A Theoretical Study of Vapour Phase Nucleation of the Rocket
Propellant N2O4
P Pal
Royal Holloway, University of London,
Egham Hill, Egham, Surrey TW20 0EX. United Kingdom.
E-mail: P.Pal@RHUL.AC.UK
Abstract.
The residual vapour of a rocket fuel at the venting stage develops a potential aerodynamic problem which is
linked with the vapour phase nucleation phenomena of the propellant. This study, based entirely on molecular treatment,
addresses the problem by focusing specifically on the N2O4 propellant which is used in the ARIANE flight. The phenomenon is
examined by considering the thermodynamic free energies of N2O4 clusters, leading to the evaluation of nucleation flux rates of
critical nuclei at incipient nucleation. Preliminary examinations of the kinetics of flux pulses provide basic explanation from a
molecular perspective.
INTRODUCTION
A potential problem is known [1] to arise which is associated with the condensation of Nitrogen Tetroxide (N2O4)
vapour at the upper stage of the flight of ARIANE 5 as the residual fuel travels through the rocket nozzle. This may be
attributed to the thermodynamic behaviour of the vapour at pre-condensation stage. As the residual gas escapes along
the exit passage, vapour phase nucleation process is initiated at suitable thermodynamic conditions. The process is
characterised by aggregation of the propellant molecules forming clusters of different sizes. Simultaneously some of the
clusters decay by loss of molecules. The net result of the growth and decay process is that the exhaust gas becomes a jet
of n-molecule clusters with a distribution of thermodynamic free energy. These clusters remain at meta-stable
equilibrium with the vapour. The process leads to a sharp increase in the nucleation rates of clusters of specific sizes
(critical nuclei) at right temperature and pressure conditions. Evidence of similar phenomena is observed in nucleation
experiments on various gases with supersonic nozzles in terrestrial laboratory conditions.
BASIC ELEMENTS OF STUDY
In the molecular approach, the propellant vapour is treated as a mixture of gaseous clusters. Hence clusters of N2O4
molecules form the basic elements of the present study. Each cluster is essentially a collection of individual molecules
that combine under some weak interaction potential force. Models of these clusters are based on the physical clusters as
defined by Stillinger [2].
The stereogram of a single N2O4 molecule exhibits a convex structure [3-5], which belongs to the Vh symmetry group
[6] similar to ethylene. The molecule has some unusual structural features: (a) its extreme thinness (planarity) and (b)
extraordinary length of its N – N bond. The bond is represented as a thin rod. This length is known to be 1.78 A from
molecular orbital calculations. Each nitrogen atom links with two O-atoms. The structure of the molecule is shown in
Figure 1. The convex envelope represents the structural boundary of the molecule. Since the two NO2 components are
symmetrically located at either end of the N - N bond, total mass of the N2O4 molecule (92.015 amu) is treated as being
concentrated at the mid-point the bond. On this premise, we have considered the molecules aggregating in a cluster as
mass points. The black circle in the figure represents the mass point for the single molecule.
In the aggregation process, the distance of closest approach for a pair of molecules is set by the intermolecular forces
of a pair potential. This specific distance defined by our chosen potential function is sufficiently large to allow the
convex envelope of each molecule to remain totally separate from its neighbour.
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
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0
0
N
N
1.78
135°
112°
0
0
FIGURE 1 Stereogram of the N2O4 molecule.
For the purposes of this study, computer models of clusters have been built in the size range 3 ≤ n ≤100. Unlike the
molecular configurations in an infinite lattice structure, clusters containing finite number of molecules form noncrystalline configurations. Similarly the N2O4 molecules are expected to aggregate in a cluster in non-crystalline
configurations. Furthermore in view of the fact that these clusters are formed at an altitude under near zero gravity
conditions, these are likely to have a degree of compactness for any size In the small size range of the growth sequence
(e.g. 3 ≤ n ≤ 20 ), we have grown the clusters on computer incrementing the cluster size by 1 molecule. In this range, a
selection of geometrical shapes [7,8] have been used as the initial configurations. for subsequent Monte Carlo
simulation. In the next range (20 < n ≤ 100), we have used a growth scheme of spherical symmetry (e.g. icosahedral
family) incrementing by 5 molecules. Essential thermodynamic energy values have been calculated from the
microscopic behaviour of the constituent molecules in these clusters.
