Efficient formulation of crystal shape evolution equations

Morphological population balance modelling of the effect of crystallisation environment on the evolution of crystal size and shape of para-aminobenzoic acid

“…The aspect was first addressed by , followed by other researchers (e.g., (Borchert et al, 2009, Borchert and Sundmacher, 2012, Chakraborty et al, 2010, Kwon et al, 2013, Kwon et al, 2014, Liu et al, 2013, Liu et al, 2010b, Liu et al, 2010a, Ma et al, 2016, Ma and Wang, 2012, Wan et al, 2009, Majumder and Nagy, 2013). A crystal has its face forms identified as {hkl}.…”

Simulations of crystal aggregation and growth: Towards correct crystal area

“…This method, as well as the involved crystallization phenomena and challenges related to shape control have been reviewed in the literature . One of the main challenges lies in the complexity of the crystal shape and the changes it undergoes during crystallization . Crystals may change the set of visible faces through growth and dissolution, as well as undergo breakage or form aggregates of several crystals.…”

An optimization-based approach to extract faceted crystal shapes from stereoscopic images

“…1. Depending on the scaling vector t not all of the m facets might be visible on the crystal, so that domains of t with a common set of constituting facets and edges can be identified, which can be accomplished using the analytical expressions reported by Borchert and Sundmacher (2013) and Singh and Ramkrishna (2013) or numerically as shown in Hours et al (2014). An example of such a morphology map is given in Fig.…”

“…Due to the nonlinearity and nonconvexity of the problem, we choose to solve the optimization problem Eq. (6) for every region in the morphology map separately by putting constraints on the scaling vector t. These constraints can be precomputed either numerically (as shown in Hours et al, 2014) or analytically (as shown in Borchert and Sundmacher, 2013;Singh and Ramkrishna, 2013). Furthermore, we warm-start a nonlinear solver (ipopt, see Wächter and Biegler (2006)) from different initial conditions for the rotation matrix R. The number of initial points that need to be considered for the rotation matrix in order to arrive at a good reconstruction of the particle morphology is dependent on the complexity of the crystal, its rotation, as well as on the optical properties of the imaging system (the quality of the images).…”