The state-of-the-art computational methods applied in the atmospheric nucleation studies, namelyab initio12quantum chemistry, often yield very accurate information about the configurational energies and molecular geometries of clusters atT = 0 K.
However, these methods do not directly provide any thermodynamical information.
After the minimum energy structure is found, the effect of temperature is included in ad hocfashion using statistical mechanics. Essentially this means that the clusters are assumed to be structurally solid or crystal-like, but at the same time the clusters have liquid-like ability to find the global energy minimum very rapidly after the cluster is formed.
Such a molecular model has been used already in 1970’s to study water, ice and Lennard-Jones clusters (Plummer and Hale, 1972; Hale and Plummer, 1974;
Hoare and Pal, 1975). Nowadays such clusters with relatively simple intermolecular interactions are treated with more sophisticated methods (e.g. using Monte Carlo or molecular dynamics simulations, both discussed later in this thesis). The statistical mechanics approach, however, is still widely used to estimate the thermochemical properties of rather complex atmospherically relevant clusters due to computational limitations.
First of all, it is assumed that the different modes of motion areuncoupled, and therefore the partition function of ann-cluster (or a monomer) can be divided into sub-partition functions of the translational motion of the centre of mass,zntr, and the internal motion (including the configurational energy) that conserve the position of the centre of mass of the cluster, znint. Due to the uncoupling, the sum in Eq. (47) can be written as
11As in the case of the hypothetical spring obeying Hooke’s law, the vibrational frequenciesω are higher for strong bonds, and the relation betweenz andωis given further in Eq. (87).
12Meaning from first principles. Ab initio methods are based on solving the Schr¨odinger (or Kohn-Sham) equations without any empirical or experimental input.
Statistical mechanics approach to nucleation 27 In same manner, the internal partition function is written as
znint=zrotn znvibznc, (52) where rigid rotor (RR) and harmonic oscillator (HO) approximations are commonly used to calculate the rotational and vibrational partition functions, zrot and zvib. The partition function related to the configurational energy Enc of an n-cluster is expressed as
znc= exp − Enc kBT
#
. (53)
According to Eqs. (46), (49) and (50), the Helmholtz free energy of a system13 ofN identical clusters can be related to the cluster’s partition functionzas
F
kBT =−lnZ =−Nln z
N − N , (54)
where Eq. (48) and the Stirling approximation14 have been used. The second law of thermodynamics states that the entropy of the system reaches its maximum at equilibrium. This is equivalent to the system reaching its free energy minimum.
Thus for a one-component case, a system consisting ofN1 andNnfree monomers and n-clusters, respectively, is at equilibrium with respect to the number of n-clusters when
Using the Stirling approximation and keeping the total number of monomers (i.e.
Ntot=N1+nNn) constant results in thelaw of mass action:
Assuming that ΔG1 = 0, the equilibrium distribution of a one-component cluster can be given as
Nn=N1exp −ΔGn kBT
#
. (57)
According to Eqs. (56) and (57), the formation free energy can be expressed as ΔGn
kBT =nlnz1−lnzn−(n −1) lnN1. (58) The translational partition function is proportional to the occupied volume of the system:
zntr= V
Λ3n, (59)
13 As a population of clusters is considered, the partition function in Eqs. (49) and (50) isZ instead ofz. Note that for free energies of some population calligraphic symbol is used. In case of
“free energy per cluster” a normal capital letter is used instead.
14lnx!≈ xlnx − x, whenx 1.
where Λnis the thermal de Broglie wavelength of the cluster, defined as pressure-dependency of the formation free energy becomes evident:
ΔGn
kBT =nln Λ−31 z1int−ln Λ−3n znint−(n −1) ln P1
kBT. (61)
The last term of the equation resembles the (n−1)kBTlnSterm of Eq. (36), here the equilibrium vapour pressure is virtually included in the internal partition functions.
In addition, several previous studies (see,e.g., Hoare and Pal (1975); McClurg and Flagan (1998)) have demonstrated that the other leading term of the formation free energy expression is roughly proportional ton2/3.
Whereas the presented derivation is valid for one component systems only, the law of mass action can be used to determine the fragmentation rate coefficients for multicomponent clusters just as easily. At equilibrium, the forward (from reactants to product) and the backward (from product to reactants) fluxes are equal:
βi,jNieqNjeq=αi+jNi+jeq. (62) By considering the number of reactants (clusters i and j) and products ((i+j )-cluster) in the same volumeV (i.e. Ni=NieqV), Eq. (56) can be expressed as Gibbs free energy per monomer/cluster can be written as
G
kBT = F
N kBT + PV
N kBT =−ln z
N. (64)
If the Gibbs free energies of the reactants and the product are calculated for some equalN (=Ni=Nj =Ni+j), Eq. (63) can be expressed as
where the numerator of the exponent is the free energy of the reaction of adding the reactants together. The reason for expressing the fragmentation ratesviaGibbs free energies (instead of partition functions) is purely practical as these rates are often used as coefficients in GDM related to atmospheric processes, and in at-mospheric computational chemistry, Gibbs free energy is the standard thermody-namic potential. The Gibbs free energies are calculated at reference pressure of P=N kBT /V = 101325 Pa by default.
Finally, as mentioned already on page 24, one cluster can have multiple min-imum energy structures; two clusters containing exactly the same monomers are