EVALUATION OF THERMODYNAMIC FREE ENERGY
Thermodynamic properties are evaluated in terms of entropy related statistical quantities of the n-body systems such
as the Helmholtz free energy A(n,T) and the Gibbs free energy of formation ∆G(n,T,p). These n-body systems are
represented by clusters of variable sizes (n ≥1). Their thermodynamic free energies are dependent on the temperature
and pressure of the environment.
Helmholtz Free Energy
Major contribution to the Helmoholtz free energy of a cluster at a temperature T comes from its overall
configurational energy which is essentially the total potential energy under an interaction potential corresponding to
specific positions occupied by n molecules in a cluster. With the displacement of a molecule to any other position
within the boundary of a physical cluster, the cluster configuration changes together with its overall potential energy.
Thus random displacements amount to random walks by the n-particle system on a 3n-dimensional potential energy
surface (PES). To evaluate the expected value of Helmholtz free energy of an n-molecule system, canonical average of
its configurational energies is required. The configurational energy values are determined for appropriate energy states
by comprehensive exploration of the PES which in turn involves an enormous number of random displacements of the
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constituent molecules. This stochastic process is simulated using the Monte Carlo (MC) technique starting with an
initial configuration. We have used our models of compact clusters as the initial configurations for each size. The MC
simulation provides an effective method to sample all the energy states accessible by the n-particle system. The
canonical average of the configurational energy is a key component of the Internal energy of a cluster.
The Helmholtz energy <A(n,T)> is determined indirectly by evaluating its derivative with respect to temperature
which gives the Internal energy E as given by the following expression.
∂A(n, T ) / T
= E = <U(n) > + 3/2 (n – 1) kT
∂ (1 / T )
(1)
The quantity < U(n) > represents canonical average of the configurational energy of an n-molecule system. Generally
the Helmholtz free energy is evaluated by integrating the derivative given by eqn (1) along a line of constant density at
two different temperatures. For absolute free energies, the integration is extended to a reference state temperature Tref ,
where the free energy is known exactly.
Conventionally in the MC simulation process, equilibrium configuration of a cluster for a particular T is used as the
initial configuration to determine the <A(n,T)> at another temperature T+∆T. The process is repeated to evaluate
<A(n,T)> values for a wide range of temperature. The MC process inherently includes anharmonicity and entropy due to
the change in configuration of the of the n-particle system. This is reflected in the thermal expansion of volume of a
physical cluster at each temperature level.
MONTE CARLO SIMULATION
Machine simulation of the PES exploration was based on the Monte Carlo method using the Metropolis [9] algorithm.
For this purpose, a stochastic chain of configurational states was generated by random displacement of each
constituent molecule sequentially in a cluster of size n.. .
One molecule (the ith molecule) of the cluster was displaced to a neighbouring position with equal probability [10,11]
resulting in a change in the energy state of the n-particle system. This overall potential energy of the cluster U(n) was
calculated by using the additive property of the Lennard-Jones potential. This is expressed as follows.
12
6

n −1 n



 
U(n)= ∑
∑ U (rij ) = ∑ ∑ 4U o  σ / rij  −  σ / rij  



 
i = 1 j = i +1
i j 
(2)
In equation (2) Uo is the L-J energy corresponding to the separation of the pair and σ is the distance of closest approach
of the two molecules where the potential energy vanishes. Two molecules of the pair are prevented from coming any
closer by the repulsive force of the potential. It is seen that the separation defined by the L-J parameter σ is large
enough to allow any possible orientation of an individual molecule. The pair potential energy reaches the L-J minimum
value Uo, at a separation rij = 2 1/6 σ.
As mentioned before, we have treated each molecule of pair as a mass unit in which the molecular mass is centred at
the mid-point (core) of the N-N bond. The separation rij of a pair of N2O4 molecules i and j is measured between the
mid-points of their respective N-N bonds. Thus an n-molecule cluster has been treated as a connected set of mass points
with a single link between each pair of constituent molecules under the L-J interaction potential. This assumption
preserves the integrability of the second virial coefficient. The implication of treating of the interacting molecules as
mass points is that the potential well depth is independent of the orientation of the molecules, which is consistent with
the Kihara prescription [12]. With this simplification, the interaction potential becomes a single link potential for each
pair of mass units in a cluster and provides a practical tool to evaluate the configurational energy of clusters in any
configurations.
