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02/07/2006 Final Technical Report Feb. 15, 2003 - Dec. 31, 20054. TITLE AND SUBTITLE 5.. CONTRACT NUMBER

Final Report for Excitation Models Grant Closing Dec. 2005

Project title, "Development of Models and Computational Tools for the 5b. GRANT NUMBER

Study of Space-Based Micro Propulsion Systems" F49620-03-1-0144

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S) 5d. PROJECT NUMBERDeborah Levin, Associate Professor of Aerospace Engineering

5e. TASK NUMBER

5f. WORK UNIT NUMBER

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) B. PERFORMING ORGANIZATION

The Pennsylvania State University REPORT NUMBER

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12. DISTRIBUTIONIAVAILABILITY STATEMENT AFRL-SR-AR-TR- 0 6 _03 10Approved for public release; distribution in unlimited.

13. SUPPLEMENTARY NOTES

14. ABSTRACT

This final report presents the major technical accomplishments of this grant: 1) Implementation of a Kinetic Nucleation Model forDSMC, and 2) Use of Quasiclassical Trajectory Methods to Model OH Production Mechanisms in a Side-Jet RCS Flow. Specifictechnical details are provided in the actual report.

15. SUBJECT TERMS

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF RESPONSIBLE PERSON

a. REPORT b. ABSTRACT c. THIS PAGE ABSTRACT OFa,.. PAGESPAGS3 19b. TELEPHONE NUMBER ,inckide area code)

Standard Form 298 (Rev. 8198)

Prescribed by ANSI Std. Z39.1B

2 0 0 60727330

I Jo

PENNSTATE

Department of.Aerospace Engineering

Inter-office Memorandum

TO Dr. Mitat A. Birkan

FROM Deborah Levin, Assoc. Prof. of Aerospace Engineering

SUBJECT Final Report for Excitation Models Grant Closing Dec. 2005

DATE 7 February 2006

This report presents the major technical accomplishments of the above mentioned grant.The report is subdivided into two sections that discuss the highlights of the research thathas been performed.

I. Implementation of a Kinetic Nucleation Model for DSMC

Comparison of First-time Cluster NucleationModels for DSMC Kinetic Nucleation vs CNT

NormaliztIA Cluster cumnber density Nnrmaliizd Cluster Size

or's AlA 1 2 34 56 7 0 9SD 16.2O Dl3 4 0.5 O 6 o.7r 0. 0.1

Aziul~Ca p• |)•JP m

IL A

IK

cur CT ' CC2 ~ O

Figure. 1

1. Explanation of Figure 1

Cluster numbers and size distributions are shown for a simple expansion of an

Argon gas into a vacuum through a sonic nozzle with a diameter of 3.2 mm, a stagnation

temperature of 280 K, and a stagnation pressure of 125 torr. The clusters are formed

during the non-equilibrium expansion through the process of homogeneous nucleation.

Two sets of comparisons are presented for cluster number density and average cluster

size, with the top contours showing the kinetic nucleation result and the bottom contours

showing the results obtained from Classical Nucleation Theory (CNT). In the first

comparison of cluster number density, the Kinetic and CNT results are normalized to 8

x10 2° and 8 x 1017 m"3 , respectively. In the comparison of the cluster size, on the right

hand side, the contour levels are normalized to 50 and 1000 for the kinetic and CNT

models, respectively.

The most important conclusion from the viewgraph is that there is a significant

change in the physics of a homogeneous condensation flow depending on which cluster

nucleation model is used. The kinetic nucleation process predicts three to four orders of

magnitude more clusters are formed than CNT. Also, the kinetic nucleation model

predicts that most of the clusters will be smaller than if the CNT is used. There are also

important computational consequences when switching from the CNT to the kinetic

nucleation model. The cluster size and number density contour distributions are

smoother for the kinetic nucleation calculations even though both DSMC calculations

have the same numerical parameters - number of gas and cluster particle F,, number of

cells and computational domain. The shakiness of the CNT contours is due to the

necessity of obtaining the local gas temperature in order to calculate the nucleation rate.

This is an inherently noisy process in a DSMC calculation.

Finally, these results are presented for Argon because there is a data base of

laboratory measurements to which we can validate these results. This activity is in

progress. When the method is extended to more realistic propellant mixtures of water

and inorganic impurities, the amount of nucleation and the size of the clusters will affect

plume signatures and the degree of backflow contamination.

2. Background explanations:

a. DSMC Implementation of cluster formation: A two-dimensional,

axisymmetric DSMC simulation begins from the imposed starting surface, located at

approximately one orifice diameter downstream of the orifice, and is composed of 45 jets.

The entire DSMC computational domain is 4 by 10 orifice diameters squared, with 100 x

250 cells, each cell can be divided up to 16 subcells. The simulated vapor and cluster

particles represent 1.0x 10+10 and 5.0 x 10+7 real particles respectively, and a total of

about 860,000 simulated particles were used. A time step of 2.0 x 10-9 sec is used with

about 24,000 steps to reach the steady state and the simulation results are sampled for

200,000 steps.

b. Failure of CNT:

The DSMC/CNT has been successful in predicting average cluster properties for flows

that exhibit very low amount of nucleation. However, it fails to predict cluster

distribution properties. Since homogeneous condensation had never been incorporated

directly into DSMC it was still a major advance to have a flow coupled condensation

model. However, there are important problems with the inherent CNT assumptions when

modeling, supersonic, expanding flows. CNT assumes that the initial clusters are born in

a constant vapor environment and can only grow to a maximum size. The initial clusters,

known as the critical clusters, are assumed to be born in an equilibrium isothermal vapor

environment. The classical nucleation rate, J, is a function of macroscopic parameters

as:

2a p2 -4m-Y2 aP •T•'f exp( )

imn p. 3kT

where a, the cluster surface tension and pw, the cluster mass density, are the most

problematic terms for small clusters.

c. Implementation of a full kinetic nucleation model: The molecular dynamics

(MD) approach was used to simulate a I-D Argon condensation plume along the

centerline. A geometry criteria and history tracking method were used to analyze how

the clusters are created in the plume and it was validated that the initial dimers were

created through triple collision processes, during which the third monomer takes away the

additional energy. The triple collision can be expressed as,

A+A+A kf >A2 +A

A and A2 are atoms and stable dimers and

n ~- t, = c c 0 k T r ,U0

where P, is the probability of creating a stable dimer from a triple collision and f is the

energy equilibrium distribution function.

To be implemented into a DSMC scheme, the triple collision is decomposed into two

successive binary collision steps:

A + A -> A;

A; +A -- A2 + A

where A2" is a temporary collision complex with a lifetime. Complex-monomer

collisions are referred as the triple collisions. In the full DSMC simulation, we need both

P, and the collision complex lifetime. A Hybrid DSMC/MD technique was employed to

obtain P, in the full DSMC simulations. In addition, the acceptance-rejection principle is

used to determine whether a binary collision occurs. Each successful binary collision => a

temporary complex. During the complex lifetime, complex-atom pairs are selected and

the acceptance-rejection principle is applied assuming a hard-sphere like interaction

between clusters and Argon atoms. The collision outcome is evaluated according to the

probability, Pt. and a "successful" complex-atom collision generates a stable dimer.

