MONTE CARLO SIMULATIONS OF SMALL H2SO4-H2O CLUSTERS*
B.N. HALE AND S.M. KATHMANN
Department of Physics and Cloud and Aerosol Sciences Laboratory
University of Missouri-Rolla, Rolla, Mo 65401, USA
Abstract - Small binary clusters of water and sulfuric acid are simulated with effective atomatom pair potentials modeled after empirical and quantum mechanical studies of the H2SO4H2O system. The effective potentials assume rigid H 2O and SO 4-- structures with two
unconstrained H+ ions free to bond with either species. The Monte Carlo simulations provide
information about size dependent cluster structure, interaction energies, free energies, RMS
displacements, and specific heats of the molecules in the cluster. The goals of this work are
to generate Helmholtz free energy differences for constant concentration, adjacent sized
clusters and estimate effective small binary cluster surface tension. Preliminary results for the
free energy differences are given.
Keywords - Monte Carlo, binary nucleation, sulfuric acid and water clusters, surface chemistry.
INTRODUCTION
Studies on the thermodynamic properties of small binary clusters are central to the understanding
of many atmospheric processes, for example, gas to particle conversion, acid rain, and ozone depletion
mechanisms involving sulfuric acid tetrahydrate (SAT) ice. At present, interpretations of most processes
rely on the classical binary nucleation formalism (Reiss 1950, Doyle 1961, Mirabel and Katz 1977 and
Wilemski 1988) and use experimental bulk liquid surface tension. Many questions persist, however,
regarding the applicability of macroscopic properties to the microscopic binary clusters. In particular, small
binary H2O-H2SO4 clusters have not been modeled on a molecular scale. In this work we present preliminary
results of a statistical mechanical study of microscopic H2O-H2SO4 clusters using effective pair potentials
and Monte Carlo simulations. The motivation for these preliminary studies has been to test the model
potentials, estimate effective binary surface tension for small clusters and to examine the binary cluster
structures for consistency with bulk property predictions.
Stratospheric conditions of interest allow formation of sulfate aerosols at temperatures from about
190 K to 240 K and sulfuric acid concentrations from about 60% to 85% wt. H2SO4 (Yue et al, 1994).
These concentrations imply that for each sulfuric acid molecule there are 1 to 4 water molecules. The
formation of sulfuric acid and the subsequent nucleation of sulfuric acid and water proceed via a series of
clustering reactions between SO3, H2SO4, HSO4-, SO4- -, H3O+, and several water molecules:
SO3 + H2O <!!
!!> H2SO4
H2SO4 + H2O <!!
!!> HSO4- + H3O+
HSO4- + H2O <!!
!!> SO4- - + H3O+
Studies imply that the above reaction schemes are not the most energetically favorable reaction paths at
stratospheric conditions (Chen et al, 1985 and Hofmann et al, 1994). A more probable scenario requires
the clustering of multiple water molecules in order to convert the sulfur trioxide molecule into the sulfuric
acid molecule and similar clustering of water molecules on each molecule/ion produced in the sequence. The
microscopic description of this process is complicated by the mobility of the H+ ions and the time scale (•
picosecond) of the formation and breaking of hydrogen bonds.
*This work supported in part by the National Science Foundation under Grant No. ATM93-07318.
MOLECULAR MODEL
In this study the rigid water molecules interact via the RSL2 (Rahman and Stillinger 1978) potential
and the free H+ and (rigid) SO4-- species interact (mutually and with the H2O) via atom-atom Coulomb plus
Lennard-Jones (LJ) potentials. The H2O-H2SO4 interaction energies and the relative size of effective atomic
charges in H2SO4 were determined from quantum chemistry calculations using the GAMESS (1993)
software. (See also Kurdi et al, 1989.) The overall scaling of the atomic charges was chosen to give the
experimental H2SO4 dipole moment, 2.72 D (Lovas et al, 1981). The LJ parameters, Fij and ,ij , for the ijth
atomic interaction were estimated from combinatorial rules and adjusted slightly to reproduce realistic H3O+
and HSO4- structures and interaction energies consistent with quantum mechanical and thermodynamic
estimates ( Mirabel et al, 1991, and Taesler et al, 1969).
STATISTICAL MECHANICAL FORMALISM FOR BINARY CLUSTERS
The present work employs the Bennett (1976) Metropolis Monte Carlo (Metropolis et al, 1953)
free energy calculation technique used previously for small argon LJ cluster (Hale 1982) and RSL2 water
clusters (Kemper 1990, Hale 1996). We calculate Helmholtz free energy differences between clusters
containing km water and m sulfuric acid molecules and clusters containing k(m-1) water and (m-1) sulfuric
acid molecules, where k = ratio of water to sulfuric acid molecules (km/m = k). The Bennett technique
allows one to calculate -kTRn[QB/QA], the free energy difference between two systems (called here B and
A) with slightly different interaction potentials. QA and QB are configurational partition functions. The Bensemble contains the normal {km,m} cluster with all molecules interacting fully whereas the A-ensemble
contains a {k(m-1),(m-1)}cluster with fully interacting molecules plus one free H2SO4 and one free H2O.
