Dynamics of liquid‐liquid mixing: A 2‐zone model

Abstract

The development of multiphase liquid-liquid morphologies during mixing at small Reynolds numbers has been modeled. The mixing process is divided into i) stretching of dispersed drops, ii) breakup of the liquid threads formed, and iii) coalescence of the final droplets upon collision. Rules and criteria of the distinct processes are presented and combined to a general 2-zone mixing model simplifying the flow field into a sequence of alternating "strong and weak zones." In a "strong zone," dispersed drops and threads are stretched unless their radius is too small: meanwhile, the stretching threads might break up into droplets. In the subsequent "weak zone," the remaining threads may disintegrate while any drops present may coalesce. After passing a number of zones, stretching, breakup, and coalescence lead to a dynamic equilibrium that could be considered as the "final" morphology. Using the 2-zone mixing model, the influence of material parameters and processing conditions on the morphology has been studied. Interestingly, increasing either viscosity (dispersed or continuous phase) yields a finer morphology due to the delay of thread breakup (allowing for further stretching) and suppression of coalescence.

Mixing of viscous immiscible liquids. Part 1: Computational models for strong–weak and continuous flow systems

“…Model I augments the ideas presented by Janssen and Meijer (1995). The basic idea is to divide the mixer into two types of zones, strong and weak.…”

“…The time for the drop to break under ow is calculated from t break = t crit + t grow , where t crit is found using Eq. (4.2) and t grow is estimated by t grow = 1:4=˙ (Janssen & Meijer, 1995). If the drop is able to break in the time allowed, s ¿ t break , the drop is replaced by the corresponding distribution of drops produced upon breakup.…”

“…The e ective viscosity, c; e , is only used in the deformation and stretching of drops and threads, but not in the ÿlm drainage stage where the presence of nearby drops is not necessarily felt. Following the approach of Janssen and Meijer (1995), we use a modiÿed version of the Krieger and Dougherty (1959) where m is the maximum packing density, which is a function of viscosity ratio and interfacial tension. Janssen and Meijer used a value of m = 0:8, which was based on a value close to 1 − perc where perc ∼0:156 is the percolation threshold.…”

“…By successively passing a distribution of solid particles through these zones, a ÿnal distribution of agglomerate sizes could be determined. A similar approach, for blending of highly viscous uids, was taken by Janssen and Meijer (1995) and Huneault, Shi, and Utracki (1995).…”

Study of morphological hysteresis in partially immiscible polymers

“…Grizzuti and Bifulco (1997), Vinckier et al (1998), and Minale et al (1997Minale et al ( , 1998 showed, by means of rheological experiments and optical microscopy (OM), that the steady-state morphology is reached only after a certain critical shearing time. However, Janssen and Meijer (1995) reported that flow conditions might exist in which more that one morphology is possible. For shear rates lower than the critical one, the dynamic equilibrium between breakup and coalescence cannot be obtained, or, at least, it cannot be reached in a reasonable shearing time.…”