Modeling of the basic processes that cover mixing of immiscible liquids starts with large dispersed drops; thus, at capillary numbers much larger than Cacril which is the critical ratio between the shear stress and the interfacial stress, above which no stable equilibrium dispersed drop shape exists. In that case, the interfacial stress is overruled by the shear stress (passive interfaces) and the (simple) principles of distributive mixing emerge, where deformation rate and time are interchangeable. Stretching and folding in a periodic flow should be realized for efficient mixing, and the occurrence of regular islands is to be avoided. The mathematical tools are available to numerically model distributive mixing even in three-dimensional transient flows, although the necessary computing time could be a constraint. Therefore the Mapping Method was developed that comprises a number of steps. First the boundaries of a large number of cells, forming a fine grid within the fluid, are accurately tracked during a short flow time. Subsequently, the deformed grid is superposed on the original undeformed one and their mutual intersections are computed reflecting, with normalized values between zero and one, the fraction of fluid that is advected from each cell to every other cell during flow time. Only the non-zero numbers (zero means that no fluid was exchanged between the cells in this flow time) are stored in a matrix that becomes huge if ...t is chosen large, the grid is fine and the problem is 3D. Using this matrix it is straightforward to compute, e.g., the concentration distribution after time M, since it follows from the mapping matrix - concentration vector (at t = 0) multiplication, which is a fast operation. The procedure is repeated during the total flow time t = N. M goingfrom step j . M to step (j + 1) . M with j = 1 ... N - 1. First the local concentration distribution on time j .M is averaged over each cell, which leads to some numerical diffusion. Next this concentration is mapped to step (j + 1) . M etc, up to N times. The method is elegant and fast, and allows for hundreds of thousands of computations in a reasonable time on a single processor PC, indeed making optimization ofdistributive mixing possible, investigating the influence of different geometries, protocols and processing parameters. As the local length scale decreases during the mixing process, the interfacial stress becomes ofthe same order as the shear stress (Ca == Cacrit ) and the long slender bodies formed disintegrate into lines of small droplets (dispersive mixing). Interfaces are active, and both deformation rate and time are important. Therefore, in transient flows the time scales of the competitive processes of deformation of the filaments, retraction, end-pinching, and growth of interfacial disturbances determine the size of the resulting dispersed fragments. Numerical models have been derived for these problems. Coalescence causes a coarsening of the morphology and the rate determining step is the drainage of the matrix fluid out of the gap between two adjacent drops. Interface mobility greatly influences the drainage rate, changing the process from pressure flow to dragflow. Surfactants are added, mainly to immobilize the interfaces, slowing down coalescence. Surfactants play an important role in break-up and coalescence, but the influence on the latter is most relevant. BIM, boundary integral methods, have been developed to study the local processes in detail. The final morphology, characterized by the average drop size, can be considered to be a result of a dynamic equilibrium between breakup and coalescence. For high volume fractions of the dispersed phase or low values of the viscosity ratio between dispersed phase and matrix, phase inversion can occur. DIM, diffuse interface methods, originally proposed by Cahn and Hilliard, allow computing these complex systems having topological changes in time and space in an elegant, easy to incorporate way. An issue remains regarding accurately resolving the thin interfaces on large domains.
|Title of host publication||Mixing and compounding of polymers : theory and practice|
|Place of Publication||Munich ; Cincinnati|
|Publication status||Published - 2009|