We have simulated small binary clusters at T = 298 K with sulfuric acid mole fraction = 0.5 (k=1). The
statistical mechanical formalism assumes that the km water and m sulfuric acid vapor monomers are in
equilibrium with the {km,m} binary cluster,
km[H2O] + m[H2SO4] <--> {[H2O]km [H2SO4]m};
and the clusters form a non-interacting mixture of ideal “gases”, so that the Law of Mass Action is valid:
[Nkm,m]
Qkm,m
'
HO
H SO
[N1 2 ]km[N1 2 4]m
HO
H SO
(km)!m![Q1 2 ]km[Q1 2 4]m
(1)
After some algebra, one obtains the following result for the cluster number distribution:
m
[Nkm,m] ' exp[Gn'1[Ckn,n & k ln I1& ln I2% k ln S1% ln S2% ln >]]
Ckn,n / ln
Qkn,n
1
Qk(n&1), n&1 [Q1 ]k
2
Q1
' ln
QB
QA
(2)
(3)
The Ckm,m values are calculated in the Monte Carlo simulations for a series of m values with k fixed and give
the free energy differences between the cluster systems A and B. I1 ( I2 ) is the ratio of experimental partial
liquid number density to equilibrium partial vapor densities of water (sulfuric acid) above the solution. S1
( S2) is the ratio of the number of ambient vapor monomers to the number of equilibrium vapor monomers
above solution for water ( sulfuric acid). Qkm,m is the {km,m} cluster canonical configurational partition
function, and Q11 (Q12) is the single molecule partition function for water (sulfuric acid). For a fixed k value
the simulations are all performed at constant density. Below are shown snapshots of the B and A ensemble
for k = 1 and m = 1. >(k,m) is a combinatorial factor which 6> 1 as m 6> 4.
Fig. 1. Snapshot of the B
ensemble for k = m = 1 at 298 K.
Fig. 2. Snapshot of the A
ensemble for k = m = 1 at 298 K
In order to analyze the configurational free energy differences we use the classical free energy of formation
for a binary cluster (see, for example, Wilemski 1988 and Oxtoby 1991) converted to the {km,m} notation:
)Fkm,m
k BT
F
' [36B]1/3
k BT
2/3
Dbulk&liq
[km%m]2/3& km ln S1& m ln S2
(4)
where F is the binary surface tension. With the assumption of Eq. (4) δ{ln[Nkm,m]} • - δm[∆Fkm,m/kBT] and
one obtains the following form for Ckm,m:
Ckm,m
k%1
• &
k I1 % I2
2
F
[km%m]&1/3%
[36B]1/3
2/3
k%1
3
kBT Dbulk&liq
(5)
Using Eq. (5) one can extract information about the effective binary surface tension, F, from the slope of
Ckm,m plotted vs. m-1/3 for fixed k. See Fig. 3. In order to calibrate our potential, calculations are performed
at 298 K where experimental surface tension and partial vapor pressures are available.
RESULTS AND CONCLUSIONS
Preliminary results for free energy differences are shown in Fig. 3 for k=1, for m = 1-6, together with
an estimate of Ckm,m from experimental surface tension data (solid line). The calculated Ckm,m values for k
= 1 are consistent with the rough experimental predictions (extrapolated to small cluster sizes) and show
some size effects. A more stringent test of the model potentials depends on Ckm,m for larger cluster sizes
which must, in the limit of large m, reproduce the experimental bulk surface tension. As part of the small
cluster results, the simulations of the k = m = 1 cluster give an average potential energy of -17.8 kcal/mole
for the H2O-H2SO4 interaction, compared to -12.8 kcal/mole enthalpy of hydration (Mirabel et al, 1991),
and -16.8 kcal/mole from ab initio results of Kurdi et al (1989). The goal of these preliminary studies has
been to test the statistical mechanical formalism, to develop a realistic potential model which can be used
to study clusters of varying compositions (k values), and to examine some general cluster properties from
the Monte Carlo simulations. Root mean square displacements of the atoms indicate that the H+ ions are
highly mobile and readily bond with both the H2O (to form H3O+) and with the SO4-- (to form HSO4 - and
H2SO4 ). In this respect the model displays the flexibility essential for modeling the binary system. In
progress are calculations for larger m values (with k=1) and for k = 2, 3, and 4.
Fig. 3. Ckm,m /[k+1] vs. m-1/3 for k=1 and T = 298 K for small binary watersulfuric acid clusters.
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