## 1. Introduction

The production of clean energy to replace the use of fossil fuels is a constant requirement, and any alternative energy sources present important opportunities. Solar energy has attracted much attention since it is both environmentally friendly and naturally abundant. In order to collect and convert scattered solar energy into electricity, different types of solar cells have been proposed, including dye-sensitized solar cells. Dye-sensitized solar cells (DSSCs), comprise four main components, namely a nanoporous semiconductor, a light-sensitive dye, an electrolyte couple and a counter electrode. The mechanism of DSSCs is based on photo electrochemical processes. Sunlight excites dye molecules to a high energy state whereupon it donates an electron to the nanoporous semiconductor. The electrons then leave DSSCs to power a load. The counter electrode reintroduces electrons back to the photosensitive dye through the redox electrolyte couple. DSSCs have advantages over other more traditional solar cells (e.g., junction solar cells) due to their simplicity in construction and consequent low manufacturing cost. Accordingly, DSSCs continue to be developed and studied in order to find ways to improve their efficiency. In this paper, we focus on the two components of DSSCs, which are a dye molecule and a nanoporous semiconductor. We comment that the aim of the paper is to determine the energetic behaviour between these two components which can lead to the optimal distance and arrangement of dye molecules on the surface of the nanoporous semiconductor. This knowledge will contribute to further our understanding of the porosity within the dye-semiconductor composite structures which is one of the key factors affecting the performance and efficiency of DSSCs. For mathematical modelling of charge transfer during the dye sensitization processes, the current density measurement and the efficiency calculation, we refer the reader to [

1,

2] for various models based on linear and non-linear diffusion equations.

In this paper, we particularly consider the van der Waals interaction between the commonly used dye molecule and semiconductor in DSSCs, which are Tris(2,2

${}^{\prime}$-bipyridyl)ruthenium(II) or Ru(bpy)

${}_{3}^{2+}$ dye molecule and TiO

${}_{2}$ semiconductor. The arrangement and the packing of the dye molecules on TiO

${}_{2}$ semiconductor impact directly on the transport of the electron density inside DSSCs and consequently, affect the efficiency of the device. With this in mind, this paper aims to provide a fundamental understanding of the interactions between dye molecules and between a dye molecule and TiO

${}_{2}$ structure in order to determine the minimum energy configurations and the equilibrium position of the dye molecules, which are useful for the design of DSSCs in order to optimise transport of the electron density inside the TiO

${}_{2}$ nanoporous semiconductor. We comment that the Ru(bpy)

${}_{3}^{2+}$ dye studied here is assumed to lack the anchor groups that covalently bind the molecule to the surface of TiO

${}_{2}$ semiconductor. While this may result in neglecting the overlap between the electronic densities of the excited state of the dye and the conduction band of the semiconductor, this assumption is made to facilitate an analytical derivation of the van der Waals interaction energy, which enables the determination of the arrangement of Ru(bpy)

${}_{3}^{2+}$ on TiO

${}_{2}$. We note that the overlap of the electron densities causes only a small electron transfer rate that is largely responsible for the small photocurrent encountered [

3], and while this is important for modelling charge transfer in DSSCs, it is not within the scope of this paper.

In DSSCs, a commonly used photosensitizer is Tris(2,2

${}^{\prime}$-bipyridyl)ruthenium(II) or Ru(bpy)

${}_{3}^{2+}$. This dye molecule possesses excellent photochemical and photophysical characteristics, such as light absorption and light emission [

3,

4], that can be utilized to improve the energy conversion efficiency of DSSCs [

5]. The structure and the properties of Ru(bpy)

${}_{3}^{2+}$ have been well studied. For example, the redox property of Ru(bpy)

${}_{3}^{2+}$ in aqueous environment has been investigated using photoemission spectroscopy and density functional molecular dynamics simulation [

6], including full quantum-mechanical and mixed quantum/classical molecular dynamics simulations [

7]. Due to the visible light absorption and the photophysical property of Ru(bpy)

${}_{3}^{2+}$, it is selected as a photosensitizer to investigate the light-driven water oxidation system, which plays an important role in the development of solar energy devices [

8,

9,

10,

11,

12,

13]. Cassone et al. [

14] study the electron transportation of LiI, NaI, and KI aqueous electrolytes and they find that the aqueous solution has an influence on the performance of DSSCs. Moreover, they propose that the hydrolytic behaviors of arsenic forms As

${}^{3+}$ and As

${}^{5+}$ occur in nature water [

15].