Subsequent cluster evolution processes, starting from dimers, are calculated using the

previously developed models for cluster sticking and evaporation.

II: Use Of Quasiclassical Trajectory Methods To Model OH ProductionMechanisms In A Side-Jet RCS Flow

Use of Quasiclassical Trajectory Methods toModel OH Production Mechanisms in a Side-Jet

RCS Flow

(+I1CI Rection Probability (TCEI

Pr > I'or morte than 10 % of the 0+110 readion

Reach,, raJichory Non-.acfl"Tre fvctory

MD/QCT calculations are used to calcualte crom sections for DSMC simulations:

Re-elon of -P.lhE for PEorO+HCI, used In the CTprodtuct. -• ' c~alculations.

OH Uo Rerono O+i4CI Ramachandran and PetersonrcactallL * R,0 Ia fixed Mth the 0 atom moving

SHCI and 0 around H-Cl molecule.

Figure 2.

1. Explanation of viewgraph:

The objective of this work is to understand why the common, readily available

Total Collisional Energy (TCE) model for chemical reactions in DSMC fails for the

exchange reaction of O+HCI --> OH+CI. This is an important chemical reaction

occurring between the efluents of a solid-non-aluminized reaction control systems (RCS)

and free stream atomic oxygen present in large quantities. Initial investigations of this

multi-scale, chemically reacting flow showed that using the baseline TCE model

approximately 10% of the reaction probabilities would be unphysically greater than unity,

if an artificial limiter was not utilized in the calculations. For this reason we propose to

develop a new chemical model that will be valid for chemically reactive, hypervelocity

collisioris.

We use the Quasi-Classical Trajectory/ Molecular Dynamics method (QCT/MD),

an accurate chemical modeling tool, to evaluate the reaction and total collision cross

sections. In this manner we can determine whether it is possible to propose a correction

to the TCE model and we investigate the sensitivity of the full DSMC simulation to the

different chemical models for this reaction for a hypersonic condition of 120 km altitude,

5 km/s, a typical free stream condition where these flows occur.

Background explanations:

The classical Hamiltonian is H=T+V=E, where T, and V are the kinetic and potentialenergy of the molecular system and Hamilton's equations of motion are

av . P!ju = -__,r~j =

"Dr. n '1

where i=1,2,3 (atoms), j=1,2,3 (x,y,z); Pij and rij are the Cartesian momenta and

coordinates, and mi is the mass of the i-th atom. We solve the 18 coupled equations for

Pij and rij using a 4th-order Runge-Kutta method. To construct the initial conditions foreach trajectory (i.e., internal energy of target HC1, relative velocity of O+HCl, offset or

impact parameter), we use the micro-canonical sampling to obtain the initial conditions of

the scattering event:

W=-(ET,,-V(R))p,.j=fRn,.j7', i,j = 1,3

where V(R) is the potential energy surface (PES) shown on the viewgraph.

To obtain final state distributions of product, we assume that the vibrational and

rotational energies may be treated classically and separable. Each trajectory outcome is

analyzed with the following classical relationships for angular momentum and vibration.

+ 4( L2" =.!ŽJdL=IuRXRJ=-2+ 2 l+ ' l2#R 2 h

eq&

The conclusions of this work is that both the reaction and total cross sections, a,

and Urr, for O+HCI obtained using the MD/QCT method affected the OH production.

Furthermore it was found that orT was independent of the HCI internal energy and the

usual Variable Hard Sphere (VHS) model used in DSMC could be corrected for high-

temperature O-HCI collision pairs if VHS parameters of w-0.39, dref=3.9 A at 1000 K are

used. The most accurate reaction probability calculated by the QCT/MD method,

Pr(MD), was less than 0.4 at 120 km altitude for freestream velocity of 5km/s.

The use of the MD/QCT model is limited by the accuracy of the PES and it is not

typical that such high fidelity PESs are available. For this reason, the MD/QCT

calculations were repeated for the older, more readily available, but less accurate, PES

known as a LEPS. These surfaces could be more easily obtained for other propellants-

atomic oxygen collisions. We are presently analyzing the results and bow they affect the

total OH production in the complete DSMC simulation.

W 6A#AA- ______

AIAA 2005-0767

A Kinetic Model of Condensationin a Free Argon Expanding Jet

Jiaqiang ZhongThe Pennsylvania State UniversityUniversity Park, PA, 16802

Michael I. ZeifmanThe Pennsylvania State UniversityUniversity Park, PA, 16802

Deborah A. LevinThe Pennsylvania State UniversityUniversity Park, PA, 16802

43rd AIAA Aerospace Science MeetingJan. 10 - 13, 2005 / Reno, Nevada

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics1801 Alexander Bell Drive, Suite 500, Reston, VA 22091

AIAA 2005-0767

A Kinetic Model of Condensation in a Free ArgonExpanding Jet

Jiaqiang Zhong* Michael I. Zeifmant Deborah A. LevintPennsylvania State University, University Park, PA 16802

Abstract uum, may result in contamination of spacecraftsurfaces as well as affect the observance of high

The direct simulation Monte Carlo (DSMC) altitude plumes [2][3]. Extensive experimentalmethod has recently been developed to sim- studies on the condensation in real thrustersulate homogeneous condensation in a free and small free-expanding jets have been con-expansion rocket plume. However, cluster- ducted since the 1970's [4][5]. However, it ismonomer and cluster-cluster collision models difficult to measure cluster size and numberas well as the determination of cluster size density distributions in an operational plume.were simplified in the previous work which may Therefore a prediction tool which can be val-greatly impact the accuracy of numerical sim- idated with small thruster data will provideulation results. In this work, the molecular crucial information about homogeneous con-dynamics (MD) method is used to simulate densation in typical chemical rocket plumes.collision and sticking probability for Ar clus- Most of the numerical work on condensationters. The Ashrigz-Poo model is used to predict coupled to a gas expansion models the pro-the outcome of cluster-cluster collisions. These cesses in ground-based facilities and uses con-improved models are then integrated into a tinuum approaches [6]. Since free-expandingDSMC code to predict the Rayleigh scattering jets are mostly in the transitional to rarefiedintensity in a free expanding Argon condensa- regimes, a continuum approach would be in-tion plume and numerical results are compared applicable to simulate condensation in a freewith experimental data along the plume cen- expansion plume. Thus, a more appropriateterline. kinetic approach should be used. Such an ap-

proach must address the processes of cluster1 Introduction nucleation, growth, decay and collisions with

other clusters and monomers as well as theoin usual gas kinetics and is, therefore, computa-Homogeneous condensation [1], observed tiniynhlenig

