Cluster beam sources: Predictions and limitations of the nucleation theory

44

Surface Science 156 (1985) 44-56 North-Holland, Amsterdam

CLUSTER BEAM THE ~CLEATION Gilbert

D. STEIN

Northwesiern

Received

university,

SOURCES: THEORY

PREDICTIONS

AND

LIMITATIONS

OF

* Molecular

13 July 1984; accepted

Beam Laboratory,

for publication

Evanston,

25 September

Illinois

60201,

USA

1984

The goal here is to ascertain the range of cluster properties and the limits of the validity of the nucleation theory for the prediction of the dynamics of formation of the condensed phase in highly underexpanded, supersonic free jet sources. These sources will be discussed, very briefly, in context to other cluster sources, with the special aim of comparing the nucleation phenomena. Results are presented for a variety of free jet sources applied to the condensation of metals at low pressures with or without carrier gas. The current controversy over whether or not free jet pure metal expansions can produce clusters in the 1000 atom range from starting pressures in the 1 to 10 Torr regime will be discussed.

1. injection The adiabatic expansion of an unsaturated gas in a free jet has been the most common method of producing clusters in supersonic molecular beam experiments. The mean size of the cluster distribution produced can be varied from dimer to 50000 atoms per cluster or more by controlling the inlet pressure p. (very high for large sizes), temperature T, (and thus saturation ratio), and free jet orifice diameter D. Use of a carrier gas to enhance the cluster nucleation provides additional parameters such as initial mole fraction x0, sound speed, and collision cross section. In addition, the use of small Lava1 nozzles as a means of controlling the rate of expansion is yet another means of varying the nucleation kinetics. All of these features have been simulated in the computer by solving the governing equations for compressible flow in conjunction with expressions for cluster nucleation rate and growth. The complete details of these condensing flows can then be predicted at any point in the expansion including the cluster size distribution. Because the vapor mean free path can become long compared

* Joint appointment in the Department of Materiais of Mechanical and Nuclear Engineering.

Science and Engineering

and the Department

G.D. Stein / Cluster beam sources

45

to the characteristic flow length in these expansions (i.e., Knudsen numbers Z.S1), and because there can be very high rates of change of ~erm~yna~c variables (e.g. cooling rates of 106 to 10” *C/s), the nucleation theory may not always be valid. Several kinds of sources have been used to nucleate cluster and small particle: (1) Evaporation in a low pressure (1 Torr) quiescent atmosphere [l]. (2) Evaporation in a low pressure (1 Ton) flowing atmosphere 121. (3) Wilson and diffusion cloud chambers 131. (4) Adiabatic cooling in a controlled expansion Lava1 nozzle 143. (5) Adiabatic cooling in an uncontrolled expansion free jet [.5]. I. I. Cloud chamber

Well designed cloud chambers (type 3 sources above) operate in regime I, table 1, and are characterized by well known thermodyna~c processes. The Wilson cloud chamber uses the reversible adiabatic (i.e. isentropic) expansion. The diffusional process to produce supersaturation in the diffusion cloud chamber is one Dimensions and carefully computed so that the thermodynamic state is precisely known. There is no convection or turbulent transport to complicate the process. Thus these two devices meet all the requirements for a quantitative nucleation instrument and have been employed as such for numerous investigations, since the turn of the century for the Wilson chamber and the mid 1950’s for the diffusion chamber [3]. 1.2. Evaporative sources This group of cluster and particles sources covers a wide variety of designs. They all use materials that are solid (could also be liquid) at room temperature.