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The parameters Uo and σ of the L-J potential function for the N2O4 molecule have been evaluated from the boiling
temperature and critical volume data.. The empirical relationship between the L-J parameter Uo and the boiling
temperature Tb is given by
U
0 = 1.15T
b
k
(3a)
By using the Van der Waals excluded volume principle, the relationship between σ and the critical volume Vc is
estimated as follows.
1
4  σ 3
V = π 
3 c 3 2
(3b)
For N2O4, Uo/k = 338.44 K and σ = 4.7x 10 -8 cm by using Tb = 294.3K and Vc = 165 cm3 mole in equations (3a) and
(3b). It is seen that the distance of closest approach for two molecules as defined by σ is larger than the separation
distance between their mass centre.
PES Exploration
Exploration of the potential energy surface of an n-molecule system presents a formidable challenge. Machine
simulation may lead to searches in unimportant regions of the energy surface. To avoid this problem, important criteria
are applied in the transition of a cluster from one state to another.
In our exploration, the key criterion was determined from the associated Boltzmann factor of the energy difference
between two states at T. This factor is defined by exp (-β ∆Uba) = ρb/ ρa where β = 1/kT. The relationship is readily
expressed as follows.
ρ b exp(− βU b )
=
= exp(− β∆U ba )
ρ a exp(− βU a )
(4)
In the PES search using MC method, the following practical steps were used..
(a) If ∆Uab ≤ 0 (downhill move), the transition was accepted since this represented a decrease in the configurational
energy of the cluster.. In probability terms, it implied that state b had a probability higher than state a. i.e. the cluster
was in a more stable state in terms of its overall potential energy in its random walk on the potential energy surface.
(b) If ∆Uab > 0 (uphill move), the new state was accepted with the probability given by exp (-β ∆Uab), provided the
probability exp (-β ∆Uba ) was less than ξ where ξ was a uniformly generated random number in the range 0 ≤ ξ ≤ 1.
Otherwise the move was rejected. and the system was returned to its previous configuration.. In the exploration, each
move regardless of it being accepted or rejected was counted for the calculation of the canonical average of the overall
potential energy and hence the internal energy
(c) For an efficient convergence of the configurational energy of the n-particle system, another practical consideration
was made on the maximum displacement element of each molecule. A virtual box of suitable dimension was defined at
the centre of each molecule which set the limit of its displacement.
The calculation of U(n) for the L-J potential at each configuration is relatively simple. For each move of a single
molecule, only 2(n-1) calculations of the pair potential are required, instead of evaluations along all the [1/2 n(n-1)]
bonds of the cluster..
The algorithm of directed random walk set the necessary volume condition. The use of compact configuration
increased the effectiveness of the PES exploration Temperature for the reference state was chosen at 350 K, the ideal
gas state of N2O4. .This temperature is far removed from the boiling point of propellant (294.15K). Over 10000 random
599
displacements were carried out on each coordinate direction of a molecule in a cluster. A constraining volume is often
defined as the boundary of movement of the constituent units of a physical cluster under consideration. However, this
definition is arbitrary and it has been found that the free energy of a cluster remains fairly insensitive at the peripheral
regions of the constraining volume.
Gibbs Free Energy of Formation
The Gibbs free energy is the energy that appears on formation of a cluster by aggregation of n single molecules. It is
dependent on pressure as well as temperature of the surrounding environment and includes the Helmholtz free energy
<A(n,T)> minus the energy expended due to loss these molecules.
The propellant gas venting out of the combustion chamber through the diverging section of a rocket nozzle, experiences
a rapid isentropic expansion [12] which provides an ideal environment for nucleation to occur. In terms of these
conditions, the free energy ∆G(n,T,p) of the n-molecule cluster is expressed as follows.
∆G (n, T , p ) = ∆G 0 + (1 − n)kT log( p )
(5)
where ∆G0 is the free energy difference between the cluster and the n individual molecules, including the reversible
work done [14] in the formation of the cluster. The ∆Go component includes the canonically averaged Helmholtz free
energy <A(n,T)> of the cluster together with the translational components of the cluster and the n individual molecules.
The free energy of formation ∆G(n) reaches a maximum value at appropriate p and T and the cluster corresponding to
the maximum value is the critical nucleus which plays a key role in the nucleation rate determining process. The
computed results of ∆G(n,T,p) are plotted as a function of the cluster size n at different p and T (Fig. 2). Each trajectory
in the graph exhibits an approximate profile analogous to that of the classical liquid drop model. However it is
emphasised that unlike the classical model, the present study is based entirely on molecular parameters of N2O4 and
makes no use of the bulk state properties (e.g. surface tension) of the specimen. This is consistent with the microscopic
size range of clusters used in the study.