various types of plumes expanding into a vac-

Recently, the direct simulation Monte Carlo*Graduate student, Department of Aerospace Engi- (DSMC) [7] method has been applied to

neering, AIAA student member. simulate homogeneous condensation in free-tResearch Associate, Department of Aerospace En- g

gineering, AIAA member. expansion plumes [8][9]. The previous work [8]lAssociate Professor, Department of Aerospace En- can be summarized as follows. Classical nu-

gineering, AIAA Associate Fellow. cleation theory (CNT) [1] was used to model

the generation of initial nuclei, which are as- review the microscopic models developed insumed to be created at the local critical con- our previous work [8] and explain the simi-ditions. Microscopic models, consistent with larities and differences of the previous modelCNT theory, have been developed from gen- with the corrected models. These models areeral conservation equations to model sticking then used in Sec. 4 to simulate a conden-and non-sticking collisions, as well as evapora- sation plume in the direct simulation Montetion processes. These models were then inte- Carlo (DSMC) method of a free expandinggrated into the DSMC method to simulate con- Argon plume through a sonic orifice. Us-densation coupled flow. The developed DSMC ing the DSMC results, the Rayleigh scatter-condensation model was numerically validated ing intensity is then calculated and comparedby comparing the simulation results with ana- with experimental data along the plume cen-lytical solutions in one dimensional test cases, terline [151.further validated by comparison with Hagana'sscaling laws [4], and finally applied to predictwater homogeneous condensation in a rocket 2 Molecular Dynamicsplume. Simulation

In the previous work [8], clusters were mod-eled as hard spheres, with radius proportional 2.1 Methodologyto i½, where i is the number of molecules in thecluster. The probability of a sticking collision Molecular dynamics (MD) is a computer sim-between a cluster and a condensible monomer ulation technique which allows one to predict

was simply assumed to be unity [8] or 0.1 [9] the time evolution of a system of interacting

without validation. Cluster-cluster collisions particles, e.g., atoms or molecules. Each real

were assumed to be elastic without any al- atom in MD is usually modeled as a point-size

lowance for coalescence, although it was shown particle, and the interaction between them is

that for cases considered in Ref. [8] the number modeled by an interaction potential. Once the

of cluster-cluster collisions was small. Thus, system initial conditions are defined in terms

there is room for improvement of the previ- of particle coordinates, and velocities, the sys-

ous numerical simulations. The purpose of this tem evolution in time is followed by solving the

work is to improve the fidelity of the models Hamilton equations.

used in the condensation flow by use of the The traditional 6-12 Lennard-Jones poten-

molecular dynamics (MD) method. tial, valid for interactions among many species,

The paper is organized as follows. To un- is chosen as the interaction potential:

derstand basic cluster model and interactions [(r)12 (U)6]

between a cluster and a molecule, we choose V = 4e - (1)

a molecular dynamics [10] (MD) approach tosimulate cluster-molecule collision and sticking where the potential parameters for Argon areprocesses in Sec. 2. In order to derive gen- c = 0.0103 eV and a = 3.405 A . Note that ineral conclusions from MD simulations, a 6-12 this section, we specifically focus on the inter-Lennard-Jones potential, applicable to many actions among Argon atoms.species, is used to simulate collision processes Trajectories are integrated numerically usingbetween Argon clusters and Argon monomers. the Nordsieck fifth-order predictor-correctorUsing reduced units, the MD simulation results algorithms [10], which can be summarized asof Argon species are compared with those of follows. First, new positions as well as timenickel from Ref. [11]. In Sec. 3, we briefly derivative terms up to fifth order are predicted

2

at a new time step using a Taylor series expan- time could last more than 30ps in the plume,sion. Then, the force acting on each particle and they usually originated from a tri-atomis computed for the predicted positions, and collision process during which one of the atomserror terms are calculated by comparing force takes away extra energy, while the other twoterms between the previous and current time form a stable dimer.steps. Finally, the above error terms are used To evaluate the cluster cross section andto correct the particle positions and derivatives cluster-monomer sticking probabilities, we alsofor the next time step. need to prepare clusters of size greater than

two for cluster configurations, typical of ex-panding flows. Trimers were created through

2.2 Initial Clusters a dimer-monomer sticking collision, tetramers

To simulate cluster collision outcomes using through trimer-monomer sticking collisions,

a molecular dynamics approach, the initial and so on. To efficiently generate larger clus-

conditions (coordinates and velocities) of each ters, cluster-cluster sticking collisions may be

molecule within a cluster must be specified be- used.

fore the simulation. The condition of dimers, Molecular dynamics simulation results of

obtained from previous modeling of a free ex- cluster-monomer collisions will be discussed in

pansion plume [12], will be used to characterize the following subsection 2.3 and 2.4. Each MD

the initial clusters for the MD simulations dis- simulation case is repeated for various trajec-

cussed in the subsections 2.3 and 2.4. tories by randomly rotating the initial cluster

To explore the physics of the cluster for- around its center of mass and the results av-

mation processes in an expanding plume, a eraged over the trajectories are shown in the

molecular dynamics approach was used to sim- subsections 2.3 and 2.4.

ulate a free argon expansion along the coreplume. One challenge in the MD modeling 2.3 Cluster Modelof such a system is the selection of a crite-rion to distinguish clusters, especially dimers, The cluster-monomer collision process may be

from atoms that are just collision pairs. The used to define the cluster cross section, as

Stillinger criterion [13] and a history tracking the monomer-monomer collision defines molec-

approach were used to find stable clusters as ular cross section in the traditional DSMC ap-

follows. First, the collision duration time be- proach. To characterize the cluster model,tween two atoms were simulated by the MD the collision processes of cluster-monomers are

approach for conditions of various impact pa- simulated with the MD method, under the con-rameters and relative velocities, and the MD dition of various impact parameters, b, andresults were compared with Bunker's approxi- relative velocities, ve,. Sticking collisions aremate formula[14J. It was found that the molec- obviously counted as effective collisions, while

ular collision duration time is usually less than for non-sticking collisions, only trajectories in

10ps. Then each molecular position in the which the deflection angle is at least 10 de-plume was output every 10ps during the MD grees are counted as collisions. Note that the

simulation, and a pure geometric Stillinger cri- colliding molecules are initially separated suffi-terion was used to identify the clusters in the ciently far from the target cluster such that theplume. In this way, the observed clusters were effective interaction potential between them is

tracked using the MD output, i.e., every 10ps, zero.until the atoms in the clusters had separated The collision probability for each case (b,into monomers. We found that the dimer's life- Vrel) is defined as the ratio of the number of

