J. Phys. Chem. 1986, 90, 2302-2305
2302
TABLE I: Comparison of Solutions for Diffusion into Cubic Particles tanh (6.7703{(Dt/~')'/~1) vs. General Solution (Dt/w2)'1*
0.001 0.036 0.056 0.076 0.096 0. I26 0.166
0.186 0.226 0.256 0.296
general solution
tanh
0.006 0.224 0.333 0.43 1 0.519 0.633 0.755 0.804 0.88 1 0.923 0.960
0.006 0.239 0.361 0.473 0.571 0.692 0.808 0.850 0.910 0.939 0.964
ratio (tanh/general) 1.001 1.064 1.086 1.097
1.100 1.093 1.070 1.057 1.032 1.017 1.004
vs. In the preceding application, experimental values of time were available. Generally, values of weight gain (or loss) vs. time at constant temperature and pressure are obtained experimentally. Typical data are shown in Figure 2. Equation 3
was used to fit the data; values of B at the different tempeatures yield a calculated activation energy of 4.21 0.12 kcal/mol which agrees well with the reported value.5 In addition, values of Qm as a function of temperature decrease with an increase in temperature as expected. Furthermore, the calculated values of Q m are obtained from relatively short-time experimental data whereas, experimentally, much longer time periods would be needed to obtain reasonable equilibrium values. In conclusion, the relationship
Qt= Qm tanh [B(t'i2- A ) ] provides an alternative method for calculating a diffusion parameter, B, and the limiting amount sorbed, Qm,for processes at constant temperature and pressure. A single, constant diffusion parameter is sufficient to describe the entire range of data, i.e. at short and long times. (5) Hill, S. G.; Seddon, D. Zeolites 1985, 5, 173.
Thermal Collision Rate Densitles of Small Clusterst William H. Marlowl Department of Applied Science, Brookhaven National Laboratory, Upton, New York 11973 (Received: January 28, 1986)
Model calculations of the thermal collision rate densities between C02 molecular clusters consisting of 1 to 13 monomers are presented. Cluster morphologies are chosen according to the minimal-energy configurations of Hoare and Pal and the attractive part of the intercluster potential energy is calculated by summing dipole-induced dipole interactions to all orders. To model the collision rate density for each pair of clusters, an average is taken over the rates for four orientations computed in body-fixed coordinates. Results of these calculations have implications for the interpretation of experimental results on nucleation, cluster growth, and cluster mass-frequency distributions.
Gas-phase atomic and molecular clusters are important in numerous areas of science and technology. They are involved in vaporization, deposition, and nozzle-beam expansions and constitute an important stage in the formation of condensed matter. Collisions lead to the formation and evolution of clusters in both critical',2 and activationless nucleation3 and in heterogeneous growth processes. In deriving classical nucleation theory, the cluster collision rate density is initially assumed to have a simple form independent of composition and related to morphology only via cluster surface area. Treatments of cluster coagulational growth4 assume collisions occur according to a liquid drop model (with unit sticking probability) that incorporates no compositional or morphological characteristics of the colliding species. Despite their apparant successes in accounting for monomer-dominated ,~ theory of nucleation), processes ( C 0 2cluster g r ~ w t hclassical liquid drop and spherical, square-well potential models raise several questions for both monomer and multimer collision rate densities because such models are known to be inadequate for molecular collisions. Current practice gives no guidance as to why these methods are successful, how far they can be extended, or when they should fail. In addition, these methods give no hint of how the collision rate densities involved in cluster coagulational evolution are related to those for free-molecule aerosol coagulational evolution where significant size and composition dependences of 'A preliminary version of this work was presented at the First International Aerosol Conference, Sept 17-21, 1984, Minneapolis, MN. f Address for Sept 1985-Aug 1986: Environmental Engineering Division, Civil Engineering Department, Texas A&M University, College Station, TX 77843-3136. Permanent address as of Sept 1, 1986: Department of Nuclear Engineering and Applied Science, Texas A & M University, College Station, TX 77843-3133.
