*Existing Knowledge / State-of-the-art*

Hybrid complex materials are amazingly diverse systems ranging from polymer nanocomposites and multi-component biomolecular systems to colloids, emulsions, surfactants, etc. and varying in their applications over a range of industrial systems such as novel nanomaterials, foodstuff, biological materials, plastic coatings, cosmetics, etc. A common theme of all these systems is that they are characterized by a very broad range of length and time scales [1,2]. In the following we will give a short overview of state-of-the-art of existing knowledge.

Out of the more general class of hybrid complex materials this proposal will focus on *hybrid multi-phase macromolecular materials (here graphene based polymer nanocomposites). *Polymers are characterized by the presence first, of chemical bonds joining the atoms into chains and, second, of intermolecular forces binding the macromolecular chains to one another. This feature (*energy*), together with the flexibility of the chains due to internal rotation of the units (*entropy*), and the interaction between different components-phases (*interfaces*) determines the properties of such hybrid systems. Various technological issues are related with the coupling of interfaces with the macroscopic properties (mechanical, structural, dynamical, rheological, etc.) of such materials. For example, polymer and/or biopolymer/solid interfaces play a key role in applications involving polymer/particle nanocomposites, since the strength of chain attachment at interfaces (a quantitative thermodynamic measure of which is the work of adhesion) governs the mechanical properties of the structure. In addition the organic/inorganic nature of their properties indicates their potential application as structural materials in renewable energy, drug design, aerospace, automotive, electronic and packaging industries [1].

In this aspect, computation methodologies have the advantage that they can be aimed at either molecular or micro-structural length scales. Therefore, nowadays, with the rapid development of computers, computational methods have become valuable tools for fundamental and applied research of complex fluids. These methods varying from ab-initio quantum calculations, where very small systems for short times are modelled in the highest possible detail, to classical microscopic (atomistic) simulations, to mesoscopic coarse graining techniques, up to macroscopic approaches where systems are modelled in the continuum description [2-17]. A typical *time-scale vs length-scale* graph with an overview of simulation methods in different scales is shown in Figure 1. For hybrid multi-component materials ab-initio calculations cannot be applied, due to extremely computational demands, while continuum approaches are not proper when a coupling between microstructure at the molecular level and properties needed. Atomistic simulation techniques, such as Monte Carlo (MC) and Molecular Dynamics (MD), have the advantaged of detailed representation of microstructure and direct comparison with experimental data and with predictions of theoretical approaches [3,4]. However MD methods, typically track the evolution of model systems for times up to about one μs (10^{-6} sec), and their straightforward application on complex systems is problematic (problem of long relaxation times).

A well-known method in the literature is to abandon chemical detail and to use coarse graining (CG) models. Ad hoc coarse grained models, like simple bead spring or lattice models [2], are very useful to study generic scaling properties but lack a link to specific systems, i.e. the physical length and time scales, which are actually simulated with these methods, are not defined due to the *lack of relation with microstructure and the chemical detail of the simulated system*. Therefore, the last few years a few CG molecular models developed directly from the chemistry have been proposed in the literature [2,4-6]. Such CG models are obtained by lumping groups of chemically connected atoms into 'superatoms' and deriving the effective, coarse-grained interaction potentials from the microscopic details of the atomistic models. However, up to now such models lack a clear quantitative prediction of the dynamical properties of the modelled systems. Very recently we have developed a hierarchical approach, that combines atomistic and CG dynamics simulations, and allows us to correctly define the time scale in the CG description and to *quantitatively* study large realistic complex systems *without any adjustable parameter* [6,7].

**Figure 1:** Simulation methods in different length and time scales.

An additional aspect concerns the nature of these effective CG interactions. These interactions are in principle result of very complicated many body effects. But, for computational reasons, for most of the cases up to now *pair effective* CG interaction potentials are assumed and many-body effects are only partially described through numerical approaches [4-6]. What is clear missing in the literature is a *rigorous mathematical CG approach* that will give us a way to treat many-body effects for the cases where these are important and also provide accurate error estimation.

It is obvious, from all the above, that although there are a few mathematical approaches and simulation techniques for the study of complex materials, there is still much progress needed. A big challenge is to introduce *novel computational schemes for the (virtual) design of hybrid complex materials* that combine different levels of modelling into a coherent hierarchy such as to minimize loss of information in going from one level to other and to maximize predictive ability and versatility. This requires an *innovative synergy between mathematical approaches and computational schemes*. This is exactly the aim of the present proposal.

*Scientific Innovation and Originality*

As stated above, this work refers to a new research paradigm that is now followed internationally. It concerns understanding the basic underlying principles that give rise to important scientific and technological applications of *hybrid complex systems*. The ultimate goal is to control and design specific novel nanomaterials by predicting, using simulation approaches (*virtual experiments*), their macroscopic properties. This cost effective way of designing complex systems is also a novel scientific and technological direction followed worldwide.

The innovation level of the proposed work is clear: We propose a new hierarchical methodology that involves *ab-initio* calculations, *microscopic* and *mesoscopic* simulation techniques as well as *mathematical approaches in a consistent way*. All simulations will be based on very powerful state-of-the-art algorithms developed by us during the last years. In the proposed work all these algorithms will be extended for studying hybrid complex systems (here graphene/polymer nanocomposites) and their macroscopic properties will be directly predicted without any adjustable parameter.

In overall, implementation of the proposed research will extend the existing knowledge towards:

- The prediction of macroscopic properties of graphene based polymer nanocomposites.
- Studying the fundamental coupling between microstructure and macroscopic properties of such composite materials.
- Development of multi-scale modeling approaches used for the design of complex materials.
- Obtaining rigorous mathematical coarse-graining methodologies.
- Examine theoretical predictions for the dynamics (e.g. Rouse theory, reputation dynamics) of polymer composite systems.

Furthermore, the area of hierarchical modelling of hybrid complex condensed matter is quite new, so there are still many principal and technical problems in the field that have not been successfully addressed. Several mathematical and numerical techniques and procedures, involved in combining different length and time scales, are still in their infancy. We believe that our research will have an important impact since several of our approaches will enhance the power of these methods. Considering also the variety of problems they can be applied to, our project has a life-span exceeding the duration of the proposal setting the foundations of a long term activity in the area.

From all the above it is clear that the proposed work will *open up new horizons and opportunities for science and technology*. The broad inter-disciplinary nature of the project bridge the gap between different methods and combines *different kinds of expertise: mathematics, materials science, chemistry, physics and engineering*. There are certainly many challenging and unconventional aspects in the proposed work: For example the combined approach of multi-scale simulations and consistent mathematical strategies is a highly novel and challenging area. Furthermore, the study of new novel innovative materials, as those proposed here (i.e. graphene based polymer nanocomposites), will be of extreme scientific and technological importance. Finally it should be stretched the **multi-tool** and **multi-disciplinary** paradigm of the proposed approach, which is essential for a direct study of hybrid complex materials across multiple length and time scales.