∆ G(n,T,p) x (10
-12
ergs)
∆ G(n,T,p) vs cluster size
4
3
2
1
0
-1
-2
-3
-4
-5
e
d
b
c
a
0
10
20
30
40
50
60
70
80
n (cluster size)
Figure 2. Gibbs free energy as a function of n at a selection of pressure p and temperature T.
(a) T = 150 K , p = 0.15 Mpa , (b) T = 180 K, p = 0.5 Mpa, (c) T = 200 K , p = 0.8 Mpa,
(d) T = 200 K , p = 2.5 Mpa, (e) T = 350 K , p = 2.0 Mpa.
600
In the temperature range (10K< T < 350K) , maxima in the ∆G function appeared at n < 40.(trajectories a-d in the
graph). The function showed no maxima within n < 100 at T=350 K and p = 2.0 Mpa (see trajectory (e)) The maximum
level of the ∆G function becomes relatively flat which indicates that several clusters with comparable free energy of
formation share the plateau region
NUCLEATION RATES
The nozzle of a rocket is designed typically with a diverging section from the throat to the final exit according to the
kinematics of the propellant thrust. As a result, the propellant gas experiences progressively decreasing temperature and
pressure along this passage. Since this gas is not a stream of single N2O4 molecules but composed of discrete clusters of
variable sizes (n ≥ 1), they are the potential embryos of the nucleating phase representing fluctuations in vapour density
[15-18]. From the knowledge of the free energy of formation of these clusters, the rate of nucleation of clusters of
different sizes is estimated using the gas kinetic theory with some degree of accuracy. From the classical nucleation
model, the rate of flux J of nuclei containing n molecules, is the combined effects of accretion and evaporation of single
molecules.
The rates of nucleation at specific temperatures and pressures were determined by assuming steady state conditions.
With these assumptions, the nucleation rate [19] is simplified as follows.
J=
αnβn
n
(6)
*
∑ (z o c )
n =1
−1
n n n
In this equation, αn is the accommodation coefficient (assumed to be unity), βn = p/(2πmkT)1/2. and the
Zeldovich factor zn = [∆G(n,T,p)/(3πkTn2] 1/2 The surface area of the cluster is given by on = (4π)1/3 (3nv)2/3 where v is
the molar volume.. Finally the equilibrium concentration Cn of the n-molecule cluster is given by
C = C exp− [∆G (n, T , p ) / kT ]
n
1
(7)
Of the quantities appearing in the rate equation, the most significant contribution to the nucleation rate is made by its
equilibrium concentration Cn. The summation is used to determine the total flux up to any critical size limit n*
including the sub-critical sizes. Alternatively the flux of nuclei of a specific size n (e.g. the critical nucleus n*) can be
calculated without the summation. It is pointed out that all the parameters appearing in the rate equation (eqn (7)) are
determined from the data at the molecular level and no approximation is made with respect to the bulk property.
The rates of flux of a selection of critical nuclei as determined from our molecular approach are tabulated below, at
associated temperatures and pressures.
TABLE 1
________________________________________________________________________________
Critical nucleus (n*)
Temperature (K) Pressure (Mpa)
Rate (nuclei sec-1)
________________________________________________________________________________
7
100
0.12
2.0 x 102
12
150
0.15
1.5 x 102
10
150
0.20
1.6 x 102
15
200
0.50
1.0 x 102
13
200
0.60
1.2 x 102
17
280
0.70
1.0 x 102
20
350
1.00
1.0 x 10
________________________________________________________________________________
We have calculated the nucleation flux up to a temperature limit T = 350K and pressure p = 3.0 Mpa. Beyond that
temperature, N2O4 has a tendency to decompose into NO2 progressively at increased proportion (N2O4 = 2 NO2) .
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However, Pearson [20] has compiled flux rates above this temperature limit. At T < 350, there is a qualitative agreement
between the two works.
The results from the above table show several characteristic features of nucleation. Critical nuclear size (n*.)
increases with the rise in temperature and pressure. In addition, larger nuclei have a slower growth rate compared with
the smaller ones. Nuclei of small size are favoured at low temperature and pressure conditions whereas large nuclei
tend to form at high temperature and pressure. At a fixed T, a rise in pressure favours an increase in the critical
embryonic size. For example, at T =150K, at 0.15 Mpa pressure, the embryo size is 12 and at 0.20 Mpa the size
decreases to 10. Similarly at T = 200K., critical nuclear size decreases from 15 at p = 0.5 Mpa to 13 at p = 0.6 Mpa.