3

trajectories that resulted in a collision to the known. The MD results are shown in Fig. 2total number of trajectories. For each case, a with a relative collision velocity of 182m/s andsample of 200 trajectories is used with the ori- are fitted to an analytic general cluster radiusentation of the target cluster to the monomer equation [16]:randomly chosen. The impact parameter b in-creases from a head-on collision value of zero to P- = AiO + B (3)

a value in which the cluster-monomer collision where i is the number of molecules in the clus-probability is close to zero. Relative collisionvelcites romaproxmatly 50to 225 rn/s ter, and A and B are constants. It can be seenvelocities from approximately 150 tthat the MD simulation results for Argon clus-are chosen, and the cluster sizes selected for ters fit this equation quite well by choosing Astudy in the MD simulations ranges from a 2.3A and B 3.4A (dashed line in Fig. 2).dimer up to 482-mers. The range of cluster The molecular dynamics simulations alsosize and collision impact parameter are typi- the cler-monomics lsi on al-cal in a condensation flow [8], and the collision show that cluster-monomer collision probabil-relative velocity corresponds to a translational velocities. Results of collision probabilitiesthermal temperature which is also representa- for different cluster sizes and relative clustertive of a typical nondimensiona] temperature,the ratio of the collision thermal temperature monomer velocities are shown in Fig. 3. Itcan be seen that the critical impact parame-to the species freezing temperature. ter increases as the relative collision velocity

Representative MD simulation results are deshown in Fig. 1. It can be seen that the dcreases, and therefore, the cluster cross sec-collision probability. s clbse en t ntyf the tion would also increase as the relative velocitycollision p rob ab ility is close to u n ity for th e d e r a s. T m o l th d p n e ce f c u -decreases. To model the dependence of clus-cases with relatively small impact parameter. ter diameter on the relative collision velocityFor the cluster collision probability of dimer-monomer collisions, for example, it can be seen moweuthat for impact parameters b less than b, -- 7.5 modelA , the probability is almost independent of d = dTef(cr )v (4)the impact parameter b. For collisions with where the subscript ,f refers to a referenceb greater than b5, the collision probability de- value and v is a constant related to speciescreases rapidly to zero as the impact parameter viscosity. The MD results for cluster radii, cal-increases. It can also be seen that b,, defined culated for various relative collision velocities,as the critical impact parameter, increases as are shown in Fig. 4, and the simulation resultsthe cluster size increases, fitted to Eq. (4) are shown as dashed lines. Fig-

The maximum impact parameter bin, cho- ure 4 shows that the cluster radii are approx-sen here to correspond to a collision probabil- imately inversely proportional to the relativeity of 0.1, can be used to calculate the cluster- collision velocity, and the slopes for differentmonomer collision cross section, oa, as a = 7rb•. cluster sizes are similar. The reference clusterIt is related to the cluster and monomer diam- diameter, dref, may be estimated from Eq. (3).eters as [7] For larger cluster size, the dcf is higher so that

bm. de + do (2) a smaller v is required to correct its diameter2 for the impact of relative collision velocity. It

where d, and do are the cluster and molecule di- can be seen that as the cluster size increasesameters, respectively. Thus, the cluster radius, from dimer to 147-mers, the constant v de-P- = -, can be determined from the MD sim- creases approximately from 0.3 to 0.07. Noteulation results, since the monomer diameter is that for Argon vapor, the constant v is about

4

0.31. Thus, the impact of collision relative ve- shows the sticking probability as a function oflocity on the cluster cross section becomes less impact parameter for different size clusters allas the cluster size increases, which indicates with a relative collision velocity of 182m/s. Itthat the hard sphere model may be used to can be seen that the sticking collision probabil-model relatively large clusters in the DSMC ity, similar to the collision probability, exhibitssimulation. As Fig. 4 shows, the error in us- a critical impact parameter dependence. Foring the hard sphere model increases for small collisions with impact parameters less than theclusters. critical impact parameter, the sticking proba-

bilities are almost constant, and are unity forcluster sizes larger than 10-mers. While for col-

2.4 Sticking Collision Probabil- lisions with impact parameters larger than the

ity critical impact parameter, the sticking proba-bilities decrease to zero more quickly than theThe sticking probability for a collision process collision probabilities. As can be seen from

between a cluster and a molecule can be simu- Fig. 6, the critical collision impact parameter

lated in a manner similar to the collision prob- for the sticking collision probability increases

ability. Initially, the target Argon cluster is at as the cluster size increases.

the origin in a static state', while the collid- In the DSMC simulation of an expanding

ing Argon atom is approximately 50 A away flow, consisting of gas and clusters, each col-

from the cluster. It takes about 25 ps before lision is not modeled in detail and only aver-

the atom collides with the target cluster. The lso sntmdldi ealadol vrtheulationtim efora collidesg wth arget custpr. -T age collision properties are used. For this rea-simulation time for a colliding case is approx- son, the dependence of the sticking probabilityimately 100 ps, which is also long enough to on impact parameter must be averaged overobserve whether the outcome is a sticking or the collision cross section. The average stick-non-sticking collision. To get good statistics, ing probability, P~i, can be calculated from thewe perform 2500 trajectories for each colliding MD results for each cluster based on a b2 dis-case by randomly rotating the target cluster, tribution of collision pairs,as discussed in Sec. 2.3.

It can be expected that cluster-monomer 1 b•'m

sticking probability is also dependent on the P -i J_ I P(b)db2 (5)relative collision velocity, especially for small i,m 0clusters. Similarly to the collision probability, where b is the cluster maximum impact pa-the impact of the relative velocity on the stick- wher or ision radiu m iven pa-ingproabiit dereaesas the cluster size in- rameter or collision radius, given by Eq. (2).ing probability decreases the cD si zeain- Figure 7 shows the averaged sticking proba-crease. Figure 5 shows the MD simulation re- biltPiasafnioofcuerizfrasuits of sticking probabilities for various cluster blity, P• as a function of cluster size for asizes and relative velocities as a function of im- relative collision velocity of 182m/s. It can be

seen that for cluster size smaller than 10, thepact parameter. This figure indicates that the sticking collision probability increases quicklysensitivity of the sticking probability decreases sthe cllisize increases quiclyas te custe sie inreaes, nd t ma be as the cluster size increases; while for clusterneglected for the clusters larger than 10-mers size larger than 10, the sticking collision proba-

with good approximation. To highlight the size bility is independent of cluster size and is verydependence of the sticking probability, Fig. 6 close to unity.Therefore, the MD simulation results show

'A static state indicates one with no translational that for clusters larger than 10-mers, cluster-velocity, and only thermal motion or internal energy. monomer interactions may be modeled approx-

5

imately with the hard sphere model, with a section radius can be expressed as:constant sticking coefficient of unity. The MD epsimulation study confirms the selection of these ri + r1 Oc af (E, m) Oc -c (6)models utilized in earlier work [81- mq

where ri is the radius of a cluster consisting of

i monomers, r, is the monomer's radius, a andc are the Lennard-Jones parameters, m is the

2.5 Comparison of MD results molecular mass, and p and q are constants.The nickel cluster cross sections are simu-

for different Lennard-Jones lated in Ref. [11] with two Lennard-Jones po-

Systems tentials, the bulk fitted (LJB) and dimer fit-ted (LJD) potential. The simulation results