0022-3654/86/2090-2302$01.50/0
the aerosol particle coagulation rate densities have been identified both theoreticallySand experimentally.6 This Letter will show that including intercluster attractive energies and cluster morphologies into a cluster collision model can help address these questions and the details of cluster size distributions as well as provide physical insight elsewhere. While the system of reference for this sutdy is C 0 2 cluster^,^ the objective is to derive general properties of the collision rate densities which are fundamental to describing all gas-phase cluster collisional growth processes. Therefore, the interaction potential energy used below for C 0 2 arises solely from the dipole-induced dipole interaction (the "van der Waals" interaction common to all matter). Inclusion of the quadrupole energy might sharpen the results for C 0 2 but would render questionable any general conclusions regarding other substances. The C 0 2 molecule is assumed to be effetively spherically symmetric, a not unreasonable assumption above the dimer dissociation temperature.' For the sake of definiteness, cluster morphologies are selected according to the maximally stable structures for spheres with pairwise interactions as determined by Hoare and Pal.* Molecular cen~~~
~~~~
~
(1) Abraham, F. F. Homogeneous Nucleation Theor); Academic Press New York, 1974. (2) McGraw, R.; Marlow, W. H. J . Chem. Phys. 1983, 78, 2542. (3) Friedlander, S. K. J . Colloid Interface Sci. 1978, 67, 388. (4) Soler, J. M.; Garcia, N.; Echt, 0.;Sattler, K ; Rechnagel, E Phys Rev Lett. 1982, 49, 1857. (5) Marlow, W. H. J. Chem. Phys. 1980, 73, 6288. (6) Okuyama, K.; Kousaka, Y . ;Hayashi, K. J. Colloid Interface Sei. 1984, 101. 98. (7) Etters, R. D.; Flurchick, K.; Pan, R. P. Chandrasekharan, V. J. Chem. Phys. 1981, 75, 929. ( 8 ) Hoare, M. R.; Pal, P. Adu. Phys. 1971, 20, 161.
0 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 11 1986 2303
Letters ter-of-mass separation at “contact”, s, is estimated to be 0.35 nm at the 50 K cluster temperature c h o ~ e n . ’ , ~ The * ~ ~ repulsive part of the interaction energy is assumed to be pairwise additive and proportional to rfI2 where rii = lri - rj( is center-of-mass separation between molecules i and j located respectively at ri and rj. Its constant of proportionality is determined by minimizing the potential
V(rij)= Vwii(2)(rij)- C/rij12
(1)
for rij = s and gives a value C = 1.1 X erg cml* which in ref 9 and 4.2X compares with the values of 6.6 X in ref 7. Here, Vco,l(2)(rij)is the collective, zero-point, dipolar interaction energy for two molecules discussed next. The attractive part of the cluster interaction energy AE was modeled as if it were due to a set of coupled, interacting dipole oscillators.” Since clusters represent the beginning of the transition from dissociated to condensed matter, we sought a description of the interaction energies of clusters that would be consistent with Lifshitz-van der Waals theory that has been successful in accounting for the long-range interaction potentials for condensed matter. Langbein has shown12how the continuum Lifshitz theoryi3 may be “derived” beginning with the iterated dipolar coupling of discrete oscillators, an approach that in the limit of a single pair of molecular oscillators reduces to the molecular van der Waals interaction. Expressed as the zero-point energy (Le. lim,+ AE, where t is the equilibrium temperature of intracting species), the total energy of the system of J K monomers, with J in one cluster and K in the other, is
+
where
Here, CJ+K(i[)is the matrix of dipole polarizability tensors of the rflolecules of clusters J and K at circular frequency w = i t and T is the matrix of dipolar coupling tensors which couples all the molecules of the two clusters” and is a function of their molecular coordinates. Subtracting the isolated-cluster self-energy terms from (PE,K)TOTleaves the long-range interaction energy between clusters K and J:
J
J
c
J+K
J+K
E X - + j=J+1 C X j-1 k = l r j k k=J+l P k
C -(4) rikI2
F”
CJand GKcontain only couplings within each distinct cluster, the indices 1 I i I J are for monomers in J , and J 1 5 i 5 J + K are for the monomers of K. For the purpose of this study, the relative positions of the molecules in each cluster were taken as fixed. If the molecules are allowed to rearrange under the many-body potential (e.g. eq 2), exactly as has been done under pair potentials,*J4 intracluster dynamics and structure could be modeled. In principle, molecular dynamics simulation modeling of cluster collisions14 could be performed utilizing the energy expression of the eq 2 with K = 1 in Newton’s equation of motion for each monomer in place of the usual sum over Lennard-Jones pair potentials. Energy accommodation and the resulting sticking probabilities could then
+
(9) Clifford, A. A.; Gray, P.; Platts, N. J . Chem. Soc., Faraday Trans. 1 1977, 73, 381.
