Energy and free energy

Our simulation results of the zero-temperature energetic potential of the center-of-mass coor-dinate of thep-6P inx, y-direction as well as upon rotation\theta versus directionxare presented in figure 4.1. They qualitatively agree with the previous ab-initio DFT calculations [18] but are quantitatively off by maximal 80\%. Responsible for these deviations are the approxima-tions in both methods, the quantum DFT as discussed in the previous work [18] as well as the MD simulations, for which the assignment of LJ parameters and partial charges to the ZnO surface is based on empirical mappings. However, all qualitative features rigorously

Uz[kJ/mol]

0.20 0.40 0.60 0.80 1.00 1.20

(a)

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60

(b)

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60

(c)

Figure 4.1: Zero temperature energy landscape between the p-6P molecule and the ZnO surface in (a) z-direction atx= 0, y= 0, (b) x-direction aty= 0,z= 0.27 nm, (c)y-direction atx= 0,z= 0.27nm, and (d) for the angle\theta between the LMA of the COM and the x-direction atx= 0, y= 0, z= 0.27 nm. Panel (e) gives an overview of the entire energy landscape in one unit-cell (averaged over all unit-cells) in x and y direction atz= 0.27nm for a molecule oriented with its LMA towardx, i.e. perpendicular to the[0001]

axis. Reprinted with permission from [132]. Copyright 2014 American Chemical Society.

agree between the various methods. In particular, the much stronger energetic corruga-tion in y-direction (\Delta Uy \simeq 125 kJ/mol) than in x (\Delta Ux \simeq 1.3 kJ/mol) suggests that at non-vanishing temperature, the molecule will diffuse significantly faster in x-direction with a weaker T-dependence. The angular corrugation suggests that it will do so in a highly directed fashion, where the LMA favorably points into the x-direction.

Figure 4.2a displays the real-space translational pathways the molecule takes on the surface over the course of unconstrained simulations at temperatures T = 440 K, T = 670 K, and

T = 800 K. It is indeed visible that at the lower investigated temperatures the motion in

y-direction is significantly hampered in contrast to the motion inx-direction. We find from the simulation trajectories that the p-6P molecule mostly slides along the rows of oxygen atoms, jumping, from time to time, across the potential energy barriers in y-direction. At the highest temperature (800 K), the jumps iny-direction appear much more often while the

4.1. Anisotropic diffusion of a p-6P molecule on the ZnO \bigl(

Figure 4.2: (a) Illustration of the real-space diffusion pathways of thep-6P molecule (center-of-mass motion) across the charged ZnO surface for three different temperatures as displayed in the legend. (b) Corresponding probability distribution of the orientation\theta of the LMA towards thex-direction. Reprinted with permission from [132]. Copyright 2014 American Chemical Society.

figure 2.5 and conjectured from the energy surface, we indeed find that the organic molecule translates in x in a directed fashion most of the time (> 85\%) with its LMA pointing perpendicular to the (polar [0001]) y-direction within its variance. This is quantified in figure 4.2b, where the average orientation distribution P(\theta )strongly peaks at \theta = 90^\circ for all three temperatures. The square root of the variance of the distribution is small and about

\sqrt{}

\theta ² = 2.8\pm 2^\circ . Additional peaks in figure 4.2b at higher temperatures are corresponding to the local minima in the angle-resolved zero temperature energy landscape (due to the atomic surface roughness) shown in figure 4.1d. These configurations are very unstable but oftentimes they serve as stepping stones for the molecule on its way to cross the high energy barrier in y direction.

The x- and y-dependent (Helmholtz) free energy landscapes sampled over the course of

a long (t\mathrm{t}\mathrm{o}\mathrm{t} = 1 \mu \mathrm{s}) unconstrained simulation at temperature T = 723 K are plotted in figure 4.3. They are calculated according to equation 2.26. We also display in figure 4.3 the entropy contributions to the free energy, the 0-K potential energies and the electrostatic energy parts, calculated in accordance with the description in section 2.6.2.

For the potential energy barriers, we find \Delta Uy = 130 \pm 5 kJ/mol and \Delta Ux = 19.3\pm 1kJ/mol, which is 4\% and 1400\% higher, respectively, then the 0-K potential barriers. The electrostatic energy contributes at least 80\% to the total potential energy. The free energy barriers in both directions are 75\% to 85\% smaller than the potential energy barriers. The

Energy[kJ/mol]

0.00 0.05 0.10 0.15 0.20 0.25 0.30 (a)

0.00 0.10 0.20 0.30 0.40 0.50

(b)

Figure 4.3: Free energyA(\alpha ) =U(\alpha ) - T S(\alpha ), energyU(\alpha ), and the entropic contributionT S(\alpha )resolved in direction\alpha =x(a) and \alpha =y (b) for a temperatureT = 723 K. Also shown is the electrostatic surface-COM interaction part of the energyUC. Reprinted with permission from [132]. Copyright 2014 American Chemical Society.

entropy contributions to the free energy are substantial and almost cancel out the potential energy contributions.

The reason why the potential energy landscapes in the unconstrained simulation differ from the idealized 0-K energy landscapes must be attributed to the idealized pathways of the p-6P in the latter case. In reality, under the influence of temperature, the molecular motion is governed by conformational and positional fluctuations which change the average interaction energies. Such a behavior was observed before for functionalized organic truxenes on insulating KBr surfaces [23] and large organic molecules with polar binding groups on the perfect TiO2 (110) surface [72]. In both studies, detailed investigations by molecular simulations demonstrated that the diffusional pathway sensitively depends on the details of the molecular structure, such as flexibility and cooperative motions of intramolecular groups. Interestingly, we find in our study that these excursions from the idealized pathways are small: during its motion along the surface in x, for instance, the standard deviation of the center-of-mass position both in the y- andz-direction is less than 0.05 nm and in\theta only less than 2.8°. The average torsional angle of the molecule is \varphi \mathrm{a}\mathrm{v} = 29° (at T = 670 K) with a fluctuation of \Delta \varphi \mathrm{a}\mathrm{v} = 15°, comparable to the values calculated in a liquid crystal composed of p-6P molecules in a smectic A phase [104].

Finally, we determine the average binding energy, as defined in section 2.6.3. The binding energy amounts toU\mathrm{Z}\mathrm{n}\mathrm{O}+6\mathrm{P}

\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{d} = 29.6\pm 10.3kJ/mol/monomer and is comparable to calculated

4.1. Anisotropic diffusion of a p-6P molecule on the ZnO \bigl(

Figure 4.4: The MSD of the center-of-mass of thep-6P molecule on the electrostatically charged ZnO\bigl(

1010\bigr)

surface. (a) The MSD in x-direction. (b) The MSD in y-direction, i.e. in the polar [0001] direction, see figure 2.5. Reprinted with permission from [132]. Copyright 2014 American Chemical Society.

binding energies of polythiophenes on ZnO [156].