Nonlinear Transient Viscoelasticity The Doi-Ohta Theory

If flow is continuous, or strains are large, droplets or domains deform greatly or burst, and the above expressions for the moduli are inapplicable. Although there have been some experimental and theoretical studies of the droplet size and size distributions in such flows (see Section 9.2.2), there seem to be few theoretical results that allow prediction of the stresses in such flows. These flows are immensely complex the stresses depend not only on the viscoelastic characteristics of the two components of the emulsion, but also on the micromorphology of the phases, which, in turn, depends on the history of the flow. 

Doi and Ohta define the area of interface (A) per unit volume (V) of sample by Q = A/V Q is inversely proportional to the characteristic domain size a. Q gradually increases in response to an increase in shear rate, and it decreases when the shear rate is reduced. 

Shearing not only increases the surface area of interface between the two phases, but also orients the interface, so that it becomes more nearly parallel to the shearing surfaces. To quantify the interfacial orientation, Doi and Ohta define an interface tensor q  

To derive Eq. (9-44), a decoupling approximation was used so that a fourth rank tensor could be replaced by a product of second rank tensors. 

Equations (9-43) and (9-44) account for the effects of flow on interfacial area and orientation, but omit interfacial shrinkage, or coarsening, which is driven by interfacial tension. Additional terms to account for coarsening must be added to the right sides of Eqs. (9-43) and (9-44). On dimensional grounds, such terms are proportional to r/t]o, Doi and Ohta chose arbitrary, but simple, terms consistent with this  

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