(10) Mannik, L.; Stryland, J. C.; Welsh, H. L. Can. J . Phys. 1971, 49, 3056. (1 1) Langbein, D. Theory of Van der Waals Attraction; Springer-Verlag: Heidelberg, 1974; particularly pp 36-38. (12) Langbein, D. J . Phys. Chem. Solids 1971, 32, 133. (13) Lifshitz, E. M. Sou. Phys. JETP 1956, 2, 73. (14) Gay, J. G.; Berne, B. J. J . Colloid Interface Sci. 1986, 109, 90.
-
~
be estimated. If the t 0 limit is not taken, temperature-dependent effects on the potential” would be included. In addition, if the molecular polarizability tensor were not assumed to be spherically symmetric, the effects of relative orientations of the molecules could also be accommodated. The polarizability for C02used here was determined by fitting a Drude oscillator model to spectroscopic data at six frequencies in IRIS and UV’6,17and adjusting the oscillator strengths so the nm3 polarizability sum rule was satisfied with a(0) = 2.7 X (ref 18, Table 5). This procedure is to be compared with an alternative approach utilized by PackI9 who employed a higher nm3) and more extensive specstatic polarizability (2.9X troscopic data than in the current work. The resultant van der Waals C, coefficient Pack calculates is 1.83 X cm8 g/s2 while cm8 g/s . These that coefficient from this work is 1.21 X values are to be contrasted with C, values ranging from 0.58 X (ref 7) to 3.80 X cm8 g/s2 (ref 9) that appear in the literature. Unlike the aerosol cases, no accounting for retardation of the interaction energy need be made here. Consequently, the relative contributions of different spectral ranges do not change with separation which is why the molecular polarizability and the dipole coupling tensor are factorizable. Each collision rate density discussed below is derived from a set of component collision rate densities. A component rate is calcualted by first choosing an orientation for the colliding pair of clusters and then assuming that as they collide they rotate to maintain their original orientations relative to each other (i.e. they collide in body-fixed coordinates20). In this model, the polar and azimuthal dependences of the interaction potential are eliminated so the collision becomes that for a pair of spheres whose radii of collision are the clusters’ projections on the line segment joining their centers of mass. The radius of collision for a cluster of J molecules in an orientation y is denoted rJ(J”.Therefore, the model collision rate density for a J-mer with a K-mer whose respective orientations are y and z is
This is the usual formula for the collision rate density for thermalized spheres (e.g., ref 21) of masses M j , MK,at translational temperature T which interact via a spherically symmetric potential V(R)where R is the center-of-mass separation. Here, Vis given by hEjK in eq 4,k is Boltzmann’s constant, and MI = I-M, where M I is the C 0 2molecular mass. In this study, T = 50 K. Following Monchick and Mason,22 we assume the transport quantity of interest can be expressed as an average over different body-fixed coordinate collisions. In the current case, the collision rate density between clusters J and K is assumed to be the simple average over four component rate densities from eq 5: 4
~ J = K
(1 /4) C
p= 1
(6)
The orientations p = b,z) were established by two radii of collision for each cluster. The maximum radius rJ(x)corresponds to the largest distance of the surface of the cluster from its center of (15) Bishop, D. M.; Cheung, L. M.. J . Phys. Chem. Ref. Dafa 1982, I / , 119. (16) Inn, E. C. Y.; Watanabe, K.; Zelikoff, M. J . Chem. Phys. 1953, 21, 1648. (17) Sun, H.; Weissler, G. L. J . Chem. Phys. 1955, 23, 1625. (18) Bottcher, C. J. F. Theory ofElectric Polarization, Vol. I, 2nd ed; Elsevier: Amsterdam, 1973. (19) Pack, R. J. J . Chem. Phys. 1974, 61, 2091. The author gratefully acknowledges a referee for bringing this very useful article to his attention. (20) Hornig, J. F.; Hirschfelder, J. 0. Phys. Reu. 1956, 103, 908. (21) Weston, R. E.; Schwartz, H . A. Chemical Kinetics; Prentice-Hall: Englewood Cliffs, 1973. (22) Monchick, L.; Mason, E. A. J . Chem. Phys. 1961, 35, 1676.