For the semiconductor employed in DSSCs, commonly used materials are metal oxides, and among these, titanium dioxide (TiO

${}_{2}$) is the most commonly used. This is due to its large surface to volume ratio, high porosity and wide band-gap energy. We note that different crystal structures of TiO

${}_{2}$, which include brookite, anatase, and rutile, have different surface areas and porosities. Park et al. [

16] find that the surface area of the anatase structure of TiO

${}_{2}$ is higher than that of the brookite and rutile TiO

${}_{2}$. Thus, the anatase TiO

${}_{2}$ semiconductor has more surface area to bind with dye molecules [

17,

18,

19,

20,

21,

22]. Utilizing the solvothermal method using tetrabutylammonium hydroxide as a morphology controlling agent, Liu and co-workers [

23] design the single-crystalline anatase TiO

${}_{2}$ nanorod which shows a high energy conversion performance for DSSCs. In addition, Chu et al. [

24] and Cui et al. [

25] synthesize the anatase TiO

${}_{2}$ microspheres in order to improve the efficient energy conversion.

In this paper, we determine the arrangement and distribution of dye molecules on the surface of the anatase TiO

${}_{2}$ by considering the van der Waals interactions between dye molecules, and between dye molecules and the constituent nanoporous semiconductor. To evaluate the interaction energy, the Lennard-Jones function is utilized together with a continuum approach. This approach has been successfully used to determine the van der Waals interaction energy between nanostructures in various applications [

26,

27,

28,

29,

30,

31,

32].

Here, we use this approach to investigate the interaction between Ru(bpy)${}_{3}^{2+}$ molecules and the interaction between Ru(bpy)${}_{3}^{2+}$ and TiO${}_{2}$. We consider two distinct models for Ru(bpy)${}_{3}^{2+}$ which are as layers of spherical shells of different atomic distribution and as a solid sphere. For the TiO${}_{2}$ molecule, we assume the structure of a solid sphere. These modelling assumptions of uniform atomic density of continuous structures are made to facilitate the derivation of analytical expressions for the total energy of the system. In addition, we perform molecular dynamics simulations for these interactions as a comparison for our analytical models. We comment that analytical expressions for the interaction energies can provide considerable insight into a complex problem, leading to the determination of benchmark behaviours for the system considered and saving computational time and resources, compared to fully extensive computational techniques, such as molecular dynamics simulations and ab initio calculations.

This paper is organized as follows. The basic model formation and assumptions are given in the following section. The procedures to determine the interaction energies between dye molecules and between dye molecules and the TiO

${}_{2}$ semiconductor are described in

Section 3. Results and discussions together with molecular dynamics simulations are presented in

Section 4. Finally, a brief summary of the paper is made in

Section 5, and three appendices summarize the derivation details for the interaction energies.

## 4. Results and Discussion

To validate the analytical models, we compare our results with molecular dynamics (MD) simulations, which are performed using the LAMMPS Molecular Dynamics Simulator [

37,

38]. In the simulations, the potential energy of the molecular system is computed using the discrete Lennard-Jones potential,

where

${\rho}_{\alpha \beta}$ represents the distance between atom

$\alpha $ and atom

$\beta $ of two molecules. The values of the Lennard-Jones parameters used in MD simulations are assumed to be the same to those used in the continuum models, which are given in

Table A1. Only the atoms at distances less than the cut-off radius

${\rho}_{c}=22$ Å are assumed to interact with each other.

In the case of the interaction between two Ru(bpy)

${}_{3}^{2+}$ molecules, one of the molecules is assumed to be fixed and the other is at distance

Z from the first molecule, where

Z is defined as the centre-to-centre distance.However, as shown in

Figure 1, the dye molecule is not perfectly symmetric, and therefore to calculate the interaction energy, we first rotate the molecule about its three axes to compute the energy at different angles and then take an average to obtain the final result. In

Figure 3, we plot the interaction energy between two dye molecules when the first molecule rotates about

$x-$,

$y-$ and

$z-$axis for different values of

Z. It can be seen that the energy is smooth when the first dye molecule rotates only about the

x-axis. This is because the configuration of the dye is almost symmetric about this axis, and the energy fluctuation is larger when the molecule rotates about the

y- and

z-axis, which demonstrates a less symmetric structure of the dye in these directions. Since the molecule is symmetric in the

$x-$direction, we only need to consider the rotation of the molecule about the

$y-$ and

$z-$ axis, which corresponds respectively to the rotation angles

$\theta $ and

$\varphi $, defined in

Figure 1.

