Wulff rule

Crystallite shape transformations, due to adsorbed reactants, may affect the steady-state kinetics of catalytic reactions. Such effects can be studied phenomenologically by employing the Wulff rule [39] in order to find the optimum crystallite shape. The use of this rule implies that the crystallite reshaping time scale is shorter than the experimental time scale. The latter can be very long in practical systems. 

An interesting example of application of the Wulff rule is given by Ovesen et al. [61]. They have analyzed the kinetics of methanol synthesis on nm Cu particles supported by ZnO. The generalized surface tension for the particle-substrate interface was assumed to be dependent on the reduction potential of the gas phase. The latter resulted in the dependence of the areas of the (111), (100), and (110) facets on the gas-phase concentrations (such changes were observed by using EXAFS). The total reaction rate, represented as a sum of the reaction rates on different facets, was found to be affected by the changes in particle morphology. 

In our work [22], the Wulff rule was employed to analyze adsorbate-induced reshaping of crystallites during the A- -B2 reaction mimicking CO oxidation. 

From Eq. (27), using the Gibbs-Wulff rule hi/ei = const, the size of the critical nucleus, described by the distances hi from the Wulff point, is obtained hi/si = s/zeoVc- When the equilibrium form is a regular polygon, the distance h is identical with the radius of the inscribed circle p, used very often as a measure for the size of the nucleus. Hence... 

Making use of Eqs. (2) and (3) the condition for the equilibrium form dO = 0, dV = 0 leads to Wulff s rule (Gibbs-Curie-Wulff theorem [iii-v]) generalized by - Kaischew [i] to account for the crystal-substrate interaction ... 

Wulff s rule - equilibrium form of crystals and droplets... 

As mentioned above, consideration of the thermodynamic equihbrium is not very well suited to describe the crystalhzation pathways observed in biomin-eraUzation and biomimetic minerahzation processes. Instead, it can be stated that these systems are usually far from equiUbrium so that kinetic effects prevail over thermodynamic control, with the consequence that it is still almost impossible to predict the morphology of a crystal with the currently existing thermodynamic equUibriiun models, which are more or less a reflection of Wulff s rule [58]. 

Finally, the Wulff s rule enables us to construct the equilibrium form of a crystal given the values of the specific free surface and adhesion energies. For that purpose perpendiculars with lengths proportional to the corresponding specific free energies are drawn from the Wulff s point to all possible crystallographic planes. The most inner from all obtained closed polyhedrons represents the equilibrium form (Figure 1.11). 

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