2304 The Journal of Physical Chemistry, Vol. 90, No 11, 1986 L-P3J\'
cj,'DRCE-
H V ? C
ip-Fi;-
JL
, > 3'
Letters 4 - P O I N T RVERRGE- C O L L I S I O N RRTES
;;--?
,-
?T
Figure 1. Cluster collision rate densities (cgs units) omitting intercluster potential energies. The numbers to the left and right of the lines are the number of monomers in the cluster and the abscissa (A') is the size of the cluster it is colliding with to give the collision rate in the ordinate. Lines connecting the points are given only for visual convenience.
mass (s/2 plus distance of molecular center from cluster center); the minimum radius rJ'")corresponds to the smallest such distance and is generally located on the surface of the plane determined by three surface molecules. In these terms, the four cluster orientations involved in the collision calculations can be represented by the pairs (x,x), (n,n),( x , n ) ,and (n,x) corresponding to four values of p. Cooper and Birge23 have presented calculations of cluster self-energies via a determinant method which is essentially equivalent to that used here; where applicable, the results are comparable. The energy results of this paper were computed for all orders of the dipole-induced dipole interaction and therefore include van der Waals, Axilrod-Teller (triple dipole), and higher-order multiple dipole energies. For the sake of comparison, simple sums of dipole and triple dipole interactions together with the repulsive r-12 potentials between clusters were performed and as expected showed that the dipole energy made the dominant contribution. Depending upon the relative orientation parameter @ and size of cluster, the dipole sum differed from the full energy by no more than 10%; adding the sum on triple dipole enrgies reduced the maximum error to less than 7% in the = (x,x) configuration and esssentially eliminated it in the (3 = (n,n)case. Figure 1 gives the collision rate densities (eq 6 ) when the contact potentials V(rJ(-")rK(z))= 0 in eq 5. These essentially geometrical results appear to be fairly irregular with no readily discernible trends. While they are morphology dependent, these curves are reminiscent of the liquid-drop model where the rates are proportiona14 to [(l/J) + (l/K)]'/z(J'/3 + K1/3)2,a result that depends only upon reduced mass and surface area. Note that a priori this result is unphysical because it scales purely with geometry and therefore treats molecules exactly the same as macroscopic objects. Figure 2 gives the collision rate densities including the longrange energies. The monotonicity of the collision curves in Figure 2, as compared with those in Figure 1, is due to incorporating compositionally specific quantities, Le., due to turning on the interaction. As the number of molecules per cluster increases, the strength of the intercluster attractive energy also increases, thereby leading to a systematic progression of enhancements to the geometrical rates which accounts for the disaggregation of the curves. The specific effect of the long-range interaction potential can be visualized by computing the enhancements to the collision rates. These are derived by dividing the rate including the potential by the rate with zero potential (Le., Figure 2 divided by Figure 1)
IJ I
10
6.00
3 .OO
9.00
12.00
15.00
CLUSTER1 N I
Figure 2. Cluster collision rate densities including intercluster potential energies. Notation same as in Figure 1. 4-PO!NT
AVERf4CE- COLLISION RATE ENHRNCEMENl
+
( 2 3 ) Cooper, T. M.; Birge, R. R. J . Chem. Phys. 1981, 74, 5669
3.00
rl
6 .OO
9.00
12.00
15.00
I I S T F R I k1 1I
L L Y Y , L l / \ I.
