Doi-Ohta theory

Although this experimentally observed scaling behavior is correctly predicted by the Doi-Ohta theory, the shape of the transient response curve—in particular, the overshoot and undershoot in the shear stress—are not predicted. This implies that the relaxation expressions chosen by Doi and Ohta, Eqs. (9-46) and (9-47), are inaccurate. This is not very surprising, since Eqs. (9-46) and (9-47) were chosen rather arbitrarily from many possible forms that satisfy the scaling relationship. Optical microscopy suggests that the overshoot and undershoot are caused by elongation of droplets followed by their breakup (Takahashi and Noda 1995). Vinckier et al. (1997) have shown that the stress growth after start-up or... 

Doi Edwards theory Doi Ohta theory Doi Onuki theory Domain size... 

Guenther, G.K. and Baird, D.G. (1996) An evaluation of the Doi-Ohta theory for an immiscible polymer blend. 

The relation between rheology and morphology during late stages of SD in PS/PVME blends was investigated by means of several techniques [Polios et al., 1997]. The results were interpreted using Doi-Ohta [1991] theory. 

Doi-Ohta s theory was also compared to the experimental data of semi-concentrated mixtures of PIB in PDMS [Vinckier et ai, 1997]. The theory described reasonably well the transient effects at the startup of steady state shearing. The scaling laws were also obeyed by these slightly viscoelastic blends. 

Experimental verification of Eqs 7.94 indicated that the scaling relationships are valid, but the shape of experimental transient stress curves, after step-change of shear rate, did not agree with Doi-Ohta s theory [Takahashi et al., 1994]. Similar conclusions were reported for PA-66 blends with 25 wt% PET [Guenther and Baird, 1996]. For steady shear flow the agreement was poor, even when the strain-rate dependence of the component viscosities was incorporated. Similarly, the... 

It should be noted that the Doi and Ohta theory predicts oifly an enhancement of viscosity, the so called emulsion-hke behavior that results in positive deviation from the log-additivity rule, PDB. However, the theory does not have a mechanism that may generate an opposite behavior that may result in a negative deviation from the log-additivity rule, NDB. The latter deviation has been reported for the viscosity vs. concentration dependencies of PET/PA-66 blends [Utracki et ah, 1982]. The NDB deviation was introduced into the viscosity-concentration dependence of immiscible polymer blends in the form of interlayer slip caused by steady-state shearing at large strains that modify the morphology [Utracki, 1991]. 

Equations [84] and [85] suggest that the droplet size in the steady state is essentially determined by competition of the interfacial tension and the viscous shear stress, as considered in the classical Taylor theory " for an instability criterion of a single droplet. However, the behavior of the normal stress difference (eqn [83b]) is not fully understood from this theory. Concerning this point, Doi and Ohta proposed a phenomenological model for the interface anisotropy and spedfic interfacial area in blends having the characteristic length determined only by the shear. The predictions of the Doi-Ohta model are consistent with the experimental observation (eqns [83]-[85]) as well as the scaling behavior observed for... 

Inspired by a mesoscopic theory for mixtures of immiscible liquids (Doi and Ohta 1991 see also Section 9.3.3), Larson and Doi (1991) have derived mesoscopic equations for polydomain nematics by multiplying the Leslie-Ericksen equation (10-13) by the director... 

The first steps toward such a theory of blend flow behavior were proposed by Helfand and Fredrickson [1989], then by Doi and Onuki [1992]. A greatly simplified constitutive equation for immiscible 1 1 mixture of two Newtonian fluids having the same viscosity and density was also derived [Doi and Ohta, 1991]. The derivation considered time evolution of the area and orientation of the interface in flow, as well as the interfacial tension effects. The relation predicted scaling behavior for the stress and the velocity gradient tensors ... 

Following Doi and Ohta s work, a more general theory was derived for immiscible polymer blends by Lee and Park [1994]. A constitutive equation for immiscible blends was proposed. The model and the implied blending laws were verified by comparison with dynamic shear data of PS/LLDPE blends in oscillatory shear flow. 

In this respect, a theory that takes into account the deformation of one droplet (Doi and Ohta 1991) can be applied to describe the shear and normal stress transients. According to this model, blend morphology is characterized by a scalar (referring to a specific interfacial area) and a tensor (characterizing interface anisotropy). These parameters may be expressed in two equations—one describing the stresses of the interfacial structures and the other for the evaluation of the scalar and interface tensor. For immiscible blends with Newtonian or weakly viscoelastic fluids and an increase in shear, the droplets deform into fibrils while maintaining their initial diameter, d. In comparison, in a highly elastic matrix where droplet shape is... 

A second class of models directly relates flow to blend structure without the assumption of an ellipsoidal droplet shape. This description was initiated by Doi and Ohta for an equiviscous blend with equal compositions of both components [34], Coupling this method with a constraint of constant volume of the inclusions, leads again to equations for microstructural dynamics in blends with a droplet-matrix morphology [35], An alternative way to develop these microstructural theories is the use of nonequilibrium thermodynamics. This way, Grmela et al. showed that the phenomenological Maffettone-Minale model can be retrieved for a specific choice of the free energy [36], An in-depth review of the different available models for droplet dynamics can be found in the work of Minale [20]. 

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