Collision processes between argon clusters and for clusters of size from dimer to 13-mers aremolecules have been simulated in the previous given in Ref. [11], e.g. a 12-mers cluster crosssections, and important results of cluster size section is about 250 A 2 for the LJB potential,and sticking coefficients are shown in Fig. 2 and and 200 A 2 for the LJD potential. The con-Fig. 7 respectively. Since Lennard-Jones po- stant p in Eq. (6) can be estimated by compar-tential is generally used to describe molecular ing those two simulated 12-mers cross sectionsinteraction for many other species, the simu- and was found to be approximately 0.18. Ex-lated results for Argon may be able to roughly amination of Fig. 2 shows that, the 12-mers ar-estimate other species without conducting nu- gon cluster cross section is approximately 363merically identical simulations using various A 2 which compared with the Nickel clusterparameters. To validate our MD simulations, computed with the LJB potential, allows onewe would like to compare our numerical results to estimate a q, value of approximately 1.56.with other independent data [11]. By using Eq. (6), the Argon cluster colli-

Nickel cluster collision cross sections and sion cross sections can be used to predict nickel

sticking collision process has been simulated cluster collision cross sections. A comparison

by the MD method in the Ref. [11], using a of the Ar derived predicted Ni cross sections

Lennard-Jones potential. First we may di- with those obtained in Ref. [11] is shown in

rectly compare Ni and Ar cluster sticking prob- Fig. 8. It can be seen that the argon cross sec-

abilities, which is a nondimensional parameter. tions can predict nickel cluster cross sections

Note that the nickel cluster sticking probabil- well for both Ni interaction potentials. Thus,ity is defined as the ratio of the cluster stick- the Argon cluster sticking probability and col-

ing cross section to the total cross section and lision cross section are consistent with those

both quantities are given in Ref. [17]. The of nickel clusters, and these results may be ex-

comparison of Argon and Nickel cluster stick- tended in the future to approximately estimate

ing probability is shown in Fig. 7, represented other species described by theLennard-Jones

by open circles and solid triangles respectively, potential, such as water.

Figure 7 suggests that the trends for small clus-ter sticking probability discussed in Sec. 2.4 is 3 Microscopic Modelssimilar for both argon and nickel clusters.

To compare cluster collision cross sections Based on the general conservation equations,for different species, we have to use reduced the developed microscopic models had been in-units to analyze the MD simulation results of tegrated into a DSMC computational schemeRef. [17]. We assume that the cluster cross to predict homogeneous condensation in a

6

rocket plume. Detail descriptions of these as:models can be found in the previous work [8]. / 2a ½ p2 ( 47rra.rTo predict Argon condensation in a free ex- J = -- exp (7)panding plume in Sec. 4, we briefly review \ 33 p1 3-mBT

these models, and discuss the improvements in where ao is the cluster surface tension, m isdetail as follows. molecular mass, Pv is vapor density, pw is clus-

ter density, and r. is the local critical clusterradius. Thus, during a time step At in a cell

3.1 Simulated Molecular Models with volume AV, the number of new simulatednuclei Nc is calculated as

To simulate condensation coupled flow, all

species including monomers and clusters are N = JCAVAt/F, (8)represented by simulated molecules in theDSMC approach. The variable hard sphere where Fc is the number of real molecules

(VHS) model is used to model simulated represented by a simulated cluster molecule.

molecules representing real atoms or molecules The latent heat, generated during the nucle-

in the plume, and the Larsen-Borgnakke model ation process, is distributed to the ambient gas

is used to redistribute molecule-molecule colli- molecules in a cell according to energy equipar-

sion energy, based on the number of degrees of tition principle. Using the bulk theory, the la-

freedom. Note that both rotation and vibra- tent heat, EL, is given by

tion energy transfer are considered in molecule- EL = LM, (9)molecule collisions.

An analytic cluster hard sphere model is de- where L is the specific latent heat of evapora-scribed in Ref. [16], providing a general asymp- tion, and M, is the mass of a cluster consistingtotical equation for cluster radius similar to of i monomers.Eq. (3). Briehl et al [19] applied this clustermodel to simulate small Cu cluster growth and 3.3 Collision Modelsdecay processes using the DSMC approach.This cluster model has been validated by our The collision process between a cluster and aMD simulations for Argon cluster, discussed in foreign molecule is referred to as non-stickingSec. 2.2. It was further shown that Argon clus- collisions, while the outcome of a collision be-ter model is consistent with other independent tween a cluster and its own molecule maystudies of Nickel clusters [11] in Sec. 2.5. Thus, be sticking, or non-sticking, depending on thewe will use the hard sphere cluster model in our sticking collision probability. Based on molec-DSMC simulation. ular dynamics simulations, a general expres-

sion for both argon and water homogeneoussticking coefficient, q, has been summarized

3.2 Nucleation Model as[211

Following CNT theory, initial clusters, called (I V1 ( o (10)nuclei, are introduced into the computational q 1- KBTdomain. The nuclei have the same size as the

local critical clusters, and they are in an equi- where VL and V9 are the specific volumes of gaslibrium state with the surrounding gas. The and liquid and E0 is the energy difference of thenucleation rate, J, is given in the CNT theory minimum quantum level between the activated

7

complex and gas. Equation (10) shows that want to first compare the improved DSMCsticking coefficient is close to unity in the low simulation results with previous results.temperature region and decreases as the tem- A free argon expanding flow through a sonicperature increases. As shown in our MD simu- nozzle with a convergent angle of 30 degreeslations, sticking coefficient for clusters smaller into a vacuum environment was simulated inthan ll-mers may not agree with the predic- Ref. [8]. The plume has a stagnation pres-tion of Eq. (10) since it is based on the con- sure of 5.30 x 104 Pa, and a stagnation tem-dition of a flat liquid surface. The impact of perature of 180 K. The DSMC simulation pa-correctly modeling the sticking probability of rameters and techniques for this case were de-clusters smaller than ll-mers on the conden- scribed in detail in Ref. [8]. Comparisonssation results will be discussed in Sec. 4. between improved (new) and previous results

In the previous work, cluster-cluster colli- (old) are shown in Fig. 9. Since the MD cor-sions are assumed to be elastic collision, with- rected cluster-monomer collision cross sectionsout considering the coalescence effect. In this are larger than the previous ones, the compar-work, the semiempirical Ashgriz-Poo model[22] ison of average cluster size, shown in the topis chosen to predict the outcome of cluster- of the figure, suggests that the cluster size ob-cluster collisions that may lead to coalescence tained by the improved model is larger thanor separation. The Weber number, defined as the previous size. However, after the clusterthe ratio of collision kinetic energy to droplet size is normalized to the corresponding termi-surface energy, provides a cluster-cluster col- nal cluster size, the middle figure of Fig. 9lision separation-coalescence boundary. The shows that the normalized curves of the im-Weber number, We, is given by: proved and previous models are almost the

same. The middle figure suggests that the im-

We =pddc (11) proved model still validates the scaling lawsad of Hagena [4], as was shown in the previous

where Pd is the cluster density, di is the diam- work [8]. Finally, the cluster number densities

eter of the smaller cluster, c, is the cluster- are compared in the bottom of Fig. 9 and the

cluster relative velocity, and Ud is the clus- results suggest that the cluster number densityobtained by the improved model is slightly lesster surface tension. The Ashgriz-Poo model yg~

has been verified for water droplets by MD than the previous value. The main reason of

simulations using a 6-12 Lennard-Jones po- the decreased cluster number density is that

tential, and was used to predict the outcome small clusters in the improved model have had

of droplet collisions in an inductively coupled more chance to evaporate into separate atoms

plasma [23]. The detail description of the due to relative smaller sticking coefficients.