Figure 3. Collision rate enhancement
and they are displayed in Figure 3. Because this procedure reinforces local artifacts of the modeling or computational method, only overall trends of the curves should be focussed upon. For N > 3, note that the monomer collision rate enhancement is essentially constant. This is consistent with the assumptions of the classical theory of nucleation' since a constant enhancement is essentially equivalent to no enhancement. In constrast, the multimer-multimer enhancements show increasing sensitivity to the interaction potential since their slope increases as cluster size
J. Phys. Chem. 1986, 90, 2305-2308 increases. Unlike the case for monomer collisions, a constant enhancement cannot correct for overall uncertainties in these rates. In both monomer and multimer cases, these results are consistent with the fit done by Soler et aL4 to their data on C02. Those authors found that, in monomer-dominated growth, their liquid drop model collisional growth rates24were capable of being used to compute the envelope of their measured size distributions. In contrast, those same growth rates did not accurately account for the cluster size distribution when the monomer was depleted and growth occurred via multimer-multimer collisions. For these larger clusters, the measured size distribution was narrower and less symmetric than the computed one. Beyond the question of multimer coagulational growth, these results imply that the multimer long-range interaction energy must be included in formulating a theory of homogeneous nucleation that is valid at supersaturations which are sufficiently high that multimermultimer collisional growth processes are important. A final observation based on these results arises from the nonuniform separation (or bunching) of the collision rates, Figure 2. This is due to a combination of cluster morphology and interaction potential and may have observable consequences. For example, small-cluster size distributions generally are not smooth (e.& Figure 1 of ref 25) indicating the possibility that bunching
2305
of the collision rates could play an important role in the formation of such size distributions. We may conjecture that these collision rate effects could also play a role, in addition to purely self-energy considerations, in cluster size distributions at all sizes. In this picture, the formation and evolution of the cluster size distribution is subject to two factors. The morphology and stability of a cluster of N monomers is determined by the customary shape and electronic structure factors, i.e., by “short-range ‘forces’ ”. Accommodation or sticking factors enter cluster formation questions at this point. The probability density that a cluster of N monomers will be “formed” in gas-phase collisions is dependent upon the long-range intercluster potential energies and the morphologies of all the possible collision pairs J , K such that J + K = N . Thus, questions of cluster mass and composition26 distributions may require examination from the viewpoint that the collision rate densities themselves can impose a preferred cluster size distribution prior to the clusters’ stabilization and assumption of their minimal energy configurations.
Acknowledgment. This research was performed under the auspices of the United States Department of Energy under Contract No. DE-ACO2-76CH00016. (25) Geusic, M. E.; Morse, M. D.; Smalley, R. E. J . Chem. Phys. 1985, 82, 590.
(24) In that study, all rates were normalized to the monomer-monomer rate and a unit sticking coefficient was assumed.
(26) Jonkman, H. T.; Even, U.; Kommandeur, J. J . Phys. Chem. 1985,89, 4240.
Transition-Metal Cation Chemistry in 1 Torr of He: M+ -t C,H, Reaction Rates Russ Tonkyn and James C. Weisshaar* Department of Chemistry, University of Wisconsin-Madison, Madison, Wisconsin 53706 (Received: February 24, 1986)
We have used laser vaporization of solid metal targets to create first transition series gas-phase metal cations M+ in a fast flow reactor with 1 Torr of He buffer gas. In contrast with single-collision results, all ten first transition series M+ ions, Sc+ through Zn’, react with C2H6 in our multicollision experiment. The major primary product is usually the adduct ion MC2H6+arising from third-body collisional stabilization of long-lived intermediates. The primary reaction rates vary a factor of 250 across the series. M+ ions having ground-state or low-energy 3d” electron configurations react the fastest with alkanes in 1 Torr of He.