Table 1 Cooling rate regimes Regime

Cooling rate dT/dt (“C/s)

Nucleation rate (clusters/cm3 s)

Remarks

I II III

CSXlO” 5x106 to5x10’ 5x1o’to5xlOs

J=J, JaJ% J = J,

No nucleation lag time J, c J,, but nearly equal J, fi: @(IO-‘) J,, lag time

IV

5x10s tosx10’

J = J,

J, -x J,,, lag time calculation

J not applicable

tenuous Relaxation or lag time too large for use of nucleation theory

calculation stilt valid

V

> 5x109

46

C.D. Stein / Cluster beam

SOWC~S

Metals are the most common species studied. They are subdivided here as quiescent (type 1) and flowing systems (type 2). Both have some kind of heater or oven to vaporize the material of interest. This vapor then mixes with an inert gas and cools as a consequence, thus producing the supersaturation necessary for cluster nucleation. In virtually all cases the cooling rate is low, i.e. within regime I of table 1. Thus the steady state nucleation theory would obtain. However, there is a problem in defining the thermodynamic state for the mixing process. Both type 1 and 2 particle sources can have either laminar or turbulent convection and mixing which is not a predictable process in space and time. Thus, although the steady state nucleation theory is germane, the thermodynamic state is unknown (e.g. pressure p, temperature T, mole fraction x, as a function of position and time). And since the nucleation rate expressions require these thermodynamic properties, they cannot be applied to the evaporative sources in general. 1.3. Shock tube driver section This is essentially a tube at high pressure (driver section) containing the vapor of interest unsaturated. A diaphragm is burst to a low pressure chamber and expansion waves enter the driver section, reflect off its end wall and initiate a secondary expansion of the gas. The time-dependent gas dynamics is well developed so that the thermodynamic state of the gas (mixture) is well known. Depending on the location of the observation station with respect to the diaphragm, the cooling rates are G 5 X 105”C/s. The steady state nucleation rate is valid at room densities and above (regime I of table 1). 1.4. Lava1 nozzle sources Supersonic flow in Lava1 nozzles, type 4 above, has been used to study particle or droplet nucleation since the extensive pioneering work of Stodola in 1927 [6]. For nozzle throat (i.e. minimum cross section) sizes of 1 cm diameter (or height) or larger, and continuum (i.e. collision dominated) flow, the cooling rate, for geometrically similar nozzles, scales inversely with the throat size. However, even for throats as small as 0.1 mm, the deviation from steady state nucleation theory is not large (i.e. regime II of table 1) [7]. Lava1 nozzles were first used as a molecular beam source in 1951 [8]. Subsequent use has been primarily as cluster sources [4] since the viscous boundary layer effects can be deleterious to the performance of unclustered molecular beams. These viscous effects often traverse all the way to the flow centerline for small (0.1 mm diameter) nozzles. resulting in exit Mach numbers (velocity divided by local sound speed i.e. M = u/a) far below their inviscid values [7]. Whereas this may be fatal for monomeric (unclustered) beams, it is still of tremendous advantage as a cluster source [4].

G. D. Stein / Cluster beam sources

41

In summary, Lava1 nozzles operated in continuum flow are excellent devices to study nucleation phenomena and to produce cluster beams. The nucleation theory obtains, with perhaps small corrections due to the so-called nucleation lag time (i.e. the time-dependent nucleation theory of regime II). However, due to viscous effects some gasdynamic measurements are necessary to predict the actual thermodynamic state within the nozzle in order to apply the nucleation rate theory.

2. Free jet expansions as cluster sources The simplest and most used source in molecular beams is the orifice free jet isentropic expansion, although it is most often operated as an unclustered source. (In fact one of its attractions, for those conducting molecular beam research, is that it does not readily produce clusters, an unwanted species for many investigations.) There has been some speculation that large clusters (N = 500 to 1000) could be generated in free jet sources at starting pressures in

--M ___ X* FREE JET

PO,T&%

Flow

X_

u,B 0

F-7

”

11 0

I

I

20

1

Mor x*

I

I

40

Fig. 1. Knudsen number ratio Kn* = Kn/K X* =x/D for y = 5/3 and 7/5. (Kn =X/D before expansion.)

