On the Prediction of Crystal Morphology. I. The Hartman–Perdok Theory Revisited

Abstract

The over 40 year old Hartman-Perdok (HP) [Hartman & Perdok (1955). Acta Cryst. 8, 49-52, 521-524, 525-529] theory for predicting crystal morphology is reconsidered. The new approach, which gives a physical foundation to the theory, is based on F faces having a roughening transition temperature higher than 0 K. The aim of this paper is to confront the field of crystal growth and in particular the classical HP theory with modern statistical thermodynamical treatments of models of surfaces of relatively simple crystal structures. It is shown that crystal faces (hkl) containing multiple connected nets with a relatively high energy content may have a very low roughening temperature and an unexpectedly high growth rate. In some cases, crystal faces become rough at OK because of multiple connected nets related by symmetry giving rise to symmetry roughening. The use of connected nets in the HP theory offers the possibility of extrapolating the results of statistical thermodynamical models of simple crystal surfaces to more complex crystals as encountered in practice. The role of the step free energy in understanding crystal morphology is emphasized.

On the prediction of crystal morphology. III. Equilibrium and growth behaviour of crystal faces containing multiple connected nets

“…A limiting case is the situation where È p È q . Then the model reduces to the (110) face of a simple cubic Kossel model which has a roughening temperature of 0 K. Note that for that case formally no valid connected net is present for this orientation (Grimbergen, Meekes, Bennema, Strom & Vogels, 1998). Consequently, such a facet will grow in a continuous (rough) mode and is in fact already rough at zero supersaturation.…”

“…This step energy can be found by calculating the energy difference of the surface bounded by and 49 (Fig. 5) (Grimbergen, Meekes, Bennema, Strom & Vogels, 1998). For this case, 1 È q À È p for both step 4 and step 9 .…”

“…In this paper, simple crystal surface models, which are in our opinion generic for many types of crystal, will be used to illustrate the equilibrium and growth behaviour of a crystal face as a function of the connected net structure. In some cases, the presence of multiple connected nets gives rise to a very small step energy which results in a roughening temperature for the orientation hkl T R hkl lower than the calculated two-dimensional Ising transition temperature T C hkl based on the connected net with the highest slice energy (Grimbergen, Meekes, Bennema, Strom & Vogels, 1998). For other cases, a so-called disordered¯at (DOF) phase is found which is known from simple statistical thermodynamical surface models with next-nearest-neighbour interactions (Rommelse & den Nijs, 1987;den Nijs & Rommelse, 1989).…”

“…Recently, a physical foundation for the description of the stability of faces (hkl) of crystals in terms of connected nets and periodic bond chains (PBCs) was derived (Grimbergen, Meekes, Bennema, Strom & Vogels, 1998). It was shown that application of the PBC theory to a crystal structure often leads to many connected nets for an orientation (hkl) and that speci®c combinations of connected nets may even cause a face (hkl) to roughen at 0 K T R 0 .…”

The structure of the agrochemical fungicidal 4-chloro-3-(3,5-dichlorophenyl)-1H-pyrazole (RPA 406194) and related compounds

“…24 Moreover, when the crystal grows in spiral (BCF model), 26 that is, when the supersaturation is weak or moderate, V hkl is an analytical function of E hkl . 28,29 For a narrow range of attachment energies, the following relationship holds: 27 V hkl -lE hkl l. This relationship implies that the lower the attachment energy of a (hkl) face in absolute value, the larger this face will be.…”