Introduction With the advent of novel ion sourcesl~zand the steady development of sensitive mass spectrometric3q4 and ion beam5v6techniques, gas-phase organometallic ion chemistry has become a lively area of research. Fundamental interest in such work stems from the possibility of studying intrinsic molecular interactions unperturbed by solvent effects. As compared to the solution phase, gas-phase ion studies may permit clean isolation and mass identification of mrdinatively unsaturated (radical) species, of primary photochemical products, and of intriguing reaction intermediates. A wide range of single-collision chemistry of first and second linear alkanes,&lo~zz alkenes,” transition series metal ions with H2,’ ~~~
Jones, R. W.; Staley, R. H. J . Am. Chem. SOC.1982, 104, 1235. Allison, J.; Ridge, D. P. J. Am. Chem. SOC.1979, 101, 4998. Cody, R. B.; Freiser, B. S. Anal. Chem. 1982, 54, 1431. Gross, M. L.; Chess, E. K.;Lyon, P. A.; Crow, F. W.; Evans, S.; Tudge, H.In?. J . Mass Soectrom. Ion Phvs. 1982. 42. 243. (5) ArmentroG, P. B.; Beauchimp, J. L: J . Chem. Phys. 1981,74, 2819. (6) Ervin, K.; Loh, S. K.; Aristov, N.; Armentrout, P.B. J . Phys. Chem. 1983.87. 3593. ( 7 j (a) Elkind, J. L.; Armentrout, P. B. J . Phys. Chem. 1985, 89, 5626. (b) Armentrout, P. B.; Beauchamp, J. L. Chem. Phys. 1980, 50, 37. (1) (2) (3) (4)
0022-3654/86/2090-2305$01 .50/0
cyclic alkanes,12 and other organic species has been studied by ion cyclotron resonance and ion beam techniques. The reactions of M+ with alkanes are remarkable in that selective cleavage of C-H and C-C bonds is frequently 0b~erved.l~The first transition series single-collision M+ reactions with linear alkanes at thermal energies can be summarized as follow^.^-'^^'^^^^ Sc’, Ti+, and (8) (a) Allison, J.; Freas, R. B.; Ridge, D. P. J . Am. Chem. SOC.1979,101, 1332. (b) Freas, R. B.; Ridge, D. P. J . Am. Chem. Sor. 1980,102, 7129. (c) Larsen, B. S.;Ridge, D. P. J. Am. Chem. SOC.1984,106, 1912. (d) Peake, D. A.; Gross, M. L.; Ridge, D. P. J. Am. Chem. Sor. 1984, 106, 4307. (9) (a) Armentrout, P. B.; Beauchamp, J. L. J. Am. Chem. Sor. 1981,103, 784. (b) Houriet, R.; Halle, L. F.; Beauchamp, J. L. Organometallics 1983, 2, 1818. (c) Mandich, M. L.; Halle, L. F.; Beauchamp, J. L. J . Am. Chem. SOC.1984, 106, 4403. (10) (a) Byrd, G. D.; Burnier, R. C.; Freiser, B. S. J. Am. Chem. Sor. 1982, 104, 3565. Byrd, G. D.; Freiser, B. S. J. Am. Chem. SOC.1982, 104, 5944. (b) Jacobson, D. B.; Freiser, B. S.J . Am. Chem. SOC.1983,105, 5197. ( c ) Uppal, J. S.; Staley, R. H. J . Am. Chem. SOC.1982, 104, 1235. (1 1) (a) Armentrout, P. B.; Beauchamp, J. L. J. Chem. Phys. 1981, 74, 2819. (b) Jacobson, D. B.; Freiser, B. S. J . Am. Chem. SOC.1983, 105, 7484. (12) (a) Jacobson, D. B.; Freiser, B. S.J. Am. Chem. SOC.1983, 105,7492. (b) Armentrout, P. B.; Beauchamp, J. L. J . Am. Chem. Sor. 1981, 103,6628. (13) Halle, L. F.; Houriet, R.; Kappes, M. M.; Staley, R. H.; Beauchamp, J. L. J. Am. Chem. Sor. 1982, 104, 6293. (14) Tolbert, M. A.; Beauchamp, J. L. J . Am. Chem. Sor. 1984,106,8117.
0 1986 American Chemical Society
