The raw scanning data of the monolayer, obtained with STM, has to be corrected with the aid of an undistorted image of the underlying substrate by using an image analysis software. For that purpose a transformation is determined that maps the substrate surface to a e.g. hexagonal grid. The parameters of this mapping are stored and applied to the STM-image of the monolayer thus resulting in an undistorted image of the adsorbate. In order to determine the unit cell structure of the adsorbate layer, one could use Fourier transforms of the substrate and the adsorbate layer images in principle. However, as large structures in real space map to small structures in Fourier space (and vice versa) the autocorrelation images of the original images are much better suited to determine the size and orientation of the unit cell of large structures. Therefore an autocorrelation analysis is performed for the adsorbate image.
From the autocorrelation image of the substrate one can determine the unit cell vectors g1 and g2 . The same procedure can be applied to the autocorrelation image of the adsorbate resulting in two unit cell vectors a and b . These vectors can now be used in order to determine whether the grids formed by the adsorbate and the underlying substrate form a commensurable or coincident system or neither of both. In order to determine this question one is searching for a unit cell matrix which maps the unit cell vectors of the substrate g1 and g2 into the unit cell vectors of the adsorbate a and b . This unit cell matrix M is defined by:
a = Mg1
b = Mg2
By defining the matrices
the unit cell matrix is defined by:
A = M·G
M = A·G-1
which can be solved by standard methods. For the adsorbate-substrate system one obtains for example the following unit cell matrix:
M=((4.98, 2.93),(-0.79, 6.27))
which means that the unit cell vector a is almost an integer multiple of the unit vectors g1 and g2 while the unit cell vector b is only a rational multiple of the unit vectors of the substrate. The unit cell of the adsorbate is thus commensurable in one direction (the adsorbate unit cell vector a ) and coincident in the other direction ( b ). The unit cell of the adsorbate is uni-axial commensurable and uni-axial coincident to the unit cell of the substrate.
One obtains for the size of the unit cell of the substrate: | a| = 10.7 Å and | b| = 16.5 Å .
Molecular Mechanics Simulation
As it is not possible to simulate a macroscopic sized region of the adsorbate-substrate system, several approximations have to be applied. The two-dimensional monolayer system was modelled as an infinite surface in x- and y-direction by using periodic boundary conditions. The simulation is realized with the CERIUS 2 software package running on a SGI workstation. The Dreiding II force field is used for modelling the molecule interaction. Periodic boundary conditions in x- and y- direction are applied. The total energy ET of the adsorbate-monolayer system is given as the sum of the two-, three- and four-body terms in CERIUS 2:
ET = Ebond+Eangle+Etorsion+Einversion+EvdW+ECoulomb+EH-bonds
where Ebond, Eangle, Etorsion, Einversion, EvdW, ECoulomb and EH-bonds are the bond stretching energy, the angular distortion energy, the dihedral angle torsion energy, the umbrella inversion energy, the van der Waals energy, the electrostatic energy and the energy of the H-bonds, respectively. In z-direction we are using two layers of the substrate as they seemed sufficient to model the substrate. The electrostatic interaction is modelled using partial atomic charges which were derived using the Gasteiger-Model. The partial atomic charges of the substrate atoms were set to zero. Individual molecules are placed manually on the substrate model surface to generate a plausible start configuration.
By using the energy minimization algorithm of the CERIUS 2 software an equilibrium state is derived. The energy minimization algorithms is applied to the proposed start configurations. During the minimizations, atomic motion constraints are applied only to the atoms of the substrate. The minimizations are terminated when the total energy did not change more than 0.1 kcal/mol per atom.