Resultados y análisis

N,V

, (27)

and by combining several simulations performed at different temperatures, it is possible to obtain the change in entropy ∆S by integration

∆S = Z T

T0

C_V(T⁰)

T⁰ dT⁰. (28)

Similar integration schemes can be constructed for other entropic quantities as well.

However, as it currently takes roughly a month of continuous computations to obtain one data point needed for the integration, it is clear that the first-principles molecular dynamics simulations cannot be the main tool to obtain free energies. In the atmo-spheric formation free energy context, these simulations could be used to study the general trends and characteristics arising from the more complete phase-space sam-pling – for small illustrative clusters.

4.3 Collision simulations describe the formation dynamics in detail

It is tempting to contemplate that the first-principles molecular dynamics simulations show us what one would see through an imaginary (and very powerful) magnifying glass observing the small atmospheric clusters. In this picture, the canonical simulations

19Typically, entropy is proportional to the logarithm of the volume of the entire accessible phase-space.

would represent some sort of a time-lapse movie, showing an average view of the clusters where the air molecules maintaining the equilibrium molecular movement have been abstracted away²⁰. However, the microcanonical simulations would show us the real thing!

The microcanonical simulations, governed by the Hamiltonian (22), model the dy-namics in a most unperturbed fashion. This renders the microcanonical simulations well-suited for investigating the dynamics of atmospheric clustering on a molecular level.

Clustering is a non-equilibrium process In the static ∆G calculations, it is as-sumed that both the formed cluster and the molecules forming the cluster are perfectly equilibrated in accordance with a given temperature throughout the process. Also, in the equilibrium canonical simulations the thermostatting ensures that the nuclear ve-locities correspond to the given temperature. However, molecular clustering is in fact a rather dynamic non-equilibrium process. By definition, in the clustering the system finds a new, more negative state of potential energy – and the kinetic energy must increase accordingly. Colloquially, the clustering process releases heat. As described earlier, the atmosphere is likely to absorb the released heat but this might take numer-ous collisions with the air molecules. Even in the case of a relatively fast thermalization, the newly-formed cluster has to cope with the “extra” kinetic energy for several hun-dreds of picoseconds. What happens to the clusters during this time? In paper IV, this was investigated in the case of (sulfuric acid)1(water)0,1 + (dimethylamine)1 clustering by direct collision simulations in the microcanonical ensemble.

According to the simulations, in the clustering of sulfuric acid with dimethylamine, a proton transfer takes place – this is in agreement with static calculations as well.

The transfer lowers the potential energy of the complex by several kcal/mol, increasing the kinetic energy by the same amount. The inability to dissipate the released energy leads to a very dynamic cluster configuration after the initial transfer; on average, the dimethylamine controlled the proton only 76 % of the time. Correspondingly, also the

20In fact, the botanist Robert Brown must have seen something similar with an actual powerful mag-nifying glass (optical microscope) while discovering the random, brownian motion of pollen particles in water – and indirectly confirming the existence of molecules. In reality, the imaging of covalent bond structure in single-molecule chemical reactions starts to be possible via atomic force microscopes (de Oteyza et al., 2013). However, with bare human eyes this will not be possible without rather curious future evolution.

structure of the cluster was continuously evolving, and in general, it was different from the static or dynamical equilibrium geometry. The addition of one water molecule, initially bound to the acid, altered the formation dynamics by introducing additional easily accessible degrees of freedom. Based on the simulations, the slightly larger system is better able to accommodate the released energy, leading to a less dynamic structure. In this case the amine controlled the proton for 88 % of the simulation time.

However, it should be kept in mind that the released kinetic energy still remains in the cluster. Perhaps this entails that the phase-space is explored more effectively, and that the inevitable collisions with the carrier gas then lead to the dynamical equilibrium in an orderly fashion, somewhat akin to the simulated annealing process described in section 4.1. On the other hand, the “kinetically exited” clusters might be more prone to evaporation/fragmentation upon a collision with a carrier gas molecule with a suitable speed and direction. Based on both the equilibrium and collision simulations, the spontaneous evaporation of a sulfuric acid molecule from the small cluster of (sulfuric acid)₁(dimethylamine)₁ is likely to be a very rare event – free energy change in the evaporation is large and positive. Such an event would raise the potential energy of the system significantly and consequently radically lower the kinetic energy. From the phase-space point-of-view, the evaporation would effectively freeze the molecular movement of the small cluster and correspond to a vanishingly improbable state of the system. It is likely that most of the evaporation events are connected with external kinetic energy change, at least in the case of the small cluster of sulfuric acid and dimethylamine.

Sticking factor & cluster rearrangement Most of the collisions in atmospheric clustering are not likely to be perfectly inelastic. Microcanonical collision simula-tions provide means to investigate the sticking factor in molecular collisions, at least for small systems. In paper IV, the sticking factor in (sulfuric acid)₁(water)_0,1 + (dimethylamine)₁ head-on collisions was observed to be unity. These collision simula-tions are particularly susceptible for sticking factor investigasimula-tions: for these systems there are only a handful of primary collision geometries and the proton transfer reac-tion constitutes a good metric for the sticking by clearly indicating complex formareac-tion.

Both of these alleviating factors are lost in the subsequent cluster growth. What is more, the cluster rearrangement is very likely going to have an even more important role for larger clusters. Especially in the cluster-cluster collisions, or in the cases where sulfuric acid collides with a cluster which is already satisfied with respect to proton transfers, the rearrangement into the global dynamical minimum energy cluster

con-figuration might be very slow – or might not happen at all.

These possibilities and dynamical features serve to remind us on the very complex nature of the clustering process. To deepen our understanding of the clustering on the elementary, molecular level, the described dynamical effects must be fully accounted for. Currently, first-principles molecular dynamics simulation is perhaps the most promising tool to achieve this in practice.