I

I

60 n, is plotted as a function of Mach number M and is the local Knudsen number while Kn, is the value

48

G. D. Stein / Cluster beam sources

the range p,, = 1 to 10 Torr. Also other qualitative features on free jet nucleation have been speculated upon. It is the purpose of this paper to explore the predictions and limitations of the classical nucleation theory applied to both pure vapor and vapor-carrier-gas free jet expansions. Free jet expansions to very low pressure, i.e. pa/p, >> 1, referred to as underexpanded (see fig. l), has been the subject of many investigations published in the Rarefied Gas Dynamics [9] Series and was first solved, using the method of characteristics, in 1948 [lo]. This work showed the r?rost uniuersaf feature of these geometrically similar flows, i.e. that Mach number M is a unique function of dimensionless centerline distance from the orifice, x/D = x*, D being orifice diameter, for a given specific heat ratio y = c,/cV ( cP and cy are the gas specific heats at constant pressure and volume respectively)*. 2.1. Free jet Knudsen number Knudsen number is defined as the atomic mean free path divided by diameter, Kn = X/D. The Knudsen number ratio Kn(x)/Kn, = Kn* is plotted both as a function of M and x/D = x* in fig. 1. Note there is a change in Kn* versus y for fixed M, while the difference with x* is minimal. Finally observe the tremendous increase in Kn along the flow direction, with a factor of lo4 in only 30 orifice diameters. It is this rapid drop in density or increase in Kn frequency (z = u,,,,nti) that (Kn = 1/2”2uC,,, nD) or decrease in collision accounts for the precipitous drop in the nucleation rate (see next section). Again note the universality of this figure. Here a,.,, is the collision cross section. n the atomic number density and fi the mean gas velocity. 2.2. Free jet cooling rate Another important feature of the free jet expansion is the rapid cooling rate of the gas (governed by the predominant mole fraction of the carrier species). For given initial conditions and hole size the free jet expanded to pressures < fpo represents the fastest possible adiabatic cooling rate. The rate scales with the hole size and the geometric nozzle contour in the case of solid wall Lava1 nozzles. In virtually all cases the maximum cooling rate occurs at or just downstream of the orifice or throat (i.e. minimum area), see fig. 5 of Abraham, Kim and Stein [7] for details.

* Respecting long standing convention the star * will be used to denote dimensionless variables. while the asterisk * is used for the gas-dynamic properties at Mach number = 1 and for the cluster properties at the so-called critical size in a supersaturated environment at which the cluster is neutrally stable. Dimensionless presentation of these particular types of flows is apropos due to the universality of M versus x* as mentioned above.

C.D. Stein / Clurter beam satmes

49

The maximum cooling rate for free jets as a function of orifice diameter, T, and the particular gas with its variable molecular weight, AWV, and specific heat ratio, y, are presented in fig. 2. These values were obtained from computer simulations including cluster nucleation and growth equations. For a given gas it is seen that the cooling rate indeed scales inversely with orifice size, i.e. three orders increase in D results in three orders of magnitude decrease in dT/dt. However, there is a stronger dependence on temperature as illustrated by the Ar carrier gas curves where an order of magnitude increase in Ta results in nearly two orders of ma~itude increase in dT/dt. Also note that for the smallest orifices used in molecular beams, cooling rates can exceed lO”‘C/s! These rates can only be sustained by very high collision rates. Thus, as the density drops rapidly, resulting in large Knudsen numbers, these rates fall off drastically as with any phenomena requiring collisions, and in particular nucleation rates. As a guideline the cooling rates have been divided, somewhat arbitrarily, into five-nucleation regimes-in fig. 2 and table 1. They are very typical of nucleation by adiabatic expansion, but are of course subject to the initial density p. (or number density no).