Ashgriz-Poo model can be found in Ref. [22]. Next, the improved condensation modelis applied to a new expanding condensationplume experimentally studied in the Arnold

4 Results and Discussions Engineering Development Center (AEDC) inthe 1970's [15]. Extensive sets of Rayleigh

The MD simulation results, cluster collision scattering intensity data were measured, whichcross sections and cluster-monomer sticking co- may be directly compared with numerical re-efficients, will be integrated into the DSMC sults. A brief summary of the scattering inten-simulations. To understand the impact of sity is given below and detail information canthese improvements and check the simulation be found in Ref. [15].results presented in the previous work [8], we Assuming that the condensing flow field

8

is composed of a collection of gas phase hours on 18 parallel AMD Athlon processorsmonomers and i-mers with the respective num- of CPU 1526 MHz.ber density Ni and polarizability ac, the ratio The DSMC simulation results for the con-of scattered intensity, I, to the incident inten- densation plume are shown in Fig. 10, includ-sity, 10, is given by: ing vapor number density contours (top), clus-

=o KNa 2 /A' (12) ter number density contours (middle), and av-erage cluster size contours (bottom). It can be

where K is a coefficient containing transmis- seen that the initial cluster or nuclei appear in

sion and calibration factors, and A0 is the wave- a nucleation region where cluster number den-

length. Note that polarizability ai for i-mers sity increases quickly along the flow directionis assumed to be i times larger than molecular consistent with the increased degree of super-

polarizability a, [15]. saturation in the free expanding process. Up-Here we will use the DSMC method to simu- stream of this region, the gas is essentially an

late condensation in a free expanding pure ar- expanding plume without condensation. Fur-

gon plume through a sonic orifice, which had ther downstream of the nucleation region, clus-experimentally done in AEDC [15] with a stag- ter number density decreases mainly due to thenation pressure of 250 torr, stagnation temper- expansion. Since the cluster number density

ature of 280 K and orifice diameter of 3.2 mm. is only on the order of 1017 per M 3 , there areThe details of DSMC simulation models and few cluster-cluster collisions so that the impor-

techniques are developed in Ref. [8], which can tance of the coalescence process on the cluster

be briefly summarized as follows. The origi- number density is limited. The average clusternal 2D axisymmetric SMILE code [24] is mod- size is usually below 10-mers in the nucleation

ified to simulate nucleation, cluster-molecule region, while it increases downstream of the

and cluster-cluster collision, and evaporation nucleation region due to the sticking collision

processes. The simulation domain extends 4 process between cluster and vapor molecules

and 10 times the orifice diameter in the radial and cluster evaporation processes.

and axial directions, with 100 and 250 cells re- The distributions of average cluster size,spectively. Each cell can be divided into up to cluster and vapor number density along the16 subcells according to the flow gradients ob- core flow are shown in Fig. 11. The figuretained during the simulation, and a time step shows that vapor number density is about 5of 2.0 x 10-1 s is used. To decrease the compu- orders of magnitude greater than the clus-tational cost, radial weights are used for dis- ter number density, while the maximum clus-tributing the number of simulated molecules ter size in the computational domain is aboutevenly in the radial direction for monomers 300-mers. The average cluster size seems toand clusters. A cluster weighting factor of grow continuously beyond the computational1.0 x 10-5 is used in the simulation due to the domain until the vapor number density be-several orders of magnitude difference in the comes so low that the contribution of stickingcluster and vapor number density. Since the collisions to cluster growth process counteractsnon-condensation region close to the orifice is the cluster evaporation effect. The Rayleightoo dense to be modeled by DSMC, the simu- scattering intensity is calculated along the corelation begins from a starting surface which is flow using Eq. (12), and is compared with ex-created with the continuum solver GASP [25]. perimental data in Fig. 12. Note that theAbout 3,200,000 simulated molecules are used dashed line in Fig. 12 indicates the changeto represent clusters and vapor molecules at in Rayleigh scattering intensity along the coresteady state. A typical simulation takes 72 of a plume for which condensation is ignored.

9

Upstream of the nucleation region, there is 5 Summaryno condensation in the plume which leads toa Rayleigh scattering intensity decrease along The general molecular dynamics (MD) ap-

the flow direction due to rarefaction. How- proach is used to verify an analytical clus-

ever, the emergence of clusters and the cluster ter model, and cluster sticking collision model.

growth process counteracts the decrease of va- The semiempirical Ashriz-Poo model is used to

por number density in the condensation region, predict the outcome of cluster-cluster collision.

leading to a scattering intensity downstream The use of LJ potential reduced units shows

of the nucleation region that is larger than its that MD simulation results of Argon cluster

corresponding value in the non-condensation agree well with those of Nickel cluster done in

plume. Figure 12 also shows that our numer- an other independent work, indicating that the

ical results agree reasonably well with the ex- MD results in this work may be approximately

perimental data, thus, we expect the DSMC transferred to other species.

condensation model may predict the conden- The MD simulation results are applied into

sation phenomenon in a free expanding plume the DSMC model, which is developed to sim-

correctly. Note that the appearance of nuclei ulate homogeneous condensation in a free ex-

in the DSMC simulation is delayed 1.2 x 10-i panding Argon plume through a sonic orifice.

sec to make the initiation of condensation in According to the DSMC simulation results,the simulation coincide with the onset of con- Rayleigh scattering intensity is estimated and

densation at the position indicated in the ex- compared with the experimental data suggest-

periment. ing that the condensation model developed inthis work may reasonably well predict homoge-neous condensation in a free expanding plume.The Rayleigh scattering contours in a conden-sation plume are predicted, which may need tobe validated by experimental data in the fu-

Figure 13 compares the Rayleigh scatter- ture.

ing intensity contours between a condensation(top) and non-condensation (bottom) plume. 6 AcknowledgementsFor the non-condensation case, the plume scat-tering intensity decreases monotonically in the The authors wish to express their gratitude toexpanding directions, due to the decrease in Professor M.S. Ivanov for inspiring this studythe vapor number density. However, for the and to Professor L.V. Zhigilei for providingcondensation plume, the scattering intensity the computer code. This work was supporteddecreases more slowly in the axial direction by the Air Force Office of Scientific Researchafter the initiation of condensation. Consis- Grant No. F49620-02-1-0104 administered bytent with the results shown in Fig. 11, there is Dr. Mitat Birkan and the Army Research Of-a counteracting effect between the increase of fice Grant No. DAAD19-02-1-0196, adminis-the cluster size and the decrease of the cluster tered by Dr. David Mann.number density. Our numerical simulation re-sults show that there would be a long tail inthe Rayleigh scattering intensity distribution Referencesin the free expansion condensation flow. Theexact spatial distribution should be further val- [1] Farid F. Abraham, Homogeneous Nucle-idated by future experiments. ation Theory, Academic Press, 1974, 263

10

pages. pansion and Condensation in SupersonicJets", AIAA paper 2004-2587, Jun. 2004.

