LIBRO VII [DE LOS LIBROS]

The behaviour of polymer molecules and molecular ensembles are statistical in their nature: the typical quantities of interest are distributions and expectation values of given thermodynamic or configurational properties. These properties are amenable to simulation methods. With the aim of simulating polymer conformations and molecular morphologies, the two major approaches are those based on Monte-Carlo modelling and molecular dynamics. Monte-Carlo methods generate ensembles of possible configurations based on an underlying physical description of a system’s energetic landscape by taking an initial configuration and varying it over the course of several steps [135, 136]. At each step, the change in configuration or ‘move’ is permitted with a probability corresponding to the Boltzmann factor exp(_−∆E/kBT )

where ∆E is the change in energy resulting from the move. Statistical ensembles of possible geometries are obtained by performing several runs over a sufficient number of steps. Molecular dynamics, on the other hand, is a dynamical simulation regime which simulates real-time dynamics based on a set of configurational co-ordinates and a description of the potentials governing these co-ordinates. If a dynamical system can be assumed to be in a state of thermodynamic equilibrium and is sampled over sufficiently long time steps so as to avoid correlated samples, a set of ‘snapshots’ is obtained which are built into a statistical ensemble [135].

Each of these simulation methods requires parameterisation based on the inter- actions between atoms and molecules. How this parameterisation is performed can be described either as atomistic or coarse-grained with the former covering param- eterisation which assigns a physical description to individual atoms and the latter to groups of atoms or blocks in a system [135]. In this work, we focus on molecular dynamics approaches at the atomistic level. Utilising atomistic simulations offers several notable advantages over coarse-grained simulations. Atomistic simulations can capture the microscopic detail underpinning phenomena such as the role of alkyl side-chains in conjugated polymers and the nature of solvent-polymer interactions which have contributions from solvent molecule shape, polarity, as well as van der Waals terms. Also, conformations resulting from atomistic simulations may be di- rectly utilised in quantum chemical calculations (which require the specification of individual atomic co-ordinates) in addition to the possibility of being mapped to coarse-grained models. For polymer dynamical simulations, Monte Carlo methods are often coarse-grained to some extent, e.g. by considering block solvent-monomer

interactions [131]. In fully atomistic cases, different atomic motions are specified with different rates at which moves are performed corresponding to the different time scales of different degrees of freedom e.g. backbone moves occurring on longer time scales than side-chain moves [137].

Molecular dynamics simulations require a description of potentials which, in the atomistic regime, means a description of the interactions between covalently bound atoms and non-covalently paired atoms (Van der Waals (VdW) forces). Essentially, this is a Born-Oppenheimer simulation regime in that the electronic motion is con- sidered to be sufficiently fast as to be separable from nuclear motion. The potentials in the simulation govern the nuclear co-ordinates and are representative of the quan- tum mechanical behaviour of the molecular electrons. Thus, the simulation itself is of the effective Newtonian dynamics of the nuclei.

The manner in which the basic potentials are described form two distinct method- ological regimes. First, a realistic scenario would involve calculating an effective force-field utilising a fully quantum mechanical representation of the electronic states. This regime involves incorporating quantum chemical methodology following a basic algorithm of: defining the electronic energy landscape for a given molecu- lar configuration using quantum chemistry; calculating effective forces on the nuclei based on the energy landscape and allowing these forces to act on the nuclei over a given timestep; and recalculating the energy landscape based on the new molecu- lar configuration. Approaches which fall into this category are often referred to as ab-initio-molecular dynamics (ai MD) [106, 138, 139]. Secondly, in a system where it can be assumed that there is zero probability of the occurence of high-energy phenomena, such as the breaking of bonds, molecular ionisation or, indeed, pho- ton absorption, one may make the approximation that each molecular configuration corresponds exactly to a given potential energy i.e. that, for a given temporal resolu- tion, the electronic structure for a given molecular configuration closely corresponds to the ground state electronic structure of said configuration. Invoking this approx- imation leads essentially to a classical description of the dynamics. This regime is known as classical molecular dynamics [135].

Currently, the utility of ai MD methods is limited to very small molecules [106, 139] due to the fully quantum mechanical treatment embedded in the method. How- ever, as we discuss in Section 1.3.3, the development of these methods offers numer- ous possibilities for observing excited-state behaviour in a dynamical environment. For dynamical simulations of systems of large molecules (i.e. L _lp) in a fully

atomistic manner, classical MD (herein referred to simply as MD) is the only viable option of the two.

Provided an appropriate description of the set of interaction potentials (referred to as a ‘force-field’), which we discuss in the following subsection, MD simulation methods are capable of modelling large volumes of molecules in solvents or aggre-

gates of molecules with each treated with explicit interactions. As such, it is an ideal candidate for understanding the subtle behaviour of conjugated material conforma- tion and morphology and its scope has been recognised in numerous works ranging from studies of crystalline and amorphous aggregate structures [44,46,132,140–143], the formation of single molecule ordering in solution [144], and the role of solvent additives on aggregation in fullerenes [145]. The ability to directly implement the results of MD simulations for quantum mechanical calculations has also been recog- nised in studying the formation of trap states due to conformational disorder [44], the effect of conjugated molecule stacking on absorption spectra and excitonic cou- pling [142, 143], and variations in absorption spectra between different phases of P3HT [47].

The scope of MD simulation for understanding conformation and morphology in conjugated materials is apparent from the above and is the focus of the work presented. As is to be expected, there are a number of technical considerations for carrying out these simulations which are discussed in greater detail in Chapter 2. One key consideration, which we introduce in the following, is forming a set of force-field parameters for conjugated molecular systems.