Fig. 2. The maximum free jet cooling rate dT/dt carrier gas.

varies with diameter D, To and the specific

G. D. Stein / Cluster beam sources

50

2.3. Free jet nucleation The free jet supersonic, compressible flow may be treated as similar to nozzle expansions in which the flow can be modeled in two ways: (1) Solution to the coupled set of equations of (i) continuity, (ii) momentum, (iii) energy, (iv) state, cluster, (v) nucleation and (vi) growth to solve for p, p. T, u and g (consisting of a size distribution f(N) with cluster concentration C,) - a set of 6 equations in 6 unknowns [ll]. Here u is velocity, g is the mass fraction condensed. (2) For the case where the latent heat release 4 = Lg +z c,T= h then the gas-dynamic equations (i)-(iv) (now isentropic) can be decoupled from the cluster formation and growth equations, (v) and (vi). However, we still have 6 equations in 6 unknowns. Here L is the latent heat and h is enthalpy. As alluded to above, the free jet expansion, especially from small orifices and low second stage pressures, can lead to additional complications due to a rapid drop in density. 2.4. A “typical” specific exumple

- gold in argon

We consider the case of Au, p,v = 1 Torr, expanded in an Ar carrier gas, pO = 1 bar and an orifice diameter D = 1 mm. As with all cases treated in this paper, the starting temperature is 50°C above saturation, 7;, = 7I, + 50. (This temperature criterion is typical of experimental requirements needed to avoid clogging.) The results are presented in fig. 3. The steady state nucleation I,, and mass fraction condensed ghs look typical of collision dominated Lava1 nozzles. However, on the rapid timescale of these free jets the rate is not high enough to generate sufficient new condensation sites and/or the growth rate is too low to condense a significant fraction of the flow (i.e. no onset of massive condensation, in contrast to larger, high density nozzle flows). Whereas in the Lava1 nozzle flows the nucleation is shut off by a fall in supersaturation, S. in these free jets S continues to increase, but the shut off occurs due to falling density. Secondly, one notices a substantial difference (Lo(l0) to cO(lO’)) between the time-dependent nucleation rate. J,, and J,,. The classical nucleation rate used here is that due to Volmer: J\, = ( p,./kT)'

(2c~/am)“~

u, exp( -AG*/kT).

where the thermodynamic properties are as given previously in the paper and p, is the condensable vapor pressure, k Boltzmann’s constant. u surface tension, ~1~volume per molecule in the condensed phase and AG* the critical size Gibbs free energy of formation. The time-dependent nucleation rate is related to the steady state rate by a nucleation lag time 7, which is a function

G.D. Stein / Cluster beam sm4)“ces

51

Au in Ar D=lmm PO,=1 Torr P,=lBar

’

b

--I,-. T

IO3 T-K and

-

10

\YM

t

S

S

M

lo2 f

B I

10’ 0

I

x’

2

I

I 4

IO.3 5

Fig. 3. The nucleation rate and mass fraction condensed are shown in (A) for the steady state and T and the the time-dependent cases versus x* for Au in Ar. (B) shows the drop in temperature rise in Mach number ,44 and supersaturation S with x*.

G. D. Stein / Cluster beam sources

52

of the thermodynamic

4 = 4,

state variables

(see ref. [7] and references

therein):

11- exp(-t/r, )I

t denotes time. (The correct nucleation rate is J, until it crosses the falling J\,, after which J,, is valid.) Thus, the actual mass fraction condensed g, -X g,, +C w0 _ the initial mass fraction of vapor (Au in this case). The lower half of fig. 3 shows M, T and S up to only x* = 5. Notice that in this relatively short distance the temperature has fallen from about 2000 to 75 K! 2.5. Comparison

of sodium, silicon, and gold in argon free jets

Fig. 4 emphasizes the significant reduction in J due to the nucleation lag time effect. A non-dimensional nucleation rate is defined where J* = J/J,,. Defined at the maximum J,, (see fig. 3) it is then denoted J,*,, and is plotted as a function of orifice diameter for poL = 1 Torr Au, p0 = 1 bar and T, = q + 50 as usual. At large sizes the correction is less than one order of magnitude and falls about linearly with D. The correction increases in the order Na, Si, Au. It may be noteworthy that the surface tension u, which has such a dramatic effect on the nucleation rate, increases in that same order.

Fig. 4. The nucleation rate ratio at the maximum in J_ (see fig. 3) is .I,*,, = J,/J,, and is presented as a function of D for Na. Si and Au. each with initial pressure of 1 Torr in Ar at 1 bar. The nucleation lag time calculation used to compute T” is J,. see eq. (2). is only an approximate theory. (See Abraham. Kim and Stein [7] for details.) Thus use of .I* -C10e2 would be quite suspect in the author’s judgement.