These trends have implications when the kinetics of these nuclei along the passage of a rocket nozzle are considered.
Beyond the nucleation rates of n* shown in the table, flux rates rise sharply for a relatively small increase in pressure
p when the temperature T is held constant. Similar sharp rise is indicated as the temperature T decreases, when the
pressure is held constant. This phenomenon is typical at incipient nucleation. Sharp rate rise is not restricted to the
critical nucleus. Neighbouring clusters around the critical nucleus (sub-critical size) with comparable free energy of
formation ∆G, show similar sharp rise. The limit of nucleation represents the deepest penetration of a fluid into the
domain of metastable states. Physical phase transition occurs when the limit is reached.
KINETICS OF NUCLEATION FLUX PULSES
Nucleation rate is dependent, among other factors, on the surrounding pressure p and temperature T as indicated by
equation (6). These p and T values are determined directly from thermodynamic investigations. Moving beyond
thermodynamics, it is possible to estimate cartesian locations at which the critical nuclei emerge inside the rocket
nozzle, from the p and T values derived from thermodynamics. This is where we leave thermodynamics and enter the
aerodynamic regime, although in the context of rocket propulsion, the term "aerodynamic" is not strictly true in its
traditional sense. Within the confines of the nozzle, an interplay is likely to develop between thermodynamics and
kinetics of the nucleation flux pulses. Only the basic kinetics of these fluxes is outlined here.
Since along the length of the nozzle, a temperature and pressure gradient exists from very high (at the throat end) to an
ambient value (at the exit end), the cluster traffic is expected to emerge at different spots of the nozzle. Larger nuclei are
formed near the throat end, together with smaller (sub-critical) nuclei and near the exit region, only the relatively small
ones nucleate. As a result, the flow density becomes a variable quantity.
In addition, a proportion of the N2O4 molecules is expected to dissociate into NO2. molecules which must have equally
susceptibility to clustering and nucleation, like the parent molecules. Thus from the molecular perspective, the
propellant flow may be seen as a traffic of nucleation flux of two compounds (N2O4 and NO2) of variable sizes.
To estimate relative velocities of different elements of this traffic, we make a simple assumption that these velocities
are within the classical limit and they show a Maxwell Boltzmann type distribution. With this assumption, the velocity
of a unit of mass M at a temperature T is calculated by using the Newtonian dynamics (non-relativistic). This is given
by
v =
2kT
M
(8)
For an estimate of velocities acquired by nuclei of size n* at a temperature T, the mass M is replaced by n*m in eqn
(8) , where m is the mass of a molecule under consideration.. By inserting values for n* and T derived from
thermodynamic study, velocities of n* nuclei can be determined
The ratio of relative velocities of two sizes n1 and n2 at temperatures T1 and T2 is evaluated using the following
expression.
v
1 =
v
2
n T
2 1
nT
1 2
(9)
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Considering a specific case, when n2 = n1 and T2 > T1, it is seen from eqn (9), that clusters forming at high
temperature near the throat of the nozzle move relatively fast and therefore can catch up with those nucleating at lower
temperature. Calculations of relative speeds of N2O4 and NO2 clusters of same size at a temperature T show that the
NO2 clusters move 1.4 times faster than the N2O4 clusters.
These preliminary examinations of relative speeds and momenta indicate the possibility of a multiple pile up.
Potential problems are likely to emerge, in the event the pile up occurring within the confines of the rocket nozzle. It is
possible to examine it further by applying a binary nucleation approach, including both N2O4 and NO2 components.
CONCLUSIONS
The molecular treatment transforms the gas flow into a traffic of discrete clusters and focuses on their thermodynamic
properties leading to the so-called aerodynamic regime. In the development of the treatment approximations have been
made at different stages that are likely to attract criticism.. The L-J function provides a simple 2-body interaction
potential with a single link between a pair of molecules, its adoption may be questionable. Multi-link potentials that use
interaction links between the constituent atoms of each molecules in a cluster are cumbersome for practical application
to n-body systems.
A proper investigation of the kinetics of the cluster traffic is likely to provide answers to potential problems of rocket
propulsion at high altitude. Further examinations using a binary nucleation treatment are in progress.
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