Contours for the interaction energy between two dye molecules at

$Z=18$ Å are plotted in

Figure 4. The horizontal and vertical axes are the rotational angles of the second molecule about the

y- and

z-axis, respectively. We observe that the peak energy for the interaction is obtained for

$\theta ={64.8}^{\circ}$ and

$\varphi ={180}^{\circ}$ where

${E}_{Dye-Dye}^{dis}=-0.05040$ kcal/mol. The variation of the energy under the various rotations of the dye molecule reflects the face that Ru(bpy)

${}_{3}^{2+}$ is not perfectly spherical (see

Figure A1 in

Appendix C).

For the interaction between a dye molecule and a sphere of TiO

${}_{2}$, in the MD simulation, the dye molecule is rotated and moved toward the fixed TiO

${}_{2}$ sphere, assuming that the radius of the TiO

${}_{2}$ sphere is 50 Å (see

Figure A2 in

Appendix C). The energy profiles for the dye molecule, rotated in the three directions, interacting with TiO

${}_{2}$ are depicted in

Figure 5. We see that the energy exhibits fluctuations similar to those of the two dye molecules but with higher energy values. As shown in

Figure 6 for

$Z=60$ Å, two configurations corresponding to

$\theta ={64.8}^{\circ}$ and

$\varphi ={180}^{\circ}$, and

$\theta ={122.4}^{\circ}$ and

$\varphi ={129.6}^{\circ}$ give the highest energy of

$-0.47335$ kcal/mol and the lowest energy of

$-0.78032$ kcal/mol, respectively.

Since the dye molecule is not perfectly symmetric, the energy values obtained from the analytical models, under the assumption of spherical dye molecule, are compared with averaged values obtained from the MD simulations. For the dye-dye interaction, the distance of the two molecules is varied from 13 Å to 20 Å and the interaction energy is recorded and plotted in

Figure 7a. Similarly, comparison between the analytical models and the MD simulation for a dye molecule and a TiO

${}_{2}$ is shown in

Figure 7b. Some discrete values of the interaction energy for the two cases are also reported in

Table 1. It is observed that the values of the interaction energies obtained by the continuum models are close to those obtained by the MD simulation. We may conclude that the Ru(bpy)

${}_{3}^{2+}$ dye molecule can be modelled either as a spherical shell or as a solid sphere in order to study the interaction and the arrangement of two dye molecules, and between the dye molecule and the TiO

${}_{2}$ structure.

We note that the atoms in the continuum model are assumed to be smeared over the molecule which is different from MD where the discrete positions of each atom are required. These differences lead to the different values for the energy between the continuum models and the MD simulation. On the other hand, the continuum approximation for the atoms has only a minor impact on the energy of the dye molecule interacting with TiO

${}_{2}$ because a semiconductor with the radius

$b=50$ Å is much larger than the size of the dye molecule. Hence, the energy profiles of the continuum models are hardly different from the discrete model in the case of the dye-TiO

${}_{2}$ interaction as illustrated in

Figure 7b.

Furthermore, we focus on the equilibrium distance which is defined by

${Z}_{min}$ arising from the interaction between two molecules. This position gives rise to the minimum interaction energy implying the equilibrium configuration. The lowest value of the interaction energy indicates the location where the two nano-objects are most stable. The distance where the molecules can overcome the energy barrier and attract each other is also investigated and denoted by

${Z}_{0}$. Therefore, the dye molecule can be attached to the other dye molecules or to the TiO

${}_{2}$ semiconductor when

$Z\ge {Z}_{0}$.

Table 2 shows the numerical values of

${Z}_{0}$ and

${Z}_{min}$ for the continuum and the discrete models. We comment that the Lennard-Jones potential appears not to give an accurate prediction of the energy level when the molecules are at a very short distance apart. This is a limitation of the Lennard-Jones potential that is shared by other currently-used potentials, and all molecular level models adopting the Lennard-Jones potential are not able to accurately predict short-range interactions between molecules. The energy value at

$Z<{Z}_{0}$ shown in

Figure 7 is an estimation from the molecular-level model and it should be taken only as a reference value. For a more precise evaluation for short-ranged interactions, ab-initio calculations at sub-atomic scale are required, which presents an opportunity for future research.

In

Figure 8a,b we plot the energy gradients corresponding to the two interactions shown in

Figure 7a,b. We note that the critical distance

Z where the energy is minimum in

Figure 7a,b corresponding to the values of

${Z}_{min}$ where the energy gradient is zero in

Figure 8a,b, respectively. We also observe a strong increase in energy near the equilibrium, and the rate of the change of the energy decreases as the distance between two molecules increases.