G.D.Stein / Cluster beam sources

53

When all is said and done what cluster concentrations (and sizes) are predicted? Is there enough to do an experiment? Is there enough to produce thin films? Fig. 5 shows the calculated cluster ~ncentrations at x’ = 12 and M = 15 in the free jet for pO= I bar argon, and Na, Si, and Au as the condensable vapors. In fig. 5a the concentration varies over 5 orders of magnitude for a two order of magnitude change in J+,“. Note also the decrease

Fig. 5. The theoretical cluster concentration at x* = 12 (x* = x/D) is computed as a function of pOu in (a). and diameter D in (b) for Na, Si and Au. The carrier gas is Ar in all cases except the dashed curve in (a) where He is used.

G.D. Stein / Cluster beam sources

54

in C, in the same order Na, Si, Au. A similar effect is seen for fixed poV = 1 Torr and variable diameter D. Clustering scales with pot, and D. Cluster sizes are not plotted here but are predicted in the size range N = 1 to 10 due to the extremely high supersaturations at the peak nucleation and thus the small starting (critical) size predicted from the Gibbs-ThompsonHelmholtz equation r * = 2au /kTlns where Us is the volume per atom in the condensed phase, and r* is ‘the so-called critical radius for formation of a stable nucleation site in the supersaturated vapor. 2.6. Effect of carrier gas on vapor nucleation It is well known in Lava1 nozzle cluster beam sources, that for a given partial pressure of condensable vapor, nucleation and growth can be greatly enhanced through use of a “non-condensable” carrier gas. There is no reason to expect that this effect will not obtain for free jet expansions as well. The effect has been observed dramatically in Na-Ar experiments [12] and is predicted theoretically here, albeit for Au-Ar expansions. One aspect of these results, the effect of initial mole fraction, x0, on cluster concentration, is presented in fig. 6 for poV = 1 Torr (Au) in Ar varying from 760 to 7.6 X 10-j Torr. There is nearly a linear drop in C, with p0 (or correspondingly a linear

Au

in

Ar

Pov = 1 Torr D=lmm

loo 10-3

1’

1’ 10.’

P;rsAr I 760

I

I 7.6

1’

’ X0 I

Po-Tow

10’

1

I

103

P”X I 7.6d

I

I 7.6&

Fig. 6. Cluster concentration at x* = 12 is shown as a function pressure pO. The Au pressure is held constant here.

of initial mole fraction

x0 and Ar

G.D. Stein / Cluster beam sources

55

increase in x0). Notice that even though D is relatively large as beam sources go (1 mm), and we know that C, increases with D, the predicted Au cluster concentrations (at M = 15 in the jet) are very low, particularly at low pO. Cluster sizes are also calculated to be quite small with N < 10 atoms/cluster, mean size. Although our intuition. based on years of nucleation studies, and with others conducting mass spectrometric experiments with metal free jet sources having pov = 1 to 10 Torr, has indicated the presence of small cluster sizes, the explicit nucleation calculation presented here substantiates these notions. It seems most unlikely that mean cluster sizes in the range of N = 500 to 1000 can occur in a 1 mm free jet source, especially with pure vapor expansions. However, experience has taught us to be open-minded enough to allow for the possibility of totally unexpected results, but so far the overwhelming evidence is for small cluster sizes in low pressure free jet sources. (Note that if To is lowered toward the saturation value, T, -B T,,,, nucleation will be enhanced, if clogging can be avoided.) 3. Summary One can briefly sum up the results of these calculations. (1) To apply the nucleation theory to condensing vapor one requires that the thermodynamic state be known and in equilibrium except for phase. (2) One also requires that the rate of change of state (e.g. dT/dt) be slow enough to both maintain thermodynamic equilibrium and a quasi-equilibrium cluster distribution up to the critical size for nucleation. Here J = J,,, the steady state rate - no nucleation lag time. (3) For faster expansion rates (see table 1) J = J,, the time-dependent rate, determined using a nucleation lag time calculation (see e.g. refs. 25-30 of Abraham, Kim and Stein [7]). (4) For extremely high rates of change J Z J,, Z J, and no nucleation theory is valid. Here the collision rate may not be high enough to maintain thermodynamic equilibrium and is definitely not high enough to preserve the quasi-equilibrium cluster distribution, due to a rapid fall off in density (Knudsen number becoming B 1). (5) Free jets with low vapor pressures yield low cluster concentrations and small sizes relative to phase changes characterized by a “massive” onset of nucleation and a significant fraction of vapor condensed. (6) Due to the extremely high supersaturations calculated for these free jet sources the starting or critical sizes for the nucleation theory are sometimes unphysically small. (7) For item (4) above, i.e. extremely rapid changes of state, these cluster nucleation, growth and stability considerations should be modeled with molecular dynamics and/or Monte Carlo simulations to provide further insight into this phenomenon.