[2] J. W. L. Lewis and W. D. Williams, "

Profile of an Anisentropic Nitrogen Nozzle [10] M. P. Allen, and D. J. Tildesley, Corn-

Expansion", The physics of Fluids, Vol. puter Simulation of Liquids, Oxford Uni-

19, No. 7, July 1976, pp. 951-959. versity Press, 1987. 385 pages.

[3] M. S. Ivanov, G. N. Markelov, Yu. 1. [11] R. Venkatesh, W. H. Marlow, R. R. Luc-

Gerasimov, A. N. Krylov, L. V. Michina, chese, and J. Schulte, "The effect of the

and E. I. Sokolov, " Free-Flight Experi- nature of the interaction potential on clus-

ment and Numerical Simulation for Cold ter reaction rates", J. Chem. Phys., Vol.

Thruster Plume", Journal of Propulsion 104, No. 22, pp. 9016-9026, Jun. 1996.

and Power, Vol. 15, No. 3, May-June, [12] Michael I. Zeifman, Jiaqiang Zhong, and1999. Deborah A. Levin, "A hybrid MD-DSMC

Approach to Direct Simulation of Conden-[4] 0. F. Hagena, and W. Obert, "Cluster sation in a Supersonic Jets", AIAA paper

Formation in Expanding Supersonic Jets:

Effect of Pressure, Temperature, Nozzle 2004-2586, Jun. 2004.

Size, and Test Gas", the Journal of Chem- [13] R. Soto, and P. Cordero, " Cluster Birth-ical Physics, Vol. 56, No. 5, 1972. death Processes in a Vapor at Equilib-

rium", J. Chem. Phys., Vol. 110, No. 15,[5] W. D. Williams and J. W. L. Lewis, pp. 7316-7325, Apr. 1999.

"Summary Report for the CONTESTProgram at AEDC", AEDC-TR-80-16, [14] V. Bernshtein, and I. Oref, " Dependence

Sept. 1980. of Collision Lifetimes on Translational En-ergy", J. Phys. Chem. A, Vol. 105, No. 14,

[6] E. R. Perrel, W. D. Erickson, and G. pp. 3454-3457, 2001.V. Candler, " Numerical Simulation ofNonequilibrium Condensation in a Hyper- [15] J. W. L. Lewis and W. D. Williams, "Ar-sonic Wind Tunnel ", JSME International gon Condensation in Free-jet Expansion",Journal, Series B, Vol. 39, No.2, pp. 277- AEDC-TR-74-32, July, 1974.283, Apr.-Jun. 1996. [16] H. Muller, H. G. Fritsche, and L. Skala,

" Analytic Cluster Models and Interpola-[7] G. A. Bird, Molecular Gas Dynamics tion Formulae for Cluster Properties", in

and the Direct Simulation of Gas Flows, Clusters of Atoms and Molecules I: The-

Oxford Science Publications, 1994. 458ory, Experiment, and Clusters of Atoms,

pages. edited by H. Haberland, pp. 115-138,

[8] Jiaqiang Zhong, Sergey F. Gimelshein, Springer-Verlag, 1995, 422 pages.

Michael I. Zeifman, and Deborah A. [17] Ian M. Withers, "A Computer SimulationLevin, "Modeling of Homogeneous Con- Study of Tilted Smectic Mesophases",densation in Supersonic Plumes with the Sheffield Hallam University, Ph.D thesis,DSMC Method", AIAA paper 2004-0166, 2000.Jan. 2004.

[18] C. M. Benson, J. Zhong, S. F. Gimelshein,[91 Quanhua Sun, lain D. Boyd, and Kenneth D. A. Levin and A. Montaser, " Simula-

E. Tatum, "Particle Simulation of Gas Ex- tion of Droplet Heating and Desolvation

11

in Inductively coupled Plasma - Part II:Coalescence in the Plasma ", Spectrochim-ica Acta - Part B Atomic Spectroscopy,Vol. 58, n 8, page 1453-1471, Aug. 2003.

[19] B. Briehl and H. M. Urbassek, " MonteCarlo Simulation of Growth and DecayProcesses in a Cluster Aggregation Source", Journal of Cacuum Science & Technol-ogy A, Vol. 17, n 1, page 256, Jan.-Feb.1999.

[201 G. K. Schenter, S. M. Kathmann, and B.C. Garrett, " Dynamical Benchmarks ofthe Nucleation Kinetics of Water ", Jour-nal of Chemical Physics, Vol. 116, n 10,pp. 4275-4280, Mar. 2002.

[21] Gyoko Nagayama, Takaharu Tsuruta " Ageneral expression for the condensationcoefficient based on transition state the-ory and molecular dynamics simulation",Journal of Chemical Physics, Vol. 118, n3, pp. 1392-1399, Mar. 2003.

[22] Ashgriz, N. and Poo, J. Y., "Coalescenceand Separation in Binary Collisions ofLiquid Drops," Journal of Fluid Mechan-ics, Vol. 221, 1990, pp. 183-204.

[23] Benson, C. M. "An Advanced Modelfor the Determination of Aerosol DropletLifetime in an Inductively CoupledPlasma", Ph.D Dissertation, Dept. ofChemistry, George Washington Univer-sity, 2002.

[24] M. S. Ivanov, G. N. Markelov, and S. F.Gimelshein, "Statistical simulation of re-active rarefied flows: numerical approachand application", AIAA Paper 98-2669,June 1998.

[25] The General Aerodynamic SimulationProgram, GASP version 4.1, Computa-tional Flow Analysis Software for theScientist and Engineer, User's Manual,Aerosoft, Inc. Virginia.

12

-A-- 2 - mem

4 -Amo--- 7-mers0.8~~ b- e-- 6Mors 0.8 -e-- 8-m1em'

H -0----9-mers

e e0.

0.

U.4 .0.40

0~ 0.4'

7( ým9 10impact Parameter (A) Impact Parameter (A)

Cluster Size-A- 18 -meom

0.8 --- 10-mere . --- 24 - mem0.8 e I I-mar- -- 27 -mere1---A-- 32 -meom

2 J2

0S 0

0 0

Impact Parameter (A) Impact Parameter (A)

Cluster Size 8me

0.8--- 4e - mare .

