular weight of a polymer through a relationship such as the Mark-Houwink-Sakurada equation. Earlier studies of polyacrylamide (PAM) systems and details of the complexity of the characterization of high-molecular-weight water-soluble systems can be found in references 9, 13, and 14. 

A typical result of a calculation [127] of the complex viscosity rf(co) is shown in Fig. 11. The real part of the viscosity, / (w), which describes the dissipation of energy when the fluid is sheared, is approximately frequency-independent for small cu, i.e., the fluid behaves as a Newtonian fluid. There is a characteristic frequency co where f/ (o>) drops rapidly. The imaginary part of the viscosity, rf"(o)), which describes the elastic response of the fluid to an external perturbation, increases linearly for small co and reaches a maximum at CO = CO. This behavior is not specific to microemuisions but has been observed in other complex fluids as well, such as in suspensions of spherical colloidal particles [128,129] and in dilute polymer solutions [130]. 

The zero shear viscosity scales with Nf" to contrast Af dependence for isotropic polymers [20] So far, we have examined the dynamics of rod-Uke macromolecules in isotropic semi-dilute solution. For anisotropic LCP solutions in which the rods are oriented in a certain direction, the diffusion constant increases, and the viscosity decreases, but their scaling behavior with the molecular weight is expected to be unchanged [2,17], Little experimental work has been reported on this subject. The dynamics of thermotropic liquid crystalline polymer melts may be considered as a special case of the concentrated solution with no solvent. Many experimental results [16-18] showed the strong molecular weight dependence of the melt viscosity as predicted by the Doi-Edwards theory. However, the complex rheological behaviors of TLCPs have not been well theorized. 

Many simulations of dynamic phenomena such as adsorption, electrophoretic motion, complexation, dilute solution viscosity, and electric birefringence have also been reported for polyelectrol5des, often for the purpose of gaining insight into experimental situations too complex for analytical theory. These... 

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