56

G.D. Stein / Cluster beam

sources

(8) Even though these free jet sources do not, apparently, produce massive nucleation and cluster growth, they are nevertheless a very important device for the study of cluster properties and the production of matrix isolatable species and even new thin film materials.

Acknowledgements the author is most grateful for the partial financial support of the Chemistry Division of the U.S. Office of Naval Research and the Engineering Energetics Program of the U.S. National Science Foundation. He is also very appreciative of the computer programming skills of three undergraduate work-study students, David Schuller, Arthur Molin and Bradford Friedman.

References [l] C.G. Granqvist and R.A. Bhurman, J. Appl. Phys. 47 (1976) 2200; N. Wada, Japan. J. Appl. Phys. 6 (1967) 553. [2] K. Sakurai, S.E. Johnson and H.P. Broida, J. Chem. Phys. 52 (1970) 1625: A. Yokozeki and G.D. Stein, J. Appl. Phys. 49 (1978) 2224; K. Sattler, in: Proc. 13th Intern. Symp. on Rarefied Gas Dynamics, Novosibirsk, 1982. [3] C.T.R. Wilson, Phil. Trans. Al89 (1897) 265; J.P. Franck and H.G. Hertz, Z. Physik 143 (1956) 559. [4] O.F. Hagena, in: Molecular Beams and Low Density Gas Dynamics, Ed. P.P. Wegener (Dekker, New York, 1974); 0. Abraham, J.H. Binn, B.G. DeBoer and G.D. Stein, Phys. Fluids 24 (1981) 1017. [5] J. Farges, J. Crystal Growth 31 (1975) 79; S.B. Ryali and J.B. Fenn, Ber. Bunsenges. Physik. Chem. 88 (1984) 245. [6] A. Stodola, Steam and Gas Turbines (McGraw-Hill, New York, 1927). Vol. I, pp. 117-128; Vol. II, pp. 1039-1073. [7] 0. Abraham, S.S. Kim and G.D. Stein, J. Chem. Phys. 75 (1981) 402; S.S. Kim, D.C. Shi and G.D. Stein, in: Proc. 12th Intern. Symp. on Rarefied Gas Dynamics, Ed. S.S. Fisher (Am. Inst. Aeronaut. and Astronaut., New York, 1981). [8] A. Kantrowitz and J. Grey, Rev. Sci. Instr. 22 (1951) 328; G.B. Kistiakowsky and W.P. Slichter, Rev. Sci. Instr. 22 (1951) 333. [9] H. Ashkenas and F.S. Sherman, in: 4th Rarefied Gas Dynamics, Vol. 2 (1964) p. 84. [lo] P.L. Owen and C.K. Thornhill, Aeronautical Res. Council, Great Britain, R&M 2616 (1948). [ll] G.D. Stein and C.A. Moses, J. Colloid Interface Sci. 39 (1972) 504. (121 G. Delacretaz. G.D. Stein and L. Waste, to be submitted.