50 - mar

J2 A2.

0 .4oo -

r-4

I

U

AAA 41 16 1 0 2 4 2

Impact Parameter (A) Impact Parameter (A)

Figure 1: Comparison of collision probability for some clusters from dimer to 482-mers under thecondition of various impact parameters, with a relative collision velocity of 182m/s.

13

30

A MD Simulation A---.-.-- Approximated Equation

25/

/

< Au 20 /

A(

I..15a)

10 -A

-A-~

5I ' ' 1 I I I I t llI I I , , , l

10, 101 103Cluster Size

Figure 2: Average cluster radius from MD simulations with a relative collision velocity of 182m/sis fitted to an approximated Equation, P• = 2.3i0 + 3.4,

14

Dimer- 82--- 158 Mds

-0.-A-- 223 MI/S 0.8 -s-(D 204 nI/SX~k 9 -- A---223 MI/S

e 2

CL IL

0. -e-- 0nr/ 0.8

02 02

01 . .\it I

7L 078

Impact Parameter (A) Impact Parameter (A)

13-mers-A-- 158 MI/S 15MI

0. 2 8 I D 8 I

0 204 mx 0.20

Impa.At Paaee (A Impa3 Paamte (A)m

Figue 3:Comprisn ofcollsio proabilty or sme custes uder he cndiion2f3vaiou

impac paaetr an0eaiecliinvlcte.

15

16 - 147-mers,=0.07

141�- - m ers, = 0.08

.12- 32-mers, - =0.1

" 10

- 13-mers, u 0.24

SA- trimer, - = 0.306

dlim er, 1 -- 0.3-0 - .. . ..-

, . . I . I . I . I . ,160 180 200 220

Relative Velocity (m/s)

Figure 4: Cluster radius simulated from MD are fitted with equations: r -lr,(ef f c, )I•

16

Dimer0.2-Trimer

1-0------ 158 MIS0 24 wls-0'-- 182 m/ls

223 its---- 204 mis

~ o.~ -0---223 m~s

010-2 0

U)GE > O ey

0 5 10 150Impact Parameter (A) Impact Parameter (A)

0.25, -es055-mer 6-mers

0.2 ----- 182 m/s 0.4, -A-- 158 M/S-e---- 204 m/s G-.--e --- 182 m/s

Go0 oQ 90t- 223 m/s plc d 0 s- 204 m~sQ0 0

GO _ -- o-- 223 m/s

~0.05 0.1I ~'

0.05m/ 0.81

00 5 10 10 A 158 ~

< -~---e - 182 mls 0.618 /0~--- 204 m~s

I>I

0051010mpa4- Paaetr(A.ma2-armteA

Fiue5:Msiuainrslsostcigcliincefcetoprbbltefosoecues

0117

0.8 0.8 •••

0.6 0.6"

0.4 ,0.4

80.2 "0.2 -

- - m ,s0 -A-- 11-Mers --- i8-mers

e 12-Mers 24 ----. . -mers

-.... >--- 14 - Mers

, I, )

5 10 15 0 5 10 15Impact Parameter (A) Impact Parameter (A)

Figure 6: MD simulation results of sticking collision coefficient for clusters of different sizes as afunction of impact parameter, with a relative collision velocity of 182m/s.

100 7 ----- 0

6

V..

.00C 10"

/I..

U)

---- MD,ArgonV LJB, Nickel

1o0; , , , ,I , . . IS10 102

Cluster Size

Figure 7: Average cluster sticking probabilities, for argon clusters from dimer to 76-mers with arelative collision velocity at 182m/s, are indicated by open circles compared with nickel clustersticking probabilities as solid symbols.

18

300

1200

.2A

U)

o ioo A

I- ----- Predicted, LJBA A MD Simulation, UB

-.- 0-- - Predicted, LJD0 MD Simulation, LID

0 0 I . . I I . , , . I , . I I0 5 10 15

Cluster Size

Figure 8: Comparison of predicted nickel cluster cross section from simulated argon clusters usingEq. (6) with MD simulation results.

19

1000

.N

S800 Old

W 600

0_400

1250 200

0 0 5 10 15 20

1.N ~New -

0.8

U)S0.6

0 0.4.N

LE 0.2

oz /

0 5 10 15 20

1020

U) New

., 1019 - Old

I-)E

S10184)

1 o 1 7 . ..0 .. . .. . . . 1 , .. . .. . . . 2 ,10 0 5 10 15 20

Axial Position, X/R

Figure 9: Contours of Ar cluster size (top), normalized cluster size (middle), and cluster numberdensity (bottom) along the plume centerline between the improved (new) and previous (old)results. Number density is given in per mi3 , Axial position is normalized to the nozzle throatradius R.

20

4

Vapor8 2E+247 8E+23

5 2E+23

4 1 E+231,3 5E+22

2 3E+221 I E+22

2 4 6 8 10

4Cluster

2 6 1.5e+187 160e+18

24 4 4.5e+17" "9 3 .5e+1 7

5 3 0e+17--- 4 1 5.0e+16

Nuceaio Regioin_

2210S 9 24000• 48 200

1160"2 61 2 .9 120

•4 60' j 3 40S<j• 2 20.1 8

002 4 6 8 10Axial Position (X/D)

Figure 10: Contours of Ar vapor number density (top), cluster number density (middle) andaverage cluster size (bottom) in the condensation plume, discussed in Sec. 4. Number density isgiven in per m.

21

10"2 400

lA024AA- 300SVapor NumberDenstey

.~23 200

<10 2 1 A 100E U)

0 2 6 8 100

i o21 oClusterSize

1020

Elo"~' Custer Number Denstiy

101? 6 1

•.~ 10.

1010 I . . I , ' ' I , , ,0 2 4 6 8 10

Axial Position, X/D

Figure 11: The distribution of Argon gas and cluster number density, and average cluster sizealong the center line of the condensation plume, discussed in Sec. 4.

10-1

-10-2

--- Experiment--t--Simulation

c-t No.Condensation

10-4 1 I0 2 4 6 8 10

Axial Position, XID

Figure 12: Comparison of Rayleigh scattering intensity between experimental data and simulationresults along the plume center line. The dashed line indicates intensity of a flow solution withoutcondensation.

22

4

3

0 6'I.''

010 1.0E-01"ýi 0 . ....---- ............................... 9 8.0E-02

a. 8 4.OE-02

S-1 7 2.OE-02S 6 1.OE-02

5 6.OE-03-2 4 4.OE-03

3 2.OE-03S2 1.OE-03

1 1.OE-04-4

2 4 6 8 10Axial Position, X/D

Figure 13: Comparison of Rayleigh scattering intensity contours, I,/1o, between condensation(top) and plume without condensation (bottom).

23