Dynamical Picture of Molecular Ensembles

To predict the thermodynamic equilibrium condition of an ensemble of molecules at constant volume and temperature, two driving forces must be considered: energy and entropy. The first quantity tends to push the collection of molecules to the lowest energy configuration and the latter to the highest number of distinct microscopic states continuously explored by the system. At each macroscopic state corresponds, generally, an enormous number of microscopic ones consistent with the constraints operating on the system and physical laws. Within the molecular model and the uncertainty principle, each microstate is characterized by the specific position and energy of every molecule of the system. In turn, the state of each molecule is defined by its librational, rotational (internal and as a whole), vibrational, and electronic states.

The mechanisms by which molecules change their energy are intermolecular collisions and the absorption and spontaneous or stimulated emission of photons. This involves that molecules coexist and are in dynamic equilibrium with a spectrum of photons continuously created and annihilated. Intermolecular collisions and absorption/emission of photons are also the mechanisms by which systems out of equilibrium reach the thermal, mechanical, composition, and chemical equilibrium through the superposition of several relaxation processes. However, it must be stressed that this is a simplified picture because it does not consider that the occurrence of intermolecular interactions prevents the assignment of well-defined energy packets to each molecule.

The ensemble of microscopic states can be represented in the configuration space by the potential energy of the N particles constituting the system as a function of their 3N spatial coordinates. The resulting hypersurface of each system is characterized by the density of minima and the depth of the potential energy barriers. Fragile systems display a density of minima larger than strong ones. It must be stressed that the specific topology of the hypersurface controls the system dynamics. The probability (pi) and the time fraction (tí) that a system can be found in the generic microscopic state with total energy Ei is given by the equation

Ei e~kT

Ee kT

where the summation is extended to all the distinct system microstates. This equation can be also applied to small pockets of molecules as well as to a single molecule or even to a specific freedom degree of the molecule. It says that pi and Ti decrease exponentially with the energy and that significant values of these quantities can be reached at Ei up to kT, i.e., at about 4 10-21 J at room temperature. On the other hand, the multiplicity or degeneracy of microscopic states generally increases steeply with Ei and the number of particles composing the system leading to a maximum of pi and Ti corresponding to the mean, i.e., thermodynamic, value of the system energy.

It is well-known that, at constant volume and temperature, the macroscopic state thermodynamically favoured is that characterized by the lowest Helmholtz free energy (A). This quantity is related to the possible Ei values of the system by the equation

However, if the activation energy barrier associated with the transition to a specific microscopic state from neighbouring ones is greater than 5-10 kT, then that state is practically unreachable by spontaneous fluctuations.

The existence of large energetic barriers between microscopic states accounts for the formation of kinetically but not thermodynamically stable systems. This is a question frequently encountered in microheterogeneous systems making experimentally arduous the distinction between these two conditions. It occurs also frequently that, by the external input of energy (mechanical, electric, magnetic, etc.), some microheterogeneous systems can be trapped in a spectrum of microscopic states separated by large energetic barriers from those corresponding to the thermodynamic stable system. This leads to a quite surprising situation, i.e., the realization with the same substances in the identical experimental conditions (concentrations, temperature, etc.) of systems showing different macroscopic physico-chemical properties and different behavior. To describe this phenomenology, it is useful to build up ideally a multidimensional diagram where the free energy of each possible thermo-dynamic state of the system is reported as a function of the macroscopic variables characterizing the system. Looking to the topology of the resulting hypersurface, it can be observed the presence of hills, mountains, and valleys as well as local minima and only one absolute minimum. One of the points of the hypersurface represents the initial state of the system: the possibility to reach the thermodynamic stable state (the absolute minimum) depends on the existence of a path joining both states along which by spontaneous thermal fluctuations the system can walk.

At the molecular level, the interplay of energy and entropy leads to an endless dynamics among all the accessible microscopic states characterizing the life of macroscopic systems in thermodynamic equilibrium. Microscopic states can be ideally grouped according to their energy (degenerate or quasi-degenerate states) or their close similarity in the spatial arrangements of molecules (degenerate or quasi-degenerate snapshots). Within characteristic system-dependent timescales, the molecular dynamics determines the formation and the breakage of transient supramolecular structures that can be identified as short-living building blocks of the macroscopic system.

In the case of microheterogeneous systems, it will be found that the constant peculiarity of the supramolecular structures is the coexistence of two nano-size polar and apolar pseudophases separated by an huge interface and characterized by local orientational order at short times and fluidity at long times. It can be easily understood that these properties are essential for building up molecular machines with complex functionalities such as self-organization, self-replication and recognition. In fact, the same features can be observed in the most complex biological systems of which they are the "artificial" counterpart. It is quite astonishing to be aware that the wide variety of supramolecular structures observed in microheterogeneous systems is the expression of a simple structural property of the molecules invariably present within such systems, i.e., surfactant molecules.

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